[    {        "title": "1502.03718v1.Electronic_states_induced_by_nonmagnetic_defects_in_two_dimensional_topological_insulators.pdf",        "content": "Electronic states induced by nonmagnetic defects in two-dimensional topological\ninsulators\nVladimir A. Sablikov and Aleksei A. Sukhanov\nV.A. Kotelnikov Institute of Radio Engineering and Electronics,\nRussian Academy of Sciences, Fryazino, Moscow District, 141190, Russia\nWe study in-gap electronic states induced by a nonmagnetic defect with short-range potential in\ntwo-dimensional topological insulators and trace their evolution as the distance between the defect\nand the boundary changes. The defect located far from the boundary is found to produce two\nbound states independently of the sign of its potential. The states are classi\fed as electronlike and\nholelike. Each of these states can have two types of the spatial distribution of the electron density.\nThe \frst-type states have a maximum of the density in the center and the second-type ones have\na minimum. When the defect is coupled with the boundary, the bound states are transformed\ncorrespondingly into resonances of two types and take up the form of the edge states \rowing around\nthe defect. Under certain conditions, two resonances interfere giving rise to the formation of a\nbound state embedded into the continuum spectrum of the edge states \rowing around the defect.\nWe calculate the spatial distribution of the electron density in the edge states \rowing around the\ndefect and estimate the charge accumulated near the defect. The current density \feld of the edge\nstates \rowing around the defect contains two components one of which \rows around the defect and\nthe other circulates around it.\nI. INTRODUCTION\nThe presence of gapless edge states at the interface of\ntopologically non-equivalent crystals is a hallmark of two-\ndimensional (2D) topological insulators (TIs)1. In these\nstates the electrons move along the boundary and their\nspin is locked to the momentum because of strong spin-\norbit interaction. Such helical edge states are protected\nagainst scattering by weak non-magnetic impurities and\ndisorders. Nevertheless, experiments reveal a noticeable\nbackscattering of electrons2{8, the mechanism of which is\nnot yet known9,10.\nBackscattering of electrons in the edge states can oc-\ncur as a result of an inelastic process due to electron-\nelectron interactions and the presence of a defect po-\ntential11. The e\u000bect of the electron-electron interaction\nin the vicinity of the defect essentially depends on the\ncharge and spin structure of the electron cloud which\nforms near it. In this regard, of great importance is the\nquestion about the electronic states induced by impuri-\nties and other structural imperfections, especially in the\ncase where the defect is located near the boundary. One-\ndimensional models of coupling between the edge states\nand the defect turn out to be insu\u000ecient to describe the\nexperiments11{13.\nElectronic states induced by a defect were studied for\nthree dimensional (3D) TIs where the defect is located\non the surface. In this case, the electron cloud around\nthe defect is formed by 2D electronic states propagating\nalong the surface. Their interference leads to a variety of\nthe electron density con\fgurations14{17and even to the\nchanges in the surface state spectra18,19.\nIn 2D TIs, the electron cloud around a defect also exists\nbut its structure is substantially di\u000berent since the elec-\ntron density con\fguration is formed mainly by evanescent\nmodes decaying in the plane. It is essential that the elec-\ntron cloud can not be described within a one-dimensional(1D) model. Electronic structures formed in this case are\ncurrently poorly understood.\nDefect-induced electronic states in 2D TIs were stud-\nied mostly in the case where the defect is located deep in\nthe bulk and decoupled from the boundary. In Ref. 20,\nthe defect was considered as a hole, at the edges of which\nthe wave function is zero. In this case, the bound states\nare in essence the edge states circulating around the hole\nwith quantized angular momentum. Although this model\ncaptures some properties of the defect-induced states, it\nis far from reality. Under realistic conditions the wave\nfunction is not zero in the defect. The bound states ap-\npearing in the Gaussian potential were investigated nu-\nmerically for a number of material parameters21,22, but\nno general conclusions were made about their spectra,\nthe electronic structure, and the conditions under which\nthey exist.\nDefects interacting with the boundaries were studied\nin the case of a slab of 2D TI. In this case, the defect\nis coupled with two boundaries. Numerical calculations\nwith using Green's function method combined with tight-\nbinding approach23have shown that the bound state\nspectrum di\u000bers from that in the continuous model. Par-\nticularly, it contains two states bound on one defect with\nshort range potential rather than one state as in the con-\ntinuous model21.\nIn recent work24we investigated analytically the\nbound states induced by a non-magnetic defect in the\nbulk of 2D TIs for defects with short-range potential.\nIt turned out that the defect creates two bound states\nwhich are classi\fed as electronlike and holelike. This is\nin contrast to the defects in topologically trivial insula-\ntors where only one bound state exists in a short-range\npotential. The bound states exist for both positive and\nnegative potentials. In turn these states can be also of\ntwo types depending on whether the electron density has\na maximum or a minimum in the point of the defect lo-arXiv:1502.03718v1  [cond-mat.mes-hall]  12 Feb 20152\ncation. Another interesting feature of the 2D TIs is an\nunusual dependence of the bound-state energies on the\ndefect potential. As the potential increases, the energies\nof both electronlike and holelike states tend correspond-\ningly to two di\u000berent limiting values, which lie within the\ngap.\nIn this paper we address the general problem of a de-\nfect coupled with the edge states in 2D TIs. We clarify\nhow the bulk bound states are modi\fed with decreasing\nthe distance between the defect and the boundary and\nhow the edge states are distorted by the defect. It turns\nout that the edge states and the bulk bound states trans-\nform into a set of eigenstates which have the form of the\nedge states \rowing around the defect. These states have\nresonances of the electron density in the vicinity of the\ndefect when the energy is close to the energy of the bulk\nbound states. Correspondingly, there are two types of\nthe resonances.\nUnder certain conditions two resonances of di\u000berent\ntypes can interfere with each other giving rise to the for-\nmation of a bound state with localized wave function in\nthe continuum of the edge states.\nWe study the spatial distribution of the electron den-\nsity and current density in the states \rowing around the\ndefect and estimate the charge accumulated near the de-\nfect at a given Fermi energy.\nThe outline of the paper is as follows. In Section II\nwe present analytical calculations showing the presence\nof the states \rowing around the defect and the bound\nstates in the continuum. Section III gives the detailed\nresults for the bound states in the bulk of 2D TI. Sec-\ntion IV deals with the electron states in the case where\nthe defect is located at a \fnite distance from the bound-\nary. In Sec. V, we study the electron density distributions\nfor resonant states, estimate the excess electron density\naccumulated near the defect and consider the patterns of\nthe electron current. We \fnish the paper with a discus-\nsion and conclusions in Sec. VI.\nII. BOUND STATES AND EDGE STATES\nFLOWING AROUND THE DEFECT\nOur study is based on the model of the 2D TIs\nproposed by Bernevig, Hughes, and Zhang (BHZ) for\nHgTe/CdTe quantum wells25. The 2D TI is described\nby the Hamiltonian\nH0=\u0012\nh(k) 0\n0h\u0003(\u0000k)\u0013\n; (1)\nwhere kis momentum operator and\nh(k) =\u0012\nM\u0000(B+D)k2A(kx+iky)\nA(kx\u0000iky)\u0000M+(B\u0000D)k2\u0013\n;(2)\nwithM,A,BandDbeing the model parameters.\nThe topological phase is realized when MB > 0.25,26\nIn the case of the HgTe/CdTe wells, the parameters\nxy\n0y0FIG. 1. (Color online) A schematic view of a defect located at\nthe distance y0from the boundary of the 2D TI. The darkened\narea shows the particle cloud near the defect. Lines represent\nthe particle \rows.\nM;B;D< 0, andA>0. The basis set of wave functions\nisfjE1\"i;jH1\"i;jE1#i;jH1#igwherejE1\"iandjE1#i\nare superpositions of the electron states of s-type and\nlight-hole states of p-type with spin up and spin down;\njH1\"iandjH1#iare the heavy-hole p-type states with\nopposite spins. In what follows we will restrict ourselves\nby considering the symmetric model where D= 0.\nLet us use the Cartesian coordinates, with the xaxis\ncoinciding with the boundary (Fig. 1). The TI lies at\ny >0 and the defect is located in the point x= 0;y=\ny0. We consider the defect described by a potential\nV(x;y\u0000y0) localized in a small region. Since the defect\nis non-magnetic, the total Hamiltonian H0+V(x;y\u0000y0)\nis separated into spin blocks. For spin-up electrons, the\nSchr odinger equation reads as\n[E\u001b0\u0000h(k)] \t(x;y) =\u001b0V(x;y\u0000y0)\t(x;y);(3)\nwhere\u001b0is a 2\u00022 unit matrix, \t( x;y) is a spinor\n( 1(x;y); 2(x;y))T. The wave functions are supposed\nto vanish at y!1 and equal zero at y= 0.\nIn what follows we will use dimensionless variables\n\"=E=jMj;fx0;y0g=fx;ygp\nM=B; a =A=p\nMB;\nv(x0;y0) =V(x;y)=jBj; b=y0p\nM=B;(4)\nand for convenience will omit the prime in the variables\nx0;y0.\nThe 2D problem (3) can be solved by using the Fourier\nand Laplace transforms over xandy:\ne\t(k;p) =Z1\n\u00001dxe\u0000ikxZ1\n0dye\u0000py\t(x;y); (5)\n\t(x;y) =1Z\n\u00001dk\n2\u0019eikxc+i1Z\nc\u0000i1dp\n2\u0019iepye\t(k;p): (6)\nWhen applying this transformation to Eq. (3) one needs\nto calculate the Fourier and Laplace transforms of the\nproductv(x;y)\t(x;y). We suppose that the region,\nwhere the defect potential is localized, is small compared\nwith the characteristic length scale of the wave function.\nIn this case the integral can be approximated as\n1Z\n\u00001dxe\u0000ikx1Z\n0dye\u0000pyv(x;y)\t(x;y)\u0019ev(k;p)e\u0000bp\t;(7)3\nwhere \t = \t(x= 0;y=b) is the wave function at\nthe defect position and ev(k;p) is the Fourier and Laplace\ntransforms of v(x;y). In such a way we arrive at the\nfollowing equation:\n[\"\u0000h(k;p)]e\t(k;p) =\u001bz\b(k) +\u001b0ev(k;p)e\u0000bp\t:(8)\nHere, [\"\u0000h(k;p)] is the matrix with elements aij(\";k;p ):\na11=\"+ 1\u0000k2+p2; a12=\u0000a(k+p);\na21=\u0000a(k\u0000p); a 22=\"\u00001 +k2\u0000p2;(9)\n\u001bzis the Pauli matrix, \b( k) is the Fourier transform of\nthe normal derivative of \t( x;y) at the boundary:\n\b(k) =Z1\n\u00001dxe\u0000ikx@\t(x;y)\n@y\f\f\f\f\ny=0: (10)\nWe are going to get a system of linear equations for\nthe components of the spinor \t which will allow one to\ndetermine the eigenenergy spectrum. This idea is imple-\nmented as follows.\nSolving Eq. (8) with respect to e\t(k;p) and using\nEq. (6), we obtain the following expression for the wave\nfunction:\n\t(x;y) =1Z\n\u00001dk\n2\u0019eikxc+i1Z\nc\u0000i1dp\n2\u0019iepy\n\u0001(\";k;p )\n\u0002\u0002\nD0(\";k;p )\b(k) +v(k;p)e\u0000bpD1(\";k;p )\t\u0003\n;(11)\nwhere \u0001(\";k;p ) is the determinant of the matrix in the\nleft-hand side of Eq. (8) which has the form:\n\u0001(\";k;p ) =\u0000\u0002\np2\u0000p2\n1(\";k)\u0003\u0002\np2\u0000p2\n2(\";k)\u0003\n;(12)\nwith\np1;2(\";k) =q\nk2+a2=2\u00001\u0006p\na2(a2\u00004)=4 +\"2\n(13)\nand Rep1;2(\";k)\u00150;D0(\";k;p ) andD1(\";k;p ) are the\nfollowing matrices:\nD0=\u0012\na22(\";k;p )a12(\";k;p )\n\u0000a21(\";k;p )\u0000a11(\";k;p )\u0013\n; (14)\nD1=\u0012\na22(\";k;p )\u0000a12(\";k;p )\n\u0000a21(\";k;p )a11(\";k;p )\u0013\n: (15)\nLet us now turn to the requirement that \t( x;y) should\nnot diverge in the limit y!1 . Equation (11) shows\nthat \t(x;y! 1 )91when the expression in the\nsquare brackets equals zero at p=p1;2(\";k). This gives\nus two equations that relate \b( k) and \t. Correspond-\ningly, there are four equations for their spinor compo-\nnents. However, one can show that only two of these\nequations are independent because the matrix elements\naij(\";k;p ) atp=p1andp=p2are connected by jointequation \u0001( \";k;p 1;2) = 0. In such a way we arrive at the\nfollowing equation:\nA(\";k)\b(k) +B(\";k)\t = 0; (16)\nwhereA(\";k) andB(\";k) are matrices\nA(\";k) =\u0012\na22(\";k;p 1)a12(\";k;p 1)\na22(\";k;p 2)a12(\";k;p 2)\u0013\n; (17)\nB(\";k) =\u0012\nv(k;p1)a22(\";k;p 1)e\u0000bp1\u0000v(k;p1)a12(\";k;p 1)e\u0000bp1\nv(k;p2)a22(\";k;p 2)e\u0000bp2\u0000v(k;p2)a12(\";k;p 2)e\u0000bp2\u0013\n:\n(18)\nSolving Eq. (16) with respect to \b( k) we obtain an\nexplicit expression for \b( k):\n\b(k) =\u0000A0(\";k)B(\";k)\n\u00011(\";k)\t +C(\")\u001f(\";k)\u000e[k\u0000k0(\")];\n(19)\nwhere \u0001 1(\";k) is the determinant of the matrix A(\";k)\nandA0(\";k) is the following matrix:\nA0(\";k) =\u0012\na12(\";k;p 2)\u0000a12(\";k;p 1)\n\u0000a22(\";k;p 2)a22(\";k;p 1)\u0013\n: (20)\nThe second term in Eq. (19) arises because of the singu-\nlarity of the matrix A(\";k) in accordance with the general\ntheory of singular matrices27. It describes the contribu-\ntion of the edge states in the pure TI into the electronic\nstates formed in the presence of the defect. k0(\") is a\nroot of the determinant \u0001 1(\";k) which gives exactly the\nspectrum of the edge states in the absence of the defect:\nk0(\") =\u0000\"\na: (21)\nFurther in Eq. (19), the coe\u000ecient C(\") is a normal-\nization constant, \u001f(\";k) is a spinor which is expressed\nvia the matrix elements aij(\";k;p ) atp=p1;2. Using\nEqs (13) and (21) it is easy to show that \u001f(\";k) coin-\ncides with the spinor of the edge states:\n\u001f=\u00121\n\u00001\u0013\n: (22)\nLet us now apply Eq. (11) to calculate \t. To this end,\nwe setx= 0 andy=band exclude \b( k) using Eq. (19).\nFinally, we obtain the equation which determines \t:\n(\u001b0\u0000K(\"))\t =C(\")F(\")\u001f; (23)\nwhereK(\") andF(\") are the following matrices\nK(\") =1Z\n\u00001dk\n2\u0019\u00141\n4a\"\u00011(\";k)D0(\";k)A0(\";k)B(\";k)\n+i1Z\n\u0000i1dp\n2\u0019iv(k;p)\n\u0001(\";k;p )D1(\";k;p )\u0015\n;(24)4\nF(\") =1\n4a\"D0\u0000\n\";k0(\")\u0001\n: (25)\nHerea\"=p\na2(a2=4\u00001) +\"2andD0(\";k) denotes the\nmatrix\nD0(\";k)=e\u0000bp1\np1D0(\";k;\u0000p1)\u0000e\u0000bp2\np2D0(\";k;\u0000p2):(26)\nEquation (23) has solutions of two kinds depending on\nthe determinant of the matrix ( \u001b0\u0000K(\"))\n\u0001\t(\") =\u0000\n1\u0000K 11(\")\u0001\u0000\n1\u0000K 22(\")\u0001\n\u0000K 12(\")K21(\") (27)\nFirst, if \u0001 \t(\")6= 0, the root of Eq. (23) is\n\t(\") =C(\")\n\u0001\t(\")[\u001b0\u0000K0(\")]F(\")\u001f; (28)\nwhere\n\u001b0\u0000K0(\") =\u0012\n1\u0000K 22(\")K12(\")\nK21(\") 1\u0000K 11(\")\u0013\n: (29)\nAn alternative is the case where\n\u0001\t(\") = 0: (30)\nLet\"0is a root of this equation. When \"=\"0, Eq. (23)\nhas a solution if C= 0. This solution reads as\n\t(\") =Cbs\u00121\n(1\u0000K 11)\u000e\nK12\u0013\f\f\f\f\f\n\"=\"0; (31)\nwith the constant Cbsbeing determined by the normal-\nization.\nIt is worth noting that in the \frst case, C(\") should\nturn to zero when \"tends to\"0. Otherwise, \t( x;y) will\nnot be normalized. Thus \t(\") is determined by Eq. (28)\nfor any\"6=\"0. But ifC(\") is exactly zero, the solution\n\t is given by Eq. (31).\nIn order to clarify the nature of these solutions we con-\nsider the asymptotics of \t( x;y) atx!\u00061 . The asymp-\ntotic behavior of \t( x;y) is easily found from Eqs (11) and\n(19). It has the following form:\n\t(x!1;y)'ieikx\n8a\"@\u00011\n@k\u0014e\u0000p2y\np2D0(\";k;\u0000p2)\n\u0000e\u0000p1y\np1D0(\";k;\u0000p1)\u0015\nA0(\";k)B(\";k)\t(\")\f\f\f\f\f\nk=k0(\"):\n(32)\nIn the case where \t(\") is determined by Eq. (28), one\ncan show that \t( x! 1;y) never equals zero and is\nproportional to exp[ ik0x]. Hence, these states propagate\nalong the edge and \row around the defect. We will call\nthem the edge states \rowing around the defect. They\nhave the continuous spectrum de\fned by Eq. (21), which\ncoincides with the spectrum of the edge states withoutdefects. The constant C(\") can be found by appropriate\nnormalization.\nAt a discrete energy \"=\"0the wave function should\nbe square integrable and hence the amplitude in its\nasymptotics, given by Eq. (32), should be zero. Using\nthe speci\fc expressions for matrices A0(\";k) andB(\";k),\ngiven by Eqs. (20) and (18), one can easily show that\nA0(\";k)B(\";k)\t = 0 if\n\t = \u00121\n1\u0013\n: (33)\nThus, when \t(\"0) satis\fes Eq. (33), a bound state\ncan arise in the continuum of edge states. Comparing\nEq. (31), which de\fnes \t(\"), and Eq. (33) we arrive at\nthe following equation for the elements of the Kmatrix:\n1\u0000K 11(\")\u0000K 12(\") = 0: (34)\nImportantly, this equation must be satis\fed together\nwith Eq. (30) that gives the necessary condition for the\nbound state to exist. At this point, one should take into\naccount that the elements of the Kmatrix depend not\nonly on the energy \", but also on the defect potential\nv(x;y). Therefore, the system of Eqs. (30) and (34) de-\ntermines the energy \"bsof the bound state in the con-\ntinuum and the defect potential vbsat which this state\narises.\nFollowing, we present the results of speci\fc calculations\nof the bound states and the states \rowing around the\ndefect.\nIII. BOUND STATES IN THE BULK\nWe start by considering the limit of b!1 , which de-\nscribes the bound states for a defect located in the bulk.\nWhenb!1 , the right-hand side of Eq. (23) goes to\nzero and the nondiagonal components of the K(\") ma-\ntrix de\fned in Eq. (24) also vanish. As a result, Eq. (23)\ndecouples into two independent homogeneous equations\nfor the components of the spinor \t = ( 1; 2)T. Cor-\nrespondingly, there are two kinds of bound states with\ndi\u000berent pseudospin components of the wave function at\nthe defect.\nThere is a solution in which  16= 0 and 2= 0. Since\n 1corresponds to the jE1icomponent of the basis set of\nwave functions, the states of this kind can be convention-\nally called the electronlike states. In another solution, in\ncontrast 1= 0 and 26= 0. We call them the holelike\nstates.\nThe eigenenergies of the states of both species are de-\ntermined by Eq. (30). In the limit b!1 , Eq. (30) de-\ncouples into two equations. Correspondingly, there are\ntwo solutions for electron-like and hole-like states: \"e\nand\"h. The bound-state energies depend on the defect\npotentialv(x;y) =vf(x;y). Although the particle-hole\nsymmetry is broken due to the defect potential, the fol-\nlowing symmetry relation holds for the energies of the5\nelectron-like and hole-like bound states:\n\"e(v) =\u0000\"h(\u0000v): (35)\nSpeci\fc calculations of the bound-state energies and\nthe electron density were carried out for the defect po-\ntential of two forms: the Gaussian function v(x;y) =\nv\u00032=\u0019exp[\u0000\u00032(x2+y2)] with the characteristic radius\n\u0003\u00001, and thev(x;y) =v=\u0019\u000e\u0000\nx2+y2\u0001\nwith regularizing\ncut-o\u000b at \u0003 when integrating over the wave vector. Both\ncases give similar results.\nUnusual properties of the bound states in 2D TIs be-\ncome apparent from the dependence of the bound state\nenergies on the defect potential amplitude v. They are\nillustrated in Fig. 2(a). The energies of both electronlike\nand holelike states have two branches with the quite dif-\nferent dependence of the energy on v. To be speci\fc, we\nconsider the electronlike states. One branch, \"e1(v), ap-\npears when the potential is attractive for electrons, v<0.\nAsjvjincreases, the bound state je1iappears with the\nenergy at the top of the gap. Thereafter, its energy goes\nto the bottom of the gap, reaching asymptotically a lim-\niting value \"e. We call such states the states of the \frst\ntype.\nWhen the potential is repulsive for electrons, there is\nanother branch \"e2(v), which represents the bound states\nof the second type, je2i. With increasing v, the energy\n\"e2changes from the bottom of the gap to the limit-\ning energy \"e. The holelike states jh1iandjh2ibehave\nsymmetrically with respect to the electron-like states in\naccordance with Eq. (35): \"h(1;2)(v) =\u0000\"e(1;2)(\u0000v).\nA physical di\u000berence between the states of the \frst\nand second types is seen from the spatial distribution of\nthe electron density \u001a= \ty\t and the pseudospin com-\nponents of the density j 1j2andj 2j2. Graphs of the\nradial distribution of the total electron density and the\npseudospin components are shown in Figs. 2(b){2(e). In\nthe states of the \frst type the density \u001ahas a maximum\nin the point of the defect location, while in the second-\ntype states the density reaches a minimum at the defect.\nNevertheless, one should note that in the general case\nthe maximum of the density in the center is not neces-\nsarily the highest maximum in the radial distribution of\nthe density. Under certain conditions, another maximum\nmay appear at some distance from the center.\nIt is remarkable that in the \frst-type states, the pseu-\ndospin component (electronlike or holelike one) which\nreaches a maximum in the center is exactly that for which\nthe defect potential is attractive. The opposite situation\noccurs for the states of the second type. The compo-\nnent of the spinor which vanishes in the center is that for\nwhich the potential is repulsive.\nThe existence of two states in a short-range potential\nis a feature of the 2D TIs. In the topologically trivial\ncase, where MB < 0, the calculations carried out by\nthe same method show that there are also electronlike\nand holelike states, but only one state arises in a given\npotential. The electronlike state exists only at v>0 and\nthe holelike state exists at v <0. Moreover, the states\nband states\nband statesenergy,(a)\n(b) (c)defect potential,\n(d) (e)FIG. 2. (Color online) A schematic view of (a) the energy of\nthe bound states in the bulk of 2D TI as a function of the\ndefect potential, (b{e) the radial distribution of the electron\ndensity\u001a(thick lines) and the densities of the spinor compo-\nnentsj 1;2j2for electronlike and holelike bound states of the\n\frst and second types.\noccur in a \fnite range of jvj. In both states, the density\n\ty\t reaches the maximum in the center, i.e. both states\nare the states of the \frst type in our classi\fcation. The\nsecond-type states are absent.\nThese facts allow one to interpret the presence of two\nbound states in a given potential as a result of a simul-\ntaneous action of two mechanisms of bound state forma-\ntion. The \frst mechanism is universal: the bound states\ncan be formed by the potential attracting the quasipar-\nticles of one of the bands. Another mechanism is speci\fc\nfor TIs. It is caused by the formation of an edge state\ncirculating around the defect similarly to the edge states\nnear the boundary. In a certain sense, the defect e\u000bec-\ntively creates a boundary condition for the wave function.\nThis mechanism was discussed in the literature20{22.\nThe existence of two states agrees qualitatively with\nrecent numerical calculations with using a tight-binding\napproach combined with the Green's function method23,\nbut there are essential discrepancies. The contradictions6\nare clearly seen in the spatial distributions of the total\ndensity\u001aand the pseudospin components of the densi-\nties, as well as in the dependence of the densities on the\nimpurity potential. The results we have obtained here\nvery well agree with the direct calculations within the\ncontinuous model of the isolated defect24.\nThe bound state energies depend also on the parameter\na, de\fned through the parameters of the BHZ model by\nEq. (4). When a<21=2, the energy gap is less than jMj.\nIn this case, two bound states with the energy within this\nreduced gap exist for all values of the defect potential and\nthe graphs of \"(e;h)(1;2)(v) look as in Fig. 2(a). In the\ncase where a > 21=2, a qualitative di\u000berence arises for\nthe energy of the \frst-type states. These states appear\nwhenjvjexceeds a threshold value. Below the threshold,\nonly the second-type states exist.\nThe approach we use, does not allow us to investi-\ngate the e\u000bect of the shape of the defect potential on\nthe bound states. One can only trace how the bound\nstate energies change with the localization radius when\nit is small. We considered the case where the localization\nlength changes together with the potential amplitude so\nthat the integral of the potential over the area remains\nconstant. It was found that as the localization length de-\ncreases, the limiting energies \"e;hshift slowly (logarith-\nmically) to the nearest edges of the gap. In the limiting\ncase of \u0003!1 , the potential shape becomes the \u000efunc-\ntion and the bound states disappear in agreement with\nthe theory of singular potentials.28.\nThe energy of the bound states does not depend on the\nspin, which means that there are two states with oppo-\nsite spins. In these states, the electron current circulates\naround the defect in opposite directions, just as in the\nedge states. However, in contrast to them, each state\ncan be occupied by one electron with some spin because\nthe capture of another electron is hindered by Coulomb\nrepulsion. The possibility of capturing another electron\nwith opposite spin requires a separate consideration, tak-\ning into account the interaction between electrons.\nIV. DEFECT-INDUCED STATES NEAR THE\nBOUNDARY\nWhen the defect is coupled with the boundary, the\nelectronic states are classi\fed as the edge states \rowing\naround the defect and the bound states in the continuum.\nIn this section, we consider the resonances of edge states\n\rowing around the defect and \fnd out the conditions\nunder which the bound state arises in the continuum of\nthe edge states.\nThe speci\fc calculations are performed for the \u000e-\nlike potential of the defect in the form v(x;y\u0000b) =\n(v=\u0019)\u000e\u0002\nx2+ (y\u0000b)2\u0003\nwith using a cuto\u000b at \u0003 in the mo-\nmentum space. This simpli\fcation allows us to calculate\nanalytically the integrals over pin Eqs. (11) and (24).\nThe subsequent integration over kis done numerically.\nIt is natural to expect that when the defect is locatedfar from the boundary, the mixing of the bound state\nand the edge states leads to the formation of a resonance\nwith the energy close to the bound-state energy. At a\n\fnite distance bbetween the defect and the boundary,\nthe resonance broadens and the resonant energy deviates\nfrom the bound-state energy. It is this mixture of the\nstates that forms the edge states \rowing around the de-\nfect. They are exactly described by Eqs (11), (19), and\n(28).\nOur analysis shows that the resonant energy is very\nclose to the energy \"0de\fned by the roots of Eq. (30).\nIn the limit b!1 , these roots describe the bulk bound\nstates. In the case of arbitrary distance b, it becomes\nessential that the roots are functions of three variables:\nthe defect potential v, the distance b, and the material\nparametera. Particularly, the dependence of \"0(v;b;a )\nonvis much more complicated than in the case of the\nbulk position of the defect. In the following, we con-\nsider the dependence of the resonant energy on all three\nquantities.\nWhena > 21=2, general patterns of the behavior of\n\"0(v) with decreasing bare as follows [see Fig. 3(a) for\nillustration]. The limiting energies \"e;hshift from their\npositions at b!1 to the edges of the gap so that at\nsome \fnite distance b=bmthe limiting energy \"ecrosses\nthe bottom of the gap and \"hcrosses the top of the gap.\nSince our calculations are not justi\fed for the energy out-\nside the gap, we can only conclude that the roots \"(e;h)2\ncorresponding to the second-type resonances disappear\nin the gap when b<b m. Nevertheless, the \frst-type res-\nonances continue to exist, but in a \fnite interval of v,\nwhich diminishes with decreasing b.\nThe minimum distance bm, above which the resonant\nenergy of the second-type states lies in the gap, depends\non the parameter a. Numerical calculations show that\nbmvaries slowly between values 2 and 3 when jaj.1:5\npassing through a minimum at jaj'1. With increasing\njajabove 1.5, the distance bmgrows sharply, reaches 15\natjaj= 2:5, and continues to grow further.\nA special situation arises in the region near the inter-\nsection point of the curves \"(e;h)1(v) and\"(h;e)2(v) cal-\nculated in the limit b!1 . At \fnite b, the coupling of\nthe bound states with the boundary results in a mixing of\nthe states of the \frst and second types. This leads \fnally\nto an avoided crossing of the energy levels \"(e;h)1(v) and\n\"(h;e)2(v). An example of such an anticrossing is shown\nin Fig. 3(b) for b= 5. The anticrossing of the levels\n\"(e;h)1(v) and\"(h;e)2(v) occurs only when the distance b\nis large enough. The minimum distance b, above which\nthe anticrossing occurs, depends upon the parameter a.\nWith decreasing athe minimum distance diminishes.\nWhena<2, the wave vectors of the evanescent states,\np1;2(\";k) de\fned by Eq. (13) become complex. How-\never, until the imaginary part of p1;2is small, no new\ne\u000bects appear in the behavior of \"0with changing v. A\nqualitatively new behavior appears when a\u001c1 and the\nevanescent states strongly oscillate with distance. Their\ninterference in the segment between the defect and the7\n-20 -10 0 10 20-1.0-0.50.00.51.0energy, ε(a)εh2\n1 2 3 4-1.00-0.98-0.96\ndefect potential, (b)\nb=5.\n3.5\n2.6\n2.energy, ε\nb=5.\n3.5b=5., 3.5b=5., 3.5, 2.6, 2.εe1\nεe2εh1b=2., 2.6, 3.5, 5.\nFIG. 3. (Color online) (a) Resonant energies \"(e;h)1(v) and\n\"(h;e)2(v) as functions of the defect potential for a vari-\nety of distances between the defect and the boundary, b=\n2;2:6;3:5;5. The dotted line shows the region magni\fed in the\nlower part of the \fgure. (b) Graphs \"(e;h)1(v) and\"(h;e)2(v)\nnear the anticrossing point. Dashed lines show the crossing\ncurves\"h1(v) and\"e2(v) for the bulk states, b=1. The\nasterisk shows the energy of the bound state \"bs=\u00000:984\nwhich exists at vbs= 1:959 when the defect is located at the\ndistanceb= 5. The calculations were carried out for a= 21=2.\nboundary results in a nonmonotonic dependence of \"0on\nv, as it is illustrated in Fig. 4. An oscillatory compo-\nnent of the dependence of \"0onvis seen to arise when\nthe defect approaches the boundary. The e\u000bect can be\nso strong that multiple resonant states at a given vcan\nexist under certain conditions.\nThese e\u000bects could be realized in quantum wells\nInAs/GaSb where the parameters of the BHZ model are\nsuch thata\u00180:2{0:5 and the imaginary part of p1;2\ngreatly exceeds the real part.\nAn unexpected e\u000bect of the coupling between the de-\nfect and the boundary consists in the appearance of the\nbound state in the continuum of the edge states. In\nSec. II, we have shown that the energy \"bsof this state\nand the defect potential vbs, at which it appears, are de-\ntermined by the system of Eqs. (30) and (34). It is not\ndi\u000ecult to analyze these equations in the case of large\nbwhere the defect is weakly coupled with the boundary\nand therefore the non-diagonal elements of the Kmatrix\ngiven by Eq. (24) are small. In this case, the non-diagonal\nterms can be treated perturbatively. Finally, we come to\nthe following conclusions: (i) the bound state can arise at\nthose\"andvwhich are located on the plane ( \";v) in the\nvicinity of the intersection point of the curves \"(e;h)1(v)\n0.4\n0.2\n0.0\n-0.2\n-0.4\n2 4 0 6 8 10 12 -2 -8 -10 -12 -4 -6\ndefect potential, energy,  ε\nεe2εh1εh2\nεe1b=3\n102.5\nb=2.510\n3 b=10b=10FIG. 4. (Color online) The resonant energies \"(e;h)1(v) and\n\"(h;e)2(v) as functions of the defect potential for a variety\nof the distance between the defect and the boundary, b=\n2:5;3;10, in the case where a= 0:4.\nand\"(h;e)2(v) calculated for b!1 . (ii) in this region\nthere is only one solution, which exists if bis large enough.\nThus, the bound state arises at a de\fnite potential vbs\nwhen the defect is located at a distance larger than a\nthreshold value. The energy of this state \"bsand the de-\nfect potential vbsare close to point where two resonances\nare degenerate. The results of speci\fc numerical calcula-\ntions forb= 5 anda= 21=2are shown in Fig. 3(b). The\nbound-state energy is indicated by the asterisk.\nThis qualitatively new property of the defect-induced\nstates is caused by the presence of two types of the reso-\nnant states in 2D TIs. The bound state in the continuum\narises due to the interference of two resonances tuned by\nchanging the defect potential so that they can be driven\ninto degeneracy. This mechanism is consistent with the\ntheory by Friedrich and Wintgen29.\nV. ELECTRONIC STRUCTURE OF STATES\nFLOWING AROUND THE DEFECT\nIn this section, we present the results of our calcu-\nlations of the spatial distribution of the electron density\nand current density in the resonant states \rowing around\nthe defect.\nThe electron density in the edge state \rowing around\nthe defect at a given energy,\n\u001a\"(x;y) = \ty\n\"(x;y)\t\"(x;y); (36)\nis calculated using \t \"de\fned by Eq. (11), where \b( \";k)\nand\t(\") are given by Eqs. (19) and (28). The results\nobtained for the defect located at the distance b= 3 from\nthe boundary are presented in Fig. 5. The calculations\nwere carried out for the \u000e-like defect potential. The am-\nplitude of the potential, v= 3:6, was chosen such that\none of the resonances \"h1was lying deep in the gap, and8\nρ, arb. units\nρ, arb. units(a)\n(b)x\nx\nyy42\n4-2-4\n026\n2\n4-2-40\n0\n2\n046\nFIG. 5. (Color online) 3D plots of electron density distribu-\ntion in 2D TI with the defect located at the distance b= 3\nfrom the boundary for two states with di\u000berent energies close\nto the resonances of the \frst and second types: (a) \"= 0:7\n(\frst-type resonance) and (b) \"=\u00000:955 (second-type reso-\nnance). Other parameters: v= 3:6,a= 21=2, \u0003 = 10.\nthe other resonance \"e2was shallow (see Fig. 3). For this\npotential, we used two energies, one of which \"= 0:7 was\nclose to the deep resonance and the other \"=\u00000:955\nwas close to the shallow resonance.\nFigure 5 clearly shows that the resonances retain the\nmain properties of the corresponding bound states in the\nbulk. In the resonances originating from the bound states\nof the \frst type, the density has a maximum in the cen-ter, while in the second-type resonances the density has a\nminimum. In contrast to the bound states in the bulk, no\ncomponent of the spinor \t equals exactly zero because\nthe resonant states are a mixture of the bulk bound state\nand the edge states. Nevertheless, one of the spinor com-\nponents remains much smaller than the other.\nNow, it is interesting to clarify what charge of electrons\nis accumulated near the defect. The excess electron den-\nsity at an energy level \"is evaluated as\n\u0001\u001a\"(x;y) =\u001a\"(x;y)\u0000\u001a\"(x;y)\f\f\nv=0; (37)\nwhere the second term in the right hand side is the den-\nsity in the absence of the defect. To simplify the pre-\nsentation of the results, we will characterize the excess\nnumber of electrons by an integral value \u0001 N(\") de\fned\nas follows:\n\u0001N(\") = lim\nL!1L=2Z\n\u0000L=2dx1Z\n0dy\u0001\u001a\"(x;y): (38)\n\u0001N(\") gives the spectral density of excess electrons in\nthe states \rowing around the defect.\nDirect calculations show that \u0001 N(\") has a maximum\nat the resonant energies. The results of speci\fc calcula-\ntions of \u0001N(\") are presented in Fig. 6 for di\u000berent posi-\ntions of the defect at a given amplitude of the potential.\nWhen the defect is located far from the boundary, the\nshape of the peaks on the curve \u0001 N(\") is well described\nby the Breit-Wigner formula. When the distance bis less\nthan the characteristic lengths of the evanescent states,\nthe shape of the peak substantially changes. Neverthe-\nless, the integral of \u0001 N(\") over\"within the gap is close\nto unity if the peak lies far from the gap edges.\nThis fact allows one to estimate the charge accumu-\nlated near the defect. If the Fermi energy is close to the\nresonant energy, the accumulated charge is of the order\nof one electron charge. If one assumes that this charge is\nlocalized in the area of the radius of about 30 nm (this\nis a typical estimation of the decay length of the in-gap\nstates in HdTe quantum wells), the potential created by\nthis charge can be of the order of 10 mV, which is com-\nparable with the gap energy.\nThe electron \row in the edge states is strongly dis-\nturbed in the vicinity of the defect. To clarify the struc-\nture of the current \feld in the presence of a defect one\nneeds to have an explicit expression for the particle cur-\nrent density jin the quantum states de\fned by the Hamil-\ntonian of the BHZ model with the defect. Since the\nHamiltonian is block diagonal with respect to the spin,\nit is enough to \fnd the current for one of the spin blocks.\nThe current density is determined by the term of a diver-\ngence in the continuity equation. For the spin-up elec-\ntrons the current j\"is expressed via the spinor compo-9\n0.70 0.75 0.80 0.85 0.90 0.95 1.00050100150200250300\nenergy, εb = 4\n-1.00 -0.99 -0.98 -0.97 -0.96 -0.95 -0.94020040060080010001200\n3.5\n3b  = 4\n3.5\n3\nenergy,εdensity, ΔN\ndensity, ΔN(a) (b)\nFIG. 6. (Color online) Spectral density of excess electrons\n\u0001N(\") as a function of the energy for a variety of b=3, 3.5,\n4 in two energy ranges: (a) near the resonance of the sec-\nond type, and (b) near the resonance of the \frst type. The\nparameters used in the calculations: a= 21=2,v= 5, and\n\u0003=10.\nnents 1(x;y) and 2(x;y) as follows:\nj\"x=\u00002i\n~Im\u0014\n(B+D) \u0003\n1@ 1\n@x\u0000(B\u0000D) \u0003\n2@ 2\n@x\u0015\n+2A\n~Re [ 1 \u0003\n2];(39)\nj\"y=\u00002i\n~Im\u0014\n(B+D) \u0003\n1@ 1\n@y\u0000(B\u0000D) \u0003\n2@ 2\n@y\u0015\n+2A\n~Im [ 1 \u0003\n2]:(40)\nWhen the defect is located in the bulk, a circular elec-\ntron current is present in each bound state with a given\nspin24. Its direction is locked to the spin as in the edge\nstates. However, there is an essential di\u000berence from\nthe edge states. The edge state can be occupied by two\ncounter-moving electrons with opposite spins so that the\ntotal current of the \flled state is zero. In contrast, in\nthe case of a point defect, the problem of the electronic\nstructure of the bound state with two electrons, and even\nits very existence, requires a separate study taking into\naccount the electron-electron interaction. This issue is\nbeyond the scope of this paper. Within the present ap-\nproach we study the one-electron states. If one makes\nthe natural assumption that the two-electron state has a\nhigher energy than the one-electron state, then the one-\nelectron bound state is realized. In this state, there is the\nelectron current, whose direction depends on the spin of\nthe trapped electron.\nIf the defect is coupled with the boundary, the \feld\nof the current density in the edge state \rowing around\nthe defect includes both the current circulating around\nthe defect and the edge current. The con\fguration of the\ncurrent \feld is calculated using Eqs. (39) and (40). The\nresults are presented in Fig. 7 for two states with the en-\nergies lying near the resonances of the \frst and second\ntypes. In both cases there is a current circulating around\nthe defect and an edge current \rowing around it. As\nthe defect approaches the boundary, the circulating cur-\nrent decreases and the current \rowing around the defect\nincreases.\n0123456\ny(a)\n-4 -2 0 2 4 -3 3 -1 1 -5 5\nx(b)\n0123456\nyFIG. 7. (Color online) Vector plot of the electron current \row\nof spin-up electrons in the presence of the defect located at the\ndistanceb= 3 from the boundary for two states with energy\n(a)\"=0.7 near the resonance of the \frst type and (b) \"=-0.955\nnear the resonance of the second type. Other parameters are\nv= 3:6,a= 21=2, \u0003=10.\nVI. SUMMARY AND CONCLUDING\nREMARKS\nWe have studied the in-gap electronic states induced\nby a nonmagnetic defect with short-range potential in 2D\nTIs and trace their evolution as the distance between the\ndefect and the boundary changes.\nIf the defect is located far from the boundary in the\nbulk, there are two bound states localized at the defect.\nThey exist for both positive and negative potentials of\nthe defect. The states are classi\fed as electron-like and\nhole-like states depending on the pseudospin (orbital)\ncomponent of the wave function which vanishes at the\ncenter: in the electron-like states  16= 0 and 2= 0\nwhile in hole-like states  1= 0 and 26= 0. In their\nturn these states are classi\fed into two types depending\non the spatial distribution of the particle density and the\ndensities of the pseudospin components. In the states of\nthe \frst type the particle density has a maximum in the\ncenter while the second-type states are characterized by\nthe presence of a minimum of the density in the center.\nIn the states of both types there is a particle current cir-\nculating around the defect. Its direction is locked to the\nelectron spin.\nThe presence of two bound states of di\u000berent types10\nat a given defect potential is a feature of 2D TIs. In\ntopologically trivial insulators, there is only one state.\nThe existence of two states raises an interesting question\nabout the two-particle bound state.\nAnother interesting property of the bound states in the\nbulk of 2D TIs is the singular dependence of their energy\non the defect potential amplitude jvj. The energies of the\nelectronlike and holelike states tend to the corresponding\nlimiting values \"eand\"hasjvj!1 . This fact could lead\nto a non-trivial consequence in the case where the crys-\ntal contains many di\u000berent defects with potentials that\nare scattered over a wide range. Since the energy of the\nstrong defects slowly changes with their potential, the\nbound state energies are concentrated in narrow spec-\ntral bands near \"eand\"h. The states may overlap and\nform a hopping system, which could manifest itself in the\ntransport.\nWhen the defect is coupled with the boundary, the\nedge states of the host crystal and the bound states trans-\nform into a unique set of the edge states \rowing around\nthe defect. These states have two resonances correspond-\ning to the two types of the bound states in the bulk. The\nresonances retain distinctive properties of the bulk bound\nstates. Particularly, they are classi\fed as the resonances\nof the \frst and second types. The states \rowing around\nthe defect with the energy close to the \frst-type reso-\nnance have a maximum of the electron density at the\ndefect location point, while in the states with the en-\nergy near the second-type resonance the density reaches\na minimum at the defect.When electrons occupy the edge states \rowing around\nthe defect, the excess charge is accumulated in the vicin-\nity of the defect. Its magnitude can be as large as one\nelectron charge. The potential created by such a charge\ncan be comparable with the energy gap in 2D TIs. This\nestimation shows that the electron-electron interaction\ncan essentially modify the defect-induced states.\nIn the edge states \rowing around the defect there is an\nelectron current for each spin orientation. The current\ndensity \feld has two components one of which circulates\nnear the defect and another \rows around it. The total\ncurrent \rowing around the defect depends on the \flling\nof the states. In the case of a narrow resonance, even\na small di\u000berence in the population of the states with\nopposite spins can lead to a noticeable current in the\nloop around the defect and the formation of a magnetic\nmoment. Estimations show that the magnetic moment\ncan be as large as one Bohr magneton.\nAn interesting feature of the bulk bound states is the\npresence of a point where the bound states of di\u000berent\ntypes are degenerate. This happens at a certain potential\nof the defect. The degeneracy of the resonances is lifted\ndue to the interaction of the defect with the boundary.\nThe interference of the resonances results in the forma-\ntion of a bound state embedded into the continuum of\nthe edge states \rowing around the defect.\nACKNOWLEDGMENTS\nThis work was partially supported by Russian Foun-\ndation for Basic Research (Grant No 14-02-00237) and\nRussian Academy of Sciences.\n1C. L. Kane, in Topological Insulators edited by M. Franz\nand L. Molenkamp (Elsevier, Amsterdam, 2013).\n2M. K onig, S. Wiedmann, C. Br une, A. Roth, H. Buhmann,\nL. W. Molenkamp, X.-L. Qi, and S.-C. Zhang, Science 318,\n766 (2007).\n3M. K onig, H. Buhmann, L. W. Molenkamp, T. Hughes,\nC.-X. Liu, X.-L. 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A 32, 3231\n(1985)."    },    {        "title": "1502.07731v2.Excited_state_geometry_optimization_with_the_density_matrix_renormalization_group_as_applied_to_polyenes.pdf",        "content": "Excited state geometry optimization with the density\nmatrix renormalization group as applied to polyenes\nWeifeng Hu1and Garnet Kin-Lic Chan1\nDepartment of Chemistry, Princeton University, Princeton, New Jersey 08544,\nUSA\nWe describe and extend the formalism of state-speci\fc analytic density matrix renor-\nmalization group (DMRG) energy gradients, \frst used by Liu et al (J. Chem. Theor.\nComput. 9, 4462 (2013)). We introduce a DMRG wavefunction maximum overlap\nfollowing technique to facilitate state-speci\fc DMRG excited state optimization. Us-\ning DMRG con\fguration interaction (DMRG-CI) gradients we relax the low-lying\nsinglet states of a series of trans -polyenes up to C 20H22. Using the relaxed excited\nstate geometries as well as correlation functions, we elucidate the exciton, soliton,\nand bimagnon (\\single-\fssion\") character of the excited states, and \fnd evidence for\na planar conical intersection.\nI. INTRODUCTION\nThe density matrix renormalization group1{7, introduced by White1, has made large ac-\ntive space multireference quantum chemistry calculations routine. In the chemistry context,\nthere have been many improvements to White's algorithm, including orbital optimization8{11,\nspin-adaptation12,13, dynamic correlation treatments14{16, and response theories17,18, to name\na few. The DMRG has been applied in many di\u000berent electronic structure problems, ranging\nfrom benchmark exact solutions of the Schr odinger equation for small molecules16,19{23, to\nactive space studies of conjugated \u0019-electron systems24{27, the elucidation of the ground- and\nexcited states of multi-center transition-metal clusters28{33, computation of high-order corre-\nlation contributions to the binding energy of molecular crystals34, relativistic calculations35,\nand the study of curve crossings in photochemistry36.\nEnergy gradients are crucial to electronic structure as they de\fne equilibrium structures,\ntransition states, and reaction trajectories. Analytic energy gradients, introduced by Pu-\nlay37{40, are preferred to numerical gradients, due to their low cost and numerical stability,\nand are now implemented for many single- and multi-reference quantum chemistry meth-arXiv:1502.07731v2  [physics.chem-ph]  2 Mar 20152\nods41{47.\nAnalytic DMRG energy gradients were \frst used by Liu et al.36in a study of the pho-\ntochromic ring opening of spiropyran. The theory, although simple, was not fully discussed.\nA contribution of the current work is to provide a more complete exposition of the theory\nbehind DMRG gradients. A second contribution is to discuss their practical implemention\nin excited state geometry optimization. The simplest formulation of the gradients arises\nwhen the excited states are treated in a state-speci\fc manner (that is, without orthogonal-\nity constraints to lower states). However, such DMRG calculations can be susceptible to\nroot-\ripping, for example, near conical intersections. Furthermore, DMRG wavefunctions\nare speci\fed both by a choice of active space as well as by discrete sets of quantum numbers\nassociated with each orbital (used to enforce global symmetries, such as the total particle\nnumber). During an energy minimization, it is important that the wavefunction changes\nsmoothly. Here, we present a state-speci\fc DMRG analytic gradient algorithm that uses a\nmaximum overlap technique both to stably converge excited states, and to ensure adiabatic\nchanges of both the orbitals and the DMRG wavefunction during geometry changes.\nTrans-polyenes are well-known examples of molecules with interesting ground- and excited\nstate structure, and form the central motifs for a large set of of biological compounds, such\nas retinals and the carotenoids. Many calculations using semi-empirical models (such as the\nPariser-Parr-Pople (PPP) model) as well as with ab-initio methods such as multi-reference\nself-consistent-\feld (MC-SCF), have been carried out to identify the low-lying electronic and\ngeometric features48{77. These studies, in general, give the following qualitative picture for\nlong even polyenes: 1) Excitations, coupled to lattice relaxation, break the dimerization of\nthe ground state and lead to local geometrical defects78. For example, adiabatic relaxation\ngives rise to a central polaronic feature in the bright 11Bustate65,69,75,76, as well as two-soliton\nor four-soliton structures for the dark 21Agstate54,56,65. 2) For short polyenes ranging from\nethylene to octatriene, studies of low-lying excited state relaxation pathways suggest that\nnon-planar molecular conformations are important at energy crossings79{95. This opens up\nthe question of the nature of energy crossings and their associated geometries in the excited\nstates of longer polyenes. Here, using an ab-initio Hamiltonian and DMRG-CI analytic\nenergy gradients, we revisit these questions in the excited states of relatively long trans -\npolyacetylenes, aiming for a more quantitative picture.\nIn Sec. II and III, we start by reviewing analytic energy gradient theory and DMRG3\ntheory. In Sec. IV we discuss the formulation of DMRG analytic gradients. In Sec. V, we\ndiscuss the DMRG maximum overlap method for stable state-speci\fc excited state calcula-\ntions without orthogonality constraints. In Sec. VI we apply our DMRG gradient algorithm\nto characterize the geometry and nature of the low-lying singlet states of the trans-polyenes.\nII. GENERAL ENERGY GRADIENT THEORY FOR VARIATIONAL METHODS\nFor completeness, we brie\ry recall analytic energy gradient theory for variational wave-\nfunctions. The energy referred to is the Born-Oppenheimer potential, the sum of the elec-\ntronic energy and the nuclear-nuclear repulsion. The nuclear-nuclear repulsion gradient is\ntrivial. The electronic energy h\tj^Hj\tiis the expectation value of the electronic Hamiltonian\n^H,\n^H=X\nij\u001btijcy\ni\u001bcj\u001b+1\n2X\nijkl\u001b\u001b0vijklcy\ni\u001bcy\nj\u001b0ck\u001b0cl\u001b; (1)\nwherei;j;k;l are orthogonal (for example, molecular) orbital indices, and \u001b;\u001b0=f\";#g.^H\ndepends on the orbital functions through the integrals tijandvijkl. In a typical implemen-\ntation, these orthogonal orbitals are represented as a linear combination of atomic orbitals\n(AO's) with orbital coe\u000ecients C:jii=P\n\u0016Ci\u0016j\u0016i, where we use Greek letters ( \u0016;\u0017) to\ndenote AO orbitals. The geometry dependence of the integrals arises from the AO func-\ntions, which (as Gaussian type basis functions) explicitly depend on the nuclear positions,\nas well as through the LCAO coe\u000ecient matrix C, which also changes with geometry. The\nHamiltonian may be regarded as a function of the nuclear coordinates faigand the coef-\n\fcient matrix C:^H(faig;C). As the electronic wavefunction j\tidepends on variational\nparametersfcig, the electronic energy is a function of faig,C, andfcig:E(faig;C;fcig).\nThe energy gradient with respect to nuclear coordinate atakes the form\ndE\nda=@E\n@a+X\ni\u0016@E\n@Ci\u0016dCi\u0016\nda+X\ni@E\n@cidci\nda: (2)\nIfj\tiis determined variationally, the energy is stationary to changes of fcig, thus the third\nterm vanishes. The gradient then only depends on the change in nuclear coordinates and\norbital coe\u000ecients,\ndE\nda=@E\n@a+X\ni@E\n@CidCi\nda: (3)4\nIt is convenient to rewrite the energy gradient in terms of density matrices. The energy\nis expressed as\nE=X\nijtij\rij+X\nijklvijkl\u0000ijkl; (4)\nwhere\rij=P\n\u001bh\tjcy\ni\u001bcj\u001bj\tiand \u0000ijkl=1\n2P\n\u001b\u001b0h\tjcy\ni\u001bcy\nj\u001b0ck\u001b0cl\u001bj\tiare the one- and two-\nparticle density matrices. Since the second-quantized operators have no dependence on\ngeometry, and the wavefunction depends only on fcig, it follows from Eq. (3) that the\nenergy gradient is expressed in terms of the one- and two-electron derivative integrals and\ndensity matrices,\ndE\nda=X\nijdtij\nda\rij+X\nijkldvijkl\nda\u0000ijkl: (5)\nThe one- and two-electron derivative integrals involve the orbital derivative (response),\ndC=da. Writing\ndCi\u0016\nda=X\njUa\nijCj\u0016 (6)\ngives\ndE\nda=X\nij@hij\n@a\rij+X\nijkl@vijkl\n@a\u0000ijkl\u0000X\nijXij@Sij\n@a\n+X\nijUa\nij(Xij\u0000Xji);(7)\nwhereSij=hijjiis the overlap matrix of the orthogonal orbitals iandj,\nSij=X\n\u0016\u0017Ci\u0016S\u0016\u0017Cj\u0017; (8)\nandS\u0016\u0017=h\u0016j\u0017iis the overlap matrix in the underlying AO basis. Xis the Lagrangian\nmatrix in the Generalized Brillouin Theorem (GBT)96, given as\nXij=X\nmhim\rmj+ 2X\nmklvimkl\u0000jmkl; (9)\ncharacterizing the energy cost of electronic excitation from ith orbital to jth orbital. The\ngradient formula can be rewritten entirely in terms of AO quantities42,\ndE\nda=X\n\u0016\u0017\r\u0016\u0017dh\u0016\u0017\n@a+X\n\u0016\u0017\u001a\u001b\u0000\u0016\u0017\u001a\u001bdv\u0016\u0017\u001a\u001b\n@a\n\u00002X\n\u0016\u0017X\ni>j(1\u0000\u000eij\n2)Ci\n\u0016Cj\n\u0017XjidS\u0016\u0017\n@a+ 2X\ni>jUa\nij(Xij\u0000Xji)(10)5\nwhereh\u0016\u0017,\r\u0016\u0017,v\u0016\u0017\u001a\u001b, \u0000\u0016\u0017\u001a\u001b are the one- and two- particle integrals and reduced density\nmatrices, respectively, in the AO basis.\nIn general, the orbital derivative dCi\u0016=darequires the solution of equations which couple\nthe wavefunction coe\u000ecients cito the orbital coe\u000ecients Ci\u0016. However, there are two com-\nmon situations where the orbital response is simpli\fed. The \frst is when the orbitals are\nde\fned independently of the correlated wavefunction, for example, for Hartree-Fock (HF)\nor Kohn-Sham (KS) orbitals. Using the Hartree-Fock canonical orbitals as an example, the\norbital response Uais de\fned by the Hartree-Fock convergence condition,\ndFij\nda= 0(i6=j): (11)\nwhich leads to the de\fnition of the Uamatrix\nUa\nij=1\n(\u000fj\u0000\u000fi)(virX\nkd:o:X\nlAij;aiUa\nkl+Ba\nij); (12)\nwhere\nAij;kl= 4vijkl\u0000vikjl\u0000viljk\nBa\nij=Fa\nij\u0000Sa\nij\u000fj\u0000X\njkSa\nkl(2vijkl\u0000vikjl);(13)\nwithFa\nij=@Fij=@a,Sa\nij=@Sij=@a, and the various \u000fare HF orbital energies. Eq. (12) is\nthe coupled-perturbed Hartree-Fock (CPHF) equation97, and uniquely de\fnes the Uamatrix\nelements for canonical orbitals. In a similar way, other types of orbital response, for example\nfor the Kohn-Sham orbitals, or localized Hartree-Fock orbitals, can be computed from the\ncorresponding coupled-perturbed single-particle equations41,98.\nThe second simplifying case is when the correlated wavefunction energy is itself stationary\nwith respect to orbital variations. In this case Xij\u0000Xji= 0, and the orbital response is\nnot required, even though it is formally coupled to the correlated variational wavefunction\ncoe\u000ecients. The energy gradient reduces to the simpler form,\ndE\nda=X\n\u0016\u0017\r\u0016\u0017dh\u0016\u0017\n@a+X\n\u0016\u0017\u001a\u001b\u0000\u0016\u0017\u001a\u001bdv\u0016\u0017\u001a\u001b\n@a\n\u00002X\n\u0016\u0017X\ni>j(1\u0000\u000eij\n2)Ci\n\u0016Cj\n\u0017XjidS\u0016\u0017\n@a+ 2X\ni>jUa\nij(Xij\u0000Xji):(14)6\nIII. GENERAL DMRG THEORY\nThe DMRG is a variational wavefunction method99. For a set of Lorthogonal orbitals\n(where the states for the ith orbital arej\u001bii=fj0i;j\"i;j#i;j\"#ig ) we choose a partitioning of\nthe orbitals into a left block, single site, and right block, consisting of orbitals f1:::l\u00001g,flg\nandfl+ 1:::Lg, respectively. The corresponding canonical \\one-site\" DMRG wavefunction\ntakes the matrix product form\nj\ti=X\n\u001b1\u001b2:::\u001bLL\u001b1L\u001b2:::L\u001bl\u00001\n\u0002C\u001blR\u001bl+1R\u001bl+2:::R\u001bLj\u001b1\u001b2:::\u001bLi:(15)\nThe (rotation) matrices L\u001biandR\u001biare of dimension M\u0002M, except for the \frst and\nlast which are of dimension 1 \u0002MandM\u00021 respectively. They satisfy the left- and\nright-canonical conditions\nX\n\u001biL\u001biTL\u001bi=1\nX\n\u001biR\u001biR\u001biT=1 (16)\nwhile the C\u001bl(wavefunction) matrix satis\fes the normalization condition\ntrX\n\u001blC\u001blTC\u001bl= 1: (17)\nTogether,fL\u001big,fC\u001blgandfR\u001bigcontain the variational parameters. As in other variational\nmethods, the coe\u000ecients of the matrices are determined by minimizing the energy. In\nprinciple, a direct gradient minimization of the energy with respect to all the matrices,\nsubject to the canonical conditions Eq. (16), (17), may be performed. In practice, the DMRG\nsweep algorithm is normally used. Here, at a given step lof the sweep, corresponding to the\nblock partitioning f1:::l\u00001g,flgandfl+ 1:::Lg, the energy is minimized only with respect\nto the C\u001blwavefunction matrix, with the fL\u001big,fR\u001bigrotation matrices held \fxed. The\nminimizing C\u001blis obtained from an e\u000bective ground-state eigenvalue problem\nHc=Ec (18)\nwherecdenotes C\u001bl\rattened into a single vector, and Hdenotes ^Hexpressed in the basis\nof renormalized basis states de\fned by the fL\u001big,fR\u001bigmatrices99. In the next step of the7\nsweep, the single site is moved from ltol+ 1 (orltol\u00001 in a backwards sweep). To satisfy\nthe new canonical form with the single site at l+ 1, where the C\u001blmatrix is replaced by\nanL\u001blmatrix, and the l+ 1 site is associated with a new C\u001bl+1matrix, we use the gauge\nrelations,\nC\u001bl=L\u001bl\u0003\nC\u001bl+1=\u0003R\u001bl+1: (19)\nBy sweeping through all the partitions l= 1:::L, and minimizing with respect to the C\u001bl\nmatrix at each partition, the DMRG sweep algorithm ensures that all the variational degrees\nof freedom in the DMRG wavefunction are optimized.\nAn important aspect of DMRG calculations is the enforcement of symmetries, including\nglobal symmetries such as the total particle number and spin. In the DMRG wavefunction,\nAbelian global symmetries (such as total particle number) are enforced by local quantum\nnumbers. For example, to enforce a total particle number of Nin the wavefunction, each\nvalue of the 3 indices \u001b,i;jin the matrix elements L\u001b\nij,R\u001b\nij,C\u001b\nijcan be associated with\nan additional integer Ni,Nj,N\u001b. (These values can be interpreted in terms of the particle\nnumbers of the renormalized states (for NiandNj) and for the states of the single site (for\nN\u001b)). Then, a total particle number of Nis enforced with the rules:\nL:Ni+N\u001b=Nj\nR:Nj+N\u001b=Ni\nC:Ni+N\u001b+Nj=N: (20)\nApplying these conditions to L\u001b\nij,R\u001b\nij,C\u001b\nijmeans that the matrices have a block-sparse\nstructure, which is important to maintain during geometry optimization.\nIV. STATE-SPECIFIC DMRG ANALYTIC ENERGY GRADIENTS\nAt convergence of the above (one-site) DMRG sweep algorithm, the contribution of the\nwavefunction coe\u000ecients to the gradient ( dci=dain Eq. (2)) vanishes, as expected for a\nvariational wavefunction method. Thus the analytic energy gradient theory for variational\nwavefunctions described in Sec. II can be applied.8\nWe will consider energy gradients for two kinds of DMRG calculations. The \frst are\nDMRG con\fguration interaction (DMRG-CI) analytic gradients, using HF canonical or-\nbitals. In this case, the orbital response is given by the CPHF equations, presented in\nSec. II. The DMRG calculations are carried out within an active space, chosen as a subset\nof the canonical orbitals. Because the DMRG wavefunction is not invariant to rotations of\nthe active space orbitals for small M, the contribution of the active orbital response must\nbe computed specifying a particular orbital choice (rather than just their manifold), such as\nthe canonical HF orbitals.\nThe algorithm to compute the DMRG-CI analytic gradient with HF canonical orbitals is\nas follows:\n1. Solve the HF equations for the canonical orbital coe\u000ecient matrix C.\n2. Select an active space, and solve for the DMRG wavefunction in this space. Compute\nthe one- and two-particle reduced density matrices \rijand \u0000ijklat the convergence of\nthe single-site sweep algorithm.\n3. Compute the AO derivative integrals dh\u0016\u0017=da,dv\u0016\u0017\u001a\u001b=daanddS\u0016\u0017=da, and the X\nmatrix in Eq. (9).\n4. Use the derivative integrals to construct the CPHF equation in Eq. (12) (or the equiv-\nalentZ-vector equation42) and solve for Uafor all nuclear coordinates.\n5. Compute the energy gradient by contractions of all the above integrals and matrices\naccording to Eq. (7) or (10).\nThe second kind of DMRG calculation we consider is a DMRG complete active space\nself-consistent \feld (DMRG-CASSCF) calculation. For DMRG-CASSCF wavefunctions,\nthe DMRG energy is stationary to any orbital rotation, thus\nXij\u0000Xji= 0 (21)\nand by Eq. (7) and (10) this means that the orbital response is not required even though it\nis coupled to the response of the DMRG wavefunction. However, because the DMRG wave-\nfunction is not invariant to active space rotations for small M, it is necessary to optimize\nthe active-active rotations also, unlike in a traditional CASSCF calculation. Alternatively,9\nif active-active rotations are omitted, the DMRG-CASSCF gradient can be viewed as an ap-\nproximate gradient with a controllable error from active-active contributions (which vanishes\nasMis extrapolated to 1.)\nThe algorithm for the DMRG-CASSCF gradient is:\n1. Solve for the DMRG-CASSCF orbitals with the one-site DMRG wavefunction. In each\nmacroiteration:\n(a) Solve for the one-site state-speci\fc DMRG wavefunction, and compute the one-\nand two-body reduced density matrices \rijand \u0000ijkl.\n(b) Using\rijand \u0000ijkl, compute the orbital gradient and Hessian, both of which\ninclude elements for active-active rotations.\n(c) Update the orbitals with the orbital rotation matrix.\n2. Compute the AO density matrices \r\u0016\u0017and \u0000\u0016\u0017\u001a\u001b at the convergence of DMRG-\nCASSCF.\n3. Compute the AO derivative integrals dh\u0016\u0017=dA,d(\u0016\u0017j\u001a\u001b)=dA anddS\u0016\u0017=dA.\n4. Contract all the above integrals and matrices using Eq. (14) to obtain the energy\ngradients.\nV. ADIABATIC ORBITAL AND WAVEFUNCTION PROPAGATION AND EXCITED\nSTATE TRACKING\nGeometry optimization requires adiabatically propagating along a potential energy sur-\nface. For a DMRG calculation, this means that in each geometry step, the orbitals de\fning\nthe active space should change continuously, and the quantum numbers and associated\nblock-sparsity pattern of the matrices should not change. The former can be achieved using\nmaximum overlap techniques, while the latter can be done by \fxing the quantum numbers\nat the initial geometry. For state-speci\fc excited state calculations, the maximum over-\nlap technique is further important to prevent root-\ripping. Root \ripping in state-speci\fc\nDMRG calculations arises because the matrices optimized in the wavefunction for one state\n(15) are not optimal for another state100. (Note that the gradient formalism presented above\nis only valid for state-speci\fc, rather than state-averaged, DMRG calculations).10\nA. Orbital maximum overlap\nThe maximum overlap technique for the orbitals involves computing the overlap matrix\nbetween MO's of the ( m\u00001)th andmth step\nSm\u00001;m\nij =h m\u00001\nij m\nji\n=X\n\u0016\u0017Cm\u00001\ni\u0016Cm\nj\u0017h\u001em\u00001\n\u0016j\u001em\n\u0017i(22)\nwhereh\u001em\u00001\n\u0016j\u001em\n\u0017iare the AO overlap matrix elements of ( m\u00001)th andmth steps. For the\nactive space, we choose the orbitals at step mwith maximum overlap with the active space\norbitals at step m\u00001. Eq. (22) also allows us to align the MO phases for adjacent geometry\noptimization steps.\nB. Excited state tracking in DMRG\nWe further use maximum overlap of the DMRG wavefunctions to target and track the cor-\nrect state-speci\fc excited state solution. Within the standard ground-state sweep algorithm\nat a given geometry, the desired excited state can usually be found in the eigenspectrum at\nthe middle of the sweep (when the renormalized Hilbert space is largest) but can be lost at\nthe edges of the sweep when the renormalized Hilbert space is small (if it is generated for\nthe incorrect eigenvector). To keep following the excited state across the sweep by generat-\ning the appropriate renormalized Hilbert space, we ensure that at each block iteration we\nalways pick the Davidson solution with maximum overlap with the excited state solution at\nthe previous block iteration. Between geometries, we ensure that we are tracking the correct\nexcited state by computing the overlap between the DMRG wavefunctions at the di\u000berent\ngeometries. In principle, this requires multiplying the overlaps between the L\u001b,R\u001bmatrices,\nandcvectors. However, we \fnd it is su\u000ecient (and of course cheaper) to only compute the\noverlap between the cvectors for the two geometries, at the middle of the sweeps.\nThe state-speci\fc DMRG wavefunction maximum overlap scheme is:\n1. At the initial geometry, use a state-averaged DMRG algorithm to obtain initial guesses\nfornstates100. (The more robust two-site DMRG algorithm may be used here101, and\na highly accurate initial guess for a small M can be obtained by running back sweeps\nfrom large M102). Store the wavefunction vectors fcig(fori= 1;2;:::;n ) at the middle11\nof the sweep. Note that in the state-averaged procedure all nstates share the same\nleft and right rotation matrices fL\u001bgandfR\u001bg.\n2. At a new geometry optimization step (=initial geometry in the \frst step), restart\nthe DMRG sweep with the same M from the previous solution for the targeted ex-\ncited state, and use state-speci\fc DMRG with the one-site sweep algorithm to get\nthe new solution for the targeted excited state. (Note that any noise in the DMRG\nalgorithm should be turned o\u000b). At each block iteration, apply the following steps in\nthe Davidson solver:\n(a) Perform DMRG wavefunction prediction by Eq. (19) from the previous block\niteration, to obtain guess vectors fci\nguessgfor the current block iteration.\n(b) Perform the Block-Davidson algorithm to obtain solutions fci\nsolg.\n(c) Compute overlaps between vectors fci\nsolgandfci\nguessg, and align the phases when\nneeded.\n(d) Choose the new solution cx\nsolinfci\nsolgfor the targeted excited state, from the\nlargest overlap between cx\nsolandcn\nguess.\n(e) Store the vector cx\nsolas the new solution.\n3. Repeat Step. 2 in further geometry optimization steps.\nVI. EXCITED STATE GEOMETRIES OF TRANS-POLYENES\nExcited state geometry optimization in linear polyenes serves as a starting point to un-\nderstand the photophysical and photochemical behaviour of analogous systems, such as\nthe carotenoids, in biological processes. We take as our systems, the trans -polyacetylenes\nC2nH2n+2, withn= 5\u000010. We modeled the excited states and geometry relaxation as fol-\nlows: 1) We obtained ground state S 0(11Ag) geometries with DFT/B3LYP103. 2) We then\nused the DFT ground state geometries as initial guesses to perform ground state geometry\noptimization with DMRG-CI analytic energy gradients. 3) We recomputed excited states at\nthe DMRG optimized ground state geometries. 4) We then further relaxed the excited state\ngeometries with the DMRG-CI gradients. All calculations were performed with the cc-pVDZ\nbasis set104{106. The active spaces were chosen as ( ne,no), wherenis the total number of \u001912\nelectrons. We identi\fed the \u0019active spaces consisting of carbon 2 pzorbitals from the L owdin\nMO population analysis at the initial geometry, and tracked the active spaces through the\ngeometry relaxation with the orbital maximum overlap method in Sec. V A. We also car-\nried out additional calculations with a second \\energy-ordered\" active space, consisting of\nthe lowest \u0019and\u001borbitals to make up an ( ne,no) active space. We clearly distinguish\nwhen we are referring to the second active space in the discussion below. The initial ground\nstate DFT/B3LYP geometry optimizations were carried out with the Molpro quantum\nchemistry package107. State-speci\fc DMRG wavefunctions were obtained with the Block\nDMRG program3,4,13,108, using the state-speci\fc and adiabatic wavefunction tracking by\nwavefunction maximum overlap in Sec. V B. DMRG-CI gradients were implemented in the\nORCA quantum chemistry package. All calculations worked in the canonical HF orbital\nbasis (no localization). To improve the geometry optimization we employed approximate\nnuclear Hessians, updated by the BFGS method109{112.\nTo simplify the analysis, in this work we only considered geometry optimization in the\nplane . Non-planar geometries are of course relevant to polyene excited states but even at\nplanar geometries, important features of the electronic excited state geometries (e.g. the\nsolitonic structure) appear and remain to be understood at an ab-initio level. The planar\noptimization was not enforced explicitly other than through a planar initial guess, and other-\nwise the coordinates were allowed to relax in all degrees of freedom. Consequently, electronic\nwavefunctions were computed within C1spatial point group symmetry. We used three dif-\nferent numbers of renormalized states M=100, 500, 1000 to obtain DMRG wavefunctions for\nall states, to examine the in\ruence of wavefunction accuracy on the geometries. Converg-\ning DMRG wavefunctions to a high accuracy ensures the accuracy of the particle density\nmatrices, which then ensures that the correct geometric minima can be reached. However,\nwhen the magnitude of gradients was much larger than the unconverged DMRG error, (for\nexample, when the geometry was far from the equilibrium) loose DMRG convergence and\nfewer sweeps were used to decrease the computational time.\nTo further characterize the low-lying excited states, we analyzed the exciton and bi-\nmagnon character of state transitions using the transition particle density matrices. The\n\frst-order transition density matrix element in the MO basis between the ground (GS) and\nexcited states (ES) isD\n\tES\f\f\fcy\nicj\f\f\f\tGSE\n(23)13\nwhereiandjdenote spatial MO indices. We used the \frst-order transition density matrix\nto locate the \frst optically dark and bright states by the following well established state\nsignatures: 1) A single large element where i=LUMO,j=HOMO, indicating the \frst op-\ntically bright state. 2) Two dominant elements where i= LUMO + 1, j= HOMO and i\n= LUMO,j= HOMO - 1 indicating the \frst optically dark state. Real space particle-hole\nexcitation patterns were further analyzed by the real space \frst-order transition density\nmatrix, which was obtained by transforming the vir-occ block of the MO \frst-order transi-\ntion density matrix to the orthogonal 2 pzbasis. Real space particle-hole excitation patterns\nwere characterized by excitations of an electron from an orbital at R\u0000r=2 to an orbital\ntoR+r=2, whereRwas set at the centre of a polyene chain, and ris the particle-hole\nseparation length. We illustrate the excitons graphically by plotting\ncy\npcn\u0000p\u000b\n, wherepis\nthe index of the carbon 2 pzorbital, and nis the total number of 2 pzorbitals in the chain.\nSimilarly, the real space bimagnon character is characterized by the real-space double-spin\n\rip transition densityD\n\tES\f\f\fcy\np;\u001bcp;\u0000\u001bcy\nn\u0000p;\u0000\u001bcn\u0000p;\u001b\f\f\f\tGSE\n(24)\nwhere\u001b=f\";#g. The real space second-order transition density matrix was transformed\nfrom the vir-vir-occ-occ block of the MO basis second-order transition density matrix.\nAnalogously to previous studies, we further examined bond orders and geometrical defects\n(solitons) through the bond length alternation (BLA) function \u000en\n\u000en= (\u00001)n+1(xn+1\u0000xn) (25)\nwheren= 1;:::;Nbond, andxdenotes bond lengths. For even-site trans -polyacetylenes, the\ntwo edge bonds at the ground state are always double bonds, thus \u000enwill always be positive.\nConsequently, negative values of \u000enindicate a reversed bond order, and a vanishing ( \u000en= 0)\nvalue comes from two equal bond lengths, i.e., an undimerized region.\nA. State signatures and geometries\n1. Ground state S 0\nThe ground state of polyenes is denoted by the symmetry label 11Ag(here we are using\nsymmetry labels characteristic of idealized C2hsymmetry) and the relaxed ground state14\n0 2 4 6 8 10 12 14 16 180.000.010.020.030.040.050.060.070.080.090.100.11\nBond index DMRG\n DFT/B3LYPδn\nFIG. 1. Bond length alternation function \u000enfor relaxed ground state geometries of C 20H22, from\nleft to right.\ngeometries are planar and dimerized. For the ground state, DMRG wavefunctions with M\n= 100 are su\u000ecient to achieve qualitative accuracy in bond lengths of our studied polyenes.\nM= 500 is su\u000ecient for quantitative accuracy. For example, M= 100 produced errors of\nno more than 0.006 \u0017A for C 20H22, whileM= 500 converged the bond lengths to an error\nof 0.0003 \u0017A, as compared to bond lengths using M= 1000 (near exact). This \fnding is\nconsistent with the ground state wavefunction of even-carbon trans-polyenes being mostly\na single-determinant, and thus accurately described by DMRG in the canonical molecular\norbital basis with small M.\nThe BLA function \u000enof the relaxed ground state geometry of C 20H22from DMRG and\nDFT is shown in Fig. 1. The BLA functions from both DMRG and DFT give the same\npattern, showing a weaker dimerization in the middle region compared to the edges of\nthe carbon chain. Compared to the dimerization in DFT, the dimerization in the DMRG\ncalculations is suppressed, indicating a smaller dimerization gap. Compared to DFT, a \u0019-\nactive space Hamiltonian (as used in the DMRG calculations) is associated with a larger\ne\u000bective Coulumb interaction Udue to the lack of dynamic correlation. A suppression of\nthe dimerization can then be expected, as the dimerization magnitude behaves as U\u00003=2in\nthe strongly interacting limit113.15\n0.100 .120 .140 .160 .180 .202.53.03.54.04.55.05.56.0Energy gap (e.V.)1\n/n\nFIG. 2. Vertical and relaxed excitation energies from DMRG optimized geometries: vertical S 0-S1\n(opened squares), vertical S 0-S3(opened circles), relaxed S 0-S1(solid squares), relaxed S 0-S3(solid\ncircles).\n2. Excited states\nThe \frst optically bright state in the single- \u0019complete active space is the third excited\nstate S 3, and denoted by the symmetry label 21Bu. The corresponding MO based \frst-order\ntransition density matrix between S 3and S 0(de\fned by Eq. (23)) possesses an element \u00181.0,\nwherei= LUMO and j= HOMO, along with other elements \u00140.1. This signals a (HOMO\n!LUMO) single particle-hole transition, characteristic of the \frst optical transition.\nThe 11Bustate corresponds to the second excited state S 2in the single- \u0019active space.\nNotable \frst-order excitations in the S 0/S2transition, for instance in C 10H12, are (HOMO!\nLUMO + 2) and (HOMO - 2 !LUMO) excitations, both with elements \u00180.5 at the ground\nstate equilibrium geometry. A large (HOMO !LUMO) excitation is missing for the S 0/S2\ntransition for all the polyenes, ruling it out as the usual bright state. Note that the order\nof excited states depends on the choice of active space, i.e., the e\u000bective Hamiltonian. If\none changes from the single- \u0019active space to an energy-ordered active space which includes\nboth\u001band\u0019orbitals within the ( ne,no) active space window, one \fnds that the 11Bustate\nis an S 2state corresponding to the physical optically bright HOMO !LUMO transition.\nThis demonstrates the well-known strong e\u000bect of dynamical correlation on the low-lying\nexcited state order in linear polyenes.\nThe \frst optically dark state is the S 1state, denoted by 21Ag. The S 0/S1transition16\nexhibits dominant (HOMO !LUMO + 1) and (HOMO - 1 !LUMO) single excitations,\nalong with a dominant (HOMO, HOMO !LUMO, LUMO) double excitation. The position\nof this low-lying excited state remains as the S 1in an energy-ordered active space.\nFor the bright state, optimized bond lengths were not strongly dependent on M. For a\nsmall system such as C 10H12,M= 100 produced a largest error of 0.0003 \u0017A in the bond\nlengths, as compared to the M= 1000 result. For a larger system such as C 20H22, the\nbond lengths at M= 100 di\u000bered the ones at M= 1000 by no more than 0.005 \u0017A, and\nthe largest error at M= 500 was only 0.0006 \u0017A. For the dark state, however, the precision\nof the optimized geometry was more sensitive to the choice of Mfor the longer polyenes.\nThis may not be surprising, as the \frst optically bright state is mainly a single-reference\nstate, while the lower dark state has more challenging multi-reference character114. For all\nthe polyenes considered, if we use small M, the largest error in the bond lengths of the dark\nstate occurs for bonds around the geometrical defects (solitons). In C 20H22, the largest error\natM= 100 is about 0.025 \u0017A, coming from the bonds C 3-C4and C 16-C17which are around\nthe solitons (see in Sec. VI C). On the other hand, central bonds in the dark state are much\nless dependent on M, e.g.M= 100 yields errors \u00140.012 \u0017A for bonds from C 6-C7to C 13-C14\nin C 20H22. In a localized real space view, this behaviour re\rects the strong localization of\nmulti-reference electronic structure around the geometrical defects.\nB. Excitation energy\nWe show vertical and relaxed excitation energies as a function of 1 =nfor the \frst optically\ndark (21Ag) and \frst optically bright (21Bu) states for all considered C 2nH2n+2in Fig. 2.\nCompared to the experimental excitation energies for C 10H12to C 14H16in hydrocarbon\nsolutions115, our relaxed excitation energies are 0.3 eV higher for the relaxed dark state, and\n1.7 eV higher for the bright state. This is in part due to the lack of dynamic correlation in\nour calculations as well as basis and solvent e\u000bects.\nThe dark state is always observed as below the bright state. We observe relaxation\nenergies for all the polyenes of about 0.35 eV for the bright state and about 1.20 eV for the\ndark state. The substantial relaxation energy for the dark state is consistent with the much\nlarger geometry relaxation as compared to the bright state67.\nOur calculations \fnd the 11Bustate to lie relatively close to the 21Bustate at the ground17\n02 4 6 8 1012141618-770.06-770.05-770.04-770.03-770.02-770.01-770.00 \nS2 \nS3Energy (a.u.)G\neometry optimization step\nFIG. 3. Energies of the S 2and S 3states in the S 2geometry optimization of C 20H22, computed with\nthe energy-ordered active space, as a function of the geometry relaxation step. At the ground state\nequilibrium (step 0), the S 2and S 3states are 11Bu(\frst bright state) and 21Bustates respectively.\nAt step 4 the molecule gives a S 2and S 3gap of 0.019 eV, strongly indicative of a conical intersection.\nAfter this step the 11Buand 21Bustates are swapped in terms of the state energy order. (Note\nthat the S 3state energy oscillates as only the S 2state energy is being minimized). The molecular\ngeometry remains planar along the relaxation.\nstate equilibrium geometry, with a 11Bu-21Buenergy gap consistently about 0.27 eV for all\nthe polyenes. Given the small magnitude of this energy gap, it seems likely that there can be\nan energy crossing between 21Buand 11Bustates. If we use the energy ordered active space\nwe do \fnd an energy crossing between these states for C 20H22at a planar geometry, near the\nFranck-Condon region (Fig. 3). Of course we also expect non-planar conical intersections,\nas previously found in butadiene93,94and octatriene95.\nC. Solitons\nThe BLA\u000enfunctions for the \frst optically dark (21Ag) and \frst optically bright (21Bu)\nstates are shown in Figs. 4 and 5. These curves are almost parallel across all the polyenes\nfor the dark and bright state respectively, indicating generally similar behaviour across the\nsystems.\nFor the 21Agstate, the BLA in short polyenes C 10H12and C 12H14is completely reversed\nfrom the ground-state, as shown by the all negative \u000envalues along the chain. The reversal18\n0123456789 1 0-0.06-0.04-0.020.000.020.040.060.080.10 C10H12\n C12H14\n C14H16\n C16H18\n C18H20\n C20H22\nBond indexδn\nFIG. 4. Bond length alternation function \u000enfor relaxed \frst dark state geometries, from edge (left)\nto center (right).\n0123456789 1 00.000.020.040.060.080.10 C10H12\n C12H14\n C14H16\n C16H18\n C18H20\n C20H22\nBond indexδn\nFIG. 5. Bond length alternation function \u000enfor relaxed \frst bright state geometries, from edge\n(left) to center (right).\nof BLA in 21Agin short polyenes has previously been understood in terms of the dominant\nvalence bond con\fgurations58with reversed BLA. For long polyenes, undimerization emerges\nnear the edges as shown by changes in the sign of the \u000enfunctions, and the BLA is opposite\non the two sides of the undimerized regions. This result is in agreement with earlier semi-\nempirical studies on long polyenes66,67, and our result shows the two-soliton structure in the\nrelaxed 21Agstate.\nFor the 21Burelaxed geometry, \u000ensystematically shows a polaronic defect in the chain\ncentre. This is also consistent with previous semi-empirical studies66,67. For short polyenes,19\nthe vanishing dimerization in the central region can be understood in terms of ionic VB\ncon\fgurations along the chain58. In terms of excitons, the polaronic geometry is also viewed\nas evidence of a bound particle-hole excitation localized near the chain centre78.\nD. Excitons\nWithin the one-electron manifold, we can visualize the excitons with the real space\nparticle-hole excitation density\ncy\npcn\u0000p\u000b\n. As we relax the geometry, we can observe the\nshape of the exciton change. Geometry relaxation is important to overcome the exciton self-\ntrapping78, e.g., in a polyene chain in its dimerized ground-state geometry. The real space\nparticle-hole excitation densities of C 20H22are shown in Fig. 6 and Fig 7, for the bright and\ndark state respectively.\nAt the ground state equilibrium geometry, i.e., a dimerized geometry, the particle-hole\nexcitations of the bright state are strongly bound, as seen in Fig. 6(a). This is similar to as\nseen in the single-peak real space exciton structure from DFT-GWA-BSE calculations116, as\nwell as the n= 1 Mott-Wannier exciton pattern in the weak-coupling limit66. For the dark\nstate, particle-hole pairs are slightly separated at the dimerized geometry, as illustrated\nby the double-peak real space exciton structure in Fig. 6(a). This has been identi\fed in\nprevious studies66,116, as ann= 2 Mott-Wannier exciton. However, the amplitudes of the\ndensities are ten times smaller as compared to that of the bright state, essentially suggesting\nneglegible exciton character for the dark state reached by a vertical transition.\nAfter geometry relaxation, the particle-hole separation in the bright state increases, al-\nthough the particle-hole pair remains bound at the bright state equilibrium geometry, as\nshown in Fig. 6(b). For the dark state, however, geometry relaxation seems to unbind the\nparticle-hole pair, as shown by largely separated peaks in Fig. 7(b). Along with the enhanced\ntransition density amplitude, this suggests the emergence of a long-distance charge-transfer\ncharacter associated with the dark state equilibrium geometry.\nE. Bimagnons and singlet \fssion in 21Ag\nThe relaxed dark state 21Aggeometry possesses a separated two-soliton structure as dis-\ncussed in Sec. VI C. The locally undimerized regions in the relaxed 21Agstate can be thought20\n(a)\n12345678910111213141516171819200.000.010.020.030.040.050.06Excitation densityC\narbon index\n(b)\n12345678910111213141516171819200.000.010.020.030.040.050.06Excitation densityC\narbon index\nFIG. 6. Real space particle-hole excitation density of C 20H22between ground state and \frst bright\nstate, computed at relaxed geometries of (a) ground state (b) \frst bright state.\nto arise from a form of \\internal singlet \fssion\"78, i.e., forming local triplets (magnons) while\nthe total spin remains a singlet. The local triplets can be identifed from the local peaks of\nthe real space spin-spin correlation function of the 21Agwavefunction as in Ref.66.\nHere, we can also characterize the bimagnon character by the real space double-spin \rip\ntransition density between the S 0and 21Agstates (see Eq. (24)). We show the real space\ndouble-spin \rip transition density of C 20H22as a function of the site index in Fig. 8. At the\nground state equilibrium, the bimagnons are con\fned near the chain centre, as indicated\nby the local central double peaks. However, the bimagnons are highly mobile, and with\ngeometry relaxation, the singlet \fssion character becomes much more delocalized. The\ntransition density distribution possesses two peaks at carbon 3 and 17 at the dark state21\n(a)\n12345678910111213141516171819200.0000.0010.0020.0030.0040.0050.006Excitation densityC\narbon index\n(b)\n12345678910111213141516171819200.000.010.020.030.040.05Excitation densityC\narbon index\nFIG. 7. Real space particle-hole excitation density of C 20H22between ground state and \frst dark\nstate, computed at relaxed geometries of (a) ground state (b) \frst dark state.\nequilibrium geometry, which is consistent with the positions of the solitons shown in Fig. 4.\nVII. CONCLUSIONS\nWe presented the detailed formalism for state-speci\fc DMRG analytic energy gradients,\nincluding a maximum overlap algorithm that facilitates state-speci\fc excited state geom-\netry optimizations. We employed these techniques to study the ground and excited state\nelectronic and geometric structure of the polyenes at the level of DMRG-CI. Our quanti-\ntative results are consistent with earlier qualitative semi-empirical studies of the exciton,\nbimagnon, and soliton character of the excited states. In addition to complex bond-length22\n12345678910111213141516171819200.00000.00050.00100.00150.00200.00250.0030Double-spin flip densityC\narbon index at ground state equilibrium geometry \nat first dark state equilibrium geometry\nFIG. 8. Real space double-spin \rip density between the ground state and the \frst dark state.\nalternation patterns, we \fnd evidence for a planar conical intersection.\nDMRG analytic energy gradients provide a path towards the dynamical modeling of\nexcited state and highly correlated quantum chemistry. The interaction of dynamic and\nnon-adiabatic e\u000bects with strong electron correlation remains an open issue, which can now\nbe explored with the further development of the techniques described here.\nVIII. ACKNOWLEDGEMENT\nWeifeng Hu thanks Gerald Knizia, and Bo-Xiao Zheng for discussions on the gradient\ntheory, and thanks Sandeep Sharma for help with the Block DMRG code. This work was\nsupported by the US National Science Foundation through CHE-1265277 and CHE-1265278.\n1White, S. R.; Martin, R. L. J. Chem. Phys. 1999 ,110, 4127{4130.\n2Mitrushenkov, A. O.; Fano, G.; Ortolani, F.; Linguerri, R.; Palmieri, P. J. Chem. Phys. 2001 ,115, 6815.\n3Chan, G. K.-L.; Head-Gordon, M. J. Chem. 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Finite density quantum \feld theories have evaded \frst principle Monte-Carlo\nsimulations due to the notorious sign-problem. The partition function of such theories appears as\nthe Fourier transform of the generalised density-of-states, which is the probability distribution of\nthe imaginary part of the action. With the advent of Wang-Landau type simulation techniques\nand recent advances [1], the density-of-states can be calculated over many hundreds of orders\nof magnitude. Current research addresses the question whether the achieved precision is high\nenough to reliably extract the \fnite density partition function, which is exponentially suppressed\nwith the volume. In my talk, I review the state-of-play for the high precision calculations of\nthe density-of-states as well as the recent progress for obtaining reliable results from highly\noscillating integrals. I will review recent progress for the Z3quantum \feld theory for which\nresults can be obtained from the simulation of the dual theory, which appears to free of a sign\nproblem.\n1. Introduction\nThe properties of strongly interacting matter has sparked many important investigations using\naccelerator experiments and large scale theoretical studies. In an astrophysical context, the\ntheory of interactions, QCD, dominates the area around 10\u00006seconds after the Big Bang when\nquark matter con\fnes to colour neutral hadrons. Since QCD captures the impact of the strong\nnuclear forces, it is widely believed that QCD plays an essential role to understand compact\nstar matter as it may exist nowadays e.g. in neutron stars (see \fgure 1 for an illustration). The\nso-called QCD phase diagram characterises the states of matter as a function of the density and\ntemperature in a 2d graph. Completing this diagram in its extreme regions and in particular\nfor cold and dense matter is an outstanding problem which still triggers model building and\nnew techniques for computer simulation after 30 years of intense research. Under moderate\nconditions, quarks and gluons are con\fned to colour-neutral hadrons, while this con\fnement\nfeature of QCD is believed to cease to exist under extreme conditions, temperatures and/or\ndensities. In the early universe before the QCD phase transition roughly at time 10\u00006seconds,\nmatter has not yet clustered due to gravity and, as a result, the density is very low compared to\ne.g. nuclear matter density. Similarly, the conditions in heavy ion collision experiments carried\nout at RHIC (located at Brookhaven National Laboratory (BNL) in Upton, New York) or LHC\n(built by the European Organization for Nuclear Research (CERN)) produce very hot matterarXiv:1503.00450v1  [hep-lat]  2 Mar 2015QCD physics\nNIC, FZ−JuelichFigure 1. QCD impact on the evolution of the early Universe.\ncompact  stars\nquark  gluon plasma\nsuperconductorconfinementcolour\ncolour\ndensitytemperature\nnuclear  matterearly  universe\n−610  secLHC\nRHIC\nFigure 2. Sketch of the QCD phase\ndiagram.\ncompact  stars\n densitycolour\nconfinementtemperature\nnuclear  matterquarkyonic\n\"chrial spirals\"\nFermi EinsteinLangfeld, Wipf\n2011Fukushima, 2009McLarren, Pisarski\n2010\ncondensationFigure 3. QCD phase diagram:\n\frst principle calculations and model\nbuilding.\nat low densities. From an experimental point of view, very little is known even at moderate\ndensities let alone the cold and dense regime of compact star matters (see \fgure 2).\nLattice gauge simulations o\u000ber \frst principle non-perturbative results with good control over\nthe errors. Markov chain Monte-Carlo simulations can be very successfully applied and reliable\nresults for the states of matter can be obtained at all temperatures and small baryon chemical\npotentials (shaded area in \fgure 3, see e.g. [2] for a review). Over the recent years, many new\nmodels have been developed to describe QCD matter at high densities: quarkyonic matter is\nmotivated by the so-called 't Hooft limit of the hypothetical theory with a large number of\ncolours and sees the quarks decon\fned inside the Fermi sphere while only the baryons at the\nsurface of the Fermi sphere undergo colour con\fnement|[3]. The chiral magnetic e\u000bect [4]\ntakes into account the strong magnetic \feld that are produced by the colliding charges in heavy\nion experiments and uses an e\u000bective quark model to estimate their impact on the QCD phase\nstructure. In the con\fnement phase, quarks in a certain gluonic background can change their\nstatistics from Fermi to Bose type. At moderate densities, these so-called centre dressed and\ncon\fned quarks therefore might undergo Bose condensation leading to to new state of cold andcon\fned matter [5]. None of these ideas have yet been tested by \frst principle calculations.\nAt a \fnite baryon chemical potential, the QCD action acquires an imaginary part, and, hence,\nstandard Monte-Carlo techniques cannot be applied for the lattice simulation of QCD at \fnite\ndensities. This has become known as the notorious sign problem. Early attempts pursued\nMonte-Carlo simulations with the modulus of the quark determinant and included the phase\nfactor to the observable to be calculated. It turns out that the expectation value of the phase\nfactor is very small (see [6] for an illustration) showing that the such generated Monte-Carlo\ncon\fgurations contain very little information on \fnite density QCD. The sign problem has\nturned into an overlap problem.\nThe recent past has seen a renaissance of ideas targeting dense and cold fermionic matter.\nSome of the methods were proposed some time ago, but have now reached an unprecedented\nlevel of sophistication. The reweighting approach proposed by Fodor and Katz [7] uses a multi-\nparameter action to optimise the overlap. This method is particularly suited for intermediate\ntemperatures and might possess a reach that covers the critical endpoint of the QCD phase\ndiagram [8]. Langevin simulations of lattice gauge theories avoid the positivity constraint of\nthe Gibbs factor, which lies at the heart of Monte-Carlo simulations, and might therefore be\nsuitable for \fnite density simulations [9, 10]. This technique regained a lot of interest when Aarts\nshowed that stochastic quantisation can evade the sign problem at least for the relativistic Bose\ngas [11, 12]. Although the conceptional question whether the approach converges to the correct\nanswer [13] is still under active investigations, the approach is one of the few methods that\nare currently applied to \fnite density QCD [14]. It might appear that integrating the gluonic\ndegrees of freedom before the fermion \felds alleviates the sign problem. This could be done\ne.g. in the strong coupling limit [15] leading to a description of Nuclear Physics suiting lattice\nsimulations [16]. It was observed in 1d QCD that even integrating part of the gluonic degrees\nof freedom leads to substantial improvements [17]. Finally, a reformulation of the theory might\nimprove on the sign the problem or remove it altogether. Indeed, theories the dual of which\nare real are then accessible by standard Monte-Carlo techniques. Example for complex action\nspin models that are real upon dualisation are Z3spin model [18] or the O(2) model at \fnite\ndensities [19, 20]. These models can be e\u000eciently simulated using worm type algorithms [21].\nAlternative lattice discretisations [22] and spin blocking techniques in combination with the\n(tensor) renormalisation group approach [23] might be equally successful to eliminate the sign\nproblem from Yang-Mills theories. Also not hinging on a real dualisation of the theory is\nthefermion bag approach approach by Chandrasekharan [24] for which the sign problem is\nrelegated to \fnite size fermion bags. This approach has been seen to be very e\u000ecient for\nfermion theories with four-fermion coupling such as the Thirring model with massless fermions\non large lattices [25].\nAn e\u000ecient alternative to conventional Monte-Carlo simulations is based upon the numerical\ncomputation of the density of states using the multi-canonical algorithm [27] or a Wang-Landau\ntype strategy [26]. A modi\fed version of the Wang-Landau method is the LLR algorithm [1],\nwhich is e\u000bective for theories with continuous degrees of freedom as opposed to spin models (see\nalso [28]). The latter algorithm has been extended from the calculation of the action distribution\nto accessing probability distributions of other extensive quantities such as the SU(2) Polyakov\nline [29]. Furthermore, it has been proposed in [30] to use LLR techniques for a high precision\ncalculation of the distribution of the imaginary part of the action. Once this quantity has been\ndetermined, the partition function of the complex theory can be computed semi-analytically by\ncarrying out the Fourier transform of the corresponding probability distribution [30]. Below, we\nwill summarise the state of a\u000bairs concerning density-of-states methods and the LLR algorithm\nin particular to simulate theories with a sign problem. An overview on selected new methods\nsolving the sign problem can be found in the recent review by Aarts [31].2. Density-of-states approach to complex action systems\n2.1. Density-of-states and the overlap problem\nLet us consider the partition function Zof a theory of one degree of freedom \u001ewith a complex\naction:\nZ=Z\nD\u001eexpf\fSR[\u001e] +i\u0016SI[\u001e]g (1)\nwhere\u0016is the \\chemical potential\", and SR=Iare real and imaginary parts of the action. We\nintroduce the generalised density of states [30] by\nP\f(s) =Z\nD\u001e\u000e\u0010\ns\u0000SI[\u001e]\u0011\nexpf\fSR[\u001e]g: (2)\nFor\f= 0,P0(s) just counts the number of states with the constraint that the imaginary part\nof the action is given by s. At \fnite \f, the number count is weighted by the \\Gibbs\" factor\nexpf\fSR[\u001e]g. OnceP\f(s) is known, the task to calculate the partition function boils down to\nevaluate the integral\nZ(\f;\u0016) =Z\ndsP\f(s) expfi\u0016sg: (3)\nSince the integrand in (2) is perfectly real, the di\u000eculties with the sign problem are relegated\nto the 1-dimensional integral (3). In fact, P\f(s) could be estimated by performing a standard\nMonte-Carlo simulation with the Gibbs factor exp f\fSR[\u001e]gand to bin the values for SIin a\nhistogram. The LLR algorithm will provide us with \f= 0,P0(s) over hundreds of orders of\nmagnitude (see below for an example). The aim of this paper is to discuss the options for a\ncalculation of the highly oscillating integral (3).\nLet us scope the amount of di\u000eculty that resides with this task. We \frstly note that the\nactionSiis an extensive quantity SI(\u001e) =VsIwithVthe number of degrees of freedom (volume)\nand with the action density siof order one. A good qualitative choice (see e.g. [20]) is given by\nP\f(s) = expn\n\u0000s2\nVo\nleading to Z=Z\ndse\u0000s2=Vexpfi\u0016sg / expf\u0000\u00162\n4Vg:\nFor a chemical potential \u0016of order one, Zis exponentially suppressed with the number of degrees\nof freedom V. On the other hand, P\f(s) is only known numerically and of order one at least for\nsmalls. For a successful evaluation of the integral in Z(3), any numerical method for obtaining\nP\f(s) must have the properties\n\u000fexponential error suppression for extensive quantities\n\u000ffor the whole domain of support for SI.\nThe LLR algorithm proposed in [1] just delivers that.\n2.2. TheZ3spin model as showcase\nThe key question is whether the quality of the result for P\f(s) obtained by the LLR algorithm\nis good enough to admit a reliable calculation of the partition function Zvia the integral (3).\nThe answer to this question is model dependent. The 3-dimensional Z3spin model on a cubic\nlattice at \fnite density maps onto a real action system upon dualisation and is thus open to\nstandard Monte-Carlo simulations. It serves as a \frst benchmark test in the feasibility study\nfor our approach to theories with a sign problem. Here, we will review our \fndings for the\n3-dimensional case. Degrees of freedom are the centre elements z(x) that take values from the\ngroupZ3\nz(x)2f1;z+;z\u0000g; z \u0006= (1 +ip\n3)=2:-1000 -500 0 500 1000\nN+ - N-0200400600800 density-of-statesFigure 4. Histogram count for N+\u0000N\u0000/\nSI(linear scale); 243lattice,\u001c= 0:17,\u0014=\n0:05.\n-1000 0 1000 2000 3000 4000 5000 6000\nN+ - N-1e-601e-451e-301e-151\nhistogram\ndensity-of-statesFigure 5. Same histogram on\na logarithmic scale with the LLR\nresult now reaching beyond N+\u0000\nN\u0000\u00195;500.\nThe action is given by\nS[z] =\u001cX\nx;\u0017[zxz\u0003\nx+\u0017+cc] +X\nx[\u0011zx+ \u0016\u0011z\u0003\nx]; \u0011 =\u0014exp(\u0016);\u0016\u0011=\u0014exp(\u0000\u0016):(4)\nThis model is inspired by \fnite density QCD in the heavy quark limit, and the parameter \u001cis\nreminiscent of the temperature and \u0014re\rects the quark hopping parameter [18]. Apparently,\nthe action becomes complex as soon as \u00166= 0. If for a given con\fguration z(x) the quantity N\u0006\nrepresents the number of spins on the lattice with z=z\u0006, the imaginary part of the action can\nbe written as:\nSI=\u0011\u0000\u0016\u0011\n2iX\nx[z\u0000z\u0003] =p\n3\u0014sinh(\u0016) [N+\u0000N\u0000]: (5)\nWe have performed a standard Monte-Carlo simulation using a 243lattice,\u0014= 0:17 and\u0014= 0:05\nto obtain a histogram for N+\u0000N\u0000(see [30] for details). The result is shown in \fgure 4 on a\nlinear scale (see \fgure 5 with the y-axis on a logarithmic scale). Note that we have very little\n\\events\" with N+\u0000N\u0000>1000. For the calculation of the Fourier transform to obtain the\npartition function Zin (3), the tails with jN+\u0000N\u0000j\u001d1000 signi\fcantly contribute for \u0016\u00191.\nWe recover the overlap problem in the light of the density-of-states approach. Our result for\nP(N+\u0000N\u0000) using the LLR method is also shown in the \fgures 4 and 5. We \fnd a reassuring\nagreement with the standard simulation result and moreover we have obtained the distribution\nP(N+\u0000N\u0000) for values as large as N+\u0000N\u0000= 5000 and over sixty orders of magnitude.\n3. The partition function from highly oscillating integrals\n3.1. Polynomial \ft\nOur task is now to carry out the Fourier transform of the generalised density of states P(s)\nin order to obtain the partition function Z(see (3)). One advantage of our approach is that\nwe can use sophisticated integration techniques, which converge like 1 =np,p > 1, wherenis\nthe number of evaluations of the integrand. Note that any Monte-Carlo integration that sub-\nsums the Fourier transform necessarily converges at best like 1 =pn. Note, however, that even\nthe sophisticated integrations techniques would fail for sizeable values of the chemical potential\nwithout further knowledge of the function P(s). We have so far studied the density-of-states forthe theories U(1),SU(2),SU(3) andZ3and a common feature has been that logP (s) is indeed\nvery smooth and, as expected, monotonic functions of its variable s. In the present case, we also\nhave the symmetry under re\rection P(s) =P(\u0000s). The \\smoothness\" of P(s) is summarised\nby the fact that the Taylor expansion\nlnP(s) =qX\ni>0;evencisi; q = 2;4;6;8;::: (6)\ncan produce results that are indistinguishable from the numerical \fndings for P(s) within error\nbars. Depending on the region of the parameter space \f= (\u001c;\u0014),qas small as 4 might be\nsu\u000ecient. As soon as an acceptable representation of the numerical data in terms of the ansatz\n(6) is found, the calculation of the partition function can be performed in a \\semi-analytic\" way\nusing advanced numerical integration techniques:\nZ(\u0016) = 2Z\ndsP(s) cos(\u0016 s) = 2Z\ndsexp(qX\nicisi)\ncos(\u0016s): (7)\n3.2. Asymptotic referencing\nSimilar to the scenario in the previous subsection, it might be useful to describe the gross features\nofP(s) by an analytic function, say Pasy(s) especially for large values s. Decomposing\nP(s) = \u0016P(s)Pasy(s); (8)\nthe function \u0016P(s) might have a moderate range of values although P(s) spans many orders\nof magnitude. For a technical side-remark, we point out that the LLR algorithm [1] can be\neasily adapted to directly produce \u0016P(s) for a given choice for Pasy(s). The partition function is\nobtained by Fourier transformation:\nZ(\u0016) = FT[P](\u0016) =Z\ndsP(s) ei\u0016s=Z\ndxFT[\u0016P](x) FT[Pasy](\u0016\u0000x): (9)\nThe idea is that FT[ Pasy] is analytically available and has already incorporated a good deal of\nthe cancellations. For moderate values of \u0014and\u001c, a sensible choice is\nPasy(s)/expf\u0000\u000bs2g leading to FT[ Pasy](\u0016\u0000x)/exp(\n\u0000(\u0016\u0000x)2\n4\u000b)\n:(10)\nDepending on the size of the intrinsic scale \u000b, we only need to numerically calculate \u0016P(s) for\ns\u0019\u0016for the chemical potential \u0016of interest.\n3.3. Eigenfunction expansion\nLet us expand the generalised density-of-states P(s) in terms of a complete set of eigenfunctions\n(in the L2 sense) that are eigenfunctions of a di\u000berential operator:\nP(s) =X\nncn n(ks); (11)\nwherekis a parameter that will be adapted to the intrinsic scale of the theory under studies.\nWe need not necessarily adopt an eigensystem with an all discrete set of eigenfunctions. Here,\nwe have indeed the eigenfunctions of the harmonic oscillator in mind having this property:\n\u0000d2\nds2 n(ks) +k4s2 n(ks) =k2(2n+ 1) n(ks): (12)The ortho-normal eigenfunctions are the well-known Hermite functions:\n n(x) =1q\n2nn!p\u0019e\u0000x2=2Hn(x); H n(x) = (\u00001)nex2dn\ndxne\u0000x2; (13)\nwhere theHn(x) are the Hermite polynomials. Using the \\Schr odinger equation\" (12), one\nproves that the eigenfunctions are \fxed points of the Fourier transformation:\nFT[ n](\u0016) =Z\nds n(ks) ei\u0016s=p\n2\u0019\nkin n\u0012\u0016\nk\u0013\n: (14)\nWe therefore \fnd for the partition function\nZ(\u0016) = FT[P](\u0016) =X\nncnFT[ n](\u0016) =p\n2\u0019\nkX\nnincn n\u0012\u0016\nk\u0013\n: (15)\nIf the chemical potential \u0016is larger than the intrinsic scale k, we observe that we attain\nsmall values for Zalready from the asymptotic behaviour of the Hermite functions, i.e.,\n n(x)\u0019expf\u0000x2=2g.\n4. TheZ3spin model - numerical results\n0 0.5 1 1.5 2µ1e-161e-081O(µ)DS L=24\nLLR L=24\nDS L=22\nDS L=20\nDS L=18\nDS L=16\nLLR L=22\nLLR L=20\nLLR L=18\nLLR L=16\nFigure 6. The overlap O(\u0016) (17) from a direct Monte-Carlo simulation of the dual theory\n(DS) and from the LLR approach for several lattice sizes L3.\u001c= 0:1,\u0014= 0:01.\nIn order to quantify the in\ruence of the imaginary part, we introduce the partition function\nZmodthat features the Z3action (4) from which we have dropped the imaginary part:\nSmod[z] =\u001cX\nx;\u0017[zxz\u0003\nx+\u0017+cc] +\u0014cosh(\u0016) [2N1+N++N\u0000]; (16)whereN1is the number of z= 1 elements on the lattice. The partition function of the modi\fed\ntheory does depend on the chemical potential, but note that it can be simulated by standard\nMonte-Carlo techniques since its Gibbs factor is real and positive by construction. We then\nde\fne the overlap between the full theory and the modi\fed theory by\nO(\u0016) =Z(\u0016)\nZmod(\u0016): (17)\nWe point out that being able to calculate the overlap O(\u0016) provides access to the observables\nof the full theory. For instance, the density \u001a(\u0016) acquires two parts:\n\u001a(\u0016) =dlnZ(\u0016)\nd\u0016=dlnO(\u0016)\nd\u0016+dlnZmod(\u0016)\nd\u0016; (18)\nwhere the latter part is free of a sign problem and is calculable by standard Monte-Carlo\nsimulations.\nAn appealing feature of the Z3spin model is that its dual is a real theory open for Monte-Carlo\nsimulations. In fact, the theory can be e\u000eciently simulated by a worm type algorithm (see [18])\nwhere the \\worms\" are conserved \rux lines of the dual theory (see [20] for this interpretation).\nIt is still di\u000ecult to calculate the partition function itself since theories with di\u000berent chemical\npotentials di\u000ber in the free energy density f(\u0016) leading to poor overlap:\nZ(\u0016+ \u0001\u0016)\nZ(\u0016)= expn\n\u0000[f(\u0016+ \u0001\u0016)\u0000f(\u0016)]Vo\n: (19)\nWe used a variant of the \\snake algorithm\" [32] to calculate ratios of the partition function and\nto reconstruct the partition function from those:\nZ(k\u0001\u0016) =Z(0)kY\n`=1Z(`\u0001\u0016)\nZ((`\u00001)\u0001\u0016); (20)\nwhere \u0001\u0016must be chosen small enough (depending on the number of degrees of freedom V) to\nensure a good enough signal-to-noise ratio. We here point out an advantage of the LLR approach:\nksimulations of the dual theories are necessary to arrive at the target value \u0016t=k\u0001\u0016, while\nthe LLR approach can aim directly at the chemical potential \u0016tof interest.\nWe have simulated the Z3theory with non-zero chemical potentials using the LLR\nmethod [30]. We point out that the method is exact implying that the numerical results need to\nagree within error bars with the exact values. Note, however, that sophisticated techniques for\nthe error analysis (e.g. bootstrap) might be needed and that standard Gaussian error analysis\nmight fail at least in certain regions of parameter space [33]. Our results from the direct\nsimulation (DS) of the dual theory are shown in \fgure 6 in comparison with our results from\nthe LLR approach. In the latter case, we have used the method of the polynomial \ft from\nsubsection 3.1 for the evaluation of the highly oscillating integral. We indeed encounter a strong\nsign problem since the overlap is as small as 10\u000016for\u0016\u00192. So far, we have only explored a\nlimited region of the parameter space ( \u001c;\u0014), but our results serve as a proof of concept: for a\nrange of volumes and for the case of a strong sign problem, the simulation of the theory in its\noriginal degrees of freedom is feasible using the LLR techniques.5. Conclusions\nQuantum \feld theory at \fnite density or, more general, statistical systems with complex\nactions such as the imbalanced Fermi gas [34] still await \frst principles results from computer\nsimulations. In the latter case, theoretical \fndings can be scrutinised against experiments,\nand, given the level of abstraction that went into model building, agreement would signal an\nunderstanding of the materials at hand (see e.g. [35]). In the context of QCD at \fnite baryon\ndensities, a variety of mechanisms have been proposed over the last couple of years that should\ndescribe the states of baryon matter in the intermediate temperature and density range with or\nwithout a strong magnetic \feld. Proposals feature \\quarkyonic matter\" [3], suggested on the\nbasis of the 't Hooft limit, the \\chiral magnetic e\u000bect\" [4] or the \\Fermi-Einstein condensation\"\nof quarks [5], and this is not meant to be a complete list. Lattice simulations would provide a\ngenuine non-perturbative approach with good systematic control of the errors, but are hampered\nby the notorious sign problem: for a non-vanishing chemical potential, the Gibbs factor is\ncomplex (or, at least, not positive de\fnite) and the action based importance sampling, which is\nat the heart of the Monte-Carlo simulations, are impossible.\nAlongside the new theory proposals for the potential state of matter a \fnite densities, the last\ndecade has seen promising progress for the simulation of theories with a sign problem. In fact,\nmany of the related ideas are rooted in the literature for decades, but techniques have reached an\nunprecedented level of sophistication. A good example are the Langevin simulation of complex\nactions systems, which date back to the early works by Parisi [9] and Karsch and Wyld [10] from\nthe mid eighties, but underwent a Renaissance when it was realised the Silver-Blaze problem\ncan be avoided for the case of a relativistic Bose gas [11].\nSimilarly, the LLR algorithm [1] emerged from a modi\fcation of Wang-Landau type\nalgorithms [26] and have progressed along the lines of the so-called density-of-states methods\n(see e.g. [27] or [37, 38, 39, 40]). The LLR algorithm copes with continuous degrees of freedom\nand is designed to numerically calculate the probability distribution of an extensive quantity,\nthe action [1] or e.g. the Polyakov line [29], to an unprecedented precision and for regions of\nthe variable (e.g. action) that would never be visited by an action based importance sampling\nMonte-Carlo approach. Thus, the LLR algorithm naturally solves overlap problems. Given the\nbelief of a correspondence between the overlap and the sign problem in \fnite density quantum\n\feld theory, it was natural to explore its readiness for complex action theories [30]. Here, the\nLLR algorithm provides a high quality probability distribution for the imaginary part of the\naction, and the partition function emerges as the Fourier transform of this distribution with\nthe chemical potential as its frequency (see (3)). Details and advances of the LLR methods\nhave e.g. been reported in [1, 28, 33, 30]. In this paper, we have focused on possible techniques\nto extract an signal, which is exponentially small with the volume, from the highly oscillating\nFourier integral. As a proof of concept, we have studied the Z3spin model [30]. For this model,\nwe have used (for a limited range of the parameter space) the Polynomial Fit technique from\nsubsection 3.1. Despite of a severe sign problem (as quanti\fed by a phase factor expectation\nvalue at the 1016level; see \fgure 6), we were able to obtain reliable results by simulating the\ntheory in its original formulation using the LLR techniques. An analysis of the full parameter\nspace of the Z3model, the LLR simulation of more involved theories (e.g. the O(2)-model) and\nthe exploration of the techniques outlined in section 3 to carry out the Fourier transform are\ncurrently work in progress.\nAcknowledgments\nWe are indebted to Arieh Iserles for fruitful discussions on highly oscillating integrals. This work\nis supported by STFC under the DiRAC framework. We are grateful for the support from the\nHPCC Plymouth, where the numerical computations have been carried out. KL and AR aresupported by the Leverhulme Trust (grant RPG-2014-118) and STFC (grant ST/L000350/1).\nBL is supported by the Royal Society (grant UF09003) and by STFC (grant ST/G000506/1).\nReferences\n[1] K. Langfeld, B. 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Goulko and M. Wingate, PoS LATTICE 2010 (2010) 187 [arXiv:1011.0312 [cond-mat.quant-gas]].\n[35] M. Wingate and O. Goulko, PoS LATTICE 2012 (2012) 057.\n[36] A. Gocksch, Phys. Rev. Lett. 61 (1988) 2054.\n[37] V. Azcoiti, A. Cruz, A. Tarancon and G. Di Carlo, Nucl. Phys. Proc. Suppl. 17(1990) 727.\n[38] K. N. Anagnostopoulos and J. Nishimura, Phys. Rev. D 66(2002) 106008 [hep-th/0108041].\n[39] V. Azcoiti, G. Di Carlo, A. Galante and V. Laliena, Phys. Rev. Lett. 89(2002) 141601 [hep-lat/0203017].\n[40] V. Azcoiti, E. Follana and A. Vaquero, Nucl. Phys. B 851(2011) 420 [arXiv:1105.1020 [hep-lat]]."    },    {        "title": "1503.08336v1.Synchrotron_and_Compton_Spectra_from_a_Steady_State_Electron_Distribution.pdf",        "content": "arXiv:1503.08336v1  [astro-ph.HE]  28 Mar 2015Draft version April 16, 2021\nPreprint typeset using L ATEX style emulateapj v. 5/2/11\nSYNCHROTRON AND COMPTON SPECTRA FROM A STEADY-STATE ELECTR ON DISTRIBUTION\nY. Rephaeli1\nSchool of Physics and Astronomy, Tel Aviv University, Tel Av iv, 69978, Israel\nand\nM. Persic2\nINAF/Osservatorio Astronomico di Trieste and INFN-Triest e, via G.B.Tiepolo 11, I-34143 Trieste, Italy\nDraft version April 16, 2021\nABSTRACT\nEnergydensitiesofrelativisticelectronsandprotonsinextendedg alacticandintraclusterregionsare\ncommonly determined from spectral radio and (rarely) γ-ray measurements. The time-independent\nparticle spectral density distributions are commonly assumed to ha ve a power-law (PL) form over\nthe relevant energy range. A theoretical relation between energ y densities of electrons and protons is\nusually adopted, and energy equipartition is invoked to determine th e mean magnetic field strength in\nthe emitting region. We show that for typical conditions, in both sta r-forming and starburst galaxies,\nthese estimates need to be scaled down substantially due to significa nt energy losses that (effectively)\nflatten the electron spectral density distribution, resulting in a mu ch lower energy density than de-\nduced when the distribution is assumed to have a PL form. The stead y-state electron distribution\nin the nuclear regions of starburst galaxies is calculated by account ing for Coulomb, bremsstrahlung,\nCompton, and synchrotron losses; the corresponding emission sp ectra of the latter two processes are\ncalculated and compared to the respective PL spectra. We also det ermine the proton steady-state\ndistribution by taking into account Coulomb and πproduction losses, and briefly discuss implications\nof our steady-state particle spectra for estimates of proton en ergy densities and magnetic fields.\nSubject headings: galaxies: spiral — galaxies: starburst; radiation mechanisms: non- thermal\n1.INTRODUCTION\nNonthermal phenomena in galaxies and galaxy clusters are importan t for a more complete understanding of these\nsystems, and for knowledge of the origin of relativistic particles and magnetic fields. Current and future measurement\ncapabilities necessitate fairly detailed modeling of these quantities, a nd inclusion of the impact of interactions of\nrelativistic particles in neutral and ionized magnetized media.\nMeasurements of radiative yields of relativistic electrons and proto ns across a wide spectral range provide the\nobservational basis for determining their spectral density distrib utions (e.g., Schlickeiser 2002). Basic properties of\nthese particles and their broad energy ranges are determined fro m measurements of radio, X-ray, and γ-ray emission\nfrom primary electrons and secondary electrons and positrons (p roduced in π±decay) in synchrotron, bremsstrahlung,\nand Compton processes, as well as γ-ray emission from protons via π0decay. These spectral bands do not fully\ncover the particle very broad energy ranges, particularly at low en ergies. Consequently, there is an appreciable degree\nof indeterminacy in quantitative estimations of integrated quantitie s, most important of which are particle energy\ndensities. Due to the approximate decreasing (with energy) PL for m of the spectral density, the generally unknown\nvalue of the low energy cutoff introduces substantial uncertainty in both proton and electron energy densities. Yet, as\nwe discuss in this paper, the implications of this uncertainty are quite significant.\nRelativistic particles are one of the consequences of the formation and evolution of high-mass stars, so phenomena\nrelated to these particles are of interest in all star-forming galaxie s. Reliable estimates of particle energy densities are\nvery much needed for a quantitative evaluation of acceleration mod els and comparison with other measures of stellar\nevolutionary processes that give rise to high-energy phenomena, such as SN explosions, compact remnants of core\ncollapse, and SN shocks, all of which are largely gauged by formation rates of high-mass stars (Torres et al. 2012,\nPersic & Rephaeli 2010).\nFor the purpose of assessing the significance of realistic estimation of their particle energy densities, galaxies in\nwhich high-energy phenomena are dominated by stellar activity are o f particular interest. Among these, well-observed\nnearby galaxies are especially suited, such as the two nearby starb urst (SB)galaxies M82 and NGC253, for which the\ngas properties in the nuclear SB region are well determined and their radio emission well mapped. Moreover, γ-ray\nemission from these galaxies has been detected by the FermiLarge Area Telescope at GeV energies (Ackermann et\nal. 2012), and at TeV energies by the Cherenkov arrays VERITAS a nd H.E.S.S (Acciari et al. 2009, Acero et al.\n2009), at flux levels that are in agreement with theoretical predict ions (Domingo-Santamaria & Torres 2005, Persic,\nRephaeli, & Arieli 2008, de Cea et al. 2009, Rephaeli et al. 2010).\nIn this paper we assess the reliability of estimation of electron energ y densities from measurements of synchrotron\nyoelr@wise.tau.ac.il\n1Center for Astrophysics and Space Sciences, University of C alifornia, San Diego, La Jolla, CA 92093-0424\n2INFN-Trieste, via A.Valerio 2, I-34127 Trieste, Italy2 Rephaeli & Persic\nradio emission, quantifying the significant over-estimation of partic le energy densities in SBGs due to the common\nassumption of a PL form for the spectral density of the emitting ele ctrons. In Section 2 we show that when all\nrelevant electron energy losses are accounted for, the electron spectral density can obviously deviate significantly from\na PL form, as had been quantitatively demonstrated long ago (e.g, R ephaeli 1979, Pohl 1993). This necessitates\nre-calculation of the synchrotron and Compton emissivities (Sectio n 3). As we demonstrate in the latter section,\nnormalization of the electron spectrum by comparison with radio mea surements has significant consequences also for\nthe predicted Compton X-ray and γ-ray spectra. The revised electron spectra yield significantly differ ent estimates of\nthe electron energy density than those obtained when the standa rd formula is used, as shown specifically (Section 4)\nfor conditions in SB nuclei. We briefly discuss our results in Section 5.\n2.STEADY-STATE ELECTRON DISTRIBUTION\nIt is commonly assumed that steady-state energetic particle (mos tly electrons, protons, and helium nuclei) distribu-\ntions can be well approximated by a PL form for a range of values [ γ1,γ2] of the Lorentz factor, γ; for electrons, the\nspectral density is\nNpl(γ) =N1γ−qpl, (1)\nwhereN1is a normalization constant and qplis the spectral index. Since this commonly used form has no tempora l\ndependence, it is implicitly assumed to be a steady-state distribution. This is the form used in the deriva tion of the\nstandard formulae for electron synchrotron, bremsstrahlung, and Compton spectra. While this approximate single-\nindex PL form may be sufficiently accurate for some purposes, it cou ld be very inadequate when the relevant electron\nenergy loss processes have different energy dependence.\nIn a SB nuclear (SBN) region intense star formation yields a high SN ra te and consequently efficient particle\nacceleration. A galactic dynamical process, the SB phase is long, O( 108) year, and since energetic electrons and\nprotons sustain significant energy losses in the relatively small gas- rich, magnetically and radiatively intense SBN\nsource region, steady-state is expected. The theoretically pred icted single index PL spectral density in the acceleration\nregionevolvesto becomeacurvedsteady-stateprofileasaresult ofenergylossprocesses. Todeterminethe steady-state\nelectron spectral density, we assume that the initialspectrum of primary accelerated electrons is a PL in momentum.\nSince we are mostly interested in the radiative yields of relativistic elec trons with γ≫1, we express the single-index\nspectrum in terms of γ, and write for the injection rate per unit volume\n/parenleftbiggdN\ndt/parenrightbigg\ni=kiγ−qi, (2)\nwherekiis a normalization constant. When an exact description of the electr on spectrum is needed (also) at low\nenergies, γ∼1, the exact relation between energy and momentum has to be used , for which the γdependence of the\ninjection spectrum is γ(γ2−1)−(qi+1)/2. Whereas the dependence of the electron density on γ1can be appreciable (as\ndiscussed below), and since typically γ2>106, the exact value of this upper cutoff is of little significance for releva nt\nvalues of qi.\nEnergetic particle propagation out of their source region and thro ughout interstellar (IS) space is typically described\nas a combination of convection and diffusion. We assume that in the re latively small SBN region spatial density\ngradients are not significant, so that the particle density is roughly uniform. While particle escape out of the SBN\nis essentially a catastrophic loss that would generally be included as a s ink term in the kinetic equation describing\nthe spectral distribution, when this term is energy independent its overall impact essentially amounts to an overall\nconstant factor in the expression for the (steady state) spect ral density. Since in our treatment here the overall\nnormalization of the electron density is set by the measured level of radio emission, this escape term does not have to\nbe explicitly included in the equation for the steady state distribution . For our purposes here we therefore focus only\non the spectral dependence of the steady-state electron dens ity,N(γ), which is determined by the energy dependence\nof the total energy loss rate, −dγ/dt=b(γ). In this case the electron steady-state spectral density is det ermined by\nsolving the simple kinetic equation\n−d[b(γ)N(γ)]\ndγ=kiγ−qi. (3)\nTherelevantenergylossprocessesinamagnetized(H-He)gasper meatedbyintense(IR)radiationfieldareionization,\nelectronic excitation (or Coulomb), bremsstrahlung, and synchro tron-Compton. The corresponding energy loss rates\nfor these well-known processes are b0(γ),b1(γ), andb2(γ), respectively. In a medium which consists of ionized, neutral,\nand molecular gas with densities ni,nH, andnH2, respectively, the lower order energy loss of (a charged energet ic\nparticle) is by exciting plasma oscillations and by ionization. The expres sions for the exact loss rate for electron\nenergies down to the sub-relativistic regime were calculated by Gould (1972, 1975; see also Schlickeiser 2002); these\nyield the approximate total rate\nb0(γ)≃1.1×10−12\nβ/bracketleftbigg\nni/parenleftbigg\n1.0−lnni\n74.6/parenrightbigg\n+0.4(nH+2nH2)/bracketrightbigg\ns−1, (4)\nwhereβ= (1−γ−2)−1/2. Inthestrongshieldinglimit, anapproximateexpressionfortheclos elyrelatedbremsstrahlungSteady-State Synchrotron Spectra 3\nloss rate is (Gould 1975)\nb1(γ)≃1.8×10−16γ[ni+4.5(nH+2nH2)] s−1. (5)\nThe synchrotron-Compton loss rate of an electron in a magnetic fie ldBand in a radiation field with energy density\nρris (e.g., Blumenthal & Gould 1970)\nb2(γ)≃1.3×10−9γ2/parenleftbig\nB2+8πρr/parenrightbig\ns−1. (6)\nWe note that the range of electron energies relevant for our discu ssion does not extend to energies beyond the validity\nof the Thomson limit, i.e., the incident photon energy, ǫ0, in the electron frame, γǫ0, satisfies the inequality γǫ0≪mc2\nfor all values of ǫ0<0.01 eV and γ <106of interest to us here. Obviously, the full Klein-Nishina cross sectio n has to\nbe used at much higher energies (e.g., Schlickeiser & Ruppel 2010). A side from the weak (logarithmic) γdependence\nin the expressions for b0andb1, the value of the subscript in each of the three loss rates corresp onds to the power\nof itsγdependence, with the relative magnitude of each term determined b y the gas densities, B, and the radiation\nfield. At low energies the Coulomb rate dominates, whereas at high en ergies synchrotron-Compton losses dominate.\nThe steady-state spectral electron density is then\nN(γ) =kiγ−(qi−1)\nb(γ)(qi−1). (7)\nClearly, the different energy dependence of the loss processes re sults in a curved steady-state spectral density, with\nthe value of the effective PL index changing from qi−1 at energies for which Coulomb losses begin to dominate over\nthe combined losses by bremsstrahlung and synchrotron-Compto n, at energies well below\nγ≤3.0×103/parenleftbigni\n100cm−3/parenrightbig1/2/parenleftbigB\n100µG/parenrightbig−1, (8)\ntoqi+ 1 at higher energies, for which synchrotron-Compton losses dom inate over the combined Coulomb and\nbremsstrahlung losses. To quantitatively compare the two distribu tions, and in order to assess the physical impli-\ncations of using a PL form for the electron spectral density instea d of the more realistic steady-state form, we first\nneed to set the relative normalization. A physically meaningful way of doing so is by normalizing to the same energy\ndensity,ρ=ρpl,\nki\nqi−1/integraldisplayγ2\nγ1γ−(qi−2)dγ\nb(γ)=N1/integraldisplayγ2\nγ1γ−(qpl−1)dγ. (9)\nSince the electron density is usually deduced from the synchrotron emission, a more observationally based choice is\nnormalization to the measured radio flux. In the next section we com pare results obtained based on these two different\nnormalizations.\n 1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1 10 100\n100101102103104Normalized Spectral Density\nγPL, qpl=2.3\nSteady-state\nFig. 1.— Electron steady-state and PL spectral densities for typica l conditions in a SBN, normalized to the same energy density. Spectral\ndensities were calculated with SBN parameters qpl= 2.3,qi= 2.0,B= 100µG,ρ≃10eV/cm−3,ni= 200 cm−3,nH= 30cm−3, and\nnH2= 150cm−3.\nIn our numerical estimates we take the theoretically motivated (an d observationallysupported) value of the injection\nPL index qi= 2, and the observationally deduced value of qpl. An estimate of the latter quantity is obtained from the4 Rephaeli & Persic\nmeasured PL index of the radio spectrum, typically in the range 0 .7±0.05 in the central SB region (as in the nearby\nSBGs M82 & NGC253; for references to the radio measurements, s ee Rephaeli, Arieli, & Persic 2010, and Persic &\nRephaeli 2014). Due to the significant (nonlinear) dependence of t he electron energy density, ρe, on the spectral index,\nwe adopt the lower (1 σ) end of the measured radio index of 0 .65, which corresponds to the fiducial value qpl= 2.3.\nDoingsowe(conservatively)underestimatethe impactonestimatio nofρedue toapproximatingasteady-statespectral\ndensity with a PL distribution. The PL and steady-state spectral d ensities are shown in Figure 1 for typical conditions\nin the dense and intense SBN environment, with B= 100µG, a total radiation field energy density which is estimated\nto be about 10 eV/cm−3(e.g., Porter & Strong 2005), or ρ≃40ρ0, whereρ0≃4.2×10−13erg/cm−3is the (present\nvalue of the) CMB energy density. Assumed ionized and neutral gas densities are ni= 200cm−3,nH= 30cm−3, and\nnH2= 150cm−3.\nThe spectral densities plotted in Figure 1 clearly demonstrate that Coulomb losses dominate up to relatively high\nenergies, ∼1 GeV in a SBN. In fact, this is so not only in the high density, high magne tic field environment of a\nSBN, but also across the disk of star-forming galaxies for which bot h the gas density, O(1) cm−3, and magnetic field,\nB <10µG, are (correspondingly) lower than in a SBN, so that the ratio n1/2\ni/Bis roughly comparable to that in a\nSBN. However, the highest characteristic frequency ( ∝B) at this critical value of γis obviously much higher in a SBN\nthan its typical value across the disk of a star-forming galaxy.\n3.SYNCHROTRON AND COMPTON SPECTRA\nThe standard formula for the synchrotron emissivity by an isotrop ically distributed population of electrons with PL\nspectral density N1γ−qplis (Blumenthal & Gould 1970)\njpl(ν) =√\n3e3BN1\n4πmc2/integraldisplay\nsin(θ)dΩ/integraldisplayγ2\nγ1γ−qpldγν\nνc/integraldisplay∞\nν/νcK5/3(ξ)dξ, (10)\nwheree, m, chave their usual meaning, Bis the mean magnetic field strength across the emitting region, θis the pitch\nangle,νc=ν0γ2sin(θ) is the characteristic synchrotron frequency, and ν0= 3eB/(4πmc) is the cyclotron frequency.\nThe last integrand is the modified Bessel function of the 2nd kind, wit h an integral representation\nK5/3(ξ) =/integraldisplay∞\n0exp−ξcosh(t)cosh(5t/3)dt. (11)\nIt is usually assumed that for all frequencies of interest νc(γ1)≪ν≪νc(γ2), so that the γintegration can be well\napproximated by integration over the interval [0 ,∞]. Doing so yields the standard formula\njpl(ν) =4πe3\nmc2a(qpl)N1B/parenleftbiggν0\nν/parenrightbigg(qpl−1)/2\n, (12)\nwherea(qpl) is expressed in terms of a ratio of Γ functions (eq. 4.60 of Blumenth al & Gould 1970).\nFor the steady-state electron density specified in eq. 7, the sync hrotron emissivity is\nj(ν) =√\n3e3Bki\n2mc2(qi−1)/integraldisplayπ\n0sin(θ)dθ/integraldisplayγ2\nγ1γ−(qi−1)\nb(γ)dγν\nνc/integraldisplay∞\nν/νcK5/3(ξ)dξ. (13)\nUsing the above integral representation for K5/3(ξ), affecting the change of variable x=ν/[ν0γ2sin(θ)], and extending\nthe limits of the γintegration to [0 ,∞], we obtain the following 3D integral\nj(ν) =√\n3ki\n4(qi−1)e3B\nmc2/parenleftbiggν0\nν/parenrightbiggqi\n2/integraldisplayπ\n0sin(θ)qi+4\n2dθ/integraldisplay∞\n0xqi/2dx\nb(ν,θ,x)/integraldisplay∞\n0e−xcosh(t)cosh(5t/3)\ncosh(t)dt, (14)\nwhere\nb(ν,θ,x) =b0(ν0/ν)sin(θ)x+b1[(ν0/ν)sin(θ)x]1/2+b2. (15)\n(In a randomlyoriented magneticfield the θintegralcanbe put in a closedform in terms ofa combinationofWhitta ker\nfunctions [Crusius & Schlickeiser 1986].)\nTocomparetheabovecurvedsteady-statespectrumtothatof aPLdistribution, wedeterminethespectralindexthat\nfits the most relevant range of the measured radio spectrum, and select a relation between the normalization factors N1\nandki. Given that energetic electron spectra are usually determined fro m radio emission, the relative normalization\nof the two distributions can be based on the measured flux at some c haracteristic frequency νc,jpl(νc) =j(νc). Below\nwe compare results obtained with energy density normalization (spe cified in the previous section) to those obtained\nwith the radio flux normalization.\nNote that since the decay of charged pions (produced in proton-p roton interactions) yields secondary electrons and\npositrons, which obviously contribute to the total synchrotron a nd Compton emission, the contributions of secondary\nelectrons and positrons are essentially accounted for approximately by normalizing the electron spectral density to the\nmeasured (i.e., total emitted) radio flux. In our estimates of partic le energy densities (in the next Section) this is\nquantified through the inclusion of the secondary-to-primary rat io,χ.Steady-State Synchrotron Spectra 5\nSince the most relevant range of the measured radio spectrum is ∼1−10 GHz, and given that typically galactic\nradio spectra in the inner disk region are well fit by ∼0.65, the implied PL distribution is characterized by qpl= 2.3.\nOf particular interest is the theoretically favored value qi= 2, which for the most relevant frequency range is also in\naccord with a mean value of α∼0.65. To assess the significance of employing the more realistic steady -state spectral\ndistribution in the analysis of radio measurements, we consider the in tense star-forming environment of a nuclear\nSB region in which the gas density and mean magnetic field are much high er than typical values across the galactic\ndisk. We calculate jpl(ν) withqpl= 2.3, andj(ν) withqi= 2,B= 100µG,ni= 200cm−3,nH= 30cm−3, and\nnH2= 150cm−3, typical values in the nearby SBN of M82 & NGC253 (e.g., Persic & Reph aeli 2014). The spectra\nare shown in Figure 2 (in arbitrary units); the PL spectrum is shown f or the case when the two distributions are\nnormalized to the same flux at 5 GHz, and for the case when the distr ibutions are normalized to the same energy\ndensity.\nThe different energy dependence of the loss processes results in a curved radio spectrum, with nearly a flat spectrum\nat verylowfrequencies to nearlya PL with an asymptotic index qi/2 = 1 at high frequencies, correspondingto emission\nfrom high energy electrons whose losses are synchrotron-Compt on dominated. As is evident from Figure 2, the overall\nimpact of strong magnetic fields and high Coulomb losses on the synch rotron spectrum is a flattening that extends to\nrelatively high frequencies well into the measured GHz range. Clearly , the synchrotron spectra shown in Figures 2 & 3\ndo not include the effect free-free absorption which decreases th e emergent emission at very low frequencies (Condon\n1982). Absorption is not included here primarily because our treatm ent is focused on the impact of spectral flattening\ndue to electron energy losses, rather than on detailed modeling (an d parameter extraction) in specific SBNs. In actual\nspectral fitting to the observed spectrum of a SBN, free-free a bsorption has to be modeled and accounted for before\nthe spectral flattening due to particle losses can be determined, a s was done by, e.g., Yoast-Hull et al. (2013, 2014a,\n2014b).\n 0.001 0.01 0.1 1 10 100\n 0.1  1  10  100  1000Normalized Spectral Luminosity\nFrequency (GHz)Steady-state\nPL: Radio norm.\nPL: Energy density norm.\nFig. 2.— Normalized synchrotron spectra for steady-state and PL den sity distributions in a SBN region. The steady-state spectr um is\nshown by the green line; PL spectra are shown by the red and blu e lines, with the relative normalization set to either the sa me energy\ndensity or to the same 5 GHz emissivity as that of the steady-s tate population, respectively. Spectral indices are qpl= 2.3, andqi= 2.0,\nandB= 100µG.\nClearly, when the two distributions are normalized to the same energ y density, the spectral emissivity of the PL is\nlower than that of the steady-state distribution. Since the stead y-state spectrum was derived by explicitly accounting\nfor electron losses at low energies, the spectral density can be ex tended down to the gas thermal energy, which is\ntypically O(1) keV. For our selected values of the PL index qpl= 2.3 and source injection spectrum with index qi= 2,\nthe PL to steady-state energy density ratio reaches a value of 7 .7 at kinetic energy of 511 keV ( γ1= 2), and attains\nits maximal value 9 .2 already at kinetic energies that are still much higher than the ther mal gas energy ( kT∼1keV).\nThe steeper the radio spectrum, the higher is the energy density r atio, which equals 17 .4 and 33 .6 forα= 0.7,0.75,\ni.e., for PL spectral densities with indices 2 .4 and 2.5, respectively.\nSince our treatment here is strictly spectral with no account take n of the spatial variation of the particle density\nacross the galactic disk, extending the above description to the fu ll disk is obviously unrealistic. However, our main\nobjective here is to assess the impact of just replacing a PL distribu tion which, after all, is commonly assumed in\ncharacterizing particle spectra in galaxies, we carry out a similar calc ulation also for the full galactic disk. We realize,\nof course, that this is at best only a rough approximation of the mor e realistic description which necessarily involves\na solution of a spectro-spatial kinetic equation. Repeating then th e calculation with B= 5µG,ni= 1cm−3, and\nρ= 4ρ0, which are typical mean values across the disk of a star-forming ga laxy, and assuming a typical best-fit radio\nspectrum with α= 0.75 (i.e.,qpl= 2.5), we obtain a value of 31 .0 for the energy density ratio.6 Rephaeli & Persic\n 100000 1e+06 1e+07 1e+08 1e+09\n 0.1  1  10  100  1000Spectral Synchrotron Luminosity, 1030erg/(s GHz)\nFrequency (GHz)Steady-state\nPL: radio norm.\nFig. 3.— Synchrotron spectral luminosity from electrons with stead y-state (green line) and PL (blue line) distributions in a SB N. The\ndistributions were normalized to the same spectral luminos ity at 5 GHz, Lν≃2×1034erg/(s Hz), with qpl= 2.3,qi= 2.0, andB= 100µG.\nTheabovelargevaluesoftheenergydensityratioreflectthe fact thatmost ofthe energydensityofthe PLpopulation\nisinlowerenergyelectrons. Therefore, whenthemeasuredspect rumisfitbyaPL,thedensitynormalizationisstrongly\nweighted by the emissivity in the measured spectral range, and con sequently the deduced energy density is much higher\nthan that of the (curved) steady-state spectrum.\nNormalizing the electron spectral density by the measured radio sp ectrum allows a direct prediction of the elec-\ntron spectral Compton luminosity. Of particular interest is a compa rison of the predicted X-ray and γ-ray spectral\nluminosity of a PL distribution in a SBN to that of a steady-state distr ibution. Integrations of the spectral Compton\nemissivity of an electron scattering off a diluted Planckian radiation fie ld (in the Thomson limit; see, e.g., eq. 2.42\nin Blumenthal & Gould 1970) over the electron PL and steady-state distributions, yield the spectra shown in Fig. 3.\nThe distributions were normalized to the same spectral radio luminos ity at 5 GHz, fiducially set to the measured flux\nfrom the SBN region of NGC253 at a distance of 3 Mpc.\n 100 1000 10000 100000 1e+06\n 0.1  1  10  100  1000Spectral Compton Luminosity, 1030erg/(s keV)\nEnergy (keV)Steady-state\nPower-law\nFig. 4.— Compton spectral luminosity of electrons with steady-stat e (green line) and PL (blue line) distributions in a SBN. The\ndistributions were normalized to the same spectral radio lu minosity at 5 GHz, , Lν≃2×1034erg/(s Hz), with qpl= 2.3,qi= 2.0,\nB= 100µG, and diluted Planckian radiation field at a temperature T= 40 K and energy density ρ= 40ρ0.\nComparison of the synchrotron and Compton spectra in Figures 3 & 4 demonstrates an important consequence of\nthe observationally-basednormalization of a steady-state electr on distribution: The Compton yield of this distribution\nis significantly higher than that of the PL distribution that is normalize d to the same radio flux. As is obvious from\nthe latter figure, the spectral luminosity in the hard X-ray ( ǫ >10 keV) and γ-ray is more than a factor ∼5 higherSteady-State Synchrotron Spectra 7\nthan that of the PL distribution. This is due to the fact that the elec tron spectrum is flatter at (a mean) energy\nǫ= (4/3)γ2ǫ0, for which the Compton boost of the incident IR photon energy, ǫ0, yields an outgoing photon energy\nǫ >10 keV.\n4.PARTICLE AND MAGNETIC FIELD ENERGY DENSITIES FROM SYNCHROT RON EMISSION\nIn most cases of interest radio synchrotron measurements are a nalyzed for the purpose of determining energetic\nelectron properties, such as density and energy density, and the mean strength of the magnetic field in the emitting\nregion. As argued above, the common assumption of a PL distributio n which is fit to the radio data can lead to very\ninaccurate values for these densities. Such estimates are also quit e meaningless since in most cases the contribution\nof low energy particles is altogether ignored: For typical electron P L indices 2 .3−2.5, the fractional energy density\nat low energies (but still relativistic) below 1 GeV is ∼90%−97%, without even accounting for (sub-relativistic)\nsupra-thermal electrons. Moreover, in many applications energy equipartition is assumed in order to determine the\nvalue of the mean magnetic field; for this purpose the proton energ y density needs to be estimated. This is usually\ndetermined by adopting a theoretical relation for the proton-to- electron number density or energy density ratio.\nTherefore, overestimation of the electron energy density leads t o overestimation also of the proton energy density and\nthe field strength.\nQuantifying the impact of the more realistic estimate of the electron energy density on estimates of the latter\nquantities necessitates a revised formulation of the common assum ption of energy equipartition between energetic\nparticles and magnetic fields: Since a state of equipartition is reache d after a sufficiently long period of tight coupling\nbetween these nonthermal quantities, the asymptotic relation be tween their respective energydensities should be based\non their steady-state distributions. Self-consistent calculation o f these distributions (e.g., Rephaeli & Silk 1995, Lacki\n& Beck 2013) requires accounting for all energy loss processes of both electrons and protons, and for a description of\nthe distributions across the full disk, also of their propagation mod es. The quantitative description of such a more\nphysically based equipartition state is outside the scope of our work here. As a first step towards this goal we describe\nhere an approximate procedure that is based on the use (also) of a steady-state spectral density for the protons.\nThe relevant proton energy loss processes in ionized and neutral g as are Coulomb (i.e., electronic excitations and\nionization), and pion production in proton-proton interactions. Th e respective loss rates, bp,C(γ) andbπ(γ), can be\nexpressed by the simplified formulae (adopted from Mannheim & Schlic keiser 1994)\nbC≃3.32×10−16niβ2/bracketleftbigg1\nη3+β3+1.2(nHI+2nH2)\nni(1+0.0185lnβ)\nβ3\n0+2β3/bracketrightbigg\ns−1, (16)\nforβ≥β0= 0.01, and the temperature-dependent factor η= 0.002(T/104K)1/2. The loss rate due to pion production\nis\nbπ≃5.85×10−16ni(γ−1) Θ(γ≥γπ) s−1(17)\nwhere Θ( γ≥γπ) is a step function , and γπ= 2.3 corresponds to the threshold kinetic energy for pion production ,\n1.22GeV. At low energies electronic excitations is the only loss process , whereas at energies above 1 .22GeV losses are\nmostly due to pion production. An example for this spectral change is the drop in the local Galactic proton spectrum\nat low energies recently measured by the Voyager I spacecraft (S tone et al. 2013); as argued by Schlickeiser et al.\n(2014) this is due to Coulomb losses.\nThe proton injection spectrum is assumed to be PL in momentum which , when transformed to γ, can be written as\nkp,iγ(γ2−1)−(qp,i+1)/2./parenleftbiggdNp\ndt/parenrightbigg\ni=kp,iγ(γ2−1)−(qp,i+1)/2, (18)\nwhich is valid also at low energies ( γ≃1). At steady-state the spectral density is simply\nNp(γ) =kp,i(γ2−1)−(qp,i−1)/2\n(qp,i−1)(bC+bπ). (19)\nThe resulting spectrum is Np(γ)∝(γ2−1)−(qp,i−1)/2at energies for which the loss rate is dominated by electronic\nexcitations, steepening to Np(γ)∝(γ−1)−(qp,i+1)/2(γ+1)−(qp,i−1)/2at energies (above 1.22GeV) for which the loss\nrate is dominated by pion production.\nThe energetic proton density is usually related to that of the electr ons by assuming a near equality in the rates\nenergetic protons and electrons escape the acceleration region ( e.g., Pohl 1993, Schlickeiser 2002). In the spirit of our\napproach here (whose objective is to estimate the impact of replac ing a PL with a steady-state distribution), we adopt\nthe charge neutrality assumption. In context of our approximate spectral (rather than spectro-spatial) steady-state\ntreatment, the very different particle densities in the SBN and galac tic disk would be explained as a consequence of the\nhigher density of acceleration sites in the SBN, on the one hand, and the higher energy loss rates there on the other\nhand. If so, and equipartition is indeed attained, radio measuremen ts then provide the observable needed to determine\nthe proton energy density and mean strength of the magnetic field . Use of the more realistic particle steady-state\ndistributions, and consideration of the full spectral range of par ticle energies, appreciably improve on the standard8 Rephaeli & Persic\ncalculation which is commonly based on PL spectra with a relatively high lo wer energy cutoff. Specifically, we extend\nour numerical example representative of conditions in the SBN of M8 2 and NGC253 (specified above) by computing\nalso the proton energy density.\nThe proton source spectrum is a PL with nearly the same index as for the electrons, i.e. qp,i≃2, and the best-fit PL\nspectrum with index qp,pl= 2.2, a value deduced from γ-ray measurements (Persic & Rephaeli 2014, and references\ntherein). Repeating the calculations in the latter work (where all pa rameter values are specified) but with the steady-\nstate spectra obtained here, we compute ρp≃75 eV cm−3andB≃56µG, as compared with ∼330eV/cm−3and\n∼120µG when assuming PL spectral density distributions. Even though ap proximate, these values clearly show\nthat the assumption of PL density spectra leads to significant over estimation of the electron energy density, and\nconsequently also to overestimation of the proton and field energy densities (assuming equipartition).\n5.DISCUSSION\nThe results presented in the previous section make it clear that the assumption of a PL distribution for energetic\nelectronsandprotonsleadstolargeoverestimationoftheirenerg ydensitiesincomparisonwiththevaluesdeducedwhen\nmore realistic steady-state spectral density distributions are us ed. We derived these distributions (self-consistently)\nby accounting for all relevant energy losses. Our quantitative est imates of the factors by which the electron energy\ndensities are overestimated when deduced from radio measuremen ts are in the range ∼10−30 for radio spectral\nindices in the range 0 .65−0.75, when calculated for typical conditions in a SBN and the disk of a st ar-forming galaxy.\nThe mere fact that in some of the analyses based on a PL distribution a high value of γ1is assumed (supposedly\ncircumventing the need for explicit accounting for the spectral fla ttening due to Coulomb losses) does not reduce\nthe inadequacy of the (single-index) PL model: Clearly, consideratio ns of energy equipartition are not valid when\nthe particle energy density is calculated over only part of the energ y range, particularly so when the relatively large\ncontribution of the lower energy particles is ignored.\nCurved synchrotron spectra may also be a consequence of spatia lly inhomogeneous magnetic fields in the emitting\nregion (e.g., Hardcastle 2013). Of course, particle spectra are int rinsically curved when the distribution is either\ntime-varying (e.g., Torres et al. 2012) or before a steady-state is reached. In our treatment here (which is essentially\ncomplementary to the aforementioned works) we calculate self-co nsistently the synchrotron spectrum emitted by\nelectrons whose spectral density is curved due to all the relevant energy losses sustained while traversing a region with\na uniformly distributed gas, magnetic, and radiation fields. The impac t of this more realistic modeling of the particle\nspectrum is usually more important than that of spatial variation of the gas density and magnetic field, especially for\nestimation of the particle total energy density in high-density envir onments (such as a SBN).\nCompton scattering of relativistic electrons by IR and optical radia tion fields in SBGs (Rephaeli, Gruber & Persic\n1995, Wik et al. 2014a), and by the CMB in galaxy clusters (Rephaeli 1 979, Wik et al. 2014b) is likely to contribute\nappreciably to their X-and- γ-ray spectra. As is clear from the spectra in Figure 4, estimating th e level of hard X-\nray emission by extrapolating the best-fit PL to the radio data can s ubstantially under-predict the likelihood of its\ndetection (and obviously affect motivation for its search). Moreov er, the accuracy of joint analyses of radio and X-ray\ndata can be improved substantially by using the properly normalized s teady-state distribution for the electrons.\nAs specified in the previous section, low energy protons lose energy effectively by ionization (in neutral media) and\nenergy excitation (in ionized gas), in addition to catastrophic losses in proton-proton interactions. Here we accounted\nfor these energy losses under typical conditions in a SBN, as has be en done for conditions in galaxy clusters (Rephaeli\n& Silk 1995) and SB galaxies (Lacki & Beck 2013). The effectiveness o f Coulomb interactions of low energy electrons\nand protons with gaseous media obviously result in much lower steady -state spectral densities of low energy particles\nthan what would be predicted from PL spectra. Overall, this implies th at gas heating rates are correspondingly lower\nthan estimated when assuming PL forms for the particle spectral d ensity.\nWe are indebted to the referee for an expert and constructively c ritical review of the original version of this paper.\nThe referee’s suggestions and the changes made in their implementa tion considerably improved the paper. Work at at\nUCSD was supported by a JCF grant. MP acknowledges the hospitalit y extended to him during a visit to UCSD.\nREFERENCES\nAcciari, V.A., et al. (VERITAS Collaboration) 2009, Nature , 462,\n770\nAcero, F., et al. (HESS Collaboration) 2009, Science, 326, 1 080\nAckermann, M., et al. (Fermi Collaboration) 2012, ApJ, 755, 164\nBlumenthal, G.R., & Gould, R.J. 1970, Rev. Mod. Phys, 42, 237 ,\nRev. Mod. 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Ben-Aryeh   \nPhysics Department Technion-Israel Institute of Tec hnology, Haifa, 32000, Israel \nE-mail: phr65yb@physics.technion.ac.il \nKeywords:  Entanglement;  Strong  and weak  separab ility;  Two qubits density matrices  \nABSTRACT \nExplicit separable density matrices, for mixed–two- qubits states, are derived by using Hilbert-Schmidt  \n(HS) decompositions and Peres-Horodecki criterion. A “strongly separable” two-qubits mixed state is \ndefined by multiplications of two density matrices,  given with pure states, while “weakly separable” \ntwo-qubits mixed state is defined, by multiplicatio ns of two density matrices, which includes non-pure  \nstates.   We find the sufficient and necessary cond ition for separability of two qubits density matric es \nand show that under this condition the two-qubits d ensity matrices are strongly separable.  \n \n1.    Introduction  \nFor systems with many subsystems, and Hilbert space s of large dimensions, the “separability \nproblem” becomes quite complicated [1-2]. In the si mple cases of two-qubits states, it is \npossible to give a measure of the degree of quantum  correlations by using the partial-\ntranspose  (PT) of the density matrix [1-3]. Accord ing to Peres-Horodecki criterion [2-3], if the \npartial transpose of the two qubits density matrix leads to negative eigenvalues of the PT \ndensity-matrix ( ) AB PT ρ ,  then the density matrix is entangled, otherwise it is separable.  \nOne should take into account, that the density oper ator of a given mixture of quantum \nstate has many ensemble decompositions. The separab ility problem for two-qubits states is \ndefined as follows: A bipartite system is separable  if the density matrix of this system can be \ntransformed into the form: \n  () () j j \nj A B \njp ρ ρ ρ = ⊗ ∑        .       (1)  2 \n Here:  0jp≥,  and  1j\njp=∑ . The density matrix ρis defined on Hilbert space A B Η ⊗ Η  \nwhere A and B are the two parts of a bipartite system. ()j\nAρ , and ()j\nBρ  are density matrices \ndescribed, respectively, for the A and B systems. T he interpretation for such definition is that \nfor bipartite separable states the two systems, giv en by  ()j\nAρ , and ()j\nBρ are completely \nindependent of each other.  The summation over j could include large numbers of density \nmatrices multiplications, but it is preferred to li mit this number to smaller ones, as far as it is \npossible.  \n The usual analysis of separability for two-qubits mixed states does not show, however, \nthe explicit expressions for separable density matr ices. I find the interesting distinction \nbetween “strong -separability”, and “weak-separabil ity”. I define   strong separability as given \nby Eq. (1) when  \n ( )()( )()2 2 \n1 , ( 1,2,...) j j \nA B Tr Tr j ρ ρ = = =    .                (2) \nWeak-separability is defined by Eq. (1) when some o f the density-matrices  ()j\nAρ , and/or  ()j\nBρ  \nsatisfy the relations \n ( )()( )()2 2 \n1 / 1 j j \nA B Tr and or Tr ρ ρ < <   .    (3) \nIn more general terms conditions (2) and (3) are re ferred, respectively, as pure density matrices \nand mixed density matrices. The interesting point h ere is that while we assume ρ to be a \nmixed state, the strong separability condition migh t still be valid. The explicit expressions for \nthe density-matrices  ()j\nAρ , and ()j\nBρ  might turn to be very complicated in general cases  [4], \nbut I restrict the discussion to two-qubits correla tion density matrices which can be written in \nthe Hilbert-Schmidt decomposition [5] as: \n  ( ) ( ) ( ) ( )3\n14AB i i i A B A B \niI I t ρ σ σ \n=  = ⊗ + ⊗     ∑          .                   (4) 3 \n Here  ( 1,2,3) it i =  are real parameters, ()iAσ  and ()iBσ  are Pauli matrices (i=1,2,3), ()AI and \n()BI  are 2 2 × unit matrices, given ,respectively, for the A and B subsystems.  We described \n(4) in a frame in which the general matrix  ,( , 1,2,3) m n t m n =  has a diagonal form.  Under the \nsymmetry condition: , , m n n m t t = , a straight forward transformation to the diagonal  form (4)  can \neasily be made. For the case for which ,m n t is not symmetric ,m n t matrix can be diagonalized by \nthe use of singular value decomposition [10].  I fi nd that the 2-qubits Bell states and the special \nWerner state analyzed in [6] are of this form. One should take into account:  that ( 1,2,3) it i =\nare real parameters which can be both positive and negative. Also the multiplications in (4) are \nover Pauli matrices which are not density matrices.  Our aim in the present article is to find \nrelations between the two-qubit density matrix desc ribed by (4) and separable density \nmatrices given by (1), and show that we get the str ong separability condition although the total \ndensity matrix might be mixed.  \n \n2.  Separability  of  two-qubits mixed states analy zed by Hilbert-Schmidt decomposition and \nPeres-Horodecki criterion \n In the standard basis of states 00 , 01 , 10 , 11 , the density matrix (4) is given as: \n  3 1 2 \n3 1 2 \n1 2 3 \n1 2 3 1 0 0 \n0 1 0 40 1 0 \n0 0 1 AB t t t \nt t t \nt t t \nt t t ρ+ −   \n  − +   =  + − \n  − +      .  (5) \nThe Partial-Transpose of the density matrix (5) is given by [7] as \n  3 1 2 \n3 1 2 \n1 2 3 \n1 2 3 1 0 0 \n0 1 0 4 ( ) 0 1 0 \n0 0 1 AB t t t \nt t t PT t t t \nt t t ρ+ +   \n  − −   =  − − \n  + +     .  (6)  4 \n The eigenvalues of ( ) AB PT ρ  are given by: \n1 1 2 3 2 1 2 3 3 1 2 3 4 1 2 3 4 1 , 4 1 , 4 1 , 4 1 t t t t t t t t t t t t λ λ λ λ = + − − = − + − = − − + = + + +   .   (7)  \nAccording to Peres-Horodecki criterion [1-3], if an y one of the eigenvalues ( 1,2,3,4) iiλ =  is \nnegative then the density matrix of (4) is entangle d. I will give here explicit results for \nseparability and entanglement as function of the ab solute values of the constants: it. We \ndistinguish between two cases: \nCase A:  The sign of the triple product 1 2 3 t t t  is   -1 ( 1 2 3 ( ) 1 sign t t t = −  ). \nWe treat this case by 4 different conditions: \nCondition a: If the three parameters 1 2 3 , , t t t are negative then the minimal value of \n( 1,2,3,4) iiλ = is given by        \n 4 1 2 3 1 2 3 4 1 1 t t t t t t λ= + + + = − − −   .              (8) \nCondition b:  If  1t and 2tare positive and 3t is negative then the minimal value ( 1,2,3,4) iiλ = is \ngiven by: \n3 1 2 3 1 2 3 4 1 1 t t t t t t λ= − − + = − − −   .               (9) \nCondition c:  If  1t and 3tare positive and 2t is negative then the minimal value ( 1,2,3,4) iiλ = is \ngiven by: \n 2 1 2 3 1 2 3 4 1 1 t t t t t t λ= − + − = − − −    .               (10) \nCondition d:    If  2t and 3tare positive and 1t is negative then the minimal value ( 1,2,3,4) iiλ =\nis given by: \n 1 1 2 3 1 2 3 4 1 1 t t t t t t λ= + − − = − − −   .                (11) \nIn order to get entanglement at least one of the ei genvalues should be negative. 5 \n We find that for cases with  1 2 3 ( ) 1 sign t t t = −  , the condition for entanglement  is given by: \n  1 2 3 1 t t t + + >  .                  (12) \nWe have found according to Peres-Horodecki criterio n for two-qubits system, that this \ncondition is both sufficient and necessary.  On the  other hand if 1 2 3 1 t t t + + ≤ , then we get \nseparable states.  These results are the same for e ach of the four conditions a, or b or c or d. \nCase B:  The sign of the triple product 1 2 3 t t t  is +1 ( 1 2 3 ( ) 1 sign t t t = +  ).  \nWe notice that any exchange of it (i=1, 2 or 3) with ( ) jt j i ≠is equivalent to a corresponding \nexchange of two eigenvalues. For simplicity of nota tion, we assume: \n  1 2 3 t t t ≥ ≥  ,                   (13) \nWe redefined the subscripts so that (13) is satisfi ed.  We treat also this case for different \nconditions: \nCondition a: If the three parameters 1 2 3 , , t t t are positive then the minimal value of \n( 1,2,3,4) iiλ = is given by \n 3 1 2 3 1 2 3 4 1 1 t t t t t t λ= − − + = − − +   .               (14) \nCondition b:  If the two parameters 1t and 2t are negative and 3t is positive then the minimal \nvalue of ( 1,2,3,4) iiλ = is given by \n 4 1 2 3 1 2 3 4 1 1 t t t t t t λ= + + + = − − +   .             (15)  \nCondition c: If the two parameters 1t and 3t are negative and 2t is positive then the  minimal \nvalue of ( 1,2,3,4) iiλ = is given by \n1 1 2 3 1 2 3 4 1 1 t t t t t t λ= + − − = − − +  .               (16) 6 \n Condition d:  If the two parameters  2t, and  3t, are negative and 1tis positive then the minimal \nvalue of ( 1,2,3,4) iiλ = is given by \n 2 1 2 3 1 2 3 4 1 1 t t t t t t λ= − + − = − − +   .               (17) \nWe find that if 1 2 3 ( ) 1 sign t t t = +  under the condition  \n  1 2 3 1 t t t + − >  ,                (18) \nthe density matrix becomes entangled. It has been s hown [10], however, that a necessary and \nsufficient condition for the density matrix to be e ntangled, also in case B, is given by (12). For \nboth cases the separable density matrices can becom e strongly entangled.  \n \n3.  Explicit expressions for the density matrices ()j\nAρ , and ()j\nBρ  of Eq. (1), for separable two \nqubits density matrices given by  the density matri x (4), corresponding to cases A and B  \nWe will analyze the explicit expressions for separa ble density matrices which will be given here \nfor the two cases derived above, by Peres-Horodecki  criterion. We discuss the corresponding \nconditions for entanglement.  \nCase A :  We study here the transformation of the density matrix (4) to a separable density \nmatrix, given as a special case of (1), under the c ondition   1 2 3 1 t t t + + ≤ , for case A , defined \nby the relation 1 2 3 ( ) 1 sign t t t = − . Such transformation will breakdown when 1 2 3 1 t t t + + > , so \nthat the results will be in agreement with the prev ious analysis made in Sec.2, by the use of \nPeres-Horodecki criterion.  In order to find the en tanglement properties of the density matrix \n(4), for case A: we define a matrix AB Sby:   \n( ) ( ) ( )3 3 \n1 1 4 ) ( AB i i i A i A B B \ni i S t I I t σ σ \n= = = ⊗ + ⊗ ∑ ∑ .                  (19) \nThe matrix AB Shas the following properties: 7 \n a)  ( ) ( )3\n14 4 1 AB AB i A B \niS I I t ρ\n=    − = ⊗ −       ∑     .          (20) \nHere AB ρ, and AB S have been defined, respectively, in (4) and (19). The right side of (20) \nrepresents a separable density matrix (up to normal ization) under the condition:  3\n11i\nit\n=≤∑  \nwhere 3\n11i\nit\n=  −    ∑ can be considered as a probability, but such repres entation breaks down \nwhen   3\n11i\nit\n=>∑ , as we cannot have a negative probability. \nb)  AB Scan be transformed into a form similar to (1) by us ing the following  transformation: \n( ) ( ) ( ) ( ) ( ) ( )\n( ) ( )( )3\n1\n3\n14\n22 2 2 2 \n2AB \ni i i i i i A B A B \ni\ni\ni i i \niS\nI sign t I sign t I I t\ntσ σ σ σ \nρ ρ =\n− + \n==\n      − + − +     ⊗ + ⊗       \n            \n= + ∑\n∑      .       (21) \nWe defined here for positive it (negative it) () () ( ) 1 1 i i sign t sign t = = − . Each multiplication in \nthe curled brackets on the right side of (21) repre sents a pure separable density matrix. We \nhave defined, ()\niρ− and ()\niρ+ (i=1, 2, 3) as the multiplication terms in the fir st and second curled \nbracket of (19). ()\niρ+, and ()\niρ−are pure density matrices as they satisfy \n( )()( )() ( )2 2 \n1 1,2,3 i i Tr Tr i ρ ρ + −     = = =          . 2it, can be considered as a probability for each \npure separable density matrix.  \nWe can complement (21) with (20), obtaining the mat rix   AB ρby \n ( ) ( )3\n14 4 1 AB AB i A B \niS I I t ρ\n=    = + ⊗ −       ∑     .          (22) 8 \n We have shown here how AB ρ of (1), in case A, is a separable density matrix w ith the use of \npure density matrix multiplications, under the cond ition: \n            3\n11i\nit\n=≤∑             .                        (23)  \nThe most interesting point is that we get explicit separable density matrics, by superposition of \npure density matrices (“strong separability”). \nCase B:  We study here the transformation of the density mat rix (4) to a separable density \nmatrix, given as a special case of (1). We studied the eigenvalues of (4) for case B defined by the \ncondition: 1 2 3 ( ) 1 sign t t t = +  , so that the results will be in agreement with th e previous analysis \nmade in Sec.2, by the use of Peres-Horodecki criter ion. The condition 1 2 3 1 t t t + − > ,  is only a \nsufficient condition for entanglement. We find that  sufficient and necessary condition for \nentanglement is given also for case B by 1 2 3 1 t t t + + >  [10].  I find that strong separability can \nbe obtained also for case B . The separable density  matrices can be given again by (21) where \nit are considered as probabilities. The agreement   b etween the separable density matrices, \nand (4) can be obtained by adapting the ()i sign t notations so that agreement will be obtained. \nWe have shown that under the sufficient and necessa ry condition 1 2 3 1 t t t + + > for \nentanglement for both cases A and B strongly separa ble states are obtained. Although I have \ntreated various separability problems in previous a rticles [8-9], I have not analyzed there the \nexplicit form of the separable density matrices.  \n \n4.   Summary, discussion  and conclusion   \nIn the present work separability and entanglement p roperties of mixed two-qubits states have \nbeen analyzed by using Hilbert-Schmidt  (HS) decomp ositions and Peres-Horodecki criterion. \nWe have used a special form of two-qubits density m atrix given by (4), depending on three \ndiagonal constants ( 1,2,3) it i =  . I have found that the eigenvalues of the two-qub its density 9 \n matrix depend on the plus or minus sign of  1 2 3 t t t  . For the case of minus sign of this \nmultiplication, referred in the article as case A, we obtained the condition ( ) 1 2 3 1 t t t + + >  for \nentanglement , while for the case with the plus sig n for the above multiplication, referred in the \narticle as case B, we get sufficient condition for entanglement given by \n( ) 1 2 3 1 2 3 1 ( t t t t t t + − ≤ ≥ ≥ . In order to get both sufficient and necessary con ditions for \nentanglement one gets again for case B the same con dition for entanglement ( ) 1 2 3 1 t t t + + >  \nas for case A [10]. These results follow from rigor ous analysis of various cases by Pere-\nHorodecki criterion. \n Explicit expressions for separable density matrice s have been obtained by (21) and (22) \nfor case A. For case B (21) and (22) can be used fo r describing separable density matrices by \nadapting the signs of it given by the notations ()i sign t . Although we analyzed special ensemble \ndecompositions,  these  results  seem to be quite g eneral for these two cases.  It is interesting \nto note that although we treat mixed two-qubits den sity matrices their decomposition for \nseparable states included multiplications of two pu re density matrices defined hare as strong \nseparability conditions.  \n \n \n \n \n \n \n \n 10 \n  \nReferences \n1.  R. Horodecki,   P. Horodecki,  M. Horodecki  and  K .  Horodecki ”, Quantum entanglement”,  Rev. \nMod. Phys .  81 ,   865-942 (2009).  \n2.  A. Peres, “Separability criterion for density matri ces”, Phys.  Rev. Lett. 77, 1413-1415 (1996). \n3.  M. Horodecki , P. Horodecki  and R. Horodecki , “Se parability of mixed states: necessary and \nsufficient conditions”,  Phys. Lett. A 223 , 1-8 (1996). \n4.  J.M. Leinaas, J. Myrheim and E.Ovrum , “Geometrical  aspects of entanglement” , Phys. Rev. A  \n74 ,  012313, 1-13 (2006). \n5.  Y. Ben-Aryeh, A. Mann  and B.C. Sanders , “ Empiric al state determination of entangled two-level \nsystems  and its relation to information theory”,  Foundations of Physics  29 , 1968-1975 (1999). \n6.  6. G. Beneti, G.  Casati  and G. Strini,    Principles of Quantum Computation and Information  \n(World Scientific, Singapore,  2007). \n7.  I. Bengtsoon and K.  Zyczkowski, Geometry of Quantum States (Cambridge University Press, \nCambridge, 2008). \n8.  Y.Ben-Aryeh, “Entanglement and separability of qubi ts systems related to measurements, \nHilbert-Schmidt decompositions and general Bell -st ates”, arxiv: 1412.0213v1[quant-ph] (2014). \n9.  Y.Ben-Aryeh, “Entangled states implemented by Bn gr oup operators, including properties based \non HS decompositions, separability and concurrence” , Int. J. Quant. Inf. 13 , 1450045 (2015). \n10.   Y.Ben-Aryeh and A. Mann, to be published. \n "    },    {        "title": "1505.02871v1.Lyapunov_based_Stochastic_Nonlinear_Model_Predictive_Control__Shaping_the_State_Probability_Density_Functions.pdf",        "content": "Lyapunov-based Stochastic Nonlinear Model Predictive Control:\nShaping the State Probability Density Functions\nEdward A. Buehler1, Joel A. Paulson1;2, Ali Akhavan1, and Ali Mesbah1;y\nAbstract — Stochastic uncertainties in complex dynamical\nsystems lead to variability of system states, which can in turn\ndegrade the closed-loop performance. This paper presents a\nstochastic model predictive control approach for a class of\nnonlinear systems with unbounded stochastic uncertainties. The\ncontrol approach aims to shape probability density function\nof the stochastic states, while satisfying input and joint state\nchance constraints. Closed-loop stability is ensured by designing\na stability constraint in terms of a stochastic control Lya-\npunov function, which explicitly characterizes stability in a\nprobabilistic sense. The Fokker-Planck equation is used for\ndescribing the dynamic evolution of the states’ probability\ndensity functions. Complete characterization of probability\ndensity functions using the Fokker-Planck equation allows\nfor shaping the states’ density functions as well as direct\ncomputation of joint state chance constraints. The closed-loop\nperformance of the stochastic control approach is demonstrated\nusing a continuous stirred-tank reactor.\nI. I NTRODUCTION\nThe need to account for system uncertainties in model\npredictive control (MPC) of complex dynamical systems has\nled to extensive investigation of robust MPC approaches\n(e.g., see [1] and the references therein). The majority\nof work on robust MPC considers bounded, deterministic\nuncertainty descriptions with the goal to design MPC control\nlaws that are robust to worst-case system uncertainties. The\ndeterministic robust MPC approaches may, however, result in\nconservative closed-loop control performance, as worst-case\nsystem uncertainties are likely to have a small probability\nof occurrence [2]. This consideration has recently motivated\nthe development of stochastic MPC (SMPC) approaches\nthat directly use probabilistic descriptions of the stochastic\nsystem uncertainties (i.e., parametric uncertainties, uncertain\ninitial conditions, and exogenous disturbances). In particular,\nSMPC approaches allow for defining chance constraints in\nthe stochastic optimal control problem to systematically seek\ntradeoffs between the control performance and robustness to\nsystem uncertainties.\nThe formulation of a SMPC approach largely depends on\nthe complexity of system dynamics, properties of stochastic\nuncertainties, and solution method for the stochastic pro-\ngramming problem. SMPC approaches have been proposed\nfor linear systems with multiplicative noise [3], [4] and\nadditive noise [5], [6], [7], [8]. The latter approaches mainly\n1Department of Chemical and Biomolecular Engineering, University of\nCalifornia, Berkeley, CA 94720, USA.\n2Department of Chemical Engineering, Massachusetts Institute of Tech-\nnology, MA 02139, USA.\nyCorresponding author: mesbah@berkeley.edu .use affine parameterizations of control inputs for finite-\nhorizon linear quadratic problems to transform the stochastic\nprogramming problem into a deterministic one. Randomized\nalgorithms have also been used to develop SMPC approaches\nfor linear systems [9], [10]. Recently, a SMPC approach\nhas been proposed for nonlinear systems with time-invariant\nprobabilistic uncertainties using the generalized polynomial\nchaos framework [11] (also see [12] for SMPC for nonlinear\nsystems in the absence of input constraints). Generally, the\ncharacteristics of stochastic uncertainties (e.g., boundedness,\nadditive/multiplicative, and time-varying/time-invariant na-\nture of stochastic uncertainties) have important implications\nfor closed-loop stability and recursive feasibility properties\nof SMPC approaches with input constraints and chance\nconstraints. In addition, the complexity of system dynamics\nlargely affects the computational complexity of probabilistic\nuncertainty propagation (through system dynamics) as well\nas chance constraint handling.\nThis paper presents a stochastic nonlinear MPC (SNMPC)\napproach for a class of nonlinear systems with probabilistic\nuncertain initial conditions and unbounded stochastic dis-\nturbances. The proposed SNMPC approach includes input\nconstraints and joint state chance constraints. To ensure\nclosed-loop stability of the SNMPC approach, a stochastic\nLyapunov-based feedback control law that explicitly charac-\nterizes stability in a probabilistic sense is used (e.g., [13],\n[14]). The Lyapunov-based feedback control law allows for\ndesigning a stability constraint in terms of a stochastic con-\ntrol Lyapunov function, which guarantees that the origin of\nthe closed-loop system is asymptotically stable in probability\n(Section II).\nThe Lyapunov-based SNMPC approach is intended to\nshape probability density functions (PDFs) of the stochastic\nstate variables. This necessitates characterizing the complete\nPDFs of states. The Fokker-Planck equation [15] is used to\ndescribe the dynamic evolution of the (multivariate) PDFs as-\nsociated with the stochastic nonlinear system (Section III-A).\nComplete characterization of the states’ PDFs also allows for\ndirect computation of joint state chance constraints without\napproximation. This work uses the Hellinger distance [16] to\nquantify the similarity between the predicted (multivariate)\nPDFs of states and user-specified reference PDFs for shaping\nthe probability density functions (Section III-B). Note that\ncontrol of PDFs of system states/outputs [17], [18], [19]\nand MPC of stochastic systems using the Fokker-Planck\nequation [20], [21] have been reported in the literature.\nWhat distinguishes this work is the generic formulation of\nthe Lyapunov-based SNMPC approach in terms of PDFarXiv:1505.02871v1  [math.OC]  12 May 2015shaping as well as input and joint state chance constraints\nhandling, while ensuring closed-loop stability (Section III-\nC). The presented Lyapunov-based SNMPC approach is\ndemonstrated for stochastic optimal control of a continuous\nstirred-tank reactor in the presence of stochastic uncertainties\n(Section IV).\nII. P RELIMINARIES\nNotation\nThroughout this paper, boldface symbols (e.g., x) denote\nvectors and subscripts denote vector elements (e.g., xi).\nRndenotes the n-dimensional Euclidean space with R+=\n[0;1). The transpose of a vector or a matrix will be\ndenoted by superscript >.Trf\u0001gdenotes the trace operator\non a square matrix. For a vector x2Rn,kxkdenotes the\nEuclidean norm of x, andkxk2\nQdenotes the weighted norm\nofxdefined bykxk2\nQ=x>Qx withQbeing a positive\ndefinite symmetric matrix. Pxdenotes the (multivariate)\nprobability density function of x2Rn.(\n;F;P)denotes a\nprobability space defined by the sample space \n,\u001b-algebra\nF, and probability measure Pon\n.Prf\u0001gdenotes the\nprobability of satisfaction of an expression. LfXdenotes\nthe Lie derivative of a scalar function X(\u0001)with respect to\na vector function f(\u0001). A continuous function V:Rn!R\nis said to be Ckif it isk-times differentiable. A continuous\nfunction\u000b:R+!R+is said to belong to class Kwhen it\nis strictly increasing and \u000b(0) = 0 . The function \u000bis said to\nbelong to classK1when\u000b2K and\u000b(a)!1 asa!1 .\nSystem description\nConsider a class of stochastic nonlinear systems described\nby the stochastic differential equation (SDE)\ndx(t) =f(x(t))dt+g(x(t))u(t)dt+h(x(t))dw(t)\nx(t0)\u0018Px0;\n(1)\nwhere x(t)2Rndenotes the stochastic state variables with\nthe known initial multivariate PDF Px0;u(t)2Rmdenotes\nthe system inputs; w(t)denotes aq-dimensional standard\nWiener process (i.e., stochastic disturbances) defined on the\nprobability space (\n;F;P); and f:Rn!Rn,g:Rn!\nRn\u0002m, and h:Rn!Rn\u0002qdenote the Borel measurable\nfunctions that describe the system dynamics. The functions\nf,g, andhare assumed to be locally bounded and locally\nLipschitz continuous in x(t);8t2R+, and f(0) = 0\n(i.e., the origin is the steady-state point of the unforced and\nundisturbed system). The latter conditions ensure uniqueness\nand local existence of solutions to the SDE (1) [22]. The\nsystem inputs u(t)in (1) are constrained to lie in a nonempty\nconvex set U\u0012Rmdefined by\nU:=fu(t)2Rmjumin\u0014u(t)\u0014umaxg; (2)\nwhere umin2Rmandumax2Rmdenote the lower and\nupper bounds on u, respectively. In addition, the stochastic\nsystem states x(t)should satisfy hard inequality constraints\nk(x(t))\u00140; (3)where k:Rn!Rpdenotes (possibly) nonlinear functions\nthat describe the state constraints, and k(0) = 0 .\nNote that the stochasticity of system (1) arises from the\nprobabilistic nature of uncertain initial states x(t0)(de-\nscribed by Px0) and the stochastic disturbances w. In (1),\nthe terms f(x(t)) +g(x(t))u(t)andh(x(t))correspond to\nthe drift and diffusion terms in the Ito stochastic process , re-\nspectively [23]. Any general (deterministic) nonlinear model\ncan be represented in terms of the control-affine deterministic\ndrift term f(x(t)) +g(x(t))u(t)[24].\nStochastic optimal control with state chance constraints\nThis paper investigates stochastic MPC of the nonlinear\nsystem (1) such that stability of the closed-loop system is\nguaranteed. The proposed SNMPC approach should allow\nfor shaping the multivariate PDF Px(t)of system states in\nan optimal manner, while the system inputs u(t)lie in the\nsetU. In addition, the stochastic optimal control approach\nshould ensure satisfaction of the (possibly nonlinear) state\nconstraints (3) with at least probability \fin the presence of\nsystem stochasticity. This requires incorporating joint chance\nconstraints of the form\nPrfk(x(t))\u00140g\u0015\f (4)\ninto the stochastic optimal control problem. This paper\nconsiders receding-horizon implementation of the SNMPC\napproach in a full state feedback control scheme, where the\nPDF Px(tk)is assumed to be known at every measurement\nsampling time instant tk.\nThe key challenges that will be addressed in this paper\nfor solving the above described stochastic optimal control\nproblem are: (i) describing the dynamic evolution of the\nmultivariate PDF Px(t)associated with the SDE (1), (ii)\nconverting the joint chance constraint (4) to computationally\ntractable expressions, and (iii) designing the SNMPC control\nlaw such that it ensures closed-loop stability of the stochastic\nsystem. Next, the main result of stochastic Lyapunov stability\n(see [13], [25]) that will be used for designing a Lyapunov-\nbased SNMPC approach is summarized.\nLyapunov-based controllers\nFor stochastic nonlinear systems, Lyapunov-based stabi-\nlizing control laws that explicitly characterize region of\nattraction of the closed-loop system in a probabilistic sense\nhave been proposed (e.g., see [13], [26], [27], [14], and the\nreferences therein). In this paper, a Lyapunov-based control\nlaw is designed for the stochastic optimal control problem\npresented above. It is assumed that there exists a nonlinear\nfeedback control law u(t) = p(x);8x2 X \u0012 Rn,\nwhere p:Rn!Rmdenotes a nonlinear function and X\ndenotes a compact set that contains the origin x= 0. The\nfeedback control law p(x)is intended to make the closed-\nloop system asymptotically stable (in probability) about the\norigin, while the input and state constraints (i.e., (2) and (4))\nare satisfied. According to the converse Lyapunov theorem\n[28], the existence of the feedback control law p(x)impliesthe existence of a stochastic control Lyapunov function V(x)\nthat is defined as in Thm. 1.\nTheorem 1 (Asymptotic stability in probability [13]):\nConsider the stochastic nonlinear system (1) and assume\nthat there exists a C2-function V:Rn!R+, classK1\nfunctions\u000b1and\u000b2, and a classKfunction\u000b3, such that\n8x2X;8t\u00150\n\u000b1(jxj)\u0014V(x)\u0014\u000b2(jxj);\nLfV(x) +LgV(x)u(t)ju(t)=p(x)+\n1\n2Trfh(x)>@2V\n@x2h(x)g\u0014\u000b3(jxj):\nThen the stochastic control Lyapunov function V(x)ensures\nthat the origin is asymptotically stable in probability. \u0004\nThm. 1 indicates that the stochastic Lyapunov-based con-\ntrol techniques allow for defining feedback control laws that\nwill lead to\nLfV(x) +LgV(x)u(t)ju(t)=p(x)+1\n2Trfh(x)>@2V\n@x2h(x)g\n+\rV(x)\u00140;8x2\u0005;\n(5)\nwhere the set \u0005is defined by\n\u0005:= sup\nc2Rfx2Rnju2U;x2X;V(x)\u0014cg;\nand\r > 0is a constant. Next, the stochastic control\nLyapunov function V(x)will be used for designing a control\nlaw for the stochastic optimal control problem such that\nclosed-loop stability of the proposed SNMPC approach is\nguaranteed.\nIII. S TOCHASTIC NONLINEAR MODEL\nPREDICTIVE CONTROL\nThis section presents the formulation of the Lyapunov-\nbased SNMPC approach with joint state chance constraints.\nThe propagation of probabilistic system uncertainties (i.e.,\nuncertain initial states and stochastic disturbances) through\nsystem dynamics is described by the Fokker-Planck equation .\nThe Hellinger distance is used as a measure of similarity of\nmultivariate PDFs to formulate the objective function of the\nstochastic optimal control problem for PDF shaping.\nA. Fokker-Planck Equation for Uncertainty Propagation\nThe Fokker-Planck (FP) equation describes the dynamic\nevolution of the PDF of stochastic states x(t)in the uncertain\nnonlinear system (1) [15]. The FP equation readily char-\nacterizes the complete multivariate PDF Px(t)arisen from\nthe stochastic system uncertainties in initial states x(t0)and\ndisturbances w. This is in contrast to uncertainty propagation\ntechniques that describe merely certain statistics of the PDFs\n(e.g., see [11] and the references therein). Characterizing the\ncomplete PDF of states using the FP equation enables the\nproposed SNMPC approach to: (i) shape the PDF Pxwith\nrespect to any desired (multivariate) PDF, and (ii) compute\nchance constraints of any complexity directly without con-\nservative approximations.The FP equation associated with the SDE (1) that describes\nthe evolution of the multivariate PDF Px(t)is defined by\n@Px\n@t+nX\ni=1@\n@xi\u0012\u0000\nfi(x) +gi(x)u\u0001\nPx\u0013\n\u00001\n2nX\ni=1nX\nj=1@2\n@xi@xj\u0012\nDij(x)Px\u0013\n= 0(6)\nwith the initial condition\nPx(0) = Px0;\nwhere D=h(x)h(x)>(i.e., the diffusion matrix). The FP\nequation (6) is a parabolic partial differential equation, whose\nsolution should be nonnegative and satisfy\nZ\n\nxPx(t)dx= 1;8t\u00150:\nThe existence and uniqueness of a solution to (6) under\nmild assumptions have been established (see [15], [29]).\nNote that the FP equation (6) can be used to compute the\nunivariate PDFs Pxifor every state xiand joint PDFs for\nany combination of stochastic states.\nSolving the FP equation is generally challenging for\nnonlinear systems, in particular systems with high state\ndimension [15]. Various numerical methods such as finite dif-\nference and finite element methods have been used to solve\nthe FP equation for nonlinear systems (e.g., see [30] and the\nreferences therein). In this work, finite volume method with\nfirst order upwind interpolation scheme is used to solve (6)\n[31]. The finite volume method allows for effectively dealing\nwith the convective nature of the FP equation (due to the drift\nterm), as well as suppressing numerical instability problems.\nB. Metric for Similarity of Probability Density Functions\nThe SNMPC approach aims to shape the PDF Pxac-\ncording to a predetermined (arbitrarily-shaped) PDF. Hence,\na metric is required to quantify the similarity between the\npredicted and the reference PDFs at each time instant t.\nThis paper uses the Bhattacharyya coefficient [32], a\nmeasure closely related to the Bayes error [33], to establish\na measure for the similarity of PDFs. The Bhattacharyya\ncoefficient quantifies the degree of overlap between two\n(multivariate) PDFs, and is defined by\nB(x):=Z\n\nxq\nPx(t)Prefxdx; (7)\nwhere Pref\nxdenotes a reference PDF. The Bhattacharyya\ncoefficient will be larger when the overlap between the PDFs\nis larger. B(x) = 0 if the PDFs do not overlap, whereas\nB(x) = 1 when the PDFs are identical. Explicit forms of\nthe Bhattacharyya coefficient for various PDFs are given in\n[32], [34].\nThe Bhattacharyya coefficient (7) is used to define a metric\nfor the similarity between Px(t)andPref\nxin the objective\nfunction of the stochastic optimal control problem. Themetric, which is known as the Hellinger distance [16], is\ndefined by\n\u0001(x):=p\n1\u0000B(x): (8)\nNote that the metric (8) is near optimal due to its relation to\nthe Bayes error, and can be used for arbitrary PDFs [35].\nC. Formulation of the Lyapunov-based SNMPC Approach\nThe Fokker-Planck equation (6) and the Hellinger dis-\ntance (8) are now used to formulate the Lyapunov-based\nSNMPC problem for the stochastic nonlinear system (1).\nProblem 1 (Lyapunov-based SNMPC with input and\njoint state chance constraints): Suppose that the PDF\nPx(tk)is known at every sampling time instant tk.1The\nstochastic optimal control problem at each time instant tkis\nstated as\nu\u0003(Px(tk)):= arg min\nuZTp\n0\u0012\n\u0001(\u0016x(t0)) +ku(t0)k2\nR\u0013\ndt0\ns.t.:@P\u0016x\n@t+nX\ni=1@\n@\u0016xi\u0012\u0000\nfi(\u0016x) +gi(\u0016x)u\u0001\nP\u0016x\u0013\n\u00001\n2nX\ni=1nX\nj=1@2\n@\u0016xi@\u0016xj\u0012\nDij(\u0016x)P\u0016x\u0013\n= 0;8t2[0; Tp]\nLfV(\u0016x) +LgV(\u0016x)u(t)+\n1\n2Trfh(\u0016x)>@2V\n@\u0016x2h(\u0016x)g+\rV(\u0016x)\u00140;8t2[0; Tp]\nPrfk(\u0016x(t))\u00140g\u0015\f; 8t2[0; Tp]\nu(t)2U; 8t2[0; Tc]\nP\u0016x(0) = Px(tk);\n(9)\nwhere u\u0003denotes the optimal inputs (i.e., control policy)\nover the control horizon [0; Tc];\u0016xdenotes the states\npredicted by the system model; Tpdenotes the prediction\nhorizon; and Rdenotes a strictly positive definite matrix. The\nclosed-loop stability of the SNMPC approach is ensured in\na probabilistic sense by incorporating the stability constraint\n(defined in terms of the stochastic control Lyapunov function\nV(\u0016x)) into (9). The asymptotic stability properties of the\nclosed-loop system are characterized in Thm. 1 and (5). The\nproposed stochastic control approach possesses the stability\nproperties of Lyapunov-based controllers when applied in\na sample-and-hold fashion (e.g., see [14]). Note that the\nstochastic optimal control problem (9) is implemented in\na receding-horizon mode, where merely the optimal inputs\nu\u0003(0)are applied to the stochastic nonlinear system (1) at\nevery sampling time instant tk.\nThe Lyapunov-based SNMPC approach in Problem 1\nallows for shaping the multivariate PDF of states, while\nsatisfying input constraints and joint chance constraints\nimposed on the stochastic states. This is due to using\n1Note that this work considers a full state feedback control scheme. The\nPDFs Px(tk)arise from measurement errors.the FP equation for probabilistic uncertainty propagation,\nas the FP equation enables explicit characterization of the\nstates’ PDFs. The objective function in (9) is stated in its\nmost general form in terms of the Hellinger distance to\nquantify the difference between the multivariate PDFs Px(t)\nand Pref\nxover the prediction horizon t2[0; Tp].2The\nobjective function can be simplified by considering only\ncertain statistics (e.g., the expected value and variance) of\nthe PDFs. Furthermore, the joint state chance constraint\nin (9) can be readily computed through explicit integration\nwithout any approximation since the knowledge of the full\nmultivariate PDF is available. When merely individual state\nchance constraints are considered in (9), the FP equation\ncan be adapted to compute only the univariate PDFs for\nthe respective states. This will reduce the computational\ncomplexity of the stochastic optimal control problem. Next,\nthe application of the Lyapunov-based SNMPC approach to\na continuous stirred-tank reactor is investigated.\nIV. C ASE STUDY : STOCHASTIC OPTIMAL CONTROL OF A\nCONTINUOUS STIRRED -TANK REACTOR\nConsider a continuous stirred-tank reactor (CSTR), in\nwhich the exothermic reaction Ak\u0000 !Boccurs. The system\ndynamics are described by\ndCA=\u0012F\nV(CA0\u0000CA)\u0000k0e\u0000E\nRTCA\u0013\ndt\n+\u001bCAdwCA(t); C A(0)\u0018B(0;2;320;320)\ndT=\u0012F\nV(T0\u0000T) +\u000eH\n\u001acpk0e\u0000E\nRTCA+Q\n\u001acpV\u0013\ndt;\nT(0) = 315:0;\nwhereCAdenotes the concentration of species A(kmol/m3);\nTdenotes the reactor temperature (K); CA0(with the mean\nvalue of 0:702kmol/m3) andT0denote the concentration of\nspeciesAand the temperature in the inlet reactor stream, re-\nspectively;Fdenotes the inlet flow rate (m3/min);Vdenotes\nthe reactor volume (m3);Bdenotes the four-parameter beta\ndistribution; Qdenotes the heat removed from the reactor\n(kJ/min); and wCA(t)denotes a standard Wiener process\nacting onCA. The model parameters are listed in Table I.\nNote that the CSTR under study is a stochastic system due\nto the probabilistic uncertainties in the initial concentration\nCA(0), as well as the stochastic disturbances wCA(t). The\nsystem inputs that can be manipulated for control are CA0\nandQ. It is assumed that the temperature Tand the PDF\nofCA(which is of beta-distribution type) are measured at\nevery sampling time instant tk(i.e., 2min). The process is\nrun for 30min.\nThe Lyapunov-based SNMPC approach presented in Prob-\nlem 1 is applied to stabilize the CSTR around the steady-\nstate point ( \u0016CA= 0:57kmol/m3,\u0016T= 317 K), while shaping\nthe probability density of CAto take the form of the Normal\n2When the control objective is to achieve desired PDFs for individual\nstates, the objective function in (9) can be defined in terms of weighted\nsum of Hellinger distances pertaining to the univariate PDF of states.TABLE I\nCSTR MODEL PARAMETERS .\nV 0:1m3\u000eH 4:78\u0002105kJ/kmol\nF 100\u000210\u00003m3/mink0 72\u0002109min\u00001\nT0 315:0K cp 0:239 kJ/kgK\nE 8:314\u0002104kJ/kmol\u001a 1000 kg/m3\nR 8:314 kJ/kmol K \u001bCA0:32\nDistributionN(\u0016CA;4\u000210\u00004)(i.e., Pref\nCA=N(\u0016CA;4\u000210\u00004)\nin (7)). The objective function in (9) is defined as\nZTp\n0\u0012\n\u0001(CA(t0)) +\r\rE[T(t0)]\u0000\u0016T\r\r2\u0013\ndt0; (10)\nwhere E[\u0001]denotes the expected value. The prediction hori-\nzon and the control horizon are selected to be Tp= 30\nmin andTc= 20 min, respectively. Note that the objective\nfunction (9) is defined such that the SNMPC approach\nminimizes the difference between the PDF of CAand the\nreference PDF Pref\nCAover the prediction horizon [0; Tp]. To\ndesign the stability constraint in (9), the quadratic Lyapunov\nfunction V(x)is defined as V(\u0016x) = \u0016x>P\u0016x, where \u0016x=\n[CA\u0000\u0016CAT\u0000\u0016T]>andP=\u00143:18 0:93\n0:93 0:58\u0015\n. The matrix\nPis obtained by solving the Lyapunov equation for the\n(nominal) linearized system dynamics around the steady-\nstate point.\nIn the stochastic optimal control problem (9), the inputs\nto the system are constrained to lie in the ranges 0\u0014\nCA0\u00142kmol/m3andjQj\u001410kJ/min. In addition, the\nconcentration CAshould remain above a threshold in the\npresence of probabilistic system uncertainties. Hence, the\nindividual chance constraint\nPrfCA(t)\u00140:53g\u00140:05 (11)\nis incorporated into (9), suggesting that the state constraint\nCA(t)\u00140:53should be satisfied with at least probability\n95% in a stochastic setting. Since the individual chance\nconstraint and the univariate PDF shaping in the objective\nfunction have been defined merely in terms of the con-\ncentrationCA, the FP equation (6) is considered only for\nFig. 1. Histograms of the concentration CAat various times based on 130\nclosed-loop simulations with different disturbance realizations wCA. The\nSNMPC approach is intended to shape the probability distribution of CAin\nthe course of the process according to the reference probability distribution\n(red solid distribution).\nFig. 2. Probability of state constraint violation at various times during the\nprocess. The probability of state constraint violation always remains below\n5%(red solid line) due to the state chance constraint (11).\nCAwith the diffusion coefficient D= 0:001. The FP\nequation is solved using the finite volume method with 200\ndiscretization points over the concentration support [0;2].\nThe above constrained nonlinear optimal control problem is\nsolved using the MATLAB subroutine fmincon , where the\nset of model equations is integrated using the solver ODE45 .\nThe control inputs are discretized in a piecewise constant\nfashion in 5intervals over the control horizon.\nTo evaluate the performance of the SNMPC approach,\nMonte Carlo simulations of the closed-loop system are\nperformed based on 130 realizations of the Wiener process\nwCA(the disturbance realizations are used for simulating\nthe CSTR model to which the optimal control inputs are\napplied at every sampling time instant tk). Fig. 1 shows the\nevolution of the probability distribution of the concentration\nCA(i.e., true plant outputs) in the course of the process.\nThe SNMPC approach stabilizes the stochastic CSTR system\naround the steady-state point \u0016CA= 0:57kmol/m3; the CSTR\ntemperature is also stabilized around its steady-state value\n\u0016T= 317 K (not shown here). In addition, Fig. 1 shows\nthat the SNMPC approach can effectively shape the PDFs\nofCAaccording to the reference PDF Pref\nCA. For instance,\nthe mean and variance of the PDF of CAat time 30min\nare0:576 and4:3\u000210\u00004, respectively, which are aligned\nwith those of the reference PDF Pref\nCA=N(0:57;4\u000210\u00004).\nThe probability of violation of the state constraint over the\nprocess time is shown in Fig. 2. The probability of constraint\nviolation remains below the predetermined probability level\n5%(i.e.,\f= 0:05in (11)) at all times during the process.\nEffective constraint handling is due to the state chance\nconstraint (11), which ensures constraint satisfaction with\na desired probability level in the presence of stochastic\nuncertainties. Note that simulation results (not shown here)\nrevealed that by decreasing the desired probability of chance\nconstraint satisfaction, the closed-loop control performance\nin terms of the PDF shaping can be further improved. This\nindicates the capability of the SNMPC approach to system-\natically seek tradeoffs between the closed-loop performance\nand robustness to system stochasticities.\nV. C ONCLUSIONS\nA stochastic model predictive control approach is pre-\nsented for a class of nonlinear systems with stochastic un-certainties. The closed-loop stability of the control approach\nis ensured by explicitly characterizing stability in a proba-\nbilistic sense using a stochastic control Lyapunov function.\nA key challenge in SMPC is efficient propagation of uncer-\ntainties through the system dynamics to fully characterize\nthe probability distribution of the stochastic states. This paper\nuses the Fokker-Planck equation for uncertainty propagation,\nwhich allows for describing evolution of the probability\ndistributions of the stochastic states for general probabilistic\nuncertainty descriptions. This work demonstrates that char-\nacterization of the complete probability distribution enables\nshaping the states’ probability distributions with respect to\narbitrarily-shaped reference distributions, as well as direct\ncomputation of chance constraints without any approxima-\ntion.\nREFERENCES\n[1] D. Q. 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Based on the observation\nthat a \fnite energy density state, if localized, can be viewed as the ground state of a local Hamil-\ntonian, we propose a simple (though possibly incomplete) rule for many-body localization of SPT\nHamiltonians: If the ground state and top state (highest energy state) belong to the same SPT\nphase, then it is possible to localize all the \fnite energy density states; If the ground and top state\nbelong to di\u000berent SPT phases, then most likely there are some \fnite energy density states which\ncan not be fully localized. We will give concrete examples of both scenarios. In some of these\nexamples, we argue that interaction can actually \\ assist \" localization of \fnite energy density states,\nwhich is counter-intuitive to what is usually expected.\n1. INTRODUCTION\nSymmetry protected topological (SPT) states and\nmany-body localization (MBL) are two striking phenom-\nena of quantum many-body physics. A d\u0000dimensional\nSPT state is the ground state of a local Hamiltonian\nwhosed\u0000dim bulk is fully gapped and nondegenerate,\nwhile its (d\u00001)\u0000dim boundary is gapless or degenerate\nwhen and only when the system preserves a certain sym-\nmetryG[1, 2]. An SPT state must have \\short range en-\ntanglement\"; meaning that the entanglement entropy of\nits subsystems scales strictly with the area of the bound-\nary of the subsystem: SA\u0018Ld\u00001[3], whereLis the\nlinear size of the subsystem A. MBL refers to a phe-\nnomenon of the entire spectrum of a local Hamiltonian\nwith disorder, including all of the highly excited states\nwith \fnite energy density. Localization of single parti-\ncle states under quenched disorder is well-understood [4],\nand recent studies suggest that localization can survive\nunder interaction [5, 6]. In our current work the phrase\nMBL refers to systems whose all many-body eigenstates\nare localized, namely the entanglement entropy of all \f-\nnite energy density states obey the same area law as SPT\nstates instead of the usual volume law typically obeyed\nby \fnite energy density states.\nThese observations imply that in a many-body local-\nized system, any \fnite energy density state actually be-\nhaves like the ground state of a local parent Hamiltonian.\nIndeed, it was proposed that phenomena such as stable\nedge states and spontaneous symmetry breaking [3, 7{\n9], which usually occur at the ground state of a sys-\ntem, can actually occur in \fnite energy density states\nof MBL systems. In fact, we can de\fne a MBL sys-\ntem as a system for which any \fnite energy den-\nsity eigenstate is a short range entangled ground\nstate of a local parent Hamiltonian . And if the\nsystem preserves a certain symmetry, then any \fnite\nenergy density state of the MBL system should\nalso obey the classi\fcation of SPT states . Then we\ncan view energy density \"as a tuning parameter betweenSPT states. Of course, in the thermodynamic limit, be-\ncause there are in\fnite states in an in\fnitesimal energy\ndensity interval ( \";\"+d\"), we expect there exists many\n1dcurves in the spectrum parameterized by \"with one\nstatej i\"at each\", which is the ground state of an ef-\nfective SPT Hamiltonian H\". And on each such curve\nj i\"is (roughly speaking) continuous in the sense that\nj i\"andj i\"+d\"are similar (despite being orthogonal),\nnamely physical quantities averaged over the entire sys-\ntem change continuously with \"on this curve. In an\nergodic system, the eigenstate thermalization hypothe-\nsis [10] implies that most states with similar energy den-\nsity\"are similar (their reduced density matrices all be-\nhave like a thermal density matrix); in a MBL state,\nalthough states with the same energy density can in prin-\nciple be very di\u000berent, we still expect (assume) that the\ncontinuous curves mentioned above exist, although states\nin di\u000berent curves can be very di\u000berent.\nWithin one of these curves mentioned above, tuning\n\"is just like tuning between the ground states of local\nHamiltonians. Furthermore, by tuning \"there may or\nmay not be a phase transition. In particular, if all excited\nstates belong to the same SPT phase for arbitrary energy\ndensity\", then there does not have to be any quantum\nphase transition when tuning \", which implies that all\nof the excited states have short range correlations and\narea-law entanglement entropy, i:e:all the \fnite energy\ndensity states are localized; on the other hand, if states\nwith di\u000berent energy density \"on the same curve belong\nto di\u000berent SPT phases, then there must be at least one\nphase transition at certain critical energy on this curve\nwhen tuning \". This phase transition behaves just like\nan ordinary zero temperature quantum phase transition\nbetween di\u000berent quantum ground states under disorder.\nFor 1dsystems this \\critical\" energy density state could\nbe in the \\in\fnite-randomness\" phase [11{14], whose en-\ntanglement entropy scales logarithmically with the sub-\nsystem size [15], hence it is not fully localized. The exis-\ntence of the \\in\fnite-randomness\" states at \fnite energy\ndensity have already been observed in Ref. 16.arXiv:1505.05147v2  [cond-mat.str-el]  31 May 20152\nDue to the fact that in a generic nonintegrable Hamil-\ntonianH, the ground state jGiand top statejTi(highest\nenergy state of Hand also ground state of \u0000H) are usu-\nally the easiest states to analyze, the most convenient\nway to determine the existence of \\critical\" states in the\nspectrum is to check whether the ground and top states\nbelong to the same SPT phase or not. In summary, if jGi\nandjTibelong to di\u000berent SPT phases, and if we under-\nstand that these two SPT states are separated by one or\nmultiple continuous phase transitions (this will depend\non the type of SPT phases jGiandjTibelong to), then\nthere must be some \\critical\" excited states in the spec-\ntrum which cannot be fully localized [46]. We will apply\nthis rule to various examples in the next section.\n2. EXAMPLES\n2A. Kitaev's chain: localization\nWe \frst apply our argument to the Kitaev's chain:\nH=X\nj\u0000\u0000\nt+ (\u0000)j\u000et+ \u0001tj\u0001\ni\rj\rj+1; (1)\nwhere\rjare Majorana fermions and \u0001 tjis a random\nhopping parameter with zero mean and standard devia-\ntion\u001b\u0001t. The topological superconductor phase ( \u000et> 0)\nand the trivial phase ( \u000et< 0) can both be fully localized\nby disorder, because for either sign of \u000et, the ground state\njGiand top statejTiboth belong to the same phase (we\nchoose the convention that (2 j\u00001;2j) is a unit cell). This\ncan be seen in the clean limit with \u0001 t= 0. In momentum\nspaceH=P\nkdx(k)\u001cx+dy(k)\u001cz, and~dis a nonzero O(2)\nvector in the entire 1 dBrillouin zone with \u000et6= 0. For\neither sign of \u000et,Hand\u0000Hhave the same topological\nwinding number n1=1\n2\u0019R\ndk^da@k^db\u000fab; thusjGiand\njTibelong to the same phase. Based on our argument,\nall the \fnite energy states with either sign of \u000etcan be\nfully localized by random hopping \u0001 t. The only states\nnot fully localized in the two dimensional phase diagram\ntuned by\"and\u000etare located at the critical line \u000et= 0.\nThe critical line \u000et= 0 is in a \\in\fnite-randomness\" \fxed\npoint, and it can be understood through the strong dis-\nordered real space renormalization group [13{18].\nHere we con\frm the conclusions in Ref. 3, 7, 9 that\nthe \fnite energy density excited states of the Kitaev's\nchain with \u000et > 0 are still \\topological\". Since the en-\nergy level spacing between two eigenstates vanishes in\nthe thermodynamic limit, the best way to determine if\nan excited state is topological or not is to compute its en-\ntanglement spectrum (the system is de\fned on a periodic\n1d lattice). And because the system is noninteracting,\nwe will compute the single-particle entanglement spec-\ntrum introduced in Ref. 19 for each excited state. The\nsingle-particle entanglement spectrum for the topologi-\ncal phase (\u000et > 0) is shown in Fig. 1(a), where two zeroenergy modes can be observed in the spectrum (corre-\nsponding to the Majorana zero modes at both entangle-\nment cuts respectively). This topologically non-trivial\nfeature persists for all energy eigenstates in the many-\nbody spectrum, including the ground/top states and the\n\fnite energy density states in between. However at the\ncritical line \u000et= 0, as shown in Fig. 1(b), the zero en-\nergy modes are lifted by the long-range entanglement,\nand the single-particle entanglement levels become gap-\nless around \u000fE= 0 which leads to the logarithmic scaling\nof the entanglement entropy.\nThe Kitaev's chain itself is just a free fermion model.\nBut our argument indicates that under interaction, as\nlong asjGiandjTiare still both in the topological su-\nperconductor phase, all of the excited states can still be\nlocalized. Such a generalization is justi\fed given that\nthe non-interacting Anderson localized states can be adi-\nabatically connected to the many-body localized states\nunder interaction, as proven in Ref. [3].\n×� ×� ×�×�×�×�×�\n���������������-���-������������\nε����ϵ �\n���������������������-���-������������\nε����ϵ �(�) (�)\nFIG. 1: Single-particle entanglement spectrum for many-body\neigenstates of the random Kitaev's chain, at (a) \u000et= 0:5tand\n(b)\u000et= 0. In both cases \u001b\u0001t= 0:3t. We take a 128-site sys-\ntem with periodic boundary condition, which is partitioned\ninto two 64-site subsystems for the entanglement calculation.\n\u000fEis the single-particle entanglement energy (s.t. the reduced\ndensity matrix \u001aA= exp(\u0000cy\u000fEc), as shown in Ref. 19). The\nspectrum of \u000fEis shown as tanh \u000fE, and is calculated for sev-\neral many-body eigenstates: including the ground and the\ntop states and other 5 randomly picked \fnite energy density\nstates, which are arranged in order of their energy density \".\nThe shading denotes the standard deviation of the entangle-\nment energy levels under a disorder average over the system.\nOf note are the topologically non-trivial, two-fold degenerate,\nzero energy modes throughout the entire spectrum \"in the\ntopological phase (a).\n2B. Modi\fed Kitaev's chain: critical states and\ninteraction assisted localization\nIn this subsection we consider a modi\fed Kitaev's\nchain:\nH=X\nj\u0000\u0000\nt\u0000(\u00001)jt0\u001bz\nj+ \u0001tj\u0001\ni\rj\rj+1\u0000h\u001bz\nj;(2)3\nwhere again \u0001 tjis random and t;t0;h> 0. In this model\n\u001bz\njcommutes with the Hamiltonian, which implies that\nany energy eigenstate will also be an eigenstate of \u001bz\nj.\nIn the clean limit, the ground state jGiof the system\nhas\u001bz\nj= 1 everywhere, and the fermions are in the triv-\nial phase; in contrast, jTimust have \u001bz\nj=\u00001 every-\nwhere, and hence jTiis in the topological superconduc-\ntor phase. With disorder, both states can be localized,\nand their entanglement entropy shows the area-law scal-\ning ( i.e.S \u0018 const. for 1 d) as in Fig. 2(a). But since\nthe ground state and the top state belong to di\u000berent\nSPT phases, based on our argument, there must be some\n\fnite energy density states which cannot be fully local-\nized. In this model it is easy to visualize these delo-\ncalized excited states. An excited state of the system\nhas a static background con\fguration of \u001bz\njwhich does\nnot satisfy \u001bz\nj= 1. If we consider a random con\fgura-\ntion of\u001bz\njthat has the average \u001bz\nj= 0, then one can\nsimply absorb \u001bz\njinto the random numbers \u0001 tj, and\nthe e\u000bective Hamiltonian for Majorana fermions \rjreads\nHe\u000b=P\nj\u0000\u0000\nt+ \u0001t0\nj\u0001\ni\rj\rj+1, which is precisely the\nrandom hopping Majorana fermion model Eq. (1) tuned\nto the critical point \u000et= 0. And according to Ref. 14, 15,\nthe ground state of He\u000b(which is a highly excited state of\nthe original Hamiltonian Eq. (2) due to the hterm) has a\npower-law correlation after disorder average, and its en-\ntanglement entropy scales logarithmically with the sub-\nsystem size:S\u0018log`[15]. So the delocalization happens\nright at the energy scale E\u001b\u0011\u0000hP\nj\u001bz\nj= 0. In deed\nour numerical calculation shows that as long as E\u001b6= 0,\nthe eigen states are all localized with area-law entangle-\nment entropy as in Fig. 2(a,b); but for E\u001b= 0, the eigen\nstates are delocalized with logarithmically-scaled entan-\nglement entropy as in Fig. 2(c). Thus the model Eq. (2)\ncannot be fully many-body localized, which is consistent\nwith our statement made in the introduction.\nThe model Eq. 2 has a time-reversal symmetry T:\n\rj!(\u0000)j\rjand\u001bz\nj!\u001bz\nj. It is known that with\nthis time-reversal symmetry and without interactions,\nthe Kitaev's chain has Zclassi\fcation [20{22]; that is\nwith an arbitrary number of \ravors of Eq. 2, jTiis al-\nways a nontrivial topological superconductor, while jGi\nis always a trivial phase. However under certain \ravor\nmixing four-fermion interaction [23, 24], the classi\fcation\nof Kitaev's chain with time-reversal symmetry reduces\ntoZ8. Namely under this four-fermion interaction, for\neight copies of Eq. 2, jGiandjTibecome the same triv-\nial phase, which implies that there does not have to be\nany phase transition when increasing \", and all of the\n\fnite energy density excited states can be fully localized\nunder the interplay between disorder and interaction.\nIn model Eq. 2, the logarithmic entanglement entropy\nat the critical excited state comes from the long range\ne\u000bective hopping under renormalization group [13{15].\nWe can assume that the four-fermion interaction on each\nsite is random, then when and only when there are 8 k\n2 4 6 8log2L\nπsinπ\nLℓ0.20.40.60.8Sℓ(a) entanglement entropyS ℓvs\nsubsystem lengthℓand fermion energyE γ\nL=1024,t′=t\n2,σΔt=t,E σ= -h L\nEγ/hL\n0\n-1/16\n-2/16\n-3/16\n-4/16\n-5/16\n-6/16\nlowest\n2 4 6 8log2L\nπsinπ\nLℓ0.20.40.60.81.0Sℓ(b) Eσ= -hL/2\nEγ/hL\n0\n-1/16\n-2/16\n-3/16\n-4/16\n-5/16\n-6/16\nlowest\n2 4 6 8log2L\nπsinπ\nLℓ0.20.40.60.81.01.21.4Sℓ(c) Eσ=0\nEγ/hL\n0\n-1/16\n-2/16\n-3/16\n-4/16\n-5/16\n-6/16\nlowestFIG. 2: Entanglement entropy S`vs log subsystem length\nlog2`vs fermion energy E\r(energy of the \frst term in Eq. (2))\nfor various boson energies E\u001b\u0011\u0000hP\nj\u001bz\nj=\u0000hL;\u0000hL=2;0\n(a,b,c) (second term in Eq. 2). Calculations are done on a\nrandom Majorana chain with L= 1024 sites, and the stan-\ndard deviation of \u0001 tjis\u001b\u0001t=t. States with E\u001b= 0 are\nthe critical excited states which are delocalized. All states\nwith di\u000berent E\ratE\u001b= 0 have logarithmic entanglement\nentropy, and hence are delocalized.\ncopies of Eq. 2, under interaction each site independently\npossesses a random set of many-body spectrum without\ndegeneracy . Let\u000eVbe the typical energy level spacing\nof the interaction Hamiltonian on each site. To cre-\nate entangled pairs between distant sites, the e\u000bective\nlong-range coupling te\u000bgenerated under RG must over-\ncome the energy scale of \u000eVto hybridize the many-body\nstates. However the e\u000bective coupling strength actually\nfalls rapidly with the distance[13{15] as te\u000b\u0018te\u0000pr, so\nthe long-range coupling can only lead to exponentially\nsmall entanglement \u0001 S \u0018 (te\u000b=\u000eV)2\u0018(t=\u000eV )2e\u00002pr.\nTherefore even with weak interaction, all of the eigen-\nstates are short-range entangled area-law states, and can4\nbe fully localized. In contrast, without interaction, no\nmatter what kind of fermion-bilinear perturbations we\nturn on in Eq. 2, as long as these terms preserve the time-\nreversal symmetry de\fned above and the topological na-\nture ofjGiandjTi, there must necessarily be some \f-\nnite energy density states which cannot be fully localized.\nThus in this case interaction actually \\assists\" many-\nbody localization , which is opposite from what is usually\nexpected for weak interaction, in for example Ref. 25, and\nis also di\u000berent from the strong interaction reinforced lo-\ncalization studied in Ref. 26{28.\nNotice that this \\interaction assisted localization\" is\nonly possible with 8 kcopies of the Kitaev's chain with\ntime-reversal symmetry. With 4 copies of the Kitaev's\nchain, the spectrum on each site contains two sets of\ntwo-fold degenerate states even under interaction that\npreserves time-reversal, then the e\u000bective long-range cou-\nplingte\u000bgenerated under RG will still lead to maximal\nentanglement between distant sites. A detailed RG anal-\nysis about this will be given in another paper [29].\n2C. Bosonic SPT states, Haldane phase\nMany bosonic SPT parent Hamiltonians can be written\nas a sum of mutually commuting local terms. For exam-\nple, the \\cluster model\" for the 1 dSPT withZ2\u0002Z2\nsymmetry [9], the Levin-Gu model [30] and the CZX\nmodel [31] for the 2 dSPT states with Z2symmetry,\nand the 3dbosonic SPT state with time-reversal sym-\nmetry [32] are all a sum of commuting local operators;\nthus their ground states are a product of eigenstates of\nlocal operators [47]. SPT Hamiltonians written in this\nform are very similar to the \\universal\" Hamiltonian of\nMBL state proposed in Ref. 33, which is also a sum of\nmutually commuting local terms, because a MBL system\nhas an in\fnite number of local conserved quantities.\nAll of the idealized SPT models mentioned above have\naZ2classi\fcation, and their ground and top states belong\nto the same SPT phase. Obviously there should be no\nphase transition while increasing energy density \". This\nstatement is still valid with small perturbations which\nmake these models nonintegrable as long as the nature of\njGiandjTiare not a\u000bected by the perturbations. Thus\nthese models (and their nonintegrable versions) can all\nbe fully localized by disorder.\nHowever, some other bosonic SPT models can not be\nfully localized. In the following we will give one such\nexample for the Haldane phase [34, 35]:\nH=X\nj(\u00001)j(J+ \u0001Jj)Sj\u0001Sj+1+\u0001\u0001\u0001 (3)\nSjare spin-1/2 operators. The ellipsis includes perturba-\ntions that break the system's symmetry down to a smaller\nsymmetry (such as time-reversal or Z2\u0002Z2) that is suf-\n\fcient to protect the Haldane phase, but do not leadto degeneracy in the bulk spectrum, namely only the\nboundary transforms nontrivially under symmetry. If\nthe random coupling \u0001 Jjis not strong enough to change\nthe sign of J, then the ground state and top state of\nthis model correspond to two opposite dimerization pat-\nterns of the spin-1/2s. Thus one of them is equivalent\nto the Haldane's phase while the other is a trivial phase\nas long as we pick a convention of boundary. If we as-\nsume the random Heisenberg coupling \u0001 Jis su\u000ecient to\nlocalize most of the excited states, then there must be\nan unavoidable phase transition while increasing energy\ndensity\". According to our argument in the introduc-\ntion, this phase transition should behave just like an or-\ndinary quantum phase transition at zero temperature. It\nis known that the quantum phase transition between a\nHaldane phase and a trivial phase is a conformal \feld\ntheory, and it is equivalent to a spin-1/2 chain without\ndimerization. With strong disorder, this quantum criti-\ncal point will be driven into the in\fnite-randomness spin\nsinglet phase [11{14] with a power-law decaying disorder\naveraged spin-spin correlation function and a logarithmic\nentanglement entropy [15].\n2D. 2dinteracting topological superconductor:\ncritical states and interaction assisted localization\nIn this subsection we will discuss the nonchiral 2 dp\u0006ip\ntopological superconductor, i:e:p +ippairing for spin-up\nfermions, and p\u0000ippairing for spin-down fermions. On\na square lattice this TSC can be written in the Majorana\nfermion basis:\nH=X\nk\u001ft\n\u0000k(\u001cxsinkx+\u001cz\u001bzsinky)\u001fk\n+\u001ft\n\u0000k\u001cy(e\u0000coskx\u0000cosky)\u001fk; (4)\nwhere\u001bz=\u00061 represents spin-up and down, while\n\u001cz=\u00061 represents the real and imaginary parts of\nthe electron operator. Without any symmetry, this sys-\ntem is equivalent to the trivial state, i:e:its bound-\nary can be gapped out without degeneracy. However,\nwhen 0< e < 2, with aZ2symmetry which acts as\nZ2:\u001f!\u001bz\u001f, the system is a nontrivial TSC. This sys-\ntem can also have another time-reversal symmetry, which\nis unimportant to our analysis. The boundary of this sys-\ntem reads: H=R\ndx \u001ft(\u0000i@x\u001bz)\u001f,Z2:\u001f!\u001bz\u001f. The\nZ2symmetry forbids any single particle backscattering at\nthe boundary for arbitrary copies of the system, thus the\np\u0006ipTSC with the Z2symmetry has a Zclassi\fcation\nwithout interaction.\nWithout any interaction, for n\u0000copies of the p\u0006ip\nTSC,jGiandjTibelong to di\u000berent SPT phases. This\nis because for either spin-up or down fermions, the Chern\nnumber ofjGiandjTiare opposite. And because the sys-\ntem has a Zclassi\fcation,jGiandjTimust belong to5\ndi\u000berent SPT states. Using our argument in the intro-\nduction, this implies that under disorder that preserves\ntheZ2symmetry, there must be some \fnite energy den-\nsity states which cannot be fully localized. This is not\nsurprising, considering that even at the single particle\nlevel there are likely extended single particle states un-\nder disorder. The existence of extended single particle\nstates is well-known in integer quantum Hall state [36],\nand recently generalized to quantum spin Hall insulator\nwith a Z2index [37, 38].\nThe situation will be very di\u000berent with interactions.\nOnce again a well-designed interaction will reduce the\nclassi\fcation of this p\u0006ipTSC from ZtoZ8[39{42].\nNamelyn\u0000copies of Eq. 4 is topologically equivalent to\n(n+ 8k)\u0000copies. This implies that under interaction jGi\nandjTiactually belong to the same phase when n= 4k.\nThus when n= 4k, the phase transition in the noninter-\nacting limit will be circumvented by interaction above a\ncertain critical value. Thus once again interaction assists\nMBL in this case. When n= 8,jGiandjTiare both\ntrivialized by interaction, namely interaction can adia-\nbatically connect both states to a direct product of local\nstates. When n= 4, Ref. 43 showed that interaction can\ncon\fne the fermionic degrees of freedom, and drive four\ncopies of the p\u0006ipTSC into a 2 dbosonic SPT state\nwithZ2symmetry, which as we discussed in the previous\nsection, can also be fully many-body localized.\nPlease note that in the noninteracting limit the quan-\ntum phase transition between 2 dTSC and trivial state\nis described by gapless (2 + 1) dMajorana fermions, and\nsince a weak short range four-fermion interaction is ir-\nrelevant for gapless (2 + 1) dDirac/Majorana fermions,\nonly strong enough interaction can gap out the quantum\nphase transition. Thus unlike the 1 danalogue discussed\nin section 2B, we expect that in this 2 dsystem only strong\nenough interaction can \\assist\" disorder and localize all\nthe excited states even for n= 4k.\n3. SUMMARY\nIn this work we propose a simple rule to determine\nwhether a local Hamiltonian with symmetry can be\nmany-body localized. Since MBL is a phenomenon for\nthe entire spectrum, we need to start with a lattice\nHamiltonian for our analysis. Therefore the low energy\n\feld theory descriptions and classi\fcation of SPT states\nsuch as the Chern-Simons \feld theory [44] and the non-\nlinear sigma model \feld theory [45] will not be able to\naddress this question. Instead, our argument is based\non the nature of the ground and top states of the same\nlattice Hamiltonian. Our argument is general enough,\nthat it can be applied to both free and interacting sys-\ntems, bosonic and fermionic SPT systems. And coun-\nterintuitively, we found that because interactions change\nthe classi\fcation of fermionic topological insulators andtopological superconductors, in some cases interactions\nactually assists localization, rather than delocalization.\nThe authors are supported by the the David and Lucile\nPackard Foundation and NSF Grant No. DMR-1151208.\nThe authors are grateful to Chetan Nayak for very helpful\ndiscussions.\n[1] X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, Phys. Rev.\nB87, 155114 (2013).\n[2] X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, Science\n338, 1604 (2012).\n[3] B. Bauer and C. Nayak, Journal of Statistical Mechanics:\nTheory and Experiment 9, 09005 (2013), 1306.5753.\n[4] P. Anderson, Phys. Rev. 109, 1492 (1958).\n[5] D. Basko, I. Aleiner, and B. Altshuler, Annals of Physics\n321, 1126 (2006).\n[6] I. V. Gornyi, A. D. Mirlin, and D. G. Polyakov, Phys.\nRev. Lett. 95, 206603 (2005).\n[7] D. A. Huse, R. Nandkishore, V. Oganesyan, A. Pal, and\nS. L. Sondhi, Phys. Rev. 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Lud-\nwig, AIP Conf. Proc. 1134 , 10 (2009).\n[21] S. Ryu, A. Schnyder, A. Furusaki, and A. Ludwig, New\nJ. Phys. 12, 065010 (2010).\n[22] A. Kitaev, AIP Conf. Proc 1134 , 22 (2009).\n[23] L. Fidkowski and A. Kitaev, Phys. Rev. B 81, 134509\n(2010).\n[24] L. Fidkowski and A. Kitaev, Phys. Rev. B 83, 075103\n(2011).\n[25] R. Nandkishore and A. C. Potter, Phys. Rev. B 90,\n195115 (2014).\n[26] Y. Bar Lev, G. Cohen, and D. R. Reichman, Phys. Rev.\nLett. 114, 100601 (2015).\n[27] V. P. Michal, I. L. Aleiner, B. L. Altshuler, and G. V.\nShlyapnikov, ArXiv e-prints (2015), 1502.00282.\n[28] M. Schreiber, S. S. Hodgman, P. Bordia, H. P. L uschen,\nM. H. Fischer, R. Vosk, E. Altman, U. Schneider, and\nI. Bloch, ArXiv e-prints (2015), 1501.05661.\n[29] Y.-Z. You and C. Xu, In progress (2015).6\n[30] M. Levin and Z.-C. Gu, Phys. Rev. B 86, 115109 (2012).\n[31] X. Chen, Z.-X. Liu, and X.-G. Wen, Phys. Rev. B 84,\n235141 (2011).\n[32] F. J. Burnell, X. Chen, L. Fidkowski, and A. Vishwanath,\narXiv:1302.7072 (2013).\n[33] M. Serbyn, Z. Papi\u0013 c, and D. A. Abanin, Phys. Rev. Lett.\n111, 127201 (2013).\n[34] F. D. M. Haldane, Phys. Lett. A 93, 464 (1983).\n[35] F. D. M. Haldane, Phys. Rev. Lett. 50, 1153 (1983).\n[36] B. I. Halperin, Phys. Rev. B 25, 2185 (1982).\n[37] M. Onoda, Y. Avishai, and N. Nagaosa, Phys. Rev. Lett.\n98, 076802 (2007).\n[38] A. A. Soluyanov and D. Vanderbilt, Phys. Rev. B 83,\n035108 (2011).\n[39] X.-L. Qi, New J. Phys. 15, 065002 (2013).\n[40] H. Yao and S. Ryu, Phys. Rev. B 88, 064507 (2013).\n[41] S. Ryu and S.-C. Zhang, Phys. Rev. B 85, 245132 (2012).\n[42] Z.-C. Gu and M. Levin, arXiv:1304.4569 (2013).\n[43] Z. Bi, A. Rasmussen, Y.-Z. You, M. Cheng, and C. Xu,\narXiv:1404.6256 (2014).\n[44] Y.-M. Lu and A. Vishwanath, Phys. Rev. B 86, 125119\n(2012).\n[45] Z. Bi, A. Rasmussen, and C. Xu, Phys. Rev. B 91, 134404(2015).\n[46] In this work, the phrase \\SPT states\" also include direct\nproduct states, we view direct product states as \\trivial\"\nSPT states. So far not all quantum phase transitions be-\ntween SPT states have been completely studied, and it\nis possible that some SPT states are separated by a \frst\norder transition. Our statement only applies to the cases\nthatjGiandjTibelong to two di\u000berent SPT phases that\nwe know are separate by a continuous phase transition,\nfor example the transition between the topological su-\nperconductor and the trivial state of the Kitaev's chain\n(section 2B).\n[47] Ref. 1 actually proposed a general way of constructing\nparent Hamiltonians for all bosonic SPT states within the\ngroup cohomology classi\fcation. However, in Ref. 1 the\nlocal Hilbert space is labeled by group elements, which\nimplies that for a system with continuous symmetry the\nlocal Hilbert space in Ref. 1's construction already has\nin\fnite dimension, and hence its excited states can also\nhave in\fnite local energy density. In this work we only\ndiscuss systems with a \fnite dimensional Hilbert space\nand \fnite energy density."    },    {        "title": "1506.06461v1.Reassessing_nuclear_matter_incompressibility_and_its_density_dependence.pdf",        "content": "arXiv:1506.06461v1  [nucl-th]  22 Jun 2015Reassessing nuclear matter incompressibility and its dens ity dependence\nJ. N. De, S. K. Samaddar, and B. K. Agrawal\nSaha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolka ta700064, India\nExperimental giant monopole resonance energies are now kno wn to constrain nuclear incompress-\nibility of symmetric nuclear matter Kand its density slope Mat a particular value of sub-saturation\ndensity, the crossing density ρc. Consistent with these constraints, we propose a reasonabl e way to\nconstruct a plausible equation of state of symmetric nuclea r matter in a broad density region around\nthe saturation density ρ0. Help of two additional empirical inputs, the value of ρ0and that of the\nenergy per nucleon e(ρ0) are needed. The value of K(ρ0) comes out to be 211 .9±24.5 MeV.\nPACS numbers: 21.10.Re,21.65.-f,21.65.Mn\nKeywords:\nI. INTRODUCTION\nThe nuclear incompressibility parameter K0defined\nfor symmetric nuclear matter (SNM) at saturation den-\nsityρ0stands out as an irreducible element of physical\nreality. It has an umbilical association with the isoscalar\ngiant monopole resonances (ISGMR) in microscopic nu-\nclei; it also underlies in a proper understanding of super-\nnova explosion in the cosmic domain [1]. From careful\nmicroscopic analysis of ISGMR energies with suitably\nconstructed energy density functional (EDF) E(ρ) in a\nnon relativistic framework as applicable to finite and in-\nfinite nuclear systems, its value had initially been fixed\natK0≃210±30MeV [2, 3]. In microscopic relativistic\napproachesonthe otherhand, ahighervalue of K0∼260\nMeVwasobtained[4]. After severalrevisionsfrom differ-\nent corners, however, its value settled to K0≃230±20\nMeV [5–7]. It gives good agreement with the experimen-\ntally determined centroids of ISGMR, in particular, for\n208Pb ,90Zr and144Sm nuclei, calculated both with non-\nrelativistic [8, 9] and relativistic [7] energy density func-\ntionals. Thenear-settledproblemwas, however,leftopen\nwith the apparent incompatibility of the said value of\nK0with the recent ISGMR data for Sn and Cd-isotopes\n[6, 10–17]. These nuclei showed remarkable softness to-\nwards compression, the ISGMR data appeared explained\nbest with K0∼200 MeV [6].\nA plausible explanation was recently put forward by\nKhanet.al[18] for the apparent discrepancy. It is ar-\ngued that there may not be an unique relation between\nthe value of K0associated with an effective force and\nthe monopole energy of a nucleus predicted by the force\n[19]. The region between the center and the surface of\nthe nucleus is the most sensitive towards displaying the\ncompression as manifested in the ISGMR. The ISGMR\ncentroid EGis related to the integral of incompressibility\n(/integraltext\nK(ρ)dρ) overthewholedensityrange[20]. Asaresult,\na larger value of K(ρ0) for a given EDF can be compen-\nsated by lower values of K(ρ) at sub-saturation densities\nsoastopredictasimilarvalueofISGMRenergyinnuclei.\nIt is seen that the incompressibility K(ρ) calculated with\na multitude of energy density functionals when plottedagainst density cross close to a single density point [18],\nthis universality possibly arising from the constraints en-\ncoded in the EDF from empirical nuclear observables.\nThis crossing density ρc[= (0.71±0.005)ρ0] [21] seems\nmore relevant as an indicator for the ISGMR centroid.\nBecause of the incompressibility integral, the centroid\nseems more intimately correlated to the derivative of\nthe compression modulus (defined as M= 3ρK′(ρ) )\nat the crossing density rather than to K0. The value of\nKc(=K(ρc)) is seen to be ∼35±4 MeV [21]. From var-\nious functionals, the calculated values of Mc(=M(ρc))\nare found to be linearly correlated with the correspond-\ningly calculatedvalues ofISGMR centroidsfor208Pb and\nalso for120Sn. From the known experimental ISGMR\ndata for these nuclei, a value of Mc≃1050±100 MeV\n[21] is then obtained, revised from an earlier estimate of\n1100±70 MeV [18]. Using a further assumption of a\nlinear correlation between K0andEGcalculated from\ndifferent EDF, a value for K0≃230 MeV with an un-\ncertainty of ≃40 MeV is reported, the uncertainty being\ninferred from the spread of K0values obtained with the\ndifferent functionals used.\nThe universality of the crossing point ρcand the val-\nues ofKcandMccan be readily acknowledged; Mcis\nseen to be well correlated to EG. The Pearson correla-\ntion coefficient r[22] ofMcwithEGfor120Sn is 0.80\nand is 0.94 for208Pb. However, assumption of a lin-\near correlation between K0andEGmay not be justi-\nfied, they seem to be very weakly correlated ( r=0.67 for\n120Sn and 0.79 for208Pb) [21]. The inferred value of\nincompressibility around saturation may then be called\ninto question. One can see that a linear Taylor ex-\npansion K0(ρ0) =K(ρc) + (ρ0−ρc)K′(ρc) yields for\nK0≃185±14.3 MeV, noting that K′(ρc) =Mc/(3ρc).\nThe absence of a strong linear correlation between K0\nandEGcalculated from different effective forces prompts\none to think that KcandMcalone are not sufficient to\nyield the correct value of K0. Further empirical informa-\ntion is possibly needed to arrive at that. In this paper,\nwe show that with given values of only KcandMcalong\nwith some time-tested values of empirical nuclear con-\nstants, it is possible to address to a proper assessment of\nthe value of incompressibility Kand its density depen-2\ndence. Theempiricalconstantsarethesaturationdensity\nρ0, taken as 0.155 ±0.008 fm−3for SNM and the energy\nper nucleon at that density e(ρ0), taken as −16.0±0.1\nMeV [23, 24]. An acceptable value of the effective nu-\ncleon mass m∗/m, which lies in the range m∗/m∼0.8\n±0.2 [25] at saturation density is also used.\nThis paper is structured as follows. In Sec. II, we in-\ntroduce the theoretical elements to calculate the nuclear\nequationofstatefrom Kcandρcwith theaidofempirical\ninputs mentioned. Results and discussions are presented\nin Sec. III. Sec. IV contains the concluding remarks.\nII. THEORETICAL EDIFICE\nWe keep the discussions pertinent for SNM at any den-\nsityρat zero temperature ( T= 0). The chemical poten-\ntial of a nucleon is given by the single-particle energy at\nthe Fermi surface,\nµ=εF=p2\nF\n2m+U (1)\nwherepF(ρ) isthe Fermimomentum and U(ρ) thesingle-\nparticle potential. Assuming the nucleonic interaction to\nbe momentum and density dependent, the single-particle\npotential separates into three parts [26]\nU=V0+p2\nFV1+V2. (2)\nThe last term V2is the rearrangement potential that\narises only for density-dependent interactions, and the\nsecondisthemomentum-dependenttermthatdefinesthe\neffective mass m∗,\np2\nF\n2m∗=p2\nF\n2m+p2\nFV1 (3)\nso that\n1\nm∗=1\nm+2V1. (4)\nThe energy per nucleon at density ρis given by,\ne=<p2\n2m>+1\n2< p2> V1+1\n2V0\n=1\n2(1+m∗\nm)<p2\n2m∗>+1\n2V0 (5)\nFrom Gibbs-Duhem relation,\nµ=e+P\nρ, (6)\nwherePis the pressure. Keeping this in mind, from\nEqs. (1),(5) and (6), we get\ne(ρ) =p2\nF\n10m[3−2m\nm∗]−V2+P\nρ, (7)\nwhere we have put < p2>=3\n5p2\nF.The density dependence of the effective mass [27] can\nbe cast asm\nm∗= 1+kρ, the rearrangement potential can\nbe written in the form V2=aρα. This is the form that\nemerges for finite range density-dependent forces [26] in\na non relativistic framework or for Skyrme interactions.\nThe quantities a, αandkare numbers. Ifm∗\nm(ρ0) is\nchosen,kis known.\nAtρ=ρ0,P= 0, then from Eq. (7), writing forp2\nF\n2m=\nbρ2/3withb=(3\n2π2)2/3/planckover2pi12\n2m,\ne0=e(ρ0) =b\n5ρ2/3\n0[1−2kρ0]−aρα\n0. (8)\nSinceP=ρ2∂e\n∂ρ, from Eq. (7) again we get,\nP=b\n15ρ5/3−1\n3bkρ8/3−1\n2αaρα+1+1\n2ρ∂P\n∂ρ.(9)\nAtρ0, this yields (since K0= 9∂P\n∂ρ|ρ0),\n1\n2αaρα\n0+1\n3bkρ5/3\n0−(K0\n18+b\n15ρ2/3\n0) = 0.(10)\nFurthermore, Eq. (9) gives\nK(ρ) = 9∂P\n∂ρ= 2bρ2/3−16bkρ5/3\n−9α(α+1)aρα+9ρ∂2P\n∂ρ2.(11)\nDefining M= 3ρdK\ndρ= 27ρ∂2P\n∂ρ2, this leads, at ρ=ρcto\n9α(α+1)aρα\nc+16bkρ5/3\nc−(2bρ2/3\nc+Mc\n3−Kc) = 0.(12)\nSincekis a given entity and ρcand (Mc/3−Kc) are\nknown, eqs. (8) and (12) can be solved for aandα,\neq. (10) then gives the value of the nuclear incompress-\nibilityK0. Once K0is obtained, M0(=M(ρ0)) is\nevaluated from eq. (12) by choosing ρ0forρc. Then\nQ0= 27ρ3\n0∂3e\n∂ρ3|ρ0is also known from M0= 12K0+Q0.\nThe structure ofeq. (9) showsthat the pressureand its\nfirst derivative are interrelated. One can then get higher\ndensity derivatives of Por of energy erecursively from\neq. (9) as is evident from eq. (11). For the present, we\nshow that\n9ρ∂3P\n∂ρ3= 9α2(α+1)aρα−1+80\n3bkρ2/3−4\n3bρ−1/3.(13)\nSince\n∂3P\n∂ρ3= 6∂2e\n∂ρ2+6ρ∂3e\n∂ρ3+ρ2∂4e\n∂ρ4, (14)\nwe find\n9ρ2\n0∂3P\n∂ρ3|ρ0= 6K0+2Q0+1\n9N0. (15)3\nwhere we have defined N0= 81ρ4\n0∂4e\n∂ρ4|ρ0. From eq. (13)\nand (15), knowing K0andQ0,N0can be calculated.\nSimilarly, one can calculate the fifth density derivative of\nenergy (R0= 243ρ5\n0∂5e\n∂ρ5|ρ0) by exploiting eqs. (13) and\n(14) from\n9ρ3\n0∂4P\n∂ρ4|ρ0= 4Q0+8\n9N0+1\n27R0. (16)\nThese help to find the density variation ofthe energy and\nalso of the incompressibility, as is seen,\ne(ρ) =e(ρ0)+1\n2K0ǫ2+1\n6Q0ǫ3\n+1\n24N0ǫ4+1\n120R0ǫ5+... , (17)\nwhereǫ= (ρ−ρ0\n3ρ0) (counting terms only up to ǫ5is seen to\nbe a very good approximation in the density range of ∼\nρ0/4< ρ <2.0ρ0, we retain terms up to them). Eqs. (7)\nand (17) give\nP(ρ)\nρ=e(ρ0)+1\n2K0ǫ2+1\n6Q0ǫ3+1\n24N0ǫ4\n+1\n120R0ǫ5−b\n5ρ2/3[1−2kρ]+aρα.(18)\nand eq. (9) gives\nK(ρ) = 9dP\ndρ= 18[P\nρ−b\n15ρ2/3+1\n3bkρ5/3+1\n2αaρα].(19)\nWe have thus the equation of state (EOS) of symmetric\nnuclearmatterinareasonablyspread-outdensitydomain\naround the saturation density.\nThe incompressibility Kat any density ρcan be cal-\nculated directly from eq. (19) or it may be calculated in\nterms of K(ρc) and its higher density derivatives as\nK(ρ) =K(ρc)+(ρ−ρc)K′(ρc)+(ρ−ρc)2\n2K′′(ρc)\n+(ρ−ρc)3\n6K′′′(ρc)+.... (20)\nThe different derivatives can be calculated from eq. (19).\nWith given values of ρ0,e0,m∗\nm(ρ0),andρc, one notes\nthat the solutions for aandαdo not depend separately\nonKcandMc, but on ( Mc/3−Kc).\nIII. RESULTS AND DISCUSSIONS\nThe values of the empirical constants ρ0,e0andm∗\nm\nneeded for our calculation have already been mentioned.\nAs for the crossingdensity, we choose ρc= 0.110±0.0008\nfm−3. With given inputs of McandKc, it should be\nnoted that the output values for McandKcmay come\nout to be different, but ( Mc/3−Kc) remains invariant.-1 -0.5 0 0.5 1180200220240K0 (MeV)ρ0\nMc\nKc\nρc\nFIG. 1: (Color online) The sensitivity of the incompressibi lity\nK0at the saturation density ( ρ0) on the values of the incom-\npressibility Kc(green dash-dotted line), its density slope Mc\n(red dashed line), the crossing density ρc(blue dotted line)\nand the value of ρ0(black full line). The abscissa extends\nfrom -1 to +1. These end points refer to the scaled lower and\nupper limits of Kc,Mc,ρcandρ0, respectively (see text).\nWith inputs Mc=1050 MeV and Kc=35 MeV, the out-\nputMcandKcare found to be 1051.8 MeV and 35.46\nMeV, respectively. Since they are very close to the input\nvalues, they were not tinkered with for exact matching\nof the output and input values. The value of incompress-\nibility at ρ0turns out to be K0=211.9±24.5 MeV either\nfrom eq. (19) or eq. (20). We note that in eq. (20), at sat-\nuration, the value of the second term on the right hand\nside is 143.3 MeV, the third term is 35.9 MeV, the fourth\nterm is−3.2 MeV, the fifth term (not shown in eq. (20))\nis 0.55 MeV and so on,which adds up to ∼211.9 MeV.\nThe uncertainty in an observable X(likeK,Metc) is\ncalculated from ∆ X2=/summationtext\ni(∂X\n∂yi∆yi)2where ∆yiare the\nuncertainties in the empirically known entities yi. The\nsensitivity of K0on these entities that influence the in-\ncompressibility most is displayed in Fig.1. The abscissa\nis scaled such that 0 refers to the central value of these\nentitiesMc,Kc,ρcandρ0;±1 refer to the extrema of\ntheirdomain( ±100MeV, ±5MeV,±0.005ρ0and±0.008\nfm−3fromthecentralvaluesoftheentities, respectively).\nThe value of K0is seen to be very sensitive with changes\nin either Mcorρ0when all other input entities are kept\nfixed. Its sensitivity to Kcorρcis weak; onm∗\nmor to the\nenergy per nucleon e0, it is rather insensitive. The near-\ninsensitivity of incompressibility to the effective mass is\nobserved for Skyrme density functionals also. From the\ndata base for these functionals as tabulated by Dutra et.\nal[25], the correlation coefficient between K0andm∗is\ncalculated to be only ∼ −0.2.4\n950 1000 1050 1100 1150 1200\nMc (MeV)050100150200250Kn (MeV) K0\nK1\nK2\n-K3\nFIG. 2: (Color online)\nThe incompressibility and its different density derivative as\ndefined in the text plotted as a function of Mc.\nThe near-perfect linear correlation of K0withMcas\nseen in Fig.1 is very startling. From Eq. (20), one\nmay expect that the second and higher order deriva-\ntives ofK(ρc) would destroy this correlation. However,\nwe find that both K′′andK′′′are also linearly corre-\nlated with Mcand thus K(ρ0) retains its linear corre-\nlation with Mc. This is displayed in Fig.2, where we\ndefineK1= (ρ0−ρc)K′(ρc),K2=(ρ0−ρc)2\n2K′′(ρc) and\nK3=(ρ0−ρc)3\n6K′′′(ρc). The weakcorrelationbetween K0\nandMcthat can be inferred from the calculated correla-\ntion structure of ( Mc−EG) and (K0−EG) in refs.[18, 21]\npossibly results from the use of different EDFs in getting\nthe various relevant observables.\nFigures 3 and 4 display the functional dependence of\nthe nuclear EOS on density. The panels (a) and (b) in\nFig.3 show the energy per nucleon and the pressure, the\nones in Fig.4 show the incompressibility and its density\nderivative M, respectively. As one sees, the uncertainty\nin energy and pressure grows as one moves away from\nthe saturation density, similarly the uncertainty in in-\ncompressibility or its density derivative increases with\ndistance from the crossing density.\nIV. CONCLUSIONS\nTo sum up, we have made a modest attempt to re-\nassess the value of K(ρ0) consistent with the new-found\nconstraint on the incompressibility K(ρc) and its den-\nsity slope M(ρc) at a particular value of density at sub-\nsaturation, the crossing density ρc. We have relied on\nsome empirically well-known values of nuclear constants.\nWe have further made the assumption of linear density-15-10-50e(ρ) (MeV)\n0 0.5 1 1.5 2\nρ/ρ0051015P(ρ) (MeV fm-3)(a)\n(b)\nFIG. 3: (Color online) The nuclear EOS as a function of den-\nsity. The panels (a) and (b) show the energy per nucleon and\npressure, respectively in a selected range around the satur a-\ntion density.\n050010001500K(ρ) (MeV)\n0 0.5 1 1.5 2\nρ/ρ00500010000M(ρ) (MeV)(a)\n(b)\nFIG. 4: (Color online) The nuclear EOS as a function of den-\nsity. The panels (a) and(b)show theincompressibility and i ts\ndensity derivative M, respectively in a selected range around\nthe saturation density.\ndependence of the effective mass and the power law de-\npendence of the rearrangement potential which happens\nto be generally true for non-relativistic momentum and\ndensity dependent interactions. In relativistic models,\nthe density dependence of the effective mass may not be\nlinear [28]. The rearrangement potential appears explic-\nitly there only in the case of density dependent meson5\nexchange models [29].\nThe value of incompressibility K(ρ0) turns out to be\n211.9±24.5 MeV. This is somewhat lower than the cur-\nrent value in vogue, K0∼230±20 MeV. From recursive\nrelations, ourmethod allowsalsoestimatesofhigher den-\nsity derivatives of energy or of pressure and thus helps\nin constructing the nuclear EoS e(ρ) at and around thesaturation density.\nThe authors gratefully acknowledge the assistance of\nTanuja Agrawal in the preparation of the manuscript.\nOne of the authors (JND) acknowledges support from\nthe Department of Science & Technology, Government of\nIndia.\n[1] G. E. Brown, Phys. Rep. 163, 167 (1988).\n[2] J. P. Blaizot, Phys. Rep. 64, 171 (1980).\n[3] M. Farine, J. M. Pearson, and F. Tondeur, Nucl. Phys.\nA615, 135 (1997).\n[4] D. Vretenar, T. Niksic, and P. Ring, Phys. Rev. C 68,\n024310 (2003).\n[5] B. G. Todd-Rutel and J. Piekarewicz, Phys. Rev. Lett\n95, 122501 (2005).\n[6] P. Avogadro and C. A. Bertulani, Phys. Rev. C 88,\n044319 (2013).\n[7] T. Niksic, D. Vretenar, and P. Ring, Phys. Rev. C 78,\n034318 (2008).\n[8] B. K. Agrawal, S. Shlomo, and V. K. Au, Phys. Rev. C\n72, 014310 (2005).\n[9] J. Kvasil, D. Bozhik, A. Repko, P.-G. Reinhard,\nV. Nesterenko, and W. Kleinig, arXiv:1412.1997 [nucl-\nth] (2014).\n[10] Y. W. Lui, D. H. Youngblood, Y. Tokimoto, H. L. Clark,\nand B. John, Phys. Rev. C 70, 014307 (2004).\n[11] D. H. Youngblood, Y.-W. Lui, B. John, Y. Tokimoto,\nH. L. Clark, and X. Chen, Phys. Rev. C 69, 054312\n(2004).\n[12] T. Li and et. al,, Phys. Rev. Lett. 99, 162503 (2007).\n[13] J. Piekarewicz, Phys. Rev. C 76, 031301(R) (2007).\n[14] U. Garg and et al. T. Li, Nucl. Phys. A788, 36c (2007).\n[15] V. Tselyaev, J. Speth, S. Krewald, E. Litvinova,\nS. Kamerdzhiev, N. Lyutorovich, A. Avdeenkov, and\nF. Grmmer, Phys. Rev. C 79, 034309 (2009).\n[16] T. Li and et. al,, Phys. Rev. C 81, 034309 (2010).[17] D. Patel and et. al,, Phys. Lett. B 718, 447 (2012).\n[18] E. Khan, J. Margueron, and I. Vida˜ na, Phys. Rev. Lett.\n109, 092501 (2012).\n[19] G. Col` o, N. V. Giai, J. Meyer, K. Bennaceur, and\nP. Bonche, Phys. Rev. C 70, 024307 (2004).\n[20] E. Khan, J. Margueron, G. Col` o, K. Hagino, and\nH. Sagawa, Phys. Rev. C 82, 024322 (2010).\n[21] E. Khan and J. Margueron, Phys. Rev. C 88, 034319\n(2013).\n[22] S. Brandt, Statistical and Computational Methods in\nData Analysis (Springer, New York, 3rd English edition,\n1997).\n[23] B. K. Agrawal, J. N. De, and S. K. Samaddar, Phys. Rev.\nLett.109, 262501 (2012).\n[24] N. Alam, B. K. Agrawal, J. N. De, and S. K. Samaddar,\nPhys. Rev. C 90, 054317 (2014).\n[25] M. Dutra, O. Lourenco, J. S. S´ aMartins, A. Delfino, J. R.\nStone, and P. D. Stevenson, Phys. Rev. C 85, 035201\n(2012).\n[26] D. Bandyopadhyay, C. Samanta, S. K. Samaddar, and\nJ. N. De, Nuclear Physics A 511, 1 (1990).\n[27] A. Bohr and B. R.Mottelson, Nuclear Structure , vol. Vol.\nI (Benjamin, New York, 1969).\n[28] B. D. Serot and J. D. Walecka, Adv. Nucl. Phys. 16, 1\n(1986).\n[29] S. Typel and H. H. Wolter, Nucl. Phys. A656, 331\n(1999)."    },    {        "title": "1506.07740v1.Ground_state_densities_from_the_Rayleigh__Ritz_variation_principle_and_from_density_functional_theory.pdf",        "content": "arXiv:1506.07740v1  [physics.chem-ph]  25 Jun 2015Ground-state densities from the Rayleigh–Ritz variation p rinciple and from\ndensity-functional theory\nSimen Kvaal∗and Trygve Helgaker†\nCentre for Theoretical and Computational Chemistry,\nDepartment of Chemistry, University of Oslo,\nP.O. Box 1033 Blindern, N-0315 Oslo, Norway\n(Dated: May 24, 2022)\nThe relationship between the densities of ground-state wav e functions (i.e., the minimizers of\nthe Rayleigh–Ritz variation principle) and the ground-sta te densities in density-functional theory\n(i.e., the minimizers of the Hohenberg–Kohn variation prin ciple) is studied within the framework of\nconvex conjugation, in a generic setting covering molecula r systems, solid-state systems, and more.\nHaving introduced admissible density functionals as funct ionals that produce the exact ground\nground-state energy for a given external potential by minim izing over densities in the Hohenberg–\nKohnvariation principle, necessary sufficient conditions o n such functionals are established toensure\nthat the Rayleigh–Ritz ground-state densities and the Hohe nberg–Kohn ground-state densities are\nidentical. We apply the results to molecular systems in the B orn–Oppenheimer approximation. For\nany given potential v∈L3/2(R3)+L∞(R3), we establish a one-to-one correspondence between the\nmixed ground-state densities of the Rayleigh–Ritz variati on principle and the mixed ground-state\ndensities of the Hohenberg–Kohn variation principle when t he Lieb density-matrix constrained-\nsearch universal density functional is taken as the admissi ble functional. A similar one-to-one\ncorrespondence is established between the pure ground-sta te densities of the Rayleigh–Ritz variation\nprincipleandthepureground-statedensities obtainedusi ngtheHohenberg–Kohnvariation principle\nwith the Levy–Lieb pure-state constrained-search functio nal. In other words, all physical ground-\nstate densities (pure or mixed) are recovered with these fun ctionals and no false densities (i.e.,\nminimizing densities that are not physical) exist. The impo rtance of topology (i.e., choice of Banach\nspace of densities and potentials) is emphasized and illust rated. The relevance of these results for\ncurrent-density-functional theory is examined.\n∗simen.kvaal@kjemi.uio.no\n†t.u.helgaker@kjemi.uio.noI. INTRODUCTION\nThe concept of modeling an electron gas and more generally all electr onic systems using only the density\nρgoes back to the early days of quantum mechanics and the independ ent work of Tomas and Fermi in 1927\n[1, 2]. However, it was only with the seminal work of Hohenberg and Ko hn in 1964 that density-functional\ntheory (DFT) was shown to be—in principle, at least—an exact theor y [3]. In 1983, Lieb formulated DFT\nrigorously for electronic systems in R3, using concepts from convex analysis [4]. Today, DFT is the most\npopular computational method for many-electron systems in chem istry and solid-state physics.\nIn an external electrostatic scalar potential v, the ground-state energy E(v) of anN-electron system is\nobtained from the Rayleigh–Ritz variation principle\nE(v) := inf\nψ/an}b∇acketle{tψ|ˆH(v)|ψ/an}b∇acket∇i}ht, (1)\nwhereˆH(v) is the Hamiltonian and the infimum extends over all L2-normalized wave functions for which\nthe expectation value makes sense. With each minimizing ground-sta te wave function ψin Eq. (1), there is\nan associated ground-state density ρ. In DFT, the ground-state energy E(v) is instead obtained from the\nHohenberg–Kohn variation principle ,\nE(v) = inf\nρ/parenleftbig\nF0(ρ)+(v|ρ)/parenrightbig\n, (2)\nwhereF0(ρ) is theuniversal functional and the minimization is over all densities for which the interaction\n(v|ρ) =/integraltext\nv(r)ρ(r)dris meaningful. Note that we have not specified the spatial domain nor the nature of the\nexternal potential v. Thus, the minimization problems (1–2) are so far not fully defined fr om a mathematical\nperspective. Moreover, the functional F0is not unique—any functional F0that gives the correct ground-\nstate energy in the Hohenberg–Kohn variation principle is said to be ‘a dmissible’. Common expositions\nof DFT use the Levy–Lieb constrained-search functionalFLL[4, 5], the density-matrix constrained-search\nfunctionalFDM[4], and the Lieb functional F, all giving the same ground-state energy E(v) for a large\nclass of potentials. Specializing to electrons in R3and assuming that the external potential vis sufficiently\nwell-behaved, these functionals are given by the following expressio ns:\nFLL(ρ) := inf/braceleftBig\n/an}b∇acketle{tψ|ˆH0|ψ/an}b∇acket∇i}ht/vextendsingle/vextendsingleψ∈H1\nS(R3N),/ba∇dblψ/ba∇dbl2= 1,ψ/mapsto→ρ/bracerightBig\n, (3)\nFDM(ρ) := inf/braceleftBig\nTr(ˆH0Γ)/vextendsingle/vextendsingleΓ =/summationtext\nkλk|ψk/an}b∇acket∇i}ht/an}b∇acketle{tψk|,/summationtext\nkλk= 1,λk≥0,ψk∈H1\nS(R3N),/ba∇dblψk/ba∇dbl2= 1,ψ/mapsto→ρ/bracerightBig\n,(4)\nF(ρ) := sup\nv∈X′(E(v)−(v|ρ)). (5)\nHereH1\nS(R3N) is the first-order Sobolev space H1\nS(R3N) with proper permutation symmetry due to spin,\nˆH0=ˆT+ˆWis the sum of the kinetic and inter-electron interaction operators, andψ/mapsto→ρand Γ/mapsto→ρ\nindicate that the wave function ψand density matrix Γ, respectively, have density ρ. In the definition of\nthe Lieb functional, Xis a Banach space in which densities are embedded, with X′as its dual, consisting\nof potentials. We note that the Lieb functional is by construction lo wer semi-continuous and convex, being\nthe conjugate function to Ein the sense of convex analysis.\nWhereas the density functionals FLL,FDMandFare all admissible and therefore give the same total\nenergy in the Hohenberg–Kohn variation principle, they may have diff erent minimizing densities. In this\narticle, we study the relationship between the densities in the Hohen berg–Kohn and Rayleigh–Ritz variation\nprinciples. In particular, we establish what conditions must be impose d on an admissible density functional\nto ensure that the ground-state densities (i.e., the minimizing densit ies) in the Hohenberg–Kohn variation\nprinciplearethesameasthedensitiesoftheground-statewavefu nctions(i.e., theminimizingwavefunctions)\nin the Rayleigh–Ritz variation principle.\nWhile our treatment covers the standard DFT setting outlined abov e, it is motivated by a study of\ncurrent-density-functional theory (CDFT), where the N-electron system is subject to an external magnetic\nvector potential Ain addition to the scalar potential v[6–8], introducing the paramagnetic current density\njp∈L1(R3) as an additional variable in the Hohenberg–Kohn variation principle:\nE(v,A) = inf\nρ,jp/parenleftbig\nF0(ρ,jp)+/parenleftbig\nv+1\n2A2|ρ/parenrightbig\n+(A|jp)/parenrightbig\n, (6)\n2where(A|jp) =/integraltext\nA(r)·jp(r)dr. TheCDFTgeneralizationoftheLevy–Liebfunctionalisthe Vignale–Rasolt\nfunctionalFVR(ρ,jp); there is likewise a generalization of the density-matrix constraine d-search functional\nFDMand the Lieb functional Fto include a current dependence. The mathematical properties of these\nfunctionals are not as well understood as in the standard DFT sett ing. The results presented in this article\nare in part meant to be a step towards such an understanding.\nThe article is structured as follows. In Sec.II, we describe DFT fro m a generic and abstract point of view,\nintroducing arbitrary admissible density functionals F0and some concepts of convex analysis. We obtain\nseveral results that characterize the relationship between minimiz ers of the Rayleigh–Ritz variation principle\nand of the Hohenberg–Kohn variation principle. In Sec.III we apply our findings to the standard DFT\nof atoms and molecules in the Born–Oppenheimer approximation. The importance of the topology of the\nunderlying density space is emphasized. In Sec.IV we discuss CDFT, and finally, in Sec.V we summarize\nand draw some conclusions.\nII. DFT FROM AN ABSTRACT POINT OF VIEW\nA. Admissible universal density functionals\nExcept for the trivial case of one electron, an explicit formula for a ny of the standard density functionals\nused in the Hohenberg–Kohn variation principle in Eq. (2) is not known . Moreover, the mathematical\nanalysis of Eq. (2) is difficult without further assumptions. In his wor k [4], Lieb placed DFT for electronic\nsystems in R3on a firm mathematical ground using the language of convex analysis [9, 10], the natural\nsetting for problems such as Eq. (2). The starting point is to embed the densities in a Banach space Xand\nto consider potentials in the dual space X′, thereby making the interaction ( v|ρ) well-defined and continuous\nin both arguments. We then obtain the ground-state energy as a m apE:X′→R∪{−∞} given by\nE(v) = inf\nρ∈X(F0(ρ)+(v|ρ)),∀v∈X′, (7)\nwhereF0:X→R∪{+∞}is the universal density functional. Being the pointwise infimum of a co llection\nof affine maps of the form v/mapsto→(v|ρ)+F0(ρ), the ground-state energy in Eq. (7) is automatically concave\nand upper semi-continuous with respect to the topology on X′. Note that we allow EandF0to be infinite.\nTheeffective domain of a function is the set where the function is finite—for example, dom (E) ={v∈X′|\nE(v)>−∞}.\nIn Ref. [4], Lieb (and Simon) proved several important results in the context where X=L1(R3)∩L3(R3),\nwith dualX′=L3/2(R3)+L∞(R3). However, this choice of Xis not unique, and the derived results may\ndepend on X. We emphasize that, even if our notation suggests the standard D FT setting, it covers also\nCDFT and other settings.\nA function F0:X→R∪{+∞}is said to be an admissible density functional if, for each potential v∈X′,\nit gives a correct ground-state energy by the Hohenberg–Kohn v ariation principle in Eq. (7)—that is, if it\nproduces identical results with the Rayleigh–Ritz variation principle in Eq. (1). Note that we do not require\nthat there exists a minimizing density ρfor a given potential v, even in those caseswhere vsupports a ground\nstate in the Rayleigh–Ritz variation principle in Eq. (1). It is sufficient t hat a minimizing sequence {ρn}\ncan be found. Clearly, for any pair ( ρ,v)∈X×X′, an admissible density functional and the ground-state\nenergy satisfy the Fenchel inequality\nE(v)≤F0(ρ)+(v|ρ). (8)\nWe are here interested in characterizing those pairs ( ρ,v) that saturate Fenchel’s inequality, E(v) =F0(ρ)+\n(v|ρ); in other words, those ρ∈Xthat are minimizers in Eq. (7) for a given v∈X′.\nSince the universal functional is in general not differentiable (see R efs. [4, 11] for the standard DFT case),\nwecannotwritedownanEulerequationforthe solutionofthe minimiza tionproblemEq.(7). Moreover,even\nifF0were differentiable, the Euler equation would in general be a necessa ry but not sufficient condition for\na global minimum in Eq. (7). Instead, we use the concept of subdifferentiation to characterize the solution.\n3The admissible density functional F0is said to be subdifferentiable atρ∈XifF0(ρ)∈Rand if there exists\nu∈X′, known as a subgradient ofF0atρ, such that\nF0(ρ′)≥F0(ρ)+(u|ρ′−ρ),∀ρ′∈X, (9)\nmeaning that F0touches the affine map ρ′/mapsto→F0(ρ′) + (u|ρ′−ρ) atρand lies nowhere below it. The\nsubgradient is thus a generalization of the concept of the slope of a tangent to the graph of F0atρ. The\nsubdifferential ∂−F0(ρ) is the (possibly empty) set of all subgradients of F0atρ:\n∂−F0(ρ) ={u∈X′|F0(ρ′)≥F0(ρ)+(u|ρ′−ρ),∀ρ′∈X, F0(ρ)∈R}. (10)\nThis set is convex [9]. Note that the subdifferential is the empty set w heneverF0(ρ) = +∞. Assuming that\n−vis a subgradient of F0atρ, we obtain by simple rearrangements\n−v∈∂−F0(ρ)⇐⇒F0(ρ′)≥F0(ρ)+(−v|ρ′−ρ)\n⇐⇒F0(ρ)+(v|ρ)≤F0(ρ′)+(v|ρ′)\n⇐⇒F0(ρ)+(v|ρ) = inf\nρ′∈X(F0(ρ′)+(v|ρ′)) =E(v), (11)\nWe have thus shown the following sufficient and necessary condition f or a minimizing density in the\nHohenberg–Kohn variation principle in Eq. (7):\nProposition 1. LetF0:X→R∪{+∞}be an admissible density functional so that Eq. (7)holds. Let\nv∈X′andρ∈Xbe given. Then,\nE(v) =F0(ρ)+(v|ρ)⇐⇒ −v∈∂−F0(ρ). (12)\nB. Lieb’s universal density functional\nFrom the Fenchel inequality in Eq. (8), we obtain by a trivial rearran gement the equivalent inequality\nF0(ρ)≥E(v)−(v|ρ), (13)\nstating that each admissible density functional F0(ρ) is an upper bound to ρ/mapsto→E(v)−(v|ρ) with respect to\nall variations in v∈X′. We now define the Lieb universal density functional F(ρ) as theleast upper bound\ntoE(v)−(v|ρ) for allv∈X′:\nF(ρ) = sup\nv∈X′(E(v)−(v|ρ)),∀ρ∈X, (14)\nwhich in the following will be referred as the Lieb variation principle . The Lieb functional is clearly a lower\nbound to all admissible density functionals:\nF(ρ)≤F0(ρ). (15)\nSince the Lieb functional by construction also satisfies the Fenche l inequality in Eq. (8), we obtain:\nProposition 2. The Lieb functional is an admissible density functional:\nE(v) = inf\nρ∈X(F(ρ)+(v|ρ)),∀v∈X′. (16)\nProof.E(v)≤infρ∈X(F(ρ)+(v|ρ))≤infρ∈X(F0(ρ)+(v|ρ)) =E(v).\nThe Lieb functional is related to the ground-state energy in a spec ial, symmetrical manner—compare the\nLieb variation principle in Eq.(14) with the Hohenberg–Kohnvariation principle Eq.(16). In the language of\nconvex analysis, EandFare said to be skew conjugate functions [12]: the function Eis concave and upper\nsemi-continuous, whereas its skew conjugate function Fis convex and lower semi-continuous. Convexity of\n4Fand concavity of Emean that, for each pair ρ1,ρ2∈X, each pair v1,v2∈X′, and each λ∈(0,1), we\nhave\nF(λρ1+(1−λ)ρ2)≤λF(ρ1)+(1−λ)F(ρ2), (17)\nE(λv1+(1−λ)v2)≥λE(v1)+(1−λ)E(v2), (18)\nwhereas lower semi-continuity of Fand upper semi-continuity of Eimply that\nliminf\nρ→ρ0F(ρ)≥F(ρ0), (19)\nlimsup\nv→v0E(v)≤E(v0). (20)\nThese properties follow straightforwardly from Eqs. (14) and (16 ). It is a fundamental result of convex\nanalysis that there is a one-to-one correspondence between all lower semi -continuous convex functions on X\nand all upper semi-continuous concave functions on X′[9, 10, 12], the correspondence being as in Eqs. (14)\nand (16). It follows that the Lieb functional Fis not only a lower bound to all admissible density functionals\nbut also the only admissible density functional that is lower semi-cont inuous and convex with respect to the\ntopology on X. It is a trivial but important observation that each property and e ach feature of Eare, in\nsome manner, exactly reflected in the properties and features of Fand vice versa. Note, however, that the\nLieb functional depends explicitly on X, just likeEdepends on X′.\nJust like there may happen to be no minimizing density in the Hohenberg –Kohn variation principle, there\nmay happen to be no maximizing potential in the Lieb variation principle. To characterize maximizing\npotentials, we introduce superdifferentiability by analogy with subdiff erentiability. The ground-state energy\nEis said to be superdifferentiable atv∈X′if there exists an element ρ∈X, known as a supergradient of\nEatv, such that\nE(v′)≤E(v)+(v′−v|ρ),∀v′∈X′. (21)\nThesuperdifferential ∂+E(v) is the (possibly empty) convex set of all supergradients of Eatv:\n∂+E(v) ={ρ∈X|E(v′)≤E(v)+(ρ|v′−v),∀v′∈X′, E(v)∈R}. (22)\nIn exactly the same manner that we proved Eq. (12), we obtain the following necessary and sufficient\nconditions for the existence of a maximizing potential in the Lieb varia tion principle:\nE(v) =F(ρ)+(v|ρ)⇐⇒ρ∈∂+E(v). (23)\nNote carefully that this result holds only for the Lieb functional, not for an arbitrary admissible density\nfunctional. Combining Eqs. (12) and (23), we arrive at the following c haracterization of the optimality\ncondition in Eqs. (14) and (16):\nProposition 3. IfF:X→R∪{+∞}is the Lieb functional and E:X′→R∪{−∞} is the ground-state\nenergy, then\nE(v) =F(ρ)+(v|ρ)⇐⇒ −v∈∂−F(ρ)⇐⇒ρ∈∂+E(v). (24)\nFor a general admissible density functional F0, these conditions are not equivalent: there may exist\nρ∈∂+E(v) such that−v /∈∂−F0(ρ) whenF0/ne}ationslash=F. On the other hand, the converse statement, −v∈\n∂−F0(ρ) =⇒ρ∈∂+E(v), holds for any admissible density functional F0. To prove this, assume that\n−v∈∂−F0(ρ). According to Eq. (12), we then have F0(ρ) =E(v)−(v|ρ). At the same time, the Fenchel\ninequality holds for any admissible density functional F0:\nE(u)≤F0(ρ)+(u|ρ),∀u∈X′. (25)\nSubstituting F0(ρ) =E(v)−(v|ρ) in this inequality, we obtain\nE(u)≤E(v)+(u−v|ρ),∀u∈X′, (26)\nimplying that ρ∈∂+E(v). We have thus proved the following:\n5Proposition 4. IfF0:X→R∪{+∞}is an admissible functional and E:X′→R∪{−∞} the ground-state\nenergy, then\nE(v) =F0(ρ)+(v|ρ)⇐⇒ −v∈∂−F0(ρ) =⇒ρ∈∂+E(v). (27)\nIn general, we have F≤F0. However, according to the following proposition, an admissible dens ity\nfunctionalF0(ρ) can differ from F(ρ) only when ∂−F0(ρ) =∅:\nProposition 5. IfF0:X→R∪{+∞}is an admissible functional and F:X→R∪{+∞}the Lieb\nfunctional, then\n−v∈∂−F0(ρ)⇐⇒F0(ρ) =F(ρ)∧ −v∈∂−F(ρ). (28)\nProof.Let us assume that −v∈∂−F0(ρ). From Eq. (27), we then have E(v) =F0(ρ)+(v|ρ). Substituting\nthis result into Eq. (24), we then find that F(ρ) =F0(ρ) and−v∈∂−F(ρ). Conversely, if F(ρ) =F0(ρ) and\n−v∈∂−F(ρ) hold, then we have E(v) =F0(ρ) +(v|ρ) by Eq. (24), from which −v∈∂−F0(ρ) follows by\nEq. (27).\nC. Constrained-search density functionals\nSo far, we have not established a connection between the minimizing d ensities in the Hohenberg–Kohn\nvariation principle in Eq. (2) and the minimizing wave functions in the Ray leigh–Ritz variation principle\nin Eq. (1). For instance, we havenot shown that each ρ∈∂+E(v) is the ground-state density associated with\nsome wave function ψ. Introducing a generic (abstract) constrained-searchdensity functional, we consider in\nthis section the relation between minimizers in the Hohenberg–Kohn a nd Rayleigh–Ritz variation principles.\nLetSbe a set with elements scalledstatesand letd:S→Xbe a map from Sto the Banach space of\ndensitiesX. The image\nρs=d(s) (29)\nis called the densityofs. However, no physical meaning is attached to s∈Sor toρ∈Xat this point; they\nare mathematical objects—for example, ρmay be a (current) density or a reduced density matrix, while s\nmay be a pure state or a density operator. As before, X′is the dual space of X, meaning that v∈X′if\nand only if v:X→Ris a linear and continuous operator. Thus, the pairing ( v|ρ) is separately linear and\ncontinuous in vandρ. We callvapotential but again no physical meaning is attached at present.\nAssume next that, for every v∈X′, we are given a map Ev:S→Rof the form\nEv(s) =E0(s)+(v|ρs), (30)\nwhereE0:S→Ris bounded below, meaning that there exists an M∈Rsuch that\nE0(s)≥M,∀s∈S. (31)\nWe callEv(s) theexpectation value of the total energy when the system is in the state sand influenced by\nthe potential v. This setting covers all standard formulations of DFT and CDFT, inc luding both pure-state\nand density-matrix formulations—in particular, we do not assume line arity ofE0(s) andd(s) ins.\nNext, define the ground-state energy E:X′→R∪{−∞} by the Rayleigh–Ritz variation principle:\nE(v) := inf\ns∈SEv(s) = inf\ns∈S(E0(s)+(v|ρs)). (32)\nObserving that vcouples toρsonly during minimization, it makes sense to define the constrained-search\nfunctionalFCS:X→R∪{+∞}as the map\nFCS(ρ) = inf{E0(s)|s/mapsto→ρ}, (33)\n6which takes the value + ∞whenever there is no s∈Swithρs=ρ. Combining Eqs. (32) and (33), we obtain\nthe Hohenberg–Kohn variation principle\nE(v) = inf\nρ∈X(FCS(ρ)+(v|ρ)). (34)\nThis result proves:\nProposition 6. The constrained-search functional FCSin Eq.(33)is an admissible density functional for\nthe energyEin Eq.(32).\nD. Characterization of ground states\nSuppose that v∈X′is such that there exists s∈Sfor which the infimum in Eq. (32) is a minimum:\nE(v) = inf\ns′Ev(s′) =Ev(s), (35)\nmeaning that there exists a ground state s∈argmins′Ev(s′) forv. Let now v′∈X′be arbitrary, and\ncompute\nE(v′) = inf\ns′∈SEv′(s′)≤Ev′(s) =E0(s)+(v′|ρs) =Ev(s)+(v′−v|ρs) =E(v)+(v′−v|ρs),(36)\nwhere we have used Eqs. (30) and (35). We have thus proved the f ollowing proposition:\nProposition 7. Ifs∈Sis a ground state for vin Eq.(30), thenρsis a supergradient of Ein Eq.(32)at\nv:\ns∈argmin\ns′Ev(s′) =⇒ρs∈∂+E(v). (37)\nThe following proposition establishes an important relationship betwe en ground states and subgradients\nofFCS:\nProposition 8. LetFCS:X→R∪{+∞}be the constrained-search functional (33). Ifsis the ground\nstate for some potential v∈X′with density ρs, then−v∈∂−FCS(ρs).\nProof.Lets′∈Swithρs′=ρs. According to Eq. (32), we then have\nE(v) =E0(s)+(v|ρs)≤E0(s′)+(v|ρs). (38)\nSubtracting ( v|ρs) from both sides, taking the infimum on the right-hand side, and usin g the definition\nin Eq. (33), we obtain\nE0(s)≤inf\ns′/mapsto→ρsE0(s′) =FCS(ρs)≤E0(s). (39)\nThus, if there is a ground state s∈Sforv∈X′, then\nFCS(ρs)+(v|ρs) =E0(s)+(v|ρs) =Ev(s) =E(v)≤FCS(ρ)+(v|ρ),∀ρ∈X. (40)\nRearranging, we obtain\nFCS(ρ)≥FCS(ρs)−(v|ρ−ρs),∀ρ∈X, (41)\ndemonstrating that −v∈∂−FCS(ρs) and completing the proof.\nConversely, does the subgradient relation −v∈∂−FCS(ρ) imply that there exists a ground state s/mapsto→ρ?\nTo answer this question affirmatively, we must assume that FCSisexpectation valued , defined as follows:\n7Definition 1 (Expectation-valued constrained-search functional) .A constrained-search functional FCSis\ncalledexpectation valued if, for every ρwithFCS(ρ)<+∞, there exists an sρ∈Ssuch that\nFCS(ρ) = inf\ns/mapsto→ρE0(s) =E0(sρ), sρ/mapsto→ρ. (42)\nIn other words, if FCSis expectation valued, then the infimum in Eq. (33) is a minimum if FCS(ρ)<+∞,\nimplying that FCS(ρ) is the expectation value E0(s) of some state s/mapsto→ρ. IfFCSis expectation valued, it\nfollows immediately that\n−v∈∂−FCS(ρ) =⇒E(v) =FCS(ρ)+(v|ρ) =⇒E(v) =E0(sρ)+(v|ρ) =Ev(sρ) =⇒sρ∈argmin\ns′Ev(s′)\n(43)\nand therefore that there exists a ground state s/mapsto→ρofv. Summarizing, we have proved the following:\nProposition 9. Suppose that FCS:X→R∪{+∞}is an expectation-valued constrained-search functional.\nThen, for each v∈X′, we have\nsρ∈argmin\ns′Ev(s′)⇐⇒ −v∈∂−FCS(ρ) =⇒ρ∈∂+E(v). (44)\nRemark: Proposition 9 for an expectation-valued constrained-se arch functional should be compared with\nProposition 4 for a general admissible density functional. Whereas, for a general admissible density func-\ntionalFCS, the condition E(v) =FCS(ρ) + (v|ρ) does not imply the existence of a ground state s/mapsto→ρ,\nthis implication does hold for for an expectation-valued constrained-search functional . Compare also with\nProposition 3, where the admissible density functional is assumed to be convex and lower semi-continuous\n(i.e., the Lieb functional).\nProposition 10. Suppose that an expectation-valued constrained-search fu nctionalFCS:X→R∪{+∞}is\nconvex and lower semi-continuous so that FCS=F. Then, for each v∈X′, we have\nsρ∈argmin\ns′Ev(s′)⇐⇒ −v∈∂−FCS(ρ)⇐⇒ρ∈∂+E(v). (45)\nProof.Combine Propositions 3 and 9.\nE. Differentiability of E(v)\nWe consider here conditions for differentiability of E:X′→R∪{−∞}. From Eq. (34), it follows that E\nis a concave and upper semi-continuous function. Upper semi-cont inuity is, however, a rather weak property.\nWhat is needed to make Econtinuous ? It is a fact that a concave (convex) map over a Banach space is\ncontinuous on the interior of its domain [9]. Hence, if we can show that E(v) is finite for every v∈X′,\nthen the domain dom( E) =X′and continuity follows. The trick to show finiteness of E(v) in the context of\nstandard DFT is to observe that s/mapsto→Vv(s) = (v|ρs) isrelatively bounded with E0-bound smaller than one ,\nmeaning that, for each v∈X′, there exist ǫ∈(0,1) andCǫ≥0 such that\n|Vv(s)|≤ǫE0(s)+Cǫ,∀x∈S. (46)\nAssuming thatVvis relatively bounded with ǫ<1 and with no assumption on Cǫ, we obtain:\nEv(s) =E0(s)+Vv(s)≥E0(s)−ǫE0(s)−Cǫ= (1−ǫ)E0(s)−Cǫ. (47)\nTaking the infimum over s∈Sand using the fact that E0(s) is by definition below bounded, we find\nE(v)≥(1−ǫ)M−Cǫ>−∞. (48)\nProposition 11. LetE0:S→R,E:X′→R∪{−∞}, andVv:S→Rbe as described above. If Vvis\nrelatively bounded with respect to E0with bound ǫ<1for eachv∈X′, thenEis continuous on X′.\n8Proof.Eis everywhere finite, so dom( E) =X′. Therefore Eis continuous at any point v∈X′.\nHaving determined sufficient conditions for continuity of E, what about differentiability? By definition,\nEis superdifferentiable at v∈X′if the superdifferential ∂+E(v) is non-empty. Clearly, if Eis differentiable\natv, the superdifferential is a singleton, ∂+E(v) ={∇E(v)}. The converse is in general almosttrue [9]:\nTheorem 1. LetXbe a Banach space. A convex (concave) map f:X→R∪{−∞,+∞}is Gˆ ateaux\ndifferentiable at x∈Xif and only if it is continuous at xwith a unique subgradient (supergradient).\nHence a unique sub- or supergradient does not by itself guarantee continuity and therefore not (Gˆ ateaux)\ndifferentiability: A subdifferential may be a singleton even if the function is no n-differentiable. This subtle\npoint illustrates the limitations of finite-dimensional intuition and has b een the source of misunderstanding\nin the DFT literature. For a discussion, see Ref. [11] and referenc es therein.\nWe can finally establish a useful statement on the differentiability of E:X′→R∪{−∞}:\nProposition 12. LetE:X′→Rbe everywhere finite and given by the Rayleigh–Ritz principl e in Eq. (32)\nand assume that FCSin Eq. (33) is expectation valued and equal to the Lieb functi onal (i.e., convex and\nlower semi-continuous). Then Eis differentiable at vif and only if all ground states s∈argmins′∈SEv(s′)\nsupported by vhave the same ground-state density, s/mapsto→ρ∈X. In particular, Eis differentiable at vif there\nexists a unique ground state sforv.\nProof.Being continuous, Eis differentiable at vif and only if ∂+E(v) is a singleton. But by Proposition 10,\n∂+E(v) is a singleton if and only if all ground states of vhave the same density, using the assumption that\nFCSis the Lieb functional.\nRemark: The message here is that given the Rayleigh–Ritz variation p rinciple, it is not sufficient to know\nthat a potential v∈X′has a unique ground-state density ρ=ρsto prove differentiability of Eatv. To\nidentify the unique supergradient ρ∈∂+E(v) with a ground-state density, FCSmust be identical with the\nLieb functional (convex and lower semi-continuous) and expectat ion valued. If the Lieb functional happens\nto be different from every possible constrained-search functiona l, differentiability may fail, even if vhas a\nunique ground-state density ρs. A counterexample is given in Sec.III, where we consider the Hilbert space\nXH=L2(R3) rather than the usual Banach space XL=L1(R3)∩L3(R3). UnlikeE:XL→R, the function\nE:XH→Ris not differentiable.\nStarting from a Hohenberg–Kohn variation principle with an arbitrar y admissible density functional, it\nseems hard to prove differentiability from the sole assumption that vhas a unique ground-state density ρs.\nIdentifying the proper constrained-search functional, based on the Rayleigh–Ritz variation principle, seems\nthe only way out of the problem.\nIII. APPLICATION TO DFT FOR MOLECULAR SYSTEMS\nA. Ground states and ground-state densities\nForN-electron systems, each normalized wave function has a density ρ∈L1(R3) withρ≥0 almost\neverywhere and/integraltext\nρ(r)dr=N. The set of states Scan be taken to be either the L2normalized states ψ∈\nH1\nS(R3N) (pure states) or the set of density matrices Γ =/summationtext\nkλk|ψk/an}b∇acket∇i}ht/an}b∇acketle{tψk|constructed as convex combinations\nfrom an orthonormal set {ψk}⊂H1\nS(R3N) withλk≥0 and/summationtext\nkλk= 1 (mixed states). In the pure-state\nand mixed-state cases, respectively, constrained search gives t he functionals FLLdefined in Eq. (3) and FDM\ndefined in Eq. (4):\nFLL(ρ) = inf\nΨ/mapsto→ρE0(ψ),E0(ψ) =/an}b∇acketle{tψ|ˆT+ˆW|ψ/an}b∇acket∇i}ht,Vv(ψ) =/an}b∇acketle{tψ|ˆV|ψ/an}b∇acket∇i}ht= (ρψ|v), (49)\nFDM(ρ) = inf\nΓ/mapsto→ρE0(Γ),E0(Γ) = Tr[( ˆT+ˆW)Γ],Vv(Γ) = Tr(Γ ˆV) = (ρΓ|v), (50)\nwhereˆVis the multiplication operator associated with v. It is obvious that FDM≤FLL. It is less obvious,\nbut true, that there are ρ∈L1for whichFDM(ρ)<FLL(ρ).\n9Inaclassicpaper[13], Katodemonstratedthat, foreach v∈L2(R3)+L∞(R3), theN-electronHamiltonian\nˆH=ˆT+ˆW+ˆVis self-adjoint on L2(R3N) with domain H2\nS(R3N) and bounded below. More generally,\nSimon demonstrated that, for each v∈L3/2(R3) +L∞(R3) (which includes all Coulomb potentials), Vvis\nrelativelyE0-bounded with an arbitrarily small bound [14]:\nTheorem 2. Letv∈L3/2(R3)+L∞(R3)and letˆVbe the corresponding multiplication operator. Then, for\nanyǫ>0, there exists a constant Cǫ≥0, such that for all ψ∈H1\nS(R3N)with/ba∇dblψ/ba∇dbl2= 1,\n|/an}b∇acketle{tψ|ˆV|ψ/an}b∇acket∇i}ht|≤ǫ/an}b∇acketle{tψ|ˆT+ˆW|ψ/an}b∇acket∇i}ht+Cǫ. (51)\nProof.See Ref. [14].\nIn the pure-state case, therefore, each v∈L3/2(R3) +L∞(R3) gives aVvrelatively bounded by E0,\nwith bound ǫarbitrarily small. Lieb proved that FLLandFDMare both expectation valued [4]. In the same\npublication, a proof(due to Simon) that FDM:L1(R3)→R∪{+∞}is convexand lowersemi-continuouswas\ngiven. Using the Sobolev embedding theorem, it can be shown that fo r eachψ∈H1\nS(R3N), the corresponding\ndensityρ←/mapsfromcharψbelongs toL3(R3) [4]. Thus, it is appropriate to consider the Banach space\nXL:=L1(R3)∩L3(R3) (52)\nwith norm/ba∇dbl·/ba∇dbl=/ba∇dbl·/ba∇dblL1+/ba∇dbl·/ba∇dblL3, whose dual space is\nX′\nL=L3/2(R3)+L∞(R3) (53)\nwith the topology induced by /ba∇dbl·/ba∇dbl[15]. Since convergence in XLimplies convergence in L1,FDM:XL→\nR∪{+∞}is lower semi-continuous as well. Applying Proposition 10 for FDM(which is expectation valued\nand lower semi-continuous convex) and Proposition 9 for FLL(which is expectation valued but not lower\nsemi-continuous convex), we obtain\nCorollary 1. Letv∈X′\nL=L3/2(R3)+L∞(R3). Then,\nΓ∈argmin\nΓ′/parenleftBig\nE0(Γ′)+(ρΓ′|v)/parenrightBig\n⇐⇒ −v∈∂−FDM(ρΓ)⇐⇒ρΓ∈∂+E(v), (54)\nψ∈argmin\nψ′/parenleftBig\nE0(ψ′)+(ρψ′|v)/parenrightBig\n⇐⇒ −v∈∂−FLL(ρψ) =⇒ρψ∈∂+E(v). (55)\nNote thatρ∈∂+E(v) does not imply the existence of a ground-state wave function for vsuch thatψ/mapsto→ρ,\nonly the existence of a mixed ground state Γ /mapsto→ρ. For example, if v∈X′\nLhas a two-fold degeneracy with\npure ground states ψ1andψ2, thenλρ1+(1−λ)ρ2∈∂+E(v) but there may be no pure ground state with\nthis density. On the other hand, λ|ψ1/an}b∇acket∇i}ht/an}b∇acketle{tψ1|+(1−λ)|ψ2/an}b∇acket∇i}ht/an}b∇acketle{tψ2|is a mixed ground state with this density.\nB. Topology dependence of the Lieb functional\nRecall that the Lieb functional F:X→R∪{+∞}depends explicitly on the Banach space X, including\nits topology. We now demonstrate this using an explicit example. Motiv ated by the set inclusion\nXL=L1(R3)∩L3(R3)⊂XH:=L2(R3) (56)\nand the fact that XHis Hilbert space with a simpler structure than the non-reflexivespac eXL, it is tempting\nto consider FDMas a function on X=XH. However, the embedding XL⊂XHis not continuous: Conver-\ngence inXLdoes not imply convergence in XH. Even ifFDMis lower semi-continuous with respect to XL,\nit may fail to be so in XH. To see this, we note that, since XHis a Hilbert space, X′\nH=XH, which does not\nadmit constant potentials: If v∈XH, thenv+c /∈XHfor each constant c/ne}ationslash= 0. This observation allows us\nto prove the following result:\n10Proposition 13. For eachv∈XH, we haveE(v)≤0. Ifv≥0almost everywhere, then E(v) = 0.\nProof.See Appendix A.\nThe proof is based on the following idea. Write v=v+−v−wherev±≥0 almost everywhere. The\nnegative part v−can only lower the energy, whereas v+∈XHimplies that vdecays at infinity (in an average\nsense), thereby allowing the electrons to lower their energy to zer o by “escaping to infinity”. Note how the\nfact that nonzero c /∈XHaffects this argument.\nConsider now ρ≡0, for which FDM(ρ) = +∞since no Γ/mapsto→0. Evaluating the conjugate (Lieb) functional\nwith respect to XHatρ= 0, we obtain\nF(0) = sup\nv∈XH(E(v)−(v|0)) = sup\nv∈XHE(v) = 0. (57)\nNote that a different result is obtained using XL, which admits constant potentials v(r)≡c∈R:\nF(0) = sup\nv∈XL(E(v)−(v|0)) = sup\nv∈XLE(v) = +∞=FDM(0). (58)\nAs an immediate consequence, we arrive at the following result on XH:\nProposition 14. For anyv∈XHsuch thatv≥0almost everywhere, it holds that,\nE(v) =F(0)+(v|0)∧ −v∈∂−F(0)∧0∈∂+E(v), (59)\nwhereFis conjugate to E. There exists no ground state ψ∈H1\nS(R3N)for anyv≥0.\nProof.Eq. (59) follows from Proposition3. It remains to show that there e xists no ground state if v≥0\nalmost everywhere. Since E(v) = 0, it is sufficient to show that, for each ψ∈H1\nS(R3N), it holds that\nEv(ψ)>0. We haveEv(ψ)≥/an}b∇acketle{tψ|ˆT|ψ/an}b∇acket∇i}ht. But ifψhas zero kinetic energy, then ∇ψ= 0, from which it follows\nthatψ= 0 almost everywhere, contradicting /ba∇dblψ/ba∇dbl2= 1. ThusEv(ψ)≥/an}b∇acketle{tψ|ˆT|ψ/an}b∇acket∇i}ht>0.\nOur discussion illustrates how the choice of Banach space Xof densities affects the properties of the\nconjugate universal functional F:X→R∪{+∞}. Clearly, the Lieb functional FDMfor the space XLis\nvery different from the Lieb functional for XH, since, in the latter case, for any non-negative potential v, the\nunphysical density ρ≡0 is a minimizer of the Hohenberg–Kohn variation principle. But vdoes not even\nhave a ground state and certainly not a ground state with density ρ≡0.\nConnecting with the discussion following Proposition12, we can see ho w topology influences differentia-\nbility ofE. The map E:XH→Ris pointwise identical to E:X′\nL→R: it is by definition the ground-state\nenergy of the system in the external potential v, defined via the Rayleigh–Ritz variation principle. However,\nthe topology on XHis different and FDMis no longer equal to the Lieb functional. We then cannot conclude\nfrom Proposition12 that E:XH→Ris differentiable at vifv∈XHhas a unique ground state.\nIV. APPLICATION TO CDFT FOR MOLECULAR SYSTEMS IN MAGNETIC FI ELDS\nConsider an N-electron system subject to an external magnetic vector poten tialA:R3→R3, with\nassociated magnetic field B(r) =∇×A(r) (in the distributional sense). Considering Aas a variable on\nthe same footing as the scalar potential v, we arrive at CDFT, where the paramagnetic current density\njp∈L1(R3) appears as a variable together with ρ[6, 8, 16]. The mathematical foundation of CDFT is\nnot as well developed as that of DFT. Indeed, part of the motivatio n for the present work stems from a\nstudy of CDFT [17]. On the other hand, Laestadius [8] has taken imp ortant steps—proving, for example,\nProposition 17 below.\nFollowing Ref. [8], we assume for simplicity that the components of AareL∞(R3) functions. The single-\nelectron momentum operator −i∇is then replaced by −i∇+A, transforming the N-electron kinetic-energy\noperator ˆTinto the corresponding ˆTA. SinceA∈L∞(R3), the kinetic-energy expectation value is still well\n11defined for each ψ∈H1\nS(R3N). In terms of the density ρ∈L1(R3) and the paramagnetic current density\njp∈L1(R3), we obtain\n/an}b∇acketle{tψ|ˆTA|ψ/an}b∇acket∇i}ht=/an}b∇acketle{tψ|ˆT|ψ/an}b∇acket∇i}ht+1\n2/integraldisplay\n|A(r)|2ρ(r)dr+/integraldisplay\nA(r)·jp(r)dr. (60)\nThe ground-state energy is thus given by\nE(v,A) = inf\nψ,/bardblψ/bardbl2=1/an}b∇acketle{tψ|ˆTA+ˆW+ˆV|ψ/an}b∇acket∇i}ht= inf\n(ρ,jp)/parenleftbig\nFVR(ρ,jp)+(v+1\n2A2|ρ)+(A|jp)/parenrightbig\n,(61)\nwhere we have introduced the Vignale–Rasolt (VR) functional [6]\nFVR(ρ,jp) := inf/braceleftBig\n/an}b∇acketle{tψ|ˆT+ˆW|ψ/an}b∇acket∇i}ht/vextendsingle/vextendsingleψ∈H1\nS(R3N),/ba∇dblψ/ba∇dbl2= 1, ψ/mapsto→(ρ,jp)/bracerightBig\n. (62)\nSimilarly, we may define a density-matrix constrained-search funct ional\nFDM(ρ,jp) = inf/braceleftBig\nTr((ˆT+ˆW)Γ)/vextendsingle/vextendsingleΓ/mapsto→(ρ,jp)/bracerightBig\n, (63)\nwhereρΓ=/summationtext\nkpkρψkandjpΓ=/summationtext\nkpkjpψk. Whereas FDMis convex by construction, the presence of the\nnonlinear A-dependent term makes E(v,A) nonconcave. Following Ref. [16], it is therefore natural instead\nto work with ˜E(u,A) =E(u−1\n2A2,A), defined in Proposition 16 below. We note that\n(ρ,jp)∈XL×L1(R3), (64)\na Banach space with norm /ba∇dbl(ρ,jp)/ba∇dbl=/ba∇dblρ/ba∇dbl+/ba∇dbljp/ba∇dblL1and dualX′\nL×L∞(R3), the latter containing all ( v,A).\nFirst, we prove finiteness of the ground-state energy:\nProposition 15. For every (v,A)∈XL×L∞, the ground-state energy is finite, E(v,A)>−∞.\nProof.For anN-electron system influenced by a magnetic field, the diamagnetic inequality [18] gives\n/an}b∇acketle{tφ|ˆT|φ/an}b∇acket∇i}ht≤/an}b∇acketle{tψ|ˆTA|ψ/an}b∇acket∇i}ht,∀ψ∈H1\nS(R3N), (65)\nwhereφ=|ψ|be the pointwiseabsolutevalue. Next, using the identity ρψ=ρφandthe relativeboundedness\nofv∈XLwith respect to ˆT, we find that ˆVis relatively bounded with respect to ˆTA:\n|/an}b∇acketle{tψ|ˆV|ψ/an}b∇acket∇i}ht|=|/an}b∇acketle{tφ|ˆV|φ/an}b∇acket∇i}ht|≤ǫ/an}b∇acketle{tφ|ˆT|φ/an}b∇acket∇i}ht+Cǫ≤ǫ/an}b∇acketle{tψ|ˆTA|ψ/an}b∇acket∇i}ht+Cǫ. (66)\nSimilarly, ˆWis relatively bounded with respect to ˆTA. It follows that/an}b∇acketle{tψ|ˆH|ψ/an}b∇acket∇i}htis below bounded, and thus\nthatE(v,A)>−∞.\nNext, we note that, for each A∈L∞(R3), we have|A|2∈L∞(R3)⊂X′\nL. Therefore, for each pair\n(v,A)∈XL, we have (v±1\n2|A|2,A)∈XL×L∞, making the following proposition easy to prove:\nProposition 16. IfF0:XL×L1is eitherFVRorFDM, then˜E:X′\nL×L∞(R3)→Rdefined by\n˜E(u,A) = inf\n(ρ,jp)(F0(ρ,jp)+(u|ρ)+(A|jp)) (67)\nis concave, finite, and therefore continuous. The ground-st ate energyEand˜Eare related by\nE(v,A) =˜E(v+1\n2|A|2,A),˜E(u,A) =E(u−1\n2|A|2,A). (68)\nProof.We leave the details to the reader.\n12The map (v,A)/mapsto→(v+1\n2|A|2,A) defined on X′\nL×L∞(R3) is smooth and invertible, with smooth inverse\n(u,A)/mapsto→(u−1\n2|A|2,A). Thus, the properties of Eare reflected in properties ˜Eand vice versa. If F0is an\nadmissible functional for ˜E:X′\nL×L∞→R, then\nE(v,A) =F0(ρ,jp)+(v+1\n2|A|2|ρ)+(A|jp)⇐⇒ − (v+1\n2|A|2,A)∈∂−F0(ρ,jp)\n=⇒(ρ,jp)∈∂+˜E(v+1\n2|A|2,A)(69)\nLaestadius proved the following result [8]:\nProposition 17. FVR:L1(R3)×L1(R3)→R∪{+∞}is expectation valued.\nProposition 9 therefore gives\nψρ∈argmin\nψ′/an}b∇acketle{tψ|ˆH|ψ/an}b∇acket∇i}ht ⇐⇒ E(v,A) =FVR(ρ,jp)+(v+1\n2|A|2|ρ)+(A|jp). (70)\nThus, using the Vignale–Rasolt functional and the space XL×L1as density space, there are no minimizers\nof the pure-state Hohenberg–Kohn variation principle for CDFT th at do not correspond to ground-state\nwave functions.\nOn the other hand, it is not known whether the density-matrix func tionalFDMin (62) is expectation\nvalued, but it seems likely. If FDMisnotexpectation valued, there may be ( ρ,jp) such that\nE(v,A) =FDM(ρ,jp)+(v+1\n2|A|2|ρ)+(A|jp) (71)\nbut such that there is no Γ /mapsto→(ρ,jp), i.e., (ρ,jp) must be considered an unphysical minimizer of the density-\nmatrix Hohenberg–Kohn variation principle for CDFT.\nNeither is it known whether FDMis lower semi-continuous in the L1×L1topology; the proof for the\nordinary DFT case in Ref. [4] is not easy to generalize to the present situation. Thus, there could be\n(ρ,jp)∈∂+˜E(v+1\n2|A|2,A) such that−(v+1\n2|A|2,A)/∈∂−FDM(ρ,jp), so that ( ρ,jp) does not satisfy\nEq. (71).\nFinally, we may consider the Lieb functional F:XL×L1→R∪{+∞}, defined as\nF(ρ,jp) := sup\nu,A/parenleftBig\n˜E(u,A)−(u|ρ)−(A|jp)/parenrightBig\n= sup\nv,A/parenleftbig\nE(v,A)−(v+1\n2|A|2|ρ)−(A|jp)/parenrightbig\n.(72)\nThis functional is convex and lower semi-continuous by constructio n. However, unlike in standard DFT, it\nis unknown whether F(ρ,jp) =FDM(ρ,jp).\nFrom the perspective of applying Proposition 10, thereby establish ing a result like Corollary 1 for CDFT,\none must establish boththatFDMis expectation valued andlower semi-continuous. Thus, we do not know at\nthe present time, whether E(v,A) is Gˆ ateaux-differentiable when the ground-state density ( ρ,jp) is unique.\nV. CONCLUSION\nWe have analyzed the relationship between ground-state densities obtained from the Rayleigh–Ritz varia-\ntion principle (by minimizing over pure-state wave functions or mixed- state density matrices) and from the\nHohenberg–Kohn variation principles (by minimizing over densities usin g an admissible density functional\nF0(ρ)). For standard DFT for molecular systems, we established, for e ach potential v∈L3/2(R3)+L∞(R3),\na one-to-one correspondence between the mixed ground-state densities of the Rayleigh–Ritz variation princi-\nple and the mixed ground-state densities of the Hohenberg–Kohn v ariation principle with the Lieb density-\nmatrix constrained-search functional FDM:L1(R3)∩L3(R3)→R∪{+∞}. A similar one-to-one correspon-\ndence is established between the pure ground-state densities of t he Rayleigh–Ritz variation principle and\nthe pure ground-state densities of the Hohenberg–Kohn variatio n principle with the Levy–Lieb functional\nFLL:L1(R3)∩L3(R3)→R. In other words, all physical ground-state densities (pure or mix ed) are recov-\nered with these functionals and there are no false densities (i.e., minim izing densities not associated with a\nground-state wave function).\n13We also noted how the topology of the underlying Banach space Ximpinges on the results—in particular,\nwe noted that the Lieb functional F:X→R∪{+∞}depends explicitly on X. As an illustration, F/ne}ationslash=FDM\nonX=L2(R3) butF=FDMonL1(R3)∩L3(R3). Finally, CDFT was discussed and some open problems\nwere pointed out—for example, it is unknown whether FDM(ρ,jp) is lower semi-continuous in any useful\ntopology such as L1(R3)×L1(R3).\nACKNOWLEDGMENTS\nThis work was supported by the Norwegian Research Council throu gh the CoE Centre for Theoretical and\nComputational Chemistry (CTCC) Grant No. 179568/V30 and the G rant No. 171185/V30 and through the\nEuropean Research Council under the European Union Seventh Fr amework Program through the Advanced\nGrant ABACUS, ERC Grant Agreement No. 267683.\nAppendix A: Proof of Proposition 13\nProof of Proposition 13. Writingv=v+−v−, we obtain\nE(v) = inf\nρ(FDM(ρ)+(v+|,ρ)−(v−|,ρ))≤inf\nρ(FDM(ρ)+(v+|ρ) =E(v+)). (A1)\nIt is therefore sufficient to show that E(v+)≤0. In fact, we show that E(v+) = 0.\nLetv≥0 almost everywhere. Let λ >0 be arbitrary and let Ω k,λwithk∈Nbe disjoint cubes of side\nlengthλsuch that∪kΩk,λ=R3. Each Ωk,λcan be obtained by translation of Ω 1,λ. Sincev∈L2(R3),\n/integraldisplay\nR3v(r)2dr=/summationdisplay\nk/integraldisplay\nΩk,λv(r)2dr<+∞,\nimplying that/integraltext\nΩk,λv(r)2dr→0 ask→∞. Letψ∈C∞\nc(R3N) be arbitrary but with support in ΩN\n1,1so that\nρψ∈C∞\nc(R3) has support contained in Ω 1,1. By translating ψproperly (denoting the result by ψk), the\nsupport ofψkis inside ΩN\nk,1and\nFDM(ρψk) =FDM(ρψ)≤/an}b∇acketle{tψ|T+W|ψ/an}b∇acket∇i}ht≡/an}b∇acketle{tT/an}b∇acket∇i}ht+/an}b∇acketle{tW/an}b∇acket∇i}ht, (A2)\nindependent of k. We obtain\nE(v)≤lim\nk/parenleftBigg\n/an}b∇acketle{tT/an}b∇acket∇i}ht+/an}b∇acketle{tW/an}b∇acket∇i}ht+/integraldisplay\nΩk,1v(r)ρk(r)dr/parenrightBigg\n=/an}b∇acketle{tT/an}b∇acket∇i}ht+/an}b∇acketle{tW/an}b∇acket∇i}ht, (A3)\nwhere we have used the fact that\n/integraldisplay\nΩk,1v(r)ρk(r)dr≤/parenleftBigg/integraldisplay\nΩk,1v(r)2dr/parenrightBigg1/2\n/ba∇dblρ/ba∇dbl2→0 (A4)\nask→∞.\nWe now increase the size of the boxes Ω k,λby varying λ>0. By dilating ψin the manner\nψ(r1,···)/mapsto→λ3N/2ψ(λr1,···), (A5)\nthe support is still inside Ω∞\n1,λand the density is scaled as ρψ(r)→λ−3ρψ(λ−1r), conserving the number of\nparticles. We obtain the scaling\n/an}b∇acketle{tT/an}b∇acket∇i}ht+/an}b∇acketle{tW/an}b∇acket∇i}ht→λ−2/an}b∇acketle{tT/an}b∇acket∇i}ht+λ−1/an}b∇acketle{tW/an}b∇acket∇i}ht. (A6)\nBy repeating the above argument for λ= 1 and letting λ→∞, we obtain E(v)≤0. On the other hand,\nE(v)≥0 since the Hamiltonian H(v) is positive with v≥0, yieldingE(v) = 0.\n14[1] L. Thomas, Proc. Camb. Philos. Soc. 23, 542 (1927).\n[2] E. Fermi, Rend. Accad. Naz. Lincei 6, 602 (1927).\n[3] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).\n[4] E. H. Lieb, Int. J. Quant. Chem. 24, 243 (1983).\n[5] M. Levy, Proc. Natl. Acad. Sci. 76, 6062 (1979).\n[6] G. Vignale and M. Rasolt, Phys. Rev. Lett. 59, 2360 (1987).\n[7] G. Vignale and M. Rasolt, Phys. Rev. B 37, 10685 (1988).\n[8] A. Laestadius, Int. J. Quantum Chem. 114, 1445 (2014).\n[9] J. van Tiel, Convex Analysis, an Introductory Text (Wiley, Chichester, 1984).\n[10] I. Ekeland and R. T´ emam, Convex Analysis and Variational Problems (SIAM, Philadelphia, 1999).\n[11] P. Lammert, Int. J. Quant. Chem. 107, 1944 (2005).\n[12] R. Rockafellar, Pac. J. Math. 25, 597 (1968).\n[13] T. Kato, Trans. Amer. Math. Soc. 70, 195 (1951).\n[14] B. Simon, Commun. Math. Phys. 21, 192 (1971).\n[15] T.-S. Liu and J.-K. Wang, Math. Scand. 23, 241 (1968).\n[16] E. Tellgren, S. Kvaal, E. Sagvolden, U. Ekstrm, A. Teale , and T. Helgaker, Phys. Rev. A 86, 062506 (2012).\n[17] S. Kvaal and T. Helgaker, (2015), in preparation.\n[18] E. Lieb and M. Loss, Analysis, 2nd ed. (American Mathematical Society, Providence, Rhod e Island, USA, 2001).\n15"    },    {        "title": "1508.04749v1.Ground_states_of_stealthy_hyperuniform_potentials__I__Entropically_favored_configurations.pdf",        "content": "Ground states of stealthy hyperuniform potentials: I. Entropically favored\ncon\fgurations\nG. Zhang and F. H. Stillinger\nDepartment of Chemistry ,Princeton University , Princeton, New Jersey 08544, USA\nS. Torquato\u0003\nDepartment of Chemistry, Department of Physics,\nPrinceton Institute for the Science and Technology of Materials,\nand Program in Applied and Computational Mathematics ,\nPrinceton University , Princeton, New Jersey 08544, USA\nSystems of particles interacting with \\stealthy\" pair potentials have been shown to possess in-\n\fnitely degenerate disordered hyperuniform classical ground states with novel physical properties.\nPrevious attempts to sample the in\fnitely degenerate ground states used energy minimization tech-\nniques, introducing algorithmic dependence that is arti\fcial in nature. Recently, an ensemble theory\nof stealthy hyperuniform ground states was formulated to predict the structure and thermodynamics\nthat was shown to be in excellent agreement with corresponding computer simulation results in the\ncanonical ensemble (in the zero-temperature limit). In this paper, we provide details and justi\f-\ncations of the simulation procedure, which involves performing molecular dynamics simulations at\nsu\u000eciently low temperatures and minimizing the energy of the snapshots for both the high-density\ndisordered regime, where the theory applies, as well as lower densities. We also use numerical sim-\nulations to extend our study to the lower-density regime. We report results for the pair correlation\nfunctions, structure factors, and Voronoi cell statistics. In the high-density regime, we verify the the-\noretical ansatz that stealthy disordered ground states behave like \\pseudo\" disordered equilibrium\nhard-sphere systems in Fourier space. The pair statistics obey certain exact integral conditions with\nvery high accuracy. These results show that as the density decreases from the high-density limit,\nthe disordered ground states in the canonical ensemble are characterized by an increasing degree\nof short-range order and eventually the system undergoes a phase transition to crystalline ground\nstates. In the crystalline regime (low densities), there exist aperiodic structures that are part of the\nground-state manifold, but yet are not entropically favored. We also provide numerical evidence\nsuggesting that di\u000berent forms of stealthy pair potentials produce the same ground-state ensemble\nin the zero-temperature limit. Our techniques may be applied to sample the zero-temperature limit\nof the canonical ensemble of other potentials with highly degenerate ground states.\nI. INTRODUCTION\nThere has been long-standing interest in the phase\nbehavior of many-particle systems in d-dimensional Eu-\nclidean spaces Rdin which the particles interact with\nsoft, bounded pair potentials [1{12]. Considerable at-\ntention has been devoted to the determination of the\nclassical ground states (global energy minima) of such\ninteractions [3, 6, 11, 12]. While typical interactions lead\nto unique classical ground states, certain special pair po-\ntentials are characterized by degenerate classical ground\nstates|a phenomenon that has attracted recent atten-\ntion [12{22].\nOne family of such pair interactions are the \\stealthy\npotentials\" because their ground states correspond to\ncon\fgurations that completely suppress single scattering\nfor a range of wave numbers. The Fourier transforms of\nthese potentials are bounded and non-negative and have\ncompact support [12], and hence they have correspond-\ning direct-space potentials that are bounded and long\nranged. Because of their special construction in Fourier\n\u0003torquato@electron.princeton.eduspace, \fnding the ground states of stealthy potentials\nis equivalent to constraining the structure factor to be\nzero for wave vectors kcontained within the support of\nthe Fourier transformed potential [12], as will be sum-\nmarized in Sec. II. In the case when the constrained\nwave vectors lie in the radial interval 0 <jkj\u0014K, the\nstealthy ground states fall within the class of hyperuni-\nform states of matter [23] and can be tuned to have\nvarying degrees of disorder. Disordered hyperuniform\nsystems in general are of current interest because they\nare characterized by an anomalously large suppression\nof long-wavelength density \ructuations and can exist as\nequilibrium or nonequilibrium states, either classically or\nquantum mechanically [24{37]. Moreover, because disor-\ndered hyperuniform states of matter have characteristics\nthat lie between a crystal and a liquid [12], they are en-\ndowed with novel physical properties [18, 19, 38{46].\nWhen a dimensionless parameter \u001f, inversely propor-\ntional to the number density \u001aand proportional to Kd\n(size of the constrained region) is su\u000eciently small, the\nhyperuniform ground states are in\fnitely degenerate and\ncounterintuitively disordered (i.e., isotropic without any\nBragg peaks) [12]. However, when \u001fis large enough ( \u001ais\nsu\u000eciently small), there is a phase transition to a regime\nin which the ground states are crystalline or highly or-arXiv:1508.04749v1  [cond-mat.stat-mech]  19 Aug 20152\ndered [13{15, 19]. For each spatial dimension d, there is\na special value of \u001f,\u001f\u0003max, at which the ground state is\nunique [47]. The unique ground state is the dual (recipro-\ncal lattice) of the densest Bravais lattice packing in each\ndimension [12]. In two and higher dimensions, as soon as\n\u001fdrops below \u001f\u0003max, the set of the ground states become\nuncountably in\fnite and gradually includes progressively\nless ordered structures [12]. Similarly to stealthy poten-\ntials, a family of two-, three-, and four-body potentials\nthat lead to disordered ground states has also been de-\n\fned in Fourier space and studied [17, 18, 21].\nDue to the complexity of the problem, almost all pre-\nvious investigations of the ground states employed com-\nputer simulations. Such numerical studies were carried\nout in one, two and three dimensions [13, 14, 17{19]. The\nground states were sampled by minimization of poten-\ntial energy at \fxed densities starting from random initial\nconditions in a d-dimensional cubic simulation box under\nperiodic boundary conditions. A few optimization tech-\nniques were employed to \fnd the global energy minima\nwith very high precision [14, 17].\nGenerally, a numerically obtained ground-state con\fg-\nuration depends on the number of particles Nwithin the\nfundamental cell, initial particle con\fguration, shape of\nthe fundamental cell, and particular optimization tech-\nnique used [12]. Adding to the complexity of the problem\nis that the disordered ground states are highly degener-\nate with a con\fgurational dimensionality that depends\non the density, and there are an in\fnite number of dis-\ntinct ways to sample this complex ground-state manifold,\neach with its own probability measure. These nontrivial\naspects had made the task of formulating a statistical-\nmechanical theory of stealthy degenerate ground states\na daunting one. Recently, we have formulated such an\nensemble theory that yields analytical predictions of the\nstructural characteristics and other properties of stealthy\ndegenerate ground states [12]. A number of exact results\nfor the thermodynamic and structural properties of these\nground states were derived that applied to general en-\nsembles. We then specialized our results to the canonical\nensemble (in the zero-temperature limit) by exploiting an\nansatz that stealthy disordered ground states (for su\u000e-\nciently small \u001f) behave remarkably like \\pseudo\" disor-\ndered equilibrium hard-sphere systems in Fourier space.\nOur theoretical predictions for the pair correlation func-\ntiong2(r) and structure factor S(k) of these entropi-\ncally favored disordered ground states were shown to be\nin agreement with corresponding computer simulations\nacross the \frst three space dimensions. We also made\npredictions for the corresponding excited states for su\u000e-\nciently small temperatures that were in agreement with\nsimulations.\nBecause the focus of that previous investigation was\nthe development of ensemble theories, few simulation\ndetails were presented about how the canonical ensem-\nble was sampled to produce stealthy disordered ground\nstates. One aim of the present paper is to provide a com-\nprehensive description of the numerical procedure thatwe used to produce the simulation results in Ref. 12.\nMoreover, here we also extend those results by apply-\ning the simulation procedure to study numerically the\nground states in the canonical ensemble for all allowable\nvalues of\u001fand thus investigate the entire phase dia-\ngram for the entropically favored states across the \frst\nthree space dimensions. In the second paper of this se-\nries, we will study the exotic aperiodic \\wavy phases\"\nidenti\fed in previous numerical work [14] (or \\stacked-\nslider phases,\" as called in the sequel to this paper [48]),\na special part of the ground-state manifold. An analyti-\ncal model will enable an even more detailed study of this\nphase.\nAs a justi\fcation of sampling the canonical ensemble\ninstead of minimizing energy, we also demonstrate here\nhow a variety of di\u000berent optimization techniques a\u000bect\nthe ground states that are sampled, which was not previ-\nously investigated [14, 17, 18]. This investigation reveals\nthat the pair statistics of the ground-state con\fgurations\nindeed generally depend on the algorithm. Moreover, we\nshow here that the energy minimization results depend\non the initial conditions as well. We also provide the rea-\nson why the simulations in Ref. 12 and this paper employ\nnoncubic, possibly deforming, simulation boxes for d\u00152.\nBecause almost all previous numerical simulations were\nperformed using some speci\fc form of stealthy potentials,\nwe show here that di\u000berent forms of stealthy potentials\nproduce identical pair correlation functions, suggesting\nthat the speci\fc choice of the potential form does not\na\u000bect the ensemble being sampled.\nAmong our major \fndings, we show that energy min-\nimizations starting from random initial conditions may\nlead to clustering of particles, the degree of which de-\npends on the algorithm for a \fnite range of \u001fbelow 1/2\nacross the \frst three space dimensions. When minimiz-\ning the energy starting from con\fgurations equilibrated\nat some temperature TE, the ground-state con\fgurations\ndiscovered depend on TE. However, the algorithm depen-\ndence diminishes in the TE!0 limit. We also demon-\nstrate that the pair statistics [ g2(r) andS(k)] in this limit\ndo not depend on the particular form of the stealthy po-\ntential. The similarity between the structure factor in\nthis limit and the pair correlation function of an equilib-\nrium hard-sphere system in direct space [12] is valid for\n\u001fup to some dimension-dependent values between 0.25\nand 0.33 in the \frst three space dimensions. Beyond\nthis range of \u001f, the hard-sphere analogy in Fourier space\nundergoes modi\fcation. As \u001fincreases further (to the\nvalue of about 0.4 in two dimensions, for example), the\n\frst peak in the structure factor diminishes while second\npeak in the structure factor grows and engulfs the \frst\npeak. Our simulated pair statistics obey certain exact\nintegral conditions in Ref. 12 with very high accuracy, in-\ndicating the high \fdelity of the numerical results. In the\nin\fnite-system-size limit, at \u001f= 0:5, the entropically fa-\nvored ground states undergo a transition from disordered\nstates to crystalline states. Depending on the dimension,\nthis phase transition can occur when aperiodic structures3\nstill are part of the ground state manifold, demonstrat-\ning that crystalline (ordered) structures can have a higher\nentropy than disordered structures.\nThe rest of the paper is organized as follows: In Sec. II,\nwe brie\ry summarize the numerical collective-coordinate\nprocedure and other details of the simulation that we em-\nploy in the present paper with justi\fcations. In Sec. III,\nwe study the dependence of the results on a variety of en-\nergy minimization algorithms, initial conditions, and the\nforms of the stealthy potentials. In Sec. IV, we provide\npair correlation function, structure factor, Voronoi cell-\nvolume distribution, and con\fguration snapshots of the\nstealthy hyperuniform ground states obtained from the\ncanonical ensemble in the zero-temperature limit. We\nprovide concluding remarks and discussion in Sec. V, in-\ncluding suggestions for sampling the canonical ensemble\nin the zero-temperature limit of other potentials with de-\ngenerate disordered ground states.\nII. MATHEMATICAL RELATIONS AND\nSIMULATION PROCEDURE\nAs detailed in Sec. II of Ref. 12, we simulate point\nprocesses in periodic fundamental cells (i.e. simulation\nboxes) with a pairwise additive potential v(r) such that\nits Fourier transform exists. Under nearest image con-\nvention, the total potential energy can be calculated by\nsumming over all pairs of particles:\n\b(rN) =X\ni<jv(rij); (1)\nwhereNis the number of particles, rN\u0011r1;r2;:::;rNis\nthe locations of the particles in d-dimensional Euclidean\nspace, and rij=ri\u0000rj. Instead of summing over all pair-\nwise contributions in the real space, the potential energy\ncan also be represented in Fourier space:\n\b(rN) =1\n2vF\"X\nk~v(k)j~n(k)j2\u0000NX\nk~v(k)#\n;(2)\nwherevFis the volume of the fundamental cell, ~ v(k) =R\nvFv(r) exp(\u0000ik\u0001r)dris the Fourier transform of the\npair potential, ~ n(k) =PN\nj=1exp(\u0000ik\u0001rj) is the complex\ncollective density variable [with ~ n(k= 0) =N], and\nboth summations are over all reciprocal lattice vector k's\nappropriate to the fundamental cell. For every k6=0,\n~n(k) is related to the structure factor, S(k), via\nS(k) =j~n(k)j2\nN: (3)\nGiven a ~v(k), the corresponding real-space pair potential\nis\nv(r) =1\nvFX\nk~v(k) exp(ik\u0001r): (4)In a \fnite-sized system, the real-space pair potential has\nthe same periodicity as the fundamental cell. Therefore,\nin the in\fnite-volume limit, the cell periodicity disap-\npears.\nA family of \\stealthy\" potentials, which completely\nsuppress single scattering for all wave vectors within\na speci\fc cuto\u000b in their ground states, are de\fned as\n[13, 14, 17{20]:\n~v(k) =(\nV(k);ifjkj\u0014K,\n0; otherwise,(5)\nwhereV(k) is a positive isotropic function and Kis a\nconstant. In this paper we always take K= 1, which\nsets the length scale. We will also use V(k) = 1 unless\notherwise speci\fed. In the in\fnite-system-size limit, the\nisotropic ~v(k) correspond to an isotropic real-space pair\npotentialv(r) [12]. However, for \fnite systems, the cor-\nresponding v(r) is anisotropic. In Appendix A, we com-\npare the in\fnite-system-size limit v(r) with the \fnite-size\nv(r)'s in di\u000berent-shaped simulation boxes and select the\nsimulation box shape to be used in this paper based on\nwhichv(r) is closest to the in\fnite-size-limit v(r).\nFrom Eqs. (2) and (5), one can see that a con\fguration\nis a stealthy ground state if ~ n(k) = 0 for all kpoints such\nthat 0<jkj\u0014K. Therefore, \fnding a ground state of a\nstealthy potential is equivalent to constraining ~ n(k) = 0\nfor all of those kpoints. However, in a simulation, one\ndoes not need to check all of the constraints. As detailed\nin Ref. 12, if there are (2 M+ 1) kpoints within the\nconstrained radius, only Mof them are independent and\nneeded to be constrained to zero. Equation (2) can be\nsimpli\fed as [49]:\n\b(rN) =1\nvFX\nk~v(k)j~n(k)j2+ \b 0; (6)\nwhere the sum is over all independent constraints, and\n\b0= [N(N\u00001)\u00002NX\nk~v(k)]=(2vF) (7)\nis a constant independent of the particle positions rN.\nWe now introduce a parameter\n\u001f=M\nd(N\u00001); (8)\nwhich determines the degree to which the ground states\nare constrained, and therefore the degeneracy and disor-\nder of the ground states [14]. Note that the constraints\ndepend onKand the fundamental cell but are indepen-\ndent of the speci\fc shape of ~ v(k) as long as ~ v(k)>0\nfor all 0<jkj\u0014K. Therefore, changing ~ v(k) does not\nchange the set of the ground states. However, there is\nno proof that changing ~ v(k) does not change the relative\nsampling weights of the ground states.\nIn this paper we study various systems with di\u000berent\n\u001f's andN's. One numerical complication is that these4\nnumbers cannot be chosen arbitrarily, since M=\u001fd(N\u0000\n1) must be an integer consistent with the speci\fc shape\nof the simulation box. (For example, a list of the allowed\nMvalues for a two-dimensional square box is given in\nTable II of Ref. 14.) This constraint is especially hard\nto meet when simulating multiple systems at the same \u001f\nvalue across dimensions. In fact, both \u001fandNin Table I\n(see Appendix C) had to be chosen carefully to meet this\nconstraint.\nTaking the gradient of Eq. (6) yields the forces on par-\nticles:\nFj=\u00005j\b(rN) =2\nvFX\nkk~v(k) Im[~n(k) exp(ik\u0001rj)];\n(9)\nwhere the sum is also over all independent constraints.\nThis equation enables us to perform both energy mini-\nmizations and molecular dynamics (MD) simulations. In\nan energy minimization, a derivative-based algorithm is\nused. The \frst term on the right side of Eq. (6) is pro-\nvided to the algorithm as the objective function and the\nnegative of the force in Eq. (9) is provided as the deriva-\ntive. In order to minimize energy, we have tried dif-\nferent algorithms including the MINOP algorithm [50],\nthe steepest descent algorithm allowing large steps [51],\nthe low-storage BFGS (L-BFGS) algorithm [52{54], the\nPolak-Ribiere conjugate gradient algorithm [51, 55], and\nour \\local gradient descent\" algorithm described in Ap-\npendix B. When \u001f <0:5, the objective function always\nends up being very close to zero (the minimum). The\nmaximum ending objective function for di\u000berent algo-\nrithms varies from as high as 10\u00007for a conjugate gra-\ndient algorithm to 10\u000017for the local gradient descent\nand steepest descent algorithms to 10\u000020for the L-BFGS\nalgorithm, and to as low as 10\u000025for the MINOP algo-\nrithm. From our practical point of view, all of these\nalgorithms are precise enough, since an error of 10\u00007or\nlower is indiscernible from any results presented below.\nBecause the L-BFGS algorithm is the fastest, we will use\nit unless otherwise speci\fed.\nThe energy minimizations, if started from random ini-\ntial con\fgurations, will sample an algorithm-dependent,\nnonequilibrium ensemble. To sample the canonical en-\nsemble at a given equilibrium temperature TEwe use\nMD simulations. One important parameter in MD sim-\nulations is the integration time step. Since the optimal\nchoice of the time step depends on the temperature, and\nthe latter varies across several orders of magnitude in\nthis paper, we desire a systematic way to determine the\noptimal time step. Starting from an energy minimized\ncon\fguration and a very small time step (0.01 in dimen-\nsionless units), we repeat the following steps 104times to\nequilibrate the system and \fnd a suitable time step:\n\u000fAssign a random velocity from Boltzmann distri-\nbution atTEto each particle.\n\u000fCalculate the total (kinetic and potential) energy\nof the system E1.\u000fEvolve the system 1500 time steps using the veloc-\nity Verlet algorithm [56].\n\u000fCalculate the total energy of the system E2.\n\u000fIfjlnE1\nE2j>1\u000210\u00005, then the time step is too\nlarge and errors will build up quickly. Therefore,\nwe decrease the time step by 5%. On the other\nhand, ifjlnE1\nE2j<4\u000210\u00006, there is still some room\nto increase the time step. Since increasing the time\nstep increases the e\u000eciency of MD simulations, we\nincrease the time step by 5%.\nAfter the system is equilibrated and the time step is cho-\nsen, we perform constant temperature MD simulations\nwith particle velocity resetting [57]. A randomly cho-\nsen particle is assigned a random velocity, drawn from\nMaxwell-Boltzmann distribution, every 100 steps. We\ntake a sample con\fguration every 3000 time steps until\nwe have sampled 20 000 con\fgurations unless otherwise\nspeci\fed. This amounts to an implementation of the gen-\neration of con\fgurations in the canonical ensemble.\nThe above MD procedure works well for \u001f<0:5. How-\never, two new features arise when it is applied to \u001f\u00150:5\nin all dimensions. First, the potential energy surface de-\nvelops local minima and energy barriers that can trap the\nsystem ifTEis too small. We address this problem by\nusing simulated annealing, employing a thermodynamic\ncooling schedule [58] which starts at T= 2\u000210\u00003and\nends at 10\u00006. Note that, by adopting a cooling schedule,\nwe concede that we may only take one sample at the end\nof each MD trajectory, whereas a \fxed-temperature MD\ntrajectory produces multiple samples.\nThe second new feature is that the entropically favored\nground states are crystalline for \u001f\u00150:5. Unlike disor-\ndered structures, a crystalline structure has long-range\norder and may not \\\ft\" in simulation boxes with certain\nshapes. To overcome the second problem, we simulate an\nisothermal-isobaric ensemble with a deformable simula-\ntion box. Every 20 MD time steps, 10 Monte Carlo trial\nmoves to deform the simulation box are attempted. The\npressure is calculated from Eq. (41) of Ref. 12.\nWe employed the Wang-Landau Monte Carlo [59] to\nattempt to determine the entropically favored ground\nstates for\u001f > 0:5 in two and three dimensions. The\nWang-Landau Monte Carlo is used to calculate the mi-\ncrocanonical entropy S(\b) as a function of the potential\nenergy \b. We limit our simulations to the energy range\n3\u000210\u000010<\b\u0000\b0<10\u00009(in dimensionless units),\nwhere \b 0is the ground state energy, by rejecting any\ntrial move that violates this energy tolerance. This en-\nergy range is evenly divided into 1000 bins. Starting from\na perfect crystal structure in a simulation box shaped like\na fundamental cell, small perturbations are introduced so\nthe energy is within the range. After that, 60 stages of\nMonte Carlo simulations are performed, each stage con-\ntaining 3\u0002107trial moves. The \\modi\fcation factor\" in\nRef. [59] is f= exp[5=(n+ 10)], where nis the number\nof stages.5\nIII. DEPENDENCE ON ENERGY\nMINIMIZATION ALGORITHM, MD\nTEMPERATURE, AND ~v(k)\nIn this section, we present numerical simulation results\ndemonstrating that:\n\u000fEnergy minimizations starting from Poisson initial\ncon\fgurations using di\u000berent algorithms can yield\nground states with di\u000berent pair correlation func-\ntions.\n\u000fEnergy minimizations starting from MD snapshots\nat di\u000berent temperatures can yield ground states\nwith di\u000berent pair correlation functions.\n\u000fFor con\fgurations obtained by minimizing energy\nstarting from MD snapshots at su\u000eciently small\ntemperature, pair correlation functions do not de-\npend on the minimization algorithm and the form\nof the stealthy potential.\nThese results motivate the reason why we ultimately\nstudy and report results in Sec. IV in the canonical en-\nsemble in the zero-temperature limit. For concreteness\nand visual clarity, we present results here in two dimen-\nsions. However, we have veri\fed that all of the conclu-\nsions here also apply to one and three dimensions.\nWe performed energy minimizations starting from\nPoisson initial con\fgurations (i.e., TE!1 state at \fxed\ndensity) using each of the \fve numerical algorithms men-\ntioned in Sec. II at \u001f= 0:2 and\u001f= 0:4. The results are\nshown in Figs. 1 and 2. At \u001f= 0:2, the pair correla-\ntion functions produced by the MINOP algorithm and\nthe L-BFGS algorithm are almost identical. However,\nthe pair correlation function produced by the conjugate\ngradient algorithm noticeably di\u000bers. The steepest de-\nscent algorithm and our local gradient descent algorithm\nproduce a signi\fcantly di\u000berent pair correlation function\nwith a much weaker peak at r= 0. The pair correla-\ntion functions produced by some algorithms appear to\nhaveg2(r)/log(r) divergence near the origin. Since\nthis divergence means particles have a tendency to form\nclusters, we call it a \\clustering e\u000bect.\" At \u001f= 0:4, the\nclustering e\u000bect disappears, but the pair statistics pro-\nduced by di\u000berent algorithms still di\u000bers. The fact that\ndi\u000berent optimization algorithms produce di\u000berent pair\nstatistics means that they sample the ground-state mani-\nfold with di\u000berent weights. In other words, di\u000berent opti-\nmization algorithms are sampling di\u000berent ground-state\nensembles.\n0 5 10 15 20r00.511.52g2(r)MINOP\nSteepest Descent\nL-BFGS\nConjugate Gradient\nLocal Gradient Descent\n0.001 0.01 0.1 1 1000.511.52\n0 0.5 1 1.5 20.250.30.350.40.450.5FIG. 1. (Color online) Pair correlation function as obtained\nfrom di\u000berent optimization algorithms (as described in the\nlegend) starting from Poisson initial con\fgurations in two di-\nmensions at \u001f= 0:2. Each curve is averaged over 20 000\ncon\fgurations of 136 particles each. The left inset zooms in\nnear the origin, showing the di\u000berences between the \fve al-\ngorithms more clearly. The right inset uses a semilogarithmic\nscale to show g2(r)/log(r) near the origin.\n0 5 10 15 20r00.511.52g2(r)MINOP\nSteepest Descent\nL-BFGS\nConjugate Gradient\nLocal Gradient Descent\n6.2 6.4 6.6 6.8 70.880.90.920.94\nFIG. 2. (Color online) As in Fig. 1, except that \u001f= 0:4\nand each curve is averaged over 20 000 con\fgurations of 151\nparticles each. The inset zooms in near the \frst well, showing\nthe di\u000berences between the \fve algorithms more clearly.\nIn order to avoid the complexity caused by the details\nof various optimization algorithms, we turn our interest\nto the canonical ensemble in the T!0 limit. To sam-\nple this ensemble, we perform MD simulations at su\u000e-\nciently small temperature TE, periodically take \\snap-\nshots,\" and then use a minimization algorithm to bring\neach snapshot to a ground state. To determine a \\suf-\n\fciently small\" TE, we calculated the pair correlation\nfunctions at various TE's and present them in Fig. 3. The\nenergy minimization result starting from TE!1 initial\ncon\fgurations clearly display the \\clustering e\u000bect\" at\n\u001f= 0:2. WhenTEgoes to zero, the \\clustering e\u000bect\"\nalso diminishes. At \u001f= 0:4, particles develop hard cores6\n[g2(0) = 0], therefore there is no clustering even if TEis\nlarge or in\fnite. However, the peak height of g2(r) be-\ncomes dependent on TEat this\u001fvalue. For both \u001fval-\nues, the pair correlation functions of the two lowest TE's\nare almost identical, verifying that the TE!0 limit ex-\nists. These results show that TE= 2\u000210\u00006is su\u000eciently\nsmall in two dimensions. Similarly, we have found that\nTE= 2\u000210\u00004andTE= 1\u000210\u00006are su\u000eciently small\nin one and three dimensions, respectively. These temper-\natures are used in generating all of the results presented\nin Sec. IV A.\n0 5 10 15 20r00.511.52g2(r)L-BFGS\nMD at TE=2×10-2, then L-BFGS\nMD at TE=2×10-4, then L-BFGS\nMD at TE=2×10-6, then L-BFGS\n(a)\n0 5 10 15 20r00.511.52g2(r)L-BFGS\nMD at TE=2×10-2, then L-BFGS\nMD at TE=2×10-4, then L-BFGS\nMD at TE=2×10-6, then L-BFGS\n(b)\nFIG. 3. (Color online) Pair correlation function produced by\nL-BFGS algorithm starting from snapshots of MD at di\u000berent\nequilibration temperatures TE, (a)\u001f= 0:2 and (b)\u001f= 0:4.\nEach curve is averaged over 20 000 con\fgurations of 136 par-\nticles each or 151 particles each.\nThe energy minimization result starting from Poisson\ninitial con\fgurations di\u000bers for di\u000berent algorithms, but\nthe canonical ensemble in the T!0 limit should not\ndepend on any particular algorithm. After \fnding that\nTE= 2\u000210\u00006is su\u000eciently small, we con\frm the dis-\nappearing of algorithmic dependence by calculating the\npair correlation function produced by di\u000berent energy\nminimization algorithms starting from MD snapshots atTE= 2\u000210\u00006. Figure 4 shows the results. The curves\nfor all algorithms almost coincide.\n0 5 10 15 20r00.511.52g2(r)MINOP\nSteepest Descent\nL-BFGS\nConjugate Gradient\nLocal Gradient Descent\nFIG. 4. (Color online) Pair correlation function produced by\nthe \fve di\u000berent algorithms starting from snapshots of MD\nat equilibration temperature TE= 2\u000210\u00006at\u001f= 0:2. Each\ncurve is averaged over 20 000 con\fgurations of 136 particles\neach.\nLast, the function V(k) in Eq. (5) can have di\u000berent\nforms. This paper mainly use V(k) = 1 but we also\nwant to know if the results obtained using this form\nare equivalent to those generated using other positive\nisotropic forms of V(k) as well. In principle, stealthy po-\ntentials of any form should have the same set of ground-\nstate con\fgurations, but the form of the stealthy poten-\ntial could theoretically a\u000bect the curvature of the poten-\ntial energy surface near each ground-state con\fgurations\nand thus also a\u000bect their relative weights. Figure (5)\nshows the pair correlation function produced by di\u000berent\nV(k)'s. The pair correlation functions for V(k) = 1 and\nV(k) = (1\u0000k)2atTE= 2\u000210\u00006are almost identical.\nForV(k) = (1\u0000k)6, we initially tried TE= 2\u000210\u00006\nbut found that the \\clustering e\u000bect\" is still noticeable.\nWe further lowered the temperature to TE= 2\u000210\u000010\nto completely suppress the \\clustering e\u000bect\" to produce\na pair correlation function identical to that of V(k) = 1\nandV(k) = (1\u0000k)2potentials. This result suggests that\nthe functional form of V(k) does not produce noticeable\ndi\u000berences in the ground-state ensembles in the T!0\nlimit of the canonical ensemble.7\n0 5 10 15 20r0.40.60.81g2(r)TE=2×10-10, V(k)=(1-k)6\nTE=2×10-6, V(k)=(1-k)2\nTE=2×10-6, V(k)=1\nFIG. 5. (Color online) Pair correlation function produced by\ndi\u000berent potentials starting from snapshots of MD at su\u000e-\nciently low temperature at \u001f= 0:2. Each curve is averaged\nover 20 000 con\fgurations of 136 particles each.IV. CANONICAL ENSEMBLE IN THE T!0\nLIMIT\nWe will show here that the entropically favored ground\nstates in the canonical ensemble in the T!0 limit for\nthe \frst three space dimensions di\u000ber markedly below\nand above\u001f= 0:5. For\u001f<0:5, the entropically favored\nground states are disordered while for \u001f\u00150:5 the entrop-\nically favored ground states are crystalline. Therefore, we\nwill characterize them di\u000berently. For \u001f < 0:5, we will\nreport the pair correlation function, structure factor, and\nVoronoi cell statistics. For su\u000eciently small \u001f, we will\nshow that the simulation results agree well with theory\n[12]. For\u001f\u00150:5, we will report the crystal structures.\nThe numbers of particles in all of the systems reported\nin this section are collected in Appendix C.\nA.\u001f<0:5region\nRepresentative entropically favored stealthy ground\nstates in the \frst three space dimensions at \u001f= 0:1 and\n\u001f= 0:4 are shown in Figs. 6-8. As \u001fincreases from 0.1 to\n0.4, the stealthiness increases, accompanied with a visu-\nally perceptible increase in short-range order. This trend\nin short-range order is consistent with previous studies\n[14, 17, 18].\n(a)\n(b) \nFIG. 6. (Color online) Representative one-dimensional entropically favored stealthy ground states at (a) \u001f= 0:1 and (b)\n\u001f= 0:4.\n(a)\n(b) \nFIG. 7. (Color online) Representative two-dimensional entropically favored stealthy ground states at (a) \u001f= 0:1 and (b)\n\u001f= 0:4.8\n(a)\n(b)\nFIG. 8. (Color online) Representative three-dimensional entropically favored stealthy ground states at (a) \u001f= 0:1 and (b)\n\u001f= 0:4.\n0 1 2 3 4\nk00.511.5S(k)d=1, Theory\nd=2, Theory\nd=3, Theory\nd=1, Simulation\nd=2, Simulation\nd=3, Simulationχ=0.05\n0 1 2 3 4\nk00.511.5S(k)d=1, Theory\nd=2, Theory\nd=3, Theory\nd=1, Simulation\nd=2, Simulation\nd=3, Simulationχ=0.1\n0 1 2 3 4\nk00.511.5S(k)d=1, Theory\nd=2, Theory\nd=3, Theory\nd=1, Simulation\nd=2, Simulation\nd=3, Simulationχ=0.143\n0 1 2 3 4\nk00.511.5S(k)d=1, Theory\nd=2, Theory\nd=3, Theory\nd=1, Simulation\nd=2, Simulation\nd=3, Simulationχ=0.2\n0 1 2 3 4\nk00.511.5 S(k) d=1\nd=2\nd=3χ=0.25\n0 1 2 3 4 5\nk01234S(k)d=1\nd=2\nd=3χ=0.33\nFIG. 9. (Color online) Structure factors for 1 \u0014d\u00143 for 0:05\u0014\u001f\u00140:33 from simulations and theory [12]. The smaller \u001f\nsimulation results are also compared with the theoretical results in the in\fnite-volume limit [12]. For \u001f\u00140:1, the theoretical\nand simulation curves are almost indistinguishable, and the structure factor is almost independent of the space dimension.\nHowever, simulated S(k) in di\u000berent dimensions become very di\u000berent at larger \u001f. Theoretical results for \u001f\u00150:25 are not\npresented because they are not valid in this regime.9\n0 5 10r00.511.5g2(r) d=1, Theory\nd=2, Theory\nd=3, Theory\nd=1, Simulation\nd=2, Simulation\nd=3, Simulationχ=0.05\n0 5 10r00.511.5g2(r) d=1, Theory\nd=2, Theory\nd=3, Theory\nd=1, Simulation\nd=2, Simulation\nd=3, Simulationχ=0.1\n0 5 10r00.511.5g2(r) d=1, Theory\nd=2, Theory\nd=3, Theory\nd=1, Simulation\nd=2, Simulation\nd=3, Simulationχ=0.143\n0 5 10r00.511.5g2(r) d=1, Theory\nd=2, Theory\nd=3, Theory\nd=1, Simulation\nd=2, Simulation\nd=3, Simulationχ=0.2\n0 5 10r00.511.5g2(r)d=1\nd=2\nd=3χ=0.25\n0 5 10r00.511.52g2(r)\nd=1\nd=2\nd=3χ=0.33\nFIG. 10. (Color online) Pair correlation functions for 1 \u0014d\u00143 for 0:05\u0014\u001f\u00140:33 from simulations and theory [12]. The\nsmaller\u001fsimulation results are also compared with the theoretical results in the in\fnite-volume limit [12]. For \u001f\u00140:1, the\ntheoretical and simulation curves are almost indistinguishable. Theoretical results for \u001f\u00150:25 are not presented because they\nare not valid in this regime.\nWe have calculated the pair correlation functions and\nthe structure factors for various \u001fvalues. Results for\n0:05\u0014\u001f\u00140:33 are shown in Figs. 9 and 10. The \u001f<0:2\nresults are in excellent agreement with the \\pseudo-hard-\nsphere ansatz,\" which states that the structure factor\nbehaves like pseudo equilibrium hard-sphere systems in\nFourier space [12]. However, the theory gradually be-\ncomes invalid as \u001fincreases.\nThe pair correlation functions of the entropically fa-\nvored stealthy ground states are shown in Fig. 10. When\n\u001f\u00140:2, since the structure factor is similar to the pair\ncorrelation function of the hard-sphere system, inversely\nthe pair correlation function is also similar to the struc-\nture factor of the hard-sphere system. As \u001fgrows larger,\nthe pseudo hard-sphere ansatz gradually deviates from\nthe simulation result.\nWe have checked that these pair statistics are consis-\ntent with four theoretical integral conditions of the pair\nstatistics in the in\fnite-volume limit [12]. The \frst three\nconditions are Eqs. (58), (59), and (63) of Ref. 12, which\nareZ\nRdP(r)dr= 0; (10)\nZ\nRdP(r)v(r)dr= 0; (11)\nand\ng2(0) = 1\u00002d\u001f+ 2d2\u001fZ1\nKkd\u00001~Q(k)dk; (12)\nwhereP(r) is the inverse Fourier transform of \u0002( k\u0000\n1)~Q(k), \u0002(x) is the Heaviside step function, and ~Q(k) =\nS(k)\u00001.The fourth condition is that the pressure calculated\nfrom the \\virial equation\" [12] has to be either noncon-\nvergent or convergent to the pressure calculated from the\nenergy route [12]. All pair statistics in Figs. 9 and 10 were\ngenerated using the step-function potential [the V(k) = 1\ncase of Eq. (5)], but this potential does not lead to a con-\nvergent virial pressure. However, as we have shown ear-\nlier, the stealthy ground states that we generated here\nare also the ground states of other stealthy functional\nforms ~v(k). In one dimension, to test our simulation\nprocedure, we used the potential form V(k) = (1\u0000k)\nto calculate the pressure from both the virial equation\n(Eq. (43) of Ref. 12) and the energy equation (Eq. (41)\nof Ref. 12). The pressure from the virial equation con-\nverges and agrees with the exact pressure from the energy\nequation, thus con\frming the accuracy of our numerical\nresults. These checks involve integrals of g2(r) andS(k)\nthat are only slowly converging. Therefore, passing them\ndemonstrates that our results have very high precision.10\n0 1 2 3 45\nk00.511.522.5 S(k)/unif063=0.33 \n/unif063=0.35 \n/unif063=0.38 \n/unif063=0.4 \n/unif063=0.43 \n/unif063=0.46 \n0 1 0 0r0123g2(r)/unif063=0.33\n/unif063=0.35\n/unif063=0.38\n/unif063=0.4\n/unif063=0.43\n/unif063=0.46\n2\nFIG. 11. (Color online) Structure factor and pair correlation\nfunction for d= 2 for 0:33\u0014\u001f\u00140:46, as obtained from\nsimulations.\nFor smaller \u001fvalues, the maximum of the structure\nfactor is at the constraint cuto\u000b k=K+. However, for\nhigher\u001fvalues, the maximum of S(k) is no longer at\nk= 1+. To probe this transition we have calculated the\nstructure factor in two dimensions for 0 :33\u0014\u001f\u00140:46.\nThe results are shown in Fig. 11. As \u001fincreases, the\npeak atk= 1+gradually decreases its height, while the\nsubsequent peak gradually grows and engulfs the \frst\npeak.\nBesides pair statistics, other widely used characteri-\nzation of point patterns include certain statistics of the\nVoronoi cells [14, 60{62]. A Voronoi cell is the region\nconsisting of all of the points closer to a speci\fc parti-\ncle than to any other. We have computed the Voronoi\ntessellation of the entropically favored stealthy ground\nstates using the dD Convex Hulls and Delaunay Trian-\ngulations package [63] of the Computational Geometry\nAlgorithms Library [64]. Since the number density of\nthe stealthy ground states depends on the dimension and\n\u001f, we rescaled each con\fguration to unity density for\ncomparison of the Voronoi cell volumes. The probability\ndistribution function p(vc) of the Voronoi cell volumes\n(wherevcis the volume of a Voronoi cell) are shownin Fig. 12. In the same dimension, as \u001fincreases, the\ndistribution of Voronoi cell volumes narrows. This is\nexpected because the system becomes more ordered as\n\u001fincreases. For the same \u001f, the distribution also nar-\nrows as the dimension increases, consistent with theo-\nretical results that at \fxed \u001f, the nearest-neighbor dis-\ntance distribution narrows as dimension increases [12]. In\nFig. 12, we additionally show the Voronoi cell-volume dis-\ntribution of saturated random sequential addition (RSA)\n[65{67] packings, the sphere packings generated by ran-\ndomly and sequentially placing spheres into a large vol-\nume subject to the nonoverlap constraint until no addi-\ntional spheres can be placed. Saturated RSA packings\nare neither stealthy nor hyperuniform [66, 67]. However,\nthe Voronoi cell-volume distributions of saturated RSA\npackings look similar to that of the entropically favored\nstealthy ground states. This is not unexpected because\nVoronoi cell statistics are local characteristics, and hence\nare not sensitive to the stealthiness, which is a large-scale\nproperty.11\n0 0.5 1 1.5 2 2.5 3vc00.511.522.53p(vc)χ=0.05\nχ=0.1\nχ=0.143\nχ=0.2\nχ=0.25\nRSAd=1\n0 1 2 3vc012345p(vc)χ=0.05\nχ=0.1\nχ=0.143\nχ=0.2\nχ=0.25\nRSAd=2\n0 1 2vc02468p(vc)χ=0.05\nχ=0.1\nχ=0.143\nχ=0.2\nχ=0.25\nRSAd=3\nFIG. 12. (Color online) Voronoi cell-volume distribution for\n1\u0014d\u00143 for 0:05\u0014\u001f\u00140:25. For the same dimension,\nthe Voronoi cell-volume distribution becomes narrower when\n\u001fincreases. For the same \u001f, the Voronoi cell-volume distri-\nbution also becomes narrower when dimension increases. We\nalso present Voronoi cell-volume distributions of RSA pack-\nings at saturation here.\nOne interesting phenomenon is that as \u001fincreases and\napproaches 1/2, systems that are not su\u000eciently large\ncan become crystalline. In Fig. 13, we show two snap-\nshots of MD simulations at \u001f= 0:48. The smaller con\fg-\nuration is crystalline. However, systems that are 4 times\nlarger remain disordered at the same \u001fand temperature.\nTherefore, this strongly indicates that crystallization is a\n\fnite-size e\u000bect for \u001ftending to 1/2 from below.\n(a)\n(b) FIG. 13. (Color online) (a) Low-temperature MD snapshot\nof a 126-particle system at \u001f= 0:48; the ground-state con-\n\fguration is crystalline. (b) MD snapshot of a 504-particle\nsystem at the same TEand\u001f; the system does not crystallize\nand is indeed disordered without any Bragg peaks.\nB.\u001f\u00150:5region\nAs explained in Sec. II, we perform MD-based sim-\nulated annealing with Monte Carlo moves of the simu-\nlation box for \u001f > 0:5, since this method works bet-\nter with rough potential energy surface and can mitigate\nthe \fnite-size e\u000bect. We performed this simulation at\n\u001f= 0:55,\u001f= 0:73, and\u001f= 0:81 in two dimensions.\nThe results are shown in Fig. 14. The resulting con-\n\fguration is always triangular lattice. Even though the\nground-state manifold in this \u001fregime contains aperiodic\n\\wavy\" phases discovered previously [14] [but which are\ncalled \\stacked-slider\" phases in the sequel to this pa-\nper [48], since they are aperiodic con\fgurations with a\nhigh degree of order in which rows (in two dimensions)\nor planes (in three dimensions) of particles can slide past\neach other] as well as crystals other than the triangular\nlattice, the entropically favored ground state is always a\ntriangular lattice. This means that the triangular lattice\nhas a higher entropy than stacked-slider phases, although\nthe latter appear to be more disordered [68].\nAlthough we will show analytically that crystals are\nmore entropically favored than stacked-slider phases in\nthe upcoming paper of this series, we still need simula-\ntion results to determine which crystal structure has the\nhighest entropy. The results of MD-based simulated an-\nnealing with Monte Carlo moves of the simulation box\nsuggest that triangular lattice has the highest entropy\nin two dimensions. It seems natural to apply the same\ntechnique to three dimensions to determine the entropi-12\ncally favored crystal structure. However, we were unable\nto crystallize the system in three dimensions. Even the\nlongest cooling schedule that we tried resulted in stacked-\nslider phases.\n(a) \n(b) \n(c)\nFIG. 14. (Color online) MD-based simulated annealing result\nat (a)\u001f= 0:55, (b)\u001f= 0:73, and (c) \u001f= 0:81. The ending\ncon\fguration is triangular lattice except for small deforma-\ntions in the \u001f= 0:55 case.\nAnother way to \fnd the entropically favored crystal is\nto use Wang-Landau Monte Carlo to directly calculate\nthe entropy of di\u000berent crystal structures as a function\nof the potential energy. We have performed this simula-\ntion on two-dimensional triangular lattice, square lattice,\nand three-dimensional body-centered cubic (BCC) lat-\ntice, face-centered cubic (FCC) lattice, and simple cubic\n(SC) lattice. The results are shown in Figs. 15 and 16. In\nall cases the entropy decreases as the energy decreases. In\ntwo dimensions, the entropy of the square lattice clearly\ndecreases faster than that of the triangular lattice at\nevery\u001fvalue, con\frming that the triangular lattice is\nentropically favored over the square lattice in the zero-\ntemperature limit. In three dimensions at \u001f= 0:58, the\nentropy of the FCC lattice decreases more slowly than\nthat of the BCC and SC lattice, suggesting that the en-\ntropically favored ground state in three dimensions at\n\u001f= 0:58 is the FCC lattice. At higher \u001fvalues, the scal-\ning of the entropy of the FCC lattice and the BCC latticebecome very close to each other, preventing us from de-\ntermining the entropically favored ground state at these\n\u001fvalues.\n3 4 5 6 7 8 9 10\n1010(Φ−Φ0)-600-500-400-300-200-1000S(Φ)-S(Φ0+10-9)\nTriangular\nSquareχ=0.51\n3 4 5 6 7 8 9 10\n1010(Φ−Φ0)-600-500-400-300-200-1000S(Φ)-S(Φ0+10-9)\nTriangular\nSquareχ=0.6\n3 4 5 6 7 8 9 10\n1010(Φ−Φ0)-600-500-400-300-200-1000S(Φ)-S(Φ0+10-9)\nTriangular\nSquareχ=0.67\n3 4 5 6 7 8 9 10\n1010(Φ−Φ0)-600-500-400-300-200-1000S(Φ)-S(Φ0+10-9)\nTriangular\nSquareχ=0.75\nFIG. 15. (Color online) Microcanonical entropy as a function\nof energy S(\b) calculated from Wang-Landau Monte Carlo of\ntriangular lattice and square lattice at various \u001f's. Here \b 0\ndenotes the ground-state energy.\n3 4 5 6 7 8 9 10\n1010(Φ−Φ0)-700-600-500-400-300-200-1000S(Φ)-S(Φ0+10-9)\nBCC\nFCC\nSCχ=0.58\n3 4 5 6 7 8 9 10\n1010(Φ−Φ0)-700-600-500-400-300-200-1000S(Φ)-S(Φ0+10-9)\nBCC\nFCCχ=0.68\n3 4 5 6 7 8 9 10\n1010(Φ−Φ0)-800-600-400-2000S(Φ)-S(Φ0+10-9)\nBCC\nFCCχ=0.77\n3 4 5 6 7 8 9 10\n1010(Φ−Φ0)-800-600-400-2000S(Φ)-S(Φ0+10-9)\nBCC\nFCCχ=0.845\nFIG. 16. (Color online) Microcanonical entropy as a function\nof energy S(\b) calculated from Wang-Landau Monte Carlo of\nBCC lattice, FCC lattice, and SC lattice at various \u001f's. A\ncurve for SC lattice is not presented for \u001f\u00150:68 because the\nlatter is not a ground state at such high \u001fvalues. Here \b 0\ndenotes the ground-state energy.\nV. CONCLUSIONS AND DISCUSSION\nThe uncountably in\fnitely degenerate classical ground\nstates of the stealthy potentials have been sampled previ-\nously using energy minimizations. We demonstrate here\nthat this way of sampling the ground states to produce\nensembles of con\fgurations introduces dependencies on\nthe energy minimization algorithm and the initial con-\n\fguration. Such arti\fcial dependencies are avoided in13\nstudying the canonical ensemble in the T!0 limit. We\nsample this ensemble by performing MD simulations at\nsu\u000eciently low temperatures, periodically taking snap-\nshots, and minimizing the energy of the snapshots.\nThe con\fgurations in this ensemble become more or-\ndered as\u001fincreases and obey certain theoretical condi-\ntions on their pair statistics [12], similarly to previous\nenergy minimization results. However, other properties\nof this ensemble are unique. First, our numerical results\ndemonstrate that the pair statistics of this ensemble dis-\nplays no \\clustering e\u000bect\" [divergence of g2(r) asr!0]\nfor any\u001fvalue, and is independent of the functional form\nof the stealthy potential. Second, we numerically verify\nthe theoretical ansatz [12] that for su\u000eciently small \u001f\nstealthy disordered ground states behave like \\pseudo\"\ndisordered equilibrium hard-sphere systems in Fourier\nspace, i.e.,S(k) has the same functional form as the pair\ncorrelation function for equilibrium hard spheres for suf-\n\fciently small densities. Third, when \u001fis above the crit-\nical value of 0.5, our results strongly indicate that crystal\nstructures are entropically favored in both two and three\ndimensions in the in\fnite-volume limit. Our numerical\nevidence suggests that the entropically favored crystal\nin two dimensions is the triangular lattice. However,\nwe could not determine the entropically favored crystal\nstructure in three dimensions. For \fnite systems, the\ndisordered-to-crystal phase transition can happen at a\nslightly lower \u001f. A theoretical explanation of this phe-\nnomenon remains an open problem.\nBesides ground states of stealthy potentials, other dis-\nordered degenerate ground states of many-particle sys-\ntems have been studied using energy minimizations.\nSpeci\fcally, previous researchers have constrained the\nstructure factor to have some targeted functional form\nother than zero for prescribed wave vectors [17, 18, 21].\nFinding the con\fgurations corresponding to such tar-\ngeted structure factors amounts to \fnding the ground\nstates of two-, three- and four-body potentials, in con-\ntrast to the two-body stealthy potential studied in the\npresent paper. This situation is the most general appli-\ncation of the collective-coordinate approach. It will be\ninteresting to study the resulting pair statistics of the\nground states for these more general interactions in the\nzero-temperature limit of the canonical ensemble.\nThe collective-coordinate approach is an independent\nand fruitful addition to the basic statistical mechan-\nics problem of connecting local interactions to macro-\nscopic observables. One important feature of collective-\ncoordinate interactions is that it has uncountably in-\n\fnitely degenerate classical ground states [12]. In the\ncase of isotropic pair interactions, the only other sys-\ntem that we know with this feature is the hard-sphere\nsystem. However, there are two important di\u000berences\nbetween hard-sphere systems and collective-coordinate\nground states. First, while the dimensionality of the\ncon\fguration space of equilibrium hard-sphere systems\nconsisting of Nparticles within a periodic box is \fxed\n[simply determined by the nontrivial number of degreesof freedom, d(N\u00001)], the dimensionality of the collective-\ncoordinate ground-state con\fguration space decreases as\n\u001fincreases and, on a per particle basis, eventually van-\nishes [12]. The decreased dimensionality of the ground-\nstate con\fguration space creates challenges for accurate\nsampling of the entropically favored ground states us-\ning numerical simulations and hence the development of\nbetter sampling methods is a fertile ground for future\nresearch.\nSecond, while the probability measure of the equi-\nlibrium hard-sphere system is uniform over its entire\nground-state manifold, that of the stealthy ground states\nis not uniform. To illustrate this point, imagine a one-\ndimensional energy landscape that has a double-well po-\ntential behavior in a portion of the con\fguration space,\nas shown in Fig. 17. Each minimum represents a degen-\nerate ground state (as we \fnd with stealthy potentials)\nand therefore the well depths of the minima are the same.\nLet us now consider harmonic approximations of the two\nwells in the vicinity of x1andx2, respectively,\nV1(x) =a1(x\u0000x1)2;\nand\nV2(x) =a2(x\u0000x2)2;\nwherexis the con\fgurational coordinate. At very low\ntemperature, to a good approximation, the system can\nonly visit the part of the con\fguration space with energy\nless than\", and\"!0 asT!0. Solving Vi(x)< \",\nwherei= 1;2, one \fnds the feasible region of con\fgura-\ntion space associated with both wells:\nx1\u0000p\n\"=a 1<x<x 1+p\n\"=a 1;\nand\nx2\u0000p\n\"=a 2<x<x 2+p\n\"=a 2:\nWhena16=a2, we see that the feasible regions associated\nwith the two potential wells have di\u000berent ranges. There-\nfore, the weights associated with the two minima, i.e., the\nrelative probabilities for \fnding the system in the vicinity\nof those minima, will also di\u000ber. Similarly, in the stealthy\nmultidimensional con\fguration space that we are study-\ning, the magnitude of the eigenvalues of the Hessian ma-\ntrix will determine the relative weights. Therefore, the\nprobability measure of the stealthy ground states is not\nuniform over the ground-state manifold, unlike the de-\ngenerate ground states of classical hard spheres. Our\nlow-temperature MD simulations sample ground states\nwith this nonuniform probability measure. It would be\nuseful to devise theories to estimate the weights of dif-\nferent portions of the ground-state manifold. However, a\nfeature that complicates the problem is that the Hessian\nmatrix has zero eigenvalues. In the associated directions\nof the eigenvectors of the con\fguration space, the energy\nscales more slowly than quadratically (harmonically) but\nwe do not know the speci\fc form.14\nx1x2\nConfigurational Coordinate, x0/unif065Potential Energy, V(x)\nFIG. 17. (Color online) A model one-dimensional energy land-\nscape with two wells located at x1andx2of the same depth\nbut di\u000berent curvatures. The \\feasible regions,\" i.e., regions\nwhereV(x)<\", is marked by red dashed lines.\nThis paper, which investigates the entropically favored\nground states, is the \frst of a two-paper series. In the\nsecond paper, we will study aspects of the ground-state\nmanifold with an emphasis on con\fgurations that are not\nentropically favored for \u001fabove 1/2 (the ordered regime).\nIn particular, we will more fully investigate the nature\nof so-called \\wavy\" crystals or \\stacked-slider\" phases,\ndiscovered in Ref. 14. Using an analytical description of\nsuch states, we will demonstrate that they are part of\nthe ground state but are not entropically favored. Our\nanalytical model will also demonstrate that stacked-slider\nphases exist in three and higher dimensions.\nACKNOWLEDGMENTS\nG. Z. thanks Steven Atkinson for his careful reading\nof some parts of the manuscript. This research was sup-\nported by the U.S. Department of Energy, O\u000ece of Basic\nEnergy Sciences, Division of Materials Sciences and En-\ngineering under Award No. DE-FG02-04-ER46108.\nAppendix A: Real-space potential in \fnite systems\nIn the in\fnite-system-size limit, an isotropic ~ v(k) cor-\nrespond to an isotropic real-space pair potential v(r).\nHowever, for \fnite systems, the corresponding v(r) is\nanisotropic. To illustrate the \fnite-size e\u000bect, we com-\npare the two-dimensional real-space potential v(r) in the\nin\fnite-system-size limit to corresponding potentials as-\nsociated with \fnite-sized fundamental cells of square and\nrhombic shapes of di\u000berent volumes in Fig. 18. The real-\nspace potential in the rhombic simulation box with a 60\u000e\ninterior angle is appreciably more isotropic than the real-\nspace potential in a square simulation box. Therefore, inthis paper, we will henceforth use rhombic fundamental\ncells in two dimensions. Similarly, in three dimensions,\nwe always use a simulation box shaped like a fundamen-\ntal cell of a body-centered cubic (BCC) lattice since BCC\nlattice is the unique ground state at \u001f\u0003max.\nAppendix B: Local Gradient Descent Algorithm\nMost optimization algorithms are designed for e\u000e-\nciency. They use complex rules to determine the direc-\ntion of the next step and take as large steps as possible.\nThese features make their path less obvious. To mini-\nmize energy in the path following the gradient vector, we\ndesigned a \\local gradient descent algorithm\" with the\nfollowing steps:\n1. Start from an initial guess, x, and \fnd the function\nvaluef(x) and derivative f0(x).\n2. Start from a relatively large (10\u00003times the simula-\ntion box side length) step size, s, and calculate the\nvector to the next step \u0001 x=\u0000sf0(x)\njf0(x)j. Find the\nfunction value at the next step f(x+ \u0001x). Calcu-\nlate the change of function value \u0001 f=f(x+\u0001x)\u0000\nf(x).\n3. If we are following the path of steepest descent ac-\ncurately, the change of the function value should\nbe close to f0(x)\u0001\u0001x. If the di\u000berence between\n\u0001fandf0(x)\u0001\u0001xis less than 1%, we accept this\nmove. Otherwise, we abort this move and half the\nstep sizes.\n4. Repeat the above steps until a minimum is found\nwith enough precision.\nAppendix C: Number of particles of every system in\nSec. IV\nTABLE I. The number of particles Nof each systems shown\nin Figs. 9 and 10.\n\u001f N ford= 1Nford= 2Nford= 3\n0.05 1001 541 261\n0.1 501 270 131\n0.143 351 190 92\n0.2 251 136 66\n0.25 201 109 53\n0.33 151 181 191\nIn this appendix we report the number of particles Nin\neach system in Sec. IV. Both con\fgurations in Fig. 6 con-\nsist of 51 particles. Con\fgurations (a) and (b) in Fig. 7\nconsist of 271 and 151 particles, respectively. Those in\nFig. 8 consist of 131 and 161 particles, respectively.15\n(a) \n(b) \n(c) \n(d) \n(e) \n(f) \n(g) \nFIG. 18. (Color online) A portion of the real-space potential v(r) around the origin for the stealthy potential (5) with K= 1\nandV(k) = 1. (a)-(f) Real-space potential in a periodic simulation box that is [(a), (c), and (e)] square or [(b), (d), and (f)]\nrhombic in shape; the latter has a 60\u000einterior angle. 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E 88, 053312\n(2013).\n[68] In Fig. 13, we show that crystallization for \u001f <0:5 is a\n\fnite-size e\u000bect. So is crystallization for \u001f\u00150:5 also a\n\fnite-size e\u000bect? In the upcoming paper of this series,\nwe will present an analytical model of \\stacked-slider\"\nphases for\u001f\u00150:5, which are relatively ordered but ape-\nriodic con\fgurations. Using the model, we will show that\nthe entropy of stacked-slider phases is smaller than that\nof the crystalline structures. Therefore, crystallization for\n\u001f\u00150:5 is not a \fnite-size e\u000bect."    },    {        "title": "1509.00494v2.Onsager_rule__quantum_oscillation_frequencies__and_the_density_of_states_in_the_mixed_vortex_state_of_cuprates.pdf",        "content": "Onsager rule, quantum oscillation frequencies, and the density of states in the\nmixed-vortex state of cuprates\nZhiqiang Wang and Sudip Chakravarty\nDepartment of Physics and Astronomy, University of California Los Angeles, Los Angeles, California 90095-1547\n(Dated: June 21, 2018)\nThe Onsager rule determines the frequencies of quantum oscillations in magnetic \felds. We\nshow that this rule remains intact to an excellent approximation in the mixed-vortex state of the\nunderdoped cuprates even though the Landau level index nmay be fairly low, n\u001810. The models\nwe consider are fairly general, consisting of a variety of density wave states combined with d-wave\nsuperconductivity within a mean \feld theory. Vortices are introduced as quenched disorder and\naveraged over many realizations, which can be considered as snapshots of a vortex liquid state. We\nalso show that the oscillations ride on top of a \feld independent density of states, \u001a(B), for higher\n\felds. This feature appears to be consistent with recent speci\fc heat measurements [C. Marcenat,\net al. Nature Comm. 6, 7927 (2015)]. At lower \felds we model the system as an ordered vortex\nlattice, and show that its density of states follows a dependence \u001a(B)/p\nBin agreement with the\nsemiclassical results [G. E. Volovik, JETP Lett. 58, 469 (1993)].\nI. INTRODUCTION\nA breakthrough in the area of cuprate supercon-\nductivity is the observation of quantum oscillations in\ncuprates [1, 2]. In these experiments a strong mag-\nnetic \feld is applied to suppress the superconductivity,\nwhich most likely reveals the ground state [3] without\nsuperconductivity. However, the understanding of this\n\\normal state\" may be a crucial ingredient in the the-\nory high temperature superconductivity. Standing in the\nway are at least two important issues: (1) Does the quan-\ntum oscillation frequencies substantially deviate from the\nclassic Onsager rule for which the oscillation frequency\nF= (~c=2\u0019e)A(\u000fF), whereA(\u000fF) is equal to the ex-\ntremal Fermi surface area normal to the magnetic \feld?\nIf so, it would lead to considerable uncertainty in the in-\nterpretation of the experiments. (2) Do the oscillations\nride on top of a magnetic \feld dependence of the density\nof states(DOS) \u001a(B)\u0018p\nB? [4] If so, it might indicate\nthe presence of superconducting \ructuations even in high\nmagnetic \felds at zero temperature, T= 0, from an ex-\ntrapolation of a result of Volovik, [5] which is supposed\nto be asymptotically true as B!0. Therefore high \feld\nbehavior requires careful analyses. Quantum oscillations\nrequire the existence of Landau levels. If this is true,\nthey might indicate the existence of normal Fermi liquid\nquasiparticles [6].\nIt has been argued from a theoretical analysis that the\nOnsager rule could be violated by as much as 30% [7]. We\n\fnd that under reasonable set of parameters, to be de-\n\fned below, the violation is miniscule, \u001810\u00004. Even for\nextreme situations discussed in Ref. 7, it is less than 2%.\nIf we are correct, one can use the Onsager rule to inter-\npret the experiments with impunity. The second encour-\naging result is that \u001a(B) saturates in the regime where\noscillations are present. We interpret this to mean that\nthere are generically no superconducting \ructuations in\nhigh \felds. A recent speci\fc heat measurement [8] shows\nthat the speci\fc heat indeed saturates at high \felds, sig-nifying that the normal state is achieved.\nTo put our paper in the context, note that in conven-\ntionals-wave superconductors, previous work has shown\nthat for higher Landau level indices, and within coher-\nent potential approximation, vortices mainly damp the\noscillation amplitude, but the shift in the oscillation\nfrequency [9, 10] is negligible; however, for d-wave un-\nderdoped cuprate superconductors with small coherence\nlength and high \felds with Landau level indices \u001810\nthis calculation should not hold [7]. A more recent semi-\nclassical analysis based on an ansatz of gaussian phase\n\ructuations of thed\u0000wave pairing [11] indicates that\nthe oscillation frequency is unchanged, as here. However,\nrelatively undamped quantum oscillations riding on top\nofp\nHwas found in this dynamic Gaussian ansatz that\ndoes not account for vortices, which must necessarily be\npresent, as in Ref. [7], and the branch cuts introduced by\nthe vortices must also be taken into account.\nWe consider the vortices explicitly in the Bogoliubov-\nde Gennes (BdG) Hamiltonian, as in Ref. [7], and model\nthe vortex liquid state as quenched, randomly distributed\nvortices, paying special attention to branch cuts. There\nare other important di\u000berences as well, as we shall dis-\ncuss below. To have a complete picture, we also consider\nthe low \feld regime where the quantum oscillations dis-\nappear. In this regime the vortices arrange themselves\ninto a vortex solid state and should be modeled as an\nordered lattice instead. We compute the DOS of such\nvortex lattices explicitly and \fnd that \u001a(B)/p\nBin the\nasymptotically low \feld limit, consistent with Volovik's\nsemiclassical analysis [5].\nIn Section II we de\fne the model Hamiltonian that in-\ncludesd-wave superconducting order parameter as well as\na variety of density wave states. Our numerical method,\nthe recursive Green function method adapted for the\npresent problem is discussed in Sec. III. The results are\ndiscussed in Sec. IV and Sec. V contains discussion.\nThere are three appendices.arXiv:1509.00494v2  [cond-mat.supr-con]  27 Jan 20162\nII. THE MODEL HAMILTONIANS\nThe starting point is the Bogoliubov-de Gennes (BdG)\nHamiltonian\nH=\u0012H\u0000\u0016 \u0001ij\n\u0001y\nij\u0000H+\u0016\u0013\n(1)\nde\fned on a square lattice. Here \u0016is the chemical po-\ntential.His the Hamiltonian that describes the normal\nstate electrons; while the o\u000b-diagonal pairing term \u0001 ij\nde\fnes the superconducting order parameter. For sim-\nplicity, we ignore self consistency, as we believe that it\ncannot change the major striking conclusions.\nA. The diagonal component H\nBesides the hopping parameters, the normal state\nHamiltonian Hcontains a variety of mean \feld order pa-\nrameters de\fned below. Although many di\u000berent orders\nare suggested to explain the normal state of the high-\nTcsuperconductivity, for our purposes it is su\u000ecient to\nconsider three di\u000berent types: a period-2 d\u0000density wave\n(DDW) [6, 12, 13], a bi-directional charge density wave\n(CDW), and a period \u00008 DDW model [13]. We believe\nthat our major conclusions in this paper do not depend\non the nature of the density wave that is responsible for\nFermi surface reconstruction. The DDW is argued to be\nable to account for many features of quantum oscillations,\nas well as the pseudogap state [14, 15] in the cuprates.\nAmong the many di\u000berent versions of density waves of\nhigher angular momentum [16], the simplest period-2\nsinglet DDW, also the same as staggered \rux state in\nRef. [7], and also a period \u00008 DDW order, proposed by\nus previously to explain quantum oscillations[13], have\nbeen chosen here for illustration.\nRecently a bi-directional CDW has been observed\nubiquitously in the underdoped cuprates [17{21]. It\nhas ordering wavevectors Q1\u00192\u0019\na(0:31;0) and Q2\u0019\n2\u0019\na(0;0:31), which are incommensurate. This order has\nalso been used to explain the Fermi surface reconstruc-\ntions and quantum oscillation experiments [22, 23], al-\nthough a recent numerical work [24] has demonstrated\nthat the strict incommensurability of the CDW can de-\nstroy strict quantum oscillations completely. For the pur-\npose of illustration we chose, instead, commensurate vec-\ntorsQ1=2\u0019\na(1\n3;0) and Q2=2\u0019\na(0;1\n3).\nTherefore without the magnetic \feld, B,His given by\nH=\u0000tP\nhi;jicy\nricrj+t0P\nhhi;jiicy\nricrj\n\u0000t00P\nhhhi;jiiicy\nricrj+ h:c:+Hd:w:; (2)\nwheret,t0, andt00are the 1st, 2nd and the 3rd nearest\nneighbor hopping parameters respectively. Hd:w:is var-\nious density wave orders speci\fed below. The external\nuniform magnetic \feld B=B^zis included into Hvia thePeierls substitution: cy\nricrj)exp[\u0000ie\n~cRri\nrjA\u0001dl]cy\nricrj\nwith the vector potential A=Bx^ychosen, for simplic-\nity, in the Landau gauge.\n1.Two-fold DDW order\nHd:w:=X\nri;\u000eiW0\n4(\u00001)xi+yi\u0011\u000ecy\nri+\u000ecri; (3)\nwhere\u000e= ^x;^ydenote the two nearest neighbors.\n\u0011\u000e= 1 for\u000e= ^xwhile\u0011\u000e=\u00001 for\u000e= ^yindicates\nthe DDW order has a local d\u0000wave symmetry.\n2.Bi-directional CDW order\nHd:w:=VcX\nri;\u000e\u0011\u000efcos[Q1\u0001(ri+\u000e=2)]\n+ cos[ Q2\u0001(ri+\u000e=2)]gcy\nri+\u000ecri: (4)\nAgain\u0011\u000e=\u00061 is the local d\u0000wave symmetry factor\nof the CDW order.\n3.Period-8 DDW order\nHd:w:=X\nkiGkcy\nkck+Q+Vccy\nkck+2Q+ h:c:: (5)\nwhere theiGkterm is the period \u00008 DDW order\n<cy\nk0ck>=iGk\u000ek0;k+Q\u0000iGk0\u000ek;k0+Q (6)\nHereGk= (Wk\u0000Wk+Q)=2 with the DDW\ngapWk=W0\n2(coskx\u0000cosky), and the ordering\nwavevector is Q= (3\u0019\n4a;\u0019\na). TheVcterm in Eq. (5)\nrepresents a period \u00004 unidirectional CDW with an\nordering wavevector 2 Q, which is consistent with\nthe symmetry of the period \u00008 DDW order. Notice\nthat this CDW is di\u000berent from the bi-directional\nCDW we considered in the previous section. Exper-\nimentally whether the observed CDW in cuprates\nis unidirectional or bi-directional is still not fully\nresolved.\nFourier transformed to the real space, the Hamil-\ntonianHd:w:becomes\nHd:w:=X\nr;r0iW0\n2sinQ\u0001(r\u0000r0)\n2sinQ\u0001(r+r0)\n2\n\u0002f\u000er;r0+a^x+\u000er;r0\u0000a^x\u0000\u000er;r0+a^y\u0000\u000er;r0\u0000a^ygcy\nrcr0\n+2VcX\nrcos[2Q\u0001r]cy\nrcr: (7)\nIn the above, W0controls the overall magnitude of\nthe period-8 DDW order parameter, isinQ\u0001(r\u0000r0)\n2\nindicates that the order is a current, sinQ\u0001(r+r0)\n2\nshows that the magnitude of this order is modu-\nlated with a wavevector Q, and the last factor in\nthe curly bracket f:::gexplicitly exhibits its local\nd\u0000wave symmetry. If Q= (\u0019=a;\u0019=a ), then the3\norder reduces to the familiar two-fold DDW order.\nIn that casejsinQ\u0001(r+r0)\n2j= 1 and the order pa-\nrameter magnitude is a constant. The last term\nin Eq. (7) gives the 2 Qcharge modulation. Note\nthat this CDW is de\fned on sites, di\u000bering from\nthe bi-directional CDW de\fned on bonds.\nB. The o\u000b-diagonal component \u0001ij\nThe o\u000b-diagonal pairing term \u0001 ijin the BdG Hamil-\ntonian is de\fned on each bond connecting two nearest\nneighboring sites iandj. \u0001ij=j\u0001ijjei\u0012ij\u0011ij, where\n\u0011ij= +1 if the bond is along x\u0000direction and \u0011ij=\u00001\nif it is along y\u0000direction so that \u0001 ijhas a local d\u0000wave\nsymmetry. The pairing amplitude is taken to be\nj\u0001ijj= \u0001re\u000bp\nr2\ne\u000b+\u00182(8)\nwhere \u0001 is the pairing amplitude far away from any vor-\ntex center. \u0018is the vortex core size. In our calculation\n\u0018= 5ais adopted, where ais the lattice spacing. In the\npresence of a single vortex, re\u000bin the above is simply\nthe distance from the center of our bondri+rj\n2to the\ncenter of that vortex. While in the presence of multiple\nvortices, following the ansatz used in Ref. [7] we choose\n(\u0018\nreff)q=P\nn(\u0018\nrn)qwherernis the distance from the bond\ncenter to the nth vortex center and q >0 is some real\nnumber.\nIn this ansatz, re\u000bis a monotonic increasing function of\nthe parameter qfor a given vortex con\fguration. There-\nfore ifqis large, the calculated re\u000bas well asj\u0001ijjis\nalso larger, which means the vortex scattering is stronger.\nHowever our conclusions do not depend on the di\u000berent\nchoices ofq(for more details see the appendix section C).\nTherefore in this paper, if not speci\fed otherwise, q= 2\nwill be chosen.\nThe bond phase variable \u0012ijcontains the information\nof our quenched random vortex con\fguration, but for\nthe purpose of our calculation we need the site phase\nvariables. We use the ansatz for \u0012ijgiven in Refs. [25, 26]\nei\u0012ij=ei\u001ei+\u001ej\n2sgn[cos\u001ei\u0000\u001ej\n2] (9)\nwhere\u001eiis the pairing order parameter phase \feld de-\n\fned on a site. In the above, without the \\sgn[...]\" factor\n\u0012ijis simply the arithmetic mean of \u001eiand\u001ej. However\nusing\u0012ij=\u001ei+\u001ej\n2is not enough because whenever the\nbondijcrosses a vortex branch cut, the phase factor\nei\u0012ijwill be incorrect and di\u000berent from the correct one\nby a minus sign. This can be corrected by the additional\n\\sgn[...]\" factor(see the appendix section A).\nThen\u001eican be further computed from the super\ruid\nvelocity \feld vs(ri) by\n\u001ei\u0000\u001e0=Rri\nr0[m\u0003vs(r)\n~+e\u0003\n~cA(r)]\u0001dl (10)withm\u0003= 2mande\u0003=\u00002eare the mass and the charge\nof the Cooper pairs respectively. The path for this inte-\ngral is chosen such as to avoid the branch cuts of all the\nvortices so that the phase \feld \u001eiis single valued on every\nsite, as illustrated in Fig. 1.\nr0r\nFIG. 1. Illustrations of the vortices (circle) on the lattice\n(dashed lines). The arrows show the path of the integral we\nhave chosen in de\fning our phase \feld \u001e(r). To make this\nphase de\fnite, the branch cuts of all the vortices are chosen to\nextend from the vortex center to the positive in\fnity ( x=1),\nrepresented by the magenta horizontal lines.\nWe still need to compute the super\ruid velocity vs(r).\nThis can be done by following Ref. [27]\nmvs(r) =\u0000i\u0019~Zd2k\n(2\u0019)2k\u0002^z\nk2+\u0015\u00002X\nneik\u0001(r\u0000Rn);(11)\nwheremis the electron mass, \u0015is the penetration depth,\nandRngives thenth random vortex position. In this\nintegrand, because k\u0002^zis odd in k, only the imaginary\npart ofeik\u0001(r\u0000Rn)will survive after the integration, so the\nwhole expression on the right hand side becomes real. We\nalso make an approximation \u0015=1so that we can ignore\nthe\u0015\u00002term in the denominator. This is equivalent to\nreplacing the magnetic \feld B(r) by its spatial average,\nwhich is equal to the external magnetic \feld B=B^z.\nIt is a good approximation when B\u001dHc1, whereHc1\nis the lower critical \feld. This condition is well satis\fed\nin the quantum oscillation experiments of cuprates. Also\nthis approximation is consistent with our initial choice of\nthe vector potential A=Bx^y, given completely by the\napplied external \feld B.\nFor our square lattice calculation we discretize the\nabove kintegral and choose 2 \u0019=\u0018as its upper cuto\u000b,\nsince the vortex is only well de\fned over a length scale\nlarger than the vortex core size \u0018. Therefore in the limit\n\u0015\u001d\u0018>a ,vs(r) can be rewritten as follows\nmvs(r) =\u0019~\nLMa2X0\n(kx;ky)k\u0002^z\nk2X\nnsin[k\u0001(r\u0000Rn)]:\n(12)\nIn this summation kx=\u00002\u0019\n\u0018;\u00002\u0019\n\u0018+2\u0019\nLa;::::;2\u0019\n\u0018\u00002\u0019\nLa;2\u0019\n\u0018,\nky=\u00002\u0019\n\u0018;\u00002\u0019\n\u0018+2\u0019\nMa;::::;2\u0019\n\u0018\u00002\u0019\nMa;2\u0019\n\u0018. The prime super-\nscript in the summation means the point ( kx;ky) = (0;0)4\nis excluded to be consistent with our approximation\n\u0015=1.\nIII. THE RECURSIVE GREEN FUNCTION\nGiven the BdG Hamiltonian Hde\fned above, we use\nthe recursive Green's function method [28] to compute\nthe local DOS(LDOS). We attach our central system,\nwhich has a lattice size L\u0002M, to two semi-in\fnite\nleads in the\u0006xdirections. The leads are normal met-\nals described by t;t0;t00only. Then we can compute\nthe retarded Green's function Gi(j;j0;E+i\u000e) at an en-\nergyEfor theith principal layer(see the appendix sec-\ntion B). Here each ith principal layer contains two adja-\ncent columns of the original square lattice sites. So there\nareL=2 principal layers and each of them contains 2 M\nnumber of sites. Therefore Gi(j;j0;E+i\u000e) is a 4M\u00024M\nmatrix, with j;j0= 1;2;:::;4M, because it has both an\nelectron part and a hole part. In calculating the LDOS\nat thejth site of the ith layer only the imaginary part\nof thejth diagonal element in the electron part of Gi\nis included. This is equivalent to treating the random\nvortices as some o\u000b-diagonal scattering centers for the\nnormal state electrons. To see smooth oscillations of the\nDOS we also average the calculated LDOS over di\u000berent\nsites and realizations of uncorrelated vortices. In other\nwords the quantity of our central interest is\n\u001a(B) =*\n1\nLML=2X\ni=12MX\nj=1(\u00001\n\u0019)ImGi(j;j; 0 +i\u000e)+\n;(13)\nwhere the angular brackets denote average over indepen-\ndent vortex realizations. In the Green's function we have\nalready set the energy to the chemical potential E= 0.\nFor all the numerical results presented in the following,\nan in\fnitesimal energy broadening \u000e= 0:005twill be cho-\nsen, if not speci\fed otherwise, and the periodic boundary\ncondition is imposed in the y\u0000direction.\nIV. RESULTS\nA. The Onsager rule for quantum oscillation\nfrequencies\n1. The two-fold DDW order case\nWith the parameters: t= 1;t0= 0:30t;t00=\nt0=9:0;\u0016=\u00000:8807t;W 0= 0:26t;Vc= 0, the hole dop-\ning level isp\u001911%. Without vortices we can diagonalize\nthe Hamilontian Hin the momentum space and obtain\nthe normal state Fermi surface. This Fermi surface con-\nsists of two closed orbits, see the inset of Fig. 2a. The\nbigger one centered around the node point (\u0019\n2;\u0019\n2) is hole\nlike. It has an areaAh\n(2\u0019=a)2\u00193:47%. This corresponds\nto an oscillation frequency Fh=Ah\n(2\u0019=a)22\bs\na2= 966Tfrom the Onsager relation, where \b s=hc=2eis the\nfundamental \rux quanta and the two lattice spacings\na2= 3:82\u0017A\u00023:89\u0017Aare chosen for YBCO. At the antin-\nodal point (0 ;\u0019) there is an electron pocket with an area\nAe\n(2\u0019=a)2\u00191:9%, corresponding to a frequency Fe= 525T\n(electron). We should notice that the fast oscillation Fh\n(hole) is not observed in the experiments in cuprates.\nThis problem can be resolved if we consider a period \u00008\nDDW model [13]; see below.\nWe compute the \u001a(B) as a function of the inverse of\nthe magnetic \feld 1 =Bin the presence of various \u0001. In\nthese calculations, the number of vortices are chosen such\nthat the total magnetic \rux is equal to \b = BLMa2.\nFrom the oscillatory part of \u001a(B) we perform Fast Fourier\nTransform(FFT) to get the spectrum. The result is\nshown in Fig. 2d. In this spectrum the two oscillation\nfrequencyFe= 525T and Fh= 966T calculated from the\nnormal state Fermi surface areas via the Onsager relation\nare also shown by the two vertical dashed lines. We see\nclearly that as we increase \u0001 the oscillation amplitudes\nare damped. However, remarkably, the oscillation fre-\nquencies remain the same within numerical errors. Thus,\neven in the presence of vortices, the Onsager rule still\nholds to an excellent approximation.\n2. The bi-directional CDW order case\nWe choose the following parameters: t= 1;t0=\n0:2t;t00=t0=8;Vc= 0:12t;\u0016=\u00000:73tso that we can pro-\nduce the right oscillation frequencies that are observed\nin experiments. The hole doping level is p\u001911%. The\nFermi surface of the normal state is plotted in the inset\nof Fig. 2b (open orbits are not shown for clarity). There\nare two closed Fermi surface sheets. Centered around\nthe point (\u0019\n3;\u0019\n3) and other symmetry related positions\nthere are diamond shaped electron pockets, highlighted\nin orange. This pocket has an areaAe\n(2\u0019=a)2= 1:9%. It\ncorresponds to a frequency Fe= 529T from the Onsager\nrelation. Besides this electron pocket, there is an oval\nshaped hole pocket centered around (\u0019\n3;2\u0019\n3), highlighted\nin blue. The area of this hole pocket isAh\n(2\u0019=a)2= 0:33%.\nThis corresponds to an oscillation frequency Fh= 92T.\nThe oscillation spectrum of the \u001a(B) is shown in\nFig. 2e. From the spectrum we see that when the vortex\nscattering is absent, \u0001 = 0, the oscillation amplitudes\npeak at the two frequencies Fe;Fh, as denoted by the\ntwo vertical dashed lines. These results agree with our\nFermi surface calculation, as we expected. When the\nvortices are included the oscillation amplitude is gradu-\nally damped as the vortex scattering strength is increased\nby increasing \u0001. However whenever the oscillation fre-\nquency can be clearly resolved, we see that their positions\ndo not change with \u0001. Again this means that the On-\nsager rule survives in the presence of vortex scattering.5\n-3 -2 -1 0 1 2 3-3-2-10123\nkxakya\n(a) FS for the two-fold DDW\n0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.51.01.52.02.53.0\nkxakya (b) FS for the bi-directional CDW\n0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.51.01.52.02.53.0\nkxakya (c) FS for the period-8 DDW\n●●●●●●●●●●\n●\n●●●●●●●●●●●●●●●●●\n●\n●●●\n●●●●●■■■■■■■■■■■\n■■■■■■■■■■■■■■■■■\n■\n■■■■■■■■▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲\n▲\n▲▲▲▲▲▲▲▲●Δ=0\n■Δ=0.05 t\n▲Δ=0.10 t\n400 600 800 1000 120001234\nF(T)Amplitudes(arb. units)\n(d) FFT spectrum for the two-fold DDW\n●●●●●●●●\n●\n●\n●●●●●●\n●\n●●●●●●●\n●\n●●●●●●\n●\n●●●●●●●●●●●●\n●\n●●●●■■■■■■■■■\n■\n■■■■■■\n■\n■■■■■■■\n■■■■■■■■■■■■■■■■■■■■\n■\n■■■■▲▲▲▲▲▲▲▲▲\n▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲●Δ=0\n■Δ=0.05 t\n▲Δ=0.10 t\n0 100 200 300 400 500 600012345\nF(T)Amplitudes(arb. units) (e) FFT spectrum for the bi-directional\nCDW\n●●●●●●\n●\n●●●●●●\n●\n●●●●●●●●\n●\n●●●●\n●\n●■■■■■■\n■\n■■■■■■\n■\n■■■■■■■■\n■\n■■■■\n■■▲▲▲▲▲▲\n▲\n▲▲▲▲▲▲\n▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲●Δ=0\n■Δ=0.05 t\n▲Δ=0.10 t\n100 200 300 400 500 600 70005101520\nF(T)Amplitudes(arb. units)(f) FFT spectrum for the period-8 DDW\nFIG. 2. The upper panel shows the plots of Fermi surfaces(FS), with electron pockets shaded in orange and hole pockets in blue.\nIn Fig. 2a, the area enclosed by the dashed lines gives the reduced Brillouin zone. In Fig. 2b, the open orbits are not shown for\nclarity. And the dashed lines denote the positions kxa;kya=\u0019=3;2\u0019=3. In the lower panel we show the corresponding oscillation\nFFT spectrums for various values of \u0001. The two vertical dashed lines in each plot denote the two fundamental oscillation\nfrequencies calculated from the Fermi surface area via the Onsager relation. They are Fe= 525T;Fh= 966T in Fig. 2d,\nFe= 529T;Fh= 92T in Fig. 2e, and Fe= 523T;Fh= 159T in Fig. 2f. Other parameters used are L= 1000;M= 100;\u000e= 0:005t\nin Fig. 2d,L= 1000;M= 102;\u000e= 0:005tin Fig. 2e, and L= 1000;M= 200;\u000e= 0:002tin Fig. 2f.\n3. The period\u00008DDW order case\nIn this subsection we present our quantum oscillation\nresults for the period \u00008 DDW model. In this model the\nperiod\u00008 stripe DDW order is considered as the major\ndriving force behind the Fermi surface reconstructions;\nwhile a much weaker unidirectional period \u00004 CDW is\nincluded as a subsidiary order.\nWe choose the parameter set t0= 0:3t;t00=\nt0=2:0;W0= 0:70t;Vc= 0:05t;\u0016=\u00000:70tand estimate\nthe hole doping level to be p\u001911:2%. We also obtain\na Fermi surface similar to the one we had in Ref.[13].\nIt has a large pocket of electron like with a frequency\nFe= 523T, a smaller pocket of hole like with a frequency\nFh= 159T, and also some open orbits which do not con-\ntribute to quantum oscillations.\nThe corresponding oscillation spectrum is presented in\nFig. 2f, where we see the oscillation amplitude decreases\nas we increase \u0001, however, the frequencies do not change\nwith \u0001. In other words the presence of vortex scatteringdoes not alter the oscillation frequencies.\nThe observations here, combined with the other two\ncases, strongly suggest that the Onsager's relation being\nintact in the presence of vortex scattering is generic and\nindependent of the order parameters that reconstruct the\nFermi surface.\nB. The Density of states at high \felds\nIn the above we have examined the e\u000bects of random\nvortex scattering on the quantum oscillations. Now we\ngive an overview of the Bdependence of the DOS for\n\feldsB&10T, at a representative value of \u0001 = 0 :1t.\nAt lower \felds, the vortex liquid model is not valid any\nmore, since vortices should order into a solid instead.\nTherefore we should use a vortex lattice to model such a\nstate. In the following we focus on the high \feld regime\n\frst and defer our vortex lattice discussions for the low\n\feld regime to the later section IV C.6\n1. The period-2 DDW order case\nIn Fig. 3a we plot \u001a(B)=\u001an(0) as a function of the \feld\nBfor the two-fold DDW order case, where \u001an(0) is the\nnormal state DOS at zero \feld. In the following, the nor-\nmal state should be understood as a state, which does not\nhave any superconductivity but can have a particle-hole\ndensity wave order. And all the DOS value calculated is\nfor one electron in a single CuO plane, without includ-\ning the spin degeneracy. From Fig. 3a we see that as\nBdecreases, the DOS oscillation gets suppressed gradu-\nally. This is because the orbital quantization of electrons\nbecomes dominated by the vortex scattering.\nA noticeable feature of this plot is that when the \feld\nbecomes large, the oscillation of \u001a(B) in 1=Bgradually\ndevelops on top of a constant background. This constant\nbackground value of \u001a(B) is suppressed from the normal\nstate DOS \u001an(0). The size of this suppression depends\non the vortex scattering strength. For the parameters\nused in Fig. 3a it is \u001815%. This constant background\nof\u001a(B) is di\u000berent from the previous results obtained in\nFig. 3(b) of the Ref. [11] in the absence of vortices.\n2. The bi-directional CDW order case\nThe constant background of the DOS oscillation is not\nrestricted to the two-fold DDW order case. As we can\nsee in Fig. 3b, for the bi-directional CDW order, the \u001a(B)\noscillation background is again a constant at high \felds.\n3. The period\u00008DDW order case\nWe also con\frm this constant \u001a(B) background feature\nin the oscillation regime for the period \u00008 DDW order\ncase in Fig. 3c.\nTherefore we can conclude that the high \feld \u001a(B)\noscillation background being a constant is generic.\nC. Vortex solid at low \felds\nNow we move on to the low \feld regime. In this regime\nwhen the \feld is low enough, the vortices order into a lat-\ntice. Whether the lattice is square or triangular requires\na self-consistent computation of the system's free energy,\nwhich is far beyond the scope of this paper. Instead we\nsimply take a square lattice for illustration. But none of\nthe following qualitative features should depend on the\nvortex lattice type.\n1. Implementation of the square vortex lattice\nTo put the square vortex lattice onto our original CuO\nlattice so that each vortex sits at the CuO lattice pla-\nquette center and the periodic boundary condition is stillpreserved along the transverse direction, we require the\nvortex lattice to be commensurate with our original CuO\nlattice, as schematically shown in Fig. 4. Namely, if the\nvortex lattice spacings are ( lx;ly), and the corresponding\nvortex lattice size is ( Nx;Ny), we require that the original\nCuO lattice size ( L;M ) satis\fesL=Nxlx;M=Nyly.\nFor a particular value of ( L;M ), this restricts the pos-\nsible values of ( lx;ly) and also the possible values of\nthe magnetic \feld, because the vortex lattice spacings\n(lx;ly) are connected to the magnetic \rux density via\nB= \bs=(lxlya2), where \b s=hc=2eis the fundamental\n\rux quanta. In our following calculation we pick a partic-\nular value of the system size ( L;M ), \fnd all the possible\ncompatible values of the vortex lattice spacings lx=ly,\nand then for each of them calculate the magnetic \feld B\nas well as the corresponding DOS.\nHowever, we should calculate the DOS of the Bogoli-\nubov quasiparticles instead of the electrons, because the\nsystem is far from being in a normal state in such a low\n\feld regime. Therefore now \u001a(B) is computed from the\nfollowing formula instead\n\u001a(B) =1\n21\nLML=2X\ni=14MX\nj=1(\u00001\n\u0019)ImGi(j;j; 0 +i\u000e):(14)\nThe major di\u000berences here from the one we used in our\nquantum oscillation calculations are: (1) the summation\nof the Green's function's diagonal matrix elements in-\ncludes both the electron part and the hole part: jruns\nfromj= 1 toj= 4Minstead ofj= 2M; (2) there is no\naveraging over di\u000berent vortices con\fgurations because\nthe vortex lattice is ordered; (3) an additional prefactor\nof 1=2 is added to avoid double counting of degrees of\nfreedoms.\nFor such a vortex lattice calculation, the summation\nover di\u000berent vortex positions in the super\ruid velocity\ncalculation in Eq. (12) can be done exactly by using\nX\nneik\u0001(r\u0000Rn)=NxNyX\n(n1;n2)eiGn1;n2\u0001r; (15)\nwhere Gn1;n2= (2n1\u0019\nlxa;2n2\u0019\nlya) is a reciprocal Bragg vector\nof the square vortex lattice, with n1;n22Z. Then the\nEq. (12) of vsbecomes\nmvs=\u0019~1\nlxlya2X\n(n1;n2)0Gn1;n2\u0002^z\njGn1;n2j2sin[Gn1;n2\u0001r] (16)\nThe summations of ( n1;n2) are restricted to those values\nthat satisfy 0\u00142n1\u0019\nlxa<2\u0019\na;0\u00142n2\u0019\nlya<2\u0019\na. Again\nthe prime superscript in the summation means the point\n(n1;n2) = (0;0) is excluded.\n2. DOS numerical results\nAccording to Volovik [5], for a dx2\u0000y2\u0000wave vortex, the\nmajor contribution to the low energy DOS comes from7\n●●●●●●●●●●●●●●●●●●●●●●●●●●●\n●●●●●\n●●●●●\n●●●\n●●●\n●\n●●●\n●\n●●●\n●●●●\n●\n●●●●\n●\n●●●●\n●\n●●●●●\n●\n●●●●●\n●\n●\n●●●●●●Δ=0.1 t\n20 30 40 50 600.70.80.91.01.1\nB(T)ρ(B)/ρ n(0)\n(a) The two-fold DDW order\n●●●●●●●●●●●●\n●●●●●\n●\n●●●\n●●\n●●●●\n●\n●●\n●●●\n●●●●\n●●●●\n●●●●●\n●\n●\n●●●●Δ=0.1 t\n10 20 30 40 50 600.50.60.70.80.91.01.11.2\nB(T)ρ(B)/ρ n(0) (b) The bi-directional CDW order\n●●●●●●●●●\n●●\n●●●\n●●●●\n●\n●●●●\n●\n●\n●\n●●●●●\n●\n●\n●\n●●●●●●●●●●\n●\n●\n●\n●\n●●\n●●●Δ=0.1 t\n10 20 30 40 50 600.40.50.60.70.80.91.01.11.2\nB(T)ρ(B)/ρ n(0) (c) The period-8 DDW order\nFIG. 3. The DOS \u001a(B),normalized to the normal state DOS \u001an(0) at zero \feld B= 0 for di\u000berent cases. The estimated values\nof the normal state DOS are \u001an(0)\u00190:23 states=t for the two-fold DDW, \u001an(0)\u00190:25 states=tfor the bi-directional CDW,\nand\u001an(0)\u00190:18 states=tfor the period-8 DDW case. The data is averaged over 108, 120,40 di\u000berent vortices con\fguration\nrealizations respectively.\nlxly\nFIG. 4. Schematic diagram of a square vortex lattice. The\ndashed lines represent the original CuO lattice; while the full\nlines stand for the vortex lattice, with each vortex, repre-\nsented by the grey disks, sitting at the CuO plaquette center.\nAnd (lx;ly) are the vortex lattice spacings, in units of the\noriginal CuO square lattice spacing a.\nthe extended states along the nodal direction. In his\nsemiclassical analysis this contribution is computed from\nthe Doppler shift of the quasiparticle energy. The conclu-\nsion is that the DOS for a single vortex is \u001a(B)/1=p\nB.\nIn the limit that the number of vortices is proportional\ntoB, which is not valid if Bis near the lower critical\n\feldHc1, multiplying it by the number of vortices gives\n\u001a(B)/p\nB. Extrapolating this result to the high \feld\nregime and using the fact that near the upper critical\n\feldHc2,\u001a(B) should roughly recover the normal state\nDOS\u001an(0), he concluded that \u001a(B)=\u001an(0) =\u0014p\nB=Hc2,\nwith\u0014some constant of order unity. This type of analy-\nsis is applicable only in the small \feld limit in the sense\nthatB\u001cHc2so that each vortex is far apart from any\nothers. This is exactly the \feld regime where the vortex\nsolid state develops. In the following we compute the\nDOS for ad\u0000wave vortex lattice, for the cases both with\nand without an additional particle-hole density wave or-\nder, and test them against Volovik's results. For our\nfollowing comparisons we slightly rewrite the above \felddependence of \u001a(B) as follows\n\u001a(B)\n\u001an(0)=\u0014p\nBpHc2=\u0014s\n2\u0019\u00182\n\bsp\nB\u00190:1\u0014p\nB; (17)\nwhere\bs\n2\u0019\u00182\u001990T, if\u0018= 5aanda\u00193:83\u0017Aare used.\n1. First we consider a square vortex lattice without\nany other additional density wave order. We choose\nthe band structure parameters to be t0=t= 0:3;t00=\nt0=9:0;\u0016=\u00001:01tso that the estimated normal\nstate hole doping level p\u001915% is at the optimal\ndoping. The computed DOS is shown in Fig. 5a.\nAt low enough \felds, all the data points follow\nthe\u001a(B)=\u001an(0) = 0:3p\nBline, although there is\nsome small scatter in the data, which comes from\nthe \fnite size e\u000bects of our vortex lattice. This\n0:3p\nBcorresponds to \u0014\u00193 in Eq. (17). To make\na comparison with the speci\fc heat measurements\non YBCO123 at the optimal doping [29], we es-\ntimate the \feld dependent electronic speci\fc heat\n\r(B) from our DOS \u001a(B) as follows\n\r(B)\n\rn=\u001a(B)\n\u001an(0)\u00190:1\u0014p\nB: (18)\nThe normal state speci\fc heat can be estimated as\n\rn= 4\u00192\n3k2\nB\u001an(0). Here the additional prefactor of\n4 comes from the spin degeneracy and the fact that\none unit cell of YBCO123 contains two CuO planes.\nIf we taket= 0:15eV, then \rn\u001915:7mJ=mol\u0001K2\nand\r(B) =Ap\nBwith the coe\u000ecient A\u0019\n4:7 mJ=mol\u0001K2\u0001T1=2. Compared with the exper-\nimental value of A\u00190:9 mJ=mol\u0001K2\u0001T1=2from\nRef. [29], our numerical value is greater by a factor\nof about 5. This quantitative discrepancy is not\nsigni\fcant given our approximations. In fact, it is\nquite reasonably consistent.8\n●●●●●●●\n●L=960,M=240\n0.3 B\n0 5 10 150.00.20.40.60.81.0\nB(T)ρ(B)/ρ n(0)\n(a) The pure d-wave vortex lattice case without any\nother density wave order at the optimal hole doping\n●●●●●●\n●L=1680,M=240\n0.4 B\n0 2 4 6 8 100.00.20.40.60.81.0\nB(T)ρ(B)/ρ n(0)(b) The coexistence of a vortex lattice and a two fold\nDDW order in the underdoped regime\nFIG. 5. The DOS of vortex solids for \u0001 = 0 :1t. In Fig. 5a, \u001an(0)\u00190:25 states=tand in Fig. 5b \u001an(0)\u00190:23 states=t.\n2. Next we consider the coexistence of a square vor-\ntex lattice and an additional two-fold DDW order\nin the underdoped regime. The parameters are\nthe same as those in our high \feld quantum os-\ncillation calculations: t0=t= 0:3;t00=t0=9:0;\u0016=\n\u00000:8807t;W 0= 0:26t, so the estimated normal\nstate, with the DDW order but no superconductiv-\nity, hole doping level is p\u001911%. Fig. 5b shows the\ncorresponding DOS results. The small \feld data\nfollows\u001a(B)=\u001an(0) = 0:4p\nB, corresponding to a\nvalue of\u0014\u00194 in Eq. (17).\nThe above two values of \u0014are consistent with the fact\nthat in Volovik's formula \u0014is of order unity. Of course its\nprecise value depends on the vortex lattice structure, on\nthe slope of the gap near the gap node (in the current case\nboth the parameters \u0001 and q), and also on the normal\nstate band structure.\nFrom the above two scenarios we can conclude that\nirrespective of the existence of an additional density wave\norder, the DOS of a clean vortex lattice always scales as\n\u001a(B)/p\nBin the low \feld limit.\nV. CONCLUSION\nIn summary we have shown that in the quenched vor-\ntex liquid state the quantum oscillations in cuprates can\nsurvive at large magnetic \felds. Although the oscillation\namplitude can be heavily damped if the vortex scatter-\ning is strong, the oscillation frequency is given by the\nOnsager rule to an excellent approximation. Of course,\nwhen the \feld is small the quantum oscillations are de-\nstroyed by the vortices and \u001a(B) gets heavily suppressed\ndue to the formation of Bogoliubov quasiparticles. When\nthe \feld is small enough, a vortex solid state forms in-\nstead and it can be modeled by an ordered vortex lattice.\nWe show the \feld dependence of the vortex lattice's den-sity of states follows \u001a(B)/p\nBin the asymptotically\nlow \feld limit, in agreement with Volovik's semiclassical\npredictions. However in contrast to the previous sugges-\ntion our results show that this small \feld limit does not\nextend to the high \feld oscillatory regime of the vortex\nliquid state. Instead when the oscillations can be re-\nsolved, the non-oscillatory background of \u001a(B) \rattens\nout, and becomes \feld independent consistent with the\nmore recent speci\fc heat measurements [8].\nACKNOWLEDGMENTS\nThis research was supported by funds from David S.\nSaxon Presidential Term Chair at UCLA. We thank P.\nA. Lee for suggesting that we should explicitly check the\nresults in Ref. [7]. These are shown in the Appendix D.\nWe used the Ho\u000bman2 Shared Cluster provided by the\nUCLA Institute for Digital Research and Education pro-\ngram. We also thank Brad Ramshaw for discussion.\nAppendix A: Bond phase \feld \u0012ijof\u0001ij\nThe phase \feld \u0012ijis de\fned on the bond ij, which\nconnects two nearest neighboring sites iandj. Therefore\nit is natural to use the phase \felds \u001eiand\u001ej, on the site\niand sitejrespectively, to de\fne \u0012ij=\u001ei+\u001ej\n2. However\nthis de\fnition does not guarantee that whenever a closed\npath encloses a vortex, \u0012ijalong that path will pick up\na 2\u0019phase as the vortex is winded once. Therefore this\n\u0012ijcan not give the correct vortices con\fguration. It is\nincorrect whenever a vortex branch cut is crossed. To\nsee this clearly, we map the phase \feld \u001eialong a closed\npath that encloses a vortex onto a unit circle since \u001eiis\nde\fned only modulo 2 \u0019, as schematically shown in Fig. 6.\nIn this \fgure, the blue arc segment corresponds to the9\nbondijon the closed path. Therefore an appropriate \u0012ij\nshould be equal to some value of the phase \feld on this\nsegment. When the bond ijdoes not cross any branch\ncut,\u0012ij=\u001ei+\u001ej\n2is indeed on the blue segment and can be\na good de\fnition of \u0012ij, as illustrated in Fig. 6a ; however,\nif the bond ijcrosses a branch cut, we see that\u001ei+\u001ej\n2,\nindicated by the red arrow in Fig. 6b, is not on the blue\nsegment and can not be an appropriate de\fnition of \u0012ij.\nIn this latter case, \u0012ij=\u001ei+\u001ej\n2\u0000\u0019instead can be a\ngood de\fnition, since it falls onto the blue arc segment,\nas indicated by the blue arrow in Fig. 6b.\nϕi\nϕjΘij=ϕi+ϕj\n2\n(a) Bondijdoes not cross the\nbranch cut:j\u001ei\u0000\u001ejj<\u0019\nϕi\nϕjΘij ϕi+ϕj\n2(b) Bondijcrosses the branch\ncut:j\u001ei\u0000\u001ejj>\u0019\nFIG. 6. The black dot in the center represents a vortex. The\ndashed line, extended to the in\fnity, is its branch cut. The\nblue arc segment corresponds to the bond ijon a closed path.\nThe phases \u001ei;\u001ejare measured counter-clock wisely from the\nupper side of the branch cut. In Fig. 6a, the bond ijdoes\nnot cross the branch cut, and ( \u001ei+\u001ej)=2 is a good de\fnition\nfor\u0012ij; however, if the bond ijcrosses a branch cut, as in\nFig. 6b, then ( \u001ei+\u001ej)=2 can not be a correct de\fnition of\n\u0012ij. Instead (\u001ei+\u001ej)=2\u0000\u0019gives an appropriate de\fnition of\n\u0012ij.\nBased on these two scenarios, a good de\fnition of \u0012ij\nwill be\nei\u0012ij=ei\u001ei+\u001ej\n2sgn[cos\u001ei\u0000\u001ej\n2] (A1)\nThis de\fnition of \u0012ijguarantees that whenever the phase\n\feld\u001eialong a closed path crosses a branch cut once,\nthe de\fned \u0012ijcrosses the same branch cut once as well.\nWhen there are multiple vortices enclosed, we only need\nto linearly superpose the contributions from each vortex\ntogether to the \feld \u001eiand\u0012ijrespectively. It is not\ndi\u000ecult to see that the above de\fnition of \u0012ijis still good\nin these cases. For our numerical calculation convenience,\nwe rewrite the above de\fnition of \u0012ijin a slightly di\u000berent\nway\nei\u0012ij=ei\u001ei+ei\u001ej\njei\u001ei+ei\u001ejj(A2)\nAppendix B: Recursive Green's function method\nThe recursive Green's function method studies a quasi-\none-dimensional system, which is a square lattice with a\nvery long axis of length Laalong thex\u0000direction and a\nshorter axis of width Maalong they\u0000direction in our\niℋi,i-1 Gi-1Lℋi-1,i ℋi,i+1 Gi+1Rℋi+1,iFIG. 7. Schematic diagram of the recursive Green's function\ncalculation. The sites enclosed by the blue dashed rectan-\ngle de\fne the ith principal layer. The exact Green's func-\ntionGihas two self-energy contributions from both the left\nsemi-in\fnite stripe and and the right one. The left stripe is\ncharacterized by its surface Green's function GL\ni\u00001with the\n(i\u00001)th layer its surface; while the right one is characterized\nby another surface Green's function GR\ni+1with the (i+ 1)th\nlayer its surface.\nproblem. The system can be built up recursively in the\nx\u0000direction by connecting many one-dimensional stripes\ntogether. Each stripe has a direct coupling only to its\nnearest neighboring ones. This property is essential for\nthe recursion. In our Hamiltonian Hthe third nearest\nneighbor hopping t00provides the farthest direct coupling\nalong thex\u0000direction. It connects two sites 2 aapart.\nTherefore each stripe necessarily contains two columns\nof the square lattice sites so that the direct coupling ex-\nists only between two adjacent stripes. The blue dashed\nrectangle in Fig. 7 shows one such stripe. We de\fne each\nof such stripes as a principal layer, so each layer contains\n2Msites.\nOur goal is to compute the diagonal matrix elements\nof the exact Green's function Gin order to get the DOS.\nFor this purpose we \frst calculate Gi\u0011< ijGji >for\neach layeri. The ketji >represents a state where the\nBogoliubov quasiparticles are found in the ith principal\nlayer. It has 4 Mcomponents, of which the \frst 2 Mones\ngive the electron part wavefunction, while the rest 2 M\nones de\fne the hole part. Therefore Gi=Gi(j;j0) is a\n4M\u00024Mmatrix with j;j0= 1;2;:::;4M. For brevity\nwe will suppress the matrix element indices hereafter, if\nthere is no confusion.\nThe exact Green's function Gican be computed by\n(for derivations see Ref.[28])\nGi= [G0\ni\u00001\u0000Hi;i\u00001GL\ni\u00001Hi\u00001;i\u0000Hi;i+1GR\ni+1Hi+1;i]\u00001;\n(B1)\nas schematically shown in Fig. 7. Here G0\ni\u0011[E\u0000<\nijHji >]\u00001is the bare Green's function of the isolated\nith principal layer, with the superscript 0 indicating it\nis de\fned as if all other layers are deleted. The matrix\nHi;i\u00001\u0011<ijHji\u00001>contains all the Hamiltonian ma-\ntrix elements connecting sites in the layer i\u00001 to the layer\ni. Similarly GL\ni\u00001\u0011<i\u00001jGLji\u00001>is a matrix de\fned\non the (i\u00001)th principal layer, where GLis the exact\nGreen's function of a subsystem of our original lattice10\nwith all layers to the right of the ( i\u00001)th layer deleted,\nas shown in Fig. 8. The superscript \\ L\" here means\nthat this subsystem, including a left lead, is extended to\nthex=\u00001. Since the ( i\u00001)th layer is the surface of\nthis subsystem, we will call GL\ni\u00001the left surface Green's\nfunction. Similarly GR\ni+1\u0011<i+ 1jGRji+ 1>is another\nsurface Green's function of a subsystem of our original\nlattice with all the layers to the left of the ( i+ 1)th layer\ndeleted. Once GL\ni\u00001;GR\ni+1are known, Gican be com-\nputed immediately from Eq. B1.\nThe central task is then to compute GL\ni\u00001andGR\ni+1.\nThis can be done recursively. Take GL\ni\u00001as an exam-\nple. We start with the leftmost layer i= 1. There our\ncentral system is connected to a semi-in\fnite lead, which\ncontains in\fnite number of layers of the same width M,\nnumbered by i=:::;\u00002;\u00001;0. We denote this left lead's\nsurface Green's function as GL\ns, whose computation will\nbe presented in the following appendix subsection B 1.\nThen we add the i= 1st layer of our central system,\nbut not other layers, to this lead so that we get a new\nsemi-in\fnite stripe. This new stripe has a new surface\nGreen's function denoted as GL\n1, which can be computed\nfromGL\nsby\nGL\n1= [G0\n1\u00001\u0000H 1;0GL\nsH0;1]\u00001(B2)\nwhereH1;0connects sites in the surface layer i= 0 of the\nleft lead to the i= 1st layer of our central system. Simi-\nlarly we can repeat this process by adding one more layer\nof our central system to the semi-in\fnite stripe each time,\nand build up the whole system. In general at an inter-\nmediate stage, we may have a semi-in\fnite stripe, whose\nsurface is, say, the ( i\u00002)th layer with a surface Green's\nfunctionGL\ni\u00002. Then the ( i\u00001)th layer is connected to\nthat stripe to form a new semi-in\fnite system, which has\na new surface Green's function GL\ni\u00001. AndGL\ni\u00001can be\ncalculated from GL\ni\u00002by the following recursive relation\nGL\ni\u00001= [G0\ni\u00001\u00001\u0000Hi\u00001;i\u00002GL\ni\u00002Hi\u00002;i\u00001]\u00001(B3)\nThis is schematically illustrated in Fig. 8.\ni-1ℋi-1,i-2 Gi-2Lℋi-2,i-1\nFIG. 8. Schematic diagram for the GL\ni\u00001computation. The\nsites enclosed by the blue dashed rectangle belong to the ( i\u0000\n1)th principal layer, which is also the surface layer of this\nsemi-in\fnite stripe.Similarly the right surface Green's function GR\ni+1can\nbe computed from GR\ni+2via\nGR\ni+1= [G0\ni+1\u00001\u0000Hi+1;i+2GR\ni+2Hi+2;i+1]\u00001(B4)\nThis recursive relation starts with GR\nL=2at the rightmost\nlayeri=L=2 of our central system, where it is connected\nto another semi-in\fnite stripe lead extended to x=1.\nNote that the central system has only L=2 principal layers\nbecause each layer contains two columns of the sites, and\nthere are only Lcolumns in total. The layers in this right\nlead are numbered by i=L=2+1;L=2+2;::::We denote\nthe right lead's surface Green's function as GR\ns. Then\nGR\nL=2can be computed from GR\nsby\nGR\nL=2= [G0\nL=2\u00001\u0000HL=2;L=2+1GR\nsHL=2+1;L=2]\u00001(B5)\nwhereHL=2;L=2+1connects our central system to the\nright lead and contains t;t0;t00only.\n1. Surface Green's function GL\ns;GR\nsof the leads\nGL\ns;GR\nscan be computed by solving a self-consistent\n2\u00022 matrix equation. We now give a detail discussion on\nhow to compute GL\ns, but only brie\ry mention the \fnal\nresults forGR\nsat the end.\nThe left lead Hamiltonian contains only the hopping\nparameters t;t0;t00\nHlead=0X\ni=\u00001MX\nj=1f\u0000t[cy\ni\u00001;jci;j+cy\ni;j+1ci;j]\n+t0[cy\ni\u00001;j+1ci;j+cy\ni\u00001;j\u00001ci;j]\n\u0000t00[cy\ni\u00002;jci;j+cy\ni;j+2ci;j] + h:c:\u0000\u0016cy\ni;jci;jg\n(B6)\nTo be compatible with our central system Hamiltonian,\nwhich contains superconductivity, our lead Hamiltonian\nshould have both an electron part and a hole part so that\nthe full Hamiltonian Hleadis\nHlead=\u0012\nHlead 0\n0\u0000Hlead\u0013\n: (B7)\nCorrespondingly the surface Green's function takes a\nblock diagonal form\nGs(E+)\u0012\n[E+\u0000Hlead]\u000010\n0 [E++Hlead]\u00001\u0013\n(B8)\nwhere for brevity we have introduced E+=E+i\u000e. We\nwill denote the two diagonal terms as Gee= [E+\u0000\nHlead]\u00001andGhh= [E++Hlead]\u00001. Apparently\nGhhcan be obtained from Geeby simple substitutions:\nft;t0;t00;\u0016g ) f\u0000t;\u0000t0;\u0000t00;\u0000\u0016g. Therefore we only\nneed to discuss how to compute Gee.11\nBecause of the periodic boundary condition along the\ny\u0000direction, we can decompose Gee(E+) into di\u000berent\nmomentum kychannels\nGee(E+) =X\nkyj\u001fky><\u001fkyjg(ky;E+) (B9)\nwithj\u001fky>=PM\nj=1eikyja\np\nMjj >; andky=2n\u0019\nMawithn=\n1;2;3;:::;M . Each channel is described by a semi-in\fnite\none dimensional chain e\u000bective Hamiltonian He\u000b(ky)\u0011<\u001fkyjHleadj\u001fky>, given by\nHe\u000b(ky) =0X\ni=\u00001f(\u00002tcosky\u00002t00cos 2ky\u0000\u0016)cy\nici\n+ [(\u0000t+ 2t0cosky)cy\nici\u00001\u0000t00cy\nici\u00002+ h:c:]g:(B10)\nAndg(ky;E+) is the corresponding surface Green's func-\ntion of this one dimensional chain.\nTo compute g(ky;E+) we group every two adjacent\ncites (cy\ni\u00001;cy\ni) of the one dimensional chain together into\na cell, indexed by the cell number n, so thatHe\u000b(ky) can\nbe rewritten in a form such that direct couplings exist\nonly between two nearest neighboring cells\nHe\u000b(ky) =0X\nn=\u00001(cy\n2n\u00001; cy\n2n)\u0014\n\u0000t00\u0000t+ 2t0cosky\n0\u0000t00\u0015\u0012\nc2n\u00003\nc2n\u00002\u0013\n+ h:c:\n+0X\nn=\u00001(cy\n2n\u00001; cy\n2n)\u0014\n\u00002tcosky\u00002t00cos 2ky\u0000\u0016\u0000t+ 2t0cosky\n\u0000t+ 2t0cosky\u00002tcosky\u00002t00cos 2ky\u0000\u0016\u0015\u0012\nc2n\u00001\nc2n\u0013\n(B11)\nSinceg(ky;E+) is a surface Green's function, it should\nsatisfy the same recursive relation given in Eq. (B3),\nwhich is rewritten here as\ng= [G0\n0\u00001\u0000[He\u000b]0;\u00001GL\n\u00001[He\u000b]\u00001;0]\u00001(B12)\nThe only di\u000berence from there is now all the matrix ele-ments are de\fned between di\u000berent cells instead of layers.\nFor clarity we have suppressed the kyandE+depen-\ndence of all the quantities in this equation. G0\n0is the\nbare Green's function of the isolated single cell n= 0.\nBecause each cell contains two sites, G0\n0is a 2\u00022matrix,\ngiven by\nG0\n0\u00001\u0011E+\u0000[He\u000b]0;0=E+\u0000\u0014\n\u00002tcosky\u00002t00cos 2ky\u0000\u0016\u0000t+ 2t0cosky\n\u0000t+ 2t0cosky\u00002tcosky\u00002t00cos 2ky\u0000\u0016\u0015\n: (B13)\nSimilarly the e\u000bective hopping matrices between the\ncelln= 0 and cell n=\u00001 can be read o\u000b directly from\nEq. (B11)\n[He\u000b]0;\u00001=\u0014\n\u0000t00\u0000t+ 2t0cosky\n0\u0000t00\u0015\n; (B14)\n[He\u000b]\u00001;0= [He\u000b]y\n0;\u00001 (B15)By de\fnition GL\n\u00001in Eq. (B12) is the surface Green's\nfunction of the same chain but with the cell n= 0 deleted.\nHowever, since the chain is semi-in\fnite, deleting the sur-\nface cell only gives another identical semi-in\fnite chain.\nThereforeGL\n\u00001should be the same as g. Then Eq. (B12)\nbecomes a self-consistent equation of gas\ng\u00001=\u0014\nE++ 2tcosky+ 2t00cos 2ky+\u0016 t\u00002t0cosky\nt\u00002t0cosky E++ 2tcosky+ 2t00cos 2ky+\u0016\u0015\n\u0000\u0014\n\u0000t00\u0000t+ 2t0cosky\n0\u0000t00\u0015\ng\u0014\n\u0000t000\n\u0000t+ 2t0cosky\u0000t00\u0015\n: (B16)\nWith this 2\u00022 matrix equation, for each ky, we solve for g numerically by iterations until the results converge. Then12\nthe computed g(ky;E) is substituted back into Eq. (B9)\nofGee(E+) to getGL\ns.\nSimilar derivations can be carried out for the right lead\nGreen's function GR\ns. It turns out GR\ns= (GL\ns)T, where T\nis the transpose operation. This result is a manifestation\nof the fact that the two semi-in\fnite leads can be con-\nnected to each other by a re\rection symmetry operation\nalong thex\u0000direction.\nAppendix C: The ansatz (\u0018\nreff)q=P\nn(\u0018\nrn)q\nThe pairing amplitude on the bond, that connects two\nnearest neighboring sites riandrj, is calculated by the\nfollowing ansatz:\nj\u0001ijj= \u0001re\u000bp\n\u00182+r2\ne\u000b(C1)\nwithre\u000bgiven by\n(\u0018\nre\u000b)q=NvX\nn=1(\u0018\nrn)q(C2)\nwherern=jri+rj\n2\u0000Rnjis the distance from the bond\ncenterri+rj\n2to thenth vortex center Rn,qis some pos-\nitive number, and Nvis the total number of vortices.\nIf we consider a special case that there is only one vor-\ntex, for instance the nth vortex, then Eq. (C2) is reduced\ntore\u000b=rn, and\nj\u0001ijj= \u0001rnp\nr2n+\u00182(C3)\nIn other words, we can de\fne the pairing amplitude j\u0001nj\nfor the case when only the nth vortex is present as follows\nj\u0001nj\u0011\u0001rnp\nr2n+\u00182(C4)\nso thatj\u0001ijj=j\u0001nj.\nWhen more than the nth vortex is present, j\u0001ijjshould\nbecome smaller than j\u0001nj. This requires re\u000b< rnbe-\ncausej\u0001ijjis an increasing function of re\u000b, as seen in\nEq. (C1). We sum the contributions from each vortex to\nj\u0001ijjsimply by adding the qth inverse moment( q > 0)\nof allrntogether to de\fne an e\u000bective distance re\u000bas in\nEq. (C2). Using the qth inverse moment, instead of the\nqth moment guarantees that re\u000b<rnwhen there is more\nthan one vortex. Furthermore it ensures that the terms\n(\u0018\nrn)qwith small rnon the right hand side of Eq. (C2)\ncontribute more signi\fcantly than those with larger rn.\nThis is consistent with the physical intuition that vor-\ntices nearby are more important in determining re\u000b, and\nthereforej\u0001ijj, than those that are far away. Also when\nthere are more vortices present, Nvbecomes larger and\nthe resultantj\u0001ijjfrom Eq. (C2) becomes smaller. This\nagain agrees with our expectation.Thej\u0001ijjde\fned above increases monotonically with\nthe parameter qfor a given vortices con\fguration. To see\nthis we only need to show re\u000bincreases with q. For that\npurpose we can rewrite Eq. (C2) as follows\nlogrmin\nre\u000b=1\nqlogf1 +X0\nn(rmin\nrn)qg (C5)\nwherermin= minfrngis the distance between the closest\nvortex and the bond, and the prime sign in the summa-\ntion means this closest vortex is excluded. The right\nhand side of Eq. (C5) is a monotonic decreasing function\nofqbecause in the summation eachrmin\nrn<1. Therefore\nre\u000bincreases monotonically with q, so doesj\u0001ijj.\n●●●●●●●●●●\n●\n●\n●●●●●●●●●●●●●●●●\n●\n●●●\n●●●●●■■■■■■■■■■■\n■■■■■■■■■■■■■■■■■\n■\n■■■■■■■■▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲\n▲\n▲▲▲▲▲▲▲▲●q=1\n■q=2\n▲q=3\n400 600 800 1000 12000.00.51.01.5\n0.01.02.03.0\nF(T)Amplitudes(arb. units)q=1\nq=2,3\nFIG. 9. Oscillation spectrum of the DOS for the two fold\nDDW order model with di\u000berent qvalues but a \fxed \u0001. Note\nthat the vertical axis scale for q= 1 is di\u000berent from those for\notherqvalues. The two vertical dashed lines mark the two\nfrequencies Fe= 525T;Fh= 966T.\nAn appropriate value of qcan not be determined with-\nout solving the whole problem self-consistently, therefore\nwe performed simulations for di\u000berent qto see if our con-\nclusions depend on qor not. One example data of the\noscillation spectrum for the two-fold DDW order case is\nshown in Fig. 9. From this \fgure, we observe that the os-\ncillation amplitude decreases as qis increased from q= 1\ntoq= 3. This is consistent with the analyses that j\u0001ijj\nis a monotonic increasing function of q, since larger q\ngives largerj\u0001ijj, which means stronger vortex scatter-\ning and therefore stronger suppression of the oscillation\namplitudes.\nAlthough the oscillation amplitudes can depend signif-\nicantly onq, the oscillation frequencies remain una\u000bected\nby varying the qvalues, therefore the conclusion of On-\nsager's relation being robust against the vortex scattering\ndoes not depend on the value of q.\nAppendix D: Check of the results in Ref.[7]\nWe have checked the Fig.1 and Fig.3 of Ref.[7], using\nthe same parameter sets, and \fnd our conclusions remain13\nthe same. For both these two cases, the normal state,\nwithout magnetic \feld, can be described by the following\nHamiltonian\nH=\u0000tX\n<i;j>cy\nicj+t0X\n<<i;j>>cy\nicj\n+X\n<i;j>i(\u00001)xi+yi\u0011ijW0\n4cy\nicj\u0000\u0016X\nicy\nici:(D1)\nIn this Hamiltonian the third term is a two-fold DDW\norder(or the staggered \rux state order), and \u0011ij=\u00061 is\nagain the local d\u0000wave symmetry factor.\n1. First consider the Fig.1 of Ref.[7]. We use the\nsame parameters t= 1;t0= 0:3t;W 0= 1:0t;\u0016=\n\u00000:949t. The normal state Fermi surface consists\nof four hole pockets with an areaAF\n(2\u0019=a)2\u00192:5%\neach. Fig. 10 shows the computed DOS. We see\nthere is no noticeable shift in the oscillation fre-\nquency when the vortex scattering is present.\n●●●●●●●●●●●●●●●\n●\n●●●● ●● ●● ●●●●●●\n●\n●\n●●●●● ● ●● ●●●●●●\n●\n●●●●●●●●●●●●●●\n●\n●●●●● ●● ●●●●●●\n●\n●\n●●●● ● ●●●●●●●●\n●\n●\n●●●● ●●●●●●●●●\n●\n●\n●●●●●● ●●●●●●●\n●\n●\n●●●●●● ●●●●●●●\n●\n●●●●●●●●●●●●●●\n●\n●●●●●●●●●●●●●\n●\n●\n●●●●● ●● ●●●●●●\n●\n●\n●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●● ● ● ● ●\n■■■■■■ ■■■■■■■■\n■\n■\n■■■■ ■ ■■■■■■■■\n■\n■\n■\n■■■■ ■ ■■■■■■■■\n■\n■\n■\n■■■■ ■ ■■■■■■■■\n■\n■\n■\n■■■■ ■■■■■■■■■\n■\n■\n■\n■■■■■■■■■■■■■\n■\n■\n■\n■■■■ ■■■■■■■■■\n■\n■\n■\n■■■■■■■■■■■■\n■\n■\n■\n■\n■■■■■■■■■■■■\n■\n■\n■\n■\n■■■■■■■■■■■■\n■\n■\n■\n■\n■■■■■■■■■■■■\n■\n■\n■\n■■■■ ■■■■■■■■■\n■\n■\n■\n■■■■■■■■▲▲▲▲▲▲ ▲▲▲▲▲▲▲▲\n▲\n▲\n▲▲▲▲▲▲▲▲▲▲▲▲▲\n▲\n▲\n▲\n▲▲▲▲▲▲▲▲▲▲▲▲▲\n▲\n▲\n▲\n▲▲▲▲▲▲▲▲▲▲▲▲▲\n▲\n▲\n▲\n▲▲▲▲ ▲▲▲▲▲▲▲▲▲\n▲\n▲\n▲\n▲▲▲▲▲▲▲▲▲▲▲▲▲\n▲\n▲\n▲\n▲▲▲▲ ▲▲▲▲▲▲▲▲ ▲\n▲\n▲\n▲\n▲▲▲▲▲▲▲▲▲▲▲▲▲\n▲\n▲\n▲\n▲▲▲▲▲▲▲▲▲▲▲▲▲\n▲\n▲\n▲▲▲▲ ▲▲▲▲▲▲▲▲▲▲\n▲\n▲▲▲▲▲ ▲▲▲▲▲▲▲▲▲▲\n▲▲▲▲▲▲▲▲▲▲▲▲▲▲ ▲▲▲▲▲▲▲▲▲▲▲▲▼▼▼▼▼▼ ▼▼▼▼▼▼▼▼\n▼\n▼\n▼▼▼▼▼ ▼▼▼▼▼▼▼▼\n▼\n▼\n▼\n▼▼▼▼▼▼▼▼▼▼▼▼▼\n▼\n▼\n▼\n▼▼▼▼ ▼▼▼▼▼▼▼▼▼\n▼\n▼\n▼\n▼▼▼▼ ▼▼▼▼▼▼▼▼▼\n▼\n▼\n▼\n▼▼▼▼▼▼▼▼▼▼▼▼▼\n▼\n▼\n▼\n▼▼▼▼▼▼▼▼▼▼▼▼▼\n▼\n▼\n▼▼▼▼▼▼▼▼▼▼▼▼▼▼\n▼\n▼\n▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ ▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ ▼▼▼▼▼▼▼▼▼▼▼▼●normal\n■Δ=0.08 t\n▲Δ=0.15 t\n▼Δ=0.25 t\n20 40 60 80 1000.00.10.20.30.40.5\n1/BDOS\nFIG. 10. DOS oscillation for t= 1;t0= 0:3t;W 0= 1:0t;\u0016=\n\u00000:949t. The unit for the \feld Bis\b0\n2\u0019a2, with \b 0=hc=e the\nfull \rux quantum and athe lattice spacing. And the DOS\nunit is states =t. In the legends the \\normal\" means \u0001 = 0.\nThe lattice size is L= 2000;M= 80, and in the Eq. (C2) of\nre\u000b,q= 1 rather than q= 2 has been chosen here.2. Then consider the Fig.3 of Ref.[7]. In this case, the\nnormal state does not have DDW, so W0= 0. For\nt= 1;t0= 0:14t;\u0016=\u00002:267t, the obtained Fermi\nsurface contains only a large hole pocket with an\nareaAF\n(2\u0019=a)2\u001914% at the Brillouin zone center.\nFig. 11 shows the corresponding DOS results. We\nsee the oscillation amplitude gets heavily damped\nas \u0001 increases. Moreover, a small frequency shift\n\u000eF=F\u00192% becomes noticeable. However, this is\ndi\u000berent from a large 30% shift found in Ref.[7].\nAlso this 2% shift does not contradict our previ-\nous conclusion of no noticeable frequency shift. Be-\ncause the shift here is obtained at magnetic \felds\nthat are larger than the experimentally applied\n\felds(\u001850T) by an order of magnitude. In Fig. 11,\n1\nB= 10 corresponds to B=1\n10\b0\n2\u0019a2\u0019450T, since\n\b0=2\u0019a2\u00194500T if we take a= 3:83\u0017Afor YBCO.\n●●●\n●\n●\n●●●●●● ●●● ●●●●●●●●●●\n●\n●\n●\n●●●●● ●● ●●●●●●●●●●●●\n●\n●\n●●●●●●● ● ● ●●●●●●●●●●\n●\n●\n●\n●●●●●●●●● ●●●●●●●●●●\n●\n●\n●\n●●●●●●●●●●●●●\n■■■\n■\n■\n■\n■\n■■■■■■■■■■■■■■■■■■■\n■\n■\n■\n■\n■■■■■■■■■■■■■■■■■■■\n■\n■\n■\n■■■■■■■■■■■■■■■■■■■■\n■\n■\n■\n■■■■■■■■■■■■■■■■■■■■\n■\n■\n■■■▲▲▲▲\n▲\n▲\n▲\n▲▲▲▲▲▲▲ ▲ ▲▲▲▲▲▲▲▲▲▲▲\n▲\n▲\n▲\n▲\n▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲\n▲\n▲\n▲▲▲▲▲▲▲▲ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▼▼▼▼\n▼\n▼\n▼\n▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼\n▼\n▼\n▼▼▼▼▼▼▼▼ ▼ ▼▼▼▼▼▼▼▼▼▼▼ ▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ ▼▼▼▼▼▼▼▼▼▼ ▼▼▼▼▼●normal\n■Δ=0.32 t\n▲Δ=0.45 t\n▼Δ=0.57 t\n5 6 7 8 9 100.00.10.20.30.40.50.60.7\n1/BDOS\nFIG. 11. DOS oscillation for t= 1;t0= 0:14t;W 0= 0;\u0016=\n\u00002:267t. 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The\nbanded state extends over a range of density, and the scalar order parameter of the system shows a\nplateau like behaviour, similar to that of the magnetic systems. In the large density limit the active\nnematic shows a bistable behaviour between a homogeneous ordered state with global ordering and\nan inhomogeneous mixed state with local ordering. The study of the above phases with density\nvariation is scant and gives signi\fcant insight of complex behaviours of many biological systems.\nPACS numbers: 87.10.Rt, 05.65.+b, 64.60.Bd\nIntroduction :| Active systems are composed of self-\npropelled particles where each particle extracts energy\nfrom its surroundings and dissipates it through motion\ntowards a direction determined by its orientation. These\nkind of systems are ubiquitous in nature, ranging from\nvery small scale systems inside the cell to larger scales [1{\n9], vibrated granular media [10, 11] etc., and have been\nstudied extensively through experiments, theories and\nsimulations [12{14]. A collection of head-tail symmetric\n`apolar' active particles with an average mutual parallel\nalignment is said to be in a `nematic' state, whereas in an\n`isotropic' state particles remain randomly oriented. An\nactive system where \ruid media do not play important\nrole in emergence of ordered state, and thus the hydrody-\nnamic interactions can be ignored, is called a `dry active\nsystem' [5, 10, 15{18].\nActive nature of particles introduces a nonequilibrium\ncoupling between density and orientation \feld, as rep-\nresented in terms of curvature coupling current in litera-\nture [12, 19, 20]. Such coupling in active nematic induces\nunusual properties like large density \ructuation [19, 21]\nand growth kinetics faster than 1 =3 as in usual conserved\nmodel [22]. Recent studies of the active nematic found\na defect-ordered nematic state [23{25] as opposed to the\nequilibrium nematic for high particle densities. Recent\nexperiment on amolyiod \ribrils [26] also found a phase\nwith coexisting aligned and disordered \fbril domains,\nsimilar to the defect-ordered state obtained in simula-\ntions. But few investigations have been done on the be-\nhaviours of the active nematic in various density limits,\nespecially at low densities. Here we introduce a minimal\nmodel for two-dimensional active nematic and compare\nvarious ordering phases of active and equilibrium nematic\nin di\u000berent density limits. The ordering in the system\nis characterised in terms of a scalar order parameter S\nwhich is the positive eigen value of nematic order param-\neterQ[1] in two-dimensions. In the low density limit\nboth active and equilibrium systems are in the isotropic(I) state with particles randomly oriented throughout the\nwhole system (see Fig. 1(b) - I), resulting in a small S.\nThe Phase diagram of the active nematic as a function\nof packing density C(see Fig. 1(a)) shows a jump in\nSclose toC= 0:37, whereas in the equilibrium case S\ngoes continuously to larger values and an isotropic to ne-\nmatic (I-N) transition occurs close to C= 0:58. In the\nequilibrium nematic (EN) state particles remain homo-\ngeneously oriented in the system (see Fig. 1(b) - EN). At\nC= 0:37 the active system goes from the isotropic to a\nbanded state (BS) where particles cluster and align in the\nperpendicular direction to the long axis of the band (see\nFig. 1(b) - BS-1). With increasing density more number\nof particles participate in band formation (see Fig. 1(b)\n- BS-2) and Sfollows a plateau over a range of density.\nIn the large density limit active system shows bistability\nbetween a homogeneous ordered (HO) (see Fig. 1(b) -\nHO) and an inhomogeneous mixed (IM) or local ordered\nstate (see Fig. 1(b) - IM). This IM state is very similar\nto defect-ordered nematic state in ref. [23{25].\nModel :| We consider a two dimensional square lat-\ntice. At each vertex ` i' we de\fne an occupation variable\nni, which can take values 1 (occupied) or 0 (unoccupied),\nand an orientation variable \u0012i, which lies between 0 and \u0019\nbecause of the apolar nature of the particles. Each parti-\ncle interacts with its nearest neighbours through modi\fed\nLebwohl - Lasher Hamiltonian [28]\nH=\u0000\u000fX\n<ij>ninjcos 2(\u0012i\u0000\u0012j) (1)\nwhere\u000fis the interaction strength between two neigh-\nbouring particles. This model is analogous to the diluted\nXY-model with nonmagnetic impurities [29], where im-\npurities and spins are analogous to vacancies and parti-\ncles respectively in the present model.\nOrientation evolves through Monte - Carlo (MC) up-\ndates [30] following the Hamiltonian in Eq. 1. Unlike\nthe diluted XY-model, particles also move on the lat-arXiv:1509.05166v1  [cond-mat.stat-mech]  17 Sep 20152\n0.2 0.4 0.6 0.8 1\nC00.20.40.60.8S\nIBSIMHO\nEN\n(a)\n(IM)\n(IM) (HO)\n(IM) (HO) (EN)\n(IM) (HO) (EN)(I)\n(IM) (HO) (EN)(I) (BS-1)\n 0 0.2 0.4 0.6 0.8 1\n(IM) (HO) (EN)(I) (BS-1) (BS-2) (b)\nFIG. 1: (Color online) (a) Plot of scalar order parameter Svs. packing density Cfor active (circles and triangles) and equilibrium\n(continuous line) nematic for system size 512 \u0002512. Equilibrium system goes smoothly from isotropic (I) to nematic (EN)\nstate. Active system goes from isotropic (I) to banded state (BS) (small jump in S) followed by either an inhomogeneous\nmixed (IM) (triangles) or a homogeneous ordered (HO) (circles) state. (b) Snapshots of particle inclination towards the\nhorizontal direction. Color bar ranging from zero to one indicates parallel and perpendicular inclinations respectively towards\nthe horizontal direction. White regions signify unoccupied sites. (I) is isotropic state at low density ( C= 0:36), (BS-1)\n(C= 0:38) and (BS-2) ( C= 0:56) are two banded state con\fgurations, (IM) is inhomogeneous mixed, (HO) is homogeneous\nordered and (EN) is equilibrium nematic state at high density ( C= 0:76).\ntice. Depending on the type of movement we de\fne two\nkinds of models. (i) `Equilibrium model' (EM) - a par-\nticle can di\u000buse to any unoccupied nearest-neighbouring\nsite, and therefore satis\fes the detailed balance condi-\ntion. (ii) `Active model' (AM) - a particle can move to\nonly those unoccupied nearest-neighbouring sites which\nare in the direction that makes the least inclination with\nthe particle orientation. Details of the model and particle\nmovement are shown in Supplemental Material [31] sec-\ntion I. The asymmetric move of the active particles does\nnot staisfy the detailed balance condition and arises in\ngeneral because of the self-propelled nature of the parti-\ncles in many biological [15, 32] and granular systems [10].\nThese moves produce an active curvature coupling cur-\nrent in coarse-grained hydrodynamic equations of motion\n[19, 20].\nNumerical details :| We consider a collection of N\nparticles with random orientation \u0012i2[0;\u0019], homoge-\nneously distributed on a L\u0002Lsquare lattice ( L=\n150;256;512) with periodic boundary condition. The\npacking density of the system is C=N=(L\u0002L). We\nchoose a particle randomly and move it to an unoccupied\nneighbouring site, followed by an orientation updation\nthrough MC. We use 106MC steps to evolve the system\nto its steady state and all the results have been obtained\nby averaging over next 2 \u0002106MC steps. Twenty four\nrealizations have been used for better statistics.\nWe calculate the scalar order parameter\nS=s\n(1\nNX\ninicos(2\u0012i))2+ (1\nNX\ninisin(2\u0012i))2(2)which is small in the isotropic state and close to 1 in the\nordered state. First we calculate Sfor EM as a function\nof inverse temperature \f= 1=kBTfor di\u000berent densities.\nAs shown in Supplemental Material [31] section II, the\ncritical temperature Tcis approximated as Tc(S= 0:4).\nCritical temperature Tc(C) decreases with the lowering\nof the packing density C, similar trend is found in the\nstudy of diluted XY-model [29] for varying nonmagnetic\nsite density. In rest of our calculations temperature is\nkept \fxed at \f\u000f= 2:0 and packing density Cis varied\nfrom small values to complete \flling C= 1:0.\nPhase diagram :| At low densities, C < 0:37, the\nactive system is in the isotropic state where the parti-\ncles with random orientation remain homogeneously dis-\ntributed throughout the system, and therefore Sholds\nvanishingly small values. The jump occurs in Sat\nC= 0:37. ForC\u00150:37 particles cluster in, and both or-\ndered state with high local density and disordered state\nwith low local density coexist (see Fig. 1(b) - BS-1).\nMean alignment inside the band is perpendicular to the\nlong axis of the band. In the moderate density lim-\nits, band formation in more favourable than lane for-\nmation (mean alignment parallel to the long axis of the\nstructure) because the number of particles that can have\ntranslational motion is much larger in the banded state,\nand therefore entropy favours band formation. Similar\nmechanism create the bending instability close to order-\ndisorder transition in the active polar systems [33, 34].\nThe above transition from I state to BS occurs at density\nlower than the corresponding equilibrium I-N transition3\ndensityC'0:58 (see Fig. 1(a)).\nAs we further increase C, unlike the equilibrium sys-\ntem where Sincreases monotonically with C, the active\nsystem shows very small change in Sfor a range of den-\nsity. This plateau like appearance of Swith variation\ninCis very similar to plateau phase in magnetization\nversus \feld curve of magnetic systems [35]. If an energy\ngap exists between two consecutive magnetic states, a \f-\nnite \feld is required for the magnetic system to go from\nthe lower to the higher state. So until that \fnite \feld\nis applied, the increasing \feld keeps the system magne-\ntization to be unchanged, and the system is called to be\nin the plateau phase. With increasing packing density in\nthe plateau regime of the active nematic more particles\nparticipate in band formation (see Fig. 1(b) - BS-1 and\nBS-2). On further increment of density, close to equi-\nlibrium I-N transition C'0:58, transverse \ructuations\nlead the system to a mixed state [20, 36].\nIn the large Climit active system shows a bistable\nbehaviour with two distinct steady states; \frst, a state\nwhereSis large and real space con\fguration is `homoge-\nneous ordered' (HO), and the second, an `inhomogeneous\nmixed' (IM) state where Srealizes some moderate val-\nues. In the HO state though the particle orientation is\nhomogeneous, large density inhomogeneity exists in the\nsystem (see Fig. 1(b) - HO). IM state is a local ordered\nstate with many aligned clusters of high particle den-\nsity. The system consists of many such aligned clusters\nof high density separated from low density disordered re-\ngions and mean alignment in each cluster is in di\u000berent\ndirections (see Fig. 1(b) - IM). IM state is similar to\nthe defect-ordered state recently found in the study of\nref. [23{25], with large number of \u00061=2 defects in high\ndensity active nematic.\nRenormalised mean \feld study for small S:| We also\ncalculate the jump in the scalar order parameter Sand\nthe shift in the transition density using the Renormalised\nmean \feld (RMF) method of the coupled coarse-grained\nhydrodynamic equations of motion for the number den-\nsity\u001a(r;t) =P\n\u000b\u000e(r\u0000R\u000b(t)) and the order parame-\nterwij(r;t) =\u001a(r;t)Q(r;t) =P\n\u000b(mi\u000bmj\u000b\u00001\n2\u000eij)\u000e(r\u0000\nR\u000b(t)) for active nematic [19, 20].\n@t\u001a=a0rirjwij+D\u001ar2\u001a (3)\nand\n@twij=f\u000b1(\u001a)\u0000\u000b2(w:w)gwij\n+\f\u0012\nrirj\u00001\n2\u000eijr2\u0013\n\u001a+Dwr2wij (4)\nwhere, m\u000b= (cos(\u0012\u000b);sin(\u0012\u000b)) is the unit vector along\nthe orientation \u0012\u000bandR\u000b(t) is the position of parti-\ncle\u000b. We can obtain the number density \u001aby coarse-\ngrainingCover small subvolume. Eqs. 3 and 4 can\nbe obtained either from symmetry arguments as in ref.\n0.0010.010.11g2(r)\n1 10 100r0.0010.010.11C=0.30C=0.36C=0.38C=0.52\n(a) Active\nC=0.78\nC=0.58\nC=0.56 C=0.30 Equilibrium (b)C=0.82 (IM)C=0.82 (HO)FIG. 2: (Color online) g2(r) vs.ron log-log scale at di\u000berent\ndensities. (a) Active nematic: ( \r) and ( \u0003) at low densities\ng2(r) decays exponentially, ( \u0005) and (+) at intermediate den-\nsityg2(r) decays algebraically, ( 4) homogeneous ordered (al-\ngebraic decay), and inhomogeneous mixed (abrupt decay) at\nhigh density. (b) Equilibrium nematic ( \r) and ( \u0003) exponen-\ntial decay of g2(r) at low densities and ( \u0005) and (+) algebraic\ndecay ofg2(r) at high densities.\n[19] or from microscopic rule based model [20]. Den-\nsity equation 3 is a continuity equation @t\u001a=\u0000r\u0001 J,\nbecause the total number of particles is conserved. Cur-\nrentJi=\u0000a0rjwij\u0000D\u001ari\u001a, where the \frst term con-\nsists of two parts, an active curvature coupling current\nJa/a0\u001arjQijand anisotropic di\u000busion Jp1/Qijri\u001a,\nwhich can also be present in the equilibrium model. The\nsecond term in density equation is an isotropic di\u000busion\nJp2/r\u001aterm. First two terms in the order param-\neter equation wijis the alignment term. We choose\n\u000b1(\u001a) = (\u001a\n\u001aIN\u00001) as a function of density which changes\nsign at some critical density \u001aIN. Third term is coupling\nto density and last term is di\u000busion in order parameter\nand written for equal elastic constant approximation for\ntwo-dimensional nematic.\nA homogeneous steady state solution of Eqs. 3 and 4\ngives a mean \feld transition from isotropic to nematic\nstate at density \u001aINwhere\u000b1(\u001a) changes sign. Us-\ning RMF method we calculate the e\u000bective free energy\nfeff(S) close to order-disorder transition when Sis small.\nWe consider density \ructuations \u000e\u001aand neglect order pa-\nrameter \ructuations. The e\u000bective free energy\nfeff(S) =\u0000b2\n2S2\u0000b3\n3S3+b4\n4S4(5)\nwhereb2=\u000b1(\u001a) +\u000b0\n1(\u001a)c, wherecis a constant.\n\u000b0\n1(\u001a) =@\u000b1=@\u001aj\u001a0,b3=a0\u000b0\n1(\u001a)\n2D\u001aandb4=1\n2\u000b2. Both\nb3andb4are positive. A detail calculation for feffis\nshown in Supplemental Material [31] section III. The den-\nsity \rcutuations introduce a new cubic order term pro-\ntortional to the activity strength a0, in the free energy4\nfeff(S). The presence of such term produces a jump\n\u0001S=Sc=2b3\n3b4at a lower density \u001ac=\u001aIN(1\u00002b2\n3\n9b4).\nThis type of jump and shift in transition because of \r-\ncutuations are also called as \ructuation dominated \frst\norder phase transition in statistical mechanics [37] and\nwidely studied in many systems [38]. The jump in Sand\nthe shift in \u001acis proportional to the activity parameter\na0and fora0= 0 we recover the equilibrium transition.\nTwo-point orientation correlation function\n:| To further characterise the system we also\ncalculate the two-point orientation correlation\ng2(r) =<P\ninini+rcos[2 (\u0012i\u0000\u0012i+r)]=P\ninini>at\ndi\u000berent packing densities, where < : > signi\fes an\naverage over many realisations. In Fig. 2 we plot g2(r)\nvs.ron log-log scale, for C= 0:30, 0:36, 0:38, 0:52 and\n0:82 for active model and C= 0:30, 0:56, 0:58 and 0:78\nfor equilibrium model. For very small packing density\nC < 0:37,g2(r) decays exponentially in the active\nsystems. Therefore the system is in short-range-ordered\n(SRO) isotropic state. At C= 0:38,g2(r) decays fol-\nlowing the power law g2(r)'1=r\u0011(C)and therefore the\nsystem is in a quasi-long-range-ordered (QLRO) state.\nAt high packing densities correlation functions con\frm\nthe bistability in the active systems. For C= 0:82\n(see Fig. 2(a)) g2(r) shows power law decay in HO\nstate, whereas in IM state g2(r) decays abruptly after\na distance r. The abrupt change in g2(r) at a certain\ndistance indicates the presence of local ordered clusters.\nIn contrast, the equilibrium systems show a transition\nfrom SRO isotropic state at low density C<\u00180:56 to\nQLRO nematic state at C>\u00180:58, similar to Berezinskii\n- Kosterlitz - Thouless (BKT) transition [39, 40] in the\ndiluted XY-model [29].\nOrientation distribution :| We also compare the\nsteady state properties of active and equilibrium mod-\nels in the high density limit. First we calculate the\nsteady state (static) orientation distribution P(\u0012) from\none snapshot of particle orientation. Both active HO and\nequilibrium nematic show a Gaussian distribution of ori-\nentation (see Fig. 3(a)). Hence in HO state orientation\n\ructuations of particles are of same kind as in equilib-\nrium model and the system is in QLRO state. Distribu-\ntionP(\u0012) in the IM state is very broad and spanning the\nwhole range of orientation. Hence the system has many\nlocal ordered regions with di\u000berent orientations.\nWe also calculate the time averaged distribution of\nmean orientation of all the particles P(\u0012m) in active HO\nand equilibrium nematic states. P(\u0012m) is calculated from\nmean of all particle orientaions averaging over a long time\n(from 106to 3\u0002106) in the steady state. P(\u0012m) for HO is\nnarrow in comparison to the broad distribution for equi-\nlibrium model (see Fig. 3(b)). Narrow distribution of\nP(\u0012m) implies that orientation autocorrelation in active\nsystem decay slowly as comapared to the corresponding\nequilibrium model which is in agreement with the slow\n0 0.2 0.4 0.6 0.8 1θ / π00.0050.010.0150.02P(θ)\n0 0.2 0.4 0.6 0.8 1\nθm / π00.010.020.030.04P(θm)EN HO\nHO\nEN(b)(a)\nIMFIG. 3: (Color online) (a) Steady state orientation distribu-\ntionP(\u0012) of particles for HO, IM (active) and EN (equilib-\nrium) states at high density. Lines are \ft to Gaussian dis-\ntribution for both HO and EN states. IM state shows very\nbroad distribution of P(\u0012). (b) Plot of mean orientation dis-\ntributionP(\u0012m) averaged over a long time in the steady state\nfor HO (active) and EN (equilibrium) states. P(\u0012m) is very\nbroad for EN in comparison to HO.\ndecay of velocity autocorrelation in bacterial suspension\n[41].\nSummary :| In this letter we have introduced a min-\nimal model for active nematic and found three distinct\nphases with the variation in density. At low densities\nthe active nematic is in disordered isotropic state with\nvery small correlation between the particles. With in-\ncreasing density active nematic undergoes a \ructuation\ninduced \frst order phase transition from the isotropic to\nthe banded state where large number of particles partic-\nipate in band formation. Large density \ructuations in\nthe active systems change the nature of the transition\nand shift the transition density to smaller value as com-\npared to the equilibrium isotropic nematic transition. At\nlarge densities equilibrium nematic is in QLRO nematic\nstate, whereas active nematic goes from the banded state\nto either the homogeneous ordered (high S) or the inho-\nmogeneous mixed (moderate S) state. This inhomoge-\nneous mixed state is similar to the phase with coexist-\ning aligned and disordered \fbril domains found in recent\nexperiment [26]. Experiments on thin layer of agitated\ngranular rods, elongated living cells, bacterial colony of\napolar B. subtilis etc. at di\u000berent densities can realize\nthe di\u000berent phases we found here. In the present model\nwe have frozen the motion of the active particles in the\ntransverse direction, i.e. the activity strength is kept\nlarge. 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Lubensky and S. K. Ma, Phys. Rev.\nLett.32, 292 (1974); J. H. Chen, T. C. Lubensky and D.\nR. Nelson, Phys. Rev. B 17, 4274 (1978).\n[39] V. L. Berezinskii, Sov. Phys. JETP 34(3), 610 (1972).\n[40] M. Kosterlitz and D. Thouless, J. Phys. C 6, 1181 (1973).\n[41] Xiao-Lun Wu and A. Libchaber, Phys. Rev. Lett., 84,\n3017 (2000).Supplemental Material\nI. MODEL FIGURE\n(a) \n𝑟 ,𝜃 0≤𝜃≤𝜋 \n𝑟 ,𝜃 \n𝑟 ,𝜃 0≤𝜃<𝜋4  \nor \n3𝜋4 <𝜃≤𝜋 \n𝜋4 ≤𝜃≤3𝜋4  (c) (b) \nFIG. 4: (a) Two dimensional square lattice with occupied ( n= 1) or unoccupied ( n= 0) sites. Filled circles signify the\noccupancy of respective sites. Inclination of the rods towards the horizontal direction show respective particle orientation\n\u0012i2[0;\u0019]. (b) Equilibrium move : particle can move to any of four neighbouring sites with equal probability 1 =4, (c) Active\nmove: particle can move to either of its two neighbouring sites with probability 1 =2, if unoccupied, in the direction it is more\ninclined to.7\nII. ESTIMATE OF CRITICAL TEMPRATURE Tc(C)IN EQUILIBRIUM MODEL\n0 1 2 3 4\nβε00.20.40.60.81S0.60.81\nC0.81.62.4\nβcεC=1.00\nC=0.80\nC=0.60\nC=0.50\nC=0.30\nFIG. 5: Plot of Svs. inverse temperature \f\u000ffor di\u000berent densities Cfor equilibrium model. System goes from isotropic (small\nS) to nematic (large S) state. Vertical dotted line shows the variation in Sfor \fxed\f\u000f= 2:0 at di\u000berent densities. Crtical\ntemperature is approximated as Tc(S= 0:4). Inset: change in Tcas a function of density C.\nIII. RENORMALISED MEAN FIELD (RMF) STUDY OF ACTIVE NEMATIC FOR SMALL SCALAR\nORDER PARAMETER S\nIn this section we will write an e\u000bective renormalised mean \feld free energy for scalar order parameter Sfor smallS.\nWe keep the density \ructuations and ignore the order parameter \ructuations in the coupled hydrodynamic equations\nof motion for active nematic. Density \ructuation produces a cubic order term in the e\u000bective free energy for scalar\norder parameter Sand such term produces a jump in Sat a new transition density \u001aclower than equilibrium I-N\ntransition point \u001aIN. Shift in transition density and jump \u0001 Sis directly proportional to the activity parameter a0\nand we recover equilibrium limit for zero a0.\nWe write coupled hydrodynamic equations of motion for density \u001aand order parameter w=\u001aQwhere nematic order\nparameter [1]\nQ(r;t) =S\u0012cos 2\u0012(r;t) sin 2\u0012(r;t)\nsin 2\u0012(r;t)\u0000cos 2\u0012(r;t)\u0013\n(6)\n\u0012being the coarse grained angle at position rand timet. Density equation\n@t\u001a=a0rirjwij+D\u001ar2\u001a (7)\nand order parameter equation w\n@twij=f\u000b1(\u001a)\u0000\u000b2(w:w)gwij+\f\u0012\nrirj\u00001\n2\u000eijr2\u0013\n\u001a+Dwr2wij (8)\nDensity Eq. 7 is a continuity equation @\u001a=@t =\u0000r\u0001J, where Jhas two parts, active and di\u000busive. Details of these\ntwo currents are given in the main text. a0is the activity parameter, present because of self-propelled nature of\nthe particles, \fis the coupling of density in wequation.D\u001aandDware the di\u000busion coe\u000ecients in density and\norder parameter equations respectively, \u000b1(\u001a) and\u000b2are the alignment terms and ingeneral depends on the model\nparameters. For metric distance interacting models [2] \u000b1(\u001a) is a function of density and changes sign at some critical\ndensity. We choose \u000b1(\u001a) =\u001a\n\u001aIN\u00001 and\u000b2= 1. Steady state solution of homogeneous Eq. 7 is \u001a=\u001a0, we add small\nperturbation to mean density \u001a=\u001a0+\u000e\u001a. In the staedy state density \ructuation \u000e\u001acan be obtained from Eq. 7,\na0rirjwij+D\u001ar2\u000e\u001a= 0\n)a0rjwij+D\u001ari\u000e\u001a=constant =c1 (9)8\nwhere w11=\u0000w22=S\n2cos(2\u0012) and w12=w21=S\n2sin(2\u0012) and keep the lowest order terms in Sand\u0012\n@x\u000e\u001a=\u0000a0\nD\u001a@xS!\u000e\u001a(x) =\u0000a0\nD\u001aS+c (10)\nand\n@y\u000e\u001a=a0\nD\u001a@yS!\u000e\u001a(y) =a0\nD\u001aS+c1 (11)\nHere we assume nematic is aligned along one direction and there is variation only along the perpendicular direction.\nHence we can choose either of equations 10 or 11. Two constants candc1are the value of density where nematic\norder parameter is zero.\nWe use Eq. 10 and substitute the solution for density in equation for wijand obtain an e\u000bective equation for S.\n@tS=\u001a\n\u000b1(\u001a)\u00001\n2\u000b2S2\u001b\nS+O(r2S) +O(r2\u001a) (12)\nWe neglect all the derivative terms and keep only polynomial in S, i.e. we neglect higher order \ructuations. We\ncan do taylor expansion of \u000b1(\u001a) about mean density \u001a0,\u000b1(\u001a) =\u000b1(\u001a0+\u000e\u001a) =\u000b1(\u001a0) +\u000b0\n1\u000e\u001a, where\u000b0\n1=@\u000b1\n@\u001aj\u001a0.\nSubstitute the expression for \u000e\u001afrom Eq. 10 hence\n@tS=\u001a\n\u000b1(\u001a0) +\u000b0\n1\u000e\u001a\u00001\n2\u000b2S2\u001b\nS (13)\nWe can write an e\u000bective free energy for S\n@tS=\u0000\u000efeff(S)\n\u000eS(14)\nhence\n\u0000\u000efeff\n\u000eS=S\u001a\n\u000b1(\u001a0) +\u000b(\u001a0)\u0012a0\n2D\u001aS+c1\u0013\n\u00001\n2\u000b2S2\u001b\n(15)\nTherefore\nfeff(S) =\u0000b2\n2S2\u0000b3\n3S3+b4\n4S4(16)\nwhereb2=\u000b1(\u001a0) +\u000b0\n1c,b3=a0\u000b0\n1\n2D\u001aandb4=1\n2\u000b2andcis a conatant. Hence \ructuation in density introduces a\ncubic order term in e\u000bective free energy feff(S). E\u000bective free energy in Eq. 16 is similar to Landau free energy\nwith cubic order term [3]. We calculate jump \u0001 Sand new critical density from coexistence condition for free energy.\nSteady state solutions of order parameter ( S= 0 for isotropic and S6= 0 for ordered state) are given by\n\u000efeff\n\u000eS=\u0000\n\u0000b2\u0000b3S+b4S2\u0001\nS= 0 (17)\nNon-zeroSis given by\u0000b2\u0000b3Sc+b4S2\nc= 0. Coexistence condition implies\nfeff(Sc) =\u0012\n\u0000b2\n2\u0000b3\n3Sc+b4\n4S2\nc\u0013\nS2\nc=feff(S= 0) = 0 (18)\nhence we get the solution\nSc=\u00003b2\nb3(19)\nand\nbc\n2=\u00002b2\n3\n9b4(20)9\nHence the jump at new critical point is \u0001 S=2b3\n3b4. Sinceb4>0 and hence bc\n2<0, the new critical density\n\u001ac=\u001aIN\u0012\n1\u00002b2\n3\n9b4\u0013\n<\u001aIN (21)\nis shifted to lower density in comparison to equilibrium \u001aIN. Eq. 21 gives the expression for new transition density\nas given in main text. Hence using renormalised mean \feld theory we \fnd a jump \u0001 Sat a lower density as compared\nto equilibrium I-N transition density.\n\u0003shradha.mishra@bose.res.in\n[1] P. G. de Gennes and J. Prost, The Physics of Liquid Crystals (Clarendon Press, Oxford, 1995).\n[2] E. Bertin, H. Chat\u0013 e, F. Ginelli, S. Mishra, A. Peshkov and S. Ramaswamy, New J. of Phys. 15, 085032 (2013).\n[3] P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics (Cambridge University Press, Cambridge,\n2000)."    },    {        "title": "1510.07843v1.Long_range_interaction_induced_density_modulated_state_in_a_Bose_Einstein_condensate.pdf",        "content": "arXiv:1510.07843v1  [cond-mat.quant-gas]  27 Oct 2015Long range interaction induced density modulated state in a Bose-Einstein condensate\nAbhijit Pendse∗and A. Bhattacharyay†\nIndian Institute of Science Education and Research\n(Dated: June 25, 2021)\nWe consider a Gross-Pitaevskii model of BEC with non-local s -wave scattering to study the\ndensity modulated state in 1D. We resort to a perturbative Ta ylor series expansion for the order\nparameter. By perturbative calculations, we show that unde r long range s-wave scattering a density\nmodulated state is energetically favourable as compared to the uniform density state. We obtain\ndensity modulated state as a solution to the perturbative no n-local GP equation, rather than the\nconventional approach of introducing amplitude modulatio ns on top of the uniform density state\nand lowering the roton minimum.\nINTRODUCTION\nThe Bose-Einstein condensate (BEC) is a superfluid\nmacroscopic quantum phase of matter. It has tremen-\ndous importance in applications across a wide range\nof physics. To mention a few, BEC has applica-\ntions in the field of Quantum Information[1], Quantum\nMetrology[2], Atomic Lasers[3, 4], Atom Holography[5],\nInterferometry[6], Slow Light[7], Atom Clocks[8], Ana-\nlogue Gravity[9] and Quantized Vortex Dynamics[20].\nNormally, the considered uniform ground state of this in-\nherently unstable gas phase at nano-kelvin temperatures\nisgivenbythe complexorderparameter ψ0=√ne−iµt//planckover2pi1,\nwhere,nis the density of the condensate and µis the\nchemical potential. In such a system, one considers\ntwo-body s-wave scattering to be the means of interac-\ntion between bosons and the s-wave scattering length\nato be much smaller than the average inter-particle\nseparation n−1/3. Small amplitude excitations of the\nuniform ground state determine the thermodynamics of\nthe system and in this respect the work done by Fet-\nter et al. is interesting[21]. The ground state is dy-\nnamically stable to small amplitude fluctuations of the\nformθ(r,t) =/summationtext\ni[ui(r)e−iωit\n/planckover2pi1+v∗\ni(r)eiωit\n/planckover2pi1]e−iµt//planckover2pi1pro-\nvided/integraltextdr|ui|2/negationslash=/integraltextdr|vi|2. These small amplitude exci-\ntations are important in determining the thermodynam-\nics of this short lived ground state of BEC.\nBecause of the superfluid character of the BEC, for\nthe velocities below the velocity of sound in it, a modu-\nlated density phase is of particular interest. Supersolid\nis a state of matter with a crystalline order flowing with-\nout dissipation. Penrose and Onsager (PO) [10] have\nshown the impossibility of having such a phase (consid-\nering superfluid helium). Since then, many have con-\ntended this result and tried to circumvent the PO ob-\nservations by postulating the presence of a lattice of va-\ncancies in the solid and considering a super-flow of these\nvacancies [11, 12]. There are situations where there are\nnot always particles sitting at each lattice site as has\nbeen modelled by PO. Some work in this direction on\nBEC havebeen done by considering loweringof the roton\nminimum[15, 16]. There also exist some other recentlygiven interesting proposals based on dynamical creation\nof super-solid in optical lattice [13] and using Rydberg-\nexcited BEC [14].\nThe present work is motivated by the idea of obtaining\na density modulated state with lower energy than that\nof the uniform density state. The usual approach for\nobtaining such a state is working in the vicinity of a uni-\nform ground state, in other words introducing modula-\ntions on top of a uniform density state and then lowering\nthe roton minimum. But, we take a different approach\nthan the usual one and look for a pure density modulated\nstate which is notthe same as modulations on top of the\nunifrom density state. We show that long range s-wave\nscattering length can indeed support such a pure den-\nsity modulated state. Feshbach Resonance in BEC[17]\ncould be of use in getting the proposed state here, since\nit lets us have control over the s-wave scattering length\nby tuning the scattering length in the interval ( −∞,∞).f\nn0.5 gn2--->\nC\nB\nA\nFIG. 1. Figure shows a schematic diagram of free energy\nversusdensityfor theuniformdensitystate andforthedesi red\ndensity modulated state.\nThe zero temperature free energy density( f) vs.\ndensity(n) plot is shown schematically for a uniform den-\nsityphasebythecontinuouslineinFig.1. Thefreeenergy\ndensityf=1\n2gn2isthefreeenergydensityoftheuniform\nBEC with contact interactions between particles where2\nnis the density of the condensate and gis the s-wave\ninteraction strength. At a density n, the uniform phase\nhangs on this energy curve, for example, at point Bas\nshown in the figure. A single particle phase characterized\nby a wavenumber(spatial order) would havea kinetic en-\nergy cost on top of this free energy density and would be\nsomewhat at a higher energy for the same density( for\nexample, the point Cin Fig.1). Now, in the presence\nof long range s-wave scattering between the particles, we\nare interested here in getting a free energy curve which\nis schematically shown by the broken line in Fig.1. On\nthis, so far as the energy associated with the wave num-\nber dominates at low density, the curve remains over the\nf=1\n2gn2curve. At a higher density where the kinetic\nenergy cost is relatively smaller than interactions, we ex-\npect to see the curve to cross the f=1\n2gn2curve and\ncome below it such that there is now a lower energy state\nas shown by the point Awith a periodic order. This is\na different approach than the conventional one aimed at\nlowering the roton minimum obtained on the dispersion\ncurve in the vicinity of the uniform phase. Here, we are\ninterested in looking directly at the role of the nonlin-\near interaction term in getting us to such a lower energy\nphase based on long range interactions. In what follows,\nwe would see that in the presence of long range scatter-\ning, there would exist a range of boundary conditions\nwhich can lower the free energy curve as shown in Fig.1.\nTHE MODEL\nThe general Gross-Pitaevskii(GP) equation for a con-\ndensate is given as\ni/planckover2pi1∂ψ(r,t)\n∂t=−/planckover2pi12\n2m∇2ψ(r,t)\n+ψ(r,t)/integraldisplay∞\nrdr′ψ∗(r′,t)V(r′−r)ψ(r′,t),\n(1)\nwhich can be derived from the energy functionalE=/planckover2pi12\n2m/integraldisplay\ndr|∇ψ(r)|2\n+/integraldisplay\ndr|ψ(r)|2\n2/integraldisplay\ndr′ψ∗(r′)V(r−r′)ψ(r′)(2)\nThe local form of the GP equation considersonly contact\ntype interactions between particles. This local form of\nthe GP equation is\ni/planckover2pi1∂ψ(r,t)\n∂t=−/planckover2pi12\n2m∇2ψ(r,t)+gψ(r,t)|ψ(r,t)|2.(3)\nThese equations have been successful in explaining\nmany properties of a BEC.\nLet us look at features of Eq.(1) if the inter-particle\ninteractions are taken to be non-local. Since, we are con-\nsidering s-wave interactions only, we have the benefit of\nusing any effective, soft interparticle potential Veffas\nlong as it satisfies the criterion/integraltext\nVeffdr′=g=4π/planckover2pi12a\nm\n. We use this property and consider a potential which is\na gaussian with standard deviation equal to the s-wave\nscattering length.\nThroughout the article, we use Cartesian coordinate\nsystem. Also, we will be considering here a condensate\nin the absence of an external potential.\nLet us look at a perturbative one dimen-\nsional solution of Eq.(1) by considering the in-\nteraction potential of the form Veff(r′−r) =\ng\n(√\n2πa)3/parenleftbigg\ne−|x−x′|2\n2a2·e−|y−y′|2\n2a2·e−|z−z′|2\n2a2/parenrightbigg\n. By one\ndimensional solution, we mean that a solution of the\nformψ(r,t) =ψ(x,t)ψ(y)ψ(z) whereψ(y) andψ(z)\nare constants and only ψ(x,t) varies. Using this form,\nwe consider the integral term in Eq.(1). Since we have\na symmetric potential, the terms with odd orders of\nderivatives will vanish due to symmetry. Also, only\nderivatives of ψ(x,t) would be present as we have set\nψ(y) andψ(z) as constants. By expanding ψ(r′,t)\naroundr≡(x,y,z), we get the following expression\nψ(y)ψ(z)/parenleftbigg\ni/planckover2pi1∂ψ(x,t)\n∂t/parenrightbigg\n=ψ(y)ψ(z)/bracketleftBig\n−/planckover2pi12\n2m∂2\nxψ(x,t)+g√\n2πaψ(x,t)/parenleftBig\n|ψ(x,t)|2/integraldisplay∞\n−∞dx′e−|x−x′|2\n2a2\n+(∂2\nx|ψ(x′,t)|2)|x′=x/integraldisplay∞\n−∞dx′|x−x′|2\n2!e−|x−x′|2\n2a2+\n(∂4\nx|ψ(x′,t)|2)|x′=x/integraldisplay∞\n−∞dx′|x−x′|4\n4!e−|x−x′|2\n2a2+.../parenrightBig/bracketrightBig\n.\nEvaluating the terms in the integral and cancelling the ψ(y)ψ(z) terms from both sides, we obtain,3\ni/planckover2pi1∂ψ(x,t)\n∂t=−/planckover2pi12\n2m∂2\nxψ(x,t)+gψ(x,t)/parenleftBig\n|ψ(x,t)|2\n+a2\n2∂2\nx|ψ(x,t)|2+a4\n8∂4\nx|ψ(x,t)|2+a6\n48∂6\nx|ψ(x,t)|2+.../parenrightBig\n.(4)\nThe denominator of the numerical factors in the ex-\npansionareevendoublefactrialsandhencethenumerical\nfactors fall off. In our calculations of the amplitude mod-\nulated state, where kis the wave number of the phase,\nwe will see that k∼1\na. ThusakisO(1). Still it is safe\nto do the analysis here in a perturbative way, because\nthe coefficients of the higher order terms would fall off\nrather quickly. We take the equation thus obtained as a\nmodified GP equation and carry out our further analysis\nwith the help of this equation retaining terms upto the\n6thorder. Since the coefficient of the next term would be\nan order of magnitude smaller than that of the 6 thorder\nterm, we truncate the series at the decimal place corre-\nsponding to the 6 thorder term. This equation would\ncapture the essential features of the effects of non-local\ninteractions on the properties of a BEC.\nA perturbative approach for the GP equation used, for\nexample to determine the density profile of a single vor-\ntex line where the use of linear approximation for the\ndensity of the vortex core is made and subsequently nu-\nmerical solution of the vortex density away from the core\nhas been constructed, has given qualitative features of\nthe vortex size and density profile. In spirit with this ap-\nproximation, we propose that our perturbative approach\ntoo would be able to show the qualitative features of a\ndensity modulated state in a BEC.\nThe first thing to note is that the uniform density so-\nlutionψ0=√ne−iµt\n/planckover2pi1is also a solution of the modified\nGP equation, where µ=gnand|ψ0|2=nis the density\nof the condensate. Also note that we had to resort to a\nperturbative scheme here to study a pure density modu-\nlated state, sinceboth equations(1) and (3) donot admit\na solution of the form cos kx/sinkxeven in one dimen-\nsion as that would leave a cubic term in cos kx/sinkx\nunbalanced. Hence, the Taylor expansion is a key ingre-\ndient in the recipe of obtaining a long range order as we\nshall see in the next section.\nAMPLITUDE MODULATED PHASE\nLet us consider a solution of Eq.(4) of the form\nψ(x,t) =ψ(x)e−iωt\n/planckover2pi1whereωis the global oscillation fre-\nquency.ωgets identified as the chemical potential µof\nthe system in the case of a uniform ground state and\nwould be of the same order for the modulated density\nstate as we will see in the following. The uniform density\nGP ground state solution ψ0=√ne−iµt\n/planckover2pi1is still a solu-\ntion of Eq.(4) with the same free energy F=gN2\n2Vwherethe total number of particles N=n/integraltext/vectordr=nVandV\nis the volume. We would refer to this particular uniform\ndensity solution as the ground state frequently in what\nfollows. There could be other single particle states as\nthe solution of Eq.(1) as ψ(r,t) =ψ(x,t)ψ(y)ψ(z) =√nei(kx−ωt\n/planckover2pi1)whereψ(y) andψ(z) are unit constants.\nThese are the solutions of the local GP equation as well,\nwhere there is a kinetic energy cost which makes them\nhigher energy states as compared to the ground state at\nthe same density. These are moving solutions with ve-\nlocityv=/planckover2pi1k/m. The ground state is the k→0 limit of\nthese single particle states.\nEq.(4) also admits solution where ψ(x) =Acoskxor\nψ(x) =Asinkx. Substituting cos kxor sinkxin Eq.(4),\nwe get 1 −2a2k2+ 2a4k4−4a6k6\n3= 0 andω=/planckover2pi12k2\n2m+\ng|A|2(a2k2−a4k4+2a6k6\n3) , givingk2= 0.894/parenleftbig1\na2/parenrightbig\nand\nω=/planckover2pi12k2\n2m+(0.57)g|A|2,where we have kept terms upto\na6in the expansion.\nThe normalization condition over a length of 2 Land a\ncross section σof the condensate gives\nσ|A|2\n2/integraldisplayL\n−Ldx(1±cos2kx) =V|A|2/parenleftbigg1\n2±sin2kL\n4kL/parenrightbigg\n=N,\n(5)\nwhere the upper sign of ±is for the cos kxprofile and\nthe lower one is for the sin kxprofile (we will follow the\nsame convention in what follows) and V= 2σL.\nUsing the expression for free energy in Eq.(2) we can\ncomparethe energiesofthe density modulatedstate( Em)\nand that of the uniform density state( Eu). Fig.(2) shows\nthe difference between the energy densities of the two\nstates ∆f=Em−Eu, for certain values of the s-wave\nscattering length a. Note here that ∆ fgoing to negative\nvalues doesn’t mean that the energy Emis negative. Em\nis always positive. ∆ fbecoming negative only means\nthat as we change the density( n),Embecomes less than\nEu. In Fig.(2), the two for curves for which ∆ fbecomes\nnegative, the difference ∆ fbetweenEmandEuis always\nless thanEu=1\n2gn2, meaning if we add1\n2gn2to ∆f, the\nresultant term would never go to zero, thus stressing on\nthe point that Emis always positive.\nIn Fig.(2) , at lower density the energy of the den-\nsity modulated state Acoskxe−iωt//planckover2pi1is greater than the\nuniform density state√ne−iµt\n/planckover2pi1due to the kinetic energy\nterm−/planckover2pi12\n2m∇2ψwhich is non zero for the modulated den-\nsity state and zero for the uniform density state. The en-\nergyoftheuniformdensitystatecomesonlyfrominterac-\ntionsandhencetomaketheuniformdensitystatetobeof4\nhigher energy, we have to increase particle density which\nwould make the interaction energy increase and eventu-\nally, as seen from Fig.(2), make the modulated density\nstate energetically favourable under certain values of pa-\nrameters (s-wave scattering length ain Fig.(2)). The\ncrossover density is of the order of n∼k2\n8πa∼108cm−3.\nSo, the free energy difference decreases continuously be-\nyond the above mentioned limit with the increase in den-\nsity and, thus, fixing the density of the BEC at a suitable\nvalue one can expect to have such a state energetically\nfavourable. This figure clearly shows that there are wide\nregions over which the free energy density of the ordered\nphase is less than that of the uniform ground state. Note\nhere that although we are using a perturbative approach\nto find the kselection for the density modulated state,\nweuse the exactFreeenergyfunctional to find the energy\nof the density modulated state.\n-1.0x1013-5.0x10120.0x10005.0x10121.0x1013\n0.0x1002.0x1074.0x1076.0x1078.0x1071.0x108∆f\nna=0.56 x10-2 cm.\na=0.48 x10-2 cm.\na=0.32 x10-2 cm.\nFIG. 2. Figure shows plot of ∆ fversus the density( n) for\ncertain values of the s-wave scattering length a. L=0.02 cm.\nIf we take Lto be the length of the condensate andapply the periodic boundary conditions viz., ψ(x= 0) =\nψ(x=L) i.e., coskL= 1 , we get a selection on kand\nhence on the scattering length a. We find that the values\nofathat we get from the periodic boundary conditions\ncannot lower the energy of the density modulated state\nas compared to the uniform density state. To make the\nmodulated energy state energetically favourable, ahas\nto be about one order of magnitude less than that of L\nand this is where Feshbach resonance can come in handy.\nSince, the values of afor which the density modulated\nstate becomes energetically favourable does not obey pe-\nriodic boundary conditions, it is clear that there would\nbe an additional healing cost at the boundary. Since\nthis healing cost is also present for the uniform density\nstate hence the healing cost can be taken to be com-\nparable for both these states. We envisage that by us-\ning feshbach resonance, probably it would be possible to\nmake the boundary condition such that the energetically\nfavourable amplitude modulated state shows up.\nAlso,tonoteisthefactthathere,thermodynamiclimit\ndoes not make sense, because the values of the scattering\nlength for which the density modulated state is energeti-\ncally favourable is just 1 order of magnitude smaller and\nhence hence both Landawould always be comparable.\nAs already mentioned in the introduction, the density\nmodulated state, like the uniform density state would be\nunstable since it does not rest in an energy minima, as\nshown schematically in Fig.(1). We show this by a linear\nstability analysis similar to that for the uniform density\nstate, which goes as follows. To look at the stability of\nthese states, let us consider the specific case ψ(x,t) =\nAcos(kx)e−iωt//planckover2pi1and perturb it by the small amplitude\nmodesθ(x,t) =/summationtext\ni[ui(x)e−iωit\n/planckover2pi1+v∗\ni(x)eiωit\n/planckover2pi1]e−iωt//planckover2pi1.\nTaking the ansatz ui(x) =uieiqxandvi(x) =vieiqx,\nthe ensuing linear equations in the small amplitudes ui\nandviwould be of the form\n/parenleftBigg\n(α−Φ)a2A2gcoskx−24/planckover2pi12q2\nm+β+(α−Φ)a2A2gcoskx+48(ω−/planckover2pi1ωi)\n−24/planckover2pi12q2\nm+β+(α−Φ∗)a2A2gcoskx+48(ω+/planckover2pi1ωi) ( α−Φ∗)a2A2gcoskx/parenrightBigg/parenleftbiggui\nvi/parenrightbigg\n= 0,\n(6)\nwhere\nα=/parenleftBig\na4k6+3a2k4(−2+5a2q2)+q2(24−6a2q2+a4q4)+3k2(8−12a2q2+5a4q4)/parenrightBig\ncoskx,\nβ= 16a2A2k2(3−3a2k2+2a4k4)cos2kx,\nΦ = 2ikq/bracketleftBig\n24−12a2(k2+q2)+a4(3k4+10k2q2+3q4)/bracketrightBig\nsinkx.\nBecause of the presence of the irremovable imaginary term in the ex pression of Φ, ωiis always complex and5\nthis indicates an instability of the ordered phase arising\nout of the coupling of the small excitations to the ampli-\ntude modulated state in the correction term representing\nthe non-local interactions. Thus, within the scope of the\nperturbative approach adopted here to obtain an explicit\namplitude modulated solution of BEC using s-wave scat-\ntering, the amplitude modulated state is generically dy-\nnamically unstable but probablycould be seen on a short\nlived state in a BEC by appropriately tuning Feshbach\nresonance.\nDISCUSSIONS\nWe propose a new approach to obtain pure density\nmodulated state as a zero temperature state of the sys-\ntem by looking directly at the long range interactions be-\ntween particles of the BEC and considering their effects\non the order parameter. This approach is very different\nfrom the usual approach of looking at small amplitude\nexcitations on top of the uniform density state and low-\nering the roton minimum. By using Taylor expansion\ntechnique, we were able to obtain a density modulated\nstate as a perturbative solution to GP equation with long\nrange order. Perturbative approach had to be used as\nthe conventional GP equation does not admit modulated\ndensity solutions. Although, the solution is perturbative,\nweusetheexactfreeenergyfunctionaltoevaluateenergy.\nThus using this sort ofperturbative scheme starting from\nthe GP equation is the key idea of this article.\nBy energy calculations, we have shown that there exist\nrange of the scattering length awhere this density mod-\nulated state becomes energetically favourable as com-\npared to the uniform density state. This range of a\ndepends on the length of the condensate and is an or-\nder of magnitude smaller than the condensate length.\nThe crossover density we have obtained is of the order\nof 108cm−3. This shows that the density modulations\nwould manifest themselves even when the density is as\nlow as 108- 1010cm−3. Also, we have seen that the a\nselection obtained by the periodic boundary conditions\ndoes not enable making the density modulated state en-\nergetically favourable, meaning that the energy lowering\nas compared to the uniform density state, comes from\nthe boundaries. Since, there is only a specific range of a\nfor which this energy lowering is possible, Feshbach res-\nonance can play a key role in obtaining a density modu-\nlated state.We have shown that as such density modulated state,\neven though energetically favourable than the uniform\ndensity state, isn’t stable under small amplitude oscilla-\ntions as it does not sit in an energy minima. This insta-\nbility is present in the case of the uniform density state\ntoo. Thus, it is clear that we have to go beyond the s-\nwave interactions in hope of stabilizing such a modulated\ndensity state. In this regard, we would look to dipolar in-\nteractions within the perturbativbe scheme that we have\nproposed, in the future. We feel that by looking at the\nperturbative scheme that we have proposed, it might be\npossible to make many more predictions about the vari-\nous aspects of BEC.\n∗abhijeet.pendse@students.iiserpune.ac.in\n†a.bhattacharyay@iiserpune.ac.in\n[1] A.Sorensen, L.-M.Duan,J. I.Cirac andP.Zoller, Nature\n409, 63-66 (2001).\n[2] Max F. Reidel et al., Nature 464, 1170-1173 (2010).\n[3] V. Bolpasi et al., New J. Phys. 16033036 (2014).\n[4] W. Guerin et al., Phys. Rev. Lett. 97, 200402 (2006).\n[5] O. Zobay, E. V. Goldstein and P. Meystre, Phys. Rev. A\n60, 3999 (1999)\n[6] H. Muntinga et al., Phys. Rev. Lett. 110, 093602 (2013).\n[7] L. V. Hau, S. E. Harris, Z. Dutton and C. H. Behroozi ,\nNature397, 594-598 (1999).\n[8] D. Kadio and Y. B. Band , Phys. Rev. A 74, 053609\n(2006).\n[9] O. Lahav et al., Phys. Rev. Lett. 105, 240401 (2010).\n[10] O. Penrose and L. Onsager , Phys. Rev. 104, 576 (1956).\n[11] G. V. Chester, Phys. Rev. A 2, 256 (1970).\n[12] A. J. Leggett, Phys. Rev. Lett. 25, 1543 (1970).\n[13] T. Keilmann, I. Cirac and T. Roscilde, Phys. Reb. Lett.\n102, 255304 (2009).\n[14] N. Henkel, R. Nath and T. Pohl, Phys. Rev. Lett. 104,\n195302 (2010).\n[15] Y. Pomeau and S. Rica , Phys. Rev. Lett. 71, 247 (1993).\n[16] Y. Pomeau and S. Rica , Phys. Rev. Lett. 72, 2426\n(1994).\n[17] S. Inouye et al., Nature 392, 151-154 (1998) .\n[18] S. Sarkar and A. Bhattacharyay, J. Phys. A: Math.\nTheor.47, 092002 (2014).\n[19] Bose-Einstein Condensation, L. Pitaevskii and S.\nStringari, Oxford Science Publications (2003) .\n[20] S. Sinha and Y. Castin, Phys. Rev. Lett. 87, 190402\n(2001).\n[21] A. L. Fetter and D. Rokhsar , Phys. Rev. A , 57, 1191\n(1998)."    },    {        "title": "1510.09172v1.Density_of_states_in_gapped_superconductors_with_pairing_potential_impurities.pdf",        "content": "arXiv:1510.09172v1  [cond-mat.supr-con]  30 Oct 2015Density of states in gapped superconductors with pairing-p otential impurities\nAnton Bespalov,1,2Manuel Houzet,1Julia S. Meyer,1and Yuli V. Nazarov3\n1Univ. Grenoble Alpes, INAC-SPSMS, F-38000 Grenoble, Franc e;\nCEA, INAC-SPSMS, F-38000 Grenoble, France.\n2Institute for Physics of Microstructures, Russian Academy of Sciences, 603950 Nizhny Novgorod, GSP-105, Russia\n3Kavli Institute of NanoScience, Delft University of Techno logy,\nLorentzweg 1, NL-2628 CJ, Delft, The Netherlands.\nWe study the density of states in disordered s-wave supercon ductors with a small gap anisotropy.\nDisorder comes in the form of common nonmagnetic scatterers and pairing-potential impurities,\nwhich interact with electrons via an electric potential and a local distortion of the superconducting\ngap. A set of equations for the quasiclassical Green functio ns is derived and solved. Within one spin\nsector, pairing-potential impurities and weak spin-polar ized magnetic impurities have essentially\nthe same effect on the density of states. We show that if the gap is isotropic, an isolated impurity\nwith suppressed pairing supports an infinite number of Andre ev states. With growing impurity\nconcentration, theenergy-dependentdensityofstates evo lvesfrom asharpgapedgewithanimpurity\nband below it to a smeared BCS singularity in the so-called un iversal limit. If a gap anisotropy is\npresent, the density of states becomes sensitive to ordinar y potential disorder, and the existence of\nof Andreev states localized at pairing-potential impuriti es requires special conditions. An unusual\nfeature related totheanisotropy is anonmonotonic depende nceofthegap edgesmearing onimpurity\nconcentration.\nPACS numbers: 74.62.En\nI. INTRODUCTION\nThermodynamic and transport properties of disor-\ndered superconductors crucially depend on the symme-\ntry of superconducting pairing as well as on the nature\nof the impurities that scatter the electron waves. It is\nwidely known that conventional scatterers described by\ncoordinate-dependentpotentialshardlyaffectthe density\nof states or the order parameter in superconductors with\nconventional spin-singlet s-wave pairing1–4. Their main\neffect is the suppression of the anisotropic part of the or-\nder parameter, which is small as far as the anisotropic\npart is small. By contrast, for unconventional supercon-\nducting pairing that is essentially anisotropic, the effect\nof potential disorder on the density of states is drastic.\nFor instance, even a single potential scatterer in a d-wave\nsuperconductor brings about a quasibound state local-\nized at the defect5, while a large concentration of defects\nleads to the complete suppression of superconductivity.\nThe situation is different for magnetic impurities6. In\nans-wave superconductor, a single magnetic impurity\ninduces a localized state, known as a Yu7, Shiba8, or\nRusinov9state, with an energy below the gap edge. If\nthe exchange field of the magnetic impurity is weak, the\nimpurity state is formed close to the gap edge. At fi-\nnite impurity concentration, cimp, the Shiba states hy-\nbridize and form an impurity band that becomes wider\nwithincreasing cimp. Simultaneously,thedisordersmears\nthe BCS singularity in the density of states at the gap\nedge. The impurity band is initially concentrated around\nthe energy of the single-impurity bound state, yet widens\nwith increasing cimp. It eventually merges with the con-\ntinuum spectrum above the gap edge and fills the whole\nsuperconducting gap. This explains the phenomenon ofgapless superconductivity6.\nA separate class of disorder in superconductors is due\nto the inhomogeneities of the superconducting pairing\npotential, which can be induced, for instance, by random\nspatial variations of the coupling constant. Larkin and\nOvchinnikov10demonstrated the smearing of the BCS\nsingularity by disorder of this type (see also Refs. 11 and\n12). The shape of the smearing is essentially the same as\nformagneticdisorderandwasarguedtobe universal12–14\nfor all depairing mechanisms. The absence of impurity\nbands in Refs. 10–12 at low disorder is a property of\nthe model: here, the pairing-potential disorder was not\nassociated with distinct impurities. A different situa-\ntion, corresponding to pairing-potential impurities not\noverlapping with other impurities, has been analyzed in\nRefs. 15–18 (see also references therein for studies of d-\nwave superconductors). According to Refs. 15, 17, and\n18, a point-like impurity with suppressed pairing always\nsupports a bound state. A numerical study of impurities\nwith a size of the order of the Fermi wavelength λF16\ndid not find such a state when the ratio of the coherence\nlength toλFwas sufficiently large. The formation of an\nimpurity band at small impurity concentrations was dis-\ncussed in Ref. 15.\nGenerally, one may expect that, upon increasing the\nconcentrationof pair-breakingimpurities, there is a com-\nplex crossover in the density of states near the gap edge.\nOne starts from discrete impurity states below the gap\nedge. Theyformanarrowimpuritybandthatwidensand\nmerges with the gap edge at some critical concentration.\nUpon further increasing the concentration, the complex\nshape ofthe density ofstates near the edge simplifies, ap-\nproaching a universal one. The common potential scat-\nterers do not influence this crossover, if the anisotropy\nof the pairing potential is neglected. However, in real-2\nistic situations, the anisotropy also modifies the density\nof states near the gap edge. In this paper, we present a\ndetailed analysisofthe crossover,thus providingmorein-\nsight into the propertiesofthe bound quasiparticlestates\nnear the gap edge.\nThe impurity model we are mainly concerned with is\na nonmagnetic scatterer that brings about a variation\nof the pair potential on a scale L≪ξS, whereξSis\nthe coherence length in the pure limit. We evaluate\nthe quasiclassical Green functions using the T-matrix\napproximation19. We find that the behavior of the den-\nsity of states in the cases of pairing potential impuri-\nties andweakmagneticimpurities is essentiallythe same,\nin a given spin sector, provided the latter are polarized\nalong the same axis. This analogy has strong implica-\ntions, reproducing the sequence of the crossovers men-\ntioned above. However, in contrast to magnetic impuri-\nties, the pair breaking impurities cannot completely close\nthe superconducting gap at any realistic concentration.\nFor localized impurity states, we demonstrate that a\nspherically symmetric impurity with local suppression of\nthe order parameter gives rise to an infinite number of\nsubgap bound states. We give explicit expressionsfor the\nenergiesElof the states with orbital momentum l, and\nfor the widths of the impurity bands at small impurity\nconcentrations. The energy scale involved, ∆ 0−El, is\nof the order of ∆ 0(L/ξS)2≪∆0, where ∆ 0is the bare\nsuperconducting order parameter.\nIt is almost forgotten nowadays that real supercon-\nductors have a slightly anisotropic gap, and with this\nthe density of states is sensitive to common potential\ndisorder1–4. We derive the condition for the existence\nof impurity states, which is modified by the anisotropy.\nA qualitative feature related to the anisotropy is a non-\nmonotonic dependence of the gap-edge smearing on im-\npurity concentration.\nThe paper is organized as follows. In Sec. II we in-\ntroduce the model for the impurities and derive gen-\neralequationswithin the T-matrixapproximationforthe\nquasiclassical Green functions. In Sec. III we analyze the\ncase of isotropic pairing. In the limit of vanishing im-\npurity concentration, we elucidate the properties of the\nimpurity-bound states. At finite concentrations, we in-\nvestigate and illustrate the crossover mentioned above.\nSection IV considers the effects of a pairing anisotropy.\nWe show that, in dirty superconductors, the remaining\nanisotropy leads to a “universal” broadening of the gapedge. Thus, in general, the anisotropy affects the pres-\nence of impurity bound states, and we derive the con-\ndition for this. We analyze in detail the case of dirty\nsuperconductors at finite concentration of potential im-\npurities and illustrate this with plots. We give our con-\nclusions in Sec. V. Several technical details are relegated\nto Appendices.\nII. GENERAL RELATIONS FOR THE GREEN\nFUNCTION AND THE T-MATRIX\nA general disordered superconductor can be character-\nized by a Hamiltonian\nˆH=/summationdisplay\nα/integraldisplay\nˆψ+\nα(r)/bracketleftbigg\n−/planckover2pi12\n2m∂2\n∂r2−µ+V(r)/bracketrightbigg\nˆψα(r)d3r+ˆHS,\n(1)\nwhere\nˆHS=/integraldisplay\n∆∗/parenleftbiggp+k\n2,p−k/parenrightbigg\nˆψ↓(p)ˆψ↑(−k)d3p\n(2π)3d3k\n(2π)3\n+h.c. (2)\ndescribes electron pairing within mean-field theory. Here\nˆψα(r) andˆψ+\nα(r) are the electron field operators, α={↑\n,↓}is a spin label, µis the chemical potential, V(r) is an\nelectric impurity potential, and\nˆψα(p) =/integraldisplay\nˆψα(r)e−iprd3r,ˆψ+\nα(p) =/integraldisplay\nˆψ+\nα(r)eiprd3r.\n(3)\nNote that the pairing potential ∆ depends on two argu-\nments, which reflect the pairing strength along the Fermi\nsurface and its spatial variation, respectively.\nTo determine the density of states associated with the\nHamiltonian (1), we introduce the real-time retarded\nGreen functions defined as\nG(r,r′,t) =−i/angbracketleftBig\nˆψ↓(r,t)ˆψ+\n↓(r′,0)+ˆψ+\n↓(r′,0)ˆψ↓(r,t)/angbracketrightBig\n,\nF(r,r′,t) =i/angbracketleftBig\nˆψ↓(r,t)ˆψ↑(r′,0)+ˆψ↑(r′,0)ˆψ↓(r,t)/angbracketrightBig\n,\nF+(r,r′,t) =i/angbracketleftBig\nˆψ+\n↑(r,t)ˆψ↓(r′,0)+ˆψ↓(r′,0)ˆψ+\n↑(r,t)/angbracketrightBig\n,\n¯G(r,r′,t) =i/angbracketleftBig\nˆψ+\n↑(r,t)ˆψ↑(r′,0)+ˆψ↑(r′,0)ˆψ+\n↑(r,t)/angbracketrightBig\n(4)\natt>0 , andG=F=F+=¯G= 0 att<0. Here, the\nfield operators ˆψare in the Heisenberg representation.\nThe Green functions satisfy the conventional Gor’kov\nequation, which in momentum representation reads\n/parenleftbigg\nE+iǫ+−ξ(p) 0\n0 −E−iǫ+−ξ(p)/parenrightbigg\nˇGE(p,p′)\n−/integraldisplay\nV(p−k)−∆/parenleftBig\np+k\n2,p−k/parenrightBig\n∆∗/parenleftBig\np+k\n2,k−p/parenrightBig\nV(p−k)\nˇGE(k,p′)d3k\n(2π)3= (2π)3δ(p−p′)ˇ1, (5)3\nwhereǫ+is an infinitely small positive quantity, ξ(p) is the kinetic energy measured from the Fermi level,\nξ(p) =/planckover2pi12p2\n2m−µ=/planckover2pi12\n2m(p2−k2\nF), (6)\nwith the Fermi wave number kF= 2π/λF,V(p) is the Fourier transformed electric potential, and ˇGEis a matrix\ncomposed of the Fourier transformed Green functions,\nˇGE(p,p′) =/parenleftbiggGE(p,p′)FE(p,p′)\n−F+\nE(p,p′)¯GE(p,p′)/parenrightbigg\n=/integraldisplay/parenleftbigg\nG(r,r′,t)F(r,r′,t)\n−F+(r,r′,t)¯G(r,r′,t)/parenrightbigg\neiEt//planckover2pi1−ipr+ip′r′d3rd3r′dt\n/planckover2pi1.(7)\nIn a clean superconductor, V= 0, the order parame-\nter is spatially uniform, ∆( Q,q) = (2π)3∆0(Q)δ(q), and\nthe translation invariance of the Green function yields\nˇGE(p,p′) = (2π)3ˇG(0)\nE(p)δ(p−p′). Using Eq. (5), we\nobtain\nˇG(0)\nE(p)=/parenleftbigg\nE+iǫ+−ξ(p) ∆ 0(p)\n−∆∗\n0(p)−E−iǫ+−ξ(p)/parenrightbigg−1\n.(8)\nFor a start, let us assume that the disorder in the su-\nperconductor is induced by identical impurities with size\nL, whosepositionsaregivenbyasetofvectors Ri. Then,\nthe pairing potential and the electric potential are\n∆(Q,q) = (2π)3∆0(Q)δ(q)+∆1(Q,q)/summationdisplay\nie−iqRi,(9)\nV(q) =U(q)/summationdisplay\nie−iqRi, (10)\nwhere the functions ∆ 1(Q,q) andU(q) give the distor-\ntion of the pairing potential and the electric potential\ninduced by a single impurity, respectively. We will eval-\nuate the Green functions averaged over impurity posi-\ntions,∝an}bracketle{tˇGE(p,p′)∝an}bracketri}htav, assuming a homogeneous distribu-\ntion of the impurities. Then, the averaging procedure\nrestores translational invariance, so that ∝an}bracketle{tˇGE(p,p′)∝an}bracketri}htav=\n(2π)3ˇGE(p)δ(p−p′).\nUsually the impurity potential is taken into account\nin the second-order Born approximation20. For our pur-\nposes this is not sufficient, since this approach does not\nyield localized impurity states. Instead, we make use of\nthe more general T-matrix approximation (see Ref. 19,\nfor example), which accounts for multiple scattering off\neach impurity. Within this appoximation, we will derive\nan equation for the quasiclassical Green functions.\nThe T-matrix and the Green functions are determined\nfrom the following system of equations:\nˇTE(p,p′) =ˇVimp(p,p′) (11)\n+/integraldisplay\nˇVimp(p,k)ˇGE(k)ˇTE(k,p′)d3k\n(2π)3,\nˇGE(p) =/bracketleftBig\nˇG(0)\nE(p)−1−cimpˇTE(p,p)/bracketrightBig−1\n,(12)where\nˇVimp(p,k) =\nU(p−k)−∆1/parenleftBig\np+k\n2,p−k/parenrightBig\n∆∗\n1/parenleftBig\np+k\n2,k−p/parenrightBig\nU(p−k)\n.\n(13)\nEquations (11) and (12) can be simplified for momenta\nclose to the Fermi surface. Assuming that the Fermi\nenergy is the largest energy scale, let us introduce the\nquasiclassical Green functions,\nˇg(E,n) =i\nπ/integraldisplay\nˇGE(pn)dξ(p), (14)\nwherenis a unit vector, and integration is performed\nover a relatively small energy range, |ξ(p)| ≪µ. For sim-\nplicity, we restrict ourselves to the case of real functions\n∆0(Q) and ∆ 1(Q,q) (a phase shift between ∆ 0and ∆ 1\nwould manifest the violation of time-reversal symmetry).\nThen, the matrix ˇ ghas only two independent compo-\nnents,\nˇg(E,n) =/parenleftbigg\ng1(E,n)g2(E,n)\n−g2(E,n)−g1(E,n)/parenrightbigg\n.(15)\nThe density of states per spin is given by\nν(E) =ν0/integraldisplay\nℜ[g1(E,n)]dn\n4π, (16)\nwhereν0=k3\nF/(4π2µ) is the density of states at the\nFermi surface in the normal state for one spin direction.\nUnder the assumptions that the dependence of ∆ 0(pn)\nandˇTE(pn,pn) onpmay be neglected when pis close to\nkF, it can be proved (see Appendix A) that the matrix\nˇgsatisfies the relations\nˇg(E,n)ˇSE(n)−ˇSE(n)ˇg(E,n) = 0 (17)\nand\ng2\n1(E,n)−g2\n2(E,n) = 1, (18)\nwhere\nˇSE(n) =/parenleftbigg\nE+iǫ+∆0(n)\n−∆0(n)−E−iǫ+/parenrightbigg\n−cimp\nπν0ˇTE(n,n) (19)\nwith ∆ 0(n)≡∆0(kFn), and ˇTE(n,n′)≡\nπν0ˇTE(kFn,kFn′). Actually, Eq. (17) is the standard4\nEilenberger equation for a macroscopically homogeneous\nsuperconductor20. Equation (18) expresses the normal-\nization condition ˇ g2=ˇ1 for the quasiclassical Green\nfunction in the Eilenberger equation.\nWe assume that the spatial range of the pairing po-\ntential distortion ∆ 1and of the electric potential Uis\nmuchsmallerthanthecoherencelengthinthecleanlimit,\nξS=/planckover2pi1vF/π∆0, wherevF=/planckover2pi1kF/mis the Fermi veloc-\nity. In this case, Eq. (11) can be further simplified. To\ndothis, weintroduceanauxiliarynormal-statescattering\nmatrixˇf(n,n′) that satisfies the equation\nˇf(p,p′) =ˇVimp(p,p′)+/integraldisplay\nˇf(p,k)ˇG(k)ˇVimp(k,p′)d3k\n(2π)3,\n(20)\nwhereˇG(k) =ˇG(0)\nE(k) taken at ∆ 0= 0 andE= 0. The\ndiagonal components of ˇf(p,p′) have the meaning of the\ndimensionless electron and hole scattering amplitudes off\nan impurity in the normal state. The off-diagonal com-\nponents are the amplitudes of Andreev reflection of elec-\ntrons and holes. Within the quasiclassical approxima-\ntion, Eq. (11) can then be rewritten as (see Appendix\nA)\nˇTE(n,n′)=ˇf(n,n′) (21)\n+i/integraldisplay\nˇf(n,n′′)[ˇτz−ˇg(E,n′′)]ˇTE(n′′,n′)dn′′\n4π,\nwhere ˇτzisthethird Paulimatrixactingin Nambuspace,\nandˇf(n,n′)≡πν0ˇf(kFn,kFn′). In Appendix B the\nmatrixˇf(n,n′) is calculated for a spherically symmetric\nimpurity with\nˇVimp(p,k) =/parenleftbigg\nU(p−k)−∆1(p−k)\n∆1(p−k)U(p−k)/parenrightbigg\n.(22)\nIn the next two sections, we will solve the equations\nfor the matrices ˇTEand ˇgand analyze the resulting den-\nsity of states in the cases of an isotropic and weakly\nanisotropic gap ∆ 0(n), respectively.\nIII. SUPERCONDUCTOR WITH AN\nISOTROPIC GAP\nWe start with the case of isotropic pairing, when\n∆0(n) = const and ∆ 1(Q,q) = ∆ 1(q). Without loss\nof generality, we may then choose ∆ 0>0. If, addition-\nally, the impurities are spherically symmetric, the matrix\nˇg(E,n) will not depend on n, and the matrix ˇfwill have\nthe form\nˇf(n,n′) =/parenleftbigg\nf1(n,n′)f2(n,n′)\n−f∗\n2(n,n′)f∗\n1(n,n′)/parenrightbigg\n.(23)To solve Eq. (21), we expand ˇfandˇTEin terms of Leg-\nendre polynomials Pl:\nˇTE(n,n′) =∞/summationdisplay\nl=0(2l+1)ˇTl(E)Pl(n·n′),(24)\nˇf(n,n′) =∞/summationdisplay\nl=0(2l+1)ˇflPl(n·n′). (25)\nUsing the addition theorem for spherical harmonics\n[Eq. (B22)], on can show that this leads to separateequa-\ntions for the components ˇTlwith different orbital indices\nl. In particular, Eq. (21) yields\nˇTl(E) =/braceleftbigˇ1−iˇfl[ˇτz−ˇg(E)]/bracerightbig−1ˇfl.(26)\nTo transform the right-hand side of this relation, it is\nconvenient to use Eq. (B33), which is a corollary of a\ngeneralized optical theorem [Eq. (B32)]. We obtain from\nEqs. (24) and (26)\nˇTE(n,n) =∞/summationdisplay\nl=0(2l+1)ˇfl+i[ˇg(E)−ˇτz]ℑ[f1l]\n1−2if2lg2(E).(27)\nSubstituting Eq. (27) into Eq. (17) yields\nEg2(E)−/bracketleftBigg\n∆0−cimp\nπν0∞/summationdisplay\nl=0(2l+1)f2l\n1−2if2lg2(E)/bracketrightBigg\ng1(E) = 0.\n(28)\nThus, Eq. (28) defines the Green functions in terms of\nthe off-diagonal scattering amplitudes f2l. Note that a\nvery similar relation can be derived for weak polarized\nmagnetic impurities, see Sec. IIIC.\nNear the gap edge, when |E−∆0| ≪∆0, bothg1and\ng2are large, |g1|,|g2| ≫1, and the normalization condi-\ntion (18) gives g2≈g1−1/2g1. Thus, Eq. (28) may be\nreduced to an equation for g1only. Namely,\n(E−∆0)g1−∆0\n2g1+cimp\nπν0∞/summationdisplay\nl=0(2l+1)f2l\n1−2if2lg1g1= 0.(29)\nAn explicit calculation of the coefficients f2lis given\nin Appendix B. Under the assumption that |f2l| ≪1 and\nforl2≪kFξS, we find that these coefficients are given\nby\nf2l=−π2ν0\nk2\nF/integraldisplay∞\n0∆1(r)|ul(r)|2dr. (30)\nThe functions ul(r), defined in Appendix B, are the so-\nlutions of the Schr¨ odinger equation in the normal state\nin the presence of the electric potential U(r) only. If,\nfurthermore, l+ 1/2/lessorsimilarkFL, the amplitudes f2lcan be\nestimated as\nf2l∼∆1\n∆0L\nξS. (31)\nThus, the applicability condition of Eq. (30), |f2l| ≪1,\nis satisfied in the realistic situation when |∆1|/lessorsimilar∆0.5\nWe would like to point out that within our model, in\nfull agreement with Anderson’s theorem1, common po-\ntential impurities have no effect on the density of states,\nsincef2l= 0 for such impurities, and thus their T-matrix\ncommutes with ˇ g. Hence, Eq. (29) is not modified if the\nmaterial is in the dirty limit with respect to potential\ndisorder, i.e., ∆ 0τ≪/planckover2pi1, whereτis the mean free time\ndue to this disorder.\nA. Impurity states\nA defect with suppressed pairing, i.e., ∆ 1(r)<0, sup-\nports a set of localized Andreev states that are similar\nto the well-known Shiba states7–9generated by magnetic\nimpurities. For a point-like defect the existence of a sin-\ngle Andreev state has been predicted in Refs. 15 and\n17. Gunsenheimer and Hahn21found multiple localized\nstates for a sufficiently large pairing defect with L≫λF.\nHere, we generalize these results, demonstrating that a\ndefect with ∆ 1<0, in fact, supports an infinitenumber\nof Andreev states.\nTo calculate the energies of the localized quasiparticle\nstates, one has to determine the poles of the T-matrix\natcimp→0 (or, equivalently, solve the Bogoliubov-de\nGennes equation, see Appendix C). They are obtained\nfrom the equation\n1−2if2lg2(E) = 0, (32)\nwhere the function g2(E) is taken at cimp= 0, i.e.,\ng2(E) =−i∆0//radicalbig\n∆2\n0−E2. Since we assume |f2l| ≪1,\nthe energies Elof the bound states lie close to the gap\nedge and are given by El= ∆0−El, where\nEl= 2f2\n2l∆0. (33)\nAs stated above, Eq. (33) is applicable for l2≪kFξS.\nHowever, this does not limit the number of bound states:\nas shown in Appendix C, there are bound states at arbi-\ntrary large l. For typical impurity parameters kFL∼1,\n|∆1|/lessorsimilar∆0, we have El∼∆3\n0/µ2whenlis of the order\nof unity. At larger orbital momenta, the energies of the\nbound states quickly approach the gap edge with grow-\ningl. Explicit expressions for the quasiparticle energies\nin the particular case of a step-like function ∆ 1(r) are\nderived in Appendix C.\nB. Finite impurity concentration\nAbove we showed that a single impurity produces\nbound states. At a finite impurity concentration, one ex-\npects these states to hybridize and form impurity bands\nthat may merge with the continuum at a sufficiently high\nimpurity concentration. We employ a standard simpli-\nfying assumption of pure s-wave scattering, neglecting\nall scattering amplitudes f2l, exceptf20. We do this to\nrestrict ourselves to a single bound state, avoiding theconsideration of a complex series of bound states, cor-\nresponding to higher orbital momenta. The assumption\nof pures-wave scattering is justified if kFL∼1, so that\nf20/lessorsimilar∆1/µ. Then, it is convenient to characterize the\nimpurity concentration by the maximum scattering rate\n1\nτu=2cimp\n/planckover2pi1πν0, (34)\nproduced by these impurities in the unitary limit. To re-\nducethe numberofparametersinEq.(29), werescalethe\nGreen functions as well as energy and impurity concen-\ntrations, introducing the following dimensionless quanti-\nties:\nG1=/radicalbigg\n2E0\n∆0g1,Ω =E−∆0\nE0, P=/planckover2pi1√\n2\n4√E0∆0τu.\n(35)\nIt can be seen that the values P∼1 are achieved at\n/planckover2pi1/τu∼√E0∆0. In the new notations, Eq. (29) takes the\nform\nΩG1−1\nG1±PG1\n1∓iG1= 0, (36)\nwhere one should take the upper sign for f20>0 (cor-\nresponding to ∆ 1<0), and the lower sign for f20<0\n(corresponding to ∆ 1<0).\nWhenf20<0, one finds a renormalized gap edge with\na broadened BCS singularity. When f20>0, such that\nan individual impurity hosts bound states, the localized\nstates overlap at finite impurity concentration. At 0 <\nP≪1, they form a band centered around Ω = −1 with\na width\nW= 4√\n2P. (37)\nUpon further increasing the impurity concentration, at\nP= 8/27 the impurity band merges with the continuum.\nThe change of the energy dependence of the density of\nstates with increasing Pis illustrated in Fig. 1. When\n-3 -2 -1 00123\nP = 0.05\nP = 0.2\nP = 0.4(a)\n0 1 2 3 4 5 60123(b)\nP= 0.5\nP= 1\nP= 3\nP= 5\nP= 7\nFigure 1. Energy dependence of the density of states in\na superconductor with an isotropic gap containing pairing-\npotential impurities [Eq. (36)]. (a) f20>0, (b)f20<0.\nP≫1 the absolute value of G1becomes small at all\nvalues of Ω, and Eq. (36) takes the form\n(Ω±P)G1−1\nG1+iPG2\n1= 0. (38)6\nThis equation describes the behavior of the Green func-\ntion in superconductors with pair-breaking impurities of\ndifferent nature in the so-called universal limit, i.e., at\nsufficiently large impurity concentrations12–14. In this\nlimit, the smoothing of the BCS singularity is commonly\ncharacterized by an effective depairing rate12, which\nequals in our case\n1\nτdep=f2\n20\nτu. (39)\nIt follows from Eq. (38) that the gap edge is smeared on\na scale of the order of\nδΩ∼P2/3, (40)\nandthereisanadditionalshift ofthegapedgeby ∓Pdue\nto the average pairing suppression/enhancement by the\nimpurities. Thecharacteristicvaluesofthe Greenfuction\nare of the order of G1∼P−1/3. These observations allow\nto rewrite Eq. (38) in a form containing no parameters.\nNamely,\nΩ′G′\n1−1\nG′\n1+iG′2\n1= 0, (41)\nwhere Ω′= (Ω±P)P−2/3andG′\n1=G1P1/3. Note that\nthe universal limit is approached rather slowly: correc-\ntions toG′\n1are of the order P−1/3.\nUsing Eq. (29), now taking into account all compo-\nnentsf2l, we arrive at the following equation for g1in\nthe universal limit:\n/bracketleftBigg\nE−∆0+/planckover2pi1\n2τmin/summationdisplay\nl(2l+1)f2l/bracketrightBigg\ng1−∆0\n2g1+i/planckover2pi1\nτdepg2\n1= 0,\n(42)\nwhere the depairing rate equals\n1\nτdep=1\nτmin∞/summationdisplay\nl=0(2l+1)(f2l)2=1\nτmin/integraldisplay\nf2\n2(n,n′)dn′\n4π.\n(43)\nLet us point out that Eq. (42) was also obtained in the\nseminal paper by Larkin and Ovchinnikov10for a differ-\nent model of pairing-potential disorder. Namely, they as-\nsumed that the coupling constant exhibits fluctuations.\nThen, on the basis of the distribution of the coupling\nconstants, the distortion of the pairing potential was cal-\nculated. This implies that there are fluctuations of the\npairingpotentialonascalethatexceeds ξS. Onthe mean\nfield level, this leads to a smoothing of the gap edge with\na universal shape described by Eq. (42). They evaluated\nthe depairing rate for a superconductor with an arbitrary\nmean free path – see Eq. (21) in Ref. 10. For an infinite\nmean free path and L≪ξS, that equation yields\n/parenleftbigg1\nτdep/parenrightbigg\nLO=πν0cimp\n/planckover2pi1k2\nF/integraldisplay /integraldisplay∆1(r)∆1(r′)\n|r−r′|2d3rd3r′.(44)\nWithin our model, we find the same result in the quasi-\nclassicallimit ( L≫k−1\nF) and forU= 0, when one should\nsubstitutef2(n,n′) =πν0∆1(kF(n−n′)) in Eq. (43).C. Comparison with magnetic impurities\nLet us compare our results with the case of magnetic\nimpurities that has been extensively studied in the liter-\nature. To describe such impurities, we add the following\nspin-dependent term to the Hamiltonian (1):\nˆHM=/summationdisplay\nα,β/integraldisplay\nˆψ+\nα(r)[J(r)·ˆσ]αβˆψβ(r)d3r,(45)\nwhereJ(r) is the exchange field, and ˆσis a vector com-\nposed of Pauli matrices acting in spin space. The ex-\nchange field is given by\nJ(r) =/summationdisplay\niJ1(r−Ri)Si, (46)\nwhereJ1>0, and the unit vectors Sispecify the polar-\nizations of the impurities.\nLet us first assume that all impurities are polarized in\nthesamedirection, i.e., allvectors Siareidentical. When\nevaluating the Green functions, we may now neglect the\ndistortion of the pairing potential induced by the impu-\nrities, since its effect on the density of states in realistic\nsituations is much smaller than the influence of the ex-\nchangefield9. Then, within the T-matrix approximation,\nwe obtain the following relation [a similar calculation has\nbeen previously done in Ref. 22 for point-like impurities]:\nEg2(E)−∆0g1(E)±cimp\nπν0∞/summationdisplay\nl=0(2l+1)fM\nl\n1∓2ifM\nlg1(E)g2(E) = 0.\n(47)\nHere, the upper/lower sign corresponds to “spin-\nup”/“spin-down” electrons with respect to the polariza-\ntion direction, and fM\nlare the differences of scattering\namplitudes of “spin-up” and “spin-down” electrons. Un-\nder the constraints/vextendsingle/vextendsinglefM\nl/vextendsingle/vextendsingle≪1 andl2≪kFξSone can\nprove that the magnetic coefficients fM\nlare given by ex-\npressions similar to Eq. (30),\nfM\nl=−π2ν0\nk2\nF/integraldisplay∞\n0J1(r)|ul(r)|2dr. (48)\nIt can be seen that Eqs. (47) and (28) have almost the\nsame form, the only difference being the permutation of\ng1andg2in the last term. However, this difference is not\nessential near the gap edge, when |E−∆0| ≪∆0. As\nnoted earlier, in that case g2≈g1−1/2g1, and Eq. (47)\nyields\n(E−∆0)g1±cimp\nπν0∞/summationdisplay\nl=0(2l+1)fM\nl\n1∓2ifM\nlg1g1= 0.(49)\nComparing Eqs. (29) and (49), we see that pairing-\npotential impurities with ∆ 1>0 act like magnetic im-\npurities in the “spin-up” sectorwhereas pairing-potential\nimpurities with ∆ 1<0 act like magnetic impurities in\nthe “spin-down” sector. The full density of states in the7\ncase of magnetic impurities is obtained by summing over\nboth spin sectors.\nThe comparison with magnetic impurities can be ex-\ntended to the case of randomly oriented spins. To see\nthis, we recast the relation for the Green function de-\nrived in Ref. 9 to a form similar to Eq. (36),\nΩG1−1\nG1+iPG2\n1\n1+G2\n1= 0. (50)\nThis equation is obtained by averaging Eq. (36) over\nimpurity-spindirections, andaccordingly G1istheGreen\nfunctionaveragedoverimpurity-spindirections. Atsmall\nP, Eq. (50) gives an impurity band with a width W=\n4√\nP. The merger of this band with the continuum oc-\ncurs atP≈0.49. The plots shown in Fig. 2 demonstrate\nthe qualitative similarity of the energy dependent densi-\nties of states derived from Eqs. (36) and (50).\nWhenP≫1, Eq. (50) reduces to the relation in the\nuniversal limit, Eq. (41). Note than due to the averag-\ning over spin directions, unlike in Eq. (38), there is no\nadditional gap shift ∓P. Moreover, the universal limit\nis approached faster than in the case of pairing-potntial\nimpurities or polarized magnetic imputrities, as correc-\ntions to the Green function averaged over impurity-spin\ndirectionsG′\n1are of the order P−2/3only.\nFinally, let us mention that in the case of relatively\nstrong magnetic impurities (with J1≫∆0) a gapless\nregime can be reached. By contrast, within the field\nof applicability of our approach, a superconductor with\npairing-potential impurities is always gapped. Indeed, to\nreachthe gaplessregimewewould need cimpf20/ν0∼∆0,\nwhich requires cimp/greaterorsimilark2\nFL−1according to Eq. (31),\ni.e., at least cimp/greaterorsimilarL−3. At such large concentra-\ntions, the impurities “overlap” and our simple model\nis not valid any longer. The estimates above also indi-\ncate that the quasiclassical Green functions and the den-\nsity of states are modified only in narrow energy range,\n|E−∆0| ≪∆0at realistic impurity concentrations. As\na consequence, in contrast to magnetic impurities, a self-\nconsistent recalculation of the bulk pairing potential ∆ 0\nis not required.\nIV. SUPERCONDUCTOR WITH A WEAKLY\nANISOTROPIC GAP\nIn any realistic superconductor the pairing potential\nis at least slightly anisotropic, i.e., ∆ 0(n)∝ne}ationslash= const. The\nanisotropy used to be a subject of active theoretical and\nexperimental research23, but has been largely ignored in\nmodern models of s-wave superconductors. It is clear\nthat even a small anisotropy significantly influences the\nspectral properties of the superconducting state in the\nvicinity of the gap edge. As such, it may modify the re-\nsults of the previous section. In this section, we consider\nthese modifications.\nWe assume a weak anisotropy, so that the anisotropic\npart of the bulk pairing potential, ∆′(n)≡∆0(n)−∝an}bracketle{t∆0∝an}bracketri}ht,-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0012\nP= 0.1\n( a)\n-10 0 10 200.00.20.4(b)P= 10\nFigure 2. Densities of states vs. energy in a superconduc-\ntor containing either impurities with a local gap [solid lin e,\nEq. (36) with the upper signs] or magnetic impurities with\nrandomly oriented spins [dashed line, Eq. (50)], and the den -\nsity of states in the universal limit [dotted line in graph (b ),\nEq. (38)]. The additional shift −Pof the gap edge, appearing\nin Eqs. (36) and (38), is compensated via a positive transla-\ntion of the corresponding curves by PE0in graph (b).\nis small: |∆′| ≪ ∝an}bracketle{t∆0∝an}bracketri}ht. Here and further, the angle brack-\nets denote the average over the Fermi surface,\n∝an}bracketle{tX∝an}bracketri}ht ≡/integraldisplay\nX(n)dn\n4π. (51)\nUnder the assumption of a weak anisotropy of ∆ 0(n),\nit is reasonable to neglect the anisotropy of the impu-\nrity pairing potential, i.e., put ∆ 1(Q,q) = ∆ 1(q). The\nmatrixˇf(n,n′) does not depend on ∆ 0(n), hence, in the\ncaseofsphericallysymmetricimpuritiesitisstillgivenby\nEq. (25), and the off-diagonal components of the expan-\nsion coefficients f2lare defined by Eq. (30). In the pres-\nence of anisotropy, the T-matrix is defined by Eq. (21).\nNow, since the Green function ˇ g(ω,n) depends on n, it is\nimpractical to solve this equation using the expansion in\nterms of Legendre polynomials. To overcome this incon-\nvenience, we employ again the approximation of s-wave\nscattering: ˇf(n,n′) =ˇf0= const. In this case ˇTE(n,n′)\ndoes not depend on nandn′, and is given by\nˇTE= [ˇf−1+i∝an}bracketle{tˇg∝an}bracketri}ht−iˇτz]−1\n=ℜ[ˇf]−idet[ˇf]∝an}bracketle{tˇg∝an}bracketri}ht\n1−2if20∝an}bracketle{tg2∝an}bracketri}ht−det[ˇf](det∝an}bracketle{tˇg∝an}bracketri}ht+1).(52)\nWemadeuseofEq.(B33)toarriveatthelastline. Under\nthe approximation of s-wave scattering, Eq. (52) is valid\neven in the case of strong anisotropy.\nAs above, the energies of the impurity states corre-\nspond to the poles of the T-matrix, which are given by\n1−2if20∝an}bracketle{tg2∝an}bracketri}ht−detˇf(det∝an}bracketle{tˇg∝an}bracketri}ht+1) = 0.(53)\nIf the potential scattering is weak (/vextendsingle/vextendsingledetˇf/vextendsingle/vextendsingle≪1) or in\nthe limit of sufficiently large impurity concentrations,\nwhen the Green functions are essentially isotropic, so\nthat det ∝an}bracketle{tˇg∝an}bracketri}ht ≈ ∝an}bracketle{tdetˇg∝an}bracketri}ht=−1, we can neglect the third\nterm in Eq. (53). Then,\nˇTE≈ℜ[ˇf]−idet[ˇf]∝an}bracketle{tˇg∝an}bracketri}ht\n1−2if2∝an}bracketle{tg2∝an}bracketri}ht. (54)8\nThe third term in Eq. (53), which is proportional\nto det[ˇf], generally, can not be neglected in a strongly\nanisotropic superconductor. In d-wave superconductors,\nthis term is responsible for the quasibound states5, that\npossess a complex energy with a small imaginary part.\nSuch states are absent in the case of weak anisotropy\nunder consideration here.\nA. Ordinary potential scatterers\nBeforeaddressingthe influence ofpairing-potentialim-\npurities on the density of states, we will briefly consider\nthe effect of ordinary potential scatterers2,4.\nAt ∆1= 0, one obtains f20= 0, and Eq. (54) reduces\nto\nˇTE=ℜˇf−i|f10|2∝an}bracketle{tˇg∝an}bracketri}ht. (55)\nThe T-matrix given by Eq. (55) has no poles, so there\nare no subgap states.\nEquations (17) and (18) yield\ng1=−i˜E/radicalBig\n˜∆2(n)−˜E2, g 2=−i˜∆(n)/radicalBig\n˜∆2(n)−˜E2,(56)\nwhere\n˜E=E+i/planckover2pi1\n2τ∝an}bracketle{tg1∝an}bracketri}ht,˜∆(n) = ∆0(n)+i/planckover2pi1\n2τ∝an}bracketle{tg2∝an}bracketri}ht,(57)\nand the scattering time is given by\n1\nτ=|f10|2\nτu. (58)\nThis is equivalent to a set of equations derived in Refs. 2\nand 4.\nIn the case of weak anisotropy the density of states is\naffected by the scatterers only at energies near the gap\nedge,|E−∝an}bracketle{t∆0∝an}bracketri}ht| ≪ ∝an}bracketle{t∆0∝an}bracketri}ht. In this energy range we can\nutilize that |δg| ≪ |∝an}bracketle{tg1∝an}bracketri}ht|, whereδg=∝an}bracketle{tg1∝an}bracketri}ht − ∝an}bracketle{tg2∝an}bracketri}ht. As a\nconsequence,/vextendsingle/vextendsingle/vextendsingle˜E−˜∆(n)/vextendsingle/vextendsingle/vextendsingle≪/vextendsingle/vextendsingle/vextendsingle˜E/vextendsingle/vextendsingle/vextendsingle, and\n∝an}bracketle{tg1∝an}bracketri}ht ≈1√\n2/angbracketleftBigg/parenleftBigg˜E−˜∆(n)\n˜E/parenrightBigg−1\n2/angbracketrightBigg\n,(59)\nδg≈1√\n2/angbracketleftBigg/parenleftBigg˜E−˜∆(n)\n˜E/parenrightBigg1\n2/angbracketrightBigg\n, (60)\nor\n∝an}bracketle{tg1∝an}bracketri}ht=/radicalBig\n∝an}bracketle{t∆0∝an}bracketri}ht+i/planckover2pi1\n2τ∝an}bracketle{tg1∝an}bracketri}ht\n√\n2/angbracketleftBigg/bracketleftbigg\nE−∆0(n)+i/planckover2pi1\n2τδg/bracketrightbigg−1\n2/angbracketrightBigg\n,\n(61)δg=1\n√\n2/radicalBig\n∝an}bracketle{t∆0∝an}bracketri}ht+i/planckover2pi1\n2τ∝an}bracketle{tg1∝an}bracketri}ht/angbracketleftBigg/bracketleftbigg\nE−∆0(n)+i/planckover2pi1\n2τδg/bracketrightbigg1\n2/angbracketrightBigg\n.\n(62)\nTo understand the scaling of the Green function in the\ncase of weak anisotropy, it is instructive to rewrite these\nrelations in terms of the dimensionless quantities\nδ(n) =∆′(n)√\n/angbracketleft∆′2/angbracketright,˜G1=∝an}bracketle{tg1∝an}bracketri}ht√\n24√\n/angbracketleft∆′2/angbracketright√\n/angbracketleft∆0/angbracketright, δ˜G=δg√\n2/angbracketleft∆0/angbracketright\n4√\n/angbracketleft∆′2/angbracketright,\n˜P=/planckover2pi1\n2√\n2/angbracketleft∆0/angbracketright4√\n/angbracketleft∆′2/angbracketrightτ,˜Ω =E−/angbracketleft∆0/angbracketright√\n/angbracketleft∆′2/angbracketright.(63)\nThen we have\n˜G1=/radicalBig\n1+i˜P˜G1/angbracketleftbigg/bracketleftBig\n˜Ω−δ(n)+i˜Pδ˜G/bracketrightBig−1\n2/angbracketrightbigg\n,(64)\nδ˜G=1/radicalbig\n1+i˜P˜G1/angbracketleftbigg/bracketleftBig\n˜Ω−δ(n)+i˜Pδ˜G/bracketrightBig1\n2/angbracketrightbigg\n.(65)\nIn the limit ˜P≪1 of low concentrations of the scatter-\ners, the gap edge is rounded at an energy scale/radicalbig\n∝an}bracketle{t∆′2∝an}bracketri}ht.\nIn the opposite limit of large impurity concentrations,\nthe rounding becomes more narrow. This is due to the\nsuppression of the anisotropy of the pairing potential,\nwhich becomes essential when ˜P≫1. This condition\nis satisfied even in relatively clean superconductors, at\n∝an}bracketle{t∆0∝an}bracketri}htτ/greaterorsimilar/planckover2pi1.\nLet us consider the limit of strong suppression of the\nanisotropy, ˜P≫1. In this limit we can expand the\nexpressions in the anglular brackets in Eqs. (64) and (65)\nin terms of the small ratio δ(n)/(˜Pδ˜G). Then, we can\neliminateδ˜Gfrom the equations to arrive at the relation\n˜Ω˜G1−1\n˜G1+i\n2˜P˜G2\n1= 0. (66)\nWe observe that Eq. (66) is equivalent to the “univer-\nsal limit” equation of Ref. 12. Thus, the rounding of\nthe gap edge owing to the weak anisotropy and potential\nscattering can also be described in the framework of this\nuniversal scheme. In this limit the density of states is\nisotropic in the main order, not depending on the details\nof the shape of ∆ 0(n). The “depairing rate”, as defined\nin Ref. 12, equals\n1\nτdep=2/angbracketleftbig\n∆′2/angbracketrightbig\nτ\n/planckover2pi12. (67)\nFinally, the substitutions\n˜G1=G′\n1(2˜P)1/3,˜Ω =Ω′\n(2˜P)2/3.(68)\nreduce Eq. (66) to the form (41), containing no param-\neters. Note that the dependence 1 /τdep∝τin Eq. (67)\nreflects the gap edge sharpening with growing impurity\nconcentration, which was discussed above.9\nB. Suppressed anisotropy and pairing potential\nimpurities\nNow we will analyze the situation when the ma-\nterial contains both common potential scatterers and\npairing-potentialimpurities. The ordinaryscatterersand\npairing-potential impurities have the concentrations c1\nandc2, and scattering amplitude matrices ˇf(1)andˇf(2),\nrespectively. The corresponding T-matrices are\nˇT(1)\nE=ℜˇf(1)−i/vextendsingle/vextendsingle/vextendsinglef(1)\n10/vextendsingle/vextendsingle/vextendsingle2\n∝an}bracketle{tˇg∝an}bracketri}ht (69)\nfor ordinary scatterers and\nˇT(2)\nE=ℜˇf(2)−i/vextendsingle/vextendsingle/vextendsinglef(2)\n10/vextendsingle/vextendsingle/vextendsingle2\n∝an}bracketle{tˇg∝an}bracketri}ht\n1−2if20∝an}bracketle{tg2∝an}bracketri}ht(70)\nfor pairing-potential impurities, where f20≡f(2)\n20, sim-\nilar to Sec. III. To determine the Green functions, we\nsubstitute in Eq. (17) ˇS(E,n) in the form\nˇS(E,n) =/parenleftbigg\nE+iǫ+∆0(n)\n−∆0(n)−E−iǫ+/parenrightbigg\n−/summationdisplay\ni=1,2ci\nπν0ˇT(i)\nE(n,n).\n(71)\nThen,g1(n) andg2(n) are given by Eq. (56) with\n˜E=E+i/planckover2pi1\n2τ(E)∝an}bracketle{tg1∝an}bracketri}ht, (72)\n˜∆(n) =∝an}bracketle{t∆0∝an}bracketri}ht(Ξ−Q)+∆′(n), (73)\n1\nτ(E)=1\nτ1+1\nτ2(1−2if20/angbracketleftg2/angbracketright),1\nτ1,2=2c1,2/vextendsingle/vextendsingle/vextendsinglef(1,2)\n10/vextendsingle/vextendsingle/vextendsingle2\n/planckover2pi1πν0,(74)\nΞ = 1+i/planckover2pi1\n2τ(E)/angbracketleft∆0/angbracketright∝an}bracketle{tg2∝an}bracketri}ht, (75)\nQ=c2f20\nπν0/angbracketleft∆0/angbracketright1\n1−2if20/angbracketleftg2/angbracketright, (76)\n1/τ1and 1/τ2being the potential scattering rates due\nto ordinary scatterers and pairing-potential impurities,\nrespectively. From Eq. (74) we can see that the contri-\nbution of pairing-potential impurities to potential scat-\ntering is enhanced at energies close to the energy of the\nbound state [see Eq. (32)], manifesting resonant scatter-\ning near this energy.\nLet us now derive simplified equations, applicable in\nthe vicinity of the gap edge ( |E−∝an}bracketle{t∆0∝an}bracketri}ht| ≪ ∝an}bracketle{t∆0∝an}bracketri}ht). To\ndo this, let us notice that the quantities Ξ and −Qin\nEq. (73) represent the renormalizations of the isotropic\npart of ∆ 0(n) due to common superconducting and po-\ntential scattering, respectively. Simplifications are possi-\nble, ifQis small. If the second fraction in Eq. (76) is of\nthe order of or smaller than unity, this statment is rather\nobvious since |Q|/lessorsimilarc2|f20|/(πν0∝an}bracketle{t∆0∝an}bracketri}ht)/lessorsimilarc2k−3\nF≪1, as\nestimated in Sec. IIIB. The danger is that the second\nfraction in Eq. (76) can become large close to the energy\nof the bound state. Hovewer, at finite concentrations of\nthe pairing-potential impurities, the largest value of this\nfraction is proportional to 1 /√c2. Hence,Q∝√c2, and\nit vanishes at c2→0. This proves that |Q| ≪1. In turn,the smallness of Qprovides the validity of Eqs. (59) and\n(60) with\n˜E−˜∆(n)\n˜E=E−∆0(n)+Q∝an}bracketle{t∆0∝an}bracketri}ht+i/planckover2pi1\n2τ(E)δg\nΞ∝an}bracketle{t∆0∝an}bracketri}ht.(77)\nA further simplification is obtained in the limit\nof strongly suppressed anisotropy, |/planckover2pi1/τ(E)| ≫\n4/radicalbig\n∝an}bracketle{t∆′2∝an}bracketri}ht√∆0. Acting like in Sec. IVA, we obtain\na generalization of Eq. (66):\n∝an}bracketle{tg1∝an}bracketri}ht[E−∝an}bracketle{t∆0∝an}bracketri}ht(1−Q)]−∝an}bracketle{t∆0∝an}bracketri}ht\n2∝an}bracketle{tg1∝an}bracketri}ht+2i/angbracketleftbig\n∆′2/angbracketrightbig\nτ(E)\n/planckover2pi1∝an}bracketle{tg1∝an}bracketri}ht2= 0.\n(78)\nThis equation can describe the bound states at low c2as\nwell as the universal smoothing with enhanced 1 /τdep, as\nwe will see below.\nC. Small concentration pairing-potential impurities\nIn contrast to the isotropic case, the pairing-potential\nimpurities with a local gap reduction (∆ 1<0) do not\nnecessarily provide bound states, even in the limit of\nsmall anisotropy. In this Section, we derive the condi-\ntion of the emergence of the bound states and evaluate\nthe width of the impurity band in the limit of small con-\ncentrations c2.\nThe energy of the possible bound state is determined\nby the pole of ˇT(2)\nEin the limit of vanishing c2,\n1−2if20∝an}bracketle{tg1(E)∝an}bracketri}ht= 0. (79)\nLet us concentrate on the limit of strongly suppressed\nanisotropy, described by Eq. (66). To satisfy Eq. (79),\n∝an}bracketle{tg1∝an}bracketri}htmust be purely imaginary. This requires that the\ndensity of states is zero at this energy, i.e., Ebelow\nthe gap edge. We notice that in the universal limit the\ngap edgeEcris shifted with respect to ∝an}bracketle{t∆0∝an}bracketri}htby a small\nenergy4,12\nEcr≡ ∝an}bracketle{t∆0∝an}bracketri}ht−Ecr=3\n2∝an}bracketle{t∆0∝an}bracketri}ht/parenleftbigg/planckover2pi1\n∝an}bracketle{t∆0∝an}bracketri}htτdep/parenrightbigg2/3\n.(80)\nAtE=Ecrthe averaged Green function equals\n∝an}bracketle{tg1(Ecr)∝an}bracketri}ht=−i/parenleftbiggτdep∝an}bracketle{t∆0∝an}bracketri}ht\n/planckover2pi1/parenrightbigg1/3\n, (81)\nreaching its maximal negative purely imaginary value.\nHence, the existence of bound states requires\n1−2if20∝an}bracketle{tg1(Ecr)∝an}bracketri}ht<0, (82)\nor\n/planckover2pi1\n∝an}bracketle{t∆0∝an}bracketri}htτdep</parenleftbigg2E0\n∝an}bracketle{t∆0∝an}bracketri}ht/parenrightbigg3/2\n, (83)10\nwhereE0= 2f2\n20∝an}bracketle{t∆0∝an}bracketri}htwould give the bound state energy\nfor an isotropic gap. This implies that even a small gap\nanisotropy can prevent the formation of a bound state at\nsufficiently small E0.\nIf the bound state exists, its energy E(counted from\n∝an}bracketle{t∆0∝an}bracketri}ht) is obtained by substituting ∝an}bracketle{tg1(E)∝an}bracketri}ht=−i/(2f20)\nfrom Eq. (79) to Eq. (66), and yields\nE=E0+/planckover2pi1\nτdep/radicalBigg\n∝an}bracketle{t∆0∝an}bracketri}ht\n2E0. (84)\nThe width of the impurity band Win the limit of small\nc2is obtained from the expansion of Eq. (78) at energies\nclose toE, see Appendix D:\nW≈Wiso\n1−/planckover2pi1\n2τdepE0/radicalBigg\n∝an}bracketle{t∆0∝an}bracketri}ht\n2E0\n1/2\n,(85)\nwhereWisois the width of the impurity band for an\nisotropic gap [Eq. (37)]. Equation (85) is valid as long\nas the width of the impurity band is much smaller than\nthe distance from an isolated bound state to the edge\nof the continuum, i. e., E −Ecr. The dependencies of the\nbound state energy [Eq. (84)], the gap edge [Eq.(80)] and\nthe impurity band width [Eq. (85)] on the depairing rate\nτ−1\ndepare shown in Fig. 3.\n0.0 0.5 1.00123\nFigure 3. The bound state energy E, the gap edge Ecrand\nthe impurity band width Wvs. the depairing rate τ−1\ndepin the\nlimit of strongly suppressed anisotropy ( ˜P≫1).\nFor comparison, we give here also the energy of the\nbound state and the width of the impurity band in the\nopposite limiting case of a vanishing concentration of or-\ndinary scatterers. In this case, the answers are not uni-\nversal,dependingontheconcreteshapeoftheanisotropic\npart of the pairing potential ∆′(n). We consider a com-\nmon model4, where the values of ∆′are uniformly dis-\ntributed in an interval [ −∆a,∆a]. Explicit equations for\n∝an}bracketle{tg1∝an}bracketri}htin this case are given in Sec. IVE. One can showthat within the approximation of weak potential scatter-\ning (|f10|2≪1) a bound state exists, provided\nE0>∆a\n2. (86)\nThe energy of the bound state is given by\nE=E0+∆2\na\n4E0. (87)\nThe width of the impurity band at small impurity con-\ncentrations is\nW=Wiso/parenleftbigg\n1−∆2\na\n4E2\n0/parenrightbigg1/2\n. (88)\nD. Universal behavior in the presence of\npairing-potential impurities\nAs we have seen, at large impurity concentrations the\nshape of the smoothing of the gap edge eventually ap-\nproaches the universal limit. We have discussed two sit-\nuations for this to occur: the disorder in the pairing po-\ntential for an isotropic gap, and the suppression of the\nanisotropy of the pairing potential by potential scatter-\ning. Here, we consider a more general situation, where\nboth an anisotropic gap and pairing-potential impurities\nare present. The universal regime then requires\n|f20∝an}bracketle{tg1∝an}bracketri}ht| ≪1,/planckover2pi1/parenleftbig\nτ−1\n1+τ−1\n2/parenrightbig\n≫4/radicalbig\n∝an}bracketle{t∆′2∝an}bracketri}ht/radicalbig\n∝an}bracketle{t∆0∝an}bracketri}ht.(89)\nIn this limit, Eq. (78) takes the form\n∝an}bracketle{tg1∝an}bracketri}ht/parenleftBig\nE−∝an}bracketle{t∆0∝an}bracketri}ht+c2f20\nπν0/parenrightBig\n−/angbracketleft∆0/angbracketright\n2/angbracketleftg1/angbracketright\n+2i∝an}bracketle{tg1∝an}bracketri}ht2/parenleftbigg\n∝an}bracketle{t∆′2∝an}bracketri}ht\n/planckover2pi1(τ−1\n1+τ−1\n2)+c2f2\n20\nπν0/parenrightbigg\n= 0.(90)\nThis reproduces the universal limit with\n1\nτdep=2/angbracketleftbig\n∆′2/angbracketrightbig\n/planckover2pi12(τ−1\n1+τ−1\n2)+2c2f2\n20\n/planckover2pi1πν0,(91)\nand an extra shift of the gap edge −c2f20/(πν0). In-\nterestingly, the depairing rate exhibits a nonmonotonic\ndependence on the concentration c2(see Fig. 4). In par-\nticular, if ordinary scatterers are absent ( P1= 0), the\ndepairing rate has a minimum at\nc2=cmin≡πν0/radicalbig\n∝an}bracketle{t∆′2∝an}bracketri}ht\n√\n2|f20|/vextendsingle/vextendsingle/vextendsinglef(2)\n10/vextendsingle/vextendsingle/vextendsingle. (92)\nAt this concentration\n1\nτdep=/parenleftbigg1\nτdep/parenrightbigg\nmin≡2/radicalbig\n2∝an}bracketle{t∆′2∝an}bracketri}ht|f20|\n/planckover2pi1/vextendsingle/vextendsingle/vextendsinglef(2)\n10/vextendsingle/vextendsingle/vextendsingle.(93)11\nAmajorconsequenceofthenonmonotonicityofthede-\npairing rate is the nonmonotonic dependence of the gap-\nedge smearing on the concentration of pairing-potential\nimpurities. To ensure that this feature is present, it is\nsufficient to provide that the universality conditions (89)\nare satisfied at concentrations close to cmin. This is the\ncase when\nE0≪ ∝an}bracketle{t∆0∝an}bracketri}ht/radicalbig\n∝an}bracketle{t∆′2∝an}bracketri}ht/vextendsingle/vextendsingle/vextendsinglef(2)\n10/vextendsingle/vextendsingle/vextendsingle2\n. (94)\nIn addition, the concentration cminshould be realistic:\nat least,cimp≪L−3. WhenL∼λF, this yields the\ncondition\n∝an}bracketle{t∆0∝an}bracketri}ht/radicalbig\n∝an}bracketle{t∆′2∝an}bracketri}ht\nµ|f2|/vextendsingle/vextendsingle/vextendsinglef(2)\n10/vextendsingle/vextendsingle/vextendsingle≪1. (95)\n/s126/s99\n/s50/s126/s99/s45 /s49\n/s50\n/s107/s45 /s51\n/s70/s99\n/s109 /s105/s110/s100/s101/s112 \n/s109/s105/s110 /s49 \n/s99\n/s50/s49 \n/s100/s101/s112 \nFigure 4. Schematic dependence of the depairing rate τ−1\ndep\non on the concentration of pairing potential impurities in t he\nuniversal limit [Eq. (91) with τ−1\n1= 0]. The dashed lines\nindicate the regions where Eq. (91) is not applicable.\nE. Numerical calculations of the density of states\nIn this section, we report some numerical calcula-\ntions to examplify the typical behavior of the density\nof states in the presence of pairing-potential impurities\nand anisotropy.\nExcept for the universal limit, the results will depend\non the details of the anisotropic part of the gap ∆′(n).\nTo be specific and keep it simple, we concentrate on the\nmodel4with the values of ∆′uniformly distributed in an\ninterval [ −∆a..∆a].\nToderiveanequationfortheGreenfunction ∝an}bracketle{tg1∝an}bracketri}htinthe\npresenceofordinarypotentialimpuritiesweuseEqs.(61)\nand (62), where the averaging over the directions of mo-\nmentum can be performed analytically. After this weeliminateδgfrom the two equations to obtain the fifth-\norder polynomial equation\n3y4−2py5\n3(1−py)2+2Eay2=−1. (96)\nHere, we made use the dimensionless variables\ny=−i∝an}bracketle{tg1∝an}bracketri}ht/radicalBig\n∆a\n∆0, E a=E−/angbracketleft∆0/angbracketright\n∆a,(97)\np=/planckover2pi1\n2√\n∆a/angbracketleft∆0/angbracketrightτ1, (98)\nwhich differ from ˜G1,˜Ω and˜P, introduced in Sec. IVA,\nby numeric factors. An equation similar to Eq. (96) has\nalready been studied by Clem4. To adjust the equation\nforthecasewhenbothpairing-potentialandordinaryim-\npurities are present, we implement Eq. (77) to show that\nthe adjustment amounts to the following substitutions:\np→p1+p2\n1+αy, Ea→Ea+p2α\n2/vextendsingle/vextendsingle/vextendsinglef(2)\n10/vextendsingle/vextendsingle/vextendsingle2\n(1+αy),(99)\nwhere\nα=/radicalbigg\n2E0\n∆a, p 1,2=/planckover2pi1\n2/radicalbig\n∆a∝an}bracketle{t∆0∝an}bracketri}htτ1,2.(100)\nWith these substitutions Eq.(96) becomes aneigth-order\npolynomial equation with respect to y. In the limit of\nstrongly suppressed anisotropy, this is simplified to a\nfifth-order polynomial equation [Eq. (78)]. We solved\nthese equations numerically.\nLet us first consider relatively large values of the pa-\nrameterα, so that the energyscale E0is of the orderof or\nlargerthan the characteristicbroadening ofthe gap edge.\nThe characteristic width ∆ Eaof the gap edge, measured\nin the units ∆ a, equals unity in the pure case and is of\nthe order of p−2/3\n1when the anisotropy is suppressed by\npotential scatterers: p1≫1. We consider the range of\nparameters where α2/greaterorsimilar∆Ea. Atα2≫∆Eathe broad-\nening of the gap edge is not relevant for the formation\nand merging of the impurity band with the continuum.\nThe situation is qualitatively the same as in the isotropic\ncase, see Sec. III. It is thereforeinterestingto concentrate\non the range of parameters α2∼∆Ea.\nWe start with the case of strongly suppressed\nanisotropy, when Eq. (78) is applicable. We choose\n/planckover2pi1\nτdep=E0\n3/radicalBigg\nE0\n3∝an}bracketle{t∆0∝an}bracketri}ht,orα2= 2.88p−2/3\n1.(101)\nThen, the smoothing shifts the gap edge to −E0/2, and\nEq. (84) predicts a bound state at the energy ǫ≈1.14E0.\nFig 5a illustrates the behavior of the density of states\nupon increasing concentration of pairing potential im-\npurities. We observe the qualitative similarity with the\nisotropiccase – see Fig. 1a. In particular, there is the for-\nmationofthe impuritybandthatmergeswith the contin-\nuum with growing c2. Moreover, the characteristic scale12\nfor the concentration c2is the same as in the isotropic\nsituation and correspondsto P∼1, where the parameter\nPis\nP=c2/radicalbig\n2E0∝an}bracketle{t∆0∝an}bracketri}htπν0. (102)\nHowever, while in the isotropic case the smoothing of the\npeak was due to pairing-potential impurities only, now\nthe pairing-potential impurities provide an extra contri-\nbution to the existing smoothing.\nThe situation changes if the scales α2and ∆Earemain\ncomparable, but the bound state is absent. For Fig. 5b\nwe choose\n/planckover2pi1\nτdep= 16E0/radicalBigg\n2E0\n∝an}bracketle{t∆0∝an}bracketri}ht,orα2= 0.12p−2/3\n1.(103)\nHere, we see no trace of the impurity band. The pairing-\npotential impurities widen the peak and shift down the\ngap edge.\nApparently, the effect of potential scattering by\npairing-potential impurities on the density of states is\nnegligible, as long as α2∼∆Ea, and if the anisotropy\nis already suppressed by ordinary potential scatterers:\nτ−1\n1≫4/radicalbig\n∝an}bracketle{t∆′2∝an}bracketri}ht/radicalbig\n∝an}bracketle{t∆0∝an}bracketri}ht//planckover2pi1. Taking/vextendsingle/vextendsingle/vextendsinglef(2)\n10/vextendsingle/vextendsingle/vextendsingle= 0 and/vextendsingle/vextendsingle/vextendsinglef(2)\n10/vextendsingle/vextendsingle/vextendsingle=\n1 (which is the maximal permitted value) produces pro-\nfiles of the density of states that are almost visually in-\ndistinguishable. To get a qualitative understand of this,\nlet us consider, e. g., the depairing rate in the universal\nlimit – see Eq. (91). Here, potential scatteing by pairing-\npotential impurities may become important only at rel-\natively large concentrations c2, whenτ−1\n2is of the order\nofτ−1\n1. However, at such values of c2the second term\nin the right-hand side of Eq. (91) becomes dominant, so\nthat the depairing rate is determined by the off-diagonal\nscattering amlitude, f20. Thus, the scattering rate τ−1\n2\ncan be neglected in Eq. (91) at all concentrations c2.\nIt is interesting to spot the similarities with the seem-\ningly different situation, when potential scatterers are\nabsent — see Figs. 5c,d. Here, we use Eqs. (96) and\n(99), putting p1= 0. The density of states at p2= 0\nexhibits a typical cusp structure, typical for the model\nwe use. However, the qualitative behavior of the density\nof states is analogous to the case of strongly suppressed\nanisotropy, if we choose E0∼∆a(α2∼1). If the shift of\nthe gap edge amounts to E0/2, the bound state is formed\natǫ= 1.06E0– see Fig. 5c. Upon increasing the impurity\nconcentration, wesee againthe impurityband formation,\nmerging with the continuum, and the extra smoothing of\nthe coherence peak at the scales P∼1 of the dimension-\nless concentration. For Fig. 5d we choose E0= ∆a/2.\nThis corresponds to the theshold of the bound state for-\nmation. Similar to Fig. 5b, we don’t see any signs of the\nimpurity band, rather the effect is a combination of the\npeak smoothing and shift.\nFinally, we have modeled the density of states in an-\nother interesting limiting case, when αis very small:α2≪/vextendsingle/vextendsingle/vextendsinglef(2)\n10/vextendsingle/vextendsingle/vextendsingle2\n. To remind, this inequality guarantees the\nnonmonotonic behavior of the peak smoothing vs. the\nimpurity concentration. In this case, at a small impu-\nrity concentration the pairing-potential impurities affect\nthe anisotropic part of ∆ 0mostly as potential scatter-\ners. One can see this in Fig. 6, where the incerease of\nthe dimensionless impurity concentration p2results in\nthe narrowing of the peak. Further increase of the con-\ncentration gives the peak shift, manifesting the pairing\npotential distortion brought by the impurities. Starting\nfromp2= 5.16 the peak also becomes wider upon in-\ncreasing concentration, in accordance with Eq. (91).\n-3 -2 -1 0 10.00.20.40.60.81.01.21.41.6 P= 0\nP= 0.0 2\nP= 0.05\nP= 0.1\nP= 0.2\nP= 0.3(a)\n-30 -20 -10 0 10 20 300.00.10.20.3(b)\nP= 0 P= 1\nP= 2 P= 4\nP= 7 P= 10\n-2.5-2.0 -1.5-1.0 -0.5 0.0 0.5 1.00.00.51.01.52.0(c)P= 0\nP= 0.05\nP= 0.1\nP= 0.2\nP= 0.4\n-4 -3 -2 -1 0 1 20.00.20.40.60.81.0(d)P= 0\nP= 0.05\nP= 0.1\nP= 0.2\nP= 0.5\nP= 1\nP= 2\nFigure 5. Density of states vs. energy in an anisotropic su-\nperconductor in the limit of strongly suppressed anisotrop y\n(a,b) [Eq. (78)] and in the absence of ordinary potential sca t-\nterers (c,d) [Eqs. (96) and (99)]. (a) E0= 2Ecr,f(2)\n10= 0;\n(b)E0=Ecr/8,f(2)\n10= 0; (c) E0= 2∆ a,/vextendsingle/vextendsingle/vextendsinglef(2)\n10/vextendsingle/vextendsingle/vextendsingle2\n= 0.1; (d)\nE0= 0.5∆a,/vextendsingle/vextendsingle/vextendsinglef(2)\n10/vextendsingle/vextendsingle/vextendsingle2\n= 0.1.\n-8 -6 -4 -2 0 20.00.20.40.60.81.01.21.4p2= 0 p2= 0.2\np2= 1 p2= 2\np2= 5.16 p2= 8\np2= 15 p2= 30\nFigure 6. Density of states vs. energy in an anisotropic su-\nperconductor containing pairing-potential impurities wi th a\nsmall parameter E0:E0= 0.025|f10|2∆a,/vextendsingle/vextendsingle/vextendsinglef(2)\n10/vextendsingle/vextendsingle/vextendsingle2\n= 0.1.13\nV. CONCLUSION\nIn conclusion, we have considered the effect of pairing-\npotential impurities on the behavior of the density of\nstates in a superconductor. We found that this behavior\nis strongly affected by the anisotropy of the bulk pairing\npotential.\nWe have considered first the limit of negligible\nanisotropy. In this case, we established an analogy be-\ntween the pairing-potential impurities and weak polar-\nized magnetic impurities. This analogy allows to extend\nthe results obtained for one type of defects to the other\ntype. Thepersistenceofboundstatesatsingleimpurities\nis typical for the isotropic case. We demonstrate that a\nspherically symmetric impurity with local suppression of\n∆givesrisetoaninfinite numberofsubgapbound states.\nUpon increasing the impurity concentration, these states\nform an impurity band that eventually merges with the\ncontinuum, resulting in a smoothed gap edge.\nEvenaslightlyanisotropicpairingpotentialforbidsthe\nformation of bound states at sufficiently small pairing-\npotential distortion at the impurities. We derived the\ncriterion of existence of the bound states and have ana-\nlyzed in detail the behavior of the density of states.\nAppendix A\nIn this Appendix we derive Eqs. (17), (18) and (21).\nEquations (12) and (14) yield\nˇg(E,n) =i/integraldisplay/bracketleftbigˇS(E,pn)−ˇ1ξ(p)/bracketrightbig−1dξ(p)\nπ,(A1)\nwhere\nˇS(E,p) =/parenleftbigg\nE+iǫ+∆0(p)\n−∆0(p)−E−iǫ+/parenrightbigg\n−cimpˇTE(p,p).\n(A2)\nLet us assume that we can keep ˇS(E,p) constant when\nintegrating over ξ, i. e., put ˇS(E,pn)≈ˇS(E,kFn)≡\nˇSE(n). Such simplification is justified if ˇS(E,pn) does\nnot change significantly while ξ(p) is of the order of or\nsmaller than the largest of the moduli of the eigenvalues\nofˇSE(n). Then, we have\nˇg(E,n)≈i−/integraldisplay+∞\n−∞/bracketleftbigˇSE(n)−ˇ1ξ/bracketrightbig−1dξ\nπ.(A3)\nFrom this relation immediately follows Eq. (17). By\nwriting Eq. (A3) in the basis where the matrix ˇSE(n) is\ndiagonal it can be proven that each eigenvalue of ˇ gcan\nbe either 1 or −1. In the pure case\nˇg(E,n) =i/radicalBig\n|∆0(n)|2−E2/parenleftbigg\n−E−∆0(n)\n∆0(n)E/parenrightbigg\n,(A4)\nand the two eigenvalues are 1 and −1. There is no reason\nfor them to change discontinuously with growing impu-\nrity concentration, hence, for any value of cimpwe have\nTrˇg= 0, and detˇ g= 1, which gives Eq. (18).To simplify Eq. (11), we want to exclude the region\nof integration far from the Fermi surface. To do this,\nwe employ the trick used in Ref. 9. Let us introduce an\nauxiliarynormal state scattering matrix ˇf(p,p′), defined\nby Eq. (20). Using Eqs. (11) and (20), we obtain\nˇTE(p,p′) =ˇf(p,p′)−/integraltextˇf(p,k)ˇG(k)ˇVimp(k−p′)d3k\n(2π)3\n+/integraltextˇf(p,k)ˇGE(k)ˇTE(k,p′)d3k\n(2π)3\n−/integraltext/integraltextˇf(p,k)ˇG(k)ˇVimp(k−k′)ˇGE(k′)ˇTE(k′,p′)d3k\n(2π3)d3k′\n(2π)3\n=ˇf(p,p′)+/integraltextˇf(p,k)(ˇGE(k)−ˇG(k))ˇTE(k,p′)d3k\n(2π)3.(A5)\nFar from the Fermi surface the difference between ˇGE(k)\nandˇG(k) vanishes. This typically happens at |ξ(k)| ≫\nmax(|∆0(n)|),|E|(strictly speaking, in this estimate the\nquantityξ(p), defined in Eq. (6), should include the\nrenormalized chemical potential due to the impurities.\nThis renormalization can be much larger than ∆ 0, how-\never, as long as it is much smaller than µ, it has a negli-\ngible effect on the results an will be disregarded further).\nFor impurities with a size much smaller than the coher-\nencelength ξSthefunctions ˇf(p,k)andˇTE(k,p′)depend\nvery weakly on |k|as long asξ(k)∼∆0(see Appendix\nB). This means that we can integrate in Eq. (A5) over\n|k|and obtain Eq. (21).\nAppendix B\nIn this appendix the matrix ˇf(p,p′) will be evaluated\nfor a spherically symmetric impurity. Considering Eqs.\n(20) and (22) with a real function ∆ 1(p) one can see that\nˇfhas the form\nˇf(p,p′) =/parenleftbigg\nf1(p,p′)f2(p,p′)\n−f∗\n2(p,p′)f∗\n1(p,p′)/parenrightbigg\n.(B1)\nThe equations for f1andf2read\nf1(p,p′) =U(p−p′)+/integraltext/bracketleftBig\nf1(p,k)G(0)\nN(k)U(k−p′)\n+f2(p,k)G(0)∗\nN(k)∆1(k−p′)/bracketrightBig\nd3k\n(2π)3,(B2)\nf2(p,p′) =−∆1(p−p′)+/integraltext/bracketleftBig\nf2(p,k)G(0)∗\nN(k)U(k−p′)\n−f1(p,k)G(0)\nN(k)∆1(k−p′)/bracketrightBig\nd3k\n(2π)3,(B3)\nwhere\nG(0)\nN(k) = (−ξ(k)+iǫ+)−1.\nThefunctions f1(p,p′)andf2(p,p′)aretheordinaryand\nAndreev scattering amplitudes, respectively, for an elec-\ntron in the normal state incident at an impurity with an\nelectric potential U(r) and pairing potential ∆ 1(r). To14\nsolve Eqs. (B2) and (B3), we assume that Andreev scat-\ntering can be taken into account within the perturbation\ntheory. This is the case when\nkF\nµ/integraldisplay\n|∆1(r)|dr≪1, (B4)\nif|U(r)|/lessorsimilarµ. Note that for |∆1| ∼∆0Eq. (B4) simply\nmeans that L≪ξS. Then, the last term in the right-\nhand side of Eq. (B2), which is second order in ∆ 1, can\nbeneglected. Asaresult, f1≈fN, wherefNisthevertex\npart of the normal-state Green function in the presence\nof a single impurity:\nfN(p,p′) =U(p−p′)+/integraldisplay\nfN(p,k)G(0)\nN(k)U(k−p′)d3k\n(2π)3.\n(B5)\nThe differential scattering cross-section dσon the po-\ntentialU(r) fromp1andp2(both lying on the Fermi\nsurface) is\ndσ\ndΩ(p1→p2) =m2\n4π2/planckover2pi14|fN(p2,p1)|2.(B6)\nTo solve Eq. (B3) we note that the function\nΓ(p,p′) =δ(p−p′)+f∗\nN(p,p′)G(0)∗\nN(p)\n(2π)3(B7)\nsatisfies the equation\nΓ(p,p′) =δ(p−p′)+/integraldisplay\nΓ(p,k)G(0)∗\nN(k)U(k−p′)d3k\n(2π)3.\n(B8)\nHence,\nf2≈/integraltext\n[−∆1(p−q)\n−/integraltext\nfN(p,k)G(0)\nN(k)∆1(k−q)d3k\n(2π)3/bracketrightBig\nΓ(q,p′)d3q\n=−∆1(p−p′)−/integraltext\nfN(p,k)G(0)\nN(k)∆1(k−p′)d3k\n(2π)3\n−/integraltext\n∆1(p−k)G(0)∗\nN(k)f∗\nN(k,p′)d3k\n(2π)3\n−/integraltext\nfN(p,k)G(0)\nN(k)∆1(k−q)G(0)∗\nN(q)f∗\nN(q,p′)d3k\n(2π)3d3q\n(2π)3.\n(B9)\nThe normal-state single-impurity Green function is\ngiven by\nGN(p,p′) =δ(p−p′)G(0)\nN(p)+G(0)\nN(p)fN(p,p′)\n(2π)3G(0)\nN(p′).\n(B10)\nHence,\nf2(p,p′) =−/integraltextG(0)−1\nN(p)GN(p,k)∆1(k−q)G∗\nN(q,p′)\n×G(0)∗−1\nN(p′)d3kd3q=−/integraltext\neipr−ip′r′G(0)−1\nN(p)\n×GN(r,r1)∆1(r1)G∗\nN(r1,r′)G(0)∗−1\nN(p′)d3rd3r1d3r′.(B11)Here we used that GN(−r1,−r′) =GN(r1,r′) due to\ninversion symmetry. The function GN(r,r′) is defined\nby the equation\n/parenleftbigg\n−/planckover2pi12\n2m∂2\n∂r2−µ+U(r)−iǫ+/parenrightbigg\nGN(r,r′) =−δ(r−r′).\n(B12)\nNow we expand GNand theδ-function in Legendre poly-\nnomialsPl:\nGN(r,r′) =/summationdisplay\nlPl/parenleftbiggr\nr·r′\nr′/parenrightbiggGl(r,r′)\nr,(B13)\nδ(r−r′) =δ(r−r′)\nr2∞/summationdisplay\nl=02l+1\n4πPl/parenleftbiggr\nr·r′\nr′/parenrightbigg\n.(B14)\nThen\n/bracketleftBig\n−iǫ++/planckover2pi12l(l+1)\n2mr2−/planckover2pi12\n2m∂2\n∂r2−µ+U(r)/bracketrightBig\nGl(r,r′)\n=−2l+1\n4πr′δ(r−r′). (B15)\nLet us denote as ul0the solution of the homogeneous\nequation(withouttheright-handside)havingtheasymp-\ntotics\nul0=eikFr−ǫ+kFr/2µ(B16)\natr→ ∞. It should be noted that at r > r′Gl(r,r′)\nis proportional to ul0, since the second linear indepen-\ndent solution of the homogeneous equation diverges at\nr→ ∞. Then, using standard methods for solving linear\ninhomogeneous equations (e. g., the method of variation\nof parameters) one can express Glin terms of ul0and\nu∗\nl0:\nGl(r,r′) =/braceleftBiggs(r′)ul0(r)−im\nkF/planckover2pi12ul0(r′)2l+1\n4πr′u∗\nl0(r), r<r′\n/bracketleftBig\ns(r′)−im\nkF/planckover2pi12u∗\nl0(r′)2l+1\n4πr′/bracketrightBig\nul0(r), r>r′\n(B17)\nHeres(r′) is some function that will be determined from\nthe boundary condition at r= 0. When deriving Eq.\n(B17) it has been used that the Wronskian\nW=/vextendsingle/vextendsingle/vextendsingle/vextendsingleul0(r)u∗\nl0(r)\nu′\nl0(r)u∗\nl0′(r)/vextendsingle/vextendsingle/vextendsingle/vextendsingle(B18)\nis almost constant: indeed, it can be demonstrated using\nEq. (B15) (without the right-hand side) that dW/dr∼\nǫ+, and hence W(r)≈ −2ikFfor not too large r.\nTo determine s(r′) we use the boundary condition\nGl(0,r′) = 0:\ns(r′) =im\n/planckover2pi12kFul0(r′)2l+1\n4πr′cl, (B19)\nwhere\ncl= lim\nr→0u∗\nl0(r)\nul0(r).15\nThen\nGl(r,r′) =im\n/planckover2pi12kF2l+1\n4πr′/braceleftbigg\nul(r′)ul0(r), r>r′\nul(r)ul0(r′), r′>r,\n(B20)\nwhere\nul(r) =clul0(r)−u∗\nl0(r). (B21)\nNowwereturntoEq. (B11). Forfurthertransformations\nwe will use the addition theorem\nPl/parenleftbiggr\nr·r′\nr′/parenrightbigg\n=4π\n2l+1m=l/summationdisplay\nm=−lYlm/parenleftBigr\nr/parenrightBig\nY∗\nlm/parenleftbiggr′\nr′/parenrightbigg\n,(B22)\nand the expansion [see Ref. 24]\neipr=∞/summationdisplay\nl=0(2l+1)iljl(pr)Pl/parenleftBigr\nr·n/parenrightBig\n.(B23)\nHereYlmarethesphericalharmonicsand jlarethespher-\nical Besselfunctions, which arerelated to ordinaryBessel\nfunctions J νvia\njl(x) =/radicalbiggπ\n2xJl+1\n2(x). (B24)\nIf one expands the Legendre polynomials in Eq. (B13)\nin spherical harmonics, one can perform integration over\nthe directions of r,r′andr1in Eq. (B11):\nf2(p,p′) =−G(0)−1\nN(p)G(0)∗−1\nN(p′)∞/summationtext\nl=0Pl(n·n′)(4π)3\n2l+1\n×/integraltext\njl(pr)jl(p′r′)Gl(r,r1)∆1(r1)G∗\nl(r1,r′)rr1r′2drdr1dr′.\n(B25)\nFurther simplifications are possible if we take pandp′\nsufficiently close to the Fermi surface: |p−kF|L≪1\nand|p′−kF|L≪1. At such parameters we may neglect\nthe contribution to the integral in (B25) from the region\nwhere either r < Lorr′< Las compared to the con-\ntribution from the region where both r > Landr′> L\n(this statement is proved by the estimates given below,\nin particular, Eq. (B28)). Then, since at r1>L∆1(r1)\nis negligible, we may put r1<L<r,r′in Eq. (B25):\nf2(p,p′) =−G(0)−1\nN(p)G(0)∗−1\nN(p′)4πm2\n/planckover2pi14k2\nF\n×∞/summationtext\nl=0(2l+1)Pl(n·n′)∞/integraltext\nLjl(pr)ul0(r)rdr\n×∞/integraltext\nLjl(p′r′)u∗\nl0(r′)r′dr′L/integraltext\n0|ul(r1)|2∆(r1)dr1.(B26)\nThe asymtotic behavior of jl(x) atx→ ∞is\njl(x)≈1\nxcos/bracketleftBig\nx−(l+1)π\n2/bracketrightBig\n.(B27)Then, using Eq. (B16), we obtain\n∞/integraltext\nLjl(pr)ul0(r)rdr≈L′/integraltext\nLjl(pr)ul0(r)rdr\n+il+1ei(kF−p)L′\n2p/bracketleftBig\nkFǫ+\n2µ−i(kF−p)/bracketrightBig−1\n,(B28)\nwhereL′is the characteristic distance at which the\nasymptotics (B27) can be used (if this distance is smaller\nthanL, we take L′=L). To ensure that f2(p,p′)\nweakly depends on |p|and|p′|whenξ(p)/lessorsimilar∆0and\nξ(p′)/lessorsimilar∆0, we need to demand that the argument of\nthe exponent in the second line of Eq. (B28) is small, i.\ne.,L′|kF−p| ∼L′/ξS≪1. According to Ref. 24, the\nasymptotics (B27) works at x≫l2. Thus,L′∼l2/kF,\nso we demand\nl2≪kFξS. (B29)\nThen, we have\n∞/integraldisplay\nLjl(pr)ul0(r)rdr≈il+1\n2p/bracketleftbiggkFǫ\n2µ−i(kF−p)/bracketrightbigg−1\n.(B30)\nFinally, Eqs. (B26) and (B30) yield\nf2(n,n′)≈ −π2ν0\nk2\nF∞/summationdisplay\nl=0(2l+1)Pl(n·n′)/integraldisplay∞\n0|ul(r)|2∆1(r)dr.\n(B31)\nNote that the condition (B4) provides the smallness of\nthe expansion coefficients: |f2l| ≪1.\nIn the end of the appendix we will derive some more\nuseful relation for ˇf. It follows from Eq. (20) that\nℑ[ˇf(p,p′)] =/integraltext\nℑ[ˇf(p,k)]ˆG(k)ˇVimp(k−p′)d3k\n(2π)3\n+i\n2/integraltextˇf∗(p,k)[ˆG∗(k)−ˆG(k)]ˇVimp(k−p′)d3k\n(2π)3\n=/integraltext\nℑ[ˇf(p,k)]ˆG(k)ˇVimp(k−p′)d3k\n(2π)3\n+πν0/integraltextˇf∗(p,kFn)/parenleftbigg\n−1 0\n0 1/parenrightbigg\nˇVimp(kFn−p′)dn\n4π.\nThen, using Eq. (20) it can be proved that\nℑ[ˇf(n,n′)] =/integraldisplay\nˇf∗(n,n′′)/parenleftbigg\n−1 0\n0 1/parenrightbigg\nˇf(n′′,n′)dn′′\n4π.\n(B32)\nSubstituting here Eq. (25), after integration over n′′one\nfinds that\nℑ(f1l) =−/vextendsingle/vextendsinglef2\n1l/vextendsingle/vextendsingle−f∗2\n2l=−/vextendsingle/vextendsinglef2\n1l/vextendsingle/vextendsingle−f2\n2l,\nℑ(f2l) =−2if∗\n1lℑ(f2l) = 2if1lℑ(f2l).\nFinally, we obtain the following relations for the imagi-\nnary parts of f1landf2l:\nℑ(f1l) =−detˇfl,ℑ(f2l) = 0.(B33)16\nAppendix C\nIn this appendix we will derive some properties of the\nbound states localized around a small ( L≪ξS) spheri-\ncally symmetric impurity, suppressing the pairing poten-\ntial. First, we prove that such impurity supports bound\nstates with arbitrary large orbital momenta l. The en-\nergiesElof such states may be determined from the\nBogoliubov-de Gennes equations20\n(ˆHl−Elˇ1)/parenleftbigg\nu(r)\nv(r)/parenrightbigg\n= 0, (C1)\nwhere\nˆHl=/parenleftbigg\nHl(r) ∆(r)\n∆(r)−Hl(r)/parenrightbigg\n, (C2)\nHl(r) =−/planckover2pi12\n2m∂2\n∂r2+/planckover2pi12l(l+1)\n2mr2+U(r).(C3)\nThe boundary conditions at the origin are u(0) =v(0) =\n0. The energy of the ground state may be estimated\nusing the variational principle\nE2\nl≤/integraltext∞\n0(u∗\nT(r)v∗\nT(r))ˆH2\nl/parenleftbigg\nuT(r)\nvT(r)/parenrightbigg\ndr\n/integraltext∞\n0/parenleftBig\n|uT(r)|2+|vT(r)|2/parenrightBig\ndr,(C4)\nwhereuT(r) andvT(r) are some trial functions satisfying\nthe boundary conditions. Let us take uT(r) =vT(r) =\nul(r)e−δr, whereul(r) is defined in Eq. (B21), and δ\nis an adjustable parameter. According to Eq. (C4), we\nhave\nE2\nl≤∆2\n0\n+/integraltext∞\n0/bracketleftbigg/parenleftBig\n/planckover2pi12δ\n2m/parenrightBig2/vextendsingle/vextendsingle/vextendsingledul\ndr−δul/vextendsingle/vextendsingle/vextendsingle2\n+[∆2(r)−∆2\n0]|ul(r)|2/bracketrightbigg\ne−2δrdr\n/integraltext∞\n0|ul(r)|2e−2δrdr,(C5)\nwhere ∆ 0is the value of the gap far from the impurity. It\ncanbeseenthatifthereisaregionwith∆( r)2<∆2\n0(and\neverywhere ∆( r)2≤∆2\n0), by taking a sufficiently small\nparameterδone can make the second line of Eq. (C5)\nnegative. Thus, E2\nl<∆2\n0, which means that a bound\nsubgap state exists for arbitrary l.\nNow we will derive a few explicit expressions for El.\nOf course, these energies can be determined by solving\nthe Bogoliubov-deGennes equations (C1), but this is not\nnecessary: at L≪ξSandl2≪kFξSthis solution will\nyield Eq. (33), which will be analyzed in the following.\nIn the case when the electric potential Uis absent or\nnegligible compared to the centifugal potential, the func-\ntionsulin Eq. (33) can be expressed in terms of the\nspherical Bessel functions jl24:\nul= 2kFrjl(kFr) (C6)\n(accordingtothedefinition(B21), ulhasacomplexphase\nfactor, but this does not matter here).We can compareour analytical result for E0with a nu-\nmerical result obtained by Flatt´ e and Byers16, who stud-\nied the eletctronic structure of defects of different nature.\nIn particular, a local Gaussian suppression of the pairing\npotential on the scale of k−1\nFhas been considered. For an\nanalytical estimate we assume that ∆( r) is proportional\nto the pairing potential, so that\n∆1(r) =−∆0e−k2\nFr2. (C7)\nSuch approximationshould workwell when L/ξS≪1, so\nthat perturbations of the Green functions at imaginary\nenergies are small. For kFξS= 10 from Eqs. (30), (33)\nand (C6) we obtain E0= 6×10−4∆0. Our value for E0\nappears to be one order of magnitude smaller than the\nnumerical value given in Ref. 16. The reason for this\ndeviation is not very clear. The states with l∝ne}ationslash= 0 have\nnot been detected in the mentioned paper, so we may\nsuppose that a higher numerical accuracy is required to\ncalculate the bound state energies when they are lying\nvery close to the gap edge.\nLet us turn to the case of relatively large defects, when\nthe energies Elcan be calculated quasiclassically. Us-\ning Eq. (B24) and Debye’s asymptotic expansion for the\nBessel functions25, we find that\njl(x)≈1/radicalbig\nxz(x)cosχ(x), (C8)\nwhere\nz(x) =/radicalBigg\nx2−/parenleftbigg\nl+1\n2/parenrightbigg2\n, (C9)\nχ(x) is the quasiclassical phase, and it is required that\nz(x)≫1, z3(x)≫/parenleftbigg\nl+1\n2/parenrightbigg2\n. (C10)\nNote thatz(x) = 0 approximately corresponds to the\nclassical turning point, where the centrifugal potential\nequals the Fermi energy. When integrating in Eq. (30),\nwe replace cos2χ(x) by 1/2, which is its average over an\noscillation period:\nf2l≈ −kF\n2µ/integraldisplayL\nk−1\nF(l+1\n2)kFr∆1(r)dr/radicalBig\nk2\nFr2−/parenleftbig\nl+1\n2/parenrightbig2.(C11)\nIt can be seen that the quasiclassical approach does not\nallow to determine the energies of the impurity states\nwith large momenta ( l/greaterorsimilarkFL−1/2). Below we will\ndemonstrate that these energies are exponentially close\nto the gap edge.\nPreviously, Gunsenheimer and Hahn21calculated the\nenergiesofthe Andreevstates onanormalspherein a su-\nperconducting continuum (which corresponds to a step-\nlike profile of ∆ 1(r): ∆1(r) =−∆0whenr < L, and\n∆1= 0 whenr >L). Their approach is essentially qua-\nsiclassical, so their results should be equivalent to Eqs.17\n(C11) and (33) when L≪ξS. Substituting a rectangular\nprofile of ∆ 1into Eq. (C11), we obtain\nf2l=∆0\n2µ/radicalBigg\nk2\nFL2−/parenleftbigg\nl+1\n2/parenrightbigg2\n.(C12)\nThis agrees well with the result from Ref. 21. Taking\nkFL≫l+1/2, we obtain the estimate (31).\nFor smaller impurities or larger lwe need to go beyond\nthe quasiclassical approximation. Using again a rectan-\ngular profile of ∆ 1(r), we find that\nf2l=∆0\nµI(l,kFL), (C13)\nwhere\nI(l,R) =/integraltextR\n0x2j2\nl(x)dx=\n=R2\n2/braceleftbig\nR[j2\nl(R)+j2\nl+1(R)]−(2l+1)jl(R)jl+1(R)/bracerightbig\n.(C14)\nTheI(l,R) vs.Rgraphs forl= 0..4 are shown in Fig. 7.\nIt can be seen that on the background of linear growth\nthese functions exhibit oscillations with a period equal to\nπ. These oscillations are a consequence of the disconti-\nnuity of ∆ 1(r) atr=L. A smooth cross-overof ∆ 1from\n−∆0to 0 on a length scale larger than k−1\nFwill remove\nthe oscillations.\n/s48 /s50 /s52 /s54 /s56/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s73 /s40 /s108 /s44/s82 /s41\n/s82/s32/s108/s32/s61/s32/s48\n/s32/s108/s32/s61/s32/s49\n/s32/s108/s32/s61/s32/s50\n/s32/s108/s32/s61/s32/s51\n/s32/s108/s32/s61/s32/s52Figure 7. The functions I(l,R) for several values of l.\nIn the limit of large l–/radicalbig\n2(2l+3)≫kFL– one can\nreplace the Bessel functions in Eq. (C14) by the first\nterm of their Taylor series. This yields\nI(l,R)≈R2l+3\n(2l+1)!!(2l+3)!!. (C15)\nSince\n(2l+1)!! =(2l+1)!\n2ll!≈exp/bracketleftbig\n−1−l(1+ln2)+1\n2ln/parenleftbig\n2+1\nl/parenrightbig\n+(2l+1)ln(2l+1)−llnl] (C16)\n(see Stirling’s formula in Ref. 24), the energies of the\nlocalized states approach the gap edge exponentially fast\nwith growing l.\nAppendix D\nIn this Appendix we calculate the width of the impu-\nrity band at a small concentration of pairing-potential\nimpurities in the limit of strongly suppressed anisotropy,\nwhen the Green function satisfies Eq. (78). We note that\nin the vicinity of the energy of the bound state E, given\nby Eq. (84), the parameter x= 1−2if20∝an}bracketle{tg1∝an}bracketri}htbecomes\nsmall:|x| ≪1. Let us rewrite Eq. (78) substituting\n∝an}bracketle{tg1∝an}bracketri}ht=−i(1−x)/(2f20):\nE\n∝an}bracketle{t∆0∝an}bracketri}ht−1 =−c2f20\nπν0∝an}bracketle{t∆0∝an}bracketri}htx−2f2\n20\n(1−x)2−/angbracketleftbig\n∆′2/angbracketrightbig\nf20/planckover2pi1∝an}bracketle{t∆0∝an}bracketri}ht/parenleftBig\nτ−1\n1+τ−1\n2\nx/parenrightBig(1−x). (D1)\nNow we will expand the right-hand side in the powers of x. We assume τ−1\n2/|x| ≪τ−1\n1, which is valid at sufficiently\nsmall concentrations c2, as we shall see further. Then, keeping terms up to the order of x2, we obtain\nδE′+/bracketleftBigg\nc2f20\nπν0∝an}bracketle{t∆0∝an}bracketri}ht−/angbracketleftbig\n∆′2/angbracketrightbig\nτ2\n1\n/planckover2pi1f20∝an}bracketle{t∆0∝an}bracketri}htτ2(1−x)/bracketrightBigg\n1\nx+/parenleftBigg\n4f2\n20−/angbracketleftbig\n∆′2/angbracketrightbig\nτ1\n/planckover2pi1f20∝an}bracketle{t∆0∝an}bracketri}ht/parenrightBigg\nx+6f2\n20x2≈0, (D2)\nwhereδE′= (E−∝an}bracketle{t∆0∝an}bracketri}ht+E)/∝an}bracketle{t∆0∝an}bracketri}ht. Using the fact that τ−1\n2≤2c2/(/planckover2pi1πν0), it is easy to prove that\nc2f20\nπν0∝an}bracketle{t∆0∝an}bracketri}ht≫/angbracketleftbig\n∆′2/angbracketrightbig\nτ2\n1\n/planckover2pi1f20∝an}bracketle{t∆0∝an}bracketri}htτ2|1−x|,18\nsince the inequality (83) holds, and τ−1\n1≫4/radicalbig\n∝an}bracketle{t∆′2∝an}bracketri}ht/radicalbig\n∝an}bracketle{t∆0∝an}bracketri}ht//planckover2pi1in the limit of suppressed anisotropy. Also, it can be\nseen that the term proportional to x2can be neglected compared to the term proportional to xwhen\n|x| ≪1−/planckover2pi1\n8∝an}bracketle{t∆0∝an}bracketri}htf3\n20τdep, (D3)\nwhereτ−1\ndep= 2/angbracketleftbig\n∆′2/angbracketrightbig\nτ1//planckover2pi12. Then, we can determine xfrom Eq. (D2):\nx=δE′\n4f2\n20+/radicalbigg/parenleftBig\nδE′\n4f2\n20/parenrightBig2\n−c2\nπν0f20/angbracketleft∆0/angbracketright/parenleftBig\n1−/planckover2pi1\n8/angbracketleft∆0/angbracketrightf3\n20τdep/parenrightBig\n2/parenleftBig\n/planckover2pi1\n8/angbracketleft∆0/angbracketrightf3\n20τdep−1/parenrightBig . (D4)\nIf the expression under the square root is negative at a\ngiven value δE′, thenxis complex, which means that theenergy lies within the impurity band. This observation\nallows to determine the width Wof this band, which is\ngiven by Eq. (88).\n1P. Anderson, J. Phys. Chem. Solids 11, 26 (1959).\n2T.Tsuneto,Progress of Theoretical Physics 28, 857 (1962).\n3P. Hohenberg, Sov. Phys. JETP 18, 834 (1964).\n4J. R. Clem, Phys. Rev. 148, 392 (1966).\n5A. V. Balatsky, M. I. Salkola, and A. Rosengren,\nPhys. Rev. B 51, 15547 (1995).\n6A. A. Abrikosov and L. P. Gor’kov, Sov. Phys. JETP 12,\n1243 (1961).\n7L. Yu, Acta Phys. Sin. 21, 75 (1965).\n8H. Shiba, Prog. Theor. Phys. 40, 435 (1968).\n9A. I. Rusinov, Sov. Phys. JETP 29, 1101 (1969).\n10A. I. Larkin and Y. N. Ovchinnikov, Sov. Phys. JETP ,\n1144 (1971).\n11J. S. Meyer and B. D. Simons,\nPhys. Rev. B 64, 134516 (2001).\n12M. A. Skvortsov and M. V. Feigel’man, Zh. Eksp. Teor.\nFiz.144, 560 (2013).\n13K. Maki, “Gapless Superconductivity,” in Superconductiv-\nity, Vol. 2, edited by R. D. Parks (Marcel Dekker, New\nYork, 1969) pp. 1035 – 1105.\n14M. Tinkham, Introduction to Superconductivity , 2nd ed.,\nDover Books on Physics (Dover Publications, USA, 2004).\n15A. Weinkauf and J. Zittartz,\nZ. Phys. B Condens. Matter 21, 135 (1975).16M. E. Flatt´ e and J. M. Byers,\nPhys. Rev. B 56, 11213 (1997).\n17A. K. Chattopadhyay, R. A. Klemm, and D. Sa,\nJ. Phys. Condens. Matter 14, L577 (2002).\n18B. M. Andersen, A. Melikyan, T. S. Nunner, and P. J.\nHirschfeld, Phys. Rev. Lett. 96, 097004 (2006).\n19A. V. Balatsky, I. Vekhter, and J.-X. Zhu,\nRev. Mod. Phys. 78, 373 (2006).\n20N. Kopnin, Theory of Nonequilibrium Superconductivity ,\nInternational Series of Monographs on Physics (Clarendon\nPress, 2001).\n21U. Gunsenheimer and A. Hahn,\nPhysica B: Condensed Matter 218, 141 (1996), pro-\nceedings of the Second International Conference on\nPoint-contact Spectroscopy.\n22Y. V. Fominov, M. Houzet, and L. I. Glazman,\nPhys. Rev. B 84, 224517 (2011).\n23B. Geilikman and V. Z. Kresin, Kinetic and nonsteady-\nstate effects in superconductors (J. Wiley, New York,\n1974).\n24M. Abramowitz and I. A. Stegun, eds., Handbook of mathe-\nmatical functions with formulas, graphs, and mathematical\ntables, tenth printing ed. (U. S. Government Printing Of-\nfice, Washington, D. C., 1972).\n25E.Jahnke, F.Emde, andF.L¨ osch, Tafeln h¨oherer Funktio-\nnen(B. G. Teubner Verlagsgesellschaft, Stuttgart, 1960)."    },    {        "title": "1511.09421v2.Dimensional_Effects_on_the_Density_of_States_in_Systems_with_Quasi_Relativistic_Dispersion_Relations_and_Potential_Wells.pdf",        "content": "Dimensional E\u000bects on the Density of States in Systems with Quasi-Relativistic\nDispersion Relations and Potential Wells\nA. Pokraka\nDepartment of Physics, University of Alberta, Centennial Centre for\nInterdisciplinary Science, Edmonton, Alberta, Canada T6G 2E1.\u0003\nR. Dick\nDepartment of Physics and Engineering Physics, University of Saskatchewan,\n116 Science Place, Saskatoon, Saskatchewan, Canada SK S7N 5E2.y\nMotivated by the recent discoveries of materials with quasi-relativistic dispersion relations, we\ndetermine densities of states in materials with low dimensional substructures and relativistic disper-\nsion relations. We \fnd that these dimensionally hybrid systems yield quasi-relativistic densities of\nstates that are a superposition of the corresponding two- and three-dimensional densities of states.\nPACS numbers: 03.65.Pm, 71.15.Rf, 73.21.Fg\nI. INTRODUCTION\nQuantum mechanics and electrodynamics in one or two\ndimensions are commonly employed to describe the be-\nhaviour of particles or quasi-particles (such as phonons)\nin low-dimensional substructures like quantum wires or\nquantum wells. However while particle wave functions\nmay be compressed and squeezed, they will always ex-\ntend beyond the boundaries of attractive potential wells.\nTherefore it is of interest to develop analytic models to\nstudy the transition between two-dimensional and three-\ndimensional behavior of particles in these systems.\nThe transition between two-dimensional and three-\ndimensional behavior of particles in the presence of low-\ndimensional substructure has been examined for non-\nrelativistic dispersion relations both for quantum wells\n[1] and for substructures where particles propagate with\ndi\u000berent e\u000bective mass (see e.g. [2] and references there).\nThe quasi-relativistic case with di\u000berent e\u000bective mass\nin a substructure has been studied in Ref. [3]. All\nthese systems exhibit inter-dimensional behavior in the\nlocal density of states (DOS) in the sense that energy\nand length scales can be identi\fed in which the density\nof states approaches the well-known two-dimensional or\nthree-dimensional limits.\nIn this paper we extend these studies to the case of\nparticles with quasi-relativistic dispersion relations in the\npresence of quantum wells. Speci\fcally, we determine the\nDOS of charged quasi-relativistic bosons in the presence\nof a thin quantum well.\nThe DOS counts the number of states per unit vol-\nume at a given energy and position, and is important for\nunderstanding various properties of materials - in partic-\nular, for estimating the availability of carriers for charge\nand heat transport. It also impacts scattering and ab-\nsorption in materials and signi\fes band gaps. Con\fning\n\u0003pokraka@ualberta.ca\nyrainer.dick@usask.capotentials change the local density of states in a way that\nmanifests the interplay between three-dimensional and\nlow-dimensional e\u000bects. For example, consider a thin in-\nterface in a three-dimensional bulk material, modeled as\na\u000ewell potential. In Schr odinger theory, the DOS at the\ninterface is [1]\n\u001a(E;z 0) =\u0014\u001an=2(E+ (~2\u00142=2m))\n+\u001an=3(E)\"\n1\u0000~\u0014p\n2mEarctan p\n2mE\n~\u0014!#\n(1)\nwhereW=ql\b0parameterizes the e\u000bective thick-\nness and depth of the well, \u0014=mW=~2is the in-\nverse penetration depth, and \u001an=2(E) = \u0002(E)m\n\u0019~2\nand\u001an=3(E) = \u0002(E)m\n\u00192~3p\n2mE are the well known\ntwo- and three-dimensional DOS with two spin states.\nThe DOS at the interface is a superposition of the\ntwo- and three-dimensional DOS. The factor 1 \u0000\n~\u0014=p\n2mEarctan(p\n2mE= (~\u0014)) smoothly turns on the\nthree-dimensional contribution to the DOS. Equation (1)\ncon\frms the intuitive assumption that bound states in\nthe well contribute to a two-dimensional DOS which is\nmade dimensionally correct in three dimensions through\nscaling by the inverse penetration depth, \u0014. Note also\nthatB=~2\u00142=2mis the binding energy of particles in\nthe well, i.e. their energy is\nE=~2k2\nk\n2m\u0000~2\u00142\n2m=~2k2\nk\n2m\u0000mW2\n2~2; (2)\nwhere ~kkis the momentum parallel to the potential well.\nThe argument of the two-dimensional contribution to the\ndensity of states in (1) is therefore the kinetic energy of\nparticles in the well.\nResearch into materials with quasi-relativistic dis-\npersion relations has exploded since the discovery of\ngraphene in 2004 [4]. These materials include two and\nthree-dimensional Dirac semi-metals [5, 6], topological\ninsulators [7, 8], topological Dirac semi-metals [9, 10] and\nsuper\ruid phases of3He [11]. All of these materials havearXiv:1511.09421v2  [cond-mat.mes-hall]  29 Apr 20162\nlow energy quasiparticle excitations described by Dirac\nHamiltonians and thus linear dispersion relations centred\naround Dirac points in momentum space [11]. In partic-\nular, these materials posses novel electronic properties\nand thus have promising applications in spintronics and\nquantum computing.\nIn this paper we study two related quasi-relativistic\ninter-dimensional systems, each implemented through\na con\fning electrostatic potential in the Klein-Gordon\nequation. In spite of the apparent relevance of quasi-\nrelativistic dispersion relations for modern materials sci-\nence, there has not been much recent activity on low-\ndimensional potentials with quasi-relativistic wave equa-\ntions. Sveshnikov and Silaev have analyzed spectra and\nwave functions in (1+1)-dimensional systems with imag-\ninary spatial translations [12], and Ananchenko et al.\nhave studied bound states in a (2 + 1)-dimensional Dirac\nequation with a spherically symmetric potential well [13].\nWe are considering Klein-Gordon type systems in 3 + 1\ndimensions where the low-dimensional substructure is a\nplanar quantum well at z=z0. We are also focusing on\nthe local density of states since this provides an excellent\nprobe for the transition between two-dimensional and\nthree-dimensional behavior [1{3]. In Section II, we cal-\nculate the inter-dimensional Klein-Gordon Green's func-\ntion for a delta well potential to \frst order in the parti-\ncle charge, q. Then using the relativistic generalization\nof the well known relation between the imaginary part\nof the Green's function and the DOS, we calculate the\ninter-dimensional DOS at the well interface. In Section\nIII we calculate, numerically to full order in q, the inter-\ndimensional DOS inside shallow \fnite square wells. In\nboth sections 2 and 3 we examine under what circum-\nstances we \fnd two and three dimensional behaviour in\nour model systems. Our conclusions are presented in Sec-\ntion IV and some mathematical details are collected in\nappendices A and B.\nII. DELTA WELL POTENTIALS\nThe relativistic generalization of the relation between\nthe imaginary part of the Green's function and the DOS\nis [3]\n\u001a(E;z)\u0000\u0016\u001a(\u0016E;z) =2E\n\u0019~2c2=hxk;zjG(E)jxk;zi (3)\nwhere \u0016\u001ais the anti-particle DOS, \u0016E=\u0000Eis the anti-\nparticle energy and =signi\fes the imaginary part of a\ncomplex number. We calculate the Green's function for\nthe quasi-relativistic quantum well and then use (3) to\ndetermine the inter-dimensional DOS.\nWe consider a three-dimensional bulk material with a\nthin interface. Inside the interface, the electrostatic po-\ntential di\u000bers by a constant factor \b 0. We approximate\nthe interface as a delta well, centred at z=z0, so that the\nelectromagnetic potential is given by A\u0016= (\b(x)=c;0)where \b(x) =\u0000l\b0\u000e(z\u0000z0). The parameter, l, has di-\nmensions of length (needed to make the potential dimen-\nsionally correct as the delta function has units of inverse\nlength) and intuitively parameterizes the thickness of the\nwell.\nThe equation of motion for coupling of charged Klein-\nGordon and electromagnetic \felds is\n\u0012\u0010\n@\u0016\u0000iq\n~A\u0016\u0011\u0010\n@\u0016\u0000iq\n~A\u0016\u0011\n\u0000m2c2\n~2\u0013\n\u001e(x0;x) = 0:\n(4)\nSubstituting A\u0016and keeping only leading order terms in\nqyields the equation of motion\n\u0012\n@\u0016@\u0016\u0000m2c2\n~2+ 2iq\b0l\nc~\u000e(z\u0000z0)@0\u0013\n\u001e(x0;x) = 0 (5)\nwhere we use the convention \u001100=\u00001 for the Minkowski\nmetric.\nSince the retarded Green's function is related to the\ndensity of states by equation (3), we look for the x-\nrepresentation of the Green's function, hxjGjx0i, that sat-\nis\fes\n\u001a\n\u0000@2\n0+r2\u0000m2c2\n~2+ 2iq\b0l\nc~\u000e(z\u0000z0)@0\u001b\nhxjGjx0i\n=\u0000\u000e(z\u0000z0)\u000e(x0\u0000x00)\u000e(xk\u0000x0\nk):(6)\nThe solution to equation (6), detailed in Appendix A,\nhas the form of a Hankel transformation\nD\nz;xkjG(E)jx0\nk;z0E\njE=~ck0=\nZdkk\n2\u0019J0\u0010\nkk\f\f\fxk\u0000x0\nk\f\f\f\u0011\nkk\nzjG(E;kk)jz0\u000b\n(7)\nwhere\n\nzjG(E;kk)jz0\u000b\njE=~ck0=\u0002\u0010\n(k0)2\u0000k2\nk\u0000m2c2\n~2\u0011\n2q\n(k0)2\u0000k2\nk\u0000m2c2\n~2\n\u0002i(\nexp\"\nir\n(k0)2\u0000k2\nk\u0000m2c2\n~2jz\u0000z0j#\n+ql\b0\nc~k0\n\u0002iexph\niq\n(k0)2\u0000k2\nk\u0000m2c2\n~2(jz\u0000z0j+jz0\u0000z0j)i\nq\n(k0)2\u0000k2\nk\u0000m2c2\n~2\u0000iql\b0\nc~k09\n=\n;\n+\u0002\u0010\nm2c2\n~2+k2\nk\u0000(k0)2\u0011\n2q\nm2c2\n~2+k2\nk\u0000(k0)2\n\u0002(\nexp\"\n\u0000r\nm2c2\n~2+k2\nk\u0000(k0)2jz\u0000z0j#\n+ql\b0\nc~k0\n\u0002exph\n\u0000q\nm2c2\n~2+k2\nk\u0000(k0)2(jz\u0000z0j+jz0\u0000z0j)i\nq\nm2c2\n~2+k2\nk\u0000(k0)2\u0000ql\b0\nc~k0\u0000i\u000f9\n=\n;:\n(8)3\nEquation (8) is the Greens function for a quasi-\nrelativistic boson in the presence of a thin quantum well,\nvalid for arbitrary position and energy. Consistently,\nin the limit l;\b0!0 equation (8) reproduces the free\nKlein-Gordon Green's function. It is not easy to identify\na two-dimensional limit from this Green's function and\ntherefore we examine the DOS in the presence of the well\nin order to explore this question further.\nSubstituting (7) into (3) yields the inter-dimensional\nDOS. However we are mostly interested in the DOS at\n(or near) the interface because this is where we expect the\nmost prominent inter-dimensional e\u000bects. The density of\nstates at the interface ( z=z0=z0) is\n\u001a(E;z 0)\u0000\u001a(E;z 0) =\n\u0002(E)\u001an=2(~E) \n1\n1 +W2\n~2c2!\nW~E\n~2c2+\u001an=3(E)\n\u0002\"\n1\u0000WE\n~cp\nE2\u0000m2c4arctan \n~cp\nE2\u0000m2c4\nWE!#\n(9)\nwhere we usedW=ql\b0as in the non-relativistic delta\nwell, the energy variable in the two-dimensional contri-\nbution is ~E=Ep\n1 + (W=~c)2, and the two- and three-\ndimensional densities of states are \u001an=2(E) = \u0002(E2\u0000\nm2c4)E\n2\u0019(~c)2and\u001an=3(E) = \u0002(E2\u0000m2c4)Ep\nE2\u0000m2c4\n2\u00192(~c)3.\nHere, like in equation (1), Wparametrizes the e\u000bective\nthickness and depth of the well.\nThe key features of (9) are analogous to those of (1).\nEquation (9) is a superposition of the standard two-\nand three-dimensional relativistic DOS, Fig. 1. The\nstates perpendicular to the interface are exponentially\nsuppressed and contribute a term proportional to the\ntwo-dimensional DOS. The argument of the two dimen-\nsional DOS is scaled by a factorp\n1 + (W=~c)2due to\nthe presence of the well. We can understand the appear-\nance of this factor in the two-dimensional contributions\nto (9) by noting that the bound energy eigenstates in the\nrelativistic delta well are\n \u0014;kk(xk;z) =p\u0014\n2\u0019exp(ikk\u0001xk\u0000\u0014jzj) (10)\nwith the transverse damping factor\n\u0014=WE\n~2c2(11)\nand energy\nE=cs\nmc2+~2k2\nk\n1 + (W=~c)2; (12)\ni.e. the argument of the two-dimensional contribution is\nactually the rest energy plus kinetic energy for motion of\nthe bound states along the well,\n~E=cq\nmc2+~2k2\nk; (13)and the dimensional factor that converts the relativistic\ntwo-dimensional DOS into the contribution from the well\nstates to the three-dimensional DOS is \u0014=p\n1 + (W=~c)2.\nThe factor 1\u0000WE\n~cp\nE2\u0000m2c4arctan\u0010\n~cp\nE2\u0000m2c4\nWE\u0011\nsmoothly turns the three-dimensional contribution on; in\nother words it is responsible for the transition between\ntwo- and three-dimensional behavior.\n1.0 1.1 1.2 1.3 1.4 1.50500100015002000250030003500\nE/mc2ρ(E)[eV-1nm-3]\nFIG. 1. The density of states for charged bosonic particles\nof massm=me=:511 MeV in the interface with W=\n0:02\u0016m\u0001eV. DotDashed: contribution from the bound states.\nDashed: three-dimensional DOS in absence of any quantum\nwells. Solid: the inter-dimensional DOS according to (9)\n0.0 0.1 0.2 0.3 0.4 0.5050010001500200025003000\nEkinetic /mc2ρ(E)[eV-1nm-3]\nFIG. 2. The density of states for charged bosonic particles of\nmassm=me=:511 MeV in the interface with W= 0:39 nm\u0001\neV which corresponds to a binding energy of 1 eV. Dashed:\nthe non-relativistic DOS given by equation (1). Solid: the\nrelativistic DOS given by (9).\nIn the limitW! 0, the inter-dimensional DOS reduces\nto the three-dimensional relativistic DOS. For W>~c\nthe contribution from the two-dimensional term is en-\nhanced so that in the limit W\u001d ~c, equation (9) tends\nto the two-dimensional term\u0010\n1 +W2\n~2c2\u0011\u00001W~E\n~2c2\u001an=2(~E)\nsince the DOS is dominated by well states. In the low\nenergy limit, E!mc2, equation (9) reduces to the non-\nrelativistic equation from Schr odinger theory (1), Fig.\n2. Furthermore, in the energy range for bound particle4\nstates\nmc2\np\n1 + (W=~c)2\u0014E <mc2(14)\nonly the two-dimensional term in (9) contributes. All\nthese limits behave as expected.\nIII. SHALLOW FINITE SQUARE WELLS\nWhile delta wells are interesting, \fnite square wells\nprovide a more realistic model for a low-dimensional\nsystems as no real physical system is in\fnitely\nthin/genuinely two-dimensional.\nAlthough the Green's function method is useful for\ndelta wells, when applied to \fnite square wells this\nmethod results in ambiguities in choosing how to shift\nthe poles of the Green's function, thus motivating us to\ntake a di\u000berent approach.\nIn general, the density of states is given by\n\u001a(E)dE=dn\nV=jhxj\u000bij2d\u000b (15)\nwhere\u000bis the set of quantum numbers representing the\nstate [1]. Therefore, to determine the DOS in systems\nwith \fnite square wells, we construct the states of the\nsystem by solving the equation of motion for our model\ninter-dimensional system and then use equation (15).\nWe consider a system consisting of a three-dimensional\nbulk with an interface of width 2 acentred at z= 0.\nThe electrostatic potential inside the interface di\u000bers\nby a constant, \b 0(i.e.,A\u0016= (\b(x)=c;0) and \b(x) =\n\u0000\b0\u0002(a2\u0000z2)). Substitution of A\u0016into the equation of\nmotion for coupling of charged Klein-Gordon and elec-\ntromagnetic \felds, equation (4), yields\n\u0012\n\u0000@2\n0+r2\u0000m2c2\n~2+\u0002(a2\u0000z2)\u00122iW\n~c@0+W2\n~2c2\u0013\u0013\n\u001e(x) = 0\n(16)\nwhereW=q\b0. Similar to the delta well, W,\nparametrizes the depth of the well. However, the factor\nlis absent because the well thickness is now contained in\nthe Heaviside step function, \u0002( a2\u0000z2).\nBreaking up the spatial vector into components par-\nallel to the interface, xk, and perpendicular to the in-\nterface,z, we let x= (xk;z). Then equation (16)\ncan be solved by invoking several separation ans atze so\nthat\u001e(x) =\u001e0(x0)\u001ek(xk)\u001e?(z). The parallel and time\ncomponents of the wave function are easily found to be\n\u001ek(xk) =1\n2\u0019eikk\u0001xkand\u001e0(x0) =e\u0000ik0x0.\nNote that\u001ekis a normalized plane wave in the plane\nparallel to the interface. Also, \u001e0does not contain a\nfactor of 1=p\n2\u0019because wave functions cannot be nor-\nmalized in the time direction.\nThe di\u000berential equation for the perpendicular wavefunction is given by\n@2\n?\u001e?\n\u001e?=\u0000(k0)2\u0000\u0002(a2\u0000z2)2W\n~ck0\u0000\u0002(a2\u0000z2)W2\n~2c2\n+m2c2\n~2+k2\nk=\u0006k2\n?: (17)\nInside the well it has the possible solutions\n\u001e?(z) =Asin(k?z) +Bcos(k?z) (18)\n\u001e?(z) =Aek?z+Be\u0000k?z(19)\nwith energies\nE=\u0000W\u0006q\nm2c4+~2c2k2\nk\u0006~2c2k2\n?; (20)\noutside the well, the solutions are\n\u001e?(z) =Asin(k0\n?z) +Bcos(k0\n?z) (21)\n\u001e?(z) =Aek0\n?z+Be\u0000k0\n?z(22)\nwith energies\nE=\u0006q\nm2c4+~2c2k2\nk\u0006~2c2k0\n?2: (23)\nHere,k?denotes the perpendicular component of mo-\nmentum inside the well while k0\n?denotes the perpendic-\nular component of momentum outside the well.\nWe consider the simplest case of shallow wells, 0 \u0014\nW \u0014mc2, which yield free particle and anti-particle,\nbound particle, and tunneling anti-particle states. These\nwave functions were calculated and are listed in appendix\nB.\nThere is no mixing of particle and anti-particle states\n(this requiresW> mc2) nor particle anti-particle pair\nproduction at the well boundaries (i.e. Klein paradox\nwhich requiresW>2mc2). Additionally, charge is con-\nserved for Klein-Gordon particles. Therefore, we can\nform a well de\fned DOS even though the Klein-Gordon\n\feld does not have a conserved probability density. The\nDOS for shallow wells (given by equation (15)) is then\nfound to be\n\u001a(E;xk;z) =\u001a(E;z) =\nX\n\u0006X\nn\u0002\u0010\nE+W\u0000p\nm2c4+~2c2\u00102n=a2\u0011E+W\n2\u0019~2c2j \u0006\nn(z)j2\n+X\n\u0006\u0002\u0010\nE\u0000mc2\u0011E+W\n2\u0019~2c2Z\u0003\ndk?j \u0006\npart(z)j2\n+X\n\u0006\u0002\u0010\n\u0000mc2\u0000(E+W)\u0011E+W\n2\u0019~2c2Z\u0003\ndk?j \u0006\nanti(z)j2\n+X\n\u0006\u0002\u0010\n\u0000mc2\u0000E\u0011E+W\n2\u0019~2c2Z\u0003\nd\u0014?j \u0006\ntunnel(z)j2\n(24)\nwhere (+) denotes the positive parity states and ( \u0000) the\nnegative parity states. The integration limits, the depen-\ndences of the wave functions on the wavenumbers, and\nthe bound state parameters \u0010nare given in Appendix B.5\nEquation (24) contains contributions from both parti-\ncle and anti-particle states. To obtain the DOS for either\nparticles or anti-particles, we simply omit the undesired\nstates from equation (24). The integrals in (24) cannot\nbe solved analytically and therefore were computed nu-\nmerically.\n1.0 1.2 1.4 1.6 1.8 2.00100020003000400050006000\nE/mc2ρ(E)[eV-1nm-3]\nFIG. 3. The particle DOS in the interface at z= 0 forW=\nmc2=10000 and mass m=me= 0:511MeV. The width was\nset to 1 micrometer. DotDashed: is the contribution from the\nstates bound inside the quantum well. Dashed: is the three-\ndimensional DOS in absence of any quantum wells. Solid: is\nthe DOS according to (24).\n1.0 1.2 1.4 1.6 1.8 2.00100020003000400050006000\nE/mc2ρ(E)[eV-1nm-3]\nFIG. 4. The particle DOS in the interface at z= 0 forW=\nmc2=300 and mass m=me= 0:511MeV. The width was set\nto 1 micrometer. DotDashed: is the contribution from the\nstates bound inside the quantum well. Dashed: is the three-\ndimensional DOS in absence of any quantum wells. Solid: is\nthe DOS according to (24).\nThe DOS in inter-dimensional systems with shallow\nwells is an interpolation between the well known two-\nand three-dimensional DOS. For extremely shallow well\ndepths,W \u001cmc2, the inter-dimensional DOS ap-\nproaches the three-dimensional limit, Fig. 3; increasing\nthe well depth brings the inter-dimensional DOS closer\ntoward the two-dimensional DOS, Fig. 4. This is a con-\nsequence of larger number of states in the well, which\ncorresponds to an increase of the relative weight of the\ntwo-dimensional contribution from the well states to the\nDOS.Using a similar procedure as outlined in this section\nwe also calculated the inter-dimensional DOS for \fnite\nsquare wells in Schr odinger theory. In the low-energy\nlimit,E!mc2, we observe agreement between the\nKlein-Gordon and Schr odinger inter-dimensional DOS.\nIV. CONCLUSIONS\nWe have calculated the density of states for quasi-\nrelativistic bosons moving in systems with thin interfaces,\nwhere we have assumed that the electrostatic potential\ndi\u000bers by a constant from the bulk material. We have\nfound that the density of states in the well reduces to the\nnon-relativistic density of states from Schr odinger the-\nory in the low energy limit, E!mc2. Also in accord\nwith intuition, the inter-dimensional density of states\ntends to the three-dimensional density of states in the\nlimitW ! 0, whereas the contribution from the two-\ndimensional density of well states increases for larger W,\ncorresponding in particular to larger well depth. Further-\nmore, we found that the dimensional conversion factor \u0014\n(inverse penetration length of the well states) which con-\nverts the two-dimensional density of states in the energy\nscale to a contribution to the three-dimensional density of\nstates, in the relativistic case becomes \u0014=p\n1 + (W=~c)2.\nACKNOWLEDGMENTS\nThis work was supported in part by NSERC Canada.\nAPPENDIX A: SOLUTION TO EQUATION (6)\nSubstitution of the Fourier decomposition of hxjGjx0i,\nhxjGjx0i=Z\nd2kkZ\nd2k0\nkZ\ndk0Z\ndk00Z\ndk?\n\u0002exph\ni\u0010\nk00x00\u0000k0x0+kk\u0001xk\u0000k0\nk\u0001x0\nk+k?z\u0011i\n\u0002\u00121\n2\u0019\u00137\n2D\nk0;kk;k?jGjk00;k0\nk;z0E\n;(25)\ninto equation (6) yields\n\u00121\n2\u0019\u00137\n2Z\nd2kkZ\nd2k0\nkZ\ndk0Z\ndk00Z\ndk?\n\u0002\u001a\n(k0)2\u0000k2\nk\u0000k2\n?\u0000m2c2\n~2+ 2ql\b0\nc~k0\u000e(z\u0000z0)\u001b\n\u0002exph\ni\u0010\nk00x00\u0000k0x0+kk\u0001xk\u0000k0\nk\u0001x0\nk+k?z\u0011i\n\u0002D\nk0;kk;k?jGjk00;k0\nk;z0E\n=\u0000\u000e(z\u0000z0)\u000e(x0\u0000x00)\u000e(xk\u0000x0\nk):(26)6\nNext, we take the inverse Fourier transform with respect\nto the variables x0;x00;xk;x0\nkand evaluate the resulting\nintegrals to get\nZ\ndk?exp (ik?z)\u0002D\n\u00140;\u0014k;k?jGj\u001400;\u00140\nk;z0E\n\u001am2c2\n~2+\u00142\nk+k2\n?\u0000(\u00140)2\u00002ql\b0\nc~\u00140\u000e(z\u0000z0)\u001b\n=p\n2\u0019\u000e(z\u0000z0)\u000e(kk\u0000k0\nk)\u000e\u0000\nk0\u0000k00\u0001\n:(27)\nTaking the inverse Fourier Transform with respect to z,\nwe get\nZ\ndk?Z\ndzexp [iz(k?\u0000\u0014?)]D\n\u00140;\u0014k;k?jGj\u001400;\u00140\nk;z0E\n\u0002\u001am2c2\n~2+\u00142\nk+k2\n?\u0000(\u00140)2\u00002ql\b0\nc~\u00140\u000e(z\u0000z0)\u001b\n=p\n2\u0019Z\ndzexp (\u0000i\u0014?z)\u000e(z\u0000z0)\u000e(kk\u0000k0\nk)\u000e\u0000\nk0\u0000k00\u0001\n:\n(28)\nSince (6) is a linear di\u000berential equation of constant co-\ne\u000ecients with respect to all variables with the exception\nofzandz0, it follows thatD\nk0;kk;k?jGjk00;k0\nk;z0E\n=\nk?jG(k0;kk)jz0\u000b\n\u000e(kk\u0000k0\nk)\u000e\u0000\nk0\u0000k00\u0001\n. Equation (28)\nbecomes\n\u0012m2c2\n~2+\u00142\nk+\u00142\n?\u0000(\u00140)2\u0013\n\u0014?jG(\u00140;\u0014k)jz0\u000b\nexp (iz0\u0014?)\n\u0000ql\b0\nc~\u00140Zdk?\n\u0019exp (iz0k?)\nk?jG(\u00140;\u0014k)jz0\u000b\n=1p\n2\u0019exp [i\u0014?(z0\u0000z0)]:(29)\nThe mixed representation,\n\u0014?jG(\u00140;\u0014k)jz0\u000b\n;appears\nboth inside and outside of the integral in equation (29).\nSince we are integrating over the perpendicular mo-\nmentum variable,Rdk?\n\u0019exp (iz0k?)\nk?jG(\u00140;\u0014k)jz0\u000b\nis\na function of \u00140;\u0014k;andz0. This suggests writing (29)\nas\n\n\u0014?jG(\u00140;\u0014k)jz0\u000b\nexp (iz0\u0014?) =\nexp[i\u0014?(z0\u0000z0)]p\n2\u0019+f(\u00140;\u0014k;z0)\nm2c2\n~2+\u00142\nk+\u00142\n?\u0000(\u00140)2\u0000i\u000f;(30)\nwhere we have shifted the poles in agreement with\nthe conventions for a retarded Green's function and\nf(\u00140;\u0014k;z0) satis\fes\nf(\u00140;\u0014k;z0) +ql\b0\nc~\u00140Zdk?\n\u0019exp (iz0k?)\n\u0002\nk?jG(\u00140;\u0014k)jz0\u000b\n= 0:(31)Solving for f(k0;kk;z0), we \fnd\nf(k0;kk;z0) =1p\n2\u0019ql\b0\nc~k0Rd\u0014?\n\u0019exp[i\u0014?(z0\u0000z0)]\n\u00142\n?\u0000\u0010\n(k0)2\u0000m2c2\n~2\u0000k2\nk\u0011\n\u0000i\u000f\n1\u0000ql\b0\nc~k0Rd\u0014?\n\u00191\n\u00142\n?\u0000\u0010\n(k0)2\u0000m2c2\n~2\u0000kk\u0011\n\u0000i\u000f:\n(32)\nSimplifying the integrals in (32) yields\nf(k0;kk;z0) =\u0002\u0010\n(k0)2\u0000k2\nk\u0000m2c2\n~2\u0011\np\n2\u0019\n\u00020\n@ql\b0\nc~\u00140iexph\ni\u0010\n(k0)2\u0000k2\nk\u0000m2c2\n~2\u0011\njz0\u0000z0ji\nq\n(k0)2\u0000k2\nk\u0000m2c2\n~2\u0000iql\b0\nc~\u001401\nA\n+\u0002\u0010\nm2c2\n~2+k2\nk\u0000(k0)2\u0011\np\n2\u0019\n\u00020\n@ql\b0\nc~\u00140exph\n\u0000\u0010\nm2c2\n~2+k2\nk\u0000(k0)2\u0011\njz0\u0000z0ji\nq\nm2c2\n~2+k2\nk\u0000(k0)2\u0000ql\b0\nc~\u00140\u0000i\u000f1\nA:\n(33)\nHence, we \fnd that\n\nk?jG(k0;kk)jz0\u000b\n=1p\n2\u00191\nm2c2\n~2+k?+k2\nk\u0000(k0)2\u0000i\u000f\n\u0002\u001a\nexp (\u0000ikz0) + \u0002\u0012\n(k0)2\u0000k2\nk\u0000m2c2\n~2\u0013\nexp (\u0000ik?z0)\n\u00020\n@ql\b0\nc~k0iexph\ni\u0010\n(k0)2\u0000k2\nk\u0000m2c2\n~2\u0011\njz0\u0000z0ji\nq\n(k0)2\u0000k2\nk\u0000m2c2\n~2\u0000iql\b0\nc~k01\nA\n+ \u0002\u0012m2c2\n~2+k2\nk\u0000(k0)2\u0013\nexp (\u0000ik?z0)\n\u00020\n@ql\b0\nc~k0exph\n\u0000\u0010\nm2c2\n~2+k2\nk\u0000(k0)2\u0011\njz0\u0000z0ji\nq\nm2c2\n~2+k2\nk\u0000(k0)2\u0000ql\b0\nc~k0\u0000i\u000f1\nA9\n=\n;:\n(34)\nTo obtain\nzjG(E;xk)jz0\u000b\njE=~ck0, we \frst preform\nan inverse Fourier transform with respect to z0on the\nmixed representation of the retarded Green's function,7\n\nk?jG(k0;kk)jz0\u000b\n, to get\n\nzjG(E;kk)jz0\u000b\njE=~ck0=\u0002\u0010\n(k0)2\u0000k2\nk\u0000m2c2\n~2\u0011\n2\u0010q\n(k0)2\u0000k2\nk\u0000m2c2\n~2\u0011\n\u0002i(\nexp\"\nir\n(k0)2\u0000k2\nk\u0000m2c2\n~2jz\u0000z0j#\n+ql\b0\nc~k0\n\u0002iexph\niq\n(k0)2\u0000k2\nk\u0000m2c2\n~2(jz\u0000z0j+jz0\u0000z0j)i\nq\n(k0)2\u0000k2\nk\u0000m2c2\n~2\u0000iql\b0\nc~\u001409\n=\n;\n+\u0002\u0010\nm2c2\n~2+k2\nk\u0000(k0)2\u0011\n2\u0010q\nm2c2\n~2+k2\nk\u0000(k0)2\u0011\n\u0002(\nexp\"\n\u0000r\nm2c2\n~2+k2\nk\u0000(k0)2jz\u0000z0j#\n+ql\b0\nc~k0\n\u0002exph\n\u0000q\nm2c2\n~2+k2\nk\u0000(k0)2(jz\u0000z0j+jz0\u0000z0j)i\nq\nm2c2\n~2+k2\nk\u0000(k0)2\u0000ql\b0\nc~\u00140\u0000i\u000f9\n=\n;:\n(35)\nPerforming a further Fourier transform with respect to\nkkyields\nhxjG(E)jx0i=Z\nd2kkexph\nikk\u0010\nxk\u0000x0\nk\u0011i\n\u0002\nzjG(E;kk)jz0\u000b\n=Z\nd\u0012Zdkk\n(2\u0019)2kkexph\nikk\f\f\fxk\u0000x0\nk\f\f\fcos(\u0012)i\n\u0002\nzjG(E;kk)jz0\u000b\n=Zdkk\n2\u0019J0(kk\f\f\fxk\u0000x0\nk\f\f\f)kk\nzjG(E;kk)jz0\u000b\n:(36)\nHowever, we are most interested in the Green's func-\ntion at the interface, z=z0, where we expect the most\nprominent inter-dimensional e\u000bects. At the interface,\nJ0(0) = 1, and\nhxjG(E)jxijE=~ck0=Zdkk\n2\u0019kk\nzjG(E;kk)jz\u000b\n:(37)\nAPPENDIX B: SUMMARY OF STATES\nIn this appendix we collect the wave functions with\ntheir appropriate energy and momentum ranges, neces-\nsary to carry out the integration in equation (24).Free Particle and Anti-Particle States\nThe wave functions for the free particle and anti-\nparticle states are\n\u001e+\npart(z) =\u001e+\nanti(z) =1=p\u0019r\ncos2(k?a) +k2\n?\nk0\n?2sin2(k?a)\n\u0002\u0014\n\u0002(a2\u0000z2) cos(k?z) + \u0002(z2\u0000a2)\n\u0002h\ncos(k?a) cos (k0\n?(jzj\u0000a))\n\u0000k?\nk0\n?sin(k?a) sin (k0\n?(jzj\u0000a))i\u0015\n(38)\nand\n\u001e\u0000\npart(z) =\u001e\u0000\nanti(z) =1=p\u0019r\nsin2(k?a) +k2\n?\nk0\n?2cos2(k?a)\n\u0002\u0014\n\u0002(a2\u0000z2) sin(k?z) + \u0002(z2\u0000a2)\n\u0002sign(z)h\nsin(k?a) cos (k0\n?(jzj\u0000a))\n+ cos(k?a) sin (k0\n?(jzj\u0000a))i\u0015\n:(39)\nThe free particle energy is given by\nE=\u0000W+q\nm2c4+~2c2(k2\nk+k2\n?)\n=q\nm2c4+~2c2(k2\nk+k0\n?2): (40)\nwhereE\u0015mc2. Therefore, for \fxed energy, k?is re-\nstricted to the interval\np\n2EW+W2\u0014~ck?\u0014p\n(E+W)2\u0000m2c4:(41)\nThe free anti-particle energy is given by\n\u0016E=W+q\nm2c4+~2c2(k2\nk+k2\n?)\n=q\nm2c4+~2c2(k2\nk+k0\n?2): (42)\nwhere \u0016E\u0015mc2+W. For \fxed energy, k?is restricted\nto the interval\n0\u0014~ck?\u0014p\n(E+W)2\u0000m2c4: (43)8\nTunneling Anti-Particle States\nThe wave functions for the tunneling anti-particle\nstates are\n\u001e+\ntunnel(z) =1=p\u0019r\ncosh2(k?a) +k2\n?\nk0\n?2sinh2(k?a)\n\u0002\u0014\n\u0002(a2\u0000z2) cosh(k?z) + \u0002(z2\u0000a2)\n\u0002h\ncosh(k?a) cos (k0\n?(jzj\u0000a))\n+k?\nk0\n?sinh(k?a) sin (k0\n?(jzj\u0000a))i\u0015\n(44)\nand\n\u001e\u0000\ntunnel(z) =1=p\u0019r\nsinh2(k?a) +k2\n?\nk0\n?2cosh2(k?a)\n\u0002\u0014\n\u0002(a2\u0000z2) sinh(k?z) + \u0002(z2\u0000a2)\n\u0002sign(z)h\nsinh(k?a) cos (k0\n?(jzj\u0000a))\n+k?\nk0\n?cosh(k?a) sin (k?(jzj\u0000a))i\u0015\n:(45)\nThe tunneling anti-particle energy is given by\n\u0016E=W\u0000q\nm2c4+~2c2(k2\nk\u0000k2\n?)\n=q\nm2c4+~2c2(k2\nk+k0\n?2) (46)\nwhere \u0016E\u0015mc2and for a \fxed energy, k?is restricted\nto the interval\n<p\nm2c4\u0000(E+W)2\u0014~ck?\u0014p\n\u0000(2EW+W2):\n(47)Bound Particle States\nThe wave functions for the bound particle states are\n\u001e+\nn(z) =e\u00100\nn\npa\u0014\n1 +1\n\u00100n\u0015\u00001=2\u0014\n\u0002(a2\u0000z2) cos(\u0010nz=a)e\u0000\u00100\nn\n+ \u0002(z2\u0000a2) cos(\u0010n)e\u0000\u00100\nnjzj=a\u0015\n(48)\nand\n\u001e\u0000\nn(z) =e\u00100\nn\npa\u0014\n1 +1\n\u00100n\u0015\u00001=2\u0014\n\u0002(a2\u0000z2) sin(\u0010nz=a)e\u0000\u00100\nn\n+ \u0002(z2\u0000a2)sign(z) sin(\u0010n)e\u0000\u00100\nnjzj=a\u0015\n:(49)\nFor the even solution, the momentum perpendicular to\nthe well is quantized by the condition\n\u0010ntan(\u0010n) =s\n2Wa\n~cr\n\u00102n+m2c2\n~2a2\u0000\u00102n\u0000W2a2\n~2c2=\u00100\nn\n(50)\nwhere\u0010=k?=aand\u00100=k0\n?=a. For the odd solution,\nthe momentum perpendicular to the well is quantized by\nthe condition\n\u0010ncot(\u0010n) =\u0000s\n2Wa\n~cr\n\u00102n+m2c2\n~2a2\u0000\u00102n\u0000W2a2\n~2c2\n=\u0000\u00100\nn(51)\nwhere\u0010=k?=aand\u00100=k0\n?=a. The allowed energies\nare given by\nEn=\u0000W+q\nm2c4+~2c2(k2\nk+\u00102n=a2) (52)\n=q\nm2c4+~2c2(k2\nk\u0000\u00100n2=a2): (53)\nwhereEn\u0015mc2\u0000W.\n[1] R. Dick, Advanced Quantum Mechanics: Materials and\nPhotons (Springer, New York, 2012).\n[2] R. Dick, Dimensional E\u000bects on Densities of States and\nInteractions in Nanostructures, Nanoscale Res. Lett. 5,\n1546 (2010).\n[3] A. C. Zulkoskey, R. Dick, and K. Tanaka, Inter-\ndimensional E\u000bects in Systems with Quasi-Relativistic\nDispersion Relations, Phys. Rev. A 89, 052103 (2014).\n[4] A. H. Catro Neto, F. Guinea, N. M. R. Peres, K. S.\nNovoselov, and A. K. Geim, The electronic properties\nof graphene, Rev. Mod. Phys. 81, 109 (2009).\n[5] S. M. Young, S. Zaheer, J. C. Y. Teo, C. L. Kane, E.\nJ. Mele, and A. M. Rappe, Dirac Semimetal in Three\nDimensions, Phys. Rev. Lett. 108, 140405 (2012).[6] S. M. Young and C. L. Kane, Dirac Semimetals in Two\nDimensions, Phys. Rev. Lett. 115, 126803 (2015).\n[7] J. E. Moore, The Birth of Topological Insulators, Nature\n464, 194 (2010).\n[8] M. Z. Hasan, C. L. Kane, Topological Insulators, Rev.\nMod. Phys. 82, 3045 (2010).\n[9] S. Borisenko, Q. Gibson, D. Evtushinsky, V. Zabolotnyy,\nB. Bchner, and R. J. Cava, Experimental Realization of\na Three-Dimensional Dirac Semimetal, Phys. Rev. Lett.\n113, 027603 (2014).\n[10] Z. K. Liu, J. Jiang, B. Zhou, Z. J. Wang, Y. Zhang, H. M.\nWeng, D. Prabhakaran, S-K. Mo, H. Peng, P. Dudin, T.\nKim, M. Hoesch, Z. Fang, X. Dai, Z. X. Shen, D. L. Feng,\nZ. Hussain, and Y. L. Chen, A Stable Three-dimensional9\nTopological Dirac Semimetal Cd3As2, Nat. Mater. 13,\n677 (2014).\n[11] T. O. Wehlinga, A. M. Black-Scha\u000berc, and A. V. Bal-\natskyd, Dirac Materials, Adv. Phys. 76, 1 (2014).\n[12] K. A. Sveshnikov and P. K. Silaev, Quasiexact Solu-\ntion of a Relativistic Finite-Di\u000berence Analogue of the\nSchr odinger Equation for a Rectangular Potential Well,\nTheor. Math. Phys. 132, 1242 (2002); Quasiexact Solu-tion of the Problem of Relativistic Bound States in the\n(1 + 1)-Dimensional Case, Theor. Math. Phys. 149, 1665\n(2006).\n[13] V. Yu. Ananchenko, A. V. Sushchevskii, M. Sh. Pevzner,\nand D. V. Kholod, Relativistic Fermions in a Spherically\nSymmetric Potential Well of Finite Depth in a Two-\nDimensional Space, Russian Physics Journal 54, 1256\n(2011)."    },    {        "title": "1512.01828v2.Spectral_properties_of_reduced_fermionic_density_operators_and_parity_superselection_rule.pdf",        "content": "arXiv:1512.01828v2  [quant-ph]  11 Dec 2016Spectral properties of reduced fermionic density operator s\nand parity superselection rule\nGrigori G. Amosov1,∗and Sergey N. Filippov2,3,4,†\n1V. A. Steklov Mathematical Institute, Russian Academy of Sc iences, Gubkina Str. 8, Moscow, 119991, Russia\n2Institute of Physics and Technology, Russian Academy of Sci ences, Nakhimovskii Pr. 34, Moscow 117218, Russia\n3Moscow Institute of Physics and Technology, Institutskii P er. 9, Dolgoprudny, Moscow Region 141700, Russia\n4P. N. Lebedev Physical Institute, Russian Academy of Scienc es, Leninskii Pr. 53, Moscow 119991, Russia\nWe consider pure fermionic states with a varying number of qu asiparticles and analyze two types\nof reduced density operators: one is obtained via tracing ou t modes, the other is obtained via tracing\nout particles. We demonstrate that spectra of mode-reduced states are not identical in general and\nfully characterize purestates with equispectral mode-red ucedstates. Suchstates are related vialocal\nunitary operations with states satisfying the parity super selection rule. Thus, valid purifications for\nfermionic density operators are found. To get particle-red uced operators for a general system, we\nintroduce the operation Φ( ̺) =/summationtext\niai̺a†\ni. We conjecture that spectra of Φp(̺) and conventional\np-particle reduced density matrix ̺pcoincide. Nontrivial generalized Pauli constraints are de rived\nfor states satisfying the parity superselection rule.\nPACS numbers: 03.65.Aa, 03.65.Fd, 03.67.Bg, 05.30.Fk\nI. INTRODUCTION\nThe physical nature of information [1] not only pro-\nvides new ways of information processing (like quantum\ncomputing [2]) but also stimulates the research of spe-\ncific information carriers — particles and quasiparticles\nwith peculiar properties. This paper deals with particular\nquantum states which represent a superposition of states\nwith different numbers of fermionic quasiparticles.\nAfter electrons were argued to possess the additional\ndegree of freedom [3] (later called spin [4]) and obey the\nPauli exclusion principle [5], the Fermi–Dirac statistics\nwas found [6, 7] and the general notion of fermion par-\nticles emerged. The antisymmetric nature of fermionic\nwavefunction [7] has provoked development of the canon-\nical anticommutation relation [8] and the general theory\nof second quantization [9], which deals with systems with\na varying number of particles. The fermionic statistics\nof half-integer spin particles was established in [10, 11].\nSince every fermion has a half-integer spin, the total spin\nof an even number of fermions must be integer, whereas\nthat of an odd number of fermions must be half-integer.\nGiven a system with a varying number of particles, it is\ntherefore possible to have a superposition of half-integer\nand integer total spins. However, it was recognized later\nthat the comparison of phases between states with half-\ninteger and integer angularmomenta cannot be reconciled\nwith the requirement of relativistic invariance [12], which\nresulted in the formulation of parity (spinor, univalence)\nsuperselection rule (see, e.g., the reviews [13–15]). Any\nsuperselection rule relies on a group of physical transfor-\nmations, for example the parity superselection rule relies\non the rotational invariance [16] whereas the mass super-\nselection rule relies on the Galilean invariance of non-\nrelativistic quantum mechanics [17]. If a system of lep-\ntons were invariant with respect to fermionic U(1) trans-\nformations, there would be a leptonic family number su-\n∗Electronic address: gramos@mi.ras.ru\n†Electronic address: sergey.filippov@phystech.eduperselection rule, however, the recent experiments clearly\nindicate the neutrino oscillations (see, e.g., [18]), which\ncontradicts to the conservation of leptonic family num-\nber (the violation was predicted in Ref. [19]). Thus, the\nexperimental investigations of nature modify our under-\nstanding of fundamental symmetries, the physical models,\nand the mathematical frameworks used for their descrip-\ntion. Recent experiments with massless Weyl fermionic\nquasiparticles [20–22] stimulate us to investigate proper-\ntiesoffermionicstatesincoherentsuperpositionsofdiffer-\nent number states. Moreover, superpositions of different\nnumber states usually emerge in fermionized models (see,\ne.g. [23]).\nIn this paper, we analyze properties of general pure\nfermionic states with a varying number of quasiparti-\ncles from the viewpoint of quantum information theory,\nnamely, we consider the spectra of two reduced operators\nobtained from such a state by two different approaches:\nreduction of modes and reduction of particles.\nSuppose a state space Hcomposed of two sets of modes\nH1andH2. The mode represents a single-particle state\nthat is either occupied or not. In quantum information\ntheory,H1andH2may correspond to distant laborato-\nries. In solidstatephysics, H1andH2maydescribediffer-\nent positions or momenta of quasiparticles. In quantum\nchemistry,H1andH2may be attributed to two distinct\nsets of spin-orbits. In this paper, we consider situations,\nwhen the number of particles may change not only in H1\nandH2but in the common space Htoo. Suppose a local\nphysical observable A1∈ A1acting on modes H1only,\nfor instance, A1can be the local energy or the popu-\nlation of modes in H1. On the one side, the quantum-\nmechanical mean value /an}bracketle{tA1/an}bracketri}ht=ω(A1) = Tr(̺A1), where\nωis a functional defined on the common algebra of opera-\ntorsA=A1×A2and̺is the total density operator. On\nthe other side, /an}bracketle{tA1/an}bracketri}ht=ω1(A1) = Tr(̺1A1), whereω1is\na functional defined on the algebra of local operators A1\nand̺1is the mode-reduced density operator. [65] Anal-\nogously,ω2and̺2represent a mode-reduced state of the\nsecond set of modes. Thus, for physical systems with two\nseparate sets of single-particle sets, mode reduced states\nnaturally occur as algebraic constructions. Such algebraic2\nconstructions are well defined for systems with a variable\nnumberofquasiparticles,soweexplorespectralproperties\nofω1andω2in this paper.\nThe inverseprocedure to reduction is known as a purifi-\ncation and takes a simple form in tensor product spaces\nbecausethetwospectraareidentical(animmediateconse-\nquence of the Schmidt decomposition). We demonstrate\nthat spectra of mode-reduced states do not necessarily\ncoincide for a general fermionic state, which can be in-\nterpreted as a violation [24] of the Araki–Lieb inequal-\nity [25]. This fact should be taken into account when the\nentanglement of pure states is quantified via the entropy\nof one of subsystems [26]. The discrepancy between spec-\ntra (entropies) of subsystems of a pure state seems to be\nunphysical (since the result depends on labelling), so we\nfurther find necessary and sufficient conditions for spectra\nto be coincident for mode-reduced states. In particular,\nwe not only find that spectra coincide for states satisfy-\ning the parity superselection rule [27] but also prove that\nall states with equispectral subsystems are the univalent\nstates subjected to local unitary transformations.\nSuppose now that one is interested in an average p-\nparticle property, for instance, the averagespin projection\n(p= 1) or the average two-electron Coulomb interaction\n(p= 2). In this case, one should trace out all irrelevant\nparticles. In quantum chemistry, a particle reduction is\nrealized by integrating over irrelevant particles’ coordi-\nnates [28–30], which results in the so-called p-particle re-\nduced density matrix (see, e.g., [31]). Such an approach\nprovides new inequalities for fermionic occupation num-\nbers and leads to the generalized Pauli constraints [32–\n35]. As far as systems with a varying number of particles\nare concerned, the construction of p-particle reduced den-\nsity matrices is non-trivial and we discuss it in this pa-\nper. Moreover, we also consider the traced out part Φp(̺)\nwith a varying number of particles. Thus, we generalize\nparticle-reduction technique for systems with a varying\nnumber of quasiparticles and compare the spectra of p-\nparticle reduced density matrix and Φp(̺).\nAs a by-product, we formalize the density matrix con-\nstruction for states with a varying number of fermions.\nOur description of fermionic density operator differs from\nthat used by Cahill and Glauber [36].\nThe problem of our interest — possible pure fermionic\nstates with equispectral reduced density operators — is\nconnected with the problem of calculating von Neumann\nentropy for the fermionic dynamical systems and esti-\nmating capacities of fermionic quantum channels [37, 38].\nOne may expect that the transfer of quantum information\nthrough a fermionic channel would be different from that\nfor bosons [39], as the fermionic theory differs from the\nqubit theory [40, 41] in both tomography [42] and quan-\ntum computation [43].\nThe paper is organized as follows. In Sec. II, we de-\nscribe the notation used and formulate the problem. In\nSec. III, the density matrix formalism is concisely clari-\nfied. In Sec. IV, the main results regarding the spectra\nof mode-reduced states are obtained. In Sec. V, the con-\nstruction and spectrum of particle-reduced operators are\nanalyzed. In Sec. VI, brief conclusions are given.II. NOTATION\nConsiderthealgebraofcanonicalanticommutationrela-\ntions (CAR algebra) Agenerated by the ladder operators\nas,a†\nssatisfying the relations [8, 44]\nasa†\nt+a†\ntas=δstI,\nasat+atas=a†\nsa†\nt+a†\nta†\ns= 0,\nwheres,t= 1,...,n. LetHnbe a Hilbert space with\nthe dimension dim Hn= 2nand{|j1...jn/an}bracketri}ht}be a fixed\northonormal basis, where indices js= 0,1. We shall sup-\npose that Ais realized as the algebra of operators in Hn\nsuch that\na†\nt|j1...jn/an}bracketri}ht\n=\n\nexp/parenleftbigg\niπt−1/summationtext\ns=1js/parenrightbigg\n|j1...jt−11jt+1...jn/an}bracketri}htifjt= 0,\n0 ifjt= 1,(1)\nandatis conjugate to a†\nt.\nA positive functional ω∈B(Hn)∗normed by the con-\nditionω(I) = 1 is called a state on the algebra of all\nbounded operators B(Hn) inHn. Given a state ω, there\nexists a unique positive unit-trace operator ̺such that\nω(x) = Tr(̺x), x∈B(Hn).A spectrum of ωis defined as\nthe spectrum of ̺. The following definition [45] will play\na major role in our analysis:\nDefinition 1. The state |ψ/an}bracketri}htsatisfies the parity superse-\nlection rule if a unit vector |ψ/an}bracketri}ht ∈Hhas the form\n|ψ/an}bracketri}ht=/summationdisplay\nj1,...,jn∈{0,1}λj1...jn|j1...jn/an}bracketri}ht, λj1...jn∈C,(2)\nwhere all the numbers/summationtext\nsjsare even or odd alternatively\nfor all non-zero λj1...jn.\nThe set of all states satisfying the superselection rule\ncontains the important subset consisting of pure states\nfor which the number of particles Nis fixed. If this is the\ncase, non-zero terms in (2) satisfy the condition/summationtext\nsjs=\nN= const. It is worth mentioning that all pure quasifree\nfermionic states have a fixed number of particles [46, 47].\nAnother important class of fermionic states is deter-\nmined by their peculiar action on monomials of the odd\norder:\nDefinition 2. The stateωis said to be even if\nω(a#\ns1...a#\ns2k+1) = 0\nfor any choice of 2 k+ 1 ladder operators a#\ns=asor\na#\ns=a†\ns.\nThe relation between the above classes becomes espe-\ncially clear in the case of pure states.\nProposition 1. The pure state ωis even iff it satisfies\nthe parity superselection rule.\nProof.Suppose that ωsatisfies the parity superselec-\ntion rule. Consider the representation (2). Since the\nodd order monomial a#\ns1...a#\ns2k+1changes even number\nof particles to the odd one and vice versa, the vectors3\na#\ns1...a#\ns2k+1|j1...jn/an}bracketri}htand|˜j1...˜jn/an}bracketri}htare orthogonal for\nany choice of a#\nsif|j1...jn/an}bracketri}htand|˜j1...˜jn/an}bracketri}htcorrespond\nto non-zero terms in Eq. (2). It follows that ωis even.\nIfωdoes not satisfy the parity superselection rule, then\nits representation in the form of (2) contains at least\ntwo non-zero terms, say |j1...jn/an}bracketri}htand|˜j1...˜jn/an}bracketri}ht, such\nthat/summationtext\nsjsand/summationtext\ns˜jshave different parity. There ex-\nists a unique monomial of the odd order as1...as2k+1,\nwhich is a partial isometrical operator mapping |j1...jn/an}bracketri}ht\nto|˜j1...˜jn/an}bracketri}htand mapping all other basis vectors to zero.\nThus, the value of ωis not equalto zero on this monomial.\nHence,ωcannot be even.\nBefore we proceed to the analysis of mode and particle\nreductions of fermionic states, we construct the density\nmatrix formalism in the convenient form.\nIII. DENSITY MATRIX FOR FERMIONS\nLet us consider nfermionic modes and the correspond-\ning CAR algebra A, dimA= 4n. The generators of Aare\nasa†\ns,as,a†\ns, anda†\nsas,s= 1,...,n. Binary representa-\ntion of numbers 0 ,...,2n−1 forms the set of multiindices\nJ ∋J= (j1,...,j n), js∈ {0,1}. Given two multiindices\nJ,K∈ J, let us define AJK∈ Aby the formula\nAJK=c†\nj1c†\nj2...c†\njnckn...ck2ck1, (3)\nwhere\nc†\njs=/braceleftbigg\nasa†\nsifjs= 0,\na†\nsifjs= 1;cks=/braceleftbigg\nasa†\nsifks= 0,\nasifks= 1.\n(4)\nProposition 2. The following relation holds:\nAJK=|j1...jn/an}bracketri}ht/an}bracketle{tk1...kn|.\nProof.Consider the operator CK=ckn...ck1, which is a\nrank one partial isometry in the sense that CK=|ϕ/an}bracketri}ht/an}bracketle{tχ|\nfor some unit vectors |ϕ/an}bracketri}ht,|χ/an}bracketri}ht ∈Hthat are either orthogo-\nnal orcoincide. Moreover, CK|k1...kn/an}bracketri}ht=|0...0/an}bracketri}ht. Then,\nAJK=C†\nJCK, which concludes the proof.\nThe construction similar to that in Eq. (3) was used in\nthe paper [48] for other purposes.\nCorollary 1. The element ̺∈ Adefines a valid quantum\nstate iff it can be represented in the form\n̺=/summationdisplay\nJ,K∈JλJKAJK, (5)\nwhere(λJK)is a positive semidefinite matrix with the unit\ntrace/summationtext\nJλJJ= 1(fermionic density matrix).\nProof.Any element ofthe algebra Acan be representedin\ntheform(5)because( AJK)arematrixunitsduetoPropo-\nsition 2. The functional ω(x) =/summationtext\nJ,K∈JλJKTr(AJKx)\ngives non-negative values for positive semidefinite opera-\ntorsxifand only if( λJK) is a positive semidefinite matrix\ntoo (as the functional is a trace of the product of two ma-\ntrices). Finally, ω(I) =/summationtext\nJλJJ= 1.The 2n×2nmatrix (λJK) is a fermionic density matrix\nandfulfilstheconventionalrequirements(Hermitian, posi-\ntive semidefinite, and unit trace). The following Corollary\nprovides a convenient description of the density operator.\nCorollary 2. The2n×2nmatrix with elements /an}bracketle{tAKJ/an}bracketri}htis\na density matrix of an n-mode fermionic state over which\nthe average is taken.\nProof.In fact,/an}bracketle{tAKJ/an}bracketri}ht= Tr(A†\nJK̺) =λJK.\nExample 1. Using Eq. (3) and Corollary 2, the 2-mode\nfermionic state can be described by the (total) density\nmatrix\n\n/an}bracketle{ta1a†\n1a2a†\n2/an}bracketri}ht /an}bracketle{ta1a†\n1a†\n2/an}bracketri}ht /an}bracketle{ta†\n1a2a†\n2/an}bracketri}ht /an}bracketle{ta†\n1a†\n2/an}bracketri}ht\n/an}bracketle{ta1a†\n1a2/an}bracketri}ht /an}bracketle{ta1a†\n1a†\n2a2/an}bracketri}ht /an}bracketle{ta†\n1a2/an}bracketri}ht /an}bracketle{ta†\n1a†\n2a2/an}bracketri}ht\n/an}bracketle{ta1a2a†\n2/an}bracketri}ht /an}bracketle{ta†\n2a1/an}bracketri}ht /an}bracketle{ta†\n1a1a2a†\n2/an}bracketri}ht /an}bracketle{ta†\n1a†\n2a1/an}bracketri}ht\n/an}bracketle{ta2a1/an}bracketri}ht /an}bracketle{ta1a†\n2a2/an}bracketri}ht /an}bracketle{ta†\n1a2a1/an}bracketri}ht /an}bracketle{ta†\n1a1a†\n2a2/an}bracketri}ht\n,\n(6)\nwhich is constructed with the help of conventional sets of\nmultiindices J,K={(0,0);(0,1);(1,0);(1,1)}and differs\nfrom symbols used in the description of fermionic Gaus-\nsian states [49] (reviewed, e.g., in [50]). /square\nIV. SPECTRA OF MODE-REDUCED STATES\nIn this section, we deal with coherent superpositions\nof different number states, so we exploit the bipartition\nbased not on the particles but rather on fermionic modes.\nFermionic modes can be thought of as nodes that can be\neither occupied or not by particles. The bipartition is\nmerely an aggregation of all nodes into two groups. For\ninstance, bipartitions with respect to different spin com-\nponents were considered recently [51]. Mathematically,\nwe deal with the algebraic bipartition in the second quan-\ntized description [52–56], which is used in the study of\nentanglement [45, 57–60].\nPartition of modes is performed among single-particle\nstates that fermions can potentially occupy, for exam-\nple, nodes of some lattice potential in coordinate con-\nfiguration space, or states with different momentum in\nmomentum configuration space. Let the first subsystem\ncontainmmodes and the second one contain n−m\nmodes. In mathematical terms, Ais obtained by join-\ning two algebras of canonical anticommutation relations\nA1andA2with the generators {a1,a†\n1,...,a m,a†\nm}and\n{am+1,a†\nm+1,...,an,a†\nn}, respectively. Fix a unit vector\n|ψ/an}bracketri}ht ∈H.\nDefinition 3. The states ωj(x) =/an}bracketle{tψ|x|ψ/an}bracketri}ht, x∈ Ajthat\nare the restrictions of a pure state |ψ/an}bracketri}ht/an}bracketle{tψ|to the algebras\nAj,j= 1,2, are said to be mode-reduced (partial) states\nof|ψ/an}bracketri}ht/an}bracketle{tψ|.\nExample 2. For a state (6) the reduced density matrices\ndescribing the first mode and the second mode read\n/parenleftbigg\n/an}bracketle{ta1a†\n1/an}bracketri}ht /an}bracketle{ta†\n1/an}bracketri}ht\n/an}bracketle{ta1/an}bracketri}ht /an}bracketle{ta†\n1a1/an}bracketri}ht/parenrightbigg\nand/parenleftbigg\n/an}bracketle{ta2a†\n2/an}bracketri}ht /an}bracketle{ta†\n2/an}bracketri}ht\n/an}bracketle{ta2/an}bracketri}ht /an}bracketle{ta†\n2a2/an}bracketri}ht/parenrightbigg\n,(7)\nrespectively. Note that both matrices (7) cannot be si-\nmultaneously obtained from the total density matrix (6)4\nby conventional partial trace methods. This is a peculiar\nproperty of fermionic states which differs them from the\nbosonic ones. /square\nRemark 1. Let us clarify the relation between the in-\ntroduced definition of the mode-reduced states and the\norbital reduced density matrices used in quantum chem-\nistry [61, 62].\nIn quantum chemistry, the mode is a composite of the\nspace orbital state and the electron spin projection ( | ↑/an}bracketri}ht\nor| ↓/an}bracketri}ht), and the whole system state vector is written in\nthe form |Ψ/an}bracketri}ht=/summationtext\nn1,...,nLψn1,...,nL|n1/an}bracketri}ht⊗...⊗|nL/an}bracketri}ht, where\nni= 0,1 indicates the occupation of the corresponding\nspin-orbit, Lis the number of active spin-orbits. The\nantisymmetric property of the electron wavefunction is\nencoded in the coefficients ψn1,...,nL, and the global basis\nstates|n1/an}bracketri}ht ⊗...⊗ |nL/an}bracketri}htalready have the tensor product\nstructure. The reduced states are then readily obtained\nby tracing out undesired orbits.\nAlternatively, one can use antisymmetric vectors\n(Slater determinants) |e1/an}bracketri}ht=|n1...ni−1/an}bracketri}ht,|e2/an}bracketri}ht=\n|ni+1...nj−1/an}bracketri}ht,|e3/an}bracketri}ht=|nj+1...nL/an}bracketri}htand expand |Ψ/an}bracketri}ht=/summationtext\nni,nj,e1,e2,e3ψni,nj,e1,e2,e3|e1/an}bracketri}ht⊗|ni/an}bracketri}ht⊗|e2/an}bracketri}ht⊗|nj/an}bracketri}ht⊗|e3/an}bracketri}ht.\nThe reduced 2-mode state reads Tr e1,e2,e3|Ψ/an}bracketri}ht/an}bracketle{tΨ|as each\nvector|e1/an}bracketri}ht ⊗ |ni/an}bracketri}ht ⊗ |e2/an}bracketri}ht ⊗ |nj/an}bracketri}ht ⊗ |e3/an}bracketri}htdoes have a tensor\nproduct structure. Using the orbit basis |ni/an}bracketri}ht={|0/an}bracketri}ht,| ↑\n/an}bracketri}ht,| ↓/an}bracketri}ht,| ↑↓/an}bracketri}ht}, one gets the 2-orbital reduced density oper-\nator [61].\nIn our approach, the global basis states |j1...jn/an}bracketri}htare\nantisymmetric automatically and coefficients λj1...jnin\nEq. (2) are arbitrary. Thus, we deal with basis states\nthat do not have the tensor product structure at all. Sim-\nilarly, the action of mode operators atanda†\ntis non-\nlocal. Since transitions between the states with different\nnumbers of particles are prohibited in quantum chemistry\n[61, 62], both quantum-chemical and our definitions of re-\nduced states do coincide for systems with a fixed number\nof electrons. The downside of the quantum-chemical def-\ninition may only appear for systems with variable num-\nber of particles, since the creation and annihilation op-\nerators cannot be represented in the tensor product form\nI⊗...⊗I⊗|1/an}bracketri}ht/an}bracketle{t0|⊗I⊗...⊗IorI⊗...⊗I⊗|0/an}bracketri}ht/an}bracketle{t1|⊗I⊗...⊗I.\n/square\nOur further goal is to characterize all the states that\nhave equispectral mode-reduced states. To anticipate\ngeneral results, let us begin with the simplest case of\ntwo fermionic modes that can be occupied by a system\nwith a varying number of quasiparticles in a pure state\n|ψ/an}bracketri}ht=c00|00/an}bracketri}ht+c01|01/an}bracketri}ht+c10|10/an}bracketri}ht+c11|11/an}bracketri}ht. According to\nExample 1 the total density matrix (6) for such a state is\n(c00,c01,c10,c11)⊤×(c00,c01,c10,c11) and corresponds to\nthe pure state indeed, whereas the reduced density matri-\nces (7) take the form\nΛ1=/parenleftbigg\n|c00|2+|c01|2c00c10+c01c11\nc00c10+c01c11|c10|2+|c11|2/parenrightbigg\nand\nΛ2=/parenleftbigg\n|c00|2+|c10|2c00c01−c10c11\nc00c01−c10c11|c01|2+|c11|2/parenrightbigg\n,\nrespectively. The spectra of Λ 1and Λ2would be identical\nif and only if Tr(Λ 1) = Tr(Λ 2) and Tr(Λ2\n1) = Tr(Λ2\n2). The\nfirst condition always holds true whereas the second onereduces to Tr(Λ2\n1−Λ2\n2) = 8Re(c00c11c01c10) = 0. This\nobservation can be summarized as follows.\nProposition 3. The two-mode fermionic state |ψ/an}bracketri}ht=\nc00|00/an}bracketri}ht+c01|01/an}bracketri}ht+c10|10/an}bracketri}ht+c11|11/an}bracketri}hthas equispectral mode-\nreduced density operators if and only if Re(c00c11c01c10) =\n0.\nExample 3. Suppose |ψ/an}bracketri}ht=1\n2(|00/an}bracketri}ht+|01/an}bracketri}ht+|10/an}bracketri}ht+|11/an}bracketri}ht),\nthen Spect( ω1) ={0,1}whereas Spect( ω2) ={1\n2,1\n2}./square\nTheorem 1. Suppose that a pure state ωsatisfies the\nparity superselection rule. Then, the spectra of ω1andω2\ncoincide.\nProof.Let us define an isometry U:Hn→Hm⊗Hn−m\nby the formula\nU|j1...jn/an}bracketri}ht=|j1...jm/an}bracketri}ht⊗|jm+1...jn/an}bracketri}ht.(8)\nThen\nUa#\nsU†=/braceleftigg\na(1)#\ns⊗Iifs= 1,...,m,\nΓ⊗a(2)#\nsifs=m+1,...,n,(9)\nwhere the fermionic operators a(1)#\nsanda(2)#\nsact on\nHilbert spaces HmandHn−m, respectively, and Γ =/producttextm\ns=1(a(1)\nsa(1)†\ns−a(1)†\nsa(1)\ns).\nConsider the pure state Ω( X) =ω(U†XU),X∈\nB(Hm⊗Hn−m). The spectra of Ω 1= TrHn−m(Ω) and\nΩ2= TrHm(Ω) are known to coincide as an immediate\nconsequenceofthe Schmidt decompositionin tensorprod-\nuct Hilbert space Hm⊗Hn−m(see, e.g., [63]). It follows\nfrom the construction that for all x∈ A1we have\nω1(x) =ω(x) = Ω(UxU†) = Ω(x(1)⊗I) = Ω1(x),(10)\ntherefore,ω1and Ω 1have the same spectra.\nTo prove that spectra of ω2and Ω 2coincide too,\nwe first notice the trivial action of these functionals\non odd order monomials, namely, ω2(a#\ns1...a#\ns2k+1) =\nω(a#\ns1...a#\ns2k+1) = 0 because ωis even state in virtue of\nProposition 1. On the other hand, for all m+1≤sp≤n\nΩ2(a(2)#\ns1...a(2)#\ns2k+1) = Ω(I⊗a(2)#\ns1...a(2)#\ns2k+1)\n=ω/parenleftiggm/productdisplay\ns=1(asa†\ns−a†\nsas)a#\ns1...a#\ns2k+1/parenrightigg\n= 0 (11)\nbecauseωis even. Thus, both ω2and Ω 2vanish on odd\nmonomials. As far as even monomials are concerned,\nω(a#\ns1...a#\ns2k) = Ω(Ua#\ns1...a#\ns2kU†)\n= Ω(Γ2k⊗a(2)#\ns1...a(2)#\ns2k) = Ω2(s(2)#\ns1...a(2)#\ns2k) (12)\nbecause Γ2k=I. Thus,ω2and Ω 2coincide on even\nmonomials too. To conclude, Spect( ω1) = Spect(Ω 1) =\nSpect(Ω 2) = Spect(ω2).\nThe obtained result shows that states satisfying the\nparity superselection rule have equispectral mode-reduced\nstates and, therefore, are physical. Note that superse-\nlected states do not necessary have a fixed number of par-\nticles, and we believe that a coherent superposition (not\na mixture) of states with different numbers of fermionic5\nquasiparticles can be observed in future experiments [20–\n22]. From a mathematical viewpoint, it is interesting to\nfind all possible states with equispectral mode-reduced\nstates. The answer to this question provides the following\ntheorem.\nTheorem 2. Suppose that for a pure state |ψ/an}bracketri}ht/an}bracketle{tψ|the\npartial states ω1andω2have identical spectra. Then,\nthere exist a state |φ/an}bracketri}ht/an}bracketle{tφ|satisfying the parity superselec-\ntion rule and unitary operators U1∈ A1, U2∈ A2such\nthat the partial states of |U2U1φ/an}bracketri}ht/an}bracketle{tU2U1φ|coincide with\nωi, i= 1,2. If the spectra of ωiare simple (nondegener-\nate), then |ψ/an}bracketri}ht=U1U2|φ/an}bracketri}ht.\nProof.Without loss of generality it can be assumed that\nbipartition of nfermionic modes into mandn−mmodes\nis such that n−m≤m(otherwise the modes can be\nrelabelled).\nConsider the state ω2. In the Hilbert space Hn−m\nchoose the orthonormal basis {|ejm+1...jn/an}bracketri}ht}such that for\nallx∈ A2\nω2(x) =/summationdisplay\njs=0,1:m+1≤s≤nλjm+1...jn/an}bracketle{tejm+1...jn|x|ejm+1...jn/an}bracketri}ht.\nPick the unitary operator U2∈ A2such that\nU2|jm+1...jn/an}bracketri}ht=|ejm+1...jn/an}bracketri}ht, js∈ {0,1}, m+1≤s≤n.\nConsider the pure state ˜ ω(x) =ω(U2xU†\n2),x∈ A. Then\nfor allx∈ A2we have\n˜ω2(x) =/summationdisplay\njs=0,1:m+1≤s≤nλjm+1...jn/an}bracketle{tjm+1...jn|x|jm+1...jn/an}bracketri}ht.\n(13)\nSinceω1andω2are equispectral by the statement of\nTheorem, the states ˜ ω1and ˜ω2are equispectral too due\nto the local action of U2. Consequently, there exists the\northonormalbasis {|ft/an}bracketri}ht}2n−m−1\nt=0∪{|gt/an}bracketri}ht}2m−1\nt=2n−minHmsuch\nthat for all x∈ A1\n˜ω1(x) =2n−m−1/summationdisplay\nt=0λjm+1...jn=bin(t)/an}bracketle{tft|x|ft/an}bracketri}ht,(14)\nwhere bin(t) is the binary representation of t. Pick the\nunitary operator U1∈ A1such that\nU1|0...0/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n2m−n\nbitsbin(t)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nn−m\nbits/an}bracketri}ht=|ft/an}bracketri}ht, t= 0,...,2n−m−1,\nU1|bin(t)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nm\nbits/an}bracketri}ht=|gt/an}bracketri}ht, t= 2n−m,...,2m−1.\nIt is not hard to see that a vector\n|φ/an}bracketri}ht=2n−m−1/summationdisplay\nt=0/radicalig\nλjm+1...jn=bin(t)|0...0/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n2m−n\nbitsbin(t)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nn−m\nbitsbin(t)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nn−m\nbits/an}bracketri}ht\ngenerates the state Ω( x) =/an}bracketle{tφ|x|φ/an}bracketri}ht,x∈ A, which sat-\nisfies the parity superselection rule as the number ofquasiparticles in t-th summand equals an even number\n2×(# of 1’s in bin(t)). Moreover, Ω has ˜ ω2and˜˜ω1(x) =\n˜ω1(U1xU†\n1) as its partialstates. In other words, the mode-\nreduced states of |ψ/an}bracketri}ht/an}bracketle{tψ|and the mode-reduced states of\n|U2U1φ/an}bracketri}ht/an}bracketle{tU2U1φ|coincide.\nNow suppose that the coincident spectra of ωi,i= 1,2\nare simple (nondegenerate). Take the unitary operators\nUand Γ determined in Eq. (8) and consider the state\n˜Ω(X) = Ω(U†XU),X∈ A1⊗A2. According to Eq. (10),\nfor allx∈ A1we have ˜Ω1(x) =˜˜ω1(x). Analogously, since\nΩiseven, fromEqs.(11)–(12) itfollowsthatforall x∈ A2\nthe relation ˜Ω2(x) = ˜ω2(x) holds.\nOn the other hand, as the spectrum of Ω iis simple\nthere exists the only state on A1⊗ A2(namely, ˜Ω) such\nthat its partial traces coincide with Ω 1and Ω 2(see, e.g.,\n[63]). Hence, the same uniqueness holds for the state on A\n(namely, Ω). Thus, there is the only possibility to recon-\nstruct the entire state from its partial states. It implies\nthatω(x) = Ω(U†\n1U†\n2xU2U1) for allx∈ A.\nTheorem 2 characterizes possible states with equispec-\ntral mode-reduced states. Such states do not have to sat-\nisfy the parity superselection rule (cf. Proposition 3) but\nthey areobtainedfrom the superselectedstates byunitary\noperators acting on corresponding parties of the bipartite\nstate.\nV. SPECTRA OF PARTICLE-REDUCED\nOPERATORS\nIn this section, we consider reductions over particles.\nWhen the number of particles Nis fixed, as it takes place\nin quantum chemistry, the reduction is performed by in-\ntegrating the density operator ̺(x1,...,x N,x′\n1,...,x′\nN)\nover some particles’ coordinates [28–31]. This results in\nthe so-called p-particle reduced density matrix ( p-RDM):\n̺p-RDM(x1,...,x p;x′\n1,...,x′\np)\n=/integraldisplay\n̺(x1,...,x p,xp+1,...,x N;x′\n1,...,x′\np,xp+1,...,x N)\n×dxp+1...dxN.(15)\nIf a pure fermionic state ̺has exactly Nparticles,\nthen the spectra of reduced density matrices ̺p-RDMand\n̺(N−p)-RDMare known to coincide [28]. Eigenvalues of\n1-RDM are called natural occupation numbers. In gen-\neralcase, natural occupationnumbers cannotbe arbitrary\nones within [0 ,1] (simple Pauli’s exclusion principle) and\nmust satisfy generalized Pauli constraints [32–35].\nLet us generalize the construction of reduced density\nmatrices for states with a varying number of particles.\nIn the second quantization formalism, the integration in\nEq. (15) takes the form\n̺p-RDM(s1,...,s p;t1,...,tp) =/an}bracketle{ta†\ns1...a†\nspatp...at1/an}bracketri}ht,\n(16)\nwheresq,tq= 1,...,n, i.e. all modes are accessible. Note\nthat allsqare to be different and all tqare to be different\nfor expression (16) not to vanish.\nThe definition (16) implies that p-RDM does not have\nunit trace (in contrast to Eq. (15)). If the number of\nparticles in the state ̺is fixed and equals N, then one6\ncan introduce a factor(N−p)!\nN!in the right-hand side of\nEq. (16) to make its trace unit [29]. As our aim is to\ndeal with general states that do not have a fixed number\nof particles, we do not introduce any factors in the right-\nhand side of Eq. (16).\nAs far as the states ̺with a variable number of parti-\ncles areconcerned, averagevalues(16) arestill meaningful\nand can be calculated. This enables one to construct the\nparticle-reduced density operator in B(Hn) as follows:\n̺p=/summationtext\ns1< ... < s p\nt1< ... < t p/an}bracketle{ta†\ns1...a†\nspatp...at1/an}bracketri}hta†\nt1...a†\ntpasp...as1\n×/productdisplay\ns/ne}ationslash=s1,...,s p,\nt1,...,t pasa†\ns.(17)\nNote that the number of particles in the operator (17)\nis fixed without regard to a possible variable number of\nparticles in the original state. Product/producttext\nsin formula (17)\nis responsible for unoccupied modes.\nExample 4. The two-mode fermionic state |ψ/an}bracketri}ht=\nc00|00/an}bracketri}ht+c01|01/an}bracketri}ht+c10|10/an}bracketri}ht+c11|11/an}bracketri}hthas 1-RDM of the\nform/parenleftbigg\n|c10|2+|c11|2c01c10\nc01c10|c01|2+|c11|2/parenrightbigg\nwhose spectrum is\n(|c01|2+|c10|2+|c11|2,|c11|2) = (1−|c00|2,|c11|2). There\narenoconstraintsonnaturaloccupationnumbers λ1,λ2∈\n[0,1]. This is in agreement with a general statement that\nin Fock space there are no constraints on the spectra of\nthe 1-RDM otherthanthose imposedbythe Pauli’sexclu-\nsion principle [64]. However, if we restrict |ψ/an}bracketri}htto satisfy\nthe parity superselection rule, then the generalized con-\nstraint on natural occupation numbers appears: λ1=λ2\nfor states with even parity, λ2= 0 for states with odd\nparity. /square\nExample 5. Odd parity 3-mode fermionic state\nc100|100/an}bracketri}ht+c010|010/an}bracketri}ht+c001|001/an}bracketri}ht+c111|111/an}bracketri}hthas natural oc-\ncupation numbers λ1=|c100|2+|c010|2+|c001|2+|c111|2=\n1,λ2=λ3=|c111|2. The requirements λ1= 1 and\nλ2=λ3may be considered as generalized Pauli con-\nstraints in this case. /square\nSuppose the physical process of tracing out 1 particle\nfrom a general fermionic state with a variable number of\nparticles. The result of this operation still contains a vari-\nable number of particles so it cannot be described by a\nreduced density matrix (16). This operation is rather de-\nscribed by the map\nΦ(̺) =/summationdisplay\njaj̺a†\nj. (18)\nIf the state ̺has a fixed number of particles, say N, then\nΦ(̺)isanN−1particleoperator,whichcoincideswiththe\nreduced density operator ̺N−1given by formula (17) (see\nthe proof below). Thus, the map Φ can be considered as a\ngeneralizationofintegrationoverone particle coordinates.\nSequentially applying this map ptimes we get Φp, which\nis nothing else but tracing out pparticles.\nProposition4. Suppose the density operator ̺has a fixed\nnumberNof particles, then ΦN−p(̺) =̺p.Proof.According to Corollary 1, ̺=/summationtext\n|J|=|K|=NλJKAJK, where|J|=/summationtext\nsjsis the number of\nparticles present in the multiindex J. Using the explicit\nform of operators (4), we get\nAJK=a†\ns1...a†\nsNatN...at1/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\ns1< ... < s N:jsq= 1\nt1< ... < t N:ktr= 1/productdisplay\ns:js=ks=0asa†\ns,(19)\nalAJKa†\nl=\n\na†\ns1...al...a†\nsNatN...a†\nl...at1\n×/producttext\ns:js=ks=0asa†\nsifjl=kl= 1,\n0 otherwise ,(20)\nwhich follows from ala†\nlala†\nl=ala†\nl. On the other hand,\nsubstituting AJKfor̺in formula (17) yields\n(AJK)N−1=/summationdisplay\nl:jl=kl=1a†\ns1...al...a†\nsNatN...a†\nl...at1\n×/productdisplay\ns:js=ks=0asa†\ns,(21)\nwhich implies that Φ( ̺) =̺N−1. Applying this relation\nN−ptimes, we get the statement of Proposition 4.\nCorollary 3. The operators ̺pandΦp(̺)have the same\nspectra for pure states ̺with a fixed number of particles.\nProof.The statement is an immediate consequence of\nProposition 4 and the fact that for pure states Spec( ̺p) =\nSpec(̺N−p) [28].\nExample 6. The spectra of ̺pand Φp(̺) may coincide\nnot only for states with a fixed number of particles. The\ncoincidence of spectra of ̺1and Φ(̺) takes place for a\ngeneraltwo-mode fermionic state |ψ/an}bracketri}ht=c00|00/an}bracketri}ht+c01|01/an}bracketri}ht+\nc10|10/an}bracketri}ht+c11|11/an}bracketri}ht. It is not hard to see from the matrix\nrepresentation (6) of Φ( |ψ/an}bracketri}ht/an}bracketle{tψ|) which reads\n\n|c01|2+|c10|2c10c11−c01c110\nc10c11|c11|20 0\n−c01c110|c11|20\n0 0 0 0\n.(22)\nEigenvaluesof (22) are the same as those of1-RDM found\nin Example 4. /square\nConsiderationofotherexamples(general3-and4-mode\npurestates) alsoresultsin equispectralparticlereductions\n̺pand Φp(̺). We can make a conjecture that spectra of\noperators̺pand Φp(̺) coincide for any pure fermionic\nstate̺.\nVI. CONCLUSIONS\nWe have constructed mode- and particle-reduced\nfermionic density operatorsfor states with a varying num-\nber of particles and analyzed their spectra. The mode\nreduction is based on the restriction functional ω(x) =\nTr(̺x) to subalgebra, whereas the particle reduction is\nrealized via two objects: p-particle operator ̺pand the\nresult of tracing out pparticles, Φp(̺). As a byprod-\nuct, we have analyzed the density matrix formalism for\ngeneral case of fermionic states and provided the explicit7\nconstruction of density matrices (Corollary 2 and Exam-\nple 1). The developed formalism clearly indicates that the\nconventional partial trace methods are not applicable to\nfermionic states.\nWe have addressed the problem of finding general pure\nfermionic states such that their mode-reduced density\nmatrices are equispectral. Equispectrality automatically\ntakesplaceforbosonicsystemsandsystemsofdistinguish-\nable particles, however, it does not necessarily hold true\nforsystemscomposedofindistinguishablefermionicquasi-\nparticles. On the other hand, equispectrality is a natural\nquantum information property that reflects the fact that\nentropies of subsystems must coincide.\nFor mode-reduced states, we have found necessary and\nsufficient conditions for their spectra to be identical. The\nresults of Proposition 3 and Theorem 2 can also be in-\nterpreted as the construction of valid purifications for\nfermionic density operators. Purification is a frequently\nused tool in quantum information theory, and we believe\nthat these results may turn out to be useful in character-\nization of fermionic channels.\nForparticle-reducedstates, wehaveshownthat Φp(̺)is\na valid reduction, which coincides with ̺N−pif the state ̺\nhas exactly Nparticles. Basing on the examples, we have\nconjectured that spectraof ̺pand Φp(̺) coincide not only\nforN-particle states but also for a general fermionic pure\nstate̺. We have demonstrated that the natural occupa-\ntion numbers λ1≥...≥λn(spectrum of 1-RDM) must\nobey generalized Pauli constraints not only for N-particle\nstates but also for states satisfying the parity superse-lection rule. For instance, even parity two-mode states\nnecessarily satisfy λ1=λ2, odd parity two-mode states\nhaveλ2= 0, and odd parity three-mode states fulfil the\nrequirements λ1= 1,λ2=λ3. The generalizationof these\nconstraints is a problem for a future research.\nAcknowledgments\nThe authors are delighted to thank A.S. Holevo for a\nmotivation of this work and illuminating discussions. The\nauthors are grateful to Christian Schilling for fruitful dis-\ncussions and bringing Ref. [64] to our attention. Proposi-\ntions 1 and 2, Corollary 1, Theorems 1 and 2 are proved\nby G.G. Amosov. Corollary 2, Proposition 3, Examples 1\nand 3, Section V are due to S.N. Filippov. Both authors\ndiscussed the results and commented on the manuscript.\nTheworkofG.G.AmosovissupportedbyRussianScience\nFoundation under grant No. 14-21-00162 and performed\nin Steklov Mathematical Institute of Russian Academy of\nSciences. S.N. Filippov’s work on Sections 3 and 4 is sup-\nported by the Russian Foundation for Basic Research un-\nder Project No. 16-37-60070 mol adk and performed in\nInstitute of Physics and Technology, Russian Academy of\nSciences. S.N. Filippov’s work on Sections 5 is supported\nby Russian Science Foundation under project No. 16-11-\n00084 and performed in Moscow Institute of Physics and\nTechnology.\n[1] Landauer, R.: The physical nature of information. Phys.\nLett. A217, 188 (1996)\n[2] Nielsen, M.A., Chuang, I.L.: Quantum Computation and\nQuantumInformation. Cambridge UniversityPress, Cam-\nbridge (2000)\n[3] Pauli, W.: ¨Uber den Einflußder\nGeschwindigkeitsabh¨ angigkeit der Elektronenmasse\nauf den Zeemaneffekt. 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PhD thesis, Albert Ludwig University of Freiburg\n(2015)\n[65] It is worth mentioning, that particular realizations o f̺1\nare known in quantum chemistry as orbital reduced den-\nsity matrices. In this case, a single orbital corresponds to\nthe algebra A1generated by operators a↑a†\n↑,a↑,a†\n↑,a†\n↑a↑\nanda↓a†\n↓,a↓,a†\n↓,a†\n↓a↓corresponding to electron spin pro-\njections +1\n2and−1\n2, respectively. In our notation, the\n1-orbital reduced density matrix ̺1is a two-mode partial\nstate as it takes into account both spin projections. Anal-\nogously, 2-orbital reduced density matrix is a four-mode\npartial state."    },    {        "title": "1512.03946v1.Quantum_energy_inequalities_in_integrable_quantum_field_theories.pdf",        "content": "August 31, 2018 8:7 WSPC Proceedings - 9.75in x 6.5in cadamuro page 1\n1\nQuantum energy inequalities in integrable quantum \feld theories\nD. Cadamuro\nMathematisches Institut, Georg-August Universit at G ottingen,\nBunsenstra\u0019e 3-5, D-37073 G ottingen, Germany\nE-mail: daniela.cadamuro@mathematik.uni-goettingen.de\nIn a large class of factorizing scattering models, we construct candidates for the local\nenergy density on the one-particle level starting from \frst principles, namely from the\nabstract properties of the energy density. We \fnd that the form of the energy density\nat one-particle level can be \fxed up to a polynomial function of energy. On the level of\none-particle states, we also prove the existence of lower bounds for local averages of the\nenergy density, and show that such inequalities can \fx the form of the energy density\nuniquely in certain models.\nKeywords : quantum integrable models; energy density; quantum energy inequalities;\nenergy conditions.\n1. Introduction\nThe role and properties of the energy density are fundamental in quantum \feld\ntheory. On curved backgrounds the energy density links to the geometry of the\nspacetime via the Einstein \feld equations. In Minkowski space, it represents a local\nobservable of particular physical signi\fcance, while in 2d conformal \feld theory the\nalgebra of the stress-energy tensor is completely \fxed up to a scalar (central charge),\nand the existence of this large symmetry constrains signi\fcantly the structure of\nthe theory.\nIn typical models of classical \feld theory the energy density is positive, e.g.,\nin electrodynamics. This has interesting consequences in general relativity, where\nit implies certain restrictions on the geometry of spacetime, excluding \\exotic\"\ncon\fgurations such as wormholes, warp drives and time machines. (See, e.g., the\nPenrose and Hawking singularity theorems1, positive mass theorems and Hawking's\nchronology protection results2.)\nIn quantum \feld theory, even in \rat spacetime (not only on curved background)\nthe energy density can be negative, while the Hamiltonian His still nonnegative.\nHowever, one expects that the smeared energy density, T00(g2) =R\ng2(t)T00(t;0),\nwith some \fxed smooth real-valued test function g, cannot be inde\fnitely negative:3\nCertain lower bounds hold, the so called quantum energy inequalities (QEIs). A\nstate-independent QEI takes the form\nh';T00(g2)'i\u0015\u0000cgk'k2(1)\nfor all (suitably regular) state vectors ', wherecg>0 is a constant depending on\nthe test function g.\nWe note that in some theories, e.g., the non-minimally coupled scalar \feld in a\ncurved spacetime, only a weaker form of this inequality can hold, a so called state-\ndependent QEI, where the right hand side of Eq. (1) depends on the total energyarXiv:1512.03946v1  [math-ph]  12 Dec 2015August 31, 2018 8:7 WSPC Proceedings - 9.75in x 6.5in cadamuro page 2\n2\nof the state '.\nState-independent QEIs have been proved for the linear scalar \feld, linear Dirac\n\feld, linear vector \feld (both on \rat and curved spacetime), the Rarita-Schwinger\n\feld, and for 1+1d conformal \felds (see Ref. 4 for a review). State-dependent QEIs\nwere established in a model-independent setting5for certain \\classically positive\"\nexpressions, but their relation to the energy density is unclear.\nOnly recently QEIs have been obtained in the case of self-interacting quantum\n\feld theories, the \frst example being the massive Ising model6. This model is an\nexample of a speci\fc class of self-interacting theories on 1+1 dimensional Minkowski\nspace, so called quantum integrable models7. Other examples include the sinh-\nGordon, the sine-Gordon and the nonlinear O(N)-invariant \u001b-models.\nThere has been recent interest into these models from the side of rigorous quan-\ntum \feld theory. In particular, a large class of these theories were constructed\nfrom a prescribed factorizing S-matrix using operator-algebraic techniques8. They\ndescribe interacting relativistic particles and are characterized by in\fnitely many\nconserved currents, implying that the particle number is preserved during the scat-\ntering process and that the full scattering matrix is completely determined by the\ntwo-particle scattering function, hence called factorizing .\nHere we consider such models with one species of massive scalar bosons and\nwithout bound states. In this large class, we are interested in the stress-energy ten-\nsorT\u000b\fevaluated in one-particle states . At that level, we investigate the existence\nof (state-independent) QEIs, but also the uniqueness of T\u000b\fitself, the existence of\nstates with negative energy density, and the lowest eigenvalue in the spectrum of\nT00(g2).\n2. Stress-energy tensor in one-particle states\nIn models derived from a classical Lagrangian, such as the sinh-Gordon model\n(see Ref. 9), a candidate for the energy density can be computed directly from\nthe Lagrangian. However, there are examples of integrable models which are not\nassociated with a Lagrangian (e.g. the generalized sinh-Gordon model in Table 1\nof Ref. 10).\nIt is therefore useful to obtain an intrinsic characterization of the stress energy\ntensorT\u000b\fat the one-particle level starting from \\\frst principles\", namely from\nthe generic properties of this operator. We write T\u000b\fat one-particle level as an\nintegral kernel operator:\nh';T\u000b\f(g2) i=Z\n'(\u0012)F\u000b\f(\u0012;\u0011) (\u0011); (2)\nwhere'; are vectors in the single-particle space L2(R). There are various re-\nstrictions on the form of the kernel F\u000b\f. The fact that the energy density is a\nlocal observable implies that F\u000b\fful\flls certain analyticity, symmetry and bound-\nedness properties11. Additional conditions come from the speci\fc properties of theAugust 31, 2018 8:7 WSPC Proceedings - 9.75in x 6.5in cadamuro page 3\n3\nstress-energy tensor, namely tensor symmetry, covariance under Poincar\u0013 e trans-\nformations and spacetime re\rections, the continuity equation ( @\u000bT\u000b\f= 0), and\nthe fact that the (0 ;0)-component of the tensor integrates to the Hamiltonian\n(R\ndxT00(t;x) =H).\nStarting from these general properties, we can determine T\u000b\fin one-particle\nmatrix elements up to a certain polynomial factor. More speci\fcally, we \fnd (see\nProposition 3.1 of Ref. 10) that functions F\u000b\fare compatible with the requirements\nabove if, and only if, there exists a real polynomial PwithP(1) = 1 such that\nF\u000b\f(\u0012;\u0011) =F\u000b\f\nfree(\u0012;\u0011)P(cosh(\u0012\u0000\u0011))Fmin(\u0012\u0000\u0011+i\u0019)| {z }\n=:FP(\u0012\u0000\u0011)eg2(\u0016cosh\u0012\u0000\u0016cosh\u0011);(3)\nwhere\u0016>0 is the mass of the particle and\nF\u000b\f\nfree(\u0012;\u0011) =\u00162\n2\u0019\u0012cosh2\u0000\u0012+\u0011\n2\u00011\n2sinh(\u0012+\u0011)\n1\n2sinh(\u0012+\u0011) sinh2\u0000\u0012+\u0011\n2\u0001\u0013\n: (4)\nHereF\u000b\f\nfreeis the well-known expression of the \\canonical\" stress-energy tensor of\nthe free Bose \feld, and Fminis the so called minimal solution of the model12, which\nis unique for a given scattering function. For example, Fmin(\u0010) = 1 for free \felds\nandFmin(\u0010) =\u0000isinh\u0010\n2in the Ising model; for the sinh-Gordon model, Fminis\ngiven as an integral expression9.\n3. Negative energy density and QEIs in one-particle states\nFirst let us investigate whether there are single-particle states with negative energy\ndensity.\nIn the case of free \felds (with P= 1), it is known that there are no such states:\nOne has to allow for superpositions, for example, of a zero- and a two-particle\nvectors to obtain negative energy densities.\nHowever, the introduction of interaction changes this situation drastically.\nSpeci\fcally, if there is a \u0012P2Rsuch thatjFP(\u0012P)j>1, then there exists a\none-particle state '2L2(R) and a real-valued Schwartz function gsuch that\nh';T00(g2)'i<0 (see Prop. 4.1 in Ref. 10.) This applies, in particular, to the\nIsing and sinh-Gordon models.\nUnder stronger assumptions on the function FPwe can actually prove a no-go\ntheorem on existence of QEIs at one-particle level (see Ref. 10, Proposition 4.2):\nTheorem 3.1. Suppose there exist \u00120\u00150andc>1\n2such that\n8\u0012\u0015\u00120:FP(\u0012)\u0015ccosh\u0012: (5)\nLetgbe real-valued and of Schwartz class, g6\u00110. Then there exists a sequence\n('j)j2NinD(R),k'jk= 1, such that\nh'j;T00(g2)'ji!\u00001 asj!1: (6)August 31, 2018 8:7 WSPC Proceedings - 9.75in x 6.5in cadamuro page 4\n4\nHereD(R) denotes the space of C1-functions with compact support. This propo-\nsition says that if the function FPgrows \\too fast\", then the operator T00(g2) (at\none-particle level) cannot be bounded below. Only under certain upper bounds on\nFP, we can establish one-particle state-independent QEIs (Thm. 5.1 of Ref. 10):\nTheorem 3.2. Suppose there exist \u00120\u00150;\u00150>0, and 0<c<1\n2such that\njFP(\u0010)j\u0014ccosh Re\u0010wheneverjRe\u0010j\u0015\u00120;jIm\u0010j<\u0015 0: (7)\nLetgbe a real-valued Schwartz function, then there exists cg>0such that\n8'2D(R) :h';T00(g2)'i\u0015\u0000cgk'k2: (8)\nThe constant cgdepends on g(and onFP, hence onPandS) but not on '.\nTo motivate the above condition on the growth property of FP, let us consider the\nexpectation value of T00(g2) in a one-particle state ',\nh';T00(g2)'i=\u00162\n2\u0019Z\nd\u0012d\u0011 cosh2\u0012+\u0011\n2FP(\u0012\u0000\u0011)eg2(!(\u0012)\u0000!(\u0011))'(\u0012)'(\u0011);(9)\nwhere!(\u0012) :=\u0016cosh\u0012.\nTo establish an inequality of the form Eq. (8), the rough idea goes as follows.\nThe relevant contributions to the integral are in the regions \u0012\u0019\u0011and\u0012\u0019\u0000\u0011;\noutside these regions, the factoreg2(!(\u0012)\u0000!(\u0011)) is strongly damping. At \u0012\u0019\u0011,\nthe factorFPis nearly constant, and hence the integral is near to the free \feld\nexpression, which is known to be positive when the smearing function is of the form\ng2. At\u0012\u0019\u0000\u0011, the factor FPmay grow but the factor cosh\u0012+\u0011\n2is nearly constant.\nIf nowFmingrows less than1\n2cosh\u0012, then the second mentioned part of the integral\nis negligible against the \frst mentioned one and the whole expression is positive,\nup to some bounded part.\nIn other words, at one-particle level the existence or non-existence of QEIs de-\npends on the asymptotic behaviour of the function FP(\u0012) =P(cosh\u0012)Fmin(\u0012+i\u0019).\nIfFP(\u0012).1\n2cosh\u0012, then a state-independent one-particle QEI holds. On the other\nhand, ifFP(\u0012)&1\n2cosh\u0012, then no such QEI holds (see Proposition 3.1).\nIn speci\fc models, the bound (7) is ful\flled for at least some choices of P: In the\nfree and sinh-Gordon models, the function Fminconverges to a constant for large \u0012,\nthus a QEI can hold only if deg P= 0;1.\nIn the Ising model, where the function Fmin(\u0012+i\u0019) grows like cosh\u0012\n2at large\nvalues of\u0012, a QEI holds if and only if P\u00111.\nIn other words, for some Sthe existence of QEIs \fxes the energy density uniquely\nat one-particle level; whereas, in the sinh-Gordon model, and even in the free Bose\n\feld, we are left with the choice\nP(x) = (1\u0000\u000b) +\u000bx with\u000b2R;j\u000bj<1\n2Fmin(1+i\u0019); (10)\nwhereFmin(1+i\u0019) := lim\u0012!1Fmin(\u0012+i\u0019). Therefore, in this second class of\nmodels the existence of a QEI strongly restricts the form of the energy density\nwithout, however, \fxing it uniquely.August 31, 2018 8:7 WSPC Proceedings - 9.75in x 6.5in cadamuro page 5\n5\n4. Numerical results\nWhile we now have upper and lower estimates for the lowest eigenvalue in the\nspectrum of T00(g2) on the one-particle level, it seems unrealistic to \fnd the actual\nlowest eigenvalue in this way to any reasonable precision. An explicit result can be\nobtained only numerically. Here we sketch some results of Ref. 10.\nFor the sake of concreteness, we \fx the smearing function gto be a Gaussian,\ng(t) =\u0019\u00001=4r\u0016\n2\u001bexp\u0010\n\u0000(\u0016t)2\n8\u001b2\u0011\n; (11)\nwhere\u001b>0 is a dimensionless parameter.\nFor the numerical treatment, we restric the one-particle wavefunctions of the ma-\ntrix elements of T00(g2) to the Hilbert space L2([\u0000R;R];d\u0012) rather than L2(R;d\u0012),\nthat is, we introduce a \\rapidity cuto\u000b\". This serves to make the kernel yield a\nbounded operator. We then discretize the operator by dividing the interval [ \u0000R;R]\ninto N subintervals and using an orthonormal system of step functions \u001ejsupported\non these intervals. We are then left with a matrix Mjk=h\u001ej;T00(g2)\u001ekifor which\neigenvalues and eigenvectors can be found by standard numerical methods, such as\nthe implicit QL algorithm.\nThe eigenvector corresponding to the lowest eigenvalue for the sinh-Gordon\nmodel is shown in Fig. 1(a). We can then analyze how the lowest eigenvalue (i.e.,\nthe best constant cgin the QEI) depends on the interaction. Fig. 1(b) shows the\ndependency on the coupling constant Bin the sinh-Gordon model. As expected, as\nthe coupling is taken to 0, we reach the limit 0 for the lowest eigenvalue (as in free\n\feld theory). Moreover, we note that the lowest possible negative energy density\nis reached when the coupling is maximal ( B= 1), which \fts with the picture that\nnegative energy density in one-particle states is an e\u000bect of self-interaction in the\nquantum \feld theory.\n-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8\n-10 -8 -6 -4 -2  0  2  4  6  8 10\nθ\n(a)\n-0.005-0.0045-0.004-0.0035-0.003-0.0025-0.002-0.0015-0.001-0.0005 0\n 0  0.5  1  1.5  2T00/µ2\nB (b)\nFig. 1. Energy density T00(g2) in the sinh-Gordon model at one-particle level. (a) Low-\nest eigenvector for B= 1. (b) Lowest eigenvalue for coupling 0 < B < 2. Parameters:\nN= 500;R= (a)10 (b)7;\u001b= 0:1;P= 1.August 31, 2018 8:7 WSPC Proceedings - 9.75in x 6.5in cadamuro page 6\n6\n5. Conclusions\nWe have investigated the properties of the energy density at one-particle level in\na large class of quantum integrable models, which include the sinh-Gordon and\nIsing models. In particular, we have determined the stress-energy tensor at one-\nparticle level up to a certain polynomial factor, and studied the existence of state-\nindependent QEIs. Moreover, we have seen that demanding the existence of QEIs\ncan, in some cases, \fx this choice uniquely.\nAn investigation of analogous results at higher particle numbers is a challenging\nopen question (except for the Ising model where results are available in Ref. 6) since\nhigher form factors Fnhave poles on the integration path, and the integral operator\nT00(g2) is hard to estimate (singular integral kernels). Numerical evidence, however,\nsuggest that a term like (9) is the dominating contribution also at two-particle level.\nApart from scalar models, existence of QEIs could be investigated also in models\nwith more then one particle species (e.g. the nonlinear O(N){invariant \u001b{models),\nor in integrable models with bound states ( Z(N){Ising and sine{Gordon models),\nand it would be highly desirable to extend this study to a curved background. This\nwould be a possible path towards a model-independent understanding of the energy\ndensity and QEIs.\nReferences\n1. S. W. Hawking and R. Penrose, The singularities of gravitational collapse and\ncosmology, Proc. Roy. Soc. London Ser. A 314, 529 (1970).\n2. S. W. Hawking, Chronology protection conjecture, Phys. Rev. D 46, 603 (1992).\n3. L. H. Ford, Constraints on negative-energy \ruxes, Phys. Rev. D 43, 3972 (1991).\n4. C. J. Fewster, Lectures on quantum energy inequalities arXiv:1208.5399, (2012).\n5. H. Bostelmann and C. J. Fewster, Quantum inequalities from operator product\nexpansions, Commun. Math. Phys. 292, 761 (2009).\n6. H. Bostelmann, D. Cadamuro and C. J. Fewster, Quantum energy inequality\nfor the massive Ising model, Phys. Rev. D 88, p. 025019 (Jul 2013).\n7. H. M. Babujian, A. Foerster and M. Karowski, The form factor program: A\nreview and new results, SIGMA 2, p. 082 (2006).\n8. G. Lechner, Construction of quantum \feld theories with factorizing S-matrices,\nCommun. Math. Phys. 277, 821 (2008).\n9. A. Fring, G. Mussardo and P. Simonetti, Form-factors for integrable Lagrangian\n\feld theories, the sinh-Gordon model, Nucl. Phys. B393 , 413 (1993).\n10. H. Bostelmann and D. Cadamuro, Negative energy densities in integrable quan-\ntum \feld theories at one-particle level arXiv:1502.01714.\n11. H. Bostelmann and D. Cadamuro, Characterization of local observables in in-\ntegrable quantum \feld theories, Commun. Math. Phys. 337, 1199 (2015).\n12. M. Karowski and P. Weisz, Exact form factors in (1 + 1)-dimensional \feld\ntheoretic models with soliton behaviour, Nucl. Phys. B139 , 455 (1978)."    },    {        "title": "1512.05673v3.Multi__Q__hexagonal_spin_density_waves_and_dynamically_generated_spin_orbit_coupling__time_reversal_invariant_analog_of_the_chiral_spin_density_wave.pdf",        "content": "Multi- Qhexagonal spin density waves and dynamically generated spin-orbit coupling:\ntime-reversal invariant analog of the chiral spin density wave\nJ. W. F. Venderbos\nDepartment of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA\n(Dated: August 16, 2021)\nWe study hexagonal spin-channel (\\triplet\") density waves with commensurate M-point propa-\ngation vectors. We \frst show that the three Q=Mcomponents of the singlet charge density and\ncharge-current density waves can be mapped to multi-component Q= 0 nonzero angular momen-\ntum order in three dimensions (3D) with cubic crystal symmetry. This one-to-one correspondence\nis exploited to de\fne a symmetry classi\fcation for triplet M-point density waves using the stan-\ndard classi\fcation of spin-orbit coupled electronic liquid crystal phases of a cubic crystal. Through\nthis classi\fcation we naturally identify a set of noncoplanar spin density and spin-current density\nwaves: the chiral spin density wave and its time-reversal invariant analog. These can be thought of\nas 3DL= 2 and 4 spin-orbit coupled isotropic \f-phase orders. In contrast, uniaxial spin density\nwaves are shown to correspond to \u000bphases. The noncoplanar triple- Mspin-current density wave\nrealizes a novel 2D semimetal state with three \ravors of four-component spin-momentum locked\nDirac cones, protected by a crystal symmetry akin to nonsymmorphic symmetry, and sits at the\nboundary between a trivial and topological insulator. In addition, we point out that a special class\nof classical spin states, de\fned as classical spin states respecting all lattice symmetries up to global\nspin rotation, are naturally obtained from the symmetry classi\fcation of electronic triplet density\nwaves. These symmetric classical spin states are the classical long-range ordered limits of chiral spin\nliquids.\nI. INTRODUCTION\nSpin-orbit coupling, an intrinsic property of electrons\nin solids, is at the root of many phenomena currently\nattracting a great deal of attention in condensed matter\nphysics. The topological insulators are a prime example\nof a novel material class of which spin-orbit coupling is\na key characteristic [1, 2]. The importance of spin-orbit\ncoupling is most clearly re\rected in the spin-momentum\nlocking of the celebrated Dirac surface states mandated\nby nontrivial bulk topology. In two dimensions graphene\nis a topological insulator in the presence of spin-orbit cou-\npling [3], however, in practice the spin-orbit interaction\nturns out to be too weak.\nWhereas the topological insulators have been success-\nfully described in terms of non-interacting electron band\ntheory, spin-orbit coupled Mott insulators [4] are mate-\nrials for which strong correlation e\u000bects are important.\nThe iridium-based oxides provide hallmark examples of\nspin-orbit coupled materials where electron interactions\nhave been proposed to give rise to intriguing phases of\nmatter such as Kitaev spin liquids [4, 5], magnetic-order-\ninduced Weyl semimetals [6], and quantum critical nodal\nnon-Fermi liquids [7]. Spin-orbit coupling is key, since it\nbreaks spin-rotation symmetry and thus allows, for in-\nstance, for Kitaev-type terms in the spin Hamiltonians\nof e\u000bective local moments describing these materials.\nIn both these cases, topological insulators and spin-\norbit coupled Mott insulators, the spin-orbit interaction\nis intrinsic and of relativistic origin. Spin-orbit coupling\ncan, however, also be dynamically generated in nonrela-\ntivistic systems by interactions. Instabilities of the Fermi\nliquid can lead to condensation of particle-hole pairs into\nnematic phases with anisotropic distortions of the Fermisurface [8, 9], possibly in combination with time-reversal\nsymmetry breaking [10], or into isotropic spin-orbit cou-\npled phases [11]. Such electronic orders belong to the\nclass of phases called electronic liquid crystals [8], which\nare quantum analogs of classical liquid crystals, as they\nexhibit symmetry breaking but remain metallic.\nIn this work, we show how aspects of these three phe-\nnomena involving spin-orbit coupling (i.e., dynamic gen-\neration of spin-orbit coupling, electron correlation, and\nnontrivial topology) come together in two-dimensional\nsystems with hexagonal symmetry. Speci\fcally, we study\ndensity wave formation, i.e., condensation of particle-\nhole pairs, at \fnite commensurate wave vectors associ-\nated with nested van Hove singularities. Our study com-\nprises a classi\fcation of density waves in the spin channel,\nreferred to as triplet density waves [12], building on pre-\nvious work which has focused on the singlet channel [13].\nOne of our main results is to show that multi-component\ndensity waves with nonzero (commensurate) wave vector\nin 2D can be mapped to and classi\fed as condensates\nof particle-hole pairs with nonzero angular momentum\nin three dimensions (3D). For the triplet case this estab-\nlishes a connection between the phenomenology of multi-\ncomponent spin density wave states and spin-orbit cou-\npled\u000band\fliquid crystal phases [11]. The latter are\nparticle-hole analogs of the AandBphases of super\ruid\n3He [14].\nVan Hove singularities connected by inequivalent com-\nmensurate wave vectors generically occur for electrons\nin simple hexagonal lattices such as the triangular and\nhoneycomb lattices when doped to band structure sad-\ndle points. Doped graphene is a notable example [15].\nThe van Hove singularities are located at the three cen-\nters of the Brillouin zone edges, i.e., the Mpoints (asarXiv:1512.05673v3  [cond-mat.str-el]  5 Mar 20162\nshown in Fig. 1), and the wave vectors connecting them\nare theM-point vectors themselves.\nA number of studies have addressed the e\u000bect of\ninteractions between the three \ravors of saddle point\nelectrons, predicting exciting unconventional correlated\nphases such as topological chiral superconductivity, as\nwell as Chern-insulating chiral spin density waves [16{\n30]. These works highlight the rich physics expected\nin a broad class of (doped) hexagonal materials, with\ndoped graphene as a concrete example. The purpose of\nthis paper is to propose a comprehensive classi\fcation\nfor density wave states that can arise when multi-\ravor\nsaddle-point electrons condense. Known phases such as\nthe chiral and uniaxial spin density wave are shown to\nbe natural products of such a classi\fcation. In addition,\nwe \fnd novel density wave states with topological quasi-\nparticle spectra and hidden order.\nWe now give a brief overview of the content of this\nwork and summarize the main results.\nOverview and main results\nIn this work, we present a symmetry classi\fcation of\nhexagonal triplet M-point order. At the heart of such\nclassi\fcation is the notion of extended point group sym-\nmetry. Extended point groups are crystal point groups\nsupplemented with those lattice translations that do not\nmap the enlarged unit cell, i.e., the unit cell de\fned by\nthe ordering vector of the density wave, to itself. Con-\nsequently, extended point groups provide a natural and\nsystematic way to study particle-hole condensation at \f-\nnite commensurate wave vector, as density waves can be\nclassi\fed in terms of extended point group representa-\ntions in the same way as angular momentum channels\nare labeled by point group representations [31, 32].\nIn previous work we have analyzed hexagonal lattice\nM-point order in the singlet channel using extended point\ngroup symmetry [13] and found a set of charge density ( s)\nwaves, in addition to a set of time-reversal odd charge-\ncurrent density ( d) waves. Both sets of orders correspond\nto nesting instabilities. On the basis of that analysis, here\nwe address the triplet variants of these orders. The tri-\nangular and honeycomb lattices will serve as examples of\nsystems with hexagonal symmetry to which our analy-\nsis and results apply, with an emphasis on the triangular\nlattice.\nA central theme of our study is dynamically gener-\nated spin-orbit coupling. By spin-orbit coupling here we\nmean the coupling of the spin and angular momentum\nof the particle-hole pairs in the condensed phase. The\nconcept of dynamically generated spin-orbit coupling is\nintroduced in more detail in the next section, where we\nconsiderQ= 0 condensation. Q= 0 condensation\nis caused by Pomeranchuk-type Fermi liquid instabili-\nties and gives rise to phases that exhibit (anisotropic)\nFermi surface distortions as a result of (rotational) sym-\nmetry breaking [8{11, 33]. In systems with hexagonalsymmetry, spin-orbit coupled phases of the general form\nh^ y\n\u001b(~k)^ \u001b0(~k)i=~\u0001(~k)\u0001~ \u001b\u001b\u001b0with~\u0001(~k) a linear com-\nbination of degenerate d-wave form factors (relevant at\nvan Hove doping), can be distinguished by total angu-\nlar momentum quantum numbers, as we explain in more\ndetail in the following. One of such orders, the d+id\n\f-phase [11, 33], is favored when nesting is weak [34].\nThe discussion of Q= 0d-wave orders, highlighting the\ncoupling of degenerate orbitals to spin, will set the stage\nfor the classi\fcation and study of triplet M-point order\nof the general form h^ y\n\u001b(~k+~M\u0016)^ \u001b0(~k)i=~\u0001\u0016(~k)\u0001~ \u001b\u001b\u001b0\n(\u0016= 1;2;3, see Fig. 1). As a \frst step, we will consider\nthe nesting instabilities at the Mpoints in the spin chan-\nnel, i.e., the triplet M-point instabilities. Based on the\nnesting instabilities and their symmetry properties, we\nwill derive and discuss three main results.\n(i)We show that the s-wave and d-wave nesting in-\nstabilities map to sets of L= 2 and 4 angular momenta,\nrespectively, transforming as partners of representations\nof the cubic group Oh. This result is established by con-\nstructing a mapping between elements of the extended\nhexagonal point group C000\n6vand elements of the cubic\ngroup, realizing an isomorphism between the two groups.\nUsing this mapping, we will demonstrate that coupling\ntheL > 0 orbitals to spin ( S), in the same spirit as\ntheQ= 0d-waves, de\fnes a symmetry classi\fcation of\nhexagonal triplet M-point order: total angular momen-\ntumJ=L+Sbecomes a symmetry label for density\nwaves. We argue that, from the perspective of symmetry\nand phenomenology, we can interpret distinct triplet M-\npoint orders as electronic liquid crystal ( \u000band\f) phases\nof a 3D Fermi liquid with cubic symmetry. In addition,\nwe present a dual interpretation in which electrons in\na cubic crystal with intrinsic orbital degrees of freedom\nspontaneously develop orbital order.\n(ii)Within the framework of the symmetry classi\f-\ncation, we identify two distinct spin-orbit coupled cu-\nbic crystal \f-phases, which are in correspondence with\nwhat we call scalar M-point density wave orders. They\nare scalar orders in the sense that they can be viewed\nas total angular momentum J= 0 terms. The \frst\noriginates from the set of charge density or swaves\nand corresponds to a full spin-rotation symmetry bro-\nken spin density wave state, the so-called chiral spin\ndensity wave [26, 27, 35, 36], associated with a gapped\nmean-\feld spectrum. The mean-\feld ground state is a\nChern insulator. The second is a time-reversal invariant\ntripletd-wave or spin-current state, breaking no symme-\ntries other than spin-rotation symmetry. We show that\nthe mean-\feld ground state is a symmetry-protected 2 D\nDirac semimetal [37], protected by symmetries closely re-\nlated to non-symmorphic crystal symmetry, and sits at\nthe boundary between a trivial and a topological insula-\ntor. Using simple symmetry arguments, we discuss how\nsymmetry breaking perturbations can drive the Dirac\nsemimetal into either a trivial or topological electronic\nstate. As such, the scalar (i.e., J= 0) triplet d-wave\nstate realizes a novel electronic phase, the dynamically3\ngenerated 2D Dirac semimetal.\n(iii)The symmetry classi\fcation of triplet M-point or-\nder can be used to obtain the symmetric spin states of a\ngiven (hexagonal) lattice. Symmetric spin states are clas-\nsical spin states that respect all symmetries of the crys-\ntal lattice up to a global rotation of all spins [38]. Such\nstates are relevant in the context of magnetic materials,\ni.e., systems described by pure spin model Hamiltonians,\nas well as materials in which itinerant carriers are coupled\nto (large) localized spins. We will demonstrate that the\nsymmetry classi\fcation provides a straightforward and\nconstructive derivation of symmetric spin states.\nTo summarize the organization of the paper, Sec. II\nwill introduce dynamically generated spin-orbit coupling.\nIn Sec. III, triplet M-point order is considered, by \frst\ndiscussing nesting instabilities and then proceeding to in-\ntroduce the mapping from hexagonal to cubic symmetry.\nUsing the mapping to de\fne the symmetry classi\fcation,\nwe identify and study two types of scalar triplet density\nwave states. Sec. IV aims at understanding the proper-\nties of the scalar triplet orders by focusing on low-energy\nelectronic degrees of freedom. In Sec. V, we point out the\nconnection to classical spin liquid states, and \fnally, in\nSec. VI we summarize and discuss the results presented.\nII.Q= 0TRIPLET d-WAVES\nIn case of hexagonal C6vsymmetry, the d-wave\nchannel is two-fold degenerate, with the d-wave or-\nbitals (dx2\u0000y2;dxy) transforming as partners of a two-\ndimensional representation. Therefore, if the leading in-\nstability is in the Q= 0d-wave channel, a general linear\ncombination of the two d-wave orbitals is allowed. This\nsituation has been shown to occur for the triangular and\nhoneycomb lattices electrons at van Hove doping, when\nnesting is weak [34, 39]. In the singlet channel only real\nsuperpositions of the two degenerate orbitals are allowed.\nIn the triplet channel, however, both real and complex\nor \\chiral\" linear combinations are possible.\nIn the triplet channel, real and chiral linear combina-\ntions of the dwaves are lattice analogs of the \u000band\felec-\ntronic liquid crystal phases with dynamically generated\n\\spin-orbit coupling\" in continuum Fermi liquids [11, 33].\nBoth in the \u000band\fphases spin-rotation symmetry is\nspontaneously broken. In the \u000bphase the spin order is\nuniaxial and spin rotation is only partially broken. Spa-\ntial rotations are broken due to d-wave nature of the or-\nbital angular momentum. In contrast, spin-rotation sym-\nmetry is fully broken in the \fphase, yet the \fphase is\nisotropic due to the coupling of spin and orbital angular\nmomentum: combined spin and spatial rotations leave\nthe state invariant. The isotropy can be thought of as a\nconsequence of two angular momenta adding to form a\nrotationally invariant singlet state.\nLet us show this more explicitly. We collect the two\nd-wave orbitals in a vector ~\u0015(~k) = (\u0015d1;\u0015d2) (explicit\nexpressions of lattice form factors can be found in Table Vof Appendix D) and then write the \u000bphase as\nh^ y\n\u001b(~k)^ \u001b0(~k)i=~\u0001\u000b\u0001~\u0015(~k)\u001b3\n\u001b\u001b0: (1)\nThe two-component order parameter ~\u0001\u000bis real, which\nfollows from the requirement of Hermiticity. As a result,\n~\u0001\u000bis a nematic order parameter breaking lattice rota-\ntional symmetry [33]. Due to the uniaxial spin polar-\nization along the zaxis, spin-rotation symmetry is only\npartially broken.\nInstead, the \fphase takes the form of a dot product\nofdwaves and spin, ~\u0015\u0001~ \u001b, given by\nh^ y\n\u001b(~k)^ \u001b0(~k)i= \u0001\f[\u0015d1(~k)\u001b1\n\u001b\u001b0+\u0015d2(~k)\u001b2\n\u001b\u001b0]:(2)\nThis is a chiral superposition of d-waves, as is easily seen\nby considering the o\u000b-diagonal elements h^ y\n#(~k)^ \"(~k)i=\n\u0001\f(\u0015d1+i\u0015d2)\u0018d+id. As a result of the spin-orbit\ncoupling~\u0015\u0001~ \u001bmomentum and spin are locked together in\nsuch a way that (properly chosen) simultaneous rotations\nmake the\fphase isotropic.\nAs a spoiler for the next section, the coupling of d-\nwaves and spin in the \f-phase can be obtained more for-\nmally by the standard recipe for addition of angular mo-\nmenta in a crystal. In hexagonal symmetry the d-waves\ntransform as the nematic doublet E2, and the spin com-\nponents (\u001b1;\u001b2) as the vector doublet E1. Addition of\nangular momenta is then given by E2\u0002E1=B1+B2+E1.\nThe \frst two terms are scalars, ~\u0015\u0001~ \u001band ^z\u0001~\u0015\u0002~ \u001b, which\nboth correspond to isotropic \f-phases.\nIt should be noted that even though the \u000bphase is\nnon-chiral whereas the \fphase is chiral, both break time-\nreversal symmetry since both are triplet d-wave states. In\naddition, it should be noted that unitary rotations in spin\nspace will yield equivalent states in case of both types of\nphases, since the normal state is spin-rotation invariant.\nWhen the dominant instability is in the Q= 0d-wave\nparticle-hole channel, it was shown that the chiral d+id\n\f-phase is favored [34]. This result mirrors the result\nfound in the Cooper channel, in which case chiral d+id\nsuperconductivity is favored [17, 18]. These \fndings are\nrooted in hexagonal symmetry and therefore apply to\nboth the triangular and honeycomb lattices. An expres-\nsion similar to Eq. (2), taking the sublattice structure\ninto account, can be written for the honeycomb lattice.\nWe summarize this section by noting that hexagonal\nQ= 0 triplet states are examples of dynamically gener-\nated spin-orbit coupling. In particular, the \f-phase of\nEq. (2) is a spin-orbit coupled scalar, analogous to total\nangular momentum J=L+S= 0 states arising from\naddition of two angular momenta LandS.\nIII. HEXAGONAL TRIPLET Q=MSTATES\nParticle-hole condensation at \fnite wave vector is ex-\npected when the Fermi surface is strongly or perfectly\nnested by that wave vector. For hexagonal lattices such4\nas the triangular and honeycomb lattices, Fermi surface\nnesting can occur at three inequivalent wave vectors ~M\u0016.\nIn this section, we study triplet density waves of hexago-\nnal lattices with commensurate ordering vectors ~M\u0016hav-\ning the general form\nh^ y\n\u001b(~k+~M\u0016)^ \u001b0(~k)i=~\u0001\u0016(~k)\u0001~ \u001b\u001b\u001b0; (3)\nwhere~\u0001\u0016(~k)\u0001~ \u001b\u001b\u001b0is the triplet version of the singlet\nterm \u0001\u0016(~k)\u000e\u001b\u001b0(suppressing sublattice indices). As a\n\frst step towards this goal, we derive all distinct nest-\ning instabilities using a simple algebraic approach. We\nestablish the symmetry of the nesting instabilities using\nextended point group representations. Based on that we\nde\fne a symmetry classi\fcation of triplet nesting insta-\nbilities in terms of spin-orbit coupling, in close analogy\nwith the spin-orbit coupled Q= 0 phases. To this end,\nwe introduce a mapping from hexagonal symmetry to cu-\nbic symmetry. We then analyze a speci\fc class of spin-\norbit coupled orders, the total angular momentum J= 0\norders, in more detail.\nA. Nesting instabilities\nThe analysis of hexagonal lattice nesting instabilities\nin the presence of spin is a straightforward extension of\nthe analysis without spin degree of freedom [13]. The\ngoal is to construct an algebra of the van Hove electrons\nand use that algebra to determine the nesting instabil-\nities. For lattices with hexagonal symmetry, the three\nvan Hove singularities are located at the inequivalent M-\npoint momenta ~M\u0016, shown in Fig. 1. The algebra of the\nM-point electrons can be expressed in terms of the van\nHove electron operators ^\b,\n^\b =0\nB@^ \u001b(~M1)\n^ \u001b(~M2)\n^ \u001b(~M3)1\nCA\u00110\nB@^ 1\u001b\n^ 2\u001b\n^ 3\u001b1\nCA; (4)\nwhere\u001blabels the spin. The full hexagonal van Hove\nalgebra is then de\fned by the bilinears ^\by\u0003i\u001bj^\b, where\n\u0003iis the set of Gell-Mann matrices, i.e., the generators\nof SU(3) (we also include the identity matrix), and \u001bj\nare Pauli matrices [generators of SU(2)] which act on the\nelectron spin. The matrices \u0003iact on the species index\n\u0016corresponding to ~M\u0016. The explicit form of the set of\neight \u0003iis given in Appendix A. We \fnd it convenient to\ngroup them into three subsets, denoted by ~\u0003a,~\u0003b, and\n~\u0003c.\nFrom the set of SU(3) generators we can form three\nSU(2) sub-algebras, each of which acts in the subspace\nof a pair of van Hove electrons. In this way the sub-\nalgebras are associated with the three ways of connecting\ntwoM-points, i.e., coupling the van Hove electrons. For\ninstance, the pair of van Hove electrons ^ 1and ^ 2is con-\nnected by wave vector ~M3. The SU(2) algebra speci\fed\nkxky~M1~M2~M3kxkyM01M02M03FIG. 1. (Left) Brillouin zone (BZ) of hexagonal lattices with\nspecialM-point momenta ~M\u0016. Red hexagon represents the\nFermi surface at Van Hove \flling; small black hexagon is the\nfolded BZ of the quadrupled unit cell. (Right) Folded BZ and\nhigh symmetry M0-points; red line segments are the folded\nFermi surface. Dashed lines are the unfolded BZ (black) and\nFermi surface (red).\nby (\u00031\na;\u00031\nb;\u00031\nc) acts in the subspace ( ^ 1;^ 2) and governs\nthe nesting instabilities at wave vector ~M3. The matrix\n\u00031\ncis diagonal and describes the population imbalance of\nvan Hove electrons ^ 1and ^ 2. All bilinears that do not\ncommute with ^\by\u00031\nc^\b give rise to nesting instabilities.\nThe non-commuting set of matrices is given by \u00031\naand\n\u00031\na\u001bj, corresponding to charge and spin density waves,\nrespectively, in addition to \u00031\nband \u00031\nb\u001bj, which corre-\nspond to charge currents and spin currents.\nNesting instabilities at the wave vectors ~M1and~M2\nare obtained by either explicitly constructing the van\nHove sub-algebras of the doublets ( ^ 2;^ 3) and ( ^ 3;^ 1),\nor more directly by using rotational symmetry. One \fnds\nthat the bilinears speci\fed by the matrices ~\u0003acorrespond\nto three degenerate charge density wave instabilities at\nthe three di\u000berent wave vectors, whereas the set ~\u0003bcorre-\nsponds to three degenerate charge-current density waves.\nIn the spin channel, the set ~\u0003a\u001bjdescribes spin density\nwaves and the set ~\u0003b\u001bjdescribes spin-current density\nwaves.\nLet us consider the symmetry properties of the nest-\ning instabilities. Charge density wave order given by ~\u0003a\nrespects time-reversal symmetry while orbital current or-\nder~\u0003bis odd under time reversal. Since spin is odd under\ntimereversal, the triplet orders ~\u0003a\u001bj(spin density waves)\nand~\u0003b\u001bj(spin-current density waves) are odd and even\nunder time reversal, respectively. As far as spatial sym-\nmetries are concerned, the components of ~\u0003atransform\nas the representation F1of the extended point group C000\n6v,\nand the components of ~\u0003basF2(see also Appendix D).\nTheF1representation describes s-wave form factors at\neach wave vector and the F2representation describes d-\nwave form factors, so we may use these symmetry labels\nto refer to charge and orbital current order (see also Ta-\nble I). In particular, we refer to ~\u0003a\u001bjas triplets-wave or\ntripletF1order, and similarly for ~\u0003b\u001bj. We assume ab-5\nsence of spin-orbit coupling in the normal state leading to\na full SU(2) rotation symmetry in spin space. Therefore,\nwe simply distinguish singlet and triplet instabilities. We\nthus obtain\nF1!~\u0003a(+);~\u0003a\u001bj(\u0000);\nF2!~\u0003b(\u0000);~\u0003b\u001bj(+); (5)\nwhere (\u0006) indicates even or odd under time-reversal.\nB. Global spin rotation and mapping to cubic L>0\nangular momentum phases\nThe purpose of this section is to establish the corre-\nspondence between symmetries of the density waves in\n2D and symmetries of angular momenta in 3D. This re-\nquires studying the action of symmetries of the hexagonal\nlattice on the density waves in more detail. To proceed,\nwe therefore explore the properties of the M-point rep-\nresentation of hexagonal symmetry. The M-point rep-\nresentation is simply given by the action of symmetries\non the three Mpoints (or, equivalently, the three den-\nsity wave components). It can be de\fned in terms of a\nvector~ v, the components of which are the linearly inde-\npendent functions describing modulations with M-point\npropagation vectors. It is given by\n~ v(~ x) =0\nB@cos~M1\u0001~ x\ncos~M2\u0001~ x\ncos~M3\u0001~ x1\nCA; (6)\nwhere~ xlabels the sites of the triangular Bravais lattice.\nThe e\u000bect of elementary translations T(~ ai) (i= 1;2;3) on\n~ vis given by~ v(~ x+~ ai) =Gi~ v(~ x). Here,~ a1;2= (1;\u0006p\n2)=2\nand~ a3= (\u00001;0). We de\fne the action of the group gen-\neratorsC6and\u001bvon~ vasX~ v(~ x) andY~ v(~ x), respectively,\nwhich are given by\nX=0\nB@0 1 0\n0 0 1\n1 0 01\nCA; Y =0\nB@0 0 1\n0 1 0\n1 0 01\nCA; (7)\n(more details in Appendix A). All matrices Giare diag-\nonal, an example is G1= diag(\u00001;\u00001;1). Consequently,\ntheO(3) matrices Gi,X, andYgenerate a representa-\ntion of the extended hexagonal point group C000\n6v.\nA general modulation of electron density with M-point\npropagation vectors can be written as a linear combina-\ntion of the three components of ~ v,~ w\u0001~ v(~ x), where~ w\nis an arbitrary vector. Except for certain special cases,\nsuch linear combinations will not respect lattice symme-\ntries. This is certainly true for the translations: Gi~ w6=~ w\nfor at least one Giand general ~ w. When spin is taken\ninto account, however, a more careful consideration of\nthe symmetries of M-point order is required. We now\ndemonstrate this based on the simplest case of M-pointmodulations described by single ~ w[i.e.,~ w\u0001~ v(~ x)], hence\navoiding unessential details such as sublattice structure.\nGeneral modulations of spin density, with spin repre-\nsented by the Pauli matrices ~ \u001b, are written in terms of a\nmatrixWas\n~ \u001b\u0001W\u0001~ v(~ x): (8)\nThis can be thought of as a vector ~ wfor each spin di-\nrection. The e\u000bect of a translation is then expressed as\n~ \u001b\u0001W\u0001Gi~ v(~ x), where the Giact onWfrom the right.\nNote that matrices acting from the right act on the M-\npoint indices \u0016, whereas matrices acting from the left of\nWact on the spin indices i(i.e.,~ \u001b\u0001W\u0001~ v=\u001biWi\u0016v\u0016).\nDue to the spin degree of freedom, it can occur that in\nsome cases multiplication by Gifrom the right is equal\nto multiplication by some O(3) matrixRifrom the left,\n~ \u001b\u0001W\u0001Gi~ v=~ \u001b\u0001RiW\u0001~ v: (9)\nIn this way, the action of translations is carried over to\nspin space, since Riis a global spin-rotation matrix. A\ntrivial example of this is the identity matrix, i.e. W=I,\nin which caseRi=Gi[40].\nThe key implication of the relation WGi=RiWis the\nequivalence of translations and spin rotations. Speci\f-\ncally, it implies that the term ~ \u001b\u0001W\u0001~ v=~ \u001b\u0001~W(with\n~W\u0011W\u0001~ v) can be made invariant under translations\nwith the help of a unitary matrix U2SU(2) associated\nwithR, expressed as\n~ \u001b\u0001~W=Uy~ \u001b\u0001(R~W)U: (10)\nThe unitary matrix Uacts on the matrix components\nof the\u001bj, and through the correspondence with Rim-\nplements the global spin rotation. Hence, even though\nM-point modulations would seem to break translational\nsymmetry, in certain cases (for certain W) they are in-\nvariant up to global spin rotation [26].\nThe signi\fcance of these global spin rotations is the\nresulting invariance of the Hamiltonian H(~k). A trans-\nlation combined with a global spin rotation leaves the\nHamiltonian invariant and this e\u000bective symmetry will\nlead to degeneracies of the spectrum.\nThe same argument applies to the point group gen-\neratorsC6and\u001bv, in which case the global spin rota-\ntion matrices are denoted as RXandRY. It is impor-\ntant, however, to distinguish proper and improper R,\nsince one can only associate an SU(2) matrix to a proper\nR. Improper rotations acquire an extra minus sign, i.e.,\n~ \u001b\u0001(R~W) =\u0000U~ \u001b\u0001~WUy, as a consequence of inversion.\nAgain taking the simplest case W=Ias an example, it\nis easy to see that all re\rections, which are composed of\nY, are improper. As a result, M-point spin density mod-\nulations of the form ~ \u001b\u0001~ v(~ x) are odd under re\rections.\nThese considerations demonstrate that in order to\nproperly analyze the symmetry properties of triplet M-\npoint density waves, it is important to take the global\nspin rotations into account.6\nThe expression ~ \u001b\u0001W\u0001~ v=~ \u001b\u0001~Win (8) bears a sugges-\ntive resemblance to the familiar spin-orbit coupling term\n~S\u0001~L, where~Sis spin and~Lis orbital angular momentum.\nIndeed, as is clear from the preceding discussion, ~ \u001b\u0001~W\nimplies an entangling of spatial symmetries and spin. We\nnow show how this resemblance can be formalized by ex-\nploiting the mapping between the extended hexagonal\npoint group C000\n6vand the cubic group Oh. We then ex-\nplain how such mapping provides a way to classify and\ndetermine the symmetry of triplet M-point density wave\norders.\nWe start by explicitly constructing the mapping from\nthe groupC000\n6vto the cubic group Oh. It is easy to see\nthat the matrices Gi,X, andY, associated with the gen-\nerators ofC000\n6v, areO(3) rotation matrices. The key ob-\nservation is that they correspond to rotations that leave a\ncube invariant. For instance, the matrices Giare two-fold\nrotation about the principal axes, and Xis a three-fold\nrotation about the body diagonal. As a result, the M-\npoint representation also realizes a representation of Oh.\nThis is, however, not a faithful representation, since the\ntwo-fold rotation C2=C3\n6inC000\n6vis represented by the\nidentity, i.e., X3= 1. A faithful representation is ob-\ntained by rede\fning the generator matrix of C6as\u0000X.\nIn this way, the twofold rotation in C000\n6vis mapped to\nthe inversion in Oh: (\u0000X)3=\u00001. This establishes a\none-to-one correspondence between elements of C000\n6vand\nOh. In particular, the representations of the two groups\nand their basis functions are in one-to-one correspon-\ndence. For instance, the cubic representation generated\nbyfGi;\u0000X;Ygis given byT1usymmetry, and it is simple\nto check that the representation generated by fGi;X;Yg\nisT2g. The three-component nature of representations\nofC000\n6vis rooted in nonzero ( M-point) linear momenta,\nwhereas the three-component nature of Ohrepresenta-\ntions is due to nonzero angular momenta. Angular mo-\nmentum orbitals with T1uandT2gsymmetry are given\nby (kx;ky;kz) and (kykz;kzkx;kxky).\nFrom this follows one of the main results of this pa-\nper: hexagonal density waves with M-point wave vectors\ncan be mapped to Q= 0 nonzero angular momentum\n(L6= 0) order in a three-dimensional cubic crystal. An-\nother way of stating it is that particle-hole pairs with\n\fnite momentum in 2D can be uniquely associated with\nparticle-hole pairs with nonzero angular momentum in\n3D. The latter belong to the class of liquid crystal phases.\nWhat is the implication of this correspondence? To an-\nswer that question, we \frst relate the C000\n6vrepresentations\nF1andF2, which describe the symmetry of the charge\nand charge-current density waves, to representations of\nthe cubic group. The former corresponds to T2g, whereas\nthe latter is generated by fGi;X;\u0000Ygand corresponds\ntoT1g. The cubic representations T2gandT1gdescribe\nthree-fold degenerate orbitals coming from L= 2 and\n4 angular momenta, respectively. Evidently, the charge\ndensityswaves, which have F1symmetry, are mapped\nto theT2gorbitals (kykz;kzkx;kxky). The charge-current\nd-density waves with F2symmetry are mapped to T1gor-Order type Extended group C000\n6v Cubic group Oh\nCharge\n(s-wave)F10\nB@1\n1\n11\nCAT2g0\nB@kykz\nkzkx\nkxky1\nCA\nCharge-current\n(d-wave)F2i0\nB@k2\n3\u0000k2\n1\nk2\n1\u0000k2\n2\nk2\n2\u0000k2\n31\nCAT1g0\nB@kykz(k2\ny\u0000k2\nz)\nkzkx(k2\nz\u0000k2\nx)\nkxky(k2\nx\u0000k2\ny1\nCA\nTABLE I. Table showing the correspondence of representa-\ntions of the extended hexagonal point group C000\n6vand the cu-\nbic groupOh. In addition, we list the angular momentum\nfunctions transforming as these representations. Hexagonal\nF1andF2order is shown to be of s- andd-wave type, re-\nspectively. Their cubic equivalents are composed of L= 2 (d\nwave) and L= 4 orbitals. Note that hexagonal d-waveM-\npoint order (i.e., F2) is imaginary and breaks time-reversal\nsymmetry. Here, k1;2;3=~k\u0001~ a1;2;3, where~ a1;2;3are the lattice\nvectors of the triangular Bravais lattice.\nbitals, which are listed in Table I.\nThe usefulness of the mapping between hexagonal and\ncubic symmetry becomes apparent once spin is consid-\nered. It can be stated as follows: spin-orbit coupled cu-\nbic liquid crystals, i.e., spin-rotation symmetry broken \u000b\nand\fphases formed from T1gandT2gorbitals, can be\nused to de\fne a classi\fcation of hexagonal triplet den-\nsity wave states. Since each spin-orbit coupled \u000band\f\nphase uniquely corresponds to a density wave state, a\nclassi\fcation of the former implies a classi\fcation of the\nlatter. This is a second key result of this paper and we\nnow demonstrate this in detail.\nThe spin-orbit coupled cubic liquid crystal phases with\nT1gandT2gangular momenta are direct analogs of the\n\u000b- and\f-phases discussed in the previous section. Con-\nsider \frst the \fphase case. Spin angular momentum ~ \u001b\ntransforms as T1gunder cubic symmetry. The \fphase\nis a spin-orbit coupled state for which only total angular\nmomentum is a good quantum number. Taking the T2g\norbitals as a \frst example, the good quantum numbers in\na cubic crystal are obtained from the product representa-\ntionT1g\u0002T2gwhich is decomposed as A2g+Eg+T1g+T2g.\nThis is the lattice analog of the addition of a pair of an-\ngular momenta L= 1 (orbital) and S= 1 (spin) in the\npresence of full rotational symmetry, giving total angular\nmomentum J=L+S= 0;1;2. The term A2gshould\nbe interpreted as the J= 0 case and corresponds to an\nisotropic\f-phase. Collecting the T2gorbitals in a vector\n~d(~k) = (kykz;kzkx;kxky), and denoting electron opera-\ntors of a 3D Fermi liquid (with cubic crystal anisotropy)\nby ^\u001f(~k), the spin-orbit coupled \fphase takes the form\nh^\u001fy\n\u001b(~k)^\u001f\u001b0(~k)i= \u0001\f~d(~k)\u0001~ \u001b\u001b\u001b0; (11)\nwhich is a 3D analog of Eq. (2). Other terms in the de-\ncomposition of T1g\u0002T2gcorrespond to multi-component\nanisotropic spin-orbit coupled liquid crystals.7\nThe terms in the decomposition are symmetry labels\n(quantum numbers) for spin-orbit coupled liquid crystals\nand provide a way to classify these phases, in the same\nway asJ= 0;1;2 classi\fes total angular momentum.\nThen, as a consequence of the one-to-one correspondence\nbetween the cubic orbitals and hexagonal M-point den-\nsity waves, this implies a symmetry classi\fcation of the\ntripletM-point density waves. Each spin-orbit coupled\nliquid crystal state can be mapped back to a unique den-\nsity wave state, and its symmetry follows directly from\nthe one-to-one correspondence. A key property of the\nclassi\fcation obtained in this way is that it manifestly\ntakes global spin-rotation invariance into account.\nTo illustrate this, let us consider the T2g\f-phase with\nA2gsymmetry. Comparing the character tables of Oh\nandC000\n6vwe \fnd that A2gsymmetry in the former corre-\nsponds toA2symmetry in the latter. Importantly, the\nA2representation is a translationally invariant represen-\ntation. This shows that global spin-rotation equivalence\nis manifest, since the only way to preserve translations\nforM-point order is to combine them with global spin ro-\ntations. Note that a spin density wave with A2symmetry\nbreaks re\rection symmetry.\nSimilar to the case of T2gorbitals, we can obtain the\n\f-phase in case of the T1gorbitals. Spin-orbit coupling\nleads to the decomposition\nT1g\u0002T1g=A1g+Eg+T1g+T2g: (12)\nHere theA1gterm corresponds to a J= 0\f-phase.A1g\nsymmetry implies full invariance under cubic symmetry\nand therefore full invariance under hexagonal symmetry.\nAs a result, the triplet spin-current d-density wave which\nuniquely corresponds to the A1g\f-phase breaks no spa-\ntial symmetries.\nIn addition to the \fphase we can also consider the ana-\nlog of the\u000bphase. For instance, the cubic \u000bphase with\nconstituent dwave orbitals ~d(~k) = (kykz;kzkx;kxky) is\nwritten as\nh^\u001fy\n\u001b(~k)^\u001f\u001b0(~k)i=~\u0001\u000b\u0001~d(~k)\u001bj\n\u001b\u001b0; (13)\nwhich should be compared to Eq. (1). Here, jis a given\ndirection in spin space, for instance, the global zaxis,\nmaking it a uniaxial spin ordered state with only par-\ntially broken spin-rotational symmetry. We will see in\nthe following that the \u000bphase corresponds to a uniaxial\nspin density wave.\nLet us summarize the main result of this section. By\nestablishing a one-to-one correspondence between the cu-\nbic groupOhand the hexagonal group C000\n6v, we have refor-\nmulated hexagonal M-point order associated with nest-\ning instabilities in terms of electronic liquid crystal states\nwith nonzero angular momentum in a cubic crystal. Cou-\npling spin and orbital angular momentum allowed us to\nassign a symmetry label to hexagonal triplet ordering in\na way that gives the correct symmetry of the orders.\nSpeci\fcally, we obtain two special triplet density wave\nstates (\\\fphases\") labeled by nondegenerate representa-\ntions and transforming as scalars. These can be viewedas isotropic total angular momentum J= 0 states. Due\nto this, in what follows we will refer to these orders as\nscalar triplet states, or alternatively a scalar spin-orbit\ncoupled states. In the remainder of this section we focus\non these two scalar triplet states with high symmetry and\nstudy them in more detail based on their realization on\nthe triangular and honeycomb lattices.\nC.s-wave triplet states: Spin density waves\nTriplets-wave states are the spin density waves de-\nrived from the F1representation and correspond to the\nnesting instabilities given by ~\u0003a\u001bjin Eq. (5). The sym-\nmetry classi\fcation of triplet swaves gives rise to a scalar\ntriplet state with A2symmetry, which is the T2g\fphase\nmapped back to a density wave state. Here, we con-\nstruct this scalar order explicitly for both the triangu-\nlar and honeycomb lattices and study its properties. In\nboth cases, the scalar order is obtained by pairing each\nM-point component ~M\u0016with a spin component \u001bj(i.e.,\nPauli matrix). This is schematically shown in Fig. 2. For\nthe triangular lattice, this directly leads to\nh^ y\n\u001b(~k+~M\u0016)^ \u001b0(~k)i= \u0001A2\u001b\u0016\n\u001b\u001b0: (14)\nIn case of the honeycomb lattice the components of F1or-\nder are speci\fed by two vectors ~ wAand~ wBwhich collect\nthe order-parameter components for each of the two sub-\nlattices (see Ref. 13 for details). To construct the scalar\ntriplet state, we pair each component with a distinct spin\ncomponent, leading to\nh^ y\ni\u001b(~k+~M\u0016)^ j\u001b0(~k)i= \u0001A2w\u0016\ni\u000eij\u001b\u0016\n\u001b\u001b0: (15)\nThe vectors ~ wAand~ wBare given by ~ wA= (\u00001;\u00001;1)\nand~ wB= (1;\u00001;\u00001).\nThese two triangular and honeycomb triple- Mspin\ndensity wave states are examples of non-coplanar spin\norder. In fact, these spin density waves, which we have\nderived from symmetry principles here, are nothing but\nthe chiral spin density waves found both in itinerant clas-\nsical magnets and mean-\feld Hubbard model calculations\non the triangular [26, 41{43], honeycomb [27, 29, 44]\nand kagome lattices [30, 35, 36]. The chiral spin den-\nsity wave was also found in a honeycomb Hubbard\nmodel study using advanced quantum many-body tech-\nniques [18, 19, 23].\nChiral spin density waves owe their name to the prop-\nerty of having nonzero chirality \u0014, de\fned as \u0014=~\u00011\u0001\n~\u00012\u0002~\u000136= 0 [where ~\u0001\u0016\u0018P\nkh^ y(~k+~M\u0016)~ \u001b^ (~k)i=N].\nThe chiral spin density waves of (14) and (15) were shown\nto induce a full spectral gap and the mean-\feld ground\nstate is a Chern insulator with spontaneous quantum Hall\n(QH) e\u000bect [26, 27]. This is consistent with A2symme-\ntry, i.e., the breaking of all re\rections, and broken time-\nreversal symmetry caused by non-coplanar spin order.\nThe mean-\feld spectra of these scalar s-wave states\nare presented in Fig. 3. They show the spectral gap at8\n~M1~M2~M3~M1~M2~M3\u00003\u00002\u00001\u00003\u00003\u00003↵\u0000phase(J= 0)phase\nFIG. 2. Schematic representation of F1s-wave triple- M\norder. (Left) Each order parameter component (i.e., M-point)\nis associated with an s-wave orbital and the same spin matrix\n\u001b3giving a uniaxial state. This state is related to the cubic\n\u000bphase of Eq. (13). (Right) Same as on the left, except\nthat eachM-point is associated with a di\u000berent spin matrix,\nresulting in the chiral state of (14) and (15). The chiral states\nare related to the cubic \f-phase of Eq. (11).\nvan Hove \flling. Note that the spectra exhibit a manifest\ntwo-fold degeneracy, i.e., each band is fully two-fold de-\ngenerate. This can be traced back to the e\u000bective trans-\nlation invariance of the chiral spin density wave: trans-\nlations are preserved when followed by a global rotation.\nAs a consequence of A2symmetry there can be no fur-\nther degeneracies. To obtain the mean-\feld spectra we\nused a tight-binding mean-\feld Hamiltonian H0+H\u0001,\nwhereH0contains a nearest-neighbor hopping t= 1 and\nH\u0001contains the mean-\felds de\fned above.\nThe scalar triplet s-wave states are the spin density\nwaves obtained from mapping the T2g\f-phase back to a\ndensity wave. A natural question then is: What is the\ndensity wave state corresponding to the T2g\u000b-phase of\nEq. (13)? Clearly, this must be a uniaxial spin density\nwave, so each ( M-point) component is associated with\nthe same spin matrix (see Fig. 2). Let us consider a\nparticular \\ \u000bphase\", where each component has equal\namplitude ~\u0001\u000b\u0018\u0001\u000b(1;1;1). In the spin density wave\nlanguage this is a triple- Muniaxial spin density wave\ngiven by\nh^ y\n\u001b(~k+~M\u0016)^ \u001b0(~k)i= \u0001 uniaxial\u001b3\n\u001b\u001b0; (16)\nwith uniform order parameter magnitude \u0001 uniaxial but\n\u0014= 0. This state is the uniaxial spin density wave of\nRef. 29.\nThe connection between the uniaxial triplet states and\nthe chiral triplet states has been demonstrated from the\nperspective of their mean-\feld spectra [30]. The former\nare semi-metallic with a (spin-\fltered) quadratic band\ncrossing at the reduced zone center [29]. Smoothly de-\nforming the uniaxial spin state so as to give it nonzero\nspin chirality \u0014gaps out this quadratic band crossing\npoint and results in a state adiabatically connected to the\ngapped scalar spin-orbit coupled state of Eq. (14) [30].\nWe thus \fnd that by mapping the cubic spin-orbit cou-\npled liquid crystal phases back to density wave states, we\nare naturally led to the chiral and uniaxial spin density\n-3.0-2.5-2.0-1.5-1.0-0.5 0\n-6-5-4-3-2-1 0 1 2 3\n\u0000K0+K0\u0000M0\u0000M01M02K0\u0000K0+\u0000K0+K0\u0000M0\u0000Energy (t)FIG. 3. . Mean-\feld spectra of the scalar triplet s-wave states\nof the triangular lattice (left) given in Eq. (14), and of the\nhoneycomb lattice (right) given in Eq. (15). In both cases\nwe show spectra for \u0001 A2= 0:25. The inset on left shows\nthe reduced BZ with the path along which bands are plotted.\nNote that all bands are doubly degenerate, as explained in\nthe text. In case of the honeycomb lattice (right) we only\nshow the lower half of the spectrum, i.e., up to E= 0. At\n\fllingsn= 3=4 (triangular) and n= 3=8 (honeycomb) the\nmean \feld ground state is insulating and has nonzero Chern\nnumber.\nwave states. The upshot of this mapping is that their\nsymmetries are made transparent.\nD.d-wave triplet states: Spin-current density\nwaves\nThe triplet d-wave states have d-wave form factors as-\nsociated with each of the three ordering components ~M\u0016.\nAs a consequence, they are imaginary, preserve time-\nreversal symmetry, and are described by the nesting in-\nstability matrices ~\u0003b\u001bjin Eq. (5). We found that the\nscalar triplet order constructed from d-wave components\nhasA1symmetry: it is the T1g\fphase mapped back to\na density wave state.\nIt is obtained by pairing each d-wave component at\nwave vector ~M\u0016with a di\u000berent spin component \u001bj. In\ncase of the triangular lattice, writing the condensate in\nterms of \u0001 \u0016(~k) ash^ y\n\u001b(~k+~M\u0016)^ \u001b0(~k)i= [\u0001\u0016(~k)]\u001b\u001b0, we\n\fnd\n\u00011(~k) =i\u0001A1(cosk3\u0000cosk1)\u001b1;\n\u00012(~k) =i\u0001A1(cosk1\u0000cosk2)\u001b2;\n\u00013(~k) =i\u0001A1(cosk2\u0000cosk3)\u001b3: (17)\nHere, \u0001A1is a real order parameter and ki=~k\u0001~ aiwith\n~ aithree (triangular) lattice vectors related by three-fold\nrotations. These triplet density waves preserve time-\nreversal symmetry, since the combination of complex con-\njugation and spin \rip leaves the state invariant. In that\nsense, it represents genuine dynamically generated spin-\norbit coupling and is therefore the time-reversal invariant\nanalog of the chiral spin density wave.\nWe note that this is di\u000berent for the cubic equivalent\nof the scalar hexagonal d-wave state, i.e., the \fphase of\nL= 4T1gorbitals. Collecting the T1gorbitals listed in9\n\u0000K0+K0\u0000M0\u0000M01M02K0\u0000K0+\u0000K0+K0\u0000M0\u0000Energy (t)-6-5-4-3-2-1 0 1 2 3\n-3.0-2.5-2.0-1.5-1.0-0.5 0\nFIG. 4. . Mean-\feld band structure of the scalar triplet d-\nwave state of Eq. (17) (left), and the equivalent state on the\nhoneycomb lattice (right). Inset on the left shows the path\ntaken in the reduced BZ. On the left we plot \u0001 A1= 0:25\n(black) and \u0001 A1=\u00000:25 (red), whereas on the right we plot\n\u0001A1=\u00060:15 (black/red). All bands are two-fold degenerate.\nThe key feature of the d-wave band structures shown here is\nthe symmetry protected degeneracy at the M0points, and the\nresulting the Dirac semimetal mean-\feld state.\nTable I in the vector ~ g(~k), the cubic L= 4\fphase takes\nthe form\nh^\u001fy\n\u001b(~k)^\u001f\u001b0(~k)i= \u0001\f~ g(~k)\u0001~ \u001b\u001b\u001b0: (18)\nHermiticity requires \u0001 \fto be real, implying that the\ncubic spin-orbit coupled \fphase and the scalar M-point\nd-wave state are equivalent with respect to all spatial\nsymmetries, yet di\u000ber with respect to time reversal.\nLet us study the mean-\feld spectrum of the density\nwave state of Eq. (17), which is shown in Fig. 4. We \frst\nnote a full two-fold degeneracy of each band, as was the\ncase for the scalar s-wave state. The reason for the de-\ngeneracy is the same: translations (in combination with\nspin rotations) are good symmetries. In the present case,\nhowever, the presence of both time-reversal symmetry\nand inversion also mandates a two-fold degeneracy, which\nwill therefore be preserved even if translations are bro-\nken. We further observe that the spectrum depends on\nthe sign of \u0001 A1, as the black (red) curves correspond\nto positive (negative) sign. For \u0001 A1>0 (black bands),\nwe \fnd that the mean-\feld ground state is a semimetal,\nwith linearly dispersing nodal points located at the M0\npoints of the reduced BZ. Due to the two-fold degener-\nacy of each band, the Dirac nodes come in pairs at each\nM0point, giving rise to three \ravors of four-component\nDirac nodes.\nThese nodal points are protected by crystal symme-\ntries, and as a result the semi-metallic mean-\feld state\nis a symmetry-protected Dirac semimetal in two dimen-\nsions [37, 45]. We show this explicitly in the next section.\nAs such, the scalar triplet d-wave state with A1symmetry\nshould be contrasted with graphene [15]. Graphene has\nfully spin-degenerate Dirac nodes which can be gapped\nby symmetry-preserving spin-orbit coupling [3]. In con-\ntrast, in the present case the mean-\feld Dirac quasipar-\nticles can only become massive by breaking symmetries,\nleading to either a trivial insulating state or a topological\ninsulator. Therefore, the Dirac semimetal state originat-\ning from scalar triplet d-wave state sits at the boundary\n~M1~M2~M3\ndx23\u0000x21dx21\u0000x22dx22\u0000x23~M1~M2~M3\ndx23\u0000x21dx21\u0000x22dx22\u0000x23\u00003\u00003\u00003\u00002\u00001\u00003\n↵\u0000phase(J= 0)phaseFIG. 5. Schematic representation of F2d-wave triple- Mor-\nder. TheM-points are associated with d-wave orbitals rotated\nby\u00062\u0019=3 with respect to each other. (Left) Each M-point is\nassociated with the same spin matrix giving a uniaxial QSH\nstate. (Right) Each M-point is associated with a di\u000berent spin\nmatrix, leading to the scalar triplet d-wave state with Dirac\nsemimetallic spectrum. The pairings on the left and right\nmay be interpreted as cubic \u000band\fphases, respectively.\nbetween a trivial and topological insulator [37].\nThe symmetry protected 2D Dirac semimetal is a\ngeneric feature of triplet M-point order, since it is rooted\nin hexagonal symmetry. This is con\frmed by Fig. 4,\nwhich shows the mean-\feld spectrum of the honeycomb\nlattice scalar triplet d-wave state. The key characteris-\ntics of the honeycomb lattice spectrum are identical to\nthe triangular lattice spectrum, notably the Dirac nodes\nat theM0points.\nHexagonal lattice triplet d-wave order is schematically\nshown and summarized in Fig. 5. On the right side, each\nwave vector ~M\u0016is paired with its corresponding d-wave\nform factor and a di\u000berent spin matrix. This depicts spin-\norbit coupling. On the left each wave vector is paired\nwith itsd-wave form factor, however, each wave vector\ncarries the same spin. We refer to this as uniaxial order,\nin analogy with s-wave order in Fig. 2. The uniaxial\norder corresponds to an L= 4 cubic\u000bphase, similar to\ntheL= 2\u000bphase of Eq. (13). The \u000b-phase is interesting\nin its own right, since it corresponds to a quantum spin\nHall (QSH) phase [3, 46]. It can be viewed as two copies\nof ad-wave Chern insulator, one for each spin species\nwith opposite sign.\nIV. LOW-ENERGY PROPERTIES OF\nHEXAGONAL TRIPLET ORDERS\nThe aim of this section is to give a more elaborate\nanalysis of the spectral properties of states discussed in\nthe previous section. In particular, we examine to what\nextent the characteristics of the mean-\feld ground state\ncan be understood from a low-energy description. Such\na description is independent of a given lattice model and\ntherefore helps to put the result on a more general foot-\ning. First, we study the lifting of energy level degenera-\ncies at the Mpoints where the van Hove electrons are\nlocated. We then derive the low-energy Dirac theory of10\nthe nodal degeneracies that arise in the triplet d-wave\nstate of Eq. (17).\nA. Electrons at the Mpoints\nWe start from the electron operator ^\b of Eq. (4) and\nconsider the action of symmetries on ^\b. The action of\nthe symmetry group C000\n6von the three \ravors of M-point\nelectrons is given by the M-point representation matri-\ncesfGi;X;Ygintroduced in the previous section (see\nalso Appendix A). Speci\fcally, for the symmetry group\ngenerators we have\nT(~ a1) :^\b!G1^\b;\nC6:^\b!X^\b\n\u001bv:^\b!Y^\b: (19)\nDensity wave ordering is expressed in terms of fermion\nbilinears, ^\by\u0003i\u001bj^\b, as discussed in the beginning of the\nprevious section.\nIn case of singlet (spin-rotation invariant) order, i.e.\n^\by\u0003i^\b, the transformation properties of the M-point\nelectrons can be used to show that the set of Gell-Mann\nmatrices~\u0003ahasF1symmetry and the set ~\u0003bhasF2sym-\nmetry [13]. Based on that, we found that the symmetry\nof triple-Morder, ordering of each of the M-point com-\nponents simultaneously with equal amplitude, is A1for\nthe former and A2for the latter. The energy levels of\ntriple-Morder are given by the two Gell-Mann matrices\n\u00032\ncand \u00031\nc, respectively, in the corresponding eigenbasis.\nThe e\u000bect of spin structure is best accounted for by\nexplicitly distinguishing the two types of triplet order,\nuniaxial and spin-orbit coupled, and analyzing them sep-\narately. In the \frst case, that of uniaxial triplet order,\nthe analysis is a straightforward extension of singlet or-\nder [13]. In the second case, that of scalar spin-orbit\ncoupled order with full SU(2) symmetry breaking, the\nanalysis of energy level splittings at the Mpoints is more\nsubtle and requires the notion of double groups. It turns\nout that, as a consequence of the intimate connection to\ncubic symmetry, energy levels are governed by the double\ncubic group, as we will explain in what follows.\nLet us \frst focus on the uniaxial triplet orders. To\nbe speci\fc, we take the spin polarization axis to be the\nzaxis.M-point electron bilinears can then be written\nas a simple product of Gell-Mann matrices and \u001b3. In\nparticular, the two bilinears describing uniaxial triple- M\norder are given by \u00032\nc\u001b3(uniaxial spin density wave) and\n\u00031\nc\u001b3(uniaxial orbital spin currents) [47]. As a result,\nthe analysis of the spin-rotation invariant case e\u000bectively\napplies to each spin sector separately. For the uniaxial\nspin density waves of Eq. (16), this implies a two-fold\ndegeneracy in each spin sector. However, due to the rel-\native sign di\u000berence ( \u0018\u001b3), these two-fold degenerate\nlevels in each spin sector are split with respect to each\nother. In addition, there are two non-degenerate levels,one for each spin species. Importantly, both spin-\fltered\ntwo-fold degeneracies constitute a spin-\fltered quadratic\nband touching (QBT) protected by rotational symme-\ntry and an e\u000bective time-reversal symmetry [29, 30, 48].\nThis directly follows from the symmetry of triple- Mor-\nder [13, 49]. Furthermore, since such argument only re-\nlies on symmetry, it proves that the \\half-metal\" state\nof Ref. 29, i.e., a metal with fully spin-polarized Fermi\nsurface electrons, is a generic feature of uniaxial M-point\nspin density waves.\nThe gap matrix \u00031\nc\u001b3, which corresponds to orbital\nspin currents, leads to a double degeneracy of all three\nenergy levels of \u00031\nc: each level occurs once for each spin\nprojection. The matrix \u00031\ncimplies time-reversal symme-\ntry breaking, but in combination with \u001b3time-reversal\nsymmetry is preserved. Moreover, since the gap matrix\n\u00031\nccorresponds to a spontaneous QH phase, \u001b3promotes\nthe ground to a QSH phase [3, 46].\nNext, we consider the case of spin-orbit coupled scalar\norder. Contrary to uniaxial order, spin-orbit coupled or-\nder does not decouple into two separate spin sectors. For\ninstance, scalar s-wave order of Eqs. (14) and (15) is writ-\nten in terms of van Hove electron bilinears as ~\u0003a\u0001~ \u001b. As a\nresult, one needs to consider the combined e\u000bect of sym-\nmetries on spin and M-point degrees of freedom. We now\nshow that degeneracies in the subspace given by ^\b can\nbe derived by exploiting the mapping between hexagonal\ntripletM-point order and cubic orbital order.\nTo demonstrate that the energy levels of van Hove\nelectrons ^\b are e\u000bectively governed by the cubic dou-\nble group, we construct an operator ^\u0007 so that its orbital\ndegree of freedom transforms in the same way under Oh\nas^\b underC000\n6v. The operator ^\u0007 is given by the T2g\norbitals, ^\u0007 = ( ^ yz\u001b;^ zx\u001b;^ xy\u001b) (see also Appendix A).\nFor instance, under a two-fold rotation about the z-axis\n(equivalent to the translation T(~ a1) inC000\n6v)^\u0007 transforms\nasG1^\u0007. Similarly, other elements of Ohact on ^\u0007 as\nproducts offGi;X;Yg, the generators of the M-point\nrepresentation.\nWith spin-orbit coupling symmetries act on ^\u0007 as dou-\nble group elements. The action of symmetries such as\nthe rotation G1is thenUG1G1^\u0007, where the SU(2) ma-\ntrixUG1implements the rotation G1in spin space. In\ngeneral, the matrix Ugimplements the symmetry g2Oh.\nSymmetry-mandated degeneracies of ^\u0007 follow from rep-\nresentations of the double group. The cubic double group\nadmits 2D and 4D spin-orbit coupled representations,\ncorresponding to total angular momenta j= 1=2 and\nj= 3=2. It is known that under cubic symmetry the\nT2gorbitals split into a two-fold degenerate j= 1=2 dou-\nblet and a four-fold degenerate j= 3=2 quadruplet. This\nsituation applies to the scalar order with A1gsymmetry\n(i.e., symmetric under all elements of the cubic group) of\nEq. (12). Instead, the scalar order with A2gsymmetry\nis symmetric under all elements of Th. With spin-orbit\ncoupling the double group of Thonly admits 2 Drepre-\nsentations, and as a result degeneracies will be two-fold.11\nRep. Type triple- M GS triple- M GS triple- M GS\nsinglet uniaxial SOC scalar\nF1s-waveA1(+;\u0000) insulator/QBT A1(\u0000;\u0000) spin-\fltered QBT A2(\u0000;+) Chern insulator\nF2d-waveA2(\u0000;\u0000) Chern insulator A1(+;\u0000) QSH insulator A1(+;+) spin-locked Dirac SM\nTABLE II. Summary of hexagonal lattice s- andd-density wave states at the M-points. Table lists the symmetry and nature\nof the mean-\feld ground state (GS) of singlet triple- M, uniaxial triple- M, and spin-orbit coupled (SOC) scalar triple- Morder.\nThe symmetry label A1;2refers to point group symmetry, and ( \u0006;\u0006) refers to preserved/broken time-reversal (\frst entry) and\ntranslational symmetry (second entry). Note that in case of uniaxial order we give the label A1since one can consider each\nspin species separately, and invert the spin if necessary.\nThe key result is that degeneracies of ^\u0007 carry over to\n^\b. The splitting of M-point electrons is identical to the\nsplitting of the T2gorbitals. This is a direct consequence\nof the mapping between cubic and hexagonal C000\n6vsym-\nmetry: symmetries acting on ^\u0007 must act in the same\nway on ^\b and therefore degeneracies are preserved. The\nmean-\feld spectra of scalar triplet M-point order con-\n\frm this. The spectra of the scalar triplet d-wave states\nshown in Fig. 5 exhibit a two-fold and four-fold degener-\nacy at \u0000, the order of which depends on the sign of \u0001 A1\n(i.e., black and red bands). Instead, all levels at \u0000 are\ntwo-fold degenerate in case of scalar s-wave triplet order\nshown Fig. 2 (recall that all bands are two-fold degener-\nate).\nIn addition to explaining degeneracies, the electron op-\nerator ^\u0007 gives rise to a dual description of the map-\nping from hexagonal density waves to nonzero angular\nmomentum condensation in 3 D. Instead of associating\nthe angular momentum with the condensed particle-hole\npairs, as we have done so far, it can be associated with an\ninternal electronic orbital degree of freedom given by the\nT2gorbitals of ^\u0007. Condensation in the s-wave (L= 0)\nchannel then corresponds to spontaneous orbital order,\nwhich is expressed by the bilinears\nh^\u0007y~\u0003a\u001bj^\u0007i;h^\u0007y~\u0003b\u001bj^\u0007i: (20)\nSince ^\u0007 and ^\b transform equivalently under cubic and\nhexagonal symmetry, respectively, these orbital order pa-\nrameters are symmetry-equivalent to the spin density and\nspin-current density waves.\nThe considerations based on a description in terms\nof low-energy M-point electrons are summarized in Ta-\nble II. The two sets of M-point order components, swave\n(F1symmetry) and dwave (F2symmetry), can condense\nin singlet or triplet channel. In the latter case, assuming\ntriple-Mordering, there is a uniaxial phase and a spin-\norbit coupled scalar phase. In the uniaxial state trans-\nlational symmetry is broken and the mean-\feld ground\nstate is a spin-\fltered QBT semimetal or a QSH insula-\ntor. In the scalar spin-orbit coupled state translational\nsymmetry is preserved and the mean-\feld ground state\nis a Chern insulator or a symmetry protected Dirac semi-\nmetal.B. Low-energy theory at the M0point: 2D Dirac\nsemimetal\nThe main characteristic of the scalar triplet d-wave\nstate is the nodal Dirac degeneracy at the M0points of\nthe reduced BZ, as shown in Fig. 4. We argued that\nthese Dirac points are symmetry protected and we will\nnow prove this. To this end we choose the ~M0\n2point\nand write the electron operator at ~M0\n2(in case of the\ntriangular lattice) as\n^\bM0\u00110\nBBBB@^ \u001b(~M0\n2)\n^ \u001b(~M0\n2+~M1)\n^ \u001b(~M0\n2+~M2)\n^ \u001b(~M0\n2+~M3)1\nCCCCA: (21)\nTo facilitate expressing the action of lattice symmetries,\nwe de\fne a set of Pauli matrices \u001ciacting within the\nblocks (~M0\n2;~M0\n2+~M1) and (~M0\n2+~M2;~M0\n2+~M3), in addi-\ntion to a set of matrices \u0017iacting on the block degree of\nfreedom (e.g., exchanging blocks). The ~M0\n2point is left\ninvariant by the inversion C2, the re\rection \u001bv, and no-\ntably the translations T(~ ai), all of which are symmetries\nof the scalar triplet d-wave state (in combination with\nglobal spin rotations).\nWe \fnd that the inversion acts as \u00171^\bM0, whereas\nthe translations T(~ a2) andT(~ a3) act as\u001b1\u00173^\bM0and\n\u001b2\u001c3^\bM0(see Appendix B). The Hamiltonian matrix at\n~M0\n2is linear combination of matrices \u001bi\u0017j\u001ck(i;j;k =\n0;1;2;3) and must be invariant under the symmetry op-\nerations. Taking into account the constraints coming\nfromC2\u001bvand time-reversal Twe \fnd only two allowed\nterms at~M0\n2, which are \u001c3and\u001c2(\u001b1\u0000\u001b3\u00171). These two\nterms anti-commute, implying two eigenvalues at ~M0\n2,\u000f,\nand\u0000\u000f, each four-fold degenerate. We conclude that the\ndouble Dirac node at ~M0\n2, and thus at all M0points, is\nmandated by symmetry.\nBased on this conclusion, we ask whether symmetry\nbreaking perturbations can gap out the Dirac nodes in\ninteresting ways. First, we expand the mean-\feld band\nstructure corresponding to the Hamiltonian H0+H[\u0001]\n[with \u0001\u0011\u0001A1of Eq. (17)] around ~M0\n2, assuming we are\nin the semi-metallic state shown in Fig. 4 (black bands).12\nWe take this node as an example, the theory around ~M0\n1;3\nmay be developed in a similar way. Details of the deriva-\ntion are presented in Appendix B, and we simply quote\nthe result here:\nH(~ q) = (v1q\u0000+v0q+)~\u00171~\u001c2+ (v2q+\u0000v0q\u0000)~\u00173~\u001c2:(22)\nThe Pauli matrices ~ \u0017iand ~\u001ciact on an e\u000bective valley and\npseudospin degree of freedom of the double Dirac node,\nspeci\fed in Appendix B together with Fermi velocity co-\ne\u000ecientsv1;2;v0. We have de\fned q\u0006=q1\u0006q2, where\nqi=~ q\u0001~ ai. As shown in Fig. 6, q+\u0018qxandq\u0000\u0018qy,\nimplying that q\u0000is along the direction of the undistorted\nFermi surface (see Fig. 1), whereas q+is orthogonal to\nit. We expect that when \u0001 A1!0 the Hamiltonian H(~ q)\nonly depends on q+. This is veri\fed by checking the\nbehavior of v1;2andv0, which become v1;v0!0, and\nv2!2t. As a result, the Hamiltonian (22) describes the\ndouble Dirac node with linear dispersion in q\u0006as func-\ntion of the order parameter \u0001 A1.\nThe symmetry protection of the double Dirac node (at\n~M0\n2) critically relies on the invariant translations T(~ ai).\nTo study the fate of the Dirac node, we therefore consider\na perturbation that breaks translational symmetry. Such\na perturbation is given by the charge modulation\n\u000eH=mX\n\u0016;\u001b;~k^ y\n\u001b(~k)^ \u001b(~k+~M\u0016) + H.c.; (23)\nand we \fnd that \u000eHgaps out the Dirac nodes at the M0\npoints. The gapped state respects time-reversal and in-\nversion symmetries, and we calculate the Fu-Kane invari-\nant\u00170[50] to determine the nature of the ground state.\nQuite remarkably, we \fnd that the topological invariant\n\u00170depends on the sign of m, i.e., (\u00001)\u00170= sgn(m). This\nresult may be understood as follows, starting from dou-\nble Dirac node at ~M0\n2. The double Dirac node consists of\ntwo Kramer's doublets with opposite inversion eigenval-\nues. The time-reversal invariant perturbation \u000eHsplits\nthe double node, leaving only one of the Kramer's dou-\nblets occupied. The sign of mcontrols which Kramer's\ndoublet, and consequently which inversion eigenvalue, is\noccupied. If the even eigenvalue is occupied at ~M0\n2, the\nsame must be true for the other M0points, implying that\nthe product of inversion eigenvalues of occupied bands\nat time-reversal invariant momenta, and thus \u00170, isodd\n(since the product of the \u0000\u000fsubspace is odd). This shows\nthat sgn(m) determines whether the gapped state is a\ntopological or trivial insulator. In Appendix B, we o\u000ber\nan alternative interpretation of this result.\nThe main features of the mean-\feld Dirac semi-metal\nstate are summarized in Fig. 6. In particular, Fig. 6 high-\nlights that the Dirac semimetal sits at the boundary be-\ntween a trivial and topological insulator. Both phases are\naccessible by a single perturbation parameter m, given by\nEq. (23). We stress that this applies in general to lattices\nwith hexagonal symmetry. We leave a comprehensive in-\nvestigation of the double Dirac node theory, including a\nclassi\fcation of all possible mass terms, for future study.\nM01M02M03\n0!mITIq1\u0000q2q1+q2FIG. 6. (Left) Schematic representation of the double Dirac\nnodes of the scalar triplet d-wave state of Eq. (17) and Fig. 4,\nlocated at the three inequivalent M0points. (Right) The\nDirac points can be gapped out by a charge density mod-\nulation [Eq. (23)] controlled by mass parameter m, the sign\nof which determines whether gapped state is a topological\ninsulator (TI) or a trivial insulator (I).\nV. GENERAL SYMMETRIC SPIN STATES\nThe symmetry classi\fcation of spin density waves has\na connection to a special class of classical spin states, the\nsymmetric classical spin states. Symmetric classical spin\nstates are con\fgurations of classical spins that respect\nall symmetries of the crystal lattice, up to a global O(3)\nrotation. The concept of symmetric classical spin states\nwas introduced in Ref. 38, where they were referred to\nasregular magnetic orders . The signi\fcance of the global\nspin rotations lies in the fact that most spin Hamiltonians\nare functions of O(3) invariant bilinears such as ~Si\u0001~Sj. In\nRef. 38 it was shown that symmetric classical spin states\nare good variational classical ground states of those spin\nHamiltonians.\nThe scalar triplet s-wave states of (14) and (15) are\nexamples of such symmetric spin states, if we interpret\nthem as classical spin states. Formally, we can identify\nthese triplet density waves with classical spin con\fgura-\ntions~S(~ x) =P\n~ q~S(~ q)e\u0000i~ q\u0001~ xby taking the Fourier com-\nponents~S(~M\u0016) equal to\nSi(~M\u0016) =1\nNX\n\u001b\u001b0~k\u001bi\n\u001b\u001b0h^ y\n\u001b(~k+~M\u0016)^ \u001b0(~k)i: (24)\n(For simplicity we have suppressed the sublattice index.)\nThe resulting spin con\fguration respects all lattice sym-\nmetries up to global spin rotation. To see this, we recall\nthat the scalar triplet states transform as scalars under\nC000\n6v, including the translations, since global spin rota-\ntion invariance is a manifest feature of the classi\fcation\nof spin density waves. The scalar triplet states are in-\nvariant up to sign, which implies that the classical spin\nstates derived from them are indeed symmetric.\nWe can ask the following question: Can we obtain\nall symmetric spin states with M-point modulation on\na given hexagonal lattice using the symmetry classi\fca-\ntion? The answer is yes. To demonstrate this, it is help-\nful to review how the scalar triplet orders of Eqs. (14)\nand (15) are constructed. The starting point is three-13\ncomponent charge order with F1symmetry, which is then\nidenti\fed with T2gorbital angular momenta in a cubic\ncrystal. Coupling the ( \u0018L= 1) angular momenta to\nspin (\u0018S= 1) and decomposing them into total an-\ngular momentum states yields an e\u000bective J= 0 state\ntransforming as a scalar. This scalar non-coplanar spin\ndensity wave corresponds to a symmetric con\fguration\nof classical spins.\nApart from charge order with F1symmetry, hexago-\nnal lattices can support other charge ordered states with\nM-point modulations (the honeycomb lattice is an ex-\nample), with di\u000berent symmetry [13] (we brie\ry review\nthis in Appendix C). Since representations of C000\n6vmap\nto representations of the cubic group, distinct charge or-\nders map to distinct angular momenta, such as porf\norbitals, which are three-fold degenerate in cubic sym-\nmetry. Coupling these angular momenta to spin, as dis-\ncussed in Section III B, and decomposing into total angu-\nlar momentum states will result in a scalar J= 0 state.\nThat state, when transformed back to spin density wave\nmust correspond to a symmetric classical spin state, as\nit is invariant under all lattice symmetries up to a global\nspin rotation.\nWe illustrate this using the honeycomb and kagome\nlattices as examples. In addition to charge order with F1\nsymmetry, the honeycomb lattice supports charge order\nwithF4symmetry. The equivalent of F4symmetry in\nthe cubic group is T2usymmetry (i.e., forbitals), and\nspin-orbit coupling yields T2u\u0002T1g=A2u+:::(we are\nonly interested in the scalar representation). The A2u\nterm implies the existence of a triplet density wave state\nwithB1symmetry, which corresponds to a symmetric\nspin state using Eq. (24). As a result, the honeycomb\nlattice admits two symmetric classical spin states with\nM-point wave vectors. Applying the same method to the\nkagome lattice we \fnd three sets of M-point charge or-\nder:F1,F3, andF4symmetries (see Appendix C). These\nare mapped to d-,p- andf-wave angular momenta, re-\nspectively. Spin-orbit coupling yields three distinct scalar\nJ= 0 states, from which we conclude that the kagome\nlattice admits three symmetric spin states with M-point\nmodulations. This is summarized in Table III. Explicit\ncomparison of the spin con\fgurations shows that our re-\nsult is in agreement with Ref. 38.\nThe constructive derivation presented here provides\na straightforward route to obtain the set of symmetric\nspin states of a given lattice. It is limited only in the\nsense that the modulation wave vectors are \fxed ahead\nof time. Here we explicitly constructed M-point sym-\nmetric spin states. Consequently, symmetric spin states\nwithK-point modulation, for instance, are inevitably\nmissed. This may be remedied, however, by simply de-\nriving allK-point charge order representations and as-\nsociating them with spin similar in spirit to spin-orbit\ncoupling. Indeed, in case of the kagome lattice, symmet-\nric spin states with K-point wave vector can be extracted\nfromK-point charge order.\nSymmetric classical spin states are good variationalTriangular Honeycomb Kagome\nCharge order F1F1+F4F1+F3+F4\nCubic symmetry T2gT2g+T2uT2g+T1u+T2u\nOrbitals fdg fdg+ffg fdg+fpg+ffg\nScalar (J= 0)A2A2+B1A2+B2+B1\nTABLE III. List of the M-point modulated symmetric clas-\nsical spin states on the triangular, honeycomb and kagome\nlattices. Top row shows the symmetry of the charge ordered\nstates they are derived from, and the second row lists the\ncubic representations corresponding to charge order with Fi\nsymmetry. The angular momenta (i.e., p;d;f -waves) trans-\nforming as those representations of the cubic group are listed\nin the third row. The bottom row lists the symmetry of the\nelectronic scalar spin density wave state (i.e., the total angular\nmomentum J= 0 state) that follows from coupling orbitals\nand spin.\nground states of a large class of spin Hamiltonians, in\nsome cases saturating rigorous lower bounds on ground\nstate energies [38]. Apart from lattice magnets described\nby spin Hamiltonians, symmetric spin states are also rel-\nevant for materials described by itinerant carriers cou-\npled to localized (classical) spins. For instance, mag-\nnetic states on the kagome lattice, stabilized by itinerant\nelectron-mediated interactions at speci\fc electron densi-\nties [35, 36], were found to be the symmetric spin states.\nIn general, the itinerant carrier density, tuned to com-\nmensurate doping, sets the magnetic modulation wave\nvectors. Therefore, the present approach is particularly\nsuited for deriving variational magnetic states of coupled\nspin-electron models, as the ordering vectors are prede-\ntermined. An added bene\ft is that the symmetry of the\nelectronic Hamiltonian directly follows from the deriva-\ntion.\nIn this section we have shown that the symmetric spin\nstates correspond to the cubic J= 0 singlets obtained\nfrom spin-orbit coupling. We conclude by noting that the\nfull set of multiplets (i.e., all representations, including\nthe multicomponent representations) may be viewed as\nan exhaustive symmetry classi\fcation of all (classical)\nspin states on a given hexagonal lattice.\nVI. DISCUSSION AND CONCLUSION\nIn this work we introduced a classi\fcation of hexago-\nnal triplet density waves on the basis of a mapping from\n2D order at \fnite wave vector to 3D Q= 0 order with\nnonzero angular momentum. The mapping follows from\nthe isomorphism between the extended hexagonal point\ngroupC000\n6vand the cubic point group Oh. Let us sum-\nmarize a number of key consequences of the mapping\nto cubic symmetry. Two important results directly fol-\nlow. First, in order to correctly determine the symme-\ntry of hexagonal triplet M-point order, it is necessary14\nto consider composites of spatial symmetries and global\nspin rotations. These composites are naturally obtained\nby mapping to cubic symmetry. Global spin rotation\nequivalence, which, for instance, mandates the double\ndegeneracy of electronic bands, is manifestly built into\nthe classi\fcation in terms of cubic L>0 orders. There-\nfore, the correct symmetry of the hexagonal triplet den-\nsity waves follows naturally from the mapping to cubic\nsymmetry. This is particularly important since the 2D\nDirac semimetal is protected by composite symmetries.\nSecond, in order to understand the splitting of energy\nlevels in the ^\b subspace [see Eq. (4)], at \u0000, one needs to\ninvoke the double group of Oh, as we demonstrated in\nSec. IV A. The fact that ^\b splits into a j= 1=2 doublet\nandj= 3=2 quadruplet, or three j= 1=2 doublets, is\ninextricably linked to the equivalence of ^\b and ^\u0007.\nA third consequence deserving a comment, is that the\nsymmetry equivalence of hexagonal triplet orders and cu-\nbic liquid crystal phases implies a common phenomeno-\nlogical Ginzburg-Landau description. As a result, from\nthe perspective of e\u000bective theories based on the sym-\nmetry of the order parameter, the two seemingly di\u000ber-\nent classes of orders have shared properties. A thorough\nsurvey of the features of such a Landau theory is left\nfor future study, but we expect that the connection be-\ntween 2D and 3D orders gives rise to additional insight.\nAn interesting aspect which is worth mentioning is that\nmulti-component orders, such as the M-point spin and\nspin-current density waves, can in general give rise to dis-\ntinct types of composite orders. These composites may\norder at temperatures above the transition temperature\nof the primary order, leading to, for instance, nematic\nor charge density wave order. The latter possibility has\nbeen addressed for the case of the uniaxial spin density\nwave in Ref. 51. Condensation of composite order pa-\nrameters has been studied to great extent in the context\nof the iron-pnictide materials, with a focus on nematic-\nity [52{57] and more recently charge density wave and\nvector chiral order [58], the latter constituting a triplet\nd-wave order.\nFourth, the classi\fcation introduced in this work leads\nto one of our key results: the identi\fcation of a new set of\nphases, the time-reversal invariant scalar triplet d-wave\nstates. In addition, it follows naturally from classi\fcation\nthat the scalar triplet dwave is the time-reversal invari-\nant analog of the chiral spin density wave. The scalar\ntripletdwave has remarkable symmetry properties: it\ndoes not break any spatial symmetries, is time-reversal\ninvariant, but does break spin-rotation symmetry. As\nsuch, it bears similarity to spin nematic order, and due\nto its peculiar symmetry may be called a \\hidden order\"\nstate.\nIt is important to mention that, even though these\ntwo types of spin-orbit coupled scalar orders can be in-\nterpreted as 3D electronic liquid crystal phases, there is\na signi\fcant di\u000berence between the two. Whereas elec-\ntronic liquid crystal phases exhibit Fermi surface distor-\ntions or reconstructions but remain metallic, the elec-tronic (mean-\feld) states corresponding to the scalar\ntriplet orders are insulating ( swave) and semi-metallic ( d\nwave). Indeed, the two systems (i.e., a 2D nested Fermi\nsurface and a 3D Fermi liquid) are di\u000berent. Therefore,\nin spite of the correspondence, the nature of both the\nnormal state and the condensate is di\u000berent for the two\ncases. There is, however, a similarity in the following\nsense. We have seen that the scalar s-wave state, i.e., the\nchiral spin density wave, gives rise to a fully gapped spec-\ntrum, whereas the spectrum of the scalar d-wave state has\nnodal degeneracies due to higher symmetry. The cubic\nT2g\fphase, in contrast to a p-wave (orT1u)\fphase, ex-\nhibits nodes. The cubic T1g\fphase has the same nodes,\nbut as a result of full cubic symmetry has an extra set of\nnodes: higher symmetry mandates an extra set of nodes.\nThe scalar spin density wave and scalar spin-current\ndensity waves, i.e., the \f-phase density waves, both con-\nstitute topological phases, in the sense that their mean-\n\feld band structures are topological. The chiral spin den-\nsity wave has nonzero Chern number in the ground state.\nIt is an example of so-called topological Mott (Chern) in-\nsulators [59]. The scalar triplet d-wave state is a novel\ntype of semimetal. It realizes a dynamically generated\nDirac semi-metal in two dimensions, protected by crys-\ntal symmetry. Perturbations can gap out the Dirac nodes\nand drive the system into either the trivial electronic in-\nsulator, or the topological insulator. This is achieved by\na very simple perturbation: charge density modulations\nthat only break translational symmetry. Interestingly,\ntheZ2topological index depends on the sign of the charge\ndensity perturbation. This transition, controlled by the\nsign of the mass perturbation, may be understood as a\nband inversion at an odd number of time-reversal invari-\nant momenta, i.e., all the M0points. In this respect, the\nDirac semimetal state signi\fcantly di\u000bers from graphene,\nwhich has double Dirac nodes (counting spin) at each of\nthe twoKpoints. The present spin-orbit coupled Dirac\nsemimetal has a double node at the threeM0points, i.e.,\nthree \ravors of double nodes. In this respect it bears\nsome similarity to the (111) surface states of topological\ncrystalline insulators in the SnTe material class [60{62],\nwhich are located at the M-points of the surface BZ. It\nwill be interesting to further develop the Dirac theory\nof the three M0points and consider the e\u000bect of various\nsymmetry breaking perturbations.\nWe have shown that the classical spin state analogs\nof non-coplanar \fphase spin density waves are symmet-\nric spin states. Recently, the latter were shown to be\nthe classical long-range ordered limits of time-reversal\nsymmetry broken or chiral spin liquid phases [63]. Very\nrecently, it was shown that quantum disordering the non-\ncoplanar chiral spin state realized in a chiral spin model\non the honeycomb lattice can result in a chiral spin liq-\nuid [64]. Furthermore, recent theoretical work has con-\nsidered quantum disordering the electronic chiral spin\ndensity wave state of a quarter doped honeycomb lat-\ntice model, and found that the resulting spin-charge liq-\nuid state is topologically ordered [23]. This establishes15\nan exciting connection between the electronic \f-phase\nspin density waves and spin liquid physics. In particu-\nlar, it will be interesting to explore to connection of the\nscalar spin-current d-wave state, the time-reversal invari-\nant analog of the chiral spin density wave, to the physics\nof spin(-charge) liquids. In this respect, we note that,\nsince thed-wave state is time-reversal invariant, it can be\nconverted into a triplet spin correlation function as [12]\nh~S(~k+~Q\u0016)\u0002~S(~k)i= \u0001~d(~k); (25)\nwhere the~d(~k) vector lives in spin space and the com-\nponents are given by Eq. (17). Therefore, the triplet\ndensity waves studied in this work give rise to intriguing\nquestions to be addressed in future work.\nACKNOWLEDGMENTS\nI have bene\fted greatly from helpful and stimulating\nconversations and with L. Fu, C. Ortix, J. van Wezel, J.\nvan den Brink, M. Daghofer, J. Hoon Han, Y. Ran, T.\nIadecola and C. Chamon. I gratefully acknowledge K.\nBarros, G.W. Chern and C.D. Batista for collaborations\non related subjects. I am particularly indebted to C.\nOrtix and J. van den Brink for a careful reading of the\nmanuscript and thoughtful comments. This work was\nsupported by the Netherlands Organization for Scienti\fc\nResearch (NWO).\nAppendix A: M-point representation of hexagonal\nsymmetry\nTheM-point representation of hexagonal symmetry is\nspeci\fed by the action of elements of the symmetry group\non the vector ~ v=~ v(~ x) de\fned as\n~ v(~ x) =0\nB@cos~M1\u0001~ x\ncos~M2\u0001~ x\ncos~M3\u0001~ x1\nCA: (A1)\nThe translations T(~ ai), where~ ai(i= 1;2;3) are the ele-\nmentary lattice vectors, are represented by the matrices\nGide\fned through the equation\n~ v(~ x+~ xi)\u0011Gi~ v(~ x); i= 1;2;3: (A2)\nExplicitly,G1andG2are given by\nG1=0\nB@\u00001\n\u00001\n11\nCA; G2=0\nB@1\n\u00001\n\u000011\nCA: (A3)\nThe matrices Giinherit the algebraic properties of the\ntranslations. They satisfy G2\ni= 1, they mutually com-\nmute and multiplication of two of them gives the third,\ni.e.G1G2=G3.The rotations and re\rections can be expressed in terms\nof the generators C6and\u001bv. The action of C6is de\fned\nas~ v0(~ x) =~ v(C\u00001\n6~ x) and is given by the matrix X,\n~ v(C\u00001\n6~ x) =X~ v(~ x); X =0\nB@0 1 0\n0 0 1\n1 0 01\nCA (A4)\nNote thatXhas the property X3= 1 and thus X\u00001=\nX2. In addition, one has X\u00001=XT, whereXTis the\ntranspose. It thus follows that ~ v(C\u00001\n3~ x) =X2~ v(~ x) =\nXT~ v(~ x). For the re\rection \u001bvwe de\fne the matrix Yas\n~ v(\u001b\u00001\nv~ x) =Y~ v(~ x); Y =0\nB@0 0 1\n0 1 0\n1 0 01\nCA: (A5)\nAll rotations and re\rections can be represented by a\nproduct of powers of XandY, i.e.XmYn. An arbitrary\nstring ofXandYmatrices can be brought into this form\nusing (XY)2= 1, which is equivalent to XY=YXT.\nThe set of generator matrices fG1;X;Yg(translations\nG2andG3can be written as products of the generators)\nde\fnes an embedding of the group C000\n6vinO(3), the group\nof orthogonal matrices in three dimensions. Clearly, this\nmapping is not invertible, as the inversion C2=C3\n6is\nmapped to identity through X3= 1. An invertible em-\nbedding is obtained by rede\fning the set of generators as\nfG1;\u0000X;Yg, i.e., associating the O(3) matrix\u0000Xwith\nC6. In this way, two-fold rotation C22C000\n6vis mapped\nto the inversion P2O(3).\nAs we explained in the main text, the key property of\nsuch mapping is that it establishes an exact isomorphism\nbetweenC000\n6vand the cubic group Oh, a subgroup of O(3).\nIndeed, direct inspection shows that the character tables\nof both groups are identical, which is a manifestation of\nthe isomorphism. When mapped onto elements of the\ncubic group, the translations T(~ ai) (represented by Gi)\nare given by two-fold rotations around the x,yandz\naxes. The six-fold rotations C6are interpreted as three-\nfold rotations about body diagonals of the cube combined\nwith inversion (i.e., \u0000X). The re\rection Yis mapped to\na two-fold rotation about the axis ^ x\u0000^zcombined with\nrotation.\nThe mapping to the cubic group implies that the rep-\nresentations de\fned by fG1;\u0000X;YgandfG1;X;Ygcan\nbe labeled using the Ohcharacter table. The faithful\nrepresentation generated by fG1;\u0000X;Ygis equal to the\nmatrix representation of ( x;y;z ) under cubic symme-\ntry and hence given by T1u. The representation gener-\nated byfG1;X;Ygcorresponds to the cubic representa-\ntionT2g, which describes the symmetry of the d-orbitals\n(yz;zx;xy ). In general, the 3D representations of the\ncubic group,fT1g;T2g;T1u;T2ug, are in correspondence\nwith the 3D representations of C000\n6v,fF2;F1;F3;F4g(in\nthat order). The latter describe translational symmetry\nbrokenM-point modulations. The mapping between cu-\nbic and hexagonal symmetries is summarized in Table IV.16\nHexagonalC000\n6v CubicOh\nReps. F1,F2,F3,F4T2g,T1g,T1u,T2u\nG1,G2,G3 Translations T(~ ai) Twofold rotations C2i\n\u0000I3 Twofold Rotation C2 InversionP\nX,XTTwofold rotations Threefold rotations\n(PrincipalC3) (Body diagonal C3)\nY Re\rection\u001bv Roto-re\rection PC0\n2\nTABLE IV. Table summarizing the mapping between the\nhexagonal group C000\n6vand the cubic group Oh. HereI3is\nthe identity matrix.\nAn equivalent de\fnition of the M-point representation\nfollows from considering the action of symmetry opera-\ntions on the van Hove electron operator ^\b introduced in\nthe main text [see Eq. (4)] and given by\n^\b =0\nB@^ \u001b(~M1)\n^ \u001b(~M2)\n^ \u001b(~M3)1\nCA: (A6)\nEvaluating the action of the generators of C000\n6von the\nM-point index \u0016, i.e.~M\u0016one \fnds\nT(~ x1) :^\b!G1^\b;\nC6:^\b!X^\b\n\u001bv:^\b!Y^\b; (A7)\nwhereG1,XandYare the matrices given in (A2){(A5).\nWe stress that here we only consider the action of symme-\ntries on the M-point index, and for the moment disregard\nthe action on the internal spin degree of freedom.\nStarting from ^\b, the mapping to cubic symmetry can\nbe formulated in terms of a d-orbital electron operator ^\u0007\nde\fned as,\n^\u0007 =0\nB@^ yz\u001b\n^ zx\u001b\n^ xy\u001b1\nCA; (A8)\nTheM-point representation generated by fG1;X;Yg\narises from considering the action of symmetries on ^\b,\nand as a result of the isomorphism between C000\n6vandOh,\nit arises equivalently from considering the action of Oh\nelements on ^\u0007. Speci\fcally, one \fnds that\nC2:^\u0007!G1^\u0007;\nPC3:^\u0007!X^\u0007;\nPC0\n2:^\u0007!Y^\u0007; (A9)\nwhereC2is a two-fold rotation about the principal z-axis,\nPis the inversion, C3is a three-fold rotation about a\nbody diagonal, and C0\n2is another (inequivalent) two-foldrotation. We conclude that ^\u0007 transforms in exactly the\nsame way under Ohsymmetry as ^\b underC000\n6vsymmetry.\nNote that the representation matrices act on the orbital\ndegree of freedom and not on spin, and the equivalence\nof^\u0007 and ^Phipertains to the spatial (i.e., orbital) degree\nof freedom.\nInsofar as spin is concerned, ^\u0007 transforms according to\nthe cubic double group. Speci\fcally, each element g2Oh\nis associated with Ug2SU(2) such that ^\u0007!UgOg^\u0007,\nwhereOgthe matrix representation of gobtained from\nfG1;X;Yg. For instance, the three two-fold rotations\nabout the principal axis have fUC(x)\n2;UC(y)\n2;UC(z)\n2g=\nf\u0000i\u001b1;\u0000i\u001b2;\u0000i\u001b3g. In addition, the rotation Xis ac-\ncompanied with UX=ei\u0019\u001b3=4ei\u0019\u001b2=4, and the rotation\n\u0000YwithUY=ei\u0019\u001b2=4e\u0000i\u0019\u001b3=2=\u0000iei\u0019\u001b2=4\u001b3.\nWe conclude this appendix by providing explicit ex-\npression for the matrices of fermion bilinears ^\u0003 given by\n^\u0003 = ^\by\n\u0016\u0003\u0016\u0017^\b\u0017. Here \u0003 is an Hermitian matrix and the\nspace of these M-point Hermitian matrices is spanned by\nthe Gell-Mann matrices, the generators of SU(3). In this\nwork we choose to group them in three sets de\fned by\n~\u0003a,~\u0003band~\u0003c. They are given by\n\u00031\na=0\nB@0 1 0\n1 0 0\n0 0 01\nCA;\u00032\na=0\nB@0 0 0\n0 0 1\n0 1 01\nCA;\u00033\na=0\nB@0 0 1\n0 0 0\n1 0 01\nCA\n\u00031\nb=0\nB@0\u0000i0\ni0 0\n0 0 01\nCA;\u00032\nb=0\nB@0 0 0\n0 0\u0000i\n0i01\nCA;\u00033\nb=0\nB@0 0i\n0 0 0\n\u0000i0 01\nCA\n\u00031\nc=0\nB@1 0 0\n0\u00001 0\n0 0 01\nCA;\u00032\nc=1p\n30\nB@1 0 0\n0 1 0\n0 0\u000021\nCA:(A10)\nWith the help of the symmetry transformation properties\nof Eq. (A7), it is straightforward to establish that ~\u0003a\ntransforms as F1and~\u0003basF2.\nAppendix B: Low-energy Dirac theory at the\nM0-points\n1. Proof of degeneracy\nThe proof of the symmetry protected denegeracy at\nthe~M0\n2point of the folded BZ requires evaluating the\ne\u000bect of symmetries leaving ~M0\n2invariant on the electron\noperator of Eq. (21). These symmetries are inversion C2,\nre\rection\u001bvand translations T(~ ai). The action of C2is\ngiven by\nC2!0\nBBBB@^ \u001b(\u0000~M0\n2)\n^ \u001b(\u0000~M0\n2+~M1)\n^ \u001b(\u0000~M0\n2+~M2)\n^ \u001b(\u0000~M0\n3+~M3)1\nCCCCA=\u00171^\bM0: (B1)17\nFrom this we conclude that the Hamiltonian at ~M0\n2can\nonly have terms \u001bi\u001cjor\u001bi\u001cj\u00171, where it is understood\nthati;j= 0;1;2;3. From Appendix A we know that the\ntranslation T(~ a2) is associated with G2, i.e., a rotation\naround the xaxis by\u0019, and as a result the symmetry\nT(~ a2) acts as [disregarding U(1) phases]\nT(~ a2)!\u001b1\u00173^\bM0: (B2)\nThis leaves us with the allowed terms \u001cj,\u001b1\u001cj,\u001b2\u001cj\u00171\nand\u001b3\u001cj\u00171. Similarly, the translation T(~ a3) is associated\nwithG3and therefore e\u0000i\u0019\u001b2=2. Hence, the action of the\ntranslation symmetry is\nT(~ a3)!\u001b2\u001c3^\bM0; (B3)\nwhich leaves us with the following allowed terms\n\u001c3; \u001b1\u001c1; \u001b1\u001c2; \u001b2\u00171; \u001b2\u001c3\u00171;\n\u001b3\u001c1\u00171; \u001b3\u001c2\u00171:\nWe are left with two re\rections leaving ~M0\n2invariant. We\nconsiderC2\u001bv, the action of which on ^\bM0is\nC2\u001bv!ei\u0019\u001b2=4\u001b30\nBBBB@1 0 0 0\n0 0 0 1\n0 0 1 0\n0 1 0 01\nCCCCA^\bM0: (B4)\nThe spin rotation UY=\u0000iei\u0019\u001b2=4\u001b3is the global spin\nrotation associated with Y(see also Appendix A). This\ntransformation property immediately leads to the exclu-\nsion of\u001b2\u001c3\u00171and\u001b2\u00171. The term \u001c3is clearly left\ninvariant. The remaining four terms must be combined\nin order to represent invariant terms, and in the end we\n\fnd the following three terms allowed by symmetry\n\u001c3; \u001c1(\u001b1\u0000\u001b3\u00171); \u001c2(\u001b1\u0000\u001b3\u00171):\nApplying a basis transformation e\u0000i\u0019\u001b1\u00171=4ei\u0019\u001b3=8brings\nthem into the form \u001b1\u001c1,\u001b1\u001c2and\u001c3. Clearly, these\nthree matrices mutually anti-commute and as result any\nlinear combination of these terms, i.e., the most general\nHamiltonian allowed by spatial symmetry, can only have\ntwo eigenvalues \u000fand\u0000\u000f. Each eigenvalue must be four-\nfold degenerate, proving the symmetry protection of the\ndouble Dirac nodes at ~M2. Clearly, the same is true for\nthe otherM0points.\nWe can exclude one more term using time-reversal\nsymmetryT. Time-reversal acts as\nT ! \u0000 i\u001b2\u00171^\bM0; (B5)\nfrom which we conclude that the only allowed terms are\n\u001c3and\u001c2(\u001b1\u0000\u001b3\u00171).2. Low-energy Dirac theory\nThe low-energy theory of the mean-\feld Dirac nodes\nat~M0\n2is constructed by expanding the mean-\feld band\nstructure to linear order around ~M0\n2. The \frst step is\nto diagonalize the Hamiltonian H(~M0\n2), which is given\nbyH(~M0\n2) =\u00002t\u001c3+ \u0001(\u0000\u001b1\u001c2+\u001b3\u001c2\u00171). We per-\nform a basis transformation UyH(~M0\n2)UwithU=\nei\u0019\u001b1\u00171=4ei\u0019\u001b3=8e\u0000i\u0019\u001b1=4. This yields the Hamiltonian\nH(~M0\n2) =\u00002t\u001c3+p\n2\u0001\u001c2\u001b3\u0011\u0000\u0018\u001c3+\u0011\u001c2\u001b3(B6)\nAs proven earlier, the Hamiltonian has two eigenvalues,\n\u000f\u0006=\u0006\u000f\u0011\u0006p\n\u00182+\u00112. The eigenvectors corresponding\nto\u000f+, i.e. the subspace of the relevant Dirac nodes, are\ngiven byj'mni,\nj'11i= (u;0;\u0000iv;0;0;0;0;0);\nj'12i= (0;0;0;0;u;0;\u0000iv;0);\nj'21i= (0;u;0;iv;0;0;0;0);\nj'22i= (0;0;0;0;0;u;0;iv); (B7)\nwhereuandvare de\fned as\nu=1p\n2r\n1\u0000\u0018\n\u000f; v =1p\n2r\n1 +\u0018\n\u000f: (B8)\nNote that if \u0001!0 (i.e.,\u0011!0) one hasv= 1 andu= 0,\nas expected.\nThe next step is to expand the mean-\feld Hamilto-\nnianH(~k) in small~ qwith respect to ~M0\n2, retaining only\nthe linear terms. We then perform the same basis trans-\nformationUand project the expanded Hamiltonian into\nthe subspace given by j'mni. Theq-linear part of the\nHamiltonian at ~M0\n2is\nHq=\u0018(q2\u00173\u0000q1\u00173\u001c3) +\u0011(q1\u00173\u001c2\u001b1\u0000q1\u00172\u001c3\u001b2)=p\n2\n+\u0011(q2\u00172\u001c1\u001b3\u0000q2\u00172\u001b2)=p\n2: (B9)\nTo express the resulting Dirac Hamiltonian in terms of\ne\u000bective valley and pseudospin degrees of freedom, we\nde\fne two sets of Pauli matrices, ~ \u0017and ~\u001c, which act on\nmandnofj'mni, respectively. In addition, we take\nq\u0006=q1\u0006q2, whereqi=~ q\u0001~ ai. We \fnd the Hamiltonian\nH(~ q) = (v1q\u0000+v0q+)~\u00171~\u001c2+ (v2q+\u0000v0q\u0000)~\u00173~\u001c2(B10)\nwithv1=\u0018u2+\u0011v2=p\n2,v2=\u0018v2+\u0011u2=p\n2, andv0=\n2\u0011uv=p\n2. As a result, in the limit \u0001 !0, i.e., the\nabsence of any order, v2=\u0018andv1;v0= 0. This is as\nexpected, since q\u0000is in the direction of the undistorted\nFermi surface, implying there is no dispersion in that\ndirection.\nThe inversion C2is given by\u00171^\bM0. Projecting \u00171into\nthe Dirac spinor subspace de\fned by j'mniwe \fnd ~\u001c1.\nProjecting the perturbation \u000eHgiven in Eq. (23) into\nthe same subspace yields m~\u001c1. This is consistent with18\nthe fact that \u000eHis symmetric under inversion. More\nimportantly, this demonstrates that mcontrols what the\ninversion eigenvalue of the occupied Kramer's doublet is.\nAn alternative way to understand the dependence of\nthe topological invariant (in this case given by the Fu-\nKane formula [50]) on sgn( m) is to start from the charge\ndensity modulations \u000eHand consider the M-point elec-\ntrons at \u0000:\n^\b\u0000=0\nB@^ \u001b(~M1)\n^ \u001b(~M2)\n^ \u001b(~M3)1\nCA\u00110\nB@^ 1\u001b\n^ 2\u001b\n^ 3\u001b1\nCA: (B11)\nIn the presence of the perturbation \u000eH(assuming \u0001 =\n0), the ^\b\u0000states split into non-degenerate level and a\ndegenerate doublet ( ^\t1\u001b;^\t2\u001b) [13], the latter given by\n^\t1\u001b= (^ 1\u001b+^ 2\u001b\u00002^ 3\u001b)=p\n6;\n^\t2\u001b= (\u0000^ 1\u001b+^ 2\u001b)=p\n2: (B12)\nThe sign of mdetermines the relative energetic ordering\nof the doublet and the non-degenerate level.\nIfm< 0, the doublet is higher in energy, and the Fermi\nlevel is at the semimetallic quadratic band crossing point\nde\fned by ( ^\t1;^\t2) and governed by the quadratic band\ncrossing Hamiltonian H(~ q)\u0018(q2\nx\u0000q2\ny)~\u001c3+ 2qxqy~\u001c1(see\nRef. 13). Here, ~ \u001c3=\u00061 labels the states ^\t1;2.\nWe can now consider \fnite \u0001, i.e., \fnite \u0001 A1in\nEq. (17). Projecting the \\perturbation\" coming from \u0001\ninto the subspace given by ( ^\t1\u001b;^\t2\u001b) we \fnd \u0001 ~ n\u0001~ \u001b~\u001c2,\nwith~ n= (1;1;1). This is recognized as a QSH gap of\na quadratic band crossing: ~ \u001c2constitutes the quantum\nanomalous Hall gap, and ^ n\u0001~ \u001bgives it opposite sign for\nthe two spin projections. This proves that the result-\ning state, which is adiabatically connected to the Dirac\nsemimetal with gap m < 0 at theM0points, is a topo-\nlogical insulator state.\nIn contrast, had we assumed m > 0, the quadratic\nband crossing point de\fned by ( ^\t1;^\t2) would be fully\noccupied (i.e., the Fermi level would notbe precisely at\nthe quadratic band crossing point) and the insulating\nstate with \fnite \u0001 would be adiabatically connected to\nthe trivial insulator de\fned by \u000eHwithm> 0.\nAppendix C: Derivation of symmetric classical spin\nstates\nWe brie\ry review the derivation of charge order repre-\nsentations, introduced in Ref. 13, based on the example\nof the honeycomb and kagome lattices discussed in Sec-\ntion V.\nModulations with M-point propagation vectors lead to\na quadrupling of the unit cell. In case of the honeycomb\nlattice the enlarged unit cell contains ns= 4\u00022 = 8\nsites, whereas in case of the kagome lattice the enlarged\nunit cell contains ns= 4\u00023 = 12. We label all sites andcollect them in a vector ~ sgiven byfsigns\ni=1. Extended\npoint group operations will permute the sites and the\npermutation matrices de\fne a representation of extended\npoint group. We write the representation as PM\nsand\nit has dimension ns. The superscript Mis meant to\nindicateM-point modulation (unit cell quadrupling).\nThe representation PM\nsis reducible and can be decom-\nposed into irreducible representations of the extended\npoint group. In case of the honeycomb one \fnds\nPM\ns=A1+B2+F1+F4; (C1)\nwhereas the kagome lattice yields\nPM\ns=A1+E2+F1+F3+F4: (C2)\nTheFisignal translational symmetry breaking and con-\nstitute the M-point modulated content of the decompo-\nsition. These representations are listed in Table III. We\nobserve that the honeycomb lattice has two independent\nrepresentations F1andF4, and the kagome lattice admits\nthree,F1,F3, andF4. From this we conclude that the\nformer admits two classical spin liquid states, whereas\nthe latter admits three.\nAppendix D: Extended point groups and character\ntables\nHere, we provide a basic review of the essentials of\nextended point group symmetry used in the main text.\nThe crystal point group of 2D hexagonal materials is C6v,\nwhich is identical to the dihedral group D6for spinless\nelectrons. For spinful electrons the point groups D6and\nC6vare technically distinct, however, for the purpose of\nthis work we consider them equivalent, as our results are\nindependent of technical di\u000berences, and focus on the\npoint group C6v. Note that time reversal acts as T=\ni\u001b2K(Kis complex conjugation).\nThe group C6vis generated by a six-fold rotation C6\nand a re\rection \u001bv, where the re\rection is de\fned as\n(x;y)!(x;\u0000y). The space group Sis given by all point\ngroup elements and all translations over lattice vectors\n~ x, i.e.,T(~ x), where the lattice vectors are generated by\n~ a1and~ a2. As a result, the translation subgroup Tis\ngenerated by T(~ a1) andT(~ a2).\nIn general, a point group Gcan be obtained as the\nfactor group of the space group Sand the translation\nsubgroupT. The extended point groups are obtained as\nthe factor group of the space group and a modi\fed trans-\nlation subgroup ~T, de\fned as the group of translations\ncompatible with ordering vectors, or equivalently, with\nthe enlarged unit cell. All translation in ~Tmap the en-\nlarged unit cell to itself. The enlargement is \fxed by the\nordering vectors, in our case the M-point vectors. Hence,\nthe extended point group is de\fned as ~G=S=~T. Clearly,\n~Gis larger than G, as the translations in Tbut no longer\nin~Tare now in ~G.19\nRep. Type Label Expression\nA1s \u0015 s(~k) (cosk1+ cosk2+ cosk3)=p\n3\nE2dx2\u0000y2\u0015d1(~k) (cosk1+ cosk2\u00002 cosk3)=p\n6\ndxy\u0015d2(~k) (cos k1\u0000cosk2)=p\n2\nTABLE V. Lattice angular momentum form factors trans-\nforming as representations of C6v, corresponding to near-\nest neighbor hopping on the triangular lattice. We de\fned\nki=~k\u0001~ aiwith~ aigiven in Sec. III. Note that ( \u0015d1;\u0015d2)\u0018\n(k2\nx\u0000k2\ny;2kxky) when expanded in ~k.\nIn this work, we consider hexagonal symmetry, C6v,\nand ordering at the Mpoints. The latter implies trans-\nlational symmetry breaking such that eTis generated\nbyT(2~ a1) andT(2~ a2). The translations T(~ a1)\u0011t1,\nT(~ a2)\u0011t2andT(~ a1+~ a2)\u0011t3are added to the point\ngroup. Since three translations are added to the point\ngroup, we denote the extended point group of C6vas\nC000\n6v. The character table of C000\n6vis given in Table VI.In the main text, we use an isomorphism between the\nhexagonal extended point group C000\n6vand the cubic point\ngroupOh. This connection is discussed in more detail in\nAppendix A. The character table of the cubic group is\ngiven in Table D 2.\n1. Lattice angular momentum basis functions\nThe triangular lattice angular momentum form factor\nfunctions, used to express condensate functions, are given\nin Table V. Table V lists the functions that transform\nas representations of C6vand correspond to form factors\noriginating from triangular lattice nearest-neighbor (e.g.,\n~ ai) hopping.\n2. Character tables\nFor completeness and convenience, here we reproduce\nthe character tables of the extended point groups C000\n6v\n(see Table VI) and Oh(see Table D 2).\n[1] Z. Hasan, and C. L. Kane, Rev. Mod. Phys. 82, 3045\n(2010).\n[2] X.-L. Qi., and S.-C. Zhang, Rev. Mod. Phys. 83, 1057\n(2011).\n[3] C. L. Kane, and E. J. Mele, Phys. Rev. Lett. 95, 226801\n(2005).\n[4] G. Jackeli, and G. Khaliullin, Phys. Rev. Lett. 102,\n017205 (2009).\n[5] J. Chaloupka, G. Jackeli, and G. Khaliullin, Phys. Rev.\nLett. 105, 027204 (2010).\n[6] X. Wan, A. M. Turner, A. Vishwanath, and S. Y.\nSavrasov, Phys. Rev. B 83205101 (2011).\n[7] L. Savary, E.-G. Moon, and L. Balents, Phys. Rev. X 4,\n041027 (2014).\n[8] S. Kivelson, E. Fradkin, and V. J. Emery, Nature 393,\n550 (1998).\n[9] V. Oganesyan, S. Kivelson, and E. Fradkin, Phys. Rev.\nB 64, 195109 (2001).\n[10] K. Sun and E. Fradkin, Phys. Rev. B 78, 245122 (2007).\n[11] C. Wu and S.-C. Zhang, Phys. Rev. Lett. 93, 036403\n(2004).\n[12] C. Nayak, Phys. Rev. B 62, 4880 (2000).\n[13] J. W. F. Venderbos, arXiv:1512.05766 (2015).\n[14] A. J. Leggett, Rev. Mod. Phys. 47, 331 (1975)\n[15] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S.\nNovoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109\n(2009)\n[16] J. Gonzalez, Phys. Rev. B 72, 205431 (2008).\n[17] R. Nandkishore, L. S. Levitov, and A. V. Chubukov, Nat.\nPhys. 8, 158 (2012).\n[18] M. L. Kiesel, C. Platt, W. Hanke, D. Abanin, and R.\nThomale, Phys. Rev. B 86, 020507 (2012).\n[19] W.-S. Wang, Y.-Y. Xiang, Q.-H. Wang, F. Wang, F.\nYang, and D.-H. Lee, Phys. Rev. B 85, 035414 (2012).\n[20] S.-L. Yu and J.-X. Li, Phys. Rev. B 85, 144402 (2012)[21] M. L. Kiesel, C. Platt, and R. Thomale, Phys. Rev. Lett.\n110, 126405 (2013).\n[22] W.-S. Wang, Z.-Z. Li, Y.-Y. Xiang, and Q.-H. Wang,\nPhys. Rev. B 87, 115135 (2013).\n[23] K. Jiang, A. Mesaros, and Y. Ran, Phys. Rev. X 4,\n031040 (2014).\n[24] R. Nandkishore, R. Thomale, and A. V. Chubukov, Phys.\nRev. B 89, 144501 (2014).\n[25] A. Black-Scha\u000ber and C. Honerkamp, J. Phys.: Condens.\nMatter 26, 423201 (2014).\n[26] I. Martin and C. D. Batista, Phys. Rev. Lett. 101, 156402\n(2008).\n[27] T. Li, Eur. Phys. Lett. 97, 37001 (2012).\n[28] D. Makogon, R. van Gelderen, R. Roldan, and C. Morais\nSmith, Phys. Rev. B 84, 125404 (2011).\n[29] R. Nandkishore, G. W. Chern, and A. V. Chubukov,\nPhys. Rev. Lett. 108, 227204 (2012).\n[30] G. W. Chern and C. D. Batista, Phys. Rev. Lett. 109,\n156801 (2012).\n[31] M. Sigrist and K. Ueda, Phys. Mod. Phys. 63, 239 (1991).\n[32] C. Platt, W. Hanke, and R. Thomale, Advances in\nPhysics 62, 453 (2013).\n[33] C. Wu, E. Fradkin, K. Sun, and S.-C. Zhang, Phys. Rev.\nB75, 115103 (2007).\n[34] A. V. Maharaj, R. Thomale, and S. Raghu, Phys. Rev.\nB88, 205121 (2013).\n[35] K. Barros, J. W. F. Venderbos, G. W. Chern, and C. D.\nBatista, Phys. Rev. B 90, 245119 (2014).\n[36] S. Ghosh, P. OBrien, M. J. Lawler, and C. L. Henley,\narXiv:156401 (2014).\n[37] S. M. Young and C. L. Kane, Phys. Rev. Lett. 115,\n126803 (2015).\n[38] L. Messio, C. Lhuillier, and G. Misguich, Phys. Rev. B\n83, 184401 (2011).20\nConjugacy class C000\n1C000\n2C000\n3C000\n4C000\n5C000\n6C000\n7C000\n8C000\n9C000\n10\nPoint group t1,t2t1C2,t2C2tiC3,tiC\u00001\n3tiC6,tiC\u00001\n6 3\u001bv,t1\u001bv2t1\u001bv,t2\u001bv3\u001bd,t2\u001bd1t1\u001bd1,t3\u001bd1\nC000\n6vI t 3C2t3C2C3,C\u00001\n3C6,C\u00001\n6t2\u001bv3,t3\u001bv1t2\u001bv2,t3\u001bv2t3\u001bd2,t1\u001bv3t1\u001bd2,t2\u001bd2\nt1\u001bv3,t3\u001bv3 t2\u001bd3,t3\u001bd3\nA1 1 1 1 1 1 1 1 1 1 1\nA2 1 1 1 1 1 1 \u00001\u00001\u00001\u00001\nB1 1 1\u00001\u00001 1 \u00001 1 1 \u00001\u00001\nB2 1 1\u00001\u00001 1 \u00001\u00001\u00001 1 1\nE1 2 2\u00002\u00002\u00001 1 0 0 0 0\nE2 2 2 2 2 \u00001\u00001 0 0 0 0\nF1 3\u00001 3\u00001 0 0 1 \u00001 1 \u00001\nF2 3\u00001 3\u00001 0 0 \u00001 1 \u00001 1\nF3 3\u00001\u00003 1 0 0 1 \u00001\u00001 1\nF4 3\u00001\u00003 1 0 0 \u00001 1 1 \u00001\nTABLE VI. Character table of the point group C000\n6v[65]. Translations t1andt2correspond to T(~ a1) andT(~ a2), respectively.\nt3=T(~ a1+~ a2). The irreducible representations that arise as a consequence of the added translations are F1,F2,F3andF4,\nall three-dimensional.\nPoint group OhI 3C2\n4 6C4 6C0\n2 8C3P 3PC2\n4 6PC 4 6PC0\n2 8PC 3\nKoster Mulliken\n\u0000+\n1 A1g 1 1 1 1 1 1 1 1 1 1\n\u0000+\n2 A2g 1 1 \u00001\u00001 1 1 1 \u00001\u00001 1\n\u0000+\n3 Eg 2 2 0 0 \u00001 2 2 0 0 \u00001\n\u0000+\n4 T1g 3\u00001 1 \u00001 0 3 \u00001 1 \u00001 0\n\u0000+\n5 T2g 3\u00001\u00001 1 0 3 \u00001\u00001 1 0\n\u0000\u0000\n1 A1u 1 1 1 1 1 \u00001\u00001\u00001\u00001\u00001\n\u0000\u0000\n2 A2u 1 1 \u00001\u00001 1 \u00001\u00001 1 1 \u00001\n\u0000\u0000\n3 Eu 2 2 0 0 \u00001\u00002\u00002 0 0 1\n\u0000\u0000\n4 T1u 3\u00001 1 \u00001 0 \u00003 1 \u00001 1 0\n\u0000\u0000\n5 T2u 3\u00001\u00001 1 0 \u00003 1 1 \u00001 0\nTABLE VII. Character table of the point group Oh.\n[39] B. Valenzuela and M. A. H. Vozmediano, New J. Phys.\n10, 113009 (2008).\n[40] Examples of when it is not possible to \"transfer\" a trans-\nlation to spin space are the uniaxial spin density wave or\n\u000b-phases (to be considered later in this section). These\nmanifestly break translational symmetry.\n[41] Y. Akagi and Y. Motome, J. Phys. Soc. Jap. 79(2010).\n[42] S. Kumar and J. van den Brink, Phys. Rev. Lett. 105,\n216405 (2010).\n[43] J. W. F. Venderbos, S. Kourtis, J. van den Brink, and\nM. Daghofer, Phys. Rev. Lett. 108, 126405 (2012).\n[44] K. Jiang, Y. Zhang, S. Zhou, and Z. Wang, Phys. Rev.\nLett. 114, 216402 (2015).\n[45] S. M. Young, S. Zaheer, J. C. Y. Teo, C. L. Kane, E. J.\nMele, and A. M. Rappe, Phys. Rev. Lett. 108, 140405\n(2012).[46] C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 146802\n(2005).\n[47] Note that this is in the triple- Meigenbasis and not in\nthe basis ^\b.\n[48] K. Sun, H. Yao, E. Fradkin, and S. Kivelson, Phys. Rev.\nLett. 103, 046811 (2009).\n[49] Due to uniaxial spin order, time-reversal symmetry,\nwhich inverts all spin, can be compensated by a global ro-\ntation. This is equivalent to de\fning time-reversal simply\nas complex conjugation in each spin sector.\n[50] L. Fu and C. L. Kane, Phys. Rev. B 76, 045302 (2007).\n[51] G.-W. Chern, R. M. Fernandes, R. Nandkishore, and A.\nV. Chubukov, Phys. Rev. B 86, 115443 (2012).\n[52] Q. Si and E. Abrahams, Phys. Rev. Lett. 101, 076401\n(2008).\n[53] C. Fang, H. Yao, W.-F. Tsai, J. P. Hu, and S. A. Kivelson,\nPhys. Rev. B 77, 224509 (2009).21\n[54] C. Xu, M. M uller, and S. Sachdev, Phys. Rev. B 78,\n020501(R) (2008).\n[55] Y. Qi and C. Xu, Phys. Rev. B 80, 094402 (2009).\n[56] R. M. Fernandes, L. H. Van Bebber, S. Bhattacharya, P.\nChandra, V. Keppens, D. Madrus, M. A. McGuire, B.\nC. Sales, A. S. Sefat, and J. Schmalian, Phys. Rev. Lett.\n105, 157003 (2010).\n[57] R. M. Fernandes, A. V. Chubukov, J. Knolle, I. Eremin,\nand J. Schmalian, Phys. Rev. B 85, 024534 (2012).\n[58] R. M. Fernandes, S. A. Kivelson, and E. Berg, Phys. Rev.\nB93, 014511 (2016).[59] S. Raghu, X.-L. Qi, C. Honerkamp, and S.-C. Zhang,\nPhys. Rev. Lett. 100, 156401 (2008).\n[60] T. H. Hsieh, H. Lin, J. Liu, W. Duan, A. Bansil, and L.\nFu, Nat. Comm. 3, 982 (2012).\n[61] J. Liu, W. Duan, and L. Fu, Phys. Rev. B 88, 241303\n(2013).\n[62] X. Li, F. Zhang, and A. H. MacDonald, Phys. Rev. Lett.\n116, 026803 (2016).\n[63] L. Messio, C. Lhuillier, and G. Misguich, Phys. Rev. B\n87, 125127 (2013).\n[64] C. Hickey, L. Cincio, Z. Papic, A. Paramekanti,\narXiv:1509.08461 (2015).\n[65] M. Hermele, Y. Ran, P. A. Lee, and X.-G. Wen, Phys.\nRev. B 77, 224413 (2008)."    },    {        "title": "1512.07456v1.Exact_Maps_in_Density_Functional_Theory_for_Lattice_Models.pdf",        "content": "Exact Maps in Density Functional Theory for Lattice Models\nTanja Dimitrov,1, 2,a)Heiko Appel,1, 2,b)Johanna I. Fuks,3,c)and Angel Rubio1, 2,d)\n1)Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin-Dahlem,\nGermany\n2)Max-Planck-Institut f ur Struktur und Dynamik der Materie, Luruper Chaussee 149, 22761 Hamburg,\nGermany\n3)Department of Physics and Astronomy, Hunter College and the Graduate Center of the City University of New York,\n695 Park Avenue, New York, New York 10065, USA\n(Dated: 5 April 2022)\nIn the present work, we employ exact diagonalization for model systems on a real-space lattice to explicitly\nconstruct the exact density-to-potential and for the \frst time the exact density-to-wavefunction map that\nunderly the Hohenberg-Kohn theorem in density functional theory. Having the explicit wavefunction-to-\ndensity map at hand, we are able to construct arbitrary observables as functionals of the ground-state density.\nWe analyze the density-to-potential map as the distance between the fragments of a system increases and\nthe correlation in the system grows. We observe a feature that gradually develops in the density-to-potential\nmap as well as in the density-to-wavefunction map. This feature is inherited by arbitrary expectation values\nas functional of the ground-state density. We explicitly show the excited-state energies, the excited-state\ndensities, and the correlation entropy as functionals of the ground-state density. All of them show this exact\nfeature that sharpens as the coupling of the fragments decreases and the correlation grows. We denominate this\nfeature as intra-system steepening . We show that for fully decoupled subsystems the intra-system steepening\ntransforms into the well-known inter-system derivative discontinuity. An important conclusion is that for e.g.\ncharge transfer processes between localized fragments within the same system it is not the usual inter-system\nderivative discontinuity that is missing in common ground-state functionals, but rather the di\u000berentiable\nintra-system steepening that we illustrate in the present work.\nI. INTRODUCTION\nOver the last decades ground-state density-functional\ntheory (DFT) has become a mature tool in material\nscience and quantum chemistry1{5. Provided that the\nexact exchange-correlation (xc) functional is known,\nDFT is a formally exact framework of the quantum\nmany-body problem. In practice, the accuracy of\nobservables in DFT highly depends on the choice of\nthe approximate xc-functional. From the local density\napproximation (LDA)6, to the gradient expansions such\nas the generalized gradient approximations (GGAs),\ne.g. Perdew-Burke-Enzerhof (PBE)7and the hybrid\nfunctionals such as B3LYP8, to the orbital-functionals\nsuch as optimized e\u000bective potentials9and to the\nrange-separated hybrids such as HSE0610, the last\ndecades have seen great e\u000borts and achievements in\nthe development of functionals with more accurate and\nreliable prediction capability.\nNonetheless, available approximate functionals such as\nthe LDA, the GGA's and the hybrid functionals have\nknown shortcomings to model gaps of semiconductors11,\nmolecular dissociation curves12, barriers of chemical\nreactions13, polarizabilities of molecular chains14,15, and\ncharge-transfer excitation energies, particularly between\na)Electronic address: dimitrov@fhi-berlin.mpg.de\nb)Electronic address: appel@fhi-berlin.mpg.de\nc)Electronic address: johannafuks@gmail.com\nd)Electronic address: angel.rubio@mpsd.mpg.deopen-shell molecules16.\nRecent advances in functional development such as\noptimally-tuned range separated functionals17, ensem-\nble density functional theory18,19and local scaling\ncorrections20, logarithmically enhanced factors in\ngradient approximations21and the particle-particle\nrandom-phase approximation22can diminish or even\ncure some of the above mentioned shortcomings but not\nall of them.\nShortcomings of approximate functionals indicate that\nsome important qualitative features of the exact func-\ntional are not (su\u000eciently well) captured. A common\nexample is the delocalization error as in the case of\nstretched molecules, where approximate functionals\nsuch as LDA and GGA's tend to arti\fcially spread out\nthe ground-state electron density in space23. Since in\nDFT every observable is a functional of the ground-\nstate density the delocalization error transmits into\nall observables as functional of the density and in\nparticular to the ground-state energy functional. As a\nconsequence most approximations for the ground-state\nenergy as functional of the particle number Nare either\nconcave or convex functions between integer N's20,24\nand hence, violate the exact Perdew-Parr-Levy-Balduz\ncondition25which states that the ground-state energy\nas a function of the particle number E(N) is a linear\nfunction between integer N. The linearity of E(N)\nleads to the commonly known derivative discontinuity25\nand is one exact condition on the xc-functional. Exact\nconditions on the xc-functional are a very useful tool\nin the development of new, improved functionals.\nIn this paper we discuss an exact condition on thearXiv:1512.07456v1  [physics.chem-ph]  23 Dec 20152\nxc-functional that is relevant for systems consisting\nof well separated but mutually-interacting fragments,\nsuch as in stretched molecules. Among the approaches\nto model the limit of strongly correlated, low density\nsystems with DFT we highlight the long range corrected\nhybrids26, the generalization of the strictly correlated\nelectron functional to fractional electron numbers27{30\nand the recently introduced local scaling correction,\nwhich imposes the linearity condition to local regions of\nthe system, correcting both energies and densities and\na\u000erming the relevance of modelling fractional electron\ndistributions to reduce the delocalization error20.\nExactly solvable model systems have shown to provide\nuseful insight essential to understand the failures of\napproximate xc-functionals and to develop new and\nimproved approximations. For example, by studying\none-dimensional model systems of few electrons it was\nshown that in the dissociation limit of molecules, the\nexact xc-potential as function of the spatial coordi-\nnate develops steps and peaks31{37. Such features are\nmanifestations of strong-correlation and the absence\nof such features in approximate functionals results in\ndelocalization errors.\nStudies of exact ground-state xc-functionals for lattice\nmodels include the exact one-to-one map between\nground-state densities and potentials computed for a\nhalf-\flled one-dimensional Hubbard chain in Ref.38\nusing the Bethe Ansatz, for the one-site and double-site\nHubbard models in full Fock space in Ref.39,40and\nfor the two-electron Hubbard dimer via constraint\nsearch in Ref.41, among others. For such lattice models\nthe Hohenberg-Kohn theorem42can be generalized by\nreplacing the real-space potentials and densities by\non-site potentials and on-site occupations43,44. The\n\fnite Hilbert space of lattice models permits the con-\nstruction of the exact density-to-potential map. The\nquestion arises what can be learned about realistic\nthree-dimensional systems by studying one-dimensional\nlattice models. Recently it was shown45,46that the time-\ndependent exact xc-functional of the one-dimensional\nHubbard dimer in the strongly-correlated limit develops\nthe same step feature as the real-space one-dimensional\nmodel studied in Ref.47. Reference calculations of Ref.48\nshow that one-dimensional model systems capture the\nessence of three-dimensional systems when studying\nstrong-correlation in DFT.\nIn this work, we study the exact density-to-potential\nand density-to-wavefunction map of a one-dimensional\nlattice model with a system size that still allows to\nexactly diagonalize the Hamiltonian in full Fock space.\nFor di\u000berent values of the external potential in the\nHamiltonian we perform exact diagonalization of the\nHamiltonian. Each diagonalization gives us all eigen-\nfunctions and eigenenergies of the system, where the\neigenstate with lowest eigenenergy corresponds to the\nground state. We use the ground-state of each exact di-\nagonalization corresponding to a \fxed external and \fxed\nchemical potential to construct both one-to-one maps,i.e. the map between on-site potentials and ground-state\non-site occupations (ground-state densities), and the\nmap between ground-state densities and ground-state\nwave-functions. To illustrate the latter, we numerically\nconstruct the con\fguration-interaction (CI) coe\u000ecients\nof the wave-function expansion as functionals of the\nground-state density. We study the exact features of\nthese maps for systems with di\u000berent ratio of discrete\nvalues of the kinetic hopping probability \u0015tto the\nelectron-electron interaction strength \u0015w. This allows\nus to study the exact maps from the non-interacting to\nthe strictly-localized electron limit while we gradually\nchange the correlation of the system. We illustrate how\nthe distinctive features of the exact density-to-potential\nmap transmit into the wavefunction-to-density map, and\nfurther into expectation values and transition matrix\nelements of arbitrary operators as functionals of the\ndensity.\nWe show that in approaching the limit of strongly\ncorrelated electrons, i.e.\u0015t\n\u0015w!0, the gradient of the\nexact density-to-potential map steepens. We denote\nthis feature as intra-system steepening which gradually\nbuilds up within the system as the hopping probability\nfavoring the delocalization of electrons decreases and\nthe electron-electron interaction favoring the localiza-\ntion increases. In the strictly localized electron limit,\nwhere\u0015t= 0, we see that the intra-system steepening\ntransforms into the step-like inter-system derivative\ndiscontinuity.\nWe \fnd that qualitative features such as the intra-system\nsteepening and the inter-system derivative discontinuity\nof the density-to-potential map are already captured\nby a two-site lattice model. In the case of a two-site\nmodel, each site can be regarded as a subsystem.\nWith increasing distance between the subsystems of\nthe system, the hopping probability decreases and the\nlocalization of the electrons on each site increases. If\nthe sites are in\fnitely apart, the subsystems are truly\nseparated and the electrons are strictly localized on\neach site. We simulate the in\fnite separation in the\ntwo-site model by setting the hopping parameter in the\nkinetic operator \u0015tstrictly to zero. Since the kinetic\nenergy is strictly zero, this limit is the classical limit.\nHowever, setting \u0015tequal to zero allows us to imitate\nthe in\fnite bond-stretching of the molecular model,\nwhere the distance of the molecular wells dgoes to1.\nIn this limit, \u0015t= 0 implying d!1 , the intra-system\nsteepening of the density-to-potential map becomes the\nstandard step-like inter-system derivative discontinuity.\nArbitrary observables and transition-matrix elements are\na\u000bected by the presence of the intra-system steepening\nand the inter-system derivative discontinuity, and in\nparticular by the lack of it in approximate functionals.\nWe illustrate how both features are transmitted to\nthe ground- and excited-state energy, the excited- and\ntransition-state density and to the correlation entropy\nfunctionals.3\nThe paper is organized as follows. In section II we\npresent the exact maps between local potentials,\nground-state wavefunctions and ground-state densities.\nIn section III we introduce the lattice model and the\nmethodology that we employ in the present work.\nSection IV is dedicated to the study of the intra-system\nsteepening in the exact density-to-potential map when\napproaching the strictly-localized limit (i.e. the strongly\ncorrelated limit) and its transition into a real inter-\nsystem derivative discontinuity for truly separated\nsubsystems. In section V we use the potential-to-density\nmap to construct the CI coe\u000ecients of the ground-\nand excited-state wavefunction expansions as explicit\nfunctionals of the ground-state density. Ground-state\ndegeneracies leave topological scars in the electron\ndensity49. We illustrate how these degeneracies and\nfurthermore also near-degeneracies of the eigenenergies\nof the system a\u000bect the ground- and excited-state expec-\ntation values and transition matrix elements of relevant\noperators as functionals of the ground-state density.\nFinally, in section VI, we summarize our \fndings and\ngive an outlook for future work.\nII. EXACT MAPPINGS\nTo understand which features approximate function-\nals are missing, it is instructive to explicitly construct\nand to analyze the exact maps between the ground-\nstate wavefunction \t 0, the local potential V, and the\nground-state density n00, sketched in Fig. 1. For\n\fxed electron-electron interaction ^W, the many-body\nSchr odinger equation,\n(^T+^W+^V)\tk=Ek\tk, (1)\n1-to-1-map\nB\nAC\tk n00\nV\nFIG. 1. Schematic illustration of the exact mapping between\nN-electron wavefunctions \t k, local potentials V, and ground-\nstate electron densities n00. The maps are depicted as red\narrows, whereAmapsVonto \tk,Bmaps \tkonton00and\nCmapsn00ontoV. Black arrows indicate the bijectivity of\neach of these one-to-one maps. Note that every element in V\nhas an exact one-to-one equivalent in n00and \t 0.de\fnes a unique map between the set of local potentials V\nand the set of energy eigenstates \t k, depicted as map A\nin Fig. 1. The ground-state density n00can be computed\nas usual according to\nn00(~ r) =NZ\nd~ r2:::d~ rNj\t0(~ r2::;~ rN)j2, (2)\nwhich establishes a unique map from the set of N-\nelectron ground-state wavefunctions \t 0to the set of\nN-electron ground-state densities n00. Hohenberg and\nKohn42proved that the map Cin Fig. 1 between n00\nandVis one-to-one and unique if V-representability\nis ful\flled50{54. Assuming existence of this one-to-one\ndensity-to-potential map allows in principle to construct\nany ground-state observable as a unique functional of the\nground-state density n00,\nO00[n00] =h\t0[n00]j^Oj\t0[n00]i. (3)\nNote that as a consequence of the one-to-one V-to-n00\nmap, the Schr odinger equation additionally establishes a\none-to-one map between n00and the excited-state wave-\nfunctions \t k;k6= 0. As a consequence, excited-state\nexpectation values with k=l>0, and transition matrix\nelements with k6=l, can be computed as functionals of\nthe ground-state density using\nOkl[n00] =h\tk[n00]j^Oj\tl[n00]i. (4)\nThe ground-state energy E0and the ground-state density\nn00can be accessed using the variational principle,\nE0= min\nnEV[n]; E 0<EV[n]; n6=n00:(5)\nGiven an external potential Vthe total energy is com-\nputed asEV[n] =FHK[n] +R\nn(~ r)V(~ r)d3r. In the Levy-\nLieb constrained search formulation55,56the Hohenberg-\nKohn energy functional FHKis found as the minimum\nover all possible N-electron densities n, of the expec-\ntation value of kinetic plus electron-electron interaction\noperator\nFHK[n] = min\n\t!nh\t[n]j^T+^Wj\t[n]i. (6)\nIn the following, we illustrate the features of the exact\ndensity-to-potential and density-to-wavefunction maps\nexplicitly for our model systems.\nIII. LATTICE MODEL\nIn the present work, we restrict ourselves to one-\ndimensional lattice systems for which the construction of\nexact functionals via exact diagonalization is numerically\nfeasible. On a lattice, the potential becomes an on-site\npotentialV(x)!v(xi), the density a site-occupation,\nn(x)!n(xi), and the integral becomes a sum over sites\ni,R\ndx!P\ni43. Furthermore, the kinetic energy opera-\ntor becomes a nearest-neighbor hopping term.4\nFIG. 2. Illustration of the kinetic hopping probability \u0015tand\nthe strength of electron-electron interaction \u0015win polar rep-\nresentation.\nA. Lattice Hamiltonian\nForNinteracting electrons in one spatial dimension\nwe consider Hamiltonians of the form\n^H'=^H(') =\u0015t(')^T+\u0015w(')^W+^V+\u0016^N, (7)\nwhere the parameter \u0016, connected to the particle num-\nber operator ^N, acts as a Lagrange multiplier shifting the\nstate with lowest energy to blocks with di\u000berent particle\nnumberNin Fock space. To switch between di\u000berent\ncoupling limits, we introduce the amplitude of the kinetic\nhopping\u0015t=rcos(') and the strength of the electron-\nelectron interaction \u0015w=rsin(') as parameters in polar\nrepresentation with radius rand angle', see Fig. 2. The\nlimit\u0015w!0, i.e.'= 0, corresponds to non-interacting\nelectrons and the limit \u0015t!0, i.e.'=\u0019\n2, to site-\nlocalized electrons. Without loss of generality we choose\nr=p\n2 and'2[0;\u0019\n2]. Throughout this work we use\natomic units \u0016 h=m=e= 1.\nNext, we introduce the operators of the lattice model\nwithMsites and lattice spacing dx. In second-order \f-\nnite di\u000berence representation the kinetic energy operator\nwith nearest neighbor hopping t0=1\ndx2reads\n^T=\u0000t0\n2MX\nl=1X\n\u001b^cy\nl;\u001b^cl+1;\u001b+ ^cy\nl+1;\u001b^cl;\u001b\u00002^cy\nl;\u001b^cl;\u001b, (8)\nwhere ^cy\nl;\u001band ^cl;\u001bdenote creation and annihilation oper-\nators of an electron placed on site lwith spin projection\nonto the z-axis \u001b. Usually, the hopping t0changes with\nthe lattice spacing dx. However, we choose to leave dx\n\fxed and use the parameter \u0015tand\u0015winstead. The last\nterm in Eq. 8 corresponds to on-site hopping. For model\nsystems this term is usually not taken into account. Here\nwe keep the term to allow for a consistent \frst and sec-\nond quantized treatment of the Hamiltonian. We study\nthe non-local soft-Coulomb electron-electron interaction,\n^WSC=X\nl;m;\u001b;\u001b0^cy\nl;\u001b^cy\nm;\u001b0^cm;\u001b0^cl;\u001b\n2p\n(dx(l\u0000m))2+a, (9)acting on particles located at sites landmwith spins\u001b\nand\u001b0. Throughout this work the Coulomb interaction is\nsoftened by the parameter a= 1. The external potential\n^V=MX\nl=1vl^cy\nl^cl, (10)\nintroduces a potential di\u000berence between the sites in the\nlattice, which depending on its strength, shifts the elec-\ntron density among the sites in the lattice. We restrict\nourselves to two di\u000berent scenarios for which exact diago-\nnalization is still possible, similar to Ref.57,58. In case (i)\nwe consider two spin-singlet electrons on M= 206 sites.\nThe particles are con\fned in a box from x=\u000010:25 a.u.\ntox= +10:25 a.u. with zero boundary conditions and\na lattice spacing of dx= 0:1 a.u. and \u0015t=\u0015w= 1.\nTo mimic the bond-stretching in molecular systems, we\nconsider an external potential\nvl=Z1(\u000b)q\n(xl\u0000d\n2)2+ 1+Z2(\u000b)q\n(xl+d\n2)2+ 1+Z1(\u000b)Z2(\u000b)p\n(d2+ 1);\n(11)\nZ1(\u000b) =\u0000\u000b; Z 2(\u000b) =\u0000(2\u0000\u000b)\n(12)\nwith two atomic wells separated by distances ranging\nfromd= 2 tod= 8 a.u.. The depth of the wells is\ngiven by the nuclear charges Z1andZ2which we modu-\nlate with the parameter \u000b2[0;2]. We will see that the\nessence of such a system is already captured by a two-site\nmodel. As case (ii), we consider M= 2 sites in the lattice\nwith a distance dx=1p\n2, where we vary the parameters\n\u0015tand\u0015w. In this case, the system size allows to perform\nexact diagonalization in the full Fock space of the model.\nB. Methodology\nTo explicitly construct the one-to-one map between ex-\nternal potentials and ground-state densities, we diago-\nnalize the Hamiltonian introduced in Eq. (7) for di\u000ber-\nent external potentials vm, but \fxed '. The external\npotential take values vm=m\u0001v, where \u0001 vis the nu-\nmerical step size, and mis the step numbers. For com-\npleteness, although not shown in the present work, sim-\nilarly the Hamiltonian can be diagonalized for di\u000berent\nchemical potentials \u0016kwith the chemical potential val-\nues\u0016k=k\u0001\u0016, the numerical step size \u0001 \u0016and the step\nnumberk. In this way all functionals are constructible\nas function of the particle number Nand can be studied\nin complete Fock-space. Here we \fx the chemical poten-\ntial and select a discrete and uniformly distributed set of\npotentials from the continuous set V of possible external\npotentials in Fig. 1. In the next step we use exact diago-\nnalization to compute the ground-state wavefunction \t'\n0\nand energy E'\n0corresponding to each value of viand\u0016i\n(but \fxed'). For each ground-state wavefunction, we5\ncompute the corresponding on-site ground-state density\nn'\n00(xj) =h\t'\n0j^n(xj)j\t'\n0i, (13)\nwherejis the site subindex, and the spin-summed den-\nsity operator reads\n^n(xj) = ^cy\nj;\"^cj;\"+ ^cy\nj;#^cj;#. (14)\nIn addition to the ground-state wavefunction, the exact\ndiagonalization gives us access to the excited-state wave-\nfunctions \t'\nk6=0, which allows us to compute excited-state\nobservables and transition matrix elements of operators\nas functionals of the ground-state density according to\nEq. 4. On a lattice with Msites, the continuous one-\ndimensional ground-state density n00(x) becomes a vec-\ntor (n00(x1);n00(x2);:::;n 00(xM)). Hence, expectation\nvalues and transition matrix elements as functionals of\nthe density become rank Mtensors\nO'\nkl(n00(x1);:::;n 00(xM)) =\nh\t'\nk(n00(x1);:::;n 00(xM))j^Oj\t'\nl(n00(x1);:::;n 00(xM))i.\n(15)\nIn case (ii) where two sites in the lattice are con-\nsidered, all functionals depend on the on-site densi-\ntiesn00(x1) andn00(x2), i.e.j\t'\nk[n00(x1);n00(x2)]iand\nV'[n00(x1);n00(x2)]. Instead of expressing all func-\ntional dependencies in terms of the variables n00(x1) and\nn00(x2), we rotate the coordinate system to the total\nparticle number N=n00(x1) +n00(x2) and the occu-\npation di\u000berence \u000en00=n00(x1)\u0000n00(x2) between the\nsites41. To illustrate the wavefunction-to-density map,\nwe expand the ground ( k= 0) and the excited ( k >0)\neigenstatesj\t'\nkiof the system in a complete set of Slater\ndeterminantsj\bqi,\nj\t'\nk[\u000en00;N]i=X\nq\u000b';k\nq[\u000en00;N]j\bqi, (16)\nwhere we have chosen j\bqito be the eigenstates of the\nkinetic operator ^T. This gives rise to the CI coe\u000ecients\n\u000b';k\nq[\u000en00;N] =h\bqj\t'\nk[\u000en00;N]i: (17)\nBy writing the CI-coe\u000ecients \u000b';k\nq[\u000en00;N] as explicit\nfunctionals of \u000en00, we gain access to all ground- and\nexcited-state expectation values or transition matrix ele-\nments of any operator, i.e.\nO'\nkl[\u000en00;N] =h\t'\nk[\u000en00;N]j^Oj\t'\nl[\u000en00;N]i\n=X\nqX\nq0\u000b'k\u0003\nq0[\u000en00;N]\u000b'l\nq[\u000en00;N]h\bq0j^Oj\bqi.\n(18)\nA prime example is the Hohenberg-Kohn energy func-\ntional de\fned in Eq. 6, which is the expectation value of^H'\nvl=0=\u0015t(')^T+\u0015w(')^W, i.e.\nF'\n00[\u000en00;N] =h\t'\n0[\u000en00;N]j^H'\nvl=0j\t'\n0[\u000en00;N]i\n=X\nq;q0\u000b\u0003\nq0[\u000en00;N]\u000bq[\u000en00;N]h\bq0j^H'\nvl=0j\bqi.\n(19)\nFor the two-particle singlet states, we compute the\nHohenberg-Kohn functional for di\u000berent values of '2\n[0;\u0019\n2]. Note the explicit dependence of F'\n00[\u000en00;N] on\nthe angle', since the Hohenberg-Kohn proof can only\nbe established for \fxed and given kinetic energy and\nparticle-particle interaction. By changing the angle ',\nwe construct the exact energy functional F'\n00[\u000en00;N] for\ndi\u000berent electron-electron interactions and kinetic terms,\nwhere'= 0 is the non-interacting and '=\u0019\n2the in-\n\fnitely correlated limit. Although the Hohenberg-Kohn\nground-state energy functional is a very important exam-\nple, our approach allows to construct the exact density\nfunctionals for any observable of interest. We illustrate\nthis for a few selected examples in the following sections.\nAlso we emphasize that throughout this work all func-\ntionals are constructed in the zero-temperature limit.\nIV. FEATURES OF THE EXACT\nDENSITY-TO-POTENTIAL MAP\nWe start our analysis for the Hamiltonian of case (i),\nwhere we consider a diatomic molecule with di\u000berent in-\nteratomic separations. While the full density-to-potential\nmap is a high-dimensional function for M= 206 sites and\nimpractical to visualize, the essence of the bond stretch-\ning can be captured by the integrated densities of frag-\nments of the system. A natural choice to partition the\nsystem into its fragments, is to divide the total molecu-\nlar charge distribution at its minima into di\u000berent Bader\nbasins59,60. By integrating the density over each of these\nBader basins, the high dimensionality of the density in\nreal-space reduces drastically. For our diatomic model\nthe partitioning reduces the dimensionality from 206 to\ntwo, by mapping the sites in the grid onto the basins. We\ncan then refer to each basin as a e\u000bective site in real space\nand regard the density di\u000berence between the basins as\ndensity di\u000berence between the two sites. For the simple\ndiatomic molecule in one dimension, we simply divide\nthe system in two equal half-spaces, and construct the\ndensity-di\u000berence according to\n\u000en00=M=2X\ni=1n00(xi)\u0000MX\ni=M=2+1n00(xi). (20)\nTo obtain the potential di\u000berence between the two basins,\nwe take the di\u000berence between the maximum depth of the\nmolecular potential wells of each basin and de\fne the po-\ntential di\u000berence as \u000ev=Z1(\u000b)\u0000Z2(\u000b). Note, the po-\ntential di\u000berence can be tuned by changing the nuclear6\nFIG. 3. Exact density-to-potential map for a one-dimensional diatomic molecule with nuclear charges Z1andZ2, where we vary\nthe potential di\u000berence \u000ev=Z1\u0000Z2from\u00005 to 5 for di\u000berent atomic separations d= 2\u00008 a.u. The density di\u000berence \u000en00\ncorresponds to the electronic density summed over the left half-space minus the density summed over the right half-space as\nde\fned in Eq. 20. The graph illustrates the in\ruence of electron localization on the ground-state density-to-potential map. From\nleft to right the distance of the molecular wells increases while the gradient of the density-to-potential steepens with increasing\ndistancedand hence, decreasing coupling of the fragments of the system. We denote this feature of the density-to-potential\nmap as intra-system steepening (see text for details).\ncharge of the two atoms continuously with the parameter\n\u000bof Eq. 12. The resulting e\u000bective density-to-potential\nmap for our diatomic model is shown in Fig. 3. Starting\nfrom left to right we increase the distance between the\nmolecular wells. The e\u000bective density-to-potential map\nstarts out with a smooth monotonic shape. When the\ndistance of the atoms is increased the gradient of the\ndensity-to-potential map steepens, leading ultimately to\nsteps in the density values for the in\fnitely separated\nlimit.\nThe very same qualitative behavior can be found for a\nsimple two-site lattice system. As a second example\nwe consider therefore the Hamiltonian of case (ii). In\nthis case we construct the exact density-to-potential map\nfor the two-particle singlet states of the two-site lattice\nmodel. The results are shown in Fig. 4. In addition\nto the ground-state density-to-potential map in the \frst\nrow of Fig. 4, the second and third row show the \frst\nand second excited-state density-to-potential map, and\nthe fourth row shows the eigenenergies E0,E1andE2as\nfunction of the external potential di\u000berence between the\ntwo sites in lattice. From left to right, 'increases, i.e.\nthe electron-electron interaction favoring the localization\nof the electrons increases, whereas the kinetic energy fa-\nvoring the delocalization decreases, i.e.\u0015t\n\u0015w!0. This lo-\ncalization is re\rected by the steep gradient of the ground-\nand excited-state densities \u000en00,\u000en11and\u000en22as func-\ntion of the external potential di\u000berence \u000ev=v1\u0000v2.\nFor the potential di\u000berence we select values from -5 to 5,\nshifting the electron density from one site in the lattice to\nthe other. Setting '= 0 in Eq. (7) corresponds to non-\ninteracting electrons, where the eigenfunctions are single-\nparticle Slater-determinants. In this limit the density-to-potential map for our model can be found analytically\n\u000en'=0\n00(\u000ev;\u0015t;dx) =\u00004dx2\u000evp\n4dx4\u000ev2+\u00152\nt('= 0):(21)\nThe map behaves smoothly as can be seen in the leftmost\n\fgure in the \frst row of Fig. 4.\nApproaching the strictly-localized limit, i.e. '!\u0019\n2, the\nslope of the exact density-to-potential map sharpens until\nthe map develops a characteristic feature, which we de-\nnote as intra-system steepening. The intra-system steep-\nening of the gradient of the density-to-potential map cor-\nresponds to the localization of the electrons in the respec-\ntive subsystems. Near the strictly-localized limit, e.g.\n'=\u0019\n2\u00001\n100, the electrons are highly-localized on the\nsites.\nIn the strictly-localized electron limit '=\u0019\n2the hop-\nping parameter is equal to zero. In this limit the system\n'breaks' into two physical disconnected sites of integer\noccupation, the Hamiltonian reduces to ^H'=\u0019\n2=^W+^V\nand hence commutes with the position operator ^ x=PM\nm=1P\n\u001bxm^cy\nm;\u001b^cm;\u001bwithxm=\u0000(M+1)=2dx+mdx.\n[^H'=\u0019\n2;^x] = 0, (22)\nand the eigenfunctions of ^Hare diagonal in the eigen-\nbasis of the position operator. The three two-particle\nsinglet states correspond to the physical situations\nwhere both electrons are located on site one, i.e.\f\f\f\t'=\u0019\n2\n0[\u000en00= +2]E\n= ^cy\n1#^cy\n1\"j0i, both electrons are\non site two, i.e.\f\f\f\t'=\u0019\n2\n0[\u000en00=\u00002]E\n= ^cy\n2#^cy\n2\"j0i,\nor where the electrons are delocalized over both sites,\ni.e.\f\f\f\t'=\u0019\n2\n0[\u000en00= 0]E\n=1p\n2\u0010\n^cy\n1#^cy\n2\"\u0000^cy\n1\"^cy\n2#\u0011\nj0i. De-\npending on the ratio between the external potential dif-\nference\u000evand the electron-electron repulsion strength7\n−202δn00ϕ= 0\n−202δn11\n−202δn22\n−5 0 5\nδv−5051015Ej\nE0\nE1\nE2ϕ=π/4\n−5 0 5\nδvϕ=π/2−1/10\n−5 0 5\nδvϕ=π/2−1/100\n−5 0 5\nδv\n−0.50.0 0.5\nδv1.141.161.181.201.22E0,E1Avoided Crossing\nϕ=π/2\n−5 0 5\nδv\n−0.50.0 0.5\nδvCrossing\n−0.50.0 0.5\nδv−202δn(00,11)δn00zoom:ϕ=π/2\n−0.50.0 0.5\nδvδn11zoom:ϕ=π/2\nFIG. 4. Exact density-to-potential map for a two-site lattice model using soft-Coulomb interaction. Despite its reduced\ndimensionality essential features of the density-to-potential map of the molecular model system are already captured by a\ntwo-site model, as can be seen by comparing Fig. 3 and Fig. 4. The graphs illustrate how the electron localization is captured\nin the ground- and excited-state density-to-potential maps and in the eigenenergies. Upper panel: exact ground-state density\nas function of the external potential, i.e. \u000en00(\u000ev). Second panel: exact \frst excited-state density as function of the external\npotential, i.e. \u000en11(\u000ev). Third panel: exact second excited-state density as functional of the external potential, i.e. \u000en22(\u000ev).\nLower panel: eigenenergies of the two-particle singlet states as functional of the external potential Ej(\u000ev), whereEjcorresponds\nto the eigenstate j\tjiand to the density di\u000berences \u000enjj=h\tjj\u000e^nj\tji. Inset at the bottom on the left-hand side: Detailed\nview of the ground-state and the \frst excited-state density functionals \u000en00(\u000ev) and\u000en11(\u000ev) in the strictly localized limit.\nInset at the bottom at the right-hand side: avoided and real crossings of eigenenergies. From left to right the angle 'increases\nthe correlation in the system going from the non-interacting ( '= 0) to the strictly-site-localized electron limit ( '=\u0019\n2). In\nthe molecular model system of Fig. 3 this corresponds to an increasing distance dof the molecular wells. The gradient of all\nthree densities steepens whenever the corresponding eigenstate as functional of the external potential comes close to an avoided\ncrossing. We denote this exact feature of the density-to-potential map as intra-system steepening. In the strictly localized\nlimit ('=\u0019\n2) the intra-system steepening transitions into the inter-system derivative discontinuity while the avoided crossing\ntransitions into a real-crossing with degenerate eigenenergies.\n\u0015w, one of these three eigenstates is energetically more\nfavorable and becomes the ground-state of the electronic\nsystem, see lower panel of Fig. 4. Using the strictly-\nlocalized ground-state wavefunction, the density di\u000ber-ence\u000en00transitions from a continuous variable to a dis-\ncrete set of integer values. Namely, the only possible\nvalues for the ground-state density di\u000berences are the in-8\nteger values\n\u000en'=\u0019\n2\n00(\u000ev) =8\n><\n>:\u00002;\n0;\n+2:\nIn this limit di\u000berent values of the external potential\nlead to the same density di\u000berence \u000en00as can be seen\nin the map for '=\u0019\n2in Fig. 4. Therefore, the one-\nto-one map between \u000en00and\u000evbreaks down and the\nintra-system steepening transitions into the inter-system\nderivative discontinuity, since the two sites decouple and\nare becoming two separate systems. Functionals in the\ndistributional limit are a linear combination of the func-\ntionals of the degenerate densities as has been shown for\nthe ground-state energy functional as functional of the\nparticle number25,61. Therefore, we connect the distri-\nbutional points for all functionals via straight lines, i.e.\n\u000en00=\u00062(1\u0000!) and 0\u0014!\u00141. In a physical pic-\nture each one of the disconnected sites can be seen as\na system in\fnitesimally weakly connected to a grand-\ncanonical particle reservoir.\nContrary to the widely discussed inter-system derivative\ndiscontinuity, which describes the piece-wise linear be-\nhavior of the energy as a function of the particle number\nE[N], the intra-system steepening describes the smooth\nbehavior of the energy as functional of the density di\u000ber-\nence between fragments within the system E[\u000en00]. Both\nfeatures already show up in the density-to-potential map\nand transmit to all observables. The Hohenberg-Kohn\nenergy functional is therefore only one speci\fc example\nfor the appearance of inter-system derivative discontinu-\nity and intra-system steepening. The smooth behavior\nof the intra-system steepening is a consequence of the\nmixing of di\u000berent quantum eigenstates around avoided\ncrossings, and the steps related to the inter-system\nderivative discontinuity directly result from intersections\nof eigenenergies, thus real crossings, see lower panel and\ninset of Fig. 4. The inter-system derivative discontinu-\nity appears when electrons are strictly-localized in states\nwith di\u000berent particle number. Note that the steepening\nof the gradient for \u000en00as well as for \u000en11and\u000en22arises\nwhenever the eigenvalues of the Hamiltonian in Eq. (7)\nas function of the external potential become nearly de-\ngenerate. The connection between the avoided crossing\nand the steepening of the gradients functional is closely\nrelated to the \fnding of Ref.36, i.e. that the step feature\nof the exact xc-potential in space arises in the vicinity\nof the avoided crossing, when the bonding and antibond-\ning orbitals become nearly degenerate. Without this step\nfeature (and the peaks) of the exact xc-potential, the non-\ninteracting electron density would arti\fcially smear out\nover both basins and lack the intra-system steepening of\nthe exact electron density-to-potential map. For '= 0\nall eigenvalues are non-degenerate, hence the density-to-\npotential map of all eigenstates behaves smoothly. When\nwe approach the strongly-correlated limit at '!\u0019\n2,\nthe \frst and second excited-state energies approach eachotherE1[\u000ev]!E2[\u000ev] and for'=\u0019\n2they become de-\ngenerate for \u000ev= 0, i.e. E1[\u000ev] =E2[\u000ev] (see inset\nFig. 4). Caused by a real crossing of the eigenenergies\nin the strictly-localized limit, the one-to-one correspon-\ndence with an external potential breaks down for all den-\nsities, i.e. the ground-state and the excited-state den-\nsities. The density-to-potential map becomes a distri-\nbution in this limit and the Hohenberg-Kohn theorem\ndoesn't apply.\nV. FEATURES OF THE EXACT\nDENSITY-TO-WAVEFUNCTION MAP\nThe inter-system derivative discontinuity and the\nintra-system steepening discussed in the previous section\nare exact properties of the density-to-potential map. As\na consequence, also the exact wavefunction and hence,\nall exact observables - here, in particular the ground-\nstate Hohenberg-Kohn energy functional- as function of\nthe exact density inherit the intra-system steepening and\nthe inter-system derivative discontinuity. In the follow-\ning sections we illustrate this fact. In particular, we show\nhow these features show up in the CI-coe\u000ecients, and\nconsequently in the energy, the excited-densities and in\nthe correlation entropy functional.\nA. Exact Con\fguration Interaction Coe\u000ecients as\nFunctionals of the Ground-State Density\nTo construct the density-to-wavefunction map, we ex-\npand the correlated ground- and excited-state wavefunc-\ntions from the exact diagonalization of the Hamiltonian\nin a complete set of Slater determinants j\bqi. This gives\nrise to CI coe\u000ecients as functionals of the ground-state\ndensity as de\fned in Eq. 17. Clearly, each choice for the\nset of Slater determinants j\bqiinduces a di\u000berent set of\nCI functionals. Here we choose as basis set the determi-\nnants which are eigenfunctions of the free kinetic energy\noperator. More speci\fcally, we project the two-particle\nsinglet ground-state wavefunction of the Hamiltonian in\nEq. 7 onto the three two-particle singlet eigenstates of the\nkinetic operator ^Tto construct one of these sets for each\ndi\u000berent'. The results are summarized in Fig. 5. Each\nrow in the \fgure displays one of the ground-state CI coef-\n\fcients as function of the density di\u000berence between the\nsites,\u000bq[\u000en00] =\n\bqj\t0[N= 2;S2= 0;Sz= 0;\u000en00]\u000b\n.\nFor non-interacting electrons, the CI coe\u000ecients can be\nevaluated analytically. In our chosen basis the coe\u000ecients9\nFIG. 5. CI coe\u000ecients of the two-particle ground-state wavefunction in the kinetic operator basis. From left-to-right we\napproach the strictly-localized limit ( '=\u0019\n2) and the gradient of all three CI coe\u000ecients steepens. For '=\u0019\n2the CI coe\u000ecients\ntake only discrete values which can be interpolated linearly (dashed lines) due to the degeneracy of the eigenstates in the\nstrictly localized limit.\nhave no direct dependency on \u0015t,\n\u000b'=0\n1[\u000en00] =\u0000\u0010\n\u000en2\n00\u00002\u0010\n2 +p\n4\u0000\u000en2\n00\u0011\u0011\n(2 +j\u000en00j)\n4p\n\u0000(\u00004 +\u000en2\n00)(4 +\u000en2\n00+ 4j\u000en00j)\n(23)\n\u000b'=0\n2[\u000en00] =\u0000\u000en00\n2p\n2(24)\n\u000b'=0\n3[\u000en00] =\u0000\u0010\n\u00004 +\u000en2\n00+ 2p\n4\u0000\u000en2\n00\u0011\n(2 +j\u000en00j)\n4p\n\u0000(\u00004 +\u000en2\n00)(4 +\u000en2\n00+ 4j\u000en00j).\n(25)\nThe CI coe\u000ecients of the non-interacting electrons are\nshown in the leftmost column of Fig. 5, where '= 0. Ap-\nproaching the strictly-localized electron limit, i.e. from\nleft to right in Fig. 5, the gradient of the CI coe\u000ecients\nsharpens. This sharpening corresponds to the intra-\nsystem steepening of the \u000en00-to-\u000evmap introduced in\nsection IV and is inherited by the CI coe\u000ecients. Fur-\nthermore, the inter-system derivative discontinuity shows\nup in the CI coe\u000ecients for '=\u0019\n2and the CI functionalsbecome distributional points,\n\u000b'=\u0019\n2\n1[\u000en00] =8\n><\n>:1\n2;for\u000en00=\u00002\n1p\n2;for\u000en00= 0\n1\n2;for\u000en00= +2,\n\u000b'=\u0019\n2\n2[\u000en00] =8\n><\n>:1p\n2;for\u000en00=\u00002\n0; for\u000en00= 0\n\u00001p\n2;for\u000en00= +2,\n\u000b'=\u0019\n2\n3[\u000en00] =8\n><\n>:\u00001\n2;for\u000en00=\u00002\n1p\n2;for\u000en00= 0\n\u00001\n2;for\u000en00= +2,\nwhich are connected via straight lines due to the degen-\neracy of the ground-state.\nB. Exact Ground-State and Excited-State Energy\nFunctionals\nSince the CI coe\u000ecients \u000b'\nqof the wavefunction in-\nherit the intra-system steepening and the inter-system10\nFIG. 6. Exact energy functionals Fjj=h\tjj\u0015t(')^T+\u0015w(')^Wj\tjiof the ground-, the \frst- and second-excited state for\ndi\u000berent strengths of the electron localization '. First row: second excited-state energy F22as functional of the ground-state\ndensity. Second row: \frst excited-state energy F11as functional of the ground-state density. Third row: ground-state energy\nF00as functional of the ground-state density, i.e. the Hohenberg-Kohn functional. From the non-interacting limit (left) to the\nstrictly-localized limit (right), the gradient of all energy functionals steepens. In the highly localized limit, where '=\u0019\n2\u00001\n100,\nall energy functionals are continuous. In particular, the ground-state energy functional shows a convex behavior as can be seen\nin the detailed view of the intra-system steepening of highly-localized electrons and the inter-system derivative discontinuity of\nstrictly-localized electrons at the bottom of the \fgure. Note, that here the x-axis has been scaled by one order of magnitude.\nIn the strictly-localized limit, for all energy functionals only the three distributional points \u000en00=\u00062 and\u000en00= 0 exist.\nDue to the degeneracy of the eigenstates in the strictly localized limit which is shown in the lower panel of Fig. 4, these three\ndistributional points connect via straight lines indicated by a black-dashed line.\nderivative discontinuity, arbitrary ground-state expecta-\ntion values, de\fned in Eq. 18, also inherit the intra-\nsystem steepening and the inter-system derivative dis-\ncontinuity. Note, the excited-state CI coe\u000ecients also\nshow the same exact features, which are then inherited\nby excited-state functionals in the respective limit. As\nparticular examples for this inheritance, we illustrate in\nFig. 6 the intra-system steepening and the inter-system\nderivative discontinuity for the exact Hohenberg-Kohn\nfunctional ( j= 0) and the excited-state energy function-als (j= 1;2)\nF'\njj[\u000en00] =\n\t'\n2s;j\f\f\u0015t(')^T+\u0015w(')^W\f\f\t'\n2s;j\u000b\n, (26)\nfor the two-particle singlet states\f\f\t'\n2s;j\u000b\n=\f\f\t'\nj[\u000en00;N= 2;S2= 0;Sz= 0]\u000b\n. The third row\nof Fig. 6 shows the exact Hohenberg-Kohn functional\n(j= 0) discussed previously in literature38{41, the \frst\nand second row show the \frst and second excited-state\nenergy functional ( j= 1;2), respectively. The gradient\nof all three functionals Fjj[\u000en00] steepens approaching\nthe limit of strictly localized electrons, just as previously11\nobserved for the density-to-potential map in Sec. IV\nand the density-to-wavefunction map in Sec. V A.\nHowever, if 'di\u000bers in\fnitesimally from the strictly\nlocalized limit, all energy functionals are continuous.\nIn particular, the ground-state energy functional F00\nis convex. The di\u000berence between the highly localized\nand the strictly localized limit, is displayed in an inset\nat the bottom in Fig. 6, which contains a zoom of the\ncritical region of the ground- and \frst excited-state\nstate functional. Again, in the limit of strictly localized\nelectrons, the intra-system steepening transitions into\nthe inter-system derivative discontinuity. As already\ndiscussed for the density-to-wavefunction map, the\ndistributional points can be connected via straight lines\ndue to the degeneracy of the eigenstates in the strictly\nlocalized limit.\nC. Exact Excited- and Transition Density Functionals\nTo illustrate the fact that all observables inherit the\nintra-system steepening and the inter-system derivative\ndiscontinuity, we also show the excited- ( k=j= 1;2)\nand transition-state densities( k6=j= 0;1;2)\n\u000enkj[\u000en00] =h\tk[\u000en00;N]j^Oj\tj[\u000en00;N]i (27)\nas functionals of the ground-state density \u000en00. The\nexcited-state density functionals are shown in the second\nand third row of Fig. 7 respectively. For completeness,\nalso the trivial linear behavior of the ground-state den-\nsity as functional of the ground-state density is shown\nin the \frst row of the \fgure. From the non-interacting\n(left) to the strictly-localized limit (right), the gradient\nof the excited-state density functionals steepens up to\nthe strictly-localized limit where the excited-state den-\nsity functionals obey the straight-line condition due to\nthe degeneracy of the ground-state. To highlight the dif-\nference of the intra-system steepening and inter-system\nderivative discontinuity of the excited-state density func-\ntionals a detailed view of the critical region can be found\non the right-hand side of Fig. 7.\nTransition densities are an important ingredient for linear\nresponse calculations in time-dependent DFT (TDDFT).\nIn TDDFT, the transition densities are often approxi-\nmated by the ones computed from Kohn-Sham determi-\nnants. For our model system, we show the exact tran-\nsition densities as functionals of the ground-state den-\nsity. In contrast to the excited-state density functionals,\nthe transition density functionals are phase-dependent.\nFig. 8 shows the absolute value of the transition density\nas functional of the ground-state density. The \frst and\nsecond row of Fig. 8 show the absolute value of the tran-\nsition density from the \frst to the second and from the\nground- to the second excited state, respectively. Ap-\nproaching the strictly localized limit, both transition-\nstate densities show clearly the intra-system steepening.\nIn the strictly localized limit, there is no transition be-tween the eigenstates of the system and the transition-\nstate densities are zero, see '=\u0019\n2in panel one and two.\nD. Exact Correlation Entropy Functional\nAs \fnal example we illustrate the functional behavior\nof the correlation entropy. The correlation entropy , dis-\ncussed in detail in Ref.62measures the correlation and\nentanglement present in a many-body system. It can be\nunderstood as well as a measure of the Slater rank62,63as\ncan be seen if we compare the correlation entropy plot-\nted in Fig. 9 with the mixing of the eigenstates in lower\npanel and inset of Fig. 4 for the di\u000berent values of the\nparameter'. In the two-site model, where we have ac-\ncess to all eigenvectors and eigenvalues, we can compute\nthe correlation entropy of the system,\nS=1X\nj=1njlnnj, (28)\nwherenjare the eigenvalues of the reduced one-body\ndensity matrix\n\u001a00(j\u001b;j0\u001b0) =h\t0j^cy\nj\u001b^cj0\u001b0j\t0i. (29)\nThe correlation entropy is zero for pure states, and has its\nmaximum for maximally mixed states62{64. In Fig. 9 we\nsee that the correlation entropy increases with increasing\ncorrelation while the gradient of the correlation entropy\nfunctional obeys the intra-system steepening and tran-\nsitions into the inter-system derivative discontinuity for\n'=\u0019\n2. In the limit of non-interacting electrons, where\nthere is no correlation, the correlation entropy vanishes.\nThe maximum value of the correlation entropy is reached\nin the strictly localized limit for \u000en00= 0 where all three\neigenenergies are degenerate.\nVI. SUMMARY\nIn the present work we have illustrated how the\nintra-system steepening, an exact feature of the ground-\nstate density-to-potential map, develops gradually\nwith increasing decoupling between fragments of a\nsystem and transforms into the well-known inter-system\nderivative discontinuity for fully decoupled systems. As\na consequence of the Hohenberg-Kohn theorem, the\nwavefunction-to-density map inherits the exact features\nof the density-to-potential map. Furthermore, the exact\nfeatures of the density-to-potential map transmit to\nground- and excited-state observables and transition-\nmatrix elements. We illustrated the inheritance of\nthese features by showing the ground- and excited-state\nenergy, the excited- and transition-state densities and\nthe correlation entropy ground-state density functionals.\nAlthough both exact features are linked to the lo-\ncalization of the electrons, we carved out that the12\nFIG. 7. Density functionals for ground- and excited-singlet states. First panel: ground-state density as functional of the\nground-state density. Second panel: \frst excited-state density as functional of the ground-state density. Third panel: second\nexcited-state density as functional of the ground-state density. From the non-interacting limit (left) to the strictly-localized\nlimit (right), the gradient of all excited-state density functionals steepens. A detailed view of the intra-system steepening for\nhighly-localized electrons ( '=\u0019\n2\u00001\n100) and the inter-system derivative discontinuity is given on the right.\nintra-system steepening and the inter-system derivative\ndiscontinuity are conceptually di\u000berent features within\ndensity functional theory. The inter-system derivative\ndiscontinuity corresponds to the electron localization\ninfully decoupled systems with \fxed particle number.\nIn the decoupled limit, the Hohenberg-Kohn theorem\nis not applicable by construction, and the one-to-one\ndensity-to-potential map breaks down. The intersystem\nderivative discontinuity coincides with a real crossing\nof the eigenenergies of the system as function of the\nexternal potential. Ground-state density functionals in\nthe decoupled limit are straight lines between di\u000berent\nvalues for the particle number Ndue to mixture of\nstates in degenerate subspaces, F= (1\u0000!)FN+!FN+1\nwith the mixing parameter 0 \u0014!\u00141. The intra-system\nsteepening instead corresponds to the electron local-\nization in coupled fragments of a system, where one\nfragment can be seen as the particle reservoir (bath) of\nthe other, but the particle number of the total system\nis \fxed. The intra-system steepening coincides with\nan avoided crossing of the eigenenergies as function of\nthe external potential and sharpens when approaching\nthe real crossing. Ground-state density functionalsresult directly from the one-to-one correspondence\nof the Hohenberg-Kohn theorem, such as the convex\nground-state energy as function of the density di\u000berence\nbetween the fragments of the system.\nThe inter-system derivative discontinuity plays a\ncrucial role whenever the particle number of the total\nsystem changes which is the case for observables such as\nthe electron a\u000enity A=E[N]\u0000E[N+ 1], the ionization\nenergyI=E[N\u00001]\u0000E[N], the fundamental gap\nwhich is the di\u000berence of ionization energy and a\u000enity\nEgap=I\u0000A, and the chemical hardness \u0011=\u0010\n@2E\n@N2\u0011\nvof a system. The intra-system steepening is linked\nto processes where particles are transferred from one\nfragment to another within a system of \fxed particle\nnumber such as stretched molecules, charge-transfer\nprocesses and any problem involving highly-localized\nelectrons. Approximate functionals fail to describe such\nproblems not due to the lack of the inter-system deriva-\ntive discontinuity but due to the lack of the intra-system\nsteepening. Given the relevance of the above mentioned\nproblems it is crucial to develop improved density13\nFIG. 8. Transition matrix elements of the density operator between di\u000berent excited many-body states as functional of the\nground-state density \u000enjk=h\tjj\u000e^nj\tki. First row: Absolute value of the exact transition density from the \frst and second\nexcited-state as functional of the ground-state density \u000en12(\u000en00). Second row: Absolute value of the exact transition density\nfrom the ground-and the \frst excited-state as functional of the ground-state density \u000en01(\u000en00). From the non-interacting (left)\nto the strictly localized limit (right), approaching the strictly localized limit the gradient of both transition density functionals\nsteepens. In the strictly localized limit, the sites are disconnected. Therefore, there are no transitions between the three\ntwo-particle singlet states, and the transition densities are zero.\nFIG. 9. Correlation entropy as functional of the ground-state density indicating the correlation within the system. For non-\ninteracting electrons the correlation entropy is zero. From left to right, approaching the strictly localized limit, the correlation\nand the mixing of the eigenstates and hence the correlation entropy increases. Furthermore, the gradient of the functional\nobeys the intra-system steepening and the inter-system derivative discontinuity for '=\u0019\n2.\nfunctionals that capture this exact condition of the\nexact density-to-potential and density-to-wavefunction\nmaps. In the highly localized electron limit the exact\nxc-functional does not present a straight line behavior\nas inE(N) but rather a sharp but di\u000berentiable one\nas in E(\u000en), where\u000enrepresents the density di\u000berencebetween the fragments.\nOur work illustrates those fundamental concepts of\ndensity functional theory. To improve the accuracy\nof DFT observables, approximate functionals should\ncapture both, the inter-system derivative discontinuity14\nand the intra-system steepening respectively. Work\nabout how to generalize the present results from lattice\nHamiltonians to real continuous systems is currently in\nprogress.\nOur results also allow to get insight about spin DFT\nfunctionals as the magnetization of the N electron system\ncan be written in terms of the ground-state density (as\nall other observables we discussed in this paper). This\nis a way to solve the known problems of spin DFT65,66\n(however it would require going beyond present adiabatic\nfunctionals, work along those lines is in progress).\nACKNOWLEDGMENTS\nThe authors thank Professor Matthias Sche\u000fer for his\nsupport, Johannes Flick, Jessica Walkenhorst and Vik-\ntor Atalla for very useful discussions and comments, and\nNicola Kleppmann, Teresa Reinhard and Anne Hodgson\nfor comments on the manuscript. 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Our simulations cover a wide density-temper ature range of\n1−100 gcm−3and 104−109K. By combining results from PIMC and DFT-MD, we\nare able to compute pressures and internal energies from first-p rinciples at all tem-\nperatures and provide a coherent equation of state. We compare our first-principles\ncalculations with analytic equations of state, which tend to agree fo r temperatures\nabove 8×106K. Pair-correlation functions and the electronic density of states reveal\nan evolving plasma structure and ionization process that is driven by temperature\nand density. As we increase the density at constant temperature , we find that the\nionization fraction of the 1s state decreases while the other electr onic states move to-\nwards the continuum. Finally, the computed shock Hugoniot curves show an increase\nin compression as the first and second shells are ionized.\na)Electronic mail: kdriver@berkeley.edu; http://militzer.berkeley.edu/ ˜driver/\n1I. INTRODUCTION\nElemental oxygen is involved in a wide range of physics and chemistry t hroughout the\nuniverse, spanning fromambient biological processes to extreme g eological and astrophysical\nprocesses. Createdduringstellarnucleosynthesis, oxygenisthe thirdmostabundantelement\nin the universe and the most abundant element on Earth. In addition to its importance for\nlife-sustainingprocesses, itsthermodynamic, physical, andchemic alpropertiesareimportant\nto numerous fields of science. As such, oxygen has inspired a vast n umber of laboratory\nexperiments and theoretical studies, which have revealed an exot ic phase diagram with a\nnumber of interesting anomalies in its thermal, optical, magnetic, elec trical, and acoustic\nproperties due to its molecular and magnetic nature1.\nAt ambient conditions, oxygen exists as a diatomic molecular gas with e ach molecule hav-\ning two unpaired electrons, resulting in a paramagnetic state. X-ra y diffraction and optical\nexperiments reveal that oxygen condenses to a molecular solid with a rich phase diagram\nmade up of at least ten different structural phases1–6. Static compression experiments on the\nsolidhavebeenperformedupto1.3Mbar and650K1. First-principles simulations have been\nused to search for structural phases up to 100 Mbar6. The transition to the highest-pressure\nphase discovered so far occurs at 96 GPa, which also drives the solid to become metallic7–10.\nA superconducting phase has also been found at 0.6 K near 100 GPa11. In addition, the\nsolid phases exhibit a complex magnetic structure with various degre es of ordering due to a\nstrong exchange interaction between O 2molecules that becomes suppressed under pressure\nand acts in tandem with weak van der Waals forces holding the lattice t ogether1,12,13.\nWarm, dense, fluid states of oxygen have also been of great intere st due to the presence of\noxygen-rich compounds in inner layers of giant planets19–23, stellar interiors24,25, astrophys-\nical processes26–28, and detonation products29. Oxygen is produced via helium burning30in\nthe late stages of Sun-like star’s life as well as in more massive stars. The larger weight of\noxygen relative to hydrogen and helium drives its settlement toward s the deepest regions of\na star. An accurate equation of state (EOS) is needed to properly describe the behavior of\nthe core of the star as well as the timing of the different nuclear pro cesses that are highly\nsensitive to temperature30,31. Eventually, intermediate mass stars evolve into white dwarfs,\nwhich have most of their hydrogen and helium depleted, leaving a remn ant composed mostly\nof carbon and oxygen. The core density of a white dwarf16is likely higher than 105g/cm3.\n2102103104105106107108109\nPressure (GPa )104105106107108109T emperature (K)\n9 Gyr300 Myr50 MyrStar with 25 Msun\nSolar interior\nWhite dwarf interiors\nComputed hugoniot\nComputed isochore\nPIMC\nDFT-MD\nFIG. 1. Temperature-pressure conditions for the PIMC and DF T-MD calculations along six iso-\nchores corresponding to the densities of 2.48634, 3.63046, 7.26176, and 14.8632, 50.00, and 100.00\ngcm−3. Thedash-dotted lineshows theHugoniot curvefor an initia l density of ρ0= 0.6671 gcm−3.\nFor comparison, we also plotted the interior profile of the cu rrent-day Sun14as well as the profile\nof a 25 M ⊙star at the end of its helium burning time15. The green dashed lines show the interior\nprofile of a 0.6 M ⊙carbon-rich white dwarf at three different stages of its cooli ng process16–18\nThe cooling process of the white dwarf is very similar from one white dw arf to another and\nthe luminosity is used for cosmological chronology32,33. However, the accuracy of chronol-\nogy measurements depends on a proper description of the thermo dynamic behavior of both\ncarbon and oxygen34. Moreover, as the third most abundant element in the solar system35,\noxygen has a significant presence in planet interiors and can exist in a partially ionized\nstate in giant planets. Therefore, the electronic and thermodyna mic behavior of oxygen at\nhigh pressures and temperatures is important for obtaining the co rrect fluid and magnetic\nbehavior in planetary, stellar, and stellar remnant models36.\nShock-compressed fluid states of oxygen have been measured un der dynamic compression\nup to 1.9 Mbar (four-fold compression) and 7000 K, which revealed a metallic transition\nin the molecular fluid at 1.2 Mbar and 4500 K37. Density functional theory molecular\ndynamics (DFT-MD) simulations suggest that disorder in the fluid lowe rs the metallization\npressure to as low as 30 GPa with molecular dissociation above 80 GPa38. Measurements of\n3Hugoniotshavereached 140GPa39–41andindicatethat oxygenmolecules becomedissociated\nin a pressure range of 80-120 GPa at temperatures over several thousand Kelvin. Using\nclassical pair-potential simulations42–44, some general agreement is found with the measured\nHugoniots, however, a fully quantum-mechanical treatment is nee ded to accurately simulate\nthe electronic and structural behavior of the fluid.\nHistorically, a lack of development in first-principles methodology for the warm dense\nmatter regime has largely prevented highly accurate theoretical e xploration of fluid oxygen\natextremeconditions, and, hence, furtherimprovements inEOSa ndHugoniotcurves. DFT-\nMD has been used to explore the structural and electronic behavio r of the fluid state38,45up\nto temperatures of 16 ×103K and densities up to 4.5 gcm−3. Massacrier et al.46investigated\nthepropertiesofoxygenforadensity-temperature rangeof10−3−104gcm−3and105−106K,\nusinganaverageionmodel. Theyshowed, forinstance, thattheco mpletepressure-ionization\nof fluid oxygen cannot be expected until the system reaches a den sity of 1000 gcm−3.\nIn order to address the challenges of first-principles simulations fo r warm dense matter,\nwe have been developing the path integral Monte Carlo (PIMC) meth odology in recent\nyears for the study of heavy elements in warm, dense states47–50. Here, we apply our PIMC\nmethodology along with DFT-MD to extend the first principles explora tion of warm dense\nfluid oxygen to a much wider density-temperature range (1–100 gc m−3and 104–109K) than\nhas been previously explored by DFT-MD alone.\nIn Section II, we cover details of the PIMC and DFT-MD methodology specific to our\noxygen simulations. In Section III, we discuss the EOS constructe d from PIMC and DFT-\nMD and show that both methods agree for at least one of temperat ure in the range of\n2.5×105–1×106K. In section IV, we characterize the structure of the plasma and the ion-\nization process by examining pair-correlation functions of electron s and nuclei as a function\nof temperature and density. In section V, we discuss the electron ic density of states as a\nfunction of density and temperature to provide further insight int o the ionization process.\nIn section VI, we discuss predictions for the shock Hugoniot curve s. Finally, in section VII,\nwe summarize and conclude our results.\n4II. SIMULATION METHODS\nPIMC47,51is currently the state-of-the-art first-principles method for sim ulating mate-\nrials at temperatures in which properties are dominated by excited s tates. It is the only\nmethod able to accurately treat all the effects of bonding, ionizatio n, exchange-correlation,\nand quantum degeneracy that simultaneously occur in the warm den se matter regime52.\nPIMC is based on thermal density matrix formalism, which is efficiently c omputed with\nFeynman’s imaginary time path integrals. The density matrix is the nat ural operator to use\nfor computing high-temperature observables because it explicitly in cludes temperature in a\nmany-body formalism.\nThe PIMC method stochastically solves the full, finite-temperature quantum many-body\nproblem by treating electrons and nuclei equally as quantum paths t hat evolve in imaginary\ntime without invoking the Born-Oppenheimer approximation. For our PIMC simulations,\nthe Coulomb interaction is incorporated via pair density matrices der ived from the eigen-\nstates of the two-body Coulomb problem51,53appropriate for oxygen. Furthermore, in con-\ntrast to DFT-MD as described below, the efficiency of PIMC increase s with temperature as\nparticles behave more classical-like and fewer time slices are needed t o describe quantum\nmechanical many-body correlations, scaling inversely with tempera ture.\nPIMC uses a minimal number of controlled approximations, which beco me vanishingly\nsmall with increased temperature and by using appropriate conver gence of the time-step\nand system size. The only uncontrolled approximation is the employme nt of a fixed nodal\nsurface to avoid the fermion sign problem54. Current state-of-the art PIMC calculations\nemploy a free-particle nodal structure, which would perfectly des cribe a fully ionized system.\nHowever, we have shown PIMC employing free-particle nodes even p roduces reliable results\natsurprisinglowtemperaturesinpartiallyionizedhydrogen55, carbon48, water48,andneon50.\nAs a general rule, we find free-particle nodes aresufficient for sys tems comprised of partially-\nionized 2s states48.\nAsufficientlysmallPIMCtimestepisdeterminedbyconvergingtotale nergyasafunction\nof time step until the energy changes by less than 0.5%, which is show n in supplemental\nmaterial56Table SI. We use a time step of 1/256 Ha−1for temperatures below 4 ×106K\nand, for higher temperatures, we decrease the time step as 1 /T, as the efficiency of PIMC\nincreases linearly with T as path lengths decrease. The number of tim e slices we use in\n5−1.0−0.50.0\n2.48634 g/cm3\nPIMC\nDFT\nDebye\nChabrier-\nPotekhi \n−1.0−0.50.0\n3.63046 g/cm3\n−1.0−0.50.0\n7.26176 g/cm3\nTemperature (K)−1.0−0.50.0\n14.8632 g/cm3\nTemperature (K)−1.0−0.50.0\n50.0 g/cm3\n104105106107108\nTemperature (K)−0.7−0.4−0.1100.0 g/cm3(P-P0)/P0\nFIG. 2. Comparison of excess pressure relative to the ideal F ermi gas plotted as a function of\ntemperature for oxygen.\nthe path integral range from 323 at lowest temperature to 5 at th e highest temperature. In\norder to minimize finite size errors, the internal energy and pressu re is converged to better\nthan 0.4% when comparing 8- and 24-atom simple cubic simulation cells, w hich is shown in\nsupplemental material56Table SII. We therefore perform all PIMC calculations in 8-atom\ncells, as PIMC scales as N2, where N is the number of particles. A typical calculation uses\na bisection level51of 5 and achieves a statistical error in the energy and pressure th at is less\n6−16−802.48634 g/cm3\nPIMC\nDFT\nDebye\nChabrier-\nPote hin\n−16−803.63046 g/cm3\n−9−6−30\n7.26176 g/cm3\nTemperature (K))7)4)1 14.8632 g/cm3\nT emperature (K))3)2)10\n50.0 g/cm3\n104105106107108\nT emperature (K))2)10\n100.0 g/cm3(E-E0)/E0\nFIG. 3. Comparison of excess internal energies relative to t he ideal Fermi gas plotted as a function\nof temperature for oxygen.\nthan 0.1%.\nFor lower temperatures (T <1×106K), DFT-MD57is the most efficient state-of-the-\nart first-principles method. DFT formalism provides an exact mappin g of the many body\nproblem onto a single particle problem, but, in practice, employs an ap proximate exchange-\ncorrelation potential to describe many body electron physics. In t he WDM regime, where\n7temperatures are at or above the Fermi temperature, the exch ange-correlation functional is\nnot explicitly designed to accurately describe the electronic physics58. However, in previous\nPIMC and DFT-MD work on helium47carbon48, and water48, and neon50, DFT functionals\nare shown to be accurate even at high temperatures.\nDFT incorporates effects of finite electronic temperature into calc ulations by using a\nFermi-Dirac function to allow for thermal occupation of single-part icle electronic states59.\nAs temperature grows large, an increasing number of bands are re quired to account for\nthe increasing occupation of excited states in the continuum, which typically causes the\nefficiency of the algorithm to become intractable at temperatures b eyond 1×106K. Orbital-\nfree density functional methods aim to overcome such thermal ba nd efficiency limitations,\nbut several challenges remain to be solved60. In addition, pseudopotentials, which replace\nthe core electrons in each atom and improve efficiency, may break do wn at temperatures\nwhere core electrons undergo excitations.\nDepending on the density, we employ two different sets of DFT-MD sim ulations for\nour study of oxygen. At densities below 15 g cm−3, the simulations were performed with\nthe Vienna Ab initio Simulation Package (VASP)61using the projector augmented-wave\n(PAW) method62. The VASP DFT-MD uses a NVT ensemble regulated with a Nos´ e-Hoov er\nthermostat. Exchange-correlation effects are described using t he Perdew-Burke-Ernzerhof63\ngeneralized gradient approximation. Electronic wave functions are expanded in a plane-\nwave basis with a energy cut-off of at least 1000 eV in order to conve rge total energy. Size\nconvergence tests up to a 24-atom simulation cell at temperature s of 10,000 K and above\nindicate that total energies are converged to better than 0.1% in a 24-atomsimple cubic cell.\nWe find, at temperatures above 250,000 K, 8-atom supercell resu lts are sufficient since the\nkinetic energy far outweighs the interaction energy at such high te mperatures. The number\nof bands in each calculation is selected such that thermal occupatio n is converged to better\nthan 10−4, which requires up to 8,000 bands in a 24-atom cell at 1 ×106K. All simulations\nare performed at the Γ-point of the Brillouin zone, which is sufficient f or high temperature\nfluids, converging total energy to better than 0.01% relative to a c omparison with a grid of\nk-points.\nFor densities above 15 g cm−3, we had to construct a new pseudopotential in order to\nprevent the overlap of the PAW-spheres. We therefore used the ABINIT package64for which\nitispossibletobuildaspecificPAW-pseudopotential usingtheAtomPA Wplugin65. Webuilt\n8a hard all-electron PAW pseudopotential with a cut-off radius of 0.4 B ohr. We checked the\naccuracy of the pseudopotential by reproducing the results pro vided by the ELK software in\nthe linearized augmented plane wave (LAPW) framework66. With this pseudopotential we\nperformed DFT-MD with ABINIT for a 24-atom cell up to 100 g cm−3and 1×106K. The\nhardness of the pseudopotential required an plane-wave energy cut-off of at least 6800 eV.\nIII. EQUATION OF STATE RESULTS\nIn this section, we report our EOS results for six densities of 2.4863 4, 3.63046, 7.26176,\nand 14.8632, 50.00, and 100.00 gcm−3and for a temperature range of 104−109K. The\nsix isochores are shown in Figure 1 and are discussed in more detail in s ection VI. These\nconditions are relevant for the modeling of stars and white dwarfs a s can be seen in Figure 1.\nFigure2comparespressuresobtainedforoxygenfromPIMC,DFT -MD,andfromanalytic\nChabrier-Potekhin67andDebye-H¨ uckel68models. Pressures, P, areplottedrelativetoafully\nionized Fermi gas of electrons andions with pressure, P0, in order to compare only the excess\npressure contributions that result from particle interactions. In general PIMC and DFT-MD\npressures differ by at most 2%, and often much less for at least one temperature in the range\nof 2.5×105−1×106K. PIMC converges to the weakly interacting plasma limit along with\nthe Chabrier-Potekhin and Debye-H¨ uckel models.\nFigure 3 compares internal energies, E, plotted relative to the internal energy of a fully\nionized Fermi gas, E 0. PIMC and DFT-MD results for excess internal energy differ by\nat most 2%, and much less in most cases for at least one temperatur e in the range of\n2.5×105−1×106K. PIMC extends the energies to the weakly interacting plasma limit at\nhigh temperatures, in agreement with the Potekhin and Debye-H¨ u ckel models68.\nTogether, Figs. 2 and 3 show that the DFT-MD and PIMC methods fo rm a coherent\nequation of state over all temperatures ranging from the regime o f warm dense matter to\nthe weakly interacting plasma limit. The agreement between PIMC and DFT-MD indicates\nthat DFT exchange-correlation potential remains valid even at high temperatures and that\nthe PIMC free-particle nodal approximation is valid for a sufficient ion ization fraction of\nthe 2s state. The analytic Chabrier-Potekhin and Debye-H¨ uckel models agree with PIMC to\ntemperatures as low as 8 ×106K. The Debye-H¨ uckel model appears to have better agreement\nwith PIMC at low densities, while the Chabrier-Potekhin model agrees better with PIMC at\n9high densities. Neither analytic model includes bound states and, th erefore, cannot describe\nlow temperature conditions.\nTable VII provides the densities, temperatures, pressures, and energies used to construct\nour equation of state. The VASP DFT-MD energies have been shifte d by 74.9392 Ha/atom\nin order to bring the PAW-PBE pseudpotential energy in alignment wit h all-electron ener-\ngies that we report with PIMC computations. The shift was calculate d by performing an\nall electron atomic calculation with the OPIUM code69and a corresponding isolated-atom\ncalculation in VASP.\nComparison of the PIMC and DFT-MD pressures and internal energ ies in Table VII\nindicates that there is roughly a 2% discrepancy in their predicted va lues at temperatures of\n1×106K. Potential sources of this discrepancy include: (1) the use of fr ee particle nodes in\nPIMC; (2) the exchange-correlation functional in DFT; and (3) th e use of a pseudopotential\nin DFT. While it is difficult to determine the size of the nodal and exchang e-correlation\nerrors, comparison of our VASP calculations with all-electron, PAW A BINIT calculations\nat 1×106K indicates that roughly one third of the discrepancy is due to the us e of frozen\n1s core in the VASP DFT-MD pseudopotential, which leaves out effect s of core excitations.\nIV. PAIR-CORRELATION FUNCTIONS\nIn this section, we study pair-correlation functions70in order to understand the evolution\nof the fluid structure and ionization in oxygen plasmas as a function o f temperature and\ndensity.\nFigure 4 shows the nuclear pair-correlation functions, g(r), computed with PIMC over\na temperature range of 2 ×106−1.034×1012K and a density range of 2 .486−100.0\ngcm−3. Atoms are kept farthest apart at low temperatures due to a com bination of Pauli\nexclusion among bound electrons and Coulomb repulsion. As tempera ture increases, kinetic\nenergy of the nuclei increases, making it more likely to find atoms at c lose range, and, in\naddition, the atoms become increasingly ionized, which gradually minimiz es the effects of\nPauli repulsion. As density increases, the likelihood of finding two nuc lei at close range is\nsignificantly increased. For the highest density and lowest tempera ture, the peak in the\npair-correlation function reaches a value of 1.2, indicating a modera tely structured fluid.\nFigure 5 compares the nuclear pair-correlation functions of PIMC a nd DFT at a temper-\n10r (Å)0.00.51.02.48634 g/cm3\nr (Å)0.00.51.03.63046 g/cm3\nr (Å)0.00.51.07.26176 g/cm3\n0.00.51.014.8632 g/cm3\nr (Å)0.00.51.050.00 g/cm3\n0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8\nr (Å)0.00.51.0100.00 g/cm3\n2×106 K\n16×106 K\n100×106 K\n1034×106 KgO−O(r)\nFIG. 4. Nuclear pair-correlation functionsforoxygen from PIMCover awiderangeof temperatures\nand densities.\nature of 1 ×106K in an 8-atom cell at a density of 14.8632 gcm−3. The overlapping g(r)\ncurves verify that PIMC and DFT predict consistent structural p roperties.\nFigure 6 shows nucleus-electron pair correlation functions. Electr ons are most highly\ncorrelated with the nuclei at low temperature and high density, refl ecting a lower ionization\nfraction. As temperature increases, electrons are thermally exc ited and gradually become\n110.0 0.5 1.0 1.5\nr (Å)0.00.51.0gO−O(r)\nPIMC\nDFT\nFIG. 5. Comparison of PIMC and DFT nuclear pair-correlation functions for oxygen at a temper-\nature of 1 ×106K and a density of 14.8632 gcm−3.\nunbound, decreasing their correlation with the nuclei. As the densit y is increased, the\nelectrons are more likely to reside near the nuclei, indicating that the ionization of the 1s\nstate is suppressed with increasing density.\nFigure 7 shows the integral of the nucleus-electron pair correlatio n function, N(r), which\nrepresents theaveragenumber ofelectronswithinasphereofra diusraroundagivennucleus,\nN(r) =/angbracketleftBigg\n1\nNI/summationdisplay\ne,Iθ(r−|/vector re−/vector rI|)/angbracketrightBigg\n, (1)\nwhere the sum includes all electron-ion pairs and thetarepresents the Heaviside function.\nAt the lowest temperature, 1 ×106K, we find that the 1s core state is always fully occupied,\nas it agrees closely with the result of an isolated 1s state. As temper ature increases, the\natoms are gradually ionized and electrons become unbound, causing N(r) to decrease. As\ndensity increases, an increasingly higher temperature is required t o fully ionize the atoms,\nconfirming that the 1s ionization fraction decreases with density as seen in Fig. 6. The 1s\nstate is thus not affected by pressure ionization in the density rang e of consideration. As we\nwill explain the density of states section, the ionization of the 1s sta te is suppressed because\nwith increasing density, the Fermi energy increases more rapidly th an energy of the 1s state.\n12r (Å)1011021032.48634 g/cm3\nr (Å)1011021033.63046 g/cm3\nr (Å)1011021037.26176 g/cm3\nr (Å)10110210314.8632 g/cm3\nr (Å)10210350.0 g/cm3\n0.00 0.05 0.10 0.15 0.20 0.25 0.30\nr (Å)102103100.0 g/cm3 1×106 K\n2×106 K\n4×106 K\n8×106 K\n16×106 Kρ gO−e(r)\nFIG. 6. The nucleus-electron pair-correlation functions f or oxygen computed with PIMC.\nFigure8 shows electron-electron pair correlationsforelectrons h aving oppositespins. The\nfunction is multiplied by the particle density, ρ, in units of g cm3, so that the integral under\nthe curves is proportional to the number of electrons. The electr ons are most highly cor-\nrelated for low temperatures, which reflects that multiple electron s occupy bound states at\none nucleus. As temperature increases, electrons are thermally e xcited, decreasing the cor-\nrelation among each other. Correlation at short distances increas es with density, consistent\nwith a lower ionization fraction.\n13r (Å)0123\n2.48634 g/cm3\n01214.8632 g/cm3\nr (Å)01250.0 g/cm3\n0.0 0.1 0.2 0.3 0.4\nr (Å)012100.0 g/cm3\n1s core state\n1×106 K\n2×106 K\n4×106 K\n8×106 KN(r)\nFIG. 7. Number of electrons contained in a sphere of radius, r, around an oxygen nucleus. PIMC\ndata at four temperatures is compared with the analytic 1s co re state.\nFigure 9 shows electron-electron pair correlations for electrons w ith parallel spins. The\npositive correlation at at ∼2.5˚A forT≤2×106K reflects that different electrons with\nparallel spins are bound to a single nucleus. For short separations, Pauli exclusion takes\nover and the functions decay to zero. The ordering of the g(r) cu rves changes with respect\nto temperature as density increases due to a competition between Coulomb and kinetic\neffects, coupled with the effects of ionization. When the density is 50 and 100 gcm−3,\npressure ionization causes the correlation to approach that of an ideal fluid, and increasing\ntemperature further only strengthens kinetic effects. We interp ret this change as pressure\nionizationofthesecondandthirdelectronshells. Astemperaturein creases, electronsbecome\nless bound, which also causes the correlation to become more like an id eal fluid.\n14r (Å)100101102 2.48634 g/cm3\nr (Å)101102 3.63046 g/cm3\nr (Å)101102 7.26176 g/cm3\nr (Å)10110214.8632 g/cm3\nr (Å)10250.0 g/cm3\n0.0 0.1 0.2 0.3\nr (Å)102100.0 g/cm31×106 K\n2×106 K\n4×106 K\n8×106 K\n16×106 Kρ ge↑−e↓(r)\nFIG. 8. The electron-electron pair-correlation functions (multiplied by ρ) for electrons with oppo-\nsite spins computed with PIMC.\nV. ELECTRONIC DENSITY OF STATES\nIn this section, we report DFT-MD results for the electronic densit y of states (DOS) of\nfluid oxygen as a function of temperature and density in order to ga in further insight into\nthe temperature- and pressure-ionization.\nIn order to closely examine the physics of pressure-ionization of th e 1s and higher states,\n15r (Å)012\n2.48634 g/cm3\nr (Å)012\n3.63046 g/cm3\nr (Å)012\n7.26176 g/cm3\nr (Å)0.00.51.0\n14.8632 g/cm3\nr (Å)0.00.51.0\n50.0 g/cm3\n0.0 0.1 0.2 0.3 0.4 0.5\nr (Å)0.00.51.0\n100.0 g/cm31×106 K\n2×106 K\n4×106 K16×106 K\n100×106 Kge↑−e↑(r)\nFIG. 9. The electron-electron pair-correlation functions for electrons with parallel spins computed\nwith PIMC.\nwe computed DOS curves using the all-electron, PAW potential we cr eated for use with the\nABINIT code. Figure 10 shows examples of the DOS for oxygen at de nsities between 2.49\nand 100 gcm−3at a fixed temperature of 100,000 K. For comparison, we show the r esult\nfor an isolated oxygen atom. Since we used the all-electron pseudo- potential we can see the\nbands related to the 1s or K shell. For the isolated atom, we also clear ly see the 2s or L Ias\n16well as the L IIand LIIIstates. The locations of the K and L Ishells for the isolated atom are\nconsistent with the binding energies of 19.97 and 1.53 Ha respectively that can be found in\nthe literature71.\nAs density increases, the L sub-shells are shifted towards higher e nergy, merging together\nas they shift into the continuum. This effect is referred to as the pr essure ionization of\noxygen, also described by Massacrier et al.46. As the density increases, the K shell is also\nshifted to higher energies and broadens significantly. Nevertheles s, the K shell remains a\nwell defined state even at 100 gcm−3. The Fermi energy is also shifted towards higher\nenergy values as the density increases. We observe that the Ferm i energy shifts more than\nthe K-shell energy, and, hence, the energy difference between t he 1s states and unoccupied\nstates increases with the density. Therefore, it is more difficult to t emperature-ionize the K\nshell at higher density and no pressure-ionization occurs for the 1 s state. This is consistent\nwith the observations we made for the electron-nuclei pair distribu tion function in Fig. 6.\nFigure 11 shows the temperature dependence of the DOS at a fixed density of 7.26176\ngcm−3. Results were obtained from VASP by averaging over at least 10 unc orrelated snap-\nshots chosen from a DFT-MD trajectory. Smooth curves were ob tained by using a 4 ×4×4\nk-point grid and applying a Gaussian smearing of 2 eV. The eigenvalues of each snapshot\nwere shifted so that the Fermi energies align at zero, and the integ ral of the DOS is normal-\nized to 1. The DOS curves show a large peak representing the atomic -like 2s and 2p states,\nfollowed by a dip in states, which is then followed by a continuous spect rum of conducting\nstates. The Fermi energy plays the role of the chemical potential in the Fermi-Dirac distri-\nbution, which shifts towards more negative values as the temperat ure is increased. Because\nwe subtract the Fermi energy from the eigenvalues, the peak shif ts to higher energies with\nincreasing temperature. The fact that the peaks are embedded in to a dense, continuous\nspectrum of eigenvalues indicates that they are conducting state s.\nVI. SHOCK COMPRESSION\nDynamic shock compression experiments are widely used for measur ing equation of state\nand other physical properties of hot, dense fluids. Commonly, sho ck experiments determine\nthe Hugoniot, which is the locus of final states that can be obtained from different shock\nvelocities. A few Hugoniot measurements have been made for oxyge n in an effort to under-\n17−20−15−10 −5 0510\nEnergy (Ha)0510152025Den ity of  tate  (Ha−1)\nK LILII,LIII\n18.7 Ha\n18.9 Ha\n22.1 Hai olated\n2.49 g/cm3\n14.86 g/cm3\n100 g/cm3K LILII,LIII\n18.7 Ha\n18.9 Ha\n22.1 Hai olated\n2.49 g/cm3\n14.86 g/cm3\n100 g/cm3\nFIG. 10. Electronic density of states of dense, fluid oxygen u sing an all-electron, PAW pseudo-\npotential. The solid lines represent all available states f or the isolated atom as well as three other\ndensities at a temperature of 1 ×105K. The curves are normalized such that the occupied DOS\nintegrates to 8. The K, L I, LIIand LIIIidentify the electronic shells and sub-shells for the isola ted\natoms. The open circle on each curve stands for the DOS at the F ermi energy level. The arrows\nshow the energy difference between the K-shell and the Fermi en ergy for the different densities.\nstand its metallic transition and determine its role in astrophysical pr ocesses39–41. Density\nfunctional theory has been validated by experiments as an accura te tool for predicting the\nshock compression of different materials45,72.\nIn the course of a shock wave experiment, a material whose initial s tate is characterized\nby an internal energy, pressure, and volume, ( E0,P0,V0), which changes to a final state\ndenoted by ( E,P,V) while conserving mass, momentum, and energy. This leads to the\nRankine-Hugoniot relation73,\nH= (E−E0)+1\n2(P+P0)(V−V0) = 0. (2)\nHere, we compute the Hugoniot for oxygen from the first-principle s EOS data we showed\nin Table VII. The pressure and internal energy data points were int erpolated with bi-cubic\nspline functions in ρ−Tspace. For the initial state of the principal Hugoniot curve, we\ncomputed the energy of an oxygen molecule at P 0= 0, E 0=−150.247327 Ha/O 2, and chose\nV0= 318.612 ˚A3. We chose a density of 0.6671 gcm−3for solid oxygen in the cubic, γphase.\n18−2 −1 0 1 2 3\nEnergy (Ha)0.00.51.0Density of States (Ha−1)7.26176 g/cm3 1.0×105 K\n2.5×105 K\n5.0×105 K\nFIG. 11. Total electronic DOS of dense, fluid oxygen at a fixed d ensity of 7.26176 gcm−3for three\ntemperatures (1 ×105, 2.5×105and 5×105K). Each DOS curve has had the relevant Fermi energy\nfor each temperature subtracted from it.\n101102\nDensity (g cm3)102103104105106107108109Pressure (GPa)1251025\nPIMC\nDFT\nPrincipal Hugoniot\nSecondary Hugoniot\nTertiary Hugoniot\nFIG. 12. Shock Hugoniot curves for different initial densitie s. The label on the curve specifies the\nratio of the initial density to that of solid oxygen at 0K, 0.6 671 gcm−3. Secondary and tertiary\nHugoniot curves are also plotted.\n193.0 3.5 4.0 4.5 5.0\nShock compression ratio ρ/ρ0105106107108Temperature (K)\n1251025Ionization energy of 1s core state\n1\n10of 1s\nionization\nenergy\nfirst\nionization\nenergy\nof atomnon-relativistic\nhigh T limitChabrier-Potekhin model\nwith relativistic effects\nFIG. 13. Hugoniot curves for different pre-compression densi ty ratios.\nThe resulting Hugoniot curve has been plotted in T-PandP-ρspaces in Figs. 1 and 12,\nrespectively.\nSamples in shock wave experiments may be pre-compressed inside of a diamond anvil\ncell in order to reach much higher final densities than possible with a s ample at ambient\nconditions. This technique allows shock wave experiments to probe d ensity-temperature\nconsistent with planetary and stellar interiors74. Therefore, we repeat our Hugoniot calcu-\nlation starting with initial densities ranging from a 1 to a 25-fold increa se of the ambient\ndensity. Figure 12 shows the resulting family of Hugoniot curves. Wh ile starting from the\nambient density leads to a maximum shock density of 3.5 gcm−3, a 25-fold pre-compression\nyields a much higher maximum shock density of 71 gcm−3, as expected. However, such\nextreme densities can be reached more easily with triple shock exper iments as our example\nin Fig. 12 illustrates. We used the first compression maximum on the pr incipal Hugoniot\ncurve (ρ=3.182 gcm−3,P=2535 GPa, T= 358,600 K) as the initial state of the secondary\nHugoniot curve. The compression maximum on this curve ( ρ=14.25 gcm−3,P=282000\nGPa,T= 4,819,000 K) served as initial state for the tertiary Hugoniot curv e.\nFigure 13 shows the temperature dependence of the precompres sion density ratio for the\nfive representative Hugoniot curves in Figure 12. In the high-temp erature limit, all curves\nconverge to a compression ratio of 4, which is the value of a nonrelat ivistic ideal gas. We\n20also include of the Hugoniot curve computed with the relativistic, fully -ionized Chabrier-\nPotekhin model, which shows the relativistic correction in the high-te mperature limit. In\ngeneral, theshockcompression isdeterminedbytheexcitationofin ternaldegreesoffreedom,\nwhich increases the compression, and interaction effects, which de crease the compression75.\nConsistent with our results for hydrogen, helium47, and neon50we find that an increase in\nthe initial density leads to a slight reduction in the shock compression (Figure 13) because\nparticles interact more strongly at higher density.\nThe shock-compression ratio also exhibits two maxima as a function o f temperature,\nwhich can be attributed to the ionization of electrons in the first and second shell. On\nthe principal Hugoniot curve, the first maximum of ρ/ρ0=4.77 occurs at temperature of\n3.59×105K (30.94 eV), which is above the first ionization energy of the oxygen atom, 13.61\neV, but less than the second ionization energy, 35.12 eV. A second c ompression maximum of\nρ/ρ0=5.10 is found for a temperature of 2 .87×106K (247.32 eV), which can be attributed\nto the ionization of the 1s core states of the oxygen ions. The 1s ion ization energy is\n871.41 eV. This is consistent with the ionization process we observe in Figure 7, where\ncharge density around the nuclei is reduced over the range of 2 −8×106K. Since DFT-\nMD simulations, which use pseudopotentials to replace core electron s, cannot access physics\nabout core ionization, PIMC is a necessary tool to determine the ma ximum compression\nalong the principle Hugoniot curve.\nVII. CONCLUSIONS\nIn this work, we have combined PIMC with DFT-MD to construct a coh erent EOS for\noxygen over wide range of densities and temperatures that include s warm dense matter and\nplasmas in stars and stellar remnants. The two methods validate eac h other in temperature\nrangeof2.5 ×105–1×106K,wherebothyieldconsistent results. Wecomparedourequationo f\nstate at high temperature with the analytic models of Chabrier-Pot ekhin and Debye-H¨ uckel.\nThe deviations that we identified underline the importance for new me thods like PIMC to\nbe developed for the study of warm dense matter. Nuclear and elec tronic pair-correlations\nreveal a temperature- and pressure-driven ionization process, where temperature-ionization\nof the 1s state is suppressed while other states are efficiently ionize d as density increases\nup to 100 gcm−3. Changes in the density of states confirms the temperature- and pressure-\n21ionization behavior observed in the pair-correlation data. Lastly, w e find the ionization\nimprints a signature on the shock Hugoniot curves and that PIMC sim ulations are necessary\nto determine the state of the highest shock compression. Our and Hugoniot and equation\nof state will help to build more accurate models for stars and stellar r emnants.\nACKNOWLEDGMENTS\nWe are grateful to Alexander Potekhin for helpful discussions abo ut the fully ionized\nEOS. We are also grateful to Cyril Georgy for sharing his data and k nowledge of massive\nstar evolution. We are also grateful to Jan Vorberger for discuss ions on continuum lowering.\nThis research is supported by the U. S. Department of Energy, gr ant DE-SC0010517. Com-\nputational support was provided by NERSC, NASA, and the Janus s upercomputer, which\nis supported by the National Science Foundation (Grant No. CNS-0 821794), the University\nof Colorado, and the National Center for Atmospheric Research.\nREFERENCES\n1Y. A. Freiman and H. J. Jodl, Phys. Rep. 401, 1 (2004).\n2M. Santoro, E. 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The numbers in parentheses indicate the statistical uncertainties of the\nDFT-MD and PIMC simulations.\nρ(gcm−3) T (K) P (GPa) E (Ha/atom)\n2.48634a1034730000 12031695(879) 44227(3)\n2.48634a99497670 1155684(608) 4242(2)\n2.48634a16167700 185881(73) 674.57(29)\n2.48634a8083850 91166(21) 323.90(9)\n2.48634a4041920 43037(12) 138.71(6)\n2.48634a2020960 17999(15) 16.06(7)\n2.48634a998004 7336(9) −41.43(4)\n2.48634b1000000 7339(6) −42.41(2)\n2.48634a748503 5118(11) −50.66(4)\n2.48634b750000 5119(5) −51.84(18)\n2.48634a500000 3044(11) −59.30(4)\n2.48634b500000 3049(5) −60.58(3)\n2.48634a250000 1189(12) −66.94(5)\n2.48634b250000 1183(3) −69.293(3)\n2.48634b100000 341(1) −73.635(1)\n2.48634b50000 161(1) −74.571(1)\n2.48634b30000 97(1) −74.811(1)\n2.48634b10000 38(1) −75.015(1)\n26TABLE I. ( Continued. )\nρ(gcm−3) T (K) P (GPa) E (Ha/atom)\n3.63046a1034730000 17566926(1904) 44223(5)\n3.63046a99497670 1685108(750) 4235(2)\n3.63046a16167700 269993(107) 669.34(28)\n3.63046a8083850 132427(35) 320.24(11)\n3.63046a4041920 61955(18) 132.56(6)\n3.63046a2020960 25689(28) 10.67(8)\n3.63046a998004 10569(14) −42.93(4)\n3.63046b1000000 10507(14) −44.13(2)\n3.63046a748503 7433(14) −51.81(4)\n3.63046b750000 7443(8) −52.79(5)\n3.63046a500000 4414(15) −60.15(4)\n3.63046b500000 4483(5) −61.412(6)\n3.63046b250000 1831(3) −69.658(2)\n3.63046b100000 605(2) −73.686(2)\n3.63046b50000 305(1)1 −74.565(1)\n3.63046b30000 202(2) −74.797(1)\n3.63046b10000 104(1) −74.992(1)\n27TABLE I. ( Continued. )\nρ(gcm−3) T (K) P (GPa) E (Ha/atom)\n7.26176a1034730000 35142831(2985) 44227(4)\n7.26176a99497670 3374099(1777) 4237(2)\n7.26176a16167700 538734(172) 664.43(26)\n7.26176a8083850 261808(75) 311.36(11)\n7.26176a4041920 120041(34) 119.03(5)\n7.26176a2020960 49637(51) 1.74(7)\n7.26176a998004 20964(31) −45.53(4)\n7.26176b1000000 21301(20) −46.16(4)\n7.26176a748503 15122(42) −53.17(5)\n7.26176b750000 15236(21) −54.51(2)\n7.26176a500000 9262(24) −61.27(3)\n7.26176b500000 9424(10) −62.652(7)\n7.26176a250000 4405(44) −67.78(5)\n7.26176b250000 4268(5) −70.098(2)\n7.26176b100000 1831(3) −73.613(2)\n7.26176b50000 1210(2) −74.382(2)\n7.26176b30000 986(5) −74.606(2)\n7.26176b10000 749(1) −74.813(1)\n28TABLE I. ( Continued. )\nρ(gcm−3) T (K) P (GPa) E (Ha/atom)\n14.8632a1034730000 71917073(5787) 44217(4)\n14.8632a99497670 6899765(3226) 4230(2)\n14.8632a16167700 1096035(364) 655.35(24)\n14.8632a8083850 527445(141) 299.20(10)\n14.8632a4041920 237350(67) 103.41(5)\n14.8632a2020960 99599(98) −5.97(6)\n14.8632a998004 44297(52) −47.32(3)\n14.8632b1000000 45274(64) −47.95(4)\n14.8632a748503 32595(59) −54.80(3)\n14.8632b750000 33293(69) −55.76(4)\n14.8632a500000 21447(56) −61.86(3)\n14.8632b500000 21945(35) −63.21(1)\n14.8632b250000 11803(11) −69.884(4)\n14.8632b100000 6975(7) −72.907(3)\n14.8632b50000 5705(6) −73.590(2)\n14.8632b30000 5239(4) −73.815(1)\n14.8632b10000 4626(8) −74.057(1)\n29TABLE I. ( Continued. )\nρ(gcm−3) T (K) P (GPa) E (Ha/atom)\n50.0000a1034730000 241912168(8061) 44208(1)\n50.0000a99497670 23165568(7204) 4215(1)\n50.0000a16167700 3638714(751) 633.85(14)\n50.0000a8083850 1721016(318) 272.08(6)\n50.0000a4041920 768044(164) 78.29(3)\n50.0000a2020960 351315(214) −13.11(4)\n50.0000a998004 185345(210) −46.12(4)\n50.0000c1000000 187281(611) −47.36(11)\n50.0000c500000 118441(752) −60.27(11)\n50.0000c250000 91835(1078) −65.16(15)\n50.0000c100000 77796(541) −67.49(7)\n50.0000c50000 75320(609) −67.90(8)\n30TABLE I. ( Continued. )\nρ(gcm−3) T (K) P (GPa) E (Ha/atom)\n100.000a1034730000 483702750(18188) 44193(2)\n100.000a99497670 46258880(13163) 4201(1)\n100.000a16167700 7213882(1458) 617.35(13)\n100.000a8083850 3396956(706) 254.73(7)\n100.000a4041920 1553594(378) 68.31(4)\n100.000a2020960 793543(497) −10.07(5)\n100.000a998004 490625(1050) −40.28(10)\n100.000c1000000 490505(1367) −41.78(12)\n100.000c500000 369913(2987) −52.88(24)\n100.000c250000 326893(1556) −56.75(12)\n100.000c100000 302710(1091) −58.79(8)\n100.000c50000 298808(1064) −59.13(8)\naPIMC\nbVASP-MD\ncABINIT-MD with a small-core, PAW pseudopotentials\n31arXiv:1601.05782v1  [physics.plasm-ph]  21 Jan 2016First-Principles Equation of State and Electronic Propert ies of Warm Dense\nOxygen\nK. P. Driver,1F. Soubiran,1Shuai Zhang,1and B. Militzer1,2\n1)Department of Earth and Planetary Science, University of Ca lifornia, Berkeley, California 94720,\nUSAa)\n2)Department of Astronomy, University of California, Berkel ey, California 94720,\nUSA\n(Dated: 29 June 2021)\nI. CONVERGENCE TESTS\nIn this section, we provide raw data from our PIMC\nand DFT-MD time-step and finite-size convergence cal-\nculations. Table SI shows the results of static path inte-\ngral Monte Carlo (PIMC) calculations for a 8-atom as a\nfunction of time-step for a fixed density. For a time-step\nof 0.00390625 Ha−1, which we used in our production\ncalculations, the results are well converged. The pres-\nsure has 0.3% error and internal energy has 0.3% error\nrelative to the smallest time-step.\nTable SII shows the comparison of pressures and in-ternal energies for a 24-atom and 8-atom simulation cell\nas a function of temperature at a fixed density. Results\nare shown for both PIMC and density functional theory\nmolecular dynamics (DFT-MD). The absolute difference\nbetween the 8-atom and 24-atom pressures and internal\nenergies is only a fraction of a per cent of the total val-\nues, and often within the statistical error. As expected in\nDFT, the agreementbetween 8- and 24-atomresults gen-\nerally improves with temperature. Above 1 ×105K, the\n8-atom cell size is sufficient as the gamma-only k-point\napproximation becomes irrelevant.\na)Electronic mail: kdriver@berkeley.edu;http://militzer.berkeley.edu/˜driver/2\nTABLE SI. Convergence of oxygen energy and pressure with res pect to PIMC time-step for static calculation of an 8-atom ce ll\nat a fixed density and temperature.\nρ(gcm−3) T(K) Time-step (Ha−1) P (GPa) E (Ha/atom)\n7.26176 1010479 0.015625 16700(45) -51.30(4)\n7.26176 1010479 0.0078125 17030(20) -50.60(2)\n7.26176 1010479 0.00390625 17200(30) -50.17(3)\n7.26176 1010479 0.00195312 17260(45) -50.00(6)\nTABLE SII. Comparison of oxygen pressures and internal ener gies computed for 24- and 8-atom simulations cells as a funct ion\nof temperature at a fixed density and their relative absolute (ABS) errors. The numbers in parentheses indicate the one-s igma\nstatistical uncertainties of the DFT-MD and PIMC simulatio ns.\nρ(gcm−3) T (K) P (GPa) P (GPa) ∆ P (GPa) E (Ha/atom) E(Ha/atom) ∆ E (Ha/a tom)\n24-atom cell 8-atom cell ABS error 24-atom cell 8-atom cell A BS error\n7.26176a1034730000 35139936(3207) 35142831(2985) 2895(4381) 442 26(4) 44227(4) 1(6)\n7.26176a16167700 537967(253) 538734(172) 767(305) 664.3(3) 664.3 (2) 0.0(4)\n7.26176a8083850 262087(87) 261808(75) 279(115) 312.2(1) 311.4(1) 0.79(3)\n7.26176a2020960 49881(56) 49637(51) 245(78) 2.3(1) 1.74(7) 0.52(1 )\n7.26176a998004 21158(46) 20964(31) 195(55) -44.95(5) -45.53(4) 0. 59(7)\n7.26176b1000000 21387(48) 21301(20) 85(52) -46.11(6) -46.16(4) 0. 05(7)\n7.26176a750000 15033(54) 15122(42) 88(69) -53.19(7) -53.17(5) 0.0 3(8)\n7.26176b750000 15272(29) 15236(21) 37(36) -54.49(3) -54.51(2) 0.0 1(3)\n7.26176b500000 9433(14) 9424(10) 9(17) -62.65(1) -62.652(7) 0.00( 1)\n7.26176b250000 4292(5) 4268(5) 24(7) -70.089(3) -70.098(2) 0.009( 4)\n7.26176b100000 1831(3) 1765(6) 66(7) -73.613(2) -73.663(4) 0.050( 4)\n7.26176b50000 1210(2) 1107(4) 103(4) -74.382(1) -74.448(2) 0.066( 2)\naPIMC\nbDFT-MD"    },    {        "title": "1603.06565v2.Systematic_construction_of_density_functionals_based_on_matrix_product_state_computations.pdf",        "content": "arXiv:1603.06565v2  [cond-mat.str-el]  25 Aug 2016Systematic construction of density functionals based\non matrix product state computations\nMichael Lubasch1‡, Johanna I Fuks2, Heiko Appel3,4, Angel\nRubio3,4, J Ignacio Cirac1and Mari-Carmen Ba˜ nuls1\n1Max-Planck-Institut f¨ ur Quantenoptik, Hans-Kopfermann-St raße 1, 85748\nGarching, Germany\n2Department of Physics and Astronomy, Hunter College and the Gra duate Center of\nthe City University of New York, 695 Park Avenue, New York, New Yo rk 10065, USA\n3Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradaywe g 4-6, 14195\nBerlin-Dahlem, Germany\n4Max-Planck-Institut f¨ ur Struktur und Dynamik der Materie, Lur uper Chaussee\n149, 22761 Hamburg, Germany\nE-mail:michael.lubasch@mpq.mpg.de\nAbstract. We propose a systematic procedure for the approximation of dens ity\nfunctionalsindensityfunctionaltheorythatconsistsoftwopart s. First,fortheefficient\napproximation of a general density functional, we introduce an effic ient ansatz whose\nnon-locality can be increased systematically. Second, we present a fitting strategy\nthat is based on systematically increasing a reasonably chosen set o f training densities.\nWe investigate our procedure in the context of strongly correlate d fermions on a one-\ndimensional lattice in which we compute accurate training densities wit h the help of\nmatrix product states. Focusing on the exchange-correlation en ergy, we demonstrate\nhow an efficient approximation can be found that includes and system atically improves\nbeyond the local density approximation. Importantly, this system atic improvement is\nshown for target densities that are quite different from the trainin g densities.\nPACS numbers: 31.15.E-, 31.15.X-, 71.15.Mb, 05.10.Cc\n‡Present address: Department of Physics, University of Oxford, Parks Road, Oxford OX1 3PU, UK.Systematic construction of density functionals based on MP S computations 2\n1. Introduction\nThe formulation of quantum mechanics in terms of density functiona ls instead of wave\nfunctions, following the ground-breaking works of Hohenberg, Ko hn, and Sham [1,\n2], made numerical simulations of quantum mechanical systems rang ing from the\nmicroscopic to the macroscopic world feasible [3, 4, 5]. The usefulnes s of density\nfunctional theory (DFT) is certified by the number of works based on the original\npublications [1, 2] and on later improvements of the exchange-corr elation (xc) energy\ndensity functional Exc[6, 7, 8, 9, 10, 11, 12].\nDFT in its most widely used form, namely Kohn-Sham (KS) DFT [2], requ ires\nthe xc density functional in order to be able to compute ground sta te energies and\ndensities. ByvirtueoftheHohenberg-Kohntheorem [1], allground stateobservables are\nfunctionals of the ground state density n, and soExc=Exc[n]. The ground state energy\nE=E[n] of a system can be decomposed into a kinetic, an interaction and a p otential\npart. By means of a fictitious non-interacting system, namely the K S system, the non-\ninteracting part of the kinetic energy, Ts=Ts[n], can be obtained efficiently, which\nrepresents a large contribution to the full interacting kinetic ener gyT. Further, part\nof the interaction energy is accounted for by the Hartree energy EH[n]. The potential\npartEV[n] can be exactly computed efficiently for any ground state density n. Finally,\nthe remaining part of the total ground state energy defines the x c density functional,\nExc[n] :=E[n]−Ts[n]−EH[n]−EV[n]. DFT is in principle exact, but in practice\ndetermining the precise form of the xc density functional is QMA-ha rd [13]. Therefore,\nKS DFT can only make use of approximations of Exc. The enormous success of DFT is\nthus deeply connected to the successful construction of good a pproximations for the xc\nenergy density functional.\nIn the history of DFT and quest for a universally applicable approxima teExc[14],\nmainly two different paths have been followed: one is the non-empirica l approach\npioneered by Perdew [15] and the other is the semi-empirical approa ch initiated by\nBecke [8]. The non-empirical approach makes use of exact condition s, that a physical\nsystem must fulfill, to find approximations for the xc density functio nal. Within this\napproach a “Jacob’s ladder” of functionals was built where each fun ctional on a higher\nrung of the ladder is supposed to improve upon the ones on the lower rungs [16, 17].\nOn the lowest rung of the “Jacob’s ladder” resides the local density approximation\n(LDA), which was already introduced by Kohn and Sham in [2]. The high er rungs are\nsupposed to systematically improve upon the LDA, which, in practice , does not always\nhappen [17]. Additionally, at the moment, the more precise functiona ls on the higher\nrungsaresomuch moredifficult tocomputethatfurtherimproveme nts ofDFTfollowing\nthis non-empirical approach seem very hard to achieve. In the sem i-empirical approach,\nan ansatz for the functional form of Excis fitted using experimental data, accurate\ntheoretical reference data, or other constraints. However, o ften relatively small training\nsets are used in these fits and then the resulting functionals can be biased towards their\ntraining [14].Systematic construction of density functionals based on MP S computations 3\nAlternatively, we might obtain further improvements of Excaway from but using\nconcepts of both the semi-empirical and the non-empirical approa ch, e.g. by using a\nlarge set of accurate training densities and corresponding values o fExc, and by fitting\nan efficient ansatz to these data that includes some exact condition s. Obviously, a\ndifficulty of this alternative scheme is that it requires a possibly large n umber of\naccurate solutions for the quantum many-body problem. However , nowadays, tensor\nnetwork states provide precise results for quantum many-body s ystems, e.g. [18, 19],\nin particular with respect to ground state properties. We remark t hat tensor network\nmethods are currently limited to low-dimensional, i.e. one- and some tw o-dimensional,\nquantum lattice problems while DFT usually handles three-dimensional continuous\nquantum systems. Since DFT can be applied to a wide range of realistic quantum\nsystems, it is a useful algorithm for a large community and thus wort h improving.\nIn this article, we want to analyze the feasibility of constructing an a pproximate xc\ndensity functional of a specific form, when large training sets of gr ound state densities\nand corresponding values of Excare available. The specific form for the ansatz of our\napproximationisinspired bythenon-empirical approach[17]: itinclude s theLDA[2,20]\nand allows a systematic improvement beyond it. For this feasibility stu dy, we focus on\ndiscretelatticeproblemsandtheone-dimensional case, andweuse matrixproductstates\n(MPS) for the computation of accurate ground state energies an d densities [21, 22]. The\nspecific discrete lattice problem considered here can be derived fro m discretization of\ncontinuous space, i.e. the usual scenario of DFT. Then our ansatz can be seen as the\ndiscretized version of acontinuous function. Althoughwe couldapp roachthecontinuum\nsolution by successively decreasing the discretization, taking the c ontinuum limit is\nbeyond the scope of this work.\nThe structure of this article is as follows. In section 2 we introduce t he considered\nHamiltonian and observables. The corresponding exact LDA is prese nted in section 3.\nWethen propose, fit, andassess ouransatz insection 4. Finally, ins ection 5 we conclude\nthis work and give an outlook.\n2. Model\nIn the following, we consider two species of fermions with long-range d soft-Coulomb\ninteraction on a finite one-dimensional lattice of length Lwith hard-wall boundary\nconditions, as represented by the Hamiltonian:\nˆH:=ˆT+ˆW+ˆV (1)\nwith\nˆT:=−tL−1/summationdisplay\nl=1/summationdisplay\nσ=↑,↓(c†\nl,σcl+1,σ+c†\nl+1,σcl,σ) (2 a)\nˆW:=UL/summationdisplay\nl=1/parenleftBig\nˆnl,↑ˆnl,↓+L/summationdisplay\nm=l+1ˆnlˆnm/radicalbig\n(m−l)2+1/parenrightBig\n(2b)Systematic construction of density functionals based on MP S computations 4\nˆV:=L/summationdisplay\nl=1(vext\nl−µ)ˆnl. (2c)\nHere,c†\nl,σcreates and cl,σannihilates a fermion of species σ=↑,↓on lattice site l,\nˆnl,σ:=c†\nl,σcl,σis the corresponding occupation number operator and ˆ nl:= ˆnl,↑+ ˆnl,↓.\nThe total particle number is denoted by N:=/angbracketleft/summationtextL\nl=1ˆnl/angbracketright. We obtain ground states with\ndifferent total particle number by choosing different values for the chemical potential\nµ, which plays the role of a Lagrange multiplier fixing N. Such a Hamiltonian can\nalso describe the discretized continuous problem with lattice spacing ∆ when in (2 a)\ntis replaced by 1 /(2∆2) and in (2 b) the denominator/radicalbig\n(m−l)2+1 is replaced by/radicalbig\n(m−l)2∆2+1. The solution for different discretizations can be very precisely\ncomputed with MPS [23, 24] and so our approach should yield highly acc urate training\ndensities. If we would like to obtain the solution for continuous space , we would have\nto run our computations repeatedly with decreasing lattice spacing ∆ and extrapolate\nour results to ∆ = 0. Here we see (1) as the Hamiltonian of the problem , and not a\ndiscrete version of a more fundamental one, thus we set t= 1/2 andU= 1 and fix the\nnumber of lattice sites to L= 21 from now on.\nOn a finite lattice, densities nl:=/angbracketleftˆnl/angbracketright=/angbracketleftˆnl,↑+ ˆnl,↓/angbracketrightcan be written into a vector\nn:= (n1,n2,...,n L)T- where T denotes the transpose - such that every density\nfunctional Fcan be written as a function of such density vectors F=F(n). In\nthe following, we will consider the universal Hohenberg-Kohn functional FHK(n), the\nHartree-energy EH(n), and the non-interacting kinetic energy Ts(n), e.g. [20]. For the\nabove Hamiltonian (1) these functionals read:\nFHK(n) :=E(n)−L/summationdisplay\nl=1(vext\nl−µ)nl (3a)\nEH(n) :=UL/summationdisplay\nl=1/parenleftBign2\nl\n4+L/summationdisplay\nm=l+1nlnm/radicalbig\n(m−l)2+1/parenrightBig\n(3b)\nTs(n) :=Es(n)−L/summationdisplay\nl=1(vs\nl−µs)nl, (3c)\nwhereE(n) denotes the ground state energy of an interacting density n, i.e.\ncorresponding to (1) with ˆW, andEs(n) denotes the ground state energy of a non-\ninteracting density n, i.e. corresponding to (1) without ˆW. Knowing the values of these\nfunctionals (3 a), (3b), and (3 c) for a particular density nallows to calculate the xc\nenergyExcfor that density:\nExc(n) :=FHK(n)−EH(n)−Ts(n). (4)\nHowever, given an arbitrary density vector n, onlyEH(n) is trivial to compute:\nFHK(n) requires the knowledge of the external potential as a function o f the density,\nvext\nl=vext\nl(n), andTs(n) requires the knowledge of the effective non-interacting Kohn-\nSham potential vs\nl=vs\nl(n). This process of calculating the external potential vl(n),\nin which the ground state has the given density n, is called inversion and can beSystematic construction of density functionals based on MP S computations 5\nperformed efficiently only in the non-interacting case or for two fer mions [25]. In\ngeneral, there exists no efficient inversion procedure for the inter acting case and we\nuse a slight modification of the iteration proposed in [26, 27]: Aiming at t he target\ndensityntar, we iterate vl(i+ 1) = vl(i) +γ(i)(nl(i)−ntar\nl) until||n(i)−ntar||-\nwhere||...||denotes Euclidean norm - is below a desired precision threshold. Here ,\nn(i) is the ground state density in the external potential v(i) at the iteration step\ni, andγ(i)>0 is adjusted during the iterations to speed up the convergence.\nSince interacting inversion necessitates several ground state co mputations to attain an\napproximate solution, it is not efficient. Even more sophisticated iter ation schemes\ncannot circumvent that some densities require incredibly many itera tions, i.e. ground\nstate computations, until convergence [28]. Therefore, in gene ral, interacting inversion\nrepresents a computationally demanding task.\n3. Exact LDA\nAs the exact form of Excfrom (4) is not known, in practice, approximations are used.\nOne of the simplest and most successful approximations is the LDA [ 2, 20].\nThe exact LDA eLDA\nxcis defined via the homogeneous electron gas, i.e. via exactly\nhomogeneous densities n= (n1,n2,...,n L)T= (n,n,...,n )Tin the thermodynamic\nlimitL→ ∞[2, 20]:\neLDA\nxc(n) := lim\nL→∞Exc(n,n,...,n )/L . (5)\nThis quantity is then used to approximate the xc energy of a finite sy stem by\nExc(n)≈ELDA\nxc(n) :=L/summationdisplay\nl=1eLDA\nxc(nl). (6)\nBecause ELDA\nxc(n) is the exact xc energy for exactly homogeneous densities in the\nthermodynamic limit, it represents a good approximation for relative ly homogeneous\ndensities on large lattices L >>1.\nOur feasibility study here assumes a relatively small lattice of size L= 21\nwith hard-wall boundary conditions, such that finite size and bound ary effects play\na role. We therefore derive our own LDA for this system and do not m ake use of the\nexisting results in [29]. Figure 1 shows our exc(n) obtained from numerically exactly\nhomogeneousdensitiescomputedbymeansofnon-interacting(fo rTs(n))andinteracting\n(forFHK(n)) inversions on L= 21 lattice sites for all possible total particle numbers\nN= 0,1,...,42. To allow for an efficient evaluation of ELDA\nxc, we parametrize the\nfunction exc(n) using a finite number of parameters. A simple way to achieve this is to\nassume a polynomial form pd(n) of certain degree dand to fit our results using different\nvalues of d. In the fit of each polynomial pd(n), we impose the physically reasonable\nconstraint pd(0) = 0 = pd(2), which trivially holds for the exact Exc, as can be seen\nin (4): Obviously exc(0) =Exc(0,0,...,0)/L= 0 because every term in (4) vanishes\nindependently for zero total particle number, and exc(2) =Exc(2,2,...,2)/L= 0Systematic construction of density functionals based on MP S computations 6\nFigure 1. Our exact LDA exc(n) (crosses) and polynomial interpolations pd(n) of\ndegreed= 2 (dotted), 4 (dashed), and 8 (solid).\nbecause FHK(2,2,...,2) =EH(2,2,...,2) andTs(2,2,...,2) = 0 due to impossible\ntunneling. Apparently, our exact LDA excis well approximated by polynomials of low\ndegreedsince, on the scale of figure 1, the d= 8 fit seems indistinguishable from the\nd= 4 fit.\nThe LDAresides onthelowest rung of“Jacob’s ladder” [17] andthe most successful\napproximations of Excbeyond the LDA were built on top of it [6, 7, 8, 9, 10, 11, 12].\nAnalogously, we will use the LDA computed above as our reference, and we will try to\nimprove upon it with a more general ansatz for the functional.\n4. Our ansatz\nOur approach for the construction of an improved xc energy appr oximation consists of\ntwo parts. Firstly, it requires an efficient variational density funct ional ansatz, denoted\nbyG, to approximate Exc. Secondly, a set of Mexternal potentials has to be specified,\nwhich will becalled training scenario , such that thecorresponding Mexact groundstate\ndensities nt, calledtraining densities , and exact values Exc(nt) are used to determine\nGby minimizing a cost function\nd(G) :=M/summationdisplay\nt=1|Exc(nt)−G(nt)|2(7)\nover the variational parameters of G.\nWe are interested in an ansatz that, firstly, includes the LDA and, s econdly, allows\nfor a systematic improvement over it by including non-local terms. I n this spirit, we\npropose a two-site ansatz of the following form:\nGX(n) :=X/summationdisplay\nk=0Gk(n) (8)Systematic construction of density functionals based on MP S computations 7\nwith\nGk=0(n) :=L/summationdisplay\nl=1g0(nl) (9a)\nGk>0(n) :=L−k/summationdisplay\nl=1gk(nl,nl+k). (9b)\nForX= 0, we have G(n) =GX=0(n) =Gk=0(n), which is completely analogous to\nthe LDA (6). And for X >0, thek >0 terms allow for a more general dependence on\nthe density with two-site functions over a range limited by X. In this way, increasing\nXallows us to systematically include more non-local information and to g o beyond the\nlocal LDA.\nIn order to have a practical functional, we want to write it in terms o f a discrete\nset of variational parameters. Thus we need to restrict the form of the functions gk. For\nsimplicity we choose here a polynomial form for each term, as we did in t he previous\nsection 3 for the reference LDA. Additionally, in all following numerica l experiments,\nwe simply fix the degree of the polynomial to d= 4.\nWhenG(n) is assumed to be a polynomial of the nl, the variational parameters of\nGare the polynomial coefficients. Then the desired argminGd(G), i.e. the argument of d\nthat minimizes the cost function, results from the solution of linear e quations Ac=Exc\nwherethepolynomial coefficients of Garevectorized in c, theexact valuesarevectorized\ninExc, and the elements in the matrix Aestablish the correct connection to the cost\nfunction (7): d( G) =/summationtextM\nt=1|Exc(nt)−G(nt)|2=/summationtextM\nt=1|(Exc)t−(Ac)t|2.\nWe want to emphasize that this approach does not need any comput ationally\ndemanding interacting inversion. Because the training densities ntfollow from the\ntraining potentials, i.e. from Mdifferent choices of vextin(3a), we know FHK(nt). While\nthe calculation of EH(nt) is trivial, Ts(nt) is computed via efficient non-interacting\ninversion. Thus, allfurther ingredients for Exc(nt) of(4) arethenefficiently computable.\nThe first step in our approach is to consider the ansatz G=G0, withg0(nl) :=/summationtextd\ns=0c0\nsns\nla polynomial in nlof degree dwithcoefficients c0\ns(andd= 4 inthe following).\nThe simplest possible, “homogeneous”, training scenario amounts t o setting vext\nl= 0,\nin which case we have one training density for each possible total par ticle number\nN= 1,2,...,2L, i.e. at most M= 42 for L= 21. Figure 2 demonstrates how training\nourlocal term g0withsuch groundstatesreproduces theexact LDA.Remarkably, a very\ngood match between our g0and the exact LDA is achieved already with M= 30 ground\nstate computations. This has to be compared to the several thou sands of ground state\ncomputations that were required for figure 1 due to the interactin g inversion iteration.\nWe can understand that the “homogeneous” training scenario lead s to the exact\nLDA because this scenario contains relatively homogeneous training densities and\nbecause the exact LDA is constructed from exactly homogeneous densities. Now we\nwant to consider more inhomogeneous densities. For that purpose we propose the\nsimplest possible extension of the “homogeneous” training scenario : the “step” training\nscenario shown in figure 3 (a). This scenario contains the simplest ex ternal potentialsSystematic construction of density functionals based on MP S computations 8\nFigure 2. Local terms g0(n) from the “homogeneous” training scenario with M=\nN= 6 (dotted), 12 (dashed), and 30 (solid), compared to the exact LDA (crosses).\nHere,g0is a polynomial of degree d= 4.\nthat give rise to inhomogeneous ground state densities, see figure 3 (b) for some example\ndensities. The “step” training scenario allows us to generate much la rger training sets\nsince we define it by free choice of: a) the step position (from l= 2,3,4,...,21), b)\nthe step height (from h= 0,0.1,0.2,...,2.0), c) the step orientation ( leftorright),\nand d) the total particle number (from N= 1,2,3,...,30). We do not include total\nparticle numbers Nlarger than 30 in this training set because we want it to contain\nsufficiently inhomogeneous densities that become more inhomogeneo us when the step\nheight increases; clearly, for large total particle numbers such as more than 30 fermions\non 21 lattice sites, an increasing step height quickly creates large ho mogeneous regions\nof maximum filling in the density, i.e. having 2 fermions per lattice site.\nWe have investigated two different ways of converging Gwith this “step” training\nscenario. In the first way, we pick Mground states randomly and study the convergence\nofGas a function of M. In the second way, we fix the total particle numbers considered\ntoN= 1,2,...,12, take all possible step positions and orientations, and increase M\nsystematically together with the step height. In both schemes, co nvergence is quantified\nby comparison of the solution for Mwith the solution for the largest considered Mmax,\nwhich we fixed to 12800 for the random and to 9612 for the systema tic densities. We\ncan then look at the quantity\nǫ(M) :=/angbracketleft|Ci(M)−Ci(Mmax)|/|Ci(Mmax)|/angbracketright (10)\nwhereCi(M) denotes the ith parameter of Gafter training with Mdensities and\n/angbracketleft.../angbracketright:= 1/P/summationtextP\ni=1...denotes taking the mean value over all Ppossible values of i.\nFigure 4 shows our results for randomdensities. Interestingly, th is training scenario\ngives rise to local terms g0that are very similar to the exact LDA, although many\ntraining densities of this scenario are very inhomogeneous. Furthe rmore, we can read\noff from the inset of figure 4 that convergence occurs rapidly.\nTo go beyond the LDA, we now include the longer-range two-site ter ms withk >0Systematic construction of density functionals based on MP S computations 9\nFigure 3. (a) “Step” training scenario: external potentials vext\nlare characterized by\na step of certain height at a certain position. (b) Ground state den sitynlforN= 12\n(main) and 6 (inset), for a step at position l= 10 of height h= 0 (dotted), 0 .3\n(dash-double dotted), 0 .5 (dash-dotted), 1 .0 (dashed), and 2 .0 (solid).\nin (8) using a general polynomial ansatz:\ngk(nl,nl+k) :=d/summationdisplay\ns0,sk=0ck\ns0,skns0\nlnsk\nl+k, (11)\ni.e. these terms are general degree dpolynomials of the density values on 2 lattice sites\nseparated by distance k(and, again, we simply fix d= 4 in the following). While, as\ndiscussed above, for X= 0, our ansatz G0(n) =/summationtextL\nl=1g0(nl) is completely analogous\nto the LDA of (6), for X >0, it contains additional non-local terms, such that, by\nsystematically increasing Xin our ansatz GX(n), we can systematically increase its\nnon-locality beyond the local LDA-like term.\nWeenforceinourdesiredsolution GXthattheterms gkforincreasing kareobtained\none after another such that each additional non-local term (i.e. c orresponding to the\nnext larger value of k) is a correction to the previous solution. This means that, for\ngivenX, we first minimize (7) only via the parameters c0which gives G0. Then we\nminimize (7) for the remainder E1\nxc(nt) :=Exc(nt)−G0(nt) only via the parameters c1\nwhich together with the previous solution c0givesG1. Then we minimize (7) for the\nremainder E2\nxc(nt) :=Exc(nt)−G1(nt) only via the parameters c2which together with\nthe previous solutions c0andc1givesG1. We continue the scheme until we have reached\ncXand thus GX. This procedure ensures that each longer-range term is built on to p of\nall previous shorter-ranged ones, in the same way as the function als on higher rungs of\n“Jacob’s ladder” are more non-local and are built on top of the more local functionals\non the lower rungs [17].Systematic construction of density functionals based on MP S computations 10\nFigure 4. Localterms g0(n) fromthe “step”trainingscenariowith M= 100(dotted),\n400 (dash-dotted), 1600 (dashed), 6400 (solid), and 12800 (cr osses). Inset: Mean\nrelative difference ǫ(M) between the coefficients of g0after training with Mdensities\nand the coefficients of g0after training with Mmax= 12800 densities. Here, g0is a\npolynomial of degree d= 4.\nInordertoanalyzetheperformanceoftheansatz, weadoptnow adifferentstrategy.\nWe will now always fit our ansatz GXwith the “step” training scenario of figure 3 and\nthen we will apply it to completely different target densities. For the la tter, we choose\nground states of the H2dissociation problem, i.e. Hamiltonian (1) with total particle\nnumberN= 2 and external potential\nˆV(R) :=L/summationdisplay\nl=1vext\nl(R)ˆnl+1√\nR2+1(12a)\nvext\nl(R) :=−1/radicalbig\n(l−(l0−R/2))2+1−1/radicalbig\n(l−(l0+R/2))2+1(12b)\nwhereRdenotes the separation between the two Hatoms placed in the middle of the\nlattice such that we set l0= 11 for L= 21. Because this problem represents a realistic\nphysical application that is significantly different from our training sc enario, we consider\nit to be a very good benchmark for our approach.\nFrom now on, the local terms g0(n) in our two-site polynomial ansatz (8) are fixed\nto be the exact LDA from figure 1. And the non-local terms gk>0(n1,n2) are enforced to\nfulfillgk>0(0,0) = 0 = gk>0(2,2) as well as gk>0(n1,n2) =gk>0(n2,n1). These properties\nare physically reasonable and they reduce the number of variationa l parameters, which\nturned out to be beneficial for the convergence of our fit. In par ticular, this helped us\nto avoid the effect known as overfitting. We distinguish two different versions of our\nansatz, namely constrained andunconstrained . In our constrained two-site polynomial\nansatz we determine gk>0(n1,n2) under the constraint gk>0(n,n) = 0: then GX(n) is\nexact for exactly homogeneous densities n= (n,n,...,n )T. In our unconstrained two-\nsite polynomial ansatz we do not impose this constraint on the polyno mial coefficients:\nthe unconstrained ansatz has thus more variational parameters than the constrained\nansatz.Systematic construction of density functionals based on MP S computations 11\nFigure 5. H2dissociation energy E(R) as a function of the separation R: from\nexact LDA (dotted) and from our constrained (a) and unconstra ined (b) ansatz with\nX= 1 (dash-double dotted), 2 (dash-dotted), 5 (dashed), compa red to the exact\nsolution (solid). Insets: Mean relative difference ǫ(M) between the coefficients of gk>0\nfork= 1 (dash-double dotted), 2 (dash-dotted), and 5 (dashed) aft er training with\nMsystematic densities and the corresponding coefficients of gk>0after training with\nMmax= 9612 systematic densities - the insets show our results for M= 1212, 2412,\nand 4812. Here, our ansaetze for Gare polynomials of degree d= 4.\nFigure 5 shows our results after convergence with Msystematic densities. As we\ncan see in (a), the constrained ansatz leads to a visible improvement over LDA close to\nR= 0, but not for larger R. A convergence of the non-local terms can be concluded\nfrom the inset. In (b), the unconstrained ansatz leads to an impro vement over LDA for\nalmost all values of R, but it produces too low energy values close to R= 0. The inset\nof (b) demonstrates that convergence of the non-local terms o ccurs, however, for k >1\nthis convergence is slower than in (a). With increasing X, our ansatz systematically\nimproves the LDA result at specific values of R: around R= 0 when the constrained\nversion is used, and at larger Rwhen the unconstrained version is used. Both versions\nof our ansatz show a systematic improvement over LDA with increas ingXwhen the\nmean energy for all values of Ris considered. In fact, such a mean value is the correct\nfigure of merit because the cost function (7), minimized for “step” training densities bySystematic construction of density functionals based on MP S computations 12\nFigure 6. Exchange-correlationpotential vxc\nlfor the exact ground state density of the\nH2dissociation curve at R= 5: from exact LDA (dotted) and from our constrained\n(a) and unconstrained (b) ansatz with X= 1 (dash-double dotted), 2 (dash-dotted),\n5 (dashed), compared to the exact solution (solid). Our ansaetze forGhere are the\nsame as the ones in figure 5.\nour ansatz, is also a mean value of many xc energies.\nClearly, we would like to use GX(n) to compute densities self-consistently via\nthe KS cycle. A first step in this direction is the calculation of the xc po tential\nvxc\nl(n) :=−∂Exc/∂nl|nfor the exact ground state density n. In the H2dissociation\nproblem, the xc potential for larger values of Ris particularly interesting, since its\nexact form exhibits a characteristic peak that cannot be reprodu ced by LDA alone [30].\nFigure 6 shows our results for R= 5. While our constrained ansatz leads to a potential\nthat basically coincides with the one from LDA, our unconstrained an satz leads to a\nsmall systematic improvement with increasing X.\n5. Conclusions\nWe have analyzed the feasibility of constructing semi-empirical appr oximations for the\nxc density functional in the context of a long-range interacting ma ny-electron system on\naone-dimensional lattice. Using numerically exact groundstatesfr omMPSsimulations,Systematic construction of density functionals based on MP S computations 13\nwe proposed to fit an ansatz that includes an LDA-like part plus addit ional terms of\nincreasing non-locality, by means of reasonably chosen training den sities. We observed\nthat our ansatz converges systematically within the training scena rio. Additionally,\nwhen applied to completely different target densities, namely of the H2dissociation\nproblem, our fitted ansatz improved upon the LDA systematically. T his systematic\nimprovement was demonstrated for the ground state energies of theH2problem and for\na xc potential corresponding to a stretched H2molecule.\nIn this work, we have tested the effect of a systematic inclusion of n on-local\ningredients in the functional using very simple ansaetze for the non -local terms, namely\nonlytwo-sitedependences, andforthefunctionalform, namelyo nlypolynomials, andwe\nconsidered only one training scenario. Our results show that by sys tematically including\nnon-local terms, the approximation can without doubt be improved beyond LDA. Our\nquantitative results are nevertheless limited to the specific form of the ansatz and\ntraining set used. For instance, thefact that the dissociation cur ve doesnot significantly\nimprove by including terms of longer range after some point seems to indicate that other\nnon-local contributions may be more relevant. Likewise, considerin g functional forms\nfor each term that go beyond a polynomial may improve the power of the ansatz. We\nhave already run initial tests to study the effect of terms that dep end on three variable\ndensities, but we observe no clear convergence with as many as ≈20000 densities from\nthe “step” training scenario. This clearly indicates the necessity fo r a different training\nscenario and possibly additional physical constraints that effectiv ely reduce the number\nof variational parameters. The question itself of how to choose th e training densities\noptimallyis, ingeneral, averyimportantonethatshouldbefurthere xplored, asabetter\ntraining scenario would always further improve our results. All in all, a lthough such\nimproved ansaetze and training scenarios must definitely achieve be tter results than the\nones reported here, a careful analysis is beyond the scope of this proof-of-principle work.\nIt would be very interesting to combine our procedure with concept s from recent\nworks on machine learning of density functionals [31, 32, 33, 34, 35 , 36]. On the one\nhand, these works typically required less training densities than our approach. On the\nother hand, our work constructs systematic corrections to a st andard approximation,\nnamely the LDA, that can be applied in general, i.e. to other types of s ystems beyond\nthose that were originally used for the fit. Thus, a combination of th e good aspects of\nour procedure with the good aspects of the previous machine learn ing concepts could\nbe the ultimate solution to some of the problems that both approach es currently have\nindependently from each other.\nIn this article, we focused exclusively on approximations of the dens ity functional\nfor the xc energy. However, in principle, any ground state observ able can be written as\na functional of the ground state density [1]. Our proposed scheme allows in principle to\nalso construct systematically density functionals for observables other than the ground\nstate energy. This fact might now be useful for ultracold atoms in o ptical lattices since\nthe densities of these systems have become experimentally access ible with remarkable\nresolution [37, 38, 39, 40, 41, 42, 43, 44]. Measurements ofa cert ainobservable which areSystematic construction of density functionals based on MP S computations 14\ndifficult to perform with the current techniques might be easier to ca rry out now when\na density functional would be provided for that observable. Partic ularly interesting\nmeasurements regard two-dimensional systems with special obse rvables, such as e.g.\nspecial order parameters. The corresponding density functiona ls could be constructed\nwith the help of projected entangled pair states (PEPS) [45]. Becau se experiments are\nrestricted to finite system sizes, PEPS algorithms are perfectly su ited for the ground\nstate simulation of such finite two-dimensional systems, for which t hey can produce\naccurate numerical results [46, 47, 48, 49, 50, 51].\nAcknowledgments\nM L is very grateful to Neepa T Maitra and Garnet K-L Chan for discu ssions. He\nacknowledges funding by the EU through SIQS grant (FP7 600645) and the DFG (NIM\ncluster of excellence). 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B90064425-1–064425-16"    },    {        "title": "1603.06646v1.Engineering_chiral_density_waves_and_topological_band_structures_by_multiple__Q__superpositions_of_collinear_up_up_down_down_orders.pdf",        "content": "APS/123-QED\nEngineering chiral density waves and topological band structures by\nmultiple- Qsuperpositions of collinear up-up-down-down orders\nSatoru Hayami,1Ryo Ozawa,2and Yukitoshi Motome2\n1Department of Physics, Hokkaido University, Sapporo 060-0810, Japan\n2Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan\nMagnetic orders characterized by multiple ordering vectors harbor noncollinear and noncoplanar\nspin textures and can be a source of unusual electronic properties through the spin Berry phase\nmechanism. We theoretically show that such multiple- Qstates are stabilized in itinerant magnets\nin the form of superpositions of collinear up-up-down-down (UUDD) spin states, which accompany\nthe density waves of vector and scalar chirality. The result is drawn by examining the ground state\nof the Kondo lattice model with classical localized moments, especially when the Fermi surface\nis tuned to be partially nested by the symmetry-related commensurate vectors. We unveil the\ninstability toward the multiple- QUUDD states with chirality density waves, using the perturbative\ntheory, variational calculations, and large-scale Langevin dynamics simulations. We also show that\nthe chirality density waves can induce rich nontrivial topology of electronic structures, such as the\nmassless Dirac semimetal, Chern insulator with quantized topological Hall response, and peculiar\nedge states which depend on the phase of chirality density waves at the edges.\nPACS numbers: 71.10.Fd, 71.27.+a, 75.10.-b\nI. INTRODUCTION\nNoncollinear and noncoplanar magnetic orderings have\nattracted much interest in condensed matter physics, as\nthey often lead to intriguing phenomena and topologi-\ncally nontrivial electronic states. These orders can si-\nmultaneously activate secondary order parameters, in ad-\ndition to the primary magnetic ordering. For instance,\na noncollinear magnetic order carries the vector chi-\nrality, which is de\fned by a vector product of spins,\nhSi\u0002Sji, while a noncoplanar magnetic order the scalar\nchirality, which is represented by a triple scalar product,\nhSi\u0001(Sj\u0002Sk)i. Such chirality degrees of freedom generate\nan emergent electromagnetic \feld for electrons through\nthe spin Berry phase mechanism, and hence, have a huge\npotential to induce and control novel electronic states\nand transport phenomena, such as the anomalous (topo-\nlogical) Hall e\u000bect1{3, orbital and spin current4{6, and\nmagnetoelectric e\u000bect5. Exploring such unusual states\nwith chirality degrees of freedom is expected to bring a\nmajor advance in the \feld of magnetism and stimulate\nfurther possibilities for \\chiraltronics\".\nAmong them, several noncoplanar magnetic orders\nhave been examined by focusing on the emergence of\nthe anomalous Hall e\u000bect. Skyrmion crystals are one\nof the most prominent examples, where the relativistic\nspin-orbit coupling plays an important role7{10. Another\nexamples have been extensively discussed for 3 d- and 4f-\nelectron compounds, on the basis of the Kondo-type spin-\ncharge coupling on several lattice structures: triangu-\nlar11{16, honeycomb16,17, kagome3,18,19, square20,21, cu-\nbic22, face-centered-cubic23, and pyrochlore lattices24. In\nparticular, the noncoplanar magnetic orders with ferroic\nordering of the scalar chirality have attracted much inter-\nest, as the spatially uniform scalar chirality generates a\ncoherent spin Berry phase for itinerant electrons and may\nlead to a quantized anomalous Hall e\u000bect3,11,12,23. On theother hand, the magnetic states with stripy patterns of\nthe scalar chirality have recently been proposed25,26. In\nthese states, the value of the scalar chirality is modulated\nin real space, and even canceled out in the whole system\n(the net chirality is zero). Thus, these states are regarded\nas antiferroic-type scalar chirality orderings. Given the\nvariety, it is a natural question what is essential for non-\ncollinear and noncoplanar orderings with the chirality de-\ngrees of freedom and what determines the spatial pattern\nof chirality density waves (ChDW). It will also be inter-\nesting to ask how the di\u000berent ChDW a\u000bect the electronic\nproperties, in both bulk and nanoscale structures, such\nas topological nature of the band structure, edge states,\ndomain walls, and local electric/spin currents.\nIn this paper, we present a systematic theoretical study\nof vector and scalar ChDW in itinerant electron sys-\ntems. The key ingredient in our study is an up-up-down-\ndown (UUDD) collinear magnetic order [see Fig. 1(b)].\nWe demonstrate that a variety of ChDW can be con-\nstructed by superpositions of such UUDD orders, which\nwe call multiple- QUUDD states. We examine the insta-\nbility toward such multiple- QUUDD states in a minimal\nmodel for itinerant magnets, the Kondo lattice model\nwith classical localized moments in two dimensions, using\nan analytical perturbative expansion with respect to the\nexchange coupling between itinerant electron spins and\nlocalized moments. We \fnd that, at particular \fllings\nwhere the di\u000berent portions of the Fermi surface are con-\nnected by commensurate vectors, the system is unstable\ntoward the multiple- QUUDD states. The higher-order\ncontributions beyond the second-order Ruderman-Kittel-\nKasuya-Yosida (RKKY) interaction27play a key role in\nthis instability. While similar mechanisms were discussed\nfor other noncoplanar states25,26,28,29, our construction\nhas the advantage of extending the variety of ChDW pat-\nterns to superstructures beyond one-dimensional stripy\nones. We carefully con\frm the perturbative argumentsarXiv:1603.06646v1  [cond-mat.str-el]  21 Mar 20162\nby two numerical calculations: variational calculations\nfor several candidates of the ground state and large-scale\nLangevin dynamics simulations enabled by the kernel\npolynomial method (KPM-LD)30. Furthermore, we \fnd\nthat the ChDW may bring about a topologically nontriv-\nial nature in the itinerant electrons, re\recting oscillations\nof chirality in real space. We show that the system be-\ncomes a Dirac semimetal and magnetic Chern insulator\ndepending on the chirality superstructures. We also re-\nveal that peculiar edge states appear in these topological\nstates, in di\u000berent forms depending on where the ChDW\nare terminated at the edges.\nThe rest of the paper is organized as follows. In Sec. II,\nafter introducing the Kondo lattice model, we brie\ry dis-\ncuss how the RKKY interaction fails to determine the\nground state of the Kondo lattice model in some particu-\nlar situations. As the candidate for the ground state, we\npropose multiple- Qmodulations of speci\fc UUDD spin\nstates, which result in ChDW. In Sec. III, we examine\nthe instability toward the multiple- QUUDD states by\ncombining the perturbative expansion with respect to\nthe exchange coupling between itinerant and localized\nspins, variational calculations by using the direct diag-\nonalization of the full Hamiltonian, and the KPM-LD\nsimulations for large-size clusters. In Sec. IV, we discuss\nthe electronic structure of the multiple- QUUDD states,\nwith emphasis on the topological properties of bulk band\nstructure and the edge states induced by ChDW. We\nsummarize our results in Sec. V, with making some re-\nmarks on the comparison between our multiple- QUUDD\nstates and other ChDW states.\nII. MULTIPLE- QUUDD STATE\nIn this section, we present the fundamental concept\nof the multiple- QUUDD states with ChDW, whose sta-\nbility is examined in the later sections. First, we in-\ntroduce the Kondo lattice model consisting of classi-\ncal localized spins and itinerant electrons in Sec. II A.\nThen, in Sec. II B, we brie\ry review the RKKY interac-\ntion, which is an e\u000bective magnetic interaction appear-\ning in the second-order perturbation with respect to the\nexchange coupling in the Kondo lattice model. After\npresenting the magnetic structure for the single- Q(1Q)\nUUDD state in Sec. II C, we describe how to construct\nthe multiple- QUUDD states on the square and trian-\ngular lattices in Sec. II D. We show that the multiple- Q\nUUDD states are energetically degenerate with the 1 Q\none at the level of the RKKY interaction. In Sec. II E, we\nshow the multiple- QUUDD states possess the real-space\nsuperstructures of vector and scalar chirality (ChDW).\nA. Kondo Lattice Model\nWe consider a model consisting of noninteracting elec-\ntrons coupled with localized spins, called the Kondo lat-tice model, on the square and triangular lattices. The\nHamiltonian is given by\nH=\u0000t1X\nhi;ji\u001b(cy\ni\u001bcj\u001b+ H:c:)\u0000t2X\nhhi;jii\u001b(cy\ni\u001bcj\u001b+ H:c:)\n+JX\ni\u001b\u001b0cy\ni\u001b\u001b\u001b\u001b0ci\u001b0\u0001Si\u0000\u0016X\ni\u001bcy\ni\u001bci\u001b; (1)\nwherecy\ni\u001b(ci\u001b) is a creation (annihilation) operator of an\nitinerant electron at site iand spin\u001b,\u001b= (\u001bx;\u001by;\u001bz) is\nthe vector of Pauli matrices, Siis a classical localized spin\nat siteiwhose amplitude is normalized as jSij= 1,Jis\nthe exchange coupling constant (the sign is irrelevant for\nclassical localized spins), and \u0016is the chemical potential.\nThe sums ofhi;jiandhhi;jiiare taken over the nearest-\nneighbor and second-neighbor sites, respectively, on the\nsquare and triangular lattices. Hereafter, we take t1= 1\nas an energy unit.\nIn the wave number representation, the Hamiltonian\nin Eq. (1) is transformed into\nH=X\nk\u001b(\"k\u0000\u0016)cy\nk\u001bck\u001b+JX\nkq\u001b\u001b0cy\nk\u001b\u001b\u001b\u001b0ck+q\u001b0\u0001Sq;\n(2)\nwherecy\nk\u001bandck\u001bare the Fourier transform of cy\ni\u001band\nci\u001b, respectively. Sqis the Fourier transform of Si. In\nEq. (2),\"kis the free electron dispersion, which is given\nby\n\"k=\u00002X\nl=1;2(t1cosk\u0001el+t2cosk\u0001e0\nl); (3)\nfor the square lattice [ e1=^x= (1;0),e2=^y= (0;1),\ne0\n1=^x+^y, and e0\n2=^x\u0000^y] and\n\"k=\u00002X\nl=1;2;3(t1cosk\u0001el+t2cosk\u0001e0\nl); (4)\nfor the triangular lattice ( e1=^x,e2=\u0000^x=2 +p\n3^y=2,\ne3=\u0000^x=2\u0000p\n3^y=2,e0\n1=e2+ 2e3,e0\n2=e3+ 2e1, and\ne0\n3=e1+ 2e2). We set the lattice constant a= 1 as the\nlength unit.\nB. RKKY Interaction\nIn general, the Kondo lattice model in Eq. (2) ex-\nhibits magnetic ordering in the ground state. The sta-\nble spin pattern depends on the electron \flling n=\n(1=N)P\ni\u001bhcy\ni\u001bci\u001bias well as the exchange coupling con-\nstantJ(Nis the number of lattice sites). When J\nis much larger than t1andt2, the system shows a\nferromagnetic order for general electron \flling, by the\ndouble-exchange ferromagnetic interaction between lo-\ncalized spins induced by the kinetic motion of itinerant\nelectrons31. On the other hand, when J\u001ct1andt2, the\nmagnetic ordering in the ground state is predominantly3\n(a) (b)\n(c)\n(e)UUDD\n1Q-UUDD\n2Q-UUDDHelical\n(f)\n(g)2Q-UUDD\n3Q-UUDD\n(d) 1Q-UUDD\n(h) 3 Q-UUDD\nFIG. 1. (Color online) Schematic pictures of (a) helical and\n(b) UUDD magnetic structures in the one-dimensional repre-\nsentation, (c) collinear single- Q(1Q) UUDD, (d) 1 QUUDD\nwith di\u000berent ordering vectors from (c) (see the text in detail),\n(e) coplanar double- Q(2Q) UUDD consisting of a superpo-\nsition of (c), (f) 2 QUUDD consisting of a superposition of\n(d) on the square lattice, and (g) noncoplanar triple- Q(3Q)\nUUDD magnetic structures on the triangular lattice. The ar-\nrows denote the directions of localized moments. The inset\nof (g) shows the directions of magnetic moments in the 3 Q\nUUDD state; each spin points along the local [111] directions\nin the cubic representation. In all cases, a global spin rotation\ndoes not alter the consequences due to the SU(2) symmetry\nin the system. (h) The square lattice with diagonal bonds,\nwhich is topologically equivalent to the triangular lattice in\n(g). In (e) and (f) [(g) and (h)], the red and blue plaque-\nttes show the positive and negative vector (scalar) chirality\nde\fned in Eq. (12) [Eq. (13)].\ndetermined by the RKKY interaction, which is also a\nkinetically-induced e\u000bective magnetic interaction27. The\nexpression of the RKKY interaction is obtained by thesecond-order perturbative expansion with respect to Jas\nH(2)=\u0000J2X\nq\u001f0\nqjSqj2; (5)\nwhere\u001f0\nqis the bare susceptibility of itinerant electrons,\n\u001f0\nq=1\nNX\nkf(\"k)\u0000f(\"k+q)\n\"k+q\u0000\"k: (6)\nHere,f(\"k) is the Fermi distribution function. As the\nbare susceptibility depends on the transfer integrals (non-\ninteracting band structure) and chemical potential (elec-\ntron \flling), these two factors play a decisive role in de-\ntermining the magnetic state in the Kondo lattice model\nforJ\u001ct1;t2.\nIn general, the RKKY interaction in Eq. (5) favors a\ncoplanar helical magnetic order, whose spin pattern is\ngiven by\nSi= (cos Q1\u0001ri;0;sinQ1\u0001ri): (7)\nNote that the helical plane, which is taken as the xz\nplane, is arbitrary in the model with isotropic exchange\ninteractions. The ordering vector Q1is determined by\nthe maximum of \u001f0\nq, and therefore, depends on the band\nstructure and electron \flling. The preference of the heli-\ncal spin state is understood from the constraint jSij= 1\nand the sum ruleP\nqjSqj2=N. Other complicated mag-\nnetic structures, such as the superpositions of the helical\nspin patterns, need higher harmonics in order to satisfy\nthe constraint of jSij= 1, which leads to an energy cost\nunder the sum ruleP\nqjSqj2=N.\nC. UUDD Magnetic Structure\nFor particular ordering vectors, however, magnetic pat-\nterns, which are more complicated than the helical one,\nare allowed without introducing higher harmonics. An\nexample is the multiple- Qstate which is composed of\na superposition of di\u000berent 1 Qstates. For instance, the\ndouble-Q(2Q) state with Q1= (\u0019;0) and Q2= (0;\u0019) on\nthe square lattice, whose spin structure is given by20,29\nSi=^xcosQ1\u0001ri+^ycosQ2\u0001ri; (8)\nsatis\fes the constraint jSij= 1. The important point is\nthat the modulation with the second component Q2is\nintroduced in the perpendicular direction to that of Q1;\nthis guarantees no additional energy cost at the RKKY\nlevel in Eq. (5). Thus, the 1 Qhelical state in Eq. (7)\nand the 2Qstate in Eq. (8) are energetically degenerate\nwithin the RKKY level. This indicates that the RKKY\ninteraction is not su\u000ecient to determine the ground state,\nwhen Q1= (\u0019;0) and Q2= (0;\u0019) maximize the bare\nsusceptibility. Note that Q1andQ2are related with\neach other by the fourfold rotational symmetry, which is\ncompatible with the square lattice. The stability of this4\ntype of multiple- Qstates was discussed in Refs. 29 and\n32.\nA similar but di\u000berent situation can occur for Q1=\n(\u0019=2;0). Interestingly, for this particular wave number,\nthere are energetically-degenerate states even within the\n1Qstates: The helical order in Eq. (7) with Q1= (\u0019=2;0)\n[Fig. 1(a)] has the same energy as a collinear UUDD or-\nder, whose spin texture is represented by\nSi= [cos Q1\u0001ri+ cos( Q1\u0001ri\u0000\u0019=2)]^x: (9)\nOne can easily \fnd that Eq. (9) satis\fes jSij= 1 as\nthe helical spin structure does. The one- and two-\ndimensional examples are shown in Figs. 1(b) and 1(c),\nrespectively. In the two-dimensional case, we can also\n\fnd another UUDD state with Q1= (\u0019=2;\u0019), as shown\nin Fig. 1(d). These UUDD states can be regarded as the\nsuperpositions of the helical states, i.e., the spin struc-\ntures in the UUDD states are decomposed into a pair\nof exp(iQ1\u0001r) and exp(\u0000iQ1\u0001r). In the following, we\nwill discuss the stability and nature of 1 QUUDD and\nmultiple-QUUDD states, which are introduced in the\nnext section, in comparison with the helical state.\nD. Multiple- QUUDD States\nWe can extend the UUDD states by considering the\nmultiple-Qsuperpositions. In a similar way to Eq. (8),\nwe can de\fne the 2 QUUDD state as\nSi=1p\n20\n@cosQ1\u0001ri+ cos( Q1\u0001ri\u0000\u0019=2)\ncosQ2\u0001ri+ cos( Q2\u0001ri\u0000\u0019=2)\n01\nA:(10)\nIn the case of the square lattice, there are two combina-\ntions of ordering vectors, which are allowed for the 2 Q\nUUDD state while keeping jSij= 1: One is Q1= (\u0019=2;0)\nandQ2= (0;\u0019=2), and the other is Q1= (\u0019=2;\u0019) and\nQ2= (\u0019;\u0000\u0019=2). Note that, in each combination, Q1\nandQ2are connected by the fourfold rotational opera-\ntion compatible with the square lattice. When we choose\nQ1= (\u0019=2;0) and Q2= (0;\u0019=2), the real-space spin\ncon\fguration is schematically shown in Fig. 1(e), while\nthat for Q1= (\u0019=2;\u0019) and Q2= (\u0019;\u0000\u0019=2) is shown in\nFig. 1(f). Their spin con\fgurations are noncollinear but\ncoplanar.\nMeanwhile, we can also consider the triple- Q(3Q)\nUUDD state on the triangular lattice, whose spin con-\n\fguration is given by\nSi=1p\n30\n@cosQ1\u0001ri+ cos( Q1\u0001ri\u0000\u0019=2)\ncosQ2\u0001ri+ cos( Q2\u0001ri\u0000\u0019=2)\ncosQ3\u0001ri+ cos( Q3\u0001ri\u0000\u0019=2)1\nA;(11)\nwhere Q1= (\u0019=2;0),Q2= (0;\u0000\u0019=2), and Q3=\n(\u0000\u0019=2;\u0019=2). Here and hereafter, we regard the trian-\ngular lattice as a topologically equivalent square lattice\nwith diagonal bonds, as shown in Fig. 1(h). The spincon\fguration given by Eq. (11) is noncoplanar, whose\noriginal geometry is shown in Fig. 1(g) ( Q1,Q2, and Q3\nare connected by the sixfold rotational operation com-\npatible with the triangular lattice).\nAs exempli\fed in Sec. II C for the case of Q1= (\u0019;0)\nandQ2= (0;\u0019), the RKKY energy for a multiple- Q\nstate remains the same as that in the 1 Qstate when\nthere are no higher harmonics in the spin con\fgurations.\nHence, the multiple- QUUDD states introduced in this\nsection have the same RKKY energy as those for the\n1Qhelical and UUDD states. The degeneracy is lifted\nby higher-order contributions beyond the RKKY inter-\naction, as discussed in Sec. III.\nE. Chirality Density Waves\nThe multiple- QUUDD states exhibit nonzero chirality.\nWe de\fne the vector and scalar chirality as\n\u001fp\nv=1\n4(Si\u0002Sj+Sj\u0002Sk+Sk\u0002Sl+Sl\u0002Si);(12)\n\u001fp\ns=Sm\u0001(Sn\u0002So); (13)\nrespectively, where i,j,k, andl(m,n, ando) are sites on\neach square (triangle) plaquette pin a counterclockwise\ndirection. In the multiple- QUUDD states, the value of\nchirality depends on the spatial position of the plaquette,\nwhich we call ChDW.\nIndeed, in the 2 QUUDD on the square lattice\n[Eq. (10)], the zcomponent of the vector chirality \u001fp\nvos-\ncillates from a positive to negative value on each plaque-\ntte, as shown in Figs. 1(e) and 1(f); see also in Figs. 5(a)\nand 7(a) for the chirality distribution in a larger scale.\nThus, this state is an antiferroic-type vector ChDW with\nvanishing net vector chirality. Note that the scalar chi-\nrality is zero everywhere because of the coplanar spin\ncon\fgurations. Meanwhile, in the 3 Q-UUDD state, the\nscalar chirality takes a positive value or zero in a periodic\nway, as shown in Figs. 1(g) and 1(h). This is a ferri-type\nscalar ChDW with a nonzero net scalar chirality. (In\nthis noncoplanar 3 Qcase, we do not discuss the vector\nchirality.)\nThus, these ChDW states are distinct from the ferroic\nchirality orders in the previous studies, where every pla-\nquette possesses the same value of vector or scalar chiral-\nity, as mentioned in the introduction11,23. Furthermore,\nthey have richer superstructures in the chirality than the\none-dimensional stripy ones in the previous studies25,26.\nRe\recting the distinct aspect, intriguing edge-dependent\nelectronic structures are obtained as discussed in Sec. IV.\nIII. INSTABILITY TOWARD MULTIPLE- Q\nUUDD STATES\nIn this section, we examine the instability toward the\nmultiple-QUUDD states from the energetic point of5\n 0(a) (b)\n 0 0(c) (d)\n 0.12 0.16 0.20 0.24 0.28 0.32 0.10 0.14 0.18 0.22\n0\n0.1\n 0.1\n 0.1\n 0.2 0\n0.1\n 0.1\n 0.2\n 0.2\n 0.2\n 0.3\n 0 0\n0\n 0 0\nFIG. 2. (Color online) The Fermi surface for (a) the square\nlattice model at t2= 0 and\u0016=\u0000p\n2 and (c) triangular\nlattice model at t2= 0:5 and\u0016= 2. The triangular lattice\nis regarded as a topologically equivalent square lattice with\ndiagonal bonds, as shown in Fig. 1(h). Q\u0017(\u0017= 1, 2, and 3)\nare the vectors connecting the Fermi surfaces. (b), (d) The\ncontour plots of the bare susceptibility corresponding to (a)\nand (c), respectively. The bare susceptibility exhibits maxima\natQ\u0017and the symmetry-related wave numbers.\nview. In Sec. III A, we show the results from higher-\norder perturbative expansion with respect to Jbeyond\nthe second-order RKKY contribution. In Sec. III B, we\nevaluate the energy di\u000berences between the multiple- Q\nUUDD, 1QUUDD, and helical states by variational cal-\nculations. Finally, we examine the ground state in an\nunbiased way by performing the KPM-LD simulation in\nSec. III C. The results provide complementary evidences\nthat the system has the instability toward the multiple- Q\nUUDD states at particular electron \fllings.\nA. Perturbative Analysis\nAs described in the previous section, when particular\nsymmetry-related wave numbers maximize the bare sus-\nceptibility, the RKKY interaction is not su\u000ecient to de-\ntermine the ground state, since it gives the same energy\nfor helical, 1 QUUDD, and multiple- QUUDD orders.\nThis occurs when the ordering vectors satisfy jQ\u0018j=\u0019,\n\u0019=2, or 0 [ Q= (Qx;Qy)6=0or (\u0019;\u0019)], and they are re-\nlated with each other by the rotational operation proper\nto the lattice structure.\nAn example is found for the square lattice model at\nt2= 0 and the chemical potential \u0016=\u0000p\n2. The Fermi\nsurface is drawn in Fig. 2(a). As shown in the \fgure,the Fermi surface is connected by two wave numbers,\nQ1= (\u0019=2;\u0019) and Q2= (\u0019;\u0000\u0019=2), which satisfy the\nabove condition. Note that the connections are not only\nwithin the same Brillouin zone but also between di\u000ber-\nent Brillouin zones. These connections maximize the bare\nsusceptibility at Q1,Q2, and the symmetry-related wave\nnumbers,\u0000Q1and\u0000Q2, as shown in Fig. 2(b). This\nindicates that the magnetically ordered states with these\nordering vectors are the candidates for the ground state.\nSpeci\fcally, the plausible ground states are the 1 Qheli-\ncal state with Q1, the 1QUUDD with Q1, and the 2 Q\nUUDD with Q1andQ2; these three states are energeti-\ncally degenerate for the RKKY Hamiltonian in Eq. (5).\nNote that we do not need to consider the 3 QUUDD state\nin this square lattice case, because its RKKY energy is\nhigher than that for the 1 QUUDD state; the 3 Qstate\nbecomes relevant in the triangular-lattice system with\nsixfold rotational symmetry.\nIn fact, another example including the possibility of 3 Q\nUUDD ordering is found for the triangular lattice model\natt2= 0:5 and\u0016= 2. Figure 2(c) shows the Fermi\nsurface. In this case, there are three vectors connecting\nthe Fermi surface, Q1= (\u0019=2;0),Q2= (0;\u0000\u0019=2), and\nQ3= (\u0000\u0019=2;\u0019=2), which lead to the six maxima in the\nbare susceptibility shown in Fig. 2(d). Thus, the possible\nground states in this case include the 3 QUUDD with Q1,\nQ2, and Q3in addition to the 1 Qand 2Qstates.\nSimilar connections of the Fermi surface occur for\nQ1= (\u0019=2;0) and Q2= (0;\u0019=2) at\u0016=\u0000p\n2(1 +p\n2)\non the square lattice model with t2= 0, and for Q1=\n(\u0019=2;0),Q2= (0;\u0000\u0019=2), and Q3= (\u0000\u0019=2;\u0019=2) at\n\u0016=\u00002(1 +p\n2) on the triangular lattice model with\nt2= 0. In these two cases, although \u001f0\nqhas maxima at\n\u0006Q\u0017, it shows less qdependence and the peaks are not\nclearly visible (not shown here), compared to the pre-\nvious cases shown in Figs. 2(b) and 2(d). Nevertheless,\nwe will discuss these two cases in addition to the former\ntwo, as the perturbative arguments below indicate that\nthe instability toward the multiple- Qstates appears in a\ncommon manner.\nFor the situations above, the RKKY energy is degen-\nerate for the ground state candidates. Hence, in or-\nder to discriminate them, we need to take into account\nthe higher-order contributions in the perturbative anal-\nysis28,29. The degeneracy at the second-order RKKY\nlevel is lifted by the fourth-order contribution with re-\nspect toJ(note that the odd-order terms vanish by\nsymmetry). When we expand the free energy F=\n\u0000TP\n\u0016log [1 + exp(\u0000E\u0016=T)] (E\u0016are the eigenvalues of\nHandTis temperature) as F=F(0)+F(2)+F(4)\u0001\u0001\u0001,\nthe fourth-order contribution F(4)for the four possible\ncandidates are analytically obtained as\nF(4)\n\u0017=J4\n2\u001a\u0012\n1\u00001\n\u0017\u0013\nA+1\n2\u0017B\u001b\n; (14)\nF(4)\nhelical=J4\n2TX\nk!pG2\nk+QG2\nk; (15)6\n-0.2-0.1 0.0 0.1 0.2 0.3 0.4\n-3.8 -3.7 -3.6 -3.5 -3.4 -3.3 -3.2 -3.1 -3.0-0.2 0.0 0.2 0.4 0.6\n-1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1.0\n-0.1 0.0 0.1 0.2\n-5.2 -5.1 -5.0 -4.9 -4.8 -4.7 -4.6 -4.5-0.2 0.0 0.2 0.4 0.6\n 1.6  1.7  1.8  1.9  2.0  2.1  2.2  2.3  2.4helical\n1Q-UUDD\n2Q-UUDD\n3Q-UUDDhelical\n1Q-UUDD\n2Q-UUDD\n3Q-UUDDhelical\n1Q-UUDD\n2Q-UUDDhelical\n1Q-UUDD\n2Q-UUDD(a) (b)\n(c) (d)\nFIG. 3. (Color online) The fourth-order contributions to\nthe free energy, F(4)\n\u0017andF(4)\nhelical in Eqs. (14) and (15), re-\nspectively, divided by J4, as functions of the chemical poten-\ntial\u0016for the ground state candidates at the RKKY level on\n(a), (b) square, and (c), (d) triangular lattices. The param-\neters are (a) t2= 0,Q1= (\u0019=2;0), and Q2= (0;\u0019=2), (b)\nt2= 0, Q1= (\u0019=2;\u0019), and Q2= (\u0019;\u0000\u0019=2), (c)t2= 0,\nQ1= (\u0019=2;0),Q2= (0;\u0000\u0019=2), and Q3= (\u0000\u0019=2;\u0019=2),\nand (d)t2= 0:5,Q1= (\u0019=2;0),Q2= (0;\u0000\u0019=2), and\nQ3= (\u0000\u0019=2;\u0019=2). The data are calculated at T= 0:03.\nThe vertical dashed lines point to the optimal chemical po-\ntential where the bare susceptibility has maxima at the cor-\nresponding wave numbers: (a) \u0016=\u0000p\n2(1 +p\n2), (b)\u0000p\n2,\n(c)\u00002(1 +p\n2), and (d) 2.\nwhere\nA=TX\nk!p(G2\nkGk+Q\u0011Gk+Q\u00110+GkG2\nk+Q\u0011Gk+Q\u0011+Q\u00110\n\u0000GkGk+Q\u0011Gk+Q\u00110Gk+Q\u0011+Q\u00110); (16)\nB=TX\nk!p(G2\nkG2\nk+Q\u0011\u0000GkGk+Q\u0011Gk+2Q\u0011Gk+3Q\u0011\n+ 2GkG2\nk+Q\u0011Gk+2Q\u0011): (17)\nIn Eq. (14), \u0017= 1, 2, and 3 stand for the 1 Q, 2Q, and 3Q\nUUDD states, and Gk(i!p) = [i!p\u0000(\"k\u0000\u0016)]\u00001is non-\ninteracting Green's function, where !pis the Matsubara\nfrequency. In Eqs. (16) and (17), ( \u0011;\u00110) = (1;2) for 2Q\nand (\u0011;\u00110) = (1;2), (1;3), and (2;3) for 3Q.\nFigure 3 shows F(4)in Eqs. (14) and (15) while\nchanging the chemical potential \u0016around the values for\nwhich the RKKY interaction favors magnetic orders with\nQ1= (\u0019=2;0) or Q1= (\u0019=2;\u0019) (the vertical dashed\nlines in each \fgure). For the square lattice case, we\n\fndF(4)\n2< F(4)\n1< F(4)\nhelical, as shown in Figs. 3(a) and\n3(b). The results indicate that the fourth-order con-\ntribution favors the 2 QUUDD states near the partic-\nular \fllings, where the Fermi surfaces are connected by\nQ\u0017. Speci\fcally, the 2 QUUDD states are favored at\nthe electron \flling n\u00180:097 [\u0016\u0018 \u0000p\n2(1 +p\n2)] inFig. 3(a) and n\u00180:506 (\u0016\u0018p\n2) in Fig. 3(b). On\nthe other hand, in the triangular-lattice case, we obtain\nF(4)\n3< F(4)\n2< F(4)\n1< F(4)\nhelical, as shown in Fig. 3(c) for\nt2= 0 and Fig. 3(d) for t2= 0:5. Hence, in these cases,\nthe fourth-order contribution prefers to the 3 QUUDD\nstates: atn\u00180:113 [\u0016\u0018\u00002(1 +p\n2)] in Fig. 3(c) and\nn\u00181:618 (\u0016\u00182) in Fig. 3(d).\nThus, in all cases, higher multiple- Qstates are favored\nby the fourth-order perturbation beyond the RKKY in-\nteraction. The instability toward the multiple- Qstates\nis understood by the (local) gap formation in the band\nstructure of itinerant electrons due to (partial) nesting of\nthe Fermi surfaces. In general, a magnetic order by the\nFermi surface nesting opens a gap at the Fermi surfaces\nconnected by the ordering vector. The multiple- Qorders\nhave more connections than the 1 Qorders, as exempli-\n\fed in Figs. 2(a) and 2(c). The connections therefore lead\nto a larger energy gain owing to the gap opening at the\nmultiple points on the Fermi surfaces. A similar mech-\nanism was discussed for other noncoplanar multiple- Q\norders26,28,29. Therefore, in the small Jlimit, a 2Q(3Q)\nUUDD state is expected to be realized on the square\n(triangular) lattice. However, it is worth noting that, al-\nthough the perturbative arguments above indicate the in-\nstabilities toward the multiple- Qorders, the fourth-order\ncorrections in Eqs. (14) and (15) diverge in the limit of\nT!0. This suggests the breakdown of the perturba-\ntive theory, and hence, we need to carefully check the\nvalidity by complementary methods, such as numerical\nsimulations, as we will discuss in the following sections.\nLet us remark on the comparison between the 1 Q\nUUDD and helical states. The fourth-order perturba-\ntive analysis shows that the energy for the UUDD state\nis always lower than the helical one near the particu-\nlar electron \fllings, as shown in Fig. 3. This is be-\ncause the UUDD state has more perturbative processes\nthan the helical one, as shown in Eqs. (eq:F4UUDD) and\n(eq:freeenergy). The di\u000berence is related with the inver-\nsion symmetry breaking by the helical order. The UUDD\norder opens a local gap in the band structure owing to\nthe multiple processes, whereas the helical one does not.\nThe preference of the 1 QUUDD state with Q=\u0019=2\nthan the helical one was indeed found in the study for\nthe one-dimensional Kondo lattice model33,34. Our per-\nturbative arguments not only support the preference but\nalso show that the tendency is general irrespective of the\nsystem dimensions.\nB. Variational Calculation\nIn order to con\frm the perturbative analysis, we nu-\nmerically examine the ground state of the model in\nEq. (1). In this section, we perform a variational calcu-\nlation: We compare the grand potential at zero tempera-\nture, \n =E\u0000\u0016n(E=hHi=Nis the internal energy per\nsite), for variational states with di\u000berent magnetic orders\nin the localized spins, and determine the lowest energy7\n-3.8 -3.7 -3.6 -3.5 -3.4 -3.3 -3.2 -3.1 -3.0 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1.0\n-5.2 -5.1 -5.0 -4.9 -4.8 -4.7 -4.6 -4.5  1.6  1.7  1.8  1.9  2.0  2.1  2.2  2.3  2.41Q-UUDD\n2Q-UUDD\n3Q-UUDD1Q-UUDD\n2Q-UUDD\n3Q-UUDD1Q-UUDD\n2Q-UUDD(a) (b)\n(c) (d)-5-4-3-2-1 0 1 2\n1Q-UUDD\n2Q-UUDD\n-8-6-4-2 0 2 4 6\n-2-1 0 1\n-6-4-2 0 2 4 6\nFIG. 4. (Color online) The grand potential at zero temper-\nature as functions of the chemical potential \u0016calculated by\nthe variational calculation for the Hamiltonian in Eq. (2) at\nJ= 0:1 on (a), (b) square, and (c), (d) triangular lattices.\nThe grand potential for the 1 Q, 2Q, and 3QUUDD magneti-\ncally ordered states is measured from that for the helical state.\nThe model parameters in each panel correspond to those in\nFig. 3.\nstate. For the variational states, we assume the helical,\n1Q, 2Q, and 3QUUDD states. We consider these ordered\nstates in a sixteen-site unit cell (4 \u00024), and compute \nin the system with 1024 supercells under the periodic\nboundary conditions.\nFigure 4 shows \u0016dependences of the grand potential\natJ= 0:1 for di\u000berent variational states measured from\nthat for the helical ordering. The results support the per-\nturbative results in Fig. 3: In Figs. 4(a)-4(d), the grand\npotential for the 2 Q(3Q) UUDD state gives the lowest\nenergy for the square (triangular) lattice model, in the \u0016\nregions where the fourth-order free energy F(4)becomes\nlowest for the corresponding state, as shown in Figs. 3(a)-\n3(d). We note that the 3 QUUDD states are also fa-\nvored at\u0016\u00182:1 and 2:25, but they may be taken over\nby other states with slightly di\u000berent ordering vectors\ndetermined by the Fermi surface at these values of the\nchemical potential. Thus, the multiple- QUUDD states\nare variationally stable near the electron \fllings where\nthe fourth-order perturbation signals their instabilities.\nC. Langevin Dynamics Simulation\nFor further con\frmation of the multiple- Qstates, we\nperform the KPM-LD simulation. This is an unbiased\nnumerical simulation based on Langevin dynamics30, in\nwhich the kernel polynomial method35is utilized for en-\nabling the calculations for larger system sizes than the\nstandard Monte Carlo simulation combined with the di-\n 0.00 0.05 0.10 0.15 0.20 0.25\n2Q-UUDD\n 0.0\n 0.0\n 0.1\n 0.1\n 0.2\n 0.2\n2\nQ\n-UUDD\n 0.00 0.05 0.10 0.15 0.20 0.25\n 15  16  17  18 11 12 13 14 0  5  10  15  20  25  30  35  40  45 0 5 10 15 20 25 30 35 40 45\n-1.0-0.5 0.0 0.5 1.0 (a)\n(b) (c)\n2Q-UUDD\nFIG. 5. (Color online) (a) Real-space con\fgurations of lo-\ncalized spins and the zcomponents of vector chirality, ( \u001fp\nv)z\n[see Eq. (12)], in the optimized ground state obtained by\nthe KPM-LD simulation for the model in Eq. (1) on the\nsquare lattice. The simulation is done for a 482-site cluster at\n\u0016=\u00001:39 (n'0:522) fort2= 0 andJ= 0:3. Green arrows\nat the vertices of the square lattice represent the directions\nof localized spins, and the color for each square plaquette in-\ndicates the value of ( \u001fp\nv)z. (b) Enlarged picture of (a) in the\ndotted square. (c) Spin structure factor divided by the system\nsize obtained from the spin con\fguration in (a).\nrect diagonalization. We here apply the method to the\nsquare lattice model at \u0016\u0018\u0000p\n2, where the perturba-\ntive and variational calculations coherently point to the\n2QUUDD state, as shown in Figs. 3(b) and 4(b), re-\nspectively. The simulation is done at zero temperature\nfor a 482-site cluster of the square lattice with periodic\nboundary conditions. In the kernel polynomial method,\nwe expand the density of states by up to 4000th order of\nChebyshev polynomials with 144 random vectors which\nare selected by a probing technique36. In the Langevin\ndynamics, we use a projected Heun scheme37for 1000\nsteps with the time interval \u0001 \u001c= 10.\nThe results are shown in Fig. 5. In the simulation, we\ntake a slightly large value of J(J= 0:3) for ensuring\nthe convergence of numerical optimization. Figures 5(a)\nand 5(b) show a snapshot of the con\fgurations of local-\nized spins and vector chirality [Eq. (12)] in the optimized8\nstates. The obtained state coincides well with the 2 Q\nUUDD order with vector ChDW in Fig. 1(f). Indeed, it\nshows eight (two independent) peaks in the spin struc-\nture factor, S(q) = (1=N)P\ni;jSi\u0001Sjeiq\u0001(ri\u0000rj), as shown\nin Fig. 5(c). Thus, the unbiased numerical simulations\nalso support the emergence of multiple- QUUDD states\nin the ground state.\nIV. ELECTRONIC STRUCTURE\nIn this section, we present the electronic band struc-\ntures of the multiple- QUUDD states with ChDW. In\nSec. IV A, we show that ChDW may bring topological\nnature in the band structure; e.g., the massless Dirac\nsemimetal and Chern insulator. We also show that the\nedge states in these ChDW states appear in a peculiar\nway depending on how the ChDW is terminated at the\nedges in Sec. IV B. To elucidate the nontrivial electronic\nstructures by the multiple- Qstates, hereafter, we assume\nthat the spin con\fgurations in Eqs. (10) and (11) remain\nstable for larger J.\nA. Bulk Dispersion\n-4-2 0 2 4Energy\n-6-4-2 0 2 4Energy(a) (b)\n+1+3-3-2+2\n+1\nFIG. 6. (Color online) Typical energy dispersions of (a)\nthe 2QUUDD state on the square lattice and (b) the 3 Q\nUUDD state on the triangular lattice. The parameters are\nt2= 0,J= 1,Q1= (\u0019=2;0), and Q2= (0;\u0019=2) for the 2 Q\nstate, while t2= 0,J= 2,Q1= (\u0019=2;0),Q2= (0;\u0000\u0019=2),\nandQ3= (\u0000\u0019=2;\u0019=2) for the 3 Qstate. The dispersions\nare shown along the symmetric lines in the folded Brillouin\nzones. The lowest thick curves show the occupied bands at\nn= 0:125. In (a), the massless Dirac node appears at k=\n(kx;ky) = (\u0019=4;\u0019=4) atn= 0:125 as well as several other\n\fllings. Meanwhile, the band gap opens at n= 0:125 in (b),\nin addition to several other commensurate \fllings. In (b),\nthe values of the quantized topological Hall conductivity in\nunit ofe2=h, which are obtained when the Fermi level locates\ninside the gap, are also shown in the right side of the panel.\nThe ChDW associated with the multiple- QUUDD\nstate may modulate the electronic structure in a non-\ntrivial way through the spin Berry phase mechanism.\nThis is demonstrated in Fig. 6. Figure 6(a) shows a\ntypical energy dispersion for the 2 QUUDD state withQ1= (\u0019=2;0), and Q2= (0;\u0019=2) on the square lattice\n(t2= 0 andJ= 1). In this case, the spin texture in-\nduces an antiferroic-type ChDW, as shown in Fig. 1(e)\n[see also Fig. 7(a)]. There are sixteen bands and each\nband is doubly degenerate. As shown in the \fgure, the\nlowest band plotted by thick curves touches the higher\nband at the single point at ( \u0019=4;\u0019=4), forming a linear\ndispersion. This is a massless Dirac node, whose electron\n\flling corresponds to n= 0:125. Thus, the 2 QUUDD\nstate with the antiferroic ChDW is a Dirac semimetal\natn= 0:125. The formation of the Dirac node implies\nthat the 2QUUDD state may be stabilized at this com-\nmensurate \flling at nonzero J, although the instability\nin the weak Jlimit occurs at a slightly smaller \flling,\nn\u00180:097, as discussed in the previous sections. Similar\nstabilization with forming Dirac semimetal was discussed\nfor square and cubic lattices20,22,32.\nOn the other hand, the other 2 QUUDD state in\nFig. 1(f) does not show such Dirac nodes; the bands are\nseparated by energy gaps, and the system is a trivial band\ninsulator at n= 0:5, 1:0, and 1:5 (not shown here). The\ndi\u000berence is understood by considering the J!1 limit:\nin the case of Fig. 1(f), itinerant electrons are con\fned in\neach four-site plaquette with nonzero vector chirality be-\ncause of the antiparallel spin con\fguration between the\nplaquettes, whereas they are not in Fig. 1(e).\nFigure 6(b) represents a typical energy dispersion for\nthe 3QUUDD state with Q1= (\u0019=2;0),Q2= (0;\u0000\u0019=2),\nandQ3= (\u0000\u0019=2;\u0019=2) on the triangular lattice ( t2= 0\nandJ= 2), which accompanies a partially ferroic-type\nChDW, as shown in Figs. 1(g) and 1(h). As shown in the\n\fgure, the partially ferroic ChDW leads to a gap open-\ning at the Fermi level for n= 0:125, which is close at\nn\u00180:113 where the instability toward the 3 QUUDD is\nanticipated in the small Jlimit. Similar to the square-\nlattice case above, the gap opening suggests that the\n3QUUDD state may be stable at n= 0:125 for \fnite\nJ. Similar stabilization by gap opening was discussed in\nRefs. 12 and 28. We \fnd that the insulating state is a\ntopologically nontrivial Chern insulator; the lowest band\nacquires the Chern number +1, leading to the quantized\ntopological Hall conductivity, \u001bxy=e2=h(eis the ele-\nmentary charge and his the Planck constant). Hence,\nthe 3QUUDD state with partially ferroic ChDW pro-\nvides a Chern insulator. Note that there are other gaps in\nstates with higher \fllings, and the bands separated by the\ngaps are assigned by the corresponding Chern numbers,\nas shown in the right side of Fig. 6(b). Similar Chern in-\nsulators were discussed for ferroic ChDW in noncoplanar\n3Qstates11,12,32.\nB. Edge States\nRe\recting the topologically nontrivial nature induced\nby ChDW in the multiple- QUUDD states, peculiar edge\nstates are observed, as demonstrated in Fig. 7. Here,\nwe consider the systems with the (100) edges: the 2 Q9\nxy(100) edge\nunit cell(a)\n(d)\n-7-6-5-4\n-4.0\nEnergy(b)\n(c)\nEnergy\n-4.4-3.6-3.2-2.8\nFIG. 7. (Color online) Schematic picture of the system\nwith the (100) edges for (a) square lattice and (b) triangular\nlattice. The dashed boxes represent unit cells used for the\ncalculations of the edge states; the unit cell contains 256 or\n260 sites depending on where the edges are terminated. The\nred (blue) plaquettes represent positive (negative) values of\n(a) vector and (b) scalar chirality. (c), (d) Energy dispersions\nnear the Fermi level at n= 0:125 for (c) the 2 Qand (d) 3Q\nUUDD states for the systems with open edges shown in (a)\nand (b), respectively. In (c)[(d)], the solid and dashed lines\nrepresent the dispersion in the systems with the (A l, Ar) [(Al,\nAr)] and (B l, Br) [(Al, Br)] edges, while the thick red lines\nthe edge states in the systems with the (A l, Ar) [(Al, Ar)].\nUUDD state on the square lattice [Fig. 7(a); see also\nFig. 1(e)] and the 3 QUUDD state on the triangular lat-\ntice [Fig. 7(b); see also Fig, 1(h)]. We adopt the periodic\nboundary condition in the (010) direction. There are\nfour choices for the edges depending on where we cut the\nChDW (two for each edge): A lor Blfor the left edge and\nAror Brfor the right edge, as shown in the \fgures. The\nedge states appear in the electronic structure in a dif-\nferent form depending on the choices, as demonstrated\nbelow.\nFigure 7(c) shows the band dispersions near the Fermi\nlevel atn= 0:125 for the 2 QUUDD state with antifer-\nroic ChDW on the square lattice with the (100) edges. In\nthis case, the edge states, which are represented by the\nthick red lines, appear around the Dirac nodes when we\ntake the (A l, Ar) edges. This is presumably owing to the\nnonzero vector chirality in the plaquettes on the edges.\nIn fact, such edge states do not appear for the (B l, Br)\nedges, where the vector chirality vanishes in the plaque-\nttes on the edges. We note that the two edge states are\ndoubly degenerate each in this case. Meanwhile, for the\n(Al, Br) or (Bl, Ar) edges, one of the two edge states\nshows a similar dispersion to that in the (A l, Ar) edge\nrepresented by thick red lines in Fig. 7(c), while the other\nedge state is similar to that in the (B l, Br) edge (notshown here).\nOn the other hand, the 3 QUUDD state on the trian-\ngular lattice shows gapless chiral edge states traversing\nthe energy gap of the Chern insulator, irrespective of the\nchoice of the edges, as shown in Fig. 7(d). These are\ntopologically-protected edge states, as the partially fer-\nroic ChDW is a Chern insulator with nonzero net compo-\nnent of the scalar chirality. Even in this situation, how-\never, the chiral edge states behave di\u000berently depending\non the choice of the edges. For instance, as shown in\nFig. 7(d), when we change the right edge from A rto Br\nwith keeping the left edge A l, the chiral edge dispersion\nwith a positive slope shows a drastic change. The result\nindicates that we can control the edge currents by the lo-\ncation of the edges, namely, by the phase of ChDW. Such\nphase-dependent edge states suggest a new possibility of\ncontrolling the electronic structures and transport prop-\nerties by nanostructure of ChDW.\nV. SUMMARY AND CONCLUDING REMARKS\nTo summarize, we have investigated the possibility of\nvector and scalar ChDW in itinerant magnets, focusing\non the construction from multiple- Qsuperpositions of the\nUUDD collinear spin structures. We have examined the\nstability of the multiple- QUUDD states in the Kondo\nlattice model with classical localized moments on square\nand triangular lattices, using three complementary meth-\nods: perturbative analysis, variational calculations, and\nLangevin dynamics simulations. Contrary to the com-\nmon belief that the RKKY interaction stabilizes a he-\nlical state, all the results consistently indicate that the\nitinerant systems exhibit the multiple- QUUDD states in\nthe limit of weak spin-charge coupling. This occurs when\nthe Fermi surface is connected by the commensurate or-\ndering vectors that are related with each other by rota-\ntional symmetry compatible to the lattice structure. Al-\nthough they share the stabilization mechanism with the\npreviously studied multiple- Qstates, we showed that the\nmultiple-QUUDD states have greater \rexibility; for in-\nstance, they can accommodate two-dimensional textures\nof vector and scalar chirality. We also found that ChDW\nassociated with the multiple- QUUDD states bring about\nnontrivial topology in the electronic structures, such as\nmassless Dirac semimetals and Chern insulators. In ad-\ndition, we clari\fed that, re\recting the spatial modulation\nof vector and scalar chirality, the peculiar edge states ap-\npear in the topologically nontrivial states, which depend\non how the ChDW are terminated at the edges. The re-\nsults suggest the controllability of edge currents by the\nphase of ChDW.\nFinally, let us comment on the competition between\nthe multiple- QUUDD states and multiple- Qhelical\nstates with one-dimensional stripy ChDW25,26In the\ncurrent study, we found that the multiple- QUUDD\nstate is more stabilized than the multiple- Qhelical ones\nwith stripy ChDW by KPM-LD numerical simulations10\nforQ1= (\u0019=2;\u0019). We also con\frmed that the situa-\ntion is also similar for Q1= (\u0019=2;0) by the variational\ncalculations (not shown here). Such preference is re-\nstricted to particular ordering vectors, Q1= (\u0019=2;0)\norQ1= (\u0019=2;\u0019); for other commensurate wave vec-\ntors, e.g., Q1= (\u0019=l;0) (lis an integer larger than two),\nwe need higher harmonics to constitute collinear orders,\nwhile we do not for helical ones. Let us take an exam-\nple of Q1= (\u0019=3;0). In order to construct a UUUDDD\ncollinear state, one needs an additional ordering vector,\nQ0\n1= (\u0019;0), in addition to Q1= (\u0019=3;0), which leads\nto an energy cost even at the RKKY level. Thus, the\nmultiple-Qhelical states with stripy ChDW will be more\nstable than the multiple- QUUUDDD states, at least, in\nthe weakJlimit. The situation might be turned over\nwhen considering large Jand taking into account other\ncontributions, such as the spin anisotropy and spin-orbitcoupling. Such extensions are left for future study.\nACKNOWLEDGMENTS\nThe authors thank H. 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Matter 22,\n176001 (2010)."    },    {        "title": "1604.01629v1.Constraining_the_supersaturation_density_equation_of_state_from_core_collapse_supernova_simulations___Excluded_volume_extension_of_the_baryons.pdf",        "content": "arXiv:1604.01629v1  [astro-ph.HE]  6 Apr 2016EPJ manuscript No.\n(will be inserted by the editor)\nConstraining the supersaturation density equation of stat e from\ncore-collapse supernova simulations?\nExcluded volume extension of the baryons\nTobias Fischer\nUniversity of Wroc/suppress law, Pl. M. Borna 9, 50-204 Wroclaw, Pola nd\nReceived: date / Revised version: date\nAbstract. In this article the role of the supersaturation density equa tion of state (EOS) is explored in\nsimulations of failed core-collapse supernova explosions . Therefore the nuclear EOS is extended via a one-\nparameter excluded volume description for baryons, taking into account their finite and increasing volume\nwith increasing density in excess of saturation density. Pa rameters are selected such that the resulting\nsupernova EOS represent extreme cases, with high pressure v ariations at supersaturation density which\nfeature extreme stiff and soft EOS variants of the reference c ase, i.e. without excluded volume corrections.\nUnlike in the interior of neutron stars with central densiti es in excess of several times saturation density,\ncentral densities of core-collapse supernovae reach only s lightly above saturation density. Hence, the impact\nof the supersaturation density EOS on the supernova dynamic s as well as the neutrino signal is found to be\nnegligible. It is mainly determined from the low- and interm ediate-density domain, which is left unmodified\nwithin this generalized excluded volume approach.\nPACS. 26.50.+x Nuclear physics aspects of novae, supernovae, and other explosive environments –\n26.60.Kp Equations of state of neutron-star matter – 97.60. Bw Supernovae\n1 Introduction\nA neutron star is born in the violent event of a core-\ncollapse supernova explosion of a star more massive than\nabout 9 M ⊙. It is associated with the revival of the stalled\nbounce shock, which forms when the initially imploding\nstellar core bounces back at supersaturation density, lead-\ning to the ejection of the stellar mantle (for recent reviews\nc.f.Refs.[ 1,2]).Severalscenariosfortheshockrevivalhave\nbeen discovered [ 3,4,5,6]. Among them the most promis-\ning one is due to neutrino heating. However, neutrino-\ndrivenexplosionsgenerallyrequiremulti-dimensionalsim-\nulations where in the presence of convection and poten-\ntially hydrodynamics instabilities the neutrino heating ef-\nficiency increases [ 7,8,9] in comparison to the spherically\nsymmetric case. The exception is the low-mass range, in\nparticular the O-Ne core progenitor of 8.8 M ⊙[10,11,12]\nand the zero-metallicity Fe-core progenitorof 9.6 M ⊙[13].\nThisisahotandactivesubjectofresearch[ 14].Moremas-\nsive progenitors have largely extended silicon layers above\nthe stellar core which make it more difficult to revive the\nstanding bounce shock. The impact of the core-collapse\nsupernova progenitor structure on the supernova dynam-\nics is generally not answered yet [ 15,16].\nSend offprint requests to :The role of the EOS in core-collapse supernova simu-\nlations was explored even in the multi-dimensional frame-\nwork [17,18,19], as well as in failed core-collapse super-\nnova explosions in spherical symmetry [ 20,21,22,23]. Re-\ncently, the role of the nuclear symmetry energy has been\nreviewed [ 27]. Particular focus has been devoted to study\nthe role of the high-density behavior exploring therefore\nthe EOS of Ref. [ 24] which is available for three differ-\nent (in)compressibility modulus [ 26,25]. Unlike in multi-\ndimensional simulations where small variations as initial\nperturbations can grow to large scale effects, e.g., due to\nthe turbulent cascade, in spherically symmetric simula-\ntions the role of the high-density EOS was never reported\nto be significant. Despite large variations of the nuclear\nmatter properties differences of the neutrino signal and\nthe general supernova evolution were on the order of a\nfew %.\nHowever, a systematic study of the supersaturation\ndensity EOS and the impact in core-collapse supernova\nsimulationshasnotbeenconduced,whichistheaimofthis\narticle.Previousstudies exploredselected EOSwhich usu-\nally differ in all nuclear matter properties. It was therefore\nnot possible to exclusively relate results from supernova\nsimulation to the high-density EOS. Here I follow a differ-\nent approach by modifying only the supersaturation den-\nsity EOS of a well selected nuclear relativistic mean-field\n(RMF) model [ 28,29] that has been widely explored in the2 Tobias Fischer: Constraining the supersaturation densit y equation of state from core-collapse supernova simulatio ns?\ncore-collapse supernova community (c.f. Ref. [ 30]). There-\nfore, the novel excluded volume approach introduced in\nRef. [31] is employed here for densities in excess of nuclear\nsaturation density ( ρ0). It modifies the available volume\nof the nucleon gas which effectively adjusts the baryon\nEOS. This setup will allow for a direct identification of\nthe high-density EOS impact on the supernova dynamics\nas well as the neutrino signal. A preliminary version of\nthis novel excluded volume approach has been discussed\nrecently [ 32].\nInthisstudythegeneralrelativisticneutrinoradiation-\nhydrodynamics model AGILE-BOLTZRTRAN is used for\nthesupernovasimulations.Itisbasedonthree-flavorBoltz-\nmann neutrino transport [ 33], being ideal to study the\nearly post-bounce phase prior to the possible onset of\nshock revival.The restrictionto spherical symmetry is not\nproblematic here since the focus of this study is the super-\nsaturationdensity EOSwhere multi-dimensional phenom-\nena can be neglected. Moreover, in this parametric study\nI will analyze results relativeto the reference case – claims\nabout the magnitude of potential observables render irrel-\nevant.\nThe manuscript is organized as follows. In sec. 2I will\nbriefly review the spherically symmetric supernova model,\nincluding weak reactions and EOS. The subsequent sec. 3\nbriefly introduces the excluded-volume mechanism of [ 31]\nthatisappliedtomodifythesupersaturationdensityEOS.\nThese are included in simulations of failed core-collapse\nsupernova simulations which are then analyzed in sec. 4.\nThe paper closes with a summary in sec. 5.\n2 Supernova model\nThecore-collapsesupernovamodel,AGILE-BOLTZTRAN,\nis based on spherically symmetric and general relativis-\ntic neutrino radiation hydrodynamics. It includes angle-\nand energy-dependent three flavor Boltzmann neutrino\ntransport [ 34,35,36]. The implicit method for solving the\nhydrodynamics equations and the Boltzmann transport\nequationonanadaptiveLagrangianmassgridagreedqual-\nitatively well with other methods, e.g., with the multi-\ngroup flux limited diffusion approximation [ 33] and the\nvariable Eddington factor technique [ 37].\nTable 1. Neutrino reactions considered, including references.\nWeak process References\n1 e−+p⇄n+νe [38,39]\n2 e++n⇄p+ ¯νe [38,39]\n3νe+ (A,Z−1)⇄(A,Z) +e−[40]\n4 ν+N⇄ν′+N [41,42,39]\n5ν+ (A,Z)⇄ν′+ (A,Z) [ 41,42]\n6 ν+e±⇄ν′+e±[41], [43]\n7 e−+e+⇄ν+ ¯ν [41]\n8N+N⇄ν+ ¯ν+N+N [44]\n9 νe+ ¯νe⇄νµ/τ+ ¯νµ/τ [45,21]\n10ν+ ¯ν+ (A,Z)⇄(A,Z)∗[46,47]\nν={νe,¯νe,νµ/τ,¯νµ/τ}andN={n,p}2.1 Weak interactions\nTable1lists the set of weak processes considered, includ-\ningreferences.Notethatweakreactionswithheavynuclei,\ne−-captures and (de)excitations, are relevant only dur-\ning the core-collapse phase. Shortly before and after core\nbounce heavy nuclei are not abundant anymore, in par-\nticular in excess of ρ0and at high temperatures on the\norder of 10 MeV. At such conditions the nuclear compo-\nsition is dominated by (partly) dissociated matter with\nfree neutrons and protons as well as light nuclear clusters.\nDuring the (early) post-bounce evolution weak reactions\nwithfreenucleonsarethemostimportantones.Scattering\non neutrons has the largest opacity in the elastic channel\nwhile charged-current absorptions on neutrons for νeand\nprotons for ¯ νehas largest opacity in the inelastic channel.\nThe latter processes also dominate neutrino heating and\ncooling, they determine the success or failure of neutrino-\ndriven explosions in multi-dimensional simulations.\n2.2 Supernova EOS\nThe EOS in supernova simulations has to handle a va-\nriety of conditions. At low densities and temperatures,\ntime-dependent nuclear burning processes determine the\nnuclear composition, for which a α-network is applied of\n20 nuclear species up to56Ni including some neutron-rich\niron-group nuclei. It is sufficient for an accurate energy\ngeneration.Above T∼0.5MeVnuclearstatisticalequilib-\nrium (NSE) is achieved and the EOS depends only on the\nthree independent variables temperature T, density ρ(or\nequivalently the baryon density nB), and electron fraction\nYe. At low densities, the nuclear composition matches the\nidealgasof56Feor56Ni, depending on Ye. With increasing\ndensity and temperatures bulk nuclear matter is reached\ncomposed of free nucleons and light nuclear species, in\nparticular4He. The transition to uniform nuclear matter\nnearρ0is usually modeled via a phase transition within\nthe nuclear EOS intrinsically.\nAGILE-BOLTZTRANhas a flexible EOS module that\ncan handle many currently available baryon EOS. Here I\nselecttherelativisticmean-filed(RMF)EOSfromRef.[ 29]\nwith the RMF parametrization DD2 [ 28,48], henceforth\ndenotedasHS(DD2).InadditiontotheRMFpartHS(DD2)\nis based on the modified NSE for nuclei including the de-\ntailed nuclear composition for several thousand species.\nA comparison with other nuclear statistical models can\nbe found in Ref. [ 49]. The HS EOS have been explored\nin supernova simulations in spherical symmetry for a va-\nriety of RMF parameterizations [ 30,23,27]. In addition\nto the baryons, contributions from e±and photons are\nadded [50].\n3 Excluded volume extension of the nuclear\nEOS at finite TandYe\nIn order to study the super-saturation density EOS in\nsupernova simulations systematically the standard DD2Tobias Fischer: Constraining the supersaturation density equation of state from core-collapse supernova simulation s? 3\n1 1.5 2 2.5 3 3.502468101214\nρ  [1014 g cm−3]P  [MeV fm−3]HS(DD2−EV) (v = +8.0) \nHS(DD2) − ref. EOS (v=0) \nHS(DD2−EV) (v = −3.0)0.080.10.120.140.160.180.2nB  [fm−3]\nρ0\n234560.60.81.01.21.4cs  [c]\nFig. 1. (color online) High-density supernova EOS under con-\nsideration at selected conditions ( T= 5 MeV, Ye= 0.3). The\nvertical dotted line marks saturation density above which t he\nexcluded volume modification is active.\nEOS is extended by taking into account the composite na-\nture of the nucleon. This is hardly possible at the level of\nthe actual degrees of freedom, quarks and gluons. Never-\ntheless, it can be modeled via an excluded-volume mech-\nanism as discussed in Ref. [ 51] in the context of RMF\nmodels. Usually it is based on a linear functional of the\nfollowing form, φi= 1−/summationtext\njvjnj. It depends on the par-\nticle densities njand the volume parameter v j, such that\nthe available volume for the particle species ireduces as\nfollows,Vi=Vφi, withVbeing the total volume of the\nsystem.\nHere, in order to guarantee for a smooth transition the\nGaussian type functional of Ref. [ 31] is employed,\nφ(nB;v) =/braceleftBigg1 ( nB≤ρ0)\nexp/braceleftBig\n−v|v|\n2(nB−ρ0)2/bracerightBig\n(nB> ρ0)(1)\nwhereφn=φp=φis assumed. The only parameter is\nthe effective excluded volume v. This formalism is applied\nfor densities in excess of nuclear saturation density, i.e.\nthe sub-saturation density EOS DD2 remains unmodified\nbeing in excellent agreement with current experimental\nconstraints (c.f. Ref. [ 52]). The modified volume avail-\nable for the nucleons generally affects their particle densi-\nties (nn,np) and consequently also their pressure ( pn,pp)\nand all other EOS quantities. Further details are given in\nRef. [31].\nFurthermore meson and lepton contributions have to\nbe added, both of which are not affected from the ex-\ncluded nucleon volume modification. The other nuclear\nEOS quantities, e.g., the energy per baryon and entropyare modified accordingly in order to ensure thermody-\nnamic consistency as well as being still consistent with\nsaturation properties of nuclear matter. It results in a\nsmooth transition from the reference EOS DD2 to the\nDD2-EV EOS for ρ > ρ0as illustrated in Fig. 1. Here we\nhave the flexibility of choosing the excluded volume pa-\nrameter arbitrarily. It results in stiff and soft EOS with\nhigher and lower pressures at supersaturation densities\nfor v>0 and v <0, compared to the reference case\n(v=0). I select the parameters v=+8.0 fm3(red dashed\nline) and v=–3.0 fm3(black dash-dotted line) as two ex-\ntreme cases, relative to the reference one (blue solid line),\nas illustrated in Fig. 1. The EOS with excluded volume\nmodifications are henceforth denoted as DD2-EV. Note\nthat the nuclear saturation properties remain unmodified,\ne.g., with ρ0= 0.149 fm−3and symmetry energy J=\n31.67MeV.However,quantitieswhichrelatetoderivatives\nare indeed modified, e.g., the (in)compressibility modu-\nlus varies from K≃541 MeV (v=+8.0 fm3) toK≃\n201 MeV (v=–3.0 fm3) compared to the reference case\nK≃243 MeV (v=0).\nBoth DD2-EV versions explored here reach maximum\nneutronstarmassesinagreementwiththecurrentlylargest\nand most precise observational pulsar mass constraints of\nPSR J1614-2230 (1 .97±0.04 M⊙) [53] and PSR J0348-\n043 (2.04±0.04 M⊙) [54]. Let me remark here that the\nparameter v=+8.0 fm3represents indeed the upper limit\nin terms of stiffness, since the speed of sound ( cs) exceeds\nthe speed of light ( c) above some densities (for details\nsee Fig.1). The presence of superluminal speed of sound\nin this model should not become problematic since such\nhigh densities are not obtained during the supernova sim-\nulations considered here, as will be shown in the following\nsection4.\nThisgeneralizedexcludedvolumeapproachaffectsboth\nthe symmetric and asymmetric parts of the EOS above\nsaturation density. Quantities which relate to the sym-\nmetry energy are particularly important for weak interac-\ntionsandthe neutrino transportin supernovasimulations.\nHowever the neutron and proton single particle energies,\nand in particular their difference, are affected from the ex-\ncluded volume only mildly as compared to the reference\nEOS HS(DD2). The same holds for the charged chemical\npotential, i.e. the difference of neutron and proton chemi-\ncal potentials. Note further that the here employed elastic\napproximationfortheexpressionsofcharged-currentweak\ninteractions (reactions (1) and (2) in Table 1) use only\nthese quantities, i.e. difference of the neutron-to-proton\nsingle particle potentials as well as the charged chemical\npotential [ 38,55,56]. Hence we cannot expect any impact\nfrom the inclusion of the excluded volume on the weak\nequilibrium obtained at high densities where the neutri-\nnos are trapped. It is determined from the competition\nof reactions (1) and (2) in Table 1. Towards low densities\nwhere neutrinos decouple the excluded volume is inactive.\nI will return to this point when analyzing results from\ncore-collapse supernova simulations in sec. 4.4 Tobias Fischer: Constraining the supersaturation densit y equation of state from core-collapse supernova simulatio ns?\n0.10.20.30.40.50.60.70.80.91234ρ [1014 g cm−1]\n0.10.20.30.40.50.60.70.80.9−6−5−4−3−2−10u [104 km s−1]\n0.10.20.30.40.50.60.70.80.90.250.30.350.4Ye , YL\nYeYL\n00.10.20.30.40.50.60.70.80.91.002468101214T  [MeV]\nMB      [M⊙]HS(DD2−EV) (v = +8.0) \nHS(DD2) − ref. case (v=0) \nHS(DD2−EV) (v = −3.0) \nFig. 2. (color online) Radial profiles of selected quantities at\ncore bounce with respect to the enclosed baryon mass, com-\nparing the different EOS under investigation.\n4 Simulation results of the early post-bounce\nevolution\nIn the following paragraphs core-collapse supernova sim-\nulation results will be analyzed. Focus is on the early\npost-bounce evolution, i.e. prior to the possible onset of\nshock revival and subsequent explosion. The simulations\nstart from the 18 M ⊙pre-collapse progenitor model form\nRef.[57].Itwasevolvedconsistentlythroughcorecollapse,\nbounce and post bounce phases using AGILE – BOLTZ-\nTRAN. I apply the above introduced EOS with the ex-\ntremely stiff and soft high-density behaviors, HS(DD2-\nEV) with v=+8.0 fm3(red dashed lines) and v=–3.0 fm3\n(black dash-dotted lines) respectively, as well as the ref-\nerence EOS HS(DD2) for which v=0 (blue solid lines). In\nthe following text and figures the units for the excluded\nvolume parameter v will be skipped for simplicity.\nIn Fig.2the first-order impact of the high-density\nEOS on the dynamics of the collapsing stellar core can\nbe identified. For the stiff EOS (v=+8.0) lowest core den-\nsities and temperatures areobtained, compared to the ref-\nerence case, while the soft EOS (v=–3.0) reaches higher\ncore densities and temperatures. Note the shock position\nin the velocity profiles in Fig. 2, separating high-density\nand low-density domain of the central PNS. The high-−0.1 00.10.20.30.40.500.511.522.533.544.5\nt − tbounce  [s]ρcentral  [10    g cm14           −3]HS(DD2−EV) (v = +8.0) \nHS(DD2) − ref. case (v=0) \nHS(DD2−EV) (v = −3.0)\n00.10.20.30.48101214161820Tcentral  [MeV]\nFig. 3. (color online) Post-bounce evolution of central density\nand temperature.\ndensity differences have only little consequences for the\ncore electron fraction and lepton fraction YeandYLre-\nspectively. YLis determined at the moment when neutri-\nnos become trapped, mainly via neutrino scattering on\nheavy nuclei. Since the same weak rates were used for all\nsimulations and since the low-density part is identical for\nall EOS under investigation, the core lepton fraction is\nexpected to remain unaffected (see Fig. 2). The further\ndecrease of Yebeyond neutrino trapping is determined\nfromthenuclearfreesymmetryenergy(forarecentreview\nc.f. [27]). Therefore, the excluded volume modifications of\nthe symmetry energy and associated EOS quantities at\nsuper-saturation density are small. This includes, e.g., the\nneutron-to-proton single particle potential difference and\nthe charged chemical potential. Both of which enter the\nrate expressions used for the weak reactions (1) and (2)\nin Table 1which determine the evolution of the core elec-\ntron fraction Ye. Hence also the evolution towards weak\nequilibrium remains unmodified as compared to the refer-\nence EOS, which is shown via Yeat core bounce in Fig. 2.\nNote the tiny differences of the shock position in Fig. 2\nwhich are due to a slight mismatch in determining the\ncore bounce.\nTable 2. Central density and temperature at selected times.\nt−tbounce vTC ρC\n[s] [fm3] [MeV] [1014g cm−3]\n0 +8.0 12.2 2.69\n0 13.0 3.08\n-3.0 14.0 3.22\n0.5 +8.0 15.8 3.05\n0 17.0 3.71\n-3.0 18.7 4.17Tobias Fischer: Constraining the supersaturation density equation of state from core-collapse supernova simulation s? 5\n0 0.1 0.2 0.3 0.4 0.520406080100120140160\nt − tbounce  [s]Rshock , Rν  [km]\nRshock\nRνe\nRνµ/τHS(DD2−EV) (v = +8.0)\nHS(DD2) − ref. case (v=0) \nHS(DD2−EV) (v = −3.0)\nFig. 4. (color online) Post-bounce evolution of shock radii and\nneutrinospheres.\nThe post-bounce evolution of central density and tem-\nperature is illustrated in Fig. 3for all EOS under inves-\ntigation, and in Table 2they are listed at selected times\nfor better comparison. Note that differences between stiff\n(v=+8.0) and soft (v=–3.0) EOS obtained at core bounce\nremain also during the post-bounce phase, i.e. with signif-\nicantly lower and higher core densities, respectively, com-\npared to the reference case. Note in particular the slow-\ndown of the central density rise for the extremely stiff\nEOS (v=+8.0) which is due to the very steep slope of the\npressure gradient for densities in excess of ρ0. Note that\nunlike inside neutron stars extremely high densities – sev-\neral times ρ0– are generally not obtained during the early\npost bounce phase of core-collapse supernovae. Even for\nthe very soft EOS with v=–3.0 the central density reaches\nonly 4.17×1014g cm−3(1.7×ρ0) att−tbounce>0.5 s.\nMoreover, the central temperature shows only a marginal\nresponse to the excluded volume modification of the high-\ndensity EOS. Temperature differences on the order of 1–\n3 MeV are obtained.\nDespite the large differences of the central density ob-\ntained for the different excluded volume parametrization\nduring the post-bounce simulations, it has only little im-\npact on the PNS structure. Enclosedmass as well as shock\npositions andneutrinosphereradii areonly mildly affected\ntowards later times, t−tbounce>0.3 s (see Fig. 4). The\nrelevant physics of core-collapse supernovae takes place\nat sub-saturation density, i.e. where the evolution of PNS\ncontractions and supernova shock dynamics is determined\nfrom neutrino heating and cooling. In particular, the con-\ntraction behavior of the PNS is driven by the accretion\nof low-density material onto its surface, from the gravi-\ntationally unstable layers above the stellar core. It also\ndefines the neutrino luminosities and spectra of νeand0.1 0.2 0.3 0.401234567Lν  [1052 erg s−1]νe\n¯νe\nνµ / τ\n00.1 0.2 0.3 0.4 0.567891011121314151617\nt − tbounce  [s]〈 Eν 〉 [MeV]\nνe¯νeνµ / τ¯νµ / τHS(DD2−EV) (v = +8.0)\nHS(DD2) − ref. case (v=0) \nHS(DD2−EV) (v = −3.0)\nFig. 5. (color online) Post-bounce evolution of neutrino lumi-\nnosities and average energies, sampled in the co-moving fra me\nof reference at 500 km.\n¯νewhich decouple inside this layer of low-density accu-\nmulated material at the PNS surface. Neutrinos trapped\nat higher densities inside the PNS interior cannot diffuse\nout on timescale on the order of 100 ms. Hence, the con-\ntraction of the high-density part of the PNS cannot affect\nsignificantly the supernova dynamics nor the neutrino sig-\nnal (see therefore Fig. 5).\nOnlyafter t−tbounce>0.3sthe high-densityPNScon-\ntraction starts to affect the low-density envelope, mainly\ndue to a somewhat stronger(weaker) gravitational poten-\ntial for the soft(stiff) EOS which reach higher(lower) cen-\ntral densities. This leads to a slightly faster(slower) shock\nwithdraw (see Fig. 4). Differences can be also identified on\nthe order of less than100 keV lower(higher) average neu-\ntrino energies for the electron flavors for v= +8.0(v=\n−3.0) compared to the reference simulation. Heavy lep-\nton flavor neutrinos, which decouple at generally higher\ndensities, are consequently less affected (see Fig. 5).6 Tobias Fischer: Constraining the supersaturation densit y equation of state from core-collapse supernova simulatio ns?\n5 Summary and conclusions\nIn this article the impact of the high density EOS on the\ndynamics of core-collapsesupernova simulations as well as\non the potentially observable neutrino signal is studied.\nStandard nuclear EOS with hadrons and mesons as de-\ngrees of freedom are based on the point-like quasi-particle\npicture, e.g., within the RMF framework. Such nuclear\nmodel DD2 was employed here as the reference case. Ex-\ntending the simple quasi-particle picture by considering\nthe composite nature of the baryons is not feasible at the\nlevel of quark and gluon degrees of freedom, especially\nunder supernova conditions. In this study their impact is\napproximated via the novel generalized excluded volume\napproachofRef. [ 31]. It results in modificationsofthe well\ncalibrated RMF EOS DD2 at supersaturation densities.\nThe excluded volume supernova EOS versions, HS(DD2-\nEV),dependonlyontheexcludedvolumeparameter.Here\nI select two variationswhich result in a extremely stiff and\nin another extremely soft EOS at supersaturation density.\nHowever, they are adjusted to be still in agreement with\nnuclear constraints, e.g., nuclear matter properties at ρ0\nas well as with observations of ∼2 M⊙neutron stars.\nThe EOS HS(DD2-EV) are explored in simulations of\nfailed core-collapse supernova explosions during the early\npost-bounce phase. This phase is determined by mass ac-\ncretion onto the central PNS, where a thick layer of low-\ndensity material accumulates at the PNS surface. The\nlatter contracts accordingly on a timescale on the order\nof several 100 ms. Unlike initial expectations this study\nconfirms that the high-density domain of the PNS has a\nnegligible impact on the PNS contraction behavior. De-\nspite large differences at supersaturation density the su-\npernova evolution in terms of shock dynamics as well as\nthe neutrino luminosities and energies are affected only\nmarginally and in particular only towards late times. Pre-\nvious studies of the nuclear EOS role in supernova sim-\nulations were based on models which differ in many (if\nnot all) nuclear matter properties. This made it difficult\nto identify the high-density EOS impact on potential su-\npernova observables. With this novel excluded volume ap-\nproach only the supersaturation density EOS is affected\nand in particular the low density EOS of HS(DD2-EV)\nremain unmodified. For the first time this allows for the\ndirect identification of the supersaturation density EOS\ninfluence, despite the non-linearity of hydrodynamics and\nneutrinotransport.Inadditiontothe18M ⊙intermediate-\nmass progenitor discussed above, a low mass progenitor\nof 11.2 M ⊙was considered for the same EOS HS(DD2-\nEV). For this one differences of the PNS evolution are\neven smaller, mainly because central densities are gener-\nally somewhat lower.\nNote that qualitatively similar conclusions were ob-\ntained in previous studies [ 26,25] which were based on\nthe commonly used supernova EOS from Ref. [ 24]. It is\navailable to the community for three different values of\nthe (in)compressibility modulus, K= 180/220/375 MeV.\nIn this sense they explore the stiffness of the EOS, how-\never,also at subsaturation density. The conclusionsdrawn\nform the present analysis are due to significantly largervariations of the (in)compressibility modulus ( K= 201−\n541 MeV). Note that the very soft version with K=\n180 MeV is violating several constraints, e.g., the max-\nimum neutron star mass is too low and it is in large dis-\nagreement with the neutron matter EOS constraint from\nchiral EFT [ 58,59]. The latter constraint is also violated\nfor the version with K= 220 MeV, in particular at low\ndensities relevant for the supernova dynamics (c.f. Fig. 3\nof Ref. [27]).\nFocus of this study relates to conditions where to first\nordermulti-dimensionalphenomenacanbeneglected.Their\nmain contribution is at the low density regime, in terms\nof turbulent hydrodynamics, where the EOS remains un-\nmodified due to the excluded volume. The here presented\nconclusionsareunlikelytochangewhenthemulti-dimensional\nnature of hydrodynamics and neutrino transport is taken\ninto account. Even though the magnitude of here pre-\nsentedobservablesmaywell be altered,relativechangesas\nwell as conclusions are expected to remain qualitatively.\nThe central PNS densities reached during a canoni-\ncal core-collapse supernova post bounce mass accretion\nphase are significantly lower than those of cold neutron\nstars. The difference is due to temperatures in excess of\n10 MeV and the large component of trapped neutrinos of\nall flavors. This, in combination with the here presented\nanalysis, leads to the conclusion that core-collapse super-\nnova studies can be excluded as laboratories to efficiently\nprobe the supersaturation density state of matter for EOS\nthat remain continuous towards higher densities. Alterna-\ntively the presence of a discontinuety, e.g., via a (strong)\n1st-order phase transition at supersaturation density per-\nhaps to deconfined quark matter may be identified via\nthe neutrino signal [ 6,60] and/or complementary via the\ngravitational wave signal.\nAcknowledgement\nSpecial thanks belongs to S. Typel for providing the EOS\ntables. The supernova simulations were performed at the\nLOEWE-Center for Scientific Computing in Frankfurt,\nGermany. The author also acknowledgessupport from the\nPolish National Science Center (NCN) under grant num-\nber UMO-2013/11/D/ST2/02645.\nReferences\n1. H.-Th. Janka, K. Langanke, A. Marek, G. Mart´ ınez-\nPinedo, and B. M¨ uller, Phys. Rep. 442, 38, (2007)\n2. H.-Th. Janka, Annual Review of Nuclear and Particle Sci-\nence62, 407, (2012)\n3. J. M. LeBlanc and J. R. Wilson, Astrophys. J. 161, 541,\n(1970)\n4. A. Burrows, E. Livne, L. Dessart, C. Ott, and J. Murphy,\nAstrophys. J. 640, 878, (2006)\n5. H. A. Bethe and J. R. Wilson, Astrophys. J. 295, 14, (1985)\n6. I. Sagert, T. Fischer, M. Hempel, G. Pagliara, J. Schaffner -\nBielich,et al. , Phys. Rev. Lett. 102, 081101, (2009)\n7. A. Marek and H.-Th. Janka, Astrophys. J. 694, 664, (2009)Tobias Fischer: Constraining the supersaturation density equation of state from core-collapse supernova simulation s? 7\n8. B. M¨ uller, H.-Th. Janka, and A. Heger, Astrophys. J. 761,\n72, (2012)\n9. S. W. Bruenn, A. Mezzacappa, W. R. Hix, E. J. Lentz,\nO. E. 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Suppl. 194, 28 (2011)"    },    {        "title": "1604.06587v1.From_dilute_matter_to_the_equilibrium_point_in_the_energy__density__functional_theory.pdf",        "content": "arXiv:1604.06587v1  [nucl-th]  22 Apr 2016From dilute matter to the equilibrium point in the energy–de nsity–functional theory\nC.J. Yang,1M. Grasso,1and D. Lacroix1\n1Institut de Physique Nucl´ eaire, IN2P3-CNRS, Universit´ e Paris-Sud, F-91406 Orsay Cedex, France\nDue to the large value of the scattering length in nuclear sys tems, standard density–functional\ntheories based on effective interactions usually fail to rep roduce the nuclear Fermi liquid behavior\nboth at very low densities and close to equilibrium. Guided o n one side by the success of the\nSkyrme density functional and, on the other side, by resumma tion techniques used in Effective\nFieldTheories for systems withlarge scatteringlengths, a newenergy–densityfunctional is proposed.\nThis functional, adjusted on microscopic calculations, re produces the nuclear equations of state of\nneutron and symmetric matter at various densities. Further more, it provides reasonable saturation\nproperties as well as an appropriate density dependence for the symmetry energy.\nVarious properties of nuclear matter are intimately re-\nlated to nuclearphenomena. Such link stronglyguides us\nin constraining effective interactions within the energy–\ndensity functional (EDF) theory, especially close to\nthe equilibrium density of symmetric matter ρ0, which\nroughly correspondsto central densities of medium–mass\nand heavy nuclei. Reproducing simultaneously the equa-\ntion of state (EOS) of symmetric and pure neutron mat-\nter is an important step for producing interactions tai-\nlored to treat both stable and neutron–rich unstable nu-\nclei or even, in the most extreme cases, the isospin–\nasymmetric systems located in the crust of neutron stars.\nIn phenomenological EDFs, attention is usually not paid\nto correctly describe the very low–density regime and\nonly density scales of interest in nuclear phenomena are\nexplored.\nIt is known that very dilute neutron matter can be de-\nscribed as an expansion in kNagiven by Lee and Yang\nin Ref. [1], where kNis the neutron Fermi momentum\nandathe neutron-neutron1S0scattering length, equal\nto -18.9 fm. Such low–density regime is well described,\nby construction, in all ab–initio EOSs derived within ef-\nfective field theories (EFTs) [2, 3]. To our knowledge, it\nis however never reproduced by phenomenological EOSs\ndirectly adjusted around the saturation point, such as,\nfor instance, Skyrme [4, 5] or Gogny [6, 7] EOSs, even in\nthosecaseswhereaspecialcareistakeninwelldescribing\nthe EOS of neutron matter [8, 9].\nThe first two terms of Lee-Yang expansion are\nENM\nN=/planckover2pi12k2\nN\n2m/bracketleftbigg3\n5+2\n3π(kNa)+4\n35π2(11−2ln2)(kNa)2/bracketrightbigg\n,\n(1)\nwheremandNare the nucleon mass and the number\nof neutrons, respectively, and the Fermi momentum is\nrelated to the density ρbykN= (3π2ρ)1/3. However,\nthis expression is valid only in the very dilute regime,\nthat is for |akN|<<1. In the nuclear case, owing to\nthe large value of the scattering length, this limit cor-\nresponds to extremely low densities, definitely far from\ntypical nuclear densities.\nRecently, efforts were made to connect phenomenolog-\nical EDFs and EFTs based on contact interactions [10].\nSimilarly to what is done in the many–body Dyson ex-\npansion [11], the total energy computed with EDF theo-t0(MeV fm3)t3(MeV fm4)x0x3\n-1803.93 12911.00 -4.46 -139.40\nTABLE I: Values of fitted parameters x0andx3and corre-\nsponding t0andt3values.\nries can be written as an expansion\nE(kN) =E(1)(kN)+E(2)(kN)+···,(2)\nwhere the leading order E(1)is the mean–field-EDF\n(MF–EDF) energy and E(2)is the second–order pertur-\nbation energy treated with proper regularization. This\napproach was recently applied to obtain beyond–mean–\nfield functionals reproducing nuclear matter properties\n[12, 13]. Within the EDF theory, we discuss here the\npossibility of combining an explicit ρdependence in an\neffective interaction with the constraint of correctly re-\nproducing the very low–density regime, that is, of repro-\nducing the first two terms of the Lee-Yang expansion.\nTo make this, we first analyze symmetric and neutron\nmatter with a simplified Skyrme interaction which con-\ntains the minimal terms for reproducing the saturation\npoint at the leading order of the Dyson equation (MF\napproximation),\nv(/vector r) =t0(1+x0Pσ)δ(/vector r)+1\n6t3(1+x3Pσ)ραδ(/vector r),(3)\nwherePσ=1\n2(1+σ1·σ2) is the spin–exchange operator,\nandt0,t3,x0,x3, andαare parameters.\nThe interaction given by Eq. (3) generates at each\norder of the Dyson equation a given density functional.\nFromtheMFdensityfunctional, onecandeducetheMF–\nEOS for neutron matter, that we report in terms of the\nFermi momentum kN,\nE(1)\nNM\nN=3\n10/planckover2pi12\nmk2\nN+1\n12π2t0(1−x0)k3\nN\n+1\n24t3(1−x3)/parenleftbigg1\n3π2/parenrightbiggα+1\nk3α+3\nN.(4)\nThe term in k3\nNin Eq. (1) has the same kNdependence\nas the contribution generated at leading order by the t02\nterm of the interaction, with the relation\nt0(1−x0) = 4π/planckover2pi12a/m. (5)\nIt is indeed possible to mimic also the following term\n(k4\nN) of the Lee-Yang expansion still using the density\nfunctional provided by the MF. One possibility would\nbe to use the non–local interaction proposed in Ref. [14].\nWefollowhereanotherdirectionandusetheexplicitden-\nsity dependence of the effective interaction. The choice\nα= 1/3 leads to a k4\nNterm in the EOS (such direction\nwas already explored for instance in Refs. [15, 16] in the\ncase of dilute Fermi gases in the unitary regime). For\nα= 1/3, by comparing Eqs. (1) and (4), it must hold\nt3(1−x3) =/planckover2pi12\nm144\n35(3π2)1/3(11−2ln2)a2.(6)\nEqs. (5) and (6) leave in this case only two free parame-\nters in Eq. (3).\nWe write now the MF–EOS for symmetric matter in\nterms of the Fermi momentum kF(kF= (3π2ρ/2)1/3),\nE(1)\nSM\nA=3\n10/planckover2pi12\nmk2\nF+1\n4t0\nπ2k3\nF+1\n16t3/parenleftbigg2\n3π2/parenrightbiggα+1\nk3α+3\nF,(7)\nwhereAindicates the number of nucleons. We adjust\nthe remaining two free parameters of the interaction to\nreproduce a benchmark EOS for symmetric matter, that\nwe have chosen as the SLy5–MF [8] EOS. This fit was\nsuccessfully performed leading to the parameters listed\nin Table I. The incompressibility modulus associated to\nsuch EOS is 240.52 MeV. This adjustment is however\na partial success. It indeed allows us to reproduce by\nconstruction the correct (very) low–density behavior in\nneutronmatterandareasonablesaturationpoint insym-\nmetric matter but, unfortunately, the resulting neutron\nmatter EOS is completely wrong beyond the low–density\nlimit. Toobtaina satisfactoryEOSforneutronmatterat\nhigher densities, the scattering length should be treated\nas a free parameter. However, by adjusting on a bench-\nmark EOS for neutron matter, such fit would provide\nvalues of avery far from -18.9 fm as noted in Ref. [10].\nAlong the line of the expansion given in Eq. (2), we\nthen enrich the functional by adding to the MF EOSs\nthe correspondingsecond–ordercontributions, calculated\nin Refs. [12, 13], by keeping only the finite parts (af-\nter dimensional regularization). For neutron matter, the\nsecond–order contribution is\nE(2)\nNM\nN=c[t2\n0(1−x0)2k4\nN+f03k3α+4\nN+f3k6α+4\nN],(8)\nwherec=m∗(11−2ln2)/(280π4/planckover2pi12) andm∗is the ef-\nfective mass. Expressions for the coefficients f03andf3\nmay be found in Ref. [13]. The second–order correction\nfor neutron matter contains a k4\nNcontribution which is\nprovided by the t2\n0(1−x0)2term. Using the constraint of0 0.2 0.4 0.60.8 1.0 1.2 1.4 1.61.8\n| a kN |0.50.60.70.80.91.01.11.2ENM / EFG SLy5 MF\nLee-Yang \nSchaefer et al.\nQMC AV4\nVery dilute regime\nFIG. 1: (Color online) Neutron matter energy divided by the\nfree gas energy EFG, obtained with the first two terms of Lee-\nYang formula, Eq. (1), (blue dashed line), a resumed formula\nof Ref. [3] (red dotted line), the SLy5–mean–field EOS (black\nsolid line), and the QMC AV4 calculations of Ref. [19] (green\ndot–dashed line).\nEq. (5), onerecoversthesecondtermoftheLee–Yangex-\npression, and could thus guarantee a correct low–density\nbehavior. It is also interesting to observe that, to guar-\nantee that the smallest kNdependences in the EOS are\nk3\nNandk4\nN, as in the Lee-Yang expansion, the parame-\nterαshould be larger than 1/3, which is not always the\ncase for Skyrme forces. By using the second–order EOS\nfor neutron matter, we have performed the adjustment\nof the Skyrme parameters on a benchmark EOS, keep-\ning the additional constraint of reproducing the first two\nterms of the Lee-Yang expansion in the case of neutron\nmatter. In spite of the correct kNdependence provided\nby the second–order contribution, it turns out that such\nfit is not successful, except in the case where the value of\nais kept free. However, this leads to an adjusted value\nofaclose to -1 fm, definitely very far from the physical\none. This direction was then rejected.\nThe first terms of the Lee-Yang expansion provide a\ncorrect behavior for neutron matter only for |akN|<<1.\nTo produce expressions that are meaningful also at typ-\nical nuclear density scales, still keeping the good prop-\nertyofcorrectlyreproducingthe low–densityregime, var-\nious resumed expressions have been proposed in EFT\n[3, 17, 18]. In its simplest form, a resumed formula may\nbe for instance written as [3]\nENM\nN=/planckover2pi12k2\nN\n2m/bracketleftbigg3\n5+2\n3πkNa\n1−6kNa(11−2ln2)/(35π)/bracketrightbigg\n.\nIn Fig. 1, the energy obtained with this expression is\ncomparedto Eq. (1) as well as to recent Quantum Monte\nCarlo (QMC) calculations [19] based on realistic nuclear\nforces [20]. The different curves are in agreement at very\nlow densities. As an illustration, the Skyrme SLy5 MF\nEOS is also shown. It completely fails to reproduce the\nlow density behavior. On the other side, Skyrme EDFs\nare recognized for reproducing remarkably well the equi-3\n0.40.60.81.0ENM/EFG\n0 10 20 30\n|akN|\nFIG. 2: (Color online) Energy of neutron matter divided by\nthe free gas energy EFGobtained with the two fits of this\nwork, YGLO (FP) (blue dashed line) and YGLO (Akmal)\n(red dot–dashed line). The SLy5–MF EOS (black solid line)\nis also plotted together with the QMC AV4 points of Ref. [19]\n(green squares), the Friedman-Pandharipande (FP) results of\nRef. [22] (violet circles), the Akmal et al. results of Ref. [ 23]\n(blue diamonds). The Auxiliary–Field results (AFDMC) of\nRef. [28] and the N2LO calculation of Ref. [29] (HS) are also\nshown, respectively, with black open squares and black open\ncircles. As an indication, 3 values of ρare provided in the\nupper horizontal axis.\nlibriumfeaturesofsymmetricmatterandtheincompress-\nibility modulus. This is in particular due to the explicit\ntwo–body density–dependent term that is introduced in\ntheir expression.\nGuided by the fact that the second–order t0contri-\nbution leads to the correct kNdependence in neutron\nmatter (which can be associated to the second term of\nthe Lee-Yang expansion), guided on one side by the re-\nsumed formulae of Refs. [3, 17, 18], and on the other\nside by the good properties of velocity–dependent and\ndensity–dependent terms in Skyrme forces, we propose\na local density functional that includes resummation to\nall orders in an effective way. We write the energy as\nE=/integraltext\n{K(r) +V(r)}d3r, where Kis the kinetic term.\nThe functional V, to be used for symmetric ( β= 1) and\nneutron ( β= 0) matter, is given by\nV=Bβρ2\n1−Rβρ1/3+Cβρ2/3+Dβρ8/3+Fβρα+2.(9)\nThis functional leads to the following EOS,\nE\nA=Kβ+Bβρ\n1−Rβρ1/3+Cβρ2/3+Dβρ5/3+Fβρα+1,\nwhereAbecomes Nin the case of neutron matter, andKis the kinetic contribution. We denote such functional\nwith the acronym YGLO (for “Yang-Grasso-Lacroix-\nOrsay”). The two parameters BβandRβare fixed by\nimposing to recover the Lee-Yang formula at low den-\nsity (the analog of Eq. (1) for symmetric matter may be\nfound in Refs. [11, 21]). This gives the constraints\nBβ= 2π/planckover2pi12\nm(ν−1)\nνa, Rβ=6\n35π/parenleftbigg6π2\nν/parenrightbigg1\n3\n(11−2ln2)a\nwhereν= 2 (4) is the degeneracy for β= 0 (1), and a\nis the corresponding scattering length. In principle, the\nscatteringlength used forsymmetric matter should be an\naverageover all the channels. Following the discussion of\nRef. [11] (chapter XI), we take only the1S0scattering\nlengths and neglect the spin–triplet3S1neutron–proton\ncontribution. This leads to an average1S0scattering\nlengths,a≃ −20 fm for symmetric matter while for neu-\ntron matter we simply take a=−18.9 fm.\nTheC–term in the denominator would provide (in the\nTaylor expansion) an additional higher–order contribu-\ntion for the Lee-Yang expression. We have however pre-\nferred to keep such term free and use the coefficient as a\nparametertoadjust. Wehavethenaddedexplicitlyother\nterms in the functional, guided by Skyrme–type forces,\nto correctly describe the EOSs at densities of interest in\nnuclear scales. The ρ5/3term mimics a term that would\nbe produced (in a MF scheme) by a velocity–dependent\nzero–rangeinteraction. Inparticular,suchtermturnsout\nto be extremely important to improve the adjustment of\nthe neutron matter EOS in density ranges between the\nvery dilute regime and the saturation density. The ρα+1\nterm in the EOS mimics a term that would be generated\nby a density–dependent two–bodyzero–rangeinteraction\nlike in Skyrme forces.\nWe perform the adjustments, this time using as bench-\nmark microscopic EOSs, by including the constraints to\ndescribe the very low–density regime. Benchmark data\nare: i) For neutron matter, the QMC AV4 results of Ref.\n[19] for values of |akN|<10 (ρ <0.05 fm−3), and two\ndifferent sets of results for |akN|>10: the Friedman et\nal. results (FP) of Ref. [22] or the Akmal et al. results\n(stiffer EOS) of Ref. [23] [we call here the correspond-\ning parameter sets YGLO (FP) and YGLO (Akmal), re-\nspectively]; ii) For symmetric matter, the FP results and\nthose of Akmal et al. are very close from each others and\nwe made a fit using only the FP points. The values of\nthe YGLO parameters are shown in Table III.\nFigure 2 shows the results obtained with the YGLO\nfunctional for neutron matter from the low–density\nregime to densities around saturation. This evolution is\ncompared with the SLy5 MF curve together with several\nrecent ab-initio calculations. We see that, with only four\nadjustable parameters, the new functional gives results\nin agreement with ab-initio calculations over the whole\nrange of densities. In Fig. 3, the EOSs obtained for sym-\nmetric and neutron matter are shown as a function of ρ.\nIn particular, one observes that the saturation properties4\nCβDβ Fβ\n(fm2) (MeV.fm5) (MeV.fm3+3α)\nβ= 0 (FP) 100.87 -9264.18 9571.90\nβ= 0 (Akmal) 70.19 -8377.83 8743.85\nβ= 1 (FP) 8.188 -6624.87 6995.46\nTABLE II: Values of the adjusted parameters obtained for\nthe YGLO functional. In all cases, α= 0.7.\n-15-10-55101520E/A(MeV)\n0.0 0.05 0.1 0.15 0.2 0.25 0.3\nρ(fm−3)\nFIG. 3: EOSs from Akmal et al. [23] (blue diamonds) and\nFriedman et al. (purple circles) in symmetric and neutron\nmatter compared to the YGLO (Akmal) (red dot-dashed\ncurve) and YGLO (FP) (blue dashed curve) results. The dif-\nferent grey dotted curves correspond to the YGLO(FP) EOSs\nobtained for different asymmetry δfrom 0.1 to 0.9 by steps\nof 0.1 (see text).\nare well reproduced. The YGLO saturation density is\n0.1683 fm−3and corresponds to an energy E/A=−15.9\nMeV. The incompressibility modulus is equal to 261.71\nMeV.\nStarting from the two EOSs given above, one could\nuse the standard parabolic approximation to obtain the\nEOS in asymmetric matter. Introducing the asymmetry\nparameter δ= (ρN−ρP)/(ρN+ρP) (ρNandρPbeingthe\nneutron and proton densities, respectively), the energy\nEδis given by\nEδ\nA(ρ) =ESM\nA(ρ)+S(ρ)δ2,\nwhereS(ρ) is the symmetry energy, which can be com-\nputed, within the parabolic approximation, as the dif-\nference of the EOSs of neutron and symmetric matter.\nCorrections beyond such approximation are expected to\nbe small [24, 25]. As an illustration, several EOSs ob-\ntained with different isospin asymmetries, from symmet-\nric matter to pure neutron matter are displayed in Fig.20 40 60 80 100 120\nL (MeV)2832364044S (MeV)From 208Pb \nFrom 68Ni\nFrom 120Sn\n0.1 0.2\nDensity (fm-3)20304050\nS (MeV)YGLO (Akmal)\nYGLO (FP)\nYGLO (FP)YGLO (Akmal)\nFIG. 4: Symmetry energy at saturation density as a function\nof its slope L. The black lines delimit the phenomenolog-\nical area constrained by the experimental determination of\nthe electric dipole polarizability in208Pb. The blue dotted\nlines delimit the area constrained by the same measurement\nin68Ni, and the red dashed lines refer to the measurement\ndone in120Sn. The yellow area is the overlap. Inset: den-\nsity dependence of the Symmetry energy for the two YGLO\nparametrizations of this work.\n3 by employing the YGLO(FP) functional for neutron\nmatter.\nThe density dependence of Sis known to be strongly\nconnected to several nuclear properties of astrophysical\ninterest such as, for instance, the proton fraction in neu-\ntronstarsorto the thicknessofneutron skinsin neutron–\nrichnuclei. In the insetofFig. 4the density dependences\nofSare illustrated for the two YGLO parameterizations.\nThese density dependences are comparable to those re-\nported for instance in Ref. [26]. As expected, the YGLO\nparametrization adjusted on the Akmal et al. data pro-\nvides a stiffer curve. A global quantity that characterizes\nthe symmetry energy evolution around saturation is its\nslopeL= 3ρ0(dS/dρ)ρ=ρ0. We present in Fig. 4 the\npoints (S,L) found in this work with respect to the phe-\nnomenological bands provided in Fig. 5 of Ref. [27].\nThese bands are dictated by the experimental determi-\nnations of the electric dipole polarizability in the nuclei\n208Pb,68Ni, and120Sn. The yellow area is the overlap\nregion of the three bands. We observe that both points\nare located at the lower limit of the yellow area. Note\nthat many phenomenological EDFs are outside this band\n[27].\nIn the present work, inspired by resummation tech-\nniques used in EFT, we propose a local EDF that we\ncall YGLO, able to describe the EOSs of symmetric and\nneutron matter from very low densities to the satura-\ntion density. We show that YGLO describes remarkably\nwell saturation properties of symmetric matter, includ-\ning incompressibility, and leads to a density dependence\nof the symmetry energy coherent with the phenomeno-\nlogical indications provided by the measurement of the\ndipole polarizability in nuclei.5\n[1] T.D. Lee and C.N. Yang, Phys. Rev. 105, 1119 (1957).\n[2] H.W. Hammer and R.J. Furnstahl, Nucl. Phys. A 678,\n277 (2000).\n[3] T. Sch¨ afer, C.-W. Kao, and S.R. Cotanch, Nucl. Phys. A\n762, 82 (2005).\n[4] T.H.R. Skyrme, Philos. Mag. 1, 1043 (1956); Nucl. Phys.\n9, 615 (1959).\n[5] D. Vautherin and D. M. Brink, Phys. Rev. C 5, 626\n(1972).\n[6] D. Gogny, Nucl. Phys. A 237, 399 (1975).\n[7] J. Decharg´ e and D. Gogny, Phys. Rev. C 21, 1568 (1980).\n[8] E. Chabanat, P. Bonche, P. Haensel, J. Meyer, and R.\nSchaeffer, Nucl. Phys. A 627, 710 (1997); 635, 231 (1998);\n643, 441 (1998).\n[9] F. Chappert, N. Pillet, M. Girod, and J.-F. Berger, Phys.\nRev.C 91, 034312 (2015).\n[10] R.J. Furnstahl, Eft for DFT . ” Renormalization Group\nand Effective Field Theory Approaches to Many-Body\nSystems. Springer Berlin Heidelberg, 2012. 133-191.\n[11] A.L. Fetter and J.D. Walecka, Quantum Theory of\nMany–Particle Systems , (McGraw-Hill, NewYork,1971).\n[12] K. Moghrabi, M. Grasso, G. Col` o, and N.V. Giai, Phys.\nRev. Lett. 105, 262501 (2010).\n[13] C.J. Yang, M. Grasso, X. Roca-Maza, G. Col` o, and K.\nMoghrabi, arXiv:1604.06278 [nucl-th].\n[14] A. Gezerlis and G.F. Bertsch, Phys. Rev. Lett. 105,\n212501 (2010).[15] A. Bulgac, Phys. Rev. A 76, 040502R (2007).\n[16] A. Bulgac, M.M. Forbes, and P. Magierski, The Uni-\ntary Fermi Gas: From Monte Carlo to Density Function-\nals, Lecture Notes in Physics, Vol. 836 (Springer-Verlag,\nBerlin, Heidelberg, 2012), Chap. 9, pp. 305-373.\n[17] J. V. Steele, arxiv:nucl-th/0010066v2\n[18] N. Kaiser, Nucl. Phys. A 860, 41 (2011).\n[19] A. Gezerlis and J. Carlson, Phys. Rev. C 81, 025803\n(2010).\n[20] R. B. Wiringa and S. C. Pieper, Phys. Rev. Lett. 89,\n182501 (2002).\n[21] R. F. Bishop, Ann. Phys. 77, 106 (1973).\n[22] B. Friedman and V. Pandharipande, Nucl. Phys. A361,\n502 (1981).\n[23] A. Akmal, V. R. Pandharipande, and D. G. Ravenhall\nPhys. Rev. C 58, 1804 (1998).\n[24] W. Zuo, I. Bombaci, and U. Lombardo, Phys. Rev. C 60,\n024605 (1999).\n[25] W. Zuo, et al., Eur. Phys. J. A 14, 469 (2002).\n[26] A.E.L. Dieperink, Y. Dewulf, D. Van Neck, M. Waro-\nquier, and V. Rodin, Phys. Rev. C 68, 064307 (2003).\n[27] X. Roca-Maza, et al., Phys. Rev. C 92, 064304 (2015).\n[28] S. Gandolfi, A. Yu. Illarionov, S. Fantoni, F. Pederiva,\nand K. E. Schmidt, Phys. Rev. Lett. 101, 132501 (2008).\n[29] K. Hebeler and A. Schwenk Phys. Rev. C 82, 014314\n(2010)."    },    {        "title": "1606.09434v2.Negative_parity_nucleon_excited_state_in_nuclear_matter.pdf",        "content": "arXiv:1606.09434v2  [hep-ph]  13 Oct 2016Negative-parity nucleon excited state in nuclear matter\nKeisuke Ohtani,1,∗Philipp Gubler,2and Makoto Oka1,3\n1Department of Physics, H-27, Tokyo Institute of Technology , Meguro, Tokyo 152-8551, Japan\n2Institute of Physics and Applied Physics, Yonsei Universit y, Seoul 120-749, Korea\n3Advanced Science Research Center, Japan Atomic Energy Agen cy, Tokai, Ibaraki 319-1195, Japan\n(Dated: June 29, 2021)\nSpectral functions of the nucleon and its negative parity ex cited state in nuclear matter are\nstudied using QCD sum rules and the maximum entropy method (M EM). It is found that in-\nmedium modifications of the spectral functions are attribut ed mainly to density dependencies of\nthe/angbracketleftqq/angbracketrightand/angbracketleftq†q/angbracketrightcondensates. The MEM reproduces the lowest-energy peaks of both the positive\nand negative parity nucleon states at finite density up to ρ∼ρN(normal nuclear matter density).\nAs the density grows, the residue of the nucleon ground state decreases gradually while the residue\nof the lowest negative parity excited state increases sligh tly. On the other hand, the positions of\nthe peaks, which correspond to the total energies of these st ates, are almost density independent\nfor both parity states. The density dependencies of the effec tive masses and vector self-energies are\nalso extracted by assuming phenomenological mean-field typ e propagators for the peak states. We\nfind that, as the density increases, the nucleon effective mas s decreases while the vector self-energy\nincreases. The density dependence of these quantities for t he negative parity state on the other\nhand turns out to be relatively weak.\nPACS numbers: 12.38.Lg, 14.20.Dh, 14.20.Gk\nI. INTRODUCTION\nThe question of how nucleons behave in dense matter\nis of great importance both from the point of view of\nnuclear physics and QCD. In particular, the role played\nby the partial restoration of chiral symmetry in nuclear\nmatter and its influence on properties of the nucleonic\nground and excited states has attracted continued inter-\nest. In this context, it is especially worth mentioning the\npotential medium modifications of the negative parity\nnucleon state, which are interesting from the viewpoint\nof the chiral symmetry and ηmesic nuclei. Considering\nthe relation between chiral symmetry and the spectral\nfunctions of chiral partners, the symmetry requires their\nspectral functions to be degenerate if chiral symmetry is\nrestored. Chiral partners among hadronic states as well\nas hadronic spectral functions have been discussed al-\nready a long time ago [1]. Assuming that the chiral part-\nner of the positive parity nucleon ground state N(939) is\nthe lowest lying negative parity state N(1535), the rela-\ntion between the restoration of chiral symmetry and the\nmodifications of N(939) and N(1535) has been investi-\ngated within effective models such aslinear sigma models\n[2, 3]. These studies show that two assignments, namely\nthe naive and mirror assignments, of the chiral transfor-\nmation to the chiral partners lead to different character-\nistic modifications of the physical nucleon states at finite\ndensity.\nPotentially ηnuclear systems, so-called η-mesic nuclei,\nwere first investigated by Haider and Liu [4]. The for-\nmation of η-mesic nuclei is strongly related to in-medium\n∗Electronic address: ohtani.k@th.phys.titech.ac.jpmodificationsof N(939)and N(1535)since the ηNsystem\nstrongly couples to N(1535) and its threshold is close to\nthe mass of N(1535). Such nuclei have been studied both\nin theoretical [5–8] and experimental approaches [9–11].\nThestudies ofmeson-nucleusbound systemsareinterest-\ning because the hadron properties at finite density, which\nare related to the restoration of spontaneous breaking of\nthe chiral symmetry, can be investigated in laboratories.\nIn this paper, we study the spectral functions of both\nthe positive and negative parity nucleons in nuclear mat-\nter using QCD sum rules. This method was initially de-\nveloped and applied to the investigation of the meson\nproperties in vacuum by Shifman et al.[12, 13]. It was\nsubsequently used to study baryonic channels by Ioffe\n[14]. Especially for the nucleon, the analyses were there-\nafter continuously improved over the years by including\nhigherordertermsin theperturbativeWilson coefficients\n[15–20]ornon-perturbativepowercorrections[16,21,22].\nAdditionally, it was pointed out that the nucleon oper-\nator couples to both positive and negative parity states\n[23]. The combined contributions of these states make\nthe analysis complicated and especially spoil the result\nof the negative parity states. This difficulty can be reme-\ndiedbythe methodsofparityprojection, whichwerepro-\nposed by Jido et al. [24] and Kondo et al[25]. In these\nstudies, the αscorrections which are large for the nu-\ncleon channel were not considered. To include these αs\ncorrections in the parity projected sum rules, the present\nauthors have improved the parity projection for baryonic\nQCD sum rules and studied the masses of both positive\nparity and negative parity nucleon states in vacuum [26].\nQCD sum rules also have been used to investigate\nhadron properties in nuclear matter [27–29]. The gener-\nalization of nucleonic QCD sum rules in nuclear matter\nwas first proposed by Drukarev [30]. Since then, many2\nstudies have been carried out for the medium modifica-\ntions of its energy, effective mass and vector self-energy,\nwhich characterize properties of the nucleon in nuclear\nmatter [31, 32]. However, all previous studies have so far\nfocused only on the positive parity state, N(939).\nIn this work, we apply the parity-projected nucleon\nQCD sum rule with the phase-rotated Gaussian kernel,\nalready used for the vacuum previously [26], to the anal-\nyses in nuclear matter. Properties of the nucleon and\nits negative parity excited state are extracted from the\nsum rules with the help of the maximum entropy method\n(MEM). The MEM analysis combined with QCD sum\nrules can provide the most probable spectral function\nwithout any strong constraint on its form and has so far\nbeen successfully applied to the ρmeson vacuum chan-\nnel [33], the nucleon vacuum channel [26, 34] and others\n[35–37]. Assuming the peaks in the spectral functions to\nbe described by in-medium nucleon propagators, we fur-\nthermore investigate the density dependence of the effec-\ntive masses and vector self-energies of both the nucleon\nground state and its negative parity first excited state.\nThe paper is organized as follows. In Sec. II, we con-\nstruct the parity-projectedin-medium nucleon QCD sum\nrules and discuss the behavior of the resulting equations.\nThe results of the analyses are summarized in Sec. III\nwhere the density dependence of the spectral functions\nof both positive and negative parity states, the effective\nmasses and the vector self-energies are presented. Next,\neffects of the uncertainties of the condensates and their\nin-medium behavior on the results are studied in Sec. IV.\nWe additionally discuss the validity of the parity pro-\njection at finite spatial momentum in the same section.\nSummary and conclusions are given in Sec. V.\nII. PARITY-PROJECTED NUCLEON QCD SUM\nRULE IN NUCLEAR MATTER\nA. Parity projection of nucleon QCD sum rules in\nnuclear matter\nThe parity-projected QCD sum rules are constructed\nfrom the “forward-time” correlation function [24]:\nΠm(q0,|/vector q|) =i/integraldisplay\nd4xeiqxθ(x0)\n×/angbracketleftΨ0(ρ,uµ)|T[η(x)η(0)]|Ψ0(ρ,uµ)/angbracketright,(1)\nwhereη(x) is the nucleon interpolating field and\n|Ψ0(ρ,uµ)/angbracketrightrepresents the ground state of nuclear mat-\nter, which is characterized by its velocity uµand the nu-\ncleon density ρ. We assume that |Ψ0(ρ,uµ)/angbracketrightis invari-\nant under parity and time reversal transformations. In\nthe rest frame of nuclear matter, the velocity is given by\nuµ= (1,/vector0). Note that in Ref.[24], this correlator was\ncalled the “old-fashioned” correlator. The essential dif-\nference from the time-ordered correlation function is the\ninsertion of the Heaviside step-function θ(x0) before car-\nrying out the Fourier transform. This correlatorcontainscontributions only from states which propagate forward\nin time. With the help of the Lorentz covariance, par-\nity invariance and time reversal invariance of the nuclear\nmatter ground state, the correlation function can be de-\ncomposed into three components [32]:\nΠm(q0,|/vector q|) =q /Πm1(q0,|/vector q|)+Πm2(q0,|/vector q|)\n+u /Πm3(q0,|/vector q|).(2)\nThe scalar functions Π m1, Πm2and Π m3depend on two\nscalar variables q2andq·u. In what follows, we denote\n(q2,q·u) as (q0,|/vector q|) since we will only work in the rest\nframe of nuclear matter. Note that Π m1, Πm2and Π m3\ncontain information about the in-medium properties of\nboth positive and negative parity states as replacing the\noperator η(x)→γ5η(x) only changes the sign of Π m2.\nTo separate these positive and negative parity con-\ntributions, we multiply the parity projection operators\nP±=γ0±1\n2to the correlator, take the trace over the\nspinor index and thus obtain the parity projected corre-\nlation functions:\nΠ+\nm(q0,|/vector q|)≡q0Πm1(q0,|/vector q|)+Πm2(q0,|/vector q|)\n+u0Πm3(q0,|/vector q|)\nΠ−\nm(q0,|/vector q|)≡q0Πm1(q0,|/vector q|)−Πm2(q0,|/vector q|)\n+u0Πm3(q0,|/vector q|).(3)\nNote that the parity projection can be carried out in\naccordance with that in vacuum because it is based on\ntheinvarianceofthegroundstateofnuclearmatterunder\nparity transformation.\nQCD sum rules are relations between correlators com-\nputed in different regions of q0. Specifically, Π±\nmOPE\nwhich is calculated at a large −q2\n0by the operator prod-\nuct expansion (OPE) and the spectral function, ρ±\nm≡\n1\nπIm[Π±\nm] atq0>0 can be related. Making use of the\nanalyticity of the correlation function, one can construct\nthe parity projected QCD sum rules:\n/integraldisplay∞\n−∞Im[Π±\nmOPE(q0,|/vector q|)]W(q0)dq0\n=π/integraldisplay∞\n0ρ±\nm(q0,|/vector q|)W(q0)dq0.(4)\nHere we have introduced a weighting function W(q0),\nwhich is real at real q0and analytic in the upper half\nof the imaginary plane of q0. The details of the deriva-\ntion of Eq.(4) are discussed in [26].\nB. Operator product expansion in nuclear matter\nIn this subsection, we provide the explicit form of\nΠ±\nmOPEincluding all known αscorrections. For the nu-\ncleon, there are two independent local interpolating op-\nerators:\nη1(x) =ǫabc/bracketleftbig\nuTa(x)Cγ5db(x)/bracketrightbig\nuc(x),(5)3\nη2(x) =ǫabc/bracketleftbig\nuTa(x)Cdb(x)/bracketrightbig\nγ5uc(x).(6)\nHere,a,bandcare color indices, C=iγ0γ2stands for\nthe charge conjugation matrix, while the spinor indices\nare omitted for simplicity. A general interpolating field\ncan be expressed as\nη(x) =η1(x)+βη2(x), (7)\nwhereβis a real parameter. The choice β=−1 is called\nthe Ioffe current, which is widely used in sum rule anal-\nyses studying the nucleon ground state. It is straight-\nforward to obtain the imaginary part of the forward-\ntime correlatorofEq.(1) fromthe time orderedcorrelator\ngiven in the literature [18, 26, 38]. The explicit expres-\nsions are given as\n1\nπIm[q0Πm1OPE(q0,|/vector q|)]\n=C1\n211π4/bracketleftbigg\n1+αs\nπ/parenleftbigg71\n12−ln(q2\nµ2)/parenrightbigg/bracketrightbigg\n×q0(q2)2θ(q0−|/vector q|)\n+C1\n210π2/angbracketleftαs\nπG2/angbracketrightmq0θ(q0−|/vector q|)\n−C1\n2532π2/angbracketleftq†iD0q/angbracketrightm\n×/bracketleftbig\n5q0θ(q0−|/vector q|)−4|/vector q|2δ(q0−|/vector q|)/bracketrightbig\n−C1\n2932π2/angbracketleftαs\nπ(E2+B2)/angbracketrightm\n×/bracketleftbig\nq0θ(q0−|/vector q|)−2|/vector q|2δ(q0−|/vector q|)/bracketrightbig\n+1\n243/parenleftBig\nC2+αs\nπC3/parenrightBig\n/angbracketleftqq/angbracketright2\nmδ(q0−|/vector q|)\n−C4\n233παs\nπ/angbracketleftqq/angbracketright2\nmIm/bracketleftbigg\nln(2|/vector q|)|/vector q|\n(q0+iǫ)2−|/vector q|2\n+ln/parenleftbig\n|/vector q|−(q0+iǫ)/parenrightbigq0\n(q0+iǫ)2−|/vector q|2/bracketrightbigg\n+C1\n243/angbracketleftq†q/angbracketright2\nmδ(q0−|/vector q|)\n−C1\n253π2/angbracketleftq†q/angbracketrightmq2\n0θ(q0−|/vector q|)\n−1\n253π2/parenleftbigg\nC5−C1ln(q2\nµ2)/parenrightbigg\n×αs\nπ/angbracketleftq†q/angbracketrightmq2\n0θ(q0−|/vector q|)\n−C6\n2532π2/angbracketleftq†gσ·Gq/angbracketrightm|/vector q|\n2δ(q0−|/vector q|)\n−C1\n243π2/bracketleftbigg\n/angbracketleftq†iD0iD0q/angbracketrightm+1\n12/angbracketleftq†gσ·Gq/angbracketrightm/bracketrightbigg\n×/parenleftbigg\n−2|/vector q|δ(q0−|/vector q|)+2|/vector q|4\n×Im/bracketleftbig1\n4π|/vector q|2−iǫ·1\n(q0−|/vector q|+iǫ)2/bracketrightbig/parenrightbigg(8)1\nπIm[Πm2OPE(q0,|/vector q|)]\n=−1\n26π2/parenleftBig\nC2+C7αs\nπ/parenrightBig\n/angbracketleftqq/angbracketrightmq2θ(q0−|/vector q|)\n+3C8\n26π2/angbracketleftqgσ·Gq/angbracketrightmθ(q0−|/vector q|)\n−C9\n6π2/bracketleftbigg\n/angbracketleftqiD0iD0q/angbracketrightm+1\n8/angbracketleftqgσ·Gq/angbracketrightm/bracketrightbigg\n×|/vector q|\n2δ(q0−|/vector q|)\n+C2\n25π2/angbracketleftqiD0q/angbracketrightmq0θ(q0−|/vector q|)\n+C2\n233/angbracketleftqq/angbracketrightm/angbracketleftq†q/angbracketrightmδ(q0−|/vector q|)(9)\n1\nπIm[Πm3OPE(q0,|/vector q|)]\n=5C1\n2332π2/angbracketleftq†iD0q/angbracketrightmq0θ(q0−|/vector q|)\n+C1\n2732π2/angbracketleftαs\nπ(E2+B2)/angbracketrightmq0θ(q0−|/vector q|)\n+C1\n233/angbracketleftq†q/angbracketright2\nmδ(q0−|/vector q|)\n−C1\n243π2/angbracketleftq†q/angbracketrightm(q2\n0−|/vector q|2)θ(q0−|/vector q|)\n−1\n243π2/parenleftbigg\nC10−C1ln(q2\nµ2)/parenrightbiggαs\nπ/angbracketleftq†q/angbracketrightm\n×(q2\n0−|/vector q|2)θ(q0−|/vector q|)\n+C6\n263π2/angbracketleftq†gσ·Gq/angbracketrightmθ(q0−|/vector q|)\n−C1\n23π2/bracketleftbigg\n/angbracketleftq†iD0iD0q/angbracketrightm+1\n12/angbracketleftq†gσ·Gq/angbracketrightm/bracketrightbigg\n×|/vector q|\n2δ((q0−|/vector q|),(10)\nwhereq2=q2\n0−/vector q2and the coefficients Ciare defined as\nC1= 5+2β+5β2\nC2= 7−2β−5β2\nC3=325\n18+448\n9β+511\n18β2\nC4=47\n3−10\n3β−61\n3β2\nC5=49\n3+14\n3β+49\n3β2\nC6= 7+10 β+7β2\nC7=15\n2−3β−9\n2β2\nC8= 1−β2\nC9= 2−β−β2\nC10=211\n12+31\n6β+211\n12β2.(11)\nThe matrix elements /angbracketleftO/angbracketrightmstand for the expectation\nvalue of operators Oin nuclear matter.4\nC. QCD condensates at finite nucleon density\nThe correlation functions are characterized by in-\nmedium QCD condensates. While the value of the vec-\ntor quark condensate /angbracketleftq†q/angbracketrightmat density ρis3\n2ρexactly,\nthe other condensates are not precisely determined and\ntheir density dependences may be more complicated. In\nthis paper, we estimate their values in the linear density\napproximation, which is valid at sufficiently low density\n[30, 39]. The in-medium condensates /angbracketleftO/angbracketrightmare in this\napproximation expressed as /angbracketleftO/angbracketrightm=/angbracketleftO/angbracketright0+ρ/angbracketleftO/angbracketrightNwith\nthe vacuum condensates /angbracketleftO/angbracketright0and the nucleon matrix el-\nements/angbracketleftO/angbracketrightN≡ /angbracketleftN|O|N/angbracketright. Each matrix element is eval-\nuated as follows:\n/angbracketleftqq/angbracketrightm=/angbracketleftqq/angbracketright0+ρ/angbracketleftqq/angbracketrightN\n=/angbracketleftqq/angbracketright0+ρσN\n2mq\n/angbracketleftq†q/angbracketrightm=ρ3\n2\n/angbracketleftαs\nπG2/angbracketrightm=/angbracketleftαs\nπG2/angbracketright0+ρ/angbracketleftαs\nπG2/angbracketrightN\n/angbracketleftq†iD0q/angbracketrightm=ρ/angbracketleftq†iD0q/angbracketrightN=ρ3\n8MNAq\n2\n/angbracketleftαs\nπ(E2+B2)/angbracketrightm=ρ/angbracketleftαs\nπ(E2+B2)/angbracketrightN\n=ρ3\n2πMNαs(µ2)Ag\n2\n/angbracketleftqiD0q/angbracketrightm=mq/angbracketleftq†q/angbracketrightm≃0\n/angbracketleftqgσ·Gq/angbracketrightm=/angbracketleftqgσ·Gq/angbracketright0+ρ/angbracketleftqgσ·Gq/angbracketrightN\n≈m2\n0/angbracketleftqq/angbracketrightm\n/angbracketleftq†gσ·Gq/angbracketrightm=ρ/angbracketleftq†gσ·Gq/angbracketrightN\n/angbracketleftq†iD0iD0q/angbracketrightm+1\n12/angbracketleftq†gσ·Gq/angbracketrightm\n=/parenleftbig\n/angbracketleftq†iD0iD0q/angbracketrightN+1\n12/angbracketleftq†gσ·Gq/angbracketrightN/parenrightbig\nρ\n=ρ1\n4M2\nNAq\n3\n/angbracketleftqiD0iD0q/angbracketrightρN+1\n8/angbracketleftqgσ·Gq/angbracketrightm\n=/parenleftbig\n/angbracketleftqiD0iD0q/angbracketrightN+1\n8/angbracketleftqgσ·Gq/angbracketrightN/parenrightbig\nρ\n=ρ3\n4M2\nNe2,\n(12)\nwhereEandBare the color electric and color magnetic\nfields, respectively. /angbracketleftqq/angbracketrightdenotes the averagesoverup and\ndown quarks,1\n2/parenleftbig\n/angbracketleftuu/angbracketright+/angbracketleftdd/angbracketright/parenrightbig\n.\nThe quantities Aq\n2,Ag\n2,Aq\n3,e2can be expressed as mo-\nments of the parton distribution functions [32]. The val-\nuesoftheparametersappearinginEq.(12)isgiveninTa-\nbleI.Theuncertaintiesofthevaluesof mqandσNwillbe\ndiscussed in Sec. IV. Note that the higher-order density\nterms of the chiral condensate /angbracketleftqq/angbracketrighthave been computed\nusingchiralperturbationtheory[40,41]. Thesecontribu-\ntions are however small up to the normal nuclear materparameters values\n/angbracketleftqq/angbracketright0−(0.246±0.002GeV)3[42]\nmq 4.725MeV [43]\nσN 45MeV\n/angbracketleftq†q/angbracketrightm ρ3\n2\n/angbracketleftαs\nπG2/angbracketright00.012±0.0036GeV4[44]\n/angbracketleftαs\nπG2/angbracketrightN−0.65±0.15GeV [45]\nAq\n2 0.62±0.06 [46]\nAg\n2 0.359±0.146 [46]\nAq\n3 0.15±0.02 [46]\ne2 0.017±0.047 [47]\nm2\n0 0.8±0.2GeV2[44]\n/angbracketleftq†gσ·Gq/angbracketrightN−0.33GeV2[45]\nTABLE I: Values of parameters appearing in Eq.(12).\ndensity and thus we do not take them into account in\nthis study.\nD. Phase-rotated Gaussian QCD sum rules\nTo explicitly compute both the left and the right hand\nsides of Eq.(4), we have to specify the kernel W(q0).\nIn a previous study, in which the nucleon properties in\nvacuum were investigated [26], we tested several kinds of\nkernels such as the Borel and Gaussian kernels and found\nthat the phase-rotated Gaussian kernel is most suitable\nfor studying the nucleon ground state and its negative\nparity excitation. As it was pointed out in Ref.[26],\nchoosing an appropriate phase parameter θ, the kernel\nimproves the convergence of the OPE and at the same\ntime suppresses the αscorrections. Moreover, the four\nquark condensate contributions are suppressed with this\nkernel, and therefore the uncertainties caused by the four\nquark condensates, whose values are only weakly con-\nstrained, will not seriously affect the results of the QCD\nsum rule analysis.\nWe will later carry out the sum rule analysis for the\nnucleon at rest relative to nuclear matter and also at\nthe Fermi surface. There is no guarantee that the above\ndesirable features are kept when investigating the in-\nmedium nucleon properties at finite |/vector q|. We, in fact, find\nthat the suppressionofthe contributionsfromthe αscor-\nrectionsbecomes less effective as |/vector q|increases. Therefore,\nwe improve the phase rotated kernel W(q0,|/vector q|) as\nW(s,τ,θ,q 0,|/vector q|) =\n1√\n4πτ1\n2/bracketleftBigg\n(q0−|/vector q|)e−2iθexp/parenleftBig\n−(q2e−2iθ−s)2\n4τ/parenrightBig\n+(q0−|/vector q|)e2iθexp/parenleftBig\n−(q2e2iθ−s)2\n4τ/parenrightBig/bracketrightBigg\n.(13)\ns,τandθare parameters in our QCD sum rule and5\n-3-2-1 0 1 2 3G(s,τ,θ)[10-5GeV6](a)  In vacuum\nG+(s,τ,θ)\nG−(s,τ,θ)\nPerturbative\n<q-q>\n<q  ²q>(b)  ρ= 0.25ρN\n-3-2-1 0 1 2\n-2 -1  0G(s,τ,θ)[10-5GeV6]\ns[GeV2](c)  ρ= 0.5ρN\n-2 -1  0\ns[GeV2](d)  ρ= 0.75ρN\n-2 -1  0  1\ns[GeV2](e)  ρ= 1.0ρN\nFIG. 1: The density dependence of G±\nmOPE(s,τ,θ). The perturbative, chiral condensate /angbracketleftqq/angbracketright, vector quark condensate /angbracketleftq†q/angbracketright\nterms and the total are shown at τ= 0.5GeV4,β=−0.9,θ= 0.108πand|/vector q|= 0. The /angbracketleftq†q/angbracketrightterm stands for the sum of the\nterms proportional to the condensate /angbracketleftq†q/angbracketrightinGm1OPE(s,τ,θ) andGm3OPE(s,τ,θ).\nwhoseanalyzedparameterregionswillbediscussedinthe\nnext section. For |/vector q|= 0, the above kernel is equivalent\nto the one used previously in Ref.[26].\nSubstituting Eqs.(8-10) and(13) intoEq.(4), we finally\nobtain the parity-projected nucleon QCD sum rules as\nG±\nmOPE(s,τ,θ)\n≡/integraldisplay∞\n−∞1\nπIm/bracketleftBig\nΠ±\nmOPE(q0,|/vector q|)/bracketrightBig\nW(s,τ,θ,q 0,|/vector q|)dq0\n= Gm1OPE(s,τ,θ)±Gm2OPE(s,τ,θ)+Gm3OPE(s,τ,θ)\n=/integraldisplay∞\n0ρ±\nm(q0)W(s,τ,θ,q 0,|/vector q|)dq0.\n(14)\nHere, G miOPE(s,τ,θ) (i= 1,2,3) are defined as\nGm1OPE(s,τ,θ) =/integraldisplay∞\n−∞Im/bracketleftbiggq0Πm1OPE(q0,|/vector q|)\nπ/bracketrightbigg\n×W(s,τ,θ,q 0,|/vector q|)dq0\nGm2OPE(s,τ,θ) =/integraldisplay∞\n−∞Im/bracketleftbiggΠm2OPE(q0,|/vector q|)\nπ/bracketrightbigg\n×W(s,τ,θ,q 0,|/vector q|)dq0\nGm3OPE(s,τ,θ) =/integraldisplay∞\n−∞Im/bracketleftbiggΠmuOPE(q0,|/vector q|)\nπ/bracketrightbigg\n×W(s,τ,θ,q 0,|/vector q|)dq0.(15)\nThe functions G±\nmOPE(s,τ,θ) are shown in Fig.1 at τ=\n0.5[GeV4],θ= 0.108πand|/vector q|= 0 for various densities.\nThe qualitative behavior at finite spatial momentum issimilar to that at |/vector q|= 0. In this figure, also shown\nare the perturbative, chiral condensate /angbracketleftqq/angbracketrightand vector\nquarkcondensate /angbracketleftq†q/angbracketrightterms, whicharedominant. From\nEq.(14) and Fig. 1, one sees that the chiral condensate\nterm dominates in vacuum and is responsible for the dif-\nference between G+\nmOPEandG−\nmOPE. This observation\nshowsclearlythatthedifferencebetweenthe positiveand\nnegative-parity spectral functions is caused by the emer-\ngence of the chiral condensate /angbracketleftqq/angbracketright. We also find that, as\nthe density increases, G+\nmOPEbecomes small due to the\ndecrease of the absolute value of /angbracketleftqq/angbracketrightand the increase of\nthe vector quark condensate. On the other hand, G−\nmOPE\nshowsno significant changesince the modifications of the\n/angbracketleftqq/angbracketrightand/angbracketleftq†q/angbracketrightcondensates cancel each other out to a\nlarge degree.\nIII. NUMERICAL ANALYSIS OF THE SUM\nRULES\nA. Spectral functions of the positive and\nnegative-parity states\nWe first discuss the parameter regions of τ,s,θ\nandβused for the analyses of this work. Consider-\ning the form of W(s,τ,θ,q 0,|/vector q|) in Eq.(13), we expect\nthat for small τvalues,G±\nm(s,τ,θ) will retain traces of\nthe peak structures of the spectral function, while at\nlargeτ, it will be dominated by continuum contribu-\ntions. This is because τrepresents the typical energy\nscale over which the kernel W(s,τ,θ,q 0,|/vector q|) averages the6\n 0 2 4 6 8\n 0  1  2ρ(q0)\nq0 [GeV](+)9×10-4\n 0 1 2\n 0  1  2\nq0 [GeV](−)3×10-4\nVacuum\n0.25ρN0.5ρN0.75ρN1.0ρN\nFIG. 2: The positive (left) and negative (right) parity spec tral functions extracted from G±\nmOPE(s,τ,θ) of Eq.(14) by MEM.\nThe red, green, blue, magenta and light blue lines correspon d to the spectral functions at the density 0 .0ρN, 0.25ρN, 0.5ρN,\n0.75ρNand 1.0ρN, respectively. Here ρNdenotes the normal nuclear matter density.\nspectral function. We therefore use several values of\nτ(τ= 0.5,0.75,1.0,1.25,1.5,1.75,2.0 GeV4) simultane-\nously and determine the corresponding parameter region\nofsforeachτ. Theminimumvaluesof satfixedτarede-\nterminedbasedonthe criterionthatthe ratioofthe high-\nest dimensional OPE term to the total G±\nmOPE(s,τ,θ) is\nless than 0 .25. The maximum values of sare chosen to\nsatisfy the condition that the second node of the kernel\nas a function of q0is less than 2.0 GeV because it is dif-\nficult to extract information in the q0region above this\nsecond node due to the suppression and fast oscillation\nof the kernel. The specific values of the minimum and\nmaximum sfor each τare shown in TableII. The values\nofθandβare set to 0 .108πand−0.9 to suppress the\neffects of higher order αscorrections and uncertainties\nof condensates, respectively. For more details about the\nparameterdetermination, we refer the readerto Ref.[26].\nWeapplythemaximumentropymethod(MEM)tothe\nOPEdataofEq.(14)andextractthespectralfunctionsof\nboth positive and negative parity states. The advantage\nofthismethodisthatthemostprobablespectralfunction\ncan be obtained without assuming its specific form such\nasthe“pole+continuum”ansatz[33]. IntheMEManal-\nysis, we however have to introduce the so-called default\nmodelm(q0), which should include our prior knowledge\nof the spectral function. To correctly reflect the spectral\nbehavior both in the high and low energy regions, we use\nthe following default model,\nm(q0) = ¯m(β)1\n1+e(qth−q0)/δ,\n¯m(β) =5+2β+5β2\n128(2π)4(16)\nwhere the values of qthandδare chosen as 3 .0 GeV and\n0.1 GeV, respectively. The factor ¯ m(β) is determined so\nthatm(q0) agrees with the asymptotic behavior of theτ0.50.751.01.251.51.752.0\nsminofG+\nm-2.44-3.92-5.41-6.90-8.39-9.88-11.37\nsmaxofG+\nm0.90-0.10-1.20-2.20-3.40-4.50-5.70\nsminofG−\nm-1.27-2.26-3.27-4.28-5.30-6.32-7.35\nsmaxofG−\nm0.90-0.10-1.20-2.20-3.40-4.50-5.70\nTABLE II: Values of smin/max[GeV2] atβ=−0.9 and fixed\nτ[GeV4].\nspectral function at high energy. For further technical\ndetails of MEM, we refer the reader to [33, 48, 49]. The\nerrors of the OPE data in vacuum σ(s,τ)ρ=0are evalu-\nated based onthe method proposedin Ref.[50], while the\nerrorsσ(s,τ)ρ=ρNin nuclear matter are determined by\nassumingthattherelativeerrorsaredensityindependent,/parenleftbig\nσ(s,τ)/GmOPE(s,τ)/parenrightbig\nρ=ρN= (σ(s,τ)/GmOPE(s,τ))ρ=0.\nWe first analyze the in-medium spectral functions of\nthe nucleons at rest relative to nuclear matter ( /vector q=0).\nThe obtained spectral functions are shown in Fig.2. For\npositive parity, the peak appears at about 910 MeV in\nvacuum. This peak corresponds to the nucleon ground\nstate N(939). As the density increases, the height of the\npeak decreases while the peak position does not change\nmuch. For negative parity, two peaks appear at 1550\nMeV and 1870 MeV in vacuum. The first peak lies close\nto the lowest negativeparity excitation N(1535)and thus\nmostlikelycorrespondstothis state. Note, however,that\nit is generally difficult to disentangle two adjoining peaks\nin a narrow region from QCD sum rule analyses. It is\ntherefore possible that the lowest peak contains contri-\nbutions of the higher N(1650) state. The second peak is\nstatistically less significant than the first and its position\ndoes not correspond to any known1\n2−nucleon excited7\nsates. This peak may therefore be a manifestation of the\ncontinuum. Consistently with the behavior of the OPE\ndatashowninFig.1, thenegativeparityspectralfunction\nis not modified significantly at finite density. These find-\nings indicate that the energies of the lowest lying states\nof both positive and negative parity are almost density\nindependent while the coupling strength of the employed\ninterpolating field to the N(939) state decreases as the\ndensity increases.\nB. Estimation of self energies\nSo far, we have found that the peak positions, namely\nthe total energies of the nucleon and its negative parity\nexcited state, are not sensitive to matter effects up to nu-\nclear matter density. The behavior ofthe nucleon ground\nstate is consistent with the small binding energy per nu-\ncleon of nuclear matter. The results for the negative par-\nity state are on the other hand unexpected because one\nwould naively anticipate that its peak moves towards the\npeak of the positive parity spectral function as the chiral\nsymmetry is partially restored in the nuclear medium.\nThe quantum hadrodynamics (QHD) model has been\nsuccessfully applied to the investigation of nuclei and in-\nmedium nucleon properties [51, 52]. In this framework,\nthe nearly-density-independent single particle energy of\nthe nucleon in nuclei is caused by the cancellation of the\nscalar and vector self-energies. The investigation of the\nself-energies of the negative parity state, to be carried\noutinthissubsectionwithinourQCDsumruleapproach,\nwill hence be similarly helpful to comprehend its remark-\nable behavior.\nConsider the nucleon propagator in nuclear matter,\nG(q0,|/vector q|) =Z′(q0,|/vector q|)\n/negationslashq−M−Σ(q0,|/vector q|)+iǫ,(17)\nwhere Σ( q0,|/vector q|) is the nucleon self-energy and Z′(q0,|/vector q|)\nmeans the renormalization factor of the nucleon wave\nfunction. As in Eq.(2), the self-energy can be decom-\nposed as\nΣ(q) = Σs′(q0,|/vector q|)+Σv′(q0,|/vector q|)/negationslashu+Σq′(q0,|/vector q|)/negationslashq.(18)\nIt turns out to be convenient to redefine the quantities\nΣq′to Σs′, Σv′andZ′as\nM∗≡M+Σs′(q0,|/vector q|)\n1−Σq′=M+Σs(q0,|/vector q|),\nΣv(q0,|/vector q|)≡Σv′(q0,|/vector q|)\n1−Σq′,\nZ(q0,|/vector q|)≡Z′(q0,|/vector q|)\n1−Σq′,(19)\nafter which the nucleon propagator can be described as\nG(q0,|/vector q|) =Z(q0,|/vector q|)/negationslashq− /negationslashuΣv+M∗\n(q0−E+iǫ)(q0+E−iǫ),(20)where\nE= Σv+/radicalbig\nM∗2+/vector q2,E=−Σv+/radicalbig\nM∗2+/vector q2.(21)\nNowweassumethatthephenomenologicalsideofthe nu-\ncleon correlation function is constituted of several sharp\n(zero-width) positive and negative parity states and the\ncontinuum. Then, each scalar function of the forward-\ntime correlation function can be expressed by a sum of\nthe contributions from the individual states as\nq0Πm1(q0,|/vector q|) =/summationdisplay\nn|λ+\nn|2E+\nn\n2/radicalBig\nM∗2\nn++/vector q21\nq0−E+n+iǫ\n+|λ−\nn|2E−\nn\n2/radicalBig\nM∗2\nn−+/vector q21\nq0−E−n+iǫ+···,\n(22)\nΠm2(q0,|/vector q|) =/summationdisplay\nn|λ+\nn|2M∗\nn+\n2/radicalBig\nM∗2\nn++/vector q21\nq0−E+n+iǫ\n−|λ−\nn|2M∗\nn−\n2/radicalBig\nM∗2\nn−+/vector q21\nq0−E−n+iǫ+···,\n(23)\nΠm3(q0,|/vector q|) =/summationdisplay\nn|λ+\nn|2−Σv\nn+\n2/radicalBig\nM∗2\nn++/vector q21\nq0−E+n+iǫ\n+|λ−\nn|2−Σv\nn−\n2/radicalBig\nM∗2\nn−+/vector q21\nq0−E−n+iǫ+···,\n(24)\nwhere|λ±\nn|2are the residues of the n-th states. The\nphenomenological side of the parity projected correlation\nfunctions can thus be expressed as follows:\nΠ±\nm(q0,|/vector q|) =/summationdisplay\nn|λ±2\nn|\n2/radicalBig\nM∗2\nn±+/vector q2(/radicalBig\nM∗2\nn±+/vector q2+M∗\nn±)\nq0−E±n+iǫ\n+|λ∓2\nn|\n2/radicalBig\nM∗2\nn∓+/vector q2(/radicalBig\nM∗2\nn∓+/vector q2−M∗\nn∓)\nq0−E∓n+iǫ+···,\n(25)\nWe next fit the combinations Π m1(q0,|/vector q|)+Πm2(q0,|/vector q|),\nΠm1(q0,|/vector q|)−Πm2(q0,|/vector q|) and Π m3(q0,|/vector q|) to the re-\nspective OPE functions to extract the effective masses\nM∗2\n±and the vector self-energies Σv\n±. To be more\nprecise, we substitute the imaginary parts of Eqs.(22-\n24) into Eq.(15), compute the (trivial) q0integral\nand fit the result to Gm1OPE(s,τ,θ) +Gm2OPE(s,τ,θ),\nGm1OPE(s,τ,θ)−Gm2OPE(s,τ,θ) andGm3OPE(s,τ,θ).\nTo carry out this fit, we keep E±\n0and|λ±\n0|2fixed\nto the values obtained from the MEM analysis of8\n 0 250 500 750 1000\n 0 0.25 0.5 0.75 1Energy [MeV]\nρ/ρN(+)\n 0 400 800 1200 1600\n 0 0.25 0.5 0.75 1\nρ/ρN(−)\nM*\nΣv\nE\nFIG. 3: The density dependence of the effective masses and vec tor self energies of positive (left) and negative parity (ri ght)\nstates. The red, green and blue lines correspond to the effect ive masses, vector self-energies and total energies, respe ctively.\nThe dashed lines are the results in which the four quark conde nsate are assumed to be independent of the density (see IVA).\nG±\nmOPE(s,τ,θ). Specifically, E±\n0is taken at the energy of\nthe peak maximum and |λ±\n0|2is obtained by integrating\nthe spectral function in the region of the corresponding\npeak. The remaining parameters that need to be fitted\nare then the factorsE±\n0+M∗\n0±\n2√\nM∗2\n0±+/vector q2and−Σv\n0+\n2√\nM∗2\n0±+/vector q2, from\nwhich we can extract the effective masses and vector self-\nenergies.\nIn the above fit, one also needs to take the contin-\nuum states [not shown in Eqs.(22-24)] into account. For\nthis purpose, we regard the continuum obtained from\nthe MEM analysis of G+\nmOPE(s,τ,θ) andG−\nmOPE(s,τ,θ)\nas the continuum contributions from Π m1+ Πm2and\nΠm1−Πm2, respectively. Concretely, we assume the\nq0≥1050MeV ( q0≥1750MeV) region to be the contin-\nuum of Π m1+Πm2(Πm1−Πm2). We have checked that\nthe choice of the lower boundaries has no strong effects\non the fitting results. The contribution of the continuum\nstate in Π m3mayfurthermorebe neglected because there\nare no perturbative contributions to this term in the high\nenergy limit.\nThe fit results are given in Fig.3. The left figure shows\nthe behavior for the positive parity state. As the density\nincreases, the effective mass decreases, while the vector\nself-energyincreases. Thevaluesoftheeffectivemassand\nvector self-energy at normal nuclear matter density are\nabout 130 MeV and 770 MeV, respectively. These find-\nings are qualitatively similar to the results of the previ-\nous QCD sum rule analyses [31, 32], while the magnitude\nof the in-medium modifications are larger than those in\nRefs.[31, 32]. The right figure shows the negative parity\neffective mass and self-energy. The density dependences\nof both quantities clearly turn out to be much weaker\nthan those of the positive parity state.The obtained spectral function, effective mass and vec-\ntorself-energyforthenegativeparitystatedifferfromthe\npredictions of the chiral doublet models [2, 3]. The mod-\nels predict that the mass difference between the nucleon\nand its negative parity excited state is reduced at finite\ndensity as it is proportionalto the chiral condensate /angbracketleftqq/angbracketright.\nThe models furthermore predict that both the masses\nmonotonically decrease and finally become degenerate in\nthe chirally restored phase. Therefore, one expects in\nthis framework that the energy of the negative parity\nexcited state moves towards the positive-parity state as\nthe density increases. Our study, however shows that\nthe energy of the negative-parity excited state is almost\ndensity independent. The disagreement between the re-\nsults of the chiral doublet model and QCD sum rules can\nbe traced back to the /angbracketleftq†q/angbracketrightcondensate at finite density.\nThe cancellation between the changes of /angbracketleftqq/angbracketrightand/angbracketleftq†q/angbracketright\nin medium leaves the correlation function (almost) un-\nchanged for the negative-parity sum rule. The behavior\nof the negative-parity nucleon in medium may be exper-\nimentally studied from η-mesic nuclei since their struc-\ntures may be sensitive to the difference between the ener-\ngiesofthe nucleongroundstateand its first negativepar-\nity excitation [6, 7]. It will be interesting to see whether\nsuch an experiment can discriminate between the above\ntwo pictures and whether it can determine which of the\ntwo is realized in nature.\nIV. DISCUSSION\nIn this section, we discuss the uncertainties of the in-\nmedium condensates and their effects on the sum rule\nanalysis results. As we have mentioned in subsection9\nIIB, in-medium condensates are evaluated in this work\nwithin the linear density approximation and their (lin-\near) density dependencies are determined by the values\nofthe quarkmass, partondistribution functions etc. The\nvalues of these quantities have some uncertainties. Fur-\nthermore, the in-medium values of the higher-order con-\ndensates such as the four quark condensates, which are\nusually evaluated using the factorization hypothesis, are\npoorly known because factorization can only be justified\nin the large Nclimit. We also discuss the spatial momen-\ntum dependence of the results and examine the validity\nof the parity projection for the finite momentum case.\nA. Dependence on the in-medium four quark\ncondensates\nFour quark condensates can, just like the chiral con-\ndensate, be related to the spontaneous breaking of chiral\nsymmetry. Their contributions to the OPE expression of\nthe correlation function are given in Eqs.(8-10). In the\ncase of the nucleon QCD sum rules in vacuum, the four\nquark condensates give the dominant non-perturbative\ncontribution to the chiral even part Π 1(q2). The in-\nmedium values of the four quark condensates are only\npoorly constrained because we at present have to rely on\nthe factorizationhypothesis, accordingto which the four-\nquark condensates are given by the square of the chiral\ncondensate. This hypothesis may not be justified even in\nvacuum, while its validity at finite density is even more\nquestionable [16, 53, 54]. The density dependence of the\nfour quark condensates and their effects on nucleon prop-\nerties have been studied previously in Refs.[16, 55–57],\nbut its effect on the lowest negative parity excited state\nis worked out here for the first time.\nIn Eqs.(8-10), three kinds of four quark condensates,\nnamely scalar-scalar /angbracketleftqq/angbracketright2, scalar-vector /angbracketleftqq/angbracketright/angbracketleftq†q/angbracketrightand\nvector-vector /angbracketleftq†q/angbracketright2four quark condensates appear. Our\nintegral kernel of Eq.(13), in fact, eliminates the contri-\nbutions of /angbracketleftqq/angbracketright/angbracketleftq†q/angbracketrightand/angbracketleftq†q/angbracketright2at leading order in αs.\nTheαscorrections of these contributions are not consid-\nered in this study because the /angbracketleftqq/angbracketright/angbracketleftq†q/angbracketrightand/angbracketleftq†q/angbracketright2con-\ndensates are not expected to have large contributions up\nto normal nuclear matter density and thus their αscor-\nrections presumably are numerically small. We therefore\nstudy only the effects of the in-medium modification of\nthe scalar-scalarfour quark condensates to both the pos-\nitive and negative paritynucleon states. Since onlychiral\ninvariant four quark condensates appear in the nucleon\nQCD sum rule with the Ioffe current [ β=−1 in Eq.(7)]\n[56], one could expect that the medium modification of\nthe/angbracketleftqq/angbracketright2condensate may also be small in the vicinity\nof the Ioffe current (we use β=−0.9). Previous stud-\nies actually pointed out that a small density dependence\nof the four quark condensate causes realistic results with\na slightly decreasing total energy of the positive parity\nground state, consistent with our knowledge of nuclear\nphenomenology [32]. Therefore, to test this possibility,we here assume the /angbracketleftqq/angbracketright2condensates to be density in-\ndependent, repeat the previous analysis and compare the\ntwo results.\nThis comparison is shown in Fig.3 as the dashed lines.\nIn these plots, we see that the density dependence of the\n/angbracketleftqq/angbracketright2condensate mainly affects the self-energies of the\npositiveparitystate. Thedensityindependentfourquark\ncondensatecausesthe effective massto increasewhile the\nvector self-energy decreases. On the other hand, the be-\nhavior of the negative parity state remains almost com-\npletely unchanged.\nB. Dependence on the in-medium chiral\ncondensates\nAs we have seen in Fig.1, the chiral condensate /angbracketleftqq/angbracketrightm\nterm contributes dominantly to the phase-rotated QCD\nsum rule of the nucleon. Its density dependence at the\nleading order in nucleon density is determined by the\nratio of πN sigma term to the light quark mass, ξ=\nσN\n2mqare taken as σN= 45MeV, mq= 4.725MeV and\nξ∼=4.76 in Sec. III. However, both σNandmqhave\nsome uncertainties and these precise values are not well\ndetermined [58–65]. Especially for the σNvalue, a recent\ndispersion analysis of πNscattering data gives a rather\nlarge value of σπN= (59±1.9±3.0)MeV [64], while\nanother recent lattice calculation that uses quark masses\nat the physical point obtains a much smaller value of\nσπN= 38(3)(3)MeV [65].\nTo check the dependence of the self-energies on their\nvalues, we additionally consider two cases, namely ξ=\n3.5,5.5, and show the results in Fig.4. The dependence\non the factorization hypothesis for the four quark con-\ndensates is also given in this figure. One sees that the\nuncertainty of ξmainly affects the effective mass and\nvector self-energy of the positive parity state regardless\nof the density dependence of the four quark condensate.\nThe values of the effective mass and vector self-energy at\nξ= 3.5 and 5.5 are changed by about 100 MeV and 70\nMeV, respectively. On the other hand, the other quanti-\nties, namely the total energies of both parity states and\nM∗and Σ vof the negative parity state, appear to be\nfairly insensitive to ξ. The change of ξalso affects the\nheights of the first peaks in the spectral functions of both\nthe positive and negative parity states and thus the val-\nues of the respective residues are modified.\nC. Dependence on three-dimensional momentum\nAs a last point, we investigate in this subsection the\nspatial momentum dependencies of the nucleon and its\nnegative parity excited state. In the previous sections,\nwe have so far carried out the analyses at rest relative\nto nuclear matter while we shall next study the density\ndependence of the total energies, effective masses and10\n 0 250 500 750 1000\n 3.5  4.5  5.5Energy [MeV]\nσN/2mq(+)\n 0 400 800 1200 1600\n 3.5  4.5  5.5\nσN/2mq(−)\nM*\nΣv\nE\nFIG. 4: The σN/2mqdependence of the effective masses and vector self-energies of positive (left) and negative parity (right)\nstates. The red and green lines correspond to the effective ma sses and vector self-energies, respectively. The solid lin es show\nthe results with full density dependence according to the fa ctorization hypothesis for the four quark condensates, whi le the\ndashed lines correspond to the density independent /angbracketleftqq/angbracketright2case.\nvector self-energies of both positive and negative parity\nstates at the Fermi momentum |/vector qf|= (3π2ρN\n2)1\n3.\nThe results are shown in Fig.5 as solid lines, while the\ndashedlinescorrespondtothecaseofthezerospatialmo-\nmentum. The momentum dependencies ofthe total ener-\ngies, effective masses and vector self-energies of positive\nand negativeparitystates turn out to be small. The solid\ncurves are not extended to the normal nuclear matter\ndensity because the MEM analysis of the positive parity\nstates does not work well above ρ= 0.75ρN. The reason\nfor this failure is the rapid decrease of the ground state\nresidue, which is faster than for |/vector q|= 0, making it im-\npossibletoextractthepositiveparityspectralfunction at\nρ=ρN. (The lines of the negative parity state similarly\ncannotbe extendedtothenormalnuclearmatterdensity\nbecause, when extracting the values of the effective mass\nand vectorself-energies, both ofthe positiveand negative\nparity spectral function are needed.) The |/vector q|dependence\nof the OPE part originates from the Wilson coefficients\nwhich depend on the two variables q2andq·u. Only\nterms involving q·ulead to|/vector q|dependencies of the in-\nmedium spectral functions. Such contributions are small\nup to 0.75ρNand hence the solid and dashed curves in\nFig.5 show qualitatively the same behavior. Therefore,\nthe weak |/vector q|dependencies of the total energies, effective\nmasses and vector self-energies are consistent with the\nOPE side of the correlation function. To investigate the\nnucleon properties in more detail, higher order contri-\nbutions to the Wilson coefficients and the condensates\nwould be needed, which is a task that goes beyond the\nscope of this work.\nFinally, we comment on the validity of the parity\nprojection for the non-zero momentum case. As canbe understood from Eq.(25) the parity projection can\nonly be carried out exactly at |/vector q|= 0, meaning that\nΠ±\nm(q0,|/vector q|)onlyreceivescontributionsofstateswithfixed\nparity. On the other hand, the obtained spectral func-\ntionρ±\nmPhys.(q0,|/vector q| /negationslash= 0) may involve some contributions\nof the opposite parity states and thus the peak positions,\nthe effective massesand vectorself-energiescan in princi-\nple be affected by such mixings. The contamination can\nbe estimated from the coefficient of the second term in\nEq.(25). The ratios of the coefficient of the second term\nto its of first term, namely|λ∓2\nn|\n2√\nM∗2\nn∓+/vector q2(/radicalBig\nM∗2\nn∓+/vector q2−\nM∗\nn∓)/|λ±2\nn|\n2√\nM∗2\nn±+/vector q2(/radicalBig\nM∗2\nn±+/vector q2+M∗\nn±), are shown in\nFig.6. In this figure, one observes that for both positive\nand negative parity states the first term is much larger\nthanthesecondandthusthe mixingeffectcaninpractice\nbe ignored for momenta around the Fermi surface.\nV. SUMMARY AND CONCLUSION\nWe have studied the spectral functions of the nucleon\nand its negativeparityexcited state in nuclearmatter us-\ning QCD sum rules and the maximum entropy method.\nAll known first order αscorrections to the Wilson coeffi-\ncientsaretakenintoaccountandthedensitydependences\nof the condensates are treated within the linear density\napproximation. With these inputs, we have constructed\nthe parity-projected in-medium nucleon QCD sum rules\nand have analyzed them with MEM. As a result, we have\nfound that the density dependences of the OPE parts are\ndominatedbythoseofthechiralcondensate /angbracketleftqq/angbracketrightandvec-11\n 0 250 500 750 1000\n 0 0.25 0.5 0.75 1Energy [MeV]\nρ/ρN(+)\n 0 400 800 1200 1600\n 0 0.25 0.5 0.75 1\nρ/ρN(−)\nM*\nΣv\nE\nFIG. 5: The density dependence of the effective masses and vec tor self-energies of positive (left) and negative parity (r ight)\nstates at Fermi momentum. The red, green and blue lines corre spond to the effective masses, vector self-energies and tota l\nenergies, respectively. For comparison, their values at |/vector q|= 0 are shown as dashed lines.\n 0 0.03 0.06 0.09 0.12\n 0  0.25  0.5  0.75Ratio\nρ/ρN\nFIG. 6: The red (green) point corresponds to the ratio of the c oefficient of the second term to its of first term of Π+\nm(q0,|/vector q|)\n(Π−\nm(q0,|/vector q|)) in Eq.(25).\ntor quark condensate /angbracketleftq†q/angbracketright. The difference between the\npositive and negative parity OPE expressions is mainly\ncausedbythechiralcondensateterm,whosesigndepends\non the parity of the respective nucleon state. There-\nfore, the density dependences of the positive and nega-\ntive parityOPEpartsareratherdifferent. As the density\nincreases, the positive parity OPE decreases rapidly be-\ncause the in-medium modifications of /angbracketleftqq/angbracketrightand/angbracketleftq†q/angbracketrightare\nadded up to reduce the OPE part. On the other hand,\nthe negative parity OPE depends little on the density\ndue to the cancellation of these modifications.\nWe have analyzed these OPE data by MEM and ex-\ntracted the spectral functions of positive and negative\nparity, which in the vacuum exhibit sharp peaks near\nthe experimental values of the lowest lying states. The\npositions of these peaks turned out to be almost densityindependent, which means that the total energies of both\nthe positive and negative-parity states are not modified\nby nuclear matter effects up to normal nuclear matter\ndensity. On the other hand, as the density increases,\nthe residue of the positive parity nucleon ground state\ndecreases while that of the negative parity first excited\nstate remains almost unchanged.\nAssuming mean-field type phenomenological nucleon\npropagators, we have next investigated the density de-\npendencesofthe effectivemassesand vectorself-energies.\nFor positive parity, we have found that as the density in-\ncreases the effective mass decreases while the vector self-\nenergy increases. For negative parity, the medium mod-\nifications of these quantities are very small. We have ex-\namined potential effects of the uncertainties of the input\nparameters, namely, the in-medium four quark conden-12\nsates/angbracketleftqq/angbracketright2\nmand the chiral condensate /angbracketleftqq/angbracketrightm, on the re-\nsults of the analyses. It is found that these uncertainties\nmainly affect the effective mass and vector self-energy of\nthe positive-parity ground state. For larger in-medium\nmodifications of these condensates, the effective mass\nand vector self-energy become more pronounced. These\nresults suggest that the effective mass and vector self-\nenergy are strongly correlated to the partial restoration\nof chiral symmetry. For negative parity, the in-medium\nmodificationsarenotmuchaffectedbythedensitydepen-\ndences of both /angbracketleftqq/angbracketrightand/angbracketleftqq/angbracketright2, which suggests that the re-\nsults qualitatively do not depend on our specific choices\nfor the in-medium condensates. We have also investi-\ngated the spatial momentum dependence of the nucleon\nspectra and found that the |/vector q|dependence of the total\nenergies, the effective masses and vector self-energies of\nbothpositiveandnegativeparityaresmallatlowdensity.\nHere, we have restricted ourselves to momenta up to the\nFermi momentum at normal nuclear matter density. We\nhave also discussed the validity of the parity projection\nat finite |/vector q|and showed that, even though parity projec-\ntion is not exact at finite |/vector q|, the mixing contributions\nof opposite parity states are sufficiently small up to the\nFermi momentum.\nThebehaviorsofthepositiveandnegativeparitystates\ncan be attributed mainly to the modifications of the /angbracketleftqq/angbracketright\nand/angbracketleftq†q/angbracketrightcondensates and thus our results indicate that\nboth the /angbracketleftqq/angbracketrightand/angbracketleftq†q/angbracketrightcondensates are important in\ndescribing the in-medium properties of the nucleon andits negative parity excited state. It is however difficult\nto provide an intuitive physical interpretation for these\nfindings. Our results show that the spectral function,\neffective mass and vector self-energy of the negative par-\nity state are not modified significantly up to normal nu-\nclear matter density. 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Hoferichter, J. Ruiz de Elvira, B. Kubis and U.-G.\nMeißner, Phys. Rev. Lett. 115092301 (2015).\n[65] S. Durr et al., Phys. Rev. Lett. 116, 172001 (2016)."    },    {        "title": "1609.02347v1.Exact_dimensional_property_of_density_of_states_measure_of_Sturm_Hamiltonian.pdf",        "content": "arXiv:1609.02347v1  [math.DS]  8 Sep 2016EXACT-DIMENSIONAL PROPERTY OF DENSITY OF\nSTATES MEASURE OF STURM HAMILTONIAN\nYANHUI QU\nAbstract. For frequency αof bounded type and coupling λ >20,\nwe show that the density of states measure Nα,λof the related Sturm\nHamiltonian is exact upper and lower dimensional, however, in general\nit is not exact-dimensional.\n1.Introduction\nSince the work [3], the Sturm Hamiltonian has been extensive ly stud-\nied as a typical model of quasi-periodic Schr¨ odinger opera tor. The Sturm\nHamiltonian is a bounded self-adjoint operator on ℓ2(Z),defined by\n(Hα,λ,θψ)n:=ψn−1+ψn+1+λχ[1−α,1)(nα+θ(mod 1))ψn,\nwhereα∈[0,1]\\Q,λ >0 andθ∈[0,1).α,λ,θare called the frequency,\ncoupling andphase, respectively. It is well-known that the spectrum and the\ndensity of states measure (DOS) of Sturm Hamiltonian are ind ependent of\nθand we denote them by Σ α,λandNα,λ, respectively (see [3, 4] for detail).\nThe fractal dimensions of the spectrum have been studied by m any authors,\nsee [5] for a detailed review. In this paper, we focus on the di mensional\nproperties, especially the exact-dimensional properties ofNα,λ. Let us recall\nthe related definitions.\nAssumeα∈[0,1]\\Qhas continued fraction expansion α= [0;a1,a2,···]\nwithan∈N.If{an:n≥1}is bounded, αis called of bounded type . If\nan=κforn≥N,αis called of eventually constant type .ακ:= [0;κ,κ,···]\nis called of constant type . The most famous frequency of constant type is\nthe inverse of golden number α1= [0;1,1,···] = (√\n5−1)/2.The Sturm\nHamiltonian Hα1,λ,θis called Fibonacci Hamiltonian.\nAssume finite measure µis defined on a compact metric space X. Fix\nx∈X, we define the upperandlowerlocal dimensions of µatxas\ndµ(x) := limsup\nr→0logµ(B(x,r))\nlogranddµ(x) := liminf\nr→0logµ(B(x,r))\nlogr.\n1Ifdµ(x) =dµ(x), we say that the local dimension ofµatxexists and denote\nit bydµ(x). The Hausdorff and packing dimensions of µare defined as\n\n\ndimHµ:= sup{s:dµ(x)≥sforµa.e.x∈X},\ndimPµ:= sup{s:dµ(x)≥sforµa.e.x∈X}.\nIf there exists a constant dsuch thatdµ(x) =d(dµ(x) =d) forµa.e.\nx∈X, then necessarily dim Hµ=d(dimPµ=d). In this case we say that\nµisexact lower (upper) dimensional . If there exists a constant dsuch that\ndµ(x) =dforµa.e.x∈X, then necessarily dim Hµ= dimPµ=d. In this\ncase we say that µisexact-dimensional .\nFixα∈[0,1]\\Qandλ>0, write\ndH(α,λ) := dim HNα,λ, dP(α,λ) := dim PNα,λ.\nThe most prominent model among the Sturm Hamiltonian is the F i-\nbonacci Hamiltonian, which was introduced by physicists to model the qua-\nsicrystal system, see [15, 25]. The dimensional properties of its DOS have\nbeen studied in many works, see for example [28, 6, 7, 8, 26], e specially the\nrecent work [9]. We summarize the results which are related t o our paper\nas follows: Nα1,λis exact-dimensional and\nd(α1,λ) :=dH(α1,λ) =dP(α1,λ) =hνλ\nLyapuνλ, (1.1)\nwhereνλis the measure of maximal entropy of the Fibonacci trace map\nTλand Lyapuνλis the unstable Lyapunov exponent of νλ. d(α1,λ) is an\nanalytic function of λand\nlim\nλ→0d(α1,λ) = 1; lim\nλ→∞d(α1,λ)logλ=−5+√\n5\n4logα1.(1.2)\nGirand [12] and Mei [23] considered the frequency αwith eventually\nperiodic continued fraction expansion. In both papers they showed that\nlimλ→0d(α,λ) = 1 and Nα,λis exact dimensional for small λ. This gener-\nalizes the results in [7]. Munger [24] gave estimation on the optimal H¨ older\nexponent of Nακ,λand obtained asymptotic formula for it. For λ >20,\nQu [27] obtained the dimension formula of Nακ,λsimilar with (1.1), he also\nshowed that Nακ,λis exact-dimensional and obtained similar asymptotic\nbehavior as (1.2). We remark that, for all works mentioned ab ove, the dy-\nnamical method is applicable due to the special types of the f requencies.\nRecently, Jitomirskaya and Zhang [13] showed that if β(α)>0, then\ndP(α,λ) = 1, where β(α) := limsupnlogqn+1\nqnandpn/qnisthen-thcontinued\n2fraction approximation of α. They also constructed specific αwithβ(α)>0\nsuch thatdH(α,λ)<1 whenλ >20. In particular, Nα,λis not exact-\ndimensional for such α. We remark that the set {α∈[0,1]\\Q:β(α)>0}\nhas Hausdorff dimension 0.\nExcept Jitomirskaya-Zhang’s result, almost nothing is kno wn about the\ndimensional property of DOS for general Sturm Hamiltonian. The main\nmotivation of this work is to understand the dimensional pro perty of DOS\nfor general Sturm Hamiltonian. As the first step, we concentr ate on the\nfrequencies of bounded type. Let us write\nB:={α∈[0,1]\\Q:αis of bounded type }.\nOur main result is as follows.\nTheorem 1.1. (i) Assume α∈Bandλ >20, thenNα,λis exact upper\nand lower dimensional. As a consequence, Nα,λis exact-dimensional if and\nonly ifdH(α,λ) =dP(α,λ).\n(ii) There exists α∈Bsuch that Nα,λis not exact-dimensional for λ\nlarge enough.\nRemark 1.1. (i) Like the example in [13], our result reveals a new phe-\nnomenon, which does not occur for frequencies with eventual ly periodic\nexpansions. Our result says more about the regularity of the DOS: although\nit can fail to be exact-dimensional, it is still nice in the se nse that it is exact\nupper and lower dimensional.\n(ii) By [18], dP(α,λ)<1 forα∈B,so our result is different from that in\n[13]. Indeed, all the frequencies in Bare Diophantine, while the frequencies\nconsidered in [13] are Liouville. It is known that Bhas Lebesgue measure\n0, but has Hausdorff dimension 1. Thus the size of Bis relatively big.\n(iii) From fractal geometry point of view, it seems quite non -trivial to\nconstruct a finite measure such that it is exact upper and lowe r dimen-\nsional, nevertheless, it is not exact-dimensional. Here, w e obtain such kind\nof measures naturally.\nIn the following, we roughly describe our idea of proof. The k ey notion\nwe will introduce is the Gibbs-like measure , which is an analog of Gibbs\nmeasure in the non-dynamical setting.\n3It is known that Σ α,λcan be coded by a symbolic space Ω(α)through the\ncoding map πα: Ω(α)→Σα,λ.Moreover\nΣα,λ=/intersectiondisplay\nn≥1/uniondisplay\nw∈Ω(α)\nnBw,\nwhere{Bw:w∈Ω(α)\nn}are intervals related to the n-th periodic approxi-\nmation of the operator Hα,λ,0(see [28, 17, 27]). The symbolic space Ω(α)is\na generalization of subshift of finite type, which is defined b y a sequence of\nalphabets and incidence matrices. In general, there is no dy namic on Ω(α)\nsince the shift map is not invariant. If αis of bounded type, only finitely\nmany alphabets and incidence matrices are needed to constru ct Ω(α).\nOne can define a metric dαon Ω(α)by\ndα(x,y) :=|Bx∧y|.\nIt can be shown that πα: (Ω(α),dα)→(Σα,λ,|·|) is bi-Lipschitz. Define\nµα:= (π−1\nα)∗(Nα,λ),\nthenµαis supported on Ω(α)and has the same dimensional property with\nNα,λsinceπαis bi-Lipschitz. By applying [28], µαhas the following equiv-\nalent definition: define\nµn:=1\n#Ω(α)\nn/summationdisplay\nw∈Ω(α)\nnδxw,\nwherexwis any fixed point in the cylinder [ w]α. Thenµnconverge to µα\nin weak-star topology. This definition suggests that µαis the “measure of\nmaximal entropy” on Ω(α). Itiswell-known that themeasureof maximal en-\ntropy of a subshiftof finite type is a Gibbs measure, hence, we are motivated\nto generalize the notion of Gibbs measure to our non-dynamic al setting.\nBased on this observation, we will define Gibbs-like measure on certain\nsymbolic space like Ω(α)and develop the dimension theory of it. More\nprecisely, at first, we define the abstract symbolic space Ω α; next, we define\nthe notion of Gibbs-like measure and introduce a family of ni ce potentials\nFα,such that for each Φ ∈ Fαwe can associate a Gibbs-like measure; then\nwewill showthatsuchkindofmeasureis exactupperandlower dimensional.\nAs an application of this theory, we show that µαis indeed a Gibbs-like\nmeasure of the potential Φ = {φn:n≥1}defined by\nφn(x) := logqn(α),\n4whereqn(α) is the denominator of the n-th approximation of α(we will\nshow thatqn(α)≤#Ω(α)\nn≤5qn(α)). As a consequence, µαis exact upper\nand lower dimensional, so does Nα,λ.\nTo construct the DOS which is not exact-dimensional, we make use of the\nfact that dim HNα1,λ/\\e}atio\\slash= dim HNα2,λforλlarge enough (see [27]). We will\nconstructafrequency α= [0;a1,a2,···]suchthata1a2···= 1t12τ11t22τ2···.\nBy choosing tiandτicarefully, we can show that the local dimension of Nα,λ\ndoes not exist Nα,λ-a.e., hence Nα,λis not exact-dimensional.\nFinally, we say some words on notations. By an∼bn, we mean that there\nexistsC >1 such that C−1bn≤an≤Cbnfor anyn. Byan∼Dbn, we\nmean that the constant Conly depends on D. Given two measures µ,νon\na measurable space ( X,F),µ≍νmeans that there exists a constant C >1\nsuch thatC−1µ(A)≤ν(A)≤Cµ(A) for anyA∈F.\nThe rest of the paper is organized as follows. In Sect. 2, we de fine the\nsymbolic space and the potentials on it. In Sect. 3, we define G ibbs-like\nmeasure and study its exact-dimensional property. The resu lt in this section\nisof independentinterest. InSect. 4, westudythestructur eof thespectrum\nof Sturm Hamiltonian. In Sect. 5, we apply the result in Sect. 3 to prove\nTheorem 1.1.\n2.Symbolic space and potentials\nThe definitions of this section are motivated both by symboli c dynamical\nsystem and by the structure of the spectrum of Sturm Hamilton ian.\n2.1.Symbolic space. FixM≥1. Define A:={1,···,M}N.Assume\n{A1,···,AM}is a pair-wise disjoint family of alphabets. Write Ai=\n{ai1,···,aini}.We further assume that ni≥2 fori= 1,···,M.Let\nA:=/uniontextM\ni=1Ai, then # A=n1+···+nM.For any pairs ( Ai,Aj), we as-\nsociate anni×njincidence matrix Aijwith entries 0 or 1. For aik∈ Ai\nandajl∈ Aj, we say word aikajladmissible and denote by aik→ajlif\nAij(k,l) = 1,whereB(k,l) denote the ( k,l)-th entry of a matrix B.We\nalways assume the following\nStrongly primitive condition : there exists p0≥2 such that for any\nm≥p0and anya1···am∈ {1,···,M}m,\nAa1a2·Aa2a3···A(m−1)mis a positive matrix . (2.1)\nThe following property will be repeated used later. It follo ws directly\nfrom (2.1).\n5Lemma 2.1. Fixm≥p0anda1···am∈ {1,···,M}m. Take any e∈ Aa1\nandˆe∈ Aam. Then there exists an admissible word w∈/producttextm\ni=1Aaisuch that\nw1=eandwm= ˆe.\nFor eachα=a1a2··· ∈A,define the symbolic space Ωαas\nΩα:={x1x2··· ∈∞/productdisplay\ni=1Aai:xi→xi+1for alli≥1}.\nαis called the indexof Ωα.From the definition, Ω αis completed deter-\nmined by the alphabet sequence ( Aa1,Aa2,···) and incidence matrix se-\nquence (Aa1a2,Aa2a3,···). Givenx∈Ωαandn∈N, writex|n:=x1···xn.\nDefine a metric on/producttext∞\ni=1Aaias\nˆd(x,y) := 2−|x∧y|,\nwherex∧ydenotes the maximal common prefix of xandy, then/producttext∞\ni=1Aai\nbecomes a compact metric space. It is seen that Ω αis a closed subset of/producttext∞\ni=1Aai, hence also compact.\nRemark 2.2. Ifα=κ∞, then there is only one alphabet Aκ, and only one\nincidence matrix Aκκ, which is primitive. In this case, Ω αis a subshift of\nfinite type with alphabet Aκand incidence matrix Aκκ.\nFix/vector a=a1···an∈ {1,···,M}n, define\nΩ/vector a,n:={w1···wn∈n/productdisplay\ni=1Aai:wi→wi+1}.\nForα=a1a2··· ∈A, writeα|n:=a1···an,\nΩα,n:= Ωα|n,nand Ω α,∗:=/uniondisplay\nn≥1Ωα,n.\nAnyw∈Ωα,∗is called an admissible word. Forw∈Ωα,∗,|w|denotes the\nlength ofw. Ifv,w∈Ωα,∗andw=vu,vis called a prefixofwand denoted\nbyv≺w.\nFor anyw∈Ωα,nandm≥0,define the set of descendents of wofm-th\ngeneration as\nΞw,m(α) :={v∈Ωα,n+m:w≺v}. (2.2)\nBy (2.1) and the definition of admissibility, it is seen that\n1≤#Ξw,m(α)≤(#A)m. (2.3)\nGivenw∈Ωα,n, thecylinder [w]αin Ωαis defined as\n[w]α:={x∈Ωα:x|n=w}.\n6It is well-known that [ w]αis both open and closed, and {[w]α:w∈Ωα,∗}\nforms a basis of the topology on Ω α.\nWe denote by M(Ωα) the set of probability measures supported on Ω α.\n2.2.potential and weak Gibbs metric on Ωα.To introduce the Gibbs-\nlike measure and the weak Gibbs metric on Ω α, we need to consider poten-\ntials defined on Ω α.\n2.2.1.Potentials on Ωα.Given a family of continuous functions φn: Ωα→\nR. Write Φ = {φn:n≥1}.Φ is called a potential.\nWe call Φ regularif there exists a constant Crg(Φ)>0 such that for any\nn,m∈N,\n/bardblφn−φm/bardbl∞≤Crg(Φ)|n−m|. (2.4)\nWe say that Φ has bounded variation if there exists a constant Cbv(Φ)≥0\nsuch that for any n∈N,\nsup{|φn(x)−φn(y)|:x,y∈Ωα,x|n=y|n} ≤Cbv(Φ).(2.5)\nwe say that Φ has bounded covariation if there exists a constant Cbc(Φ)≥\n0 such that for any w=uv,˜w= ˜uv∈Ωα,∗and anyx∈[w]α,˜x∈[˜w]α,\n|(φ|w|(x)−φ|u|(x))−(φ|˜w|(˜x)−φ|˜u|(˜x))| ≤Cbc(Φ).(2.6)\nWe call Φ positiveifφn(x)↑+∞for anyx∈Ωα. We call Φ negative, if\n−Φ is positive.\nDenote by/hatwideFαthe set of all regular potentials with bounded variation.\nDenote by Fαthe set of all potentials in /hatwideFαwith bounded covariation.\nDefine\n/hatwideF+\nα:={Φ∈/hatwideFα: Φ is positive }andF+\nα:={Φ∈ Fα: Φ is positive }.\n/hatwideF−\nαandF−\nαcan be defined similarly.\n2.2.2.Weak Gibbs metric on Ωα.The following definition is motivated by\n[11, 14, 2]. Fix a potential Ψ ∈/hatwideF−\nα. For anyw∈Ωα,∗, define\nrw(Ψ) := sup\nx∈[w]αexp(ψ|w|(x)).\nFor anyx,y∈Ωα, define\ndΨ(x,y) :=\n\nrx∧y(Ψ)x/\\e}atio\\slash=y,\n0x=y.(2.7)\n7Lemma 2.3. (i)dΨis a ultrametric on Ωαand it induces the same topology\nonΩαasˆddoes.(Ωα,dΨ)has no isolated point.\n(ii) Define c:= exp(−Cbv(Ψ)−p0Crg(Ψ)),then for any w∈Ωα,n,\nc·rw(Ψ)≤diam([w]α)≤rw(Ψ), (2.8)\nDefineˆc:= exp(−Cbv(Ψ)), then for any x∈[w]α,\nB(x,ˆc·rw(Ψ))⊂[w]α⊂B(x,rw(Ψ)), (2.9)\nwhereBmeans closed ball.\n(iii)(Ωα,dΨ)has Besicovitch’s covering property.\nProof.(i) It is obvious that dΨ(x,y)≥0,dΨ(x,y) = 0 if and only if x=y\nanddΨ(x,y) =dΨ(y,x). Let us check the triangle inequality.\nClaim 1: Ifx∧z≺x∧y, thendΨ(x,y)≤dΨ(x,z).\n⊳Writem:=|x∧z|andn:=|x∧y|, thenm≤n.Assumet∈[x∧y]αis\nsuch thatdΨ(x,y) = exp(ψn(t)), we have ψn(t)≤ψm(t) since Ψ is negetive.\nSincet∈[x∧y]α⊂[x∧z]α, we conclude that dΨ(x,y)≤dΨ(x,z).⊲\nGivenx,y∈Ωα,without loss of generality, we assume x/\\e}atio\\slash=y.Assume\n|x∧y|=n, thenxn+1/\\e}atio\\slash=yn+1. Take any z∈Ωα.Ifz|n/\\e}atio\\slash=x∧y,then\nx∧z≺x∧y. By Claim 1, dΨ(x,y)≤dΨ(x,z).Ifz|n=x∧y, then at\nleast one of x∧zandy∧zis equal to x∧ysincexn+1/\\e}atio\\slash=yn+1. Hence\ndΨ(x,y)≤max{dΨ(x,z),dΨ(y,z)}.ThusdΨis a ultrametric on Ω α.\nWriteψ∗\nn= max{ψn(x) :x∈Ωα}andψn∗= min{ψn(x) :x∈Ωα}. Since\nΨ is negative, we conclude that ψ∗\nn↓ −∞asn→ ∞.By the definition, we\nhave\nexp(ψ|x∧y|∗)≤dΨ(x,y)≤exp(ψ∗\n|x∧y|) and ˆd(x,y) = 2−|x∧y|.\nFrom this, it is easy to show that dΨandˆdinduce the same topology.\nFix anyx∈Ωα,by Lemma 2.1, we can construct a sequence {yk:k≥1}\nsuch thatx/\\e}atio\\slash=ykand|x∧yk| ↑ ∞,thusdΨ(x,yk)→0. This means that x\nis not isolated.\n(ii) Ifx,y∈[w]α,then [x∧y]α⊂[w]α. Since Ψ is negetive,\ndΨ(x,y) = sup\nz∈[x∧y]αexp(ψ|x∧y|(z))≤sup\nz∈[w]αexp(ψ|w|(z)) =rw(Ψ).(2.10)\nOn the other hand, by Lemma 2.1, there exist x,y∈[w]αsuch thatx∧y=\nwuwithm:=|u| ≤p0.Assumet∈[x∧y]αis such that exp( ψ|w|+m(t)) =\ndΨ(x,y) and˜t∈[w]αis such that exp( ψ|w|(˜t)) = supz∈[w]αexp(ψ|w|(z)).\n8Then by the regular property and bounded variation of Ψ,\ndΨ(x,y) = exp( ψ|w|+m(t)) = exp(ψ|w|(t))exp(ψ|w|+m(t)−ψ|w|(t))\n≥exp(ψ|w|(t))exp(−mCrg(Ψ))≥exp(ψ|w|(t))exp(−p0Crg(Ψ))\n≥exp(ψ|w|(˜t))exp(ψ|w|(t)−ψ|w|(˜t))exp(−p0Crg(Ψ))\n≥exp(ψ|w|(˜t))exp(−Cbv(Ψ)−p0Crg(Ψ)).\nThen (2.8) holds.\nNow fixx∈[w]α, by (2.10), [ w]α⊂B(x,rw(Ψ)).Ify/\\e}atio\\slash∈[w]α, thenx∧yis\na strict prefix of wand hencem:=|x∧y|<|w|.Assumeτ∈[x∧y]αis such\nthatdΨ(x,y) = exp(ψm(τ)) and ˜τ∈[w]αis such that rw(Ψ) = exp(ψ|w|(˜τ)).\nSince Ψ is negative and has bounded variation, we have\ndΨ(x,y) = exp( ψm(τ)) = exp(ψm(˜τ))exp(ψm(τ)−ψm(˜τ))\n≥exp(ψm(˜τ))exp(−Cbv(Ψ))\n= exp(ψ|w|(˜τ))exp(ψm(˜τ)−ψ|w|(˜τ))exp(−Cbv(Ψ))\n≥exp(−Cbv(Ψ))rw(Ψ).\nThis means that B(x,ˆc·rw(Ψ))⊂[w]α.Thus (2.9) holds.\n(iii) At first we show the following:\nClaim 2: For anyr>0 andx∈Ωα, the closed ball B(x,r) is a cylinder.\n⊳Sincexis not isolated, there exists y∈B(x,r)\\{x}.Let us show that\n[x∧y]α⊂B(x,r).Indeed, ifz∈[x∧y]α, thenx∧y≺x∧z. By Claim 1,\ndΨ(x,z)≤dΨ(x,y)≤r,soz∈B(x,r).\nDefinen0:= inf{|x∧y|:y∈B(x,r)}. Let us show that B(x,r) = [x|n0]α.\nSinceB(x,r)\\ {x}is nonempty, there exists y∈B(x,r) such that x∧\ny=x|n0.We already showed that [ x|n0]α= [x∧y]α⊂B(x,r). If there\nexistsz∈B(x,r)\\[x|n0]α,thenx∧zmust be a strict prefix of x|n0,hence\n|x∧z|<n0, which contradicts with the definition of n0. ⊲\nNow assume A⊂ΩαandBis a family of closed balls such that for any\nx∈A, there exists a closed ball B(x,rx)∈ B.By Claim 2, B(x,rx) = [wx]α\nfor somewx∈Ωα,∗. WriteU0:={wx:x∈A}. We define two sequences\nWnandUninductively as follows. Assume Un−1has been defined, write\nmn−1:= min{|w|:w∈ Un−1}. Define\nWn:= Ωα,mn−1∩Un−1.\nDefineUnas\nUn:={w∈ Un−1:u/\\e}atio\\slash≺wfor anyu∈ Wn}.\n9IfUn=∅, then the process stops. Otherwise, mn> mn−1, thus induction\ncan continue. By induction, we finish the definition of WnandUn.\nNow define W∞:=/uniontext\nn≥1Wn.\nClaim 3: Ifw,˜w∈ W∞andw/\\e}atio\\slash= ˜w, then [w]α∩[˜w]α=∅.Moreover\nA⊂/uniondisplay\nw∈W∞[w]α.\n⊳By the way we define Wn, ifw/\\e}atio\\slash= ˜w,thenw/\\e}atio\\slash≺˜wand ˜w/\\e}atio\\slash≺w,hence\n[w]α∩[˜w]α=∅.Givenx∈A. Eitherwx∈ W∞,thenx∈B(x,rx) = [wx]α.\nOrwx/\\e}atio\\slash∈ W∞,thuswxis gotten rid of from Uk−1for somek.However, this\nmeans that there exists some w∈ Wksuch thatw≺wx.Consequently,\nx∈B(x,rx) = [wx]α⊂[w]α. ⊲\nClaim 3 obviously implies that (Ω α,dΨ) has Besicovitch’s covering prop-\nerty with multiplicity 1 (see for example [22] Chapter 2 for t he related defi-\nnition). /square\n3.Exact-dimensional property of Gibbs-like measure\nAt first, we define Gibbs-like measure and give criterion for i ts existence.\nThen we fix a weak Gibbs metric on Ω αand study the exact-dimensional\nproperties of Gibbs-like measures. We write αasα=a1a2a3···.\n3.1.Definition and existence of Gibbs-like measure. Our definition\nof Gibbs-like measure is inspired by [21, 10], where the meas ure is defined\nin a geometric way.\nGiven Φ ∈/hatwideFα,µ∈ M(Ωα) is called a Gibbs-like measure of Φ, if there\nexists a constant C >1 such that for any x∈Ωα,\nC−1exp(φn(x))/summationtext\nw∈Ωα,nexp(φn(xw))≤µ([x|n]α)≤Cexp(φn(x))/summationtext\nw∈Ωα,nexp(φn(xw)),(3.1)\nwhere for any w∈Ωα,n, xwis any fixed point in [ w]α.\nWe have the following criterion for the existence of Gibbs-l ike measure:\nTheorem 3.1. IfΦ∈ Fα, thenΦhas a Gibbs-like measure. Moreover the\nconstantCin(3.1)only depends on Crg(Φ),Cbv(Φ),Cbc(Φ),#Aandp0.\nThe proof relies on a technical lemma which we will state now.\nFrom now on, for any w∈Ωα,∗, we fixxw∈[w]αonce for all. Given a\nΦ∈ Fαandw∈Ωα,n, define\n\n\nσn=σn(Φ) :=/summationtext\nw∈Ωα,nexp(φn(xw)),\nσw\nm=σw\nm(Φ) :=/summationtext\nv∈Ξw,m(α)exp(φn+m(xv)−φn(xw)),\n10where Ξ w,m(α) is defined by (2.2).\nLemma 3.2. FixΦ∈ Fαand define σn,σw\nmas above. Then there exists a\nconstantC >1depending only on Crg(Φ),Cbv(Φ),Cbc(Φ),#Aandp0such\nthat for any n∈N,m≥0and anyw,˜w∈Ωα,n,\nC−1σ˜w\nm≤σw\nm≤Cσ˜w\nm. (3.2)\nConsequently for any w∈Ωα,n,\nC−1σnσw\nm≤σn+m≤Cσnσw\nm. (3.3)\nProof.LetC1,C2,C3be the constants in (2.4), (2.5) and (2.6), respectively.\nRecall that p0is such that (2.1) holds.\nAt first we assume 0 ≤m≤p0.Ifw≺v,by regularity and bounded\nvariation of Φ ,we have |φn+m(xv)−φn(xw)| ≤p0C1+C2. Then by (2.3),\nexp(−p0C1−C2)≤σw\nm≤(#A)p0exp(p0C1+C2).(3.4)\nThus (3.2) holds for ˜C= (#A)p0exp(2p0C1+2C2).\nNext we assume m>p0. Fix anye∈ Aan+p0and define\n\n\nXe={u∈/producttextn+p0−1\nj=n+1Aaj:wueadmissible }\n˜Xe={˜u∈/producttextn+p0−1\nj=n+1Aaj: ˜w˜ueadmissible }\nYe={v∈/producttextn+m\nj=n+p0+1Aaj:evadmissible }.\nThen we have\n\n\nσw\nm=/summationtext\ne∈Aan+p0/summationtext\nv∈Ye/summationtext\nu∈Xeexp(φn+m(xwuev)−φn(xw))\nσ˜w\nm=/summationtext\ne∈Aan+p0/summationtext\nv∈Ye/summationtext\n˜u∈˜Xeexp(φn+m(x˜w˜uev)−φn(x˜w)).(3.5)\nBy Lemma 2.1 and the definition of admissibility, we have\n1≤#Xe,#˜Xe≤(#A)p0. (3.6)\nFor anyu∈Xeandv∈Yewe have\nφn+m(xwuev)−φn(xw)\n=φn+m(xwuev)−φn(xwuev)+φn(xwuev)−φn(xw)\n≤φn+m(xwuev)−φn(xwuev)+C2\n=φn+m(xwuev)−φn+p0(xwuev)+φn+p0(xwuev)−φn(xwuev)+C2\n≤φn+m(xwuev)−φn+p0(xwuev)+p0C1+C2,\n11where the second equation is due to bounded variation of Φ ,the fourth\nequation is due to the fact that Φ is regular. By the same proof we get\nφn+m(xwuev)−φn(xw)≥φn+m(xwuev)−φn+p0(xwuev)−p0C1−C2.\nSimilarly, for any ˜ u∈˜Xeandv∈Yewe have\n|φn+m(x˜w˜uev)−φn(x˜w)−(φn+m(x˜w˜uev)−φn+p0(x˜w˜uev))| ≤p0C1+C2.\nBy bounded covariation of Φ, we have\n|φn+m(xwuev)−φn(xw)−(φn+m(x˜w˜uev)−φn(x˜w))| ≤2p0C1+2C2+C3.\n(3.7)\nDefineC= (#A)p0exp(2p0C1+2C2+C3),we get (3.2) by combining (3.5),\n(3.6) and (3.7).\nNow for any fixed w∈Ωα,n, by (3.2) we have\nσn+m=/summationdisplay\nu∈Ωα,n+mexp(φn+m(xu)) =/summationdisplay\n˜w∈Ωα,n/summationdisplay\nu∈Ξ˜w,m(α)exp(φn+m(xu))\n=/summationdisplay\n˜w∈Ωα,nσ˜w\nmexp(φn(x˜w))≤C σw\nm/summationdisplay\n˜w∈Ωα,nexp(φn(x˜w)) =Cσw\nmσn.\nσn+m≥C−1σw\nmσnfollows in the same way. /square\nProof of Theorem 3.1. For anyn∈N, we define a measure µnon the\nBorelσ-algebra generalized by the n-th cylinders as follows: for any w∈\nΩα,n,letµn([w]α) := exp(φn(xw))/σn.Assumeµis any weak-star limit of\n{µn:n≥1}. Let us show that µis a Gibbs-like measure of Φ.\nFix anyw∈Ωα,nandm≥0. By (3.3), we have\nµn+m([w]α) =/summationdisplay\nu∈Ξw,m(α)µn+m([u]α) =σ−1\nn+m/summationdisplay\nu∈Ξw,m(α)exp(φn+m(xu))\n=σ−1\nn+mexp(φn(xw))/summationdisplay\nu∈Ξw,m(α)exp(φn+m(xu)−φn(xw))\n=σ−1\nn+mexp(φn(xw))·σw\nm∼Cexp(φn(xw))\nσn.\nNotice that [ w]αis open and closed, since some subsequence µn+mlconverge\nweakly toµ, letl→ ∞we get\nµ([w]α)∼Cexp(φn(xw))\nσn.\nSince Φ has bounded variation, the above property implies th atµis a\nGibbs-like measure, and the constant in (3.1) only depends o nCrg(Φ),\nCbv(Φ),Cbc(Φ),#Aandp0. /square\n123.2.Exact-dimensional properties of Gibbs-like measure. Fix Ψ∈\nF−\nαand definethe weak Gibbs metric dΨon Ωαby (2.7). Gibbs-like measure\nbehaves well on the metric space (Ω α,dΨ) in the following sense:\nTheorem 3.3. FixΦ∈ Fαand letµbe a Gibbs-like measure of Φ,thenµ\nis exact upper and lower dimensional.\nProof.At first we show that µis exact lower dimensional. We show it\nby contradiction. If µis not exact lower dimensional, then there exist two\ndisjoint Borel subsets X,Y⊂Ωαwithµ(X),µ(Y)>0 and two numbers\nd1<d2such that\ndµ(x)≤d1(∀x∈X) anddµ(y)≥d2(∀y∈Y).\nBy Lemma 2.3 (iii), the space Ω αhas Besicovitch’s covering property. By\nthe density property for Radon measure (see for example [22] Chapter 2),\nthere exist x∈Xandy∈Ysuch that\nlim\nn→∞µ(X∩[x|n]α)\nµ([x|n]α)= 1 and lim\nn→∞µ(Y∩[y|n]α)\nµ([y|n]α)= 1.\nLetC1,C2,C3,C4,C5>0 be the constants in (2.4), (2.5), (2.6), (3.1) and\n(3.3), respectively. Recall that p0is such that (2.1) holds. By (3.4), there\nexists a constant C6>0 such that σw\np0≤C6for anyw∈Ωα,∗.\nFixδ>0 such that\nδ<min{/parenleftbig\n(4C4\n4exp(2C2+C3)+1)C2\n4C5C6exp(p0C1+C2)/parenrightbig−1,1/2}.\n(3.8)\nAssumeqis such that\n\n\nµ(X∩[x|q]α)≥(1−δ)µ([x|q]α),\nµ(Y∩[y|q]α)≥(1−δ)µ([y|q]α).(3.9)\nClaim 1: for anyv∈Ξx|q,p0(α) andw∈Ξy|q,p0(α),\n\n\nµ(X∩[v]α)≥(1−Cδ)µ([v]α),\nµ(Y∩[w]α)≥(1−Cδ)µ([w]α),(3.10)\nwhereC=C2\n4C5C6exp(p0C1+C2).\n⊳We only show the first inequality of (3.10), since the proof of the second\none is the same. We have [ x|q]α=/uniontext\nv∈Ξx|q,p0(α)[v]α. Define\nη:= max/braceleftbiggµ([v]α\\X)\nµ([v]α):v∈Ξx|q,p0(α)/bracerightbigg\n.\n13Ifη= 0,the result holds trivially. So we assume η >0 and ˆv∈Ξx|q,p0(α)\nattains the maximum. Consequently by (3.9),\nηµ([ˆv]α) =µ([ˆv]α\\X)≤µ([x|q]α\\X)≤δµ([x|q]α).\nBy Gibbs-like property of µ, we have\nµ([ˆv]α)≥C−1\n4exp(φq+p0(xˆv))\nσq+p0andµ([x|q]α)≤C4exp(φq(x))\nσq.\nBy (3.3) and (3.4), we have σq+p0≤C5σqσx|qp0≤C5C6σq. By regularity and\nbounded variation of Φ ,we have\n|φq(x)−φq+p0(xˆv)| ≤ |φq(x)−φq(xˆv)|+|φq(xˆv)−φq+p0(xˆv)| ≤C2+p0C1.\nThus\nη≤δµ([x|q]α)\nµ([ˆv]α)≤C2\n4σq+p0\nσqexp(φq(x)−φq+p0(xˆv))δ≤C2\n4C5C6exp(p0C1+C2)δ.\nThen the result follows. ⊲\nBy Lemma 2.1, we can find v∈Ξx|q,p0(α) andw∈Ξy|q,p0(α) such that\nvq+p0=wq+p0. We fix such a pair ( v,w).\nClaim 2: There exist ˜ x∈[v]α∩Xand ˜y∈[w]α∩Ysuch that ˜xand ˜y\nhave the same tail, i.e., there exist l≥q+p0andz∈/producttext∞\nj=l+1Aajsuch that\n˜x= ˜x|l·zand ˜y= ˜y|l·z.\n⊳We show it by contradiction. Assume for any ˜ x∈[v]α∩Xand any\n˜y∈[w]α∩Y, ˜xand ˜yhave no the same tail.\nBy (3.10) we have µ(X∩[v]α)>(1−Cδ)µ([v]α).Take a compact set\nˆX⊂X∩[v]αsuch thatµ(ˆX)>(1−2Cδ)µ([v]α).Consequently µ([v]α\\ˆX)≤\n2Cδµ([v]α). Notice that [ v]α\\ˆXis an open set, thus it is a countable\ndisjoint union of cylinders: [ v]α\\ˆX=/uniontext\nj≥1[vwj]α,where different wjare\nnon compatible. Thus we get\n/summationdisplay\nj≥1µ([vwj]α)\nµ([v]α)≤2Cδ. (3.11)\nSincevq+p0=wq+p0and for any ˜ x∈[v]α∩Xand any ˜y∈[w]α∩Y, ˜x\nand ˜yhave no the same tail, we must have\n[w]α∩Y⊂/uniondisplay\nj≥1[wwj]α. (3.12)\n14By the Gibbs-like property of µ,we have\n\n\nµ([vwj]α)\nµ([v]α)≥C−2\n4·exp(φq+p0+|wj|(xvwj))\nexp(φq+p0(xv))·σq+p0\nσq+p0+|wj|\nµ([wwj]α)\nµ([w]α)≤C2\n4·exp(φq+p0+|wj|(xwwj))\nexp(φq+p0(xw))·σq+p0\nσq+p0+|wj|.\nBy bounded variation and covariation of Φ,\nµ([wwj]α)\nµ([w]α)/µ([vwj]α)\nµ([v]α)\n≤C4\n4exp/parenleftig\nφq+p0+|wj|(xwwj)−φq+p0(xw)\n−(φq+p0+|wj|(xvwj)−φq+p0(xv))/parenrightig\n≤C4\n4exp(2C2)exp/parenleftig\nφq+p0+|wj|(xwwj)−φq+p0(xwwj)\n−(φq+p0+|wj|(xvwj)−φq+p0(xvwj))/parenrightig\n≤C4\n4exp(2C2+C3).\nThus by (3.12), (3.11) and (3.8), we have\nµ([w]α∩Y)\nµ([w]α)≤/summationdisplay\nj≥1µ([wwj]α)\nµ([w]α)≤2CC4\n4exp(2C2+C3)δ≤(1−Cδ)/2,\nwhich contradicts with (3.10). ⊲\nClaim 2 leads to a contradiction. Indeed we have\ndµ(˜x) = liminf\nn→∞logµ([˜x|n]α)\nlogdiam([˜x|n]α)anddµ(˜y) = liminf\nn→∞logµ([˜y|n]α)\nlogdiam([˜y|n]α).\n(We note that, due to (2.9), we can use cylinders to compute th e lower local\ndimension.) On one hand, by the Gibbs-like property of µ, we have\nµ([˜x|n]α)∼exp(φn(˜x))\nσnandµ([˜y|n]α)∼exp(φn(˜y))\nσn.(3.13)\nBy the bounded covariation of Φ, |φn(˜x)−φl(˜x)−(φn(˜y)−φl(˜y))| ≤Cbc(Φ)\nfor anyn≥l. Sincelis a fixed number, combine with (3.13) we conclude\nthat\nµ([˜x|n]α)∼µ([˜y|n]α). (3.14)\nOn the other hand, by (2.8) and bounded variation of Ψ,\ndiam([˜x|n]α)∼exp(ψn(˜x)) and diam([˜ y|n]α)∼exp(ψn(˜y)).\nBy the bounded covariation of Ψ, we conclude that\ndiam([˜x|n]α)∼diam([˜y|n]α). (3.15)\n15Combine (3.14) and (3.15) we get dµ(˜x) =dµ(˜y).However since ˜ x∈Xand\n˜y∈Y, we also have dµ(˜x)≤d1<d2≤dµ(˜y), which is a contradiction.\nThus we conclude that µis exact lower dimensional.\nThe same proof shows that µis also exact upper dimensional. /square\n4.Structure of the spectrum of Sturm Hamiltonian\nIn this section, as a preparation for the proof of the main res ult, we study\nthe structure of the spectrum of Sturm Hamiltonian.\n4.1.Facts on continued fraction and Sturm sequence. Recall that\nGauss map G: (0,1)→(0,1) is defined as G(α) := 1/α−[1/α]. Assume\nαhas continued fraction expansion [0; a1,a2,···].Letpn/qn(n >0) be the\nn-th approximation of α, given by:\np−1= 1, p0= 0, pn+1=an+1pn+pn−1, n≥0,\nq−1= 0, q0= 1, qn+1=an+1qn+qn−1, n≥0.(4.1)\nWe also use the notations an(α),pn(α) andqn(α) to emphasize the depen-\ndence of these quantities on α.\nFixM∈Nand define a subset of Bas\nBM:={α∈[0,1]\\Q:α= [0;a1,a2,···] withan≤M}.\nIt is seen that B=/uniontext\nM∈NBM.The following Lemma will be useful later.\nLemma 4.1. (i) Givenα∈[0,1]\\Q.Then for any n,m∈N,\nqn(α)qm(Gn(α))≤qn+m(α)≤2qn(α)qm(Gn(α)).\n(ii) Ifα∈BM, thenqn(α)≤(M+1)n.\nProof.(i) Writee1= (1,0),e2= (0,1).For anyβ∈[0,1]\\Qandn∈N,\nwrite\nBn(β) =/parenleftigg\nan(β) 1\n1 0/parenrightigg\n.\nBy (4.1) we have\n(qn(β),qn−1(β))t=Bn(β)···B1(β)·et\n1. (4.2)\nSinceB1(β)·et\n2=et\n1andan−1(G(β)) =an(β), we have\n(qn−1(G(β)),qn−2(G(β)))t=Bn(β)···B1(β)·et\n2. (4.3)\nWriteˆβ=Gn(α). By (4.2) and (4.3), we have\nqn+m(α) =e1·Bn+m(α)···Bn+1(α)·Bn(α)···B1(α)·et\n1\n16=e1·Bm(ˆβ)···B1(ˆβ)·Bn(α)···B1(α)·et\n1\n=e1·Bm(ˆβ)···B1(ˆβ)·(qn(α),qn−1(α))t\n=qn(α)e1·Bm(ˆβ)···B1(ˆβ)·et\n1+\nqn−1(α)e1·Bm(ˆβ)···B1(ˆβ)·et\n2\n=qn(α)qm(ˆβ)+qn−1(α)qm−1(G(ˆβ)).\nBy induction, it is ready to show that 0 < qn−1(α)≤qn(α) and 0<\nqm−1(G(ˆβ))≤qm(ˆβ).Then the result follows.\n(ii) It follows from (4.1) and induction. /square\nForα∈[0,1]\\Q, thestandard Strum sequence Sn(α) is defined by\nSn(α) :=χ[1−α,1)(nα(mod 1)),(n∈Z).\nLemma 4.2. Ifai(α) =ai(β)fori= 1,···,n, thenqn(α) =qn(β) =:qn.\nMoreoverSi(α) =Si(β)for1≤i≤qn.\nProof.The first statement follows from (4.1). See [1, 20] for a proof of the\nsecond statement. /square\n4.2.Structure and coding of the spectrum.\n4.2.1.The structure of the spectrum. We describe the structure of the spec-\ntrum Σ α,λfor fixedα∈[0,1]\\Qandλ >0, for more details, we refer to\n[29, 3, 28, 18].\nRecallthattheSturmpotentialisgivenby vn=λχ[1−α,1)(nα+θ(mod 1)).\nSince Σ α,λis independent of the phase θ, in the rest of the paper we will\ntakeθ= 0.Thusvn=λSn(α).Assumeαhas continued fraction expansion\n[0;a1,a2,···] and define qnby (4.1). For any n≥1 andx∈R, the transfer\nmatrixMn(x) overqnsites is defined by\nMn(x) :=/bracketleftigg\nx−vqn−1\n1 0/bracketrightigg/bracketleftigg\nx−vqn−1−1\n1 0/bracketrightigg\n···/bracketleftigg\nx−v1−1\n1 0/bracketrightigg\n,\nBy convention we take\nM−1(x) =/bracketleftigg\n1−λ\n0 1/bracketrightigg\nandM0(x) =/bracketleftigg\nx−1\n1 0/bracketrightigg\n.\nForn≥0,p≥ −1, leth(n,p)(x) := trMn−1(x)Mp\nn(x) and\nσ(n,p):={x∈R:|h(n,p)(x)| ≤2},\n17where trMstands for the trace of the matrix M.σ(n,p)is made of finitely\nmany bands. Moreover, for any n≥0,\n(σ(n+2,0)∪σ(n+1,0))⊂(σ(n+1,0)∪σ(n,0)) and Σ α,λ=/intersectiondisplay\nn≥0(σ(n+1,0)∪σ(n,0)).\nThe intervals in σ(n,p)are called bands. For any band Bofσ(n,p),h(n,p)(x)\nis monotone on Bandh(n,p)(B) = [−2,2].We callh(n,p)thegenerating\npolynomial ofBand denote it by hB:=h(n,p).\n{σ(n+1,0)∪σ(n,0):n≥0}form a covering of Σ α,λ. However there are some\nrepetitions between σ(n,0)∪σ(n−1,0)andσ(n+1,0)∪σ(n,0). Whenλ>4,it is\npossible to choose a covering of Σ α,λelaborately such that we can get rid of\nthese repetitions, as we will describe in the follows:\nDefinition 4.3. ([28, 18]) Forλ>4,n≥0, define three types of bands as:\n(n,I)-type band: a band of σ(n,1)contained in a band of σ(n,0);\n(n,II)-type band: a band of σ(n+1,0)contained in a band of σ(n,−1);\n(n,III)-type band: a band of σ(n+1,0)contained in a band of σ(n,0).\nAll three types of bands actually occur and they are disjoint . We call\nthese bands spectral generating bands of order n. Note that there are only\ntwo spectral generating bands of order 0, one is σ(0,1)= [λ−2,λ+2] with\ngenerating polynomial h(0,1)=x−λand type (0 ,I), the other is σ(1,0)=\n[−2,2] with generating polynomial h(1,0)=xand type (0 ,III).\nFor anyn≥0, denote by Bnthe set of spectral generating bands of order\nn, then the intervals in Bnare disjoint. Moreover ([28, 18])\n•(σ(n+2,0)∪σ(n+1,0))⊂/uniontext\nB∈BnB⊂(σ(n+1,0)∪σ(n,0)), thus\nΣα,λ=/intersectiondisplay\nn≥0/uniondisplay\nB∈BnB.\n•any (n,I)-type band contains only one band in Bn+1, which is of\n(n+1,II)-type.\n•any (n,II)-type band contains 2 an+1+1 bands in Bn+1,an+1+1 of\nwhich areof ( n+1,I)-type andan+1ofwhich areof ( n+1,III)-type.\n•any (n,III)-type band contains 2 an+1−1 bands in Bn+1,an+1of\nwhich are of ( n+1,I)-type andan+1−1 of which are of ( n+1,III)-\ntype.\nThus{Bn}n≥0form a natural covering of the spectrum Σ α,λ([19, 16]).\nRemark 4.4. From the definition, B0={[λ−2,λ+2],[−2,2]}. Forn≥1,\nBnshould depend on α, i.e.,Bn=Bn(α). On the other hand, by Definition\n184.3, Lemma 4.2 and the fact that vk=λSk(α),we conclude that Bnindeed\nonly depends on /vector a=a1···an, whereα= [0;a1,a2,···]. That is, Bn=\nBn(/vector a).\n4.2.2.The coding of the spectrum. In the following we give a coding of the\nspectrum Σ α,λbased on [28, 17, 27]. Here we essentially follow [27]. For\neachn∈N,define an alphabet Anas\nAn:={(I,j)n:j= 1,···,n+1}∪{IIn}∪{(III,j)n:j= 1,···,n}.\nThen #An= 2n+2.We order the elements in Anas\n(I,1)n<···<(I,n+1)n<IIn<(III,1)n<···<(III,n)n.\nTo simplify the notation, we rename the above line as\nen,1<en,2<···<en,2n+2.\nGivenen,i∈ Anandem,j∈ Am, we callen,iem,jadmissible , denote by\nen,i→em,j,if\n(en,i,em,j)∈ {((I,k)n,IIm) : 1≤k≤n+1}∪\n{(IIn,(I,l)m) : 1≤l≤m+1}∪\n{(IIn,(III,l)m) : 1≤l≤m}∪\n{((III,k)n,(I,l)m) : 1≤k≤n,1≤l≤m}∪\n{((III,k)n,(III,l)m) : 1≤k≤n,1≤l≤m−1}.\nFor pair ( An,Am),we define the incidence matrix Anm= (aij) of size (2n+\n2)×(2m+2) as\naij=\n\n1en,i→em,j,\n0 otherwise .(4.4)\nWhenn=m,we writeAn:=Ann.For anyn∈N, we define an auxiliary\nmatrix as follows:\nˆAn=\n0 1 0\nn+1 0n\nn0n−1\n.\nLet us show that {Anm:n,m≥1}defined by (4.4) is strongly primitive:\nProposition 4.5. For anyk≥6and any sequence a1···ak∈Nk, the\nmatrixAa1a2Aa2a3···Aak−1akis positive.\n19Proof.At first we show the result for k= 6.Fix any sequence a1···a6∈N6\nand anye∈ Aa1and ˆe∈ Aa6, we only need to show that there is an\nadmissible word e1···e6∈/producttext6\nj=1Aajsuch thate1=eande6= ˆe.We\nconstruct the desired admissible word as follows.\nIfe= (I,j)a1and ˆe= (I,l)a6or (III,l)a6, take\ne1···e6=e IIa2(III,1)a3(I,1)a4IIa5ˆe.\nIfe= (I,j)a1and ˆe=IIa6, take\ne1···e6=e IIa2(I,1)a3IIa4(I,1)a5ˆe.\nIfe=IIa1and ˆe= (I,l)a6or (III,l)a6, take\ne1···e6=e(I,1)a2IIa3(I,1)a4IIa5ˆe.\nIfe=IIa1and ˆe=IIa6, take\ne1···e6=e(III,1)a2(I,1)a3IIa4(I,1)a5ˆe.\nIfe= (III,j)a1and ˆe= (I,l)a6or (III,l)a6, take\ne1···e6=e(I,1)a2IIa3(I,1)a4IIa5ˆe.\nIfe= (III,j)a1and ˆe=IIa6, take\ne1···e6=e(I,1)a2IIa3(III,1)a4(I,1)a5ˆe.\nFor general k, we note that for any n,m∈N, each column and each array\nofAnmhas at least one 1. Then the result follows easily from the cas e\nk= 6. /square\nWe also define A0:={I,III}and writee0,1=Iande0,2=III. Fix\nm∈N.Givene0,i∈ A0andem,j∈ Am, we calle0,iem,jadmissible , denote\nbye0,i→em,j,if\n(e0,i,em,j)∈ {(I,IIm)}∪{(III,(I,l)m) : 1≤l≤m}∪\n{(III,(III,l)m) : 1≤l≤m−1}.\nFor pair ( A0,Am),we define the incidence matrix A0m= (aij) of size 2 ×\n(2m+2) as\naij=\n\n1e0,i→em,j,\n0 otherwise .\nFor any/vector a=a1···an∈Nnand any 0 ≤m≤n,define\nΩ(/vector a)\nm:={w0···wm∈ A0×m/productdisplay\nj=1Aaj:wj→wj+1}.\n20Ifα= [0;a1,a2,···], define a symbolic space Ω(α)as\nΩ(α)={x0x1x2··· ∈ A 0×∞/productdisplay\nj=1Aaj:xj→xj+1}.\nForx∈Ω(α), wewritex|n:=x0···xn.Moregenerally wewrite x[n,n+k] for\nxn···xn+k.Define Ω(α)\nn:={x|n:x∈Ω(α)}and Ω(α)\n∗=/uniontext\nnΩ(α)\nn.It is seen\nthat Ω(α)\nn= Ω(/vector a)\nn, where/vector a=a1···an.Givenw=w0···wn∈Ω(α)\nn, denote\nby|w|the length of w, then|w|=n+ 1. If the last letter wn= (T,j)an,\nthen we write tw=T,and say that whastypeT.Ifw=uw′, then we say u\nis aprefixofwand denote by u≺w.Forx,y∈Ω(α), we denote by x∧ythe\nmaximal common prefix of xandy.Givenw∈Ω(α)\nn,we define the cylinder\n[w]α:={x∈Ω(α):x|n=w}.\nIn the following we explain that Ω(α)is a coding of the spectrum Σ α,λ.\nAt first we fix /vector a=a1···an∈Nnand code the intervals in Bn(/vector a) by\nΩ(/vector a)\nn(see Remark 4.4 for the notation Bn(/vector a)). Write B0(/vector a) :=B0. For any\n1≤m≤n, writeBm(/vector a) :=Bm(a1···am). DefineBI:= [λ−2,λ+2] and\nBIII:= [−2,2], thenB0(/vector a) =B0={Bw:w∈Ω(/vector a)\n0}.AssumeBwis defined\nfor anyw∈Ω(/vector a)\nm−1. Now for any w∈Ω(/vector a)\nm, writew=w′e=w′(T,j)am.\nThen define Bwto be the unique j-th (m,T) type band in Bm(/vector a) which is\ncontained in Bw′. ThenBm(/vector a) ={Bw:w∈Ω(/vector a)\nm}.By induction, we code\nall the bands in Bn(/vector a) by Ω(/vector a)\nn.\nNow we can define a natural projection πα: Ω(α)→Σα,λas\nπα(x) :=/intersectiondisplay\nn≥0Bx|n. (4.5)\nIt is seen that παis a bijection, thus Ω(α)is a coding of Σ α,λ.\nRemark 4.6. Ω(α)is not exactly the symbolic space Ω αdefined in Sect. 2,\nindeed it is an “augment” of Ω αwith alphabet sequence {A0,Aa1,Aa2···}\nand incidence matrix sequence {A0a1,Aa1a2,···}. However, since 0 only\nappears at the first place, by checking the proof, all the resu lts developed in\nSect. 3 still holds on the symbolic space Ω(α).We denote the class of regular\n(negative) potentials on Ω(α)with bounded variation and covariation by\nFα(Fα\n−).\nWe also need the following facts later.\n21Lemma 4.7. (i) For any sequence a1···a5∈ {1,···,M}5, we have\n\n1 1 1\n1 1 1\n1 1 1\n≤ˆAa1···ˆAa5≤81(M+1)5\n1 1 1\n1 1 1\n1 1 1\n.\n(ii) Ifα= [0;a1,a2,···], then\n#Ω(α)\nn= (1,0,1)·ˆAa1···ˆAan·(1,1,1)t.\nProof.(i) Notice that if 1 ≤a≤M,thenˆA1≤ˆAa≤(M+ 1)J, whereJ\nis the matrices with all entries equal to 1. Then the result fo llows by direct\ncomputation.\n(ii) It follows from the definition and induction. /square\n4.3.Other useful facts of Sturm Hamiltonian. We collect some use-\nful results of Sturm Hamiltonian for later use. Fix α= [0;a1,a2,···] and\nconsider the operator Hα,λ,0. We write hk(x) :=hk+1,0(x) = trMk(x) and\nσk:=σ(k+1,0)={x∈R:hk(x)| ≤2}. (4.6)\nWe note that σkconsists ofqk(α) disjoint intervals.\nThe following two lemmas are implicitly proven in [28]:\nLemma 4.8. σk={B∈ Bk(α) :Bis of type (k,II)or(k,III)}.\nDefinevI= (1,0,0),vII= (0,1,0),vIII= (0,0,1) andv∗= (0,1,1)T.\nLemma 4.9. Givenw∈Ω(α)\nn, then\n#{u∈Ω(α)\nn+m:w≺u,tu=II,III}=vtw·ˆAan+1···ˆAan+m·v∗.\nThe following theorem is [10] Theorem 5.1, see also [17] Theo rem 3.3.\nTheorem 4.10 (Bounded covariation) .Assumeλ >20andα,β∈BM.\nThen there exists constant η >1only depending on λandMsuch that if\nw,wu∈Ω(α)\n∗and˜w,˜wu∈Ω(β)\n∗, then\nη−1|Bwu|\n|Bw|≤|B/tildewidewu|\n|B/tildewidew|≤η|Bwu|\n|Bw|.\nWe remark that in [10], only the case α=βis considered. However by\nchecking the proof, the same argument indeed shows the stron ger result as\nstated in Theorem 4.10.\nThe following lemma is a direct consequence of [10] Corollar y 3.1:\n22Lemma 4.11. Letλ >20andα∈BM. Then there exists constant 0<\nc=c(λ,M)<1such that for any w∈Ω(α)\nn,\nc|Bw|n−1| ≤ |Bw|and|Bw| ≤22−n. (4.7)\nThe following result is also useful later.\nLemma 4.12. qn(α)≤#Ω(α)\nn≤5qn(α).\nProof.By the definition of Bn(α),we know that\nBn(α) ={Bw:w∈Ω(α)\nn}\n={Bw:w∈Ω(α)\nn;tw=II,III}∪{Bw:w∈Ω(α)\nn;tw=I}\n={Bw:w∈Ω(α)\nn;tw=II,III}∪\n{Bu(I,j)an:u∈Ω(α)\nn−1;tu=II,1≤j≤an+1}∪\n{Bu(I,j)an:u∈Ω(α)\nn−1;tu=III,1≤j≤an}.\nRecall that σnis then-th approximation of Σ α,λ(see (4.6)), which is made\nofqn(α) bands. Combine with Lemma 4.8 and (4.1), we have\nqn(α)≤#Bn(α) = #Ω(α)\nn≤qn(α)+4anqn−1(α)≤5qn(α).\n/square\n5.Exact-dimensional property of DOS\nIn this section, we apply the result in Sect. 3 to prove Theore m 1.1. At\nfirst, we construct a potential Ψα∈ Fα\n−and define the related weak Gibbs\nmetricdα=dΨαsuch thatπαis a bi-Lipschitz homeomorphism between\n(Ω(α),dα) and (Σ α,λ,| · |). Then we construct a potential Φα∈ Fαsuch\nthat (πα)∗(µα)≍ Nα,λ, whereµαis a Gibbs-like measure of Φα.From this,\nwe conclude that Nα,λis exact upper and lower dimensional. In the last\nsubsection, we construct a frequency α∈B2, such that Nα,λis not exact-\ndimensional when λis large enough.\nThroughout the first two subsections, we always fix α∈Bandλ >20.\nChooseM∈Nsuch thatα∈BM.\n5.1.Weak Gibbs metric on Ω(α).Define Ψα={ψα\nn:n≥1}on Ω(α)as\nψα\nn(x) := log|Bx|n|. (5.1)\nLemma 5.1. Ψα∈ Fα\n−with related constants Crg(Ψα),Cbv(Ψα),Cbc(Ψα)\nonly depending on λandM.\n23Proof.Fix anyn,k∈N. Thenψα\nn+k(x)−ψα\nn(x) = log( |Bx|n+k|/|Bx|n|).\nSince Ω(α)\n0={I,III}and Ω(α)\n1contains at least one word of type II, there\nexistsu∈Ω(α)\n0∪Ω(α)\n1such thatuandx|nhave the same type. Write\nw=xn+1···xn+k, by (4.7), there exists constant 0 <c<1 only depending\nonλandMsuch that\n4c≤ |Bu| ≤4 and 4 ck+1≤ |Buw| ≤4·2−k.\nThen by Theorem 4.10, there exists constant η >1 only depending on λ\nandMsuch that\n|ψα\nn+k(x)−ψα\nn(x)|=/vextendsingle/vextendsingle/vextendsingle/vextendsinglelog/parenleftbigg|Bx|n+k|\n|Bx|n|/|Buw|\n|Bu|/parenrightbigg\n+log|Buw|\n|Bu|/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤logη+|log|Buw||+|log|Bu||\n≤logη+4log2+k(log2−3logc)\n≤k(5log2−3logc+logη).\nHence Ψαis regular with Crg(Ψα) = 5log2 −3logc+logη.\nFrom the definition, if x|n=y|n, thenψα\nn(x) =ψα\nn(y).Hence Ψαhas\nbounded variation with Cbv(Ψα) = 0.\nNow assume w=uv,˜w= ˜uv∈Ω(α)\n∗andx∈[w]α,˜x∈[˜w]α. By Theorem\n4.10, we have\n|(ψα\n|w|(x)−ψα\n|u|(x))−(ψα\n|˜w|(˜x)−ψα\n|˜u|(˜x))|=/vextendsingle/vextendsingle/vextendsingle/vextendsinglelog/parenleftbigg|Bw|\n|Bu|/|B˜w|\n|B˜u|/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤logη.\nThus Ψαhas bounded covariation with Cbc(Ψα) = logη.\nThus Ψα∈ FαandCrg(Ψα),Cbv(Ψα),Cbc(Ψα) only depend on λandM.\nSinceBx|n+1is a subinterval of Bx|n,combine with (4.7), we have\nψα\nn+1(x) = log|Bx|n+1| ≤log|Bx|n|=ψα\nn(x)≤(2−n)log2.\nThus Ψαis negative. /square\nDefine the weak Gibbs metric dα:=dΨαon Ω(α),then for any x,y∈Ω(α),\ndα(x,y) =|Bx∧y|.\nRecall that παis defined by (4.5).\nProposition 5.2. πα: (Ω(α),dα)→(Σα,λ,|·|)is bi-Lipschitz.\nTo prove this property, we do some preparation. Write Co(Σ α,λ)\\Σα,λ=/uniontext\niGi,where Co(Σ α,λ) is the convex hull of Σ α,λ.EachGiis called a gapof\nthe spectrum. A gap Gis called of ordern, ifGis covered by some band in\nBnbut not covered by any band in Bn+1.Denote by Gnthe set of gaps of\n24ordern. For anyG∈ Gn, letBGbe the unique band in Bnwhich contains\nG.The following lemma is proven in [27]:\nLemma 5.3. There exists a constant C=C(M,λ)∈(0,1)such that for\nany gapG∈/uniontext\nn≥0Gnwe have |G| ≥C|BG|.\nProof of Proposition 5.2. Givenx,y∈Ω(α).Assumex|n=y|nand\nxn+1/\\e}atio\\slash=yn+1,thendα(x,y) =|Bx|n|.Write ˆx:=πα(x) and ˆy:=πα(y). By\nthe definition of πα,we have ˆx,ˆy∈Bx|n, consequently\n|ˆx−ˆy| ≤ |Bx|n|=dα(x,y).\nOn the other hand, since xn+1/\\e}atio\\slash=yn+1,there is a gap Gof ordernwhich is\ncontained in the open interval (ˆ x,ˆy).By Lemma 5.3, there exists a constant\nC=C(M,λ)>0 such that\n|ˆx−ˆy| ≥ |G| ≥C|BG|=C|Bx|n|=Cdα(x,y).\nThusπαis bi-Lipschitz. /square\n5.2.DOS is exact upper and lower dimensional. Define Φα={φα\nn:\nn≥1}on Ω(α)as\nφα\nn(x) := logqn(α). (5.2)\nProposition 5.4. Φα∈ Fαwith\nCrg(Φα) = 2log(M+1), Cbv(Φα) = 0andCbc(Φα) = log2.\nProof.For anyn,m≥1, by Lemma 4.1 and the fact that Gn(α)∈BM,\n|φα\nn(x)−φα\nn+m(x)|=|logqn(α)−logqn+m(α)| ≤log2+log |qm(Gn(α))|\n≤log2+mlog(M+1)≤2mlog(M+1).\nThus Φαis regular with Crg(Φα) = 2log(M+1).\nFrom the definition, Φαhas bounded variation with Cbv(Φα) = 0.\nNow assume w=uv,˜w= ˜uv∈Ω(α)\n∗andx∈[w]α,˜x∈[˜w]α. Write\nn=|u|,˜n=|˜u|andm=|v|.Similar computation as above shows that\nlogqm(Gn(α))≤φα\n|w|(x)−φα\n|u|(x)≤log2+logqm(Gn(α)),\nlogqm(G˜n(α))≤φα\n|˜w|(x)−φα\n|˜u|(x)≤log2+logqm(G˜n(α)).\nSincewand ˜whave the same suffix of length m, we have\nGn(α) = [0;v1,v2,···,vm,bm+1···];G˜n(α) = [0;v1,v2,···,vm,˜bm+1···].\n25Henceqm(Gn(α)) =qm(G˜n(α)).As a consequence, we have\n|(φα\n|w|(x)−φα\n|u|(x))−(φα\n|˜w|(˜x)−φα\n|˜u|(˜x))| ≤log2.\nThus Φαhas bounded covariation with Cbc(Φα) = log2.\nBy the definition, Φα∈ Fα. /square\nLetµαbe a Gibbs-like measure of Φα.\nProposition 5.5. (πα)∗(µα)≍ Nα,λ.\nProof.By the definition of παand≍, we only need to show that\nµα([w]α)∼ Nα,λ(Bw),∀w∈Ω(α)\n∗. (5.3)\nOn one hand, by Theorem 3.1, Proposition 5.4, and Lemma 4.12, there\nexists a constant C >1 only depending on Msuch that for any w∈Ω(α)\nn,\n1\nCqn(α)≤µα([w]α)≤C\nqn(α). (5.4)\nOntheotherhand,let Hnbetherestrictionof Hα,λ,0tothebox[1 ,qn]with\nperiodic boundary condition. Let Xn={xn,1,···,xn,qn}be the eigenvalues\nofHn. Recall that σnis defined by (4.6). It is well-known that each band\ninσncontains exactly one value in Xn(see for example [29, 28]). Define\nνn:=1\nqn/summationtextqn\ni=1δxn,i,thenνn→ Nα,λweakly(see for example [4]). For any\nl>n+10, by Lemma 4.8 and Lemma 4.9, we have\nνl(Bw) =/summationdisplay\nu∈Ω(α)\nl,w≺uνl(Bu)\n=#{u∈Ω(α)\nl:w≺u,tu=II,III}\nql(α)\n=vtw·ˆAan+1···ˆAal·v∗\nql(α).\nBy Lemma 4.7 (i),\nvtw·ˆAan+1···ˆAan+5∼M(1,0,1)·ˆAan+1···ˆAan+5\nˆAal−4···ˆAal·v∗∼MˆAal−4···ˆAal·(1,1,1)t.\nConsequently, by Lemma 4.7 (ii), Lemma 4.12 and Lemma 4.1 (i) , we have\nνl(Bw)∼(1,0,1)·ˆAan+1···ˆAal·(1,1,1)t\nql(α)\n=#Ω(Gn(α))\nl−n\nql(α)∼ql−n(Gn(α))\nql(α)∼1\nqn(α).\n26Letl→ ∞, we conclude that\nNα,λ(Bw)∼1\nqn(α). (5.5)\nCombine (5.4) and (5.5), we get (5.3). /square\nProof of Theorem 1.1 (i). By Theorem 3.3, µαis exact upper and lower\ndimensional. Since παis bi-Lipschitz, and ( πα)∗(µα)≍ Nα,λ,Nα,λhas the\nsame property.\nThe statement that Nα,λis exact-dimentional if and only if dH(α,λ) =\ndP(α,λ) follows directly from the definition and the fact that Nα,λis exact\nupper and lower dimensional. /square\n5.3.Non exact-dimentional DOS. In this subsection, we prove Theo-\nrem 1.1 (ii), i.e., we construct a Sturm Hamiltonian Hα,λ,θsuch that the\nrelated DOS is not exact-dimensional. We roughly describe t he idea as\nfollows: by [27], for ακ= [0;κ,κ,···],Nακ,λis exact-dimensional with di-\nmensiond(κ,λ), andd(1,λ)<d(2,λ) forλlarge enough. We will construct\na frequency α= [0;a1,a2,···]∈B2witha1a2···= 1t12τ11t22τ2···such\nthattn≫τn−1andτn≫tn. For such a frequency α,Nα,λis, in cer-\ntain sense, a concatenation of Nα1,λandNα2,λin an alternative way. We\ncan show that for xin a subset of positive Nα,λmeasure, the quantity\nlogNα,λ(B(x,r))/logroscillates between d(1,λ) andd(2,λ),which means\nthat the local dimension does not exist at x. In particular, Nα,λcan not be\nexact-dimensional.\nProposition 5.6. There exist α∈B2andλ0>20such thatdH(α,λ)<\ndP(α,λ)for anyλ≥λ0.\nProof.By Lemma 4.1 (ii), for any β∈B2,\nqn(β)≤3n. (5.6)\nDefine Φβby (5.2) and assume µβis a Gibbs-like measure of Φβ.By (5.4),\nthere exists an absolute constant C >1 such that for any w∈Ω(β)\nn,\n1\nCqn(β)≤µβ([w]β)≤C\nqn(β). (5.7)\nBy Remark 2.2, Ω ακis a subshift of finite type with alphabet Aκand\nincidence matrix Aκκforκ= 1,2.Define Ψκ={ψκ\nn:n≥1}on Ωακby\nψκ\nn(x) := ln|B̟x1x|n|,\n27where̟x1∈Ω(ακ)\n5is some fixed word such that ̟x1x1admissible (see [27]\n(4.3)). By [27] (Theorem 10 and eq. (5.7)), there exist an erg odic measure\nνκsupported on Ω ακand a constant Cκ>1 such that (warn that in [27],\nακis defined as ακ:= [κ;κ,κ,···] )\ndimHνκ=logακ\nΨκ∗(νκ)=:d(κ,λ) andC−1\nκα|w|\nκ≤νκ([w]ακ)≤Cκα|w|\nκ,(5.8)\nwhere Ψκ\n∗(νκ) := lim n→∞1\nn/integraltext\nΩακψκ\nndνκ.\nBy [27] Proposition 6 and Remark 7, there exists λ0>20 such that\nd(1,λ)<d(2,λ) whenλ≥λ0. Sinceνκis ergodic, for νκ-a.e.x∈Ωακ,\nψκ\nn(x)\nn→Ψκ\n∗(νκ). (5.9)\nFixκ∈ {1,2},n∈Nandǫ>0, define\nFκ(n,ǫ) ={w∈Ωακ,n:/vextendsingle/vextendsingle/vextendsingle/vextendsingleψκ\nn(x)\nn−Ψκ\n∗(νκ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤ǫfor anyx∈[w]ακ}.(5.10)\nSinceψκ\nn(x) =ψκ\nn(y) ifx|n=y|n,we can replace “any” by “some” in the\ndefinition above. For each e∈ Aκ, define\nFκ(e,n,ǫ) :=Fκ(n,ǫ)∩Ξe,n−1(ακ).\nClaim: For anym∈N, there exists lm∈Nsuch that for any κ∈ {1,2},\nanye∈ Aκand anyn≥lm,\n#Fκ(e,n,2−m)\n#Ξe,n−1(ακ)≥1−C−22−m.\n⊳Fix anye∈ Aκ. By (5.9), νκ(An(e))→νκ([e]ακ) asn→ ∞, where\nAn(e) :=/uniondisplay\nw∈Fκ(e,n,2−m)[w]ακ.\nWriteηm:=C−2\nκC−22−m. Assumelm(e)∈Nis such that for any n≥lm(e),\nνκ(An(e))≥(1−ηm)νκ([e]ακ). (5.11)\nDefinelm:= max{lm(e) :e∈ A1∪ A2}. Fix anye∈ Aκandn≥lm, by\n(5.8),\nνκ([e]ακ\\An(e))≥(#Ξe,n−1(ακ)−#Fκ(e,n,2−m))C−1\nκαn\nκ.\nBy (5.11) and (5.8), we have\nνκ([e]ακ\\An(e))≤ηmνκ([e]ακ)≤ηm#Ξe,n−1(ακ)Cκαn\nκ.\n28Consequently, we have\n#Ξe,n−1(ακ)−#Fκ(e,n,2−m)\n#Ξe,n−1(ακ)≤C−22−m,\nwhich implies the claim. ⊲\nFor anyβ∈B2, define Ψβby (5.1). By (4.7), for any x∈Ω(β),\n2log2+nlogc≤ψβ\nn(x)≤2log2−nlog2, (5.12)\nwhere 0< c <1 only depends on λ. By [27] (4.4), there exists constant\nd(λ)>0 such that for any κ∈ {1,2},n∈Nandy∈Ωακ,\n−d(λ)(n+6)≤ψκ\nn(y)≤ −nlog2. (5.13)\nNow we define two sequences {tn:n≥1}and{τn:n≥1}inductively\nas follows. Define t1=τ1:=l1.Assumet1,···,tn−1,τ1,···,τn−1have been\ndefined. Write /hatwideTn−1:=/summationtextn−1\nj=1(tj+τj) with convention /hatwideT0= 0 and choose\ntn≥lnsuch that\n\n/hatwideTn−1log3+logqtn(α1)≤(1+1\nn)logqtn(α1),\n|ψβ\n/hatwideTn−1|max+ψ1\ntn(x)≤(1−1\nn)ψ1\ntn(x),(∀β∈B2,∀x∈Ωα1).(5.14)\nBy (5.12), (5.13) and the fact that qn(α1)∼α−n\n1, suchtnexists. Write\nTn:=/hatwideTn−1+tnand choose τn≥lnsuch that\n−|ψβ\nTn|min+ψ2\nτn(y)≥(1+1\nn)ψ2\nτn(y),(∀β∈B2,∀y∈Ωα2).(5.15)\nBy (5.12) and (5.13), such τnexists.\nDefineα:= [0;a1,a2,···]∈B2witha1a2···= 1t12τ11t22τ2···. Assume\nµαis a Gibbs-like measure of Φα.Let us show that\ndimHµα≤d(1,λ) and dim Pµα≥d(2,λ).\nRecall that A1={(I,1)1,(I,2)1,II1,(III,1)1}={e1,1,e1,2,e1,3,e1,4}.\nDefine a Cantor subset Cof Ω(α)as follows. Write ǫm= 2−mand define\n\nW1:={Iu:u∈F1(e1,3,t1,ǫ1)},\n/hatwiderW1:={uv∈Ω(α)\n/hatwideT1:u∈W1, v∈F2(τ1,ǫ1)}.\nAssumeWn−1,/hatwiderWn−1have been defined, define Wn,/hatwiderWnas\n\n\nWn:={uv∈Ω(α)\nTn:u∈/hatwiderWn−1, v∈F1(tn,ǫn)},\n/hatwiderWn:={uv∈Ω(α)\n/hatwideTn:u∈Wn, v∈F2(τn,ǫn)}.\nDefine Cn:=/uniontext\nw∈Wn[w]αand/hatwideCn:=/uniontext\nw∈/hatwiderWn[w]α.It is seen that Cn+1⊂\n/hatwideCn⊂Cn. Define C:= limn→∞Cn.Let us show that µα(C)>0.\n29At first,W1is nonempty by the Claim. Then by (5.7), µα(C1)>0.Next,\nwe compare µα(/hatwideC1) andµα(C1).By (5.7) and the Claim, we have\nµα(C1\\/hatwideC1) =/summationdisplay\nw∈W1/summationdisplay\ne∈A2\nw→e/summationdisplay\nu∈Ξe,τ1−1(α2)\\F2(e,τ1,ǫ1)µα([wu]α)\n≤/summationdisplay\nw∈W1/summationdisplay\ne∈A2\nw→eC(#Ξe,τ1−1(α2)−#F2(e,τ1,ǫ1))\nq/hatwideT1(α)\n≤C−12−1/summationdisplay\nw∈W1/summationdisplay\ne∈A2\nw→e#Ξe,τ1−1(α2)\nq/hatwideT1(α)\n≤2−1/summationdisplay\nw∈W1/summationdisplay\ne∈A2\nw→e/summationdisplay\nu∈Ξe,τ1−1(α2)µα([wu]α) = 2−1µα(C1).\nOr equivalently, we have µα(/hatwideC1)≥(1−2−1)µα(C1).By essentially the same\nproof as above, we can show that for any n≥2,\nµα(Cn)≥(1−2−n)µα(/hatwideCn−1) andµα(/hatwideCn)≥(1−2−n)µα(Cn).\nAs a consequence, we get\nµα(C)≥µα(C1)(1−2−1)/productdisplay\nn≥2(1−2−n)2>0.\nTake anyx∈Cand writex|/hatwideTn=Iu1v1···unvnsuch that |ui|=ti\nand|vi|=τi.Thenui∈F1(ti,ǫi),vi∈F2(τi,ǫi) andx|Tn=x|/hatwideTn−1un,\nx|/hatwideTn=x|Tnvn.By the definition of α,G/hatwideTn−1(α) = [0;1,···,1/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\ntn,···],hence\nqtn(G/hatwideTn−1α) =qtn(α1).By (5.7), Lemma 4.1 and (5.6), we have\nµα([x|Tn]α)∼1\nqTn(α)∼1\nq/hatwideTn−1(α)qtn(G/hatwideTn−1α)≥1\n3/hatwideTn−1qtn(α1).\nBy Lemma 2.3 (ii), Lemma 5.1 and (5.1),\ndiam([x|Tn]α)∼λexp(ψα\nTn(x)) =|Bx|Tn|=|Bx|Tn|\n|Bx|/hatwideTn−1|·|Bx|/hatwideTn−1|\n=|Bx|/hatwideTn−1un|\n|Bx|/hatwideTn−1|·exp(ψα\n/hatwideTn−1(x)).\nNow take any yn∈[un]α1,we have\nexp(ψ1\ntn(yn)) =|B̟yn\n1yn|tn|=|B̟yn\n1un|=|B̟yn\n1un|\n|B̟yn\n1||B̟yn\n1|.\n30Since̟yn\n1∈Ω(α1)\n5,it only has finitely many choices, combine with Theorem\n4.10, we have\ndiam([x|Tn]α)∼λexp(ψ1\ntn(yn))exp(ψα\n/hatwideTn−1(x)).\nRecall that un∈F1(tn,ǫn), thus by (5.14), (5.10), (5.8) and the fact that\nqn(α1)∼α−n\n1, we have\ndµα(x)≤liminf\nn→∞logµα([x|Tn]α)\nlogdiam([x|Tn]α)\n≤liminf\nn→∞O(1)+/hatwideTn−1log3+logqtn(α1)\n−/parenleftig\nO(1)+ψα\n/hatwideTn−1(x)+ψ1\ntn(yn)/parenrightig\n≤liminf\nn→∞(1+1\nn)logqtn(α1)\n−(1−1\nn)ψ1\ntn(yn)=d(1,λ).\nOntheotherhand,noticethat GTn(α) = [0;2,···,2/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nτn,···],thusqτn(GTn(α)) =\nqτn(α2). Hence by (5.7) and Lemma 4.1, we have\nµα([x|/hatwideTn]α)∼1\nq/hatwideTn(α)∼1\nqTn(α)qτn(GTn(α))≤1\nqτn(α2).\nTake anyzn∈[vn]α2,by similar argument as above, we also have\ndiam([x|/hatwideTn]α)∼λexp(ψ2\nτn(zn))exp(ψα\nTn(x)).\nRecall that vn∈F2(τn,ǫn), thus by (5.15), (5.10), (5.8) and the fact that\nqn(α2)∼α−n\n2, we have\ndµα(x)≥limsup\nn→∞logµα([x|/hatwideTn]α)\nlogdiam([x|/hatwideTn]α)\n≥limsup\nn→∞O(1)+logqτn(α2)\n−/parenleftbig\nO(1)+ψα\nTn(x)+ψ2τn(zn)/parenrightbig\n≥limsup\nn→∞logqτn(α2)\n−(1+1\nn)ψ2τn(zn)=d(2,λ).\nAs a result, for any x∈C,we have\ndµα(x)≤d(1,λ) anddµα(x)≥d(2,λ).\nSinceµα(C)>0 andµαis exact upper and lower dimensional by Theorem\n3.3, we conclude that\ndimHµα≤d(1,λ)<d(2,λ)≤dimPµα.\n31Now by Proposition 5.5, we have ( πα)∗(µα)≍ Nα,λ.Thus\ndH(α,λ) = dim Hµα<dimPµα=dP(α,λ).\nand the result follows. /square\nProof of Theorem 1.1 (ii). 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QU) Department of Mathematical Science, Tsinghua University, Bei-\njing 100084, P. R. China\nE-mail address :yhqu@math.tsinghua.edu.cn\n33"    },    {        "title": "1609.03814v1.Nonequilibrium_steady_states_in_a_closed_inhomogeneous_asymmetric_exclusion_process_with_particle_nonconservation.pdf",        "content": "arXiv:1609.03814v1  [cond-mat.stat-mech]  13 Sep 2016Nonequilibrium steady states in a closed inhomogeneous asy mmetric exclusion\nprocess with particle nonconservation\nBijoy Daga,1,∗Souvik Mondal,1,†Anjan Kumar Chandra,2Tirthankar Banerjee,1,‡and Abhik Basu1,§\n1Condensed Matter Physics Division, Saha Institute of Nucle ar Physics, Calcutta 700064, India\n2Department of Physics, Malda College, Malda, West Bengal, P in: 732101, India¶\n(Dated: July 16, 2018)\nWe study asymmetric exclusion processes (TASEP) on a nonuni form one-dimensional ring con-\nsisting of two segments having unequal hopping rates, or defects. We allow weak particle noncon-\nservation via Langmuir kinetics (LK), that are parameteris ed by generic unequal attachment and\ndetachment rates. For an extended defect, in the thermodyna mic limit the system generically dis-\nplays inhomogeneous density profiles in the steady state - th e faster segment is either in a phase with\nspatially varying density having no density discontinuity , or a phase with a discontinuous density\nchanges. Nonequilibrium phase transitions between them ar e controlled by the inhomogeneity and\nLK. The slower segment displays only macroscopically unifo rm bulk density profiles in the steady\nstates, reminiscent of the maximal current phase of TASEP bu t with a bulk density generally dif-\nferent from half. With a point defect, there are low and high d ensity spatially uniform phases as\nwell, in addition to the inhomogeneous density profiles obse rved for an extended defect. In all the\ncases, it is argued that the the mean particle density in the s teady state is controlled only by the\nratio of the LK attachment and detachment rates.\nI. INTRODUCTION\nThe effects of nonuniformities or disorders on the\nmacroscopic properties of equilibrium systems are well\nunderstood by now within the general framework of sta-\ntistical mechanics[1]. In contrast, the effects ofquenched\ndisorders on the dynamics and nonequilibrium steady\nstates (NESS) of out of equilibrium systems are much\nless understood [2]. Totally asymmetric simple exclusion\nprocess (TASEP) and its variants with open boundaries\ninonedimension(1D)serveassimplemodelsofrestricted\n1D transport [3]. Natural realisations of TASEP include\nmotions in nuclear pore complex of cells [4], motion of\nmolecular motors along microtubules [5], fluid flow in ar-\ntificial crystalline zeolites [6] and protein synthesis by\nmessenger RNA (mRNA) ribosome complex in cells [7];\nsee, e.g., Refs. [3] for basic reviews on asymmetric ex-\nclusion processes. Subsequently, studies on TASEP with\nparticle nonconservation in the form of on-off Langmuir\nkinetics (LK) [8] in the limit, when LK competes with\nthe hopping movement of TASEP reveals unusual phase\ncoexistencesinthe NESS, notfound in pureTASEP.Fur-\nthermore, open TASEPs with defects [9], both point and\nextended, have been studied, which investigated the ef-\nfects of the defects on the NESS and currents. Open\nTASEPwith asinglepoint defect alongwith LKisshown\nto display a variety of phases and phase coexistences as a\nresult of the competition between the defect and LK [10].\nStudies of TASEP in a closed ring are relatively few\nand far between. Translational invariance ensures that\n∗bijoy.daga@saha.ac.in\n†souvik.mondal@saha.ac.in\n‡tirthankar.banerjee@saha.ac.in\n§abhik.basu@saha.ac.in,abhik.123@gmail.com\n¶anjanphys@gmail.comTASEP in a homogeneous ring, with or without LK,\nproduces a homogeneous density profile in the steady\nstate. Nonuniform or inhomogeneous steady states are\nexpected only with explicit breakdown of the transla-\ntion invariance, e.g., by means of quench disorder in\nthe hopping rates at different sites. Exclusion processes\nin closed inhomogeneous rings with strict particle num-\nber conservation have been shown to display inhomoge-\nneous NESS [11–13]. More recently, TASEP in a ring\nwith quench disordered hopping rates together with non-\nconserving LK having equal rates of particle attachment\nand detachment has been studied in Ref. [14]. Very un-\nexpectedly, the model admits only macroscopically in-\nhomogeneous NESS, in the forms of two- and three-\nphase coexistences, regardless of extended or point de-\nfects, a feature that has been explained in general terms\nin Ref. [14]. Equal attachment and detachment rates, as\nused in Ref. [14], are actually an idealisation and sim-\nplification. In typical physical realisations of this model,\ne.g., vehicular/pedestrial traffic and ribosome transloca-\ntions along closed mRNA loops in the presence of defects\nwith particle nonconservation, attachment and detach-\nment rates are generically expected to be unequal. Thus,\nit is important to find how unequal rates will affect the\nresults of Ref. [14], or, the robustness of the results in\nRef. [14] against variation in the ratio Kof the attach-\nment and detachment rates. In this work, we set out\nto study this question by considering TASEP in a two-\nsegmentringhavingunequalhoppingrateswithLK.How\nLK dynamics with K/negationslash= 1 affects the phases of an open\nTASEP are studied in Ref. [8]. Here, we analyse how the\ninterplay of LK with K/negationslash= 1 and quenched inhomogene-\nity affects the NESS of a TASEP on a ring. The latter\nadmits only strictly uniform density due to the particle\nnumber conservation and translational invariance, and\nhas no phase transitions. In the present study, we find\nthat with an extended defect, in the NESS the faster seg-2\nmentdisplayseitheraninhomogeneous(spatiallynonuni-\nform) phase with no discontinuity in the density (here-\naftercontinuous density phase , orCDP)andaphasewith\na discontinuous change or a shock (hereafter shock phase ,\nor SP) in the density profile for all values of the defect\nstrength (i.e., the hopping rate along the extended de-\nfect) and the ratio of the attachment and detachment\nrates. Steady state density profiles in the CDP are found\ntohavethreeparts-twopartswith nonzerospatialgradi-\nents of the density and an intervening part having a con-\nstant magnitude of n0=K/(1+K), i.e., with zero slope.\nIn general, n0/negationslash= 1/2 forK/negationslash= 1, unlike the corresponding\nresult in Ref. [14]. The CDP and SP, respectively, arethe\ndirect generalisations of the three- and two- phase coex-\nistences for K= 1 [14]. By tuning the model parameters,\nnonequilibrium transitions between CDP and SP are ob-\nserved. The steady state density in the slower segment is\nalwaysmacroscopically uniform. Being controlled solely\nby the LK attachment and detachment rates, the value\nof the steady state density in the slower segment is n0.\nThis is reminiscent of the maximal current (MC) phase\nof TASEP, but with a current generically less than 1/4\nforK/negationslash= 1. In contrast, for a point defect, the model can\ndisplay spatially macroscopically uniform steady states\nfor either Kvery small or very large, yielding either low\ndensity (LD) or high density (HD) phases. For interme-\ndiate values of K, the model exhibits nonuniform phases\n-either CDPorSP, aswith anextended defect. Nonequi-\nlibrium transitions between all the three types of phases\nare again controlled by the LK attachment-detachment\nratesand inhomogeneity. Lastly, in all the phasesofboth\nextended and point defects, the mean particle density in\nthe NESS ofthe systemis arguedtobe n0, i.e., controlled\nonly by K. Our study is a close-ring analogue of the\nmodel in Ref. [8] that considered an open system. Our\nresults, as described in details below, manifestly reveals\nthe the significance of the ring geometry in controlling\nthe NESS. The rest of the article is organised as follows:\nIn Sec. II, we define our model. Then, a short review\nof the results for K= 1 is presented in Sec. IIA. In\nSecs. IIIA and IIIB we analyse the NESS of the model\nwith an extended and a point defect, respectively. We\nsummarise in Sec. IV at the end.\nII. THE MODEL\nConsideraTASEPof NsitestogetherwithLKona1D\nperiodic ring with spatial inhomogeneity. Owing to ex-\nclusion, the particle occupation, niat each site iis either\n0 or 1. In TASEP, a particle can only hop to its immedi-\nate vacant neighbour in one direction, say anticlockwise\n(see Fig. 1). Spatial inhomogeneity in the model is then\nintroduced by unequal hopping rates of the particles in\nthetwosegments,onefasterandtheotherslower,marked\nCHI and CHII respectively in Fig. 1. The hopping rates\nin CHI and CHII, respectively, are 1 and p(<1). LetN1\nandN2betherespectivesizesofCHIandCHII,suchthatA B\nCH ICH IIp\n11\nl1 –l\npω\nω\nωKωK\nFIG. 1. (Colour online) TASEP and LK dynamics on a 1D\nring with unequal hopping rates in CHI and CHII; the LK\ndetachmentand attachment rates are ωandωK, respectively;\nsee text.\nN1+N2=N. We define l=N1/Nas the fraction of the\ntotal sites in CHI; N1=Nl, N 2= (1−l)N. Note that\nsince the hopping rates do not evolve in time, the spatial\ninhomogeneities considered here are quenched. Further,\ndue to LK, a particle can detach from a given site with\nrateωorattach to a given site with rate ωK. We workin\nthe regime where TASEP competes with LK-dynamics,\nsuch that the net flux due to TASEP must be compara-\nble to the total flux due to LK. To ensure this, we apply\nscaling relations: Ω = ωN, whereNis a large but fi-\nnite number and study the system in the limit Ω ∼O(1)\n[8, 14]. For convenience, we label the sites by a contin-\nuous variable xin the thermodynamic limit, defined by\nx=i/N,0< x <1. In terms of the rescaled coordinate\nx, the lengths of CHI and CHII are land 1−l, respec-\ntively. The coordinate xrises from junction B ( x= 0) to\njunction A ( x=l) in the anticlockwise direction.\nA. Short review of the results for K= 1\nWe briefly review the results for K= 1 [14], i.e., with\nequal attachment and detachment rates. The mean-field\n(MF) equations are set up in both CHI and CHII. The\nsteady state densities are then calculated by observing\nthat current is conserved locally, and hence current con-\nservation can be applied very close to the junctions A\nandB:\nnI(y)(1−nI(y)) =pnII(y)(1−nII(y)),(1)\nwherey= junctions AorB. The steady state densities\nare given by\nnIB(x) = Ωx+1−√1−p\n2, (2)3\nnIA(x) = Ω(x−l)+1+√1−p\n2, (3)\nsuch that both Eqs. (2) and (3) depend upon the density\nvaluesatthejunctionsBandA,respectively. Incontrast,\nnIb(x) =1\n2, (4)\nand\nnII(x) =1\n2(5)\nare the bulksolutions independent of the densities at\nthe junctions, and are analogue of the MC phase in an\nopen TASEP. In CHI, let xαandxβrespectively be the\npoints where the left solution nIB(x) and the right so-\nlutionnIA(x) meets with the bulk solution nIb. It was\nshown that\nxβ−xα=l−√1−p\nΩ. (6)\nThus ifxα< xβ, the system is in a three phase (LD-\nMC-HD) coexistence. But if xα> xβ, the system can\nbe only in a two phase(LD-HD) coexistence. The critical\nline separating the two phases can be thus determined by\nsettingxα=xβ. Further, the global average density of\nthe system is always n0= 1/2, irrespective of the chosen\nvalues of pand Ω.\nIII. STEADY STATE DENSITIES\nWe perform MF analysis of our model, supplemented\nbyitsextensiveMonteCarloSimulations(MCS).Wesep-\narately present our results for an extended and a point\ndefect below. We consider only K <1; the results for\nK >1 may be obtained from those for K <1 together\nwith the particle-hole symmetry. In our MCS studies,\nwe use a random-sequential update scheme. We mea-\nsure the average site density, /angbracketleftni/angbracketrightover approximately\n2×109Monte-Carlo steps after relaxing the system for\n109Monte-Carlo steps. We separately study an ex-\ntended and a point defect. For an extended defect here,\n0< l <1, where as for a point defect, l→1. In Ref. [8],\nthe authors solved the steady state densities in the anal-\nogous open system in terms of the Lambert Wfunctions.\nHere, we use the implicit solutions of the densities in the\nsteady state that suffice for our purposes.\nA. MF analysis and MCS results for an extended\ndefect\nWe set up our MF analysis by closely following the\nlogic outlined in Ref. [14]. In our MF analysis, we de-\nscribe the model as a combination of two TASEPs - CHI\nand CHII, joined with each other at the junctions A and\nB; see Fig. 1. Thus, junctions B and A are effectiveentry (exit) and exit (entry) ends of CHI (CHII). This\nallows us to analyse the phases of the system in terms of\nthe known phases of the open boundary LK-TASEP [8].\nWithout LK, the steady state densities of a TASEP in\nan inhomogeneousring are completely determined by the\ntotal particle number (a conserved quantity) in the sys-\ntem in the steady state and the inhomogeneity configu-\nrations [11–13]. Due to the nonconserving LK dynamics,\nhowever, the particle current here is conserved only lo-\ncally, since the probability of attachment or detachment\nat a particular site vanishes as 1 /N[8, 14]. Hence, the\ntotal particle number in the NESS is not a conserved\nquantity. The steady state densities nI(x) andnII(x) in\nCHI and CHII respectively follow [8, 14]\n(2nI(x)−1)∂nI(x)\n∂x−Ω(1+K)/parenleftbigg\nnI(x)−K\n1+K/parenrightbigg\n= 0,(7)\nand\np(2nII(x)−1)∂nII(x)\n∂x−Ω(1+K)/parenleftbigg\nnII(x)−K\n1+K/parenrightbigg\n= 0.\n(8)\nBeforewe attempt to solveEqs.(7) and (8), we define an\naverage density n0in a given NESS that remains a con-\nstant on average, although the model dynamics does not\nadmit anyconservationlaw for the total particle number.\nIn the homogeneous limit of the model, i.e., with p= 1,\nCHI and CHII are identical and the steady state density\nis spatially uniform, due to the translational invariance\nforp= 1 for all K. Equation (7) or (8) then yield\nn0=K\nK+1. (9)\nThus, the average hole density is 1 −n0forp= 1. A\nglobal deviation of the mean density from n0should in-\ndicate either more particles or more holes entering into\nthe system in the steady state, than that is given by\nn0. However, even when inhomogeneity is introduced\n(p <1), there is nothing that favours either particles or\nholes, since the inhomogeneity that acts as an inhibitor\nfor the particle current, equally acts as an inhibitor for\nthe hole current. Hence, it is not expected to affect the\nvalue of n0. This is a major result of this work that is\nin agreement with the MCS studies; see below. Notice\nthat this argument does not preclude any local excess of\nparticles or holes, since the particles and the holes move\nin the opposite directions, and hence the presence of a\ndefect should lead to excess particles on one side and\nexcess holes (equivalently deficit particles) on the other.\nThis holds for any Ω and K. WhenK= 1,n0= 1/2, in\nagreement with Ref. [14]. Eqs. (7) and (8) are first order\ndifferential equations, each having one constant of inte-\ngration in their solutions. These may be determined by\nconsideringthe currentconservationor“boundarycondi-\ntions”at the junctions Aand B. In addition, Eqs.(7) and\n(8) admit a third spatially constant solution independent\nof the boundaries, given by K/(1 +K). Since CHI has\na higher hopping rate (1 > p), on physical grounds there\ncannot be pile up of particles in CHII behind junction B.\nInthe sameway, wedonotexpect an x-dependent nII(x)4\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1nI,II(x)\nxJunction-A\nxαxβxαxβnI(MCS)\nnII(MCS)\nnIB(MFT)\nnIA(MFT)\nn0  0.2 0.25\n 0.2 0.4jI(x)\nxjIBjIAj0\nFIG. 2. (Colour online) Plots of the density profiles in CDP\n(K= 0.5, p= 0.2,Ω = 4.5,l= 1/2, N= 2000). Mean-field\nsolutions nIB(x) (black dotted lines), nIA(x) (blue dotted\nlines) and the corresponding MCS results nI(x) (red squares)\nare shown. MCS results for nII(x) are shown with magenta\nsquares; xαandxβare extracted from MCS data (see text).\nn0(green dotted lines) is the average density of the system.\nInset: Mean-field values of the currents jIB(x),jIA(x) and\nj0(x) are plotted; xαandxβare extracted (see text), which\nmatch well with their corresponding MCS results.\nthatdecreaseswith xfromjunctionsAtoB.Thus, nII(x)\nshould be macroscopically uniform in the bulk. We are\nthenleftwithonlyonesolution nII(x) =K/(1+K). This\nsolution is independent ofthe boundariesAand B, and is\nthus akin to the maximal current (MC) phase of TASEP.\nThere is however a crucial difference: in an MC phase of\na TASEP, the steady state bulk density reaches its max-\nimum value of 1/2. However, with nII=K/(1+K), the\nsteady state bulk density in CHII is alwaysless than 1 /2\n(withK <1). We therefore call it the generalised MC\n(GMC) phase [16]; see also Ref. [8] for analogous results\nin a similar open system. Now consider CHI: since the\nbulk steady state density in CHII is K/(1 +K) =n0,\nthe average steady state density in CHI must also be\nK/(1 +K), in order to have n0as the mean density in\nthe whole system. Notice that a uniform nI(x) =n0\ndoes solve (7) above. However, this solution is not ad-\nmissible, as it manifestly violates current conservations\nat A and B. Thus, CHI can only admit macroscopically\nnonuniform density profiles in its NESS. If there are spa-\ntially varying LD phases in CHI (monotonically rising\nfrom B to A, remaining less than 1/2 everywhere), the\ncurrent conservation at either the junctions A and B will\nbe violated. This rules out a spatially varying LD phase\nin CHI. Clearly then, an analogous HD phase in CHI is\nalso ruled out. Therefore, on physical grounds one part\nof the solution for nI(x) should be <1/2 (near junction\nB) and another part >1/2 (near A) to maintain current\nconservation at both A and B; see Eqs. (10) and (11)\nbelow. This leaves us with two possibilities - (a) the two\nparts of the solutions matching smoothly with the bulk\nsolution nI(x) =n0,withoutany density discontinuity 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1nI,II(x)\nxJunction-A\nxwxwnI(MCS)\nnII(MCS)\nnIB(MFT)\nnIA(MFT)\nn0\n 0.2 0.21 0.22\n 0.35  0.4jI(x)\nxjIBjIA\nFIG. 3. (Colour online) Plots of the density profiles in\nSP (K= 0.5, p= 0.2,Ω = 2.25, l= 1/2, N= 2000).\nMean-field solutions nIB(x) (black dotted lines), nIA(x)(blue\ndotted lines) and the corresponding MCS results nI(x) (red\nsquares) are shown. MCS results for nII(x) are shown with\nmagenta squares; xwis extracted from MCS data (see text).\nn0(green dotted lines) is the average density of the system.\nInset: Mean-field values of the currents jIB(x),jIA(x) are\nplotted; xwhas been extracted (see text), which match well\nwith their corresponding MCS results.\n(CDP solution), or (b) the two parts do notmeet with\nthe bulk solution, leading to a density discontinuity (SP\nsolution), as discussed below. These arguments are used\nbelow to analyse the phases in the model.\n1. Phase diagrams and density profiles with an extended\ndefect\nAs pointed out in the previous section, CHII is always\nin the GMC-phase, with nII(x) =n0in the bulk. Since\na pure LD (hence an HD) phase is ruled out in CHI, it\nshould only have macroscopically inhomogeneous densi-\nties. This means there are no boundary layers in CHI\nat the junctions A and B. Then, applying the current\nconservation as given by Eq. (1) we get,\nnI(0) =α1=1\n2/bracketleftbigg\n1−1\n1+K/radicalbig\n(1+K)2−4pK/bracketrightbigg\n<K\n1+K,\n(10)\nand\nnI(l) =α2=1\n2/bracketleftbigg\n1+1\n1+K/radicalbig\n(1+K)2−4pK/bracketrightbigg\n>1\n2.(11)\nWe also obtain the general solution to Eq. (7),\n1\na/bracketleftbigg\n2nI(x)−/parenleftbigg\n1+2b\na/parenrightbigg\nln|anI(x)+b|/bracketrightbigg\n=x+C,(12)\nwherea= Ω(1 + K) andb=−ΩK. The constant of\nintegration, Cis to be determined appropriately by us-\ning either the boundary conditions (10) or (11), yielding\ngenerally two different solutions. Defining nIB(x) and\nnIA(x) as the solutions of Eq. (7) corresponding to the5\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1nI,II(x)\nxJunction-A\nxα= xβxα= xβnI(MCS)\nnII(MCS)\nnIB(MFT)\nnIA(MFT)\nn0  0.2 0.25\n 0.1 0.2 0.3 0.4 0.5jI(x)\nxjIBjIAj0\nFIG. 4. (Colour online) Plots of the density profiles in\nthe borderline case ( K= 0.5, p= 0.2,Ω = 3.75, l=\n1/2, N= 2000). Mean-field solutions nIB(x) (black dotted\nlines),nIA(x) (blue dotted lines) and the corresponding MCS\nresultsnI(x) (red squares) are shown. MCS results for nII(x)\nare shown with magenta squares; xαandxβare extracted\nfrom MCS data (see text). n0(green dotted lines) is the av-\nerage density of the system. Inset: Mean-field values of the\ncurrents jIB(x),jIA(x) andj0(x) are plotted; xαandxβare\nextracted (see text), which match well with their correspon d-\ning MCS results.\nboundary conditions ((10)) and ((11)), respectively, we\nfind\n2(nIB(x)−α1)−/parenleftbigg\n1+2b\na/parenrightbigg\nln/vextendsingle/vextendsingle/vextendsingle/vextendsingleanIB(x)+b\naα1+b/vextendsingle/vextendsingle/vextendsingle/vextendsingle=ax,\n(13)\nand\n2(nIA(x)−α2)−/parenleftbigg\n1+2b\na/parenrightbigg\nln/vextendsingle/vextendsingle/vextendsingle/vextendsingleanIA(x)+b\naα2+b/vextendsingle/vextendsingle/vextendsingle/vextendsingle=a(x−l).\n(14)\nGiven the physical expectation that nI(x) cannot de-\ncrease as xrises, we identify two points xαandxβ, at\nwhichnIB(x) andnIA(x), respectively, meet with the\nbulk solution nI(x) =n0. Defining jIB(x) andjIA(x)\nas the spatially varying currents corresponding to the\ndensities nIB(x) andnIA(x), respectively: jIB(x) =\nnIB(x)(1−nIB(x)),jIA(x) =nIA(x)(1−nIA(x)) and\nj0(x) =n0(1−n0),xαandxβmay be obtained from\njIB(x) =j0andjIA(x) =j0. As in Ref. [14], three\ndifferent scenarios are possible:\n(i)Continuous density phase (CDP) corresponding to\nxα< xβ:nI(x) rises from nI(x= 0) to reach nI(x) =n0\natxα, and then rise again from xβto reach nI(x) =\nnI(x=l). Ignoring boundary layers, nII(x) =n0, i.e.,\nGMC phase for CHII ensues. See Fig. 2 for a represen-\ntative plot with CDP for nI(x), where results from MFT\nand MCS studies are plotted together; good agreement\nbetween the MFT and MCS results are evident. Enu-\nmeration of xα, xβfrom MFT are shown in the inset.\nWhile with K= 1,nI(x) takes a very simple form\nas a function of x[14], for K/negationslash= 1 its functional form\nis more complex. Nonetheless, from the structures of 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1nI,II(x)\nxJunction-A\nxwxwnI(MCS)\nnII(MCS)\nnIB(MFT)\nnIA(MFT)\nn0  0.12 0.13 0.14 0.15\n 0.44 0.48jI(x)\nxjIBjIA\nFIG. 5. (Colour online) Plots of the density profiles in SP\n(K= 0.2, p= 0.2,Ω = 4.5,l= 1/2, N= 2000). Mean-field\nsolutions nIB(x) (black dotted lines), nIA(x) (blue dotted\nlines) and the corresponding MCS results nI(x) (red squares)\nare shown. MCS results for nII(x) are shown with magenta\nsquares; xαandxβare extracted from MCS data (see text).\nn0(green dotted lines) is the average density of the system.\nInset: Mean-field values of the currents jIB(x),jIA(x) and\nj0(x) are plotted; xαandxβare extracted (see text), which\nmatch well with their corresponding MCS results.\nEqs.(7), (13)and(14), wecanmakethefollowinggeneral\nobservations. (a) In general nIA> n0=nII(x)> nIB,\n(b) the slopes ∂nIB,A/∂x→0 asnIB,A(x)→K/(1+K)\nat some points in the bulk, (c) with K <1,∂nIB(x)/∂x\nneverdiverges,whereas ∂nIA(x)/∂xdivergesas nI(x)→\n1/2. Thus broadly, the slope nIA(x) should be steeper\nthan that of nIB(x) [15]. It is also clear that nIA(x)\nstarts from nIA(x=xβ) =n0<1/2 forK <1 to\nrise to (11) that is larger than 1/2. Thus, nIA(x) is\na combination of LD-HD phases. For K= 1,nIA(x)\nis necessarily more than 1/2, and hence fully HD, (d)\nlastly,nI(x) starts from LD (i.e., LD near junction B)\nand ends in HD (i.e., HD near junction A) always. These\nare consistent with our observations from MCS results;\nsee Fig. 2.\n(ii)Shock phase (SP) corresponding to xα> xβ[14].\nThere is no intervening flat portion in nI(x). Instead,\nnI(x) discontinuously changes from its value nIBtonIA\natx=xw, yielding a density shock or a localised do-\nmain wall (LDW) at x=xw. The condition jIB(xw) =\njIA(xw) yieldsxw. Density nII(x) remains at its GMC\nphase, i.e., nII(x) =n0, just like the CDP. See Fig. 3 for\na representative plot for SP of nI(x). Again, the agree-\nment between MFT and MCS results is close. In the\ninset, enumeration of xwfrom MFT is shown.\n(iii) The borderline case with xα=xβ: this corre-\nsponds to jIB(x) =jIA(x) =j0atx=xα=xβ. A\nrepresentative profile of nI(x) withxα=xβ, i.e., at the\nboundary between SP and CDP, is shown in Fig. 4; in-\nset shows a plot of numerical evaluation of xα=xβfrom\nthe MFT. Unlike for K= 1, when the density profiles\natxα=xβare linear [14] (because the factor 1+2b\naap-6\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1\n 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5p\nΩMFT\nMCS\nSP CDP\nFIG. 6. (Colour online) Phase diagram in the Ω −pplane\nfor extended defects, with K= 0.5: mean field result (red\ncontinuous line) and MCS results (black circles) are shown,\nwhich agree well. Here l= 1/2,N= 1000.\nK 0.2 0.5 1.0\nn0(MCS)0.166672 0.333547 0.499187\nn0(MFT)0.166667 0.333333 0.5\nTABLE I. Table comparing the average density value n0ob-\ntained from MCS and MFT for an extended defect. Here\nΩ = 2.5 andp= 0.5, andN= 2000.\npearing in Eq. (12) becomes zero for K= 1), for K <1,\nthe same factor is not zero and hence we do not get a lin-\near density profile at borderline case separating the CDP\nand SP regions. It may be noted that our above results\nhold good for all l,0< l <1; we have reported here the\nresults only for l= 1/2.\nIn Figs. 6 and 7, the phase diagrams for l= 1/2 in\nthe Ω−pplane for K= 0.5 andK= 0.2, respectively,\nas obtained from our MCS and MFT approaches, are\nshown. Unsurprisingly, there are only two phases - CDP\nand SP in both of them. The phase boundary between\nCDP and SP are determined from the condition xα=xβ\n(see above). Similar to Ref. [14], the MCS and MFT\nresults mutually agree qualitatively as well as quantita-\ntively. The phase diagrams in Fig. 6 and Fig. 7 are qual-\nitatively similar to each other, and are also similar to\nthe corresponding phase diagram for K= 1 in Ref. [14].\nStill, Fig.6andFig.7donotmatchquantitatively,nordo\nthey agree quantitatively with the corresponding phase\ndiagram for K= 1 in Ref. [14]. The region occupied by\nSP in Fig. 7 is noticeably larger than in Fig. 6, which in\nturn is larger than the two-phase region for K= 1; see\nRef. [14]. This trend may be explained in simple terms.\nFrom Eqs. (10) and (11), it is clear that for small K,\nthe jump in the values of nI(x) across the extended de-\nfect is large. On the other hand, ∂nI(x)/∂xis clearly\ncontrolled by the factor Ω(1 + K); see Eq. (7). Now in\norder for the CDP to exist, nI(x) must rise (fall) sharply\nenough from its value α1atx= 0 (α2atx=l) to match\nwithn0=K/(1 +K) in the bulk. Since ∂nI(x)/∂xis 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1\n 1 2 3 4 5 6 7 8 9 10p\nΩMFT\nMCS\nSP CDP\nFIG. 7. (Colour online) Phase diagram in the Ω −pplane\nfor extended defects, with K= 0.2: mean field result (red\ncontinuous line) and MCS results (black circles) are shown,\nwhich agree well. Here l= 1/2,N= 1000.\nsmaller for smaller Kfor a given Ω, a larger Ω is clearly\nneeded with smaller Kto ensure CDP. This explains\nwhy for K= 0.2, the CDP requires a higher Ω than\nforK= 0.5, which in turn has a higher-Ω threshold than\nthat for three-phase coexistence for K= 1 [14].\nThe changes in the phase diagrams given in Fig. 6 and\nFig. 7 with variation in Kare also reflected in the corre-\nsponding plots of nI(x) versus x. For instance, compare\nFig. 2 (K= 0.5) with Fig. 5 ( K= 0.2). Clearly, as K\nis reduced a CDP density profile for nI(x) gives way to\na SP density profile. This is consistent with the trends\nobserved in the above two phase diagrams.\nIn Table I, we compare the mean density in the NESS\nof the whole system as obtained from MCS with the pre-\ndictions from MFT for an extended defect. Clearly, good\nagreements are found. This validates our MFT argu-\nments elucidated above.\nB. MF analysis and MCS results for a point defect\nFor a point defect, junctions A and B merge. As a\nresult, the constraint from the constant bulk current in\nCHII on jI(x) =nI(x)(1−nI(x)) for an extended defect\ndoes not exist for a point defect. In fact, CHII effectively\nceases to exist, and Eq. (8) no longer holds. The steady\nstate density nI(x) of CHI now spans the whole system,\nand satisfies Eq. (7),\n(2nI(x)−1)∂nI(x)\n∂x−Ω(1+K)/parenleftbigg\nnI(x)−K\n1+K/parenrightbigg\n= 0.\n(15)\nWe now argue that (15) allows for a spatially uniform\nsteady state density in CHI, with a localised peak at the\nlocation of the point defect. Evidently, (15) allows for\na uniform solution nI(x) =K/(1+K)<1/2 (K <1),\nin addition to the space-dependent solutions, akin to the\nsolutions of nI(x) in the extended defect case, in the7\n 0.1 0.2 0.3 0.4 0.5 0.6\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1nI(x)\nxMCS\nnIB(MFT)\nnIA(MFT)\nn0\nFIG. 8. (Colour online) Plots of the density profiles in the\nLD region for a point defect located at x= 1 (K= 0.5, p=\n0.6,Ω = 1.0, N= 2000). Mean-field solutions nIB(x) (black\ndotted lines), nIA(x) (blue dotted lines) and the correspond-\ning MCS results nI(x) (red squares) are shown. n0(green\ndotted lines) is the average density of the system. We have\nextended the xaxis up to x= 1.1 to resolve the peak at x= 1\n(the location of the point defect) properly.\nNESS of the model. As argued in Refs. [10, 12], for a\nsufficiently low average density, the effect of the point\ndefect is confined to creating a localised peak (or a dip)\nthat has a vanishing width in the thermodynamic limit,\nin an otherwise homogeneous density profiles. In other\nwords, the bulk density should be uniform for sufficiently\nlow average density. For a ring geometry, this implies\nnI(x) =K/(1 +K), withnI(x) =n0+hhaving a lo-\ncalised peak of height h=n0(1−p)/pat the location\nof the point defect [10, 12]. This is consistent with our\ndiscussions above. See Fig. 8 for a representative plot of\nnI(x) asa function of xin the LD phase. As n0rises(i.e.,\nKrises), or the defect strength rises (i.e., pdecreases),\neventually this picture breaks down and the point defect\nthen leads to macroscopically nonuniform density pro-\nfiles. Following the logic outlined in Refs. [10, 12], we\nfind the threshold of inhomogeneous phases to be\nK\n1+K=p\n1+p=⇒K=p. (16)\nTherefore as Kexceedsp, spatially nonuniform density\nprofiles are to be formed in the NESS.\nForK > p, thespatiallyvaryingsolutionsof nI(x)may\nbe analysed as before. Without any loss of generality, we\nassume that the site with the defect is located at x= 1.\nAssuming that the particles hop anti-clockwise as before,\none expects that n(1−ǫ)> n(ǫ), where ǫ→0. LetρL\nandρRbe the densities at the left and right of x= 1.\nNow assume phase coexistence for nI(x) in the NESS,\ni.e., no boundary layers at x= 1. Applying the current\nconservation at x= 1, we get\nρL(1−ρL) =pρR(1−ρL) =ρR(1−ρR).(17)\nThis gives, ρL=p\n1+pandρR=1\n1+p. The CHI den- 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1nI(x)\nxxαxβxαxβMCS\nnIB(MFT)\nnIA(MFT)\nn0\n 0.1 0.2 0.3 0.4\n 0 0.2 0.4 0.6 0.8 1jI(x)\nxjIB\njIA\nj0\nFIG. 9. (Colour online) Plots of the density profiles in the\nCDP (K= 0.5, p= 0.05,Ω = 2.0, N= 2000). Mean-field\nsolutions nIB(x) (black dotted lines), nIA(x) (blue dotted\nlines) and the corresponding MCS results nI(x) (red squares)\nare shown. MCS results for nII(x) are shown with magenta\nsquares; xwis extracted from MCS data (see text). n0(green\ndotted lines) is the average density of the system. Inset:\nMean-field values of the currents jIB(x),jIA(x) are plotted;\nxwhas been extracted (see text). Qualitative agreement with\ntheir corresponding MCS results are evident.\nsity is then obtained by using Eq. (7) with the boundary\nconditions\nnI(0) =p\n1+pandnI(1) =1\n1+p.(18)\nAs for the extended defect case, there are three dif-\nferent solutions: two spatially varying depending upon\nthe boundary conditions (18) and a third bulk solution\nnI(x) =K/(1+K) =n0, that is independent of x. The\nspatially varying solutions are\n2/parenleftbigg\nnIB(x)−p\n1+p/parenrightbigg\n−/parenleftbigg\n1+2b\na/parenrightbigg\nln/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleanIB(x)+b\nap\n1+p+b/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=ax,\n(19)\nand\n2/parenleftbigg\nnIA(x)−1\n1+p/parenrightbigg\n−/parenleftbigg\n1+2b\na/parenrightbigg\nln/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleanIA(x)+b\na\n1+p+b/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=a(x−1).\n(20)\nAgain as in CDP with extended defect, we can de-\nfinexαandxβfrom the conditions nIB(xα) =n0and\nnIA(xβ) =n0. Equivalently, defining currents jIA(x) =\nnIA(x)[1−nIA(x)],jIB(x) =nIB(x)[1−nIB(x)] and\nj0(x) =n0(1−n0),xαandxβare obtained from\njIB(xα) =j0andjIA(xβ) =j0respectively. Similar\nto the extended defect case (see also Ref. [14]), three dis-\ntinct cases are possible:\n(i) CDP for xα< xβ:nI(x) is qualitatively similar to\nits analogue for the extended defect case. A representa-\ntive plot of nI(x) in its CDP as a function of xis shown\nin Fig. 9.\n(ii) SP with xα> xβyielding an LDW at x=xw. A\nrepresentative plot of nI(x) in its SP as a function of x\nwith an LDW at x=xwis shown in Fig. 10.8\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1nI(x)\nxxwxwMCS\nnIB(MFT)\nnIA(MFT)\nn0\n 0.2 0.225 0.25\n 0.85 0.9jI(x)\nxjIB\njIA\nFIG. 10. (Colour online)Plots ofthedensityprofilesin theS P\nphase (K= 0.5, p= 0.2,Ω = 0.75, N= 2000). Mean-field\nsolutions nIB(x) (black dotted lines), nIA(x) (blue dotted\nlines) and the corresponding MCS results nI(x) (red squares)\nare shown. MCS results for nII(x) are shown with magenta\nsquares; xwis extracted from MCS data (see text). n0(green\ndotted lines) is the average density of the system. Inset:\nMean-field values of the currents jIB(x),jIA(x) are plotted;\nxwhas been extracted (see text). Qualitative agreement with\ntheir corresponding MCS results are evident.\n(iii) The borderline case between CDP and SP given\nbyxα=xβ; see Fig. 11.\nTo find out the phase boundary between the CDP and\nSP regions, we proceed in the same way as in case of\nextended defects, that is we find out pand Ω values for\nwhichxα=xβ. ForK < p, the system is in a spatially\nhomogeneous LD phase. The phase diagram for the case\nof point defect showing the three distinct phase regions\nis shown in Figs. 13 and 14.\nNotice that the extent of the LD phase in Fig. 14 is\ndistinctly larger than in Fig. 13. This is consistent with\nthe fact that the boundary between LD and the spatially\ninhomogeneous phases (SP or CDP) in MFT is given by\np=K(in agreement with MCS results), that clearly\nyields a larger LD phase in the Ω −pplane for a lower\nK. Next, consider the relative preponderance of the SP\nover the CDP in Fig. 14 in comparison with Fig. 13.\nUnlike the case with an extended defect, the jump in an\ninhomogeneous nI(x) at the point defect depends onlyon\np, and not on K, regardless of SP or CDP. Nonetheless,\nas discussed above, the slope ∂nI(x)/∂xin NESS is still\ncontrolled by Ω(1 + K). Hence, for reasons similar to\nthose for an extended defect, in a Ω −pphase diagram,\nthe CDP starts for a higher Ω with a lower K. ForK=\n1, there is no LD phase even for a point defect. For a\nputative LD phase one must have p > K; forK= 1 this\ncondition rules out an LD phase.\nThe quantitative dissimilarities between the phase di-\nagram in Ref. [14] for a point defect and the inhomo-\ngeneous parts of those in Figs. 13 and 14 may be ex-\nplained following the logic outlined in the discussions on\nthe phase diagrams for an extended defect above. These 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1nI(x)\nxxα=xβxα=xβMCS\nnIB(MFT)\nnIA(MFT)\nn0\n 0.1 0.2 0.3\n 0 0.2 0.4 0.6 0.8 1jI(x)\nxjIBjIAj0\nFIG. 11. (Colour online) Plots of the density profiles in\nthe borderline case for a point defect located at x= 1\n(K= 0.5, p= 0.2,Ω = 1.25, N= 2000). Mean-field solu-\ntionsnIB(x) (black dotted lines), nIA(x) (blue dotted lines)\nand the corresponding MCS results nI(x) (red squares) are\nshown. MCS results for nII(x) are shown with magenta\nsquares; xαandxβare extracted from MCS data (see text).\nn0(green dotted lines) is the average density of the system.\nInset: Mean-field values of the currents jIB(x),jIA(x) and\nj0(x) are plotted; xαandxβare extracted (see text), which\nmatch well with their corresponding MCS results.\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1nI(x)\nxxwxwMCS\nnIB(MFT)\nnIA(MFT)\nn0\n 0.1 0.15\n 0.9  0.95jI(x)\nxjIB\njIA\nFIG.12. (Colour online) Plots ofthedensityprofilesintheS P\nfor a point defect located at x= 1 (K= 0.2, p= 0.05,Ω =\n2.0, N= 2000). Mean-field solutions nIB(x) (black dotted\nlines),nIA(x) (blue dotted lines) and the corresponding MCS\nresultsnI(x) (red squares) are shown. MCS results for nII(x)\nare shown with magenta squares; xαandxβare extracted\nfrom MCS data (see text). n0(green dotted lines) is the av-\nerage density of the system. Inset: Mean-field values of the\ncurrents jIB(x),jIA(x) andj0(x) are plotted; xαandxβare\nextracted (see text), which match well with their correspon d-\ning MCS results.\ndifferences can lead to qualitative differences in the den-\nsity profiles in the NESS. For instance, compare Fig. 9\n(K= 0.5) and Fig. 12 ( K= 0.2) to observe that the\nCDP density profile for K= 0.5 has become a SP den-\nsity profile for K= 0.2. This trend is similar to what\none finds for extended defects.9\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1\n 0.5  1 1.5  2 2.5p\nΩMFT\nMCS\nLD\nSP CDP\nFIG. 13. Phase diagram in the Ω −pplane for a point defect\nlocated at x= 1, with K= 0.5.: Mean-field results (red solid\nline) and the MCS results are shown. The green-dotted line,\nK=pseparating the LD-phase from other phase regions is\nobtained from MFT and satisfy the equation p=K. The\ncorresponding MCS data points are represented by the blue-\ntriangles. Here N= 1000.\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1\n 1  2  3  4  5p\nΩMFT\nMCS\nLD\nSP CDP\nFIG. 14. Phase diagram in the Ω −pplane for a point defect\nlocated at x= 1, with K= 0.2.: Mean-field results (red solid\nline) and the MCS results are shown. The green-dotted line,\nK=pseparating the LD-phase from other phase regions is\nobtained from MFT and satisfy the equation p=K. The\ncorresponding MCS data points are represented by the blue-\ntriangles. Here N= 1000.\nIn Table II, we compare the values of the mean den-\nsities obtained from MCS with the corresponding MFT\nresults for different values of K. Again, similar to the\nextended defect case, there is a good quantitative agree-\nment between the two, lending credence to our MFT re-\nsults.\nIV. SUMMARY AND OUTLOOK\nIn this work, we have studied an asymmetric exclu-\nsion process in an inhomogeneous ring with particle non-\nconserving LK dynamics. The attachment/detachmentK 0.2 0.5 1.0\nn0(MCS)0.166915 0.333366 0.499587\nn0(MFT)0.166667 0.333333 0.5\nTABLE II. Table comparing the average density value n0ob-\ntained from MCS and MFT for a point defect. Here Ω = 1 .0,\np= 0.5 andN= 2000.\nrates of LK are generally assumed to be unequal. We\nhave considered both extended and point defects. The\nMFT analysis is done by assuming the system to be a\ncombination of two TASEP channels CHI (of hopping\nrate 1) and CHII (of hopping rate p <1) of unequal hop-\nping rates, which are joined together at both the ends.\nFor a point defect, CHII shrinks to a point. Our MFT\nanalysis, backed up by extensive MCS studies, clearly re-\nveals that with an extended defect, nII=K/(1+K) in\nthebulkofCHII,aconstantthatisindependentof p(<1)\n(i.e., independent ofthe boundariesorjunctions at Aand\nB). In analogy with the MC phase of an open TASEP,\nwe call this the GMC phase. In contrast, CHI is found\nin either CDP or SP. Unsurprisingly, CDP is favoured\nfor larger Ω, where as SP prevails for smaller Ω for fixed\np,K, as in Ref. [14]. For a point defect, for which CHII\nshrinks to a point, for K < p,nI(x) is found in a uniform\nphase, unlike an extended defect. The variation in the\nphase diagrams with Kare explained in simple physical\nterms. As in Ref. [14], the quantitative agreement be-\ntween the MCS and MFT results for an extended defect\nis very high, as is evident in the corresponding phase di-\nagram; see Fig. 6 and Fig. 7. In contrast, the agreement\nfor the point defect case is weaker, in particular for small\nΩ. The physical reason behind these discrepancies are\nexpected to be the same as that elaborated in Ref. [14].\nOurresultsclearlybringouttherelevanceoftheringor\nclosed geometry of the system in the presence of LK with\nunequal attachment and detachment rates. The results\nhere as well as those in Ref. [14] clearly establish how\nthe ring geometry (or the lack of independent entry and\nexit events) restricts the possible phases in these mod-\nels, in comparison with the results on the corresponding\nopen system; see Refs. [8, 10]. It would be interesting\nto numerically study the crossover between the extended\nand point defect cases by varying N, while keeping the\nlength of the slower segment unchanged. As an alterna-\ntive to our simple MFT, it should interesting to extend\nthe boundary layer formalism developed in Refs. [17] for\nthe present problem. The simplicity of our model lim-\nits direct applications of our results to practical or ex-\nperimental situations. Nonetheless, our above results in\nthe context of vehicular traffic along a circular track, or\nrailway movements in series along a closed railway loop\nline with possibilities of new carriages joining or exist-\ning carriages going off the loop track in the presence of\nconstrictions (regions of slow passages due to, e.g., ac-\ncidents or damages in the tracks), or ribosome translo-\ncations along mRNA loops with defects and random at-10\ntachment/detachment, clearly demonstrate that for ex-\ntended defects, the steady state densities are in general\ninhomogeneous, where as for a point defect, globally uni-\nform densities are possible for a sufficiently low average\ndensity. We hope experiments on ribosomes using ribo-\nsome profiling techniques [18] and numerical simulations\nof more detailed traffic models should qualitatively vali-\ndate our results.\nV. ACKNOWLEDGEMENT\nSM acknowledgesthe financial support from the Coun-\ncil of Scientific and Industrial Research, India [Grant No.09/489(0096)/2013-EMR-I]. TB and AB gratefully ac-\nknowledge partial financial support from Alexander von\nHumboldt Stiftung, Germany under the Research Group\nLinkage Programme (2016).\n[1] J. Ziman, Models of Disorder: the Theoretical Physics of\nHomogeneously Disordered Systems (Cambridge Univer-\nsityPress, Cambridge, MA,USA,1979); R.Stinchcombe,\nDilute magnetism , vol. 7 of Phase Tran- sition and Crit-\nical Phenomena (Academic Press, New York, NY, USA,\n1983).\n[2] R. Stinchcombe, J. Phys.: Condens. Matter 14, 1473\n(2002).\n[3] D. Chowdhury, L. Santen, and A. Schadschneider, Phys.\nRep.329, 199 (2000); D. Helbing, Rev. Mod. Phys. 73,\n1067 (2001); T. Chou, K. 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Arava et al, Nucl. Acids Res. 33, 2421 (2005); N. T.\nIngoliaet al,Science324, 218 (2009); H. Guo et al, Na-\nture466, 835 (2010)."    },    {        "title": "1609.09835v1.Extremal_Density_Matrices_for_Qudit_States.pdf",        "content": "Extremal Density Matrices for Qudit States\nArmando Figueroa,\u0003Julio A. L\u0013 opez-Sald\u0013 \u0010var,yOctavio Casta~ nos,zand Ram\u0013 on L\u0013 opez{Pe~ nax\nInstituto de Ciencias Nucleares, Universidad Nacional Aut\u0013 onoma de M\u0013 exico,\nApartado Postal 70-543, 04510 M\u0013 exico DF, Mexico\n(Dated: October 4, 2018)\nAbstract\nAn algebraic procedure to \fnd extremal density matrices for any Hamiltonian of a qudit system\nis established. The extremal density matrices for pure states provide a complete description of\nthe system, that is, the energy spectra of the Hamiltonian and their corresponding projectors.\nFor extremal density matrices representing mixed states, one gets mean values of the energy in\nbetween the maximum and minimum energies associated to the pure case. These extremal densities\ngive also the corresponding mixture of eigenstates that yields the corresponding mean value of the\nenergy. We enhance that the method can be extended to any hermitian operator.\n\u0003armando.\fgueroa@nucleares.unam.mx\nyjulio.lopez@nucleares.unam.mx\nzocasta@nucleares.unam.mx\nxlopez@nucleares.unam.mx\n1arXiv:1609.09835v1  [math-ph]  30 Sep 2016I. INTRODUCTION\nThe density matrix approach was introduced to describe statistical concepts in quantum\nmechanics by Landau [1], Dirac [2], and von Neumann [3]. In several branches of physics like\npolarized spin assemblies or qudit systems, and cavity electrodynamics the density matrix\napproach can be cast into a su(d) description [4]. The Bloch vector parametrization was used\nto describe the 2-level problem which later on was generalized to describe beams of particles\nwith spinsin terms of what are known as Fano statistical tensors [5, 6]. In particular (2 s+1)2\nprojectors de\fning the generators of a unitary algebra have been introduced in [7] to expand\na density matrix of spin systems, even more, they established a procedure to reconstruct\nthe density matrix by a \fnite number of magnetic dipole measurements with Stern-Gerlach\nanalyzers and concluded that it was necessary to do at least 4 smeasurements to reconstruct\nthe density matrix of pure states while 4 s(s+ 1) were required for mixed states [7, 8]. An\nexperimental reconstruction of a cavity state for s= 4 using this method is given in [9].\nAnother approach uses the Moore - Penrose pseudoinverse to express the elements of the\nspin density matrix in terms of (2 s+ 1)(4s+ 1) probabilities of spin projections [10]. A\nmethod to reconstruct any pure state of spin in terms of coherent states is provided in [11]\nand by means of non orthogonal projectors on coherent states a reconstruction of mixed\nstates can be done [12]. A parametrization based on Cholesky factorization [13] was \frst\nused to guarantee the positivity of the spin density matrices in [14], and more recently, a\ntomographic approach to reconstruct them [15{18].\nIn the last twenty years, a lot of work related with parametrization of the density ma-\ntrices ofd-level quantum systems has been done [19{23]. This is due to its applications to\nquantum computation and quantum information systems [24]. The decomposition of the\ndensity matrix into a symmetrized polynomial in Lie algebra generators has been deter-\nmined in [25]. A novel tensorial representation for density matrices of spin states, based on\nWeinberg's covariant matrices, may be another important generalization of the Bloch sphere\nrepresentation [26].\nActually, there are several parametrizations of \fnite density matrices: generalizations of\nthe Bloch vector [19], the canonical coset decomposition of unitary matrices [21, 22], the\nrecursive procedures to describe n\u0002nunitary matrices in terms of those of U(n\u00001)[23, 27],\nby factorizing n\u0002nunitary matrices in terms of points on complex spheres [28], and by\n2de\fning generalized Euler angles [29]. Even in the case of composite systems there are\nparametrizations of \fnite density matrices [30, 31].\nRecently we have established a procedure to determine the extremal density matrices of\na qudit system associated to the expectation value of any observable [32]. These matrices\nprovide an extremal description of the mean values of the energy, and in the case of restricting\nthem to pure states the energy spectrum is recovered. So, apart from being an alternative\ntool to \fnd the eigensystem one has information of mixed states which minimize its mean\nvalue.\nThe aim of this work is to give another option to compute extremal density matrices in\na qudit space by means of an algebraic approach that leads to an underdetermined linear\nsystem in terms of the components of the Bloch vector \u0015= (\u00151;\u00152;:::;\u0015d2\u00001), the antisym-\nmetric structure constants fijkof asu(d) algebra, and the parameters of the Hamiltonian\noperatorfhkg. Their solution, in general, implies to get the Bloch vector in terms of a\nknown number of free components. These are determined by establishing a system of equa-\ntions associated to the characteristic polynomial of the density matrix. Finally, one arrives\nto the extremal density matrices of the expectation value of the Hamiltonian, which for the\npure case let us obtain the corresponding full spectrum or for the mixed case at most d!\nextremal mean value energies. Another goal is to bring and join di\u000berent algebraic tools in\nthe study of the behaviour of both the density matrix and hermitian operators.\nII. GENERALIZED BLOCH-VECTOR PARAMETRIZATION\nAny hermitian Hilbert-Schmidt operator acting on the d-dimensional Hilbert space can\nbe expressed in terms of the identity operator plus a set of hermitian traceless operators\nf^\u00151:::^\u0015d2\u00001gwhich are the generators of the su(d) algebra. In this basis, the Hamiltonian\noperator ^Hand the density matrix ^ \u001aare written as [33]\n^H=1\ndh0bI+1\n2d2\u00001X\nk=1hk^\u0015k; (1)\n^\u001a=1\nd^I+1\n2d2\u00001X\nk=1\u0015k^\u0015k; (2)\nwith the de\fnitions h0\u0011Tr(^H); hk\u0011Tr(^H^\u0015k) and\u0015k\u0011Tr(^\u001a^\u0015k).\n3These generators are completely characterized by means of their commutation and anti-\ncommutation relations given by\nh\n^\u0015j;^\u0015ki\n= 2id2\u00001X\nq=1fjkq^\u0015q; (3)\nf^\u0015j;^\u0015kg=4\nd\u000ejk^I+ 2d2\u00001X\nq=1djkq^\u0015q; (4)\nwheredjkqandfjkqare the symmetric and antisymmetric structure constants\ndjkq=1\n4Tr(f^\u0015j;^\u0015kg^\u0015q); (5)\nfjkq=1\n4iTr(h\n^\u0015j;^\u0015ki\n^\u0015q); (6)\nand consequently, it follows the multiplication law [34]\n^\u0015j^\u0015k=2\nd^I\u000ejk+d2\u00001X\nq=1(djkq+ifjkq)^\u0015q: (7)\nA realization of the generators can be given by the generalized Gell-Mann matrices [20],\nconsisting in s= 1;:::;d(d\u00001)\n2symmetric matrices\n^\u0015s=^Pjk+^Pkj; (8)\nplusa=d(d\u00001)\n2+ 1;:::;d (d\u00001) antisymmetric matrices\n^\u0015a=\u0000i(^Pjk\u0000^Pkj); (9)\nandl= 1;:::;d\u00001 diagonal ones\n^\u0015d(d\u00001)+l=s\n2\nl(l+ 1)(^P11+^P22+\u0001\u0001\u0001+^Pll\u0000l^Pl+1l+1); (10)\nwhere 1\u0014j < k\u0014dand ^Pjk\u0011jjihkjare matrices with 1 in the component ( j; k) and 0\notherwise.\nThis type of realization belongs to the so called generalized Bloch vector parametriza-\ntion [19]. The Fano statistical tensors [5, 6], the multipole moments [7], the Weyl matri-\nces [19], and the generalized Gell-Mann matrices [20], belong to this group. Therefore a\nvector with d2\u00001 real components de\fne the so called generalized Bloch vector [4, 20],\n\u0015= (\u00151;:::;\u0015 d(d\u00001)\n2;\u0015d(d\u00001)\n2+1;:::;\u0015d(d\u00001);\u0015d(d\u00001)+1;:::;\u0015d2\u00001); (11)\n4whose magnitude is bounded by [35]\nj\u0015j\u0014r\n2(d\u00001)\nd; (12)\nwhere the equality speci\fes a necessary condition to represent a pure state.\nIn general, a SU(d) unitary transformation acting on a hermitian matrix implies a rota-\ntion in its components, i.e.,\n^H0=^U^H^Uy=1\ndh0^I+1\n2d2\u00001X\nj=1hj^U^\u0015j^Uy\n\u00111\ndh0^I+1\n2d2\u00001X\nj=1h0\nj^\u0015j; (13)\nwhere in the last equality, one has de\fned\nh0\nk= Tr( ^H0^\u0015k) =d2\u00001X\nj=1Okjhj; (14)\nand\nOkj\u00111\n2Tr(^\u0015k^U^\u0015j^Uy); (15)\nare elements of an orthogonal matrix that belongs to the SO(d2\u00001) group, which provides\nthe adjoint representation of SU(d) [36, 37].\nIII. POSITIVITY CONDITIONS FOR THE DENSITY OPERATOR\nThe density matrix must satisfy the following three properties: (a) It is Hermitian, (b)\nit has trace one, and (c) all its eigenvalues are positive semide\fnite. While for dimension\nd= 2, the condition Tr(^ \u001a2)\u00141 implies (c), for d\u00153 that is not true.\nThe positivity conditions of the density matrix are established by the set fakgof coe\u000e-\ncients of its corresponding characteristic polynomial. This set can be obtained by means of\nthe recursive relation known as Newton-Girard formulas [22, 38]\nak=1\nkkX\nj=1(\u00001)j\u00001ak\u0000jtj; (16)\nwith the de\fnitions a0=a1= 1,ad= det ^\u001a, andtj= Tr(^\u001aj), forj= 1;:::;d . Therefore the\nallowed density matrix must satisfy the following system of d\u00001 simultaneous polynomial\nequations\nak=ck; fork= 2;3;:::;d; (17)\n5where the constants ck\fx the degree of mixing of the system. Thus, they must be in the\nregion given by [33, 39, 40]\n0\u0014ck\u00141\ndk\u0012d\nk\u0013\n; (18)\nwhere\u0000d\nk\u0001\ndenotes a binomial coe\u000ecient. The upper bound de\fnes the most mixed state\nand then it has maximum entropy, while the lower bound speci\fes pure states which have\nzero entropy. Additionally, the ck= 0 fork>rank(^\u001a) [13].\nAll of them are polynomial functions in terms of the invariants of the density matrix,\ni.e.,tj, forj= 1;:::;d . In terms of tk\u0011Tr(^\u001ak), it is de\fned the symmetric matrix called\nBezoutian [41{43]\nBd=0\nBBBBBBBBB@d t 1t2\u0001\u0001\u0001td\u00001\nt1t2t3...td\nt2t3...td+1\n.........\ntd\u00001tdtd+1\u0001\u0001\u0001t2(d\u00001)1\nCCCCCCCCCA: (19)\nA polynomial with real coe\u000ecients has reals roots i\u000b the Bezoutian matrix is positive de\f-\nnite [41]. Hence, the compatible region among the invariants is obtained with the intersection\nof the positivity conditions of the density matrix from (18) with the respective positivity\nconditions of the Bezoutian (see details in Appendix A).\nIV. RAYLEIGH QUOTIENT AND THE DENSITY MATRIX\nThe Rayleigh quotient RT( ) of a hermitian Hilbert-Schmidt operator ^Tis\nRT( ) :=h j^Tj i\nh j i; (20)\nwherej iis ad-dimensional complex vector. Since the Rayleigh quotient is invariant under\nscale transformations, in searching its maximum or minimum it su\u000eces to con\fne the search\non unit norm vectors, i.e., when h j i= 1 [44]. This leads to de\fne the numerical range\nW(^T), which is the set of all possible Rayleigh quotients RT( ) over the unit vectors:\nW(^T) =fRT( );h j i= 1g: (21)\nThe numerical range W(^T) is a closed interval on the real axis, whose end points are\nthe extreme eigenvalues of ^T[45]. This result is a particular case of the Courant-Fischer\n6Theorem [13], which states that every eigenpair (eigenvalue and eigenvector) of ^Tis the\nsolution of a optimization (max-min problem) of W(^T) in some subspace of ^T. Therefore,\neigenvectors and eigenvalues of ^Tare the critical points and critical values, respectively, of\nthe Rayleigh quotient and W(^T) is the convex hull of its eigenvalues.\nIn the density matrix formalism, the numerical range of the Hamiltonian (or any her-\nmitian operator) can be identi\fed with its mean value in an arbitrary state ^ \u001a, i.e.,h^Hi=\nTr(^H^\u001a) [46]. In this scheme, a useful theorem is the following one.\nTheorem 1 [13]. Let^Hand^\u001abed\u0002dhermitian matrices with their eigenvalues f\u000fjgand\nf\rkg, respectively, arranged in descending order, v.g., \u000f1\u0015\u000f2\u0001\u0001\u0001\u0015\u000fdand\r1\u0015\r2\u0001\u0001\u0001\u0015\rd.\nThus, one has the inequality\ndX\ni=1\u000fd\u0000i+1\ri\u0014Tr(^H^\u001a)\u0014dX\ni=1\u000fi\ri: (22)\nIf either inequality is an equality, then ^Hand^\u001acommute.\nSince the equality is easy to verify when ^Hand ^\u001aare diagonals (diagonal frame), this\ntheorem leads to the assumption that the density matrix can be adapted to get the spectrum\nof^Hif they both commute. In that way, a related fact is the following.\nProposition 1. For an arbitrary ^\u001accommuting with ^H, in any frameh^Hicdepends on\nat mostd\u00001variables parametrizing ^\u001ac.\nProof . Since ^\u001acand ^Hcommute, they are simultaneously diagonalizable therefore, in the\ndiagonal frame,\nh^Hic=1\ndh0+1\n2h0\u0001\u00150; (23)\nwhere we use the convention\nh0= (0;:::; 0;h0\nd(d\u00001)+1;:::;h0\nd2\u00001);\n\u00150= (0;:::; 0;\u00150\nd(d\u00001)+1;:::;\u00150\nd2\u00001);\nandh0= Tr( ^H). By applying the relation (14) one has \u0015=O\u00150andh=Oh0withOas the\northogonal matrix from (15). Setting aside scalar matrices, since orthogonal transformations\npreserve the dot product, h^Hicis an invariant quantity and depends on at most d\u00001 variables\nof ^\u001ac.q.e.d.\nWith the aim to provide further properties of the set of density matrices that commute\nwith ^Hand propose an algorithm to compute the extremal mean values of the energy in\n7the density matrix formalism, we consider from here on ^Has a given non-scalar matrix and\nestablish the proposition:\nProposition 2. Let^Hand^\u001abe two \fnite d\u0002dhermitian matrices where ^\u001arepresents an\narbitrary density matrix. For a \fxed degree of mixture, the critical points of h^Hidetermine\nthe extremal density matrices ^\u001ac\nmcommuting with ^Handh^Hic= Tr( ^H^\u001ac\nm), with 1\u0014m\u0014\nd!.\nProof . Suppose that ^ \u001ais unitarily related to a density matrix ^ \u001acwhich commute with ^H\nthen, ^\u001a=^U(\u0012m) ^\u001ac^Uy(\u0012m), where ^U(\u0012m) de\fne aSU(d) unitary transformation. Therefore,\nwith the mean value h^Hi= Tr( ^H^\u001a) one can de\fne the scalar function\nE(\u0012m;\u0015c\nk;hi)\u0011Tr(^H^U(\u0012m) ^\u001ac^Uy(\u0012m)); (24)\nwhere the real constants hiand\u0015c\nkare the components of the expansion of ^Hand ^\u001acrespec-\ntively, in a basis for the Hilbert space of Hermitian operators.\nOtherwise, if ^Uis su\u000eciently close to the identity, by considering f\u0012mgas in\fnitesimal\nparameters one can make a Taylor series expansion of the function as follows\nE(\u0012m;\u0015c\nk;hi) =E(0;\u0015c\nk;hi) +d2\u00001X\np=1\u0012p\u000fp+1\n2d2\u00001X\nq;p=1\u0012p\u0012q\u000fp;q+O(\u00123); (25)\nwhere we have de\fned\n\u000fq\u0011@\n@\u0012qE(\u0012m;\u0015c\nk;hi)\f\f\f\f\nf\u0012mg!0; (26)\n\u000fp;q\u0011@2\n@\u0012p\u0012qE(\u0012m;\u0015c\nk;hi)\f\f\f\f\nf\u0012mg!0: (27)\nAs theSU(d) unitary transformation is in\fnitesimal, one has that\n^\u001a\u0019^\u001ac+id2\u00001X\np=1\u0012p[^\u0015p;^\u001ac] +i2\n2d2\u00001X\nq;p=1\u0012q\u0012p[^\u0015q;[^\u0015p;^\u001ac]]: (28)\nSubstituting the last expression into (24), comparing with (25) and by applying the cyclic\nproperty of the trace, Eqs. (26) and (27) lead to\n\u000fq=iTr\u0010\n[^\u001ac;^H]^\u0015q\u0011\n; (29)\n\u000fp;q=i2Tr\u0010\n[^\u0015p;^\u001ac] [^\u0015q;^H]\u0011\n: (30)\nThe algebraic system which determines the critical points is given by equating Eq. (29)\nto zero, for q= 1;:::;d2\u00001, whereby [^ \u001ac;^H] =0. Hence,h^Hiachieves its extreme values\n8at ^\u001ac. In that sense, any density matrix which commutes with ^Hand optimizes its mean\nvalue, is extremal. Even though the commutativity is satis\fed by hypothesis, it implies that\nany state ^\u001acan be approximated at \frst-order by ^ \u001acandh^Hihas an error that vanishes to\nthe second-order in O(\u00122), i.e.,\nh^Hi\u0019h ^Hic+O(\u00122): (31)\nThus, by means of the proposition 1, ^ \u001acdepends in general on d\u00001 variables that are \fxed\nby establishing a degree of mixture through the expressions (17).\nFinally, the highest degree of the polynomial akin (17) isk. Then, by Bezout's theorem,\nthe number of solutions for the polynomial system (known as Bezout bound or Bezout\nnumber) is at most the product of the degree of all the equations, i.e., \u0005d\nk=2k=d! [47{50].\nAll this implies that the single critical matrix ^ \u001acrepresents at most d! di\u000berent critical\ndensity matrices ^ \u001ac\nm, where 1\u0014m\u0014d!.q.e.d.\nBy matching results, in the non degenerate case of ^H, if all ^\u001ac\nmare pure states, they must\ncorrespond to one-dimensional eigenprojectors of ^H.\nV. ALGEBRAIC APPROACH TO EXTREMAL DENSITY MATRICES\nIn a previous work [32] we proposed an approach to obtain information of the energy\nspectrum of a Hamiltonian by considering its mean value together with d\u00001 constraints to\nguarantee the positivity of the density matrix. This is achieved by de\fning the function\nf(\u0015k;\u0003j;hi;cl)\u0011Tr(^H^\u001a) +dX\nj=2\u0003j(aj\u0000cj); (32)\nwhich depends on d2real parametersfhigandd2\u00001 independent variables f\u0015kgassociated\nto the expansions (1) and (2), respectively. Additionally, there are d\u00001 Lagrange multipliers\nf\u0003jgandd\u00001 positive real constants fcjgto \fx the degree of purity of the density matrix\n(see the bound (18)). One can note that f(\u0015k;\u0003j;hi;cl) is a continuous function because is\nthe sum of the Rayleigh quotient RH( ), where ^\u001a\u0011j ih jwith Tr^\u001a= 1, and the positivity\nconstraints which are polynomials in the variables of the density matrix. Therefore, in order\nto reach all the eigenvalues of ^Hand its numerical range, one must \fnd the min-max sets of\nf(\u0015k;\u0003j;hi;cl) with respect to the variables f\u0015kgand the Lagrange multipliers f\u0003jg. Their\n9respective derivatives give d2+d\u00002 algebraic equations,\n1\n2hq\u0000dX\nj=2\u0003j@aj\n@\u0015q= 0; q= 1;:::d2\u00001; (33)\nap=cp; p = 2;:::d: (34)\nThese sets of algebraic equations determine the extremal values of the density matrix, i.e.,\n\u0015q=\u0015c\nqand \u0003q= \u0003c\nqfor which the expressions (33) and (34) are satis\fed. By substituting\n\u0015c\nqinto equation (2) one obtains the extremal density matrices. If we restrict the solutions to\npure statesfcp= 0g, we have shown explicitly that the energy spectrum of the Hamiltonian\nis recovered for d= 2 and 3 [32]. Extremal expressions for the mean value of the Hamiltonian\ncan be obtained with density matrices representing mixed quantum states, which determine\nalso the corresponding mixture of eigenstates of the Hamiltonian.\nFrom here on, we describe an alternative algebraic procedure to get the extremal density\nmatrices which is simpler than the one mentioned above. First of all, notice that propositions\n1 and 2 in section IV are based on the assumption of a common basis, which it is always\npossible to \fnd if ^ \u001acand ^Hcommute. Thus, in the following paragraphs, we propose for the\npure case (or mixed case) a systematic approach to get information about the Hamiltonian\nspectrum (or mean value of the Hamiltonian), i.e., its numerical range (interval of extremal\nmean values of ^H), without making use of a diagonalization procedure.\nFirst we replace into the commutator [ ^H;^\u001a] = 0 the expressions Eqs. (1) and (2) and use\nthe properties of the generators ^\u0015qof thesu(d) algebra. Then the expression (29) gives rise\nto thed2\u00001 dimensional homogeneous system of equations\nM\u0001\u0015= 0; (35)\nthat determines the critical points and where \u0015is the Bloch vector de\fned in (11). The\nmatrix elements of the skew symmetric matrix Mofd2\u00001 dimensions are given by\nMi;j=d2\u00001X\nk=1fijkhk; (36)\nwherefijkare the antisymmetric structure constants of the su(d) algebra.\nA single solution of the homogeneous system (35) can be obtained through the Gauss-\nJordan elimination method and it is identi\fed as the critical Bloch vector \u0015c, with itsnfree\nvariables equal to the dimension of the null space of M. This implies that maximal mixed\n10states are always critical for any observable because the null space always contains the zero\nvector.\nOn the other hand, notice that bwq\u0011i[bH;^\u0015q] are hermitian vectors spanning the tangent\nspace of the orbits associated with ^H, withq= 1;2;:::;d2\u00001. Then by substituting the\nHamiltonian expression (1) one gets\n^wq=X\nk;lfqklhk^\u0015l: (37)\nNote that these vectors give the rows of M, and the number of independent vectors ris\ndetermined by the rank of the Gram matrix\nGq;p= Tr( ^wq^wp) = 4X\nk1;k2;jfqk1jfpk2jhk1hk2: (38)\nThereforerdetermines the dimension of the tangent space of the Hamiltonian orbits and\nthe rank ofM, i.e.,r= rank(M) [51{56]. Furthermore, the maximal dimension of the orbit\noccurs when the Hamiltonian is non degenerate, i.e., when r=d(d\u00001) and by comparing\nd2\u00001 withrone has that the system (35) is always underdetermined with n=d2\u00001\u0000r\nfree variables (see Table I).\nTo clarify the method, we are going to discuss the non degenerate and degenerate cases\nof^Hseparately. In section VIII we shall illustrate the method for quantum systems of\ndimensions d= 2;3 and 4.\nVI. NON DEGENERATE CASE OF ^H\nIn this case, the rank of the matrix Mis given by r=d(d\u00001) and the nfree variables\nreach its minimum number, i.e., n=d\u00001. Therefore, \u0015cis given by\n\u0015c= (\u0015c\n1; :::;\u0015c\nd(d\u00001); \u0015d(d\u00001)+1;:::\u0015d2\u00001); (39)\nwhere thed(d\u00001) componentsf\u0015c\nqgare functions of the parameters of the Hamiltonian,\nthe antisymmetric structure constants and a set of d\u00001 independent free variables. It is\nnatural to choose this set from the diagonal generators (10) of su(d).\nThe substitution of \u0015cin (2) gives its associated critical density matrix denoted as ^ \u001ac.\nTherefore, the determination of the d\u00001 free variables is done by solving the system of\nd\u00001 polynomial equations (17), which by proposition 2 has at most d! di\u000berent solutions.\n11TABLE I. Manifolds and their dimension for the unitary orbits of the Hamiltonian based on their\ndiagonal form. It is supposed that \u000b>\f >\r >\u000e . The number of free parameters of Eq. (35) is\nonly determined by the expression n=d2\u00001\u0000r[55].\nHamiltonian Diagonal Manifold Manifold\ndimension representation dimension\nd r\ndiag(\u000b;\u000b) point 0\n2 diag(\u000b;\f) U(2)=[U(1)\u0002U(1)] 2\ndiag(\u000b;\u000b;\u000b ) point 0\n3 diag(\u000b;\f;\f ) U(3)=[U(1)\u0002U(2)] 4\ndiag(\u000b;\f;\r )U(3)=[U(1)\u0002U(1)\u0002U(1)] 6\ndiag(\u000b;\u000b;\u000b;\u000b ) point 0\ndiag(\u000b;\f;\f;\f )U(4)=[U(1)\u0002U(3)] 6\n4 diag(\u000b;\u000b;\f;\f )U(4)=[U(2)\u0002U(2)] 8\ndiag(\u000b;\f;\r;\r )U(4)=[U(1)\u0002U(1)\u0002U(2)] 10\ndiag(\u000b;\f;\r;\u000e )U(4)=[U(1)\u0002U(1)\u0002U(1)\u0002U(1)] 12\nThis is also in agreement with the analysis in the diagonal representation ^ \u001ac\ndiag, wherein the\naction of the permutation group of nelements produces d! matrices [37, 57]. They satisfy\nthe same polynomial system (17) but give di\u000berent mean values of ^H. Therefore, the critical\ndensity matrices are given by\n^\u001ac\nm=1\nd^I+1\n2d2\u00001X\nk=1\u0015c\nm;k^\u0015k; (40)\nwithmdenoting the Bloch vector solution, the variables f\u0015c\nm;kgare function only of the\nknown quantities, i.e., the structure constants, the parameters of the Hamiltonian ( h0;hk),\nandd\u00001 constantsfckg. The number mof solutions decreases up to dwhen (40) represent\npure states (allfck= 0g) and the extremal density matrices are one-dimensional orthogonal\nprojectors.\nAs to be expected, the expectation value of the Hamiltonian is in general given by\nh^Hic\nm\u0011Tr(^H^\u001ac\nm); (41)\nfor each critical (or extremal) density matrix ^ \u001ac\nm, withm= 1;2;:::d !. For the pure case\n12the expectation values yield the energy spectrum of the system and the extremal density\nmatrices are orthogonal projectors.\nVII. DEGENERATE CASE OF ^H\nIn this case the rank of the matrix Msatis\fes that r<d (d\u00001), whose value is associated\nto the orbits of the Hamiltonian (see Table I). As the critical Bloch vector \u0015chasnfree\nvariables with n=d2\u00001\u0000rone then has that n>d\u00001. Thus, if Ndenotes the number\nof variables appearing in the expression for the mean value of the Hamiltonian in the state\n^\u001ac,h^Hic= Tr( ^H^\u001ac), there are two cases to consider:\ni) When 1\u0014N\u0014d\u00001, one has to select n\u0000Ncomponents of the density matrix\nBloch vector to have d\u00001 free variables and then solve the polynomial system of\nequations (17).\nii) Whend\u00001<N\u0014n, one has only to pick up d\u00001 components of the density matrix\nBloch vector from the set of Nelements, and again to solve the mentioned polynomial\nequation.\nIn both cases d\u00001 free variables will be determined by the polynomial system (17). The\nremainingn\u0000(d\u00001) components can be taken equal to zero because they do not a\u000bect the\ncommutator of ^Hwith ^\u001ac. Speci\fcally, if we are interested in the eigensystem, i.e., all the\nset offck= 0g, one can apply the following method recursively:\n1) Make zero the n\u0000(d\u00001) components, to solve the polynomial system (17), whose\nsolution give at least two extremal density matrices.\n2) Take the trace of the \frst set of solutions ^ \u001ac\nkwith the commuting general density matrix\n^\u001ac, this yields a system of algebraic equations by asking the orthogonality conditions,\ni.e., Tr(^\u001ac^\u001ac\nk) = 0, where kis a label counting the number of solutions.\n3) Substitute the solutions of the linear system, to express the new critical density matrix\nin terms of the free variables, where d\u00001 are \fxed by means of the positivity conditions\n(allfck= 0g) and the rest of the components can be taken equal to zero.\n4) Return to step 1 and repeat the procedure again, until one gets dorthogonal projectors.\n13Of course, one has in this case several solutions related with the degeneracy of the Hamil-\ntonian in similar form as in the standard diagonalization procedure of a \fnite Hamiltonian\nmatrix. Although this may seem arbitrary, ultimately it is related to the codimension con-\nditions [58, 59]. This topic will be addressed in a future contribution.\nSimilarly to (41), in all cases, the energy spectrum is given by\nh^Hic\nm\u0011Tr(^H^\u001ac\nm); (42)\nfor each critical density matrix ^ \u001ac\nm, withm= 1;2;:::d .\nVIII. EXAMPLES FOR d= 2;3, AND 4\nA. Case d= 2.\nFord= 2, one has that the generators f^\u0015kgcan be realized in terms of the Pauli matrices,\ni.e.,^\u00151= ^\u001b1,^\u00152= ^\u001b2and^\u00153= ^\u001b3. Therefore the density and Hamiltonian matrices can be\nwritten in terms of the Bloch vectors \u0015= (\u00151;\u00152;\u00153) andh= (h1;h2;h3),\n^\u001a=1\n20\n@1 +\u00153\u00151\u0000i\u00152\n\u00151+i\u001521\u0000\u001531\nA;^H=1\n20\n@h0+h3h1\u0000ih2\nh1+ih2h0\u0000h31\nA; (43)\nwhere\u0015is also called the polarization vector.\nThus by substituting the expressions (43) into (29) we obtain the condition that the\nBloch vectors of Hand the density matrix are parallel,\n(\u0015\u0002h)q= 0;\nIts solution gives the critical Bloch vector\n\u0015c=\u0012h1\nh3\u00153;h2\nh3\u00153; \u00153\u0013\n: (44)\nwith a free variable \u00153, according with the dimensions of the orbits of the Hamiltonian (see\nTable I).\nFrom expressions (17), one has a single positivity condition\nc2=1\n4(1\u0000(\u0015c\n1)2\u0000(\u0015c\n2)2\u0000\u00152\n3);\n14by substituting (44) into the above equation, we obtain\n\u00153=\u0006\u000eh3\nh; (45)\nwhere we de\fne h=p\nh2\n1+h2\n2+h2\n3and\u000e=p1\u00004c2. By means of (44) and (45), we \fnd\nthe critical density matrices\n^\u001ac\n\u0006=1\n20\n@1\u0006\u000eh3\nh\u0006\u000e(h1\u0000ih2)\nh\n\u0006\u000e(h1+ih2)\nh1\u0007\u000eh3\nh1\nA; (46)\nwhich correspond exactly to the solutions given in [32]. Note that ^ \u001ac\n+^\u001ac\n\u0000=c2I2. Substituting\nthem into (41) we get\nh^Hic\n\u0006=1\n2(h0\u0006\u000eh): (47)\nWe can distinguish two types of solutions:\n\u000fPure case (when c2= 0): One has \u000e= 1, the eigenvalues \u000f\u0006=1\n2(h0\u0006h) of the\nHamiltonian, and from (46), with \u000e= 1, the corresponding orthogonal projectors.\n\u000fMixed case (when 0 < c 2\u00141=4): The extremal density matrices for the expectation\nvalue of the Hamiltonian are given in terms of the convex sum\n^\u001amixed\n\u0006 =1\n2(1 +\u000e)^\u001apure\n\u0006+1\n2(1\u0000\u000e)^\u001apure\n\u0007; (48)\nwherep\u0006=1\n2(1 +\u000e) indicates the probability of \fnding the system with eigenvalue \u000f+\nwhilep\u0007=1\n2(1\u0000\u000e) the corresponding probability of \fnding an energy \u000f\u0000.\nB. Case d= 3.\nFor the qutrit case, the generators ^\u0015k, withk= 1;2;:::8 can be realized in terms of the\nGell-Mann matrices [20]. Thus, an arbitrary density matrix is given by\n^\u001a=1\n20\nBBB@\u00157+\u00158p\n3+2\n3\u00151\u0000i\u00154\u00152\u0000i\u00155\n\u00151+i\u00154\u00158p\n3\u0000\u00157+2\n3\u00153\u0000i\u00156\n\u00152+i\u00155\u00153+i\u001562\n3\u00002p\n3\u001581\nCCCA; (49)\n15while the matrix (36) takes the form\nM=1\n20\nBBBBBBBBBBBBBBBBB@0h6h52h7\u0000h3\u0000h2\u00002h4 0\n\u0000h60h4\u0000h3h78h1\u0000h5\u0000p\n3h5\n\u0000h5\u0000h4 0h2h1h87h6\u0000p\n3h6\n\u00002h7h3\u0000h20h6\u0000h52h1 0\nh3\u0000h78\u0000h1\u0000h6 0h4h2p\n3h2\nh2\u0000h1\u0000h87h5\u0000h4 0\u0000h3p\n3h3\n2h4h5\u0000h6\u00002h1\u0000h2h3 0 0\n0p\n3h5p\n3h60\u0000p\n3h2\u0000p\n3h30 01\nCCCCCCCCCCCCCCCCCA; (50)\nwhich is a real skew symmetric matrix, and to simplify the matrix notation we de\fne h78=\nh7+p\n3h8andh87=p\n3h8\u0000h7.\nWe consider the following Hamiltonian matrix\n^H=0\nBBB@bcp\n20\ncp\n20cp\n2\n0cp\n2b1\nCCCA; (51)\nwhere the parameters bandcare real parameters. It represents a Hamiltonian written in\nterms of the angular momentum ^H=b^J2\nz+c^Jxwithj= 1. This Hamiltonian has been\nused to describe a two mode Bose-Einstein condensate where the parameter brepresents\nthe atom-atom interaction, and cis related with the tunnelling parameter or a symmetric\nsystem of two interacting qubits [60]. The Bloch vector for the Hamiltonian is given by\nh= (p\n2c;0;p\n2c;0;0;0; b;\u0000p\n3b):\nSubstituting the components of hinto (50), one \fnds that the rank of Mis 6, implying\nthat the Hamiltonian is non degenerate. Solving the system of equations (35) with the\nGauss-Jordan elimination method, one obtains the extremal Bloch vector for the density\nmatrix\n\u0015= \nc\u00152p\n2b\u0000p\n6c\u00158\nb;\u00152;c\u00152p\n2b\u0000p\n6c\u00158\nb;0;0;0;\u0000p\n3\u00158;\u00158!\n:\nThus the associated critical density matrix is found by replacing the components of \u0015c\ninto (49), which is denoted by ^ \u001ac. For this case, to guarantee the positivity of the density\nmatrix, one must consider\nc2=1\n2\u0000\n1\u0000Tr(^\u001ac)2\u0001\n; c 3= det ^\u001ac: (52)\n16Newly one has two types of solutions:\n\u000fPure case (taking c2=c3= 0): Solving the system of equations c2= 0 andc3= 0,\none arrives to 3 di\u000berent solutions for \u00152and\u00158, denoted by\n(\u00152;\u00158)0=\u0010\n\u00001;\u00001\n2p\n3\u0011\n; (53)\n(\u00152;\u00158)\u0006=1\n2\u0010\n1\u0006bp\nb2+ 4c2;p\n3\n6\u0007p\n3b\n2p\nb2+ 4c2\u0011\n; (54)\nwhich yield three independent Bloch vectors of the density matrix, namely\n\u00150=\u0010\n0;\u00001;0;0;0;0;1\n2;\u00001\n2p\n3\u0011\n; (55)\n\u0015\u0006=\u0010\n\u0006p\n2cp\nb2+ 4c2;1\n2\u0006b\n2p\nb2+ 4c2;\u0006p\n2cp\nb2+ 4c2;0;0;0;\n\u00001\n4\u00063b\n4p\nb2+ 4c2;p\n3\n12\u0007p\n3b\n4p\nb2+ 4c2\u0011\n; (56)\nwhose norm is equal to 4 =3 and the scalar products between them are equal to \u00002=3.\nThus it is straightforward to check that the corresponding extremal density matrices\nare orthogonal projectors associated to the energy eigenvalues of the Hamiltonian\n\u000f0=b,\u000f\u0006= 1=2(b\u0006p\nb2+ 4c2).\n\u000fMixed case: For any other values for c2andc3in the region shown in Fig. 1(a), one\ncan solve the polynomial system given by (52). As an example we take c2= 29=100\nandc3= 1=50. There are 6 di\u000berent solutions for \u00152and\u00158which give rise to 6 Bloch\nvectors,\n\u0015(1)\n\u0006=\u0010\n\u0006p\n2c\n10p\nb2+ 4c2;7\n20\u0006b\n20p\nb2+ 4c2;\u0006p\n2c\n10p\nb2+ 4c2;0;0;0;\n\u00007\n40\u00063b\n40p\nb2+ 4c2;p\n3\n120\u0007p\n3b\n40p\nb2+ 4c2\u0011\n; (57)\n\u0015(2)\n\u0006=\u0010\n\u00062p\n2c\n5p\nb2+ 4c2;\u00001\n10\u0006b\n5p\nb2+ 4c2;\u00062p\n2c\n5p\nb2+ 4c2;0;0;0;\n+1\n20\u00063b\n10p\nb2+ 4c2;\u0000p\n3\n60\u0007p\n3b\n10p\nb2+ 4c2\u0011\n; (58)\n\u0015(3)\n\u0006=\u0010\n\u00063p\n2c\n10p\nb2+ 4c2;\u00001\n4\u00063b\n20p\nb2+ 4c2;\u00063p\n2c\n10p\nb2+ 4c2;0;0;0;\n+1\n8\u00069b\n40p\nb2+ 4c2;\u00005p\n3\n120\u00079p\n3b\n120p\nb2+ 4c2\u0011\n; (59)\n17(a)\n0.00 0.05 0.10 0.15 0.20 0.25 0.300.0000.0050.0100.0150.0200.0250.0300.035\nc2c3 (b)\nFIG. 1. Regions of c2,c3, andc4where the positivity conditions of density matrix are satis\fed. (a)\nFor the case d= 3, with the horizontal and vertical dashed lines taking the values c3=f0;1=27g\nandc2=f1=4;1=3g, respectively. (b) For the case d= 4 one gets a solid \fgure. The pure case\nis associated to ( c2;c3;c4) = (0;0;0) while the maximal mixed state correspond to ( c2;c3;c4) =\n(3=8;1=16;1=256).\nwhose corresponding extremal expectation values of the Hamiltonian are given by\nh^Hi(1)\n\u0006=11b\n20\u0006p\nb2+ 4c2\n20;h^Hi(2)\n\u0006=7b\n10\u0006p\nb2+ 4c2\n5;\nh^Hi(3)\n\u0006=3b\n4\u00063p\nb2+ 4c2\n20: (60)\nWe \fnd the expansion of the extremal density matrices for the mixed case in terms of\nthe pure case described before,\n^\u001a(1)\n+=1\n10^\u001a0+1\n2^\u001a++2\n5^\u001a\u0000; ^\u001a(1)\n\u0000=1\n10^\u001a0+2\n5^\u001a++1\n2^\u001a\u0000;\n^\u001a(2)\n+=2\n5^\u001a0+1\n2^\u001a++1\n10^\u001a\u0000; ^\u001a(2)\n\u0000=2\n5^\u001a0+1\n10^\u001a++1\n2^\u001a\u0000;\n^\u001a(3)\n+=1\n2^\u001a0+2\n5^\u001a++1\n10^\u001a\u0000; ^\u001a(3)\n\u0000=1\n2^\u001a0+1\n10^\u001a++2\n5^\u001a\u0000: (61)\nNote that the expressions (60) can be checked by calculating the expectation value of the\nHamiltonian with the expansions given in the last expression.\n181. Degenerate case.\nNow we consider the Hamiltonian matrix given by\n^H=0\nBBB@2\u00001 +i\u00001\u0000i\n3\n\u00001\u0000i13\n31 +i2\n\u00001 +i\n31\u0000i2 31\nCCCA: (62)\nIn this case the Bloch vector characterising the Hamiltonian is given by\nh= \n\u00002;\u00002;2;\u00002;2\n3;\u00004;\u00007\n3;p\n3\n9!\n(63)\nwithh0=28\n3. Replacing this values into the matrix (50), the rank of Misr= 4, which\naccording to Table I the Hamiltonian exhibits a double degeneracy. Thus, if \u000b > \f the\ndiagonal representation is diag( \u000b;\f;\f ), or in opposite way, if \f >\u000b then diag(\f;\f;\u000b ).\nBy applying the Gauss-Jordan method to (35), it yields\n\u0015c\n1=1\n14\u0010\n6\u00155+ 8\u00156+\u00157+ 7p\n3\u00158\u0011\n;\n\u0015c\n2=1\n7\u0010\n3\u00155\u00003\u00156+ 11\u00157\u00007p\n3\u00158\u0011\n;\n\u0015c\n3=1\n42\u0010\n24\u00155\u000024\u00156+ 11\u00157\u00007p\n3\u00158\u0011\n;\n\u0015c\n4=1\n42\u0010\n\u000030\u00155+ 30\u00156\u000019\u00157+ 35p\n3\u00158\u0011\n:\nHence, the corresponding critical Bloch vector (39) is given by\n\u0015c= (\u0015c\n1; \u0015c\n2; \u0015c\n3; \u0015c\n4; \u00155; \u00156; \u00157; \u00158):\nwith 4 free parameters and its associated critical density matrix is denoted as ^ \u001ac.\nNow, in order to obtain the eigensystem of ^H, we are going to use the procedure estab-\nlished before for the degenerated case:\n\u000fThus we select the components \u00155=\u00156= 0, solve the polynomial condition (52) with\nc2=c3= 0, and we get the following Bloch vectors of the density matrix\n\u0015(1)=1\n11\u0012\n6;0;0;6;0;0;7;11=p\n3\u0013\n;\u0015(2)=1\n46\u0012\n12;36;6;0;0;0;35;19=p\n3\u0013\n:(64)\nThese Bloch vectors yield two density matrices ^ \u001a(1), and ^\u001a(2)which are not independent,\nboth by taking the trace with the Hamiltonian give an energy eigenvalue \u000f= 4=3.\n19\u000fWe establish the algebraic system of equations,\n\u001a\nTr\u0012\n^\u001a(1)^\u001ac\u0013\n;Tr\u0012\n^\u001a(2)^\u001ac\u0013\u001b\n= 0; (65)\nwhose solution together with the positivity condition gives another Bloch vector\n\u0015(3)=1\n16\u0012\n\u00006;\u00006;6;\u00006;2;\u000012;\u00007;1=p\n3\u0013\n: (66)\nTherefore we have obtained another extremal density matrix orthogonal to ^ \u001a(1), and\n^\u001a(2)and the expectation value of the Hamiltonian yields the eigenvalue \u000f2= 20=3.\nUntil now we have obtained 2 independent and orthogonal projectors, we chose ^ \u001a(1),\nand ^\u001a(3).\n\u000fWe repeat the procedure by establishing the algebraic system of equations\nn\nTr\u0010\n^\u001a(1)^\u001ac\u0011\n;Tr\u0010\n^\u001a(3)^\u001ac\u0011o\n= 0; (67)\nwhose solution give the Bloch vector\n\u0015(4)= \n\u000015\n88;3\n8;\u00003\n8;\u000015\n88;\u00001\n8;3\n4;\u000035\n176;\u000017\n16p\n3!\n: (68)\nThus one gets another orthogonal projector ^ \u001a(4)and the expectation value of the\nHamiltonian is \u000f3= 4=3.\nWe have obtained the complete eigensystem of the degenerated Hamiltonian. For the eigen-\nvalue\u000f= 4=3, we indeed have a family of projectors yielding the same eigenvalue. This\nfamily is associated to the standard problem, when there is degeneracy, of the diagonaliza-\ntion of a Hamiltonian matrix, i.e., we can take any linear combination of the corresponding\nindependent eigenstates.\nC. Case d= 4.\nFor the states space of a quartit, the density matrix is given by\n^\u001a=1\n20\nBBBBBBB@r11 \u00151\u0000i\u00157 \u00152\u0000i\u00158 \u00153\u0000i\u00159\n\u00151+i\u001571\n6(3\u00006\u001513+ 2p\n3\u001514+p\n6\u001515)\u00154\u0000i\u001510 \u00155\u0000i\u001511\n\u00152+i\u00158 \u00154+i\u0015101\n6(3\u00004p\n3\u001514+p\n6\u001515)\u00156\u0000i\u001512\n\u00153+i\u00159 \u00155+i\u001511 \u00156+i\u0015121\n2(1\u0000p\n6\u001515)1\nCCCCCCCA;(69)\n20where we de\fne r11=1\n6(3 + 6\u001513+ 2p\n3\u001514+p\n6\u001515).\nWe consider the Hamiltonian matrix\n^H=0\nBBBBB@a \u000e b +ai b +ai\n\u000e a\u0000b+ia b\u0000ia\nb\u0000ia\u0000b\u0000ia b 0\nb\u0000ia b +ai 0b1\nCCCCCA; (70)\nwitha,band\u000eas real parameters.\nIn the basis of the generalized Gell-Mann matrices ^\u0015k, withk= 1;2;:::15, the parameters\nBloch vector for the Hamiltonian, hk= Tr( ^H^\u0015k), is given by\nh= 2 \n\u000e; b; b;\u0000b; b; 0;0;\u0000a;\u0000a;\u0000a; a; 0;0;p\n3\n3(a\u0000b);r\n1\n6(a\u0000b)!\n: (71)\nwithh0= 2(a+b).\nTherefore, its associated matrix (36) is\nM=0\nBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@0\u0000a a\u0000a\u0000a0 0b\u0000b\u0000b\u0000b0 0 0 0\na0 0 0 0\u0000a b r 0\u000e 0\u0000b a \u0011 0\n\u0000a0 0 0 0\u0000a\u0000b0r 0\u000e b aap\n3\u000b\na0 0 0 0 a b \u000e 0r 0\u0000b\u0000a \u0011 0\na0 0 0 0\u0000a b 0\u000e 0r\u0000b a\u0000ap\n3\u0000\u000b\n0a a\u0000a a 0 0b b b\u0000b0 0 0 0\n0\u0000b b\u0000b\u0000b0 0\u0000a a a a 0 2\u000e0 0\n\u0000b\u0000r0\u0000\u000e0\u0000b a 0 0 0 0 a b \r 0\nb0\u0000r0\u0000\u000e\u0000b\u0000a0 0 0 0 \u0000a bbp\n3\f\nb\u0000\u000e0\u0000r0\u0000b\u0000a0 0 0 0 \u0000a b\u0000\r0\nb0\u0000\u000e0\u0000r b\u0000a0 0 0 0 \u0000a\u0000bbp\n3\f\n0b\u0000b b b 0 0\u0000a a a a 0 0 0 0\n0\u0000a\u0000a a\u0000a0\u00002\u000e\u0000b\u0000b\u0000b b 0 0 0 0\n0\u0000\u0011\u0000ap\n3\u0000\u0011ap\n30 0\u0000\r\u0000bp\n3p\n3b\u0000bp\n30 0 0 0\n0 0\u0000\u000b0\u000b0 0 0\u0000\f0\u0000\f0 0 0 01\nCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA;\nwhere to simplify the matrix notation we have de\fned r=a\u0000b,\f=p\n8=3b,\u000b=p\n8=3a,\n\r=p\n3b, and\u0011=p\n3a. We are going to consider two illustrative instances to exemplify\nthe non degenerate and degenerate cases.\n21(a)\n-1.5 -1.0 -0.5 0.5 1.0 1.5δ\n-1123〈H〉 (b)\n-1.0 -0.5 0.5 1.0δ\n-1123〈H〉\nFIG. 2.h^Hicas a function of \u000e. (a) Pure case with c2=c3=c4= 0; and (b) Mixed case with\nc2=931\n10000; c3=141\n50000; c4=27\n1000000are plotted with continuous lines. The dotted lines represent\nthe minimum and maximum eigenvalues of the pure case.\n1. Non degenerate case.\nIfa= 1,b= 1=2, and\u000e6= 0, thus the rank of Mequals tor= 12. In consequence,\nfrom Table I, for these values the Hamiltonian (70) is non degenerate. By applying the\nGauss-Jordan method to the system (35), one gets the Bloch vector of the density matrix,\n\u0015=\u0010\n\u0015c\n1;\u001511\n2; \u0015c\n3;\u0000\u001511\n2; \u0015c\n3;0;0;\u00002\u0015c\n3;\u0000\u001511;\u00002\u0015c\n3; \u001511;\u0015c\n12;0; \u001514; \u001515\u0011\n;(72)\nwhere\n\u0015c\n1=\u00001\n18(9 (1\u00002\u000e)\u001511\u00002p\n3(\u001514+ 8p\n2\u001515);\n\u0015c\n3=\u00003 (1\u00002\u000e)\u001511+ 8p\n3\u001514+ 4p\n6\u001515\n6 (1 + 2\u000e);\n\u0015c\n12=\u00004p\n3\n9(\u001514\u0000p\n2\u001515): (73)\nThe componentsf\u001511; \u001514; \u001515gare free variables, which are determined by establishing\nthe system of polynomial equations (34), where the constants c2,c3andc4must lie inside\nthe allowed region exhibited in Fig. 1(b). We consider two cases: (i) the pure case when one\nhasc2=c3=c4= 0, which has four independent solutions for the parameters ( \u0015c\n11;\u0015c\n14;\u0015c\n15).\nThe extremal density matrices are projectors de\fned by ^ \u001ac\n1\u0006and ^\u001ac\n2\u0006. They are functions\nof the parameter \u000eand the corresponding expectation values are plotted in Fig 2(b). The\nlevels are indicated by dotted lines, which indicates that for \u000e= 0 the Hamiltonian system is\ndegenerated by pairs. (ii) The mixed case is established by taking from the region exhibited\nin Fig. 1(c) the values c2=931\n10000; c3=141\n50000; c4=27\n1000000. One has 24 di\u000berent extremal\n22expectation values of the Hamiltonian, 6 for each energy level of the pure case. The results\nare shown also in Fig. 2(b) with continuous lines. Notice that the extremal expectation\nvalues are contained within the minimum and maximum eigenvalues of the Hamiltonian.\n2. Degenerate case.\nFor\u000e= 0 in the Hamiltonian (70), the rank of Misr= 8 implying, from Table I, that\n^His doubly degenerate and its diagonal representation is of the form diag( \u000b;\u000b;\f;\f ).\nBy applying the Gauss-Jordan method to the system (35), one gets the Bloch vector of\nthe density matrix with seven free components ( \u00157; \u001510; \u001511; \u001512; \u001513; \u001514; \u001515); the others\ncan be written as\n\u0015c\n1=3b(\u001510+\u001511+ 2\u001512\u00002\u00157)\u0000a\u0000\n3\u001510+ 3\u001511+ 2p\n3\u001514\u00002p\n6\u001515\u0001\n6a;\n\u0015c\n2=a2(\u001512+\u00157) +ab\u0000\n\u001510+ 2p\n3\u001514\u0001\n\u0000b2(\u001510+\u001512\u0000\u00157)\na(a\u0000b);\n\u0015c\n3=\u00003a2(\u001512+\u00157) +ab\u0000\n\u00003\u001511+ 2p\n3\u001514+ 4p\n6\u001515\u0001\n+ 3b2(\u001511+\u001512\u0000\u00157)\n3a(a\u0000b);\n\u0015c\n4=\u00003a2(\u001512+\u00157) +ab\u0000\n\u00003\u001511\u00002p\n3\u001514+ 2p\n6\u001515\u0001\n+ 3b2(\u001511+\u001512\u0000\u00157)\n3a(a\u0000b);(74)\n\u0015c\n5=\u00003a2(\u001512+\u00157) +ab\u0000\n\u00003\u001510\u00002p\n3\u001514+ 2p\n6\u001515\u0001\n+ 3b2(\u001510+\u001512\u0000\u00157)\n3a(a\u0000b);\n\u0015c\n6=\u001513\u0000a\u0000\n3\u001510\u00003\u001511+ 4p\n3\u001514+ 2p\n6\u001515\u0001\n+ 3b(\u001511\u0000\u001510)\n6a;\n\u0015c\n8=\u001511\u00002a\u0000\n2\u001514+p\n2\u001515\u0001\np\n3(a\u0000b);\n\u0015c\n9=\u00002a\u0000\n2\u001514+p\n2\u001515\u0001\np\n3 (a\u0000b)\u0000\u001510:\nThe mean value of ^Hin the state ^ \u001acdepends only on the components \u001514and\u001515; the\nremaining free variables can be chosen from the mentioned free set. By considering\nf\u00157; \u001510; \u001511; \u001512gequal to zero, the components f\u001513; \u001514; \u001515gare obtained by solving\nthe system of polynomial equations (34) in terms of the constants fc2;c3;c4g.\nFor the pure case, associated to c2=c3=c4= 0, the set of solutions for this polynomial\nsystems are\nf\u001513; \u001514; \u001515g1\u0006=\u0006(\n(a\u0000b)\u0006P\n2P;a\u0000bp\n3P;a\u0000bp\n6P)\n;\n23whereP\u0011p\n9a2\u00002ab+ 9b2. Therefore, their respective critical density matrices are\n^\u001ac\n1\u0006=\u00061\n24P0\nBBBBB@12 (a\u0000b\u0006P) 0 24(b+ia) 24(b+ia)\n0 0 0 0\n24(b\u0000ia) 0 6 (b\u0000a\u0006P) 6 (b\u0000a\u0006P)\n24(b\u0000ia) 0 6 (b\u0000a\u0006P) 6 (b\u0000a\u0006P)1\nCCCCCA;\nwhich correspond to orthogonal projectors of ^Hrelated to its degenerate eigenvalues, given\nrespectively by\nTr(^\u001ac\n1\u0000^H) =1\n2(a+b\u0000P);Tr(^\u001ac\n1+^H) =1\n2(a+b+P):\nIn order to \fnd the remaining projectors, one constructs the linear system of equations\nTr(^\u001ac^\u001ac\n1\u0000) = 0;Tr(^\u001ac^\u001ac\n1+) = 0;\nwhere the ^ \u001acis written in terms of the general density matrix that commutes with the\nHamiltonian.\nBy solving it for \u001511and\u001514in terms of \u001510; \u001513and\u001515, one \fnds\n\u001511=a(\u001510\u00008\u001513\u00004)\u0000b\u001510\na\u0000b;\n\u001514=\u00006\u001513+p\n6\u001515+ 3\n2p\n3:\nThen by replacing this solution into ^ \u001ac, setting\u00157and\u001512equal to zero, and solving the\npolynomial system (34) for c2=c3=c4= 0 in terms of \u001510; \u001513; \u001515, one has\nf\u001510; \u001513; \u001515g2\u0006=\u00061\nP(\n\u00002a;b\u0000a\u0007P\n2;a\u0000bp\n6)\n;\nwhich yield the one rank projectors\n^\u001ac\n2\u0006=\u00061\n24P0\nBBBBB@0 0 0 0\n0 12 (a\u0000b\u0006P) 24 (\u0000b+ia) 24 (\u0000ia+b)\n0\u000024 (b+ai) 6 (b\u0000a\u0006P) 6 (a\u0000b\u0007P)\n0 24 (ia+b) 6 (a\u0000b\u0007P) 6 (b\u0000a\u0006P)1\nCCCCCA:\nThe respective expectation values of the Hamiltonian are\nTr(^\u001ac\n2\u0000^H) =1\n2(a+b\u0000P);Tr(^\u001ac\n2+^H) =1\n2(a+b+P):\nFinally, it is possible to corroborate that the set f^\u001ac\n1\u0006;^\u001ac\n2\u0006gare a complete set of orthog-\nonal one rank projectors because ^ \u001ac\n1++ ^\u001ac\n1\u0000+ ^\u001ac\n2++ ^\u001ac\n2\u0000=^I4.\n24IX. SUMMARY AND CONCLUSIONS\nThe main contribution of our work is to give an algebraic procedure to \fnd extremal\ndensity matrices for a given Hamiltonian. Our approach applies to both the degenerate\nand non degenerate cases of the Hamiltonian. The examples of the procedure are given\nfor dimensions d= 2;3;4;and show that the Hamiltonian spectrum for the pure case is\nrecovered. For the mixed case, we have veri\fed that the extremal values of the expectation\nvalue of the Hamiltonian is a convex sum of the corresponding results for the pure case.\nWe want to enhance that the method can be applied by replacing the Hamiltonian for any\nobservable acting on a qudit space.\nWe established that an extremal density matrix commutes with the Hamiltonian op-\nerator and optimises its mean value. We demonstrated that at most d\u00001 variables are\nnecessary to \fnd extremal density matrices with appropriate positivity conditions, for the\nnon-degenerated case of the \fnite matrix Hamiltonian. In the degenerate pure case, one\nhas more free components of the extremal density matrix which can be selected by asking\northogonality between the projectors, which allow us to obtain the energy spectrum.\nFinally, in Appendix A following the method given in [42], we \fnd also the compatible\nregions between the coe\u000ecients of the characteristic polynomial of the density matrix in\nterms of the positivity conditions of the Bezoutian matrix in order to provide a self-contained\napproach.\nACKNOWLEDGEMENT\nThis work was partially supported by CONACyT-M\u0013 exico (under Project No. 238494) and\nDGAPA-UNAM (under Project No. IN110114). The authors would like to thank Giuseppe\nMarmo, Margarita A. Man'ko and Vladimir I. Man'ko for their valuable comments and also\nto CONACyT-M\u0013 exico for the Ph.D. scholarship to A.F.\nAppendix A: Bezoutian matrix\nFork= 1;:::;d , the elements tk\u0011Tr(^\u001ak) form an integrity basis for all polynomial U(d)\ninvariants. In terms of them, it is de\fned the symmetric matrix called Bezoutian given in\nEq. (19).\n25A polynomial with real coe\u000ecients has reals roots i\u000b the Bezoutian matrix is positive\nde\fnite [41]. Hence, the compatible region among the global invariants is obtained with\nthe intersection of the positivity conditions of the density matrix from (18) with the re-\nspective positivity conditions of the Bezoutian, mainly in its determinant det Bd\u00150 [42].\nBesides, due to det Bdis equal to the discriminant of the characteristic polynomial of ^ \u001a, the\ndegeneracy condition is obtained by the vanishing of det Bd[40, 61, 62].\nOn the other hand, the relation between the constants fcpgwithtkis established by\ntk=kX\np=1(\u00001)p+1cptk\u0000p; t 0\u0011k; (A1)\nwithk= 1;:::;d .\nNext we establish the allowed regions of the fcpgandtkfor the matrix Hamiltonians with\ndimensions d= 2;3;4. Consequently, the Bezoutian matrix for d= 2 is\nB2=0\n@2 1\n1t21\nA;\nwhere from formula (A1), it is obtained t2= 1\u00002c2. Then the condition det B2\u00150 gives\nthe positivity condition c2\u00141=4, which corroborates the maximum value (18).\nFor the case d= 3, the positivity conditions (18) of the density matrix are\n0\u0014c2=1\n2(1\u0000t2)\u00141\n3; (A2)\n0\u0014c3=1\n6(1\u00003t2+ 2t3)\u00141\n27; (A3)\nwhile the Bezoutian matrix is\nB3=0\nBBB@3 1t2\n1t2t3\nt2t3t41\nCCCA:\nBy applying the Cayley-Hamilton theorem and the formula (A1) one obtains\nt2= 1\u00002c2; t 3= 1\u00003c2+ 3c3;\nt4= 1\u00004c2+ 2c2\n2+ 4c3: (A4)\nSimilarly to the case of d= 2, detB3\u00150 is the only relevant positivity condition of B3.\nExpressed in terms of fc2; c3g, it gives\nc2\n2\u00004c3\n2+ 18c2c3\u0000c3(4 + 27c3)\u00150: (A5)\n26Thus, the inequalities system formed by (A2), (A3) and (A5) produces the compatible\nregion between c2andc3. This is shown in Fig. 1(a). The bottom line is associated with one\neigenvalue zero (yielding the condition on c2for the case d= 2) while the other curves imply\ntwo equal eigenvalues for the density matrix. The ( c2;c3) = (0;0) case is associated to density\nmatrices of pure states and the highest is the maximal mixed state (all the eigenvalues are\nequal).\nIn the case of d= 4, the positivity conditions of the density matrix are given by\n0\u0014c2=1\n2(1\u0000t2)\u00143\n8;\n0\u0014c3=1\n6(1\u00003t2+ 2t3)\u00141\n16; (A6)\n0\u0014c4=1\n24\u0000\n1 + 3t2\n2\u00006t2+ 8t3\u00006t4\u0001\n\u00141\n256;\nand the respective Bezoutian matrix is\nB4=0\nBBBBB@4 1t2t3\n1t2t3t4\nt2t3t4t5\nt3t4t5t61\nCCCCCA;\nwith\nt2= 1\u00002c2; t 3= 1\u00003c2+ 3c3;\nt4= 2(c2\u00002)c2+ 4c3+ 4c4+ 1; (A7)\nt5= 5c2(c2\u0000c3\u00001) + 5c3+ 5c4+ 1;\nt6= 9c2\n2\u00002c3\n2\u00006(2c3+c4+ 1)c2+ 3c3(c3+ 2) + 6c4+ 1:\nIn this case, det B4\u00150 is the main condition; nevertheless, the remaining ones are crucial\nto avoid fake points in the compatible region for fc2; c3; c4g. All these conditions are\n277 + 11c2\n2\u00002c3\n2+c3(10 + 3c3) + 10c4\u00006c2(2 + 2c3+c4)\u00150;\nc4\n2+ 30c3\u00004c5\n2+ 42c4+ (c3(12c3\u000017)\u000017c4)(c3+c4) + 2c3\n2(9c3\u00008\u000010c4) +\nc2\n2(33 + (4\u000019c3)c3+ 18c4)\u00002c2(19 + 6c3+ 8c2\n3+ 11(1 +c3)c4+ 12c2\n4) + 9\u00150;\n8c3\n3\u00002c3(6 +c4(53 + 36c4))\u000027c4\n3+ 2c2\n3(\u000037 + 9c4) +c4(6\u0000c4(77 + 64c4))\u0000\n2c3\n2(8 + 2(\u00009 +c3)c3+ 27c4) +c2\n2(4\u000045c2\n3+ 44c3c4\u000048(c4\u00001)c4)\u00008c5\n2 (A8)\n2c2(c3(27 +c3(9c3\u00008))\u0000(3 +c3(43 + 45c3))c4\u000023c2\n4) +c4\n2(2\u00008c4)\u00150\n18c2(c3\u00008c4)(c2\n3\u0000c4)\u0000c3\n3(4 + 27c3)\u000016c4\n2c4\u00003(9 + 64c3)c2\n4+ 4c3\n2(c4\u0000c2\n3) +\n6c2\n3c4\u0000256c3\n4+c2\n2(c2\n3+ 80c3c4\u0000128c2\n4)\u00150;\nwhere the last one is det B4\u00150.\nHence, for the set fc2; c3; c4g, the region which satis\fes the inequalities system formed\nby (A6) and (A8), is shown in Fig. 1(b). 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Phys. 75942.\n31"    },    {        "title": "1610.07820v1.Biseparability_of_3_qubits_density_matrices_using_Hilbert_Schmidt_decompositions__Sufficient_conditions_and_explicit_expressions.pdf",        "content": "1 \n  Biseparability of 3-qubits density matrices using Hilbert-Schmidt \ndecompositions: Sufficient conditions and explicit expressions \nY.Ben-Aryeh ∗ and A.Mann †  \nPhysics Department, Technion-Israel Institute of Te chnology, \nHaifa 32000, Israel  \n65 @ . . . phr yb physics technion ac il ∗     ;    †@ . . . ady physics technion ac il   \n \nAbstract \nHilbert-Schmidt (HS) decompositions and Frobenius n orms are  used   to analyze biseparability of 3-qub it systems, \nwith particular emphasis on density matrices with m aximally disordered subsystems (MDS) and on the W state \nmixed with white noise. The biseparable form of a M DS density matrix is obtained by using the Bell sta tes of a 2-\nqubit subsystem,  multiplied by density matrices of   the third qubit, which include the relevant HS pa rameters. Using \nour methods a sufficient condition and explicit bis eparability of the W state mixed with white noise are given. They \nare compared with the sufficient condition for expl icit full separability given in a previous work. \nCondensed  paper  title: Biseparability of 3 qubits .  \nKeywords:   3-qubit systems; biseparability; MDS density matric es; W state mixed with white noise; 1l and \nFrobenius  norms; Hilbert-Schmidt decompositions, B ell states. \n \n1.     Introduction \nIn previous works 1 4 −  we found that the Hilbert-Schmidt (HS) decomposit ion of a density matrix is a very \nuseful method for analyzing various properties of n -qubit systems related to quantum information and \ncomputation (see reviews 5 6 − , books 7 11 − ). We showed 3,4   that a sufficient condition for full separability  \nis given by the 1l norm 12  of the HS parameters, i.e. sum of the absolute val ues of those parameters is \nbounded by 1. The 1l norm is not invariant under unitary transformation s, and we found methods to \ndecrease it by local unitary transformations includ ing singular value decompositions (SVD) 13,14  (see Ref. \n4, Sec.  2.3). \n For 2 qubits the Peres-Horodecki (PH) criterion of  partial transpose (PT) 15,16  is necessary and \nsufficient for entanglement/seperability. For more than 2 qubits the situation is much more complicate d. If \nthe PH criterion yields a negative eigenvalue then we know that the given ρ is not fully separable. 2 \n However, if the PT yields a valid density matrix no  information is gained (see Ref.4, Sec. 2.2, Case A ). \nFor 3 qubits one should distinguish between full se parability, biseparability and genuine entanglement . 17  \nSufficient conditions for full separability of 3-qu bits were analyzed by us in previous work.4.  In the \npresent work we analyze sufficient conditions for b iseparability of 3 qubits. Fulfillment of our crite ria \nimplies that the density matrix is not genuinely en tangled.  \nWe put much emphasis in the present work on density  matrices with maximally disordered \nsubsystems (MDS), 18   i.e. density matrices for which tracing over any subsystem gives the unit density \nmatrix of the remainder. The reason for this is two fold: First:  these 3-qubits density matrices have not \nbeen studied extensively in the literature and by u sing our methods we can analyze various properties of \nsuch density matrices. Second: we find also that fo r more general 3-qubit density matrices their HS \ndecompositions include often MDS terms in addition to other terms which have an explicit fully separab le \nform. Therefore the analysis for MDS density matric es is quite useful for studying biseparability for \ndensity matrices for which the HS MDS terms are a p art of the full density matrix (such analysis is ma de \nfor the W state mixed with white noise in Sec. 4). \nNecessary conditions for biseparablty were given in  Ref. 19, equations (2) and (4). Taking into \naccount the positivity of ρ (for example 1,8 11 88 ρ ρ ρ ≤ ) it is easy to verify that these necessary \nconditions are always satisfied for any 3-qubit MDS ρ. Here we deal with sufficient conditions.  \n. The present paper is arranged as follows: \nIn Sections (2-3) we find explicit biseparable expr essions for MDS density matrices. It has been \nshown that for odd number of qubits with MDS the PH  criterion does not give any information about \nentanglement (see Ref. 4, Eq. (2.16)).  In Sec. 2 w e show explicit biseparability of a very simple MDS   \ndensity matrix with special 3 HS parameters. In thi s example of one triad of HS parameters,   \nbiseparability (say of qubit A with respect to the other two qubits BC) is achieved by representing th e BC \nsystem by its Bell states. 20  The condition for biseparability is that the Frobe nius norm 13  of the HS \nparameters is bounded by 1. In this special example , this condition is fulfilled since it is the condi tion that \nthe eigenvalues of the given matrix are non-negativ e. Similar expressions can be obtained for \nbiseparability of B with respect to AC or biseparab ility of C with respect to AB. \n Based on the special density matrix of Section 2, we show in Sec.3 that the 27 HS parameters \nmay be grouped into 9 triads, related to the triad of the special example by local unitary transformat ions. \nEach triad (with the unit matrix) by itself is a bi separable density matrix, since the Frobenius norm of its 3 3 \n HS parameters is bounded by 1, which is the necessa ry condition that it is a density matrix. For more than \none triad, we show that a  sufficient condition for  biseparability is that the sum of their Frobenius norms \nis bounded by 1. In some special examples of up to 3 triads, this condition was shown to be always \nfulfilled, as it is a necessary condition that it i s a density matrix.  \n In Sec. 4 we discuss the explicit biseparability o f the W state mixed with white noise. Using the \nmethods of the previous Sections we find that when the probability p of the W state is less than 0.1937 \nit can be written explicitly in a biseparable form.  This should be compared with our previous result 4 that \nfor 0.1111 p<  it may be written explicitly in a fully separable form. \n \n2. Explicit biseparability of a simple MDS density matrix \nWe analyze here the conditions for biseparability o f the following 3-qubit very simple MDS density \nmatrix: \n( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1\n111 222 333 8 ( ) ( ) ( ) A B C \nx x x y y y z z z A B C A B C A B C I I I \nR R R ρ\nσ σ σ σ σ σ σ σ σ = ⊗ ⊗ + \n⊗ ⊗ + ⊗ ⊗ + ⊗ ⊗  .      (2.1) \nHere, I represents the unit 2 2 × matrix, the subscripts A,B,C refer to the three qu bits, , , x y z σ σ σ  are the \nthree Pauli matrices, ⊗ denotes outer product, and 111 222 ,R R , and 333 R are real parameters. The 8 \neigenvalues of this density matrix are given by  \n ( )\n( )3\n2\n1 2 3 4 \n1\n3\n2\n5 6 7 8 \n11 / 8 1 ; \n1 / 8 1 iii \ni\niii \niR\nRλ λ λ λ \nλ λ λ λ =\n=  \n= = = = +   \n    \n  \n= = = = −   \n    ∑\n∑ .                   (2.2) \nHence it is a density matrix when iii R are within the unit sphere, i.e. when  \n3\n2\n11iii \niR\n=≤∑      .                      (2.3) \n The sufficient condition for full separability 4  is given by  4 \n  3\n1| | 1 iii \niR\n=≤∑     .             (2.4) \nThis condition limits the parameters to be within t he appropriate cube inscribed in the unit sphere. S ince \nthe condition (2.4) is sufficient but not necessary , the separability/nonseparability of ρ in the volume \nbetween the cube and the sphere is not decided by i t. We now show that ρ of (2.1) may be written in a \nbiseparable form, and therefore cannot be genuinely  entangled. \n Explicit biseparable expression for the density ma trix (2.1) is given by: \n ( ) ( ) ( ) ( )( ) ( )\n( ) ( ) ( ) ( )( ) ( )\n( ) ( ) ( ) ( )( ) ( )\n( ) ( ) ( ) ( )( ) ( )1\n111 333 222 \n111 333 222 \n111 333 222 \n111 333 222 8\nx y z A A A A BC BC \nx y z A A A A BC BC \nx y z A A A A BC BC \nx y z A A A A BC BC I R R R \nI R R R \nI R R R \nI R R R ρ\nσ σ σ \nσ σ σ \nσ σ σ \nσ σ σ ψ − − \n+ + \n+ + \n− − =\n      − + + ⊗ Φ Φ +       \n      + − + ⊗ Φ Φ +                 + + − ⊗ Ψ Ψ +       \n      − − − ⊗ Ψ           .              (2.5) \nHere   ( )\nBC −Φ ,  ( )\nBC +Φ ,  ( )\nBC +Ψ  and  ( )\nBC −Ψ  are the Bell states 20  of the qubits pair B and C.  \nWe used Bell states density matrices which are expa nded in terms of Pauli matrices as: \n( ) ( )( ) ( ) ( ) ( ) ()()( ) ( )\n( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )\n( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )\n( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )4 ; \n4 ; \n4 ; \n4 ; x x y y z z B C B C B C B C BC BC \nx x y y z z B C B C B C B C BC BC \nx x y y z z B C B C B C B C BC BC \nx x y y z z B C B C B C B C BC BC I I \nI I \nI I \nI I σ σ σ σ σ σ \nσ σ σ σ σ σ \nσ σ σ σ σ σ \nσ σ σ σ σ σ − − \n+ + \n+ + \n− −   Φ Φ = ⊗ − ⊗ + ⊗ + ⊗   \n  Φ Φ = ⊗ + ⊗ − ⊗ + ⊗   \n  Ψ Ψ = ⊗ + ⊗ + ⊗ − ⊗   \n  Ψ Ψ = ⊗ − ⊗ − ⊗ − ⊗       (2.6) \nEquations   (2.1) and (2.5) represent a biseparable  density matrix, under the condition  \n 2 2 2 \n111 222 333 1 R R R + + ≤      ,                                 (2.7) \nwhich is equivalent to the condition (2.3) for Eq. (2.1) to be a density matrix. \n Although the above biseparable form is given for t he very special density matrix (2.1) we will \nshow in the next Section that by using local unitar y transformations a sufficient criterion for a bise parable 5 \n form may be obtained for more general MDS density m atrices.. In this context one should take into \naccount that local unitary transformations of Bell states of qubits B and C will preserve the Bell states \nproperties in the transformed frames of reference. We note that similar expressions of the biseparabil ity \n(2.5) may be written in terms of the Bell states of  AB  or AC . \n \n3. Sufficient condition for biseparability for gene ral  3-qubit MDS density matrix using Bell states  \nA general 3-qubits MDS density matrix, in the HS de composition, has the form \n( ) ( ) ( )3\n, , \n, , 1 8 ( ) ( ) ( ) ( ) ( ) ( ) A B C l m n l m n A B C A B C \nl m n I I I R I I I R ρ σ σ σ \n== ⊗ ⊗ + ⊗ ⊗ ≡ ⊗ ⊗ + ∑   .       (3.1) \nWe would like to show that  \n 3\n2\n, , \n, , 1 1l m n \nl m n R\n=≤∑       .         (3.2) \nWe note first that  \n ( )32 2\n, , \n, , 1 8 8 8 l m n \nl m n Tr R ρ\n== + ∑      .       (3.3) \nOn the other hand  \n ( )82 2\n18 64 i\niTr ρ λ \n==∑         .         (3.4) \nHere iλ are the 8 eigenvalues of ρ .  Since  0 1/ 4 iλ≤ ≤   (see Ref. 4, comment after Eq. (2.4)) we write \n(recalling that the 8 ir come in 4 pairs   | | ir±4 ):  \n 8\n11 1 ; | | ; 0 8 8 i i i i \nir r r λ\n== + ≤ = ∑ .                  (3.5) \nHence \n ( )28 8 8 8 \n2 2 \n1 1 1 1 1 1 1 1 1 \n8 64 8 8 4 i i i \ni i i i r r λ\n= = = =     = + = + ≤ + =         ∑ ∑ ∑ ∑      .      (3.6)  6 \n By using  Eqs.  (3.3-3.6) we get Eq. (3.2).  Eq. (3 .2) may be generalized to any odd-n MDS density \nmatrix. \n Note that the equality in Eq. (3.6) holds only if  \n1( 1,2,3,4) ; 0 ( 5,6,7,8) 4i j i j λ λ = = = =    .                (3.7) \n   We note that local unitary transformations, on t he qubits , , A B C   in Equations.  (2.1)  and (2.5) \nwill obviously produce other simple biseparable MDS  density matrices. For example the density matrix \n( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2\n132 321 213 8 ( ) ( ) ( ) A B C \nx z y z y x y x z A B A C B C C B A I I I \nR R R ρ\nσ σ σ σ σ σ σ σ σ = ⊗ ⊗ + \n⊗ ⊗ + ⊗ ⊗ + ⊗ ⊗ ,   (3.8) \nis obtained from (2.1) by a 090  rotation, of A around x, B around y, and C around z and inverting \nthe signs of the HS  parameters . Therefore in the biseparable form, Eq. (2.5) (and Eq. (2.6)) we have  \nsimply to make the following exchanges   \n             () () () () () ()\n( ) ( ) ( ) ( ) ( ) ( )\n( ) ( ) ( ) ( ) ( ) ( ); ; ; \n; ; ; \n; ; x x y z z y A A A A A A \nx z y y z x B B B B B B \nx y y x z z C C C C C C σ σ σ σ σ σ \nσ σ σ σ σ σ \nσ σ σ σ σ σ → → → − \n→ − → → \n→ → − →    .                   (3.9) \nand invert the signs of the relevant HS parameters. \nThen, by using the corresponding parameters in Eq. (3.1) we obtain 2ρ in the biseparable form: \n( ) ( ) ( ) ( )( ) ( )\n( ) ( ) ( ) ( )( ) ( )\n( ) ( ) ( ) ( )( ) ( )\n( ) ( ) ( ) ( )( ) ( )2\n132 213 321 \n132 213 321 \n132 213 321 \n132 213 321 8\nx y z A A A A BC BC \nx y z A A A A BC BC \nx y z A A A A BC BC \nx y z A A A A BC BC I R R R \nI R R R \nI R R R \nI R R R ρ\nσ σ σ \nσ σ σ \nσ σ σ \nσ σ σ ψ − − \n+ + \n+ + \n− − =\n      − + + ⊗ Φ Φ +     \n    + − + ⊗ Φ Φ +             + + − ⊗ Ψ Ψ +     \n     − − − ⊗ Ψ     /tildenosp /tildenosp\n/tildenosp /tildenosp\n/tildenosp /tildenosp\n/tildenosp/tildenosp\n\n\n\n\n\n.            (3.10)  \nThe density matrices ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ); ; ; \nBC BC BC BC BC BC BC BC − − + + + + − − Φ Φ Φ Φ Ψ Ψ Ψ Ψ /tildenosp /tildenosp /tildenosp /tildenosp /tildenosp /tildenosp /tildenosp /tildenosp  are \nobtained from Eq. (2.6) by using the transformation s of Eq. (3.9). \nNow, suppose we are given a more complicated densit y matrix, in the form  7 \n ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )\n( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )111 222 333 \n132 321 213 8 ( ) ( ) ( ) A B C \nx x x y y y z z z A B C A B C A B C \nx z y z y x y x z A B A C B C C B A I I I \nR R R \nR R R ρ\nσ σ σ σ σ σ σ σ σ \nσ σ σ σ σ σ σ σ σ = ⊗ ⊗ + \n⊗ ⊗ + ⊗ ⊗ + ⊗ ⊗ \n+ ⊗ ⊗ + ⊗ ⊗ + ⊗ ⊗  (3.10) \nWe would like to see if it can be written in a bise parable form. For this purpose we note that we can write \nEqs. (2.1) in a different form: \n( ){ } ( )2 2 2 2 2 2 \n1 111 222 333 111 222 333 1 8 1 ( ) ( ) ( ) 8 A B CR R R I I I R R R ρ ρ ≡ − + + ⊗ ⊗ + + + /tildenosp  .     (3.11) \nHere we defined a new density matrix 1ρ/tildenosp given by  \n( ) ( ) ( )3\n1 1 2 3 31 2\n118 ( ) ( ) ( ) ; , , A B C iii i i i x y z A B C \ni\niii \niI I I R \nRρ σ σ σ σ σ σ σ σ σ \n=\n== ⊗ ⊗ + ⊗ ⊗ ≡ ≡ ≡ ∑\n∑/tildenosp  \n            (3.12) \nThis density matrix can be written in a biseparable  form: \n( )( ) ( ) ( )\n( )( ) ( )\n( )( ) ( ) ( )\n( )( ) ( )\n( )( ) ( ) ( )\n( )( ) ( )1\n111 333 222 \n2 2 2 \n111 222 333 \n111 333 222 \n2 2 2 \n111 222 333 \n111 333 222 \n2 2 2 \n111 222 333 8\nx y z A A A\nABC BC \nx y z A A A\nABC BC \nx y z A A\nABC B R R R \nI\nR R R \nR R R \nI\nR R R \nR R R \nI\nR R R ρ\nσ σ σ \nσ σ σ \nσ σ σ − − \n+ + \n+ + =\n  − + +     + ⊗ Φ Φ +     + +     \n  − +     + ⊗ Φ Φ +     + +     \n  + −   + ⊗ Ψ Ψ   + +     /tildenosp\n( )( ) ( ) ( )\n( )( ) ( ) 111 333 222 \n2 2 2 \n111 222 333 C\nx y z A A A\nABC BC R R R \nI\nR R R σ σ σ \nψ− −   \n  \n  \n  \n  \n  \n  \n  \n    \n  \n    +    \n  \n      − − −     + ⊗ Ψ       + +                 .                (3.13) \nThen, using Eq. (3.8) and the transformations (3.9)  we get  \n( ){ } ( )2 2 2 2 2 2 \n2 132 321 213 132 321 333 2 8 1 ( ) ( ) ( ) 8 A B CR R R I I I R R R ρ ρ = − + + ⊗ ⊗ + + + /tildenosp .        (3.14) 8 \n 28ρ/tildenosp  is obtained from Eq. (3.13) by using the transfor mations of Eq. (3.9) on qubit A and on  the Bell \nstates and transforming the HS  parameters as: \n          111 132 222 321 333 213 ; ; R R R R R R → → →       ,                                                   (3.15) \n With these definitions of 1 2 ,ρ ρ /tildenosp /tildenosp  we have for Eq. (3.10) the identity   \n( )\n( ) ( )2 2 2 2 2 2 \n111 222 333 132 321 213 \n2 2 2 2 2 2 \n111 222 333 1 132 321 333 2 8 ( ) ( ) ( ) 1 \n8 8 A B C I I I R R R R R R \nR R R R R R ρ\nρ ρ = ⊗ ⊗ − + + − + + \n+ + + + + + /tildenosp /tildenosp  .                      (3.16)          \nWe note that since according to Eq. (3.2) we get  2 2 2 2 2 2 \n111 222 333 132 321 333 1 R R R R R R + + + + + ≤  , the \ncoefficients of 18ρ/tildenospand 28ρ/tildenosp are smaller than 1.  But, for the right hand side to represent a biseparable \ndensity matrix we have to require  \n 2 2 2 2 2 2 \n111 222 333 132 321 213 1 R R R R R R + + + + + ≤         .       (3.17) \nEq. (3.17) represents a sufficient condition for bi separability of the density matrix given by Eqs. (3 .10), \nand  (3.16). We note that the condition (3.16) requ ires that the sum of the Frobenius norms of the two  \ntriads of MDS–parameters is not larger than 1. \n        It is worth noting that in the example for ρ of Eq. (3.10) the sufficient condition for bisepar ability \n(3.17) is the necessary condition that ρ is a density matrix. This is so, because the opera tors in one triad \ncommute with those in the other; therefore the smal lest eigenvalue of 8ρ is given by   \n2 2 2 2 2 2 \n111 222 333 132 321 213 1 0 R R R R R R − + + − + + ≥ . \n To treat the general case of MDS density matrix gi ven by (3.1) (up to 27 MDS terms), we can \ndivide  ( ) ( ) ( )3\n, , \n, , 1 l m n x y z A C Bl m n R σ σ σ \n=⊗ ⊗ ∑  into 9 groups of three triads. Starting with the t riad \n()()() ()()() ()()()111 222 333 x x x y y y z z z A B C A B C A B C R R R σ σ σ σ σ σ σ σ σ ⊗ ⊗ + ⊗ ⊗ + ⊗ ⊗       \nof   Eq.(2.1), we can apply 8 transformations (simi lar to those of  Eq. (3.9))  to the Pauli matrices of each \nqubit, obtaining  8 triads with corresponding HS pa rameters. Each triad may be treated as in Eqs. (3.1 3, \n3.14) to include together with Eq. (3.11) the 27 , , l m n R  parameters. Therefore the sufficient condition for  9 \n biseparability of Eq. (3.1) becomes that the sum of  the Frobenius norms of the 9 (at most) triads of M DS-\nparameters is not larger than 1. Such condition is sufficient for biseparability but the sufficient co ndition \nfor biseparability may be improved by other methods . \n As another simple example we add another triad to ρof (3.10) to obtain   \n( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )\n( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )\n( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )111 222 333 \n132 321 213 \n123 312 231 8 ( ) ( ) ( ) A B C \nx x x y y y z z z A B C A B C A B C \nx z y z y x y x z A B A C B C C B A \nx y z z x y y z x A C A B B C B C A I I I \nR R R \nR R R \nR R R ρ\nσ σ σ σ σ σ σ σ σ \nσ σ σ σ σ σ σ σ σ \nσ σ σ σ σ σ σ σ σ = ⊗ ⊗ + \n⊗ ⊗ + ⊗ ⊗ + ⊗ ⊗ \n+ ⊗ ⊗ + ⊗ ⊗ + ⊗ ⊗ \n+ ⊗ ⊗ + ⊗ ⊗ + ⊗ ⊗ . (3.18) \nThe sufficient condition for biseparability is \n2 2 2 2 2 2 2 2 2 \n111 222 333 132 321 213 123 312 231 1 R R R R R R R R R + + + + + + + + ≤   ,     (3.19) \nAlso in this example, the operators in any triad co mmute with those of the other triads, so that Eq. ( 3.19) \nis the necessary condition that the smallest eigenv alue of ρ (Eq. (3.18)) is nonnegative. This does not \nhold in the general case when the operators in one triad do not commute with those of other triads. In  such \ncases the condition for density matrix does not see m to necessarily imply biseparability.  \n                                    \n4. Explicit biseparability of the W state mixed wit h white noise  \n In the present Section we apply our methods to cal culate sufficient conditions for biseparability of W \nstate mixed with white noise. Assuming that the den sity matrix ( ) Wρ  for the W state with a probability \np is mixed with white noise with probability (1-p) w e get  \n ( )( )()()() () 8 ; 1 8 A B C W mixed p I I I p W ρ ρ = − ⊗ ⊗ +     .      (4.1) \nHere () 8Wρ is given by 4  10 \n  3 8 ( ) \n0 0 0 0 0 0 0 0 \n0 8 8 0 8 0 0 0 \n0 8 8 0 8 0 0 0 \n0 0 0 0 0 0 0 0 \n0 8 8 0 8 0 0 0 \n0 0 0 0 0 0 0 0 \n0 0 0 0 0 0 0 0 \n0 0 0 0 0 0 0 0 Wρ⋅ = \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n         (4.2) \nThe HS decomposition of (4.2) is quite complicated and given by  \n 3 8 ( ) 2( ) ( ) ( ) 2( ) ( ) ( ) 2( ) ( ) ( ) \n( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) \n( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) \n2( ) ( ) ( ) 2( ) ( ) ( ) 2 y A B y C A y B y C y A y B C \nz A B z C A z B z C z A z B C \nz A B C A z B C A B z C \nx A z B x C z A x B x C W I I I \nI I I \nI I I I I I ρ σ σ σ σ σ σ \nσ σ σ σ σ σ \nσ σ σ \nσ σ σ σ σ σ ⋅ = ⊗ ⊗ + ⊗ ⊗ + ⊗ ⊗ \n− ⊗ ⊗ − ⊗ ⊗ − ⊗ ⊗ \n+ ⊗ ⊗ + ⊗ ⊗ + ⊗ ⊗ \n+ ⊗ ⊗ + ⊗ ⊗ + ( ) ( ) ( ) \n2( ) ( ) ( ) 2( ) ( ) ( ) 2( ) ( ) ( ) \n2( ) ( ) ( ) 2( ) ( ) ( ) 2( ) ( ) ( ) \n3( ) ( ) ( ) 3( ) ( ) ( ) x A x B z C \nx A x B C x A B x C A x B x C \ny A y B z C y A z B y C z A y B y C \nz A z B z C A B C I I I \nI I I σ σ σ \nσ σ σ σ σ σ \nσ σ σ σ σ σ σ σ σ \nσ σ σ ⊗ ⊗ \n+ ⊗ ⊗ + ⊗ ⊗ + ⊗ ⊗ \n+ ⊗ ⊗ + ⊗ ⊗ + ⊗ ⊗ \n− ⊗ ⊗ + ⊗ ⊗ (4.3) \nIn our previous work 4 we have used equations (4.1) and (4.3) and derived  the sufficient condition \n1 / 9 p≤  for explicit full separability of the density matr ix of Eq. (4.1). \n We analyze in the present Section a sufficient con dition for explicit biseparability for the density \nmatrix of Eq. (4.1) using the HS decomposition of  () 8Wρ  given by Eq. (4.3).  Although in Eq. (4.3) \nonly a part of the density matrix is related to MDS , the use of the methods presented in previous sect ions \ncan improve the condition for biseparabily relative  to that of full separability. \nThe following six MDS terms  \n( ) ( ) ( ) , ( ) ( ) ( ) , ( ) ( ) ( ) , \n( ) ( ) ( ) , ( ) ( ) ( ) , ( ) ( ) ( ) x A z B x C z A x B x C x A x B z C \ny A y B z C y A z B y C z A y B y C σ σ σ σ σ σ σ σ σ \nσ σ σ σ σ σ σ σ σ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ \n⊗ ⊗ ⊗ ⊗ ⊗ ⊗   , \ncan be grouped into 3 pairs:  11 \n ( ) ( ) ( ) ( ) , ( ) ( ) ( ) \n( ) ( ) ( ) ( ) , ( ) ( ) ( ) \n( ) ( ) ( ) ( ) , ( ) ( ) ( ) x A z B x C y A y B z C \nx A x B z C z A y B y C \nz A y B y C y A z B y C a\nb\ncσ σ σ σ σ σ \nσ σ σ σ σ σ \nσ σ σ σ σ σ   ⊗ ⊗ ⊗ ⊗   \n  ⊗ ⊗ ⊗ ⊗   \n  ⊗ ⊗ ⊗ ⊗   . \nPair (a) is obtained from the pair ( ) ( ) ( ) , ( ) ( ) ( ) x A x B x C y A y B y C σ σ σ σ σ σ   ⊗ ⊗ ⊗ ⊗    by the local \nunitary transformation 1U which rotates B around the y axis, and C around x by  / 2 π  . Similarly \n(b) is obtained from the same pair by 2U which rotates A around x and C around y by / 2 π , and (c) \nis obtained by 3U rotating A around y and B around x by / 2 π . \nRecalling the decomposition of 1ρ  (Eq. (2.1) in a biseparable   form  (Eq. (2.5) ),   we get after some \ntedious calculations the following explicit express ion for the biseparability  of the density matrix o f Eq. \n(4.1):  \n ( )\n( ) ( ) ( ) ( ) ( ) ( )\n( ) ( ) ( )\n( ) ( ) ( ) ( ) ( ) ( ){ }\n( ) ( ) ( ) ( ) ( ) ( ){ }\n( )( ) ( )( ) ( )\n( )( ) ( )8 ; \n3\n3\n3\n2\n223z z z z z z A B C A B C \nz z z A B C \nY Y Y Y Y z A B C A B C \nx x x x x x A B C A B C \nx y A A\nABC BC \nx y A\nA\niW mixed \nI I I I I I p\nI I I \npI I I I I I \npI I I I I I \nI\nI\npUρ\nσ σ σ σ σ σ \nσ σ σ \nσ σ σ σ σ σ \nσ σ σ σ σ σ \nσ σ \nσ σ − − =\n+ ⊗ + ⊗ − + − ⊗ + ⊗ +       + + ⊗ − ⊗ +     \n+ + ⊗ + ⊗ + + − ⊗ − ⊗ − \n+ + ⊗ + ⊗ + + − ⊗ − ⊗ − \n  − +   + ⊗ Φ Φ +         \n−\n+( ) ( )\n( )( ) ( )( ) ( )\n( )( ) ( )( ) ( )3\n†\n12\n2\n2\n71 2 2 ( ) ( ) ( ) 3A\nBC BC \ni\nix y A A\nABC BC \nx y A A\nABC BC \nA B C U\nI\nI\np p I I I σ σ \nσ σ \nψ+ + \n=+ + \n− −   \n  \n  \n  \n      ⊗ Φ Φ +                  +     + ⊗ Ψ Ψ +             \n    − −     + ⊗ Ψ               \n  + − − ⊗ ⊗     ∑\n (4.4) 12 \n We get from Eq. (4.4) that for 10.1937 \n7 / 3 2 2 p≤ = \n+ , ( ) ;W mixed ρ of Eq. (4.1) is not genuinely \nentangled. This condition can be compared with the condition 10.1111 9p≤ ≈  which was obtained by us \nas a sufficient condition for full separability.4  \n \n6.  Summary  and discussion  \nIn the present work we treated in Sections 2 and 3 a  sufficient condition for biseparability   of one qu bit  \n(say A ) with respect to the other two qubits (BC). T his is obtained by using a representation, where the \nfirst qubit includes the 3 HS parameters, and it is mul tiplied by the Bell states of the other two qubits. I n \nSec. 2 we analyzed the sufficient condition for bisep arability of the density matrix (2.1) which includes  \nthree HS parameters multiplied by the three diagonal  products of Pauli matrices. Explicitly biseparable \nexpression for the density matrix was given in Eqs. (2 .5-2.6). A sufficient condition for biseparability f or \nthis special case (given by Eq. (2.7)) is that the sum o f HS parameters squared will not be larger than 1, \nwhich is equivalent to the condition of Eq. (2.1) to  be a density matrix.  In another form the condition  in \nEq. (2.7) requires the Frobenius norm 13    of the 3 HS parameters to be not larger than 1. \n In Sec. (3) we proved the relation (3.2) showing th at for odd-n MDS density matrices the sum of \nHS parameters squared cannot be larger than 1. The e quality with 1 will be obtained only for the special  \ncase described by Eq. (3.7). For treating the general  case of MDS density matrix given by Eq. (3.1) (up t o \n27 MDS terms) we can start with the triad of Eq. (2.1 ) and apply 8 local transformations (similar to those \nof Eq. (3.9)) to the Pauli matrices of each qubit, o btaining 8 triads with corresponding HS parameters. \nThe sufficient condition for biseparability of Eq. (3 .1) becomes then that the sum of the Frobenius norms \nof the 9 (or less) triads of MDS parameters is not large r than 1. We demonstrated this method for the \ncase in which the 3-qubits MDS density matrix include s two triads of three qubits products given by Eq. \n(3.10). This density matrix can be written by the exp licitly  biseparable form given by Eq. (3.16) and t he \nsufficient condition for biseparability in this case is given by Eq. (3.17) requiring that the sum of \nFrobenius norms of the two triads will not be larger than 1. We demonstrated in Eq. (3.18) a density \nmatrix which includes three triads of Pauli matrices products. The sufficient condition for bisepability i s \nthen given by Eq. (3.19) requiring here again that the sum of the three Frobenius norms will not be \nlarger than 1. 13 \n  We should mention that the present sufficient conditi on for biseparability does not preclude the \npossibility that the density matrix is fully separable . In some cases (especially for a large number of HS \nparameters) it may happen that the sufficient condit ion for full separability obtained by using SVD 4 is \nsatisfied and then the present condition is not needed .  \n In Sec. 4, we analyzed explicit biseparability of t he density matrix (4.1) for the W state mixed \nwith white noise. In this analysis we used the HS dec omposition of the W density matrix where six MDS \nterms have been grouped into 3 pairs and their bisepa rable form has been given by the methods \nanalyzed in Sections 2, 3.  All other terms in the HS  decomposition were given in a fully separable form . \nWe obtained the sufficient condition for biseparabili ty given by 0.1937 p≤ in comparison with the \nsufficient condition for full separability given in o ur previous work 4 as  0.1111 p≤  . This result can also \nbe compared with the result obtained by the PH crit erion by which for 0.209589 p>  this density \nmatrix cannot be fully separable. 4 It is interesting to note that for 0.529 p>  it has been shown 19  that \nthe mixed W state is genuinely entangled. \n \nReferences \n1.  Y.  Ben-Aryeh, A. Mann and B.C. Sanders, Foundations of Physics  29 (1999) 1963. \n2.  Y. Ben-Aryeh, Int. J. Quant. Inf. 13 (2015) 1450045; The use of Braid operators for impl ementing \nentangled large n-qubits Bell states, arXiv.org. (q uant-ph) 1403.2524. \n3.  Y. Ben-Aryeh and A. Mann, Int. J. Quant. Inf.  13 (2015) 1550061; Explicit constructions of all \nseparable two-qubits density matrices and related p roblems for three-qubits systems. arXiv.org \n(quant-ph) 1510.07222 \n4.  Y. Ben-Aryeh and A.Mann, Int. J. Quant. Inf. 14  (2016)  1650030; Separability and entanglement of \nn-qubit and a qubit and a qudit using Hilbert-Schmi dt decompositions.arXiv.org. (Quant-ph) \n1606.04304v2. \n5.  R. Horodecki,  P. Horodecki,  M. Horodecki, and K. Horodecki, Rev.  Mod. Phys.  81  (2009) 865.  \n6.   O. Guhne and G. Toth,  Physics   Reports   474   (2009)  1. \n7.  M.A. Nielsen and I.L. Chuang, Quantum computation and Quantum Information  (Cambridge \nUniversity   Press, Cambridge,   2000). \n8.  J.  Auderetsch,   Entangled   Systems   (Wiley,   Weinheim,   2007). 14 \n 9.  G. Benenti, G. Casati and G. Strini, Principles of quantum computation and information (World \nScientific, Singapore, 2005). \n10.    J.  Stolze  and  D.  Suter, Quantum   Computing  (Wiley,  Weinheim,  2008).  \n11.    I. Bengtsson and K. Zyczkowski, Geometry of Quantum states; An introduction to enta nglement \n(Cambridge University Press, Cambridge, 2006).  \n12.   R. Horn and C.R.  Johnson, Matrix analysis (Cambridge University Press, Cambri dge, 1991). \n13.    G. H.  Golub and C. F.  Van loan, Matrix Computation (Johns Hopkins University Press, Baltimore, \n2013). \n14.    L. De Lathauwer , B.  De Moroder and J.  Vanderwa lls, SIAM J. Matrix   Anal.  Appl.   21   (2000) \n1253. \n15.    A. Peres, Phys.  Rev. Lett.  77   (1996)   1413. \n16.   M. Horodecki , P. Horodecki  and R. Horodecki ,  Phys.  Lett.  A    223   (1996)   1. \n17.  J-D Bancal, N. Gisin, Y-C Liang, and S. Pironio, Ph ys. Rev. Lett. 106    (2011) 250404. \n18.   R.  Horodecki and M.  Horodecki, Phys. Rev. A   54    (1996) 1838. \n19.  G. Guhne and M. Seevinck, New J. Phys.  12 ( 2010) 053002 \n20.   S.L. Braunstein, A. Mann and M. Revzen, Phys. Rev. Lett.  68  (1992) 3259. "    },    {        "title": "1611.01194v2.Convex_set_of_quantum_states_with_positive_partial_transpose_analysed_by_hit_and_run_algorithm.pdf",        "content": "CONVEX SET OF QUANTUM STATES\nWITH POSITIVE PARTIAL TRANSPOSE\nANALYSED BY HIT AND RUN ALGORITHM\nKONRAD SZYMAŃSKI, BENOÎT COLLINS, TOMASZ SZAREK, AND KAROL ŻYCZKOWSKI\nAbstract. Theconvexsetofquantumstatesofacomposite K\u0002Ksystemwithpositivepartialtranspose\nisanalysed. Aversionofthe hit and run algorithmisusedtogenerateasequenceofrandompointscovering\nthis set uniformly and an estimation for the convergence speed of the algorithm is derived. For K\u00153\nthis algorithm works faster than sampling over the entire set of states and verifying whether the partial\ntranspose is positive. The level density of the PPT states is shown to differ from the Marchenko-Pastur\ndistribution, supported in [0;4]and corresponding asymptotically to the entire set of quantum states.\nBased on the shifted semi–circle law, describing asymptotic level density of partially transposed states,\nand on the level density for the Gaussian unitary ensemble with constraints for the spectrum we find\nan explicit form of the probability distribution supported in [0;3], which describes well the level density\nobtained numerically for PPT states.\n1.Introduction\nThe structure of the set of quantum states is a subject of a current interest form the perspective of pure\nmathematics [1] and theoretical physics [2, 3, 4, 5]. A special attention is paid to the bipartite systems,\nin which separable and entangled states can be distinguished [3] as entanglement plays a key role in the\ntheory of quantum information processing [6].\nIt is well known that any separable state of a bipartite, K\u0002Ksystem has positive partial transpose\n(PPT). In the case of a two–qubit system, K= 2, every PPT state is separable, while for higher systems\nit is not the case [7]. Description of the set of separable states for an arbitrary Kis difficult, while it is\nstraightforward for the set of PPT states.\nUp till now, we are not aware of any dedicated algorithm to obtain a random PPT states with respect\nto the flat (Hilbert–Schmidt) measure. It is not difficult to generate a random density matrix from the\nentire set \nNof quantum states of size Nusing a matrix Gof the Ginibre ensemble of complex square\nrandom matrices of order N, as the matrix \u001a=GGy=TrGGyis positive and normalized and forms thus\na legitimate random quantum state [8, 9]. However, as the relative volume of the subset of PPT states\ndecreases exponentially with the dimension [10, 11, 12], is not efficient to generate random points in the\nentire set \nNof states of size N=K2and to check a posteriori, if the PPT property is satisfied.\nThe goal of this work is to analyse the set of PPT states by generating a sequence of random points\nfrom this set according to the flat measure. In this way we are in position to analyse the level density\nof random PPT states and to show the difference with respect of the Hilbert–Schmidt density, which\ncorresponds to uniform sampling of the entire set of quantum states and asymptotically tends to the\nMarchenko-Pastur distribution. Note that analytical analysis of the set of PPT states is not an easy\ntask, as this set does not enjoy the symmetry with respect to unitary transformations. However, based\non the fact that the level density of partially transposed quantum states is asymptotically described\nby the shifted semi–circle law [13, 14] and making use of the level density of for the Gaussian Unitary\nDate: March 28, 2017.\n1arXiv:1611.01194v2  [quant-ph]  31 Mar 2017Ensemble (GUE) with constraints for the spectrum [15], we are in position to conjecture the level density\nwhich describes well numerical results obtained by sampling the set of PPT states.\nA parallel aim of the paper is to introduce an effective algorithm of generating random points in a convex\nhigh dimensional set and to analyse its features. We propose a variant of the hit and run algorithm\n[16, 17, 18, 19] and establish its speed of convergence. In each iteration of the algorithm one selects a\nrandom direction by choosing a random point from a hypersphere. In the next step one finds the longest\ninterval belonging to the set which passes through the initial point and is parallel to this direction and\njumps to a random point drawn from this interval.\nResults obtained are easily applicable for all convex sets for which it is computationally efficient to find\ntwo points in which a given line enters and exists the set. Hence this approach, suitable for the set\nof PPT states, is not easy to apply for the set of separable states of higher dimensions, for which the\nproblem of finding out whether a given state is separable is ‘hard’ [21].\nThispaperisorganisedasfollows. Insection2wepresentthealgorithmandprovideageneralestimation\nof the convergence speed. Usage of the algorithm for simple bodies in Rdis illustrated in section 3. In\nsection 4 we discuss the set of quantum states of composite systems and its subset containing the states\nwith positive partial transform. Making use of the algorithm we generate sequences of random points\nin this set and analyze numerical results for the level density for such states. Considerable deviations\nfrom the known distributions for the entire set of states are reported and it is conjectured that the\nasymptotic level density for the PPT states is described by the distribution the (4.6), different from the\nMarchenko–Pastur distribution [22].\n2.Hit and run algorithm and its convergence speed\nConsider a given convex, compact set X\u001aRd. To generate a sequence of random points distributed\naccording to the uniform Lebesgue measure in Xwe will use the following version of the “hit and run\"\nalgorithm [17, 20].\n(1) Choose an arbitrary starting point x02X,\n(2) Draw randomly a unit vector e2Sd\u00001according to the uniform distribution on the hypersphere,\n(3) Find boundary points xmin\n1(e);xmax\n1(e)2@Xalong the direction e: there exist a;b\u00150such that\nxmin\n1(e) =x0\u0000aeandxmax\n1(e) +be.\n(4) Select a point x1randomly with respect to the uniform measure in the interval [xmin\n1;xmax\n1].\n(5) Repeat the steps (ii)-(iv) to find subsequent random points x2;x3;:::.\nNote that each time the direction eis taken randomly and that the direction used in step i+ 1does not\ndepend on the direction used in step i.\nA first result of this work consists in the following estimation of the convergence speed in the total\nvariation norm. The total variation norm of two probability measures is defined by the formula\nk\u00161\u0000\u00162kTV:= ^\u0016+(X) + ^\u0016\u0000(X);\nwhere ^\u0016+\u0000^\u0016\u0000is the Jordan decomposition of the signed measure \u00161\u0000\u00162.\nLet\bbe a Markov chain [24] described by the above algorithm and let Sbe its transition kernel.\nTheorem 2.1. LetX\u001aRdbe a compact convex set. Assume that there exist r;R > 0such that for\nsomex02Xone hasB(x0;r)\u001aX\u001aB(x0;R). Then the chain \bsatisfies the condition\n(2.1) kSn(x;\u0001)\u0000LdkTV\u0014(1\u0000\u0012)nfor anyx2X;\n2where\n(2.2) \u0012= (2=d)[(R=r+ 1)d\u00001(R=r)]\u00001:\n(HereLddenotes the d–dimensional Lebesgue measure.)\nProof.At the very beginning of the proof we show that the Lebesgue measure Ldrestricted to Xis\ninvariant for the considered algorithm. To do this observe that for any x2Xn@Xand a Borel set\nA\u001aXwe have\nS(x;A) = \u0007Z\nSd\u000011\nkxmax(e)\u0000xmin(e)kZ\nR1A(x+\u0015e)d\u0015Ld\u00001(de);\nwhere\n\u0007 = (Ld\u00001(Sd\u00001))\u00001:\nTherefore, by the Fubini theorem we haveZ\nXS(x;A)Ld(dx) =Z\nX\u0007Z\nSd\u000011\nkxmax(e)\u0000xmin(e)kZ\nR1A(x+\u0015e)d\u0015Ld\u00001(de)Ld(dx)\n= \u0007Z\nSd\u000011\nkxmax(e)\u0000xmin(e)kZ\nRZ\nX1A(x+\u0015e)Ld(dx)d\u0015Ld\u00001(de)\n= \u0007Z\nSd\u000011\nkxmax(e)\u0000xmin(e)kZ\nRZ\nX1A\u0000\u0015e(x)Ld(dx)d\u0015Ld\u00001(de)\n= \u0007Z\nSd\u000011\nkxmax(e)\u0000xmin(e)kZ\nRLd(A\u0000\u0015e)d\u0015Ld\u00001(de)\n=Ld(A)\u0007Z\nSd\u000011\nkxmax(e)\u0000xmin(e)kZ\nRd\u0015Ld\u00001(de) =Ld(A)\u0007\u0007\u00001=Ld(A);\nwhich finishes the proof that the Lebesgue measure Ldis invariant.\nWe are going to evaluate S(x;\u0001)for anyx2X. First assume that X=B(x;R). It is easy to see that\nthe transition function S(x;\u0001)is absolutely continuous with respect to the Lebesgue measure Ld. Its\ndensity is equal to\n(2.3) fx(u) = (cdku\u0000xkd\u00001R)\u00001foru2B(x;R)nfxg,\nwherecd= 2\u0019d=2=\u0000(d=2)denotes the surface of the d-sphere of radius 1. Indeed, by symmetry argument\nwe see that fx(u) =~f(ku\u0000xk)for some function ~f: [0;R]!R. Then we have\nZu\n0~f(v)cdvd\u00001dv=u=R for anyu2[0;R].\nDifferentiating both sides with respect to uwe obtain (2.3).\nNow letXbe arbitrary. Since X\u001aB(x0;R)we see that for any x2Xthe transition function S(x;\u0001)\nis absolutely continuous with respect to the Lebesgue measure Ldand its density fxsatisfies\nfx(u)\u0015(cdku\u0000xkd\u000012R)\u00001foru2X.\nThus\nS(x;\u0001)\u0015Z\n\u0001\\B(x0;r)fx(u)Ld(du)\u0015(cd(R+r)d\u000012R)\u00001Ld(B(x;r))\u0017(\u0001)\n\u0015(bd=cd)[(R=r+ 1)d\u00001(R=r)]\u00001\u0017(\u0001);\nwhere\u0017(\u0001) =Ld(\u0001\\B(x0;r))=Ld(B(x0;r))andbd=\u0019d=2=\u0000(d=2 + 1)denotes the volume of a unit ball\ninRd. Sincebd=cd= 2=d, we finally obtain\nS(x;\u0001)\u0015(2=d)[(R=r+ 1)d\u00001(R=r)]\u00001\u0017(\u0001):\n3(a)\n-1 0 1-101-1 0 1\n-101\n0.050.100.150.200.250.30 (b)\n-1 0 1-101-1 0 1\n-101\n0.050.100.150.200.250.30\nFigure 1. Histograms of sampled points in two simple 2–D cases: (a) square and (b)\nunit circle. For both plots the number of sampled points is 500000, the bins are squares\nof side 0:05.\nNow the version of Doeblin’s theorem finishes the proof – see Proposition 3.1 in [20] (see also [25]). \u0003\n3.Balls, cubes and simplices in Rd.\nTo show the convergence estimate (2.1) in action, in this section we apply it for simple bodies in Rd.\nFor cubes and balls in Rdit is straightforward to generate random points according to the uniform\nmeasure, so we will not advocate to use the above algorithm for this purpose. However, it is illuminating\nto compare estimations for the parameters determining the convergence rate according to Eq. (2.1).\nFor a unit ballBdboth radii coincide, R=r, so their ratio \u0014=r=Ris equal to unity. Estimation (2.2)\ngives\u0012= 2\u0000d=dwherebddenotes the volume of a unit d–ball. This implies the convergence rate\n\u000b= 1\u0000\u0012= 1\u00002\u0000d=d:\nFor an unit cubeCdthe inscribed radius r= 1=2, and outscribed radius R=1\n2p\ndso the ratio reads\n\u0014=r=R= 1=p\nd. Hence the convergence rate reads,\n\u000b= 1\u0000\u0012= 1\u0000(2=d)[(p\nd+ 1)d\u00001p\nd]\u00001:\nTo demonstrate usefulness of the algorithm we present in Fig. 1 the density of points it generated in a\ncircle and a square.\nThe\u001f2test of the data obtained numerically does not suggests to reject the hypothesis that the data\nwere generated according to the uniform distribution with a confidence level p= 0:999.\nFor anN–simplex \u0001Nembedded in Rdwithd=N\u00001we haveR=p\n(N\u00001)=Nandr=\n1=p\nN(N\u00001)so that the ratio \u0014= 1=(N\u00001) =d\u00001. In this case the convergence rate reads,\n\u000b= 1\u0000\u0012= 1\u0000(2=d)[(d+ 1)d\u00001d]\u00001:\nNote that the simplex \u0001Ndescribes the set of classical states – N-point probability distributions.\n44.Quantum states with positive partial transpose\nLet\nNbe the set of density matrices of size N, formed by complex Hermitian and positive operators,\n\u001ay=\u001a\u00150, normalized by the trace condition Tr \u001a= 1. Due to the normalization condition the\ndimension of the set is d=N2\u00001.\nTheradiusoftheout-sphere, equaltotheHilbert–Schmidtdistancebetweenapurestatediag (1;0;:::; 0)\nand the maximally mixed state \u001a\u0003=I=N, isR=p\n(N\u00001)=N. The radius of the inscribed sphere\ngiven by the distance between \u001a\u0003and the center of a face, diag (0;1;:::; 1)=(N\u00001)is equal to r=\n1=p\nN(N\u00001), so their ratio \u0014= 1=(N\u00001) = 1=(p\nd+ 1\u00001)\u0018d\u00001=2. Hence the convergence rate of\nthe algorithm reads in this case\n\u000b= 1\u0000\u0012\u00181\u0000(2=d)[(p\nd+ 1)d\u00001p\nd]\u00001:\nConsider now a composite dimension N=K2, so the Hilbert space has a tensor product structure,\nHN=HA\nHB. In this case one defines a partial transposition, T2=I\nTand the set of PPT\nstates (positive partial transpose) which satisfy \u001aT2\u00150. This set forms a convex subset of \nNand it\ncan be obtained as an intersection of \nNand its reflection T2(\nN)[5]. The set of PPT states can be\ndecomposed into cones of the same height r= 1=p\nN(N\u00001)[23], hence \u0016= 1=(N\u00001)\u0018d\u00001=2.\nIn the case of set of PPT states the algorithm to generate random elements from this set according\nto the flat measure gives the same rate of convergence as in the case of quantum states, \u000b= 1\u0000\u0012\u0018\n1\u0000(2=d)[(p\nd+ 1)d\u00001p\nd]\u00001:\n4.1.Two–qubit system. We start the discussion with the simplest composed system consisting of two\nqubits, setting K= 2andN=K2= 4. To construct a random state from the entire 15dimensional set\n\n4we generated a random square Ginibre matrix Gof order four with independent complex Gaussian\nentries and used the normalized Wishart matrix, writing \u001a=GGy=TrGGy. In this way one obtains\nrandom states generated according to the Hilbert–Schmidt measure in \n4[8, 9].\nNumerical data, obtained from 105density matrices and represented in Fig. 2a by closed boxes, fit well\nto the analytical distribution P4(x)denoted in the plot by a solid curve. Here x=N\u0015= 4\u0015denotes\nthe rescaled eigenvalue of \u001a. In general, for any finite N, the distribution PN(x)can be represented as\na superposition of 2Npolynomial terms,\n(4.1) PN(x) =2NX\nm=2am(Nx)m\u00002(1\u0000Nx)N2\u0000m\nwith weights amgiven by the Euler gamma function [26]. Observe characteristic oscillations of the\ndensity and four maxima of the distribution related to the effect of level repulsion.\nGiven any bipartite quantum state \u001ait is straightforward to diagonalize the partial transpose, \u001aT2=\n( 1\nT)\u001a, to check, whether this matrix is positive. In the case of two qubits the relative volume of the\nset of states with positive partial transpose (PPT) with respect to the HS measure is conjectured to be\nequal to 8=33\u00190:242[27]. Thus verifying a posteriori the PPT property we could divide the random\nstates into two classes and obtain histograms of the level density for the sets of states with positive or\nnegative partial transpose (sets PPT and NPT, respectively) – see. Fig. 2a.\nFurthermore, we studied the level density of the partially transposed operators \u001aT2with eigenvalues \u001f.\nFor the states sampled from the entire set \n4, some partially transposed matrices are not positive, so\nthe level density is supported also for negative values of the variable y= 4\u001f– see. Fig. 2a. It is known\nthat for 2\u00022systems only a single eigenvalue \u001aT2can be negative and it is not smaller than \u00001=2[28],\nso one sees a single peak of the density P(y)containing the negative eigenvalues. For large systems the\n5(a)\n○\n○\n○\n○\n○\n○\n○\n○\n○\n○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○□\n□\n□\n□\n□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□△\n△\n△\n△\n△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△0 1 2 3x12P4(x) (b)\n○\n○\n○\n○\n○\n○\n○\n○○○○○○○○○○○○○○○○○○○\n○○○○\n○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□\n□\n□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△-1 0 1 2 3y0.511.5P4(y)\nFigure 2. (a) — Spectral density P(x)for two–qubit density matrices \u001awherex=\n4\u0015: numerical data representing sampling 105times over entire set \n4(light/yellow \u0003)\ncompared with analytical distribution (solid, red curve). This distribution differs from\nlevel density corresponding to the set of PPT states (blue \r), and the set of states with\nnegative partial transpose states (green 4). Panel (b) shows analogous data for partially\ntransposed states, \u001aT2: (light/yellow \u0003) denotes data for entire set \n4, (blue\r) denotes\ntransposed PPT states, while (green 4) denotes transposed states with negative partial\ntranspose.\nasymptotic density for the transposed states is known to converge to the shifted semicircle [13, 14], but\nno analytical expression for such a density in the case of K= 2is known so far.\nFigure 2b presents also histogram for the transposed states belonging to both classes of PPT and NPT\nstates. Note that for the set of PPT states the level density P(x)for the spectrum of \u001aandP(y)for the\nspectrum of \u001aT2, are the same. This is a consequence of the fact that the partial transpose is a volume\npreserving involution, so the level density will not be altered, if statistics is restricted to PPT states\nonly. Observe also that the spectrum corresponding to the PPT states of size N= 4do not display a\nsingularity at the origin.\n4.2.Larger systems. In the asymptotic case of large systems, N >> 1, the oscillations of the mean\nlevel density averaged over the entire set \nNof quantum states are smeared out and the distribution\n(4.1) tends to the Marchenko–Pastur distribution [22],\n(4.2) MP(x) =1\n2\u0019r\n4\u0000x\nx;\nwhere the rescaled eigenvalue x=N\u0015belongs to the interval [0;4]. Although there exists pure quantum\nstateswiththeoperatornormequaltounity, jj\u001ajj= 1, anormofagenericlargedimension Nmixedstate\nis asymptotically almost surely bounded from above by 4=N, since the support of the above distribution\nis bounded by xmax= 4. Due to the effect of concentration of measure in high dimensions a typical\nquantum state has spectral distribution close to the Marchenko–Pastur law (4.2), while states with\ndifferent level densities become unlikely [29].\nTo demonstrate the quality of the hit and run algorithm, we used it to generate samples of random\nstates for the entire convex set \nN. Diagonalising the generated states we obtained their eigenvalues\n\u0015, and analyzed the level density P(x)withx=N\u0015. Histograms obtained in this way for N= 9and\nN= 25coincide with the analytical expression (4.1) – see Fig. 3. Observe that the data for N= 25\nare already close to the Marchenko–Pastur distribution valid asymptotically. Although in this case the\n6(a)\n0 1 2 3 4x 012P9(x) (b)\n0 1 2 3 4x 012P25(x)\nFigure 3. Spectral densities PN(x)averaged over entire set \nNof quantum states of size:\na)N= 9and b)N= 25. Spectral density (4.2) denoted by a solid (blue) line describes\nwell the data already for N= 25.\nrandom points are generated from the set \n25of dimension d=N2\u00001 = 624, the average over a sample\nconsisting of circa 105random points provides reliable data.\nConsider now a bipartite case of size N=K\u0002K, for which the notion of partial transpose is defined. It\nis was first shown by Aubrun [13] that level density of the partially transposed bipartite random states,\nwritten\u001aT2= (I\nT)\u001a, asymptotically tends to the shifted semicircle law of Wigner,\n(4.3) PT(x)\u00181\n2\u0019p\n4\u0000(x\u00001)2;\nwith support x2[\u00001;3].\nUp to a linear shift, x!x\u00001, this distribution is that of the ensemble of random Hermitian matrices\nof GUE – see e.g. [30].\nAn extended model of random GUE matrices with spectrum of the rescaled eigenvalue restricted to a\ncertain interval [0;L(z)]was analyzed by Dean and Majumdar [15]. They derived an explicit family\nof probability distributions, labeled by a parameter z, which determines the position of the ‘hard wall\nbarrier’ for the corresponding Coulomb gas model,\n(4.4) hz(y) =p\nL(z)\u0000y\n2\u0019py(L(z) + 2y+ 2z);\nwhere the upper edge L(z)of the support of hz(y)reads\n(4.5) L(z) =2\n3\u0010p\nz2+ 6\u0000z\u0011\n:\nSince partial transpose is a volume preserving involution, the level density averaged over the set of PPT\nstates can be approximated by the density of states obtained from the shifted semicircle law of Aubrun\n(4.3), by imposing the restriction that all eigenvalues are non-negative. The approximation consists in\nan assumption that the ensemble of partially transposed density matrices asymptotically coincides with\nthe shifted GUE.\nTaking distribution (4.4), setting z= 0and normalizing the rescaled variable x=ay+bso that the\nexpectation value is set to unity as required, hxi=R\nxg(x) = 1, we arrive at the normalized probability\ndistribution\n(4.6) g(x) =4\n27\u0019r\n3\u0000x\nx(3 + 2x);\n7(a)\n0 1 2 3x12P9PPT(x) (b)\n0 1 2 3x12P25PPT(x)\nFigure 4. Spectral densities PPPT\nN(x)for PPT states of bipartite systems of size: a)\nN= 3\u00023 = 9and b)N= 5\u00025 = 25. Spectral density (Eq. (4.6)) denoted by a solid\n(blue) line describes well the data for N= 25.\nsupported in [0;3].\nTo demonstrate that this distribution can describe asymptotic level density of PPT states we need to\nrely on numerical computations. In the case of a larger bi–partite system of size N=K2the probability\nof finding a random PPT state decays exponentially with the dimension [10, 11, 12]. Therefore the\nsimplest approach to get a PPT state by generating random states from the entire set \nNand verifying,\nwhether its partial transpose is positive becomes ineffective.\nHowever, the ‘hit and run’ algorithm, advocated in this work, is still suitable for the case of the set of\nPPT states. For any two points inside the set it is easy to find where the line joining them hits the\nboundary, provided the dimensionality N=K2is low enough. We have found the numerical procedure\nto be stable for K\u00145. Level density for the subset of PPT states for bipartite K\u0002Ksystem is\nshown in Fig. 4. The number of random points generated, equal to 5\u0002104forN= 9and4:4\u0002105\nforN= 25, was found to give reliable results. Characteristic finite–size oscillations visible for N= 9\nbecome less pronounced for N= 25. As for this dimension the exact expression (4.1) is already close to\nthe Marchenko–Pastur distribution, our data obtained for the 5\u00025system support the conjecture that\nthe level density for the PPT states is asymptotically described by the distribution (4.6).\nObserve that the right edge of the support of PPPT\nN(x)is smaller than the upper bound for the MP\ndistribution, xmax= 4. Hence our numerical results support the conjecture that the operator norm of\nthe PPT states is asymptotically almost surely smaller than the norm of a generic state taken from the\nentire set \nN, which typically behaves as jj\u001ajj\u0019a:s:4=N.\nIn view of these numerical results, it is natural to conjecture that as N!1, the spectral distribution\nof\u001aPPTis different from the distribution obtained from an uniform sampling on all states. To prove such\na statement one could show that there exists \">0such thatjj\u001aPPTjj\u0014a:s:CPPT=N\u0014(4\u0000\")=N, as this\nwould result in differentiating mathematically the uniform distribution on all states from the uniform\ndistribution on PPT states. However, despite our efforts, we were not able to prove this result and we\nleave it as an open question.\nIt is slightly easier to deal with these sets of quantum states, which are invariant with respect to unitary\ntransformations. There is a natural notion of APPT(absolute positive partial transpose), which means\nthat a state satisfies the PPTproperty regardless of its (global) unitary evolution. In the case of 2\u00022\nsystemsthispropertyisequivalenttoabsoluteseparability[3], i.e. separabilitywithrespecttoanychoice\nof a two dimensional subspace embedded in H4, which defines both subsystems. Hence the property of\n8APPT of a given state cannot depend on its eigenvectors but is only a function of its eigenvalues [31].\nThis feature holds also for higher dimensions and in some cases the boundary of this set are known\n[32, 33]. Interestingly, although for higher dimensions the set of PPT states has much larger volume\nthen the set of separable states [34], it is conjectured that the set of absolutely separable states and\nAPPT states do coincide [35]. Furthermore, Proposition 8.2 from [36] concerning the largest eigenvalue\nof a quantum state belonging to the set of APPT states implies that the support of the level density for\nthe states of this set is asymptotically bounded by x= 3. Observe that this fact is consistent with our\nobservations concerning the generic behavior of the norm of a PPT state.\nNote that the above questions of separating distributions through their typical largest eigenvalues can\nbe partly addressed in the more general context of sampling from a uniform purification. It is known\n[8, 9] that if the ancilla space has the same dimension as the state space, this sampling method gives\nthe uniform distribution. It is also interesting to study other regimes, where the ancilla space is bigger\nthan the state space. Aubrun proved in [13] that the PPT property holds with probability 1asN!1\nif, for\">0the ancilla space is of dimension at least (4 +\")N. Intuitively, a bigger ancilla space means\nthat the sampling is more concentrated around the maximally mixed state. For even larger ancillas the\ngenerated states belong to the set of APPT states, as it was showed in [37] that the threshold is (4+\")N2\nand that this is essentially optimal. Unsurprisingly, this shows that the set of APPT states is much\nsmaller than the set of PPT states. But the problem of comparing these sampling probabilities with the\nuniform PPT distribution or the uniform APPT distribution remains difficult to achieve formally.\n5.Concluding Remarks\nIn this paper we analysed a universal algorithm to generate random points inside an arbitrary compact\nsetXinRdaccording to the uniform measure. Any initial probability measure \u0016transformed by the\ncorresponding Markov operator converges exponentially to the invariant measure \u0016\u0003, uniform in X.\nExplicit estimations for the convergence rate are derived in terms of the ratio \u0014=r=Rbetween the radii\nof the sphere inscribed inside Xand the sphere outscribed on it.\nThus the algorithm presented here can be used in practice to generate, for instance, a sample of random\nquantum states. In the case of states of a composed quantum system, one can also generate a sequence\nof random states with positive partial transpose. Sampling random states satisfying a given condition\nand analyzing their statistical properties is relevant in the research on quantum entanglement and\ncorrelations in multi-partite quantum systems. A standard approach of generating random points from\nthe entire set of quantum states with respect to the flat measure [8] and checking a posteriori, whether\nthe partial transpose of the state constructed is positive, becomes inefficient for large dimensions, as the\nrelative volume of the set of PPT states becomes exponentially small [10].\nObtained numerical results show that the level density for random states covering uniformly the subset\nof the PPT states differs considerably from the Hilbert-Schmidt level density corresponding to the entire\nset\nK2of mixed states of a bipartite system. Making use of the shifted semicircle law of Aubrun (4.3),\nwhich describes the asymptotic density of partially transposed states, and imposing the restriction that\nall the eigenvalues are non-negative we arrived at the probability distribution (4.6). This distribution\nwas compared with the level density obtained numerically with the ‘hit and run’ algorithm applied for\nthe set of PPT states for the K\u0002Ksystem forK= 3;4;5. The larger dimension the better agreement\nof the numerical data with the distribution (4.6), so is tempting to conjecture that it describes the level\ndensity of the PPT states in the asymptotic limit.\n9Acknowledgements\nWe are obliged to Uday Bhosale and Arul Lakshminarayan for their interest in the project and several\nfruitful discussions. It is a pleasure to thank Nathaniel Johnston and Ion Nechita for constructive\nremarks and for bringing to our attention Ref. [15]. The research of TS was supported by the National\nScience Center of Poland, grant number DEC- 2012/07/B/ST1/03320 and EU grant RAQUEL, while\nKŻ acknowledges a support by the NCN grant DEC-2011/02/A/ST1/00119. BC was supported by\nJSPS Kakenhi grants number 26800048 and 15KK0162, and the grant number ANR-14-CE25-0003.\nReferences\n[1] E. M. Alfsen and F. W. Shultz, Geometry of State Spaces of Operator Algebras , Birkhäuser, Boston (2003).\n[2] M. Adelman, J. V. Corbett and C. A. Hurst, The geometry of state space, Found. Phys. 23, 211 (1993).\n[3] M. Kuś and K. Życzkowski, Geometry of entangled states. Phys. Rev. A 63, 032307 (2001).\n[4] J. Grabowski, G. Marmo and M. Kuś, Geometry of quantum systems: density states and entanglement. J. Phys. A\n38, 10217-10244 (2005).\n[5] I. Bengtsson and K. Życzkowski, Geometry of Quantum States: An Introduction to Quantum Entanglement . Cam-\nbridge University Press, Cambridge (2006).\n[6] R. Horodecki, P. Horodecki, M. Horodecki and K. 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Math.\nSoc. Roum. Sci. 39, 57–115 (1937).\n[26] H.–J. Sommers, K. Życzkowski, Statistical properties of random density matrices, J. Phys. A 37, 8457 (2004).\n[27] P. B. Slater and C. F. Dunkl, Formulas for Rational-Valued Separability Probabilities of Random Induced Generalized\nTwo-Qubit States, Advances Math. Phys. 2015, 621353.\n[28] A. Sanpera, R. Tarrach and G. Vidal, Local description of quantum inseparability, Phys. Rev. A 58, 826 (1998).\n[29] Z. Puchała, Ł. Pawela, K. Życzkowski, Distinguishability of generic quantum states, Phys. Rev. A 93, 061221 (2016).\n[30] P. J. Forrester, Log-gases and Random matrices , (Princeton University Press, Princeton, 2010).\n[31] F. Verstraete, K. Audenaert, and B. DeMoor, Maximally entangled mixed states of two qubits, Phys. Rev. A 64,\n012316 (2001).\n[32] R. Hildebrand, Positive partial transpose from spectra, Phys. Rev. A 76, 052325 (2007).\n[33] N. Johnston, Separability from spectrum for qubit–qudit states, Phys. Rev. A 88, 062330 (2013).\n[34] S. J. Szarek, E. Werner, and K. Życzkowski, Geometry of sets of quantum maps: a generic positive map acting on a\nhigh-dimensional system is not completely positive, J. Math. Phys. 49, 032113-21 (2008).\n[35] S. Arunachalam, N. Johnston, and V. Russo, Is separability from spectrum determined by the partial transpose?\nQuant. Inf. Comput. 15 0694-0720, (2015).\n[36] M. A. Jivulescu, N. Lupa, I. Nechita and D. Reeb, Positive reduction from spectra Lin. Alg. Appl. 469, 276-304\n(2015).\n[37] B. Collins, I. Nechita and D. Ye, The absolute positive partial transpose property for random induced states, Random\nMatrices Theor. Appl. 1, 1250002 (2012).\nK.S: Institute of Physics, Jagiellonian University, Cracow, Poland\nE-mail address :konrad.szymanski@uj.edu.pl\nB.C.: Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-8502,\nJapan and CNRS, France\nE-mail address :collins@math.kyoto-u.ac.jp\nT.S.: Faculty of Physics and Applied Mathematics, Gdańsk University of Technology, ul. Gabriela\nNarutowicza 11/12, 80-233 Gdańsk, Poland\nE-mail address :szarek@intertele.pl\nK.Ż: Institute of Physics, Jagiellonian University, Cracow, Poland and Center for Theoretical Physics,\nPolish Academy of Sciences, Warsaw\nE-mail address :karol@tatry.if.uj.edu.pl\n11"    },    {        "title": "1612.08992v1.Generalized_density_functional_equation_of_state_for_astrophysical_simulations_with_3_body_forces_and_quark_gluon_plasma.pdf",        "content": "Generalized Density Functional Equation of State for Astrophysical\nSimulations with 3-body forces and Quark Gluon Plasma\nJ. Pocahontas Olson,1MacKenzie Warren,1, 2, 3,\u0003Matthew Meixner,4,y\nGrant J. Mathews,1, 2,zN. Q. Lan,1, 2, 5,xand H. E. Dalhed6\n1Center for Astrophysics, Department of Physics,\nUniversity of Notre Dame, Notre Dame, IN 46556\n2Joint Institute for Nuclear Astrophysics, Department of Physics,\nUniversity of Notre Dame, Notre Dame, IN 46556\n3Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824\n4Space Exploration Sector, Johns Hopkins University Applied Physics Laboratory,\nLaurel, Maryland 20723 USA\n5Hanoi National University of Education, 136 Xuan Thuy, Hanoi, Vietnam\n6Lawrence Livermore National Laboratory, Livermore, CA, 94550\n(Dated: June 29, 2021)\nWe present an updated general purpose nuclear equation of state (EoS) for use in simulations of\ncore-collapse supernovae, neutron star mergers and black hole collapse. This EoS is formulated in the\ncontext of Density Functional Theory (DFT) and is generalized to include all DFT EoSs consistent\nwith known nuclear and astrophysical constraints. This EoS also allows for the possibility of the\nformation of material with a net proton excess ( Yp>0:5) and has an improved treatment of the\nnuclear statistical equilibrium and the transition to heavy nuclei as the density approaches nuclear\nmatter density. We include the e\u000bects of pions in the regime above nuclear matter density and\nincorporate all of the known mesonic and baryonic states at high temperature. We analyze how a 3-\nbody nuclear force term in the DFT at high densities sti\u000bens the EoS to satisfy the maximum neutron\nstar constraint, however the density dependence of the symmetry anergy and the formation of pions\nat high temperatures allows for a softening of the central core in supernova collapse calculations\nleading to a robust explosion. We also add the possibility of a transition to a QCD chiral-symmetry-\nrestoration and decon\fnement phase at densities above nuclear matter density. This paper details\nthe physics, and constraints on, this new EoS and presents an illustration of its implementation\nin both neutron stars and core-collapse supernova simulations. We present the \frst results from\ncore-collapse supernova simulations with this EoS.\nI. INTRODUCTION\nTo describe the hydrodynamics of compact\nmatter, be it in heavy-ion nuclear collisions, su-\npernovae or neutron stars, an equation of state\n(EoS) is needed to relate the physics of the var-\nious state variables. In supernovae the EoS de-\ntermines the dynamics of the collapse and the\noutgoing shock, and in part determines whether\nthe remnant ends up as a neutron star or a black\n\u0003mwarren@msu.edu\nymatthew.meixner@jhuapl.edu\nzgmathews@nd.edu\nxnquynhlan@hnue.edu.vnhole. In a neutron star, it determines the max-\nimum mass, mass-radius relationship, internal\ncomposition, cooling timescales, and dynamics\nof neutron star mergers.\nThe two most commonly used equations of\nstate in astrophysical simulations are the EoS\nof Lattimer and Swesty (LS91) [1] and that of\nH. Shen et al. (Shen98) [2, 3]. The former uti-\nlizes a non-relativistic parameterization of nu-\nclear interactions in which nuclei are treated\nas a compressible liquid drop including surface\ne\u000bects. The latter is based upon a Relativis-\ntic Mean Field (RMF) theory using the TM1\nparameter set in which nuclei are calculated in\na Thomas-Fermi approximation. Subsequently,\nH. Shen et al. [4] released updates of the Shen98arXiv:1612.08992v1  [nucl-th]  28 Dec 20162\nEoS table. The \frst update [4], EoS2, increased\nthe number of temperature points as well as\nswitching to a linear grid spacing in the proton\nfraction. In the second update [4], EoS3, the ef-\nfects of \u0003 hyperons were taken into account. It\nshould also be noted that several extensions to\nthe Shen98 table have also been developed, ei-\nther by the implementation of hyperons [5] or, of\nparticular relevance to the present work, includ-\ning a mixed phase transition to a quark gluon\nplasma [6, 7].\nOver the last several years much progress has\nbeen made on other popular formulations of\nthe nuclear EoS for astrophysical simulations,\nwhich we brie\ry summarize here. The EoS\nof Hempel et al. [8] is described by a RMF\nin nuclear statistical equilibrium (NSE) for an\nensemble of nuclei and interacting nucleons.\nSteiner et al. [9] also constructed several EoSs\nto match recent neutron star observations. In\nthese models the nucleonic matter was parame-\nterized with a RMF model that treats nuclei and\nnon-uniform matter with the statistical model\nof Hempel et al. [8].\nThe EoS described here, the Notre Dame-\nLivermore (NDL) EoS, complements the two\nmost popular EoSs, in that it is formulated\nin the context of Density Functional The-\nory (DFT) rather than the liquid drop or RMF\nformalism. All three approaches are an approx-\nimation to the exact many-body problem with\nthe true strong interaction. Since DFT con-\nnects transparently with the many-body Hamil-\ntonian and can be constrained by nuclear struc-\nture [10, 11] it may be closer to the true many\nbody problem. Nevertheless, all three EoSs rep-\nresent di\u000berent approaches. Thus, one measure\nof the uncertainty in the supernova EoS is to\ncompare these three approaches. The purpose\nof the present paper is to summarize the formu-\nlation and \frst application of this general DFT\nEoS.\nMoreover, the present EoS is of particular in-\nterest in the context of modern supernova sim-\nulations. Even after decades of research the\nmechanism of core collapse supernova explo-\nsions is not yet understood in detail. Indeed,\nmost supernova simulations that impose spher-ical symmetry do not explode except for low-\nmass progenitor stars [12{14]. Thus, it is cur-\nrently thought that a successful explosion re-\nquires some other subtle e\u000bects such as neutrino\nheated convection [15] and the standing accre-\ntion shock instability (SASI) [16], or micro-\nturbulent heating behind the shock [17]. It\nis also worthy of note that it still possible to\nobtain an explosion in spherical symmetry ei-\nther by invoking a low mass progenitor [13, 14],\nby enhancing the \rux of neutrinos emanating\nfrom the core via convection below the neutri-\nnosphere [18{20], via a magnetic-rotation insta-\nbility [21], a second shock produced by a transi-\ntion to quark-gluon plasma within the nascent\nneutron star [22, 23], or a resonant oscillation\nbetween a\u0018keV sterile neutrino and an elec-\ntron neutrino [24, 25].\nIt has also been argued [21] that at least part\nof the reason for a successful spherical explosion\ncould be attributed the utilization of an EoS\nthat was su\u000eciently soft (high compressibility)\nnear nuclear matter density to produce a heated\nhigh-density proto-neutron star core (with asso-\nciated high-temperature neutrinos). However,\nconstraints from the observed masses [26, 27]\nand radii [28] of neutron stars require the ex-\nistence of a sti\u000b (low compressibility) EoS. In\nview of the importance of clarifying the contri-\nbutions of the EoS to the explosion mechanism\nit is important to update the physics and also to\ninclude the possibility of a transition to quark\ngluon plasma (QGP) at high density.\nIn this work we describe the new NDL EoS,\nthat is publicly available at www.crc.nd.edu/\n~astro/NDLEOS/ . This EoS evolves from the\noriginal formulation of Bowers and Wilson [29]\nand somewhat updated in Wilson and Math-\news [20]. The NDL EoS is updated to be con-\nsistent all available experimental nuclear matter\nconstraints and recent mass [26, 27] and radius\n[9] constraints from neutron stars. In particu-\nlar, we re-formulate this EoS in the context of a\ngeneralized DFT with a Skyrme force near the\nnuclear saturation density. We also add a tran-\nsition to QGP in the regime above the nuclear\nsaturation density. We also add a more real-\nistic transition through the sub-nuclear pasta3\nphases.\nWith regards to the DFT formulation we note\nthat there are already hundreds of DFT Skyrme\nparameterizations available (see for example\n[30{32]). Of these, only a small fraction can\nsatisfy both the nuclear structure constraints\nand properties of neutron stars [32]. Here, we\ndevelop a new generalized nuclear EoS capable\nof incorporating any or all of the Skyrme pa-\nrameterizations in realistic astrophysical simu-\nlation.s Moreover, this EoS can be easily up-\ndated as new data and/or Skyrme DFT param-\neterizations become available. We present an il-\nlustration of the \frst results of core collapse su-\npernova simulations based upon the set of DFT\nformulations that satisfy all of the nuclear and\nneutron-star mass-radius constraints. We show\nthat the these new EoSs lead to an enhancement\nin the supernova kinetic energy at early times\n(\u0018250 ms) compared to the earlier version of\nthe EoS.\nII. THE NDL EQUATION OF STATE\nDepending upon the density and temperature\nthere are a variety of matter components that\nmay or may not contribute signi\fcantly to the\nequation of state during various epochs of su-\npernova collapse, the interiors of neutron stars,\nand black hole collapse. These include photons,\nelectrons, positrons, neutrinos, mesons, excited\nmesonic and baryonic states [33], free neutrons,\nprotons, and atomic nuclei, and even the possi-\nbility of a transition to quark gluon plasma.\nAt low density and high temperatures we as-\nsume a meson gas; consisting of thermally cre-\nated, pair-produced mesons with zero chemical\npotential. In the high density, but low temper-\nature limit, pions are constrained by chemical\nequilibrium among the neutrons, protons and\nthe other baryonic states. Baryons are assigned\na non-zero chemical potential that guarantees\nbaryon number conservation. The inclusion of\nthe additional mesonic and baryonic states is\nyet another improvement appearing in this up-\ndated EoS.\nBelow nuclear matter density, the conditionsfor nuclear statistical equilibrium (NSE) are\nimposed in the NDL EoS above a tempera-\nture ofT\u00190:5 MeV. Below this temperature\nin dynamical astrophysical simulations the nu-\nclear matter is solved using a nine element re-\naction network which must be evolved during\ncollapse. Above this temperature, the nuclear\nconstituents are represented by free nucleons,\nalpha particles, and a single \\representative\"\nheavy nucleus. As nuclear matter density is ap-\nproached an approximation to the transitions\namong pasta phases is adopted that is consis-\ntent with the nuclear matter Skyrme density\nfunctional adopted at high density.\nThe high density hadronic phase of the EoS is\ntreated with parameterized Skyrme energy den-\nsity functionals. The e\u000bects of pions and other\nmesonic and baryonic resonances on the state\nvariables at high densities are also included as\nwell as a phase transition to a QGP.\nSince at the relevant densities the material is\noptically thick to photons, one can include pho-\ntons along with matter particles in the equa-\ntion of state. The electrons and positrons are\napproximated as a uniform background and are\ntreated as a non-interacting ideal Fermi-Dirac\ngas. Photons are approximated as a black-body\nand obey the usual Stefan-Boltzmann law. Neu-\ntrinos, however, are not necessarily con\fned and\nmust be transported dynamically. In astrophys-\nical simulations most matter except neutrinos\ncan be assumed to be in local thermodynamic\nequilibrium (one temperature in a zone) but\nnot necessarily in chemical equilibrium (i.e. in\nsupernovae the weak reactions have not neces-\nsarily equilibrated). The independent variables\ngenerally chosen for the equation of state are\nthen the temperature T, the matter rest-mass\ndensity\u001a(or alternatively, the number density\nn), and the net charge per baryon Ye=ne=nB.\nThe previous formulation [29] required Ye<0:5,\nbut we have removed that restriction in this new\nversion based upon the possibility [34] for pro-\nton rich ejecta above the proto-neutron star in\ncore-collapse supernovae.\nThe baryonic contribution to the NDL EoS is\ndivided into \fve regimes:\n1. Baryons below nuclear matter density and4\nnot in NSE;\n2. Baryons below nuclear matter density and\nin NSE, including the e\u000bects of pasta\nphases of nuclear matter;\n3. Hadronic matter above saturation density\nincluding pions;\n4. A phase transition to quark gluon plasma;\nand\n5. A pure quark-gluon plasma.\nThe description of matter is completely deter-\nmined by three input state variables: the den-\nsity (nin [fm\u00003]), temperature ( Tin [MeV])\nand the \\electron fraction\" ( Ye). We de\fne the\nelectron fraction and constituent number frac-\ntions as\nYe=ne\nnB(1)\nYi=ni\nnB: (2)\nwherenBis the baryon number density and ni\ndenotes the the number density of species i.\nA. Baryons Below Saturation\nand not in NSE\nBelow nuclear saturation density and above\nT\u00190:5 MeV nuclei are in NSE. However, be-\nlow this temperature the isotopic abundances\nshould be evolved dynamically. To achieve this,\nthe nuclear constituents are approximated by\na 9 element nuclear burn network consisting\nofn,p,4He,12C,16O,20Ne,24Mg,28Si, and\n56Ni [29].\nThe free energy per baryon is taken to\nbe the sum of contributions from an ideal\ngasFg[Eq. (3)] and a coulomb correction\nFc[Eq. (5)]. The ideal gas contribution is sim-\nply,\nFg=T\nmBX\niXi\"\n1\nAiln\u0012XinBmBA\ngiT3=2A5=2\ni\u0013#\n;(3)whereXiis the nuclear mass fraction, Tis the\ntemperature, Aiis the atomic mass number of\neach species, and giis the spin degeneracy. The\nindexiruns over the entire reaction network,\nandAis given by\nA=2\u00193=2\ne(mB)3=2: (4)\n[Note, that natural units ( ~=c=k= 1)\nhave been adopted here and throughout this\nmanuscript. We also maintain capital letters\nfor total energy per baryon [MeV/baryon] and\nlowercase for energy densities [MeV/fm3].]\nIn this regime the baryonic Coulomb contri-\nbution to the free energy is approximated by\nFC=\u00001\n3n1=3\nBe2hAi2=3Y2\ne; (5)\nwherehAiis the average atomic mass of the\ndynamic composition. From these relations the\nbaryonic pressure and energy per unit mass can\nbe calculated from the ideal gas thermodynamic\nrelations [20].\nPM=nBT X\niXi\nAi!\n\u00001\n9mBn4=3\nBe2hAi2=3Y2\ne;\n(6)\n\u000fM=3\n2T\nmB X\niXi\nAi!\n\u00001\n3mBn1=3\nBe2hAi2=3Y2\ne:\n(7)\nB. Baryons Below Saturation and in NSE\nWhen nuclear statistical equilibrium (NSE) is\nvalid, the baryonic nuclear material is approx-\nimated as consisting of a four component \ruid\nof free protons, neutrons, alpha particles and an\naverage representative heavy nucleus. This for-\nmulation is reasonably accurate and convenient\nin that it leads to fast analytic solutions for the\nNSE. One should exercise caution, however, [29]\nwhen considering detailed thermonuclear burn-\ning or a precise value of Yein NSE is desired.5\nIn such cases an extended NSE network should\nbe employed.\nIn the absence of weak interactions the neu-\ntron and proton mass fractions are constrained\nby charge conservation (i.e. constant electron\nfractionYe),\nX\ni(Zi=Ai)Xi=Ye; (8)\nand baryon conservation, i.e.\nX\niXi= 1: (9)The thermodynamic quantities are deter-\nmined from the free energy per baryon, which\nis given as a sum of the various constituents,\nF=Fn+Fp+F\u000b+FhAi; (10)\nwhereFnandFpare contributions from free\nneutrons and protons respectively, F\u000bis the free\nenergy of the alpha particles, and FhAiis the\nfree energy of heavy nuclei. These can each be\nexpanded [20] in terms of their various contri-\nbutions,\nFn=XBYn(\n\u000fn0W+\u000fN(1\u0000W) +3\n2T\"\np\n1 +\u00102n\u0000ln \n1 +p\n1 +\u00102n\n\f\u0010n!#)\n; (11)\nFp=XBYp8\n<\n:\u000fp0W+\u000fN(1\u0000W) +3\n2T2\n4q\n1 +\u00102p\u0000ln0\n@1 +q\n1 +\u00102p\n\f\u0010p1\nA3\n59\n=\n;; (12)\nF\u000b=X\u000b\u001a\n\u000f\u000b0W+\u000fN(1\u0000W) +T\n4ln\u0012X\u000bnmB\u000b\nT3=245=2\u0013\u001b\n; (13)\nFhAi=Fbulk+FS+FC+Fthermal (14)\nwhere the various terms in Eqs. (11) - (14) are\nde\fned as the following.\nThe mass number of the representative heavy\nnucleushAiis taken to be A= 100 ifhAi\u0015100,\nwhile forhAi<100 we approximate the density\ndependent mass of the average heavy nucleus\nas:\nhAi= 194:0(1\u0000Ye)2(1+X+2X2+3X3):(15)\nThis expression arises [35] from enforcing that\nthe nuclear surface energy be twice the Coulomb\nenergy.\nThe density parameter Xin Eq. (15) is de-\n\fned by\nX\u0011\u0012\u001a\n7:6\u00021013g=cm3\u00131=3\n: (16)In Eqs. (11) and (12) XBis the free baryon mass\nfraction, while in Eqs. (13) and (14) X\u000bandXA\nare the mass fractions of4He and the average\nheavy nucleus respectively. The quantities Yp\nandYnare the relative number fractions of free\nbaryons in protons or neutrons, respectively.\nThe quantity YAis the average Z=A for heavy\nnuclei determined by the minimization of the\nfree energy as described below. The quantity W\nin Eqs. (11)-(14) is a weighting factor that inter-\npolates between the low density and high den-\nsity regimes. It is de\fned by W= (1\u0000\u001a=\u001aN)2.\nThe transition from subnuclear to supra-nuclear\ndensity is expected to be continuous. The rea-\nson for this is that, as the density increases, the\nequilibrium continuously shifts to progressively\nheavier nuclei.\nThe quantity \u001aNis the density at which nu-\nclear matter becomes a uniform sea of nucleons.6\nIn the formulation of Bowers and Wilson [20],\nthis was found by \ftting the saturation density\nof nuclear matter [i.e. PM(n0;T= 0;Ye) = 0]\nas a function of \u001aandYe. The zero-temperature\nresult was chosen to simplify the problem of\nmaking a smooth transition between the three\nequation of state regimes. The result is\n\u001aN= 2:66\u00021014h\n1\u0000(1\u00002Ye)5=2i\ng cm\u00003:\n(17)\nWe caution, however, that the true crust-core\ntransition density is temperature dependent\nand the transition density is correlated with\nsome of the EOS properties such as the sym-\nmetry energy slope [31]. However, in supernova\ncollapse the the passage through this transition\nis rapid and thus has little observable a\u000bect on\nthe explosion. Also, the maximum neutron star\nmass constraint utilized here is not sensitive to\nthe crust-core transition as shown in Xu et al.\n[31].\nThe normal56Fe ground state is taken as the\nzero of binding energy. This is unlike most other\nequations of state for which the zero point is\nchosen relative to dispersed free nucleons. The\nreason for these choices is that it avoids the nu-\nmerical complication of negative internal ener-\ngies in the hydrodynamic state variables at low\ntemperature and density due to the binding en-\nergy of nuclei. The energy per nucleon required\nto dissociate56Fe into free nucleons is \fxed at\n\u000fp0= 8:37 MeV for protons, while for neutrons\nit is\u000fn0= 9:15 MeV [20]. Thus we take\nFbulk=XhAi\u000fN(1\u0000W): (18)\nThe last component of the Helmholtz free en-\nergy for heavy nuclei is the thermal contribu-\ntion,\nFthermal =XhAiT\nAln\u0012XAnmB\u000b\ngAT3=2A5=2\u0013\n:(19)\nwhereAis the mass number of the average\nheavy nucleus, as given in Eq. 15.\nThe quantities \u0010nand\u0010pin Eqs (12) and\n(11) are a measure of the degeneracy of the freebaryons. They are de\fned [20, 29] by\n\u0010n=B(\u001aYnXB)2=3\nT;\u0010p=B(\u001aYpXB)2=3\nT;\n(20)\nwhere the quantity B(\u001aYiXB)2=3is the en-\nergy per baryon of a zero-temperature, non-\nrelativistic ideal fermion gas and the constant\nBis\nB=3\n10\u00123\n8\u0019\u00132=3h2\nm5=3\nB: (21)\nThe dimensionless constant \fappearing in\nEqs. (12) and (11) is determined such that the\ntranslational part of fpandfnreduces to the\ncorrect non-degenerate limit ( T!1 ,\u0010i!0).\nThat is,\n3\n2T\u0014p\n1 +\u00102n\u0000ln\u00121 +p\n1 +\u00102n\n\f\u0010n\u0013\u0015\n!Tln\u0012XB\u001aYiA\nT3=2\u0013\n:(22)\nThis requirement implies\n\f=\u0012A\n2\u00132=3\u00123\neB\u0013\n= 0:781; (23)\nwhereAis given in Eq. (4).\nThe function b(Ye) in Eq. (14) is determined\nby the condition that the Coulomb contribution\nto the pressure at \u001a=\u001aNbe canceled by the\nterm proportional to b(Ye). This requires,\nb(Ye) =e2\n18\u0012hAi2\nmB\u00131\n3\u00141\n\u001aN+ 2\u0012@lnhAi\n@\u001a\u0013\n\u001aN\u0015\n:\n(24)\nThe expression for the statistical weight of\nthe heavy nucleus gAappearing in Eq. (14) is7\ntaken to be\n1\nAlngA=3\n2(\u0014\n1\u0000s\n1 +\u0012T\nTS\u00132\u0015T\nTS\n+ ln\u0014T\nTS+s\n1 +\u0012T\nTS\u0013\u0015)\n;(25)\nwhere\nTS= (8 MeV)\u0012\n1 + 2\u001a\n\u001aN\u0013\n: (26)\nC. Nuclear Pasta Phases\nThere is a great deal of interesting nuclear\nphysics in this regime at low temperatures\nin which the interplay between the Coulomb\nand surface energies lead to various forms of\n\\pasta\" nuclei, with growing mass number and\ngeometries varying from spherical to sheet-like\nto cylinder-like geometries [36]. Moreover, al-\nthough this regime is not important during the\ncollapse itself it does matter for the nascent\nproto-neutron star. This is because convection\nnear the surface and in this density regime of the\nstar can have a signi\fcant impact on the early\n(\u00180:1\u00000:5 sec) transport of neutrino \rux and\nits associated heating of material behind the\nshock. Additionally, pasta phases will have a\nsigni\fcant impact upon crust cooling timescale\nfor neutron stars [37, 38].\nTherefore, in the interest of providing a\ndeeper physical underpinning of the current\nEoS we include the transition among the pasta\nphases. A great deal of e\u000bort [39] has gone\ninto describing this interesting regime, however,\nin the spirit of the current phenomenological\nSkyrme-force approach of the current work, we\ncan follow the Wigner-Seitz cell derivation of\n[1, 36] updated to self-consistently transition\nthe current Skyrme parameters of the EoS em-\nployed here. This approach was based upon anadoption of the Skyrme interaction, but is ap-\nplicable to a broad class of density functionals\nsuch as the ones of interest here, and hence is\na natural means to extend the model developed\nhere.\nWithin the Wigner-Seitz cell one begins as\nabove by dividing the nuclear free energy into\ncontributions from the formation of very large\nbulk heavy nuclei that occupy a fraction of the\nvolume in addition to an exterior \ruid com-\nposed of neutrons, protons, and alpha particles.\nHeavy nuclei (Eq. (14)) are characterized by a\nbulk energy plus surface and coulomb energies.\nHence, for the free energy in Eq. (10), we re-\nplace the volume factor W= (1\u0000\u001a=\u001aN)2with\nW= (1\u0000u=XA)2; (27)\nwhere\nu=VA\nVc=XA\u001a\n\u001aN: (28)\nHere,VAis the volume of heavy nuclei and Vc\nis the cell volume. The nuclear volume is ex-\npressedVA= (4=3)\u0019r3\nAwithrAthe e\u000bective\nnuclear radius corrected for various shapes as\ndescribed below.\nIn this case, the surface and Coulomb terms\nin the free energy of the heavy nucleus FhAiare\ndescribed with modi\fed terms due to the ex-\notic shapes. For the formation of nuclear pasta\nphases in bulk nuclear matter in the Wigner-\nSeitz cell approximation one can express [1] the\nsum ofFS+FCduring the passage through this\ntransition as a simple analytic function of the\nvolume parameter u, charge to mass ratio YA\nfor the average nucleus, and the temperature T\nas:\nFS+FC=\f[c(u)s(u)2]1=3=nB=\fD(u)=nB;\n(29)\nwhereD(u) was deduced in Ref. [1], based upon\na \ft to the Thomas-Fermi Skyrme-force calcu-\nlations of Ref. [36]:8\nD(u) =u(u\u00001)\u0010\n(1\u0000u)D(u)1=3+uD(1\u0000u)1=3\u0011\nu2+ (1\u0000u)2+\u000bu2(1\u0000u)2(30)\nwhere,D(u)\u00111\u0000(3=2)u1=3+ (1=2)uis a\nCoulomb correction for spherical bubbles in the\nWigner-Seitz approximation, and \u000b= 0:6 is a\nparameter adjusted to optimize the \ft to the\nThomas-Fermi calculations of [36].\nThe normalization factor \fthen contains the\ndependence of the Coulomb correction upon the\ncharge-to-mass ratio YAand temperature T.\nThis can also be written analytically\n\f= 9\u0014\u0019\u001b(YA;T)2e2Y2\nAn2\n15\u00151=3\n: (31)\nHere,nis the nuclear number density, while\n\u001b(YA;T) is the temperature dependent surface\nenergy per unit area [1]. For a broad range of\ndensity functionals can be written [1]:\n\u001b(YA;T) =\u001b(0:5;0)h(T)\n\u000216 +q\nY\u00003\nA+q+ (1\u0000YA)\u00003;(32)\nwhere\u001b(0:5;0)\u00191:15 MeV fm\u00002is the sur-\nface tension of cold symmetric nuclear matter\ndeduced [1] from \fts to individual nuclei. The\ntemperature dependence of the surface tension\nis taken to diminish quartically up to a critical\ntemperature according to:\nh(T) =(\n[1\u0000(T=Tc(YA))2]2T\u0014Tc(YA)\n0 T >Tc(YA);\n(33)\nwhereTcis the critical temperature above which\nnuclear pasta phases do not exist and is re-\nlated the frequency of the giant monopole reso-\nnance [1]. Here, we express this in terms of the\nnuclear compressibility parameter Kdescribed\nin Section II E and density n,\nTc(YA) = 2:4344K1=2n\u00001=3\nBYA(1\u0000YA) MeV.\n(34)\nThe dimensionless quantity qin Eq. (32) re-lates to surface symmetry energy S0,\nq= 384\u0019r2\n0\u001b(0:5;0)=S0\u000016; (35)\nwithr0= (3=4\u0019n0)1=3is the nuclear radius pa-\nrameter here written in terms of the nuclear sat-\nuration density n0.\nThis speci\fes the transition to pasta nuclei\nin terms of W; YA, andT. What remains is to\nspecify the dependent variables WandYAin\nterms of the EoS variables \u001aandYe. This is ob-\ntained from the conditions of mass and charge\nbalance in the solution of the chemical poten-\ntials as described in the next subsection.\nThe quantities np=YpnB,nn=YnnB, and\nn\u000b=Y\u000bnBare the fractions of unbound pro-\ntons, neutrons, and \u000bparticles, respectively.\nThese quantities are determined from the min-\nimization of the free energy as described be-\nlow. This then provides a treatment of pasta\nphases consistent with the Skyrme parametriza-\ntion above the saturation density which we de-\nscribe in Section II E.\nD. Chemical Potentials\nHaving speci\fed the free energies above, the\nchemical potentials are found from the min-\nimization of the Helmholtz free energy per9\nbaryon (F),\n\u0016n=\u0012@F\n@XB\u0000Yp\nXB@F\n@Yp\u0013\n; (36)\n\u0016p=\u0012@F\n@XB+Yn\nXB@F\n@Yp\u0013\n; (37)\n\u0016\u000b= 4\u0012@F\n@X\u000b\u0013\n; (38)\n\u0016nA=\u0012@F\n@XA\u0000YA\nXA@F\n@YA\u0013\n; (39)\n\u0016pA=\u0012@F\n@XA+(1\u0000YA)\nXA@F\n@YA\u0013\n; (40)\nwhere\u0016p,\u0016nand\u0016\u000bare the chemical potentials\nof free protons, neutrons, and alpha particles.\nThe quantities \u0016nAand\u0016pAare the chemical\npotentials of neutrons and protons within heavy\nnuclei. These quantities are related by the Saha\nequation:\n2\u0016n+ 2\u0016p=\u0016\u000b (41)\n2\u0016nA+ 2\u0016pA=\u0016\u000b (42)\n\u0016nA\u0000\u0016pA=\u0016n\u0000\u0016p= ^\u0016: (43)\nThis set of conditions is su\u000ecient to specify\nthe relative mass fractions of the constituent\nspecies. In the current implementation, the\nthree chemical potential constraints [Eqs. (41)-\n(43)] combined with charge and baryon num-\nber conservation [Eqs. (8)and (9)] are solved self\nconsistently to determine the matter composi-\ntion. This leads to a 20% increase in the mass\nfraction of heavy nuclei when compared to the\noriginal approximation scheme of Ref. [29].\nE. Baryonic Matter Above Saturation\nDensity\nAbove nuclear matter density, the baryons\nare treated as a continuous \ruid. In this regime,\nthe free energy per nucleon is given in the form\nF=Fbulk(nB;Yp)+Ftherm (nB;T)+8:79 MeV,\n(44)where the addition of 8.79 MeV sets the zero for\nthe free energy to be the ground state of56Fe\nas discussed above.\nAbove the saturation density we include both\n2-body (v(2)\nij) and 3-body ( v(3)\nijk) interactions in\nthe many-nucleon system. The Hamiltonian of\nthis system is thus given by\n^H=X\ni^ti+X\ni<jv(2)\nij+X\ni<j<kv(3)\nijk; (45)\nwhere ^tiis the one body contribution while vij\nandvijkare the 2- and 3-body interactions, re-\nspectively. In the density functional approach\none can parametrize these interactions to de-\nscribe the ground-state properties of \fnite nu-\nclei and nuclear matter [40{42]. The micro-\nscopic interactions, such as meson exchange, are\nembedded in the parameters of the density de-\npendent forces.\nAmong the most widely used interactions are\nthose of the Skyrme type forces. In this ap-\nproach, the two-body potential is given in the\nform introduced by Vautherin and Brink [43].\nThat is, for most of the Skyrme potentials\nconsidered here the high density behavior can\nbe dominated by a 3-body repulsive interaction.\nThis term is taken to be a zero range force of\nthe formv123=t3\u000e(r1\u0000r2)\u000e(r2\u0000r3). If the\nassumption is made that the medium is spin-\nsaturated, which is valid for neutron star mat-\nter and nuclei [44], the three-body term is then\nequivalent to a density dependent two-body in-\nteraction [43] given by\nv(3)0\n12=1\n6t3\u0010\n1 +x3^Ps\u0011\n\u000e(r1\u0000r2)n\u001b\nB\u0012r1+r2\n2\u0013\n:\n(46)\nThis modi\fed Skyrme potential can then be\nwritten as in [45]. This modi\fcation has been\nintroduced [40] to enhance the incompressibility\nof nuclear matter at high densities. A value of \u001b\n= 1/3 is a common choice [46, 47], although this\nparameter varies in the range of \u001b\u00180:14\u00001:0\nas seen in Table I.\nThe main advantage of the Skyrme density\nfunctional is that the variables that character-\nize nuclear matter can be expressed as analytic10\nfunctions. We use TFto denote the kinetic en-\nergy of a particle at the Fermi surface\nTF=~2\n2m\u00123\u00192\n2\u00132=3\nn2=3\nB: (47)Then, calculating the expectation value of the\nHamiltonian [Eq. (45)] in a Slater determinant,\nthe energy per nucleon for symmetric nuclear\nmatter can be derived [32, 43],\nE\nA=3\n5TFH5=3+t0\n8nB[2(x0+ 2)\u0000(2x0+ 1)H2] +1\n483X\ni=1t3in\u001bi+1\nB[2(x3i+ 2)\u0000(2x3i+ 1)H2]\n+3\n40\u00123\u00192\n2\u00132=3\nn5=3\nB\u0000\naH5=3+bH8=3\u0001\n+3\n40\u00123\u00192\n2\u00132=3\nn5=3+\u000e\nB\u0014\nt4(x4+ 2)H5=3\u0000t4(x4+1\n2)H8=3\u0015\n+3\n40\u00123\u00192\n2\u00132=3\nn5=3+\r\nB\u0014\nt5(x5+ 2)H5=3+t5(x5+1\n2)H8=3\u0015\n;\n(48)\nwhere\na=t1(x1+ 2) +t2(x2+ 2); (49)\nb=1\n2[t2(2x2+ 1)\u0000t1(2x1+ 1)];(50)\nHm(Yp) = 2m\u00001[Ym\np+ (1\u0000Yp)m]: (51)\nNotice that for symmetric matter Hm(YP=\n1=2) = 1 for all m. We have included here\nmore non-standard terms, such as those involv-\ningt4;x4;t5andx5, which do appear in someparameterizations [32].\nAll quantities and coe\u000ecients for symmetric\nnuclear matter can be obtained from Eq. (48).\nThe pressure, is deduced from P=n2\nB@\n@nB\u0000E\nA\u0001\nand is given in Eq. (52). Eq. (53) gives the vol-\nume compressibility of symmetric nuclear mat-\nter. This is calculated from the derivative of\nthe pressure with respect to number density:\nK= 9\u0010\n@P\n@nB\u0011\n= 18P\nnB+ 9n2\nB@2\n@n2\nB\u0000E\nA\u0001\n. Finally,\nthe skewness coe\u000ecient, Q= 27n3\nB@3\n@n3\nB\u0000E\nA\u0001\n, is\ndeduced from the third derivative of the free\nenergy per nucleon [32] and is given in Eq. (54).11\nP=2\n5TFnBH5=3+t0\n8n2\nB[2(x0+ 2)\u0000(2x0+ 1)H2] +1\n483X\ni=1t3i(\u001bi+ 1)n\u001bi+2\nB[2(x3i+ 2)\u0000(2x3i+ 1)H2]\n+1\n8\u00123\u00192\n2\u00132=3\nn8=3\nB\u0000\naH5=3+bH8=3\u0001\n+1\n40\u00123\u00192\n2\u00132=3\n(5 + 3\u000e)n8=3+\u000e\nB\u0014\nt4(x4+ 2)H5=3\u0000t4(x4+1\n2)H8=3\u0015\n+1\n40\u00123\u00192\n2\u00132=3\n(5 + 3\r)n8=3+\r\nB\u0014\nt5(x5+ 1)H5=3+t5(x5+1\n2)H8=3\u0015\n(52)\nK= 6TFH5=3+9t0\n4nB[2(x0+ 2)\u0000(2x0+ 1)H2]\n+3\n163X\ni=1t3i(\u001bi+ 1)(\u001bi+ 2)n\u001bi+1\nB[2(x3i+ 2)\u0000(2x3i+ 1)H2] + 3\u00123\u00192\n2\u00132=3\nn5=3\nB\u0000\naH5=3+bH8=3\u0001\n+3\n40\u00123\u00192\n2\u00132=3\n(5 + 3\u000e)(8 + 3\u000e)n5=3+\u000e\nB\u0014\nt4(x4+ 2)H5=3\u0000t4(x4+1\n2)H8=3\u0015\n+3\n40\u00123\u00192\n2\u00132=3\n(5 + 3\r)(8 + 3\r)n5=3+\r\nB\u0014\nt5(x5+ 2)H5=3+t5(x5+1\n2)H8=3\u0015\n(53)\nQ=24\n5TFH5=3+9\n163X\ni=1t3i\u001bi(\u001bi+ 1)(\u001bi\u00001)n\u001bi+1\nB[2(x3i+ 2)\u0000(2x3i+ 1)H2]\n\u00003\n4\u00123\u00192\n2\u00132=3\nn5=3\nB\u0000\naH5=3+bH8=3\u0001\n+3\n40\u00123\u00192\n2\u00132=3\n(2 + 3\u000e)(5 + 3\u000e)(3\u000e\u00001)n5=3+\u000e\nB\u0014\nt4(x4+ 2)H5=3\u0000t4(x4\u00001\n2)H8=3\u0015\n+3\n40\u00123\u00192\n2\u00132=3\n(2 + 3\r)(5 + 3\r)(3\r\u00001)n5=3+\r\nB\u0014\nt5(x5+ 2)H5=3+t5(x5+1\n2)H8=3\u0015(54)\nEquations (48) and (52)-(54) completely de-\nscribe the properties of symmetric nuclear mat-\nter. Values for the coe\u000ecients can be con-\nstrained by \fxing the density of nuclear sat-\nuration, as well as imposing the observational\nconstraint that the maximum mass of a neutron\nstar must exceed 2 :01\u00060:04M\f[26, 27].\nTable I summarizes the Skyrme parameter-\nizations considered in this work. These ver-\nsions were identi\fed in Dutra et al [32] as theEoSs that satisfy all available constraints from\nproperties of nuclear matter and nuclei (e.g.\nsee review in Ref. [48]). The EoS's that sat-\nisfy the neutron star maximum mass constraint\nM\u00152:01\u00060:04 [27] are also identi\fed. In what\nfollows we will run core collapse simulations\nwith the EoS's compatible with those EoS's that\nare compatible with the maximum neutron star\nmass constraint.12\nTABLE I: Skyrme parameterizations considered in this work, taken from Ref. [32]. All models\nconsidered here have t4,x4,t5, andx5equal to zero.\nModel t0t1t2t31t32t33x0x1x2x31x32x33\u001b1\u001b2\u001b3\nGSkIa-1855.5 397.2 264.6 13858.0 -2694.1 -319.9 0.12 -1.76 -1.81 0.31 -1.19 -0.46 0.33 0.67 1.00\nGSkII -1856.0 393.1 266.1 13842.9 -2689.7 { 0.09 -0.72 -1.84 -0.10 -0.35 { 0.33 0.67 {\nKDE0v1a-2553.1 411.7 -419.9 14603.6 { { 0.65 -0.35 -0.93 0.95 { { 0.17 { {\nLNS -2485.0 266.7 -337.1 14588.2 { { 0.06 0.66 -0.95 -0.03 { { 0.17 { {\nMSL0a-2118.1 395.2 -64.0 12875.7 { { -0.07 -0.33 1.36 -0.23 { { 0.24 { {\nNRAPRa-2719.7 417.6 -66.7 15042.0 { { 0.16 -0.05 0.03 0.14 { { 0.14 { {\nSka25s20a-2180.5 281.5 -160.4 14577.8 { { 0.14 -0.8 { 0.06 { { 0.25 { {\nSka35s20a-1768.8 263.9 -158.3 12904.8 { { 0.13 -0.80 0.00 0.01 { { 0.35 { {\nSKRA -2895.4 405.5 -891. 16660.0 { { 0.08 { 0.20 { { { 0.14 { {\nSkT1a-1794.0 298.0 -298.0 12812.0 { { 0.15 -0.50 -0.50 0.09 { { 0.33 { {\nSkT2a-1791.6 300.0 -300.0 12792.0 { { 0.15 -0.50 -0.50 0.09 { { 0.33 { {\nSkT3a-1791.8 298.5 -99.5 12794.0 { { 0.14 -1.00 1.00 0.08 { { 0.33 { {\nSkxs20 -2885.2 302.7 -323.4 18237.5 { { 0.14 -0.26 -0.61 0.05 { { 0.17 { {\nSQMC650 -2462.7 436.1 -151.9 14154.5 { { 0.13 { { { { { 0.17 { {\nSQMC700 -2439.1 371.0 -96.7 13773.6 { { 0.10 { { { { { 0.16 { {\nSV-sym32 -1883.3 319.2 197.3 12559.5 { { 0.01 -0.59 -2.17 -0.31 { { 0.30 { {\naMaximum neutron star mass >2 M\f[27]\nF. Thermal Correction\nFor energetic environments such as core col-\nlapse supernovae or heavy ion collisions it is nec-\nessary to consider nuclear matter at \fnite tem-\nperature. This is addressed via a thermal cor-\nrection. For the thermal contribution to the free\nenergy per particle in bulk nuclear matter we\nfollow the approach described in Refs. [20, 49].\nWe assume a degenerate gas of the mesonic\nand baryonic states [33], as a function of tem-\nperatureTand baryon density n. Since the\nzero-temperature contribution to the free en-\nergy is already properly taken into account by\nthe Skyrme contribution, only the thermal por-\ntion needs to be added. We also assume that\nthe baryonic states are in chemical equilibrium.The expression for the thermal contribution is\nwritten:\nFthermal (nB;T) =\u000b\u0000\u000b0+1\nnB(!\u0000!0);(55)\nwhere\u000band!are a \fnite temperature \\chem-\nical potential\" and grand potential density, re-\nspectively. The quantities, \u000b0and!0are the\nzero-temperature limits of the \\chemical poten-\ntial\" and grand potential density and nBis the\nlocal baryon number density.\n\u000b0is constrained from the number density of\nbaryons and is determined by the relation\nnB=X\nigi\n2\u00192\u0000\n\u000b2\n0\u0000em2\ni\u00013=2; (56)\nwhile the zero-temperature limit of the grand\npotential density is [20]\n!0=\u0000X\nigi\n2\u00192\"\n\u000b0q\n\u000b2\n0\u0000em2\ni\u0012\n\u000b2\n0\u00005\n2em2\ni\u0013\n+3\n2em4\niln \n\u000b0+p\n\u000b2\n0\u0000em2\ni\nemi!#\n: (57)13\nIn Eqs (56) and (57) emiis an e\u000bective particle\nmass deduced from \fts to results from relativis-\ntic Bruckner Hartree-Fock theory [50]\nemi=mi\n1 +\u0018\u0011; (58)\nwhere\u0018= 0:027 and the sum over iincludes\nboth nucleons and delta particles.\nThe \fnite temperature \u000bis then found by en-\nforcing baryon number conservation and assum-ing an ideal Fermi gas,\nnB=X\nigi\n2\u00192Z1\n0(h(pi;\u000bi)\u0000h(pi;\u0000\u000bi))p2dp;\n(59)\nwhere,h(pi;\u000bi) is the usual Fermi distribution\nfunction\nh(pi;\u000bi) =1\nexp[(\u000fi\u0000\u000bi)=T] + 1; (60)\nandgiis the spin-isospin degeneracy factor.\nThe \fnite temperature grand potential den-\nsity is given as\n!baryon =\u0000X\nigiT\n2\u00192Z1\n0p2dp(ln (1 + exp [\u0000(\u000f\u0000\u000b)=T]) + ln (1 + exp [\u0000(\u000f+\u000b)=T])) (61)\n!meson = 2X\nigiT\n2\u00192Z1\n2\u0019~n1=3p2dpln (1\u0000exp (\u0000\u000f=T)) (62)\nwhere the index iruns over baryonic and\nmesonic resonances and the e\u000bective energy \u000fi\nis given by\n\u000fi=q\np2+em2\ni: (63)\nIt should be noted that \u000band\u000b0are only\nused to construct Fthermal and do not corre-\nspond to a real chemical potential, since they\nassume an ideal gas behavior. The actual chem-\nical potentials are found from derivatives of the\ntotal free energy [Eq. 44] with respect to den-\nsity as in Eqs. (36)-(40). For nuclear matter\nthey simplify to:\n\u0016n=F+nB\u0012@F\n@nB\u0013\nT;Yp\u0000Yp\u0012@F\n@Yp\u0013\nT;nB\n(64)\n\u0016p=F+nB\u0012@F\n@nB\u0013\nT;Yp+Yn\u0012@F\n@Yp\u0013\nT;nB\n(65)whereYn+Yp= 1.\nG. Thermodynamic State Variables\nOnce the thermal contribution to the free en-\nergy is constructed, the thermodynamic quan-\ntities can be calculated. Of particular interest\nare the total internal energy ( E), the total pres-\nsure (P), the entropy per baryon ( S) and the\nadiabatic index \u0000. The total internal energy is\ncalculated from the free energy:\nE=F\u0000T\u0012@F\n@T\u0013\nnB;Ye+Ee+E\r;(66)\nwhereEeis the electron energy determined by\nnumerically integrating over Fermi-Dirac dis-\ntributions and E\ris the photon energy con-\ntribution determined from the usual Stefan-\nBoltzmann law.\nThe pressure is calculated from14\nP=n2\nB(@F=@nB). It is important to note that\nthe thermal contribution to the pressure is not\nthe simple form of !0\u0000!as one would expect\nfrom the usual application of the thermody-\nnamic potential, but is given by a slightly more\ncomplicated form\nPtherm =!0\u0000!+nB@\n@nB[!\u0000!0]:(67)\nThis is due to the fact that the e\u000bective mass\n[cf. Eq. (58)] is density dependent. If this de-\npendence were removed we would recover the\nusual form of the pressure from the thermody-\nnamic potential.\nWe thus determine the total pressure from\nthe free energy\nP=n2\nB\u0012@Fbulk\n@nB\u0013\n+Ptherm +Pe+P\r;(68)\nwhere again the electron and photon contri-\nbutions are determined from the energies dis-\ncussed previously. The entropy per baryon\nin units of Boltzmann's constant is given by\na simple derivative S=\u0000(@F=@T ), and the\nadiabatic index is given by the usual form\n\u0000 = [@lnP=@ln (\u001a)]. We note, however, that\nthe neutrino pressure and energy contributions\nmust be independently solved and accounted for\nin a simulation.\nThe density dependence of the symmetry\nenergy beyond saturation is highly uncertain.\nFig. 1 shows the symmetry energy versus den-\nsity for various parameter sets found in Ta-\nble I and Ref. [32]. Note that, for these mod-\nels, the non-standard terms t4; t5; x4;andx5\nare all zero. The symmetry energy at satura-\ntion is known [32, 51] to lie within the range\nS0= 30\u000035 MeV [52]. For all parameter sets\nconsidered here, this constraint is met. How-\never, for many Skyrme models the symmetry\nenergy either saturates at high densities, or in\nthe worst case becomes negative. This results\nin a negative pressure deep inside the neutron\nstar core. For this work, we will only consider\nparameter sets with a fairly sti\u000b symmetry en-\nergy.\n0 0.1 0.2 0.3 0.40.50.6 0.7 0.8Density (fm  )01020304050607080Symmetry Energy (MeV)B&WGSkIGSkIIKDE0v1LNSMSL0NRAPRSka25s20Ska35s20SKRASkT1SkT2SkT3Skxs20SQMC650SQMC700SV-sym32-3\nFIG. 1: Symmetry energy versus density for\nvarious Skyrme parameter sets listed in\nTable I. Existing constraints at nuclear\nsaturation density ( n0= 0:16\u00060:01fm\u00003) [52]\nare indicated by the gray shaded region\n(S0= 30\u000035 MeV) and constraints below\nsaturation density [53] are indicated by the\nblue shaded region. The behavior of the\nsymmetry energy above saturation density is\nunconstrained. Many parameterizations\nbecome negative at supranuclear densities,\nwhich leads to erroneous behavior in neutron\nstars. (Color available online.)\nIII. PIONS IN THE NUCLEAR\nENVIRONMENT\nPions are a crucial ingredient in simulations\nof core collapse supernovae. The interior tem-\nperatures can reach to a signi\fcant fraction\n(\u001850 MeV) of the pion rest mass so that some\nthermally produced pions exist in the tail of the\nBoltzmann distrobution. The existence of these\nthermally produced pions has the e\u000bect of soft-\nening the EoS. This increases the central densi-\nties neutrino luminosities and thereby enhances\nthe explosion.\nThe contribution to the hadronic EoS from\nthe lightest mesons (i.e. the pions) has been\nconstrained [54] from a comparison between rel-\nativistic heavy-ion collisions and one-\ruid nu-\nclear collisions. In that work the formation and\nevolution of the pions was computed in the con-\ntext of Landau-Migdal theory [50] to determine15\nthe pion e\u000bective energy and momentum. In\nthis approach the pion energy is given by a dis-\npersion relation [50]\n\u000f2\n\u0019=p2\n\u0019+em2\n\u0019; (69)\nwhereem\u0019is the pion \\e\u000bective mass\" de\fned\nto be\nem\u0019=m\u0019p\n1 + \u0005 (\u000f\u0019;p\u0019;nB): (70)\nFollowing [49] and [55] the polarization pa-\nrameter \u0005 can be written,\n\u0005 (\u000f\u0019;p\u0019;nB) =p2\n\u0019\u00032(p\u0019)\u001f(\u000f\u0019;p\u0019;nB)\nm2\u0019\u0000g0m2\u0019\u00032(p\u0019)\u001f(\u000f\u0019;p\u0019;nB):\n(71)\nwhere the denominator is the Ericson-Ericson-\nLorentz-Lorenz correction [56]. The quantity\n\u0003\u0011exp(\u0000p2\n\u0019=b2) withb= 7m\u0019, is a cuto\u000b that\nensures that the dispersion relation [Eq. (69)]\nasymptotically approaches the high momentum\nlimit,\n\u000f1\u0011\u000f(p\u0019!1;nB) =q\nm2\n\u0001+p2\u0019\u0000mN:\n(72)\nFollowing [56] we take the polarizability to be\n\u001f(\u000f\u0019;p\u0019;nB) =\u00004a\u000f1nB\n\u000f21\u0000\u000f2\u0019; (73)\nwherea= 1:13=m2\n\u0019. This form for the polar-\nizability ensures that the e\u000bective pion mass is\nalways less than or equal to the vacuum rest\nmassm\u0019.\nA key quantity in the above expressions is\nthe Landau parameter g0. This is an e\u000bective\nnucleon-nucleon coupling strength. To ensure\nconsistency with observed Gamow-Teller tran-\nsition energies a constant value of g0= 0:6 was\ndeduced in [55]. However, in [54], Monte-Carlo\ntechniques were used to statistically average a\nmomentum dependent g0with particle distribu-\ntion functions. It was found that g0varies lin-\nearly with density and is approximately given\nby\ng0=g1+g2\u0011 ; (74)where\u0011\u0011nB=n0. A value of g1= 0:5 was\nchosen to be consistent with known Gamow-\nTeller transitions. A value for g2was then\nobtained [54] by optimizing \fts to a range of\npion multiplicity measurements obtained at the\nBevlac [57]. These data were best \ft for a value\nofg2= 0:06.\nThe pions are assumed to be in chemical equi-\nlibrium with the surrounding nuclear matter.\nWe consider the pion-nucleon reactions:\np$n+\u0019+; n$p+\u0019\u0000: (75)\nThis leads to the following relations among the\nchemical potentials for neutrons, protons, and\npions\n\u0016p=\u0016n+\u0016\u0019+; \u0016n=\u0016p+\u0016\u0019\u0000:(76)\nThese equilibrium conditions let us express the\npion chemical potentials in terms of the neutron\nand proton chemical potentials: ^ \u0016\u0011\u0016n\u0000\u0016p=\n\u0016\u0019\u0000=\u0000\u0016\u0019+. Using the de\fnitions of \u0016nand\n\u0016pfrom Eqs. (64) - (65), the expressions for the\npion chemical potentials are found to be\n\u0016\u0019\u0000=\u0000\u0016\u0019+=\u00001\nnB@F\n@Yp: (77)\nFor a given temperature ( T) and number den-\nsity (nB) the pion number densities are given by\nthe standard Bose-Einstein integrals\nni=Z1\n0p2\n2\u00192dp\ne(\u000f\u0019\u0000\u0016i)=T\u00001; (78)\nwhereiis forf\u0019+;\u0019\u0000;\u00190g, and\u000f\u0019is given by\nEq. (69). Note that the \u00190chemical potential\nis taken to be zero, since these particles can be\ncreated or destroyed without charge constraint.\nThe charge fraction per baryon for the\ncharged pions is de\fned as Y\u0019\u0000=n\u0019\u0000=nB.\nFrom Eq. (77) we can calculate the pion num-\nber densities from the pion chemical potentials.\nThen, electric charge conservation gives,\nYe=Yp\u0000Y\u0019\u0000+Y\u0019+: (79)\nThus, we can solve Eq. (79) for the unknown16\nquantityYp.\nOnceYpis determined, the pionic energy den-\nsities and partial pressures can be calculated\nfrom\nEi=Z1\n0p2dp\n2\u00192\u000f\u0019\nexp [(\u000f\u0019\u0000\u0016i)=T]\u00001;(80)\nand\nPi=Z1\n0p2dp\n2\u00192(1=3)p(@\u000f\u0019=@p)\nexp [(\u000f\u0019\u0000\u0016i)=T]\u00001:(81)\nIn the high temperature, low density limit the\npionic mass approaches the bare pion mass and\nother pionic excitations are become relevant.\nHence, to properly treat other density and tem-\nperature regimes, we also include all mesonic\nand baryonic states using bare masses as dis-\ncussed in Sec. II F.\nWe note that this treatment of the pions is\npreferable to including a condensate, as was\ndone in Refs. [58] and [59]. In Refs. [58] and [59],\npions were treated as a non-interacting Bose\ncondensate. In our treatment, pion-nucleon\ncouplings are accounted for.\nIV. QCD PHASE TRANSITION\nIt is generally expected [60] that for suf-\n\fciently high densities and/or temperature,\na transition from hadronic matter to quark-\ngluon plasma (QGP) can occur. Recent\nprogress [61] in lattice gauge theory (LGT)\nhas shed light on the transition to a QGP\nin the low baryon-chemical-potential, high-\ntemperature limit. It is now believed that at\nhigh temperature and low density a decon\fne-\nment and chiral symmetry restoration occur si-\nmultaneously at the crossover boundary. In par-\nticular, at low density and high temperature,\nit has been found [61] that the order param-\neters for decon\fnement and chiral symmetry\nrestoration changes abruptly for temperatures\nofT= 145\u0000170 MeV [62, 63]. However, nei-\nther order parameter exhibits the characteristic\nchange expected from a \frst order phase tran-\nsition. An analysis of many [64, 65] thermo-dynamic observables con\frms that the transi-\ntion from a hadron phase to a high temperature\nQGP is a smooth crossover.\nHowever, at low density the hadron phase can\nbe approximated as a pion-nucleon gas, while\nthe QGP phase can be approximated as a non-\ninteracting relativistic gas of quarks and glu-\nons [66]. Equating the pressures in the hadronic\nand QGP phases, the critical temperature Tc\nfor the low density transition can be approxi-\nmated [66] as:\nTc\u0019(gq\u0000gh)\u00001=4\u001290\n\u00192\u00131=4\nB1=4; (82)\nwhere the statistical weight gqfor a low-density\nhigh-temperature QGP gas with three relativis-\ntic quarks is gq\u001951:25, whilegh\u001917:25 was\nfound for the hadronic phase by summing over\nall known meson data.\nAdopting the lattice gauge theory results [61]\nthat 145<\u0018Tc<\u0018170 MeV, then implies [66]\nthat a reasonable range for the QCD vacuum\nenergy is 165 <\u0018B1=4<\u0018240 MeV. This pro-\nvides an initial range for the QCD vacuum en-\nergy to be adopted in this work. In Section V C,\nwe will further constrain this parameter by re-\nquiring that the maximum mass of a neutron\nstar exceed 2 M\f[26, 27].\nAnother parameter that impacts the thermo-\ndynamic properties of the system is the strong\ncoupling constant \u000bs. For this manuscript we\nadopt a value of \u000bs= 0:33 as this is a rep-\nresentative value for the energy regime under\nconsideration [33].\nA transition to a QGP phase during the\ncollapse can have a signi\fcant impact on the\ndynamics and evolution of the nascent proto-\nneutron star. In Ref. [67] it was shown that a\n\frst order phase transition to a decon\fned QGP\nphase resulted in the formation of two distinct\nbut quickly coalescing shock waves. More re-\ncently, it has been shown [68] that if the transi-\ntion is \frst order, and global conservation laws\nare imposed, then the two shock waves can be\ntime separated by as much as \u0018150 ms. Neu-\ntrino light curves showing such temporally sepa-\nrated spikes might even be resolvable in modern17\nterrestrial neutrino detectors [68]. Moreover,\nthe arrival of the second shock can signi\fcantly\nenhance the explosion.\nThe observation of M > 2M\fneutron stars\n[26, 27], however, highly restricts the possibility\nof a \frst order phase transition to a quark gluon\nplasma taking place inside the interiors of stable\ncold neutron stars. Nevertheless, a transition\nto quark-gluon plasma would always occur for\ninitial stellar masses >\u001820M\fthat result in\nfailed supernova events, because every phase of\nmatter must be traversed during the formation\nof stellar mass black holes. Even if neutron stars\ndo not have pure- or mixed- quark-gluon plasma\ninteriors, this transition to QGP may have an\nimpact [69] on the neutrino signals during black\nhole formation in addition to its possible impact\non core-collapse supernovae.\nA. The Quark Model\nFor the description of quark-gluon plasma\nwe use a bag model with 2-loop corrections\n[60], and construct the EoS from a phase-space\nintegral representation over scattering ampli-\ntudes. We allow for the possibility of a coex-\nistence mixed phase in a \frst order transition,\nor a simple direct crossover transition. In the\nhadronic phase the thermodynamic state vari-\nables, are calculated from the Helmholtz free\nenergyF(T;V;nB) as described in the previous\nsections. However, it is convenient to compute\nthe QGP phase in terms of the grand poten-\ntial, \n(T;V;\u0016 ). Both descriptions are equiva-\nlent and are related by a Legendre transform:\n\n =F\u0000P\ni\u0016iNi. Adopting the convention of\nLandau and Lifshitz [70], the thermodynamic\npotential can be written in terms of the parti-tion function Zas\n\n =\u00001\n\flnZ ; (83)\nwhere\f= 1=T. In the Feynman path inte-\ngral formulation the partition function is rep-\nresented as a functional integral of the expo-\nnential of an e\u000bective action integrated over all\n\felds [71].\nOnce the grand potential is speci\fed, the\nstate variables can be easily deduced, e.g.\nP=\u0000\u0012@\n@V\u0013\nT;\u0016; (84)\nn=\u00001\nV\u0012@\n@\u0016\u0013\nV;T: (85)\nThe grand potential for the quark-gluon\nplasma summed over each \ravor itakes the\nform:\n\n =X\ni(\ni\nq0+ \ni\nq2) + \ng0+ \ng2+BV (86)\nwhereq0andg0denote the 0th-order bag model\ngrand potentials for quarks and gluons, re-\nspectively, while q2andg2denote the 2-loop\ncorrections. The \ffth term BV is the usual\nQCD vacuum energy. In most calculations,\nsu\u000ecient accuracy is obtained by using \fxed\ncurrent algebra masses (e.g. mu\u0018md\u00180,\nms\u001895\u00065 MeV). For this work we choose a\nbag constant B1=4= 165\u0000240 MeV as noted\nabove.\nThe quark contributions to the grand poten-\ntial are given [60] from phase-space integrals\nover Feynman amplitudes [71]:18\n\ni\nq0=\u00002NcTVZ1\n0d3p\n(2\u0019)3h\nln\u0010\n1 +e\u0000\f(Ei\u0000\u0016i)\u0011\n+ ln\u0010\n1 +e\u0000\f(Ei+\u0016i)\u0011i\n(87)\n\ni\nq2=\u000bs\u0019NgV\"\n1\n3T2Z1\n0d3p\n(2\u0019)3Ni(p)\nEi(p)+Z1\n0d3p\n(2\u0019)3d3p0\n(2\u0019)31\nEi(p)Ei(p0)[Ni(p)Ni(p0) + 2]\n\u0002\"\nN+\ni(p)N+\ni(p0) +N\u0000\ni(p)N\u0000\ni(p0)\n(Ei(p)\u0000Ei(p0))2\u0000(p\u0000p0)2+N+\ni(p)N\u0000\ni(p0) +N\u0000\ni(p)N+\ni(p0)\n(Ei(p) +Ei(p0))2\u0000(p\u0000p0)2##\n;(88)\nwhereNcis the number of colors, and Ngis the\nnumber of gluons ( Ng= 8). The N\u0006\nidenote\nthe quark Fermi-Dirac distributions:\nN\u0006\ni(p) =1\ne\f(Ei(p)\u0007\u0016i)+ 1: (89)\nThe one- and two-loop gluon and ghost con-\ntributions to the thermodynamic potentials can\nbe evaluated in a similar fashion to that of the\nquarks.\n\ng0=2NgTVZ1\n0d3p\n(2\u0019)3ln\u0010\n1\u0000e\u0000\fjpj\u0011\n=\u0000\u00192\n45NgT4: (90)\n\ng2=\u0019\n36\u000bsNcNgT4: (91)\nFor massless quarks, Eqs. (87-88) are easily\nevaluated [60] to give\n\ni\nq0=\u0000NcV\n6\u00127\u00192\n30T4+\u00162\niT2+\u00164\ni\n2\u00192\u0013\n(92)\n\ni\nq2=Ng\u000bsV\n8\u0019\u00125\u00192\n18T4+\u00162\niT2+\u00164\ni\n2\u00192\u0013\n:(93)\nFor the massive strange quark Eq. (87) can be\neasily integrated. However, Eq. (88) cannot be\nintegrated numerically, due to the divergences\ninherent in the integral. We therefore, approx-\nimate [60] the two-loop strange quark contri-\nbution with the zero mass limit. This may\nover estimate the contribution due to a \fnitestrong coupling constant. However, sincethe\nquark mass is relatively small compared to its\nchemical potential, this is a reasonable approx-\nimation.\nB. Conservation Constraints in the Mixed\nPhase\nThe matter deep inside the core of a collaps-\ning star consists of a multi-component system\nconstrained by the conditions of both charge\nand baryon number conservation. All thermo-\ndynamic quantities in a \frst order quark-hadron\nphase transition vary in proportion to the vol-\nume fraction [ \u001f\u0011VQ=(VQ+VH)] throughout\nthe mixed phase regime, where VQis the\nvolume of material composed of quark-gluon\nplasma and VHis the volume composed of\nhadronic matter.\nFor the description of a \frst order phase tran-\nsition we utilize a Gibbs construction. In this\ncase the two phases are in equilibrium when the\nchemical potentials, temperatures, and pres-\nsures are equal. For the description of the phase\ntransition from hadrons to quarks, this con-\nstruction can be written\n\u0016p= 2\u0016u+\u0016d (94)\n\u0016n= 2\u0016d+\u0016u (95)\n\u0016d=\u0016s (96)\nTH=TQ (97)\nPH\u0000\nT;Ye;f\u0016H\nig\u0001\n=PQ\u0010\nT;Ye;f\u0016Q\nig\u0011\n;(98)19\nwheref\u0016H\nig\u0011f\u0016n;\u0016p;\u0016\u0019;\u0016e;\u0016\u0017gandf\u0016Q\nig\u0011\nf\u0016u;\u0016d;\u0016s;\u0016e;\u0016\u0017g.\nThe Gibbs construction ensures that a uni-\nform background of photons and leptons exists\nwithin the di\u000bering phases. Therefore, the con-\ntribution from the photon, neutrino, electron,\nand other lepton pressures cancel out in phase\nequilibrium. Also, with this choicethe two con-\nserved quantities, charge and baryon number,\nvary linearly in proportion to the degree of com-\npletion of the phase transition (or volume frac-\ntion\u001f), i.e.,\nnBYe= (1\u0000\u001f)nH\nBYH\nc+\u001fnQ\nBYQ\nc (99)\nnB= (1\u0000\u001f)nH\nB+\u001fnQ\nB; (100)\nwhere we have de\fned YH\nc=Yp+Y\u0019+\u0000Y\u0019\u0000\nandnQ\nBYQ\nc= 1=3 (2nu\u0000nd\u0000ns). The inter-\nnal energy and entropy densities likewise vary\nin proportion to the degree of phase transition\ncompletion,\n\u000f= (1\u0000\u001f)\u000fH+\u001f\u000fQ(101)\ns= (1\u0000\u001f)sH+\u001fsQ: (102)\nV. RESULTS AND COMPARISONS\nIn this section we compare properties of the\nnew NDL EoS with the two most commonly em-\nployed equations of state used in astrophysical\ncollapse simulations as well as the original EoS\nof Bowers and Wilson.\nFig. 2 shows the total internal energy as a\nfunction of local proper baryon density. We\ncompare the Lattimer and Swesty EoS [1] with\nK0= 220 MeV, the Shen EoS [2, 3], and the\nNDL EoS with several choices of Skyrme pa-\nrameterization as labeled from Ref. [32]. These\nplots correspond to a \fxed electron fraction and\ntemperature of Ye= 0:3 andT= 10 MeV. A\nsteep rise in the energy per baryon at high den-\nsities occurs for larger values of the compress-\nibility as expected.\nSimilarly, Fig. 3 depicts the pressure vs.\ndensity for the Shen EoS [4], the Lat-\ntimer and Swesty EoS [1], and several Skyrme\n0 0.1 0.2 0.3 0.40.50.6 0.7 0.8Density (fm  )0100200300Energy per baryon (MeV)L&SShenGsKIKDE0v1LNS\n-3FIG. 2: The energy per particle as a function\nof density comparing the Lattimer and Swesty\nEoS (with compressibility of K0= 220 MeV),\nthe Shen EoS, and the NDL EoS with several\nSkyrme parametrizations from Table I. (Color\navailable online)\nparameterizations of the NDL EoS. The Shen\nEoS consistently leads to higher pressure. This\nresults from the use of the TM1 parameter set\n[2, 3] that contains relatively high values for\nboth the symmetry energy at saturation and\nthe nuclear compressibility, typical of RMF ap-\nproaches.\nA. Pion E\u000bects on the EoS\nThe solution to the pion dispersion relation,\nEq. (69), does not lead to conditions within the\nproto-neutron star necessary to generate a pion\ncondensate. The pions considered here are ther-\nmal pionic excitations calculated from the pion\npropagator in Eq. (71). In very hot and dense\nnuclear matter the number density of pionic ex-\ncitations is greatly enhanced by the \u0019N\u0001 cou-\npling [55]. Hence, it becomes energetically fa-\nvorable to form pions in the nuclear \ruid when\nthe chemical potential balance shifts from elec-\ntrons to negative pions. This allows the charge\nstates to equilibrate with these newly formed\npions. Since the pions are assumed to be in20\n0 0.1 0.2 0.3 0.40.50.6 0.7 0.8Denisty (fm  )050100150200250300Pressure (MeV/fm  )L&SShenGsKIKDE0v1LNS\n-3-3\nFIG. 3: Pressure as a function of density\ncomparing the Lattimer and Swesty EoS (with\ncompressibility of K0= 220 MeV), the Shen\nEoS, and the NDL EoS with several Skyrme\nparameterizations as labeled from Table I.\n(Color available online)\nchemical equilibrium with the surrounding nu-\nclear \ruid, this has a profound e\u000bect on the pro-\nton fraction within the medium, particularly for\nlow electron fractions (see Fig. 4).\nThe pion charge fraction as a function of\nbaryon number density is shown in Fig. 5. For\na low \fxed Ye, the charge fraction of negative\npions can actually become greater than the elec-\ntron fraction, and the negative pions essentially\nreplace the electrons in equilibrating the charge.\nFrom the solution to the pion chemical poten-\ntials [Eq. (77)], one can see that as the density\nincreases, negative pions are created. This is\ndue to the chemical potential constraints and\nthe dispersion relation [Eq. (69)]. At the same\ntime the number density of positively charged\npions remains negligible due to the fact that\nthey have a negative chemical potential. The\npion chemical potential [Eq. (77)] increases lin-\nearly with respect to density but decreases lin-\nearly with respect to Yp, due to its dependence\non the slope of the free energy with respect\ntoYp. Therefore, for high electron fractions\nthe pion chemical potential will remain small\nand charge equilibrium can be maintained solely\n0 0.2 0.4 0.6 0.8Density (fm  )00.050.10.150.20.250.3Y  = 0.1Y  = 0.2Y  = 0.3\n-3pppFIG. 4: The proton fraction above nuclear\nsaturation showing the e\u000bects of pions in a hot\ndense astrophysical environment with\nT= 10 MeV, such as occurs in core collapse\nsupernovae. For small electron fractions more\npions are created due to the dependence of the\nchemical potential on the isospin asymmetry\nparameterI.\namong the electrons and protons.\nIt should be noted, however, that as treated\nhere, pions would not exist in the ground state\ncon\fguration of a cold neutron star. As the\ntemperature approaches zero the pionic e\u000bects\ndiminish, until only the nucleon EoS contributes\nto the neutron star structure.\nIn the hot dense medium of supernovae, how-\never, these pions tend to soften the hadronic\nEoS since they relieve some of the degeneracy\npressure due to the electrons. We have found\nthat the reduction in pressure is relatively in-\nsensitive to the temperature of the medium and\nis lowered by a few percent for all representa-\ntive temperatures found in the supernova envi-\nronment, as shown in Fig. 6.\nThis will a\u000bect SN core collapse models since\nit allows collapse to higher densities and tem-\nperatures without violating the requirement\nthat the maximum neutron star mass exceed\n2:01\u00060:04M\ffor cold neutron stars [27].21\n0 0.2 0.4 0.6 0.8Density (fm  )00.050.10.150.20.25Charge fraction-3YeYpYπ-\nFIG. 5: Pion charge fraction versus density at\na temperature T= 10 MeV using the GsKI\nSkyrme parameter set. At a density of about\nnB\u00190:45 fm\u00003, the pion charge fraction\nexceeds the electron fraction of the medium.\nB. Hadron-QGP Mixed Phase\nThe constraint of global charge neutrality ex-\nploits the isospin restoring force experienced\nby the con\fned hadronic matter phase. The\nhadronic portion of the mixed phase becomes\nmore isospin symmetric than the pure hadronic\nphase because charge is transferred from the\nquark phase to the hadron phase in equilibrium.\nFig. 7 shows the charge fractions of the mixed\nphase and hadronic phases for the GsKI Skyrme\nparameterization at T= 1 MeV and a bag con-\nstant ofB1=4= 190 MeV. From this we see that\nthe internal mixed phase region of a hot, proto-\nneutron star contains a positively charged re-\ngion of nuclear matter and a negatively charged\nregion of quark-gluon plasma until a density\nofn0\u00181:3 fm\u00003. The presence of the isospin\nrestoring force causes the thermal pionic con-\ntribution to the state variables to be negligible.\nThis is due to the dependence of the pion chem-\nical potential on the isospin asymmetry param-\neter (1\u00002Yp). As the hadronic phase becomes\nmore isospin symmetric, the pion chemical po-\ntential remains small compared to its e\u000bective\nmass.\n0 0.2 0.4 0.6 0.8Density (fm  )050100150200250Pressure (MeV fm  )With pionsWithout pions\n-3-3FIG. 6: Pressure versus baryon number\ndensity atT= 10 MeV showing a few percent\nreduction in the pressure at high densities due\nto the presence of pions. This \fgure was made\nwith the GsKI parameter set. (Color available\nonline)\n0123Density (fm  )-0.4-0.200.20.4Y  Y  \n-3hq\n\u0000=0.1\u0000=0.3\u0000=0.5\u0000=0.7\u0000=0.9\nFIG. 7: Charge fractions of the mixed quark\nand hadronic phase. Due to the redistribution\nof charge, the hadronic phase becomes isospin\nsymmetric even exceeding Yp>0.5. This has\nthe e\u000bect of lowering the symmetry energy and\nthus reducing the pressure in the hadronic\nphase. This \fgure was made using the GSkI\nSkyrme parameterization at T= 1 MeV and a\nbag constant of B1=4= 190 MeV.22\nSince the system contains two conserved\nquantities, electric charge and baryon number,\nthe coexistence region cannot be treated as a\nsingle substance, but must be evolved as a com-\nplex multi-component \ruid. It is common in\nNature to have global conservation laws and not\nnecessarily locally conserved quantities. Hence,\nwithin the Gibbs construction the pressure is a\nmonotonically increasing function of density.\nFig. 8 shows pressure versus density for var-\nious values of Yeand the GsKI Skyrme pa-\nrameterization, through the mixed phase region\ninto a phase of pure QGP. One of the features\nshown is that, as the density increases to about\n2-3 times the saturation density the slope of\nthe pressure versus density decreases as one en-\nters the mixed phase. Note that in a simple\ncrossover there is no mixed phase so that the\nonset of the QCD phase causes an immediate\njump toward the asymptotic behavior shown on\nFig. 8.\nHowever, in a simple crossover transition the\nEoS at \frst becomes sti\u000b then asymptotes to\na simple \u0000\u00194=3 EoS. The reason for this\nstraightforward to explain. In the low temper-\nature limit for Nf\ravors of massless quarks,\nEq. (92) leads to [67] the following approximate\nexpressions for the chemical potential, baryon\ndensity and pressure\n\u0016= 3\u00123\u00192\nNf\u00131=3\u0014\n1 +2\u000bs\n3\u0019\u0015\nn1=3+B (103)\nn=9\n4\u00123\u00192\nNf\u00131=3\u0014\n1 +2\u000bs\n3\u0019\u0015\nn4=3+B (104)\nP=3\n4\u00123\u00192\nNf\u00131=3\u0014\n1 +2\u000bs\n3\u0019\u0015\nn1=3\u0000B(105)\nThe adiabatic index is then\n\u0000 =n\nP\u0012@P\n@n\u0013\n=4\n3P+B\nP: (106)\nFrom this one can see that immediately after\na crossover transition when the bag pressure iscomparable to the baryonic pressure the adi-\nabatic index is sti\u000b \u00182. It then asymptotes\ntoward \u0000 = 4 =3 as the pressure increases.\nThe e\u000bect on the adiabatic index for a mixed\nphase is shown in Fig. 9. Here, one can see that\nfor for a mixed phase the EoS softens abruptly\nupon entering the mixed phase due to the fact\nthat increasing density leads to a larger vol-\nume fraction of QGP rather than an increase in\npressure. If this occurs while forming a proto-\nneutron star, its evolution will be a\u000bected as \u0000\nfalls below the stability point of \u0000 <4=3 and a\nsecond collapse can ensue.\n0123Density (fm  )050010001500Pressure (MeV/fm  )Y  = 0.1Y  = 0.25Y  = 0.4\n-33eee\nFIG. 8: Pressure as a function of baryon\nnumber density through the mixed phase\ntransition for T= 1 MeV and the GSkI\nSkyrme parameterization and a bag constant\nofB1=4= 190 MeV. The EoS softens\nsigni\fcantly upon entering the mixed phase at\nnB\u00180:6 fm\u00003due to the larger number of\navailable degrees of freedom.\nAnother feature seen in Figs. 8 and 9 is the\nYedependence of the onset density of the mixed\nphase. Fig. 10 shows a phase diagram indicat-\ning the mixed phase transition temperature as\na function of density for three values of Ye. Pi-\nons were included in the making of this \fgure.\nFor higher temperatures the onset happens at\nlower densities as would be expected. However,\nfor high electron fractions ( Ye\u00180:3) such as\nthose that can be found deep inside the cores of23\n00.511.522.53Density (fm  )11.21.41.61.822.22.4Adiabatic indexT = 10 MeVT = 25 MeVT = 50 MeV\n-3𝚪 = 4/3\nFIG. 9: \u0000 as a function of baryon number\ndensity showing the softening of the EoS as it\nenters the mixed phase regime. The EoS\npromptly sti\u000bens as it exits the mixed phase\ninto the pure quark-gluon plasma phase due to\nlosing the extra degrees of freedom supplied by\nthe hadrons. The horizontal line indicates\n\u0000 = 4=3. Systems with \u0000 >4=3 will be stable.\nThis \fgure was made using the GsKI Skyrme\nparameterization and a bag constant of\nB1=4= 190 MeV.\na proto-neutron star, the transition density re-\nmains quite high nB\u00180:7 fm\u00003. We note that\nsimilar phase diagrams have been made (e.g.\n[58, 59, 72, 73]) indicating that the phase di-\nagram is quite model dependent.\nThe supernova simulations of Refs. [67, 68]\nshow that, as the interior of the PNS reaches\nthe onset of the mixed phase, the equation of\nstate softens considerably and a secondary core\ncollapse ensues. The matter then sharply sti\u000b-\nens upon entering the pure quark phase and\na secondary shock wave can be generated. As\nthis shock catches up to the initially stalled ac-\ncretion shock, a more robust explosion ensues.\nHowever, their simulations assumed a Bag con-\nstants ofB1=4\u0014165 MeV, since they did not\nenforce the M > 2M\fmaximum neutron star\nmass limit, and thus the mixed phase had a\nmuch lower onset density ( nonset\u00180:1 fm\u00003)\nthan what we have found here.\n0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Density (fm  )020406080Temperature (MeV)Y  = 0.1Y  = 0.25Y  = 0.4MixedphasePurehadronicPurequark\n-3eeeFIG. 10: Density-temperature phase diagram\nshowing the density range of the mixed phase\ncoexistence region for three values of Ye(solid\nline forYe= 0:1, dashed line for Ye= 0:25 and\ndash-dotted line for Ye= 0:4), including pions.\nThe onset of the mixed phase is indicated by\nthe set of curves on the left, while the curves\non the right show the completion of the mixed\nphase. For low temperatures it is seen that the\nonset density is highly Yedependent.\nC. Neutron Stars with QGP Interiors\nAs stated previously, the observations [26, 27]\nof a 1:97\u00060:04M\fand a 2:01\u00060:04M\fneu-\ntron star have ruled out many soft EoSs includ-\ning many hyperonic models [51]. We explore\nhere how neutron stars with QGP interiors may\nbe constrained by this consideration. To do so,\nwe enforcen+p+ebeta equilibrium in hadronic\nmatter (and u+d+eequilibrium in quark-gluon\nplasma).\nFig. 11 compares the neutron star mass ra-\ndius relation for the GSKI Skyrme parameter\nset of Table I. Using the range of bag constants\ndetermined by Eq. (82), we \fnd that a \frst or-\nder phase transition to a QGP can be consistent\nwith the high maximum neutron star mass con-\nstraint [26, 27] the GsKI Skyrme parameteriza-\ntion in the NDL EoS. From Fig. 11 we deduce\nthat a bag constant B1=4>190 MeV is required\nto satisfy the maximum neutron star mass con-24\nstraint. This imposes a low baryon density tran-\nsition temperature of Tc>150 MeV, which is in\nthe low end of the range of crossover tempera-\ntures allowed from LGT [61]. Hence, all allowed\nvalues of the Bag constant inferred from LGT\nare consistent with the neutron star mass con-\nstraint in this formulation. For our purpose we\nwill adopt B1=4= 190 MeV (corresponding to\nTc\u0018150 MeV).\nD. Neutron Stars without QGP Interiors\nFig. 12 compares the neutron star mass ra-\ndius relation for the Skyrme parameter sets in\nTable I without a transition to QGP. Note that\nnot all Skyrme parameter sets accommodate a\nmaximum neutron star mass \u00152:01\u00060:04M\f\n[26, 27]. These we disregard in supernova sim-\nulations.\n8 9 10 11 12 13 1415Radius (km)00.511.522.5Mass (M  )NoneB  = 180 MeVB  = 190 MeVB  = 200 MeVB  = 210 MeV⦿Atoniadis et al (2013)\nCausality\n1/41/41/41/4\nFIG. 11: Neutron star mass-radius relation for\nvarious values of the bag constant B1=4using\nthe GSkI parameter set. We \fnd that a \frst\norder phase transition is consistent with the\nmaximum mass neutron star measurement for\nour adopted value of B1=4= 190 MeV.\nHorizontal lines show 2 :01\u00060:04 M\f\nmeasurement from Ref. [27].25\n9 10 11 12 13 141516Radius (km)00.511.522.53Mass (M  )GsKIGsKIIKDE0LNSMSL0NRAPRSka25s20Ska35s20SKRASkT1SkT2SkT3Skxs20SQMC650SQMC700SV-sym32LS180LS220LS375Shen⦿\nAntoniadis (2013)Causality\nFIG. 12: Mass-radius relation for the NDL EoS with the Skyrme parameterizations from Table I.\nThe horizontal lines indicate the 2 :01\u00060:04M\fmaximum neutron star mass constraint from\nRef. [27]. The grey shaded region shows the causality constraint and the blue shaded region shows\nthe 2\u001bbounds from Ref. [28]. Note that not all of these curves satisfy the maximum mass or\nradius constraints. (Color available online)\nVI. SUPERNOVA SIMULATIONS\nTo demonstrate the viability of the NDL\nEoS for astrophysical simulations, we have\nrun a series of core-collapse supernova simu-\nlations. We have utilized University of Notre\nDame/Lawrence Livermore National Labora-\ntory supernova model [20, 29], a spherically\nsymmetric general relativistic hydrodynamic\nsimulation with neutrino transport via multi-\ngroup \rux limited di\u000busion. We present herethe explosion dynamics of simulations run with\nthe Skyrme models in Table I that satisfy the\nmass maximum neutron star >2 M\fcon-\nstraint. For this study we do not consider the\ntransition to QGP. That we leave to a subse-\nquent paper.\nAs an example, Fig. 13 shows the radial evo-\nlution versus time post-bounce of mass elements\nin a simulation using the NDL EoS with the\nGSkI Skyrme parameters. For ease of compar-\nison with other simulations in the lliterature,26\nthis simulation was run using the 20 M \fpro-\ngenitor model of Ref. [74]. As one can see in this\n\fgure the late time neutrino heating at t\u0018200\nms leads to an explosion. Indeed, the expansion\nof the neutrino heated bubble is quite similar\nto the simulation in [20] based upon the Bower\nand Wilson EoS. Hence, the sti\u000ber EoS has not\ndiminished the explosion.\n-0.05 0 0.05 0.1 0.15 0.2 0.25\nTime post-bounce (s)106107108109Radius (cm)\nFIG. 13: Radius versus time post-bounce for\nthe NDL EoS with the GsKI Skyrme\nparameter.\nTo explore the impact of the new EoSs on\nthe explosion, Fig. 14 shows kinetic energy ver-\nsus time post-bounce for the various Skyrme pa-\nrameter sets compared to the Bowers and Wil-\nson equation of state as a point of reference.\nThe early evolution from core bounce ( tpb= 0\ns) to shock stagnation ( tpb\u00180:2 s) is relatively\nunchanged for di\u000berent Skyrme models, but dif-\nfers from the evolution of the Bowers and Wil-\nson equation of state. Even though the neutrino\nluminosity is greater for t>0:26 s in the Bow-\ners and Wilson EoS, this does not signi\fcantly\na\u000bect the explosion. This di\u000berence will, how-\never, have an a\u000bect on the subsequent r-process\nnucleosynthesis as will be explored in a subse-\nquent paper.\nAlthough the Bowers and Wilson EoS is\nsofter and leads to a higher neutrino luminosity\nat later times the kinetic energy of the explosion\nwith the Skyrme models is greater for t>0:2 s.However, the kinetic energy of supernova sim-\nulations with di\u000berent Skyrme parameters di-\nverge after tpb\u00180:2 s. These e\u000bects can be re-\nlated to di\u000berences in the neutrino luminosities,\nand thus in the e\u000eciency of neutrino reheating\nof the shock as illustrated in Fig. 15a-c.\nFig. 15a shows the electron neutrino luminos-\nity versus time post-bounce, while Figs. 15b-c\nshow the electron anti-neutrino and \u0016;\u001cneu-\ntrino luminosities, respectively. As is expected,\nthe neutrino luminosities of the various simula-\ntions are similar until tpb\u00180:2 s, after which\nthe di\u000berent properties of the Skyrme parame-\nter sets lead to di\u000berences in the electron neu-\ntrino luminosity. The Skyrme parameter sets\nthat result in higher electron neutrino luminosi-\nties also result in higher kinetic energies and\nshorter explosion timescales, as is expected. For\nall three neutrino \ravors, however there is a\nbump in neutrino luminosity at t\u00180:23 s. We\nattribute this to a softening of the core by the\nformation of thermal pions. This leads to an\nenhanced neutrino luminosity and a more ener-\ngetic reheated shock as shown in Fig. 14.\nThere is a general correlation among Skyrme\nparameter sets in that those which result result\nin the highest maximun neutron star mass (and\nare relatively \\sti\u000b\") lead to lower explosion en-\nergies. However, the Bowers and Wilson equa-\ntion of state, which is too soft to meet the mod-\nern neutron star mass constraint, has the lowest\nkinetic energy of all of the simulations shown.\nWe attribute this at least in part to di\u000berences\nin the density dependence in the symmetry en-\nergy and the formation of pions. Both of these\ncause the the Skyrme DFT EoSs to soften rela-\ntive to the Bowers and Wilson EoS at the high-\nest densities and temperatures. However, we\ncaution that there is not one nuclear physics\nparameter that describes the relative \\softness\"\nor \\sti\u000bness\" of the EoS at the highest tem-\nperatures and densities, as the response of the\nnuclear matter relies on an interplay among the\nsymmetry energy, compressibility, etc. Never-\ntheless, it is clear that di\u000berences in nuclear\nproperties, such as the symmetry energy, result\nin di\u000berent behavior for the Skyrme parame-\nter sets, particularly at supra-nuclear densities.27\nThis leads to divergences at later times in the\nneutrino luminosities and evolution of the su-\npernova simulation.\nWe leave a more detailed analysis of the EoS\ndependence of core-collapse supernova, includ-\ning the possible phase transition to quark-gluon\nplasma, to a later publication. Nevertheless,\nwe have shown here that the NDL EoS can beused to successfully simulate core-collapse su-\npernovae. Moreover, we show that the new EoS\nparameter sets presented here lead to an even\nmore robust explosion than the softer Bowers\nand Wilson EoS. This we believe is due in part\nto the critical roles of the density dependence of\nthe symmetry energy and thermal pion forma-\ntion in the core.\n0 0.05 0.1 0.15 0.2 0.25\nTime post-bounce (s)105010511052Kinetic energy (ergs)B&W\nGSkI\nKDE0v1\nMSL0\nNRAPR\nSka25s20\nSka35s20\nSkT1\nSkT2\nSkT3\nFIG. 14: Kinetic energy versus time post-bounce for the NDL EoS with Skyrme parameters found\nin Table I and the Bowers and Wilson (B&W) equation of state. (Color available online)\nVII. CONCLUSION\nWe have discussed a new equation of state\nfor astrophysical applications based upon a den-\nsity functional theory approach at nuclear mat-\nter density. We have shown that it is com-\nplementary to the most frequently employednuclear equations of state for astrophysics:\nShen et al. [2{4], which is based upon the RMF,\nand the Lattimer-Swesty EoS [1], which utilizes\na liquid drop model approach. We have adopted\na set Skyrme DFT parameterizations that are\nconsistent with all known constraints on nuclei,\nnuclear matter, and observed properties of neu-28\ntron stars and pulsars. We made a \frst explo-\nration of their e\u000bect on the dynamics of core\ncollapse supernovae and \fnd that the Skyrme\nEoSs all lead to a somewhat more robust explo-\nsion.\nA \frst order phase transition to a QGP phase\nwas also added to the EoS in the context of a\nGibbs construction. Applying the constraints\nfrom LGT for the range of low-baryon-density\ncrossover temperatures, we were able to match\nthe known constraints of the current maximum\nneutron star mass measurement for a bag con-\nstantB1=4\u0015190 MeV if the \fnite baryon den-\nsity transition if \frst order. On the other hand,\nif the high baryon-density QCD transition is a\nsimple crossover, then the neutron star mass\nconstraint can be easily accommodated for any\nvalue of the bag constant. In future work, we\nplan to address the possible e\u000bects of a transi-\ntion to QGP in CCSNe and failed supernovae\nand also explore possible e\u000bects of a color su-\nperconducting phase [75, 76].\nWe have also shown that the NDL EoS can\nlead to observable consequences in the late time\nneutrino luminosities. Uncertainties in nuclear\nproperties, such as the symmetry energy, cause\nthe di\u000berent Skyrme parameter sets to result in\ndiverging evolution at late times ( >250 ms) in\nthe supernova simulations. This work con\frms\nthat, in the a core-collapse explosion paradigm,\nthe equation of state may produce observable\ne\u000bects in the neutrino light curve, shock dy-\nnamics, and heavy element nucleosynthesis both\nin core-collapse supernovae and black hole for-\nmation in failed supernova events. The con-\nsequences of this new NDL EoS for the dy-namics of core collapse supernovae, including a\nQCD transition and the impact on nucleosyn-\nthesis, will be further explored in forthcoming\nmanuscripts.\nAdditionally, we have several future updates\nand improvements planned for the NDL EoS, in-\ncluding the inclusion of hyperons, an improved\ntreatment of the nuclear pasta phases, and an\nimproved treatment of the thermal component\nat supra-nuclear densities [77].\nWith the availability of the Notre Dame-\nLivermore (NDL)-DFT Equation of State, there\nis now another EoS which allows for an investi-\ngation of the dependence of the equation of state\nin astrophysical simulations { be it supernova\ncore-collapse, neutron star mergers or black hole\nformation. 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Oliviera,\nNuc.Phys.B 199(2010).\n[76] T. do Carmo and G. Lugones, Physica A 392\n(2013).\n[77] C. Constantinou, B. Muccioli, M. Prakash,\nand J. Lattimer, Phys.Rev.C 92, 025801\n(2015).31\n00.050.10.150.20.25Time post-bounce (s)105110521053Luminosity (ergs/s)Bowers & WilsonSkxs20SKRAGsKIGsKIIKDE0MSL0NDL2NRAPRSka25Ska35SkT1SkT2SkT3SQMC650SQMC700SVsym⌫e\n(a) Electron neutrino luminosity\n00.050.10.150.20.25Time post-bounce (s)105110521053Luminosity (ergs/s)Bowers & WilsonSkxs20SKRAGsKIGsKIIKDE0MSL0NDL2NRAPRSka25Ska35SkT1SkT2SkT3SQMC650SQMC700SVsym¯⌫e\n(b) Electron antineutrino luminosity\n00.050.10.150.20.25Time post-bounce (s)105110521053Luminosity (ergs/s)Bowers & WilsonSkxs20SKRAGsKIGsKIIKDE0MSL0NDL2NRAPRSka25Ska35SkT1SkT2SkT3SQMC650SQMC700SVsym⌫µ⌧\n(c)\u0016;\u001cneutrino luminosity\nFIG. 15: Neutrino luminosities versus time post-bounce for the NDL EoS with Skyrme parameters\nfound in Table I and the Bowers and Wilson (B&W) equation of state. (Color available online)"    },    {        "title": "1701.05646v1.Effect_of_nuclear_saturation_parameters_on_possible_maximum_mass_of_neutron_stars.pdf",        "content": "arXiv:1701.05646v1  [astro-ph.HE]  20 Jan 2017Effect of nuclear saturation parameters on possible maximum mass of neutron stars\nHajime Sotani1,∗\n1Division of Theoretical Astronomy, National Astronomical\nObservatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588 , Japan\n(Dated: June 17, 2021)\nIn order to systematically examine the possible maximum mas s of neutron stars, which is one of\nthe important properties characterizing the physics in hig h-density region, I construct neutron star\nmodels by adopting phenomenological equations of state wit h various values of nuclear saturation\nparameters for low-density region, which are connected to t he equation of state for high-density\nregion characterized by the possible maximum sound velocit y in medium. I derive an empirical\nformula for the possible maximum mass of neutron star. If mas sive neutron stars are observed,\nit could be possible to get a constraint on the possible maxim um sound velocity for high-density\nregion.\nPACS numbers: 04.40.Dg, 21.65.Ef\nI. INTRODUCTION\nNeutron stars are good candidates for investigating physics unde r extreme conditions. The density inside a neutron\nstar is significantly over the nuclear saturation density, ρ0. This is one reason why the neutron star structure is not\nyet fixed, i.e., the determination of the equation of state (EOS) for neutron star matter is quite difficult (for high-\ndensity region) in the terrestrial nuclear experiments. Therefor e, the observationsof a neutron star itself and/or of the\nphenomena associated with neutron stars could provide opportun ities for obtaining a constraint on the EOS and/or\nan imprint ofthe physics for high-density region. So far, there are severalattempts to extrapolate the EOS for neutron\nstarmatterfrom the densityaroundsaturationtomuch higherde nsities(e.g., [1, 2]). Moreover,the discoveriesof2 M⊙\nneutron stars [3, 4], where some of soft EOSs have been ruled out, impacted the field. In particular, the appearance\nof hyperons in high-density region might be a crucial problem, becau se EOSs with hyperons are generally soft and\ndifficult to support 2 M⊙.\nThe possible maximum mass of neutron stars is one of the important p roperties characterizing the physics in higher\ndensity region. In Ref. [5], the maximum mass of neutron stars is der ived asM≃6.8M⊙, preparing the EOS\nconstructed in such a way that the stiffest EOS allowed from the cau sality for high-density region, i.e., dp/dρ= 1, is\nconnected to the EOS given for lower density region at an appropria te transition density. Here, pandρare pressure\nand energy density (not a number density), and the transition den sity is adopted around 1014g/cm3. It is also\ndiscussed that the maximum mass could become larger with the EOS wh ich is softer for low-density region and stiffer\nfor high-density region [6]. So, since the stiffness of EOS is associate d with the sound velocity via v2\ns=dp/dρ, the\nmaximum mass should depend strongly on the possible maximum sound v elocity,vmax\ns, for high-density region. With\nrespect to the value of vmax\ns, the theoretical maximum value must be 1, which comes from the cau sality, while there\nis also a conjecture that vmax\nsshould be less than 1 /√\n3 [7]. Although this conjecture may be still uncertain, it could\nbe better to consider in the range of 1 /√\n3≤vmax\ns≤1, for discussing the dependence of possible maximum mass on\nvmax\ns. Eventually, the exact value of vmax\nswould be constrained from the observations and/or theoretical a rguments.\nOn the other hand, the properties for lower density region are rela tively well constrained from the terrestrialnuclear\nexperiments. In particular, the nuclear saturation parameters a re important for expressing lower density region. It is\npractically known that the neutron star constructed with the cen tral density lower than 2 ρ0can be described nicely\nwith parameters constructed as the combination of the incompres sibility of the symmetric nuclear matter, K0, and\nthe so-called slope parameter of nuclear symmetry energy, L, viaη= (K0L2)1/3[8, 9]. Therefore, it is expected that\nthe maximum mass of neutron stars should also depend on η, whereηhas been already constrained in some range\nvia the constraints on K0andL[10–12] and will be further constrained in the future.\nNow, it is considered that the possible maximum mass of neutron star s must depend on the two parameters, i.e.,\nvmax\nsandη. In order to see such a dependence, I systematically examine the n eutron star models by adopting the\nphenomenological EOS for lower density region with various values of η, which are connected to the EOS for higher\n∗Electronic address: sotani@yukawa.kyoto-u.ac.jp2\nTABLE I: Saturation parameters in OI-EOS and an auxiliary pa rameter η≡(K0L2)1/3.\ny(MeV fm3)K0(MeV)L(MeV)η(MeV)\n−220 180 52.2 78.9\n−220 230 73.4 107.4\n−220 280 97.5 138.6\n−220 360 146.1 197.3\n−350 180 31.0 55.7\n−350 230 42.6 74.7\n−350 280 54.9 94.5\n−350 360 76.4 128.1\n−600 230 23.7 50.6\n−600 280 30.1 63.4\n−600 360 40.9 84.4\ndensity regioncharacterizedby vmax\ns. I remarkthat the possiblemaximum massmust depend on the trans itiondensity\nwhere the EOSs for lower and higher density regions are connected . According to the result in Ref. [5], the possible\nmaximum mass can be inversely proportional to the square root of t he transition density. Meanwhile, the properties\nof nuclear matter for ρ/lessorsimilar2ρ0are relatively known experimentally and predicted well theoretically. In fact, one could\nexpect that the non-nucleonic components do not appear below ∼2ρ0, and that the uncertainties from three-nucleon\ninteractions in EOS for pure neutron matter do not become significa nt below ∼2ρ0[13]. Thus, since the EOS above\n∼2ρ0is more uncertain, I adopt the transition density to be 2 ρ0as in Ref. [7]. Hereafter I adopt the units of c= 1,\nwherecdenotes the speed of light.\nII. EOS AND SATURATION PARAMETERS\nFor any EOSs, the bulk energy per nucleon of uniform nuclear matte r at zero temperature can be expanded around\nthe saturation point of symmetric nuclear matter, for which the nu mber of proton is equal to that of neutron, as a\nfunction of the baryon number density nband neutron excess α, as discussed in Ref. [14]:\nw=w0+K0\n18n2\n0(nb−n0)2+/bracketleftbigg\nS0+L\n3n0(nb−n0)/bracketrightbigg\nα2, (1)\nwherew0andK0are the saturation energy and incompressibility at the saturation d ensity,n0, of symmetric nuclear\nmatter, while S0andLare associated with the density dependent nuclear symmetry ener gy.w0,n0, andS0, which\nare absolute values at the saturation point, are relatively constra ined well via terrestrial nuclear experiments, owing\nto the nuclear saturation. Meanwhile, since K0andLchange rapidly at the saturation point, one has to obtain\nexperimental data in a wide range of densities around the saturatio n point. Thus, it is more difficult to fix the values\nofK0andLvia the terrestrial experiments. For this reason, I focus on K0andLas parameters characterizing EOS.\nIn practice, to systematically analyze the dependence of neutron star properties on the saturation parameters K0\nandL, I adopt the phenomenological EOS proposed by Oyamatsu and Iida [15, 16]. This EOS is constructed in\nsuch a way that the energy of uniform nuclear matter reproduces to the form as Eq. (1) in the limit of nb→n0\nandα→0 for various values of y≡ −K0S0/(3n0L) andK0. For given K0andy, the other saturation parameters\nn0,w0, andS0are determined to fit the empirical data for masses and radii of sta ble nuclei [15, 16]. Hereafter, I\ncall this phenomenological EOS as OI-EOS. In particular, I focus on the parameter K0,L, andyin the range of\n180≤K0≤360 MeV, 0 < L <160 MeV, and y <−200 MeV fm3, which can reproduce the mass and radius data for\nstable nuclei well and effectively cover even extreme cases [15]. The concrete parameter sets adopted in this paper are\nshown in Table I, where ηis an auxiliary parameter defined as η= (K0L2)1/3[8]. I remark that the low-mass neutron\nstar models where central density is up to ρc= 2ρ0can be described well with the parameter ηindependently of the\nnuclear theoretical models [8, 9].\nOn the other hand, several EOSs have been suggested for the de nsity region higher than ∼2ρ0, which are based\non the different nuclear theories, interactions, and components. The theoretical constraints on EOS are only that the\nsound speed should be less than the speed of light (causality), and t hat the sound speed should be more than zero\n(thermodynamics stability). So, the stiffest EOS satisfying the the oretical constraints can be expressed in the density3\nregion of ρ > ρt, such as\np=ρ−ρt+pt, (2)\nwhereρtis a transition density and ptis the pressure determined at ρ=ρtwith the EOS for lower density region.\nAdopting this type of EOS for high-density region and connecting to the Harrison-Wheeler EOS for lower density at\nρt= 4.6×1014g/cm3, the maximum mass of neutron star is expected as Mmax≃3.2M⊙[5, 18]. However, since the\nstellar properties in the density region of ρ/lessorsimilar2ρ0strongly depend on η[8, 9], the maximum mass of the neutron star\ncould also depend on ηeven if one fixes the transition density ρt.\nIn addition, it has been conjectured that the sound velocity inside t he star should be smaller than the speed of\nlight in vacuum divided by√\n3 [7]. With this conjecture, the stiffest EOS for higher density region can be expressed\nasp= (ρ−ρt)/3 +pt. Since this EOS becomes softer than the EOS given by Eq. (2), the p ossible maximum\nmass becomes smaller. In practice, the neutron star mass was disc ussed with this conjecture for ρt= 2ρ0, where the\npossible maximum mass is ∼2M⊙[7]. So, if the neutron star more massive than ∼2M⊙were to be discovered, this\nconjecture may not be good. In fact, a candidate of massive neut ron star has been discovered in a neutron star and\nwhite dwarf binary system, where the mass of neutron star is estim ated asM= (2.74±0.21)M⊙[17].\nIn any way, the possible maximum sound velocity inside the star, which is associated with the stiffness of EOS,\nmust affect the determination of maximum mass of neutron stars. T hus, in order to examine the dependence of the\npossible maximum sound velocity inside the star (or the stiffness of EO S) on the maximum mass of neutron star, I\nconsider a general formula of EOS given by\np=α(ρ−ρt)+pt, (3)\nwhereαis an parameter associated with the possible maximum sound velocity in side the star, i.e., vmax\ns=√α[18].\nFor this examination, I adopt the OI-EOS for lower density region up toρt= 2ρ0, i.e.,ptis the pressure determined\nwith OI-EOS at the transition density ρt, and I adopt the EOS given by Eq. (3) for ρ >2ρ0. I remark that I simply\nconnect the EOS for lower and higher density regions at the transit ion density as in Ref. [7]. Thus, the EOS is almost\ncontinuous, but the sound velocity is not continuous at the transit ion density. In this paper, I focus on αin the range\nof 1/3≤α≤1. Then, I will see the dependence of the maximum mass on ηandα.\nIII. POSSIBLE MAXIMUM MASS\nThe spherically symmetric neutron star models are constructed by integrating the Tolman-Oppenheimer-Volkoff\nequation together with the appropriate EOS. As an example, in Fig. 1 , I show the relation between the stellar mass\nand radius for the cases of α= 1/3, 0.6, and 1 with η= 50.6, 74.7, and 107 .4, where open marks denote the maximum\nmasses for various EOS models. From this figure, I find that the max imum mass strongly depends on the possible\nmaximum sound velocity inside the star, while the dependence on ηis relatively weak. Additionally, the filled marks\nin the figure denote the local maximum of the stellar radius for variou s EOS models, which tells us that the local\nmaximum radius becomes larger with α.\nTo see the dependence of the maximum mass on η, in Fig. 2 I plot the maximum mass predicted from the various\nvalues of ηfor the cases of α= 1/3, 0.6, and 1. From this figure, one can observe that the maximum mass w ith fixed\nvalue ofαis well fitted as a linear function of η, such as\nMmax\nM⊙=a1+a2/parenleftBigη\n1 MeV/parenrightBig\n, (4)\nwherea1anda2are coefficients in the linear fitting, depending on the value of α. In Fig. 2, the linear fitting given\nas Eq. (4) for α= 1/3, 0.6, and 1 are shown with the solid, dashed, and dotted lines, respect ively.\nWith respect to the value of η, by adopting fiducial values of 30 /lessorsimilarL/lessorsimilar80 MeV [11] and K0= 230±40 MeV [12],\none can get a plausible range for ηas 55.5/lessorsimilarη/lessorsimilar120 MeV. This plausible range of ηis shown in Fig. 2 as the stippled\nregion, while the observations of neutron star masses, i.e., M= (1.97±0.04)M⊙[3] andM= (2.01±0.04)M⊙[4],\nare also shown in the same figure. To explain the observations of neu tron star masses, the case with α= 1/3, which\ncomes from the conjecture of Ref. [7], seems to be marginal with t he plausible range of η. In practice, in order to\nexplain the lower limit of neutron star mass of PSR J0348+0432, i.e., M= 1.97M⊙,ηshould be larger than ∼100\nMeV, which leads to the constraint of L/greaterorsimilar66 MeV with adopting the canonical value of K0= 230 MeV.\nIn the similar way to the discussion about the maximum mass, I addition ally find that the radius for the neutron\nstar with maximum mass with fixed αcan be well described as a linear function of ηas shown in Fig. 3. Thus, I can\nget a linear fit, such as\nR\n1 km=b1+b2/parenleftBigη\n1 MeV/parenrightBig\n, (5)4\n10 11 12 130123\nR (km)M/M!\n107.4\n74.7\n50.6! = 1/3 \n11 12 13\nR (km)107.4\n74.7\n50.6! = 0.6\n11 12 13 14\nR (km)107.4\n74.7\n50.6! = 1.0\nFIG. 1: Mass and radius relation for various EOSs for lower de nsity region with η= 50.6 (dashed line), 74.7 (dotted line),\nand 107.4 (solid line). The left, middle, and right panels co rrespond to different sound velocities for higher density re gion,\ni.e.,α= 1/3, 0.6, and 1, respectively. The open marks correspond to the stellar models with maximum mass for various EOS\nmodels, while the filled marks correspond to the stellar mode ls with local maximum radius. For reference, the stellar mod els\nconstructed with the central density ρc= 2ρ0denote by the double circles.\nxxxxxx\nxxxx40 80 120 160 2001.82.02.22.42.62.83.03.2\n! (MeV)Mmax/M!(220, 180)\n(220, 230)\n(220, 280)\n(220, 360)\n(350, 180)\n(350, 230)\n(350, 280)\n(350, 360)\n(600, 230)\n(600, 280)\n(600, 360)\" = 1/3\" = 0.6\" = 1\nFIG. 2: The expected maximum masses for various EOS models ar e shown with different marks, while the solid, dashed, and\ndotted lines respectively denote the fitting formula given b y Eq. (4) for α= 1/3, 0.6, and 1. In the label, I show the values of\nthe saturation parameters for the adopted EOS, such as ( −y,K0). The region between the horizontal dot-dash-lines denote s the\nmass observation of PSR J1614-2230 [3], while the horizonta l shaded region denotes the mass observation of PSR J0348+04 32\n[4]. The stippled region denotes a plausible range for ηdetermined from the current terrestrial nuclear experimen ts.\nwhereb1andb2are coefficients in the linear fit, which depend on the value of α.\nFurthermore, I plot the coefficients in the linear fit [Eqs. (4) and (5) ], i.e.,a1,a2,b1, andb2, as a function of αin\nFigs. 4 and 5. Then, I find that such coefficients can be well fitted as a function of αwith the functional forms given\nby\na1(α) =−0.356/α+2.445+0.767α, (6)\na2(α) = (0.806/α+1.098−0.393α)×10−3, (7)\nb1(α) =−0.883/α+11.548+1.388α, (8)\nb2(α) = (6.008/α+4.834−0.824α)×10−3. (9)\nIn Figs. 4 and 5, the marks denote the numerical values in linear fittin g [Eqs. (4) and (5)], while the dashed lines are\nplotted with using Eqs. (6) – (9). Now, I can get the fitting formulae expressing the maximum mass and radius of\nneutron star with maximum mass as a function of ηandα.\nFinally, adopting the linear fitting expressed by Eq. (4) together wit h Eqs. (6) and (7), one can obtain the possible\nmaximum mass of neutron star predicted with the plausible value of ηfor various values of α, which corresponds\nthe region between the solid lines in Fig. 6. In the same figure, I put th e observations of neutron star mass for\nJ1748−2021B [17] and for PSR J0348+0432 [4] with the shaded regions. As m entioned the before, the observation\nof PSR J0348+0432 is possible to explain even for α= 1/3 (orvmax\ns= 1/√\n3), but to explain the observation of5\nxxxxxx\n40 80 120 160 2001011121314\n! (MeV)R (km)(220, 180)\n(220, 230)\n(220, 280)\n(220, 360)\n(350, 180)\n(350, 230)\n(350, 280)\n(350, 360)\n(600, 230)\n(600, 280)\n(600, 360)\nFIG. 3: The expected radii for the stellar models with maximu m mass constructed with various EOS models are shown with\ndifferent marks, while the solid, dashed, and dotted lines re spectively denote the fitting formula given by Eq. (5) for α= 1/3,\n0.6, and 1. In the label, I show the values of the saturation pa rameters for the adopted EOS, such as ( −y,K0). The stippled\nregion denotes a plausible range for ηdetermined from the current terrestrial nuclear experimen ts.\n0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.01.52.02.53.0\n1.52.02.53.03.5\n!a1\n103 a2 a1\na2\nFIG. 4: The coefficients a1anda2in Eq. (4) as a function of α. The dashed lines are fitting given by Eqs. (6) and (7).\nJ1748−2021B (even though this may be rather uncertain), the value of αshould be at least lager than ∼0.57, i.e.,\nvmax\ns/greaterorsimilar0.75c. If this result is to be believed, one may need to introduce some mech anism with which the EOS for\nhigher density region makes stiff, for example introducing a vector in teraction. In any way, with future observations\nof massive neutron stars, one could put a constraint on the possib le maximum sound velocity inside a star.\n0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.09101112\n1.01.21.41.61.82.02.2\n!b1b1\nb2\n102 b2\nFIG. 5: The coefficients b1andb2in Eq. (5) as a function of α. The dashed lines are fitting given by Eqs. (8) and (9).6\nxxxxxxxx\nxxxx0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.01.82.02.22.42.62.83.03.2\n!Mmax/M! J1748-2021B\nPSR J0348+0432\nFIG. 6: The maximum mass of neutron star predicted with the pl ausible value of η, i.e., 55.5/lessorsimilarη/lessorsimilar120 MeV, is shown as a\nfunction of αin the region between the solid lines. The shaded regions cor respond to the observations of neutron star mass for\nJ1748−2021B [17] and for PSR J0348+0432 [4].\nIV. CONCLUSION\nTo describe the EOS ofneutron starmatter, the nuclear saturat ionparametersare important for low-densityregion,\nwhile the possible maximum sound velocity could be a key parameter for high-density region. In fact, the neutron star\nstructures in the density region lower than ∼2ρ0can be well described by the combination of the nuclear saturation\nparameters such as η= (K0L2)1/3[8, 9]. In order to discuss the possible maximum mass of neutron star s, I simply\nconsider the EOS constructed in such a way that the phenomenolog ical EOS with various values of ηfor lower density\nregion is connected at ρ= 2ρ0to the EOS for higher density region characterized by the possible m aximum sound\nvelocity. As a result, I find that the possible maximum mass can be exp ressed as a function of ηand the possible\nmaximum sound velocity for high-density region. With future observ ations of massive neutron stars, one could get a\nconstraint on the possible maximum sound velocity, which may give us a hint for understanding the physics in the\nhigher density region. In this paper I simply connect the EOS for lowe r density region to that for higher density\nregion, but the smooth connection at the transition density might r educe the maximum mass. In such a case, the\nconstraint on the possible maximum sound velocity inside a star may be come more severe.\nAcknowledgments\nI am grateful to K. D. Kokkotas and J. M. Lattimer for fruitful co mments, and to K. Oyamatsu and K. Iida for\npreparing the EOS table. I am also grateful to B. Balantekin for che cking the manuscript carefully. This work was\nsupported in part by Grant-in-Aid for Young Scientists (B) throug h Grant No. 26800133 provided by Japan Society\nfor the Promotion of Science (JSPS) and by Grants-in-Aid for Scien tific Research on Innovative Areas through Grant\nNo. 15H00843 provided by MEXT.\n[1] J. M. Lattimer, Annu. Rev. Nucl. Part. Sci. 62, 485 (2012)\n[2] A. W. Steiner, J. M. Lattimer, and E. F. Brown, Eur. Phys. J . A52, 18 (2016)\n[3] P. B. Demorest, T. Pennucci, S. M. Ransom, M. S. E. Roberts , and J. W. T. Hessels, Nature 467, 1081 (2010).\n[4] J. Antoniadis, et al., Science 340, 1233232 (2013).\n[5] J. B. Hartle, Phys. Rep. 46, 201 (1978).\n[6] S. Koranda, N. Stergioulas, and J. L. Friedman, Astrophy s. J.488, 799 (1997).\n[7] P. Bedaque and A. W. Steiner, Phys. Rev. Lett. 114, 031103 (2015).\n[8] H. Sotani, K. Iida, K. Oyamatsu, and A. Ohnishi, Prog. The or. Exp. Phys. 2014, 051E01 (2014).\n[9] H. O. Silva, H. Sotani, and E. Berti, Mon. Not. R. Astron. S oc.459, 4378 (2016).\n[10] M. B. Tsang, et al., Phys. Rev. C 86, 015803 (2012).\n[11] W. G. Newton, J. Hooker, M. Gearheart, K. Murphy, D. H. We n, F. J. Fattoyev, and B. A. Li, Eur. Phys. J. A 50, 41\n(2014).\n[12] E. Khan and J. Margueron, Phys. Rev. C 88, 034319 (2013).\n[13] S. Gandolfi, J. Carlson, and S. Reddy, Phys. Rev. C 85, 032801 (2012).7\n[14] J. M. Lattimer, Annu. Rev. Nucl. Part. Sci. 31, 337 (1981).\n[15] K. Oyamatsu and K. Iida, Prog. Theor. Phys. 109, 631 (2003).\n[16] K. Oyamatsu and K. Iida, Phys. Rev. C 75, 015801 (2007).\n[17] P. C. C. Freire, S. M. Ransom, S. Begin, I. H. Stairs, J. W. T. Hessels, L. H. Frey, and F. Camilo, Astrophys. J. 675, 670\n(2008).\n[18] S. L. Shapiro and S. A. Teukolsky, in Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects\n(Wiley, New York, 1983)."    },    {        "title": "1702.00543v1.Universality_of_density_of_states_in_configuration_space.pdf",        "content": "Universality of density of states in con\fguration space\nTetsuya Taikei, Kazuhito Takeuchi, and Koretaka Yuge\nDepartment of Materials Science and Engineering, Kyoto University, Kyoto 606-8501, Japan\nIn this study, we con\frm the universality of density of microscopic states in non-interacting\nsystem; this means statistical interdependence is vanished in any lattices. This enable one to\nobtain information of con\fguration of solute atoms, free energy, phase diagram with performing\n\frst-principles calculation on few special microscopic states combined with our established theory.\nI. INTRODUCTION\nFor materials design, \frst we need to know constituent\nelements with phase diagram, and composition of each el-\nements. For clarifying the properties of materials more\nin detail, we need to reveal the con\fguration of solute\natoms, which means the importance of quantitative deal\nwith con\fguration space. Especially in areas of com-\nputational materials science, the so-called Generalized\nIsing Model (GIM) [1] has been widely used to describe\ncon\fguration space with \frst-principles calculations. In\nthe GIM, the con\fguration properties are speci\fed with\na complete set of coordination (i.e., basis functions),\nfq1;q2;::::q gg. However, the conventional approaches [][]\nto get information of con\fguration space of equilibrium\nstate with GIM and Monte Carlo (MC) [2][3] simulation\nneeds a lot of \frst-principle calculations for every selected\nmaterial even with the same lattice [4][5]: they do not\nfocus on how spatial constraint (i.e., lattice) plays an im-\nportant role in equilibrium state.\nRecently, we have established new approach combined\nwith GIM to address the problem of conventional ap-\nproach: With the new approach in composition-\fxed\nsystem, we \fnd that canonical average of each physical\nquantityQu(T) is determined by con\fgurational energy\nof 'Projection State' (PS) for each physical quantities\n[8]. In composition-un\fxed system, we obtain the fact\nthat energy of PS for the coordinate describing compo-\nsition ('Grand Projection State', GPS) determines con-\nnection between composition and chemical composition.\nEspecially in binary system, GPS determines free energy,\nwhich enables one to obtain binary phase diagram for\nphase separation system with two GPSs [9]. Here, we\nemphasize that PS (including GPS) is only dependent\non spatial constraint (e.g., lattice for crystalline solids,\nvolume and density for liquid in rigid box), independent\nof constituent elements, temperature, and interactions:\nOne can know PS a priori only with the information of\nspatial constraint. This result, therefore, is not only (i)\ntheoretically interesting in describing free energy by a sin-\ngle microscopic state, GPS, even though free energy has a\nmember of entropy dependent on all possible microscope\nstates, but also (ii) practically very useful for the material\ndesign with avoiding large number of \frst-principles cal-\nculations; one need to perform \frst-principle calculations\non few structures for estimating physical quantities/free\nenergy, and constructing phase diagram.\nThese approach is based on the 'statistical independence'forfqugon a non-interacting system in thermodynami-\ncal limit for the number of atoms even though the basis\nfunctions themselves are not essentially statistically inde-\npendent: Here, statistical independence means that (A)\nideally numerical vanishment of non-diagonal elements of\ncovariance matrix for fqugand (B) the density of micro-\nscopic states for fqugon a non-interacting system can\nbe well characterized by a multi-dimensional Gaussian\ndistribution. However, the validity of this assumption is\ncon\frmed only in FCC lattice in composition-\fxed sys-\ntem (CFS) and in composition-un\fxed system (CUFS)\n[9][10]. In this study, we con\frm the universality of den-\nsity of states in con\fguration space with random ma-\ntrix (RM) constructed by Gaussian orthonormal ensem-\nble through the MC simulation on real lattice, which\nmeans the validity of our assumption in any lattices. In\nCFS, we obtain the connection of the statistical interde-\npendence and the number of atoms, Nnumerically. In\nCUFS, we demonstrate that statistical interdependence\nremains however we make another lattice from one, which\ncertainly indicate universality of density of states. In the\nfollowing, we demonstrate the validity \frst in CFS in or-\nder to con\frm that PS can describe Qu(T) in any lattices,\nthen in CUFS in order to con\frm that GPS can describe\nfree energy in any lattices.\nII. METHODOLOGY AND DISCUSSIONS\nLet us \frst explain how we have con\frmed the statisti-\ncal interdependence in FCC lattice [10]. For considering\ndensity of states on con\fguration space, we take the ma-\ntrixA. We construct Awith its (s;t) elementastdenotes\nthe value of qtat sampling time s. Therefore, when we\nsamplempoints on the con\fguration space and consider\nnkinds of basis functions, Ashould bem\u0002nmatrix.\nWithA, we construct the covariance matrix R;\nR=1\nmATA; (1)\nwhereATmeans transposed matrix of A. When we con-\nsider an ideal system where statistical interdependence\nis disappeared and the distribution of the elements at\neach column is constructed by a Gaussian distribution,\nAshould be random matrix, ARM, with a Gaussian or-\nthonormal ensemble. Therefore, our strategy is to com-\npare matrix constructed from the practical system and\nrandom matrix. In this study, we construct ARMwitharXiv:1702.00543v1  [cond-mat.dis-nn]  2 Feb 20172\nML SL\nset1 FCC BCC1,Diamond\nset2 BCC2 HCP\nTABLE I. Sets of HSL and LSLBCC1 and BCC2 are di\u000berent\nin terms of considered clusters.\nits all elements independently consist of normal random\nnumbers, with the average and variance respectively tak-\ning 0, and 1. For comparison, Aof practical system is\nnormalized so that the average and variance of each col-\numn respectively should be 0, and 1. When we compare\nmatrices, we focus on the density of eigenstates (DOE) of\nAin order to avoid excessive estimation of numerical er-\nror in the simulation to sample from large con\fguration\nspace. For more quantitative comparison of DOEs, we\nestimate moments (from 2nd to 4th) of DOEs, de\fned\nbyM1= \u0006Nd\ni=1xi=Nd,ML=n\n\u0006Nd\ni=1(xi\u0000M1)L=Ndo1=L\n,\nwhereLis an order of the moment, Ndis number of data\nandxiis each data of index i.\nFor setting calculation condition of each lattice, such as\nwhat kind of basis functions we consider, we take care\nabout how one lattice (named Son Lattice, SL) is made\nfrom other lattice (named Mother Lattice, ML). We de-\ntermine kinds of considered clusters of SL with consid-\nering how the basis functions of ML is convereted to\nthat of SL; this connection makes considered clusters of\nSL more than that of ML. We change sampling times,\nMSL=MML\u0002NSL\nNML; this change in sampling times is\nappropriate for random matrix too. Marchenko and Pas-\nture showed that the DOE of covariance matrix for ran-\ndom matrix can be analytically determined with m!1\nwhenn=m is \fxed [14]. In this study, we prepare two sets\nof ML and SL, which Table. I shows. The detail of how\nwe make SL from ML and what kind of clusters we con-\nsider in each set, and the di\u000berence between BCC1 and\nBCC2 are shown in Appendix A.\nIn this study, we consider an example for equiatomic A-\nB binary system on each lattice. We employ generalized\nIsing spin model with spin variables of \u001b=\u00061 in order to\ngetfqug. Here,qucan be de\fned as qu=hQ\ni2u\u001biilattice ,\nwhere\u001biis spin at site i,h\u0001\u0001ilattice is average over all sites\non the lattice, and uis the index indicating cluster type,\nsuch as empty, point, 1st nearest neighbor pair. With\nthis de\fnition, we can take advantage to get complete\northonormal basis functions; this means we can expect\n(A) (explained in introduction), numerical vanishment\nof non-diagonal elements, without transforming coordi-\nnation of basis functions. In order to demonstrate the\nvalidity of our PS/GPS approach in CFS/CUFS, we pre-\npare two types of covariance matrices corresponding to\nthe individual conditions, which we explain more in de-\ntail in the following.A. Universality of density of states in CFS\nFor con\frming the universality of density of states in\nCFS, we construct the matrix from the composition-\fxed\nsystem (i.e., an equiatomic system) with MC simulation.\nIn previous study for FCC lattice with random matrix\n[10], we have showed that statistical dependence of den-\nsity of microscopic states is gradually eliminated when\nthe number of atoms, N, increases. Therefore, we demon-\nstrate whether this tendency of Ndependence can be\nholed in set1 and set2. Figures. 1 shows the landscape\nof DOEs for set1 and set2. In terms of decrease of sub-\npeaks and location of highest-peak, we can easily under-\nstand that statistical interdependence is con\frmed with\nthe increase of N, which meets our previous research [10].\nFor more quantitative comparison, we estimate 2nd-4th\nmoments of DOEs de\fned above, and show Figs. 2 and\n3; Figure 2 clearly shows that when the increase of the\nsize of spatial constraint, all the moments for practical\nsystem become close to that for RM, and Fig. 3 shows\nthat all moments for practical system may be in inverse\nproportion to N;we derive the model showing this con-\nnection of 2nd moment followings, and that of 3rd and\n4th moment in Appendix B.\nFirst, we note that Rde\fned in Eq. (1) is symmetric\nmatrix, which leads ( s;t) element of R2to be equal to\n(rst)2. From the linear algebra, 2nd moment de\fned in\nthis paper, M2, can be represented as\nM2=vuuut1\nm0\n@X\n(i;j)(rij)2+X\ni(rii)21\nA\u0000M2\n1:;(2)\nwhere (i;j;k;\u0001\u0001\u0001) denotes the combination whose all ele-\nments are di\u000berent. We normalize each column of Awith\nits variance taking 1, which develop Eq. (2) as\nM2=vuut1\nmX\n(i;j)(rij)2: (3)\nIn order to proof the inverse proportion, Eq. (2) tells that\nwe need to show the rijhas the inverse proportion to N.\nWe note that rijis covariance of qi=hqiisdandqj=hqjisd\n(hereinafter, de\fned as \u0013 qiand \u0013qj). With average of each\ncolumn ofAtaking 0, we obtain\nrij=Z Z\ng(\u0013qi;\u0013qj)(\u0013qi\u0000h\u0013qii1)(\u0013qj\u0000h\u0013qji1)d\u0013qid\u0013qj\n=Z Z\ng(\u0013qi;\u0013qj)\u0013qi\u0013qjd\u0013qid\u0013qj: (4)\nHere, we consider how many spin products consisting of\nqi,Y\ni2u\u001bi, is changed with the change of \u0013 qi,\u000e\u0013qi:\n\u000e\u0013qi=\u000eCi=Ns\nhqiisd=\u000eCi\nNs\u0001hqiisd; (5)\nwhere\u000eCidenotes the the number of spin products\nchanged,sdenotes the number of spin products per site.3\nDensity of states\n1.21.11.00.90.80.7\neigenvalue/s32/s33/s32/s34/s35/s35/s32/s36/s37/s38\n/s32/s33/s32/s34/s35/s35/s32/s37/s39/s38/s40\n/s32/s33/s32/s34/s35/s35/s32/s38/s39/s40/s41\n/s32/s33/s32/s42/s43\nDensity of states\n1.21.11.00.90.80.7\neigenvalues/s32/s33/s32/s34/s35/s35/s32/s36/s37/s38\n/s32/s33/s32/s34/s35/s35/s32/s37/s39/s38/s40\n/s32/s33/s32/s34/s35/s35/s32/s38/s39/s40/s41\n/s32/s33/s32/s42/s43\nDensity of states\n1.21.11.00.90.80.7\neigenvalues/s32/s33/s32/s34/s35/s36/s37/s38/s39/s40/s32/s41/s42/s43\n/s32/s33/s32/s34/s35/s36/s37/s38/s39/s40/s32/s42/s44/s43/s45\n/s32/s33/s32/s34/s35/s36/s37/s38/s39/s40/s32/s43/s44/s45/s46\n/s32/s33/s32/s47/s48\nDensity of states\n1.10 1.00 0.90 0.80\neigenvalue/s32/s33/s32/s34/s35/s35/s32/s36/s37/s38\n/s32/s33/s32/s34/s35/s35/s32/s39/s40/s41/s42\n/s32/s33/s32/s34/s35/s35/s32/s41/s40/s42/s43\n/s32/s33/s32/s44/s45\nDensity of states\n1.10 1.00 0.90 0.80\neigenvalue/s32/s33/s32/s34/s35/s36/s37/s38/s39\n/s32/s33/s32/s34/s35/s36/s40/s41/s42/s43\n/s32/s33/s32/s34/s35/s36/s42/s41/s43/s44\n/s32/s33/s32/s45/s46\nFIG. 1. Up: DOEs for set1. Down: DOEs for set2.\nThis landscape shows that statistical interdependence is con\frmed with increase of the lattice size.\n8x10-2\n6\n4\n2\n0 2nd moment\n25002000150010005000\n atoms/s32/s33/s32/s34/s35/s36/s37/s38/s39/s40\n/s32/s33/s32/s41/s42/s42/s43\n/s32/s33/s32/s44/s42/s42\n/s32/s33/s32/s41/s42/s42/s45\n/s32/s33/s32/s46/s42/s47\n-0.12-0.10-0.08-0.06-0.04-0.020.00 3rd moment\n25002000150010005000\n atoms/s32/s33/s32/s34/s35/s36/s37/s38/s39/s40\n/s32/s33/s32/s41/s42/s42/s43\n/s32/s33/s32/s44/s42/s42\n/s32/s33/s32/s41/s42/s42/s45\n/s32/s33/s32/s46/s42/s47\n0.16\n0.12\n0.08\n0.04\n0.00 4th moment\n25002000150010005000\n atoms/s32/s33/s32/s34/s35/s36/s37/s38/s39/s40\n/s32/s33/s32/s41/s42/s42/s43\n/s32/s33/s32/s44/s42/s42\n/s32/s33/s32/s41/s42/s42/s45\n/s32/s33/s32/s46/s42/s47\nFIG. 2. 2nd, 3rd, and 4th order moments of DOEs for CFS along the number of atoms. These shows that with increase of the\nnumber of atoms, statistical independence is guaranteed.\n8x10-2\n6\n4\n2\n0 2nd moment\n2.0x10-3 1.5 1.0 0.5 0.0\n 1/atoms/s32/s33/s32/s34/s35/s36/s37/s38/s39/s40\n/s32/s33/s32/s41/s42/s42/s43\n/s32/s33/s32/s44/s42/s42\n/s32/s33/s32/s41/s42/s42/s45\n/s32/s33/s32/s46/s42/s47\n-0.12-0.10-0.08-0.06-0.04-0.020.00 3rd moment\n2.0x10-3 1.5 1.0 0.5 0.0\n 1/atoms/s32/s33/s32/s34/s35/s36/s37/s38/s39/s40\n/s32/s33/s32/s41/s42/s42/s43\n/s32/s33/s32/s44/s42/s42\n/s32/s33/s32/s41/s42/s42/s45\n/s32/s33/s32/s46/s42/s47\n0.16\n0.12\n0.08\n0.04\n0.00 4th moment\n2.0x10-3 1.5 1.0 0.5 0.0\n 1/atoms/s32/s33/s32/s34/s35/s36/s37/s38/s39/s40\n/s32/s33/s32/s41/s42/s42/s43\n/s32/s33/s32/s44/s42/s42\n/s32/s33/s32/s41/s42/s42/s45\n/s32/s33/s32/s46/s42/s47\nFIG. 3. 2nd, 3rd, and 4th order moments of DOEs for CFS along the inverse of the number of atoms. These certainly indicate\nthat moments has inverse proportion to the number of atoms.4\nEq. 5 shows that under the same \u000eCi,\u000e\u0013qiis in inverse\nproportion to Nbecausehqiisd= 1=p\nNd[12][13]. With\nusing this connection, when we consider how many spin\nproducts consisting of qiis obligatorily changed by the\nchange ofqj, which we de\fne as H(\u000e\u0013qi;\u000e\u0013qj),H(\u000e\u0013qi;\u000e\u0013qj)\nshould be\nH(\u000eqi;\u000eqj) =1\nNh(\u000eqi;\u000eqj); (6)\nwhereh(\u000eqi;\u000eqj) denotes ideal H(\u000eqi;\u000eqj) atN= 1,\nfrom the view point of the 2D space of spin products for\nqiandqj. WithH, we introduce the model to represent\ng(\u000eqi;\u000eqj) as\ng(\u000eqi;\u000eqj)'gi(\u000eqi)gj(\u000eqj) (1 +H(\u000eqi;\u000eqj) + \u0001 (\u000eqi;\u000eqj)):\n(7)\nWith this model,R\ng(\u000e\u0013qi)\u000e\u0013qid\u000e\u0013qi= 0 because the land-\nscape of DOEs certaily indicate g(\u000e\u0013qi) should be Gaus-\nsian distribution: Therefore, rijshould be inverse pro-\nportion to N: We can demonstrate the inverse propor-\ntion of moments to N. This connection certainly indicate\nthat statistical dependence can be vanished with taking\nlimit ofNeven in any lattices. Consequently, we can con-\n\frm the universality of density of states in CFS, which\nstrongly support our PS approach.\nB. Universality of density of states in CUFS\nIn CUFS, we have shown that statistical interdepen-\ndence is more eliminated than in CFS under the same N\nin FCC lattice; even in about 1000 atoms, the moments\nof CUFS successfully agree with that of RM [9]. In this\nsection, therefore, we con\frm whether this tendency of\nCUFS remains in any lattice in order to demonstrate van-\nishment of statistical interdependence in CUFS.\nIn order to construct the matrix from CUFS with MC\nsimulation, we have introduced a local system in an ide-\nally large composition-\fxed system; we can theoretically\ndetermine the probability of composition of local system\nfrom the composition of an ideally large system with bi-\nnomial distribution. Therefore, based on this probability,\nwe set the sampling times for the each composition of lo-\ncal system in set1 and set2.\nFigs. 4 shows the landscape of DOEs, which shows statis-\ntical independence is more con\frmed in CUFS under the\nsameN. Moreover, Fig. 5 shows the moments of DOEs\nfor each lattices, which clarify the moments of CUFS\nshows excellent agreement with that of RM in every sim-\nulation. These clearly meets our previous research, and\nthe tendency in FCC lattice remains. Moreover these two\nresults of set1 and set2 indicate however we make SL from\nML, statistical independence is con\frmed, which means\nstatistical independence in any lattices Consequently, we\ncan con\frm the universality of density of states in CUFS,\nwhich strongly support our GPS approach.III. CONCLUSION\nIn this study, we con\frm the universality of density of\nmicroscopic states in non-interacting system; this means\nstatistical interdependence is vanished in any lattices\neven though the basis functions themselves are not essen-\ntially statistically independent. This enable one to obtain\ninformation of con\fguration of solute atoms, free energy,\nphase diagram with performing \frst-principles calcula-\ntion on few special microscopic states combined with our\nestablished theory. Moreover, this study can open the\ndoor to new approach based on statistical independence.\nACKNOWLEDGEMENT\nThis work was supported by a Grant-in-Aid for Sci-\nenti\fc Research (16K06704), and a Grant-in-Aid for\nScienti\fc Research on Innovative Areas Materials Sci-\nence on Synchronized LPSO Structure (26109710) from\nthe MEXT of Japan, Research Grant from Hitachi\nMetals\u0001Materials Science Foundation, and Advanced Low\nCarbon Technology Research and Development Program\nof the Japan Science and Technology Agency (JST).\nAppendix A: Table\nIn this section, we show the condition of experiment,\nsuch as the detail of how we make SL from ML and what\nkind of clusters we consider in each set, and the di\u000ber-\nence between BCC1 and BCC2. Figure. I shows how we\nmake SL from ML. In ML of set 1 (FCC lattice), clusters\nconsidered are pair clusters up to 6NN, triplet clusters\nconsisting of up to 6NN pairs, resulting in 29 basis func-\ntions(i.e.,n=29). We sample 500,000 microscopic states\n(i.e.,m= 500;000) by performing MC simulation for\nthe MC-cells. The condition of the study of SL, BCC1\nand Diamond, is decided based on this FCC setting (ex-\nplained above in Sec. II); 6NN pair in FCC is extended\nto 8NN pair in BCC1 and 12NN in Diamond. Therefore,\nwe do calculation with Table. A1\nFCC BCC1 Diamond\nBasis functions considered 29 42 106\nSampling times 500,000 724,138 1,827,586\nNumber of atoms 512, 1024, 2048atoms\nTABLE A1. Condition of set1.\nBCC2 HCP\nClusters considered 17 50\nSampling times 500,000 1470588\nNumber of atoms 576, 1024, 2048 atoms\nTABLE A2. Condition of set2.5\nAlso in set2, \frst we determine the condition of ML,\nBCC2; pair clusters considered up to 5NN, triplet clus-\nters consisting of up to 5NN pairs, resulting in 17 basis\nfunctions(i.e., n=17). We sample 500,000 microscopic\nstates (i.e., m= 500;000) with MC simulation, which\ndecide condition of SL, HCP. In HCP, we consider pair\nclusters up to 10NN (except 6NN and 9NN), and triplet\nclusters up to 10NN (also except 6NN and 9NN) pairs.\nFigure. A2 shows the condition of set2 more in detail.\nAppendix B: Connection of 3rd/4th moment and\nthe number of atoms\nIn this section, we show the proof of inverse proportion\nof 3rd/4th moment to N, which is seen in Fig, 3. Thedi\u000berence from 2nd moments is Eqs. (2) and (2): the\ninverse proportion ,to N, ofrijati6=jof course remains\nwhen we consider 3rd/4th moment. Therefore, we can\nrepresent 3rd/4th moment with simple representation as\nM3=3vuut3X\nt=0a3tN\u0000t: (B1)\nM4=4vuut4X\nt=0a4tN\u0000t: (B2)\nWith this representation, we can easily obtain the inverse\nproportion of 3rd/4th moment to Nat largeN.\n[1] J. M. Sanchez, F. Ducastelle and D.Gratias, Pysica\n128A , 334 (1984).\n[2] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A.\nH. Tellerand, and E. Teller, J. Chem. Phys. 21, 1087\n(1953).\n[3] A. M. Ferrenberg and R. H. Swendsen, Phys. Rev. Lett.\n63, 1195 (1989).\n[4] A. Seko, Y. Koyama, and I. Tanaka, Phys. Rev. B. 80,\n165122 (2009).\n[5] S. Mller, J. Phys.: Condens. Matter 15, R1429 (2003).\n[6] V. Blum and A. Zunger, Phys. Rev. B 70, 155108 (2004).\n[7] V. Blum, G. L. W. Hart, M. J. Walorski, and A. Zunger,\nPhys. Rev. B 72, 165113 (2005).[8] K. Yuge, J. Phys. Soc. Jpn. 85, 024802 (2016).\n[9] T. Taikei, T. Kishimoto, K. Takeuchi, and K. Yuge, (sub-\nmitted).\n[10] K. Yuge, T. Kishimoto and K. Takeuchi, Trans. Mat.\nRes. Soc. Jpn. 41213 (2016).\n[11] A. Zunger, S.-H. Wei, L. G. Ferreira, and J. E. Bernard,\nPhys. Rev.Lett. 65, 353 (1990).\n[12] S.H. Wei, L. G. Ferreira, J. E. Bernard, and A. Zunger,\nPhys. Rev. B 42, 9622 (1990).\n[13] T. Taikei, K. Takeuchi, and K. Yuge, (under prepara-\ntion).\n[14] V.A. Marchenko and L.A. Pastur, Math. USSR Sb. 1457\n(1967).6\nDensity of states\n1.2 1.1 1.0 0.9 0.8\neigenvalueBCC 1024 atoms\n : Fixed\n :  Unfixed\n: Random matrix\nDensity of states\n1.2 1.1 1.0 0.9 0.8\neigenvalueHCP 1024 atoms\n : Fixed\n :  Unfixed\n: Random matrix\nDensity of states\n1.2 1.1 1.0 0.9 0.8\neigenvalueDiamond 1024 atoms\n : Fixed\n :  Unfixed\n: Random matrix\nFIG. 4. Density of states along eigenvalues of covariance matrix (DOE), constructed from CFS, CUFS, and RM. These shows\nthat DOE for CUFS is more similar to that for RM than for CFS in every lattice.\n-6x10-2-4-2024 moments\n/s32/s33/s34/s35/s36/s37 /s38/s39/s40/s41/s37 /s42/s35/s43/s41/s38/s39bcc 1024 atoms\n: Fixed\n : Unfixed\n : RM 42\n-4x10-2-2024 moments\n/s32/s33/s34/s35/s36/s37 /s38/s39/s40/s41/s37 /s42/s35/s43/s41/s38/s39hcp 1024 atoms\n : Fixed\n : Unfixed\n : RM 50\n-6x10-2-4-2024 moments\n/s32/s33/s34/s35/s36/s37 /s38/s39/s40/s41/s37 /s42/s35/s43/s41/s38/s39Diamond 1024atoms\n :Fixed\n : Unfixed\n : RM106\nFIG. 5. 2nd, 3rd, and 4th-order moments of DOEs for CFS, and CUFS, and RM. These show excellent agreement of RM and\nCUFS quantitatively. .\nc / a = 1.633  \nγ = 120° BCC HCP \nc / a = 1 \nγ = 90° １NN \n１NN 2NN \n2NN 1NN  \nFCC BCC \n１NN \nFCC Diamond  \n１NN １NN \n2NN 3NN \nFIG. A1. These \fgures shows how we construct SL from ML."    },    {        "title": "1703.03614v2.Density_of_States_FFA_analysis_of_SU_3__lattice_gauge_theory_at_a_finite_density_of_color_sources.pdf",        "content": "Density of States FFA analysis of SU(3) lattice\ngauge theory at a finite density of color sources\nMario Giuliani, Christof Gattringer\nUniversität Graz, Institut für Physik, Universitätsplatz 5, 8010 Graz, Austria\nAbstract\nWe present a Density of States calculation with the Functional Fit Approach (DoS FFA) in SU(3)\nlattice gauge theory with a finite density of static color sources. The DoS FFA uses a parameterized\ndensity of states and determines the parameters of the density by fitting data from restricted Monte\nCarlo simulations with an analytically known function. We discuss the implementation of DoS FFA\nandtheresultsforaqualitativepictureofthephasediagraminamodelwhichisafurthersteptowards\nimplementing DoS FFA in full QCD. We determine the curvature \u0014in the\u0016-Tphase diagram and\nfind a value close to the results published for full QCD.\n1 Introductory remarks\nThe success of numerical calculations in lattice field theory relies on the availability of probabilistic\npolynomial algorithms for computing observables in a Monte Carlo (MC) simulation. The key point is\nthe interpretation of the Boltzmann factor e\u0000Sas a probability. However, in some situations the action\nSacquires an imaginary part that spoils the probabilistic interpretation necessary for a MC simulation, a\nproblem usually referred to as ”complex action problem” or ”sign problem”.\nAn important class of systems with a sign problem are lattice field theories with non-zero chemical\npotential. In many cases the complex action problem is the main obstacle for an ab-initio determination of\nthe full phase diagram at finite density. Different methods such as complex Langevin, Lefshetz thimbles,\nTaylorexpansion, fugacityexpansion, reweighting, andworldlineformulationswereappliedtofinitedensity\nlattice field theory (see, e.g., the reviews [1] – [9] at the annual lattice conferences).\nAnother important general approach are Density of States (DoS) techniques [1], [10] – [27]. Here we\ndevelop further the Density of States Functional Fit Approach (DoS FFA) and apply it to SU(3) lattice\ngauge theory with static color sources (SU(3) LGT-SCS). The DoS FFA was already presented in depth\nin [24] – [27] and we refer to these papers for a detailed discussion of the method. Here we review only\nthe parts specific for the SU(3) LGT-SCS and the respective observables, which are related to the particle\nnumber and its susceptibility used to determine a qualitative picture of the phase diagram.\nWe stress at this point that the results presented here are not meant as a detailed systematic study\nof the phase diagram of SU(3) LGT-SCS, which would imply a controlled thermodynamical limit followed\nby extrapolating to vanishing lattice spacing. This paper serves to document the developments and tests\nof the DoS FFA in a model which is a further step towards a Density of States calculation in full lattice\nQCD at non-zero chemical potential.\n1arXiv:1703.03614v2  [hep-lat]  10 Aug 20172 Definition of the model and the density of states\nWe study SU(3) lattice gauge theory with static color charges. The dynamical degrees of freedom are\nSU(3)-valued gauge links U\u0017(n);\u0017= 1;2;3;4, wheren= (~ n;n 4)denotes the sites of a N3\ns\u0002Ntlattice\nwith periodic boundary conditions. The action is given by\nS[U] =\u0000\f\n3X\nnX\n\u0016<\u0017Reh\nTr U\u0016(n)U\u0017(n+ ^\u0016)Uy\n\u0016(n+ ^\u0017)Uy\n\u0017(n)i\n\u0000\u0011X\n~ nh\ne\u0016NtP(~ n) +e\u0000\u0016NtP(~ n)?i\n;(1)\nwhere\fistheinversegaugecoupling, \u0011thecouplingstrengthofthestaticcolorsourcesand \u0016isthechem-\nical potential. The static color sources are represented by Polyakov loops P(~ n) =1\n3TrQNt\u00001\nn4=0U4(~ n;n 4).\nIn the action (1) the chemical potential \u0016gives a different weight to charges P(~ n)and to anti-charges\nP(~ n)?, such that the theory has a complex action problem which is equivalent to the one of QCD.\nFor defining the density of states we decompose the action S[U]into real and imaginary parts and\nwrite it in the form S[U] =S\u001a[U]\u0000i\u0018\u0016X[U], where it is easy to see that\nS\u001a[U] =\u0000\f\n3X\nnX\n\u0016<\u0017Reh\nTr U\u0016(n)U\u0017(n+ ^\u0016)Uy\n\u0016(n+ ^\u0017)Uy\n\u0017(n)i\n\u00002\u0011cosh(\u0016Nt)X\n~ nRe[P(~ n)];\nX[U] =X\n~ nIm[P(~ n)]and\u0018\u0016= 2\u0011sinh(\u0016Nt):(2)\nThe functional X[U]in the imaginary part is bounded, i.e., X[U]2[\u0000xmax;xmax], withxmax=p\n3\n2N3\ns.\nForx2[\u0000xmax;xmax]we introduce the weighted density of states \u001a(x)as\n\u001a(x) =Z\nD[U]e\u0000S\u001a[U]\u000e(X[U]\u0000x); (3)\nwhereD[U]is the product of Haar measures for all link variables. Exploiting the transformation U\u0017(n)!\nU?\n\u0017(n)one finds that \u001a(x)is an even function. Thus the partition sum Zin terms of the density reads\nZ=Z\nD[U]e\u0000S[U]=Zxmax\n\u0000xmaxdx\u001a(x)ei\u0018\u0016x= 2Zxmax\n0dx\u001a(x) cos(\u0018\u0016x); (4)\nand vacuum expectation values of moments of X[U]can be computed as moments of xin the correspond-\ning integrals over the density \u001a(x). The expression (4) makes clear the emergence of the complex action\nproblem in the DoS formulation: The density \u001a(x)is integrated with the oscillating function cos(\u0018\u0016x),\nand from the definition of the coupling \u0018\u0016in (2) it is obvious that the frequency of the oscillation increases\nexponentially with \u0016(and linearly with \u0011), such that \u001a(x)has to be computed with sufficient accuracy.\nThe recent developments of DoS techniques [16] – [27] are based on new strategies for calculating \u001a(x)\nwith very high precision.\n3 Using DoS FFA for computing the density of states\nFor a DoS calculation the density \u001a(x)has to be parameterized on the interval [0;xmax]in a suitable\nway. For our parameterization we divide the interval [0;xmax]intoNsub-intervals In\u0011[xn;xn+1];n=\n0;1; :::N\u00001withx0= 0andxN=xmax. The density is then written in the form \u001a(x) =e\u0000l(x),\nwherel(x)is a continuous function that is piecewise linear on the intervals In. We normalize the density\n2using the condition \u001a(0) = 1, which corresponds to l(0) = 0. Together with the continuity of l(x)this\nimplies that only the slopes kn;n= 0;1; :::N\u00001, which determine l(x)in the intervals Inappear as the\nparameters of \u001a(x). A short calculation [24] – [27] gives the explicit form of \u001a(x)as function of the kn,\n\u001a(x) =Ane\u0000knxforx2InwhereAn=e\u0000n\u00001P\nj=0\u0001j(kj\u0000kn)\n: (5)\nHere \u0001j\u0011xj+1\u0000xjdenotes the size of the j-th interval. We stress that the intervals can be chosen\nwith different sizes such that in regions of xwhere\u001a(x)varies quickly a finer discretization can be used.\nFor computing the slopes knin the DoS FFA we use restricted vacuum expectation values hhXiin(\u0015),\nn= 0;:::N\u00001, which depend on a free parameter \u00152R. They are defined as\nhhXiin(\u0015) =1\nZn(\u0015)Z\nD[U]e\u0000S\u001a[U]+\u0015X[U]X[U]\u0012n\u0000\nX[U]\u0001\n; Zn(\u0015) =Z\nD[U]e\u0000S\u001a[U]+\u0015X[U]\u0012n\u0000\nX[U]\u0001\n;\n(6)\nwhere\u0012n(x) = 1forx2In,\u0012n(x) = 0forx62In. The free parameter \u0015, which the restricted vacuum\nexpectation values hhXiin(\u0015)depend on, can be used to probe the system. We stress that the restricted\nvacuum expectation values are free of the complex action problem and can be evaluated with standard\nMonte Carlo calculations.\nThe key observation of the DoS FFA is that with the parameterization (5) the restricted partition sum\nZn(\u0015)and thus the restricted vacuum expectation value hhXiin(\u0015) =@lnZn(\u0015)=@\u0015can be computed\nin closed form. It is convenient to shift and rescale the hhXiin(\u0015)into a new form Yn(\u0015)for which one\nfinds the explicit expression [24] – [27]:\nYn(\u0015)\u0011hhXiin(\u0015)\u0000Pn\u00001\nj=0\u0001j\n\u0001n\u00001\n2=1\n1\u0000e\u0000(\u0015\u0000kn)\u0001n\u00001\n(\u0015\u0000kn)\u0001n\u00001\n2=h\u0010\n(\u0015\u0000kn)\u0001n\u0011\n;(7)\nwhere in the last step we introduced the function h(s) = 1=(1\u0000e\u0000s)\u00001=s\u00001=2. We find that the\nshifted and rescaled expectation value Yn(\u0015)is given by h((\u0015\u0000kn)\u0001n), i.e., it depends only on one\nof the parameters of \u001a(x), the slopeknin the respective interval In. Thus we can compute Yn(\u0015)for\nseveral values of \u0015and determine knfrom a fit of the data for Yn(\u0015)withh((\u0015\u0000kn)\u0001n). The function\nh(s)approaches\u00061=2fors!\u00061, is increasing monotonically and obeys h(0) = 0. Thus it is gives\nrise to simple stable 1-parameter fits and the kncan be determined reliably [24] – [27]. Analyzing the\nquality of the fit is an important self consistency check and poor quality of the fit indicates that the size\n\u0001nof the corresponding interval should be chosen smaller [24] – [27]. Once the slopes knare computed,\nthe density\u001a(x)can be determined with (5) and from \u001a(x)we can evaluate the observables.\nBefore we come to presenting our results for observables in the next section, we have a look at the\ndensity and how it changes when varying the parameters. In Fig. 1 we show ln\u001a(x)as a function of x\nfor different values of the inverse coupling \fat fixed\u0011= 0:04,\u0016= 0:15. When comparing the different\ncurves for ln\u001a(x)over the full range of xin the lhs. plot, the different couplings seem to give rise to\nessentially the same density. However, the zoom into the small- xregion (rhs. plot) reveals that the curve\nfor the largest \fshows a quite different behavior. We will see below that this change corresponds to a\nphase transition between \f= 5:60and5:70. We conclude that inspecting the qualitative behavior of the\ndensity\u001a(x)already reveals physical properties of the system. However, we stress again that only the\nevaluation of physical observables is the true benchmark for a DoS calculation, since the density still has\nto be integrated over with the highly oscillating factors, which tests if the evaluation of \u001a(x)is sufficiently\naccurate.\n3-8000-6000-4000-20000\n0 50 100 150 200 250 300 350 400 450ln(ρ)\nxβ=5.40\nβ=5.50\nβ=5.60\nβ=5.70\n-70-60-50-40-30-20-100\n0 10 20 30 40 50 60 70 80ln(ρ)\nxβ=5.40\nβ=5.50\nβ=5.60\nβ=5.70Figure 1: Results for the logarithm of the density ln\u001a(x)as a function of x(83\u00024,\u0011= 0:04,\n\u0016= 0:15and different values of the inverse coupling \f). We show ln\u001a(x)for the full range of xin\nthe lhs. plot and a zoom into the small- xregion (rhs.).\nWe conclude this section with a short comment on the piecewise linear parameterization of the\nexponent of \u001a(x). It is clear that the exact result is obtained only in the limit where one sends the\nnumber of intervals Nto infinity and their sizes \u0001nto 0. In our studies for this paper, as well as in\n[27], where we analyzed a related model where we could systematically compare the DoS FFA results\nto reference data from a dual simulation free of the complex action problem, it was found that the\naccuracy of the Monte Carlo results has a considerably larger effect on the final results than the size of\nthe discretization intervals we use here. More specifically, our discretization intervals \u0001nwere chosen\nsuch that an optimal use of the data for Yn(\u0015)in Eq. (7) is obtained: One can show [27] that the slope\nof the fit function h((\u0015\u0000kn)\u0001n)at\u0015=knis given by \u0001n=12, and that \u0001n=12\u00180:1\u00000:5gives rise\nto an optimal fit of the data for Yn(\u0015). This choice from [27] was implemented in our study here.\n4 Observables and results\nThe observables we study are the expectation value of the imaginary part of the Polyakov loop and the\ncorresponding susceptibility. Their definitions and expressions in terms of density integrals are given by\nhIm Pi \u0011 \u00001\nN3s@\n@\u0018\u0016lnZ=1\nN3s2\nZxmaxZ\n0dx\u001a(x) sin(\u0018\u0016x)x; (8)\n\u001fIm P\u0011@\n@\u0018\u0016hIm Pi=1\nN3s2\n42\nZxmaxZ\n0dx\u001a(x) cos(\u0018\u0016x)x2+\u00122\nZxmaxZ\n0dx\u001a(x) sin(\u0018\u0016x)x\u001323\n5:(9)\nNote that in leading order we have \u0018\u0016= 2\u0011sinh(\u0016Nt)/2\u0011 \u0016Nt, such that in this order @lnZ=@\u0018\u0016/\n1=2\u0011@lnZ=@\u0016Nt, indicating thathIm Piis closely related to the particle number density, and \u001fIm Pto\nthe particle number susceptibility, which makes them suitable observables for assessing the phase diagram.\nIn our numerical simulations we use N= 256intervals for the discretization of \u001a(x)forx2[0;xmax]\n(for our 83\u00024lattices – for smaller test volumes see the discussion of the volume dependence in the next\n40.01.02.03.04.0\n0 10 20 30 40 50ρ(x)sin(ξ µx)x\nxµ=0.075\nµ=0.150\nµ=0.250\nµ=0.350\n-20.0-10.00.010.020.030.040.0\n0 10 20 30 40 50ρ(x)cos(ξ µx)x2\nxµ=0.075\nµ=0.150\nµ=0.250\nµ=0.350Figure 2: The integrand of (8) \u001a(x) sin(\u0018\u0016x)x(lhs. plot), and the integrand \u001a(x) cos(\u0018\u0016x)x2of the\nconnected part in (9) (rhs.) as a function of xfor the different values of \u0016used here.\nparagraph). For each interval we used 51 values of \u0015and a statistics of 4800 configurations generated\nwith a local restricted Metropolis algorithm. The statistical errors for the raw data were computed with\na jackknife blocking analysis. For some parameter values in the vicinity of the crossover we increased the\nnumber of intervals to N= 384and used statistics up to 30000 configurations.\nIn order to analyze the dependence of the cost on the volume we implemented a small case study\nat\f= 5:45;\u0016= 0:25: In addition to V4= 83\u00024, we also did runs at V4= 63\u00024andV4= 44,\nand adjusted the number of intervals Nsuch that the interval size \u0001nand thus the discretization of\n\u001a(x)remained constant. Since xmax=p\n3N3\ns=2is proportional to the 3-volume V3\u0011N3\ns, keeping the\ndiscretization of \u001a(x)constant gives rise to a cost factor proportional V3, which is the cost factor that\nis specific for the DoS FFA. However, as for any other Monte Carlo method, we also need to take into\naccount the longer correlation times and the increasing cost of an individual sweep when the volume\nis increased: Keeping the number of values for \u0015fixed at 51, we adjusted the number of Monte Carlo\nsweeps such that the errors for the density are the same on all volumes. This resulted in a statistics of\n350, 1500 and 4800 for V4= 44;63\u00024andV4= 83\u00024, which roughly scales like (V4)1:25. Finally, the\ncost for one local Monte Carlo sweep is proportional to V4, such that we expect that the cost of FFA\nroughly scales like V3\u0002(V4)2:25, where only the factor V3is specific for DoS FFA, while the other factor\nis more general and will also depend on the couplings. It is clear that this brief study provides only a very\nrough assessment of the cost and a dedicated analysis is necessary for conclusive cost estimates. More\ncomplicated is the analysis of the \u0016-dependence of the cost, since this is expected to very strongly depend\non the other parameters. Here we can only refer to our study [27], where this question could partly be\nassessed through a comparison with dual simulation data in a closely related model.\nBefore we come to analyzing the physical observables it is interesting to have a look at the integrands\nof (8) and (9) for the different values of \u0016used here. In Fig. 2 we show \u001a(x) sin(\u0018\u0016x)x(lhs. plot of\nFig. 2) and\u001a(x) cos(\u0018\u0016x)x2(rhs.) as a function of xfor different values of \u0016. While the integrand of (8)\nremains predominantly positive in the range of \u0016values we consider here, the integrand of the connected\npart of (9) develops essential negative regions illustrating that the complex action problem (sign problem)\nis challenging for that observable already at the values of \u0016we access here. When increasing \u0016further,\nboth integrands quickly develop a highly oscillating behavior.\n50.020.040.060.080.100.120.14\n1.0 2.0 3.0 4.0 5.0 6.0 7.0χIm[P]\nβDensity of States\nconventional simulation\n0.00.10.20.30.40.50.60.7\n2.0 3.0 4.0 5.0 6.0 7.0⟨|P|⟩\nβη=0.00\nη=0.04\nη=0.32\nη=0.64\nη=0.96\nη=1.28\nη=1.60Figure 3: Lhs. plot: Comparison of the results for \u001fIm Pversus\fcomputed with the DoS FFA and\nfrom a conventional simulation for \u0016= 0(83\u00024,\u0011= 0:04). Rhs. plot: The absolute value of the\nPolyakov loop versus \ffrom a conventional simulation \u0016= 0(83\u00024, different values of \u0011).\nFrom (8) it is obvious that hIm Pi\u00110since\u0018\u0016= 2\u0011sinh(\u0016Nt)vanishes at \u0016= 0, while \u001fIm Pis\nnon-zeroalsoatvanishing \u0016. Thuswecanuse \u001fIm PforafirstassessmentoftheDoSFFAimplementation\nat\u0016= 0where we can compare \u001fIm Pto the outcome of a conventional simulation at \u0016= 0. The lhs.\nplot of Fig. 3 shows the DoS FFA results as well as the results from a standard simulation at \u0016= 0.\nWe find very good agreement of the DoS FFA data with the conventional simulation, which reassures\nus about the correctness of the implementation of DoS FFA and its accuracy, but stress again that the\nsituation at \u00166= 0is more demanding since there the density is integrated with the oscillating factors.\nFor the runs at chemical potential \u00166= 0we first have to determine the parameters for the simulation,\ni.e., we need to identify the region with transitory behavior. For this purpose we ran a \u0016= 0conventional\nsimulation at different values of \u0011to locate a possible transition as a function of \f. Note that with\nincreasing\fwe decrease the lattice spacing a(\f), such that increasing \fcorresponds to increasing the\ntemperature T= 1=(a(\f)Nt).\nIn the rhs. plot of Fig. 3 we show the expectation value of the absolute value of the Polyakov loop\nhjPji=hjP\n~ nP(~ n)ji=N3\ns. For all\u0011we observe a fast increase of hjPjifor values of \fin the range\nbetween\f= 5:2and 5.8. For \u0011= 0this increase corresponds to the first order phase transition (in\nthe thermodynamical limit) that leads from the confined to the deconfined phase. This transition can\nbe understood as spontaneous breaking of the Z3center symmetry. A non-zero \u0011breaks this symmetry\nexplicitly and thus one expects that above some critical value of \u0011the phase transition turns into a\ncrossover. We also observe that the position of the transition is shifting towards smaller \f(i.e., towards\nsmaller temperature) when increasing \u0011. We illustrate the physical picture in the schematic plot on the\nlhs. of Fig. 4: The full red curve in the \u0016= 0plane indicates the line of first order transitions, which\nbends towards smaller \fwhen\u0011is increased. Based on the argument with the explicit breaking of center\nsymmetry discussed above, beyond some critical \u0011we expect only crossover type of behavior which we\nindicate by using a dashed instead of a full curve for that pseudo-critical line.\nIn the diagram in the lhs. plot of Fig. 4 we also display the \u0016axis. When considered in the space\nof all three parameters \f;\u0011and\u0016we have a (pseudo-) critical surface that separates the confined and\ndeconfined phases. This surface runs through the (pseudo-) critical line in the \f-\u0011plane. An interesting\nquestion is how this surface bends for \u0016>0, and in the figure we show in blue the trajectories (straight\n6lines parallel to the \f-axis at finite \u0011and different values of \u0016) along which we compute our observables\nto explore the bending of the (pseudo-) critical surface.\nThe results forhIm Pi(lhs. plot) and \u001fIm P(rhs.) as a function of \fat different values of \u0016are\npresented in Fig. 5. Both observables show a maximum at the transition between confinement and\ndeconfinement (an exception is hIm Piwhich vanishes at \u0016= 0as discussed above). We observe that\nfor increasing chemical potential the maxima and thus the transition shift towards smaller \f, i.e., towards\nsmaller temperature. In order to determine the critical line in the \u0016-\fplane we fit the data points near\nthe maxima of \u001fIm Pwith a cubic polynomial and so determine the peak positions of \u001fIm Pas a function\nof\ffor different values of \u0016, i.e., we determine \fc(\u0016). The results of this determination are used for the\nplot of the phase diagram in the rhs. plot of Fig. 4.\nFor the presentation of the results for the (pseudo-) critical line in the rhs. plot of Fig. 4 we converted\nlattice units to physical units using the scale determined for the Wilson gauge action in [28]. On the\nvertical axis we use the temperature Tin units of the critical temperature at vanishing chemical potential,\ni.e., we plot T=Tc(0). On the horizontal axis we plot the baryon chemical potential \u0016B= 3\u0016in units\nof the critical temperature at that \u0016, i.e., the combination 3\u0016=Tc(\u0016). The results of our determination\nofTc(\u0016)from the maxima of the susceptibility are shown as asterisks in the rhs. plot of Fig. 4. With\nincreasing\u0016we observe the bending of the (pseudo-) critical line towards lower temperature values as\nexpected also in full QCD. This bending can be be quantified with the curvature \u0014defined via the relation\n(again we use \u0016B= 3\u0016)\nTc(\u0016)\nTc(0)= 1\u0000\u0014\u00123\u0016\nTc(\u0016)\u00132\n+O \u00123\u0016\nTc(\u0016)\u00134!\n: (10)\nThe fit of our data with this quadratic polynomial is shown as the full curve in the rhs. plot of Fig. 4. The\nsmall-\u0016data are described reasonably well and we obtain a value of \u0014= 0:012(3)for the curvature. We\nstress that this result is of course a very crude estimate, since we work with only a single lattice spacing\nand do not attempt a thermodynamic limit. Nevertheless it is interesting to note that our result is in\nthe vicinity of the curvature values published for full QCD in different settings, e.g., \u0014= 0:0135(2)[29],\n\u0014= 0:0149(21) [30] and\u0014= 0:020(4)[31].\n5 Concluding remarks\nIn this letter we have presented an exploratory study where the Density of States Functional Fit Approach\nDoS FFA was implemented for SU(3) lattice gauge theory with static color sources. The purpose is to\nfurther develop the DoS FFA method towards its use in a full lattice QCD simulation at finite density.\nThe key challenge of DoS calculations is the determination of the density \u001a(x)with sufficient precision,\nsuch that one can reliably determine physical observables by integrating \u001a(x)with the oscillating factor,\nwhere the frequency increases exponentially with the chemical potential.\nIn our test we demonstrate that for the system of SU(3) lattice gauge theory with static color sources\nthe DoS FFA method can be implemented and the accuracy is sufficient for an evaluation of hIm Piand\n\u001fIm P. Comparing the DoS results to a conventional simulation at \u0016= 0shows good agreement and for\n\u0016 >0the observables and the critical line could be determined up to moderately large values of \u0016. A\ndetermination of the curvature \u0014in the\u0016-Tphase diagram gives a value which is surprisingly close to\nthe results published for full QCD.\nWe stress again, that the results presented here should not be considered as a final determination\nof the phase structure of SU(3) lattice gauge theory with static color sources, since infinite volume\n7µ= 0ηβµ\n0.840.880.920.961.00\n0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5T/TC(0)\n3µ/T C(µ)Figure 4: Lhs. plot: Schematic sketch of the phase diagram in the \u0016= 0plane and illustration of\nthe trajectories in coupling space for the runs at \u0016 >0. The red curve in the \f-\u0011plane at\u0016= 0\nillustrates the phase boundary between the confined (small \f) and the deconfined region. We use\na dashed line to indicate that above some critical \u0011one expects only a crossover type of behavior,\nwhile at small \u0011the deconfinement transition is of first order (full curve). The blue lines at different\nnon-zero values of \u0016illustrate the lines in parameter space along which we evaluate observables to\nprobe the curvature of the critical surface. Rhs. plot: Results for the critical line in the \u0016-Tplane\nat fixed\u0011= 0:04. The data points on the (pseudo-) critical line were determined as the maxima of a\ncubic fit of the data points for \u001fIm P. In the\u0016-Tplane we fit the data with a quadratic polynomial\nto determine the curvature \u0014as explained in the text (the result of this fit is shown as full curve).\n0.0000.0050.0100.0150.020\n5.40 5.45 5.50 5.55 5.60 5.65 5.70 5.75 5.80⟨Im[P]⟩\nβµ=0.000\nµ=0.075\nµ=0.150\nµ=0.250\nµ=0.350\n0.020.040.060.080.100.120.140.160.180.20\n5.40 5.45 5.50 5.55 5.60 5.65 5.70 5.75 5.80χIm[P]\nβµ=0.000\nµ=0.075\nµ=0.150\nµ=0.250\nµ=0.350\nFigure 5: Results for the imaginary part of the Polyakov loop hIm Pi(lhs. plot) and its susceptibility\n\u001fIm P(rhs.) as a function of \f(83\u00024; \u0011= 0:04and several values of \u0016). The symbols are the results\nfrom the DoS FFA calculation and the dashed curves near the maxima of the \u001fIm Prepresent the fit\nof the data.\n8extrapolation and continuum limit were not attempted here. The purpose of the paper is to document\nthe further development of DoS FFA and to explore the steps necessary towards getting the method ready\nfor a full QCD calculation.\nAcknowledgments: This work was supported by the Austrian Science Fund, FWF, DK Hadrons in\nVacuum, Nuclei, and Stars (FWF DK W1203-N16) and we thank Kurt Langfeld, Biagio Lucini and\nPascal Törek for discussions.\nReferences\n[1] K. Langfeld, PoS LATTICE 2016010 [arXiv:1610.09856 [hep-lat]].\n[2] H.T. Ding, PoS LATTICE 2016022 [arXiv:1702.00151 [hep-lat]].\n[3] S. Borsanyi, PoS LATTICE 2015(2016) 015 [arXiv:1511.06541 [hep-lat]].\n[4] D. Sexty, PoS LATTICE 2014(2014) 016 [arXiv:1410.8813 [hep-lat]].\n[5] C. Gattringer, PoS LATTICE 2013(2014) 002 [arXiv:1401.7788 [hep-lat]].\n[6] G. Aarts, PoS LATTICE 2012(2012) 017 [arXiv:1302.3028 [hep-lat]].\n[7] U. Wolff, PoS LATTICE 2010(2010) 020 [arXiv:1009.0657 [hep-lat]].\n[8] P. de Forcrand, PoS LATTICE 2009(2009) 010 [arXiv:1005.0539 [hep-lat]].\n[9] S. 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Günther, S. D. Katz, C. Ratti and K. K. Szabo, Phys.\nLett. B 751(2015) 559 [arXiv:1507.07510 [hep-lat]].\n[31] P. Cea, L. Cosmai and A. Papa, Phys. Rev. D 93(2016) 014507 [arXiv:1508.07599 [hep-lat]].\n10"    },    {        "title": "1703.08328v1.Spin_polarized_local_density_of_states_in_the_vortex_state_of_helical_p_wave_superconductors.pdf",        "content": "Spin-polarized local density of states\nin the vortex state of helical p-wave superconductors\nKenta K. Tanaka,1,\u0003Masanori Ichioka,1, 2,yand Seiichiro Onari1, 2\n1Department of Physics, Okayama University, Okayama 700-8530, JAPAN\n2Research Institute for Interdisciplinary Science, Okayama University, Okayama 700-8530, JAPAN\n(Dated: March 3, 2022)\nProperties of the vortex state in helical p-wave superconductor are studied by the quasi-classical\nEilenberger theory. We con\frm the instability of the helical p-wave state at high \felds and that\nthe spin-polarized local density of states M(E;r) appears even when Knight shift does not change.\nThis is because the vorticity couples to the chirality of up-spin pair or down-spin pair of the helical\nstate. In order to identify the helical p-wave state at low \felds, we investigate the structure of the\nzero-energy M(E= 0;r) in the vortex states, and also the energy spectra of M(E;r).\nI. INTRODUCTION\nThe superconductor (SC) Sr 2RuO 4has attracted much\nattention as a topological SC, since exotic quantum states\nsuch as a Majorana state are expected in the vortex and\nsurface states. A lot of experimental and theoretical\nstudies support that Sr 2RuO 4is a spin-triplet chiral p-\nwave SC1,2. On the other hand, the helical p-wave state\nalso has been suggested as another scenario3{5. This is\nbecause the detailed structure of d-vector in Sr 2RuO 4\nremains unclear. In addition, the di\u000berence of conden-\nsation energy between chiral and helical states is very\nsmall compared to the transition temperature6. There-\nfore, we need methods to distinguish between chiral and\nhelical states in experiments for Sr 2RuO 4or other can-\ndidate materials for spin-triplet SC. For the purpose, it\nis necessary that we study a unique behavior of physical\nquantity depending on the symmetry of d-vector.\nIn the bulk state of chiral SC, the time-reversal sym-\nmetry is broken because of the angular momentum of\nCooper pair Lz6=0. The chirality of chiral p-wave state,\ni.e.,Lz=\u00061 can be distinguished via coherence e\u000bect in\nthe vortex state. In fact, previous theories suggested that\nthe impurity e\u000bects on the local density of states (LDOS)\nand local NMR relaxation rate T\u00001\n1show di\u000berent behav-\niors between p+andp\u0000states7{11. This chirality depen-\ndence is caused by the interaction between the chirality\nand the vorticity, depending on whether the chirality is\nparallel (Lz= +1) or anti-parallel ( Lz=\u00001) to the\nvorticity (W= +1)12,13. On the other hand, in the bulk\nstate of helical p-wave SC, the time-reversal-invariant su-\nperconductivity appears since Lz=\u00061 are quenched\nwith the degeneracy between up-spin and down-spin\npairs. The up-spin (down-spin) pair's order-parameter\n\u0001\"\"(\u0001##) characterized by Sz= +1(\u00001) has chirality\nLz=\u00001(+1) so that the bulk condition Lz+Sz= 03.\nTherefore, in the vortex state of helical p-wave SC, spin\nstates of low-energy excitations may show a unique be-\nhavior, re\recting the vorticity coupling to the chirality of\n\u0001\"\"(Lz=\u00001) or \u0001##(Lz= +1).\nThe scanning tunneling microscopy and spectroscopy\n(STM/STS) measurement can directly detect the LDOS\nvia excitations in the vortex state14,15. Recently, theSTM/STS measurement in the vortex state of topologi-\ncal insulator-superconductor Bi 2Te3=NbSe 2heterostruc-\nture has performed16, and theoretical studies for the mea-\nsurement have supported the existence of Majorana zero-\nenergy mode in the vortex core17,18. Moreover, spin\npolarization of Majorana zero-energy modes are inves-\ntigated by the spin-polarized STM/STS measurement,\nwhich can selectively detect the spin-dependent conduc-\ntance19. The spin polarization in the vortex state of topo-\nlogical SC Cu xBi2Si3is also theoretically studied20.\nIn this paper, we study properties of the helical p-wave\nSC, and focus on the spin-polarized LDOS in the vortex\nlattice state, in order to reveal a unique behavior of the\nhelical state. In particular, we calculate the structure of\nthe zero-energy spin-polarized LDOS at low \felds, and\nalso the energy spectra. These results help to investigate\nthe vortex state of helical p-wave SC and Majorana zero-\nenergy state by spin-polarized STM/STS measurement.\nThis paper is organized as follows. After the introduc-\ntion, we describe our formulation of the quasi-classical\nEilenberger equation in the vortex lattice state and the\ncalculation method for the spin-resolved LDOS in Sec.\nII. In Sec. III, we investigate the H-dependence of order-\nparameter, and examine the instability of the helical state\nat high \felds. In Sec. IV, we show the H-dependence of\nthe zero-energy spin-polarized DOS and LDOS. The E-\ndependence of the spin-polarized LDOS is presented in\nSec. V. The last section is devoted to the summary.\nII. FORMULATION\nWe calculate the spatial structure of vortices in the\nvortex lattice state by quasi-classical Eilenberger theory.\nThe quasi-classical theory is valid when the atomic scale\nis small enough compared to the superconducting coher-\nence length. For many SCs including Sr 2RuO 4, the quasi-\nclassical condition is well satis\fed1,2. Moreover, since our\ncalculations are performed in the vortex lattice state, we\ncan obtain the structure of LDOS quantitatively.\nFor simplicity, we consider the helical p-wave pairing\non the two-dimensional cylindrical Fermi surface, k=\n(kx;ky) =kF(cos\u0012k;sin\u0012k), and the Fermi velocity vF=arXiv:1703.08328v1  [cond-mat.supr-con]  24 Mar 20172\nvF0k=kF. In the following, the symbol of hat indicates\nthe 2\u00022 matrix in spin space and the symbol of check\nindicates the 4\u00024 matrix in particle-hole and spin spaces.\nTo obtain quasi-classical Green's functions \u0014 g(i!n;r;k)\nin the vortex lattice state, we solve Riccati equation de-\nrived from Eilenberger equation21\n\u0000iv\u0001r\u0014g(i!n;r;k) =1\n2[i~!n\u0014\u001bz\u0000\u0014\u0001(r;k);\u0014g(i!n;r;k)] (1)\nin the clean limit, where ris the center-of-mass coordi-\nnate of the pair, v=vF=vF0, \u0014\u001bzis the Pauli matrix,\nand i~!n= i!n\u0000v\u0001Awith Matsubara frequency !n. The\nquasi-classical Green's function and order parameter are\ndescribed by\n\u0014g(i!n;r;k) =\u0000i\u0019\u0014^g(i!n;r;k) i ^f(i!n;r;k)\n\u0000i^f(i!n;r;k)\u0000^g(i!n;r;k)\u0015\n;(2)\n\u0014\u0001(r;k) =\u0014\n0 ^\u0001(r;k)\n\u0000^\u0001y(r;k) 0\u0015\n(3)\nwhere \u0014g2=\u0000\u00192\u00141. The spin spaces of ^ gand ^\u0001 are\nde\fned by the matrix elements g\u001b\u001b0(i!n;r;k) =\n[g0(i!n;r;k)^1 +P\n\u0016=x;y;zg\u0016(i!n;r;k)^\u001b\u0016]\u001b\u001b0 and\n\u0001\u001b\u001b0(r;k) = [iP\n\u0016=x;y;z(d\u0016(r;k)\u0001^\u001b\u0016)^\u001by]\u001b\u001b0where\n\u001b;\u001b0=\"(up-spin) or#(down-spin), and d\u0016is\u0016-\ncomponent of d-vector. In addition, the matrix elements\nof order-parameter are de\fned by\n\u0001\u001b\u001b0(r;k) = \u0001 +;\u001b\u001b0(r)\u001ep+(k) + \u0001\u0000;\u001b\u001b0(r)\u001ep\u0000(k)(4)\nwith the order-parameter \u0001 \u0006;\u001b\u001b0(r) and pairing function\n\u001ep\u0006(k) =kx\u0006ikyforp\u0006-state. Length, temperature, and\nmagnetic \feld are, respectively, measured in unit of \u00180,\nTc, andB0. Here,\u00180= \u0016hvF0=2\u0019kBTc,B0=\u001e0=2\u0019\u00182\n0with\nthe \rux quantum \u001e0.Tcis superconducting transition\ntemperature at a zero magnetic \feld. The energy E, pair\npotential \u0001 and !nare in unit of \u0019kBTc. In the following,\nwe set \u0016h=kB= 1. In this study, our calculations are\nperformed at T= 0:5Tc.\nWe set the magnetic \feld along the zaxis. The vec-\ntor potential A(r) =1\n2H\u0002r+a(r) in the symmet-\nric gauge. H= (0;0;H) is a uniform \rux density, and\na(r) is related to the internal \feld B(r) = (0;0;B(r)) =\nH+r\u0002a(r). The unit cell of the vortex lattice is set\nas square lattice1.\nTo determine the pair potential ^\u0001(r) and the quasi-\nclassical Green's functions selfconsistently, we calculate\nthe order-parameter ^\u0001\u0006(r) by the gap equation\n^\u0001\u0006(r) =gN0TX\nj!nj\u0014!cutD\n\u001e\u0003\np\u0006(k)^f(i!n;r;k)E\nk;(5)\nwhereh:::ikindicates Fermi surface average, ( gN0)\u00001=\nlnT+ 2TP\n0<!n\u0014!cut!\u00001\nn, and we use !cut= 20kBTc.\nIn Eq. (5), p-wave pairing interaction is isotropic in spin\nspace. For the selfconsistent calculation of the vector\npotential for the internal \feld B(r), we use the current\nequationr\u0002(r\u0002A) =\u00002T\n\u00142P\n0<!nhvImfg0gikwith\nFIG. 1.H-dependence of the spatial average of the order-\nparameter amplitudes hj\u0001\u0000;##jir,hj\u0001+;##jir,hj\u0001\u0000;\"\"jirand\nhj\u0001+;\"\"jirde\fned by Eq. (4). The helical p-wave state is\nunstable at H > 0:35Hc2, and changes to a chiral p-wave\nstate where d?H.\nthe Ginzburg-Landau parameter \u0014=B0=\u0019kBTcp8\u0019N0.\nIn our calculations, we use \u0014= 2:7 appropriate to\nSr2RuO 4as a candidate material for the chiral or helical\np-wave SC. We iterate calculations of Eqs. (1)-(5) for !n\nuntil we obtain the selfconsistent results of A(r),^\u0001(r)\nand the quasi-classical Green's functions in the vortex\nlattice state.\nIn the helical p-wave SCs, d-vector is given by\nd(k)/kx^x+ky^y=\u001ep+(k)d\u0000+\u001ep\u0000(k)d+in uniform state\nat a zero \feld, with d\u0006(k) =1\n2(1;\u0006i;0). Thus, when we\niterate calculations of Eq.(1)-(5), the initial value of d-\nvector is set to be d(r;k) =d(r)(kx^x+ky^y) whered(r)\nis Abrikosov vortex lattice solution.\nNext, using the selfconsistently obtained A(r) and\n\u0001(r), we calculate \u0014 g(E\u0006i\u0011;r;k) for real energy Eby\nsolving Eilenberger eq. (1) with i !n!E\u0006i\u0011.\u0011is a\nsmall parameter, and we use \u0011= 0:01 in this paper ex-\ncept for the calculations of distribution in Figs. 4(d) and\n4(e), and Figs. 5(d) and 5(e). The spin-resolved LDOS\nN\u001b(E;r) is given by\nN\u001b(E;r) =hRef[^g(E+ i\u0011;r;k)]\u001b\u001bgik: (6)\nWe de\fne the LDOS N(E;r) =N#(E;r)+N\"(E;r), and\nspin-polarized LDOS M(E;r) =N#(E;r)\u0000N\"(E;r).\nIII.H-DEPENDENCE OF\nORDER-PARAMETER\nIn order to examine the instability of helical p-wave\nstate at high H, we show the H-dependence of spatial\naverage of the order-parameter amplitude, hj\u0001\u0006;\u001b\u001b0(r)jir\nde\fned by Eq. (4) in Fig. 1. Using the initial state\nof helical states, \u0001 #\"and \u0001\"#components do not ap-\npear in the selfconsistent calculations of our model. In\nthe vortex state of helical p-wave SC at H < 0:35Hc2,\nup-spin pair has a form \u0001 \"\"(r;k) = \u0001\u0000;\"\"(r)\u001ep\u0000(k) +\n\u0001+;\"\"(r)\u001ep+(k) with sub component \u0001 +;\"\"(r). The3\nmain component \u0001 \u0000;\"\"(r) has chirality Lz=\u00001, anti-\nparallel to vorticity W= +1 asLz+W= 0. The\nsub component \u0001 +;\"\"(r) is induced around the vortex\ncore. Since the local winding number can be a value\nother than W= +1 in the induced components, the sub\ncomponent with Lz= +1 has inverse winding number\nW=\u00001 to satisfy the conservation of Lz+W= 0.11\nAccording to the previous studies for the vortex state\nof chiralp-wave SC12,13, the anti-parallel vortex state\n(Lz+W= 0) is stable compared with the parallel vortex\nstate (Lz+W= +2) by the interaction between the chi-\nrality and the vorticity. Therefore, the H-dependence of\nhj\u0001\u0000;\"\"jirandhj\u0001+;\"\"jirshow same behavior to those\nfor anti-parallel case in a chiral p-wave SC12, and the\namplitude survives until Hc2.\nOn the other hand, down-spin pair has a form\n\u0001##(r;k) = \u0001 +;##(r)\u001ep+(k) + \u0001\u0000;##(r)\u001ep\u0000(k) at low\n\felds, with sub component \u0001 \u0000;##(r). Since the chiral-\nityLz= +1 of main \u0001 +;##(r) is parallel to vorticity as\nLz+W= +2, \u0001##(r;k) is rapidly suppressed as a func-\ntion ofH, as shown in Fig. 1. In addition, at H\u00180:35Hc2,\nwe \fnd the change of chirality Lz= +1!\u00001 in \u0001##(r;k),\nwhere \u0001\u0000;##(r;k) changes to be main part of \u0001 ##(r;k)\nfrom the sub component. At H > 0:35Hc2,hj\u0001\u0000;##jiris\nequal tohj\u0001\u0000;\"\"jiras main components and hj\u0001+;##jir\nis equal tohj\u0001+;\"\"jiras sub components, so that the\norder-parameter is chiral p\u0000form. Even in this chiral\nstate, \u0001#\"= \u0001\"#= 0 so that d?H. Therefore, the he-\nlicalp-wave state becomes unstable at high \felds by the\ne\u000bect of vorticity coupling to the chirality, and changes\nto a chiral state.\nIn our model, we assume that the helical state can ap-\npear in the Meissner state H= 0, since condensation en-\nergy of the helical state is the same as chiral state. The\nhelical state can be more stable than the chiral state,\nif we consider additional mechanism such as weak spin-\norbit coupling e\u000bect4. Even when very small number of\nvortices penetrate to the helical p-wave SC, we expect\nthat the helical state can be sustained at the low \felds.\nWith increasing H, it becomes metastable state, and \f-\nnally show instability to the chiral state. The instability\n\feldHcan be shifted from our estimation of Fig. 1.\nIV.H-DEPENDENCE OF ZERO-ENERGY\nSPIN-POLARIZED DOS AND LDOS\nIn this section, to \fnd di\u000berence of observed quan-\ntities between helical and chiral states, we investigate\nthe characteristic behavior of helical state under the as-\nsumption that the helical p-wave state is sustained at low\nH(<0:35Hc2).\nFirst, we study the H-dependence of the zero-energy\nDOShN(E= 0;r)ir, the zero-energy spin-resolved DOS\nhN\u001b(E= 0;r)irand the zero-energy spin-polarized\nDOShM(E= 0;r)ir. As shown in Fig. 2(a), the H-\ndependence ofhN\"(E= 0;r)irshows the typical behav-\nior, which is same behavior in the anti-parallel vortex\nFIG. 2. (a) H-dependence of DOS hN(E= 0;r)ir=2,\nspin-resolved DOS hN\u001b(E= 0;r)irand spin-polarized DOS\nhM(E= 0;r)ir. The distributions of zero-energy (b)\nLDOSN(E= 0;r)\u00143 and (c) spin-polarized LDOS M(E=\n0;r)\u00140:3 atH'0:12Hc2. The brighter region indicates the\nlarge value of NorM.\nstate of chiral p-wave SC12. On the other hand, the\nH-dependence ofhN#(E= 0;r)iratH < 0:35Hc2is\nlarger thanhN\"(E= 0;r)ir. AtH > 0:35Hc2, since \u0001##\nand \u0001\"\"have same chirality, hN#(E= 0;r)ir=hN\"(E=\n0;r)ir. Here, contributions of the Zeeman e\u000bect are ab-\nsent since d?H. As a result, the H-dependence of DOS\nhN(E= 0;r)irshows a jump when the helical state be-\ncomes unstable in Fig. 2(a). The jump behavior may be\nobserved by the low temperature speci\fc heat measure-\nment. When the instability \feld shifts into high (low) H,\nthe jump of speci\fc heat becomes larger (smaller).\nTheH-dependence ofhM(E= 0;r)irat low \felds has\na \fnite value and shows increasing behavior, re\recting\nthehN#(E= 0;r)irbehavior in Fig. 2(a). And, it jumps\nto zero when the helical state becomes unstable. At\nhigh \felds as the vortex state of chiral p-wave SC, where\n\u0001##= \u0001\"\",Mvanishes. This H-dependence of Mis the\nunique behavior of the helical p-wave state. In addition,\nFigs. 2(b) and 2(c) show the LDOS and spin-polarized\nLDOS distributions at a low \feld H'0:12Hc2, which\nhave large amplitudes around the vortex core. Since the\nzero energy state localized around the vortex core is Ma-\njorana state in the chiral and helical SCs, Fig. 2(c) shows\nthat the Majorana state is spin-polarized in the helical\np-wave SCs. This is another type of spin-polarized zero\nenergy state than that supposed in Bi 2Te3=NbSe 218or\nCuxBi2Si320.\nNext, we present the structure of spin-polarized LDOS\nM(E;r) at low \felds to study the properties of the vor-\ntex state of helical p-wave SC. Figure 3 presents the H-\ndependence of M(E= 0;r) andN\u001b(E= 0;r) at some\npositions on a line between next-nearest-neighbor (NNN)\nvortices atH < 0:5Hc2. Atr=ax= 0:5 which is midpoint\nof between NNN vortices, N#(E= 0;H)>N\"(E= 0;H)\nand their magnitudes are small and monotonically in-4\nFIG. 3. (a), (b), (c) H-dependence of spin-resolved LDOS\nN\u001b(E= 0;r) and spin-polarized LDOS M(E= 0;r) at radius\nr=ax= 0.5, 0.1, 0.0 from the vortex center along the NNN\ndirection, respectively. axis NNN intervortex distance.\ncrease as a function of H. On the other hand, at the vor-\ntex core region in Figs. 3(b) and 3(c), M(E= 0;r) shows\na large amplitude at some \felds in the helical state. In\nparticular, at the vortex center in Fig. 3(c), M(E= 0;r)\natH=Hc2'0.02 shows much larger value than the normal\nstate DOS(= 1), while it monotonically decrease with\nraisingH. These large values of M(E= 0;r) may be\nobserved by the spin-polarized STM measurement.\nV.E-DEPENDENCE OF SPIN-POLARIZED\nLDOS\nFinally, we study the E- andr-dependences of\nN\u001b(E;r) andM(E;r) in order to investigate the be-\nhavior of LDOS spectrum of spin-polarized STM/STS\nmeasurement. When N\"(E;r=0) is compared\nwithN#(E;r=0) at a low \feld H'0:02Hc2, shown\nin Figs. 4(a)-(c), the height of zero-energy peak in\nN\"(E;r=0) is smaller, and instead the gap edges\natE\u0018\u00060:5 have small peak. Thus, M(E;r=0) is\npositive at E= 0, and negative at E\u0018\u00060:5. These\nweights cancel each other, so that total spin polarizationR0\n\u00001M(E;r)dE= 0. This condition can be extended to\n\fniteTasR1\n\u00001M(E;r)F(E;T)dE= 0 with Fermi dis-\ntribution function F(E;T) sinceM(E;r) is even func-\ntion ofE. The absence of total spin polarization corre-\nsponds to the fact that Knight shift is invariant in the\nhelicalp-wave state, where d?H. To observe the spin-\npolarized LDOS in the helical state, we have to perform\nE-resolved observation such as spin-polarized STM/STS.\nTher-dependence of spectra N\u001b(E;r) andM(E;r) are\nFIG. 4. (a), (b), (c) E-dependence of spin-resolved LDOS\nN#,N\"and spin-polarized LDOS Mat the vortex center at\nH=Hc2'0.02, respectively. (d), (e) E-dependence of N\u001b(E;r)\nfor\u001b=#;\", andM(E;r) as a function of radius r=axfrom\nthe vortex center along the NNN direction at H=Hc2'0.02,\nrespectively. N\u001b(\u0000E;r) =N\u001b(E;r). In (d) and (e), we use\n\u0011= 0:03.\npresented in Figs. 4(d) and 4(e), respectively. When\nwe focus on the dispersion curve of brighter region in\nFig. 4(d), the zero-energy peak at r= 0 evolves toward\nthe gap-edge with increasing r. Since the zero-energy\nvortex bound state connects with the gap-edge state at\nsmallerrforN\"thanN#, the e\u000bective vortex core ra-\ndius is smaller for N\". Therefore, in N\", the peaks of the\ngap edge (E\u0018\u00060:5) outside vortices can extend until the\nvortex center, as shown in Fig. 4(b). In Fig. 4(e), we see\nthat the spin-polarized state appears near the dispersion\ncurve of vortex bound state extending from the Majorana\nzero mode, in addition to gap edges.\nMoreover, we show the E- andr-dependences of\nN\u001b(E;r) andM(E;r) at a higher \feld H'0:29Hc2,\nconsidering that the helical p-wave state is still sus-\ntained at higher H. In Figs. 5(a)-(c), the hight of zero-\nenergy peak of N\"is larger than N#, resulted in negative\nM(E= 0;r=0). To compensate negative value at\nE= 0 and at the gap edge, M(E;r= 0) becomes pos-\nitive for in-gap states for 0 <jEj<0:5. As shown in\nFigs. 5(d) and 5(e), since the down-spin's in-gap states\nhave a larger value compared with the up-spin states,\nM(E;r) has \fnite distributions at 0 <jEj<0:5 even\nfar from dispersion curve of bound state.\nVI. SUMMARY\nWe studied the vortex state of helical p-wave SCs based\non the quasi-classical Eilenberger theory. We con\frmed5\nFIG. 5. (a), (b), (c) E-dependence of spin-resolved LDOS\nN#,N\"and spin-polarized LDOS Mat the vortex center at\nH=Hc2'0.29, respectively. (d), (e) E-dependence of N\u001b(E;r)\nfor\u001b=#;\", andM(E;r) as a function of radius r=axfrom\nthe vortex center along the NNN direction at H=Hc2'0.29,\nrespectively. N\u001b(\u0000E;r) =N\u001b(E;r). In (d) and (e), we use\n\u0011= 0:03.the instability of the helical p-wave state at high \felds\nand that the spin-polarized LDOS M(E;r) appears even\nwhen Knight shift does not change. This is because the\nvorticity couples to the chirality of up- or down-spin pair\nof helical state. In addition, we found that the mag-\nnetic \feld dependence of zero-energy DOS shows a jump\nwhen the helical state becomes unstable. This jump be-\nhavior may be observed by the low temperature speci\fc\nheat measurement. In order to identify the helical p-wave\nstate at low \felds, we investigated the structure of the\nzero-energy M(E= 0;r) in the vortex states. In partic-\nular, at the vortex center, the value of M(E= 0;r= 0)\nat a low \feld H=Hc2'0.02 shows much larger value\nthan the normal state DOS, while it monotonically de-\ncrease with raising \feld. Moreover, we present the E-\nandr-dependences of the spin-resolved LDOS N#(E;r),\nN\"(E;r) andM(E;r) in the vortex state. We hope\nthat these theoretical calculation results of spin-polarized\nLDOS will be examined, and will be used for detecting\nthe spin-polarized Majorana zero-energy modes by the\nspin-polarized STM/STS measurement.\nACKNOWLEDGMENTS\nThis work was supported by JSPS KAKENHI Grant\nNumber JP16J05824.\n\u0003ktanaka@mp.okayama-u.ac.jp\nyichioka@cc.okayama-u.ac.jp\n1A. P. Mackenzie and Y. Maeno, Rev. Mod. Phys. 75, 657\n(2003).\n2Y. Maeno, S. Kittaka, T. Nomura, S. Yonezawa, and K.\nIshida, J. Phys. Soc. Jpn. 81, 011009 (2012).\n3T. M. Rice and M. Sigrist, J. Phys.: Condens. Matter 7,\nL643 (1995).\n4S. Takamatsu and Y. Yanase, J. Phys. Soc. Jpn. 82, 063706\n(2013).\n5T. Sca\u000edi, J. C. Romers, and S. H. Simon, Phys. Rev. B\n89, 220510(R) (2014).\n6M. Tsuchiizu, Y. Yamakawa, S. Onari, Y. Ohno, and H.\nKontani, Phys. Rev. B 91, 155103 (2015).\n7Y. Kato and N. Hayashi, Physica C 388, 519 (2003).\n8Y. Tanuma, N. Hayashi, Y. Tanaka, and A. A. Golubov,\nPhys. Rev. 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Gao, D.-D. Guan, Y.-Y. Li, C.\nLiu, D Qian, Y. Zhou, L. Fu, S.-C. Li, F.-C. Zhang, and\nJ.-F. Jia, Phys. Rev. Lett. 116, 257003 (2016).\n20Y. Nagai, H. Nakamura, and M. Machida, J. Phys. Soc.\nJpn.83, 064703 (2014).\n21Y. Tsutsumi, T. Kawakami, K. Shiozaki, M. Sato, and K.\nMachida, Phys. Rev. B 91, 144504 (2015)."    },    {        "title": "1704.01640v1.Transition_of_the_uniform_statistical_field_anyon_state_to_the_nonuniform_one_ar_low_particle_densities.pdf",        "content": "arXiv:1704.01640v1  [cond-mat.mes-hall]  6 Apr 2017TRANSITION OF THE\nUNIFORM STATISTICAL\nFIELD ANYON STATE\nTO THE NONUNIFORM\nONE AT LOW PARTICLEDENSITIES.\nM. Hudak1\nO. Hudak2\n1Stierova 23, Kosice\n2Department of Aerodynamics and Simulations, Fac-\nulty of Aviation, Technical University Kosice, Kosice\n1Abstract\nUsing results of our exact description of the spinless fermion motion in a nonho-\nmogeneous magnetic field B=B(0,0,1/cosh2(x−x0\nδ)) we study a gas of these\nparticles moving in this field. For lower densities ν < νc(B,δ) the corresponding\ntotal energy is lower than that of the uniform field state. Thus whe n the density\nof anyons decreases a transition from the uniform statistical field state to the\nnonhomogeneous field state is predicted.\n2Non-Abeliananyonsandtopologicalquantumcomputationhasrece ntlyemerged\nas one of the most exciting approaches to constructing a fault-to lerant quan-\ntum computer [1]. Strongly correlated quantum systems can exhibit behavior\ncalled topological order which is characterized by non-local correla tions that\ndepend on the system topology. Such systems can exhibit phenome na such as\nquasi-particles with anyonic statistics and have been proposed as c andidates for\nnaturally fault-tolerant quantum computation. Despite these rem arkable prop-\nerties, anyons have not been observed in the year 2008 directly [2]. Recently\nit was presented an experimental emulation of creating anyonic exc itations in\na superconducting circuit that consists of four qubits, achieved b y dynamically\ngenerating the ground and excited states of the toric code model, i.e., four-qubit\nGreenberger-Horne-Zeilinger states. The anyonic braiding is implem ented via\nsingle-qubit rotations: a phase shift of related to braiding, the hallm ark of\nAbelian 1/2 anyons, has been observed through a Ramsey-type int erference\nmeasurement [3].\nNeverthless it is interesting to study spinless fermion motion in a nonh omo-\ngeneous magnetic field B=B(0,0,1/cosh2(x−x0\nδ)) of these particles moving in\nthis field. For lower densities ν < ν c(B,δ) the corresponding total energy is\nlower than that of the uniform field state. Thus when the density of anyons\ndecreases a transition from the uniform statistical field state to t he nonhomoge-\nneous field state is predicted. When the density of anyonspresent in this system\ndecreases a transition from the uniform statistical field state to t he nonhomo-\ngeneous field state is predicted. This may forward our observation possibilities\nto observe anyon properties.\nAn homogeneous magnetic field B= ( 0, 0, B) strongly influences states of\ncharged fermions moving in an x-y plane perpendicular to the field dire ction.\nLandau energy levels and their degeneracy characterize this motio n, [4]. A gas\nof free spinless fermions has its total energy ET(B,ν) larger or equal to its total\nenergyET(0,ν) = 2πtNν2in the zero field:\n∆Eh(n)≡ET(B,ν)−ET(0,ν) = 2πtN(νn+1−ν)(ν−νn) (1)\nwhereνn+1≥ν > νn,νn≡n.Φ\nΦ0,n= 0,1,...;Φ0is a unit of the magnetic flux,\nΦ≡Ba2,t≡¯h2\n2ma2.Hereνis the number density of the gas, ν≡Nf\nN,Nfis\nthe total number of fermions, Na2is the area of the square with the side length\nL=a√\nN,to which the motion is bounded, m is the fermion mass. Note that\nais a characteristic length of the system, its value is of the order of a lattice\nconstant value. The energy level degeneracy occurs only if a numb er of Landau\nlevels is completely filled (e.i. if for some n νn=ν).Recently it was shown in [5]\nthat in the presence of a periodic lattice potential the ground stat e energy of a\ngas of spinless fermions in an uniform magnetic field in the vicinity of the filled\nlowest Landau level is lower than that in zero field. This problem was st udied\n3further in context of commensurate flux phases, [6]. If a nonhomo geneity of\nthe field is introduced by a local field intensity decrease then compet ition of two\ntendencies is expected to occur: a decreaseof the single fermion e nergylevel due\ntodecreasedvalueofthefieldandadecreaseoftheeveryenergy leveldegeneracy\ndue to larger spacing between centers of neighboring orbits within t he region\nof smaller fields. Spectrum of 2d Bloch electrons in a periodic magnetic field\nwas studied in [7]. Using semiclassical methods authors of this later pa per\ninvestigated the case where the magnetic unit cell is commensurate with the\nlattice unit cell. Their work is in some sense extension of previous stud ies of free\nelectrons in periodic magnetic field [8] to the lattice case. Our aim in this paper\nis to present results of our study of the motion of a spinless fermion gas bounded\nto the square LxL in a nonhomogeneous static magnetic field perpen dicular\nto this plane. We neglect the lattice periodic potential influence on th e gas\nenergy spectrum in this paper. We consider in more details the limit in wh ich\nnonhomogeneity disappears and a uniform field appears. In differen ce to [7], [8]\nand [9] we do not consider a periodic magnetic field. Recently, [10], an e xact\ndescription of motion of the quantum spinless fermion in a nonhomoge neous\nmagnetic field described by the vector potential A= (0,Bδtanh (x−x0\nδ),0) was\nfound. We use these results in this paper to study the stability of th e statistical\nuniform anyon state with respect to a nonuniform field state. First ly using the\nsinglefermionenergyfrom[10]wefindatotalenergyofagasofspin lessfermions\nmoving in our nonhomogeneous field. Then we compare this energy wit h the\ntotal energy of the same gas moving in the uniform field with the same intensity\nB. We have found that at low densities ν < νc(B,δ) the nonhomogeneous field\nstate of the anyon gas is preferred. Occurence of such a kind of in stability has\nconsequences for interpretation of recent experiments [11] sea rching for the T-\nand P- symmetry breaking phenomena due to presence of particles with exotic\nstatistics - anyons.\nIn the case of motion of a quantum spinless fermion in a nonhomogene ous\nmagnetic field described by the vector potential A= (0,Bδtanh (x−x0\nδ),0) the\nenergy spectrum of the motion in the x-direction is splitted, see in [10 ], into\na discrete and a continuous parts for general values of the field B a nd of the\nnonhomogeneity parameter δ.We take x0= 0 in the following, thus field has\nits maximum intensity at x= 0.Let us consider the limit of strong fields ( F≡\n2πΦ′\nΦ>>1/2,where Φ′≡Bδ2) in which case a linear type nonhomogeneity is\nlocalized near the two edges x=±L/2,ifδ >> akeeping L finite. In this limit\nit is sufficient to take into account the lowest energy levels of the spe ctrum. The\neigenvalues of the energy corresponding to this part of the spect rum are given\nby, see in [10] :\nEn(p) =p2\ny\n2m(1−F2\n((1\n4+F2)1\n2−((1/2)+n))2)+\n4(¯h2\n2mδ2)(F2−((1\n4+F2)1\n2−(1\n2+n))2,\nwheren= 0,1,...[nmax],here [n] denotes an integer part of a real number n, py\nis the y-momentum. Let us define P≡|py|δ\n¯h. The number nmaxis defined by:\nnmax= (1\n4+F2)1\n2−(1/2)−(|P|F)1\n2,\nfor given values of P and F.\nThe limit of strong but still nonhomogeneous field is achieved for F−→\n∞keeping the nonhomogeneity parameter δfinite while increasing the field\nintensity B, B−→ ∞. ForF2>>1\n4and for small quantum numbers n the\nenergyEn(py) expanded into series of 1/F powers takes the form :\nEn(py)≈¯hω(n+1\n2)−(¯h2\n2mδ2)((n+1\n2)2+1\n4)−p2\ny\nmF(n+1\n2)+\n(¯h2\n8mFδ2)(n+1\n2)+O(1\nF3).\nwhereω≡Bc\nemis the cyclotron frequency. We see that the energy levels are\ndegenerated in the limit of strong but modulated fields if the energy e xpansion\nabove is restricted to the first two terms, which are of the F1andF0orders\nrespectively. The largest value of the third term in this expansion is n egligible\nwith respect to the second term\nmax(p2\ny\nmF(n+1\n2))<<(¯h2\n2mFδ2)(n+1\n2).\nif we take into account that there exists a natural cut-off for pymomenta,\nmax(|py|) =π¯h\na,due to the underlying crystal and if we assume that the field\nintensity B satisfies the inequality:\nΦ\nΦ0>>8π2,\nwhere Φ ≡B.a2.If the third term and the following terms are not taken into\naccount in calculations of the energy En(py) then the degeneracy of the n-th\nlevel appears due to the lost of the energy dependence on pymomentum. One\ncan say that these levels are, [10], modified Landau levels with energie s in the\nform:\nEn= ¯hω(n+1\n2)−¯h2\n2mδ2[(n+1\n2)2+1\n4]+O(1/F), (2)\nwhere\nn= 0,1,... << n m;nm≈F.\n5Note that\n¯h2\n2mδ2= 4t(L\n2δ)2/N.\nFrom (2) we see that in the strong nonhomogeneous magnetic fields the neigh-\nboring energy levels are not equidistant as in the uniform field case. E very\nenergy level Enremains degenerated within considered approximation, its de-\ngeneracy Dnis found to be:\nDn=DLtanh(L\n2δ)\nL\n2δ, (3)\nif the characteristic length L and the nonhomogeneity parameter δsatisfy\ntanh(L/2δ)<(1−2\nF(n+1\n2)).\nHereDL≡Bea2\nhcNis the Landau level degeneracy as it is given in the case\nof the uniform field. The form of the degeneracy Dngiven above holds for all\norders of F. However, the large F expansion in (2) limits its validity to t he\nregion of system parameters given by the inequality below (3). This in equality\nfollows from the usual, [12], boundary conditions: periodicity in the y- direction\nperpendicular to the x-axis and limits on the position of the orbit cent er in the\nx-direction to the region <−L/2,+L/2> .The orbit center x-coordinate xcis\ngiven, [10], by\ntanh(xc/δ) = (−pyδ\n¯h)/F.\nNote that this relation also reflects the fact that closed particle or bits of their\nmotion in our magnetic field do exist only in the limited regionofsystem pa ram-\neters and of the py−momentum such that tanh function above is note larger\n(or smaller) than 1 (than -1).\nStraightforward calculations of the ground state energy ET(B,δ,ν) for spin-\nless fermion gas with density νin the limit of strong but nonhomogeneous fields\nspecified by B,δlead to the modification of (1). We have found that the energy\ndifference between the nonhomogeneous field state and the zero fi eld state:\n∆Enh(n)≡ET(B,δ,ν)−ET(0,ν)\nis given by the following expression:\n∆Enh(n) = 2πtN[(ν−νntanh(L\n2δ)\nL\n2δ)(νn+1tanh(L\n2δ)\nL\n2δ−ν)+ (4)\n(1−tanh(L\n2δ)\nL\n2δ)(ν(2νn+ν1)−tanh(L\n2δ)\nL\n2δνn+1νn)]−\n6−ta2\nδ2N[ν(n2+n+1\n2)−νn(2n2\n3+n+1\n3)].\nThetotalenergydifference(4)isfoundassumingthattherearen levels0,1,...,n−\n1 filled and that the n-th level is filled partially. The gas density νin (4) is lim-\nited by the following inequalities:\nνn+1tanh(L\n2δ)\nL\n2δ≥ν > νntanh(L\n2δ)\nL\n2δ, (5)\nνn≡nΦ/Φ0.\nTheuniform fieldresult(1) followsfrom(4) and(5)in the limit δ−→ ∞keeping\nvalues of all the other system parameters constant.\nWhen only the lowest energy level n= 0 is filled partially we find from (4)\nand (5) that:\n∆Enh(0) = 2πtNν(ν1−ν)−Nνt(a\nδ)2/2, (6)\nwhere\nν1tanh(L\n2δ)\nL\n2δ≥ν >0.\nThe result (6) holds to the same order as the energy expansion (2) . The filled\nlowest energy level n= 0 corresponds with the density νgiven by:\nν1tanh(L\n2δ)\nL\n2δ=ν. (7)\nIt follows from (7) that there is a decrease of the number of n= 0 states with\nrespect to the uniform field case. In this later case the density at w hich the\nn= 0 state is filled is given by ν1≡Φ\nΦ0.Moreover in our limit of large but still\nnonhomogeneous fields the quantity ν1satisfies the inequality given above (2).\nLetus nowcomparetotalenergiesofourgasata givendensity νbetween the\nn= 0 state in the uniform field B and the n= 0 state in the nonhomogeneous\nmagnetic field B with a finite parameter δ. We obtain from (1) and (6) that\ntheir total energy difference is given by:\nET(B,ν)−ET(B,δ,ν) =Nνt(a\nδ)2/2>0 (8)\nfor\nν1tanh(L\n2δ)\nL\n2δ≥ν >0.\nItfollowsfrom(8)that inthisrangeofdensitiesandofsystempara metersvalues\nthe nonhomogeneous field state has lower energy than that in the h omogeneous\nfield.\n7Let us now increase the gas density νto the value ν1.The lowest energy level\noftheuniformfieldstatebecomesfilled. Letusassumethatthenon homogeneity\nparameter δis large enough and such that the following inequalities hold:\nν1tanh(L\n2δ)\nL\n2δ< ν1<2ν1tanh(L\n2δ)\nL\n2δ.\nThen the n= 0 level of the nonhomogeneous field case is filled completely and\nthen= 1 level of the same case only partially. Let us compare energies of t he\nuniform field state and of the nonhomogeneous state for the dens ityν1=ν.We\nobtain for the difference of the total energies of both states::\nEn=1(B,δ,ν)−En=0(B,ν) = 4πtNν2\n1(1−tanh(L\n2δ)\nL\n2δ)−Nν1t(a\nδ)2(5−4tanh(L\n2δ)\nL\n2δ)/2.\nThis quantity is positive for macroscopically nonvanishing density ν.Decrease\nof the single fermion energy due to the nonhomogeneity is overcomp ensated by\nthe decrease of the number of particles in the n= 0 nonhomogeneous field level\nand by their increase in the n= 1 level. The gap between energies of these two\nlevels is\n¯hω−¯h2\nmδ2+O(1/F),\nthe increase of the number of particles in the n= 1 level increases substantially\nthe total energy of the system. Thus the uniform field state beco mes preferred\nat higher particle densities. Qualitatively the same type of conclusion s holds for\nhigher densities νand higher level numbers n.\nWe conclude that the nonhomogeneous field state of our gas of spin less\nfermions is preferred with respect to the uniform field state of the same gas\nfor densities νless or equal to a critical value νc(B,δ) defined as\nνc(B,δ)≡ν1tanh(L\n2δ)\nL\n2δ.\nFor higher densities ν > νc(B,δ) the later state is preferred.\nOne may ask at which value of the nonhomogeneity parameter δthe energy\ndifference (8) takes the largest value. The difference ET(B,ν)−ET(B,δ,ν)\nfrom (8) becomes larger when (L\nδ)2=N(a\nδ)2is increasing quantity, e.i. when δ\ndecreaseswith respect to the the length L. There exists a critical value ofδgiven\nbyδc≡L/ln(πNν1),. Itisthatlimitingvalueof δforwhichtheinequalitybelow\n(3) becomes equality. Below δcthe degeneracy of every energy level becomes n-\ndependent [10] as it follows from pydependence of nmaxgiven in the beggining\nof this paper. It is possible to find that\nDn≈DL(2δ/L)(1−2\nF(n+1\n2)),\n8if\ntanh(L/2δ)>1−2\nF(n+1\n2).\nThe density νin (8) is for the n= 0 state now from the region:\nν1(2δ/L)(1−1/F)≥ν >0.\nWe have found that for such more localized nonhomogeneity of the fi eld for\nwhichδis smaller, δ < δc,the energy difference depends on ( L/2δ) in the same\nway as in (8). The maximum value of this difference for the filled lowest le vel\nn= 0 of the nonhomogeneous field state is obtained for δ=a(3/2πν1)1/2as:\nET(B,ν)−ET(B,δ,ν) = (4t√\nN/3)ν√πν1/6.\nThe gas density corresponding to this value is found to be ν= (√\n6−\n2\n3)/radicalbig\nν1/πN.This density corresponds to nonzero linear density of particles,\nNf/√\nN.Thus we conclude that in our type of the nonhomogeneous field the\nenergy difference (8) is maximized for small densities of particles. On e can say\nthat it is an edge effect which leads to the maximum of the considered e nergy\ndifference.\nLet us discuss shortly consequences of our results obtained abov e for the\nanyon gas physics. According to [9] and [13] anyons may be describe d as spin-\nless fermions moving in the statistical field generated by the statist ical potential\nAiacting on the i-th anyon and given by:\nAi= (1−ρ)(¯hc/e)zxΣj/negationslash=i(ri−rj)/(|ri−rj|)2.\nThe fractional statistics parameter is denoted here by ρ.A Hartree-Fock consid-\nerations in [9] lead to description of anyons with a single fermion Hamilto nian\nH=1\n2m[p−e\ncA]2,\nwhere the uniform average statistical field is described by the vect or potential\nA= (1−ρ)(¯hcν/a2e)zxr.This potential may be transformed into another\ngauge form:\nA= (1−ρ)(¯hcν/a2e)(0,x−x0,0).\nWe may assume that the potential of our field\nA= (0,Bδtanh ((x−x0)/δ),0)\nis a result of the averaging of the potential Aigiven above over some stationary\nconfigurationsofanyonswhicharedifferentfromthosewhichleadt otheuniform\nstatistical field. Such a modulated form of the average potential m ay results\nfromAiwhen fermions are non- homogeneously distributed within the plane.\nFrom our results described above for the gas of spinless fermions m oving in\nour nonhomogeneous field it follows that for the anyon densities νsuch that\n9inequality\nν1tanh(L\n2δ)\nL\n2δ≥ν >0\nholds ( where ν,L,δare fixed parameters ) the uniform statistical field state\nis unstable with respect to the state with nonhomogeneous statist ical field de-\nscribed by our potential. Our comparison of the total energies bet ween the non-\nhomogeneous field state and the uniform field state as given above s hows that\ntheformerispreferredatzerotemperature. Thisinstabilityeffec toftheuniform\nstatistical field state is larger for smaller densities. However it is pre sent also\nfor macroscopically nonvanishing density of particles. For densities of anyons ν\nhigher than ν1tanh(L\n2δ)\nL\n2δthe uniform statistical field state is preferred for anyons.\nIt is howevernot clear from our results whether our nonhomogene ousfield is the\nmost stable nonhomogeneous field state of anyons at lower densitie s of particles.\nIn our calculations we may consider the characteristic length L to be either the\nlinear dimension of the sample either it may be a characteristic length o f a do-\nmain within the sample. In the later case the whole sample is expected t o be\ncovered by similar domains. From (8) it follows that the most preferr ed value\nof the nonhomogeneity parameter δis that value for which ν=νc(B,δ) when\nthe enrgy difference between both considered states becomes ma ximized.\nThus we see that our results are directly related to the physics of a nyons.\nExperimental evidence for presence of these new physical pheno mena in real\nmaterials is controversial nowadays. There exists some positive ev idence for\nobservation of broken T- and/or P- symmetry in superconductor s based on\noxidic layers, [11]. Some of these experimental results are interpre ted, however,\nas a negative evidence or there is no their clear interpretation. Our results\npresented here may contribute to better understanding why the re exists variety\nof different results obtained under different physical conditions an d in different\nsamples in cited above experiments. If the statistical field varies in t he sample\nthen also measured physical quantities such as the optical axis rot ation angle\nwill vary within the CuO2plane in cuprate perovskites as it may be found f.e.\nfrom the results of analysis in [14] of rotation of polarized light reflec ted from\nT- and P- violating phases. According to results presented in this pa per in\nthose samples in which higher densities of charge carriers occur the uniform\nstatistical field anyon state may be realized. In those samples wher e the density\nof anyons is below some critical value νcthe anyon statistical field becomes\nspatially modulated. Whether this modulation is described by the stat istical\nfield considered in our calculations remains an open problem. Here we h ave\nshown only that such a transition exists, from our results we are no t able to say\nwhich type of field modulation represents the true ground state of anyons. We\nmay expect that some other than our modulation may lead to energe tically (\nwe consider T=0 ) more preferred anyon state.\nResults of this paper point to principal possibility that a phase trans ition\nbetween anyon states: uniform statistical field state and modulat ed statistical\n10field state occurs when the carrier density is decreased. Such a ph ase transition\nmay be experimentally observed under appropriate conditions. Phy sical prop-\nerties of modulated states as well as their response to external s ignals probing\ntheir nature should be established in other to improve our understa nding of the\nexperimental situation in search of broken T-/P- symmetries due t o presence of\nanyons. It is known, [15], that dynamic response of fermions in cont inuum as\nwell as on the lattice in a magnetic field may be calculated. This task is be yond\nthe scope of this paper.\nAcknowledgement:\nSome of problems considered here O.H. formulated during his visit in th e The-\noretical Physics Institute, ETH, Zurich. He acknowledges the fina ncial support\nby the ETH, which enabled the visit. He would like to express his gratitu de\nto prof. T.M. Rice, prof. G. Blatter, D. Poilblanc, Y. Hasegawa and C . Gross\nfor discussions. O.H. acknowledges also the support by the ICTP, T rieste, espe-\ncially expresses his sincere thanks to Prof. Yu Lu and Prof. E. Tosa tti for their\nkind hospitality.\nReferences\n[1] Ch. Nayak, S.H. Simon, A. Stern, M. Freedman, S.D. Sarma, Rev. Mod.\nPhys.80(2008) 1083\n[2] L. Jiang, G.K. Brennen, A.V. Gorshkov, K. Hammerer, M. Hafezi, E.\nDemler, M.D. Lukin, P. Zoller, Nature Physics 4(2008) 482 - 488\n[3] Y.P. Zhong, D. Xu, P. Wang, C. Song, et al., Phys. Rev. Lett. 117(2016)\n110501\n[4] L.D.LandauandE.M.Lifshitz, QuantumMechanics,(Theoretical Physics\nCourse, Vol. III.), Nauka, Moscow, 1974, pp. 522-525, in Russian\n[5] Y. Hasegawa,P. Lederer, T.M. Rice and P.B. Wiegmann, Phys. Rev . Lett.\n63(1989) 907-910\n[6] D. Poilblanc, Phys. Rev. B40(1989) 7376-7379\nM. Kohmoto, Phys. Rev. B39(1989) 11943-11949\nG. Montambaux, Phys. Rev. Lett. 63(1990) 1657\nY. Hasegawa, Y. Hatsugai, M. Kohmoto, G. Montambaux, Phys. Re v.\nB41(1990) 9174-9182\nA.A. Belov, Ju.E. Lozovik, V.A. Mandelshtam, JETP Lett. 51(1990)\n422-425, in Russian\nA.G. Abanov, D.V. Khveshchenko, Mod. Phys. Lett. 4(1990) 689-696\nF. Nori, F. Abrahams, G.T. Zimanyi, Phys. Rev. B41(1990) 7277\n11R. Rammal, J. Bellisard, J. Phys. France 51(1990) 2153-2165\n[7] A. Barelli, J. Bellisard, R. Rammal, J. Phys. France 51(1990) 2167-2185\n[8] B.A. Dubrovin, S.P. Novikov, Sov. Phys. JETP 52(1980) 511\nB.A. Dubrovin, S.P. Novikov, Sov. Math. Dokl. 22(1980) 240\nS.P. Novikov, Sov. Math. Dokl. 23(1981) 298\n[9] R. B. Laughlin, Phys. Rev. Lett. 60(1988) 2677\nC. B. Hanna, R. B. Laughlin, A. L. Fetter, Phys. Rev. B40(1989) 8745-\n8758\n[10] O. Hud´ ak, Zeitschrift fr Physik B Condensed Matter 2 88239-246\n[11] K.B.Lyons, J.Kwo, J.F. DillonJr., G.P.Espinosa, M.McGlashab-Po well,\nA.P. Ramirez, L.P. Schneemeyer, Phys.Rev.Lett. 64(1990) 2949\nH.J. Weber et al., Sol. St. Comm. 76(1990) 511\nS. Spielman, K. Fesler, C.B. Eom, T.H. Geballe, M.M. Fejer and A. Ka-\npitulnik, Phys.Rev.Lett. 65(1990) 123-126\nM.A.M. Gijs, A.M. Gerrits, C.W.J. Beenakker, Phys.Rev. B42(1990)\n10789\nK.B. Lyons, J.F. Dillon, E.S. Hellman, E.H. Hartford, M. McGlashan-\nPowell, Phys.Rev. B43(1991) 11408\nR.F. Kiefl et al. Phys.Rev.Lett. 64(1990) 2082\n[12] R. E. Peierls, Quantum Theory of Solids, Oxford, Clarendon Pre ss, 1955\n[13] Y-H. Chen, F. Wilczek, E. Witten, B.I. Halperin, Int.J.Mod.Phys. B3\n(1989) 1001-1067\nF. Wilczek, FractionalStatistics andAnyonSuperconductivity, Wo rld Sci-\nentific, Singapore-New Jersey-London-Hong Kong, 1991\n[14] X.G. Wen, A. Zee, Phys.Rev. B43(1991) 5595\n[15] B. Doucout, P.C.E. Stamp, Phys.Rev.Lett. 66(1991)2503\n12"    },    {        "title": "1704.08510v1.A_Lee_Yang__inspired_functional_with_a_density__dependent_neutron_neutron_scattering_length.pdf",        "content": "arXiv:1704.08510v1  [nucl-th]  27 Apr 2017A Lee-Yang–inspired functional with a density–dependent n eutron-neutron scattering\nlength\nM. Grasso,1D. Lacroix,1and C.J. Yang1\n1IPNO, CNRS/IN2P3, Universit´ e Paris-Sud, Universit´ e Par is-Saclay, F-91406, Orsay, France\nInspired by the low–density Lee-Yang expansion for the ener gy of a dilute Fermi gas of density ρ\nand momentum kF, we introduce here a Skyrme–type functional that contains o nlys-wave terms\nand provides, at the mean–field level, (i) a satisfactory equ ation of state for neutron matter from\nextremely low densities up to densities close to the equilib rium point, and (ii) a good–quality equa-\ntion of state for symmetric matter at density scales around t he saturation point. This is achieved by\nusing a density–dependent neutron-neutron scattering len gtha(ρ) which satisfies the low–density\nlimit (for Fermi momenta going to zero) and has a density depe ndence tuned in such a way that\nthe low–density constraint |a(ρ)kF| ≤1 is satisfied at all density scales.\nThe interactionbetween the constituents ofvery dilute\nFermi systems is accuratelydetermined by a few parame-\nters associated to s–wavescattering processes. An exam-\nple is given by ultracold trapped Fermi gases where the\ninteraction may be reasonably well approximated by a\nzero–rangeforcewith acouplingconstantdirectlyrelated\nto thes–wave scattering length a[1–4]. The study of a\ndilute regimeinfermionicsystemsisofparticularinterest\nfor a wide community of many–body practitioners for in-\nstance in the domains of atomic (cold fermionic trapped\natoms) and nuclear (nuclear matter) physics, as well as\nin nuclear astrophysics for the investigation of the prop-\nerties of neutron star crusts. Within such a wide frame-\nwork, bridging Effective Field Theories (EFTs), which\nby construction correctly describe low–density regimes\nwith energy–density–functional (EDF) theories, which\nare currently employed for instance in the nuclear many–\nbody problem, is a very appealing challenge which re-quires a tight interchange and connections between EFT\nand EDF expertise and competences. The dilute regime\nis characterized by the relation |akF|<1. For such a\nregime, Lee and Yang introduced in the 50s an expansion\nin (akF) for the ground–state energy [5]. It is important\nto notice that, at the unitarity limit, for example in ul-\ntracold atomic dilute gases close to Feshbach resonances,\nanother expansion is used, on 1 /(akF) (instead of akF).\nThe first terms of the Lee and Yang low–density expan-\nsion in ( akF) are reported for instance in Refs. [6–9].\nMore recently, such terms were derived in the framework\nof EFTs [10]. The first four terms contain only s–wave\nparameters, the s–wave scattering length and the asso-\nciated effective range rs(the following term appearing\nin the expansion contains the p–wave scattering length).\nWe report here the first four terms, in the case where the\nspin degeneracy is equal to 2 (for example for neutron\nmatter),\nE\nN=/planckover2pi12k2\nF\n2m/bracketleftbigg3\n5+2\n3π(kFa)+4\n35π2(11−2ln2)(kFa)2+1\n10π(kFrs)(kFa)2+0.019(kFa)3/bracketrightbigg\n, (1)\nwhereNis the number of particles. Within EFT, the\naboveequationisobtainedviadimensionalregularization\n(DR) with minimal subtraction. It is independent of the\nadopted regularizationscheme, provided that a matching\nto an effective–range expansion is performed [11].\nThe comparison of the Lee-Yang (LY) energy, Eq. (1),\nwith the energy obtained in a many–body perturbative\nexpansion indicates that the terms in ( kFa), (kFa)2, and\n(kFa)3correspond respectively to the leading–, second–,\nand third–order contributions produced by a zero–range\ninteraction with a coupling constant related to the scat-\ntering length a[12]. The term in ( kFrs)(kFa)2corre-\nsponds to the leading–order contribution provided by a\nvelocity–dependent zero–range s-wave interaction. Fur-\nthermore, it was shown in Ref. [13] that the ( kFa)2–\nterm may be alternatively obtained at leading order witha specific density–dependent zero–range force. We no-\ntice finally that the term in ( kFa)3has the same kF–\ndependence as the term in ( kFrs)(kFa)2. This implies\nthat a zero–range s–wave velocity–dependent term may\nmimic such a term at leading order. At very small den-\nsities, these first terms of the LY expansion, containing\nonlys–wave scattering parameters, are enough to cor-\nrectly describe the energy of the system.\nInspired by this expansion, we introduce here a\nSkyrme–type functional [15, 16], containing only s-wave\nterms, that leads, at the mean–field level, to an EOS for2\nneutron matter given by Eq. (1), with the relations\nt0(1−x0) =4π/planckover2pi12\nma,\nt3(1−x3) =144/planckover2pi12\n35m(3π2)1/3(11−2ln2)a2,(2)\nt1(1−x1) =2π/planckover2pi12\nm(a2rs+0.19πa3),\nwheret0,t1,t3, andx0,x1,x3areSkyrme parametersand\nthe power of the density–dependent t3-term is chosen\nequal to 1/3 [13]. We require that such a functional: (i)\ncorrectly describes neutron matter from extremely low\ndensities up to densities close to the equilibrium point\nof symmetric matter, ρ= 0.16 fm−3; (ii) provides in\naddition a correct equation of state (EOS) for symmet-\nric matter at density scales around the saturation point.\nThe requirement (i) cannot be fully satisfied by using\nthe value of -18.9 fm for the s-wave scattering length.\nSuch a very large value leads indeed to a correct descrip-\ntion of neutron matter with the first terms of Eq. (1)\nonly at extremely low densities [13]. For this reason, re-\nsumed expressions have been proposed in the literature\nwithin EFT for cases where the scattering length is very\nlarge (see for instance Refs. [17–19]). Recently, going to-\nwards this direction, a hybrid functional was introduced,\nYGLO,combiningaresumedexpression(thatguarantees\nthe correct low–density behavior) and good properties of\nSkyrme–typeforces,whichareknowntowelldescribethe\nEOS of matter close to the equilibrium point of symmet-\nric matter [13]. A resumed functional making connection\nbetween cold atoms and neutron matter was introduced\nin Ref. [14].\nIn this work, we probe the possibility of adopting a\nsimpler functional, which does not contain any resumed\nexpression. To satisfy both requirements (i) and (ii) we\nimposethattheneutron-neutronscatteringlengthisden-\nsity dependent, a(ρ), in such a way to ensure the correct\nlow–density behavior for neutron matter (for kFgoing to\nzero) and to justify the use of a LY–type EOS truncated\nat the very first terms (only parameters related to the\ns-wave scattering length are taken into account).\nOwing to the fact that the parameters xiassociated\ntos–wave terms do not appear in the mean–field EOS of\nsymmetric matter with a Skyrme force, we adjust here\nthe parameters tito have a reasonable mean–field EOS\nfor symmetric matter, and we tune the neutron-neutron\nscattering length in the following way: We impose that\nit takes the value of -18.9 fm up to a Fermi momentum\nkmax\nFsuch that kmax\nF18.9 = 1. This ensures the correct\nlow–density behavior. It turns out that kmax\nF∼0.05\nfm−1, corresponding to a maximum density ∼4×10−6\nfm−3, wherethedensityandtheFermimomentumarere-\nlatedbytherelation kF= (3π2ρ)1/3. Beyondthisdensity\nvalue, wegeneralizethelow–densityconstraint |kFa| ≤1\n(that identifies the density window where the LY for-\nmula is valid) to the case where the scattering length\nis density dependent, with the relation |kFa(ρ)| ≤1\nused to tune the density dependence of the scattering0 0.08 0.16 0.24\nDensity (fm-3)-4-3-2-10a (fm) - a(kF) kF = 0.5\n- a(kF) kF = 1\n00.51.01.52.0\nkF (fm-1)(a) (b)\nFIG. 1: Neutron-neutron s–wave scattering length as a func-\ntion of the density (a) and of the Fermi momentum (b).\nThe green region contains all the possible cases between\n|kFa(kF)|= 0.5 (red dashed line) and |kFa(kF)|= 1 (black\nsolid line).\nlength. Interestingly, the momentum dependence 1 /kF\nobtained by imposing such a constraint strongly resem-\nbles to the magnetic–field dependence of the scattering\nlength in ultracold trapped atoms close to Feshbach res-\nonances [2]. This indicates a very strong analogy with\nultra–cold atomic gases, where the scattering length is\ntuned byan appliedmagnetic field. Ourstrategyofusing\na Fermi momentum (or density)–tuned scattering length\nleads in our case to a strikingly similar behavior.\nWe plot in Fig. 1 the neutron-neutron scattering\nlength as a function of the density (a) and as a func-\ntion of the Fermi momentum (b). The lower curve corre-\nsponds to the tuning |kFa(kF)|= 1. As an illustration\nof the sensitivity to the hypothesis discussed above, the\nupper curve delimiting the green region is also shown,\ncorrespondingtoatuningobtainedbyimposingastricter\nlow–density constraint, |kFa(kF)|= 0.5. The green ar-\neasin the two panels ofthe figurecontain all the interme-\ndiate cases. When replacing abya(ρ), thexiparameters\nbecome density dependent and their expressions may be\ndeduced from Eqs. (2).\nFirst two terms of the Lee-Yang expansion.\nThe EOS obtained by dropping the last two terms\nof Eq. (1) corresponds to a mean–field EOS within a\nSkyrmet0−t3model. Within such a simplified model,\nwe adjusted in Ref. [13] the parameters t0andt3to\nhave a satisfactory EOS for symmetric matter around\nthe saturation point, providing a saturation density of\n0.16 fm−3with an energy per particle of -16.04 MeV: t0\n(t3) = -1803.93 MeV fm3(= 12911.00 MeV fm4). By us-\ning these values for t0andt3, we may deduce the density\ndependenceoftheparameters x0andx3throughtheden-\nsity dependence of the neutron-neutron scattering length\n(Fig. 2). Notice that the values of the parameters x0\nandx3vary from -4.46 and -139.40, respectively, at zero\ndensity [13], to values which are closer to typical x0,x33\n-0.500.51x0-a(kF) kF = 0.5\n-a(kF) kF = 1\n0 0.04 0.08 0.12 0.16 0.20 0.24\nDensity (fm-3)-1-0.500.51x3(a)\n(b)\nFIG. 2: Density dependence of the parameters x0(a) andx3\n(b). The green region contains all the possible cases betwee n\n|kFa(kF)|= 0.5 (red dashed line) and |kFa(kF)|= 1 (black\nsolid line).\n0 0.05 0.10 0.15 0.20 0.25 0.30\nDensity (fm-3)-20-15-10-50E/A (MeV)SLy5 mean field\nt0-t3t0-t3-t1\nFIG. 3: EOS of symmetric matter obtained within a t0−t3\nmodel, with the parameters adjusted in Ref. [13] (red dashed\nline) and within a t0−t3−t1model, with the parameters ad-\njusted in this work (blue circles). For comparison, the SLy5 -\nmean-field EOS is plotted (black solid line).\nvalues in Skyrme forces at larger densities. The corre-\nsponding EOS of symmetric matter is plotted in Fig. 3\nand compared to the SLy5–mean–field [20] EOS (used as\na benchmark for the fit in Ref. [13]).\nThe EOS for neutron matter with x0,x3displayed\nin Fig. 2 is shown in Fig. 4, where also the SLy5–\nmean–field EOS is drawn for comparison (black trian-\ngles). We also show in the same figure two alternative\nmean–field Skyrme EOSs, obtained with the parameter-\nizations SkP [21] (cyan circles) and SIII [22] (magenta\nsquares). Whereas the SLy5 parametrization was de-\nsigned to accurately reproduce a microscopic EOS for\nneutron matter even beyond the saturation point of sym-\nmetric matter, the other two Skyrme parameterizations\nwere not adjusted in the same way and are not so accu-0 0.1 0.2\nDensity (fm-3)01020E/A (MeV)-a(kF) kF = 0.5\n-a(kF) kF = 1\nSLy5 mean field\nSkP mean field\nSIII mean field\n Constant a=-18.9 fm\nrs = -4.5 fm\nFIG. 4: EOS of neutron matter as a function of the density\ncomputed in the mean–field approximation with SLy5 (black\ntriangels), SkP (cyan circles), and SIII (magenta squares) .\nThe EOSs obtained with the first two terms of the Lee-Yang\nexpansion by using the constant value a=−18.9 fm (blue\nsquares) and a density–dependent scattering length are als o\nshown. For the latter case, the black solid line represents t he\nEOS obtained with the low–density constraint |kFa(kF)|= 1\nwhereas the red dashed curve represents the EOS obtained\nby imposing |kFa(kF)|= 0.5. The green area contains the\nintermediate cases. The black dotted line represents the EO S\nobtained by imposing |kFa(kF)|= 1 and using an effective\nrange of -4.5 fm.\nrate. However, they still provide reasonable results for\nneutron matter at least up to densities around the equi-\nlibrium point of symmetric matter. One observes that\ntheupperandlowerEOSsdelimitingthegreenareadiffer\nvery weakly. To estimate how much the density depen-\ndence of the scattering length affects the EOS, we also\nplot the EOS obtained by using a constant scattering\nlength equal to the free value, -18.9 fm, (blue squares)\nwhich obviously leads to a totally wrong curve, except at\nextremely small densities.\nOne observes that the obtained EOS is still quite far\nfrom the Skyrme EOSs. The energy is systematically too\nhighindicatingthat anattractivecontributionis missing.\nIncluding the s–wavek5\nFterms.\nThe value of the effective range associated to the scat-\ntering length a=-18.9fm is 2.75 fm. In general, the term\ncontainingthe effective rangemay be neglected in the LY\nexpansion if kF|rs|<1. For higher momenta, kF|rs|/greaterorsimilar1\nand the corresponding term cannot be neglected any-\nmore. In our case, the scattering length is equal to -18.9\nfm only up to kF∼0.05 fm−1. At this value of the\nFermi momentum and for rs= 2.75 fm, kFrs∼0.14,\nwhich is still sensibly less than 1. Thus, up to kF∼0.05\nfm−1, the effective–range term may be safely neglected.\nHowever, at higher densities, the value of the scatter-\ning length changes very fast as a function of the density\nand it would be meaningless to still associate to such a\nvalue an effective rangeof 2.75fm. Furthermore, we have\nobserved in the Skyrme t0−t3model that, at ordinary4\nnuclear densities, a density–dependent neutron-neutron\nscattering length is not sufficient to reproduce a reason-\nable EOS and that an attractive contribution is missing.\nSuch a missing attractive contribution in the EOS may\nindeed be obtained by including the following two terms\nof the expansion, which are still s–wave terms and have\nboth ak5\nFdependence (Eq. (2) shows the relation with\nthet1velocity–dependent term of a Skyrme model). The\nfirst of these terms contains the effective range and we\nuse the effective range as a parameter for reproducing a\nreasonable neutron–matter EOS up to around the satu-\nration density of symmetric matter. We have found that\nrs= -4.5 fm for the case |kFa(kF)|= 1 leads to an\nacceptable EOS (black dotted line in Fig. 4).\nThe symmetric matter EOS will also be modified by\nthe inclusion of the t1term in the interaction. We pro-\nceed as done previously: we readjust the parameters t0,\nt3, andt1to have a satisfactory EOS for symmetric mat-\nter and we keep these values unchanged. The adjusted\nparameters are t0= -1818 MeV fm3,t3= 12970 MeV\nfm4,t1= 15 MeV fm5and the corresponding EOS of\nsymmetric matter is plotted in Fig. 1 within the model\nt0−t1−t3.\nLet us now investigate in more detail the very low–\ndensity sector. Figure 5 shows the energy of neutron\nmatter divided by the free gas energy as a function of\n−akF(witha=−18.9 fm). The curves corresponding\nto the first two terms of the Lee-Yang expansion with a\nconstant scattering length (-18.9 fm) and with a density–\ndependent scattering length are shown together with the\nSLy5 EOS. In addition, the dot–dashed green curve illus-\ntrates the results obtained with the inclusion of the k5\nF–\nterms with rs=-4.5 fm. We have already observed that\nthis case gives a reasonableEOS at ordinary nuclearden-\nsities (Fig. 4). We see now that, in addition, it well re-\nproduces the correct low–density behavior. For instance,\nfor−akF∼6, the value of the energy divided by the\nfree gas energy is ∼0.6, which is comparable to the mi-\ncroscopic QMC s–wave, the QMC AV4, or the AFDMC\nresults reported in Ref. [23–26].\nOur exploratory study can be summarized as follows.\nWe start with a functional based on the first two terms\nof the LY expansion, which is correct at very low den-\nsities but produces a wrong EOS for ρ >10−6fm−3.\nThen, for higher densities, we depart from the matching\nto an effective–range expansion by imposing a density-\ndependent scattering length a(kF), which is constrained\nby a dimensionless quantity C(|kFa(kF)|=C≤1).Empirical evidence shows that the effective range needs\nto enter as a free parameter in order to obtain a rea-\nsonable description of neutron matter up to the satura-\ntion density. The EOS of symmetric and neutron mat-\nter can be deduced by a simple functional which corre-\nsponds to a t0−t1−t3Skyrme interaction with density–\ndependent x′\nis. Our functional may be employed into\nvarious applications for the description of neutron–rich\nsystems at very low–density scales as well as of isospin–\nsymmetric and asymmetric systems at ordinary nuclear\n0 1 2 3 4 5 6\n-a kF0.40.60.81.0E/Free Gas EnergyConstant a=-18.9 fm \n- a(kF) kF = 1\nSLy5 mean field\nr = -4.5 fm\nQMC-s wave\nQMC AV4\nAFDMC\nFIG. 5: Energy of neutron matter divided by the energy of a\nfree Fermi gas as a function of |akF|, witha= -18.9 fm. The\nblack solid and the blue dotted curves represent the SLy5–\nmean–field and the LY (with only the first two terms and\na= -18.9 fm) EOSs, respectively. The red dashed and the\ngreen dot–dashed curves illustrate the EOSs obtained with\na density–dependent scattering length with |a(kF)kf|= 1,\nwith only the first two terms and with the full expression of\nEq. (1) ( rs= -4.5 fm), respectively. The ab-initio Quantum\nMonte-Carlo (QMC) with only s-wave and with the full AV4\ninteraction taken from Refs. [24, 25] are shown respectivel y\nwith purple squares and blue triangle. The AFDMC ab-initio\nresults are taken from Ref. [26].\ndensity scales.\nAcknowledgments\nThis project has received funding from the European\nUnions Horizon 2020 research and innovation program\nunder grant agreement No. 654002.\n[1] G. Bruun, Y. Castin, R. Dum, and K. Burnett, Eur.\nPhys. J. D 7, 433 (1999).\n[2] S. Giorgini, L.P. Pitaevskii, and S. Stringari, Rev. of\nMod. Phys. 80, 1215 (2008).\n[3] Immanuel Bloch, Jean Dalibard, and Wilhelm Zwerger,\nRev. Mod. Phys. 80, 885 (2008).\n[4] W. Zwerger, ed. The BCS-BEC crossover and the uni-tary Fermi gas. Vol.836. Springer Science and Business\nMedia, (2011).\n[5] T. D. Lee and C. N. Yang, Phys. Rev. 105, 1119 (1957).\n[6] V.N. Efimov and M.Ya. Amusya, Sov. Phys. JETP 20,\n388 (1965).\n[7] M.Ya. Amusia and V.N. Efimov, Ann. Phys. (NY) 47,\n377 (1968).5\n[8] G.A. Baker, Rev. Mod. Phys. 43, 479 (1971).\n[9] R.F. Bishop, Ann. Phys. (NY) 77, 106 (1973).\n[10] H. W. Hammer and R. J. Furnstahl, Nucl. Phys. A 678,\n277 (2000).\n[11] R.J. Furnstahl, J. V. Steele and N. Tirfessa, Nucl. Phys .\nA 671, 396-415 (2000).\n[12] C.J. Yang, M. Grasso, X. Roca-Maza, G. Col` o, Phys.\nRev. C 94, 034311 (2016).\n[13] C.J. Yang, M. Grasso, and D. Lacroix, Phys. Rev. C 94,\n031301(R) (2016).\n[14] D. Lacroix, Phys. Rev. A 94, 043614 (2016).\n[15] T.H.R. Skyrme, Philos. Mag. 1, 1043 (1956); Nucl. Phys.\n9, 615 (1959).\n[16] D. Vautherin and D.M. Brink, Phys. Rev. C 5, 626\n(1972).\n[17] J.V. Steele, arXiv:nucl-th/0010066v2.\n[18] T. Schaefer, C.-W. Kao, and S. R. Cotanch, Nucl. Phys.\nA 762, 82 (2005).[19] N. Kaiser, Nucl. Phys. A 860, 41 (2011).\n[20] E. Chabanat, P. Bonche, P. Haensel, J. Meyer, R. Scha-\neffer, Nucl. Phys. A 627, 710 (1997) ; ibid. A 635, 231\n(1998) ; ibid. A 643, 441 (1998).\n[21] J. Dobaczewski, H. Flocard, J. Treiner, Nucl. Phys. A\n422, 103 (1984).\n[22] M. Beiner, H. Flocard, Nguyen Van Giai, P. Quentin,\nNucl. Phys. A 238, 29 (1975).\n[23] S. Gandolfi, A. Gezerlis, J. Carlson, Annual Review of\nNucl. and Part. Science 65, 303 (2015).\n[24] A. Gezerlis and J. Carlson, Phys. Rev. C 81, 025803\n(2010).\n[25] J. Carlson, Stefano Gandolfi, Alexandros Gezerlis, Pro g.\nTheor. Exp. Phys. 01A209 (2012).\n[26] S. Gandolfi, A. Yu. Illarionov, S. Fantoni, F. Pederiva,\nand K. E. Schmidt, Phys. Rev. Lett. 101, 132501 (2008)."    },    {        "title": "1705.00019v3.Short_note_on_the_density_of_states_in_3D_Weyl_semimetals.pdf",        "content": "arXiv:1705.00019v3  [cond-mat.dis-nn]  18 Sep 2018Short note on the density of states in 3D Weyl semimetals\nK. Ziegler and A. Sinner\nInstitut f¨ ur Physik, Universit¨ at Augsburg, D-86135 Augs burg, Germany\n(Dated: September 19, 2018)\nThe average density of states in a disordered three-dimensi onal Weyl system is discussed in the\ncase of a continuous distribution of random scattering. Our results clearly indicate that the average\ndensity of states does not vanish, reflecting the absence of a critical point for a metal-insulator\ntransition. This calculation supports recent suggestions of an avoided quantum critical point in\nthe disordered three-dimensional Weyl semimetal. However , the effective density of states can be\nvery small such that the saddle-approximation with a vanish ing density of states might be valid for\npractical cases.\nI. INTRODUCTION\nThe existence of a metal-insulator transition in disordered three-d imensional (3D) Weyl semimetals\nhas been debated in the recent literature [1–11]. It is closely relate d to the question, whether or not the\naverage density of states (DOS) at the spectral node vanishes b elow some critical disorder strength. The\nself-consistent Born approximation provides such a critical value w ith a vanishing DOS for weak disorder.\nIt has been arguedthat rareregionsofthe randomdistribution ma y lead to a non-vanishing averageDOS,\nthough [1]. This was supported by recent numerical studies based o n the T-matrix approach, which gives\nan exponentially small DOS [8] but was questioned in a recent study ba sed on an instanton solution\n[11]. In this short note we show that, depending on the type and str ength, a continuous distribution of\ndisorder can create a substantial average DOS at the spectral n ode in 3D Weyl systems. This requires\nat least two impurities to create a resonant state between these im purities. A single impurity or a single\ninstanton does not contribute to the spectral weight at the Weyl node, though, in accordance with the\narguments in Ref. [11]. This supports the picture of an avoided quan tum critical point in the presence\nof a distribution of impurities, as advocated in Ref. [8].\nII. MODEL\nThe 3D Weyl Hamiltonian for electrons with momentum /vector pis expanded in terms of Pauli matrices τj\n(j= 0,1,2,3;τ0is the 2×2 unit matrix) as H=H0−Uτ0, where\nH0=vF/vector τ·/vector pwith/vector τ= (τ1,τ2,τ3). (1)\nvFis the Fermi velocity and Uis a disorderterm, representedby a random potential with mean /an}bracketle{tU/an}bracketri}ht=EF\n(Fermi energy) and variance g. The average Hamiltonian /an}bracketle{tH/an}bracketri}ht=H0−EFτ0generates a spherical Fermi\nsurface with radius |EF|, and with electrons (holes) for EF>0 (EF<0). Physical quantities are\nexpressed in such units that vF¯h= 1.\nThe DC limit ω→0 of the conductivity of 3D Weyl fermions depends only on the scatt ering rate η\nand the Fermi energy EF[7]:\nσ(η,EF) = 2e2\nhη2/integraldisplayλ\n0(η2+k2)2+E2\nF(2η2+2k2/3+E2\nF)\n[(η2−E2\nF+k2)2+4η2E2\nF]2k2dk\n2π2(2)\nwith momentum cut-off λ. At the node ( EF= 0) the DC conductivity in Eq. (2) is reduced to the\nexpression\nσ= 2e2\nhη2/integraldisplayλ\n0k2\n(η2+k2)2dk\n2π2=e2η\n2π2h/bracketleftbigg\narctan(1/ζ)−ζ\n1+ζ2/bracketrightbigg\n(ζ=η/λ), (3)\nwhich becomes for λ≫η\nσ∼e2\n4πhη . (4)2\nU plane\npole of the CL distribution\npole of the CL distributionpoles of the Green's function\nclosed integration contour\nFIG. 1: Poles of the one-particle Green’s function and the Ca uchy-Lorentz distribution. The contour of the\nUr–integration encloses only one pole of the Cauchy-Lorentz d istribution but not the other poles.\nThe last result was also derived by Fradkin some time ago [12]. In cont rast to the 2D case, where\nσ=e2/πh, the 3D case gives a linearly increasing behavior with respect to the s cattering rate.\nThe results in (2) – (4) clearly indicate that a metal-insulator transit ion in disordered 3D Weyl systems\nis directly linked to the scattering rate η. The latter describes the broadening of the poles of the one-\nparticle Green’s function and is proportional to the average DOS\nρr(EF) = lim\nǫ→01\nπIm/bracketleftbig¯Grr(−iǫ))/bracketrightbig\n,¯G(−iǫ) =/an}bracketle{t(H0−Uτ0−iǫ)−1/an}bracketri}ht, (5)\nwhere is ¯Grris the diagonal element of ¯Gwith respect to space coordinates. The self-consistent Born\napproximation [7, 12] at the node EF= 0 reads\nη=ηIwithI=γ[λ−ηarctan(λ/η)] (6)\nfor the effective disorder strength γ=g/2π2. There are two solutions, namely η= 0 and a solution with\nη/ne}ationslash= 0, which exists only for sufficiently large γ. Moreover, ηvanishes continuously as we reduce γ. For\nη∼0 we obtain the linear behavior\nη∼2λ\nπ(γλ−1), (7)\nwhereγc= 1/λappears as a critical point with η= 0 forγ≤γcandη >0 forγ > γc.\nIII. AVERAGE DENSITY OF STATES\nA. Few impurities: Lippmann-Schwinger equation\nAt the node EF= 0 the pure DOS ρ0;r(EF= 0) vanishes. However, a few impurities have already a\nsignificant effect on the local DOS: Assuming an impurity potential UNonNsites, we use the identity\n(lattice version of the Lippmann-Schwinger equation)\n(G−1\n0−UN)−1=G0+G0(1−UNPNG0PN)−1\nNUNG0, (8)\nwherePNis the projector on the impurity sites and ( ...)−1\nNis the inverse on the impurity sites. Although\nρ0;r(EF= 0) vanishes, the second term on the right-hand side of Eq. (8) ca n contribute with the poles of\n(1−UNPNG0PN)−1\nNto the DOS. These poles are “rare events” and require a fine-tunin g of the impurity\npotential, whereas the generic case of a general UNwould still have a vanishing DOS. In a realistic\nsituation the number of impurities is macroscopic with a non-zero den sity in the infinite system. Then\nthe identity (8) cannot be used for practical calculations and we ha ve to average over many impurity\nrealizations. This leads to the average Green’s function of Eq. (5), which will be calculated subsequently.3\nB. One vs. two impurities\nThe Green’s function G0of the system without impurities reads\nG0,r(−iǫ) =1\n|B|/integraldisplay\nBei/vectork·r\nǫ2+k2/parenleftig\niǫτ0+/vectork·/vector τ/parenrightig\nd3k≡iǫγ0τ0+/vector γ·/vector τ , (9)\nwhereBis the Brillouin zone of the underlying lattice and\nγ0=1\n|B|/integraldisplay\nBei/vectork·r\nǫ2+k2d3k , γ j=1\n|B|/integraldisplay\nBei/vectork·rkj\nǫ2+k2d3k(j= 1,2,3).\nThen the diagonal element G0,0=iǫγτ0vanishes with ǫ∼0. This implies that for a single impurity there\nis no bound state at finite impurity strength Ur, since in the impurity term of the Lippmann-Schwinger\nequation (8) the 2 ×2 matrix\n(1−UrPrG0Pr)−1=1\n1−iǫγ0Urτ0 (10)\nhas a pole at Ur∼ ∞. The latter reflects the statement that a potential well in 3D Weyl semimetals\ndoes never generate spectral density at zero energy [11]. For t wo impurities, though, there is a resonant\ninter-site bound state between the impurities, since G0,r−r′(r′/ne}ationslash=r) does not vanish for ǫ→0 but decays\nwith a power law for |r−r′|due to the Pauli matrix coefficients γjin Eq. (9):\n(1−UP{r,r′}G0P{r,r′})−1=\n1−iǫγ0Ur 0 −Urγ3−Ur(γ1−iγ2)\n0 1 −iǫγ0Ur−Ur(γ1+iγ2)Urγ3\n−Ur′γ3−Ur′(γ1−iγ2) 1−iǫγ0Ur′ 0\n−Ur′(γ1+iγ2)Ur′γ3 0 1 −iǫγ0Ur′\n−1\n.\n(11)\nThe degenerate eigenvalues of this matrix\n1\n1−iǫγ0(Ur+Ur′)/2±/radicalbig\nUrUr′(γ2\n1+γ2\n2+γ2\n3)−ǫ2γ2\n0(Ur−Ur′)2/4(12)\nhave poles for finite Ur,Ur′. Thus, the corresponding bound states contribute with a non-va nishing\ndensity of states. In the remainder of the paper this result will be g eneralized to multiple impurities with\ncorresponding resonant bound states.\nC. Distribution with simple poles\nFrom here on we consider a continuous distribution of the disorder p otentialUwith/producttext\nrP(Ur)dUrand\naverage one-particle Green’s function\n¯G(−iǫ) =/integraldisplay\n(H0−Uτ0−iǫ)−1/productdisplay\nrP(Ur)dUr. (13)\nForǫ >0 the one-particle Green’s function ( H0−Uτ0−iǫ)−1has poles for Uron the upper complex\nhalf-plane. Assuming that the distribution density P(Ur) has isolated poles in the lower complex half-\nplane, the Cauchy integration can be applied by closing the integratio n along the real axis in the lower\ncomplex half-plane, as depicted in Fig. 1. The simplest realization is the Cauchy-Lorentz distribution\nPCL(Ur) =1\nπη\n(Ur−EF)2+η2, (14)\nwhich gives\n¯G(−iǫ) = (H0−(EF+iǫ+iη)τ0)−1. (15)4\nSboundary\nFIG. 2: Dividing the system into cubes {S}of size|S|with boundary ∂S.\nThe average DOS then reads\nρr(EF) =η\nπ[(H0−EFτ0)2+η2τ0]−1\nrr. (16)\nThe Cauchy-Lorentzdistribution has an infinite second moment (i.e., gis infinite). A distributions with a\nfinite second moment can be created from the differential of the Ca uchy-Lorentzdistribution with respect\ntoη. Many distributions, like the popular Gaussian distribution\nPG(Ur) =1√πge−(Ur−EF)2/g, (17)\ndo not have a simple pole structure, though. Then another approa ch can be applied to show that there\nis a non-vanishing average DOS.\nD. Distribution without simple poles\nNow we only assume that the distribution of Uris continuous. Then the path of integration can also be\ndeformed away from the poles of the Green’s function to obtain a sim ilar result as in the case of simple\npoles. The calculation would be more complex, though. Therefore, w e use a different approach, whose\nmain idea is to divide the system into cubes {S}of finite identical size (cf. Fig. 2). Then we estimate (i)\nthe average DOS inside an isolated cube and (ii) the contribution of th e boundary ∂Sbetween the cubes.\nThis approach was used for a periodic lattice [13], for a random tight- binding model with symmetric\nHamiltonian [14] and for two-dimensional Dirac fermions with random m ass [15]. Later it was applied to\nS-wave superconductor with random order parameter [16], and t o D-wave superconductor with random\nchemical potential [17].\nFor the average local DOS\n¯ρr=/integraldisplay\nρr(U)/productdisplay\nrP(Ur)dUr (18)\nwe obtain from the estimation procedure with steps (i) and (ii) the ine quality (cf. Supplemented Mate-\nrials)\n/summationdisplay\nr∈S¯ρr≥inf\n{−a≤U′r≤a}/bracketleftigg/integraldisplayv\n−v/summationdisplay\nr∈SρS,r(U′+E)dEinf\n−v≤w≤v/productdisplay\nr∈SP(U′\nr+w)/bracketrightigg\n−¯PS|∂S|, (19)\nwhere|S|(|∂S|) is the number of sites of S(∂S) and\n¯PS= inf\n{−a≤U′r≤a},−v≤w≤v/productdisplay\nr∈SP(U′\nr+w).\n¯PS|∂S|is the contribution of the boundary of a cube and the integral is the integrated DOS on a cube S.\nThe boundary term is substracted because we have removed the b oundary. In other words, the left-hand5\nside of (19) is the average DOS on the entire lattice, the right-hand side is the average DOS on the\nisolated cube S.\nThe value of the lower bound requires an adjustment of the still und etermined parameters aandv.\nThe integrated DOS/integraltextv\n−v/summationtext\nr∈SρS,r(U′+E)dEonSis the number of eigenvalues on the interval [ −v,v]\nof theS–projected Hamiltonian H0−U′. The projected Hamiltonian is an |S|×|S|Hermitean matrix\nwith finite elements, whose eigenvalues are also finite. Thus, for a fix edawe can choose a sufficiently\nlargevsuch that all eigenvalues of the projected Hamiltonian are inside the interval [ −v,v]. In this case\nthe integrated DOS is |S|and we get from (19) the inequality\n/summationdisplay\nr∈S¯ρr≥¯PS[|S|−|∂S|]. (20)\nScan always be chosen such that the size of the cube |S|is larger than the size |∂S|of its boundary.\nThen the right-hand side of (19) is strictly positive. vshould not be too large, though, in order to avoid\nthat¯PSbecomes too small, assuming that a typical P(Ur) decays for large values. The actual value of\nPSdepends on the distribution and can be exponentially small.\nThe average DOS of the entire lattice is estimated by the sum over all cubes, normalized by its number\nN. Since all cubes have the same lower bound, this sum is bounded by th e right-hand side of (20). This\nindicates that our estimation works only for a macroscopic number o f impurities, the case of a single\nimpurity (10) would always give a lower bound zero.\nIV. CONCLUSION\nThere is a crucial difference in terms of the average DOS: For a discr ete distribution the average DOS\nis non-zero only if the disorder potential is “resonant” with the pur e Green’s function G0, according to\nthe second term in Eq. (8). In particular, a single impurity fails to cre ate spectral weight at the Weyl\nnode. On the other hand, for a dense distribution of impurities, rep resented by a continuous random\npotential, there is always a non-vanishing average DOS due to inter- impurity bound states, provided that\nthe values of Urcover the entire spectrum of H0.\nThe existence of a critical disorder strength γc, as indicated by the self-consistent approximation in Eq.\n(7), contradicts the existence of a lower non-zero bound of the a verage DOS in Sect. IIID. Therefore, the\nself-consistent calculation is not sufficiently accurate to describe t he transport properties of the 3D Weyl\nsemimetal properly. Since the lower bound of the average DOS is only a qualitative, although rigorous,\nestimation, still a reliable approximation is necessary to obtain an app roximative value for the average\nDOS. The exact result obtained for the Cauchy-Lorentz distribut ion in Sect. IIIC gives only a hint,\nbecause this distribution is not generic. A possible option is a N−αexpansion with non-integer α[18].\nAcknowledgment: This work was supported by a grant of the Julian S chwinger Foundation.\n[1] R. Nandkishore, D. A. Huse, and S. Sondhi, Phys. Rev. B 89, 245110 (2014).\n[2] B. Sbierski, G. Pohl, E.J. Bergholtz and P.W. Brouwer, Ph ys. Rev. Lett. 113, 026602 (2014).\n[3] S.V. Syzranov, V. Gurarie, L. Radzihovsky, Phys. Rev. Le tt.114, 166601 (2015).\n[4] J.H. Pixley, D.A. Huse, and S. Das Sarma, Phys. Rev. X 6, 021042 (2016).\n[5] J.H. Pixley, P. Goswami, and S. Das Sarma, Phys. Rev. B 93, 085103 (2016).\n[6] J.H. Pixley, D.A. Huse, and S. Das Sarma, Phys. Rev. B 94, 121107(R) (2016).\n[7] K. Ziegler, Eur. Phys. J. B 89, 268 (2016).\n[8] J.H. Pixley, Yang-Zhi Chou, P. Goswami, D.A. Huse, R. Nan dkishore, L. Radzihovsky, S. Das Sarma, Phys.\nRev. B95, 235101 (2017).\n[9] B. Sbierski, K.A. Madsen, P.W. Brouwer, and Ch. Karrasch , Phys. Rev. B 96, 064203 (2017).\n[10] A. Sinner and K. Ziegler, Phys. Rev. B 96, 165140 (2017).\n[11] M. Buchhold, S. Diehl and A. Altland, arXiv:1805.00018 .\n[12] E. Fradkin, Phys. Rev. B 33, 3263 (1986).\n[13] W. Ledermann, Proc. R. Soc. London 182, 362 (1944).\n[14] F. Wegner, Z. Physik B - Condensed Matter 44, 9 (1981).\n[15] K. Ziegler, Nucl. Phys. B 285[FS19], 606 (1987).6\n[16] K. Ziegler, Commun. Math. Phys. 120, 177 (1988).\n[17] K. Ziegler, M.H. Hettler, P.J. Hirschfeld, Phys. Rev. B 57, 10825 (1998).\n[18] K. Ziegler, Phys. Lett. 99A, 19 (1983).arXiv:1705.00019v3  [cond-mat.dis-nn]  18 Sep 2018Supplementary Material :\nShort note on the density of states in 3D Weyl semimetals\nK. Ziegler and A. Sinner\nInstitut f¨ ur Physik, Universit¨ at Augsburg, D-86135 Augs burg, Germany\n(Dated: September 19, 2018)\nI. LOWER BOUND OF THE AVERAGE DENSITY OF STATES\nThe main idea of calculating the DOS of an infinite system is to divide the la rge system into smaller\n(finite) cubes, calculate the DOS of these smaller cubes and estimat e the contribution of the boundaries\nbetween them. This concept was developed for a periodic lattice by L edermann [1]. Later it was extended\nto estimate the average DOS of a tight-binding model with random po tential by Wegner [2], using the\nrelation between the DOS and the integrated DOS. The calculational advantage of using a finite cube\nis its discrete spectrum. Then the corresponding DOS is a sum of Dira c Delta functions (or poles of\nthe corresponding Green’s function), which can be studied, for ins tance, by averaging with respect to a\ncontinuous disorder distribution. Then the Dirac Delta functions co ntribute to the average with their\nspectral weights.\nThis concept can be generalized by introducing a generating functio n for the local DOS, which is the\nphase of a unimodular function [3–5]. The phase has special propert ies under the change of the matrix\nelements of the underlying tight-binding Hamiltonian, which leads to a fl exible method for estimating\nthe average DOS.\nFor a diagonal matrix Uand a short-range tight-binding matrix H0with lattice sites {r}there is a\ngenerating function\nFΛ=ilog/bracketleftbiggdet(H0−U−iǫ)\ndet(H0−U+iǫ)/bracketrightbigg\nwithρr=1\n2π∂FΛ\n∂Ur(ǫ >0) (1)\nfor the local density of states ρron the lattice Λ. The specific form of H0is not important for the\nfollowing discussion, as long as it is short ranged. The latter is crucial because it allows us to obtain a\nsufficiently thin surface to disconnected cubes on the lattice Λ. Whe therH0is a symmetric tight-binding\nHamiltonian or a discrete Dirac operator with spinor states does not affect the validity of the approach.\nFΛhas some remarkable properties: It is real, since the argument of t he logarithm is unimodular, and\nit is an increasing function for any Ur, since the DOS is non-negative. FΛis bounded for the shift of a\nsingle variable U′\nr→Uras\n0≤FΛ(Ur)−FΛ(U′\nr)≤2π(Ur> U′\nr) (2)\nand fornshifted variables U′\nr1,U′\nr2,...,U′\nrn→Ur1,Ur2,...,Urnit is bounded as\n0≤FΛ(Ur1,Ur2,...,Urn)−FΛ(U′\nr1,U′\nr2,...,U′\nrn)≤2πn(Urj> U′\nrj). (3)\nFΛis additive on the lattice in the limit Ur→ ∞on∂S:\nlim\nUr→∞,r∈∂SFΛ=FS+FS′′ (4)\nfor a cube Swith the boundary ∂Sand the complement S′′outsideS∪∂S.FS(FS′′) is the function FΛ\nwithH0−U±iǫprojected onto the subspace S(S′′). The combination of (3) and (4) implies\nFS+FS′′−2π|∂S| ≤FΛ≤FS+FS′′, (5)\nwhere|∂S|is the number of lattice sites in ∂S.\nBefore we discuss the lower bound of the average DOS, an interpre tation of the generating function FΛ\nmight be useful. According to its definition in Eq. (1) FΛ/2is the phase of the determinant of H0−U−iǫ.\nIf we increase a single Urthis phase also increases, as indicated by (2). Since this happens fo r the increase\nof any of elements of U, the phase changes add up and the shift U′\nr1,U′\nr2,...,U′\nrn→Ur1,Ur2,...,Urn2\nprovides a winding number of the determinant. The change of all elem ents ofUby a constant Eleads\nto the integrated DOS on the interval [0 ,E]:\nN(0,E) =/integraldisplayE\n0/summationdisplay\nrρr(U+y)dy , (6)\nwhich is the number of eigenvalues of H0−Uon the interval [0 ,E]. In other words, the winding number\nof a global change of Uis equal to the number of eigenstates on the interval of the chang e.\nNow we return to the average DOS\n/summationdisplay\nr∈S¯ρr:=/summationdisplay\nr∈S/integraldisplay\nρr(U)/productdisplay\nr∈ΛP(Ur)dUr=1\n2π/integraldisplay/summationdisplay\nr∈S∂FΛ\n∂Ur/productdisplay\nr∈ΛP(Ur)dUr. (7)\nThe distribution density on Scan be written as an integral transform\n/productdisplay\nr∈SP(Ur) =P′(UΠS)/integraldisplayv\n−v/productdisplay\nr∈SP(Ur−u)du , (8)\nwhere Π Sis the projector onto S. This gives us for the right-hand side of (7)\n1\n2π/integraldisplay/summationdisplay\nr∈S∂FΛ(U)\n∂Ur/integraldisplayv\n−v/productdisplay\nr∈SP(Ur−u)duP′(UΠS)/productdisplay\nr∈SdUr/productdisplay\nr/∈SP(Ur)dUr\nand with the new integration variable U′\nr=Ur−uwe have\n=1\n2π/integraldisplay /integraldisplayv\n−v/summationdisplay\nr∈S∂FΛ(U′+uΠS)\n∂UrP′((U′+u)ΠS)du/productdisplay\nr∈SP(U′\nr)dU′\nr/productdisplay\nr/∈SP(Ur)dUr\n≥1\n2π/integraldisplay\ninf\n−v≤w≤vP′((U′+w)ΠS)/integraldisplayv\n−v/summationdisplay\nr∈S∂FΛ(U′+uΠS)\n∂Urdu/productdisplay\nr∈SP(U′\nr)dU′\nr/productdisplay\nr/∈SP(Ur)dUr(9)\nWith the relation\n/integraldisplayv\n−v/summationdisplay\nr∈S∂FΛ(U′+uΠS)\n∂Urdu=/integraldisplayv\n−v∂FΛ(U′+uΠS)\n∂udu=FΛ(U′+vΠS)−FΛ(U′−vΠS)\nwe obtain\n/summationdisplay\nr∈S¯ρr≥1\n2π/integraldisplay\n[FΛ(U′+vΠS)−FΛ(U′−vΠS)] inf\n−v≤w≤vP′((U′+w)ΠS)/productdisplay\nr∈SP(U′\nr)dU′\nr/productdisplay\nr/∈SP(Ur)dUr.(10)\nNow we apply the inequalities (5) and get a lower bound\n/summationdisplay\nr∈S¯ρr≥1\n2π/integraldisplay\n[FS(U′+v)−FS(U′−v)−2π|∂S|] inf\n−v≤w≤vP′((U′+w)ΠS)/productdisplay\nr∈SP(U′\nr)dU′\nr,(11)\nwhere the integration outside of Shas been performed, since the integrand does not depend on Urfor\nr/∈S. Next, we estimate the integral as\n1\n2π/integraldisplay\n[FS(U′+v)−FS(U′−v)−2π|∂S|] inf\n−v≤w≤vP′((U′+w)ΠS)/productdisplay\nr∈SP(U′\nr)dU′\nr\n≥inf\n−a≤U′r≤a,r∈S[FS(U′+v)−FS(U′−v)−2π|∂S|]1\n2πinf\n−v≤w≤vP′((U′+w)ΠS). (12)3\nFS(U′+v)−FS(U′−v) is the integrated DOS on the cube S\n/integraldisplayv\n−v/summationdisplay\nr∈SρS,r(U′+E)dE ,\nwhich counts the number of eigenstates of the |S|×|S|–matrix Π S(H0−U′)ΠSon the interval [ −v,v].\nFinally, from Eq. (8) we get\nP′((U′+w)ΠS) =/producttext\nr∈SP(U′\nr+w)/integraltextv\n−v/producttext\nr∈SP(U′r+w−u)du≥/productdisplay\nr∈SP(U′\nr+w), (13)\nwhich gives inequality (19) of the Letter.\nII. DERIVATION OF PROPERTIES (3) AND (4)\nAs discussed above, we obtain a lower bound of the average DOS ess entially through properties (3)\nand (4) of the generating function FΛ. These properties were discussed previously in Refs. [3]–[5] but for\na consistent notation we summarize them in the following.\nA. Property (3)\nThe inequality (2) for one shifted variable is directly related to the Lip pmann-Schwinger equation for\na single impurity via\nFΛ(U′\nr)−FΛ(U′′\nr) = 2π/integraldisplayU′\nr\nU′′rρr(Ur)dUr=−i/integraldisplayU′\nr\nU′′r/bracketleftbigg1\n1/γ∗r−Ur−1\n1/γr−Ur/bracketrightbigg\ndUr≤2π ,(14)\nwhereγr=G0,rris the spatial diagonal element of the Green’s function. The integra l is also non-negative\nbecause the imaginary part of γris positive for ǫ >0. A special case is that of Weyl particles because of\nγr∼0. Then the above expression is always zero for finite Ur,U′\nrbecause the pole of the integrand is at\ninfinity, as mention in the Letter.\nThen we apply two times (14) to obtain for two shifted variables\n0≤FΛ(Ur1,Ur2)−FΛ(U′\nr1,U′\nr2) =FΛ(Ur1,Ur2)−FΛ(U′\nr1,Ur2)+FΛ(U′\nr1,Ur2)−FΛ(U′\nr1,U′\nr2)≤4π .\nThis procedure can be repeated ntimes for nshifted variables to give (3). The result is justified by\ncomplete induction: Suppose (3) is correct. Then we get for n+1\nFΛ(Ur1,Ur2,...,Urn+1)−FΛ(U′\nr1,U′\nr2,...,U′\nrn+1) =FΛ(Ur1,Ur2,...,Urn+1)−FΛ(U′\nr1,U′\nr2,...,U′\nrn,Urn+1)\n+FΛ(U′\nr1,U′\nr2,...,U′\nrn,Urn+1)−FΛ(U′\nr1,U′\nr2,...,U′\nrn,U′\nrn+1)≤2πn+2π= 2π(n+1).\nB. Property (4)\nThe relation (4) can be derived by splitting U=US∪S′′+U∂Swith the projectors Π S, ΠS′′and Π ∂S\nontoS,S′′and∂S, respectively:\nUS∪S′′= ΠSUΠS+ΠS′′UΠS′′−ΠS−ΠS′′, U∂S= Π∂SUΠ∂S+ΠS+ΠS′′.\nThen we obtain the following equations\ndet(H0−U−iǫ)\ndet(H0−U+iǫ)=det(H0−US∪S′′−U∂S−iǫ)\ndet(H0−US∪S′′−U∂S+iǫ)4\n=det{U1/2\n∂S[U−1/2\n∂S(H0−US∪S′′−iǫ)U−1/2\n∂S−1]U1/2\n∂S}\ndet{U1/2\n∂S[U−1/2\n∂S(H0−US∪S′′+iǫ)U−1/2\n∂S−1]U1/2\n∂S}=det[U−1/2\n∂S(H0−US∪S′′−iǫ)U−1/2\n∂S−1]\ndet[U−1/2\n∂S(H0−US∪S′′+iǫ)U−1/2\n∂S−1].\n(15)\nThe limit Ur→ ∞on∂Sgives\nlim\nUr→∞,r∈∂SU−1/2\n∂S= ΠS+ΠS′′. (16)\nSince∂Sseparates two regions SandS′′on the lattice with Π SH0ΠS′′= 0, this leads to\nlim\nUr→∞,r∈∂SU−1/2\n∂S(H0−US∪S′′±iǫ)U−1/2\n∂S= (ΠS+ΠS′′)(H0−US∪S′′±iǫ)(ΠS+ΠS′′)\nand, because of Π SH0ΠS′′= 0, we get\n= ΠS(H0−U±iǫ)ΠS+ΠS′′(H0−U±iǫ)ΠS′′. (17)\nFor the ratio of determinants we have\nlim\nUr→∞,r∈∂Sdet(H0−U−iǫ)\ndet(H0−U+iǫ)=detS(H0−U−iǫ)\ndetS(H0−U+iǫ)detS′′(H0−U−iǫ)\ndetS′′(H0−U+iǫ), (18)\nwhere the index of the determinants refers to the projection of t he matrix. Inserting this result into FΛ\ngives Eq. (4).\n[1] W. Ledermann, Proc. R. Soc. London 182, 362 (1944).\n[2] F. Wegner, Z. Physik B - Condensed Matter 44, 9 (1981).\n[3] K. Ziegler, Nucl. Phys. B 285[FS19], 606 (1987).\n[4] K. Ziegler, Commun. Math. Phys. 120, 177 (1988).\n[5] K. Ziegler, M.H. Hettler, P.J. Hirschfeld, Phys. Rev. B 57, 10825 (1998)."    },    {        "title": "1707.07850v1.Order_disorder_transition_in_active_nematic__A_lattice_model_study.pdf",        "content": "Order-disorder transition in active nematic: A lattice model study\nRakesh Das,1,\u0003Manoranjan Kumar,1,yand Shradha Mishra1, 2,z\n1S N Bose National Centre for Basic Sciences, Block JD, Sector III, Salt Lake, Kolkata 700106, India\n2Department of Physics, Indian Institute of Technology (BHU), Varanasi 221005, India\nWe introduce a lattice model for active nematic composed of self-propelled apolar particles, study\nits di\u000berent ordering states in the density-temperature parameter space, and compare with the\ncorresponding equilibrium model. The active particles interact with their neighbours within the\nframework of the Lebwohl-Lasher model, and move anisotropically along their orientation to an\nunoccupied nearest neighbour lattice site. An interplay of the activity, thermal \ructuations and\ndensity gives rise distinct states in the system. For a \fxed temperature, the active nematic shows\na disordered isotropic state, a locally ordered inhomogeneous mixed state, and bistability between\nthe inhomogeneous mixed and a homogeneous globally ordered state in di\u000berent density regime.\nIn the low temperature regime, the isotropic to the inhomogeneous mixed state transition occurs\nwith a jump in the order parameter at a density less than the corresponding equilibrium disorder-\norder transition density. Our analytical calculations justify the shift in the transition density and\nthe jump in the order parameter. We construct the phase diagram of the active nematic in the\ndensity-temperature plane.\n(Received on 26 January, 2017 and accepted on 27 June, 2017 in Scienti\fc Reports)\nINTRODUCTION\nSelf-propelled particles compose an interesting type of the active systems [1{5] where each particle extracts energy from\nits surroundings and dissipates it through motion and collision. Their examples range from very small intracellular\nscale to larger scales [6{20]. Also many arti\fcially designed systems, e.g., vibrated granular media [21{24], active polar\ndisks [25], active colloids [26{29] imitate the physics of the active systems. If ^ nis the average alignment direction\nof a collection of such active particles, and the system remains invariant under the transformation ^ n!\u0000^ n, it is\ncalled `active nematic'. Activity introduces many interesting properties which are absent in their thermal equilibrium\ncounterparts. One of such interesting features is the presence of large density \ructuation in the ordered active\nnematic [30]. Density is a key control parameter in various experiments and numerical simulations. Earlier studies\non equilibrium nematic for a \fxed temperature show the isotropic to nematic transition at some critical density [31].\nHowever, the e\u000bect of the density \ructuation in the active nematic is not well understood.\nMost of the previous studies of the active systems are done either by using the coarse-grained hydrodynamic\nequations of motion [32] or microscopic rule based numerical simulation of agent based point particles [33] or Brownian\ndynamic simulation [34]. Here we introduce a lattice model for a two-dimensional active nematic, explore various\nstates of the system in the density-temperature plane, and compare it with the corresponding equilibrium model. In\ngeneral, lattice model itself is interesting for development of simpli\fed theories, and provides insight into complex\nsystems. Our model is analogous to the previous lattice model of polar active spins [35, 36]; but we include volume\nexclusion to avoid multiple occupancy on single site. Such volume exclusion limits the motion of particles towards\nan occupied neighbouring site, and introduces new features, e.g., typical pattern formation [20, 37], density induced\nmotility [38] in the system.\nWe construct a phase diagram for the active nematic in the density-temperature plane, as shown in Fig. 1(a). There\nwe observe - (i) disordered isotropic (I) state in low density regime, (ii) locally ordered inhomogeneous mixed (IM)\nstate in intermediate density regime, and (iii) bistability between the IM and a homogeneous globally ordered (HO)\nstate in high density regime. In contrast to the continuous isotropic to nematic (I-N) transition in the equilibrium\nsystem, the I to IM state transition in the active nematic in the low temperature regime occurs with a jump in order\nparameter, as shown in Fig. 2(a). This transition occurs at a density lower than the equilibrium critical value, and\nthe system forms clear bands (BS) in this regime. We \fnally justify the jump in the order parameter and the shift in\nthe transition density by analytical study of the coarse-grained hydrodynamic equations written for the active model.\n\u0003rakesh.das@bose.res.in\nymanoranjan.kumar@bose.res.in\nzsmishra.phy@itbhu.ac.inarXiv:1707.07850v1  [cond-mat.soft]  25 Jul 2017 0.5 0.75 1\n< C >(b)\n(b) BS\n(b) BS IM\n(b) BS IM HO\n 0 0.25 0.5 0.75 1\n|2θ - π| / π(b) BS IM HO ENβε\nC1.21.41.61.82.02.2\n 0.3  0.5  0.7  0.9(b) BS IM HO EN (a) I BS IM Bistable\nβε\nC1.21.41.61.82.02.2\n 0.3  0.5  0.7  0.9(b) BS IM HO EN (a) I BS IM Bistable\nβε\nC1.21.41.61.82.02.2\n 0.3  0.5  0.7  0.9(b) BS IM HO EN (a) I BS IM Bistable\nβε\nC1.21.41.61.82.02.2\n 0.3  0.5  0.7  0.9(b) BS IM HO EN (a) I BS IM Bistable\nβε\nC1.21.41.61.82.02.2\n 0.3  0.5  0.7  0.9(b) BS IM HO EN (a) I BS IM BistableFigure 1. Phase diagram. (a) Phase diagram for both the equilibrium and the active nematic in the density - inverse\ntemperature plane. The equilibrium system remains in the isotropic (EI) state in the low density regime (on the left of the solid\nline) and in the nematic (EN) state in the high density regime (on the right of the solid line). The active nematic goes from\nthe disordered isotropic (I) state to the locally ordered inhomogeneous mixed (IM) state with increasing density or decreasing\ntemperature. The I - IM transition occurs with the appearance of clear bands (BS) in the low temperature regime. In the high\ndensity regime the active nematic shows bistability between the IM and the homogeneous globally ordered (HO) state. (b)\nUpper panel shows particle inclination towards the horizontal direction. Colour bar ranging from zero to one indicates vertical\nto horizontal orientation, respectively. BS is the banded state con\fguration shown for ( \f\u000f;C ) = (2:0;0:38). IM, HO and EN\nstate con\fgurations are shown for ( \f\u000f;C ) = (2:0;0:78). Lower panel shows the coarse-grained density in the respective states.\n0.3 0.5 0.7 0.9\nC00.20.40.60.8 S\n0 0.04 0.08 0.12 0.16\nS00.10.2P(S)0 4 8 12 16\n106/ L200.40.8 S\n1.5 2 2.5 3\nt / 10600.20.40.60.8 SIIMHO\nE\nHO\nIMIMHO(a)\n(b)\n(d)(c)\nFigure 2. Disorder - order transition. (a) Scalar order parameter versus packing density plot for 400 \u0002400 system size at\n\f\u000f= 2:0. The equilibrium system (E, solid line) shows continuous isotropic to nematic state transition with increasing density.\nThe active system goes from the isotropic (I) state to the locally ordered inhomogeneous mixed (IM, \u0002) state. In the high\ndensity regime the system shows bistability between the IM state and the homogeneous globally ordered (HO, \u000f) state. (b) The\nI to IM transition at low temperature occurs with a jump in Swhere the particles form bands (BS). Distribution of the scalar\norder parameter near the I - BS transition at ( \f\u000f;C ) = (2:0;0:37) shows two peaks. (c) Finite size scaling of Sfor both the\nHO and the IM state at ( \f\u000f;C ) = (2:0;0:76). (d) Order parameter time series show that the active system \rips in between the\nHO and the IM state in the bistable regime. Two time series are shown for two di\u000berent parameter values in the high density\nregime.\nMODEL\nWe consider a collection of apolar particles on a two dimensional square lattice, as shown in schematic diagram Fig.\n3(a). Occupation number ` ni' of theithlattice site can take values 1 (occupied) or 0 (unoccupied). Orientation \u0012i\nof apolar particle at the ithsite can take any value between 0 and \u0019. The model follows two sequential processes at\nevery step; \frst, a particle moves to a nearest neighbouring site with some probability , and then orientation of the\n2Figure 3. Model \fgure. (a) Two dimensional square lattice with occupied ( n= 1) or unoccupied ( n= 0) sites. Filled\ncircles indicate the occupied sites. Inclinations of the rods towards the horizontal direction show respective particle orientations\n\u00122[0;\u0019]. (b) Equilibrium move: particle can move to any of the four neighbouring sites with equal probability 1 =4. (c, d)\nActive move: particle can move to either of its two neighbouring sites with probability 1 =2, if unoccupied, in the direction it\nis more inclined to, i.e., along BDin (c), and ACin (d).\nparticle is updated based on its nematic interaction with its nearest neighbours. We de\fne two kinds of models on\nthe basis of particle movement: (i) `Equilibrium model' (EM) - particle moves with equal probability 1 =4 to any of\nthe four neighbouring sites (Fig. 3(b)), (ii) `Active model' (AM) - in this model particle movement occurs in two\nsteps. First, it chooses a direction along which it is more inclined. As shown in Fig. 3(c,d), it chooses the direction\nof movement along BDif\u0019=4< \u0012\u00143\u0019=4 and along ACotherwise. In the second step, it moves to a randomly\nselected site between the two nearest neighbouring sites along the chosen direction. For example, if BDis selected as\nthe direction of movement, then the particle moves to randomly selected site BorDin the second step. In both the\nmodels, we consider volume exclusion, i.e., particle movement is allowed only if the selected site is unoccupied.\nIn both the models, the particles also interact with their nearest neighbours. The interaction depends on the relative\norientation of the particles and is represented by a modi\fed Lebwohl-Lasher Hamiltonian [39]\nH=\u0000\u000fX\n<ij>ninjcos 2(\u0012i\u0000\u0012j) (1)\nwhere\u000fis the interaction strength between two neighbouring particles. The interaction in equation (1) governs the\norientation update of the particle. We employ Metropolis Monte-Carlo (MC) algorithm [40] for orientation update of\nthe particle after the movement trial. In both the models, an order parameter de\fning the global alignment of the\nsystem does not remain conserved during the MC orientation update described above. In actual granular or biological\nsystems where mutual alignment emerges because of steric repulsion, orientation of particles need not to follow a\nconservation law. An order parameter de\fned by coarse-graining the orientation in our present model is a class of\nnon-conserved order parameter: Model A as described by Hohenberg and Halperin [41].\nBoth the models EM and AM comprise of two di\u000berent physical aspects - motion of the particles and nematic\ninteraction amongst the nearest neighbours. If the particles are not allowed to move, the models reduce to an apolar\nanalogue of the diluted XY-model with nonmagnetic impurities [42], where impurities and spins are analogous to\nvacancies and particles, respectively. However, unlike the diluted XY-model, particles in these models are dynamic.\nIn the EM, the particle di\u000buses to neighbouring sites, whereas it moves anisotropically in the AM. The anisotropic\nmovement of the active particles arises in general because of the self-propelled nature of the particles in many biological\n[43] and granular systems [21, 22]. This move produces an active curvature coupling current in the coarse-grained\nhydrodynamic equations of motion [30, 32]. The AM does not satisfy the detailed balance principle [40], because of\nthe orientation update after the anisotropic movement. The coupling of the particle movement with the orientation\nupdate in our active model is analogous to the active Ising spin model introduced by Solon and Tailleur [35, 36],\nwhere the probabilistic \rip of the spins is an equilibrium process, whereas the out-of-equilibrium aspect of the model\nis attributed to the anisotropic movement probability of the spins. However, their orientation update algorithm [35, 36]\nis similar to kinetic Monte-Carlo, whereas we use Metropolis Monte-Carlo algorithm to update particle orientation.\nNUMERICAL STUDY\nWe consider a collection of Nparticles with random orientation \u00122[0;\u0019] homogeneously distributed on a L\u0002L\nlattice (L= 256;400;512) with periodic boundary. Packing density of the system is de\fned as C=N=(L\u0002L). We\n3choose a particle randomly, move it to a neighbouring site obeying exclusion, and then update its orientation using\nMetropolis algorithm. In each iteration, we repeat the same process for Nnumber of times, and we use 1 :5\u0002106\niterations to achieve the steady state of the system. We obtain the steady state results by averaging the observables\nover next 1 :5\u0002106iterations and use more than twenty realisations for better statistics.\nThe ordering in the system is characterised by a scalar order parameter de\fned as\nS=vuut \n1\nNX\ninicos(2\u0012i)!2\n+ \n1\nNX\ninisin(2\u0012i)!2\n: (2)\nIt is proportional to the positive eigenvalue of the nematic order parameter Q[31]. It takes the minimum value 0 in\nthe disordered state and the maximum value 1 in the complete ordered state. First we study the EM as a function\nof inverse temperature \f= 1=kBTfor di\u000berent packing densities. As shown in Fig. A1, the system shows disordered\nisotropic to nematic state (I-N) transition with decreasing temperature. In contrast to the \frst order I-N transition in\nthe equilibrium Lebwohl-Lasher model in three dimensions [39, 44], we \fnd continuous transition for the EM de\fned\nin two dimensions. The observed nature of transition supports the study by Mondal and Roy [45]. Similar to the\ndiluted XY-model [42], the critical inverse temperature \fc(C) increases with density in the EM.\nPhase diagram\nWe construct phase diagram for both the equilibrium model and the active model on the density-temperature plane.\nAs shown in Fig. 1(a), two distinct states appear in the EM - (i) an equilibrium isotropic (EI) state on the left side of\nthe red boundary and (ii) an equilibrium nematic (EN) state on the right side of the red boundary. In the EI state,\nparticles remain disordered and homogeneously distributed throughout the system. Consequently, the scalar order\nparameterS'0 in this state. With increasing density or decreasing temperature the particles get mutually ordered\nand form the EN state ( S >0). As shown in Fig. 2(a), for a \fxed temperature the scalar order parameter increases\ncontinuously with increasing density, and the system enters into the nematic state. Both the particle orientation and\nthe coarse-grained density remain homogeneous in the EN state, as shown in the real space snapshot Fig. 1(b).\nSimilar to the EM, the active system remains in a homogeneous disordered isotropic (I) state in the high temperature\nand/or low packing density regime (cyan coloured regime in the phase diagram Fig. 1(a)). With increasing density\nor decreasing temperature, beyond the I state, the active system enters into an inhomogeneous mixed (IM) state\n(golden regime in the phase diagram Fig. 1(a)), where locally ordered high-density domains coexist with disordered\nlow-density regions. In the low temperature regime ( \f\u000f2[1:9;2:2]), the I to IM state transition with increasing C\noccurs with a jump in the scalar order parameter S, as shown in Fig. 2(a). In the very beginning of the IM state, as\nindicated by cross symbols in Fig. 1(a), we \fnd a banded state (BS) in the low temperature regime, where particles\ncluster and align themselves within a strip to form band. However, out of the strip the system remains disordered\nwith low local density, as shown in the real space snapshot Fig. 1(b). On further increment of the packing density C,\nbands formed in di\u000berent directions start mixing leaving the system with many locally ordered high density patches\nseparated by low density disordered regions. Typical real space snapshots for the orientation and the coarse grained\ndensity in the IM state are shown in Fig. 1(b). The jump in the S\u0000Ccurve reduces with increasing temperature,\nand no bands appear in the high temperature ( \f\u000f< 1:9) regime.\n1 10 100 1000 10000\n< N >110 ∆N / < N >1/2 ζ = 1.0\nHOIMBS\nI\nFigure 4. Density \ructuation \u0001 N=p\n<N2>\u0000<N >2. All the active ordered states show large density \ructuation obeying\nthe relation \u0001 N\u0018hNi\u0010with\u0010 >1=2. The active disordered isotropic state shows normal density \ructuation with \u0010= 1=2.\n4Figure 2(a) shows that the I to BS transition occurs in the low temperature regime with a jump in Sat a density\nlower than the corresponding equilibrium I-N transition density CIN. These bands appear because of the large\nactivity strength. A linear stability analysis, as detailed later in this paper, shows that the large activity strength\ninduces an instability in the disordered isotropic state. This instability goes away for small activity strength or at\nhigh temperature. We also do a renormalised mean \feld calculation of an e\u000bective free energy written for the active\nnematic. The calculation predicts a jump in the scalar order parameter and shows a shift in the disordered ( S= 0)\nto ordered ( S6= 0) state transition density. Both the jump in Sand the shift in the transition density reduce with\nthe activity strength or increasing temperature. The I to BS transition is a \frst order transition. The shift in the\ndisorder-order transition point is a common feature of the active systems. For large activity and low temperature, if\nthe system density is above a certain value but less than CIN, the large density \ructuation present in these systems\ncauses local alignment with local density higher than CIN. Large density \ructuation is an intrinsic feature of the\nactive systems, and as shown in Fig. 4, we also observe the same in the ordered active states in our model. Due to\nactivity these locally ordered regions move anisotropically and combine with nearby region with similar local ordering.\nSo larger ordered region forms at mean density lower than the equilibrium I-N transition density. Therefore, we \fnd\na disordered to ordered state transition at a lower density. For large activity strength, I-BS transition occurs with\nthe jump in scalar order parameter. In our numerical study we calculate the probability P(S) of the scalar order\nparameter averaging over many iterations and realisations near the I-BS transition point. Figure 2(b) shows P(S)\nhas two peaks, which further supports the \frst order I-BS transition for large activity strength.\nIn the high density regime (red coloured regime in the phase diagram Fig. 1(a)), the AM shows bistability, i.e.,\nit can be either in the locally ordered IM state or in a homogeneous globally ordered (HO) state. As shown in Fig.\n2(a), theS\u0000Ccurve for \fxed temperature bifurcates in the high density regime; the lower branch corresponds to\nthe earlier discussed IM state, whereas the higher branch indicates the existence of the globally ordered state. Figure\n1(b) shows that the system possesses less density inhomogeneity in the HO state compared to the IM state. A \fnite\nsize scaling of both the HO and the IM state, as shown in the Fig. 2(c), shows that the active nematic possesses\nnon-zero \fnite order in both these states. Order parameter time series shown in Fig. 2(d) con\frms the bistability of\nthe system in the high density regime. Bistability is not generally seen in other agent based numerical simulations of\npoint particles [33]; it appears because of \fnite \flling constraint of the model. This feature can be suppressed if we\nallow more than one particle to sit together. In the complete \flling limit C= 1:0, the AM is equivalent to the EM,\nand it shows the globally ordered HO state only.\nTwo-point orientation correlation\nWe further characterise various states on the basis of the two-point orientation correlation in the di\u000berent states of\nthe equilibrium and the active nematic. It is de\fned as g2(r) =<P\ninini+rcos [2 (\u0012i\u0000\u0012i+r)]=P\nin2\ni>wherer\nrepresents interparticle distance, and < : > signi\fes an average over many realisations. Figure 5(a, b) show g2(r)\nversusrplots on log-log scale for the AM and the EM, respectively, for a \fxed inverse temperature \f\u000f= 2:0. In the\nAM,g2(r) decays exponentially at low packing density C < 0:37, i.e., in the isotropic state. Therefore, the active\nisotropic is a short-range-ordered (SRO) state. In the BS at C= 0:38,g2(r) decays following a power law. Therefore,\nthe system is in a quasi-long-range-ordered (QLRO) state. Ordering increases with density. At high packing density,\ncorrelation function con\frms the bistability in the active system. At C= 0:82,g2(r) shows power law decay in the\n1 10 100r0.0010.010.11g2(r)\n1 10 100C=0.30C=0.36C=0.38\n(a)\nActiveC=0.78\nC=0.50\nC=0.48\nC=0.30Equilibrium(b)C=0.82 (IM)C=0.82 (HO)\nC=0.52\nFigure 5. Two-point orientation correlation shown for \f\u000f= 2:0 on log-log scale. (a) Active model: g2(r) decays exponentially\nat low density (\r,\u0003) and algebraically at high density ( \u0005,4). In the bistable regime at high density ( 4),g2(r) decays\nalgebraically in the HO state and abruptly in the IM state. (b) Equilibrium model: g2(r) decays exponentially at low density\n(\r,\u0003) and algebraically at high density ( \u0005, +,4). Continuous lines are the respective \fts, \ftted for more than one decade.\n5HO state, whereas in the IM state g2(r) decays abruptly after a distance r. The abrupt change in g2(r) at a certain\ndistance indicates the presence of locally ordered clusters in the IM state. In contrast, the equilibrium system shows a\ntransition from SRO (exponential decay) isotropic state at low density C<\u00180:48 to QLRO (power law decay) nematic\nstate at high density C>\u00180:50.\nOrientation distribution and autocorrelation of the mean orientation\nWe compare the steady state properties of the active and the equilibrium models in the high density limit. First we\ncalculate the steady state (static) orientation distribution P(\u0012) from a snapshot of particle orientation \u0012. As shown in\nFig. 6(a), both the active HO and the equilibrium nematic show Gaussian distribution of orientation. Peak position\nofP(\u0012) for both the EN and the HO state can appear at any point between 0 and \u0019because of the continuous broken\nrotational symmetry of the Hamiltonian shown in equation (1). Data shown in Fig. 6(a) is for one realisation only, and\nfor other realisations also the distribution P(\u0012) remains Gaussian with peak at other \u0012values. Therefore, orientation\n\ructuation of the particles in the active HO state is same as in the equilibrium nematic state. The distribution P(\u0012)\nin the IM state is very broad and spans over the whole range of orientation. Therefore, the system possess no global\nordering in the IM state.\nWe also calculate the time averaged distribution P(\u0016\u0012) of mean orientation of all the particles in the active HO and\nthe equilibrium nematic states. The mean orientation \u0016\u0012(t) of all particles is calculated for each iteration time tin\nthe steady state. The distribution P(\u0016\u0012) of the mean orientation is obtained from these \u0016\u0012(t) data. This distribution\nis a measure of the \ructuation in the global orientation of the particles in the steady state. As shown in Fig. 6(b),\nP(\u0016\u0012) in the active HO state is narrow in comparison to the broad distribution in the EN state. We also calculate\nthe autocorrelation of the mean orientation C\u0016\u0012(t) =<1\ntPt\n\u001c=1cos\u0002\n2\b\u0016\u0012(t0)\u0000\u0016\u0012(t0+\u001c)\t\u0003\n>in the steady state. As\nshown in Fig. 6(c), C\u0016\u0012(t) decreases with time in the EN state, but remains unchanged in the active HO state. Both\nthese results imply that the \ructuation in the global orientation direction \u0016\u0012in the active HO state is small compared\nto the EN state. We do not calculate the mean orientation \u0016\u0012in the active IM state, because the system possesses no\nglobal ordering in this state.\n0 0.2 0.4 0.6 0.8 1\nθ / π00.0050.010.0150.02P(θ)\n0 0.2 0.4 0.6 0.8 1\nθ / π00.020.040.060.08P(θ)\n104105106\nt0.40.60.81< Cθ (t) >EN HO\nHO\nEN(b)(a)\nIM\nHO\nEN(c)\nFigure 6. Steady state characteristics of high density states. (a) Orientation distribution P(\u0012) of particles calculated from one\nsnapshot in the steady state. P(\u0012) \fts with Gaussian distribution (continuous lines) for both the HO and the EN states. The\nIM state shows broad distribution of \u0012. (b) Distribution of the mean orientation P(\u0016\u0012) calculated from \u0016\u0012of each snapshot in\nthe steady state. P(\u0016\u0012) is broad for the EN state in comparison to the HO state. (c) Steady state autocorrelation C\u0016\u0012(t) of the\nmean orientation of the system. All plots are shown for ( \f\u000f;C ) = (2:0;0:80).\n6PHENOMENOLOGICAL APPROACH TO UNDERSTAND LOW DENSITY STATES OF THE ACTIVE\nMODEL\nIn this section we write the hydrodynamic equations of motion for the active model and characterise the low density\nstates of the system. The equations of motion for the slow variables of the system, i.e., the number density }(r;t) =P\nl\u000e(r\u0000Rl(t)) and the order parameter \u0005 ij(r;t) =}(r;t)Qij(r;t) =P\nl(mlimlj\u00001\n2\u000eij)\u000e(r\u0000Rl(t)) are as follows\n[30, 32]:\n@t}=a0rirj\u0005ij+D}r2} (3)\nand\n@t\u0005ij=f\u000b1(})\u0000\u000b2(\u0005 : \u0005)g\u0005ij+\f\u0012\nrirj\u00001\n2\u000eijr2\u0013\n}+D\u0005r2\u0005ij (4)\nHereRl(t) represents position of the particle l, and ml= (cos\u0012l;sin\u0012l) is the unit vector along the orientation \u0012l.\nThe total number of particles being a conserved quantity of the system, equation (3) represents a continuity equation\n@t}=\u0000r\u0001Jwhere the current Ji=\u0000a0rj\u0005ij\u0000D}ri}. The \frst term of Jiconsists of two parts: an anisotropic\ndi\u000busion current Jp1/Qijri}and an active curvature coupling current Ja/a0}rjQijwherea0is the activity\nstrength of the system. The second term represents an isotropic di\u000busion Jp2/r}. The\u000bterms in equation (4)\nrepresent mean \feld alignment in the system. We choose \u000b1(}) = (}\n}IN\u00001) as a function of density that changes\nsign at some critical density }IN. The\fterm represents coupling with density. The last term represents di\u000busion in\norder parameter that is written under equal elastic constant approximation for two-dimensional nematic. The steady\nstate solution }(r;t) =}0and \u0005( r;t) = \u0005 0, where \u0005 0=q\n\u000b1(}0)\n\u000b2, of equations (3) and (4) represents a homogeneous\nordered state for \u000b1(}0)>0 at}0>}IN, and a disordered isotropic state for \u000b1(}0)<0 at}0<}IN.\nWe study the linear stability of the disordered isotropic state (\u0005 0= 0) by examining the dynamics of spatially\ninhomogeneous \ructuations \u000e}(r;t) =}(r;t)\u0000}0,\u000e\u000511= \u0005 11(r;t), and\u000e\u000512= \u0005 12(r;t). We obtain the linearised\ncoupled equations of motion for small \ructuations as\n@t\u000e}=a0\u0000\n@2\nx\u0000@2\ny\u0001\n\u000e\u000511+ 2a0@x@y\u000e\u000512+D}r2\u000e} (5)\n@t\u000e\u000511=\u000b1(}0)\u000e\u000511+D\u0005r2\u000e\u000511+\f\n2\u0000\n@2\nx\u0000@2\ny\u0001\n\u000e} (6)\n@t\u000e\u000512=\u000b1(}0)\u000e\u000512+D\u0005r2\u000e\u000512+\f@x@y\u000e} (7)\nUsing Fourier transformation\nY(q;\u0015) =Z\neiq:re\u0015tY(r;t)drdt (8)\nwe get linear set of equations in the Fourier space as\n\u00150\n@\u000e}\n\u000e\u000511\n\u000e\u0005121\nA=M0\n@\u000e}\n\u000e\u000511\n\u000e\u0005121\nA (9)\nwhereMis the coe\u000ecient matrix as obtained from equations (5), (6), and (7) after the transformation. We solve\nequation (9) for the hydrodynamic modes \u0015. We choose qx=qy=qp\n2since both the directions are equivalent.\nTherefore, we obtain\n\u0000\n\u0015\u0000\u000b1(}0) +D\u0005q2\u0001\u001a\u0000\n\u0015+D}q2\u0001\u0000\n\u0015\u0000\u000b1(}0) +D\u0005q2\u0001\n\u00001\n2a0\fq4\u001b\n= 0 (10)\nFor small wave-vector q, we can \fnd an unstable mode\n\u0015+=\u00002D}q2+a0\fq4\n2j\u000b1(}0)j\u0000a0\fq6(D\u0005\u0000D})\n\u000b2\n1(}0)(11)\nFor smallD}and large actvitity a0this mode becomes unstable for q<qc, where\nq2\nc=j\u000b1(}0)j\n2\u0001D+1\n2s\u0012j\u000b1(}0)j\n\u0001D\u00132\n\u00008D}\u000b2\n1\n\u0001Da0\f(12)\n7provided \u0001D=D\u0005\u0000D}is positive and a0\f >8D}\u0001D. Therefore, the unstable mode \u0015+causes the I - BS transition\nfor small di\u000busivity, i.e., at low temperature, and for large activity strength a0.\nWe also calculate the jump in the scalar order parameter Sand the shift in the transition density from equations\n(3) and (4). A homogeneous steady state solution of these equations gives a mean \feld transition from the isotropic\nto the nematic state at density }INwhere\u000b1(}) changes sign. Using renormalised mean \feld (RMF) method we\ncalculate an e\u000bective free energy Feff(S) close to the order-disorder transition where Sis small. We consider density\n\ructuations \u000e}and neglect order parameter \ructuations. The e\u000bective free energy is\nFeff(S) =\u0000b2\n2S2\u0000b3\n3S3+b4\n4S4(13)\nwhereb2=\u000b1(}) +\u000b0\n1c, wherecis a constant. \u000b0\n1=@\u000b1=@}j}0,b3=a0}0\u000b0\n1\n2D}, andb4=1\n2}2\n0\u000b2. Bothb3andb4are\npositive. A detailed calculation for Feffis shown in Appendix B. The density \ructuations introduce a new cubic\norder term in the free energy Feff(S) that is proportional to the activity strength a0. The presence of such term\nproduces a jump \u0001 S=Sc=2b3\n3b4at a density }c=}IN(1\u00002b2\n3\n9b4)< }IN. Fluctuation in density produces a jump\nin order parameter and shifts the critical density. Such type of \ructuation induced transitions are called \ructuation\ndominated \frst order phase transitions in statistical mechanics [46] and are widely studied for many systems [47, 48].\nThe jump in Sand the shift in the transition density are proportional to the activity strength a0, and fora0= 0 we\nrecover the equilibrium transition.\nDISCUSSION\nIn our present work we have introduced a minimal lattice model for the active nematic and study di\u000berent ordering\nstates in the density-temperature plane. A brief summary of the results is as follows. In the low density regime,\nthe system is in the disordered isotropic (I) state with short range orientation correlation amongst the particles. In\nthe low temperature regime, large density \ructuation in the active system induces a \frst order transition from the\nisotropic to the banded state with a jump in the scalar order parameter at a density lower than the equilibrium\nisotropic-nematic (I-N) transition density. The linear stability analysis of the isotropic state shows an instability\nfor large activity strength in the low temperature regime. Such instability governs the band formation at density\nbelow the equilibrium I-N transition density. As we further increase density, bands vanish and locally ordered patches\nappear in the inhomogeneous mixed (IM) state. Renormalised mean \feld calculation con\frms the jump in the scalar\norder parameter and the shift in the transition density. With increasing temperature the shift in the transition\ndensity and the jump in scalar order parameter decreases, and no bands appear in the system. The IM state is a\nstate with coexisting aligned and disordered domains, similar to the coexisting or defect-ordered states found in Ref.\n[33, 34, 49{53].\nIn the high density regime, the active nematic shows switching between the IM (low S) and the homogeneous\nordered (HO, high S) states, i.e., the system shows bistability. In the complete \flling limit and with excluded\nvolume assumption the active model reduces to the equilibrium model. Therefore, the active model tends to show a\nhomogeneous nematic state in the high density regime. However, large activity strength makes the HO state unstable\nand leads the system to the IM state. This instability in the HO state is similar to the earlier studies in Ref. [34, 54].\nNgo et al. [33] considered a two dimensional o\u000b-lattice model for the active nematic without the exclusion constraint.\nIn the low and moderate density regime, they show a homogeneous disordered phase and an inhomogeneous chaotic\nphase, which are similar to the isotropic and the IM states, respectively. Similar to their study, the spanning area\nof the IM state (golden regime in the phase diagram Fig. 1(a)) along the density axis decreases with the increasing\ntemperature. In the high density limit, they note a homogeneous quasi-ordered phase only, which similar to the HO\nstate in our study. However, we show the bistability between the HO and the IM state in this density limit.\nIn conclusion, our lattice model for the active nematic is a simple one to design and execute numerically, and easy\nto compare with the corresponding equilibrium model. It shows new features like the BS in the low temperature\nregime and the bistability in the high density regime, as well as some of the early characterised states, e.g., the\nIM state. It also shows many basic features of the active nematic like large number \ructuation, long-time decay of\norientation correlation, transition from SRO isotropic to QLRO nematic state. The shift in the transition density\ndue to activity strength compared to the equilibrium model can be tested in experimental systems where activity can\nbe tuned. We expect the emergence of the bistability in the high density regime in a two dimensional experimental\nsystem composed of apolar particles with \fnite dimension and high activity strength. It would be interesting to study\nthe model without volume exclusion. In this study, particle orientation has continuous symmetry of O(2). Therefore,\nthe equilibrium limit of our model is an apolar analogue of the two-dimensional XY-model. One can also study the\nmodel with discrete orientation symmetry as in Ref. [20, 35, 36] and compare the results with the corresponding\nequilibrium model.\n8Appendix A: Order-disorder transition in the EM\n0 1 2 3 4\nβε00.20.40.60.81S0.4 0.6 0.8 1\nC123\nβcε\nC = 0.30C = 0.40C = 0.50C = 0.60C = 1.00\nFigure A1. Order-disorder transition in the equilibrium model. Main - scalar order parameter Sversus inverse temperature\n\f\u000fplot for di\u000berent density C. With increasing \f\u000fthe system goes from the isotropic (small S) to the nematic (large S) state.\nInset - the critical inverse temperature decreases with increasing density.\nAppendix B: Renormalised mean \feld (RMF) study of active nematic for small scalar order parameter\nIn this section we write an e\u000bective renormalised mean \feld free energy for the scalar order parameter Sunder the\nsmallSapproximation. We consider the \ructuations in the density and ignore the order parameter \ructuations in\nthe coupled hydrodynamic equations of motion for the active nematic. Density \ructuation introduces a cubic order\nterm inSin the e\u000bective free energy. Such term produces a jump in Sat a new transition density }clower than the\nequilibrium I-N transition point }IN. Shift in the transition density and the jump \u0001 Sare directly proportional to\nthe activity strength a0. We recover the equilibrium limit for zero a0.\nIn the main text we write the coupled hydrodynamic equations of motion for the density }and the order parameter\n\u0005 =}Qwhere nematic order parameter [31] is de\fned as\nQ(r;t) =S\n2\u0012\ncos 2\u0012(r;t) sin 2\u0012(r;t)\nsin 2\u0012(r;t)\u0000cos 2\u0012(r;t)\u0013\n(B1)\nHere\u0012is the coarse-grained orientation at position rand timet. These hydrodynamic equations are previously derived\nin Ref. [32], but with speci\fc coe\u000ecients. Here we retain general coe\u000ecients. The density equation is a continuity\nequation@}=@t =\u0000r\u0001J, where the current Jhas two parts - active and di\u000busive. Details of these two currents are\ngiven in the main text. The activity strength a0represents the self-propelled nature of the particles, \fis the coupling\ncoe\u000ecient of the density in the order parameter equation, D}andD\u0005are the di\u000busion coe\u000ecients in the density\nand the order parameter equations, respectively. \u000b1(}) and\u000b2represent alignment in the system, and depend on\nthe model parameters. For metric distance interacting model [32], \u000b1(}) is a function of density and changes sign\nat the critical density. We choose \u000b1(}) =}\n}IN\u00001 and\u000b2= 1. Let us consider a small perturbation \u000e}over the\nhomogeneous steady state solution of the density equation so that }=}0+\u000e}. Now from the density equation, we\nobtain\na0rirj\u0005ij+D}r2\u000e}= 0\n)a0rj\u0005ij+D}ri\u000e}=c\u0011constant (B2)\nwhere \u0005 11=\u0000\u000522=S\n2cos(2\u0012) and \u0005 12= \u0005 21=S\n2sin(2\u0012). Considering only the lowest order terms in Sand\u0012, we\nobtain\n@x\u000e}=\u0000a0}0\n2D}@xS)\u000e}(x) =\u0000a0}0\n2D}S+c1 (B3)\nand\n@y\u000e}=a0}0\n2D}@yS)\u000e}(y) =a0}0\n2D}S+c2 (B4)\n9Here we assume the system is aligned along one direction, and the variation in orientation is only along the per-\npendicular direction. Therefore, we can choose either of equations (B3) or (B4). Two constants c1andc2are the\n\ructuations in density when the nematic order parameter is zero.\nNow from the equation for \u0005 ij, we obtain an e\u000bective equation for Sas\n@tS=\u001a\n\u000b1(})\u0000}2\n2\u000b2S2\u001b\nS+O(r2S) +O(r2}) (B5)\nWe neglect all the derivative terms and retain only the polynomials in S, i.e., we neglect higher order \ructuations. The\nTaylor expansion of \u000b1(}) about the mean density }0gives\u000b1(}) =\u000b1(}0+\u000e}) =\u000b1(}0)+\u000b0\n1\u000e}where\u000b0\n1=@\u000b1\n@}j}0.\nThis gives\n@tS=\u001a\n\u000b1(}0) +\u000b0\n1\u000e}\u0000}2\n0\n2\u000b2S2\u001b\nS (B6)\nWe can write an e\u000bective free energy Feff(S) so that\n@tS=\u0000\u000eFeff(S)\n\u000eS(B7)\nSubstituting the expression for \u000e}from equation (B4), we obtain\n\u0000\u000eFeff\n\u000eS=S\u001a\n\u000b1(}0) +\u000b0\n1\u0012a0}0\n2D}S+c2\u0013\n\u0000}2\n0\n2\u000b2S2\u001b\n(B8)\nTherefore,\nFeff(S) =\u0000b2\n2S2\u0000b3\n3S3+b4\n4S4(B9)\nwhereb2=\u000b1(}0) +\u000b0\n1c2,b3=a0}0\u000b0\n1\n2D}andb4=1\n2}2\n0\u000b2. Since the free energy is a state function, we have assumed\nthe integration constant to be zero. Therefore, the \ructuation in the density introduces a cubic order term in the\ne\u000bective free energy Feff(S). E\u000bective free energy in equation (B9) is similar to the Landau free energy with a new\ncubic order term [44]. Now we calculate the jump \u0001 Sand the new critical density from the coexistence condition for\nfree energy. Steady state solutions of order parameter ( S= 0 for isotropic and S6= 0 for ordered state) are given by\n\u000eFeff\n\u000eS=\u0000\n\u0000b2\u0000b3S+b4S2\u0001\nS= 0 (B10)\nNon-zeroSis given by\u0000b2\u0000b3Sc+b4S2\nc= 0. Coexistence condition implies\nFeff(Sc) =\u0012\n\u0000b2\n2\u0000b3\n3Sc+b4\n4S2\nc\u0013\nS2\nc=Feff(S= 0) = 0 (B11)\nHence we get the solution\nSc=\u00003b2\nb3(B12)\nand\nbc\n2=\u00002b2\n3\n9b4(B13)\nTherefore, the jump at the new critical point is \u0001 S=2b3\n3b4. Sinceb4>0 and hence bc\n2<0, the new critical density\n}c=}IN\u0012\n1\u00002b2\n3\n9b4\u0013\n<}IN (B14)\nis shifted to a lower density in comparison to the equilibrium transition density }IN. 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E 77,011920 (2008).\n12"    },    {        "title": "1710.04042v1.Real_State_Transfer.pdf",        "content": "arXiv:1710.04042v1  [math.CO]  11 Oct 2017Real State Transfer\nChris Godsil∗\nCombinatorics & Optimization\nUniversity of Waterloo\nOctober 12, 2017\nAbstract\nA continuous quantum walk on a graph Xwith adjacency matrix\nAis specified by the 1-parameter family of unitary matrices U(t) =\nexp(itA). These matrices act on the state space of a quantum system,\nthe states of which we may represent by density matrices, pos itive\nsemidefinite matrices with rows and columns indexed by V(X) and\nwith trace 1. The square of the absolute values of the entries of a\ncolumnof U(t)defineaprobabilitydensityon V(X), anditisprecisely\nthese densities that predict the outcomes of measurements. There are\ntwo special cases of physical interest: when the column dens ity is\nsupported on a vertex, and when it is uniform. In the first case we\nhave perfect state transfer; in the second, uniform mixing.\nThere are many results concerning state transfer and unifor m mix-\ning. In this paper we show that these results on perfect state transfer\nhold largely because at the time it occurs, the density matri x is real.\nWe also show that the results on uniform mixing obtained so fa r hold\nbecause the entries of the density matrix are algebraic numb ers. As a\nconsequence of these we derive strong restrictions on the oc curence of\nuniform mixing on bipartite graphs and on oriented graphs.\n∗Research supported by Natural Sciences and Engineering Council of Canada, Grant\nNo. RGPIN-9439\n11 Introduction\nLetXbe a graph with adjacency matrix A. The states of a continuous\nquantum walk onXare represented by positive semidefinite matrices and\nwith trace 1. Physicists refer to such matrices as density matrices .\nLetXbe a graph with adjacency matrix A. The states of a continuous\nquantum walk onXare density matrices with rows and columns indexed by\nV(X). The behaviour of the walk is determined by its initial state and the\ntransition matrix U(t), defined by\nU(t) = exp(itA).\nThis is a symmetric and unitary matrix. If the initial state of our walk is\ngiven by a density matrix D, then the state D(t) at time tis given by\nD(t) =U(t)DU(−t).\nAdensity matrix Drepresents a purestate ifrk(D) = 1. Inthiscase D=zz∗\nfor some complex unit vector z. Ifeadenotes the standard basis vector in\nCV(X)indexed by the vertex a, then\nDa=eaeT\na\nis the pure state associated to the vertex a.\nOne question of interest to physicists is whether, for two distinct v ertices\naandb, there is a time tsuch that Da(t) =Db. In this case we say that\nthere isperfect state transfer fromatob. If perfect state transfer occurs, a\nnumber of interesting consequences have been extablished. (See e.g. [8].)\nWe will summarize these, after developing some more terminology. We\nassume that the eigenvalues of Aareθ1,...,θ m, and that the matrix Er\nrepresents orthogonal projection onto the θr-eigenspace. Then ErEr=δr,sEr\nand/summationtext\nrEr=Iand we have the spectral decomposition\nA=/summationdisplay\nrθrEr.\n(One consequence of this is that U(t) =/summationtext\nreitθrEr.) Vertices aandbofX\nare said to be cospectral if the graphs X\\aandX\\bare cospectral, that is,\nthey have the same characteristic polynomial. It is known that aandbare\ncospectral if and only if\n/bardblErea/bardbl=/bardblEreb/bardbl\n2for allr. We say that aandbarestrongly cospectral if, forr,\nErea=±Ereb.\nFor more about cospectral and strongly cospectral vertices, s ee [7].\nWe can now list the consequences of perfect state transfer. If t here is pst\nfromatobat timet, then:\n(a) There is pst from btoaat timet.\n(b)Da(2t) =Da.\n(c) IfEkPEℓ/\\e}atio\\slash= 0 and ErPEs/\\e}atio\\slash= 0 and k/\\e}atio\\slash=ℓ, then\nθr−θs\nθk−θℓ∈Q.\n(d) The vertices aandbare strongly cospectral.\nWe refer to (a), (b) respectively as symmetry and periodicity, (c) is related\nto what is known as the ratio condition.\nOnegoalofthispaperistoshowthatallthesepropertiesarecons equences\nof the fact that the density matrices DaandDbare real. Thus if D1andD2\nare real density matrices and D1(t) =D2, then strong analogs of the above\nfour properties hold. (In fact these properties will be easy corolla ries of our\nmore general results.)\nWe will also see that interesting things happen if we assume that the\nentries of D1andD2are algebraic numbers.\nFor background on state transfer and continuous quantum walks , see e.g.\n[8, 10].\n2 Real State Transfer\nAstateis realifitsdensitymatrixisreal. Werecallthatwehaveperfectstate\ntransfer at time tfrom a density matrix Pto a density matrix QifP(t) =Q.\nWesay that we have pretty goodstate transfer fromastate PtoQiffor each\npositive real number ǫthere is a time tǫsuch that /bardbl(U(t)PU(t)−Q)/bardbl< ǫ.\nWe define the eigenvalue support of a density matrix Pto be the set of\npairs (θr,θs) such that ErPEs/\\e}atio\\slash= 0. (This definition extends the one given in\nthe introduction to density matrices not of the form Da.)\n32.1 Lemma. IfPis a density matrix and Er,Esare spectral idempotents\nofAsuch that ErPEs/\\e}atio\\slash= 0then neither ErPErnorEsPEsis zero.\nProof.SincePis positive semidefinite, there is a unique positive semidefinite\nmatrixQsuch that P=Q2. Hence if ErPEs/\\e}atio\\slash= 0, then ErQ/\\e}atio\\slash= 0 and\nEsQ/\\e}atio\\slash= 0. Hence ErPEr=ErQ2Er/\\e}atio\\slash= 0 and similarly EsPEs/\\e}atio\\slash= 0.\nIfErPEs= 0 whenever r/\\e}atio\\slash=s, then\nP=/summationdisplay\nrErPEr\nandPA=AP; thusU(t)PU(−t) =Pfor allt.\nWe say that the eigenvalue support of Psatisfies the ratio conditionif, for\nany two pairs of distinct eigenvalues ( θr,θs) and (θk,θell) in the eigenvalue\nsupport of Pwithθk/\\e}atio\\slash=θℓ, we have\nθr−θs\nθk−θℓ∈Q.\nNotethat if Pispure, that is, P=xx∗forsome unit vector x, thenErPEs=\n0if andonlyif ErxorEsxis zero. (Inthis casewe could define theeigenvalue\nsupport to be the the set of eigenvalues θrsuch that ErPEr/\\e}atio\\slash= 0, which what\nwe do elsewhere.)\n2.2 Theorem. LetU(t)be the transition matrix corresponding to a graph\nX. Let=/summationtext\nreitθrErbe the spectral decomposition of A. IfPis a real\ndensity matrix, there is a positive time tsuch that U(t)PU(−t)is real if and\nonly if the eigenvalue support of Psatisfies the ratio condition.\nProof.We have\nP(t)=/summationdisplay\nr,seit(θr−θs)ErPEs\nand therefore the imaginary part of P(t)is\n/summationdisplay\nr,ssin(t(θr−θs))ErPEs.\nThe non-zero matrices ErPEsare linearly independent, and therefore the\nimaginary part of P(t)is zero if and only if sin( t(θr−θs)) = 0 whenever\nErPEs/\\e}atio\\slash= 0. Hence if ( θr,θs) lies in the eigenvalue support of Pthen there\nis an integer mr,ssuch that t(θr−θs) =mr,sπ.\n4IfErPEs= 0 whenever r/\\e}atio\\slash=s, then\nP=/summationdisplay\nrErPEr\nandPA=AP; thusU(t)PU(−t) =Pfor allt.\n2.3 Lemma. LetPbe a real density matrix. If P(t)is real, then P(2t) =P\nandU(2t)commutes with PandQ.\nProof.IfU(t)PU(−t) =QwhereQis real, then taking complex conjugates\nyields\nU(−t)PU(t) =Q\nand consequently P=U(t)QU(−t). It follows at one that U(2t) commutes\nwithP.\n2.4 Lemma. IfAis real and symmetric and we have state transfer at t\nbetween real density matrices PandQ, then\n(a)ErPEr=ErQEr.\n(b) Ift(θr−θs)is not a multiple of πthenErPEs=ErQEs= 0.\n(c) Otherwise ErPEs=±ErQEs.\nProof.IfAis real and symmetric then the idempotents Erare real and\nsymmetric. Assume\nQ=U(τ)PU(−τ) =/summationdisplay\nr,seiτ(θr−θs)ErPEs.\nIf we multiply this expression on the left by Erand on the right by Es, then\nErQEs=eiτ(θr−θs)ErPEs\nand, since all matrices here are real, eit(θr−θs)must be real.\nSince/summationtext\nr,sErPEs=P, we see that if Ais real and there is pst from P\ntoQ, there are signs ǫr,s=±1 such that\nQ=/summationdisplay\nr,sǫr,sErPEs.\nConsider the rank-one case. If P=uuTthenEruuTEs= 0 if and only\nifEru= 0 orEsu= 0. Hence the constraint in (b) gives a constraint on\nthe eigenvalue support of u. (In fact this is the usual ratio condition, so (b)\ngeneralizes this.)\n53 Pretty Good State Transfer\nWe have pretty good state transfer from a state Pto a state Qif, for each\nǫ >0, there is a time tsuch that\n/bardblU(t)PU(−t)−Q/bardbl< ǫ.\nSince the complex conjugate of\nU(t)PU(−t)−Q\nis\nU(−t)PU(t)−Q=U(−t)(P−U(t)QU(−t))U(t)\nand since U(t) is unitary,\n/bardblU(t)PU(−t)−Q/bardbl=/bardblP−U(t)QU(−t)/bardbl.\nHence if we have pretty goodstate transfer from PtoQ, then we have pretty\ngood state transfer from QtoP.\n3.1 Lemma. Suppose Ais real and we have pretty good state transfer from\nPtoQ. ThenErPEs=±ErQEs(for allrands) andErPEr=ErQEr.\nProof.By assumption there is an increasing sequence of times ( tk)k≥0such\nthat\nU(tk)PU(−tk)→Q\nastk→ ∞. Hence\nei(θr−θs)tkErPEs→ErQEs\nastk→ ∞. SinceErPEsandErQEsare real, the result follows.\n3.2 Corollary. IfAis real and Pis real, there are only finitely many real\ndensity matrices Qfor which there is pretty good state transfer from Pto\nQ.\nProof.SinceQ=/summationtext\nr,sErQEsit follows that Q=/summationtext\nr,sǫr,sErPEs, where\nǫr,s=±1.\nPretty good state transfer is treated in some detail in [2].\n64 Algebras\nBecause they are trace-orthogonal, the non-zero matrices ErPEsare linearly\nindependent. The “off-diagonal” terms ErPEsare nilpotent, indeed the ma-\ntrices\nErPEs,(r < s)\ngenerate a nilpotent algebra where the product of any two element s is zero.\nThe “diagonal” terms ErPErgenerate a commutative semisimple algebra;\ntheir sum is the orthogonal projection of Ponto the commutant of A. (See\n[7, Section 5] for further details).\nSinceU(t) is a linear combination of the spectral idempotents of A, it\nis a polynomial in Aand therefore, for each twe that P(t)∈ /a\\}bracketle{tA,P/a\\}bracketri}ht. In\nconsequence\n/a\\}bracketle{tP(t),A/a\\}bracketri}ht=/a\\}bracketle{tP,A/a\\}bracketri}ht\nfor allt.\n4.1 Lemma. If we have pretty good state transfer from PtoQ, then\n/a\\}bracketle{tA,P/a\\}bracketri}ht=/a\\}bracketle{tA,Q/a\\}bracketri}ht.\nProof.The algebra /a\\}bracketle{tA,P/a\\}bracketri}htis closed and as Qis a limit of a sequence of\nmatrices in it, it follows that Q∈ /a\\}bracketle{tA,P/a\\}bracketri}ht. If we have pretty good state\ntransfer from PtoQ, then as we noted st the beginning of Section 3, there\nis pretty good state transfer from QtoPand so/a\\}bracketle{tA,P/a\\}bracketri}ht=/a\\}bracketle{tA,Q/a\\}bracketri}ht.\nIf/a\\}bracketle{tA,P/a\\}bracketri}htis the full matrix algebra, we say that Piscontrollable . IfP\nis real and P(t) is real, then U(2t) commutes with AandP. IfPis also\ncontrollable it follows that U(2t) must be a scalar matrix, say U(2t) =ζI.\nSince det( U(t)) = 1, we have ζ|V(X)|= 1 and therfore ζis a root of unity.\n5 Timing\nWe investigate the times at which perfect state transfer can occu r.\n5.1 Lemma. LetPbe a density matrix and let Sbe given by\nS:={σ:U(σ)PU(−σ) =P}.\nThen there are three possibilities:\n7(a)S=∅.\n(b) There is a positive real number τandSconsists of all integer multiples\nofτ.\n(c)Sis a dense subset of RandU(t)PU(−t) =Pfor allt.\nProof.Suppose S/\\e}atio\\slash=∅. ThenSis an additive subgroup of Rand there are\ntwo cases. First, Sis discrete and consists of all integer multiples of its least\npositive element. Second, Sis dense in Rand there is sequence of positive\nelements ( σi)i≥0with limit 0. Since for small values of twe have\nU(t)≈I+itA\nit follows that AP=PAandU(t)PU(−t) =Pfor allt.\nIfU(t)PU(−t) =Pwhent=τ >0, but not for all t, we say that Pis\nperiodic with period τ. If a density matrix is periodic, it has a well defined\nminimum period. If there is perfect state transfer from PtoQ, thenPand\nPare both periodic with the same minimum period.\n5.2 Lemma. Suppose PandQare distinct real density matrices. If there\nis perfect state transfer from PtoQ, thenPis periodic and if the minimum\nperiod of Pisσ, then pst occurs at time σ/2.\nProof.Suppose we have pst from PtoQand define\nT:={t:U(t)PU(−t) =Q}\nAssume that the minimum period of Pisσ. Ift∈TthenPis periodic with\nperiod 2tand soTis a coset of a discrete subgroup of R. AlsoT=−T. Let\nτbe the least positive element of T. Then 2 τ≥σand thus\nτ≥1\n2σ.\nIfτ≥σthenτ−σ∈Tand since τis not a period, τ < σ. Asσmust divide\n2τ, it follows that τ=σ/2.\n5.3 Corollary. For any real density matrix P, there is at most one real\ndensity matrix Qsuch that there is perfect state transfer from PtoQ.\n85.4 Lemma. Suppose we have pst from PtoQat timetand that θ1,...,θ m\nare the distinct eigenvalues of Ain nonincreasing order. If tr(PQ) = 0then\nt≥π\nθ1−θm.\nSuppose U(t)PU(−t) =Qand tr(PQ) = 0. We have\nU(t)PU(−t) =/summationdisplay\nreiθrtU(t)ErU(−t)\nand therefore\ntr(PQ) =/summationdisplay\nreiθrttr(PU(t)ErU(−t))\n=/summationdisplay\nreiθrttr(U(−t)PU(t)Er)\n=/summationdisplay\nreiθrttr(QEr)\nSinceQand the idempotents Erare positive semidefinite, tr( QEr)≥0. Also\n/summationdisplay\nrtr(QEr) = tr(Q) = 1.\nHence if tr( PQ) = 0 then we see that 0 is a convex combination of the\neigenvalues eiθrtofU(t). IfAhasmdistinct eigenvalues\nθ1≥ ··· ≥θm.\nthisimpliesthat eitθ1,...,eitθmcannotbecontainedinanarcontheunitcircle\nin the complex plane of length less than π, and therefore t(θ1−θm)≥π.\nNote that in this lemma we do not need PandQto be real.\nAn algebraic integer is totally real if all its algebraic conjugates are real,\nequivalently, all zeros of its minimal polynomial are real.\n5.5 Theorem. LetPbe a rational state with eigenvalue support S. IfS\nsatisfies the ratio condition, then there is a square-free integer ∆such that\nifErPEs/\\e}atio\\slash= 0, thenθr−θsis an integer multiple of√\n∆.\n9Proof.LetEdenote the extension field of Qgenerated by the elements of S.\nSincePis rational, it follows that if ErPEs/\\e}atio\\slash= 0 and γ∈Γ, then\nEγ\nrPEγ\nr/\\e}atio\\slash= 0.\nWe also note that, as Ais an integer matrix, the spectral idempotents Er\nare algebraic and therefore Eγ\nris a spectral idempotent of A.\nThe product/productdisplay\n(θr,θs)∈Sθr−θs\nis fixed by Γand is consequently an integer. Given the ratio condition, we\nsee that if ( θk,θℓ)∈S, then\n/productdisplay\n(θr,θs)∈Sθk−θℓ\nθr−θs∈Q\nand therefore\n(θk−θℓ)|S|∈Q.\nAs the eigenvalues of Aare algebraic integers, this implies that ( θk−θℓ)|S|\nis an integer. The eigenvalues of Aare totally real, but if the polynomial\nts−1 has a real root, then if must be ±1 and in this case smust even. We\nconclude that ( θk−θℓ)2is an integer.\nSuppose there are integers mk,ℓandmr,sand square-free integers bandc\nsuch that\nθk−θℓ=mk,ℓ√\nb, θ r−θs=mr,s√c.\nIf √\nb√c=θk−θℓ\nθr−θs∈Q,\ntneb=c.\n5.6 Corollary. IfPis a periodic rational state, then the period of Pis at\nmost2π.\nProof.Ift= 2π/√\n∆thent(θr−θs) is an even multiple of πfor each pair\n(θr,θs) in the eigenvalue support of P. Consequently\nP(t)=/summationdisplay\nr,seit(θr−θs)ErPEs=/summationdisplay\nr,sErPEs=P.\n106 Algebraic States\nWe say that a state with density matrix Disalgebraic if the entries of Dare\nalgebraic numbers. Clearly the states Daare algebraic.\nSuppose Dis a pure state. Then D(t) is pure for all t. IfD=zz∗, then\nD(t) =ww∗, wherew=U(t)z. We say that a matrix or vector is flatif all\nits entries have the same absolute value. We see that a vector wis flat if and\nonly the diagonal entries of ww∗are all equal. (Note that these entries are\nnon-negative and real.)\nWe say that a quantum walk has uniform mixing relative to a pure state\nDif there is a time tsuch that\nD(t)◦I=1\nnI.\nWe refer to uniform mixing relative to the vertex state Daaslocal uniform\nmixing. The walk has uniform mixing if, it at some t, it admits uniform\nmixing relative to each vertex. In many of the cases where uniform m ixing is\nknown to occur, the underlying graph is vertex transitive, and the n uniform\nmixing occurs if and only if uniform mixing relative to a vertex occurs. T he\nonly examples we know of graphs that are not regular and that do ad mit\nuniform mixing are the complete bipartite graph K1,3and its Cartesian pow-\ners (an observation due to H. Zhan). If n≥2, the stars K1,nadmit uniform\nmixing relative to the vertex of degree n.\nCarlson et al. [5] observed that we have uniform mixing relative to the\nvertex of degree nin the star K1,n.\nWe say that the continuous quantum walk on Xisperiodic at aif there\nis a time τsuch that Da(τ) =Da.\n6.1 Theorem. LetAbe a Hermitian matrix with algebraic entries and let\nU(t) = exp( itA)If the density Dis algebraic and, for some t, the density\nD(t)is algebraic, then the ratio condition holds on the eigenvalue support of\nD.\nProof.Since the entries of Aare algebraic, its eigenvalues are algebraic and\ntherefore the spectral idempotents are algebraic.\nWe have\nD(t) =/summationdisplay\nr,seit(θr−θs)ErDEs.\n11The matrices ErDEsare pairwise orthogonal, and so, for all rands,\n/a\\}bracketle{tD(t),ErDEs/a\\}bracketri}ht=eit(θr−θs)/a\\}bracketle{tErDEs,ErDEs/a\\}bracketri}ht.\nThe entries of the spectral idempotents are algebraic, and if the e ntries of\nDandD(t) are algebraic, then the values of the two inner products in the\nabove identity are algebraic numbers.\nIt follows that eit(θr−θs)must be algebraic, for all rands. Now if k/\\e}atio\\slash=ℓ,\nthen\neit(θr−θs)=/parenleftbig\neit(θk−θℓ)/parenrightbigθr−θs\nθk−θℓ.\nTheGelfond-Schneider theorem tellsusthat if αandβarealgebraicnumbers\nandα/\\e}atio\\slash= 0,1 andβis irrational, then αβis transcendental, whence we deduce\nthat ifDandD(t) arealgebraic, then if k/\\e}atio\\slash=ℓandneither ErDEsnorEkDEℓ\nis zero, the ratio\nθr−θs\nθk−θℓ\nis rational.\nThe Gelfond-Schneider theorem was first used as above in [1]; the te ch-\nnique is due to Jennifer Lin. One source for the Gelfond-Schneider t heorem\nis Burger and Tubbs [3].\n7 Oriented Graphs\nWe study state transfer on oriented graphs. In this section we co nsider the\ngraph theory and linear algebra, in the next we turn to the continuo us walks.\nAn oriented graph is a structure consisting of vertices and arcs, w here\nan arc is an ordered pair of vertices, and any two vertices lie in at mos t one\narc. We can construct (a large number of) oriented graphs by cho osing a\ngraph and assigning a direction to each edge. We use arcs( X) to denote the\nset of arcs of X. The adjacency matrix SofXid the matrix with rows and\ncolumns indexed by V(X), where\nSu,v=\n\n1, uv∈arcs(X);\n−1, vu∈arcs(X);\n0,otherwise .\n12ThusSis a skew symmetric matrix. We define the degreeof a vertex vin\nXto be the number of arcs that use v. As we defined them, each oriented\ngraph has an underlying graph whose adjacency matrix is S◦S. The total\nvalency of a vertex in Xis its valency in the undirected graph that underlies\nX.\nThe matrix iSis Hermitian, with all eigenvalues real, and therefore the\neigenvalues of Sare purely imaginary. They are symmetric about the real\naxis of the complex plane, so we will assume that the r-th eigenvalue is iλr,\nwhereλris real. We can then write the spectral decomposition of Sas\nS=/summationdisplay\niλrFr\nwhere the idempotents FrareHermitian. Further, the idempotent associated\nto the eigenvalue −iλrisFr.\n7.1 Lemma. LetXbe an oriented graph with maximum total valency ∆.\nIfλis an eigenvalue of X, then|λ| ≤∆.\nProof.Ifz/\\e}atio\\slash= 0 and Sz=λz, then\nλzk=/summationdisplay\nℓSk,ℓzℓ\nand so by the triangle inequality,\n|λ||zk| ≤/summationdisplay\nℓ:Sk,ℓ/negationslash=0|zℓ|.\nBy choosing kso that|zk|is maximal, we obtain the stated bound.\nIfYis a bipartite graph and\nA(Y) =/parenleftbigg0B\nBT0/parenrightbigg\n,\nthen\nS=/parenleftbigg\n0−B\nBT0/parenrightbigg\nis skew symmetric. As\n/parenleftbigg−iI0\n0I/parenrightbigg/parenleftbigg0−B\nBT0/parenrightbigg/parenleftbiggiI0\n0I/parenrightbigg\n=i/parenleftbigg0B\nBT0/parenrightbigg\n,\n13each eigenvalue of Sis equal to iθ, for some eigenvalue θofY. The oriented\ngraph with adjacency matrix Swill be called the natural orientation ofY.\nIn a sense, the spectral theory of bipartite graphs is the interse ction of the\nspectral theory of graphs with the spectral theory of oriented graphs.\nFinally, since iSis Hermitian, the eigenvalues of a principal submatrix\ninterlace the eigenvalues of S, and therefore the eigenvalues of an induced\nsubgraph of an oriented graph Xinterlace the eigenvalues of X.\n8 Quantum Walks on Oriented Graphs\nContinuous quantum walks on oriented graphs were first studied in [4 , 6].\nIn the introduction we defined the transition matrix U(t) as exp( itA),\nwhereAwas the adjacency matrix of a graph. Thus Awas symmetric and\nreal. However all that is needed is that Ashould be Hermitian and hence, if\nSis the adjacency matrix of an oriented graph, we may define a trans ition\nmatrix\nU(t) = exp(it(−iS)) = exp( tS).\nWe note that U(t) is then real and orthogonal for all t. We have the spectral\ndecomposition\nU(t) =/summationdisplay\nreitλrFr\nwhereλr∈RandFris Hermitian.\nAs we noted in the previous section, if Fis the spectral idempotent\nassociated to the eigenvalue iλ, then the eigenvalue associated to −iλisF.\nHenceFrea= 0 if and only Fea= 0. Consequently the eigenvalue support of\na vertex is symmetric about the real axis of the complex plane and th erefore\nthe ratio condition on the eigenvalue support of a vertex is equivalen t to the\ncondition that λ/µ∈Qfor all choices of λandµ.\nIfSarises astheadjacency matrixofthenatural orientationofabipa rtite\ngraphY, with adjacency matrix A, then\n/parenleftbigg−iI0\n0I/parenrightbigg\nexp(tS)/parenleftbiggiI0\n0I/parenrightbigg\n= exp(itA).\nAccordingly we have perfect state transfer from atobrelative to exp( itA) if\nand only if it occurs from atobrelative to exp( tS); similarly we have local\nuniform mixing at ain the graph if and only if we have it in the oriented\ngraph.\n148.1 Theorem. If there is perfect state transfer on an oriented graph from a\nvertexa, then the eigenvalue support of asatisfies the ratio condition.\nProof.We simply note that DaandDbare algebraic, whence Theorem 6.1\nimplies the conclusion.\n8.2 Theorem. If there is local uniform mixing at a vertex ain an oriented\ngraph, then the eigenvalue support of asatisfies the ratio condition.\nProof.Assumen=|V(X)|. If there is local uniform mixing at aat tinet,\nthenU(t)eais flat. As U(t) is real, this implies that the entries of U(t)eaare\nall equal to ±n1/2and hence they are algebraic. Now apply Theorem 6.1.\nWe say that an oriented graph is connected if its underlying undirected\ngraph is connected.\n8.3 Lemma. LetXbe an oriented graph with adjacency matrix Sand\nsuppose a∈V(X). Then exp(tS)eais periodic if and only if the ratio\ncondition holds on the eigenvalue support of a.\nProof.Suppose U(t)ea=ea. Then\nea=U(t)ea=/summationdisplay\neeitλrFrea\nand since we also have\nea=/summationdisplay\nrFrea,\nit follows that eitλr= 1 for all rsuch that iλr∈esupp(a). Consequently tλr\nis an integer multiple of 2 πfor each r, and therefore the ratio of any two\nelements of esupp( a) is rational.\nNow assume conversely that the ratio condition holds on the eigenva lue\nsupport of a, and set m=|esupp(a)|. LetΓdenote the Galois group of the\nextension field of Qgenerated by λ1,...,λ s. Then esupp( a) is closed under\nΓ. Thereforem/productdisplay\nr=1λr\nis fixed by each element of Γ, and is thus an integer. Hence/producttext\nrλr∈Z. Now\nm/productdisplay\nr=1λs\nλr\n15is rational and accordingly λm\nsis rational. Since it also an algebraic integer,\nit must be an integer. Because it is an eigenvalue of the Hermitian matr ix\niS, all algebraic conjugates of λrare real, and therefore we must have λ2\nr∈\nZ.Therefore for each swe haveλs=as√bswhereas,bs∈Zandbsis square\nfree. Since λs/λris rational it follows that bsis independent of s. Therefore\nesupp(a) consists of integer multiples of√\nb, for some square-free integer b.\nThis shows that ais periodic, with period 2 π/√\nb.\n8.4 Theorem. There are only finitely many connected bipartite graphs with\nmaximum valency at most kwhich contain a periodic vertex.\nProof.Letcbe the eccentricity of a. The vectors\n(A+I)rea,(r= 0,...,c)\nare linearly independent, because their supports form a strictly inc reasing\nsequence. These vectors lie in the span of the non-zero vectors Frea, whence\nc+1 is bounded above by the size of the eigenvalue support of a. Since any\ntwo elements ofesupp( a)differ byaleastone, andsincethelargesteigenvalue\nofSis at most k, we have that |esupp(a)| ≤2∆+1. Hence the eccentricity\nofais bounded by a function of ∆, and therefore |V(X)|is bounded.\n8.5 Corollary. There areonly finitely many connected bipartite graphswith\nmaximum valency at most kwhich admit local uniform mixing.\nThe theorem also implies that there are only finitely many connected\nbipartite graphs with maximum valency at most kon which perfect state\ntransfer occurs, but this holds more generally for all graphs, bipa rtite or not.\nSee [9, Corollary 6.2].\n9 Prospects, Problems\nWehave established asurprising connection between behaviour ofc ontinuous\nquantum walks at certain times and the field of definition of the assoc iated\ndensity matrix. An obvious problem is to find more examples of such be -\nhaviour.\nSecondly, mostworkoncontinuous quantumwalkassumes thatthe initial\nstate is of the form ereT\nr. Our results indicate that it might be fruitful to\nconsider more general initial states. Our personal feeling is that p ure states\nwill bemost interesting, because their eigenvaluesupport tendsto besmaller.\n16References\n[1] William Adamczak, Kevin Andrew, Leon Bergen, Dillon Ethier, Peter\nHernberg, Jennifer Lin, and Christino Tamon. Non-uniform mixing of\nquantumwalkoncycles. InternationalJournal of Quantum Information ,\n5(06):781–793, 2007.\n[2] Leonardo Banchi, Gabriel Coutinho, Chris Godsil, and Simone Seve rini.\nPretty goodstate transfer inqubit chains—the Heisenberg Hamilto nian.\nJ. Math. Phys. , 58(3):032202, 9, 2017.\n[3] Edward B Burger and Robert Tubbs. Making transcendence transpar-\nent: An intuitive approach to classical transcendental num ber theory .\nSpringer Science & Business Media, 2004.\n[4] Stephen Cameron, Shannon Fehrenbach, LeahGranger, Oliver Hennigh,\nSunrose Shrestha, and Christino Tamon. Universal state transf er on\ngraphs.Linear Algebra and its Applications , 455:115–142, 2014.\n[5] WilliamCarlson, AllisonFord,ElizabethHarris, JulianRosen, Christino\nTamon, and Kathleen Wrobel. Universal mixing of quantum walk on\ngraphs.Quantum Inf. Comput. , 7(8):738–751, 2007.\n[6] Erin Connelly, Nathaniel Grammel, Michael Kraut, Luis Serazo, an d\nChristino Tamon. Universality in perfect state transfer. Linear Algebra\nand its Applications , 2017.\n[7] C. Godsil and J. Smith. Strongly Cospectral Vertices. ArXiv e-prints ,\nSeptember 2017.\n[8] Chris Godsil. State transfer on graphs. Discrete Math. , 312(1):129–147,\n2012.\n[9] ChrisGodsil. Whencanperfectstatetransferoccur? Electron. J. Linear\nAlgebra, 23:877–890, 2012.\n[10] Alastair Kay. Perfect, efficient, state transfer and its applica tion as\na constructive tool. International Journal of Quantum Information ,\n8(04):641–676, 2010.\n17"    },    {        "title": "1710.09373v1.Entropic_Updating_of_Probability_and_Density_Matrices.pdf",        "content": "Entropic Updating of Probability and Density Matrices\nKevin Vanslette\nkvanslette@albany.edu\nDepartment of Physics, University at Albany (SUNY)\nAlbany, NY 12222, USA\nNovember 6, 2018\nAbstract\nWe \fnd that the standard relative entropy and the Umegaki entropy are designed for the purpose of\ninferentially updating probability and density matrices respectively. From the same set of inferentially\nguided design criteria, both of the previously stated entropies are derived in parallel. This formulates\na quantum maximum entropy method for the purpose of inferring density matrices in the absence of\ncomplete information in Quantum Mechanics.\n1 Introduction\nWedesign an inferential updating procedure for probability distributions and density matrices such that\ninductive inferences may be made. The inferential updating tools found in this derivation take the form of\nthe standard and quantum relative entropy functionals, and thus we \fnd the functionals are designed for\nthe purpose of updating probability distributions and density matrices respectively. Design derivations\nwhich found the entropy to be a tool for inference originally required \fve design criteria (DC) [1, 2, 3],\nthis was reduced to four in [4, 5, 6], and then down to three in [7]. We reduced the number of required\nDC down to two while also providing the \frst design derivation of the quantum relative entropy { using\nthe same design criteria and inferential principles in both instances .\nThe designed quantum relative entropy takes the form of Umegaki's quantum relative entropy, and\nthus it has the \\proper asymptotic form of the relative entropy in quantum (mechanics)\" [8, 9, 10]. Re-\ncently, [11] gave an axiomatic characterization of the quantum relative entropy that \\uniquely determines\nthe quantum relative entropy\". Our derivation di\u000bers from their's, again in that we design the quantum\nrelative entropy for a purpose, but also that our DCs are imposed on what turns out to be the functional\nderivative of the quantum relative entropy rather than on the quantum relative entropy itself. The use of\na quantum entropy for the purpose of inference has a large history: Jaynes [12, 13] invented the notion of\nthe quantum maximum entropy method [14], while it was perpetuated by [15, 16, 17, 18, 19, 20, 21, 22]\nand many others. However, we \fnd the quantum relative entropy to be the suitable entropy for updating\ndensity matrices, rather than the von Neumann. The relevant results of their papers may be found using\nour quantum relative entropy with a suitable uniform prior density matrix.\nIt should be noted that because the relative entropies were reached by design, they may be interpret\nas such, \\the relative entropies are tools for updating\", which means we no longer need to attach an\ninterpretation ex post facto { as a measure of disorder or amount of missing information. In this sense,\nthe relative entropies were built for the purpose of saturating their own interpretation [4, 7].\nThe remainder of the paper is organized as follows: First we will discuss some universally applicable\nprinciples of inference and motivate the design of an entropy function able to rank probability distri-\nbutions. This entropy function will be designed such it is consistent with inference by applying a few\nreasonable design criteria, which are guided by the aforementioned principles of inference. Using the same\nprinciples of inference and design criteria, we \fnd the form of the quantum relative entropy suitable for\ninference. We end with concluding remarks.\n1arXiv:1710.09373v1  [quant-ph]  26 Oct 2017Solutions for ^ \u001aby maximizing the quantum relative entropy give insight into the Quantum Bayes'\nRule in the sense of [23, 24, 25, 26]. This, and a few other applications of the quantum maximum entropy\nmethod, will be discussed in a future article.\n2 The Design of Entropic Inference\nInference is the appropriate updating of probability distributions when new information is received.\nBayes' rule and Je\u000brey's rule are both equipped to handle information in the form of data; however,\nthe updating of a probability distribution due to the knowledge of an expectation value was realized\nby Jaynes [12, 13, 14] through the method of maximum entropy. The two methods for inference were\nthought to be devoid of one another until the work of [27], which showed Bayes' and Je\u000brey's Rule to be\nconsistent with the method of maximum entropy when the expectation values were in the form of data\n[27]. In the spirit of the derivation we will carry-on as if the maximum entropy method were not known\nand show how it may be derived as an application of inference.\nGiven a probability distribution '(x) over a general set of propositions x2X, it is self evident that\nif new information is learned, we are entitled to assign a new probability distribution \u001a(x) that somehow\nre\rects this new information while also respecting our prior probability distribution '(x). The main\nquestion we must address is: \\Given some information, to what posterior probability distribution \u001a(x)\nshould we update our prior probability distribution '(x) to?\", that is,\n'(x)\u0003\u0000!\u001a(x)?\nThis speci\fes the problem of inductive inference. Since \\information\" has many colloquial, yet poten-\ntially con\ricting, de\fnitions, we remove potential confusion by de\fning information operationally (\u0003)\nas the rationale that causes a probability distribution to change (inspired by and adapted from [7]).\nDirectly from [7]:\n\\Our goal is to design a method that allows a systematic search for the preferred posterior distribu-\ntion. The central idea, \frst proposed in [4] is disarmingly simple: to select the posterior \frst rank all\ncandidate distributions in increasing order of preference and then pick the distribution that ranks the\nhighest. Irrespective of what it is that makes one distribution preferable over another (we will get to that\nsoon enough) it is clear that any ranking according to preference must be transitive: if distribution \u001a1is\npreferred over distribution \u001a2, and\u001a2is preferred over \u001a3, then\u001a1is preferred over \u001a3. Such transitive\nrankings are implemented by assigning to each \u001a(x) a real number S[\u001a], which is called the entropy of\n\u001a, in such a way that if \u001a1is preferred over \u001a2, thenS[\u001a1]> S[\u001a2]. The selected distribution (one or\npossibly many, for there may be several equally preferred distributions) is that which maximizes the\nentropy functional.\"\nBecause we wish to update from prior distributions 'to posterior distributions \u001aby ranking, the\nentropy functional S[\u001a;'], is a real function of both 'and\u001a. In the absence of new information, there\nis no available rationale to prefer any \u001ato the original ', and thereby the relative entropy should be\ndesigned such that the selected posterior is equal to the prior '(in the absence of new information).\nThe prior information encoded in '(x) is valuable and we should not change it unless we are informed\notherwise. Due to our de\fnition of information, and our desire for objectivity, we state the predominate\nguiding principle for inductive inference:\nThe Principle of Minimal Updating (PMU): A probability distribution should only be updated\nto the extent required by the new information.\nThis simple statement provides the foundation for inference [7]. If the updating of probability dis-\ntributions is to be done objectively, then possibilities should not be needlessly ruled out or suppressed.\nBeing informationally stingy, that we should only update probability distributions when the informa-\ntion requires it, pushes inductive inference toward objectivity. Thus using the PMU helps formulate a\npragmatic (and objective) procedure for making inferences using (informationally) subjective probability\ndistributions [28].\nThis method of inference is only as universal and general as its ability to apply equally well toany\nspeci\fc inference problem. The notion of \\speci\fcity\" is the notion of statistical independence; a special\n2case is only special in that it is separable from other special cases. The notion that systems may be\n\\su\u000eciently independent\" plays a central and deep-seated role in science and the idea that some things\ncan be neglected and that not everything matters, is implemented by imposing criteria that tells us how\nto handle independent systems [7]. Ironically, the universally shared property by all speci\fc inference\nproblems is their ability to be independent of one another. Thus, a universal inference scheme based on\nthe PMU permits,\nProperties of Independence (PI): Subdomain Independence: When information is received about\none set of propositions, it should not e\u000bect or change the state of knowledge (probability distribution) of\nthe other propositions (else information was also received about them too);\nAnd,\nSubsystem Independence: When two systems are a-priori believed to be independent and we only re-\nceive information about one, then the state of knowledge of the other system remains unchanged.\nThe PI's are special cases of the PMU that ultimately take the form of design criteria in the design\nderivation. The process of constraining the form of S[\u001a;'] by imposing design criteria may be viewed\nas the process of eliminative induction , and after su\u000ecient constraining, a single form for the entropy\nremains. Thus, the justi\fcation behind the surviving entropy is not that it leads to demonstrably correct\ninferences, but rather, that all other candidate entropies demonstrably fail to perform as desired [7].\nRather than the design criteria instructing one how to update, they instruct in what instances one\nshould notupdate. That is, rather than justifying one way to skin a cat over another, we tell you when\nnotto skin it, which is operationally unique { namely you don't do it { luckily enough for the cat.\n2.1 The Design Criteria and the Standard Relative Entropy\nThe following design criteria (DC), guided by the PMU, are imposed and formulate the standard relative\nentropy as a tool for inference. The form of this presentation is inspired by [7].\nDC1: Subdomain Independence\nWe keep the DC1 from [7] and review it below. DC1 imposes the \frst instance of when one should\nnot update { the Subdomain PI. Suppose the information to be processed does notrefer to a particular\nsubdomainDof the spaceXofx's. In the absence of new information about Dthe PMU insists we\ndo not change our minds about probabilities that are conditional on D. Thus, we design the inference\nmethod so that '(xjD), the prior probability of xconditional on x2D, is not updated and therefore\nthe selected conditional posterior is,\nP(xjD) ='(xjD): (1)\n(The notation will be as follows: we denote priors by ', candidate posteriors by lower case \u001a, and the\nselected posterior by upper case P.) We emphasize the point is not that we make the unwarranted\nassumption that keeping '(xjD) unchanged is guaranteed to lead to correct inferences. It need not;\ninduction is risky. The point is, rather, that in the absence of any evidence to the contrary there is no\nreason to change our minds and the prior information takes priority.\nDC1 Implementation:\nConsider the set of microstates xi2X belonging to either of two non-overlapping domains Dor its\ncomplimentD0, such thatX=D[D0and;=D\\D0. For convenience let \u001a(xi) =\u001ai. Consider the\nfollowing constraints:\n\u001a(D) =X\ni2D\u001aiand\u001a(D0) =X\ni2D0\u001ai; (2)\nsuch that\u001a(D) +\u001a(D0) = 1, and the following \\local\" constraints to DandD0respectively are,\nhAi=X\ni2D\u001aiAiandhA0i=X\ni2D0\u001aiA0\ni: (3)\n3As we are searching for the candidate distribution which maximizes Swhile obeying (2) and (3), we\nmaximize the entropy S\u0011S[\u001a;'] with respect to these expectation value constraints using the Lagrange\nmultiplier method,\n0 =\u000e\u0010\nS\u0000\u0015[\u001a(D)\u0000X\ni2D\u001ai]\u0000\u0016[hAi\u0000X\ni2D\u001aiAi]\n\u0000\u00150[\u001a(D0)\u0000X\ni2D0\u001ai]\u0000\u00160[hA0i\u0000X\ni2D0\u001aiAi]\u0011\n;\nand thus, the entropy is maximized when the following di\u000berential relationships hold:\n\u000eS\n\u000e\u001ai=\u0015+\u0016Ai8i2D; (4)\n\u000eS\n\u000e\u001ai=\u00150+\u00160A0\ni8i2D0: (5)\nEquations (2)-(5), are n+ 4 equations we must solve to \fnd the four Lagrange multipliers f\u0015;\u00150;\u0016;\u00160g\nand thenprobability values f\u001aig.\nIf the subdomain constraint DC1 is imposed in the most restrictive case, then it will hold in general.\nThe most restrictive case requires splitting Xinto a set offDigdomains such that each Disingularly\nincludes one microstate xi. This gives,\n\u000eS\n\u000e\u001ai=\u0015i+\u0016iAiin eachDi: (6)\nBecause the entropy S=S[\u001a1;\u001a2;:::;'1;'2;:::] is a function over the probability of each microstate's\nposterior and prior distribution, its variational derivative is also a function of said probabilities in general,\n\u000eS\n\u000e\u001ai\u0011\u001ei(\u001a1;\u001a2;:::;'1;'2;:::) =\u0015i+\u0016iAifor each (i;Di): (7)\nDC1 is imposed by constraining the form of \u001ei(\u001a1;\u001a2;:::;'1;'2;:::) =\u001ei(\u001ai;'1;'2;:::) to ensures that\nchanges inAi!Ai+\u000eAihave no in\ruence over the value of \u001ajin domainDj, through\u001ei, fori6=j. If\nthere is no new information about propositions in Dj, its distribution should remain equal to 'jby the\nPMU. We further restrict \u001eisuch that an arbitrary variation of 'j!'j+\u000e'j(a change in the prior state\nof knowledge of the microstate j) has no e\u000bect on \u001aifori6=jand therefore DC1 imposes \u001ei=\u001ei(\u001ai;'i),\nas is guided by the PMU. At this point it is easy to generalize the analysis to continuous microstates\nsuch that the indices become continuous i!x, sums become integrals, and discrete probabilities become\nprobability densities \u001ai!\u001a(x).\nRemark:\nWe are designing the entropy for the purpose of ranking posterior probability distributions (for the pur-\npose of inference); however, the highest ranked distribution is found by setting the variational derivative\nofS[\u001a;'] equal to the variations of the expectation value constraints by the Lagrange multiplier method,\n\u000eS\n\u000e\u001a(x)=\u0015+X\ni\u0016iAi(x): (8)\nTherefore, the real quantity of interest is\u000eS\n\u000e\u001a(x)rather than the speci\fc form of S[\u001a;'].Allforms of\nS[\u001a;'] that give the correct form of\u000eS\n\u000e\u001a(x)areequally valid for the purpose of inference. Thus, every\ndesign criteria may be made on the variational derivative of the entropy rather than the entropy itself,\nwhich we do. When maximizing the entropy, for convenience, we will let,\n\u000eS\n\u000e\u001a(x)\u0011\u001ex(\u001a(x);'(x)); (9)\nand further use the shorthand \u001ex(\u001a;')\u0011\u001ex(\u001a(x);'(x)), in all cases.\nDC1': In the absence of new information, our new state of knowledge \u001a(x)is equal to the old state of\nknowledge'(x).\n4This is a special case of DC1, and is implemented di\u000berently than in [7]. The PMU is in principle a\nstatement about informational honestly { that is, one should not \\jump to conclusions\" in light of new\ninformation and in the absence of new information, one should not change their state of knowledge. If no\nnew information is given, the prior probability distribution '(x) does not change, that is, the posterior\nprobability distribution \u001a(x) ='(x) is equal to the prior probability. If we maximizing the entropy\nwithout applying constraints,\n\u000eS\n\u000e\u001a(x)= 0; (10)\nthen DC1' imposes the following condition:\n\u000eS\n\u000e\u001a(x)=\u001ex(\u001a;') =\u001ex(';') = 0; (11)\nfor allxin this case. This special case of the DC1 and the PMU turns out to be incredibly constraining\nas we will see over the course of DC2.\nComment:\nFrom [7]. If the variable xis continuous, DC1 requires that when information refers to points in\fnitely\nclose but just outside the domain D, that it will have no in\ruence on probabilities conditional on D.\nThis may seem surprising as it may lead to updated probability distributions that are discontinuous. Is\nthis a problem? No.\nIn certain situations ( e.g., physics) we might have explicit reasons to believe that conditions of con-\ntinuity or di\u000berentiability should be imposed and this information might be given to us in a variety of\nways. The crucial point, however { and this is a point that we keep and will keep reiterating { is that\nunless such information is explicitly given we should not assume it. If the new information leads to\ndiscontinuities, so be it.\nDC2: Subsystem Independence\nDC2 imposes the second instance of when one should not update { the Subsystem PI. We emphasize\nthat DC2 is not a consistency requirement . The argument we deploy is notthat both the prior andthe\nnew information tells us the systems are independent, in which case consistency requires that it should\nnot matter whether the systems are treated jointly or separately. Rather, DC2 refers to a situation where\nthe new information does not say whether the systems are independent or not, but information is given\nabout each subsystem. The updating is being designed so that the independence re\rected in the prior is\nmaintained in the posterior by default via the PMU and the second clause of the PI's. [7]\nThe point is not that when we have no evidence for correlations we draw the \frm conclusion that the\nsystems must necessarily be independent. They could indeed have turned out to be correlated and then\nour inferences would be wrong. Again, induction involves risk. The point is rather that if the joint prior\nre\rected independence and the new evidence is silent on the matter of correlations, then the prior takes\nprecedence. As before, in this case subdomain independence, the probability distribution should not be\nupdated unless the information requires it. [7]\nDC2 Implementation:\nConsider a composite system, x= (x1;x2)2X =X1\u0002X 2. Assume that all prior evidence led us to\nbelieve the subsystems are independent. This belief is re\rected in the prior distribution: if the individual\nsystem priors are '1(x1) and'2(x2), then the prior for the whole system is their product '1(x1)'2(x2).\nFurther suppose that new information is acquired such that '1(x1) would by itself be updated to P1(x1)\nand that'2(x2) would be itself be updated to P2(x2). By design, the implementation of DC2 constrains\nthe entropy functional such that in this case, the joint product prior '1(x1)'2(x2) updates to the selected\nproduct posterior P1(x1)P2(x2). [7]\nThe argument below is considerably simpli\fed if we expand the space of probabilities to include\ndistributions that are not necessarily normalized. This does not represent any limitation because a nor-\nmalization constraint may always be applied. We consider a few special cases below:\nCase 1: We receive the extremely constraining information that the posterior distribution for system\n1 is completely speci\fed to be P1(x1) while we receive no information at all about system 2. We treat\n5the two systems jointly. Maximize the joint entropy S[\u001a(x1;x2);'(x1)'(x2)] subject to the following\nconstraints on the \u001a(x1;x2) ,Z\ndx2\u001a(x1;x2) =P1(x1): (12)\nNotice that the probability of each x12X 1within\u001a(x1;x2) is being constrained to P1(x1) in the marginal.\nWe therefore need a one Lagrange multiplier \u00151(x1) for eachx12X 1to tie each value ofR\ndx2\u001a(x1;x2)\ntoP1(x1). Maximizing the entropy with respect to this constraint is,\n\u000e\u0014\nS\u0000Z\ndx1\u00151(x1)\u0012Z\ndx2\u001a(x1;x2)\u0000P1(x1)\u0013\u0015\n= 0; (13)\nwhich requires that\n\u00151(x1) =\u001ex1x2(\u001a(x1;x2);'1(x1)'2(x2)); (14)\nfor arbitrary variations of \u001a(x1;x2). By design, DC2 is implemented by requiring '1'2!P1'2in this\ncase, therefore,\n\u00151(x1) =\u001ex1x2(P1(x1)'2(x2);'1(x1)'2(x2)): (15)\nThis equation must hold for all choices of x2and all choices of the prior '2(x2) as\u00151(x1) is independent\nofx2. Suppose we had chosen a di\u000berent prior '0\n2(x2) ='2(x2)+\u000e'2(x2) that disagrees with '2(x2). For\nallx2and\u000e'2(x2), the multiplier \u00151(x1) remains unchanged as it constrains the independent \u001a(x1)!\nP1(x1). This means that any dependence that the right hand side might potentially have had on x2and\non the prior '2(x2)must cancel out . This means that\n\u001ex1x2(P1(x1)'2(x2);'1(x1)'2(x2)) =fx1(P1(x1);'1(x1)): (16)\nSince'2is arbitrary in fsuppose further that we choose a constant prior set equal to one, '2(x2) = 1,\ntherefore\nfx1(P1(x1);'1(x1)) =\u001ex1x2(P1(x1)\u00031;'1(x1)\u00031) =\u001ex1(P1(x1);'1(x1)) (17)\nin general. This gives,\n\u00151(x1) =\u001ex1(P1(x1);'1(x1)): (18)\nThe left hand side does not depend on x2, and therefore neither does the right hand side. An argument\nexchanging systems 1 and 2 gives a similar result.\nCase 1 - Conclusion: When the system 2 is not updated the dependence on '2andx2drops out,\n\u001ex1x2(P1(x1)'2(x2);'1(x1)'2(x2)) =\u001ex1(P1(x1);'1(x1)): (19)\nand vice-versa when system 1 is not updated,\n\u001ex1x2('1(x1)P2(x2);'1(x1)'2(x2)) =\u001ex2(P2(x2);'2(x2)): (20)\nAs we seek the general functional form of \u001ex1x2, and because the x2dependence drops out of (19) and the\nx1dependence drops out of (20) for arbitrary '1;'2and'12='1'2, the explicit coordinate dependence\nin\u001econsequently drops out of both such that,\n\u001ex1x2!\u001e; (21)\nas\u001e=\u001e(\u001a(x);'(x)) must only depend on coordinates through the probability distributions themselves.\n(As a double check, explicit coordinate dependence was included in the following computations but in-\nevitably dropped out due to the form the functional equations and DC1'. By the argument above, and\nfor simplicity, we drop the explicit coordinate dependence in \u001ehere.)\nCase 2: Now consider a di\u000berent special case in which the marginal posterior distributions for systems\n1 and 2 are both completely speci\fed to be P1(x1) andP2(x2) respectively. Maximize the joint entropy\nS[\u001a(x1;x2);'(x1)'(x2)] subject to the following constraints on the \u001a(x1;x2) ,\nZ\ndx2\u001a(x1;x2) =P1(x1) andZ\ndx1\u001a(x1;x2) =P2(x2): (22)\n6Again, this is one constraint for each value of x1and one constraint for each value of x2, which therefore\nrequire the separate multipliers \u00161(x1) and\u00162(x2). Maximizing Swith respect to these constraints is\nthen,\n0 =\u000e\u0014\nS\u0000Z\ndx1\u00161(x1)\u0012Z\ndx2\u001a(x1;x2)\u0000P1(x1)\u0013\n\u0000Z\ndx2\u00162(x2)\u0012Z\ndx1\u001a(x1;x2)\u0000P2(x2)\u0013\u0015\n; (23)\nleading to\n\u00161(x1) +\u00162(x2) =\u001e(\u001a(x1;x2);'1(x1)'2(x2)): (24)\nThe updating is being designed so that '1'2!P1P2, as the independent subsystems are being updated\nbased on expectation values which are silent about correlations. DC2 thus imposes,\n\u00161(x1) +\u00162(x2) =\u001e(P1(x1)P2(x2);'1(x1)'2(x2)): (25)\nWrite (25) as,\n\u00161(x1) =\u001e(P1(x1)P2(x2);'1(x1)'2(x2))\u0000\u00162(x2): (26)\nThe left hand side is independent of x2so we can perform a trick similar to that we used before. Suppose\nwe had chosen a di\u000berent constraintP0\n2(x2) that di\u000bers from P2(x2) and a new prior '0\n2(x2) that di\u000bers\nfrom'2(x2) except at the value \u0016 x2. At the value \u0016 x2,the multiplier \u00161(x1) remains unchanged for all\nP0\n2(x2),'0\n2(x2), and thus x2. This means that any dependence that the right hand side might potentially\nhave had on x2and on the choice of P2(x2),'0\n2(x2) must cancel out leaving \u00161(x1) unchanged. That is,\nthe Lagrange multiplier \u0016(x2) \\pushes out\" these dependences such that\n\u001e(P1(x1)P2(x2);'1(x1)'2(x2))\u0000\u00162(x2) =g(P1(x1);'1(x1)): (27)\nBecauseg(P1(x1);'1(x1)) is independent of arbitrary variations of P2(x2) and'2(x2) on the LHS above\n{ it is satis\fed equally well for all choices. The form of g=\u001e(P1(x1);q1(x1)) is apparent if P2(x2) =\n'2(x2) = 1 as\u00162(x2) = 0 similar to Case 1 as well as DC1'. Therefore, the Lagrange multiplier is\n\u00161(x1) =\u001e(P1(x1);'1(x1)): (28)\nA similar analysis can be carried out for \u00162(x2) leads to\n\u00162(x2) =\u001e(P2(x2);'2(x2)): (29)\nCase 2 - Conclusion: Substituting back into (25) gives us a functional equation for \u001e,\n\u001e(P1P2;'1'2) =\u001e(P1;'1) +\u001e(P2;'2): (30)\nThe general solution for this functional equation is derived in the Appendix, section 6.3, and is\n\u001e(\u001a;') =a1ln(\u001a(x)) +a2ln('(x)) (31)\nwherea1;a2are constants. The constants are \fxed by using DC1'. Letting \u001a1(x1) ='1(x1) ='1gives\n\u001e(';') = 0 by DC1', and therefore,\n\u001e(';') = (a1+a2) ln(') = 0; (32)\nso we are forced to conclude a1=\u0000a2for arbitrary '. Lettinga1\u0011A=\u0000jAjsuch that we are really\nmaximizing the entropy (although this is purely aesthetic) gives the general form of \u001eto be,\n\u001e(\u001a;') =\u0000jAjln\u0010\u001a(x)\n'(x)\u0011\n: (33)\nAs long asA6= 0, the value of Ais arbitrary as it always can be absorbed into the Lagrange multipliers.\nThe general form of the entropy designed for the purpose of inference of \u001ais found by integrating \u001e, and\ntherefore,\nS(\u001a(x);'(x)) =\u0000jAjZ\ndx(\u001a(x) ln\u0010\u001a(x)\n'(x)\u0011\n\u0000\u001a(x)) +C[']: (34)\n7The constant in \u001a,C['], will always drop out when varying \u001a. The apparent extra term ( jAjR\n\u001a(x)dx)\nfrom integration cannot be dropped while simultaneously satisfying DC1', which requires \u001a(x) ='(x) in\nthe absence of constraints or when there is no change to one's information. In previous versions where\nthe integration term ( jAjR\n\u001a(x)dx) is dropped, one obtains solutions like \u001a(x) =e\u00001'(x) (independent\nof whether '(x) was previously normalized or not) in the absence of new information. Obviously this\nfactor can be taken care of by normalization, and in this way both forms of the entropy are equally\nvalid; however, this form of the entropy better adheres to the PMU through DC1'. Given that we may\nregularly impose normalization, we may drop the extraR\n\u001a(x)dxterm andC[']. For convenience then,\n(34) becomes\nS(\u001a(x);'(x))!S\u0003(\u001a(x);'(x)) =\u0000jAjZ\ndx\u001a(x) ln\u0010\u001a(x)\n'(x)\u0011\n; (35)\nwhich is a special case when the normalization constraint is being applied. Given normalization is applied,\nthe same selected posterior \u001a(x) maximizes both S(\u001a(x);'(x)) andS\u0003(\u001a(x);'(x)), and the star notation\nmay be dropped.\nRemarks: It can be seen that the relative entropy is invariant under coordinate transformations. This\nimplies that a system of coordinates carry no information and it is the \\character\" of the probability\ndistributions that are being ranked against one another rather than the speci\fc set of propositions or\nmicrostates they describe.\nThe general solution to the maximum entropy procedure with respect to Nlinear constraints in \u001a,\nhAi(x)i, and normalization gives a canonical-like selected posterior probability distribution,\n\u001a(x) ='(x) exp\u0010X\ni\u000biAi(x)\u0011\n: (36)\nThe positive constant jAjmay always be absorbed into the Lagrange multipliers so we may let it equal\nunity without loss of generality. DC1' is fully realized when we maximize with respect to a constraint\non\u001a(x) that is already held by '(x), such ashx2i=R\nx2\u001a(x) which happens to have the same value\nasR\nx2'(x), then its Lagrange multiplier is forcibly zero \u000b1= 0 (as can be seen in (36) using (34)),\nin agreement with Jaynes. This gives the expected result \u001a(x) ='(x) as there is no new information.\nOur design has arrived at a re\fned maximum entropy method [12] as a universal probability updating\nprocedure [27].\n3 The Design of the Quantum Relative Entropy\nLast section we assumed that the universe of discourse (the set of relevant propositions or microstates)\nX=A\u0002B\u0002:::was known. In quantum physics things are a bit more ambiguous because many probability\ndistributions, or many experiments, can be associated to a given density matrix. In this sense it helpful\nto think of density matrices as \\placeholders\" for probability distributions rather than a probability\ndistributions themselves. As any probability distribution from a given density matrix, \u001a(\u0001) = Tr(j\u0001ih\u0001j^\u001a),\nmay be ranked using the standard relative entropy, it is unclear why we would chose one universe of\ndiscourse over another. In lieu of this, such that one universe of discourse is not given preferential\ntreatment, we consider ranking entire density matrices against one another. Probability distributions of\ninterest may be found from the selected posterior density matrix. This moves our universe of discourse\nfrom sets of propositions X!H to Hilbert space(s).\nWhen the objects of study are quantum systems, we desire an objective procedure to update from\na prior density matrix ^ 'to a posterior density matrix ^ \u001a. We will apply the same intuition for ranking\nprobability distributions (Section 2) and implement the PMU, PI, and design criteria to the ranking of\ndensity matrices. We therefore \fnd the quantum relative entropy S(^\u001a;^') to be designed for the purpose\nof inferentially updating density matrices.\n3.1 Designing the Quantum Relative Entropy\nIn this section we design the quantum relative entropy using the same inferentially guided design criteria\nas were used in the standard relative entropy.\n8DC1: Subdomain Independence\nThe goal is to design a function S(^\u001a;^') which is able to rank density matrices. This insists that\nS(^\u001a;^') be a real scalar valued function of the posterior ^ \u001a, and prior ^ 'density matrices, which we will\ncall the quantum relative entropy or simply the entropy. An arbitrary variation of the entropy with\nrespect to ^\u001ais,\n\u000eS(^\u001a;^') =X\nij\u000eS(^\u001a;^')\n\u000e\u001aij\u000e\u001aij=X\nij\u0010\u000eS(^\u001a;^')\n\u000e^\u001a\u0011\nij\u000e(^\u001a)ij=X\nij\u0010\u000eS(^\u001a;^')\n\u000e^\u001aT\u0011\nji\u000e(^\u001a)ij= Tr\u0010\u000eS(^\u001a;^')\n\u000e^\u001aT\u000e^\u001a\u0011\n:(37)\nWe wish to maximize this entropy with respect to expectation value constraints, such as, hAi= Tr( ^A^\u001a) on\n^\u001a. Using the Lagrange multiplier method to maximize the entropy with respect to hAiand normalization,\nis setting the variation equal to zero,\n\u000e\u0010\nS(^\u001a;^')\u0000\u0015[Tr(^\u001a)\u00001]\u0000\u000b[Tr(^A^\u001a)\u0000hAi]\u0011\n= 0; (38)\nwhere\u0015and\u000bare the Lagrange multipliers for the respective constraints. Because S(^\u001a;^') is a real\nnumber, we inevitably require \u000eSto be real, but without imposing this directly, we \fnd that requiring\n\u000eSto be real requires ^ \u001a;^Ato be Hermitian. At this point, it is simpler to allow for arbitrary variations\nof ^\u001asuch that,\nTr\u0010\u0010\u000eS(^\u001a;^')\n\u000e^\u001aT\u0000\u0015^1\u0000\u000b^A\u0011\n\u000e^\u001a\u0011\n= 0: (39)\nFor these arbitrary variations, the variational derivative of Smust satisfy,\n\u000eS(^\u001a;^')\n\u000e^\u001aT=\u0015^1 +\u000b^A; (40)\nat the maximum. As in the remark earlier, allforms ofSwhich give the correct form of\u000eS(^\u001a;^')\n\u000e^\u001aTunder\nvariation are equally valid for the purpose of inference. For notational convenience we let,\n\u000eS(^\u001a;^')\n\u000e^\u001aT\u0011\u001e(^\u001a;^'); (41)\nwhich is a matrix valued function of the posterior and prior density matrices. The form of \u001e(^\u001a;^') is\nalready \"local\" in ^ \u001a, so we don't need to constrain it further as we did in the original DC1.\nDC1': In the absence of new information, the new state ^\u001ais equal to the old state ^'.\nApplied to the ranking of density matrices, in the absence of new information, the density matrix\n^'should not change, that is, the posterior density matrix ^ \u001a= ^'is equal to the prior density matrix.\nMaximizing the entropy without applying any constraints gives,\n\u000eS(^\u001a;^')\n\u000e^\u001aT=^0; (42)\nand therefore DC1' imposes the following condition in this case,\n\u000eS(^\u001a;^')\n\u000e^\u001aT=\u001e(^\u001a;^') =\u001e( ^';^') =^0: (43)\nAs in the original DC1', if ^ 'is known to obey some expectation value constraint h^Ai, then if one goes\nout of their way to constrain ^ \u001ato that expectation value with nothing else, it follows from the PMU that\n^\u001a= ^', as no information has been gained. This is not imposed directly, but can be veri\fed later.\nDC2: Subsystem Independence\nThe discussion of DC2 is the same as the standard relative entropy DC2 { it is not a consistency\nrequirement, and the updating is designed so that the independence re\rected in the prior is maintained\nin the posterior by default via the PMU, when the information provided is silent about correlations.\nDC2 Implementation:\n9Consider a composite system living in the Hilbert space H=H1\nH 2. Assume that all prior\nevidence led us to believe the systems were independent. This is re\rected in the prior density matrix:\nif the individual system priors are ^ '1and ^'2, then the joint prior for the whole system is ^ '1\n^'2.\nFurther suppose that new information is acquired such that ^ '1would by itself be updated to ^ \u001a1and\nthat ^'2would be itself be updated to ^ \u001a2. By design, the implementation of DC2 constrains the entropy\nfunctional such that in this case, the joint product prior density matrix ^ '1\n^'2updates to the product\nposterior ^\u001a1\n^\u001a2so that inferences about one do not a\u000bect inferences about the other.\nThe argument below is considerably simpli\fed if we expand the space of density matrices to include\ndensity matrices that are not necessarily normalized. This does not represent any limitation because\nnormalization can always be easily achieved as one additional constraint. We consider a few special cases\nbelow:\nCase 1: We receive the extremely constraining information that the posterior distribution for system 1\nis completely speci\fed to be ^ \u001a1while we receive no information about system 2 at all. We treat the two\nsystems jointly. Maximize the joint entropy S[^\u001a12;^'1\n^'2], subject to the following constraints on the\n^\u001a12,\nTr2(^\u001a12) = ^\u001a1: (44)\nNotice all of the N2elements inH1of ^\u001a12are being constrained. We therefore need a Lagrange multiplier\nwhich spansH1and therefore it is a square matrix ^\u00151. This is readily seen by observing the component\nform expressions of the Lagrange multipliers ( ^\u00151)ij=\u0015ij. Maximizing the entropy with respect to this\nH2independent constraint is,\n0 =\u000e\u0010\nS\u0000X\nij\u0015ij\u0010\nTr2(^\u001a1;2)\u0000^\u001a1\u0011\nij\u0011\n; (45)\nbut reexpressing this with its transpose ( ^\u00151)ij= (^\u0015T\n1)ji, gives\n0 =\u000e\u0010\nS\u0000Tr1(^\u00151[Tr2(^\u001a1;2)\u0000^\u001a1])\u0011\n; (46)\nwhere we have relabeled ^\u0015T\n1!^\u00151, for convenience, as the name of the Lagrange multipliers are arbitrary.\nFor arbitrary variations of ^ \u001a12, we therefore have,\n^\u00151\n^12=\u001e(^\u001a12;^'1\n^'2): (47)\nDC2 is implemented by requiring ^ '1\n^'2!^\u001a1\n^'2, such that the function \u001eis designed to re\rect\nsubsystem independence in this case; therefore, we have\n^\u00151\n^12=\u001e(^\u001a1\n^'2;^'1\n^'2): (48)\nThis equation must hold for all choices of the independent prior ^ '2inH2. Suppose we had chosen a\ndi\u000berent prior ^ '0\n2= ^'2+\u000e^'2. For all\u000e^'2the LHS ^\u00151\n^12remains unchanged. This means that any\ndependence that the right hand side might potentially have had on ^ '2must cancel out , meaning,\n\u001e(^\u001a1\n^'2;^'1\n^'2) =f(^\u001a1;^'1)\n^12: (49)\nSince ^'2is arbitrary, suppose further that we choose a unit prior, ^ '2=^12, and note that ^ \u001a1\n^12and\n^'1\n^12are block diagonal in H2. Because the LHS is block diagonal in H2,\nf(^\u001a1;^'1)\n^12=\u001e\u0000\n^\u001a1\n^12;^'1\n^12\u0001\n(50)\nthe RHS is block diagonal in H2, and because the function \u001eis understood to be a power series expansion\nin its arguments,\nf(^\u001a1;^'1)\n^12=\u001e\u0000\n^\u001a1\n^12;^'1\n^12\u0001\n=\u001e(^\u001a1;^'1)\n^12: (51)\nThis gives,\n^\u00151\n^12=\u001e(^\u001a1;^'1)\n^12; (52)\nand therefore the ^12factors out and ^\u00151=\u001e(^\u001a1;^'1). A similar argument exchanging systems 1 and 2\nshows ^\u00152=\u001e(^\u001a2;^'2) in this case.\n10Case 1 - Conclusion: The analysis leads us to conclude that when the system 2 is not updated the\ndependence on ^ '2also drops out,\n\u001e(^\u001a1\n^'2;^'1\n^'2) =\u001e(^\u001a1;^'1)\n^12; (53)\nand similarly,\n\u001e( ^'1\n^\u001a2;^'1\n^'2) =^11\n\u001e(^\u001a2;^'2): (54)\nCase 2: Now consider a di\u000berent special case in which the marginal posterior distributions for systems 1\nand 2 are both completely speci\fed to be ^ \u001a1and ^\u001a2respectively. Maximize the joint entropy, S[^\u001a12;^'1\n^'2], subject to the following constraints on the ^ \u001a12,\nTr2(^\u001a12) = ^\u001a1and Tr 1(^\u001a12) = ^\u001a2: (55)\nHere each expectation value constraints the entire space Hi, where ^\u001ailives. The Lagrange multipliers\nmust span their respective spaces, so we implement the constraint with the Lagrange multiplier operator\n^\u0016i, then,\n0 =\u000e\u0010\nS\u0000Tr1(^\u00161[Tr2(^\u001a12)\u0000^\u001a1])\u0000Tr2(^\u00162[Tr1(^\u001a12)\u0000^\u001a2])\u0011\n: (56)\nFor arbitrary variations of ^ \u001a12, we have,\n^\u00161\n^12+^11\n^\u00162=\u001e(^\u001a12;^'1\n^'2): (57)\nBy design, DC2 is implemented by requiring ^ '1\n^'2!^\u001a1\n^\u001a2in this case; therefore, we have\n^\u00161\n^12+^11\n^\u00162=\u001e(^\u001a1\n^\u001a2;^'1\n^'2): (58)\nWrite (58) as,\n^\u00161\n^12=\u001e(^\u001a1\n^\u001a2;^'1\n^'2)\u0000^11\n^\u00162: (59)\nThe left hand side is independent of changes in of ^ \u001a2and ^'2inH2as ^\u00162\\pushes out\" this dependence\nfrom\u001e. Any dependence that the RHS might potentially have had on ^ \u001a2, ^'2must cancel out, leaving\n^\u00161unchanged. Consequently,\n\u001e(^\u001a1\n^\u001a2;^'1\n^'2)\u0000^11\n^\u00162=g(^\u001a1;^'1)\n^12: (60)\nBecauseg(^\u001a1;^'1) is independent of arbitrary variations of ^ \u001a2and ^'2on the LHS above { it is satis\fed\nequally well for all choices. The form of g(^\u001a1;^'1) reduces to the form of f(^\u001a1;^'1) from Case 1 when\n^\u001a2= ^'2=^12and similarly DC1' gives ^ \u00162= 0. Therefore, the Lagrange multiplier is\n^\u00161\n^12=\u001e(^\u001a1;^'1)\n^12: (61)\nA similar analysis can be carried out for ^ \u00162leading to\n^11\n^\u00162=^11\n\u001e(^\u001a2;^'2): (62)\nCase 2 - Conclusion: Substituting back into (58) gives us a functional equation for \u001e,\n\u001e(^\u001a1\n^\u001a2;^'1\n^'2) =\u001e(^\u001a1;^'1)\n^12+^11\n\u001e(^\u001a2;^'2); (63)\nwhich is,\n\u001e(^\u001a1\n^\u001a2;^'1\n^'2) =\u001e(^\u001a1\n^12;^'1\n^12) +\u001e(^11\n^\u001a2;^11\n^'2): (64)\nThe general solution to this matrix valued functional equation is derived in the Appendix 6.5, and is,\n\u001e(^\u001a;^') =\u0018\nAln(^\u001a)+\u0018\nBln( ^'); (65)\nwhere tilde\u0018\nAis a \\super-operator\" having constant coe\u000ecients and twice the number of indicies as ^ \u001a\nand ^'as discussed in the Appendix (i.e.\u0010\u0018\nAln(^\u001a)\u0011\nij=P\nk`Aijk`(log(^\u001a))k`and similarly for\u0018\nBln( ^')).\nDC1' imposes,\n\u001e( ^';^') =\u0018\nAln( ^')+\u0018\nBln( ^') =^0; (66)\n11which is satis\fed in general when\u0018\nA=\u0000\u0018\nB, and now,\n\u001e(^\u001a;^') =\u0018\nA\u0010\nln(^\u001a)\u0000ln( ^')\u0011\n: (67)\nWe may \fx the constant\u0018\nAby substituting our solution into the RHS of equation (63) which is equal to\nthe RHS of equation (64),\n\u0010\u0018\nA1\u0010\nln(^\u001a1)\u0000ln( ^'1)\u0011\u0011\n\n^12+^11\n\u0010\u0018\nA2\u0010\nln(^\u001a2)\u0000ln( ^'2)\u0011\u0011\n=\u0018\nA12\u0010\nln(^\u001a1\n^12)\u0000ln( ^'1\n^12)\u0011\n+\u0018\nA12\u0010\nln(^11\n^\u001a2)\u0000ln(^11\n^'2)\u0011\n; (68)\nwhere\u0018\nA12acts on the joint space of 1 and 2 and\u0018\nA1,\u0018\nA2acts on single subspaces 1 or 2 respectively.\nUsing the log tensor product identity, ln(^ \u001a1\n^12) = ln(^\u001a1)\n^12, in the RHS of equation (68) gives,\n=\u0018\nA12\u0010\nln(^\u001a1)\n^12\u0000ln( ^'1)\n^12\u0011\n+\u0018\nA12\u0010\n^11\nln(^\u001a2)\u0000^11\nln( ^'2)\u0011\n: (69)\nNote that arbitrarily letting ^ \u001a2= ^'2gives,\n\u0010\u0018\nA1\u0010\nln(^\u001a1)\u0000ln( ^'1)\u0011\u0011\n\n^12=\u0018\nA12\u0010\nln(^\u001a1)\n^12\u0000ln( ^'1)\n^12\u0011\n: (70)\nor arbitrarily letting ^ \u001a1= ^'1gives,\n^11\n\u0010\u0018\nA2\u0010\nln(^\u001a2)\u0000ln( ^'2)\u0011\u0011\n=\u0018\nA12\u0010\n^11\nln(^\u001a2)\u0000^11\nln( ^'2)\u0011\n: (71)\nAs\u0018\nA12,\u0018\nA1, and\u0018\nA2are constant tensors, inspecting the above equalities determines the form of the\ntensor to be\u0018\nA=A\u0018\n1whereAis a scalar constant and\u0018\n1is the super-operator identity over the appropriate\n(joint) Hilbert space.\nBecause our goal is to maximize the entropy function, we let the arbitrary constant A=\u0000jAjand\ndistribute\u0018\n1identically, which gives the \fnal functional form,\n\u001e(^\u001a;^') =\u0000jAj\u0010\nln(^\u001a)\u0000ln( ^')\u0011\n: (72)\n\\Integrating\" \u001e, gives a general form for the quantum relative entropy,\nS(^\u001a;^') =\u0000jAjTr(^\u001alog ^\u001a\u0000^\u001alog ^'\u0000^\u001a) +C[ ^'] =\u0000jAjSU(^\u001a;^') +jAjTr(^\u001a) +C[ ^']; (73)\nwhereSU(^\u001a;^') is Umegaki's form of the relative entropy, the extra jAjTr(^\u001a) from integration is an artifact\npresent for the preservation of DC1', and C[ ^'] is a constant in the sense that it drops out under arbitrary\nvariations of ^ \u001a. This entropy leads to the same inferences as Umegaki's form of the entropy with added\nbonus that ^ \u001a= ^'in the absence of constraints or changes in information { rather than ^ \u001a=e\u00001^'\nwhich would be given by maximizing Umegaki's form of the entropy. In this sense the extra jAjTr(^\u001a)\nonly improves the inference process as it more readily adheres to the PMU though DC1'; however now\nbecauseSU\u00150, we have S(^\u001a;^')\u0014Tr(^\u001a) +C[ ^'], which provides little nuisance. In the spirit of this\nderivation we will keep the Tr(^ \u001a) term there, but for all practical purposes of inference, as long as there\nis a normalization constraint, it plays no role, and we \fnd (letting jAj= 1 andC[ ^'] = 0),\nS(^\u001a;^')!S\u0003(^\u001a;^') =\u0000SU(^\u001a;^') =\u0000Tr(^\u001alog ^\u001a\u0000^\u001alog ^'); (74)\nUmegaki's form of the relative entropy. S\u0003(^\u001a;^') is an equally valid entropy because, given normalization\nis applied, the same selected posterior ^ \u001amaximizes both S(^\u001a;^') andS\u0003(^\u001a;^').\n123.2 Remarks\nDue to the universality and the equal application of the PMU by using the same design criteria for both\nthe standard and quantum case, the quantum relative entropy reduces to the standard relative entropy\nwhen [^\u001a;^'] = 0 or when the experiment being preformed ^ \u001a!\u001a(a) = Tr(^\u001ajaihaj) is known. The quantum\nrelative entropy we derive has the correct asymptotic form of the standard relative entropy in the sense\nof [8, 9, 10]. Further connections will be illustrated in a follow up article that is concerned with direct\napplications of the quantum relative entropy. Because two entropies are derived in parallel, we expect\nthe well known inferential results and consequences of the relative entropy to have a quantum relative\nentropy representation.\nMaximizing the quantum relative entropy with respect to some constraints h^Aii, wheref^Aigare a\nset of arbitrary Hermitian operators, and normalization h^1i= 1, gives the following general solution for\nthe posterior density matrix:\n^\u001a= exp\u0010\n\u000b0^1 +X\ni\u000bi^Ai+ ln( ^')\u0011\n=1\nZexp\u0010X\ni\u000bi^Ai+ ln( ^')\u0011\n\u00111\nZexp\u0010\n^C\u0011\n; (75)\nwhere\u000biare the Lagrange multipliers of the respective constraints and normalization may be factored\nout of the exponential in general because the identity commutes universally. If ^ '/^1, it is well known\nthe analysis arrives at the same expression for ^ \u001aafter normalization as it would if the von Neumann\nentropy were used, and thus one can \fnd expressions for thermalized quantum states ^ \u001a=1\nZe\u0000\f^H. The\nremaining problem is to solve for the NLagrange multipliers using their Nassociated expectation value\nconstraints. In principle their solution is found by computing Zand using standard methods from\nStatistical Mechanics,\nh^Aii=\u0000@\n@\u000biln(Z); (76)\nand inverting to \fnd \u000bi=\u000bi(h^Aii), which has a unique solution due to the joint concavity (convexity\ndepending on the sign convention) of the quantum relative entropy [8, 9] when the constraints are linear\nin ^\u001a. Between the Zassenhaus formula\net(^A+^B)=et^Aet^Be\u0000t2\n2[^A;^B]et3\n6(2[^B;[^A;^B]]+[^A;[^A;^B]]):::; (77)\nand Horn's inequality, the solutions to (76) lack a certain calculational elegance because it is di\u000ecult to\nexpress the eigenvalues of ^C= log( ^') +P\n\u000bi^Ai(in the exponential) in simple terms of the eigenvalues\nof the ^Ai's and ^', in general, when the matrices do not commute. The solution requires solving the\neigenvalue problem for ^C, such the the exponential of ^Cmay be taken and evaluated in terms of the\neigenvalues of the \u000bi^Ai's and the prior density matrix ^ '. A pedagogical exercise is, starting with a prior\nwhich is a mixture of spin-z up and down ^ '=aj+ih+j+bj\u0000ih\u0000j (a;b6= 0) and maximize the quantum\nrelative entropy with respect to the expectation of a general Hermitian operator. This example is given\nin the Appendix 6.6.\n4 Conclusions:\nThis approach emphasizes the notion that entropy is a tool for performing inference and downplays\ncounter-notional issues which arise if one interprets entropy as a measure of disorder, a measure of\ndistinguishability, or an amount of missing information [7]. Because the same design criteria, guided by\nthe PMU, are applied equally well to the design of a relative and quantum relative entropy, we \fnd that\nboth the relative and quantum relative entropy are designed for the purpose of inference. Because the\nquantum relative entropy is the function which \fts the requirements of a tool designed for inference, we\nnow know what it is and how to use it { formulating an inferential quantum maximum entropy method.\nA follow up article is concerned with a few interesting applications of the quantum maximum entropy\nmethod, and in particular it derives the Quantum Bayes Rule.\n135 Acknowledgments\nI must give ample acknowledgment to Ariel Caticha who suggested the problem of justifying the form\nof the quantum relative entropy as a criterion for ranking of density matrices. He cleared up several\ndi\u000eculties by suggesting that design constraints be applied to the variational derivative of the entropy\nrather than the entropy itself. As well, he provided substantial improvements to the method for imposing\nDC2 that lead to the functional equations for the variational derivatives ( \u001e12=\u001e1+\u001e2) { with more\nrigor than in earlier versions of this article. His time and guidance are all greatly appreciated { Thanks\nAriel.\nReferences\n[1] Shore, J. E.; Johnson, R. W.; Axiomatic derivation of the Principle of Maximum Entropy and the\nPrinciple of Minimum Cross-Entropy. IEEE Trans. Inf. Theory 1980 IT-26, 26-37.\n[2] Shore, J. E.; Johnson, R. W. Properties of Cross-Entropy Minimization. IEEE Trans. Inf. Theory\n1981 IT-27, 472-482.\n[3] Csisz\u0013 ar, I. Why least squares and maximum entropy: an axiomatic approach to inference for linear\ninverse problems. Ann. Stat. 1991 ,19, 2032.\n[4] Skilling, J. The Axioms of Maximum Entropy. Maximum- Entropy and Bayesian Methods in Sci-\nence and Engineering , Dordrecht, Holland, 1988; Erickson, G. J.; Smith, C. 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Phys.\nRev. B 2000 ,63, 115403, 10.1103/PhysRevB.63.115403.\n[25] Jordan, A.; Korotkov, A. Qubit feedback and control with kicked quantum nondemolition measure-\nments: A quantum Bayesian analysis. Phys. Rev. B 2006 ,74, 085307.\n[26] Hellmann, F.; Kami\u0013 nski, W.; Kostecki, P. Quantum collapse rules from the maximum relative\nentropy principle. New J. Phys. 2016 ,18, 013022.\n[27] Gi\u000en, A.; Caticha, A. Updating Probabilities. Presented at MaxEnt (2006). MaxEnt 2006, the 26th\nInternational Workshop on Bayesian Inference and Maximum Entropy Methods , Paris, France.\n[28] Caticha, A. Toward an Informational Pragmatic Realism, Minds and Machines 2014 ,24, 37-70.\n[29] Umegaki, H. Conditional expectation in an operator algebra, IV (entropy and information). K odai\nMath. Sem. Rep 1962 ,14, 59-85.\n[30] Uhlmann, A. Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an interpolation\ntheory. Commun. Math. Phys. 1997 ,54, 21-32.\n[31] Schumacher, B.; Westmoreland, M. Relative entropy in quantum information theory. AMS special\nsession on Quantum Information and Computation , January, 2000.\n[32] von Neumann, J. Mathematische Grundlagen der Quantenmechanik ; Springer-Verlag: Berlin, Ger-\nmany, 1932. [English translation: Mathematical Foundations of Quantum Mechanics ; Princeton Uni-\nversity Press, Princeton NY, USA, 1983.].\n[33] Acz\u0013 el, J. Lectures on Functional Equations and their Applications , Vol. 19; Academic Press Inc.:\n111 Fifth Ave. New York NY 10003, USA, 1966; pp. 31-44, 141-145, 213-217, 301-302, 347-349.\n6 Appendix:\nThe Appendix loosely follows the relevant sections in [33], and then uses the methods reviewed to solve\nthe relevant functional equations for \u001e. The last section is an example of the quantum maximum entropy\nmethod for spin.\n6.1 Simple functional equations\nFrom [33] pages 31-44.\nThm 1: If Cauchy's functional equation\nf(x+y) =f(x) +f(y); (78)\nis satis\fed for all real x,y, and if the function f(x)is (a) continuous at a point, (b) nonegative for small\npositivex's, or (c) bounded in an interval, then,\nf(x) =cx (79)\nis the solution to (78) for all real x. If (78) is assumed only over all positive x,y, then under the same\nconditions (79) holds for all positive x.\n15Proof 1: The most natural assumption for our purposes is that f(x) is continuous at a point (which\nlater extends to continuity all points as given by Darboux). Cauchy solved the functional equation by\ninduction. In particular equation (78) implies,\nf(X\nixi) =X\nif(xi); (80)\nand if we let each xi=xas a special case to determine f, we \fnd\nf(nx) =nf(x): (81)\nWe may let nx=mtsuch that\nf(x) =f(m\nnt) =m\nnf(t): (82)\nLetting lim t!1f(t) =f(1) =c, gives\nf(m\nn) =m\nnf(1) =m\nnc; (83)\nand because for t= 1,x=m\nnabove, we have\nf(x) =cx; (84)\nwhich is the general solution of the linear functional equation. In principle ccan be complex. The\nimportance of Cauchy's solution is that can be used to give general solutions to the following Cauchy\nequations:\nf(x+y) =f(x)f(y); (85)\nf(xy) =f(x) +f(y); (86)\nf(xy) =f(x)f(y); (87)\nby preforming consistent substitution until they are the same form as (78) as given by Cauchy. We will\nbrie\ry discuss the \frst two.\nThm 2: The general solution of f(x+y) =f(x)f(y)isf(x) =ecxfor all real or for all positive x;y\nthat are continuous at one point and, in addition to the exponential solution, the solution f(0) = 1 and\nf(x) = 0 for (x>0) are in these classes of functions.\nThe \frst functional f(x+y) =f(x)f(y) is solved by \frst noting that it is strictly positive for real x,\ny,f(x), which can be shown by considering x=y,\nf(2x) =f(x)2>0: (88)\nIf there exists f(x0) = 0, then it follows that f(x) =f((x\u0000x0) +x0) = 0, a trivial solution, hence why\nthe possibility of being equal to zero is excluded above. Given f(x) is nowhere zero, we are justi\fed in\ntaking the natural logarithm ln( x), due to its positivity f(x)>0. This gives,\nln(f(x+y)) = ln(f(x)) + ln(f(y)); (89)\nand letting g(x) = ln(f(x)) gives,\ng(x+y) =g(x) +g(y); (90)\nwhich is Cauchy's linear equation, and thus has the solution g(x) =cx. Becauseg(x) = ln(f(x)), one\n\fnds in general that f(x) =ecx.\nThm 3: If the functional equation f(xy) =f(x) +f(y)is valid for all positive x;y then its general\nsolution isf(x) =cln(x)given it is continuous at a point. If x= 0(ory= 0) are valid then the general\nsolution isf(x) = 0 . If all real x;yare valid except 0then the general solution is f(x) =cln(jxj).\nIn particular we are interested in the functional equation f(xy) =f(x) +f(y) whenx;yare positive.\nIn this case we can again follow Cauchy and substitute x=euandy=evto get,\nf(euev) =f(eu) +f(ev); (91)\nand letting g(u) =f(eu) givesg(u+v) =g(u) +g(v). Again, the solution is g(u) =cuand therefore the\ngeneral solution is f(x) =cln(x) when we substitute for u. Ifxcould equal 0 then f(0) =f(x) +f(0),\nwhich has the trivial solution f(x) = 0. The general solution for x6= 0,y6= 0 andx;ypositive is\nthereforef(x) =cln(x).\n166.2 Functional equations with multiple arguments\nFrom [33] pages 213-217. Consider the functional equation,\nF(x1+y1;x2+y2;:::;xn+yn) =F(x1;x2;:::;xn) +F(y1;y2;:::;yn); (92)\nwhich is a generalization of Cauchy's linear functional equation (78) to several arguments. Letting\nx2=x3=:::=xn=y2=y3=:::=yn= 0 gives\nF(x1+y1;0;:::;0) =F(x1;0;:::;0) +F(y1;0;:::;0); (93)\nwhich is the Cauchy linear functional equation having solution F(x1;0;:::;0) =c1x1whereF(x1;0;:::;0)\nis assumed to be continuous or at least measurable majorant. Similarly,\nF(0;:::;0;xk;0;:::;0) =ckxk; (94)\nand if you consider\nF(x1+ 0;0 +y2;0;:::;0) =F(x1;0;:::;0) +F(0;y2;0;:::;0) =c1x1+c2y2; (95)\nand asy2is arbitrary we could have let y2=x2such that in general\nF(x1;x2;:::;xn) =X\ncixi; (96)\nas a general solution.\n6.3 Relative entropy:\nWe are interested in the following functional equation,\n\u001e(\u001a1\u001a2;'1'2) =\u001e(\u001a1;'1) +\u001e(\u001a2;'2): (97)\nThis is an equation of the form,\nF(x1y1;x2y2) =F(x1;x2) +F(y1;y2); (98)\nwherex1=\u001a(x1),y1=\u001a(x2),x2='(x1), andy2='(x2). First assume all qandpare greater than\nzero. Then, substitute: xi=ex0\niandyi=ey0\niand letF0(x0\n1;x0\n2) =F(ex0\n1;ex0\n2) and so on such that\nF0(x0\n1+y0\n1;x0\n2+y0\n2) =F0(x0\n1;x0\n2) +F0(y0\n1;y0\n2); (99)\nwhich is of the form of (92). The general solution for Fis therefore\nF0(x0\n1+y0\n1;x0\n2+y0\n2) =a1(x0\n1+y0\n1) +a2(x0\n2+y0\n2) =a1ln(x1y1) +a2ln(x2y2) =F(x1y1;x2y2) (100)\nwhich means the general solution for \u001eis,\n\u001e(\u001a1;'1) =a1ln(\u001a(x1)) +a2ln('(x1)) (101)\nIn such a case when '(x0) = 0 for some value x02X we may let '(x0) =\u000fwhere\u000fis as close to zero as\nwe could possibly want { the trivial general solution \u001e= 0 is saturated by the special case when \u001a='\nfrom DC1'. Here we return to the text.\n6.4 Matrix functional equations\n(This derivation is implied in [33] pages 347-349). First consider a Cauchy matrix functional equation,\nf(^X+^Y) =f(^X) +f(^Y) (102)\nwhere ^Xand ^Yaren\u0002nsquare matrices. Rewriting the matrix functional equation in terms of its\ncomponents gives,\nfij(x11+y11;x12+y12;:::;xnn+ynn) =fij(x11;x12;:::;xnn) +fij(y11;y12;:::;ynn) (103)\n17is now in the form of (92) and therefore the solution is,\nfij(x11;x12;:::;xnn) =nX\n`;k=0cij`kx`k (104)\nfori;j= 1;:::;n . We \fnd it convenient to introduce super indices, A= (i;j) andB= (`;k) such that\nthe component equation becomes,\nfA=X\nBcABxB: (105)\nresembles the solution for a linear transformation of a vector from [33]. In general we will be discussing\nmatrices ^X=^X1\n^X2\n:::\n^XNwhich stem out of the tensor products of density matrices. In this\nsituation ^Xcan be thought of as 2 Nindex tensor or a z\u0002zmatrix where z=QN\niniis the product\nof the ranks of the matrices in the tensor product or even ^Xis a vector of length z2. In such a case we\nmay abuse the super index notation where AandBlump together the appropriate number of indices\nsuch that (105) is the form of the solution for the components in general. The matrix form of the general\nsolution is,\nf(^X) =eC^X; (106)\nwhereeCis a constant super-operator having components cAB.\n6.5 Quantum Relative entropy:\nThe functional equation is,\n\u001e\u0010\n^\u001a1\n^\u001a2;^'1\n^'2\u0011\n=\u001e\u0010\n^\u001a1\n^12;^'1\n^12\u0011\n+\u001e\u0010\n^11\n^\u001a2;^11\n^'2\u0011\n: (107)\nThese density matrices are Hermitian, positive semi-de\fnite, have positive eigenvalues, and are not equal\nto^0. Because every invertible matrix can be expressed as the exponential of some other matrix, we can\nsubstitute ^\u001a1=e^\u001a0\n1, and so on for all four density matrices which gives,\n\u001e\u0010\ne^\u001a0\n1\ne^\u001a0\n2;e^'0\n1\ne^'0\n2\u0011\n=\u001e\u0010\ne^\u001a0\n1\n^12;e^'0\n1\n^12\u0011\n+\u001e\u0010\n^11\ne^\u001a0\n2;^11\ne^'0\n2\u0011\n: (108)\nNow we use the following identities for Hermitian matrices,\ne^\u001a0\n1\ne^\u001a0\n2=e^\u001a0\n1\n^12+^11\n^\u001a0\n2 (109)\nand\ne^\u001a0\n1\n^12=e^\u001a0\n1\n^12; (110)\nto recast the functional equation as,\n\u001e\u0010\ne^\u001a0\n1\n^12+^11\n^\u001a0\n2;e^'0\n1\n^12+^11\n^'0\n2\u0011\n=\u001e\u0010\ne^\u001a0\n1\n^12;e^'0\n1\n^12\u0011\n+\u001e\u0010\ne^11\n^\u001a0\n2;e^11\n^'0\n2\u0011\n: (111)\nLettingG(^\u001a0\n1\n^12;^'0\n1\n^12) =\u001e\u0010\ne^\u001a0\n1\n^12;e^'0\n1\n^12\u0011\ngives,\nG(^\u001a0\n1\n^12+^11\n^\u001a0\n2;^'0\n1\n^12+^11\n^'0\n2) =G(^\u001a0\n1\n^12;^'0\n1\n^12) +G(^11\n^\u001a0\n2;^11\n^'0\n2): (112)\nThis functional equation is of the form\nG(^X0\n1+^Y0\n1;^X0\n2+^Y0\n2) =G(^X0\n1;^X0\n2) +G(^Y0\n1;^Y0\n2); (113)\nwhich has the general solution\nG(^X0;^Y0) =\u0018\nA^X0+eB^Y0; (114)\nsynonymous to (96), and \fnally in general,\n\u001e(^\u001a;^') =\u0018\nAln(^\u001a) +eBln( ^'): (115)\nwhere\u0018\nA;\u0018\nBare super-operators having constant coe\u000ecients.\n186.6 Spin Example\nConsider an arbitrarily mixed prior is (in the spin- zbasis for convenience) with a;b6= 0,\n^'=aj+ih+j+bj\u0000ih\u0000j (116)\nand a general Hermitian matrix in the spin-1 =2 Hilbert space,\nc\u0016^\u001b\u0016=c1^1 +cx^\u001bx+cy^\u001bx+cz^\u001bz (117)\n= (c1+cz)j+ih+j+ (cx\u0000icy)j+ih\u0000j+ (cx+icy)j\u0000ih+j+ (c1\u0000cz)j\u0000ih\u0000j; (118)\nhaving a known expectation value,\nTr(^\u001ac\u0016^\u001b\u0016) =c: (119)\nMaximizing the entropy with respect to this general expectation value and normalization is:\n0 =\u0010\n\u000eS\u0000\u0015[Tr(^\u001a)\u00001]\u0000\u000b(Tr(^\u001ac\u0016^\u001b\u0016)\u0000c)\u0011\n; (120)\nwhich after varying gives,\n^\u001a=1\nZexp(\u000bc\u0016^\u001b\u0016+ log( ^')): (121)\nLetting\n^C=\u000bc\u0016^\u001b\u0016+ log( ^') (122)\ngives\n^\u001a=1\nZe^C=UeU\u00001^CUU\u00001=1\nZUe^\u0015U\u00001\n=e\u0015+\nZUj\u0015+ih\u0015+jU\u00001+e\u0015\u0000\nZUj\u0015\u0000ih\u0015\u0000jU\u00001; (123)\nwhere ^\u0015is the diagonalized matrix of ^Chaving the real eigenvalues. They are,\n\u0015\u0006=\u0015\u0006\u000e\u0015; (124)\ndue to the quadratic formula, explicitly:\n\u0015=\u000bc1+1\n2log(ab); (125)\nand\n\u000e\u0015=1\n2r\u0010\n2\u000bcz+ log(a\nb)\u00112\n+ 4\u000b2(c2x+c2y): (126)\nBecause\u0015\u0006anda;b;c 1;cx;cy;czare real,\u000e\u0015\u00150. The normalization constraint speci\fes the Lagrange\nmultiplierZ,\n1 = Tr(^\u001a) =e\u0015++e\u0015\u0000\nZ; (127)\nsoZ=e\u0015++e\u0015\u0000= 2e\u0015cosh(\u000e\u0015). The expectation value constraint speci\fes the Lagrange multiplier \u000b,\nc= Tr(^\u001ac\u0016\u001b\u0016) =@\n@\u000blog(Z) =c1+ tanh(\u000e\u0015)@\n@\u000b\u000e\u0015; (128)\nwhich becomes\nc=c1+tanh(\u000e\u0015)\n2\u000e\u0015\u0010\n2\u000b(c2\nx+c2\ny+c2\nz) +czlog(a\nb)\u0011\n;\nor\nc=c1+ tanh\u00101\n2r\u0010\n2\u000bcz+ log(a\nb)\u00112\n+ 4\u000b2(c2x+c2y)\u00112\u000b(c2\nx+c2\ny+c2\nz) +czlog(a\nb)r\u0010\n2\u000bcz+ log(a\nb)\u00112\n+ 4\u000b2(c2x+c2y):\n(129)\nThis equation is monotonic in \u000band therefore it is uniquely speci\fed by the value of c. Ultimately this\nis a consequence from the concavity of the entropy. The proof of (129)'s monotonicity is below:\n19Proof: For ^\u001ato be Hermitian, ^Cis Hermitian and \u000e\u0015=1\n2p\nf(\u000b) is real. Further more, because \u000e\u0015is\nrealf(\u000b)\u00150 and thus \u000e\u0015\u00150. Because f(\u000b) is quadratic in \u000band positive, it may be written in vertex\nform,\nf(\u000b) =a(\u000b\u0000h)2+k; (130)\nwherea>0,k\u00150, and (h;k) are the (x;y) coordinates of the minimum of f(\u000b). Notice that the form\nof (129) is,\nF(\u000b) =tanh(1\n2p\nf(\u000b))p\nf(\u000b)\u0002@f(\u000b)\n@\u000b: (131)\nMaking the change of variables \u000b0=\u000b\u0000hcenters the function such that f(\u000b0) =f(\u0000\u000b0) is symmetric\nabout\u000b0= 0. We can then write,\nF(\u000b0) =tanh(1\n2p\nf(\u000b0))p\nf(\u000b0)\u00022a\u000b0; (132)\nwhere the derivative has been computed. Because f(\u000b0) is a positive, symmetric, and monotonically\nincreasing on the (symmetric) half-plane (for \u000b0greater than or less that zero), S(\u000b0)\u0011tanh(1\n2p\nf(\u000b0))p\nf(\u000b0)is\nalso positive and symmetric, but it is unclear whether or not S(\u000b) is also monotonic in the half-plane.\nWe may restate\nF(\u000b0) =S(\u000b0)\u00022a\u000b0: (133)\nWe are now in a decent position to preform the derivate test for monotonic functions:\n@\n@\u000b0F(\u000b0) = 2aS(\u000b0) + 2a\u000b0@\n@\u000b0S(\u000b0)\n= 2aS(\u000b0)\u0010\n1\u0000a\u000b02\na\u000b02+k\u0011\n+aa\u000b02\na\u000b02+k \n1\u0000tanh2(1\n2p\na\u000b02+k)!\n\u00152aS(\u000b0)\u0010\n1\u0000a(\u000b0)2\na\u000b02+k\u0011\n\u00150\n(134)\nbecausea;k;S (\u000b0), and thereforea\u000b02\na\u000b02+kare all>0. The function of interest F(\u000b0) is therefore monotonic\nfor all\u000b0, and therefore it is monotonic for all \u000b, completing the proof that there exists a unique real\nLagrange multiplier \u000bin (129).\nAlthough (129) is monotonic in \u000bit is seemingly a transcendental equation. This can be solved\ngraphically for the given values c;c1;cx;cy;cz, i.e. given the Hermitian matrix and its expectation value\nare speci\fed. Equation (129) and the eigenvalues take a simpler form when a=b=1\n2, because in this\ninstance ^'/^1 and commutes universally so it may be factored out of the exponential in (121).\n20"    },    {        "title": "1710.10458v2.Universality_of_Quantum_Information_in_Chaotic_CFTs.pdf",        "content": "MIT-CTP/4956\nUniversality of Quantum Information in Chaotic\nCFTs\nNima Lashkaria, Anatoly Dymarskyb, and Hong Liua\naCenter for Theoretical Physics, Massachusetts Institute of Technology\n77 Massachusetts Avenue, Cambridge, MA 02139, USA\nbDepartment of Physics and Astronomy, University of Kentucky,\nLexington, KY 40506, USA\nSkolkovo Institute of Science and Technology, Skolkovo Innovation Center,\nMoscow 143026 Russia\nAbstract\nWe study the Eigenstate Thermalization Hypothesis (ETH) in chaotic con-\nformal \feld theories (CFTs) of arbitrary dimensions. Assuming local ETH, we\ncompute the reduced density matrix of a ball-shaped subsystem of \fnite size\nin the in\fnite volume limit when the full system is an energy eigenstate. This\nreduced density matrix is close in trace distance to a density matrix, to which\nwe refer as the ETH density matrix , that is independent of all the details of an\neigenstate except its energy and charges under global symmetries. In two dimen-\nsions, the ETH density matrix is universal for all theories with the same value of\ncentral charge. We argue that the ETH density matrix is close in trace distance\nto the reduced density matrix of the (micro)canonical ensemble. We support the\nargument in higher dimensions by comparing the Von Neumann entropy of the\nETH density matrix with the entropy of a black hole in holographic systems in\nthe low temperature limit. Finally, we generalize our analysis to the coherent\nstates with energy density that varies slowly in space, and show that locally such\nstates are well described by the ETH density matrix.\nlashkari@mit :edu;a:dymarsky@uky :edu;hong liu@mit:eduarXiv:1710.10458v2  [hep-th]  13 Jun 20181 Introduction and outline\nQuantum information plays an increasingly important role in our understanding and\ncharacterization of quantum matter. The holographic duality together with the black\nhole information loss paradox give strong hints that quantum information is also likely\nto play a central role in our understanding of quantum gravity and the emergence of\nspacetime.\nIn this paper, we discuss the quantum information properties of chaotic conformal\n\feld theories (CFTs) expanding on the observations made in an earlier paper [1].\nWe provide evidence that the quantum information content of highly excited energy\neigenstates of in conformal theories exhibit a great degree of universality.\nWede\fne chaotic quantum \feld theories (QFT) to be those satisfying a local version\nof the Eigenstate Thermalization Hypothesis (ETH) [1] (see [2, 3] for ETH in generic\nquantum systems including density matrix formulation [4, 5]). More explicitly, we say\nthat a QFT on a homogenous compact space satis\fes local ETH if for a local operator\nOp(withplabeling di\u000berent operators),\nhEajOpjEbi=Op(E)\u000eab+ \u0001pab; (1)\nwherejEaiis a highly excited energy eigenstate, the diagonal element Op(E) is a\nsmooth function of E=Ea+Eb\n2, and \u0001 pab\u0018e\u0000O(S(E))whereeS(E)is the density of\nstates at energy E. IfjEaihas other quantum numbers associated with other global\nsymmetries, Op(E) can also smoothly depend on those quantum numbers. To simplify\nthe notation, we will suppress such dependence. In case of CFTs, de\fnition of ETH\n(1) will require additional clari\fcations which we explicitly described below.\nThe high-energy eigenstates of a quantum many-body system are, in general, hard\nto access, and until now essentially all discussions of ETH have been limited to direct\nnumerical diagonalizations (for instance see [6]). With the current computational re-\nsources, a direct numerical diagonalization approach to QFT seems unrealistic. In [1],\nwe advocated that CFTs provide an exciting laboratory for exploring the implications\nof ETH and potentially even proving it. In a CFT, due to the state-operator corre-\nspondence, the energy eigenstates can be represented as local operators with de\fnite\nscaling dimensions, and (1) becomes a condition on the operator product expansion\n(OPE) coe\u000ecients. This opens up many powerful analytic tools for studying ETH.\nThe previous studies of ETH in CFTs that have been inspiration for our work are\n[7, 8, 9, 10, 11, 12].\nMore explicitly, consider a ( d+ 1)-dimensional CFT on a d-dimensional sphere Sd\nwith radius L. Since a primary operator and its descendants are algebraically related,\nthe equation (1) written for CFTs should restrict only to the states jEaidual to primary\noperators [1]. In particular, for two-dimensional CFTs, jEaishould correspond to\nVirasoro primary operators.2Without loss of generality, we further restrict to scalar\n2In every two-dimensional CFT there is an in\fnite number of conserved charges associated to the\nKdV hierarchy [13]. As we will discuss later, for a Virasoro primary, all these charges are \fxed in\nterms of the conformal dimension, therefore Op(E) depends only on E.\n1primary operators \t aof dimension ha=EaL. The energy density of the system in\nsuch a state is\n\u000fa=Ea\nLd!d=ha\nLd+1!d; (2)\nwhere!dis the volume of a unit sphere Sd. For a CFT in a thermal state of temperature\nT,\u000fa\u0018dTTd+1wheredTis the normalization of the two-point function of stress tensor\n(91). This motivates us to de\fne the \\thermal\" length scale associated with jEaias\n\u0015T=\u0012\u000fa\ndT\u0013\u00001\nd+1\n\u0018T\u00001: (3)\nIn the thermodynamic limit with L!1 , while keeping energy density \u000fa\fnite,\nand hence a \fnite \u0015T, the scaling dimension hascales with Las\nha=dT!d\u0012L\n\u0015T\u0013d+1\n: (4)\nApplying the conformal transformation that maps the cylinder Sd\u0002RttoRd+1(the\nradial quantization frame) the local ETH condition (1) translates into a statement\nabout the OPE coe\u000ecient Cp\nabmultiplying the operator Opappearing in the expansion\nof two primaries \t aand \ty\nbcorresponding tojEaiandhEbj,\nCpabL\u0000hp=Op(E)\u000eab+ \u0001pab;\nh\ty\nb(1)Op(1)\ta(0)i=Cpab;\n\ta\u0002\ty\nb=X\npCp\nabOp: (5)\nWe raise and lower the pindex ofCp\nabusing the Zamolodchikov metric hOp(1)Op(0)i=\ndp. In the thermodynamic limit, under the assumption that (1) applies for any operator\nOpof dimension hp, which we keep \fxed as Lbecomes large, the equation (5) implies\nthat the OPE coe\u000ecient Cp\nabmust scale with ha!1 as\nCpab=hhp\nd+1a(dT!d)\u0000hp\nd+1\u000eabfp(E) +Rpab: (6)\nHere, the correction term Rpab=Lhp\u0001pab\u0018e\u0000O(hd\nd+1\na)+hp\nd+1loghais exponentially small\ninha, andfp(E) =\u0015hp\nTOp(E) is a smooth dimensionless function of E. Since there are\nno other dimensionfull parameters in the problem, fp(E) then has to be a constant,\nindependent of E, i.e.\nCpab=hhp\nd+1a(dT!d)\u0000hp\nd+1\u000eabfp+Rpab: (7)\nWe stress that the equation (7) encodes the following nontrivial implications of the\nlocal ETH. (i) Operators OpwhoseCpaagrow slower than hhp\nd+1awithhacannot have a\n2B\nl (\u00001) (1)Sd⇥Rt(a)\nB (1)\n(b) (0)(c)L\nB\nXhpOp(E)lhpˆOp(0)Figure 1: (a) The cylinder Sd\u0002Rtframe and the Euclidean path-integral that prepares\nthe the density matrix in the eigenstate corresponding to \t on subsystem B(b) The same\npath-integral in the radial quantization Rd+1conformal frame (c) The path-integral for  ETH\nin the radial quantization frame.\nnon-vanishing expectation value in the thermodynamic limit, while it is impossible for\nthe OPE coe\u000ecient Cp\nabto grow faster than hhp\nd+1aas that would imply thermodynamic\nlimit for such a theory does not exist. (ii) The spectrum of operators Opappearing in\nthe OPE of \t aand \ty\nais independent of speci\fc properties of \t a, and only depends\non its scaling dimension (energy).\nIntegrable systems are expected not to satisfy the local ETH. A simple example is a\ntwo-dimensional free massless boson on a spatial circle. This theory has heavy coherent\nprimary states ei\u000b\u001ej\niwith large dimension h\u000b=j\u000bj2=2\u001d1. The OPE coe\u000ecient of\nthis heavy state with a primary of dimension one, @\u001e, explicitly violates (6) since it\ngrows as\nC@\u001e\nei\u000b\u001e;e\u0000i\u000b\u001e\u0018\u000b\u0018p\nh\u000b; (8)\nwhile from the thermal expectation value of @\u001ewe know that f@\u001eon the right-hand-side\nof (6) is zero.\nNow consider a chaotic CFT in a highly excited energy eigenstate. We focus on\nthe reduced density matrix of a ball-shaped region Bof sizelinside Sdof sizeLand\nconsider the thermodynamic limit L!1 withlkept \fxed. The complement of B\ninside Sdwill be denoted as Bc. It was shown in [1] that the reduced density matrix\n a(B)\u0011TrBcjEaihEajfor the system in state jEaican be well approximated by a\ndensity matrix  ETH(B;E), to which we will refer as an ETH density matrix . ETH\ndepends only on Band energy Ea\njj a(B)\u0000 ETH(B;E =Ea)jj\u0018e\u0000O(S(Ea)); (9)\nwherek\u0001\u0001\u0001k is the trace distance. In particular, it was shown that the ETH density\nmatrix ETH(B;E) can be written as\n ETH(B;E) =X\nhpOp(E)lhp^Op(0); ^Op=UyOpU; (10)\n3whereOpdenotes the family of operators which appear in the OPE of \t aand \ty\na,Op(E)\ndenotes their expectation values (1), and Uis the unitary operator corresponding to\nthe conformal transformation from the Rindler frame to the radial quantization frame;\nsee \fgure 1(c). Equation (10) de\fnes a density matrix on Bas being prepared via a\nEuclidean path-integral over Rd+1with the speci\fed boundary conditions \\above\" and\n\\below\"Bwithin Sdof unit radius, and the sum of local operators on the right hand\nside of (10) inserted at the origin of Rd+1(see \fgure 1). We will see later that the\ndomain of convergence of this sum is \fxed by the conformal symmetry to be in\fnite.\nExpressing Op(E) in terms of constants fpof (7), we \fnd that (10) is an expansion\ninl\n\u0015T\n ETH(B;E) =X\npfp\u0012l\n\u0015T\u0013hp\n^Op(0): (11)\nIn the low temperature regimel\n\u0015T\u001c1, it is enough to keep the \frst few terms while\nin the high temperature limitl\n\u0015T!1 one has to sum the whole series, which should\nbe convergent for any large but \fnite l=\u0015T.\nIn this paper, we \frst give a general argument that the ETH density matrix (11)\nis close in trace distance to the reduced density matrix of a thermal state (there are\nsubtleties in 2d). Thus, by denoting the set of primary (quasi-primary in 2d) opera-\ntors of a CFT that have non-zero thermal one-point functions by Atherm, we can also\nwrite (11) as\n ETH(B;E) =X\np2Athermfp\u0012l\n\u0015T\u0013hp\n^Op(0): (12)\nAll (quasi-)primary operators that are not in Atherm, and all the descendant \felds drop\nout in the thermodynamic limit from the sum (12). We then discuss in detail the\nstructure of the expansion (11) in the low temperature regime.\nNote that the reduced density matrix in the eigenstate is close to the ETH density\nmatrix (11) (before we discard descendant \felds) with exponential precision in S(E),\nas dictated by local ETH. However, the convergence of the ETH density matrix to the\nreduced thermal state is controlled with corrections that are polynomially supressed in\nS(E), as is the case anytime we compare quantities calculated in the microcanonical\nand the canonical ensembles.\n1. In two dimensions ( d= 1), the only Virasoro primary operator which has non-\nzero thermal value is the identity operator. Therefore, the ETH density matrix\n ETH(B;E) of (10) is solely expressed in terms of the Virasoro descendants of\nidentity, i.e. Op(E) that are the polynomials of stress tensors and their deriva-\ntives. Allfp's that correpond to the quasi-primaries in the Virasoro indentity\nblock are \fxed by the Virasoro algebra, and hence are independent of any spe-\nci\fc properties of the 2d CFT except for the value of the central charge. The ETH\ndensity matrix in 2d is universal across all CFTs with the given value of central\ncharge, thus we refer to it as the universal density matrix . We argue that if (1)\nholds for Virasoro primaries, the subsystem density matrix in the eigenstate is\n4well approximated by the universal density matrix. Furthermore, we argue that\nthe universal density matrix in the thermodynamic limit is close to the reduced\nGeneralized Gibbs Ensemble (GGE) provided we can map all their conserved\ncharges. That is to say\n univ=1\nZtrBc\u0000\ne\u0000\fH+P\ni\u0016iQi\u0001\n+O(1=p\nL); (13)\nwhere the inverse temperature \fand the charges \u0016iare chosen such that the GGE\nhas the same value of Qicharges as the universal density matrix. The conserved\nchargesQiare the in\fnite set of Korteweg-de Vries integrals of motions in two-\ndimensional CFTs [13]. Due to the complexity of evaluating the expectation\nvalues ofQiin the GGE, we are not able to provide a direct support for (13) at\nthis point. Note that CFT formulation of ETH does not require (13) to hold.\nThe equation (13) should hold if we further assume that one can solve for \u0016isuch\nthat the GGE has the same values of charges Qias the pure state.\nIn the limit that the central charge cgoes to1, we show that all the \u0016i= 0\nand the universal density matrix becomes close in trace distance to the standard\nGibbs state. This is consistent with previous results of [7, 11].3\n2. In higher dimensional CFTs, in general, the polynomials of the stress tensor do\nnot exist in the spectrum as primary operators. Furthermore, the conformal\nsymmetry is a lot less restrictive than 2d, and any primary operator can have\nnonzeroOp(E). It is natural to expect, and we provide further support in section\n2.2, that (11) sums into the standard thermal ensemble\n ETH=1\nZtrBc\u0000\ne\u0000\fH\u0001\n+O(1=p\nL); (14)\nwhere the inverse temperature \fis again chosen such that the thermal density\nmatrix has the same energy Eas the ETH density matrix. We provide support\nfor (14) by computing the entanglement entropy of the ETH density matrix to the\norder (l=\u0015T)2(d+1)and matching the answer with the holographic entanglement\nentropy of the same subsystem as computed with the Ryu-Takayanagi formula in\na black hole background. Note that up to this order, the entanglement entropy\nexhibits universality and depends only on the energy density and dT, the two-\npoint function of stress tensor. That is why one can match the answer with\nholography.\nThe plan of the paper is as follows. In Sec. 2 we give a general discussion of the\nrelation between the ETH density matrix and that of a thermal state. In Sec. 3 we\ndiscuss the structure of the ETH density matrix for a two dimensional CFT in detail.\nIn Sec. 4 we study the subsystem ETH in CFTs of dimensions larger than two. In\n3As we explain in detail in section 3 equivalence of  ETH and the reduced Gibbs state does not\nimply that corresponding higher Renyi entropies for n > 1 would have to match, and we \fnd that\nthey, indeed, do not match.\n5Sec. 5 we consider states that have spatial and time dependence at scales much larger\nthan the subsystem size and show that the same universal density matrix remains a\ngood approximation to describe local physics.\n2 ETH density matrix and thermal states\nWe start with a brief discussion of various thermal ensembles for CFTs. The goal is\nto show that local ETH (1) implies that the expectation values of Opin eigenstates as\nde\fned in (1) coincide with the thermal averages. This enables us to show that the\nreduced density matrix of an energy eigenstate is close in trace distance to those of\nvarious thermal ensembles.\n2.1 Di\u000berent ensembles\nConsider a QFT with a number of global symmetries living on a sphere. The micro-\ncanonical ensemble \u001amicro(E0;f\u000bg) is de\fned as an equal-weight average over all energy\neigenstates lying within a narrow band around E0with a given set of quantum numbers\nf\u000bgunder various global symmetries,\n\u001amicro(E0;f\u000bg) =1\nNX\nE2(E0\u0000\u0001;E0+\u0001);givenf\u000bgjE;f\u000bgihE;f\u000bgj: (15)\nAs always, we choose the energy band width \u0001 to be much larger than the average\nlevel spacing that scales like exp( \u0000O(Ld)), but much smaller than the typical energy\nscales of interest. Here, Nis the total number of states in the band. The density\nmatrix of the canonical ensemble is\n\u001acan(\f;f\u000bg) =1\nZf\u000bge\u0000\fHPf\u000bg; Zf\u000bg= TrPf\u000bge\u0000\fH(16)\nwherePf\u000bgdenotes projection into the subspace of the Hilbert space with given f\u000bg.\nThe grand canonical density matrix is de\fned as\n\u001agrand(\f;f\u0016g) =1\nZf\u0016ge\u0000\fH\u0000P\ni\u0016iQi; Zf\u0016g= Tre\u0000\fH\u0000P\ni\u0016iQi(17)\nwhereQidenote the complete set of commuting charges and f\u0016gdenotes the collection\nof the corresponding chemical potentials.\nFor a general quantum \feld theory, in the thermodynamical limit, for a local op-\neratorOwhose quantum numbers we keep \fxed as the volume goes to in\fnity, the\nmicrocanonical, canonical, and grand canonical averages are all equivalent by the stan-\ndard arguments, provided that one chooses \fandf\u0016gto give the average energy E0and\nthe average charges f\u000bg. For example, the micro-canonical and the canonical ensemble\nwhich average over rotationally-invariant states (i.e. with J2= 0 whereJ2denotes the\n6Casimir operator of the rotation group) are equivalent to the grand canonical ensemble\nwith the corresponding \u0016i= 0.\nThe equivalence of ensembles in conformal \feld theory is more intricate since the\nrepresentations of a conformal group are in\fnite dimensional. Furthermore, the states\nwhich lie in the same representation of the conformal group in general do not have the\nsame energy. Let us \frst consider a CFT in d>2. In this case, the conformal group\nis the higher dimensional Mobius transformations, and there are no new conserved\ncharges beyond the generators of the conformal transformations. For convenience,\nlet us introduce ^ \u001a(0)\nmicro(E0;fJ2= 0g) as the (un-normalized) microcanonical density\nmatrix of scalar primaries with energies in a narrow band around E0, where one sums\nover only the energy eigenstates which are scalar primaries. Similarly we can de\fne\n^\u001a(n)\nmicro(E0;fJ2= 0g) to be the ensemble of states that descend at level nfrom primary\nstates of energy E0. A state in the subspace de\fned by ^ \u001a(n)\nmicro(E0;fJ2= 0g) has energy\napproximately equal to E0+O(n\nL). The standard microcanonical ensemble can then\nbe expressed as\n\u001amicro(E0;fJ2= 0g) =1\nNX\nn\u001a(n)\nmicro\u0010\nE0\u0000O\u0010n\nL\u0011\n;fJ2= 0g\u0011\n; (18)\nwhereNis the total number of states at energy E0including both primaries and\ndescendants.\nNow, we consider the thermodynamic limit that is L!1 withE0=Ld\fxed. In\nthis limit, from (1) we have that for any nwhich does not scale with L\nhE0jOjE0i=D\nE0\u0000O\u0010n\nL\u0011\njOjE0\u0000O\u0010n\nL\u0011E\n+O(L\u00001);=D\nE(n)\n0jOjE(n)\n0E\n+O(L\u00001)\n(19)\nwherejE0idenotes a primary state while jE(n)\n0idenotes an n-th level descendant state\nof a primary state of approximate energy E0\u0000O(n\nL); see [1]. The density of states\ngrows exponentially with energy\nlog \n(E)\u0018O(E\u000b) 0<\u000b< 1:\nThe contribution of states in (18) with nscaling asLor larger, is exponentially sup-\npressed compared to the contribution of those with n= 0; hence we neglect such states.\nWe conclude that in the thermodynamic limit for any local operator\nhE0jOjE0i= Tr\u0000\nO\u001amicro(E0;fJ2= 0g)\u0001\n+O(L\u00001) (20)\nand will also be the same as in the canonical and grand canonical ensembles.\nA CFT ind= 2 has an in\fnite number of conserved charges that commute with\nbothL0and \u0016L0. This is the KdV hierarchy of charges fQ2k+1;\u0016Q2k+1; k= 1;2;\u0001\u0001\u0001g.\nHere, the corresponding microcanonical and canonical ensembles are denoted as\n\u001amicro(E0;fQ2k+1;\u0016Q2k+1g); \u001a canonical (\f;fQ2k+1;\u0016Q2k+1g) (21)\n7and the corresponding grand canonical ensemble is the so-called Generalized Gibbs\nEnsemble (GGE)\n\u001aGGE(\f;f\u00162k+1;\u0016\u00162k+1g) =e\u0000\f(L0+\u0016L0)\u0000P\nk\u00162k+1Q2k+1\u0000P\nk\u0016\u00162k+1\u0016Q2k+1\nZ: (22)\nAgain,\u001amicro(E0;fQ2k+1;\u0016Q2k+1g) contains descendant states. By descendants we are\nnow referring to Virasoro descendants. Following the same arguments as above we\nconclude that\n\nE0;fQ2k+1;\u0016Q2k+1gjOjE0;fQ2k+1;\u0016Q2k+1g\u000b\n= Tr\u0000\nO\u001amicro(E0;fQ2k+1;\u0016Q2k+1g)\u0001\n:\n(23)\nThe same holds also for the canonical ensemble and the GGE, provided we assume an\nappropriate growth of the density of states \n as a function of Q.\n2.2 Equivalence of reduced density matrices\nWe now present a general argument showing that given (20), the reduced density matrix\nfor a region BofjE0ihE0j, and the ETH density matrix  ETHare close in trace distance\nto the reduced state \u001aof the subsystem Bof a thermal state (the two-dimensional case\nis di\u000berent and will be discussed in more depth in section 4). The argument works for\nany of the three ensembles mentioned earlier.\nThe reduced density matrix of a region Bis a map from the observables living on\nBto the expectation values. In conformal \feld theory, if Bis a topologically-trivial\nregion the set of local operators on Bprovide a basis for all operators in B. One can\ncompute the expectation value of a k-point function of operators local in the subsystem\nBin a reduced state such as \u001aor ETHby successively applying OPEs to reduce the\nk-point function to a one-point function. This is possible because neither \u001anor ETH\nhave any operator insertions in their corresponding Euclidean path-integrals that limits\nthe domain of the convergence of OPEs on the subsystem.\nConsider any two reduced density matrices \u001aand\u001bwhose Euclidean path-integral\nde\fnitions do not involve any operator insertions that limits the subsystem OPE. We\nwill now show that \u001a=\u001bif and only if they have the same expectation value for all\nthe local operators. The proof is a simple application of the Pinsker inequality:\nk\u001a\u0000\u001bk2\u00141\n2(S(\u001ak\u001b) +S(\u001bk\u001a)) = Tr ((\u001a\u0000\u001b)(K\u001b\u0000K\u001a)) (24)\nwhereK\u001aandK\u001bdenote the modular operators for \u001aand\u001b, respectively. The modular\noperators of both \u001aand\u001bcan be expanded as\nK=X\nplhp\u0000(d\u00001)Z\nxfp(x)Op(x) +X\np;qlhp+hq\u00002(d\u00001)Z\nx;yfp;q(x;y)Op(x)Oq(y) +\u0001\u0001\u0001(25)\nwherepsums over the set of all local operators. We can use the OPEs of operators\nin conformal \feld theory to reduce the expression above to an in\fnite sum over local\n8operators\nK=X\nplhp\u0000(d\u00001)Z\nx~fp(x)Op(x): (26)\nFrom (24) it then follows that if all the one-point functions of local operators match\nthen the density matrices are the same. Now, imagine that the two density matrices\nhave matching one-point functions of local operators up to precision \u000f\u001c1:\nTr ((\u001a\u0000\u001b)Op) =\u000fO\u001a;\u001b(p) (27)\nThen, from the analysis above, we claim that the relative entropy is order \u000f, which\nimplies that the density matrices are close. One might worry that the sum over in\fnite\nterms (the coe\u000ecient of \u000f) can diverge. In this case the relative entropy will diverge\nwhich implies that \u001aand\u001bhave support on unequal subspaces in the Hilbert space.\nHowever, in a continuum \feld theory we believe that all \fnitely excited energy density\nmatrices are full rank.4\nIn our case, we are comparing  ETHwith the reduced state of a thermal density\nmatrix. From (20), the one-point functions of local operators in these two states match\nup to volume suppressed corrections \u000f\u00181=L. We thus conclude that the states are\nclose in trace distance up to volume suppressed corrections.\n3 Two dimensional CFTs\nIn this section, we explore the structure of  ETH(11) for a general two-dimensional\nCFT. We show that it is universal across all CFTs of the same central charge. That is\nto say that the density matrix is comprised of only the polynomials of the stress tensor\n4If a density matrix is not full rank it means that the state where it was reduced from can be\nkilled by a local operator with support only on the subsystem, that is the projector to the eigenvector\nwith eigenvalue zero. This violates the \\separating\" property of the states of a von Neumann algebra.\nIn the algebraic formulation of quantum \feld theory, the states are often chosen to be cyclic and\nseparating [14].\nB\n (\u00001)Sd⇥Rt(a)\nB(b) (0)\nB\u00001(c) (\u0000sin✓0)\n †(1) †(1) †(sin✓0)\n✓0\nFigure 2: (a) The cylinder Sd\u0002Rtconformal frame (b) The radial quantization Rd+1con-\nformal frame. (c) The Rindler frame: the conformal frame convenient for the study of the\ndensity matrix on subsystem B.\n9and the derivative operator, and thus does not depend on any speci\fc structure of a\nCFT other than the central charge. The ETH density matrix (  ETH) enables us to\ncompute the Renyi and entanglement entropies for primary energy eigenstate. In next\nsection, we will compare these quantities with those of a generalized Gibbs ensemble.\n3.1 Universal reduced density matrix\nConsider a two-dimensional CFT on S1\u0002Rt, where the circle has radius L, in an energy\neigenstatej iof energyE. We take the subsystem Bto be an interval of length 2 l.\nWe will work with a Euclidean time and it is convenient to use complex coordinates\nw=t+i\u001bwith\u001b2[0;2\u0019L]. In radial quantization, with z=ew\nL,j iandh jare\nmapped to operators \t(0) and \ty(+1) of dimension h=EL, andBis on the unit\ncircle between\u0000\u00120and\u00120with\u00120=l\nL. The energy density is\n\u000f=E\n2\u0019L=h\n2\u0019L2: (28)\nIn the thermodynamic limit we take L!1 withland\u000f\fxed, and thus h/L2!1 .\nWe de\fne the thermal length as\n\u0015T=\u0012h jT00j i\ndT\u0013\u00001=2\n=\u00122\u0019h\ncL2\u0013\u00001=2\n; (29)\ndT= 2hT00T00i=c\n2\u00192\nwhereT00=1\n2\u0019(T+\u0016T).\nA convenient conformal frame to study the reduced density matrix of Bis the\nRindler frame in which the subsystem is mapped to the negative half-line see \fgure 2:\n!=z\u0000q\nqz\u00001; q =ei\u00120\ndzd\u0016z= \n(!)\u0016\n(\u0016!)d!d\u0016!; \n(!) =(q2\u00001)\n(q!\u00001)2: (30)\nThe operators \t(0) and \ty(+1) are mapped to !\u0000=qand!+=q\u00001, respectively.\nThis is the two dimension version of the map written introduced in [1]; see Appendix\nA. The key observation of [1] is that in the thermodynamical limit, where we take\nL!1 and keepl\fxed,!\u0006!1 and!\u0000\u0000!+= 2isin\u00120!0. The insertions of \t\nand \tycan then be replaced by their OPEs, and the reduced density matrix for region\nBin the Rindler frame can be written as5\n~ = \t(!\u0000;\u0016!\u0000)\t(!+;\u0016!+) =X\npX\nm;n\u00150(!\u0000\u0000!+)hp+m(\u0016!\u0000\u0000\u0016!+)\u0016hp+nCp;\u0016p;m;n\n\t\t@m\u0016@nOp\n(31)\n5We use tilde to denote density matrices in !coordinates: ~ =Uy UwhereUis the unitary that\nimplements the conformal transformation.\n10whereOpis a quasi-primary of dimension ( hp;\u0016hp). It should be understood that\n@m\u0016@nOpis inserted at != 1 which we have suppressed.\nThe expression (31) can be further simpli\fed with the following two observations:\n1. The ratios of the OPE coe\u000ecients\nCp;\u0016p;m;n\n\t;\t\nCp;\u0016p;0;0\n\t;\t(32)\nis \fnite (see also Appendix B for explicit expressions). Thus, in the thermody-\nnamic limit the operators with spatial derivatives are 1 =Lsuppressed as they are\nmultiplied with extra powers of ( !\u0000\u0000!+)m(\u0016!\u0000\u0000\u0016!+)n!0 form;n > 0. We\ncan keep only the terms with m=n= 0.\n2. From (5) the OPE coe\u000ecient for quasi-primary Op;\u0016pis given by\nCp;\u0016p\n\t;\t=L(hp+\u0016hp)\ndp;\u0016pOp;\u0016p(E) (33)\nwhere we have now allowed an arbitrary normalization factor dp;\u0016pfor two-point\nfunction ofOp;\u0016p. We then have\n(!\u0000\u0000!+)hp(\u0016!\u0000\u0000\u0016!+)\u0016hpCp;\u0016p\n\t\t=ihp\u0000\u0016hp(2l)hp+\u0016hp\ndp;\u0016pOp;\u0016p(E) (34)\nwhere we have used that in the thermodynamic limit 2 sin \u00120L= 2\u00120L= 2l.\nLocal ETH implies that Op;\u0016pis, up to corrections suppressed in L, the same\nas the one-point function in the canonical ensemble. The thermal one-point\nfunctions of quasi-primaries which are outside the identity Virasoro block vanish\nin theL!1 limit as they can be mapped to one-point functions on a complex\nplane.6This implies that the contribution of any operator outside of the identity\nVirasoro block vanishes.\nWe thus conclude that\n~ 'X\n(p;\u0016p)2Viraosoro identitiy blockihp\u0000\u0016hp(2l)hp+\u0016hp\ndp;\u0016pOp;\u0016p(E)OpO\u0016p: (35)\nThe Virasoro algebra \fxes the dimensions of the operators in the above sum to positive\nintegers. We can organize the sum (35) in terms of quasi-primaries of dimension kand\u0016k\nconstructed from the holomorphic (anti-holomorphic) stress tensor and its derivatives.\nMore explicitly,Opin (35) are given by T(\u000b)\nk's which can be schematically written as7\nT(\u000b)\nk=X\nk1+k2=kc(\u000b)\nk1k2@k1Tk2(36)\n6In fact, one can compute the one-point function of primaries on a torus with the modular parameter\n\f=L\u001c1, and see that the \fnite-size corrections are exponentially suppressed in volume, see Appendix\nD\n7The expression below should be understood as summing over di\u000berent ways the derivatives are\ndistributed among T's.\n11and satis\fes the quasi-primary constraint ( Lndenote the Virasoro operators)\nL1T(\u000b)\nk= 0: (37)\nAt any positive integer kthere are several linearly independent T(\u000b)\nkthat solve the above\nquasi-primary constraint, which are labeled by index \u000b. We show in Appendix C, for\nkeven (odd) only one (none) of them survives the thermodynamic limit which is the\none with the Tkterm in it. We take \u000b= 0 to be the surviving quasi-primary at each\nlevel. The same holds for the anti-holomorphic OPE coe\u000ecients. Then (35) becomes\n~ 'X\nk;\u0016k2Nik\u0000\u0016k(2l)k+\u0016k\nd2kd2\u0016kOk;\u0016k(E)T(0)\n2k\u0016T(0)\n2\u0016k(38)\nwhere\nOk;\u0016k(E) =h jT(0)\n2kT(0)\n2\u0016kj i;hT(0)\n2k(z)T(0)\n2ki=d2k\njzj4k: (39)\nOperatorT(0)\n2kis a polynomial of order kin holomorphic stress tensor Tthat starts\nwithTk\u0011(T(T:::(TT))). The \frst few T(0)\n2kare computed in Appendix C:\nT(0)\n2=T;T(0)\n4= (TT)\u00003\n10@2T\nT(0)\n6= (T(TT)) +9(14c+ 43)\n2(70c+ 29)(@T@T )\u00003(42c+ 67)\n4(70c+ 29)@2(TT)\u0000(22c+ 41)\n8(70c+ 29)@4T\nd2=c\n2; d 4=c(5c+ 22)\n10; d 6=3c(2c\u00001)(5c+ 22)(7c+ 68)\n4(70c+ 29): (40)\nFor largeh, we have\nh jT(0)\n2kj i'h jTkj i'L\u00002k(L\u00002)kh\t\ti\nh\t\ti= (h=L2)k=\u0012c\n2\u0019\u00152\nT\u0013k\n(41)\nwhere we have used (29) and all the other terms in T(0)\n2kare suppressed in h:\nh j@mTj i\nh jT1+m=2j i\u0018h\u0000m=2\u001c1: (42)\nWe thus \fnd that\n~ 'X\nk;\u0016k2Nik\u0000\u0016k\u00122lp\n2\u0019\u0015T\u00132(k+\u0016k)ck+\u0016k\nd2kd2\u0016kT(0)\n2k\u0016T(0)\n2\u0016k: (43)\nThe set of thermodynamically relevant observables are those with non-vanishing ex-\npectation value in j i. From the local ETH we know that this set does not include\nany operator outside of the Virasoro identity block. The translation-invariance of j i\n12further implies that among the operators in the identity block only quasi-primaries\nhave a chance of having a non-zero expectation value, because the descendants of\nquasi-primaries have the derivative operator which are suppressed by 1 =L. The quasi-\nprimaries of dimension kcan be organized in the orthonormal basis introduced in\nappendix C. Since only T2kappear in the universal density matrix ~ , they are the only\nquasi-primaries with non-vanishing expectation value in j i.\nTo conclude this subsection we stress that the reduced density matrix (43) is uni-\nversal across all two-dimensional CFTs.\n3.2 Renyi entropies\nRenyi entropies are invariant under unitary transformations. Hence, we can directly\ncompute them in the Rindler conformal frame. The n-th Renyi entropy of a spinless\nquasi-primary state ( h=\u0016h) is given by the Euclidean path-integral over an n-sheeted\ncomplex plane with 2 noperators inserted at qandq\u00001on each sheet.8This manifold\nis topologically a Riemann sphere, and can be uniformized to one sheet using the map\nz=!1=n. Then,\n\u0001Sn( ;l) =1\n1\u0000nlog \nn\u00004nh hQn\u00001\nj=0\t(zj;n;\u0016zj;n)\t(z0\nj;n;\u0016z0\nj;n)i\nh\t(z0;1;\u0016z0;1)\t(z0\n0;1;\u0016z0\n0;1)in!\n=4nh \n1\u0000nlog \nsin(l\nL)\nnsin(l\nnL)!\n+1\n1\u0000nlog \nhQn\u00001\nj=0\tj(zj;n;\u0016zj;n)\tj(z0\nj;n;\u0016z0\nj;n)i\nQn\u00001\nj=0h\tj(zj;n;\u0016zj;n)\tj(z0\nj;n;\u0016z0\nj;n)i!\n(44)\nwherezj;n=ei(2\u0019j+l=L)=nandz0\nj;n=ei(2\u0019j\u0000l=L)=n. Using the universal OPE of \t in the\nthermodynamic limit we \fnd\n\u0001Sn( ;l) =(n+ 1)c\n12\u0019n(2l=\u0015T)2+1\n1\u0000nlogDnY\nj=1X\nkj;\u0016kj2N\u00124cl2\n2\u0019n2\u00152\nT\u0013kj+\u0016kjT2kj(zj;n)T2\u0016kj(\u0016zj;n)\nd2kjd2\u0016kjE\n:\nFigure 3 illustrates the expansion above. The n-point functions in the vacuum block\nare universal. In appendix F we compute Renyi entropies perturbatively in subsystem\nsize up to order O((l=\u0015T)8) and \fnd\n\u0001Sn( ;l) =(1 +n)c\n12\u0019n(2l=\u0015T)2\u0000(1 +n)c\n120\u00192n(2l=\u0015T)4(n2+ 11)\n12n2\n+(1 +n)c\n630\u00193n(2l=\u0015T)6(4\u0000n2)(n2+ 47)\n144n4\u0000(1 +n)c\n2800n\u00194(2x=\u0015T)8s8(n;c) +\u0001\u0001\u0001 (45)\nwith\ns8(n;c) =88(n2\u00009)(n2\u00004) (n2+ 119) +c(\u000013n6+ 1647n4\u000033927n2+ 58213)\n5184(5c+ 22)n6:\n(46)\n8Due to the Znsymmetry of this correlator one can alternatively compute it using a 4-point\nfunction with twist operators in a Znorbifold theory. This is done in appendix F.\n13···\n      \n=X1jn\n+X1j<ln+X1j<l<qn\n j j j j\n l l\n j j\n l\n l q q+···TKTK1TK2TK2TK1TK3Figure 3: Renyi entropies correspond to 2 n-point function of the operator that creates the\nstate.\nThe authors of [10] computed the Renyi entropies above to the eighth order in the large\nclimit. The equation (45) is consistent with their result.\n3.3 Generalized Gibbs ensembles\nIn this section, we explore the relation between the ETH density matrix  ETHcom-\nputed in the last section with that of a Generalized Gibbs Ensemble (GGE). The\ncomparison of observables in these two states can be used to study distinguishibility of\nthe corresponding density matrices. Due to the complexity of computing the value of\nobservables in a GGE, our comparison is, so far, incomplete. We hope this discussion\ncan set the stage for future investigations of the properties of GGE.\nTwo-dimensional CFTs have an in\fnite number of conserved charges, which are the\nKdV hierarchy of charges fQ2k\u00001;\u0016Q2k\u00001; k= 1;2;\u0001\u0001\u0001g constructed from the polyno-\nmials of stress tensor [13, 15]\nQ2k\u00001=1\n2\u0019iI\nd!J 2k(!);\n[Q2k\u00001;Q2l\u00001] = 0; (47)\nwith the \frst few local currents given by\nJ2=T; J 4= (TT); J 6= (T(TT))\u0000c+ 2\n12(@T@T ): (48)\nOn a cylinder of circumference 2 \u0019Lthe \frst two charges are\nQ1=1\nL\u0010\nL0\u0000c\n24\u0011\nQ3=1\nL3 \n21X\nn=1L\u0000nLn+L2\n0\u0000c+ 2\n12L0+c(5c+ 22)\n2880!\n: (49)\nA Virasoro primary j iis a simultaneous eigenstate of fQ2k\u00001;\u0016Q2k\u00001; k= 1;2;\u0001\u0001\u0001g,\nwith all the eigenvalues fq2k\u00001;\u0016q2k\u00001; k= 1;2;\u0001\u0001\u0001g \fxed in terms of only the conformal\n14dimensionh =EL. For example, the charges associated to Q1andQ3are\nq1=L\u00001\u0010\nh \u0000c\n24\u0011\n; q 3=L\u00003\u0012\nh2\n \u0000c+ 2\n12h +c(5c+ 22)\n2880\u0013\n: (50)\nIn what follows we assume that the hypothesis of local ETH (1) holds for any\nsu\u000eciently excited Virasoro primary \t. As we discussed in Sec. 2, we expect that\n ETHfor the eigenstate j iprepared from \t to be close in trace distance to the reduced\ndensity matrix of the GGE\n\u001aGGE=Z\u00001exp \n\u00001X\nk=1\u0000\n\u00162k\u00001Q2k\u00001+ \u0016\u00162k\u00001\u0016Q2k\u00001\u0001!\n; (51)\nwhere the chemical potentials f\u00162k\u00001;\u0016\u00162k\u00001gare chosen to match the set of charges\nfq2k\u00001;\u0016q2k\u00001gofj i. If correct, (51) would provide a non-trivial consistency check of\nthe local ETH hypothesis. In the thermodynamic limit the KdV charges of a Virasoro\nprimary are easy to compute:\nq1\nL= 2\u0019\u000f;q3\nL= (2\u0019\u000f)2;\u0001\u0001\u0001;q2k\u00001\nL= (2\u0019\u000f)k;\u0001\u0001\u0001 (52)\nwhere\u000fis the energy density.\nTo proceed further, we assume that the central charge cis large. In the c!1\nlimit, all\u00162k\u00001except\u00161=\fvanish; thus we recover the standard Gibbs ensemble\n[16, 11]. To see this, note that in the large climit (see the next subsection for a\nderivation)\n1\nZTr\u0000\nJ2ke\u0000\fH\u0001\n=\u0012\u00192c\n6\f2\u0013k\n: (53)\nThe two-dimensional thermal energy density is\n\u000f(\f) =c\u0019\n6\f2: (54)\nMatching this with the energy density in the eigenstate, using (3) and the de\fnition of\ndTin (29), we \fnd\n\u00152=3\f2\n\u00193: (55)\nIn the next subsection, we use this change of parameters to compare the one-point\nfunctions in the energy eigenstate in (41) with those of the thermal state in (53) in the\nc!1 limit and they match exactly.\nThe reduction of the conventional Gibbs ensemble e\u0000\fH=Zis only matching the\nETH density matrix in the in\fnite central charge limit. The necessity to modify it\nwhen the central charge is \fnite is suggested by the non-zero values of KdV charges\n(52). Historically, \frst indication that the excited primary state is locally di\u000berent from\nthermal state came from the comparison of entanglement and thermal entropies in [10],\nalthough it should be noted that such a discrepancy by itself does not immediately\n15preclude the corresponding density matrices to be trace-distance close [1, 5]. A direct\ncomparison of local observables unambiguously showing that ETH density matrix can\nnot match the canonical one was soon performed in [11], with more analysis probing\n\fnite 1=ccorrections in an attempt to match ETH density matrix with the GGE\none following in [12]. In this paper we further investigate this question. The main\nunresolved challenge here is to compute the expectation value of KdV currents in the\nGGE at \fnite c. Despite the fact that for larger kcorresponding \u00162k\u00001are suppressed\nby the increasingly negative powers of c, we \fnd a strong indication that one cannot\nperform a perturbative analysis by truncating (51) to \fnite number of \u0016's even for the\nnext to the leading order in the 1 =cexpansion (see Appendix G). Hence to complete the\ncheck, one needs a truly non-perturbative expression for (51) both in terms of powers\nand numbers of included \u0016's. We leave this task for a future investigation.\nIn the limit ha\u001dc\u001d1, the universal density matrix in (43) simpli\fes and expo-\nnentiates (see Appendix C)\n~ =e(DaT+\u0016Da\u0016T); a2=(2\u0019L)2\n12\f2(56)\nDa=a2\u0000a4\n10\u00022!@2+11a6\n70\u00024!@4\u00009a8\n140\u00026!@6\u000034a10\n1925\u00028!@8+\u0001\u0001\u0001:(57)\nThis is because at large c\n1\nd2k'2k\nk!ck; (58)\nand we have used the change of parameters in (55). Note that in order to properly\nde\fne the operator ~ one has to smooth out the exponent on a circle of radius \u000faround\nz= 0 where the operator is inserted:\nh~ \u0001\u0001\u0001i=heH\nr=\u000fDaT+\u0016Da\u0016T\u0001\u0001\u0001i (59)\n3.4 Matching with thermal density matrix in the in\fnite c\nlimit\nIn two dimensions, the thermal cylinder is conformally \rat, therefore the expectation\nvalue of any operator that is outside of the Virasoro identity block vanishes in the\nGibbs state. The translation-invariance further restricts the set of observables with\nnon-vanishing thermal one-point function to the quasi-primaries. Below, we show that\nat largecthe thermal expectation value of T2kscales asck, whereas the expectation\nvalue of other quasi-primaries of the same conformal dimension scale with lower powers\nofc.\nThe current J2kis a polynomial of order kin stress tensor, where the normal-ordered\noperator (Tk) = (T(T\u0001\u0001\u0001(TT))) is de\fned by isolating the distance independent term\nin the OPE:\n(AB)(!) = lim\n!1!!(A(!1)B(!)\u0000divergent terms)\n=1\n2\u0019iId!1\n(!\u0000!1)A(!1)B(!); (60)\n16where in the second line the normal ordering is imposed by a Cauchy integral. In a\nthermal state\ntr(\u001a\f(TT)(!)) =1\n2\u0019iI\n!d!1\n!1\u0000!tr(\u001a\fT(!1)T(!))\ntr(\u001a\f(T(TT))(!)) =1\n2\u0019iI\n!I\n!d!1d!2\n(!\u0000!1)(!\u0000!2)tr(\u001a\fT(!1)T(!2)T(!)):(61)\nAt large central charge, the multi-point thermal correlators are dominated by the dis-\nconnected piece:\ntr(\u001a\fT(x1)\u0001\u0001\u0001T(xk)) =tr(\u001a\fT(x))k(1 +O(1=c)) =\u0012\u00192c\n6\f2\u0013k\n(1 +O(1=c)):(62)\nPlugging this in the right hand side of (61), and performing the Cauchy integral we\nobtain\ntr(\u001a\f(Tk)) =\u0012\u00192c\n6\f2\u0013k\n(1 +O(1=c)): (63)\nNow, consider a quasi-primary that is not Tk. The \frst non-trivial such quasi-\nprimary appears at dimension six:\nA= (@T@T )\u00002\n9@2(TT) +1\n42@4T: (64)\nThe normal-ordering is imposed by\ntr(\u001a\fA(!)) =tr(\u001a\f(@T@T )(!)) =1\n2\u0019iI\n!d!1\n!1\u0000!tr(\u001a\f@T(!1)@T(!)) =O(c);(65)\nwhere we have used the fact that the thermal state is translation-invariant in space and\ntime; hence, the disconnected piece of the expectation value on the right hand side is\nzero. The same conclusion applies to all other quasi-primary operators in the Virasoro\nidentity block that are not T2k, as they also can be considered as multi-trace operators\nwith at least one factor containing derivatives. If we rede\fne the stress tensor in the\nlargeclimit according to ~T=T=c, the expectation value of T2kbecome order one,\nwhile the expectation value of any other quasi-primary in the Virasoro identity block\nis suppressed by negative powers of c. Thus, the only operators with non-vanishing\nexpectation values in this limit are T2k.\nThe quasi-primary operators T2kand the KdV charges J2kare both polynomials of\norderkinTand start with ( Tk). The derivative terms are di\u000berent, however, as we\njust discussed the derivative terms are suppressed in large central charge. Therefore,\ntr(\u001a\fT2k) =tr(\u001a\fJ2k) =\u0012\u00192c\n6\f2\u0013k\n(1 +O(1=c)) (66)\nThis is the same answer as the one-point functions in the eigen-state (41) after we\nreplace\f2=\u00193\u00152\nT=3. Since there are no other thermodynamically-relevant observables\n17we have found that all the one-point functions of the eigenestate matches of those of the\nGibbs state in the large central charge limit. Thus, in the large climit we have proved\nthat the universal density matrix of a Virasoro primary eigenstate is indistinguishable\nfrom that of the Gibbs state.\nIt is interesting to compare the Renyi entropies in the thermal state with the eigen-\nstate in the large climit. We can take a large climit in the low temperature expansion\nin (45) the perturbation theory of small x=\u0015T:\n\u0001Sn( ;x) =(1 +n)c\n12n\u0019(2x=\u0015T)2\u0000(1 +n)c\n120n\u00192(n2+ 11)\n12n2(2x=\u0015T)4\n+(1 +n)c\n630n\u00193(4\u0000n2)(n2+ 47)\n144n4(2x=\u0015T)6\u0000(1 +n)c\n2800n\u00194(2x=\u0015T)8s8+\u0001\u0001\u0001 (67)\nwith\ns8=\u000013n6+ 1647n4\u000033927n2+ 58213\n25920n6: (68)\nIt is clear that the Renyi entropies for n>1 do not acquire thermal values given by\n\u0001Sn(\f;x) =(n+ 1)c\n6log\u0012\f\n2\u0019xlog\u00122\u0019x\n\f\u0013\u0013\n=(1 +n)c\n12n\u0019(2x=\u0015T)2\u0000(1 +n)c\n120n\u00192(2x=\u0015T)4+(1 +n)c\n630n\u00193(2x=\u0015T)6+\u0001\u0001\u0001 (69)\nwhere in the second line we have used the change of parameters in (55). This is in\ncontrast with the entanglement entropies of the states that match to the eighth order\nthat we have computed.\nIn the large climit, one can in fact compute the dominant cpiece of the en-\ntanglement entropy of the eigenstate non-perturbatively for \fnite values of l=\u0015T. In\nsection 3.2, we computed the Renyi entropies directly by constructing the partition\nfunction that represents tr(\u001a2) and uniformizing it. An alternative method to compute\nthe Renyi entropy of the eigenstate is computing the four-point function of twist oper-\nators with \tnin an orbifold theory; see (136) of Appenix F. The assumption of local\nETH tells us that only the Virasoro identity block contributes to the correlator\nG4(z;\u0016z) =h\tn(1)\u001bn(z;\u0016z)\u001bn(1)\tn(0)i: (70)\nwherez=eix=L. The leading cpiece of the contribution of the Virasoro identity\nblock to the four point function above in the large climit was found by solving the\nmonodromy equation for nnearn= 1 in [7]:\nlogG4(z;\u0016z)'c(1\u0000n)\n6log\u0012z(1\u0000\u000b )=2\u0016z(1\u0000\u0016\u000b )=2(1\u0000z\u000b )(1\u0000\u0016z\u0016\u000b )\n\u000b \u0016\u000b \u0013\n+O((n\u00001)2)\n\u000b \u0011ir\nh \n24: (71)\nThe entanglement entropy computed this way from the identity block in the large\nclimit matches the entanglement entropy in the Gibbs state for any l=\f. Note that\n18here we are working in the limit where h \u001dc\u001d1, which in the language of [7]\ntranslates to h =\u000bc,c\u001d1 and\u000b\u001d1. In our approach the assumption of local ETH\nguarantees that only the Virasoro identity block dominates. However, the authors of\n[7] assumed a sparse spectrum of low-dimension operators to truncate to the identity\nblock.\n4 Higher dimensional CFTs\nIn this section, we \frst discuss the general structure of the ETH density matrix in higher\ndimensions, and then compute the entanglement entropy to the leading nontrivial order\ninl=\u0015Texpansion. We compare the result to the holographic entanglement entropy\ncomputed using the Ryu-Takayanagi formula at this order and \fnd agreement. The\nintuition is that even though our CFT computation does not assume large Nor strong\ncoupling, at this order the answer is universal because it depends only on dTthat is the\nnormalization of the two-point function of stress tensor. To match the entanglement\nentropies we have to set the coe\u000ecient dTto be (98), as is required in a holographic\nCFT. This provides a consistency check of the local ETH.\n4.1 ETH density matrix\nWe observed that in two dimensions assuming local ETH implies that only the polyno-\nmials of stress tensor propagate in the thermodynamic limit of OPE. Here, we consider\ndensity matrices in primary energy eigenstates of higher-dimensional CFTs satisfying\nlocal ETH. A generalization of the map introduced in (30) (see appendix A) maps the\nradial quantization frame to the Rindler frame. In Rindler coordinates, the subsystem\nBis mapped to the negative half-space X1<0, and the operators that create and an-\nnihilate the state are, respectively, at X\u0000\n\u0016andX+\n\u0016. SinceX\u0006\ni>2= 0 we can use the two-\ndimensional complex coordinates to describe their location: X\u0000\n0+iX\u0000\n1=e\u0000i\u00120= 1=q\nandX+\n0+iX+\n1=ei\u00120=q.9The distance between the two operators in these coordi-\nnates is 2 sin \u00120'2l=L. The operator product expansion in the thermodynamic limit\nl=L!0 becomes\n\t(X+\n\u0016)\t(X\u0000\n\u0016)\nh\t(X+\n\u0016)\t(X\u0000)i'1X\npCp;^n\n  j~ njhpO^n\np(X\u0000\n\u0016) =1X\npf^n\np(l=\u0015T)hpO^n\np(X\u0000\n\u0016)\n(72)\nwhereX+\n\u0016= 2 sin\u00120^n\u0016+X\u0000\n\u0016, ^nis the unit vector in the X0directions, and we have\ndropped the descendant \felds because their contribution is 1 =Lsuppressed. The op-\neratorO^n\npis a primary with spin with its indices contracted with ^ naccording to\nO^n\np= (^n\u00161^n\u00162\u0001\u0001\u0001\u0000 traces) (Op)\u00161\u00162\u0001\u0001\u0001\nCp;^n\n  =h\t(1)O^n\np(1)\t(0)i\nhO^n\np(1)O^n\np(0)ih\t(1)\t(0)i; (73)\n9Note that compared to the two-dimensional map the location of !\u0000and!+are swapped.\n19and \fnally fpis de\fned by\nf^n\np= (2\u0015T=L)hpCp\n  =h jO^n\npj i\ndp;^n(2\u0015T)hp\ndp;^n=hO^n\np(1)O^n\np(0)i: (74)\nIt is customary to de\fne a coe\u000ecient dpthat is independent of ^ nin the following way:\nh(Op)\u00161\u0001\u0001\u0001\u0016m(x\u0016)(Op0)\u00171\u0001\u0001\u0001\u0017m(0)i=dp\u000epp0jxj\u00002hpI\u00161\u0001\u0001\u0001\u0016m;\u00171\u0001\u0001\u0001\u0017m; (75)\nwhere the tensor I\u00161\u0001\u0001\u0001\u0016m;\u00171\u0001\u0001\u0001\u0017mis \fxed by conformal symmetry [17, 18]. Every CFT has\na stress tensor that is a primary of dimensions d+ 1. The energy density in primary\nstatej aiis\n\u000f=E\nLd!d=ha\nLd+1!d; (76)\nwhere!dis the volume of the unit sphere Sd. As an example, consider the term in the\nOPE expansion (72) that corresponds to stress tensor\n\u000f\ndT;\u001c(2l)(d+1)(^n\u0016^n\u0017\u0000\u000e\u0016\u0017=(d+ 1))T\u0016\u0017\n=d+ 1\nd\u00122l\n\u0015T\u0013(d+1)\n(^n\u0016^n\u0017\u0000\u000e\u0016\u0017=(d+ 1))T\u0016\u0017;\n\u0015T=\u0012\u000f\ndT\u0013\u00001=(d+1)\n(77)\nwhere\u0015Tis the length associated with the energy density, and dTis the central charge\nde\fned by the two-point function of stress tensor:\nhT\u0016\u0017(u)T\u000b\f(v)i=dT\nju\u0000vj2(d+1)S\u0016\u0017;\u000b\f(u\u0000v);\nS\u0016\u0017;\u000b\f(u) =1\n2(I\u0016\u000b(u)I\u0017\f(u) + (\u0016$\u0017))\u00001\nd+ 1\u000e\u0016\u0017\u000e\u000b\f\nI\u0016\u0017(u) =\u000e\u0016\u0017\u00002u\u0016u\u0017\njuj2: (78)\nTo obtain the density matrix in the thermodynamic limit we have to study the OPE\nin (72) in more detail. From the equivalence of the microcanonical ensemble and the\nthermal ensemble we expect the coe\u000ecient\nfp'(2\u0015T)hp\ndptr(\u001aTOp) (79)\nto have the interpretation of a thermal one-point function up to volume suppressed\ncorrections, where the thermal state is chosen to have the same energy density as the\n20eigenstatej i. In two dimensions, we saw that thermal one-point functions vanish\nwhich let to a truncation of the OPE to only the Virasoro identity block. However,\nin higher than two dimensions thermal one-point functions do not vanish, and fpare,\npotentially, non-zero.\nOne way to obtain universality in higher dimensions is by restricting the class\nof higher dimensional theories we study; for instance the holographic theories. In\nholographic CFTs the thermal one-point function of conformal primaries are 1 =Nsup-\npressed except for operators constructed from the stress tensor. Tn large NCFTs\nresemble two-dimensional CFTs in the sense that they have multi-trace operators Tm\nin their spectrum that are primaries of conformal dimension m(d+ 1), up to 1 =Ncor-\nrections. In holographic theories the thermal correlator is essentially classical, that is\nto say the thermal variance of the operator Tis 1=Nsuppressed:\ntr(T2\u001aT)\u0000tr(T\u001aT)2=O(1=N) (80)\nTherefore, from local ETH and the equivalence of ensembles one expects CTm\n  \u0018hm\nwhich implies that they survive the thermodynamic limit and contribute to Atherm. In\nholographic theories, Tmare inAtherm and one needs to include them in the sum in\nthe de\fnition of the \\universal\" density matrix.10\n4.2 Entanglement entropy from ETH density matrix\nAs opposed as two-dimensional case, the ETH density matrix in (72) is not universal.\nThat is to say that at \fnite central charge we only know one operator in the set of\nthermodynamically relevant operators Atherm. If we try to repeat our low temperature\nanalysis of the ETH density matrix in d > 2 we need to make further assumptions\nabout the spectrum of the theory.\nLet us assume that there are no relevant primary operators in the set Atherm. In\nother words, we are assuming that fp= 0 for all operators Opin(72) with hp< d.\nThen, to the \frst non-trivial order the ETH density matrix is\ntr(~ \u0001\u0001\u0001)'* \n1 +\u0012d+ 1\nd\u0013\u00122l\n\u0015T\u0013d+1\n^n\u0016^n\u0017T\u0016\u0017+\u0001\u0001\u0001!\n\u0001\u0001\u0001+\n: (81)\nIn a CFT the operator T\u0016\n\u0016= 0 in \rat space. Now, we can compute the entanglement\nentropy of the ETH density matrix at this order and compare it with the reduced\n10At \fnite central charge the only primaries one can construct from Tare large spin operators of\ntype (TT)n;l\u0011T@\u00161\u0001\u0001\u0001@\u0016l(@2)nT. In fact for large lthere are operators of this type for all m2N:\n(Tm)(n1;l1)\u0001\u0001\u0001(nm;lm)= ((TT)n1;l1Tn2;l2)\u0001\u0001\u0001Tn;lm). However, every derivative su\u000bers a 1 =Lsuppression\nand hence one expects their OPE coe\u000ecients to scale, at best, as hmrather than hm+(2n+l)=(d+1)that\nis required to survive the thermodynamic limit. An explicit calculation of the OPE coe\u000ecients C[TT]n;l\n  \ncon\frms this expectation [9]. This calculation is done assuming that the spin is largest parameter.\nHowever, for our case of interest we want the conformal dimension of the operator to be much larger\nthan its spin which is much larger than one. It is plausible that in our limit of interest these operators\nsurvive the thermodynamic limit and contribute to Atherm . We thank Liam Fitzpatrick and Sasha\nZhiboedov for pointing this out to us.\n21density matrix of the Gibbs state. Renyi entropies are unitarily invariant, and it is\nmore convenient to compute the entanglement entropy in Rindler coordinates. The\nvacuum-subtracted Renyi entropy in primary state j iis given by\n\u0001Sn( ;l) =1\n1\u0000nloghQn\nj=1 j ji(2\u0019n)\nh  i(2\u0019)(82)\nwhere the subscript (2 \u0019n) refers to the angle around the boundary of B:X0=X1= 0\nin Rindler space. We denote the generator of rotation around this hypersurface as @\u001c:\nX0+iX1=!; != \u0016!=e2i\u001c: (83)\nWe are interested in entanglement entropy which is found from the n!1 limit of\n\u0001S( ;l) =\u000e(1)S+\u000e(2)S\n\u000e(1)S=\u0000@n\u0014\nnlogh  i(2\u0019n)\nh  i(2\u0019)\u0015\nn!1\n\u000e(2)S=\u0000@nlog\"\nhQn\nj=1 j ji(2\u0019n)\nh  in\n(2\u0019n)#\nn!1: (84)\nOur calculation closely follows the method used in [19], and uses the Hamiltonian\nlanguage:\nh  i(2\u0019n)=tr\u0000\ne\u00002\u0019nHP(  )\u0001\n(85)\nwherePis the path-ordering operator in the Euclidean space. The \frst term in (84)\nis the change in the expectation value of the vacuum modular operator H:\n\u000e(1)S=\u0000@ntr(e\u00002\u0019nHP(  ))\f\f\f\nn!1\nh  i(2\u0019)=2\u0019hH  i(2\u0019)\nh  i(2\u0019)=2\u0019!d\u00001\u000fld+1\nd(d+ 2)=2\u0019!d\u00001dT\nd(d+ 2)\u0012l\n\u0015T\u0013d+1\n!d\u00001=2\u0019d=2\n\u0000(d=2): (86)\nThis is the so-called \frst law of entanglement entropy; for small variation of density\nmatrix\u000eS= 2\u0019\u000eH, whereHis the generator of Euclidean rotation in the \u001cdirection.\nThe second term in (84) is the relative entropy of the eigenstate with respect to the\nvacuum reduced to the subsystem B:S( k\u001b):The task is to compute the relative\nentropy above perturbatively in powers of l=\u0015T.\nSince \t's approach each other pairwise in Rindler space, one can use the \rat space\nOPE. At the next-to-leading order the entanglement entropy is\n\u000e(2)S=(d+ 1)2\nd2\u00122l\n\u0015T\u00132(d+1)\n@n\"\nn\n2n\u00001X\nj=1G00\nn(2\u0019j)#\nn!1\nG\u0016\u0017\nn(2\u0019j) =hT\u0016\u0016(0)T\u0017\u0017(2\u0019j)i(2\u0019n): (87)\n22⌧⌧\nss\n002⇡n2⇡n\n···\n···\nC\nC(a)(b)\nFigure 4: (a) The path-integral over complexi\fed \u001cpicks upnpoles at\u001c= 2\u0019j. (b) The\ncontourCis deformed to run over ( \u00001+i(2\u0019n\u0000\u000f);1+i(2\u0019n\u0000\u000f)) and (1+i\u000f;\u00001+i\u000f) .\nwhere the index 0 signi\fes the X0in Rindler coordinates. We follow the method\nadvocated in [19] to analytically continue the expression above in n:\nA\u0016\u0017(n) =n\u00001X\nj=1G\u0016\u0017\nn(2\u0019j) =Z\nCds\n2\u0019iG\u0016\u0017\nn(\u0000is)\nes\u00001\nwheresis the complexi\fed \u001cangle. The contour Cis deformed to run over ( \u00001+\ni(2\u0019n\u0000\u000f);1+i(2\u0019n\u0000\u000f)) and (1+i\u000f;\u00001+i\u000f); see \fgure (4):\nA\u0016\u0017(n) =Z1\n\u00001ds\n2\u0019i\u0012G\u0016\u0017\nn(\u0000is+\u000f)\nes+i\u000f\u00001\u0000G\u0016\u0017\nn(\u0000is+ 2\u0019n\u0000\u000f)\nes+2\u0019in\u0000i\u000f\u00001\u0013\n(88)\nThe analytic continuation is the choice to set e2\u0019in= 1 in the denominator.\n@nG\u0016\u0017\nn(\u0000is+\u000f)\f\f\nn!1=@ntr\u0002\ne\u00002\u0019nHT\u0016\u0016(0)T\u0017\u0017(s+i\u000f)\u0003\nn!1=\u00002\u0019tr\u0002\ne\u00002\u0019HHT\u0016\u0016(0)T\u0017\u0017(s+i\u000f)\u0003\nand\n@nA(n)\u0016\u0017\f\f\nn!1=iZ1\n\u00001ds\u0014tr(e\u00002\u0019HHT\u0016\u0016(0)T\u0017\u0017(s+i\u000f))\nes+i\u000f\u00001\u0000tr(e\u00002\u0019HHT\u0016\u0016(s\u0000i\u000f)T\u0017\u0017(0))\nes\u0000i\u000f\u00001\u0015\nThe second term can be further simpli\fed using the commutator [ H;T\u0016\u0016(s)] =\u0000idT\u0016\u0016\nds\nand the KMS condition\ntr(e\u00002\u0019HHT\u0016\u0016(s\u0000i\u000f)T\u0017\u0017(0)) =tr(e\u00002\u0019H(T\u0016\u0016(s\u0000i\u000f)H\u0000[H;T\u0016\u0016(s\u0000i\u000f)])T\u0017\u0017(0))\n=tr(e\u00002\u0019HHT\u0016\u0016(0)T\u0017\u0017(s+ 2\u0019i\u0000i\u000f)) +id\ndstr(e\u00002\u0019HT\u0016\u0016(s\u0000i\u000f)T\u0017\u0017(0))\nPutting this back in A(n) gives\n@nA\u0016\u0017(n)\f\f\nn!1=iZ1\n\u00001ds\u0012G\u0016\u0017\n1(\u0000is+\u000f)\nes+i\u000f\u00001\u0000G\u0016\u0017\n1(\u0000is\u0000\u000f)\nes\u0000i\u000f\u00001\u0013\n+Z1\n\u00001ds\nes\u0000i\u000f\u00001d\ndstr(e\u00002\u0019HT\u0016\u0016(s\u0000i\u000f)T\u0017\u0017(0)) (89)\n23The term in the \frst line vanishes since there are no poles in the region encircled by\nthe contour integration. Using integration by parts we can write the second term as\n@nA\u0016\u0017(n)\f\f\nn!1=\u0000Z1\n\u00001ds\n4 sinh2((s\u0000i\u000f)=2)hT\u0016\u0016(Xs)T\u0017\u0017(X0)i (90)\nwhereX0= (1;\u0000is=2;\u0001\u0001\u0001) andXs= (1;is=2;0;\u0001\u0001\u0001) in Rindler coordinates. Therefore,\n@nA\u0016\u0017(n)\f\f\nn!1= (\u00001)d+1Z1\n\u00001dsdT\n(2 sinh(~s=2))2(d+2)S\u0016\u0016;\u0017\u0017\nwhere ~s=s\u0000i\u000fand\nhT\u0016\u0017(u)T\u000b\f(v)i=dT\nju\u0000vj2(d+1)S\u0016\u0017;\u000b\f(u\u0000v);\nS\u0016\u0017;\u000b\f(u) =1\n2(I\u0016\u000b(u)I\u0017\f(u) + (\u0016$\u0017))\u00001\nd+ 1\u000e\u0016\u0017\u000e\u000b\f\nI\u0016\u0017(u) =\u000e\u0016\u0017\u00002u\u0016u\u0017\njuj2(91)\nThen,\n@nA00(n)\f\f\nn!1=dCddT\nd+ 1\nCd= (\u00001)dZ1\n\u00001ds\n(2 sinh(~s=2))2(d+2): (92)\nOne can perform the integral explicitly\nCd=2\n(d+ 2)2F1[2(2 +d);2 +d;3 +d;\u00001] =2\n(d+ 2)\u0000(d+ 3)2\n\u0000(5 + 2d)\nTherefore,\n\u000e(2)S=\u0000(d+ 1)CddT\n2d\u00122l\n\u0015T\u00132(d+1)\n=\u00002(d+ 1)2\u0000(d+ 3)\u0000(d)dT\n2\u0000(5 + 2d)\u00122l\n\u0015T\u00132(d+1)\nNote that here dT=hT00T00i(d+ 1)=d, and ind= 1 we have dT=c=(2\u00192) therefore\n\u000e(2)S=\u00004c\n15\u00192\u0012l\n\u0015T\u00134\n(93)\nwhich is the same as the result we found in two dimensions.\nInd > 2 we do not know the entanglement entropy in the reduced state of the\nGibbs ensemble, \u001aT\nl, however, if the theory is holographic we can compare the result\n24with the prediction of the Ryu-Takayanagi formula. Next, we show that the above\nresult can be reproduced using a gravitational calculation in a black hole background.\nThe calculation of the entanglement entropy of excited states in the small size limit and\nmatching it with the black hole holographic entropy have appeared earlier in a very\nnice paper [20]11. Instead of the Renyi entropy calculation presented here the authors\nuse a replica trick that directly computes the relative entropy [21].\n4.3 Holographic theories\nConsider the thermal state of a holographic CFT in \rat space dual to the planar black\nhole\nds2=L2\nz2\u0012\n\u0000f(z)dt2+d~ x2\nd+dz2\nf(z)\u0013\n; f (z) = 1\u0000zd+1\nzd+1\nh: (94)\nHere,zhis related to the thermal wavelength zh=(d+1)\f\n4\u0019. The entanglement entropy\nof the reduced state on a ball of radius lis the area of an extremal surface in the bulk\nanchoring on the boundary of the subsystem:\nS(\u001aT;l) =LdSd\u00001\n4GZl\n0drrd\u00001\nzds\n1 +(@rz)2\nf(z)(95)\nIt is convenient to switch to the Fe\u000berman-Graham coordinates to compute the entan-\nglement entropy perturbatively in l=\f\u0018l=zh:\nds2=L2\nz2(dz2+g\u0016\u0017(z;x\u0016)dx\u0016dx\u0017);\ng\u0016\u0017(z;x\u0016) =\u0011\u0016\u0017+azd+1T\u0016\u0017+a2z2(d+1)(n1T\u0016\u000bT\u000b\n\u0017+n2\u0011\u0016\u0017T\u000b\fT\u000b\f) +\u0001\u0001\u0001(96)\nwherea=16\u0019G\n(d+1)Ld,n1= 1=2 andn2=\u00001\n8d. The bulk Ricci tensor written in these\ncoordinates with \u001a=z2=L2(dimensionless) is\nR\u001a\u001a=\u0000d\n\u001a2\u00001\n2g\u0016\u0017g00\n\u0016\u0017+1\n4(g\u0016\u0016)2(g0\n\u0016\u0016)2\nL2R\u0016\u0016=\u00002\u001ag00\n\u0016\u0016+ 2\u001ag\u0016\u0016(g0\n\u0016\u0016)2\u0000\u001ag0\n\u0016\u0016g\u0017\u0017g0\n\u0017\u0017+ (d\u00002)g0\n\u0016\u0016+g\u0016\u0016g\u0017\u0017g0\n\u0017\u0017\u0000d\n\u001ag\u0016\u0016:\nPerturbatively in lwe \fnd that the vacuum subtracted entropy is [22]12\n\u000e(1)S=2\u0019!d\u00001T00ld+1\nd(d+ 2)=2\u0019!d\u00001dT\nd(d+ 2)\u0012l\n\u0015T\u0013d+1\n\u000e(2)S=\u0000\u00193=2(d+ 1)!d\u00001\u0000(d)\n2d+2(d+ 2)\u0000(d+ 5=2)\u00128\u0019G\nLd\u0013\nT2\n00l2(d+1)\n=\u0000\u00193=2(d+ 1)!d\u00001\u0000(d)\n2d+2(d+ 2)\u0000(d+ 5=2)\u00128\u0019GdT\nLd\u0013\ndT\u0012l\n\u0015T\u00132(d+1)\n(97)\n11We thank G. Sarosi and T. Ugajin for bringing their paper to our attention.\n12Note that there is a typo in equation (3.55) of that paper.\n25where we have used T00=dT\n\u0015d+1\nTand!d=2\u0019(d+1)=2\n\u0000((d+1)=2). The \frst term is simply the \frst\nlaw of entanglement entropy. The quantityLd\n8\u0019Gis related to the two-point function of\nstress tensor as:\ndT=d+ 2\nd\u0000(d+ 2)\n\u0019(d+1)=2\u0000((d+ 1)=2)Ld\n8\u0019G: (98)\nPlugging this back in (97) gives\n\u000e(2)S=\u0000(d+ 1)2\u0000(d+ 3)\u0000(d)dT\n\u0000(2d+ 5)\u00122l\n\u0015T\u00132(d+1)\n(99)\nThis is exactly the answer we found in the \feld theory in (93) for the entanglement\nentropy of the universal density matrix in arbitrary dimension d.\nIf the local ETH hypothesis is correct in holographic CFTs, the reduced density\nmatrix in any energy eigenstate is well approximated by the ETH density matrix (81).\nAccording to holography, the gravity dual of a heavy energy eigenstate is a black\nhole of the same energy density. Therefore, if the local ETH holds the entanglement\nentropy of the ETH density matrix should match the entanglement entropy computed\nholographically in the dual black hole geometry. In this section, we checked that in\nthe same temperature limit l=\u0015T\u001c1, indeed, the local ETH hypothesis passes this\nconsistency check.\n5 Local equilibrium\nUp to this point we were only concerned with the eigenstate thermalization hypothesis.\nWe showed that the reduced density matrix of small subsystems in energy eigenstates\nare universal. Energy eigenstates are highly \fne tuned and that their time-evolution\nis given by just an overall phase. Intuitively, we expect the density matrix of small\nsubsystems to be only a function of energy not only in translationally-invariant energy\neigenstates but also in all states that have spatial and time dependence over scalecs\nmuch larger than the size of the subsystem. In this section, we establish that this\nis indeed the case by studying the reduced density matrices in two classes of time-\ndependent states: \\coherent\" states, and arbitrary superpositions of N\u001ceS(E)=2\nenergy eigenstates.\n5.1 Time-dependent coherent states\nWe de\fne \\coherent states\" j\b(~ s)ivia a Euclidean path-integral with a local operator\ninserted at ~ sinside the unit ball in the radial quantization frame:\nj\b(~ s)i=es\u0016P\u0016\b(0)e\u0000s\u0016P\u0016j\ni (100)\nWe can use the rotational symmetry of the unit ball to bring the operator insertion\nto the point ( r=e\u001c;\u00121=\u000b) and\u0012i= 0 for all i > 1. Coherent states include\n26a superposition of many energy eigenstates, and hence evolve non-trivially in time.\nMapping to the Rindler space the operators that create and annihilate the state go to,\nrespectively, Y\u0016\n\u0000andY\u0016\n+:\n(Y0\n\u0006;Y1\n\u0006) =\u0012\u0000sin\u00120sinh\u001c\u0006\ncos(\u00120+\u000b)\u0000cosh\u001c\u0006;cos\u000b\u0000cos\u00120cosh\u001c\u0006\ncos(\u00120+\u000b)\u0000cosh\u001c\u0006\u0013\n; Yi>1\n\u0006= 0 (101)\nwhere\u001c\u0006=\u0006\u001c0\u0000itand we have analytically continued to the real time to keep track\nof the time evolution of the state. The analytic continuation in time is achieved by\ntreating\u001c\u0006as a real parameter.\nThe parameter \u001c0controls the width and angular dependence of the energy pro\fle\naround Sdat timet= 0. To see that we compute the energy density in this spinless\nprimary state:\nh\u001e\u000b;\u001c0(t)jT00(\u0012;0;\u0001\u0001\u0001)j\u001e\u000b;\u001c0(t)iCyl=ha\nLd+1!d\"\nsinh2\u0000\u001c\u0000\u0000\u001c+\n2\u0001\n(cos(\u000b\u0000\u0012)\u0000cosh\u001c\u0000)(cos(\u000b\u0000\u0012)\u0000cosh\u001c+)#d+1\n2\n=ha\nLd+1!d1\n(\u0000\ncostcoth\u001c0\u0000cos(\u000b\u0000\u0012) csch\u001c0)2+ sin2t\u0001(d+1)=2\nAtt= 0 the energy density around Sdhas its peak value coth2(\u001c0=2) at the point\n(\u000b;0;\u0001\u0001\u0001). In the thermodynamic limit of small subsystem l=L\u001c1 the energy density\nis constant over the subsystem\n\u000f(t;\u00122B) =ha\nLd+1!d\u0018d+1(t)(1 +O(1=L))\n\u00182(t) = (costcoth\u001c0\u0000cos\u000bcsch\u001c0)2+ sin2t\n(Y0\n\u0006;Y1\n\u0006) =\u0012lsinh\u001c\u0006\nL(cosh\u001c\u0006\u0000cos\u000b);1\u0000lsin\u000b\nL(cosh\u001c\u0006\u0000cos\u000b)\u0013\n(102)\nThe \\local\" length scale associated to the energy density is\n\u0015T(\u001c0;\u000b;t) =\u0018(t)L\u0012!ddT\nha\u0013 1\n(d+1)\n(103)\nThen, the distance between the operator insertions is\njY+\u0000Y\u0000j2=4l2\nL2\u0018(t)(104)\nand the density matrix becomes\ntr(~ \u0001\u0001\u0001) =X\np2AthermCp;^n\n\u001e;\u001e\u0018\u0000hpj^njhpO^n\np(Y\u0000) =X\np2Athermf^n\np(l=~\u0015T(t))hpO^n\np(Y\u0000):(105)\nwhich shows that the reduced density matrix is universal with \u0015Tmultiplied by \u0018(t).\nThat is to say at any time tthe reduced density matrix is in equilibrium with a time-\ndependent thermal wavelength \u0018(t).\n275.2 Arbitrary initial states\nAn arbitrary CFT state in the Schrodinger picture expanded in the energy eigen-basis\nis\nj\u001f(t)i=NX\na=1eihat=Lcaj ai (106)\nThe reduced density matrix on a ball-shaped region in this state is a partial trace over\nthe complement region\n\u001aBR(t) =trBc\nRj\u001f(t)ih\u001f(t)j=X\nabcac\u0003\nbeit(ha\u0000hb)trBc\nRj aih bj (107)\nNow, it is straightforward to see\nk\u001aBR(t)\u0000X\najcaj2\u001auni(E=Ea)k\u0014supa6=bk\u001babk\f\f\fX\na6=bcac\u0003\nb\f\f\fj\n\u0014\u0011e\u0000S(E)=2(NX\na=1jcaj)2\u0014\u0011Ne\u0000S(E)=2(108)\nfor some\u0011=O(1). Therefore, as long as the number of superposed energy eigenstates\nNdoes not scale with entropy the reduced density matrix is well-approximated with a\nclassical mixture of universal density matrices:\nZ\ndEp(E)\u001auni(E) (109)\nwhich does not evolve in time. If the state has h\u001f(t)jHj\u001f(t)i=E0andh\u001f(t)jH2\u0000\nE2\n0j\u001f(t)i= \u0001E0then the density matrix is approximately\n\u001auni(E0) +\u0001E0\n2@2\nE\u001auni(E)jE0+\u0001\u0001\u0001 (110)\nQuenching an energy eigenstate with a local operator of energy order one is an\nexample of a state that necessarily includes a large number of energy eigenstates.\n6 Conclusions\nIn this work, we continue the study of the Eigenstate Thermalization Hypothesis (ETH)\nin the context of Conformal Field Theories initiated in [1]. In that paper, we formulated\nthesubsystem ETH in CFTs as a statement about the smooth dependence of the\nreduced density matrix of an energy eigenstate on energy. We proved that if ETH\nis satis\fed at the level of individual local operators ( local ETH ), the subsystem ETH\nfollows.\n28In [1] it was shown that the ETH density matrix exhibits a great degree of univer-\nsality provided that the subsystem in question is small compared to the total volume.\nWhen the subsystem is small in comparison to the inverse e\u000bective temperature, the\nETH density matrix admits a perturbative expansion in terms of the light primary\noperators (12). In 2d CFTs the statement of ETH implies that no operator outside of\nthe Virasoro descendants of identity contributes to the OPE of any two heavy Virasoro\nprimaries. As a result the ETH density matrix exhibits a greater degree of universality,\ndepending only on the e\u000bective temperature and the central charge, but on other detail\nof the underlying theory (43).\nIn section 2 of the paper we provided an argument based on the equivalence of\nensembles, modi\fed for the case of CFTs, to argue that the ETH density matrix\nfor a small subsystem is trace-distance close to other thermal ensembles, the reduced\ncanonical and the microcanonical ones. This general argument is further supported\nby the calculation and comparison of the eigenstate entanglement entropy with the\nholographic one in section 4. In case of two dimensions, because of the additional\nconservation laws, the canonical ensemble must be substituted by the grand canonical\nensemble that includes an in\fnite number of conserved KdV charges { the Generalized\nGibbs Ensemble. A new representation of the ETH density matrix and its equivalence\nwith the thermal one in the limit of in\fnite central charge is demonstrated in section\n3. There we also calculate the von Neumann and the Renyi entropies for the eigenstate\nand discuss the \fnite ccase.\nFinally, in section 5 we discuss the reduced density matrix of time-dependent co-\nherent states and show that their reduced density matrix on a small subsystem is\nwell-described by the universal ETH density matrix with time-dependent e\u000bective tem-\nperature.\nAcknowledgements\nWe would like to thank John Cardy, Liam Fitzpatrick, Thomas Hartman, Matthew\nHeadrick, Tarun Grover, Mark Srednicki, Matthew Walters and Sasha Zhiboedov for\nvaluable discussions. The research of NL is supported in part by funds provided by\nMIT-Skoltech Initiative. AD is supported by NSF grant PHY-1720374. This work is\nsupported by the O\u000ece of High Energy Physics of U.S. Department of Energy under\ngrant Contract Number DE-SC0012567.\nA Rindler space: a convenient conformal frame\nConsider a ( d+1)-dimensional CFT in radial quantization with a ball-shaped subsystem\nof angular size \u00120onSdatr= 1. According to the operator/state correspondence the\ndensity matrix in the subsystem is given by a path-integral over the ( d+1)-dimensional\nspace with two operators inserted, \t at r=\u000fand \tyatr= 1=\u000fwith\u000f!0, and a cut\n29open at the location of the subsystem. The initial metric in the radial quantization is\nds2=dr2+r2d\n2\nd (111)\nwith (\u00121;\u0001\u0001\u0001\u0012d) the coordinates on Sd. We perform the following conformal transfor-\nmation\nL(r2\u00001)\n1 +r2+ 2rcos\u00121=X0\n1\u00002X1+X\u0001X;2Lrsin\u00121cos\u00122\n1 +r2+ 2rcos\u00121=(1\u0000X\u0001X)=2\n1\u00002X1+X\u0001X;\n2Lrsin\u00121sin\u00122\u0001\u0001\u0001cos\u0012i+1\n1 +r2+ 2rcos\u00121=Xi\n1\u00002X1+X\u0001X; d>i> 1\n2Lrsin\u00121sin\u00122\u0001\u0001\u0001sin\u0012d\n1 +r2+ 2rcos\u00121=Xd\n1\u00002X1+X\u0001X; L=1\n2cot(\u00120=2):\nthat maps the subsystem at r= 0 and\u00121\u0014\u00120to the negative half-space, i.e. (0 ;X1<\n0;0\u0001\u0001\u00010). HereLis the radius of Sdin units where Ris set to one. The new metric in\ntheX-coordinates that we call Rindler frame is given by\nds2= \u0003(X)2dXidXi\n\u0003(X) =\u0012\nX0\u0000LV\u0000\n2\u0000V+\n8L\u0013\u00001\nV\u0006= (1\u00062X1+X\u0001X): (112)\nIn these coordinates the path-integral without operator insertions prepares the\nRindler density matrix in vacuum. The operators \t and \tyare now inserted at X\u0000\nandX+respectively.\nX\u0006= (\u0006sin\u00120;cos\u00120;0\u0001\u0001\u0001;0);\n\u0003(X\u0000) = (2 sin\u00120)\u00001;\n\u0003(X+) =\u000f\u00002(2 sin\u00120)\u00001: (113)\nUnder this map a conformal primary transforms according to\nh\t(r= 0)\u0001\u0001\u0001i \u0003(X)\u000eij= \u0003(X(r= 0))\u0000hh\t(X(r= 0)\u0001\u0001\u0001i\u000eij\nTherefore,\nh\t(1=\u000f)\t(\u000f)\u0001\u0001\u0001iradial = (2\u000fsin\u00120)2hh\t(X+)\t(X\u0000)\u0001\u0001\u0001iRind\nIn the thermodynamic limit \u00120\u001c1 the distance between \t and \tygoes to zero:\njX+\u0000X\u0000j= 2 sin\u00120\u001c1, and we use the OPE to obtain\nh\t(1=\u000f)\t(\u000f)\u0001\u0001\u0001iradial =\u000f2hX\npCp\n  (2 sin\u00120)hphOp(X0)\u0001\u0001\u0001i:\n30B Global descendants in two dimensions\nConsider the OPE of two quasi-primaries \t in CFT 2\n\t(z;\u0016z)\t(0;0)\nh\t(z;\u0016z)\t(0;0)i=X\npCp\n  X\nj;\u0016jaj\n  p\u0016a\u0016j\n  p\nj!\u0016j!zhp+j\u0016z\u0016hp+\u0016j@j\u0016@\u0016j\bp (114)\nwhere \b pare quasi-primaries and\naj\n  p=C(j;hp+j\u00001)\nC(j;2hp+j\u00001); \u0016aj\n  p=C(\u0016j;\u0016hp+\u0016j\u00001)\nC(\u0016j;2hp+\u0016j\u00001)\nCp\n  =1\nh\bp\bpih j\bpj i; C (j;h) =\u0000(h+ 1)\n\u0000(j+ 1)\u0000(h\u0000j+ 1)(115)\nIn the thermodynamic limit z=l=L,handLgo to in\fnity with \u0015T\u0018L=p\nhkept\n\fxed we have\naj\n  pzj!08j >0: (116)\nTherefore, all the derivative terms are subleading, and we have\n\t(z;\u0016z)\t(0;0)\nh\t(z;\u0016z)\t(0;0)i=X\npCp\n  zhp\u0016z\u0016hp\bp+O(1=L): (117)\nThis argument generalizes to higher dimensions. Consider a primary Opand its\n\frst descendant. Then, the OPE coe\u000ecients are the same order\nC@Op\n  \nCOp\n  =dOp\nd@Oph\t(1)@Op(1)\t(0)i\nh\t(1)Op(1)\t(0)i= 2hp(2hp\u00001)hp=O(h0\n ) (118)\nhowever, by in the OPE of \ts, the derivative term has an extra power of l=Land is\nhence more suppressed.\nC Thermodynamically relevant quasi-primaries\nIn this appendix, we expand the reduced state on an interval of length 2 kin a highly\nexcited primary energy eigenstate, and \fnd the quasi-primaries that contribute to the\nuniversal density matrix, that are T2kin (38). Consider a primary energy eigenstate\nj aiand its correponding operator \t a. In Rindler coordinates, the density matrix is\ncreated by the insertion of operator\n\ta(z;\u0016z)\ta(0)\nh\ta(z;\u0016z)\ta(0)i=X\npX\nfk;\u0016kgCpfk;\u0016kg\naazhp+K\u0016z\u0016hp+\u0016KL\u0000fkg\u0016L\u0000f\u0016kgOp (119)\n31in the Euclidean path-integral. Here, fkg=fk1\u0001\u0001\u0001klg,K=k1+\u0001\u0001\u0001+kl, andz=x=L\nwithLgoing to in\fnity in the thermodynamic limit. The OPE coe\u000ecient Cp;fk;\u0016kg\naa\n(growing with ha) competes with the vanishing coe\u000ecient ( x=L)hp+K.\nTo determine what operator survive the thermodynamic limit in (119) we need to\ninvestigate the growth of this OPE coe\u000ecient with ha. It is convenient to de\fne the\nOPE coe\u000ecient with lowered indices [23]\nCp;fk;\u0016kg\naa =X\nfk0;\u0016k0g\u0002\nM\u00001\u0003pfkgfk0g\u0002\nM\u00001\u0003p;f\u0016kgf\u0016k0gCaa;pfk0gf\u0016k0g\nCaa;pfk0gf\u0016k0g=L\u0000fk0g\u0016L\u0000f\u0016k0gh\ta(1)\ta(1)Op(y)i\f\f\f\ny=0: (120)\nThe matrixMis the Kac matrix de\fned by Mfkg;fk0g(hp;c) =hhpjLfkgL\u0000fk0gjhpi, and\nis independent of ha. We only need to consider Caa;pfkgf\u0016kg. The di\u000berential operator\nL\u0000fkg\u0011L\u0000k1\u0001\u0001\u0001L\u0000klwith eachL\u0000kacting as\nL\u0000kh\ta(1)\ta(1)Op(y)i=\nCp\naa lim\n(z;!)!(1;1)z2ha\u0000\nha(k\u00001)(z\u0000k+!\u0000k)\u0000(z1\u0000k@z+!1\u0000k@!)\u0001\n(z\u0000!)hp\u00002ha(z!)\u0000hp\n=Cp\naa(ha(k\u00001) +hp)'Cp\naaha(k\u00001): (121)\nAt orderKwe are comparing OPE coe\u000ecients of operators of the form Lk1Lk2\u0001\u0001\u0001LklOp\nwithk1+\u0001\u0001\u0001+kl=K. From (121) it is clear that the OPE coe\u000ecient of operators\nwithL\u00001does not grow fast enough with haand they drop out of the thermodynamic\nlimit, which is consistent with the result in appendix B. We only need to consider the\ncase withki>1. Then,\nCaa;pfk1;\u0001\u0001\u0001;klgf\u0016k1;\u0001\u0001\u0001\u0016kmg\u0018hl+m\na: (122)\nFor evenKthe OPE coe\u000ecient of the quasi-primary that includes LK=2\n\u00002wins over other\nterms. When Kis odd none of the OPE coe\u000ecients are large enough to compete with\n(x=L)K+\u0016K. Therefore, the sum over fk0;\u0016k0gin (120) only has one term, and\nCpfk;\u0016kg\naa =Cp\naabp;fk;\u0016kghK=2+\u0016K=2\na\nbp;fk;\u0016kg=\u0002\nM\u00001\u0003f2;\u0001\u0001\u0001;2gfkg\u0002\nM\u00001\u0003f\u00162;\u0001\u0001\u0001;\u00162gf\u0016kg(123)\nwhereKand \u0016Kare both even. Note that in two dimensions Cp\naa= 0 for all non-identity\n32Virasoro primaries p. Therefore,\n\ta(ei\u00120;e\u0000i\u00120)\ta(e\u0000i\u00120;ei\u00120)\nh\ta(e\u0000i\u00120;ei\u0012)\ta(e\u0000i\u00120;ei\u00120)i=0\n@X\nfkgbfkg(2p\nhasin\u00120)KL\u0000fkg1\nA\u0002h:c:\n= X\nm2N(2p\nhasin\u00120)2mX\nk1+\u0001\u0001\u0001kl=2m[M\u00001]f2\u0001\u0001\u00012gfk1\u0001\u0001\u0001klgL\u0000k1\u0001\u0001\u0001L\u0000kl!\n\u0002h:c:\n= X\nm2N\u00122lp\n2\u0019\u0015T\u00132mcm\nd2mT(0)\n2m!\n\u0002h:c:\n1\nd2mT(0)\n2m\u0011X\nk1+\u0001\u0001\u0001kl=2m[M\u00001]f2\u0001\u0001\u00012gfk1\u0001\u0001\u0001klgL\u0000k1\u0001\u0001\u0001L\u0000kl (124)\nwhere in the last two lines we have de\fned an operator T(0)\n2mwith the norm d2m=\nhT(0)\n2m(1)T(0)\n2m(0)i. The \frst fewT(0)\n2kare\nT(0)\n2=L\u00002;T(0)\n4=L2\n\u00002\u00003\n5L\u00004\nT(0)\n6=L3\n\u00002+93\n70c+ 29L2\n\u00003\u00003(42c+ 67)\n70c+ 29L\u00004L\u00002\u00006(10c+ 13)\n70c+ 29L\u00006\nT(0)\n8=L4\n\u00002+\u00006 (630c2+ 3471c\u0000557)L\u00004L2\n\u00002\n5c(210c+ 661)\u0000251+(5844\u00001512c)L\u00005L\u00003\n5c(210c+ 661)\u0000251+\n27(c(42c+ 265)\u0000167)L2\n\u00004\n5c(210c+ 661)\u0000251\u000024(c(150c+ 569) + 67) L\u00006L\u00002\n5c(210c+ 661)\u0000251\u00006(5c(126c+ 463)\u0000543)L\u00008\n5c(210c+ 661)\u0000251\nT(0)\n10=L5\n\u00002\u000012 (8250c2+ 58115c\u00007161)L\u00006L2\n\u00002\n25c(462c+ 3067) + 3767+\u0012\n\u000012(11650c+ 15341)\n25c(462c+ 3067) + 3767\u000018\u0013\nL\u00008L\u00002\n+36(4358\u00003225c)L\u00007L\u00003\n25c(462c+ 3067) + 3767+36(c(1650c+ 16783)\u00008405)L\u00006L\u00004\n25c(462c+ 3067) + 3767+(31032c+ 220236)L2\n\u00005\n25c(462c+ 3067) + 3767\n+9(45c(154c+ 1873) + 25133) L2\n\u00004L\u00002\n25c(462c+ 3067) + 3767+\u0012\n\u000048(5115c+ 1081)\n25c(462c+ 3067) + 3767\u00006\u0013\nL\u00004L3\n\u00002\n+30(5115c+ 1081)L2\n\u00003L2\n\u00002\n25c(462c+ 3067) + 3767\u0000924(90c+ 259)L\u00005L\u00003L\u00002\n25c(462c+ 3067) + 3767\u000018(5115c+ 1081)L\u00004L2\n\u00003\n25c(462c+ 3067) + 3767\n\u0000504(c(300c+ 1693) + 266) L\u000010\n25c(462c+ 3067) + 3767\nd2=c\n2; d 4=c(5c+ 22)\n10; d 6=3c(2c\u00001)(5c+ 22)(7c+ 68)\n4(70c+ 29)(125)\nd8=3c(2c\u00001)(3c+ 46)(5c+ 3)(5c+ 22)(7c+ 68)\n10c(210c+ 661)\u0000502\nd10=15c(2c\u00001)(3c+ 46)(5c+ 3)(5c+ 22)(7c+ 68)(11c+ 232)\n4(25c(462c+ 3067) + 3767)(126)\n33Note that\n(L\u0000n\u00002\b)(!) =1\nn!@nT(!);\nL3\n\u00002(!) = (T(TT))(!); L2\n\u00003(!) = (@T@T )(!);\n\u0000\nL2\n\u00003+L\u00004L\u00002+L\u00002L\u00004\u0001\n(!) =1\n2@2(TT)(!): (127)\nThen, we \fnd\nT(0)\n2=T;T(0)\n4= (TT)\u00003\n10@2T\nT(0)\n6= (T(TT)) +9(14c+ 43)\n2(70c+ 29)(@T@T )\u00003(42c+ 67)\n4(70c+ 29)@2(TT)\u0000(22c+ 41)\n8(70c+ 29)@4T:\nAn alternative way to construct the quasi-primary operators T2kis by choosing the\nbasis where the Kac matrix is diagonal. In this basis, it is evident that the only quasi-\nprimaries that include the term Lm\n\u00002=Tm(0) propagate. Here, Tm= (T(T(T\u0001\u0001\u0001T))).\nWe can choose our operator basis such that at even order Konly one quasi-primary\nincludesLK=2\n\u00002which becomes our operator of interest T(0)\n2k. Below, we describe how to\nconstruct it at any even order K.\n1. Consider an arbitrary superposition of L\u0000fkgwith noL\u00002;\u0001\u0001\u0001\u00002:P\nfkg6=(2;\u0001\u0001\u0001;2)akL\u0000fkg(0).\n2. Chooseafkgsuch that this state is annihilated by L1. The result is the most\ngeneric quasi-primary with no L\u00002;\u0001\u0001\u0001\u00002.\n3. Find an arbitrary superposition state with L\u00002;\u0001\u0001\u0001\u00002that is perpendicular to the\nstate above, and demand that it is killed by L1. The resulting state is T(0)\nK.\nWe end this appendix by consider the quasi-primaries T2kin the limit h\u001dc\u001d1.\nIn this limit, the expressions for the \frst T2ksimplify to\n1\nd2m=1\nm!\u00122\nc\u0013m\nT(0)\n2=L\u00002;T(0)\n4=L2\n\u00002\u00003\n5L\u00004;\nT(0)\n6=L3\n\u00002\u00009\n5L\u00004L\u00002\u00006\n7L\u00006\nT(0)\n8=L4\n\u00002\u000024\n7L\u00006L\u00002+27\n25L2\n\u00004\u000018\n5L\u00004L2\n\u00002\u000018\n5L\u00008\nT(0)\n10=L5\n\u00002\u000018L\u00008L\u00002+36\n7L\u00006L\u00004\u000060\n7L\u00006L2\n\u00002+27\n5L2\n\u00004L\u00002\u00006L\u00004L3\n\u00002\u0000144\n11L\u000010:(128)\nTherefore, the holomorphic part of the density matrix operator becomes\nX\nm2N\u00124l2\n\u0019\u00152\nT\u0013m1\nm!T(0)\n2m=X\nm2N\u00124\u00192l2\n3\f2\u0013m1\nm!T(0)\n2m (129)\n34It is convenient to write the universal density matrix in an exponentiated form in this\nlimit:\nexp X\n0<m2Na2m\u00122lp\u0019\u0015T\u00132m\nL\u00002m!\na2= 1; a 4=\u00003\n10; a 6=11\n70; a 8=\u00009\n140; a 10=\u000034\n1925;\u0001\u0001\u0001:\nD One-point functions on a torus\nThe \fnite temperature expectation value of a primary operator at \fnite volume in two-\ndimensions is a one-point function on a torus with modular parameter \u001c=i\f\nLwhere\nLand\fare the periodicities of the spatial and time circles, respectively. Modular\ninvariance related the one-point function at high temperatures to low temperatures\nhOpi\u00001=\u001c= (\u00001)hp\u0000\u0016hp\u001chp\u0016\u001c\u0016hphOpi\u001c: (130)\nTherefore, for ~\f=L\n\fwe have\ntr(\u001a\fOp) = (LT)2hptr(\u001a~\fO) (131)\nThe parameter q=e2\u0019i\u001cat\u001c=iLT becomesq=e\u00002\u0019LTand small at large LT.\nTherefore, we can expand the one-point function perturbatively in small q:\nhOpiq=X\nh;\u0016hCp\n(h;\u0016h)(h;\u0016h)qh\u0000c\u00001\n24\u0016q\u0016h\u0000c\u00001\n241\n\u0011(q)\u0011(\u0016q)1X\nN=0qNHN;h;p: (132)\nThe coe\u000ecients HNare found using a recursive relation with the \frst term H0;h;p= 1\n[24]. At large LTonly the lowest dimension primary of dimension (\u0001 ;\u0016\u0001) contributes\ntr(\u001a~\fOp)'P\nh;\u0016hCp\n(h;\u0016h)(h;\u0016h)e\u00002\u0019LT(h+\u0016h\u0000c=12)\nP\nh;\u0016he\u00002\u0019LT(h+\u0016h\u0000c=12)'Cp\n\u0001;\u0016\u0001e\u00002\u0019LT(\u0001+ \u0016\u0001); (133)\nThis conclude our estimate of the size of one-point function probes in the thermody-\nnamic limit\ntr(\u001a\fOp) = (TL)2hpe\u00002\u0019LT(\u0001+ \u0016\u0001)Cp\n\u0001;\u0016\u0001: (134)\nAs expected in the limit LT!1 the thermal one-point functions are exponentially\nsuppressed.\n35E Perturbative Renyi entropies\nIn this appendix, we compute the Renyi entropies of the universal density matrix  via\na direct calculation of tr( n). We take the subsystem to have size 2 x, and the length\nscale associated with the energy density in  to be\u0015T. The trace of  nis computed\nby sewingncopies of the path-integrals that prepares  (the path-integral in Rindler\nspace with the operator (43) on each copy). Therefore, the vacuum subtracted Renyi\nentropy of is\n\u0001Sn( ;x) =(n+ 1)c\n12n\u0019(2x=\u0015T)2+\n1\n1\u0000nlogDnY\nj=1X\nKj\u0016Kj\u00122xpcp\n2\u0019n\u0015T\u0013Kj+\u0016Kj\ne2\u0019ij(Kj\u0000\u0016Kj)=nTKj(e2\u0019ij=n)T\u0016Kj(e\u00002\u0019ij=n)\ndKjd\u0016KjE\n:\nWe expand the above expression in powers of 2 x=\u0015Tand consider the \frst few terms.\nThe \frst term corresponds to ( Kj;\u0016Kj) = (0;0) for alljexcept forK0and \u0016K0. This\nterm is equal to one by the normalization of two point functions. The \frst non-trivial\nterm appears at j= 2 and (2x=\u0015T)4:\nn\n2n\u00001X\nl=1X\nK1K2=0;2;4\n\u0016K1\u0016K2=\u00160;\u00162;\u00164\u00122xpcp\n2\u0019n\u0015T\u0013K1+\u0016K1+K2+\u0016K2e2\u0019il=n (K2\u0000\u0016K2)\nd2\nK1d2\nK2h(TK1T\u0016K1)(1)(TK2T\u0016K2)(e2\u0019il=n)i\n=n\n2n\u00001X\nl=1X\nK;\u0016K\u00122xpc\n2p\n2\u0019n\u0015T\u00132K+2\u0016Ksin(\u0019l=n)\u00002(K+\u0016K)\ndKd\u0016K=c(x=\u0015T)4\n16\u00192(n2\u00001)(n2+ 11)\n90n3:\nAtj= 3 we have 6-point functions of \t a(3-point functions of TK)\nX\n1\u0014l<m<q\u0014n\u00001X\nK1K2K3=2\u00122xpcp\n2\u0019n\u0015T\u0013P3\ni=1(Ki+\u0016Ki)e2\u0019i(l\u000eK 1+m\u000eK 2+q\u000eK 3)=n\nd2\nK1d2\nK2d2\nK3\u0002\nD\nTK1(e2\u0019il=n)TK2(e2\u0019im=n)TK3(e2\u0019iq=n)T\u0016K1(e\u00002\u0019il=n)T\u0016K2(e\u00002\u0019im=n)T\u0016K3(e\u00002\u0019iq=n)E\n=X\n1\u0014l<m<q\u0014n\u000011\n(8d2)3\u00122xpc\nnp\u0019\u0015T\u001362CTTT\ns2\nlms2\nmqs2\nql= (2x=\u0015T)6c\n32\u00193(n2\u00001)(n4\u00004)(n2+ 47)\n2835n5\nwhere\u000eKi=Ki\u0000\u0016Kiandslm= sin(\u0019(l\u0000m)=n). We have used the summation\nidentities in [25]. It is important to note that up to the order ( l=\u0015T)6the density\nmatrix depends only on the energy density of the pure state.\nTherefore, to the sixth order we \fnd\n\u0001Sn( ;x) =(1 +n)c\n12n\u0019(2x=\u0015T)2\u0000(1 +n)c\n120n\u00192(n2+ 11)\n12n2(2x=\u0015T)4\n+(1 +n)c\n630n\u00193(4\u0000n2)(n2+ 47)\n144n4(2x=\u0015T)6(135)\n36The next non-trivial one-point function h jT4j icontributes to the entanglement en-\ntropy at order ( l=\u0015T)8. In the next appendix, we result above to the sixth order and\ncompute the eighth-order term using the twist operator method.\nF Twist operators\nThe correlation function (135) that appears in the calculation of the Renyi entropy\nof the universal density matrix is Znsymmetric. That is to say that it is invariant\nunderz!e2\u0019i=nz. An alternative way to compute this correlator is by employing\ntwist operators in a Zn-orbifold theory. Here, we use the orbifold theory to reproduce\nthe result of the last subsection and extend it to the eighth order in subsystem size.\nIn the orbifold theory, the vaccum-subtracted Renyi entropy in terms of the four-point\nfunction below\n\u0001Sn( ;x) =1\n1\u0000nlogG4(z;\u0016z);\nG4(z;\u0016z) =h\tn(1)\u001bn(z)\u001bn(1)\tn(0)i\nh\t(1)\t(0)inh\u001bn(z)\u001bn(0)i(136)\nwherez=eix=L. The quasi-primaries of the orbifold theory take the formQn\ni=1O(i),\nwhereOiis the primary on the ithcopy. Local ETH implies that this correlator\nis dominated by the Virasoro identity block. Below we use perturbation theory to\ncompute Renyi entropies order by order in 2 x=\u0015T.\nThe quasi-primaries that contribute to the Virasoro identity block at even orders\nup toz6are\norderz2T(j)\norderz4T(i)T(j)(i6=j);T(j)\n4\norderz6T(i)T(j)T(l)(i6=j6=l6=i);T(j)\n4T(l)\n2(j6=l);T(j)\n6\norderz8T(i)T(j)T(l)T(m)(6=); T(i)T(j)T(l)\n4(6=)\nT(j)\n4T(l)\n4(j6=l);T(j)\n6T(l)(j6=l);T8(j)\nwhere the symbol 6= means that all pairs of indices are unequal. These operators are\nlisted in [25]. The correlator factorizes into the holomorphic and anti-holomorphic\nparts\nG4(z;\u0016z) =jF(z;n;c )j2(137)\nwhere the vacuum conformal block Fis only a function of cross ratio z, Renyi index n\n37and central charge c.\nF(z) = 1 +X\nordered\u0012\nCT(j)\n\u001bn\u001bnCT(j)\n n n(1\u0000z)2+\u0012\nCT(j)T(l)\n n nCT(j)T(l)\n\u001bn\u001bn+CT(j)\n4\u001bn\u001bnCT(j)\n4\n n n\u0013\n(1\u0000z)4\n+\u0012\nCT(j)T(l)T(q)\n n nCT(j)T(l)T(q)\n\u001bn\u001bn+ 2CT(j)\n4T(l)\n n nCT(j)\n4T(l)\n\u001bn\u001bn+CT(j)\n6\u001bn\u001bnCT(j)\n6\n n n\u0013\n(1\u0000z)6\n+\u0000\nCTTTT\n n nCTTTT\n\u001bn\u001bn+ 3CTTT4\n n nCTTT4\n\u001bn\u001bn+CT4T4\n n nCT4T4\n\u001bn\u001bn+ 2CT6T\n n nCT6T\n\u001bn\u001bn+CT8\n n nCT8\n\u001bn\u001bn\u0001\n(1\u0000z)8\n\u0001\u0001\u0001 (138)\nwhereP\nordered runs over all indices of the operator as 1 \u0014j1<j2<\u0001\u0001\u0001<jk\u0014n. At\nlargehwe haveCTk1\u0001\u0001\u0001Tkm\n n n =hk1+\u0001\u0001\u0001km, and de\fne bTk1\u0001\u0001\u0001Tkm=P\norderedCTk1\u0001\u0001\u0001Tkm\u001bn\u001bn. These\nsums are computed in [10]:\nbT=n2\u00001\n12n; bT4=(n2\u00001)2\n288n3; bT6=(n2\u00001)3\n10368n5; bT8=(n2\u00001)4\n497664n7\nbTT=(n2\u00001) (5c(n+ 1)(n\u00001)2+ 2n2+ 22)\n1440cn3;\nbTT4=(n2\u00001)2(5c(n+ 1)(n\u00001)2+ 4n2+ 44)\n17280cn5\nbTT6=(n2\u00001)3(5c(n+ 1)(n\u00001)2+ 6n2+ 66)\n622080cn7\nbT4T4=1\n5806080c(5c+ 22)n7\u0000\n175c2(n+ 1)4(n\u00001)5\n+70c\u0000\nn2\u00001\u00013\u0000\n11n3\u00007n2\u000011n+ 55\u0001\n+ 8\u0000\nn2\u00001\u0001\u0000\nn2+ 11\u0001\u0000\n157n4\u0000298n2+ 381\u0001\u0011\nbTTT=(n\u00002) (n2\u00001) (35c2(n+ 1)2(n\u00001)3+ 42c(n4+ 10n2\u000011)\u000016(n+ 2) (n2+ 47))\n362880c2n5\nbTTT4=(n\u00002) (n2\u00001)\n14515200c2n7\u0010\n175c2(n+ 1)3(n\u00001)4+ 350c\u0000\nn2\u00001\u00012\u0000\nn2+ 11\u0001\n\u0000128(n+ 2)\u0000\nn4+ 50n2\u0000111\u0001\u0001\nbTTTT =(n\u00003)(n\u00002) (n2\u00001)\n87091200c3n7\u0010\n175c3(n+ 1)3(n\u00001)4+ 420c2\u0000\nn2\u00001\u00012\u0000\nn2+ 11\u0001\n\u00004c\u0000\n59n5+ 121n4+ 3170n3+ 6550n2\u00006829n\u000011711\u0001\n+ 192(n+ 2)(n+ 3)\u0000\nn2+ 119\u0001\u0001\n38Performing the Znsums over trigonometric functions we \fnd\nF(z) = 1 +a2h(1\u0000z)2+a4h2(1\u0000z)4+a6h3(1\u0000z)6+\u0001\u0001\u0001\na2=(n2\u00001)\n12n; a 4=(n2\u00001)2\n288n2+(n2\u00001)(n2+ 11)\n720n3c\na6=(n2\u00001)3\n10368n3+(n2\u00001)2(n2+ 11)\n8640n4c+(n2\u00001)(4\u0000n2)(n2+ 47)\n22680n5c2\na8=(n3\u00003n+ 3) (n2\u00001)4\n497664n7+(n4+ 9n2\u000022) (n2\u00001)3\n207360cn7\n\u0000(n\u00002)(n\u00001)(n+ 1)(59n6+ 136n5+ 3191n4+ 6640n3\u00007279n2\u000012536n\u00007491)\n21772800c2n7\n+(n\u00003)(n\u00002)(n\u00001)(n+ 1)(n+ 2)(n+ 3) (n2+ 119)\n453600c3n7+bT4T4 (139)\nSquaring the above vacuum block we \fnd\n\u0001Sn( ;x) =(1 +n)c\n12n\u0019(2x=\u0015T)2\u0000(1 +n)c\n120n\u00192(n2+ 11)\n12n2(2x=\u0015T)4\n+(1 +n)c\n630n\u00193(4\u0000n2)(n2+ 47)\n144n4(2x=\u0015T)6\u0000(1 +n)c\n2800n\u00194(2x=\u0015T)8s8(n;c) +\u0001\u0001\u0001\ns8(n;c) =88(n2\u00009)(n2\u00004) (n2+ 119) +c(\u000013n6+ 1647n4\u000033927n2+ 58213)\n5184(5c+ 22)n6:\nThe entanglement entropy is\n\u0001S1( ;x) =c\n6\u0019(2x=\u0015T)2\u0000c\n60\u00192(2x=\u0015T)4+c\n315\u00193(2x=\u0015T)6\n\u0000c\n1400\u00194(2x=\u0015T)8\u0012\n1 +242\n9(5c+ 22)\u0013\n+\u0001\u0001\u0001 (140)\nNote again that up to the order ( l=\u0015T)6all the contributions to the entanglement\nentropy come from TandTiTjandTiTjTk. That is because bT2k\u0018(n\u00001)kand\nbTT4\u0018(n\u00001)2. Therefore, up to this order the one-point function of h jT4j idoes not\nappear. However, at the eighth order in l=\u0015Tthere is a term in bT4T4andbT4TTthat\nare proportional to the \frst power of ( n\u00001) and hence contribute to the entanglement\nentropy.\nG Failure of perturbation theory for GGE\nIn this appendix, we expand the GGE in small KdV chemical potential in a perturbative\nexpansion. We show that demanding that the one-point functions of GGE to match\nthose of the eigenstate is inconsistent in perturbation theory. All orders of chemical\npotential contribute to the one-\frst correction in 1 =c, and one needs a non-pertubative\nexpression for one-point functions of GGE to compare with the eigenstate. We choose\n39the following simplifying notation\n1\nZtr\u0000\ne\u0000\fHA\u0001\n=hAi\f\n1\nZtr\u0000\ne\u0000\fH\u0000\u0016iQiA\u0001\n=hAi\f;\u0016i\n~A=A\u0000hAi\f (141)\nwhere repeated indices are summed over. Then, assuming a perturbative expansion for\nthe GGE we have\nhAi\f;\u0016i=hAi\f\u0000\u0016ih~A~Qii\f+\u0016i\u0016j\n2h~A~Qi~Qji\f+O(\u0016i\u0016j\u0016k) (142)\nTakingAto be the KdV current J2kwe have\nhTi\f;\u0016i=hTi\f\u0000\u0016ih~T~Qii\f+\u0016i\u0016j\n2h~T~Qi~Qji\f+O(\u0016i\u0016j\u0016k)\nhJ2ki\f;\u0016i=hJ2ki\f\u0000\u0016ih~J2k~Qii\f+\u0016i\u0016j\n2h~J2k~Qi~Qji\f+O(\u0016i\u0016j\u0016k) (143)\nIn (143) it is understood that the index i= 2m\u00001 is summed over, and mruns\nover 2 to1. The \frst term in the series above hJ2ki\f\u0018ckat largec. The above\nexpansion is a valid perturbation theory if chemical potentials are suppressed at large\ncby\u00162m\u00001\u0018c\u0000\u000b(m). Since the disconnected piece of h~J2k~Q2m\u00001i\fis zero, at large\ncentral chargeh~J2k~Q2m\u00001i\f=O(ck+m\u00001). The \frst order term gives us the condition\n\u000b(m)>m\u00001, and from the second order term we \fnd \u000b(m)>m.\nIn order to match this with the energy eigenstate we should solve for \u0016isuch that\nhTik\n\f;\u0016i=hJ2ki\f;\u0016i: (144)\nIf\u0016iare suppressed by powers of c, we can try to impose the above condition by setting\n1X\nm=2\u00162m\u00001\u0010\nkhTik\u00001\n\fh~T~Q2m\u00001i\f\u0000h~J2k~Q2m\u00001i\f\u0011\n=hJ2ki\f\u0000hTik\n\f+O(ck\u00002) (145)\nThe coe\u000ecient of \u00162m\u00001in the left hand side of (145) is O(ck+m\u00001), hence the each\nterm in the sum on the left is scales at bet as ck\u00001; while on the right hand side we\nhave terms that are order ck\u00001. The only option is to take \u000b=m. According to the\nperturbation expansion (143) this means that the higher orders terms in \u0016contribute\nto the same order in c. In order to make sense of the perturbation theory we should\nbe able to truncate the sum on the left to a \fnite number of terms. Say we keep the\ncoe\u000ecients \u00162m\u00001\u0018c\u0000\u000b(m)with for\u000b(m) =mform\u0014Cand\u000b(m)<m form>C ,\nwhereCis a \fnite number. Then, we have Cunknowns ( \u00162m\u00001form\u0015C) that should\nsatisfy an in\fnite number of equations at the \frt order in 1 =cin (145). We take this\nover-constrained system of equations as an indication that the question of \fnding a\nGGE with the same one-point functions as the energy eigenstate is non-perturbative\nin nature.\n40Below, we develop the perturbation theory in small chemical potential further, even\nthough it does not shed light on our study of ETH. In the remainder of this appendix,\nwe compute some of the one-point function of J4andTin an example of a GGE with\nonly\u00163turned on. The conserved currents are T(!) and (TT)(!) =T4(!) +3\n10@2\n!T(!)\non the thermal cylinder of circumference \f. Under a conformal transformation z=f(!)\nthe currents change according to\nT(!) =f02T(f) +c\n12Schw (f)\n(TT)(!) =T4(!) +3\n10@2\n!T(!)\n=f04T(f) +(5c+ 22)\n30Schw (f)\u0010\nf02T(f) +c\n24Schw (f)\u0011\n+3\n10@2\n!\u0010\nf02T(f) +c\n12Schw (f)\u0011\nSchw (f) =f000\nf0\u00003\n2\u0012f00\nf0\u00132\n: (146)\nMapping the thermal cylinder to the complex plane by z=e2\u0019!=\fwe \fnd (see [26])\nT(!) =\u00122\u0019\n\f\u00132\u0010\nz2T\u0000c\n24\u0011\n(TT)(!) =\u00122\u0019\n\f\u00134\u0012\nz4T4(z) +D2T(z) +c(5c+ 22)\n2880\u0013\nD2=3\n10\u0012\nz4@2+ 5z3@\u00005(c\u000010)\n18z2\u0013\n: (147)\nFrom this it is immediately clear that on the complex plane\n~T(z) =\u00122\u0019\n\f\u00132\nz2T(z)\n~J4(z) =\u00122\u0019\n\f\u00134\u0000\nz4T4(z) +D2T(z)\u0001\n: (148)\nAfter some straightforward algebra we \fnd\nh~T(0)~Q3i\f=\u00122\u0019\n\f\u00133Z1\n0dz\nzhT(\u00001)\u0000\nz4T4(z) +D2T(z)\u0001\ni=\u0000\u00122\u0019\n\f\u00133c(5c+ 22)\n720\nh~T(0)~Q3~Q3i\f=\u00122\u0019\n\f\u00136Z1\n0dzdz0\nzz0hT(\u00001)\u0000\nz4T4(z) +D2T(z)\u0001\u0000\nz04T4(z0) +D2T(z0)\u0001\ni\n=\u00122\u0019\n\f\u00136c(5c+ 22)(7c+ 74)\n8640: (149)\n41and for the KdV current\nh~J4(0)~Q3i\f=\u00122\u0019\n\f\u00133c(5c+ 22)(7c+ 74)\n60480\nh~J4(0)~Q3~Q3i\f=\u00122\u0019\n\f\u00136c(5c+ 22)\n10 \u00125c+ 22\n360\u00132\n+(5c+ 43)\n300!\n(150)\nHere, we have used the following three-point functions\nhT(1)T(1)T(0)i=c;hT(1)T(1)T4(0)i=c(5c+ 22)\n10\nhT4(1)T(1)T4(0)i=2c(5c+ 22)\n5;hT4(1)T4(1)T4(0)i=c(5c+ 22)(5c+ 64)\n25:\nAfter some algebra we \fnd that the expectation value of currents in the GGE in\nthe small chemical potential limit is given by\ntr(\u001a\f;\u0016T(0)) =\u00122\u0019\n\f\u00132\u0012\n\u0000c\n24+(2\u0019)3\u00163\n\f3c(5c+ 22)\n720\u0013\n+\u00162\n2\u00122\u0019\n\f\u00136\u0012c(5c+ 22)(7c+ 74)\n8640\u0013\n+O(\u00163=\f9)\ntr(\u001a\f;\u0016(TT)(0)) =\u00122\u0019\n\f\u00134\u0012c(5c+ 22)\n2880\u0000(2\u0019)3\u0016\n\f3c(5c+ 22)(7c+ 74)\n60480\u0013\n+\u00162\n2\u00122\u0019\n\f\u00136c(5c+ 22)\n10 \u00125c+ 22\n360\u00132\n+(5c+ 43)\n300!\n+O(\u00163=\f9): (151)\nFrom which we obtain\ntr(\u001aGGEH) =L\u00122\u0019\n\f\u00132\u0012c\n12\u0000(2\u0019)3\u00163\n\f3c(5c+ 22)\n360\u0013\n+\u00162\n2\u00122\u0019\n\f\u00136\u0012c(5c+ 22)(7c+ 74)\n4320\u0013\n+O(\u00163=\f9)\ntr(\u001aGGEQ3) =L\u00122\u0019\n\f\u00134\u0012c(5c+ 22)\n2880\u0000(2\u0019)3\u0016\n\f3c(5c+ 22)(7c+ 74)\n60480\u0013\n+\u00162\n2\u00122\u0019\n\f\u00136c(5c+ 22)\n10 \u00125c+ 22\n360\u00132\n+(5c+ 43)\n300!\n+O(\u00163=\f9) (152)\nwhere we have suppressed the \u00163=\f9corrections.\nReferences\n[1] N. 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Zhang, \\On short interval expansion of Rnyi entropy,\" JHEP\n11(2013) 164, arXiv:1309.5453 [hep-th] .\n[26] M. Gaberdiel, \\A General transformation formula for conformal \felds,\" Phys.\nLett.B325 (1994) 366{370, arXiv:hep-th/9401166 [hep-th] .\n44"    },    {        "title": "1711.01216v1.Density_functional_theory_for_internal_magnetic_fields.pdf",        "content": "Density-functional theory for internal magnetic \felds\nErik I. Tellgren\u0003\nHylleraas Centre for Quantum Molecular Sciences,\nDepartment of Chemistry, University of Oslo, N-0315 Oslo, Norway\nA density-functional theory is developed based on the Maxwell{Schr odinger equation with an\ninternal magnetic \feld in addition to the external electromagnetic potentials. The basic variables\nof this theory are the electron density and the total magnetic \feld, which can equivalently be repre-\nsented as a physical current density. Hence, the theory can be regarded as a physical current-density\nfunctional theory and an alternative to the paramagnetic current density-functional theory due to\nVignale and Rasolt. The energy functional has strong enough convexity properties to allow a for-\nmulation that generalizes Lieb's convex analysis-formulation of standard density-functional theory.\nSeveral variational principles as well as a Hohenberg{Kohn-like mapping between potentials and\nground-state densities follow from the underlying convex structure. Moreover, the energy functional\ncan be regarded as the result of a standard approximation technique (Moreau{Yosida regulariza-\ntion) applied to the conventional Schr odinger ground state energy, which imposes limits on the\nmaximum curvature of the energy (w.r.t. the magnetic \feld) and enables construction of a (Fr\u0013 echet)\ndi\u000berentiable universal density functional.\n\u0003erik.tellgren@kjemi.uio.noarXiv:1711.01216v1  [physics.chem-ph]  3 Nov 20172\nI. INTRODUCTION\nDensity functional theory (DFT) is one of the most widely applied electronic structure methods in quantum chem-\nistry and solid-state physics. This theory enables the standard problem of determining a ground-state wave function\nto be recast as the problem of determining only the total electron density of the ground state. Both the mathematical\nfoundations and the density-functional approximations used in practice have reached a mature stage. Many attempts\nhave been made to extend this computational framework to novel areas of application. One type of extension deals\nwith Hamiltonians containing a magnetic vector potential representing an external magnetic \feld, which is beyond\nthe scope of standard DFT. For example, the proof of the Hohenberg{Kohn theorem relies on the assumption that\nany two Hamiltonians di\u000ber by at most a multiplicative scalar potential [1]. Likewise, the constrained-search formu-\nlation [2] and Lieb's convex analysis formulation [3, 4] also require modi\fcation to allow for external magnetic \felds.\nThe most conservative extensions to date are the magnetic-\feld density-functional theory (BDFT) due to Grayce and\nHarris [5] as well as the paramagnetic current density functional theory (CDFT) due to Vignale and Rasolt [6]. Less\nclear-cut are the attempts to formulate current density functional theories based on the physical current density due\nto Diener [7] as well as Pan and Sahni [8], which both su\u000ber from gaps or mistakes in the proposed proofs of central\nresults [9{14]. Another possible route to incorporate external magnetic \felds is via a relativistic density-functional\ntheory based on the Dirac equation, which is likely to involve the physical current density as a basic variable. However,\nstandard ground-state DFT relies on the variational principle and boundedness from below of the spectrum of the\nSchr odinger Hamiltonian, which are not available for the Dirac Hamiltonian. At present, the available formulations of\nrelativistic ground-state DFT remain incomplete and non-rigorous [15{18]. In particular, relativistic DFT is presently\ntoo incomplete to allow a clear non-relativistic limit to be constructed within the formalism. Such a limit, if it exists,\nis likely to be a type of physical CDFT.\nIn the present work, it will be shown how consideration of the energy of the induced magnetic \feld or, alternatively,\na type of current-current interactions, allows a novel density-functional framework. This is done using the Maxwell{\nSchr odinger model, rather than the conventional Schr odinger equation, as the point of departure. The resulting DFT\nframework|termed MDFT [19]|can be regarded as a CDFT formulated in terms of the physical current density.\nWithin the new formalism, the Grayce{Harris BDFT functional reappears in a di\u000berent role. It will also be shown\nthat the Maxwell{Schr odinger energy and related MDFT functionals have rich convexity properties and satisfy several\nvariational principles.\nThe article is organized as follows. Basic concepts from convex analysis are brie\ry reviewed in Sec. II. Next, in\nSec. III, Grayce and Harris' BDFT formulation, and Vignale and Rasolt's paramagnetic CDFT formulations, both\nbased on the conventional Schr odinger model, are reviewed along with basic de\fnitions. In Sec. IV, the Maxwell{\nSchr odinger model is speci\fed. A density-functional theory is then formulated and analyzed, with focus on convexity\nproperties. The main results appear in Sec. IV B, IV C, and IV D. In Sec. V, a restriction of the theory to uniform\nmagnetic \felds is considered. The reduction from in\fnite-dimensional to low-dimensional representation of the mag-\nnetic \feld enables visualization of important aspects of the theory. Finally, a few concluding remarks are made in\nSec. VI.\nII. REVIEW OF CONVEX ANALYSIS\nBefore turning to the formulation of density-functional theories, some useful concepts from convex analysis will be\nbrie\ry reviewed|for a thorough exposition see Ref. 20.\nConvex functions are an important well-characterized class of functions. In particular, this is one of the few large\nclasses of functions for which global optimization is tractable. In many practical cases, it is su\u000ecient to consider\nfunctions de\fned on a \fnite-dimensional domain. However, many results hold also for in\fnite-dimensional Banach\nspaces, a fact that proved useful in Lieb's reformulation of standard DFT [3]. For the novel aspects of the present\nwork, an intermediate level of generality is needed|it is su\u000ecient to consider functions de\fned on a Hilbert space H.\nA setD\u001aHis said to be convex if x1;x22D, and 0\u0014\u0015\u00141, entails that \u0015x1+ (1\u0000\u0015)x22D. Geometrically, a\nline connecting any two points in a convex set is itself completely contained in the set. A function f:D!R, de\fned\non a convex domain D\u001aH, is said to be convex if\nf(\u0015x1+ (1\u0000\u0015)x2)\u0014\u0015f(x1) + (1\u0000\u0015)f(x2);0\u0014\u0015\u00141: (1)\nGeometrically, this means that linear interpolation always overestimates the function. A function fis said to be\nconcave if\u0000fis convex, i.e. if linear interpolation instead underestimates the function. By choosing x1andx2to be\nglobal minimizers, it is seen that the global minima of a convex function form a convex set. This set can be empty,\ncontain a unique point as the global minimum ( x1=x2), or contain an in\fnite family of global minima. There can\nbe no additional local minima besides the global minimum.3\nA function is said to be a\u000ene when the above relation is an equality, i.e.,\nf(\u0015x1+ (1\u0000\u0015)x2) =\u0015f(x1) + (1\u0000\u0015)f(x2);0\u0014\u0015\u00141: (2)\nA\u000ene functions are both convex and concave. An important type of functions that are convex by construction are\nthose that arise from maximization over a function g:H\u0002A!Rthat is a\u000ene in the \frst argument:\nh(x) = sup\nz2Ag(x;z): (3)\nSettingx=\u0015x1+ (1\u0000\u0015)x2, convexity follows directly,\nh(x) = sup\nzg(x;z) = inf\nz\u0000\n\u0015g(x1;z)+(1\u0000\u0015)g(x2;z)\u0001\n\u0014\u0015inf\nyg(x1;z)+(1\u0000\u0015) inf\nzg(x2;z) =\u0015h(x1)+(1\u0000\u0015)h(x2):(4)\nWith trivial adaptations, one also has that minimization over a function that is a\u000ene in the \frst arguments yields a\nconcave function\nh0(x) = inf\nz2Ag(x;z): (5)\nA function f:D!Ris said to be paraconvex orweakly convex if there exists a c >0 such that f(x) +ckxk2is\nconvex. Equivalently, a paraconvex function satis\fes\nf(\u0015x1+ (1\u0000\u0015)x2)\u0014\u0015f(x1) + (1\u0000\u0015)f(x2) +c\u0015(1\u0000\u0015)kx1\u0000x2k2: (6)\nThe de\fnition of paraconcave functions is analogous. More generally, a function fis said to be di\u000berence convex if\nthere exists convex functions gandhsuch thatf=g\u0000h. By writing f= (\u0000h)\u0000(\u0000g) it is seen that a di\u000berence\nconvex function is also di\u000berence concave .\nA function f:D!Ris said to be lower semicontinuous (l.s.c.) if, for every x0and\u000f > 0 there exists a\nneighborhood U=fx2D:kx\u0000x0k< \u000egsuch thatf(x)\u0014f(x0) +\u000ffor allx2U. A function fisupper\nsemicontinuous (u.s.c.) if\u0000fis l.s.c.\nFor a generic function f(x), the Legendre{Fenchel transformation orconvex conjugate is de\fned as\nf\u0003(y) = sup\nx\u0000\n(yjx)\u0000f(x)\u0001\n; (7)\nwhere (yjx) is the inner product of the Hilbert space. Noting that the right-hand side is a\u000ene in y, it can be seen\nto be a special case of Eq. (3) above. Hence, f\u0003(y) is convex by construction. It can also be shown to be l.s.c. by\nconstruction. The double transform f\u0003\u0003is equal to the original fif and only if fis both convex and l.s.c.\nIn the present context, it is convenient to change the sign convention in the Legendre{Fenchel transformation. We\ntherefore de\fne a Legendre{Fenchel transformation that maps convex functions to concave functions,\nf^(y) = inf\nx\u0000\n(yjx) +f(x)\u0001\n; (8)\nas well as a transformation that maps concave functions to convex functions,\ng_(x) = sup\ny\u0000\ng(x)\u0000(yjx)\u0001\n: (9)\nFor a convex, l.s.c. function f, one then has f= (f^)_. Likewise, for a concave, u.s.c. function g, one hasg= (g_)^.\nA di\u000berentiability concept that is particularly well-adapted to convex functions is that of subdi\u000berentiability . Given\na pointx0in the domain of a convex function f, asubdi\u000berential at x0is any element y0that satis\fes\nf(x)\u0015f(x0) +y0(x\u0000x0); (10)\nfor allx. At points where the function is di\u000berentiable in the ordinary sense, the subdi\u000berential is unique and equal to\nthe ordinary derivative. At non-di\u000berentiable points, there may be multiple subdi\u000berentials, a unique subdi\u000berential,\nor none at all. The set of all subdi\u000berentials is called the subgradient and is denoted\n@f(x0) =fy0jf(x)\u0015f(x0) +y0(x\u0000x0) for allxg: (11)\nFor concave functions, superdi\u000berentials and supergradients are de\fned in the analogous way. That is, y0is a\nsupergradient of the concave function f(x0) if and only if y0is a subgradient of the convex function \u0000f(x0). The\nsupergradient is denoted @f(x0).4\nIn convex analysis there is an operation that is somewhat analogous to convolution in Fourier analysis. The in\fmal\nconvolution of two functions f(x) andg(x) is de\fned as\n(f2g)(x) = (g2f)(x) = inf\ny\u0000\nf(x) +g(y\u0000x)\u0001\n: (12)\nWhen both fandgare convex l.s.c. functions, the Legendre{Fenchel transform takes the particularly simple form\n(f2g)\u0003=f\u0003+g\u0003. For general, non-convex functions one has ( f2g)\u00036=f\u0003+g\u0003.\nThe in\fmal convolution often has regularizing e\u000bect. In the special case when gs(x) =1\n2skxk2, the in\fmal convo-\nlutionf2gsis called the Moreau{Yosida (MY) regularization off. In what follows, the notation\nMsf(x) = (f2gs)(x) = inf\ny\u0010\nf(y) +1\n2skx\u0000yk2\u0011\n(13)\nwill be used. When fis convex, the MY regularization indeed improves the regularity. For example, while fneed\nonly be subdi\u000berentiable, the MY regularization is always di\u000berentiable. On the other hand, when fis non-convex,\nthe function Msfis not guaranteed to be more regular. For example, it is possible that a di\u000berentiable, non-convex f\nresults in a non-di\u000berentiable Msf. A connection between MY regularization and Legendre{Fenchel transformation\nis found by expanding the square in the above expression, leading to\nMsf(x) =gs(x)\u0000(f+gs)\u0003(1\nsx): (14)\nIt follows immediately that Msfis paraconcave, that Msf\u0000gsis concave, and that the MY regularization is invertible\nwheneverf+gsis convex and l.s.c.\nAn alternative to MY regularization that is more adapted to the non-convex case is the Lasry{Lions (LL) regular-\nization [21, 22], de\fned by\nLstf(x) =\u0000(gt2\u0000(gs2f))(x) = sup\nzinf\ny\u0010\nf(y) +1\n2skz\u0000yk2\u00001\n2tkx\u0000zk2\u0011\n(15)\nand\nLstf(x) = (gt2\u0000(gs2\u0000f))(x) = inf\nzsup\ny\u0010\nf(y)\u00001\n2skz\u0000yk2+1\n2tkx\u0000zk2\u0011\n: (16)\nThese operations can be identi\fed with double MY regularizations|for example, Lstf=\u0000Mt(\u0000Msf). The \frst\nversion results in approximations of f(x) from below, whereas the second results in approximations from above. The\napproximation is furthermore tighter than a MY regularization:\nMsf(x)\u0014Lstf(x)\u0014f(x); s\u0015t: (17)\nWhens>t , LL regularization of a quadratically bounded function, f(x)\u0015\u0000const\u0001(kxk2+ 1), results in a Fr\u0013 echet\ndi\u000berentiable function with H older continuous derivative [22].\nThe degenerate case s=tis known as the proximal hull [20]. Expanding the squares and reparametrizing the\noptimization of zusingz0=z=syields\nLssf(x) =\u0000gs(x) + (f+gs)\u0003\u0003(x)\u0014f(x); (18)\nLssf(x) =gs(x)\u0000(\u0000f+gs)\u0003\u0003(x)\u0015f(x); (19)\nwith equality when f+gsis convex and l.s.c. and f\u0000gsis concave and u.s.c., respectively.\nIII. DENSITY-FUNCTIONAL THEORY IN THE SCHR ODINGER MODEL\nAt the non-relativistic level, an electronic system subject to external electromagnetic potentials is typically described\nby the Schr odinger equation, also called the Pauli equation when electron spin and the spin-Zeeman e\u000bect is explicitly\nconsidered. In SI-based atomic units, the ground-state energy is then given by\nE(v;A) = inf\n h j^TA+ ^v+^Wj i= inf\n\u0000tr(\u0000( ^TA+ ^v+^W)); (20)5\nwhere the last minimization is performed over all valid mixed states \u0000 and the Hamiltonian has been decomposed\ninto three parts,\n^TA=1\n2X\nl(\u001bl\u0001\u0019A;l)2=1\n2X\nl\u0000\n\u001bl\u0001(pl+A(rl))\u00012; (21)\n^v=X\nlv(rl); (22)\n^W=X\nk<l1\nrkl; (23)\nwhere pl=\u0000irlis the canonical momentum operator and \u0019A=p+Athe physical momentum operator. It will\nalso prove useful to introduce a kinetic energy operator that excludes the diamagnetic term,\n^Tpara\nA=1\n2X\nl(\u001bl\u0001pl)2+1\n2X\nl\b\n\u001bl\u0001pl;\u001bl\u0001A(rl)\t\n=^TA\u00001\n2X\nljA(rl)j2: (24)\nAny mixed state \u0000 =P\nkpkj kih kj, where the weights satisfy pk\u00150 andP\nkpk= 1, gives rise to a one-particle\nreduced density matrix (1-RDM),\n\r(x1;x0\n1) =X\nkpkZ\n k(x1;x2;:::;xN) k(x0\n1;x2;:::;xN)\u0003dx2\u0001\u0001\u0001dxN; (25)\nwhere xj= (rj;!j) contains both spatial and spin coordinates of particle jand summation over spin degrees of\nfreedom is implied. The 1-PRDM is diagonalized by the natural spinors (eigenvectors) \u001ek, with occupation numbers\n(eigenvalues) nk,\n\r(x;x0) =X\nknk\u001ek(x)\u001ek(x0)\u0003: (26)\nOrganizing the natural spinors as two-component column vectors \u001ek(r) = (\u001e\"\nk(r);\u001e#\nk(r))T, enables convenient expres-\nsions for the densities of interest in this work. The density and spin density are given by\n\u001a(r) =X\nknk\u001ek(r)y\u001ek(r); (27)\nm(r) =X\nknk\u001ek(r)y\u001b\u001ek(r): (28)\nFurthermore, the paramagnetic current density and magnetization current density are given by\njp(r) =1\n2X\nknk\u001ek(r)yp\u001ek(r) + c.c.; (29)\njm(r) =1\n2X\nknk\u001ek(r)y\u001b(\u001b\u0001p)\u001ek(r) + c.c.; (30)\nwhere `c.c.' denotes the complex conjugate of the preceeding expression. Both these current densities are gauge\ndependent and they di\u000ber by the spin current density jm\u0000jp=r\u0002m. These currents can be contrasted with the\ngauge invariant, physical current density\njA(r) =1\n2X\nknk\u001ek(r)y\u001b(\u001b\u0001\u0019A)\u001ek(r) + c.c. (31)\nIn what follows, the index Awill sometimes be dropped but it should be noted that this current density explicitly\ndepends on the external vector potential.\nStandard density functional theory only takes into account the external scalar potential, and gives rise to a universal\nfunctionalF(\u001a) and the Hohenberg{Kohn variational principle ,\nE(v) = inf\n\u001a2X\u0000\n(\u001ajv) +F(\u001a)\u0001\n; (32)6\nwhere the pairing integral is de\fned by\n(\u001ajv) =Z\n\u001a(r)v(r)dr: (33)\nDi\u000berent functionals F, with di\u000berent convexity properties and di\u000berent domains of densities they take \fnite values\non, can be used in this variation principle. A unique Fis obtained from the Lieb variational principle ,\nF(\u001a) = sup\nv2X\u0003\u0000\nE(v)\u0000(\u001ajv)\u0001\n: (34)\nLieb showed that \u001a2X=L1\\L3, with the corresponding restriction v2X\u0003=L1+L3=2on the potentials, is\na natural function space and minimization domain [3]. Important aspects of Lieb's formulation include, \frst, that\ndensities and potential belong to dual function spaces, i.e. the potential space is the space of all linear functionals\nde\fned on the density space via the pairing ( \u001ajv). Second, that the energy functional E(v) (in the absence of a\nvector potential) is concave in vand that there exists a universal functional F(\u001a) in standard DFT that is convex\nin\u001a, so that they can be related via Legendre{Fenchel transformations [3, 4]. Third, concave (convex) functionals\nhave either a unique global maximum (minimum) or a convex set of global maxima (minima), and no additional local\nmaxima (minima). Fourth, though the functionals are not di\u000berentiable [23], there is an applicable, rigorous notion\nof superdi\u000berential for concave functionals and subdi\u000berentials for convex functions. A potential v0with ground state\ndensity\u001a0is then a subdi\u000berential of F(\u001a0) and\u001a0is a superdi\u000berential of E(v0). The Hohenberg{Kohn theorem can\nbe reinterpreted in this setting as a mapping between super- and subdi\u000berentials, and the conventional assumption\nof non-degenerate ground states can be dropped. Fifth, the Lieb variation principle provides a path to systematically\napproximate the exact functional F(\u001a) by introducing approximations into the maximization problem [24{31].\nExternal magnetic \felds can be incorporated into density-functional theory in di\u000berent ways. In BDFT due to\nGrayce and Harris, a semi-universal functional of the pair ( \u001a;B) is introduced [5]. As a slight modi\fcation of this\nidea, the semi-universal functional\nFGH(\u001a;A) = inf\n\u00007!\u001atr(\u0000( ^TA+^W)) =1\n2(\u001ajA2) + inf\n\u00007!\u001atr(\u0000( ^Tpara\nA+^W)); (35)\ncan be introduced. One can then write the BDFT variational principle\nE(v;A) = inf\n\u001a2X\u0000\n(\u001ajv) +FGH(\u001a;A)\u0001\n: (36)\nA complementary approach was introduced in by Vignale and Rasolt, who formulated current-density functional\ntheory (CDFT) and de\fned a universal current density-dependent functional [6, 32]. When the spin Zeeman term\nis neglected, the ground state energy functional can be expressed in terms of a universal functional of ( \u001a;jp). Sub-\nsequent developments identi\fed an alternative formalism based on the magnetization current jm[33]. Rewriting the\nminimization as a nested minimization, with the inner minimization given by the Vignale{Rasolt constrained search\nfunctional,\nfVR(\u001a;jm) = inf\n\u00007!\u001a;jmtr(\u0000( ^T0+^W)) (37)\nyields\nE(v;A) = inf\n\u001a2X;\njm2Y\u0010\n(\u001ajv+1\n2A2) + (jmjA) +fVR(\u001a;jm)\u0011\n:(38)\nFor the current density and magnetic vector potential, a possible choice of function spaces is jm2Y=L1and\nA2Y\u0003=L1[13]. These function spaces guarantee a \fnite pairing ( jmjA) and a \fnite diamagnetic term1\n2(\u001ajA2). A\nsubtle point is that in order for the functional fVR(\u001a;jm) to be universal , i.e. independent of the external potentials,\nthe minimization domain of states \u0000 must be universal too. For bounded vector potentials, this issue gets a simple\nsolution, since the magnetic Sobolev space,\nH1\nA=f 2L2j(pk+A(rk)) 2L2g; (39)\nis independent of A2L1and equal to the regular Sobolev space H1=f 2L2jpk 2L2g[13]. Hence, \fniteness\nof the physical and canonical kinetic energies are equivalent, and a universal minimization domain, de\fned by the\nmixed states formed from pure states in H1, can be used.7\nThe concavity properties of E(v;A) are of crucial importance for a convex analysis formulation along the lines of\nLieb's formulation for standard DFT. However, while E(v;A) is concave in v, it is not concave (nor convex) in A.\nRelatedly, the semi-universal functional FGH(\u001a;A) is convex in \u001aand non-concave in A. For both functionals, the non-\nconcavity is a manifestation of diamagnetism, which gives rise to local convexity with respect to small perturbations\nof particular ( v;A) or (\u001a;A), for which the diamagnetic term is larger than the paramagnetic term. It would thus\nbe desirable to be able to eliminate the diamagnetic energy. In fact, FGH(\u001a;A) isdi\u000berence concave inAand the\ndi\u000berence, denoted\n\u0016FGH(\u001a;A) =FGH(\u001a;A)\u00001\n2(\u001ajA2) = inf\njm2Y\u0000\n(jmjA) +fVR(\u001a;jm)\u0001\n; (40)\nis manifestly concave in A. This subtraction of the diamagnetic energy in the ( \u001a;A)-representation corresponds to\nthe change of variables u=v+1\n2A2in the (v;A)-representation. The resulting energy functional, \u0016E(u;A), is jointly\nconcave and amendable to a convex analysis formulation that extends Lieb's formulation [10]. Moreover, the universal\nfunctionalfVR(\u001a;jm) is jointly convex in ( \u001a;jm). In the resulting theory, Legendre{Fenchel transforms connect the\nenergy functional \u0016E(u;A) to the semi-universal BDFT functional and the universal CDFT functional [34]:\n\u0016E(u;A) = inf\n\u0000tr(\u0000( ^Tpara\nA+ ^u+^W))\n= inf\n\u001a2X\u0000\n(\u001aju) +\u0016FGH(\u001a;A)\u0001\n= inf\n\u001a2X;\njm2Y\u0000\n(\u001aju) + (jmjA) +fVR(\u001a;jm)\u0001\n:(41)\nInversion of the last Legendre{Fenchel transformation yields\n\u0016fVR(\u001a;jm) = sup\nu2X\u0003\nA2Y\u0003\u0000\u0016E(u;A)\u0000(\u001aju)\u0000(jmjA)\u0001\n(42)\nAt present, it is not known whether or not the constrained-search functional fVRis lower semicontinuous. If it is, then\nby the properties of Legendre{Fenchel transforms one has that \u0016fVRandfVRare identical. If, however, fVRis not\nlower semicontinuous, it must be distinguished from \u0016fVRwhich has this property by construction. For completeness,\none may similarly construct a concave-convex functional \u0016 e(u;jm),\n\u0016e(u;jm) = inf\n\u001a2X\u0000\n(\u001aju) +\u0016fVR(\u001a;jm)\u0001\n= sup\nA2Y\u0003\u0000\u0016E(u;A)\u0000(\u001aju)\u0001\n: (43)\nThe di\u000berent transform pairs are summarized in Fig. 1.\nAlthough the outlined theory has useful, well-de\fned convexity properties, the change of variables appears to be\nunnatural from the perspective that the non-relativistic CDFT is a limiting case of a relativistic density-functional\ntheory based on the Dirac equation. At the relativistic level, no diamagnetic term1\n2A2enters the Hamiltonian and\nit is presently unclear how the potential ucould arise in the non-relativistic limit. A related issue is how a universal\nfunctional of a gauge-dependent jmmight arise in the non-relativistic limit of a theory that is formulated in terms\nof gauge invariant physical current densities. Relativistically, the decomposition of the physical current density into\nparamagnetic and diamagnetic parts is much less natural. These points merit further investigation but are beyond\nthe scope of the present work. Instead, a very di\u000berent convex formulation will be developed below.\nIV. DENSITY FUNCTIONAL THEORY IN A MAXWELL{SCHR ODINGER MODEL\nThe above Schr odinger model does not take into account that current densities associated with di\u000berent electrons\ninduce magnetic \felds that a\u000bect the other electrons. Classically, a stationary current density jinduces a magnetic\n\feldb, as described by the magnetostatic Maxwell equation r\u0002b=\u0000\u0016j. The energy of the induced \feld bis given by\n1\n2\u0016(bjb) =1\n2\u0016R\njbj2dr. In the present case, only magnetic \felds with \fnite energy are of interest. A natural function\nspace is thus the Hilbert space of square integrable functions, b2L2(R3). EveryL2function can be represented\nas a Fourier transform, which will be denoted ^b(k) =R\ne\u0000ik\u0001rb(r)dr. Adding the condition that a magnetic \feld is\ndivergence free leads to the self-dual function space\nL2\ndiv(R3) =fb2L2(R3)jk\u0001^b= 0g: (44)\nTwo function spaces will be used for the magnetic vector potentials. The larger is simply\nA=fa2L1\nlocjk^a2L2g; (45)8\nwherek=jkj, and the smaller is the subset of divergence-free vector potentials,\nAdiv=fa2L1\nlocjk^a2L2;k\u0001^a= 0g: (46)\nHere, the condition a2L1\nlocensures that a Fourier transform ^aexists at least in the distributional sense. For every\nmagnetic \feld b2L2\ndiv, one can de\fne a divergence-free current density using the above mentioned Maxwell equation.\nIn terms of the Fourier transform, ^j=\u0000k\u0002^b=\u0016. The corresponding function space is\nJdiv=fj2L1\nlocjk\u00001^j2L2;k\u0001^j= 0g: (47)\nA natural additional requirement on the current density space Jdivwould be that j2L1or^j2L1, since this\nis true of the current densities computed from a wave function or mixed state. However, in order to be able to\nwork with a self-dual space of magnetic \felds, this route will not be taken. A restriction on Jdivwould result in a\ncorresponding expansion of its dual space, and one would further need to distinguish between magnetic \felds obtained\nas^b=\u0000i\u0016k\u00002k\u0002^jand magnetic \felds obtained as ^b0=\u0000ik\u0002^a. Two such di\u000berent \felds would have a \fnite\npairing ( bjb0) and the \frst type would have \fnite self-energy ( bjb), while the second self-energy ( b0jb0) would not be\nguaranteed to be \fnite.\nFor vector potentials a2A div, the Maxwell equation \u0000r2a=r\u0002b=\u0000\u0016jcan be inverted to give\na(r1) =\u0000\u0016\n4\u0019Zj(r2)\njr1\u0000r2jdr2: (48)\nTaking the curl of this expression yields Biot{Savart's law. Introducing the notation\n(jjr\u00001\n12jj) =Zj(r1)\u0001j(r2)\njr1\u0000r2jdr1dr2 (49)\nand using partial integration establishes an inequivalence between magnetic \feld energy and current-current interac-\ntion,\n1\n2\u0016(bjb) =\u00001\n2(jja) =\u0016\n8\u0019(jjr\u00001\n12jj): (50)\nIn what follows, the lower-case symbols a;b;jwill be used to denote generic vector potentials, magnetic \felds, and\ncurrent densities. The symbol jwill also be used to denote current densities computed from wave functions or mixed\nstates. The external vector potential is denoted A2A, with associated magnetic \feld B=r\u0002Aand current density\nJ=\u0000r\u0002 B=\u0016. In addition, we introduce an internal \feld denoted by Greek letters: The internal vector potential\nis denoted \u000b2A, with associated magnetic \feld \f=r\u0002\u000band current density \u0014=\u0000r\u0002 \f=\u0016. Inclusion of the\nenergy in \fand the mean-\feld interaction of \u000bwith the electrons leads to the Maxwell{Schr odinger functional\nEM(v;A) = inf\n 2H1\n\u000b+A\n\u000b2A\u00101\n2\u0016(\fj\f) +h j^T\u000b+A+ ^v+^Wj i\u0011\n= inf\n\u0000\n\u000b2A\u00101\n2\u0016(\fj\f) + tr(\u0000( ^T\u000b+A+ ^v+^W))\u0011\n;(51)\nwhereH1\n\u000b+Ais a magnetic Sobolev space and the in\fmum over \u0000 is taken over all corresponding mixed states. It\nwill be seen below that this leads to a universal functional, despite the appearance of the external vector potential\ninH1\n\u000b+A. This functional can be regarded as a simpli\fed mean-\feld picture of the radiation degrees of freedom\nconsidered by Ruggenthaler and co-workers [35{37] in work combining non-relativistic quantum electrodynamics and\ntime-dependent density functional theory. The stationarity conditions for the Maxwell{Schr odinger functional are a\nset of coupled equations: The energy eigenvalue equation for a Schr odinger Hamiltonian,\n\u00101\n2\u0016(\fj\f) +^T\u000b+A+ ^v+^W\u0011\nj i=EM(v;A)j Mi; (52)\nand one of Maxwell's equations (the magnetostatic specialization of Amp\u0013 ere's circuital law),\n1\n\u0016r\u0002\f(r) +j\u000b+A(r) = 0; (53)9\nwhere j\u000b+Ais the physical current density arising from the state  Mand the total vector potential. The self-\nconsistent nature of the condition is more apparent when the physical current density is decomposed into diamagnetic\nand paramagnetic terms,\n1\n\u0016r\u0002\f(r) +\u001a(r)\u000b(r) +\u001a(r)A(r) +jm(r) = 0; (54)\nwhere\u001ais the density resulting from  M.\nSome further insight into the model can be gained by rewriting the ground-state energy in a form that involves\nmagnetostatic current-current interactions. Using the identity ^T\u000b+A=^TA+1\n2f\u0019A;\u000bg+1\n2\u000b2one obtains\nEM(v;A) =(\fj\f)\n2\u0016+h Mj^T\u000b+A+ ^v+^Wj Mi=\u0016\n8\u0019(\u0014jr\u00001\n12j\u0014) + (jA+1\n2\u001a\u000bj\u000b) +h Mj^TA+ ^v+^Wj Mi; (55)\nwhere jA=j\u000b+A\u0000\u001a\u000bis the current density computed using only the external vector potential. Assume now that\nthe gauge of  Mis such that \u000bis divergence free. Then the self-consistency condition \u0014=j\u000b+Acan be used to \frst\nrewrite the \feld energy and second to express \u000busing Biot{Savart's law. The result,\nEM(v;A) =\u0000\u0016\n8\u0019(jAjr\u00001\n12jj\u000b+A) +h Mj^TA+ ^v+^Wj Mi; (56)\nshows that the Maxwell{Schr odinger energy implicitly includes a type of current-current interaction between j\u000b+A\nandjA.\nAlthough the Maxwell{Schr odinger energy is di\u000berent from the Schr odinger energy, the two quantities are related\nin several ways. For example, the in\fmum over \u000bcan be overestimated by setting \u000b= 0 or \u000b=\u0000A. It immediately\nfollows that\nEM(v;A)\u0014E(v;A); (57)\nEM(v;A)\u0014E(v;0) +1\n2\u0016(BjB): (58)\nThese inequalities show that the Maxwell{Schr odinger ground state functional is a general a lower bound on the\nSchr odinger ground state functional and that its magnetic-\feld variation is bounded by the energy of the external\nmagnetic \feld. Overestimating the in\fmum in the de\fnition of EM(v;0) by taking \u000b=Asimilarly leads to\nEM(v;0)\u00141\n2\u0016(BjB) + inf\n\u0000\u0010\ntr(\u0000( ^TA+ ^v+^W))\u0011\n=1\n2\u0016(BjB) +E(v;A); (59)\nsetting a bound on how strongly paramagnetic a system can be.\nThe in\fmum de\fning EM(v;A) can also be overestimated by \fxing  to be the ground state of the Schr odinger\nHamiltonian TA+v+W. Let jSbe the corresponding current density evaluated using only the external vector\npotential. Finally, the in\fmum over internal vector potential can be overestimated by setting\n\u000b(r1) =\u0000\u0015\n4\u0019ZjS(r2)\njr1\u0000r2jdr2; (60)\nand minimizing only over the one-dimensional parameter \u00152R. The optimal value is\n\u0015=\u0000(jSj\u000b)\n\u0016\u00001(\fj\f) + (\u001aj\u000b2)=\u0000(jSjr\u00001\n12jjS)\n\u0016\u00001(jSjr\u00001\n12jjS) + 4\u0019(\u001aj\u000b2)(61)\nand the resulting energy bound is\nEM(v;A)\u0014E(v;A)\u0000\u0016\n8\u0019\u0001(jSjr\u00001\n12jjS)2\n(jSjr\u00001\n12jjS) + 4\u0019\u0016(\u001aj\u000b2)(62)\nWhen (\u001aj\u000b2) is \fnite and \u0016is su\u000eciently small,\nEM(v;A).E(v;A)\u0000\u0016\n8\u0019\u0001(jSjr\u00001\n12jjS): (63)\nHence, the Maxwell{Schr odinger energy is bounded by the standard energy E(v;A) corrected by an attractive current-\ncurrent interaction that resembles the relativistic Gaunt interaction [38].10\nA. Constrained-search formulation of MDFT\nA constrained-search formulation is obtained by introducing the total vector potential as a basic variational param-\neter. A reparametrization \u000b0=\u000b+Aleads to\nEM(v;A) = inf\n\u0000\n\u000b02A\u00101\n2\u0016(\f0\u0000Bj\f0\u0000B) + tr(\u0000( ^T\u000b0+ ^v+^W))\u0011\n:(64)\nNote that it is here crucial that both the internal and external vector potential are in A, since this guarantees that\nthe reparametrized minimization over \u000b0can be performed over the same domain. By contrast, if A=2A, then the\nminimization domain for \u000b0would be dependent on A, ruining the universality of the theory. Note also that the\nminimization over \u0000 is performed over all mixed states formed from pure states in H1\n\u000b0. Since \u000b0is a variational\nparameter, and minimization is performed over all possible \u000b02A, this does not prevent the construction of a\nuniversal density functional. Introducing a nested minimization, one may identify the functional FGHfrom Eq. (35)\nabove,\nEM(v;A) = inf\n\u001a2X\n\u000b02A\u0010\n(\u001ajv) +1\n2\u0016(\f0\u0000Bj\f0\u0000B) + inf\n\u00007!\u001atr(\u0000( ^T\u000b0+^W))\u0011\n=1\n2\u0016(BjB) + inf\n\u001a2X\n\u000b02A\u0010\n(\u001ajv)\u00001\n\u0016(\f0jB) +1\n2\u0016(\f0j\f0) +FGH(\u001a;\u000b0)\u0011 (65)\nDe\fning the universal functional\nfM(\u001a;\u000b0) =1\n2\u0016(\f0j\f0) +FGH(\u001a;\u000b0) (66)\nnow allows one to write\nEM(v;A) =1\n2\u0016(BjB) + inf\n\u001a2X\n\u000b02A\u0010\n(\u001ajv)\u00001\n\u0016(\f0jB) +fM(\u001a;\u000b0)\u0011\n:(67)\nHere, the external potentials ( v;A) have been completely isolated from the remaining, universal functional fM. More-\nover, the ground state energy is obtained from a straightforward variational principle involving global minimization\nover (\u001a;\u000b0).\nTwo di\u000berent semi-universal constrained-search functionals can also be de\fned. Nesting the minimizations so that\nminimization over \u001ais performed after \u000b0leads to the functional\nFM(\u001a;A) =1\n2\u0016(\f0\u0000Bj\f0\u0000B) + inf\n\u00007!\u001atr(\u0000( ^T\u000b0+^W))\n=1\n2\u0016(BjB) + inf\n\u000b02A\u0010\n\u00001\n\u0016(\f0jB) +fM(\u001a;\u000b0)\u0011\n;(68)\nwhile the opposite nesting yields\neM(v;\u000b0) =1\n2\u0016(\f0j\f0) + inf\n\u0000tr(\u0000( ^T\u000b0+ ^v+^W))\u0011\n= inf\n\u001a2X\u0010\n(\u001ajv) +fM(\u001a;\u000b0)\u0011\n:(69)\nIt then follows that the ground-state energy can be obtained from two di\u000berent variational principles,\nEM(v;A) = inf\n\u001a2X\u0010\n(\u001ajv) +FM(\u001a;A)\u0011\n=(BjB)\n2\u0016+ inf\n\u000b02A\u0010\n\u0000(\f0jB)\n\u0016+eM(v;\u000b0)\u0011\n:(70)\nIn the ground state energy functional EM(v;A), and the three constrained-search functionals fM(\u001a;\u000b0),eM(\u001a;\u000b0),\nandFM(\u001a;A) de\fned above, the second argument is a magnetic vector potential. By exploiting gauge invariance11\ntogether with the fact that \u00140and\f0are little more than alternative representations of \u000b0(and similarly for J;B;A),\nit is easy to reformulate the functionals so that the second argument is a magnetic \feld or a current density instead.\nFor de\fniteness, we de\fne\nFM;b(\u001a;b) = inf\na2A\nr\u0002a=bFM(\u001a;a); (71)\nFM;j(\u001a;j) = inf\na2A\nr\u0002r\u0002 a=\u0000\u0016jFM(\u001a;a); (72)\nand similarly for EM,eMandfM. It should be noted, however, that these minimizations are trivial, since the choice\nof function spaces actually allows the relation \u0000\u0016j=r\u0002bto be inverted. By gauge invariance, the remaining\nnon-invertibility of b=r\u0002adoes not matter. Speci\fcally,\nFM;b(\u001a;B) =FM(\u001a;A+r\u001f);B=r\u0002A; (73)\nfor all\u001fsuch thatr\u001f2A.\nBy abuse of notation, the same symbol will be used for all three versions of the underlying functional|e.g., FM(\u001a;A),\nFM;b(\u001a;B), andFM;j(\u001a;J) will be written FM(\u001a;A),FM(\u001a;B), andFM(\u001a;J). With this notational convention Eq. (68)\ncan alternatively be expressed as\nFM(\u001a;B) =1\n2\u0016(BjB) + inf\n\f02L2\ndiv\u0010\n\u00001\n\u0016(\f0jB) +fM(\u001a;\f0)\u0011\n; (74)\nFM(\u001a;A) =1\n2\u0016(BjB) + inf\n\u001402Jdiv\u0010\n(\u00140jA) +fM(\u001a;\u00140)\u0011\n; (75)\nFM(\u001a;J) =\u0016\n8\u0019(Jjr\u00001\n12jJ) + inf\n\u000b02A\u0010\n(\u000b0jJ) +fM(\u001a;\u000b0)\u0011\n: (76)\nB. Reformulation using convex analysis I: generalized Lieb formulation\nThe ground state functional and the constrained-search functionals de\fned in the previous section have non-trivial\nmathematical properties, which can be characterized in terms of the convex analysis concepts reviewed in Sec. II.\nWe \frst turn turn to the convexity properties of the de\fned functionals as well as their interrelations through\nLegendre{Fenchel transformations. In order to distinguish partial Legendre{Fenchel transformations a\u000becting the\n\frst and second argument, respectively, we now modify the notation slightly. Transformation with respect to the\n\frst (second) argument will be denoted by ^and_(4and5) superscripts. Moreover, the pairing term in the\ntransformation w.r.t. the second argument will be taken to be1\n\u0016(\f0jB) rather than\u0000(\f0jB). With these conventions,\none may identify the Legendre{Fenchel transform\nEM(v;B) =F^\nM(v;B) = inf\n\u001a2X\u0000\n(\u001ajv) +FM(\u001a;B)\u0001\n; (77)\nin the \frst part of Eq. (70). Lieb's original proof of convexity and l.s.c. for the universal Levy{Lieb functional in\nstandard DFT requires only trivial adaptations to apply also to FM(\u001a;B) as a functional of \u001awithB\fxed. Hence,\nthe above transformation can be inverted,\nFM(v;B) =E_\nM(\u001a;B) = inf\nv2X\u0003\u0000\nEM(v;B)\u0000(\u001ajv)\u0001\n: (78)\nBoth the above functionals are paraconcave in B. In order to exploit this fact, it is useful to shift the ground state\nenergy by the magnetic-\feld energy g\u0016(B) =1\n2\u0016kBk2,\n\u0016EM(v;B) =EM(v;B)\u00001\n2\u0016kBk2; (79)\nyielding a functional that is jointly concave in ( v;B). This can be seen by noting that Eqs. (64) and (67) are instances\nof the general minimization over a\u000ene functions in Eq. (5). The concavity properties of \u0016EMare not restricted to the\n(v;B)-representation: \u0016EM(v;A) is jointly concave in ( v;A) and \u0016EM(v;J) is jointly concave in ( v;J).\nShifting the half-universal functional FMby the magnetic \feld energy yields the convex-concave functional\n\u0016FM(\u001a;B) =FM(\u001a;B)\u00001\n2\u0016kBk2; (80)12\nthat is, \u0016FMis convex in \u001aand concave in B. Once again, the convexity properties are not speci\fc to the ( \u001a;B)-\nrepresentation| \u0016FM(\u001a;A) and \u0016FM(\u001a;J) are convex-concave too.\nBy inspection of Eqs. (67) and (70), it is seen that the shifted ground-state energy can be obtained as a full\nLegendre{Fenchel transform of the universal functional, \u0016EM=f^4\nM, and also as half-transforms of two semi-universal\nfunctionals, \u0016EM=\u0016F^\nM=e4\nM. In detail,\n\u0016EM(v;B) =f^4\nM(v;B) = inf\n\u001a;\f0\u0000\n(\u001ajv)\u00001\n\u0016(\f0jB) +fM(\u001a;\f0)\u0001\n(81)\n=\u0016F^\nM(v;B) = inf\n\u001a;\f0\u0000\n(\u001ajv) +\u0016FM(\u001a;B)\u0001\n(82)\n=e4\nM(v;B) = inf\n\f0\u0000\n\u00001\n\u0016(\f0jB) +eM(v;\f0)\u0001\n: (83)\nAlso the semi-universal functional \u0016FMhas multiple representations as a Legendre{Fenchel transformation. It is a\nfull Legendre{Fenchel transform of the other semi-universal functional, \u0016FM=e4_\nM, and half-transforms of the shifted\nground state energy and universal functionals, \u0016FM=\u0016E_\nM=f4\nM. In detail, the half-transforms are\n\u0016FM(\u001a;B) =\u0016E_\nM(\u001a;B) = sup\nv2X\u0003\u0000\u0016EM(\u001a;A)\u0000(\u001ajv)\u0001\n(84)\n=f4\nM(\u001a;B) = inf\n\f02L2\ndiv\u0000\n\u00001\n\u0016(\f0jB) +fM(\u001a;\f0)\u0001\n: (85)\nInserting Eq. (83) into Eq. (84) yields the full transformation,\n\u0016FM(\u001a;B) =e4_\nM(\u001a;B) = sup\nv2X\u0003inf\n\f02L2\ndiv\u0000\n\u0000(\u001ajv)\u00001\n\u0016(\f0jB) +eM(v;\f0)\u0001\n: (86)\nAn alternative expression with the maximization and minimization interchanged is obtained by inserting Eq. (90) into\nEq. (85),\n\u0016FM(\u001a;B) =e_4\nM(\u001a;B) = inf\n\f02L2\ndivsup\nv2X\u0003\u0000\n\u0000(\u001ajv)\u00001\n\u0016(\f0jB) +eM(v;\f0)\u0001\n: (87)\nTurning next to the properties of fM, we rewrite Eq. (66) into\nfM(\u001a;\u000b0) =1\n2\u0016(\f0j\f0) +\u0016FGH(\u001a;\u000b0) +1\n2(\u001aj\u000b02); (88)\nwhere we temporarily switch back to regarding fMas a functional of \u000b0since the last two terms are separately gauge\ndependent. The sum of the terms, however, is gauge invariant. Each term above is separately convex in \u001a, so that\nthe sum is also convex. The \frst and third terms are also convex in \u000b0. The second term, however, is concave in \u000b0,\nleaving the concavity properties of the functional unclear. Without a proof of joint convexity and l.s.c. of fM, it is\nnecessary to distinguish this functional from its double Legendre{Fenchel transform. The functionals fMandeMdo,\nhowever, constitute a transform pair w.r.t. the \frst argument,\neM(v;\f0) =f^\nM(v;\f0) = inf\n\u001a2X\u0000\n(\u001ajv) +fM(\u001a;\f0)\u0001\n(89)\nand\nfM(\u001a;\f0) =e_\nM(\u001a;\f0) = sup\nv2X\u0003\u0000\neM(v;\f0)\u0000(\u001ajv)\u0001\n: (90)\nIn order to obtain a theory with invertible Legendre{Fenchel transformations, it is useful to de\fne the semi-universal\nfunctional\n\u0016eM(v;\f0) =\u0016E5\nM(v;\f0) = sup\nB2L2\ndiv\u0000\u0016EM(v;B) +1\n\u0016(\f0jB)\u0001\n(91)\nand the universal functional\n\u0016fM(\u001a;\f0) =\u0016E_5\nM(\u001a;\f0) = sup\nv2X\u0003\nB2L2\ndiv\u0000\u0016EM(v;B)\u0000(\u001ajv) +1\n\u0016(\f0jB)\u0001\n: (92)13\nAll of the expressions from Eq. (81{87) hold with eMandfMreplaced \u0016eMand \u0016fM, respectively. In particular, one\nobtains the transform pair \u0016EM=\u0016f^4\nMand \u0016fM=\u0016E_5\nM. The cycle of partial Legendre{Fenchel transformations is\nsummarized in Fig. 2. (As a technical remark, the partially transformed functionals \u0016 eM(v;\f0) and \u0016FM(\u001a;B) are\nsaddle-functionals that satisfy dual minimax and maximin principles [39]. The general theory of saddle functionals\ndoes not guarantee that these are unique saddle functionals, only that they are two equivalence classes of functionals\nthat can be used interchangeably in the variational principles. The equalities \u0016 eM=\u0016F_4\nM=\u0016F4_\nMand \u0016FM= \u0016e^5\nM= \u0016e5^\nM\nshould thus rigorously be understood as statements about equivalence classes.)\nThe above formulation includes an explicit variational principle (in Eq. (92)) for the universal functional \u0016fMthat\ngeneralizes Lieb's variational principle in standard DFT. This variational principle can be slightly generalized and\nused to provide a practical framework for computing systematic approximations to Adiabatic Connection curves that\ncharacterize the exact functional. Several studies have employed this method in the standard DFT framework [24{31]\nand the BDFT framework [34]. This generalization is done by replacing the electron{electron repulsion ^Win Eq. (51)\nby\u0015^W, where\u0015is a scaling factor.\nC. Reformulation using convex analysis II: MY and LL regularization\nBy inspection of Eq. (64), the Maxwell{Schr odinger ground state functional is seen to be the in\fmal convolution\nw.r.t. the second argument of the Schr odinger functional and the magnetic self-energy g\u0016(B) =kBk2=2\u0016= (BjB)=2\u0016.\nEM(v;B) = (E2g\u0016)(v;B) = inf\n\f02A\u0000\nE(v;\f0) +(B\u0000\f0jB\u0000\f0)\n2\u0016\u0001\n; (93)\nwhere we let 2denote convolution w.r.t. the second argument. We have here also chosen to regard the ground state\nenergy as a functional of ( v;B), since Band\f0are elements of the Hilbert space L2\ndiv. Due to the speci\fc form of\ng\u0016, the functional EMis a Moreau{Yosida regularization of E. Previous work has considered MY regularization of\nthe scalar potential in standard DFT [40], which corresponds to the MY regularization of the \frst argument (i.e., v)\nin the present setting. Due to concavity in v, this results in improved regularity so that the functional derivative,\nin addition to the sub- or supergradient, is well-de\fned [40]. However, the present situation is more involved since\nE(v;A) is neither concave nor convex in A. The MY regularized functional EM=E2g\u0016then does not automatically\nhave improved regularity.\nFrom inspection of Eq. (68), one \fnds\nFM(\u001a;B) = (FGH2g\u0016)(\u001a;B): (94)\nConsequently, the functionals shifted by the magnetic \feld energy can be written\n\u0016EM(\u001a;B) = (E2g\u0016)(\u001a;B)\u0000g\u0016(B); (95)\n\u0016FM(\u001a;B) = (FGH2g\u0016)(\u001a;B)\u0000g\u0016(B): (96)\nFor comparison, the constrained-search functionals eMandfM, are simply shifted by g\u0016rather than being MY\nregularizations,\neM(v;\u000b0) =E(v;\u000b0) +g\u0016(\u000b0); (97)\nfM(\u001a;\u000b0) =FGH(\u001a;\u000b0) +g\u0016(\u000b0): (98)\nThe MY regularizations and additive shifts that relate the above functionals to the standard energy functional and\nthe Grayce{Harris functional are illustrated in Fig. 2.\nInserting the identity\n1\n\u0016(\f0jB) =g\u0016(B)\u0000g(B\u0000\f0) +g\u0016(\f0) (99)\ninto Eq. (91) above yields\n\u0016eM(v;\f0) = sup\nB2L2\ndiv\u0000\nEM(v;B)\u0000g(B\u0000\f0) +g\u0016(\f0)\u0001\n=\u0000(g\u00162\u0000EM)(v;\f0) +g\u0016(\f0):(100)\nFinally, inserting Eq. (93) yields\n\u0016eM(v;\f0) =\u0000(g\u00162\u0000(g\u00162E))(v;\f0) +g\u0016(\f0) =L\u0016\u0016E(v;\f0) +g\u0016(\f0): (101)14\nVery similar reasoning shows that\n\u0016fM(\u001a;\f0) =\u0000(g\u00162\u0000(g\u00162FGH))(\u001a;\f0) +g\u0016(\f0) =L\u0016\u0016FGH(\u001a;\f0) +g\u0016(\f0): (102)\nComparing to the generic Eq. (15), the above two expressions can be recognized as the proximal hull, or a LL\nregularization, of the Schr odinger functional Eand the semi-universal functional FGH.\nD. Reformulation using convex analysis III: di\u000berentiable LL regularization\nIn the above treatment, the parameter \u0016plays a dual role as it is used both to obtain EMfromEand in the\ncycle of Legendre{Fenchel transformations summarized in Fig. 2. In fact, at the cost of somewhat complicating the\nrelationship to the constrained-search functionals, it is possible to operate with two di\u000berent regularization parameters,\n\u0010and\u0015. The parameter \u0010is used in the MY regularization of the Schr odinger energy and Grayce{Harris functional:\nEM(v;B) =M\u0010E(v;B) andFM(\u001a;B) =M\u0010FGH(\u001a;B). The parameter \u0015\u0014\u0010is used to de\fne the jointly concave\nfunctional\n\u0016EM(v;B) =\u0000g\u0015(B) +M\u0010E(v;B) =\u0000g\u0010(B) +M\u0010E(v;B) +\u0000\ng\u0010(B)\u0000g\u0015(B)\u0001\n; (103)\nwhere\u0000g\u0010+M\u0010Eandg\u0010\u0000g\u0015are both concave, and the convex-concave functional\n\u0016FM(\u001a;B) =\u0000g\u0015(B) +M\u0010FGH(\u001a;B): (104)\nThe parameter \u0015is also used in the cycle of Legendre{Fenchel transforms,\n\u0016eM(v;\f0) =\u0016E5\nM(v;\f0) = sup\nB2L2\ndiv\u0000\u0016EM(v;B) +1\n\u0015(\f0jB)\u0001\n; (105)\n\u0016fM(\u001a;\f0) =\u0016F5\nM(\u001a;\f0) = sup\nB2L2\ndiv\u0000\u0016FM(\u001a;B) +1\n\u0015(\f0jB)\u0001\n; (106)\nand similarly for the inverse partial transformations. The above expressions are equivalent to the following LL\nregularizations:\n\u0016eM(v;\f0) =k\f0k2\n2\u0015+L\u0015\u0010E(v;\f0); (107)\n\u0016fM(\u001a;\f0) =k\f0k2\n2\u0015+L\u0015\u0010FGH(\u001a;\f0): (108)\nEquivalence follows by writing out de\fnition of the LL regularizations and expanding terms representing the norm of\nmagnetic \feld di\u000berences.\nBesides the additional \rexibility, the advantage of this generalization is that functional di\u000berentiability can be\nestablished. Di\u000berentiability with respect to the \frst argument (scalar potentials and densities) has been shown to\nbe a non-trivial issue and the standard DFT functionals are in fact not di\u000berentiable in the ordinary sense [23].\nFor a \fxed v, it follows from Eq. (59) that the Schr odinger energy functional is bounded by a quadratic function,\nE(v;B)\u0015\u0000const\u0001(kBk2+ 1). For such quadratically bounded functions, Theorem 4.1 due to Attouch and Aze is\napplicable [22]. When \u0015 < \u0010 , this theorem guarantees that L\u0015\u0010E, and therefore also \u0016 eM, has a H older continuous\n(Fr\u0013 echet) derivative as a function of the magnetic \feld (for a \fxed v). Because the constraint \u0000 7!\u001aonly raises the\nenergy, the bound in Eq. (59) can be straightforwardly adapted to a quadratic bound on the Grayce{Harris functional,\n(\u001ajv) +FGH(\u001a;B) =\u0000kBk2\n2\u0010+ inf\n\u00007!\u001a\u0010kBk2\n2\u0010+ tr(\u0000( ^TA+v+W))\u0011\n\u0015\u0000kBk2\n2\u0010+EM(v;0); (109)\nwhereEM=M\u0010E. Hence, for \u0015 < \u0010 and a \fxed \u001a, it follows that also \u0016fM=g\u0015+L\u0015\u0010FGHis di\u000berentiable as a\nfunction of the magnetic \feld.\nE. Current densities, subgradients, and a Hohenberg{Kohn-like mapping\nTwo obstacles that to date have prevented the formulation of a CDFT based on the physical current density get\nnatural resolutions in the present formulation. Firstly, the physical current density in Eq. (31) explicitly depends on15\nthe external potential, making it di\u000ecult to construct a universal functional of ( \u001a;j). Secondly, it has been shown that\na physical CDFT is inconsistent with the standard variational principle [14]. It is implicit in this result that the wave\nfunction (or density matrix) is the only variational parameter and that physical current densities are obtained in the\nstandard way from the wave function and the external magnetic vector potential. In the present Maxwell{Schr odinger\nformulation, the internal magnetic \feld is an additional variational parameter and the physical current density can\nbe obtained as\nj\u000b0=1\n2X\nknk\u001ek(r)y\u001b(\u001b\u0001\u0019\u000b0)\u001ek(r) + c.c.; (110)\nwherenkare natural occupation numbers and \u001eknatural spinors. Both obstacles are circumvented because the\ninternal magnetic \feld, and its associated current density \u0014=\u0000r\u0002 \f=\u0016, is a variational degree of freedom that is\ncompletely independent of the wave function. The current density j\u000b0depends directly on the wave function, but\ndoes not \fgure in the variational principle. Nonetheless, as will be detailed below, the resulting theory is physical\nCDFT because energy-minimization leads to the condition \u0014=j\u000b0. Hence, although \u0014is an independent parameter\nin the variational principle, its optimal value is always equivalent to the physical current density.\nConsider the case when a minimizer ( \u000b; ) exists in Eq. (51). Further assume that this minimizer is unique up to\ngauge transformations. Then\nEM(v;A) =1\n2\u0016(\fj\f) +h j^T\u000b+A+ ^v+^Wj i: (111)\nFor any a2A, the Hellmann{Feynman theorem yields\nd\nd\u000fEM(v;A+\u000fa)\f\f\f\n\u000f=0= (ajj): (112)\nAny pure gauge term a=r\u001fyields a vanishing contribution, since ( r\u001fjj) =\u0000(\u001fjr\u0001j) andr\u0001j= 0 for an\nenergy-optimized state. After the reparametrization \u000b0=\u000b+A, one may also write\nEM(v;A) =1\n2\u0016(B\u0000\f0jB\u0000\f0) +h j^T\u000b0+ ^v+^Wj i; (113)\nwhere ( \u000b0; ) is now the unique minimizer (up to gauge transformations) of the reparametrized minimization problem\nin Eq. (64). The functional derivative now takes the form\nd\nd\u000fEM(v;A+\u000fa)\f\f\f\n\u000f=0=1\n\u0016(r\u0002ajB\u0000\f0) =\u00001\n\u0016(ajr\u0002 \f): (114)\nThe above two forms are consistent, because any minimizer must satisfy Amp\u0013 ere's circuital law, r\u0002\f=\u0000\u0016j. In\nthe present setting, this equation is the stationarity condition for variations with respect to \u000b. Without entering into\ntechnical details about functional derivatives, we may write informally\n\u000eEM(v;A)\n\u000ev=\u001a; (115)\n\u000eEM(v;A)\n\u000eA=\u0014=j: (116)\nFor the concave functional \u0016EM(v;A), one obtains\n\u000e\u0016EM(v;A)\n\u000ev=\u001a; (117)\n\u000e\u0016EM(v;A)\n\u000eA=\u0014+J=\u00140: (118)\nand the alternative forms\n\u000e\u0016EM(v;B)\n\u000eB=\u00001\n\u0016(\f+B) =\u00001\n\u0016\f0; (119)\n\u000e\u0016EM(v;J)\n\u000eJ=\u000b+A=\u000b0; (120)16\nwhere gauge invariance of the last expression requires that \u000b02A divis divergence free.\nLet us now focus on the ( v;B)-representation as it is straightforward to obtain analogous results in the ( v;A)- and\n(v;J)-representations. Because \u0016EM(v;B) is a concave functional, the above somewhat informal functional derivatives\ncan be replaced by the rigorous notion of sub- and superdi\u000berentials. For given external potentials ( v0;B0), any\nminimizer ( \u001a0;\f0\n0) in Eq. (81) is a supergradient. The functional value at an arbitrary external potential pair always\nlies below the linear extrapolation based on such a supergradient,\n\u0016EM(v;B)\u0014\u0016EM(v0;B0) + (\u001a0jv\u0000v0)\u00001\n\u0016(\f0\n0jB\u0000B0);8v;B: (121)\nThe minimizer ( \u001a0;\f0\n0) need not be unique in order for this to hold.\nAnalogously, the functional \u0016fM(\u001a;\f0) is convex. For a given density pair ( \u001a0;\f0\n0), assume that a maximizer ( v0;B0)\nexists in Eq. (92). Then the subgradient of \u0016fM(v0;\f0\n0) is (\u0000v0;\u0000B0). The linear extrapolation based on the subgradient\nyields a global upper bound,\n\u0016fM(\u001a;\f0)\u0015\u0016fM(\u001a0;\f0\n0)\u0000(\u001a\u0000\u001a0jv0) +1\n\u0016(\f0\u0000\f0\n0jB0);8\u001a;\f0: (122)\nAgain, the maximizing potentials ( v0;B0) need not be unique for this to hold.\nThe superdi\u000berential @\u0016EM(v;B) is the set of all supergradients at ( v;B). This set can be empty, contain a single\nelement, or contain multiple elements. Any minimizing pair ( \u001a;\f0) in Eq. (81) contributes an element ( \u001a;\f0)2\n@\u0016EM(v;B) to the superdi\u000berential. Hence, more than element in @\u0016EM(v;B) implies a ground state degeneracy. When\nno minimizing pair exists, the superdi\u000berential @\u0016EM(v;B) is the empty set.\nThe subdi\u000berential @\u0016fM(\u001a;\f0) is the set of all subgradients at ( \u001a;\f0). For density pairs ( \u001a;\f0) that are not obtainable\nas the ground state density of any ( v;B), the subdi\u000berential @\u0016fM(\u001a;\f0) is the empty set. When there a multiple\npotentials that all yield the density ( \u001a;\f0) as the minimizer, the subdi\u000berential @\u0016fM(\u001a;\f0) has as many elements.\nIt is a general fact about convex functionals that there is a one-to-one mapping between non-empty subdi\u000berentials\nand non-empty superdi\u000berentials of the convex conjugate. In the present setting, there is a one-to-one mapping be-\ntween subdi\u000berentials @\u0016fM(\u001a;\f0) and superdi\u000berentials @\u0016EM(v;B). This mapping can be expressed with the following\nequivalence:\n(\u001a;\f0)2@\u0016EM(v;B)() (\u0000v;\u0000B)2@\u0016fM(\u001a;\f0): (123)\nHence, the present theory admits a mapping that generalizes the usual Hohenberg{Kohn mapping in the standard\nDFT. This mapping exists for all positive values \u0016 > 0 of the regularization parameter. In addition, the Maxwell{\nSchr odinger model tends to the standard Schr odinger model in the \u0016!0 limit in the sense that\nlim\n\u0016!0+EM(v;A) =E(v;A) (124)\nholds at least pointwise. We therefore speculate that the limit \u0016!0 produces a Hohenberg{Kohn-like mapping\nbetween density pairs ( \u001a;j) and potentials ( v;B) also in the standard Schr odinger model. This sheds new light on a\nlong-standing open problem in the CDFT literature [7, 8, 10, 12].\nV. SPECIALIZATION TO UNIFORM MAGNETIC FIELDS\nRecent work has established a specialization of paramagnetic CDFT to uniform magnetic \felds [41]. In this\ntheory, the basic variables are the density \u001a, the canonical momentum p=R\njpdr, and the paramagnetic moment\nMp=R\nr\u0002jpdr+geS. Due to the gauge-dependence and non-extensivity of the paramagnetic moment, it is\nchallenging to \fnd an approximate, practical Ansatz for the exchange-correlation functional that preserves additivity\nover well-separated, non-interacting subsystems [41].\nIn order to obtain a parallel specializiation of the above MDFT formulation to uniform magnetic \felds, we \frst\nintroduce a \fnite spatial domain VNand take the energy of a uniform \feld to be1\n2\u0016R\nVNB2dr=jVNjB2=2\u0016. The\nN-dependence of the volume will be discussed shortly. Interpreting \u0016as a regularization parameter, rather than a\nphysical parameter with a \fxed empirical value, the volume jVNjcan be made arbitrarily large while holding the ratio\n\u0017N=\u0016=jVNj\fxed. The energy can be written\nEM(v;B) = inf\n\u001a2X;\n\u000b0\u0010\n(\u001ajv) +j\f0\u0000Bj2\n2\u0017N+ inf\n\u00007!\u001atr(\u0000( ^T\u000b0+^W))\u0011\n; (125)17\nwhere the minimization is over all vector potentials \u000b0representing homogenous \felds \f0. Taking the derivative with\nrespect to Byields the physical magnetic moment,\nM= 2@EM(v;B)\n@B= 2B\u0000\f0\n\u0017N=\u00002\n\u0017N\f: (126)\nHence, the internal magnetic \feld, \f=\u0000\u0017NM=2, is essentially the magnetic moment in di\u000berent units.\nUnlike the paramagnetic moment Mp, the physical magnetic moment Mis an extensive quantity. If jVNj=jV1jis\nindependent of N, also the internal magnetic \feld \fbecomes an extensive quantity. If the volume is instead scaled\nwith the number of particles, jVNj=NjV1j, one obtains the more natural case that \fis an intensive quantity. In\nwhat follows, we suppress the dependence on Nfrom the notation and write \u0017.\nThe simpli\fed setting o\u000bered by this reduction to a three-dimensional space of external and internal magnetic \felds\ncan be viewed as a theory in its own right. Alternatively, calculations in this setting can be viewed as idealized,\nschematic illustrations of the full theory. On this note, two model cases will be considered.\nA. Example: weak-\feld regime of an atomic system\nThe weak-\feld regime of an atom with paramagnetic moment 2 \r >0 (directed anti-parallel to the magnetic \feld)\nand isotropic magnetizability \u00002\u0018<0 is described by\nE(v;B) =E(v;0)\u0000\rjBj+\u0018B2; (127)\nwherev(r) =\u0000Z=r. Due to the even symmetry of this function it is su\u000ecient to study the case B;\f0\u00150, leading to\nEM(v;B) = inf\n\f0\u00150\u0010(\f0\u0000B)2\n2\u0017+E(v;\f0)\u0011\n=E(v;0) +\u0018B2\u0000\rB\u00001\n2\u0017\r2\n1 + 2\u0017\u0018: (128)\nThe energy for B\u00140 is obtained by mirroring the positive side, EM(v;\u0000B) =EM(v;B). From the above paraconcave\nfunctional of Bwe can form the concave functional\n\u0016EM(v;B) =EM(v;B)\u0000B2\n2\u0017=E(v;0)\u00001\n2\u0017B2+\rjBj+1\n2\u0017\r2\n1 + 2\u0017\u0018=E(v;0)\u00001\n2\u0017\u0000\njBj+\u0017\r\u00012\n1 + 2\u0017\u0018: (129)\nA Legendre{Fenchel transform of the Bvariable now yields\n\u0016eM(v;\f0) = sup\nB\u0000\u0016EM(v;B)\u0000\f0B\n\u0017\u0001\n=(\nE(v;0)\u0000\u0017\n2(1+2\u0017\u0018)\r2; ifj\f0j\u0014\u0017\r=(1 + 2\u0017\u0018);\nE(v;0)\u0000\rj\f0j+ (\u0018+1\n2\u0017)\f02;otherwise:(130)\nFor comparison, the constrained-search functional is\neM(v;\f0) =E(v;0)\u0000\rj\f0j+\u0000\n\u0018+1\n2\u0017\u0001\n\f02: (131)\nThis functional is not convex, but agrees with the convex functional \u0016 eMon the subdomain j\f0j\u0015\u0017\r=(1 + 2\u0017\u0018). In the\nother part of the domain, j\f0j<\u0017\r= (1 + 2\u0017\u0018), the constrained-search functional eMis strictly larger than the convex\nenvelope \u0016eM.\nA numerical example, obtained by setting E(v;0) = 0,\u0018= 5,\r= 1, and\u0017= 0:1, is shown in Fig. 3. Except for\nthe cusp at b= 0, all functionals are di\u000berentiable. The optimization problems\n\u0016EM(v;B) = \u0016e4\nM(v;B) = inf\n\f0\u0000\n\u0000\f0B\n\u0017+ \u0016eM(v;\f0)\u0001\n; (132)\n\u0016eM(v;\f0) =\u0016E5\nM(v;\f0) = sup\nB\u0000\u0016EM(v;B) +\f0B\n\u0017\u0001\n; (133)\nare therefore solved when\n@\u0016eM(v;\f0)\n@\f0\u0000B\n\u0017= 0; (134)\n@\u0016EM(v;B)\n@B+\f0\n\u0017= 0: (135)18\nThese equations establish mapping between external \felds Band total \felds \f0that is an instance of a Hohenberg{\nKohn-like mapping. The solution pair B= 0:1 and\f0= 0:1 is illustrated in Fig. 4. At B= 0, the mapping becomes\nmany-to-one. At B= 0, the supergradient is a \fnite interval,\n@\u0016EM(v;B= 0) =f\f0jj\f0j\u0014\r=(1 + 2\u0017\r)g: (136)\nThis interval of total \felds is mapped onto the unique external \feld in the subgradients\n@\u0016eM(v;\f0) =f0g;for allj\f0j\u0014\r=(1 + 2\u0017\r): (137)\nB. Example: closed-shell paramagnetism\nIn closed-shell paramagnetic systems, the permanent magnetic moment vanishes and there is only an induced\nmagnetic moment. Moreover, the energy initially decreases with the magnetic \feld strength. A paradigmatic example\nis the boron monohydride molecule, which is paramagnetic with respect to magnetic \felds perpendicular to the bond\naxis. For weak and intermediate \feld strengths, the magnetic-\feld variation of the energy is well described by the\nfollowing two-state model Hamiltonian [42],\nH=\u0012\nE0\u00001\n2\u001f0B2\u0000im\nim E 1\u00001\n2\u001f1B2\u0013\n; (138)\nwith suitably chosen values for the model parameters E0,E1,m,\u001f0, and\u001f1. The numerical example E0= 0,\nE1= 0:01,m= 0:6,\u001f0=\u000014, and\u001f1=\u00008 captures the essential qualitative features|paramagnetism for weak\n\felds and eventually a transition to diamagnetism. The e\u000bects of varying scalar potential or density are not captured|\nvis held \fxed at the potential that corresponds to the electrostatic attraction to the boron and hydrogen nuclei. The\nresulting energy curve, E(v;b), is shown in Fig. 5 together with related energy functionals obtained with a large value\nof the regularization parameter \u0017= 0:3 in order to make the e\u000bects visible. The fact that the Maxwell{Schr odinger\nenergyEM(v;b) has a non-di\u000berentiable cusp at b= 0 illustrates that a MY regularization of a non-convex function\nsometimes results in less regularity. Moreover, the curvature of the energy EM(v;b) nearb= 0 has the opposite\nsign compared to E(v;b). This occurs because the state with vanishing internal \feld \f= 0 is unstable and it is\nenergetically favorable for the system to spontaneously break time-reversal symmetry by developing an internal \feld.\nThis e\u000bect only occurs for large enough \u0017so that\n1\n\u0017+@2E(v;B)\n@B2\f\f\f\nB=0<0: (139)\nHence, the value of the regularization parameter also sets a limit on the maximum negative curvature that can occur\ninEwithout resulting in cusps in EM. In Fig. 6, the e\u000bect on EMof decreasing the regularization parameter is\nillustrated. A value of \u0017= 0:005 is su\u000eciently small to avoid misrepresenting the sign of the curvature around b= 0.\nVI. DISCUSSION AND CONCLUSION\nThe MDFT formalism detailed above constitutes an alternative to the two most well-established extensions of\nstandard density functional theory that accomodate external magnetic \felds|BDFT [5] and paramagnetic CDFT [6].\nIt is also bene\fcial from the perspective that a non-relativistic DFT should be a rigorous limit of a relativistic DFT as\nit does not rely on the paramagnetic current density nor the change of variables u=v+1\n2A2. The key insight behind\nMDFT is that replacing the standard energy functional by the energy of a Maxwell{Schr odinger model leads to a\nparaconcave energy functional EM(v;A). Moreover, subtraction of a function with the physical interpretation as the\nmagnetic self-energy1\n2\u0016(BjB) leads to a concave functional. Universal functionals of ( \u001a;\f0)|or, equivalently, of ( \u001a;\u000b0)\nor (\u001a;\u00140)|have been constructed both as constrained-search functionals and as Legendre{Fenchel transformations.\nThe latter generalize the previous convex formulation due to Lieb [3] and the cycle of partial transforms detailed in\nSec. IV B and Fig. 2 parallels the corresponding cycle for paramagnetic CDFT in Sec. III and Fig. 1 (see also Ref. 34).\nEach transformation is a variational principle that the Maxwell{Schr odinger energy and related MDFT functionals\nsatisfy.\nAs a complement to the previous discovery that a change of variables can bring out hidden convexity properties\nofE(v;A) [10], the results in Sec. IV C show that a MY regularization brings out a di\u000berent, rich set of convexity\nproperties. The fact that the energy functionals are MY and LL regularizations of the standard Schr odinger energy19\nalso enables a reinterpretation of the theory and a rationale for using other values of \u0016than the actual vacuum\npermeability: EM(v;B) is the closest approximation to E(v;B) when curvature (w.r.t. B) is not allowed to exceed\n1=\u0016. Hence, the parameter \u0016can be used to choose a trade o\u000b between regularity (maximum curvature) and accuracy.\nWhen generalized to a two-parameter family of LL regularizations, as in Sec. IV D, the universal functional is even\na di\u000berentiable functional of the magnetic \feld. This complements the previous work by Kvaal, et al., on MY\nregularization with respect to the scalar potential and density variables [40].\nA direct consequence of the convex formulation is the existence of a Hohenberg{Kohn-like mapping between (pos-\nsibly degenerate) density pairs ( \u001a;\u00140) and potential pairs ( v;B0). Moreover, the fact that this mapping exists for\nall parameter values \u0016 > 0 sheds light on the status of Hohenberg{Kohn-like results for physical current densi-\nties within the Schr odinger model: counter-examples, if they exists at all, likely rely on pathological properties of\nE(v;B) that cannot be well approximated by any MY regularization with bounded curvature. A recent independent\nwork has established a comparable Hohenberg{Kohn-like mapping within the non-relativistic QED setting, where the\nelectromagnetic \felds are quantized [43].\nFinally, practical experience with available approximate paramagnetic CDFT functionals has shown that vorticity\nis a numerically di\u000ecult quantity to work with [44{46]. 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Chem. Phys. 11, 5489 (2009).\n[43] M. Ruggenthaler, \\Ground{state quantum-electrodynamical density-functional theory,\" (2017), arXiv:1509.01417.\n[44] W. Zhu and S. B. Trickey, J. Chem. Phys. 125, 094317 (2006).\n[45] W. Zhu, L. Zhang, and S. B. Trickey, Phys. Rev. A 90, 022504 (2014).\n[46] E. I. Tellgren, A. M. Teale, J. W. Furness, K. K. Lange, U. Ekstr om, and T. Helgaker, J. Chem. Phys. 140, 034101 (2014).21\nFIG. 1. Graphical representation of partial Legendre{Fenchel transforms, represented by double arrows, between functionals\nin the convex formulation of paramagnetic CDFT. Single arrows represent the the change of variables and additive shift by\nthe diamagnetic energy that relate the cycle of partial transforms to the standard energy functional and the Grayce{Harris\nfunctional. The designations `concave-gen.' and `convex-gen.' are used for functionals that are generic in their second argument,\ni.e., do not have any apparent convexity properties.\nFGH(⇢, A)convex-gen.¯FGH(⇢, A)convex-concaveFGH\u000012(⇢|A2)E(v,A)concave-gen.¯E(u, A)concave¯e(u,jm)concave-convex¯fVR(⇢,jm)convex±12(⇢|A2)u=v+12A2\n122\nFIG. 2. Graphical representation of the relations between functionals in the convex formulation of MDFT. Double arrows\nrepresent Legendre{Fenchel transformations, while single arrows represent MY regularization and additive shifts of the func-\ntionals.\n¯EM(v,A)concaveMµE\u0000gµ¯FM(⇢, A)convex-concaveMµFGH\u0000gµ¯eM(v,0)concave-convexLµµE+gµ¯fM(⇢, 0)convexLµµFGH+gµfM(⇢, 0)convex-gen.FGH+gµFGH(⇢, A)convex-gen.E(v,A)concave-gen.MY,\u0000gµ\nMY,\u0000gµ±gµ\n1\nFIG. 3. Functionals related to the model energy for an atom in the weak-\feld regime. MY regularization of the non-convex\nE(v;b) (black dotted curve) results in the paraconcave functional EM(v;b) (blue dash-dot curve), which is non-di\u000berentiable\natb= 0. Subtraction of the magnetic \feld energy yields the concave functional \u0016EM(v;b) (solid blue curve). Addition of the\nmagnetic \feld energy to E(v;b) yields the functional eM(v;b) (red dashed curve), which remains non-convex. Finally, a double\nconvex conjugation yields the convex envelope \u0016 eM(v;b) (solid red curve), which is constant on the interval between the minima\nofeM(v;b).\n-0.15 -0.1 -0.05 0 0.05 0.1-0.05-0.045-0.04-0.035-0.03-0.025-0.02-0.015-0.01-0.0050Energy (hartree)23\nFIG. 4. On the left, the concave functional \u0016EM(v;B) is displayed together with all the supergradients at two selected points\n(marked by circles). There is an in\fnite family of supergradients (dashed lines) at the non-di\u000berentiable point B= 0. At\nthe di\u000berentiable point B= 0:1, the supergradient (dash-dot line) is unique. On the right, the convex functional \u0016 eM(v;\f0) is\ndisplayed. The minimum is attained on a \fnite interval (black \fgure). The unique subgradient at \f0= 0:1 is also displayed\n(dash-dot line). The correspondence between the in\fnite family of subdi\u000berentials at the maximum of \u0016EMand the unique\nsubgradient at the minimum of \u0016 eMis an example of the Hohenberg{Kohn-like mapping that exists in the formalism. Another\nexample is the correspondence between the supergradient of \u0016EM(v;0:1) and the subgradient of \u0016 eM(v;0:1).\n-0.15 -0.1 -0.05 0 0.05 0.1 0.15-0.15-0.1-0.050\n-0.15 -0.1 -0.05 0 0.05 0.1 0.15-0.04-0.0200.020.040.06\nFIG. 5. Functionals related to the two-state model. MY regularization of the non-convex E(v;b) (black dotted curve) results\nin the paraconcave functional EM(v;b) (blue dash-dot curve), which is non-di\u000berentiable at b= 0. Subtraction of the magnetic\n\feld energy yields the concave functional \u0016EM(v;b). Addition of the magnetic \feld energy to E(v;b) yields the functional\neM(v;b) (red solid curve), which remains non-convex. Finally, a double convex conjugation yields the convex envelope \u0016 eM(v;b)\n(red dotted curve), which is constant on the interval between the minima of eM(v;b).\n-0.15 -0.1 -0.05 0 0.05 0.1-14-12-10-8-6-4-2024Energy (hartree)10-324\nFIG. 6. MY regularizations M\u0017E(v;B) of the two-state model energy E(v;B) (solid black curve). Approximations obtained\nwith three di\u000berent values \u0017= 0:005;0:05;0:2 of the regularization parameter are shown (blue dash-dot curves). Smaller values\nyield closer approximations and \u0017= 0:005 is su\u000eciently conservative to preserve the \fnite, negative curvature around B= 0.\n-0.06 -0.04 -0.02 0 0.02 0.04 0.06-12-10-8-6-4-202Energy (hartree)10-3"    },    {        "title": "1711.02991v3.Effect_of_density_of_states_peculiarities_on_Hund_s_metal_behavior.pdf",        "content": "arXiv:1711.02991v3  [cond-mat.str-el]  19 Mar 2018Effect of density of states peculiarities on Hund’s metal beh avior\nA. S. Belozerov,1,2A. A. Katanin,1and V. I. Anisimov1,2\n1M. N. Mikheev Institute of Metal Physics, Russian Academy of Sciences, 620137 Yekaterinburg, Russia\n2Ural Federal University, 620002 Yekaterinburg, Russia\nWe investigate a possibility of Hund’s metal behavior in the Hubbard model with asymmetric\ndensity of states having peak(s). Specifically, we consider the degenerate two-band model and\ncompare its results to the five-band model with realistic den sity of states of iron and nickel, showing\nthattheobtainedresults aremore general, providedthatth ehybridizationbetweenstatesofdifferent\nsymmetry is sufficiently small. We find that quasiparticle dam ping and the formation of local\nmagnetic moments due to Hund’s exchange interaction are enh anced by both, the density of states\nasymmetry, which yields stronger correlated electron or ho le excitations, and the larger density of\nstates at the Fermi level, increasing the number of virtual e lectron-hole excitations. For realistic\ndensities ofstates these twofactors are often interrelate d because the Fermi levelis attracted towards\npeaksofthedensityofstates. Wediscusstheimplication of theobtainedresultstovarioussubstances\nand compounds, such as transition metals, iron pnictides, a nd cuprates.\nI. INTRODUCTION\nSome components of Coulomb interaction, in partic-\nular, Hund’s exchange, play special role in multiorbital\nsystems, since they induce electronic correlations, which\nmay drive an unusual behavior of the electronic degrees\nof freedom. Hund’s metals were defined [1] as met-\nals with large quasiparticle mass, caused by Hund’s ex-\nchange interaction, cf. Refs. [2–5]. Apart from en-\nhancement of quasiparticle damping (and the possibility\nof non-quasiparticle behavior), Hund’s interaction also\nyields in the vicinity of half filling the so-called ’spin\nfreezing’ (i.e. local moment) behavior, as was shown in\nRef. [5] for semicircular density of states (DOS), and in-\nvestigated for a number of real materials [6–14]. This\nbehavior can be considered as one of the distinctive fea-\ntures of Hund’s metals and it is characterized by the\ntemperature-independent local spin correlation function\nK(τ) =∝angbracketleftSi(τ)Si(0)∝angbracketrightfor the imaginary time τnot too\nclose to 0 or β= 1/T(iis the site index, Tis the tem-\nperature). The τ-dependence of the correlation function\nK(τ) in this regime is rather weak, which then yields\n(approximate) fulfillment of the Curie law for the static\nlocalsusceptibility χloc=/integraltextβ\n0K(τ)dτ∝1/T,correspond-\ning to a local moment formation. At the transition to\nthe spin freezing (i.e. local moment) phase, the non-\nFermi-liquid ( ω1/2) behavior of electronic self-energy has\nbeen obtained [5], while inside the spin freezing phasethe\nquasiparticle damping is essentially enhanced.\nIn realistic substances and compounds only electrons\nbelonging to part of the bands may show Hund’s metal\nbehavior, which is reminiscent of the orbital-selective\nMott transitions, introduced first to explain unusual\nproperties of Ca 2−xSrxRuO4[15] and studied actively\nwithin the dynamical mean-field theory [16–22]. In this\nrespect, the “orbitalselective”Hund’s metals can be con-\nsideredasbeinginproximitytotheorbital-selectiveMott\ntransitions. The prominent example is α-iron, which\nhas a non-quasiparticle form of egstates [7], while the\nt2gstates are more quasiparticle-like, although they alsoshow some deviations from the Fermi liquid behavior [8].\nWhile for conventional orbital-selective Mott transi-\ntions, the widths of the bands and/or their fillings are\nthe decisive factors in which bands undergo the transi-\ntion, the widths and fillings ofdifferent bands in “orbital-\nselective” Hund’s metals are close to each other, and the\nstates, exhibiting Hund’s metal behavior (including the\nlocal moment formation), are determined to a large ex-\ntent by the profile of the partial density of states. In-\ndeed, the egelectrons in α-iron, for which the above de-\nscribed features of Hund’s metal behavior are especially\npronounced, havethe Fermi level almost at the top of the\npeak of the density of states; in γ-iron [12] and pnictides\n[14, 23–27] the peak of the density of states is somewhat\nshifted with respect to the Fermi level, which is accompa-\nnied by weaker (in comparison with α-iron) quasiparticle\ndamping and only partially formed local moments [12–\n14]. Therefore, the questions can be posed as to which\nfactors(apartfromthe filling)aredecisiveforthat, which\nstates (if any) show Hund’s metal behavior in these sub-\nstances, and whether the proximity of the Fermi level to\nthe peak of the density of states is a necessary/sufficient\ncondition for that.\nIn the present paper we show that the quasiparticle\ndamping and formation of local moments are in fact en-\nhanced by the asymmetry of the density of states (which\noften shifts the Fermi level towards the maximum of the\ndensity of states), as well as by the larger value of the\ndensity of states at the Fermi level. In the presence of\nthe asymmetry, the behavior of the self-energy and spin\ncorrelation function above- and below half filling is dras-\ntically different; the Hund’s metal behavior is enhanced\nfor the Fermi level being on the same side, as the maxi-\nmum of the density of states. We also find that Hund’s\nexchange interaction is necessary to make these factors\nactive, similarly to a previous observation for the sym-\nmetric density of states [5].\nThe plan of the paper is the following. In Sec. II we\nconsider the results for the self-energies and local suscep-\ntibilities of the two-band model. In Sec. III we consider2\nthe results of the five-band model and implications of\nthe obtained results to real substances and compounds.\nFinally, in Sec. IV we present conclusions.\nII. TWO-BAND MODEL\nToinvestigatetheeffects ofthepeculiaritiesofthe den-\nsity of states on Hund’s metal behavior, we perform dy-\nnamical mean-field theory (DMFT) calculations for the\ndegenerate two-band Hubbard model (we have verified\nthat the three-band model yields similar results)\nH=/summationdisplay\nijmσtijc+\nimσcjmσ+U/summationdisplay\nimnim↑nim↓ (1)\n+/summationdisplay\ni,m>m′,σ[U′nimσnim′σ+(U′−I)nimσnim′σ],\nwherecimσ(c+\nimσ) are the electron destruction (creation)\noperators with spin σ(=↑,↓) at site iand orbital m=\n1,2;nimσisthe numberoperatorofelectrons, Uisthein-\ntraorbital Coulomb interaction, Iis the Hund’s coupling,\nandU′=U−2I. Toobtainthecorrelationstrengthsim-\nilar to that in Hund’s metals such as pure iron and iron\npnictides, we consider the interaction values U= 1.5D\nandI=U/4, where Dis half of the bandwidth. The im-\npurityproblemwassolvedbythehybridizationexpansion\ncontinuous-time quantum Monte Carlo method [28].\nLet us first consider the results for the DOS ρa(ε) =\nc√\nD2−ε2/(D−aε) withc= (1+√\n1−a2)/(πD), sug-\ngested in Ref. [29] and leading to a peak in the den-\nsity of states for aclose to one (see Fig. 1a). Note\nthat in this and the following calculations, we enforce\nthe paramagnetic state by assuming spin- and site inde-\npendent self-energy, since we are interested in the for-\nmation of local moments (see below). In Fig. 1b we\npresentthe imaginarypartsofself-energiesobtainedwith\nband fillings n= 1.1 andn= 1.3 (here and below the\nband fillings are indicated per band) at the inverse tem-\nperature β= 1/T= 40D−1. One can see that for\nboth fillings increasing asymmetry parameter ayields a\nlarger absolute value of the imaginary part of the self-\nenergy; for n= 1.1 and strongest asymmetry a= 0.98\nthe absolute value of the self-energy is even increasing\nwith decreasing Matsubara frequency, corresponding to\nthe non-quasiparticle behavior, similar to that, observed\nforegstates in αiron [7]. We note that the effec-\ntive bandwidth of the density of states ρa(ε), character-\nized by the second central moment (standard deviation)\nσ2= (1/2)/integraltext\ndε(ε−ε)2ρ(ε) with respect to the mean\nvalue of the energy ε= (1/2)/integraltext\ndεερ(ε), isσ=D/2 and\ndoes not depend on a. As we argue below, the observed\nstrengthening of local correlations with increasing acan\nbe explained by two effects: increasing asymmetry itself,\nwhich can be characterized by the skewness of the DOS\nα= 1/(2σ3)/integraltext\ndε(ε−ε)3ρ(ε), changing from α= 0 at\na= 0 toα=−0.82 ata= 0.98, and increase of the\ndensity of states at the Fermi level. The former effect-1 -0.5 0 0.5 1\nε/D0123DOS (states/( D*band))a = 0\na = 0.5\na = 0.9\na = 0.98(a)\nn = 1.1n = 1.3\nn = 0.9\n00.511.5\niω/D-0.6-0.4-0.20ImΣ/D\n00.511.5\niω/Dn=1.1, a=0\nn=1.1, a=0.5\nn=1.1, a=0.9\nn=1.1, a=0.98\nn=1.3, a=0\nn=1.3, a=0.5\nn=1.3, a=0.9\nn=1.3, a=0.98\nn=0.9, a=0.98I = U/4 I = 0(b)\nFIG. 1: (Color online) (a) Density of states ρa(ε) for different\nvalues of the asymmetry parameter a. The positions of the\nrespective Fermi levels are shown by arrows in the middle of\nthe figure (for filling n= 0.9), and at the lower (for n= 1.1)\nand upper (for n= 1.3) axes (the value of aincreases from\nleft to right arrows). (b) The respective imaginary parts of\nthe self-energies on the Matsubara frequency axis with (lef t\npanel) and without (right panel) Hund’s exchange.\nyields narrowing of the holes band width Wh, defined as\na distance from the Fermi level to the upper edge of the\nband (in the case when the major weight of the density\nof states is below the center of the band, the bandwidth\nfor electrons We, corresponding to the distance from the\nFermi level to the lower edge, is narrowed instead), while\nthe latter effect increases the number of virtual particle-\nhole excitations.\nThe asymmetric form of the density of states with the\npeak also yields strong difference of the frequency de-\npendence of the self-energy above and below half fill-\ning. To illustrate this, we also present in Fig. 1b the\nresults for n= 0.9 and strong asymmetry, a= 0.98. One\ncan see that the absolute value of the imaginary part\nof the self-energy in this case is much smaller, than for\nn= 1.1,a= 0.98 and has the quasiparticle-like imagi-\nnary frequency dependence. Without Hund’s exchange\ninteraction we find that all of the above discussed pecu-\nliarities of the self-energy disappear (see Fig. 1b) and the\nself-energy depends rather weakly on nanda. There-3\n-1 -0.5 0 0.5 1\nε/D00.511.52DOS (states/( D*band))b1 (n=1.1)\nb2 (n=1.1)\nb3 (n=1.1)\nb4 (n=1.1)\nb5 (n=0.9)(a)\nb1b2b3,4b5\n0 1 2 3 4\niω/D-0.6-0.4-0.20ImΣ/Db1 (n=1.1)\nb2 (n=1.1)\nb3 (n=1.1)\nb4 (n=1.1)\nb5 (n=0.9)(b)\nFIG. 2: (Color online) The same as in Fig. 1 for the density\nof states ρb, allowing us to disentangle the effects of asymme-\ntry and the value ρ(EF). The arrows indicate the positions of\nthe corresponding Fermi levels.\nfore, similarly to the previous study of the symmetric\ndensity of states, Hund’s exchange interaction represents\na driving force of the obtained anomalies of the self-\nenergy, which is due to the decrease of the critical value\nof Coulomb interaction for the metal-insulator transition\nby Hund’s exchange near half filling [2, 3, 30–32].\nAs it is mentioned above, the considered density of\nstates at the Fermi level ρa(EF) increases with increas-\ning asymmetry. To disentangle the effects of the asym-\nmetry and increasing of the density of states, we con-\nsider one more model density of states ρb(ε), consist-\ning of two linear energy dependences at ε < E Fand\nε > E F. We fix the value of the density of states at the\nFermi level, but change the asymmetry (see dependences\nρb1,2,3on Fig.2a); we have verified that this yields an\nalmost unchanged effective bandwidth σ≈0.58D. One\ncan see that increasing the asymmetry in this case yields\na qualitatively similar enhancement of |ImΣ|as in the\nabove discussed results for the density of states ρa(ε).\nSomewhat weaker enhancement in this case can be ex-\nplained by the relatively weak asymmetry ( α=−0.22\nandα=−0.50 forρb2andρb3, respectively) and smaller\nvalue of the density of states at the Fermi level. Increas-\ning the value of the density of states at the Fermi level(seeρb4(ε) on Fig. 2a), but keeping the standard devi-\nationσand skewness αthe same as for the density of\nstatesρb3(ε) (the position of the Fermi level also almost\ndoes not change), yields further enhancement of |ImΣ|.\nTherefore, both factors, i.e., the asymmetry of the den-\nsity of states and the value of the density of states at\nthe Fermi level enhance quasiparticle damping. At the\nsame time, increasing filling yields decreasing the abso-\nlute value of the imaginary part of the self-energy (see\nn= 1.1 andn= 1.3 results on Fig. 1), despite the fact\nthat the Fermi level approaches the maximum (peak) of\nthe density of states, since the strength of correlations\ndecreases away from half filling. This shows that a shift\nof the filling away from half filling plays a more impor-\ntant role in this case than an increase of the density of\nstates. Changing the filling to n= 0.9 and keeping the\ndensity of states at the Fermi level, the second moment\nσand skewness αthe same as for the density of states\nρb3(ε) (see the density of states ρb5(ε)), we find almost\nthe same quasiparticle damping, as for the fully symmet-\nric band with filling n= 1.1, which also agrees with the\nresults for the density of states ρa(ε) shown in Fig. 1.\nThis suggests that the asymmetry enhances quasiparti-\ncle damping only for electron (hole) excitations for the\nposition of the Fermi level on the same side from half fill-\ning as the maximum of DOS and sufficiently far from the\ncenter of the band, which supports the suggested mech-\nanism of narrowing of the bandwidth for hole Wh(or\nelectron We) excitations.\nTo show the effect of the obtained self-energies on the\nformation of local moments, in Fig. 3a we show the tem-\nperature dependence of the inverse local susceptibility\nin the presence of Hund’s exchange for the density of\nstatesρaand model parameters, shown in Fig. 1. One\ncan see that above half filling χ−1\nloc(T) becomes linear for\nstrongasymmetryofthedensityofstates, a= 0.98, when\n|ImΣ(0)|has its maximal value, while χ−1\nloc(T) shows\na crossover between Pauli-like and linear behavior for\nsmallera. The size of the local moment, determined\nby the slope of χ−1\nloc, decreases going away from half fill-\ning, and almost does not depend on the asymmetry of\nthe density of states. The Kondo temperature TK, cor-\nresponding to the temperature scale, below which local\nmoments are screened by itinerant electrons (note that\nfor the considered paramagnetic state within DMFT the\nKondo effect does not compete with magnetism) can be\ndetermined from the offset of the inverse susceptibility,\nχ−1\nloc∝T+ 2TK, similarly to the single local moment\ncase [34, 35], cf. Ref. [11]. The obtained TKmoder-\nately decreases with increasing asymmetry of the density\nof states, but strongly increases going awayfrom half fill-\ning.\nStudying the τ-dependence ofthe local spin-spincorre-\nlation function (see Fig. 3b), we also observe the features\nof spin freezing (i.e. local moment formation) for those\ncases, when the inverse local susceptibility is linear in\ntemperature: the τ-dependence of the correlation func-\ntion becomes weak and away from τ= 0,βit is weakly4\n0 0.03 0.06 0.09 0.12\nT/D0.30.60.91.2χloc-1 (10D/µB2 )\n0n=1.1, a=0\nn=1.1, a=0.5\nn=1.1, a=0.9\nn=1.1, a=0.98\nn=1.3, a=0\nn=1.3, a=0.5\nn=1.3, a=0.9\nn=1.3, a=0.98\nn=0.9, a=0.98(a)\n0 0.25 0.5 0.75 1\nτ/β00.20.40.6<Sz(τ)Sz(0)>a=0, β=160/D\na=0, β=80/D\na=0, β=40/D\na=0, β=8/D\na=0.98, β=160/D\na=0.98, β=80/D\na=0.98, β=40/D\na=0.98, β=8/Dn=1.1 n=1.3(b)\n0 0.2 0.4 0.6 0.8 1a11.522.533.54c1/2(β=80/D)/c1/2(β=160/D)\nn=0.9, I=U/4\nn=1.1, I=U/4\nn=1.3, I=U/4\nn=1.1, I=0(c)\nFIG. 3: (Color online) (a) Temperature dependence of the\ninverse local susceptibility for the parameters considere d in\nFig. 1 with I=U/4. (b) Local spin-spin correlation functions\nin the imaginary-time domain. (c) The dependence of the\nratio of spin correlators c1/2(β) =K(β/2) atβ= 80D−1and\nβ= 160D−1on the asymmetry parameter a. The solid lines\nwere obtained by spline interpolation. The horizontal dash ed\nline represents thecriterion for thespin-freezing transi tion [5].\ntemperature dependent, cf. Refs. [5–7, 11, 13]. From\nthe requirement of two times difference (which corre-\nsponds to the criterion, suggested in Ref. [5]) of K(β/2)\natβ= 80/Dandβ= 160/Dwe find the transition to\nthe spin freezing (local moment) phase at a≈0.74 for\nn= 1.1 anda≈0.96 forn= 1.3 (see Fig. 3c).-4-3.5-3-2.5-2-1.5-1-0.500.51\nEnergy (eV)00.40.81.21.6DOS (states/(eV*orbital))t2g (α-Fe)\nt2g (γ-Fe)\neg (α-Fe)\neg (γ-Fe)α-Fe\nγ-Fe\nFIG. 4: (Color online) Density of t2gandegstates of α- and\nγ-Fe, obtained by GGA. The arrows at the top (bottom) axis\ncorrespond to positions of the chemical potential for vario us\nfillings in α-Fe (γ-Fe). The vertical line corresponds to the\nposition of the Fermi level in pure iron.\nThe local moments (when they exist) interact via\nRKKY-type of exchange, cf. Refs. [8, 36], that may\ninduce some type of magnetic order, which formation we\ndo not study here. The feedback of this exchange to the\nformation of the local moments is however expected to\nbe small, since the latter is provided by Hund’s exchange\non the much larger energy scale, than the non-local mag-\nnetic interactions. This allows us to neglect non-local\neffects in the present study.\nFinally, we note that in the considered model we did\nnot introduce the hybridization between different bands,\nbut weak hybridization is not expected to change the ob-\ntained results. In the presence of several types of states\nof different symmetry (e.g. egandt2gfor the cubic sym-\nmetry), the obtained results can again be qualitatively\napplied to these states separately, if the hybridization\nbetween the states of different symmetry is sufficiently\nsmall, and they are not strongly mixed by the Coulomb\ninteraction (see below).\nIII. FIVE-BAND MODEL AND IMPLICATIONS\nTO VARIOUS SUBSTANCES AND COMPOUNDS\nLet us discuss the effect of the density of states pecu-\nliarities for the substances and compounds, having par-\ntially formed local moments. Iron in αandγphases has\nalmost the same filling of the d-states 6.78 and 6.76, re-\nspectively, close values of standard deviation σoft2gand\negstates, but α-iron has stronger asymmetry of the den-\nsity of states and larger value of the density of states at\nthe Fermi level (see Fig. 4and Table I), and therefore\nmore pronounced local moment behavior. In both sub-\nstances the quasiparticle damping and spin freezing are\nstronger for the egthant2gstates because of stronger\nasymmetry and larger density of states at the Fermi\nlevel, but also the slight difference of fillings ( nα\neg= 1.24,5\n0 2 4 6 8\niω (eV)-2-10Im Σ (eV)\n∆µ = 0.25  (nt2g = 1.59)\n∆µ = 0.15  (nt2g = 1.54)\n∆µ = 0       (nt2g = 1.44)\n∆µ = -0.13 (nt2g = 1.32)\n∆µ = -0.27 (nt2g = 1.19)\n∆µ = -0.46 (nt2g = 1.06)(a)\n0 2 4 6 8\niω (eV)-2-10Im Σ (eV)\n∆µ = 0.25  (neg = 1.51)\n∆µ = 0.15  (neg = 1.37)\n∆µ = 0       (neg = 1.24)\n∆µ = -0.13 (neg = 1.16)\n∆µ = -0.27 (neg = 1.11)\n∆µ = -0.46 (neg = 1.07)(b)\nFIG. 5: (Color online) Self-energies of t2g(a) andeg(b)states\nofα-iron for different fillings and β= 10 eV−1.\nnα\nt2g= 1.44,nγ\neg= 1.28,nγ\nt2g= 1.40), such that the egor-\nbitals are closer to half filling. We note that in contrast\nto the two-band model, discussed above, in these and the\nfollowing results we account for the finite hybridization\nbetween different states. As it is usual in the ab ini-\ntiocalculations, for the states of the same symmetry we\ninterpret the results in terms of the (partial) densities\nof states. The hybridization between states of different\nsymmetry is sufficiently small for α-iron and nickel and\nmoderate for γ-iron (see the Appendix), and the con-\nsideration in terms of partial densities of states for t2g\nandegstates remains valid, especially for the two former\nsubstances.\nTo study the effect of filling on the self-energies of α-\nandγ-iron,we alsoconsiderthe resultsofDMFT calcula-\ntions ofthe five-bandmodel with the ab initio dispersion,\nwhich yields density of states, presented in Fig. 4, but a\ndifferent concentration of delectrons achieved by a shift\nof the chemical potential ∆ µwith respect to its position\ninα-orγ-iron. AsinSect. II,wealsoassumealocalform\nof the Coulomb interaction with the same parameters as\nin the previous study of Ref. [33]. We find (see Figs.\n5,6) that decreasing filling towards half filling always in-\ncreases quasiparticle damping, yielding peculiarities of\nthe local susceptibility, described above for the two-band0 2 4 6 8\niω (eV)-2-10Im Σ (eV)\n∆µ = 0.50  (nt2g = 1.58)\n∆µ = 0.26  (nt2g = 1.50)\n∆µ = 0       (nt2g = 1.40)\n∆µ = -0.14 (nt2g = 1.33)\n∆µ = -0.33 (nt2g = 1.24)\n∆µ = -0.53 (nt2g = 1.15)(a)\n0 2 4 6 8\niω (eV)-2-10Im Σ (eV)\n∆µ=0.50  (neg=1.62)\n∆µ=0.26  (neg=1.44)\n∆µ=0       (neg=1.28)\n∆µ=-0.14 (neg=1.21)\n∆µ=-0.33 (neg=1.14)\n∆µ=-0.53 (neg=1.08)(b)\nFIG. 6: (Color online) The same as in Fig. 5 for γ-iron.\nmodel. The most pronounced change with changing the\nfilling corresponds to egelectrons in γ-iron (see Fig. 6b).\nFor the filling, corresponding to the pure γ-iron, the eg\nstates are characterized by quasiparticle-like self-energy,\nwith|ImΣ(iω)|decreasing with decreasing ω(see the\nself-energy for ∆ µ= 0 in Fig. 6b, cf. Ref. [12]). A re-\nduction of number of egelectrons per orbital towards 1\n(correspondingtohalffilling)monotonicallyincreasesthe\nabsolute value of the self-energy ultimately yielding for\nneg<1.18 above half filling non-quasiparticle like self-\nenergy, despite the Fermi level shifting further from the\nSubstance States σ(eV) α\nα-Fe t2g 1.36 −0.012\neg 1.43 −0.596\nγ-Fe t2g 1.40 −0.068\neg 1.37 −0.545\nNi t2g 1.26 −0.036\neg 1.22 −0.479\nTABLE I: Standard deviation (third column) and skewness\ncoefficient (fourth column) of density of states for α-Fe,γ-Fe\nand Ni.6\n-4.5-4-3.5-3-2.5-2-1.5-1-0.500.51\nEnergy (eV)00.20.40.60.81DOS (states/(eV*orbital))t2g\neg(a)\n0 2 4 6 8\niω (eV)-3-2-10Im Σ (eV)\nt2g (∆µ= 0,      nt2g=1.75)\nt2g (∆µ=-0.57, nt2g=1.46)\nt2g (∆µ=-1.22, nt2g=1.07)\neg  (∆µ= 0,      neg=1.79)\neg  (∆µ=-0.57, neg=1.29)\neg  (∆µ=-1.22, neg=1.04)(b)\nFIG. 7: (Color online) Partial densities of states (a) and se lf-\nenergies (b) of Ni for different fillings and β= 10 eV−1.\npeak.\nBecause of stronger asymmetry and larger density of\nstates at the Fermi level (which occur due to the proxim-\nity to the peak of the density of states for egstates),\nfor comparable fillings per orbital egstates in αiron\nhave stronger quasiparticle damping (or even show non-\nquasiparticle behavior), than the t2gstates. The same\napplies to γ-iron close to half filling (see, for example,\nneg= 1.14 andnt2g= 1.15 self-energies), but for larger\nfillings the situation there is more complex, possibly be-\ncause of the substantial hybridization between egandt2g\nstates in this substance (see the Appendix).\nInnickelthepartialdensityof egstatesissmalleratthe\nFermi level, than the one for t2gstates, but has stronger\nasymmetry (see Fig. 7a and Table I). For the ab initio\nposition of the Fermi level, the filling is very far from\nhalf filling ( n= 8.54), and therefore the local moments\nare not fully formed in this substance; it is also char-\nacterized by large Kondo temperature TK∼600 K [11].\nThe dependence of the self-energy on the position of the\nchemical potential is determined mainly by the filling of\ntheegandt2gstates(seeFig. 7b), yieldingstrongerquasi-\nparticle damping on approaching half filling.\nFinally we mention some layered materials. In iron\npnictides the partial DOS are asymmetric, but the local\nmoments are partially formed mainly due to the prox-imity of some bands to half filling, at least in LaFeAsO\nmostly studied in that respect [13, 14]. For cuprates,\nthe effective two-band model, derived to account for non-\nlocal correlationsbeyond DMFT [37], has an asymmetric\ndensity of states for the non-vanishing next-nearest hop-\npingt′and the Fermi level is shifted towards the max-\nimum of the density of states, which may also enhance\nspin freezing near half filling. In this model, however,\nthe bands are not degenerate, and therefore this case re-\nquires further investigations. Since the effective model\ncorresponds to the non-local (plaquette) degrees of free-\ndom of the original model, the spin freezing in this case\ndoes not necessarily correspond to the local moment for-\nmation, and maybe accompaniedby non-trivialbehavior\nof the charge degrees of freedom (e.g., charge order, or-\nbital currents, phase separation, etc.).\nIV. CONCLUSIONS\nIn summary, we have investigated the effect of asym-\nmetry of the density of states on Hund’s metal behavior\nin multiband Hubbard models. We find, that the asym-\nmetry and larger value of the density of states at the\nFermi levelenhancethe Hund’s metal behavior, i.e., yield\nstronger quasiparticle damping (or non-quasiparticle be-\nhavior) and stronger spin freezing, corresponding to the\nformation of local moments, because of the stronger cor-\nrelated and larger number of electron or hole excitations.\nThe inverselocalsusceptibility becomeslinear in temper-\naturewiththeKondoscaledecreasingwithincreasingthe\nasymmetry. The above discussed behavior is observed\nsufficiently close to half filling. For realistic densities of\nstates the two considered factors (asymmetry and value\nof the density of states at the Fermi level) are often in-\nterrelated, since the Fermi level is attracted to the peaks\nof the density of states in a broad range of fillings due to\nboth, band structure and correlation effects.\nIn the present study we did not consider the effect of\nthe non-local correlations, but they are not expected to\nqualitatively change the obtained results, except, pos-\nsibly, for the narrow critical regions in the vicinity of\nthe ordered phases, which require further investigation.\nThe obtained features allow us, on one hand, to explain\nproperties of known materials, but on the other hand,\nthey allow one to predict the way to find new materials\nshowing Hund’s metal behavior and, therefore, unusual\nelectronic and magnetic properties.\nThe work was supported by the Russian Science Foun-\ndation (project no. 14-22-00004).\nAppendix A: Hybridization between egandt2g\nstates in iron and nickel\nAs a measure of the strength of the hybridization be-\ntweenegandt2gstates, we consider the kinetic energy7\ncontribution\nEMM′\nkin=/summationdisplay\nk,m∈M,m′∈M′,σHmm′\nk∝angbracketleftc+\nkmσckm′σ∝angbracketright(A1)\nwherec+\nkmσ(ckmσ) are the operators of the creation (an-\nnihilation) of the electron with momentum kand spin σ\nat the orbital m, Hmm′\nkis theab initio Hamiltonian in\nmomentum space (estimated with respect to the Fermi\nlevel) and M,M′denote states of different symmetries\n(egandt2gfor cubic lattice). 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B\n94, 245134 (2016)."    },    {        "title": "1712.06679v2.DecideNet__Counting_Varying_Density_Crowds_Through_Attention_Guided_Detection_and_Density_Estimation.pdf",        "content": "DecideNet: Counting Varying Density Crowds\nThrough Attention Guided Detection and Density Estimation\nJiang Liu1, Chenqiang Gao2, Deyu Meng3, Alexander G. Hauptmann1\n1Carnegie Mellon University\n2Chongqing University of Posts and Telecommunications3Xi’an Jiaotong University\n1fjiangl1,alexg@cs.cmu.edu,2gaocq@cqupt.edu.cn,3dymeng@mail.xjtu.edu.cn\nAbstract\nIn real-world crowd counting applications, the crowd\ndensities vary greatly in spatial and temporal domains. A\ndetection based counting method will estimate crowds ac-\ncurately in low density scenes, while its reliability in con-\ngested areas is downgraded. A regression based approach,\non the other hand, captures the general density information\nin crowded regions. Without knowing the location of each\nperson, it tends to overestimate the count in low density ar-\neas. Thus, exclusively using either one of them is not suf-\nficient to handle all kinds of scenes with varying densities.\nTo address this issue, a novel end-to-end crowd counting\nframework, named DecideNet (DEteCtIon and Density Es-\ntimation Network) is proposed. It can adaptively decide the\nappropriate counting mode for different locations on the im-\nage based on its real density conditions. DecideNet starts\nwith estimating the crowd density by generating detection\nand regression based density maps separately. To capture\ninevitable variation in densities, it incorporates an attention\nmodule, meant to adaptively assess the reliability of the two\ntypes of estimations. The final crowd counts are obtained\nwith the guidance of the attention module to adopt suitable\nestimations from the two kinds of density maps. Experimen-\ntal results show that our method achieves state-of-the-art\nperformance on three challenging crowd counting datasets.\n1. Introduction\nThe crowd counting task in the computer vision commu-\nnity aims at obtaining number of individuals appearing in\nspecific scenes. It is the essential building block for high-\nlevel crowd analysis, including crowd monitoring [3], scene\nunderstanding [41] and public safety management [5].\nVarious methods have been proposed to tackle this prob-\nlem. They could generally be classified into detection and\nregression based approaches. The detection based meth-\nods [8, 18, 34, 10, 43, 13] employ object detectors to lo-\n(a)                                                                                      (b)\n(c)                                                                                                   (d)\nFigure 1. Ablation studies of detection and regression based crowd\ncounting on the ShanghaiTech PartB (SHB) dataset [42]. Detec-\ntion reliability decreases along with the increased crowd density,\nresulting in underestimated counts in those areas. Counts from\ndensity estimation tend to be overestimated in scenes with low\ndensities. (a) Visualization of the detection results on a image\nfrom a Faster R-CNN [25] detector. (b) The density map on the\nsame image from a CNN regression network [28]. (c) The median\nobject detection scores from the detector used in (a) versus the\nground-truth counts. (d) The predictions from the network used in\n(b) versus the true crowd counts.\ncalize the position for each person. The number of de-\ntections is then treated as the crowd count. Early works\n[8, 34, 10] employ low-level features as region descriptors,\nfollowed by a classifier for classification. Benefiting from\nthe recent progress in object detection using deep neural\nnetworks [14, 25, 24, 12], in ideal images with relatively\nlarge individual sizes and sparse crowd densities, detection\nbased counting could surpass human performance. Dif-\nferent from crowd counting by detection, regression based\nmethods [19, 23, 40, 28, 42, 2] obtain the crowd count with-\nout explicitly detecting and localizing each individual. Pre-\n1arXiv:1712.06679v2  [cs.CV]  7 Mar 2018liminary works directly learn the mapping between features\nof image patches to crowd counts [19, 23, 40]. Recent re-\ngression based works improve the performance with Convo-\nlutional Neural Network (CNN) [2, 28, 42, 21, 22] to output\ndensity maps of image patches. Integrating over the map\nwill give the count for the patch. Regression based methods\nusually work well in crowded patches since they can cap-\nture the general density information by benefiting from the\nrich context in local patches.\nIn real-world counting applications, the crowd density\nvaries enormously in spatial and temporal domains. In\nthe spatial aspect, even in a same image, the density in\nsome regions may be much higher than those of others.\nIn some background regions, there may even be no person\npresent. Meanwhile, it is also natural for the crowd volume\nto change along with time: a business street may have very\nhigh crowd volumes during the workdays, while the week-\nends counterparts are much lower. Intuitively, here comes\na question: can crowd counting exclusively based on either\nregression or detection be enough to simultaneously handle\nhigh and low density scenes?\nTo answer this question, we study the performance of\ntwo types of approaches on the ShanghaiTech PartB (SHB)\ndataset [42] collected from real street scenes with great vari-\nation in crowd densities. The result is illustrated in Figure 1.\nFigure 1(a) gives the detections from a fine-tuned Faster R-\nCNN head detector on a specific image: with the distance to\nthe camera increasing, the crowd density and the number of\nmissed detections rises. Figure 1(c) shows the relationship\nbetween median detection scores and ground-truth counts\nfor 10,000 image patches with sizes of 256\u0002128. It is clear\nthat the score drops rapidly with the rise of the ground-\ntruth count. We may therefore find that the reliability of\ndetection based counting, reflected by the detection score,\nis highly correlated to the crowd density. In scenes with\nsparse crowds, the estimations are reliable, and the detec-\ntion scores are also higher than those of congested scenes.\nOn the other hand, in crowded scenes, the corresponding\nobject sizes tend to be very small. Detection in these scenes\nis less reliable, leading to low detection scores and recall\nrates. Consequently, the predicted crowd counts will be\nunderestimated ; while the regression based counting meth-\nods could perform better on these occasions. Figure 1(b)\nprovides the crowd density map visualization on the same\nimage in (a), outputted by a 5-layer CNN based regression\nnetwork with similar structure employed in [28]. We find\nthat the estimations in remote congested areas are quite rea-\nsonable. However, in background regions near the camera\nviewpoint, there exist false alarm hot spots on the pave-\nment. The relationship between ground-truth counts and\ncorresponding predictions is plotted in Figure 1(d). Note\nthat the prediction dots for patches with lower ground-truth\ncounts are mostly above the dashed line. This indicates thatprediction counts in these scenes are mostly larger than the\nground-truth. Hence, being not aware of the location of\neach individual, and directly applying the regression based\napproaches to low density scenes may lead to overestimated\nresults.\nBased on the above ablation analysis, we may find that\nthe detection and regression based counting approaches\nhave their different strengths on different crowd densities.\nThe regression based method is preferred for congested\nscenes. Without localization information for each person,\napplying them to low density scenes tends to overestimate\ncounts. The detection based approach could localize and\ncount each person precisely on these occasions since they\nare expected settings for object detectors. However, its re-\nliability degenerates in crowded scenes due to small target\nsizes and occlusion.\nTherefore, we may find that a conventional crowd count-\ning method which only relies on either detection or regres-\nsion is limited when handling real scenes with unavoidable\ndensity variations. An ideal counting method, on the other\nhand, should have an adaptive ability to choose the appro-\npriate counting mode according to the crowd density: in low\ndensity scenes, it is expected to count by localizing as an ob-\nject detector; whereas in congested scenes, it should behave\nin a regression manner. Motivated by this understanding,\nwe propose a novel crowd counting framework named as\nDecideNet (DEteCtIon and Density Estimation Network ),\nas shown in Figure 2. To the best of our knowledge, Deci-\ndeNet is the first framework, which is capable of perceiving\nthe crowd density for each pixel in a scene and adaptively\ndeciding the relative weights for detection and regression\nbased estimations.\nIn detail, for a given scene, the DecideNet first estimates\ntwo kinds of crowd densities maps by detecting individuals\nand regressing pixel-wise densities, respectively. To capture\nthe subtle variation in crowd densities, an attention module\nQualityNet is proposed to assess the reliability of two types\nof density maps with the additional supervision of detection\nscores. The final count is obtained under guidance from\nQualityNet to allocate adaptive attention weights for the two\ndensity maps. Parameters in our proposed DecideNet are\nend-to-end learnable by minimizing a joint loss function.\nIn summary, we make the following contributions:\n\u000fWe find that real-world crowd counting occasions are\nfrequently faced with great density variations. While\nexisting estimation methods, which either rely exclu-\nsively on detection or regression, are unable to provide\nprecise estimations along the whole density range.\n\u000fBased on the complementary property of two types of\ncrowd counting methods, we design a novel frame-\nwork DecideNet , which can capture this variation and\nestimate optimal counts by assigning adaptive weights\nfor both detection and regression based estimations.\u000fExperimental results reveal that our method achieves\nstate-of-the-art performance on public datasets with\nvarying crowd densities.\n2. Related works\nCrowd counting by detection. Early works addressing\nthe crowd counting problem major follow the counting by\ndetection framework. Region proposal generators [9, 33]\nare firstly used to propose potential regions that include per-\nsons. Low-level features [8, 18, 27, 34] are then used for\nfeature representation. Different binary classifiers including\nNaive Bayes [4], Random Forest [23] and their variations\n[40, 11] are trained with these features. The crowd count is\nthe number of positive samples outputted by the classifier\non a test image. Global detection scores are employed to\nestimate crowd densities and utilized for object tracking in\n[26]. Recent approaches seek the end-to-end crowd count-\ning solution by CNN based object detectors [14, 25, 24, 7]\nand greatly improve the counting accuracy. Though detec-\ntion based crowd counting is successful for scenes with low\ncrowd density, its performance on highly congested envi-\nronments is still problematic. On these occasions, usually\nonly partial of the whole objects are visible, posing great\nchallenge to object detectors for localization. Therefore,\npart and shape based models are introduced in [10, 20, 39],\nwhere ensembles of classifiers are built for specific body\nparts and regions. Although these methods mitigate the is-\nsue in some degree, counting in evident crowded scenes\nstill remains challenging, since objects in those areas are\ntoo small to be detected.\nCrowd counting by regression. Different from counting\nby detection, counting by regression estimates crowd counts\nwithout knowing the location of each person. Preliminary\nworks employ edge and texture features such as HOG and\nLBP to learn the mapping from image patterns to corre-\nsponding crowd counts [19, 23, 40]. Multi-source informa-\ntion is utilized [15] to regress the crowd counts in extreme\ndense crowd images. An end-to-end CNN model adopted\nfrom AlexNet is constructed [36] recently for counting in\nextreme crowd scenes. Later, instead of direct regressing\nthe count, the spatial information of crowds are taken into\nconsideration by regressing the CNN feature maps as crowd\ndensity maps [41] . Observing that the densities and ap-\npearances of image patches are of large variations, a multi-\ncolumn CNN architecture is developed for density map re-\ngression [42]. Three CNN columns with different recep-\ntive fields are explicitly constructed for counting crowds\nwith robustness to density and appearance changes. Sim-\nilar frameworks are also developed in [22], where a Hydra-\nCNN architecture is designed to estimate crowd densities in\na variety of scenes. Better performance can be obtained by\nfurther exploiting switching structures [31, 28, 17] or con-textual correlations using LSTM [29]. Though counting by\nregression is reliable in crowded settings, without object lo-\ncation information, their predictions for low density crowds\ntend to be overestimated. The soundness of such kind of\nmethods relies on the statistical stability of data, while in\nsuch scenarios the instance number is too small to help ex-\nplore the its intrinsic statistical principle.\nܫ\nܦ\n…\nGaussian\nconvolution\nܰௗ௧RegNetblock\nDetNetblockStacking\nQualityNet blockܦ\nܦௗ௧ܦௗ௧ܦ\n*+\nFigure 2. The architecture of our proposed DecideNet . Image\npatches are sent to the RegNet andDetNet blocks for two types\nof density maps Dreg\niandDdet\niestimation. The final density map\nDfinal\ni is outputted by the QualityNet , which adaptively decides\nthe attention weight between two density maps for each pixel.\nThree blocks are jointly learned on the training data.\n3. Crowd Counting by DecideNet\n3.1. Problem formulation\nOur solution formulates the crowd counting task as a\ndensity map estimation problem. It requires Ntraining im-\nagesI1;I2;\u0001\u0001\u0001;INas inputs. For a specific image Ii, a col-\nlection ofci2D points Pgt\ni=fP1;P2;\u0001\u0001\u0001;Pcigis provided\nby the dataset [6, 41, 42], indicating the ground-truth head\npositions in the image Ii. The ground-truth crowd density\nmapDgt\niofIiis generated by convolving annotated points\nwith a Gaussian kernel Ngt(pj\u0016;\u001b2)[22]. Therefore, the\ndensity at a specific pixel pofIicould be obtained by con-\nsidering the effects from all the Gaussian functions centered\nby annotation points, i.e.,\n8p2Ii;Dgt\ni(pjIi) =X\nP2Pgt\niNgt(pj\u0016=P;\u001b2):(1)\nSumming over the density values of all pixels over the en-\ntire imageIi, the total person count ciofIican be acquired:P\np2IiDgt\ni(pjIi) =ci. For a counting model parameter-\nized by\n, its objective is to learn a non-linear mapping\nforIi, whereas the difference between the prediction den-\nsity mapDout\ni(pjIi)and the ground-truth Dgt\ni(pjIi)is min-\nimized.Traditional crowd counting by density estimation meth-\nods regress density maps by minimizing the pixel-wise Eu-\nclidean loss to the ground-truth [36, 2, 42, 22]. However,\nas we have analyzed in introduction, counting by purely\nregression would result in the overestimation problem on\noccasions with low crowd densities. Oppositely, counting\nby detection works comparably better in those scenes, since\nlow crowd density is the expecting environment to an object\ndetector.\nIn practical applications, the crowd density varies both\nspatially and temporally. Hence, deciding the crowd counts\nexclusively based on either regression or detection is in-\nsufficient. DecideNet is motivated by their complementary\nproperty to address this problem. As shown in Figure 2, in-\nstead of counting people either by merely regressing density\nmaps, or applying an object detector over the whole image,\nDecideNet simultaneously estimates crowd counting with\nboth detection and regression modules. Later, an attention\nblock is utilized to decide which estimation result should be\nadopted for a specific pixel. Three CNN blocks are included\nin our framework: the RegNet , the DetNet and the Quali-\ntyNet , parameterized by \n= (\ndet;\nreg;\nqua). The pa-\nrameters for three CNN blocks could be jointly learned on\nthe training set.\n3.2. The RegNet block\nܫ\nܦ\n20\n40\n20\n10\n 1\nconv1\n7×7conv2\n5×5conv3\n5×5conv4\n5×5conv5\n1×1\nFigure 3. The RegNet block consisting of 5 fully convolutional\nlayers. It outputs the crowd density map Dreg\niof each pixel in\nimage patches without predicting the head locations.\nTheRegNet block counts crowds in the absence of local-\nizing each individual. Without knowing the specific loca-\ntion of each head in the input image patch, it directly esti-\nmates the crowd density for all the pixels in Iiwith a fully\nconvolutional network:\nFreg(Iij\nreg) =Dreg\ni(pj\nreg;Ii): (2)\nAs shown in Figure 3, the RegNet block consists of 5 convo-\nlutional layers. Because it is designed to capture the general\ncrowd density information, larger filters’ receptive fields\nwill grasp more contextual details, which is more beneficial\nfor modeling the density maps. Therefore, in our imple-\nmented RegNet block, the “conv1” layer has 20 filters with\na7\u00027kernel size. 40filters with a 5\u00025kernel size are set as\nthe “conv2” layer. In order to capture scale and orientation\ninvariant person density features, the “conv1” and “conv2”\nlayers are followed by two 2\u00022max-pooling layers. The“conv3” and “conv4” layers both have 5\u00025filter sizes with\n20 and 10 filters, respectively. Since the density estimation\nresult could be viewed as a CNN feature map with only one\nchannel, we add a “conv5” layer with only one filter and a\n“1\u00021” filter size. This layer is responsible to return the\nregression based crowd density map Dreg\ni, in which value\non each pixel represents the estimated count at that point. A\nReLU unit is applied after the “conv5” layer ensuring that\nthe output density map will not contain negative values.\n3.3. The DetNet block\nܦௗ௧\nboxandscoreROIpooling\nboxandscore\nboxandscoreGaussian\nconvolution\nFasterR‐CNNwithResNet101backbone\nܫ\n…ܰௗ௧\n*\nFigure 4. The proposed DetNet block is built upon the Faster R-\nCNN network. A Gaussian convolutional layer is plugged after\nthe bounding box outputs to generate the detection based crowd\ndensity map Ddet\ni.\nTo handle varying perspectives, crowd densities and ap-\npearances, existing density estimation methods [41, 42, 28,\n17] consist of several CNN structures like the RegNet block.\nHowever, without the prior knowledge about the exact posi-\ntion of each person in the image patches, the network purely\ndecides the crowd density based on the raw image pix-\nels. This regression methodology may be accurate in image\npatches with relatively large crowd densities, while it tends\nto overestimate the crowd counts in sparse or even “no-\nperson” (background) scenes. In our proposed DecideNet\narchitecture, the DetNet is designed to address this issue\nby generating the “location aware” detection based density\nmapDdet\ni. The motivation is intuitive and simple: sparse\nand non-crowded image patches are expected settings for\npresent CNN based object detectors. Therefore, compared\nto use regression networks to count on these patches, using\nthe prior knowledge from outputs of object detectors should\nsubstantially relieve the overestimation problem.\nThe DetNet block, illustrated in Figure 4 is built based\non the above assumption. It could be viewed as an exten-\nsion of the Faster-RCNN network [14] for head detection\non the basis of the ResNet-101 architecture. To be specific,\nwe design a Gaussian convolutional layer and plug it after\nthe bounding box outputs of the original Faster-RCNN net-\nwork. The Gaussian convolutional layer employs a constant\nGaussian function Ndet(pj\u0016=P;\u001b2), to convolve over\nthe centers of detected bounding boxes Pdet\nion the origi-\nnal image patch. The detection based density map Ddet\niis\nobtained by this layer, i.e.,\nDdet\ni(pj\ndet;Ii) =X\nP2Pdet\niNdet(pj\u0016=P;\u001b2):(3)Since the pixel values of Ddet\niare obtained by considering\nthe impact from the points in detection output Pdet\ni,Ddet\niis\na “location aware” density map. Compared to Dreg\nifrom\nthe output of RegNet , responses of Ddet\niare more concen-\ntrated on specific head locations. The difference between\nthem is obvious in Dreg\niof Figure 3 and Ddet\niof Figure 4.\n3.4. Quality-aware density estimation\n0.4 0.4 0.3 0.1\n0.4 0.5 0.4 0.2\n0.9…\n……\n…0.1\n6\n0.7 0.1 0.1 0.8\nܦ\nܦௗ௧\nܦ\nܭ\t\nܵௗ௧Quality‐aware\nloss\nܫJointdensity\nestimation\nܦ௧\nconv1\n7×7\n20\n4020\nconv2\n5×5conv3\n5×5conv4\n1×1Stacking\n+\nFigure 5. The QualityNet block: stacking two density maps and the\noriginal image Iias input, it outputs a probabilistic attention map\nKi(pj\nqua;Ii). The final density estimation Dfinal\ni is jointly de-\ntermined by Ki,Dreg\niandDdet\ni.\nHerein, we have described the details about obtaining\ntwo kinds of density maps: Dreg\niandDdet\nifor a given im-\nageIi. The detection based map Ddet\niemploys object de-\ntection results for density estimation. Therefore, it could\ncount persons precisely in sparse density scenes by local-\nizing their head positions. However, counting via Ddet\niis\nnot accurate on crowded occasions due to the low detec-\ntion confidence resulted from the small object size and oc-\nclusion. On the contrary, the regression based map Dreg\ni,\nwhich is unaware of individual locations, is the preferred\nestimation for these scenes: the full convolutional network\nis capable of capturing rich context crowd density informa-\ntion. Intuitively, one may think that fusing Dreg\niandDdet\ni\nby applying average or max pooling [37] may obtain bet-\nter results on varying density crowds. Nevertheless, even\nin the same scene, the density may differ significantly in\ndifferent parts or time intervals. Therefore, the importance\nbetweenDdet\niandDreg\nialso changes correspondingly for\ninstant pixel values in Ii. InDecideNet , we propose an at-\ntention block QualityNet , shown in Figure 5 to model the se-\nlection process for optimal counting estimations. It captures\nthe different importance weight of two density maps by dy-\nnamically assessing the qualities of them for each pixel.\nFor a given Ii, the QualityNet block firstly upsamples\nDdet\niandDreg\nito the same size of Ii. ThenDdet\ni,Dreg\ni\nandIiare stacked together as the QualityNet input with 5\nchannels. Four fully convolutional layers and a pixel-wise\nsigmoid layer is followed to output a probabilistic atten-\ntion mapKi(pj\nqua;Ii). We define the specific value of\nKi(pj\nqua;Ii)at the pixelpreflects the importance of thedetection based density map Ddet\ni, compared to the regres-\nsion counterpart Dreg\ni. As a result, the QualityNet block\ncould decide the relative reliability (i.e., the quality) be-\ntweenDdet\niandDreg\ni. A higher Ki(pj\nqua;Ii)at pixel\npmeans a higher attention we should rely on the detection,\nrather than the regression density estimation for p. Hence,\nwe could further define the final density map estimation\nDfinal\ni(pjIi)as a weighted sum between two density maps\nDreg\niandDdet\ni, guided by the attention map Ki:\nDfinal\ni(pjIi) =Ki(pj\nqua;Ii)\fDdet\ni(pj\ndet;Ii)+\n(J\u0000Ki(pj\nqua;Ii))\fDreg\ni(pj\nreg;Ii);\n(4)\nwhereas\fis the Hadamard product for two matrices and\ntheJis an all-one-matrix with the same size of Ki.\n4. Model Learning\nParameters of DecideNet\nconsist of three parts: \nreg,\n\ndetand\nqua. Hence, we generalize the training process\nas a multi-task learning problem. The overall loss function\nLdecide , is given by Eq. (5):\nLdecide =Lreg+Ldet+Lqua; (5)\nwhereas the Lreg,LdetandLquaare the losses for Reg-\nNet,DetNet andQualityNet , respectively. Ldecide could be\noptimized via Stochastic Gradient Descent with annotated\ntraining data. In each iteration, gradients for Lreg,Ldetand\nLquaare alternatively calculated and employed to update\ncorresponding parameters. To be specific, for the loss of the\nRegNet component, we employ the pixel-wise mean square\nerror as the loss function. That is:\nLreg=1\nNX\niX\np2Ii\u0002\nDreg\ni(pj\nreg;Ii)\u0000Dgt\ni(pjIi)\u00032;\n(6)\nwhereasNis the total number of training images.\nFor theDetNet block, different from the regression\ncounterpart, the responses on the density map Ddet\nimostly\nconcentrate on the detected head centers. Directly mini-\nmizing the difference between Ddet\niandDgt\niinvolves in\noverwhelmed negative pixel samples, i.e., background pix-\nels without head detections. Hence, instead of using the\npixel-wise Euclidean loss as error measurement, we employ\nthe bounding boxes as supervision. In this way, optimizing\n\ndetis equivalent to minimizing the classification and lo-\ncalization error in the original Faster R-CNN [25]:\nLdet=1\nNX\ni\u0002\nLcls(Pdet\ni;Pgt\nij\ndet) +Lloc(Pdet\ni;Pgt\nij\ndet)\u0003\n:\n(7)\nDue to the fact that only the centers of individuals’ heads\nare provided as the annotation on crowd density estimationdatasets, we manually label the bounding boxes on partial of\nthe training set points. Later, we employ the average width\nand height of them for the bounding box supervision in Eq.\n(7).\nThe loss function Lquafor the attention module Quali-\ntyNet should measure two kinds of errors. One is the differ-\nence between the final crowd density map Dfinal\ni and the\nground-truth density map Dgt\ni. This error is similar to that\nwe have defined in Lreg. The second error measures the\nquality of the output probabilistic map KiinQualityNet .\nRecall thatKi(pj\nqua;Ii)is the confidence of how reliable\nthe detection result is at pixel pin the image Ii. As we\nanalyzed in Figure 1(c), this confidence could be reflected\nby the object detection score Sdet(pjIi)atp. Therefore,\nwe employ the Euclidean distances between the probabilis-\ntic attention map Kiand object detection score map Sdet\nas the second error component in Lqua. From another per-\nspective, this error could be considered as a regularization\nterm over the QualityNet parameters\nqua, by incorporating\ndetection scores as prior information. In experiment evalu-\nation, we will show that this regularization is indispensably\nbeneficial to the performance of our proposed DecideNet\narchitecture. Since the object detection qualities are brought\ninto this loss function, we name it as the “quality-aware”\nloss. The final formulation of this loss Lquais defined as\nfollowing:\nLqua=1\nNX\niX\np2Ii\u001ah\nDfinal\ni(pj\nqua;Ii)\u0000Dgt\ni(pjIi)i2\n+\n\u0015kKi(pj\nqua;Ii)\u0000Sdet(pjIi)k2o\n; (8)\nwhere\u0015is the hyper-parameter to balance the importance\nbetween two errors.\n5. Experimental Results\n5.1. Evaluation settings\nOur proposed method is evaluated on three major crowd\ncounting datasets [6, 41, 42] collected from real-world\nsurveillance cameras. For all datasets, DecideNet is opti-\nmized with 40k steps of iterations. We set the initial learn-\ning rate at 0.005 and cut it by half in each 10k steps. Then\nthe best model is selected over the validation data. Instead\nof sending the whole image to DecideNet during training,\nwe follow the strategy used in [28, 2, 22] to crop images\ninto4\u00023patches. In this way, the number of samples for\ntraining the regression network is boosted. Each patch is\nthen augmented by random vertical and horizontal flipping\nwith a probability of 0:5. We also add uniform noise rang-\ning in [\u00005;5]on each pixel in the patch with a probabil-\nity of 0:5for data augmentation. To optimize the param-\neters for the RegNet and the QualityNet , the ground-truth\ndensity maps are obtained by applying the Gaussian kernelNgt(pj\u0016;\u001b2)with\u001b= 4:0and a window size of 15. In\neach iteration, the object detection score map Sdet(pjIi)is\nacquired by evaluating Iion the DetNet . For each pixel p\nin the detected bounding boxes, the value of Sdet(pjIi)is\nfilled with corresponding detection score. For the rest of\npixels which are not included in any bounding boxes, they\nare filled with a default value set at 0.1. The score map is\ndownsampled to the same size of Kiin order to calculate\nthe “quality-aware” loss Lqua. We follow the convention\nof existing works [32, 41, 23] to use the mean absolute er-\nror (MAE) and mean squared error (MSE) as the evaluation\nmetric. The MAE metric reveals the accuracy of the algo-\nrithm for crowd estimation, while the MSE metric indicates\nthe robustness of estimation.\n5.2. The Mall dataset\nThe Mall dataset [6] contains 2000 frames, collected\nin a shopping mall. Each frame has a fixed resolution of\n320\u0002240. We follow the pre-defined settings to use the\nfirst 800 frames as the training set and the rest 1200 frames\nas the test set. The validation set is selected randomly from\n100 images in the training set. We compare our DecideNet\nwith both detection based approaches: SquareChn Detector\n[1], R-FCN [7], Faster R-CNN [25]; and regression based\napproaches: Count Forest [23], Exemplary Density [38],\nBoosting CNN [35], MoCNN [17], Weighted VLAD [30].\nThe evaluation results are exhibited in Table 1.\nMethod MAE MSE\nSquareChn Detector [1] 20.55 439.1\nR-FCN [7] 6.02 5.46\nFaster R-CNN [25] 5.91 6.60\nCount Forest [23] 4.40 2.40\nExemplary Density [38] 1.82 2.74\nBoosting CNN [35] 2.01 N/A\nMoCNN [17] 2.75 13.40\nWeighted VLAD [30] 2.41 9.12\nDecideNet 1.52 1.90\nTable 1. Comparison results of different methods on the Mall\ndataset. The MAE and MSE error of our proposed DecideNet is\nsignificant lower than other approaches.\nFrom Table 1, we can observe the detection based ap-\nproaches [1, 7, 25] generally perform worse than the re-\ngression counterparts. Even the most recent CNN based\nobject detectors [7, 25] still have a large performance gap\nto the CNN based regression approaches [35, 17, 30]. Our\nproposed DecideNet obtains the minimum error on both\nMAE and MSE metrics. Compared to the best approach\n“Boosting CNN”, which based on regression, DecideNet\nreveals 0.49 point improvement on MAE metric. This is\nachieved without using the ensemble scheme employed by\nthe “MoCNN” and “Boosting CNN” methods. Moreover,\nthe MSE metric of the DecideNet is merely 1.90. This is sig-\nnificantly lower than other state-of-the-art methods, which\neither use detection or regression approach. This gain ratio-\nnally results from our density estimations formulated fromboth detection and regression results.\n5.3. The ShanghaiTech PartB dataset\nMethod MAE MSE\nR-FCN [7] 52.35 70.12\nFaster R-CNN [25] 44.51 53.22\nCross-scene [41] 32.00 49.80\nM-CNN [42] 26.40 41.30\nFCN [21] 23.76 33.12\nSwitching-CNN [28] 21.60 33.40\nCP-CNN [31] 20.1 30.1\nDecideNet 21.53 31.98\nDecideNet+R3 20.75 29.42\nTable 2. Comparison results of different methods on the Shang-\nhaiTech PartB dataset.\nWe also perform the evaluation experiments on the Shang-\nhaiTech PartB (SHB) [42] crowd counting dataset, which is among\nthe largest datasets captured in real outdoor scenes. It consists of\n716 images taken from business streets in Shanghai, in which 400\nof them are pre-defined training set and the rest are the test set.\nCompared to the Mall dataset, it poses very diverse scene and per-\nspective types over greatly changing crowd densities. We use 50\nrandomly selected images in the training set for validation. Since\nthe resolution of each image is 768\u00021024 , the patches are cropped\nfrom the original image with a size of 256\u0002256during training.\nOur evaluation result and the comparison to other state-of-the-art\nmethods are shown in Table 2. Due to the large variation in den-\nsity and object size on the SHB dataset, the detection based ap-\nproaches [7, 25] perform worse than the others relying on regres-\nsion. Specifically, the ensemble and fusion strategy is employed\nby the M-CNN [42], Switching-CNN [28], CP-CNN [31] in Ta-\nble 2. Compared to the Mall dataset, the challenging SHB dataset\nleads to much higher MAE and MSE on all the methods. Even\nthough, our proposed method ( DecideNet ;DecideNet+R3 , which\ntrained with an additional R3 stream in Switching-CNN) is very\ncompetitive to existing approaches.\n5.4. The WorldExpo’10 dataset\nThe WorldExpo’10 dataset [41] includes 1132 annotated video\nsequences collected from 103 different scenes in the World Expo\n2010 event. There are a total number of 3980 frames with sizes\nnormalized to 576\u0002720. The patch size we used for training\nis144\u0002144. The training set consists of 3380 frames and the\nrests are used for testing. Since the Region Of Interest (ROI) are\nprovided for test scenes (S1-S5), we follow the fashion of previ-\nous method [32] to only count persons within the ROI area. We\nuse the same metric, namely MAE, suggested by the author [41]\nfor evaluation. The results of our proposed approach on each test\nscene and the comparisons to other methods are listed in Table 3.\nMethodMAE\nS1 S2 S3 S4 S5 Ave\nCross-scene [41] 2.00 29.50 9.70 9.30 3.10 12.90\nM-CNN [42] 3.40 20.60 12.90 13.00 8.10 11.60\nLocal&Global [29] 7.80 15.40 15.30 25.60 4.10 11.70\nCNN-pixel [16] 2.90 18.60 14.10 24.60 6.90 13.40\nSwitching-CNN [28] 4.40 15.70 10.00 11.00 5.90 9.40\nDecideNet 2.00 13.14 8.90 17.40 4.75 9.23\nTable 3. Comparison results of different methods on 5 scenes in\nthe WorldExpo’10 dataset.From Table 3, we can notice that our proposed approach\nachieves an average MAE at 9.23 across all 5 scenes. This is the\nbest performance among those obtained by all compared methods,\nrevealing 0.17 improvement on the second best “Switching-CNN”\napproach. It is not that significant, because our error on S4 is\na little bit higher. The reason may lie on the fact that people in\nS4 majorly gather in crowds at remote areas, posing great chal-\nlenge for the DetNet to output meaningful estimations. There-\nfore, the estimation on S4 are mostly relied on the outputs from\nRegNet . While without the ensemble regression structure, using\ntheRegNet only may not be able to exhibit the superior counting\nprecision. We can also notice that the prediction counts of dif-\nferent state-of-the-art methods alter considerably on the 5 scenes,\nrevealing different approaches have their own strengths to specific\nscenes. However, DecideNet obtains three minimum MAE errors\nwhen compared to other approaches. This indicates DecideNet\nhaving a good generalization ability and prediction robustness on\ndifferent scenes.\nMethodMAE MSE\nMall SHB Mall SHB\nRegNet only 3.37 42.85 4.22 63.63\nDetNet only 4.50 44.90 5.60 73.18\nRegNet +DetNet (Late Fusion) 3.93 38.63 4.96 65.27\nRegNet +DetNet +QualityNet 1.83 24.93 2.27 41.86\nRegNet +DetNet +QualityNet (quality-aware loss) 1.52 21.53 1.90 31.98\nTable 4. Qualitative results of different DecideNet components on\nthe Mall and SHB dataset.\n5.5. Effects of different components in DecideNet\nTo analyze effects of each components of the proposed Deci-\ndeNet , we conduct ablation studies on the Mall and SHB dataset.\nThe qualitative results are listed in Table 4, which shows sev-\neral interesting observations. First, using the estimations exclu-\nsively from either the RegNet (“RegNet only”), or DetNet (“DetNet\nonly”) only obtains fair results on both datasets. The estimations\nfrom the RegNet have lower error than the detection counterparts.\nThis is possibly due to the fact that most of the image regions\nare with high crowd density on both datasets. Further, late fusion\nby averaging two classes of density maps (“ RegNet +DetNet (Late\nFusion)”) exhibits improvements than “ RegNet only” and “ Det-\nNetonly” on that SHB dataset. While on the Mall dataset, it only\nachieves a mediocre result between two kinds of density estima-\ntions. This indicates that direct late fusion is not robust enough\nto obtain better results across all kinds of datasets. Second, with\nDecideNet , even training without the object detection scores reg-\nularization (“ RegNet +DetNet +QualityNet ”), we obtain significant\nMAE and MSE decrease as compared with those obtained by the\nprevious methods. Compared to late fusion, it almost decreases\nthe MAE by half on two datasets, revealing the power of the atten-\ntion mechanism. Last but not least, adopting the “quality-aware”\nloss during training (“ RegNet +DetNet +QualityNet (quality-aware\nloss)”), the MAE and MSE errors are further reduced on two\ndatasets. In particular, the MSE decreases from 41.86 to 31.98 on\nSHB dataset: this shows that the loss can substantially increase the\nprediction stability on challenging datasets with great variations.\nIn Figure 6, we show the relationships between different crowd\ncount predictions and the ground-truth crowd counts on the test\nsets of two datasets. Note that the horizontal axes “image id” are\nsorted in ascending order by the number of ground-truth crowdFigure 6. Prediction and the ground-truth crowd counts on the test\nsets of the Mall (left) and SHB (right) datasets.\ncounts. Clearly, when the numbers of ground-truth crowd count\nare small, the regression based results from the RegNet overesti-\nmate the estimations: the blue lines are above the ground-truth\nred lines in the first half part of the horizontal axis in both fig-\nures. On the opposite, the detection based result curves (the green\nlines) fit the red lines well at that region on two datasets. How-\never, when the numbers of ground-truth count increase, the esti-\nmations of the detection based density map become considerably\nlower than the red lines, particularly after the second half parts of\nthe horizontal axis. The blue lines fit the ground-truth lines best\nin the middle part of the horizontal axes. This verifies our ob-\nservation that regression based estimations are more suitable for\nhigh crowded patches. Directly applying the late fusion (the pur-\nple curves) helps to a certain extent, while its predicted counts are\nnot stable along all images. At last, the cyan lines, which rep-\nresent DecideNet outputs, indicate the smallest differences to the\nground-truth curves along all parts of the horizontal axes. That is,\ntheDecideNet trained with “quality-aware” loss exhibits the best\nestimation results for all kinds of crowd densities on two datasets.\n5.6. Visualization on density maps\nTo better understand what is learned in our proposed model,\nwe visualize three categories of crowd density maps in the SHB\ndataset from three blocks: RegNet ,DetNet andQuialityNet in Fig-\nure 7 (best viewed in color).\nRegressionbaseddensitymap ࢍࢋ࢘ࡰDetectionbaseddensitymap ࢚ࢋࢊࡰFinaldensitymap ࢇࢌࡰ\nPredictedcount(G T:429)464.3  367. 2 403.1\nPredictedcount(G T:293)305.5 219. 4 284.9\nPredictedcount(G T:266)293.6 230. 0 252.5\nFigure 7. The visualization results of three types of density maps\non the SHB dataset (best viewed in color).We can discover that the outputs of regression based density\nmapsDreg\ni(on the most left column) exhibit diffused density esti-\nmations along the image regions. For the remote areas with highly\ncongested crowds, such predictions from the RegNet are reliable.\nHowever, when it comes to the nearby regions with lower crowd\ndensity, the results are not satisfactory: some single person bod-\nies are erroneously predicted with very high density. The predic-\ntion counts of the Dreg\niare also larger than the ground-truth (GT)\ncounts, implying the occurrence of overestimation issue. Com-\npared toDreg\ni, the detection based density maps Ddet\ni(the middle\ncolumn) are very different: the predicted peak regions are concen-\ntrated on the center of heads. This is resulted from the fact that\nthese maps are generated from outputs of head detectors. We can\nfurther observe that the detection based density results are pretty\ngood in nearby low density regions of the given image, while not\nall the heads are marked with high prediction peaks in the remote\nareas. The underestimated predicted counts of Ddet\nialso reflect\nthis phenomenon. With the attention information from the Qual-\nityNet , final density maps in the right column reveal very good\ncharacteristics: in the nearby region, the estimation prefers the de-\ntection results. Persons in those areas share very similar estimation\npatterns with Ddet\ni. Oppositely, in remote and congested regions,\ninstead of the “concentrated dot” patterns, the density maps are\ndiffused. DecideNet considers the regression based results Dreg\ni\nare more reliable for those cases. This confirms that the Quali-\ntyNet block is able to assess the reliability of the corresponding\ndensity map value for a specific pixel.\n6. Conclusion\nIn this paper, a novel end-to-end crowd counting architecture\nnamed DecideNet has been proposed. It is motivated by the com-\nplementary performance of detection and regression based count-\ning methods under situations with varying crowd densities. To the\nbest of our knowledge, DecideNet is the first framework to esti-\nmate crowd counts via adaptively adopting detection and regres-\nsion based count estimations under the guidance from the atten-\ntion mechanism. We evaluate the framework on three challenging\ncrowd counting benchmarks collected from real-world scenes with\nhigh variation in crowd densities. Experimental results confirm\nthat our method obtains the state-of-the-art performance on three\npublic datasets.\n7. Acknowledgment\nJiang Liu and Alexander Hauptmann are supported by the\nIntelligence Advanced Research Projects Activity (IARPA) via\nDepartment of Interior/Interior Business Center (DOI/IBC) con-\ntract number D17PC00340. The U.S. Government is autho-\nrized to reproduce and distribute reprints for Governmental pur-\nposes notwithstanding any copyright annotation/herein. The views\nand conclusions contained herein are those of the authors and\nshould not be interpreted as necessarily representing the offi-\ncial policies or endorsements, either expressed or implied, of\nIARPA, DOI/IBC, or the U.S. Government. Chenqiang Gao\nand Deyu Meng are supported by the China NSFC projects with\nNo.61571071, No.61661166011, No.61721002.References\n[1] R. Benenson, M. Omran, J. H. Hosang, and B. Schiele. Ten\nyears of pedestrian detection, what have we learned? CoRR ,\nabs/1411.4304, 2014. 6\n[2] L. Boominathan, S. S. Kruthiventi, and R. V . Babu. Crowd-\nnet: a deep convolutional network for dense crowd counting.\nInACM MM , 2016. 1, 2, 3, 6\n[3] A. B. Chan, Z.-S. J. Liang, and N. Vasconcelos. Privacy pre-\nserving crowd monitoring: Counting people without people\nmodels or tracking. In CVPR , 2008. 1\n[4] A. B. 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ITPAMI ,\n2008. 1"    },    {        "title": "1801.00373v1.Simultaneous_conduction_and_valence_band_quantisation_in_ultra_shallow__high_density_doping_profiles_in_semiconductors.pdf",        "content": "Simultaneous conduction and valence band quantisation in ultra-shallow, high density\ndoping profiles in semiconductors\nF. Mazzola,1J. W. Wells,1,∗A. C. Pakpour-Tabrizi,2\nR. B. Jackman,2B. Thiagarajan,3Ph. Hofmann,4and J. A. Miwa4\n1Center for Quantum Spintronics, Department of Physics,\nNorwegian University of Science and Technology, NO-7491 Trondheim, Norway\n2London Centre for Nanotechnology and Department of Electronic and Electrical Engineering,\nUniversity College London, 17-19 Gordon Street, London WC1H 0AH, U.K.\n3MAX-lab, PO Box 118, S-22100 Lund, Sweden\n4Department of Physics and Astronomy and Interdisciplinary Nanoscience Center (iNANO),\nAarhus University Ny Munkegade 120 DK-8000 Aarhus\n(Dated: November 10, 2021)\nWe demonstrate simultaneous quantisation of conduction band (CB) and valence band (VB) states\nin silicon using ultra-shallow, high density, phosphorus doping profiles (so-called Si:P δ-layers). We\nshow that, in addition to the well known quantisation of CB states within the dopant plane, the\nconfinement of VB-derived states between the sub-surface P dopant layer and the Si surface gives\nrise to a simultaneous quantisation of VB states in this narrow region. We also show that the VB\nquantisation can be explained using a simple particle-in-a-box model, and that the number and\nenergy separation of the quantised VB states depend on the depth of the P dopant layer beneath\nthe Si surface. Since the quantised CB states do not show a strong dependence on the dopant depth\n(but rather on the dopant density), it is straightforward to exhibit control over the properties of the\nquantised CB and VB states independently of each other by choosing the dopant density and depth\naccordingly, thus offering new possibilities for engineering quantum matter.\nThere has been a surge of interest in two-dimensional\n(2D) materials due to their remarkable quantum proper-\nties. Graphene and layered transition metal dichalco-\ngenides are just two examples of such materials that\nhave been recently studied and proffered as advanta-\ngeous for developing quantum electronic devices [1–4]. A\nrather unique branch of the 2D material family are ultra-\nshallow, high-density, doping profiles in semiconductors,\nso-calledδ-layers. In particular, phosphorus δ-layers in\nsilicon (Si:P δ-layers) combined with atomically precise\nlithography have led to recent technological successes to-\nwards scalable qubit architectures [5–8]. It has been\ndemonstrated that P donors, which can act as qubits,\nin Si have long spin lifetimes [9, 10] which are essential\nfor spin-based quantum calculations. Importantly, Si:P\nδ-layers can be readily synthesized – they are comprised\nof a Si(001) substrate with a high density P dopant profile\nsituated a few nanometers beneath an epitaxial grown Si\nencapsulation layer – and they are potentially straightfor-\nward to integrate into existing Si-based technology. Both\nthe dopant layer and the encapsulation layer can be eas-\nily modified during the growth process [11–13], and it is\nthis flexibility that makes δ-layers so promising, not only\nfor enhancing the performance of quantum electronic de-\nvices, but for engineering new 2D materials with new\ncapabilities.\nThe confinement of a high density, atomically thin\nlayer of P atoms beneath the surface abruptly changes the\npotential within the Si crystal. This brings about strong\nbending of the conduction band (CB) and valence band\n(VB) around the dopant plane, leading to strong confine-ment of the silicon CB. This strong confinement results in\nlowering and discretisation of the CB and consequently\ngives the system metallic character [14–18]. These CB\nstates have been studied in considerable detail [16, 19–\n21], and it has been determined that their binding energy\nand energy separation (so-called valley splitting) can be\neffectively controlled and tuned by varying the P doping\ndensity and/or depth profile [22].\nIt is not only essential for device operation and per-\nformance that the quantised CB states can be tuned and\ncontrolled but also their VB counterparts. We demon-\nstrate a general method based on δ-doping to realise\nsimultaneous quantisation of CB and VB electrons by\nstructuring the band bending at the nanoscale. We\nshow, using angle-resolved photoemission spectroscopy\n(ARPES), that quantised VB states arise from confine-\nment between the P dopant layer and surface of the Si\nencapsulation. We verify that these quantised VB states\ncan be tuned by varying the thickness of the Si encapsu-\nlation. This capability promises new prospects in engi-\nneering quantum matter, for example, the possibility of\ncontrolling carrier lifetimes by modifying the interaction\nbetween quantised CB and VB states.\nARPES measurements were performed at the I4 beam-\nline at the MAX-III synchrotron radiation source [23].\nThe energy and momentum resolutions were better than\n40 meV and 0 .02˚A−1, respectively. The base pressure in\nthe analysis chamber was ≈5×10−10mbar, and the tem-\nperature of the sample was maintained at room tempera-\nture throughout data acquisition. The Si:P δ-layer sam-\nples were prepared by growing epitaxial Si (thicknessesarXiv:1801.00373v1  [cond-mat.mtrl-sci]  31 Dec 20172\n(b) 4 nm (a) Control (c) (d)\nminmax\nVB\nSSSSEF\n2EB (eV)1\n00.4 -0.4\nk|| (Å-1)00.4 -0.4\nk|| (Å-1)1.19 eV0.12 eV\nVB qwCB qw\nControl \n4 nm\nEnergyEF\nSurface Bulk δ-layerVBCB\nV(z)\nqw3qw2qw11.1 eV 1Г2Г\nFIG. 1. Simultaneous quantisation for CB and VB states in\nsilicon. (a) ARPES data for the control sample, (b) corre-\nsponding ARPES data of a Si:P δ-layer sample with a 4 nm\nencapsulation thickness, CB and VB states indicated. (c)\nMomentum integrated EDCs to emphasise the differences be-\ntween the two samples. (d) Band bending schematic of a Si:P\nδ-layer. The resulting potential, V(z), is shown by the black\nline and confines the CB band electrons to give rise to the\nstates labelled 1Γ and 2Γ. Recovery of the potential well to\nthe surface leads to quantised states confined to the Si en-\ncapsulation layer: qw 1, qw 2and qw 3. The blue and green\nshaded areas represent the continuum of CB and VB bulk\nstates where these quantised states cannot form.\nfrom 1 to 4 nm) on top of ≈0.25 monolayers of P atoms\nincorporated in the topmost layer of a clean Si(001) sub-\nstrate; a detailed recipe can be found in Ref. 16. The\narrangement of incorporated P atoms in the Si substrate\nhas been investigated by combined atom-resolved scan-\nning tunnelling microscopy [24] and density functional\ntheory [25]. The results of these studies suggest the incor-\nporated P atoms exhibit some short- but no long-range\nordering. Control samples were measured for compari-\nson, and fabricated by growing a similar amount of epi-\ntaxial Si directly on the clean Si(001) substrate without\nthe inclusion of a P-rich layer.\nThe ARPES acquisitions of a control sample and a Si:P\nδ-layer sample with an approximate 4 nm thick Si encap-\nsulation layer are presented in Fig. 1(a) and (b), respec-\ntively. The two samples have similar spectral features:\nthe VB dispersion around the ¯Γ point of the 2D Brillouin\nzone and surface states (SS) are consistent with bulk and\nsurface states previously reported for electronic structure\nmeasurements of Si(001) with a 2 ×1 reconstructed sur-\nface [26, 27].\nThe Fermi level ( EF) lies within the 1.1 eV band gap\nand situated in close proximity to the conduction band\nminimum (CBM); confirming the n-type doping. There\nare notable differences between the two samples: an ad-\nditional feature near EFand extra bands in the VB re-\ngion — marked by the black rectangles — can be seen\nin the ARPES data of the Si:P δ-layer sample shown in\nFig. 1(b). These differences are prominent in Fig. 1(c)\nwhere energy distribution curves (EDCs), integrated over\na momentum range of -0.15 to 0.15 ˚A−1, are plotted. Theadditional states appear as peaks at binding energies of\n0.12 eV and 1.19 eV for the Si:P δ-layer sample (green\ncurve) and are noticeably absent in the control sample\n(black curve).\nWe use the band bending diagram of a Si:P δ-layer\nin Fig. 1(d) to illustrate the origin of these additional\nstates. As we go from bulk to surface, i.e. from right to\nleft across the diagram, the bulk CB becomes partially\noccupied in the region around the high density P dopant\nplane thereby creating a confined metallic layer. The CB\nstates which are bound by the Coulomb-like potential\nwell are labelled 1Γ and 2Γ. Whilst these electronic states\nhave already been studied in detail by ARPES [19], an\nunderstanding of the nature and origin of the extra bands\nthat are visible in the VB region is lacking. Previous\nARPES measurements have shown that both the CB and\nVB states are non-dispersing with photon energy, firmly\nestablishing the 2D character of these co-existing states\n[16]. ARPES acquisitions at photon energy of 36 eV are\nonly presented here, as the intensity of the CB states is\nknown to be enhanced at this energy [16, 21].\nIf VB states should exist between the surface and the\nδ-layer, they too must be strongly confined since both the\nsurface and the δ-layer act as a barrier (see left side of Fig.\n1(d)). Therefore, the hole-like bands of the VB become\ntrapped like a particle-in-a-box, where the confinement\nwidth is dictated by the depth of the δ-layer beneath the\nsurface and the confinement potential is dictated by the\nFermi level pinning at the surface and at the δ-layer. All\nof these parameters can be controlled during the sample\ngrowth.\nWe have explored the influence of Si encapsulation\nlayer thickness on the quantisation of CB and VB states\nusing ARPES. First of all, we want to study the thickness\ndependent photoemission intensity, as this will give infor-\nmation about whether a certain electronic state is bulk\nor surface derived. In Fig. 2(b-e) we consider the follow-\ning Si encapsulation thicknesses: 1 nm, 2 nm, 3 nm, 4 nm\nand compare them with the ARPES measurements for\nthe control sample shown in Fig. 2(a). At first glance,\nall of the ARPES spectra of the different Si:P δ-layer\nsamples appear qualitatively similar to each other. A\npronounced difference is the diminishing spectral weight\nof 1Γ near the EFfor Si:Pδ-layers with thicker encap-\nsulation. Since 1Γ originates from the P dopant layer\nsituated beneath the surface, it is expected that the sig-\nnal intensity gets weaker for thicker Si encapsulations. In\nFig. 2(f), EDCs (integrated over a momentum range of\n-0.15 to 0.15 ˚A−1) are plotted for the control sample and\nthe four Si:P δ-layers. In this manner, the peak intensi-\nties of both the CB and VB quantised states, marked by\nthe arrows, can be directly compared. For increasing Si\nencapsulation thicknesses, a decrease in the intensity of\nthe quantised CB peak corresponds to an increase in the\nintensity of the quantised VB states. This is confirmed in\nFig. 2(g) where the spectral intensity of the CB is plotted3\nFIG. 2. Evolution of the quantised VB states with encapsulation thickness. (a) ARPES data for the control Si sample. (b –\ne) Si:Pδ-layer ARPES spectra acquired for different Si encapsulation thicknesses ranging from 1 nm to 4 nm. The CB state\nat the Fermi level (1Γ) becomes gradually weaker with increasing Si encapsulation thickness while the states within the VB\nregion become more intense. For panels (c – e), the ARPES spectra are shown twice with salient features marked and labelled\non the spectra displayed in the lower panels. To enhance the visibility of the quantised VB states, the curvature method [28]\nwas applied and the results are presented on right sides of the lower panels of (d) and (e). The spectra for the 1 nm – 4 nm\nencapsulation thicknesses have been left-right symmetrised, while the control sample has not. An even sixth-order polynomial\nwas used to fit the quantised VB states, qw 1(orange) and qw 2(yellow), for the data shown in panels (c – e) [29]. (f) Momentum\nintegrated EDCs for Si:P δ-layers with different encapsulation thicknesses. The positions of the CB (1Γ) and SS are indicated.\nThe inset shows an enlarged region of the 4 nm thick Si encapsulation data where the qw 2state is readily visible. Adjacent to\nthe qw 2state, an unlabelled arrow marks the location of small peak which may be due to a third quantised valence band state.\n(g) The intensity ratio of CB to VB states for each of the Si:P δ-layer samples. (h) and (i) Numerically obtained solutions to\nthe Schr¨ odinger equation for a linear potential well ( V(z), black line) for 3 nm and 4 nm Si encapsulation layers, respectively.\nThe calculated eigenstates are marked by the green curves and the separation energies (in eV) marked by the double-headed\narrows.\n(relative to the intensity of the VB states, i.e. Iδ/IVB),\nas a function of Si encapsulation thickness and shows an\nexponential suppression for photoelectrons emitted from\ndeeper P dopant layers. Whilst the sub-surface origin\nof the quantised CB states was known, this analysis sug-\ngests that the quantised VB states exist up to the surface.\nBy increasing the thickness of the encapsulation from\n3 nm to 4 nm, the energy separation between the quan-\ntised VB states decreases; compare qw 1(orange) and qw 2\n(yellow) in Fig. 2(d and e). This trend can be explained\nby a particle-in-a-box picture: as the width of the box,\nor in this case the thickness of the Si encapsulation, is in-\ncreased, the energies of the quantum states are lowered.\nWe note that the energy separation between the quan-\ntised VB states for the 2 nm encapsulation thickness (Fig.\n2(c)) does not follow this trend. This exception may be\ndue to the complex interaction of the SS, located at Eb≈\n1 eV, with the quantised VB states. While the physical\nextent of the SS wave function is relatively shallow [30],a broadening and shifting could still be expected for a\nsufficiently small spatial separation of the SS and the\nquantised VB states.\nThe dispersion of the quantised VB states was fitted\nusing an even sixth-order polynomial (orange and yellow\ncurves in Fig. 2), and their effective masses and uncer-\ntainties estimated [31]. We expect the different encapsu-\nlation thicknesses to have a small affect on the effective\nmasses of the quantised VB states. Given the associ-\nated uncertainties, the effective masses for the qw 1state\nare in agreement with the heavy-hole state in the bulk\nVB of Si. The effective masses for the qw 2state are less,\nbut also probably derived from the bulk heavy-holes since\ntheir effective masses are more similar to that of the bulk\nheavy-hole state than the bulk light-hole state.\nAdditional confirmation that the extra features in the\nVB region are quantised VB states confined within the Si\nencapsulation layer is provided by our numerical model\nfor solving the Schr¨ odinger equation presented in Fig.4\n2(h and i). For simplicity we only consider a linear po-\ntential (V(z), black line) between the dopant layer and\nthe surface. The approximation is crude but reproduces\nthe quantised VB states seen in the ARPES measure-\nments of Fig. 2. It is worth noting that we only apply\nour model to Si:P δ-layers with the thickest encapsula-\ntion layers studied here, i.e. 3 nm and 4 nm, since the\nquantised VB states and the surface states are well sepa-\nrated for these cases thereby facilitating the comparison\nbetween data and model. The interaction of the SS with\nthe quantised VB states, for the 1 nm and 2 nm cases, is\nsimply not captured in this model that assumes a quan-\ntum well with the same boundary conditions for every\nthickness.\nIn our model by increasing the thickness from 3 to\n4 nm, the number of solutions to the Schr¨ odinger equa-\ntion increases from two to three. For the 3 nm case, the\ntwo calculated states are assigned to the qw 1and qw 2\nstates observed in the experiment; see Fig. 2(d). The ex-\nperimental data in Fig. 2(e) shows a weak hint of a third\nqw3state expected for the 4 nm case but the intensity\nof the signal is weak and comparable to the background.\nThe reduced intensity of the state may also be a result of\nits wave function being less localised at the surface (com-\npared to qw 1& qw 2) as illustrated in Fig. 2(i), or due\nto the fact that the photoionisation cross-section of this\nstate is lower at this photon energy [16] (or, most likely,\nboth effects might play a role).\nWe extracted EDCs, integrated over a finite momen-\ntum range, for Si:P δ-layers with different encapsulation\nthicknesses to investigate further this possible qw 3state.\nIn the inset of Fig. 2(f), the qw 2state is readily visible\nfor the 4 nm thick Si encapsulation data, and adjacent to\nthis state, there is a small peak where the qw 3state may\nbe expected.\nThe energy separations between the quantised VB\nstates are determined from the numerical model to be:\nqw2-qw 1=0.19 eV for the 3 nm thick Si encapsulation,\nand qw 2-qw 1=0.16 eV and qw 3-qw 2=0.13 eV for the 4 nm\nthick layer. From the experimental data we measure an\nenergy separation between the two lowest lying states to\nbe qw 2-qw 1=0.30±0.17 eV for the 3 nm thick Si encap-\nsulation and qw 2-qw 1=0.17±0.12 eV for the 4 nm thick\nlayer, respectively (3 nm: qw 1= 1.02±0.08 eV and\nqw2= 1.32±0.09 eV, 4 nm: qw 1= 1.02±0.08 eV and\nqw2= 1.19±0.04 eV). The experimental values for all\nthe quantised VB states are different from the ones ex-\ntracted numerically, however the general trend holds for\nthe thicker encapsulation thicknesses: (i) a shift of the\nquantised states toward lower binding energy and (ii) a\ndecrease in the energy separation between higher to lower\nlying states for increasing Si encapsulation thickness is\nobserved. This supports the notion that the quantised\nVB states originate from confinement in the Si encapsu-\nlation layer. We expect that a more accurate model for\nthe doping potential and its recovery to the surface, in-cluding the influence of the SS wave function, might be\nable to give more realistic energy separations.\nSimultaneous quantisation of CB and VB has not been\ndemonstrated in common semiconductors, previously, a\nspecial case of simultaneous quantisation of the CB and\nVB has been reported for the topological insulator Bi 2Se3\n[32]. The adsorption of CO gas on the Bi 2Se3surface\ninduces a similar downward band bending of the CB\nand the formation of quantised CB states. However, the\nquanitsed VB has quite another origin: Bi 2Se3has a pe-\nculiar valence electronic structure near the centre of its\nsurface Brillouin zone — in this region the upper VB\nonly exists in a narrow ( ≈200 meV) energy window —\nand thus downward bending of the VB can also lead to\nquantised states. The origin of the CB and VB quan-\ntisation is completely different in a δ-layer since the si-\nmultaneous quantisation of the CB and VB is purely ar-\ntificial: it is dictated by the type, density and profile of\nthe dopant layer and unlike Bi 2Se3is not an innate and\nunusual property of the bulk material. Artificially in-\nduced quantisation of the CB and VB by δ-doping offers\nthe realisation of the same effect in a wide spectrum of\nsemiconductor hosts.\nThe properties of the band-bending in δ-layers can be\neasily modified during the growth process, and as a re-\nsult quantisation of the CB and VB can be controlled\nand tuned. We can, for example, also occupy the 2Γ\nstate so that it is situated below the EF[19, 22], by ei-\nther increasing the P dopant density or broadening the\nP dopant profile of the δ-layer. The surface of the Si en-\ncapsulation layer will similarly impact the quantisation\nof the VB states as different surface terminations or sur-\nface adsorbates can alter the EFpinning at the surface,\nand thus modify the degree of band bending between the\ndopant layer and the surface.\nThe situation of simultaneous quantisation of electron\nand hole states is a rather unusual effect [32], never ob-\nserved before in traditional doped semiconductors. This\neffect provides the appealing prospect of controlling the\nlifetime of carriers, creating additional channels to gen-\nerate electron-hole pair recombinations in the CB, me-\ndiated by electronic transitions from the VB, potentially\ncontrolled by a biased top gate (analogous to a field effect\ntransistor). That is, to mediate transitions between the\nCB and VB by tuning the potential landscape in which\nthese states reside by modification of the dopant layer\nand the surface termination. For example, surface dop-\ning would directly influence the barrier potential respon-\nsible for the near-surface quantised VB states, and thus\ndirectly influence their energy, but would have a mini-\nmal influence on the sub-surface quantised CB (and bulk\nVB), for which the δ-layer and bulk doping densities, re-\nspectively, determine the Fermi level pinning. Thus, by\nmodifying the surface potential, it should be possible to\ndeliberately align (or misalign) the energies of the quani-\ntised VB and CB states so as to exhibit control of their5\ninteraction (and therefore lifetime). The flexibility that\ntheseδ-layers offer could be expected to play a major role\nin the performance of quantum electronic devices.\nIn summary, simultaneous quantisation of the CB and\nVB states of Si:P δ-layers has been experimentally veri-\nfied using ARPES. The origins of these quantised states\nare different: the CB states arise from the potential\nwell induced by the ultra-dense dopant layer whereas\nthe VB states originate from confinement between the\npotential well created by the dopant layer, and the\nsample surface. All of the relevant properties of both the\ndopant and encapsulation layers can be easily controlled\nand modified during the δ-layer growth process; not only\nproviding the ability to exhibit control the quantisation\nof CB and VB states, but also offering the intriguing\npossibility of influencing lifetimes within the δ-layer\nstructure, thereby opening up new possibilities for\nengineering quantum materials with new capabilities.\nAcknowledgements: We acknowledge Johan Adell for\nsupport at the I4 beamline at MAX-III. Partial funding\nfor this work was obtained through the Norwegian PhD\nNetwork on Nanotechnology for Microsystems sponsored\nby the Research Council of Norway, Division for Science\nunder contract no. 221860/F40. J.A.M. acknowledges\nsupport from the Danish Council for Independent Re-\nsearch, Natural Sciences under the Sapere Aude program\n(Grant no. DFF-6108-00409) and the Aarhus University\nResearch Foundation. This work was supported by VIL-\nLUM FONDEN via the Centre of Excellence for Dirac\nMaterials (Grant No. 11744) and partly supported by the\nResearch Council of Norway through its Centres of Excel-\nlence funding scheme, project number 262633, “QuSpin”,\nand through the Fripro program, project number 250985\n“FunTopoMat”.\n∗quantum.wells@gmail.com\n[1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang,\nY. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A.\nFirsov, Science 306, 666 (2004).\n[2] K. S. Novoselov, Z. Jiang, Y. Zhang, S. V. Morozov, H. L.\nStormer, U. 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Simmons, Phys. Rev.\nB74, 195310 (2006).\n[26] L. S. O. Johansson, P. E. S. Persson, U. O. Karlsson, and\nR. I. G. Uhrberg, Phys. Rev. B 42, 8991 (1990).\n[27] L. S. O. Johansson, R. I. G. Uhrberg, P. M˚ artensson, and\nG. V. Hansson, Phys. Rev. B 42, 1305 (1990).\n[28] P. Zhang, P. Richard, T. Qian, Y.-M. Xu, X. Dai, and\nH. Ding, Rev. of Sci. Instrum. 82, 043712 (2011).\n[29] See Supplementary Material for details regarding the fit\nanalysis.\n[30] M. Yengui, H. P. Pinto, J. Leszczynski, and D. Riedel, J.\nof Phys.: Condens. Matter 27, 045001 (2015).\n[31] See Supplementary Material for details regarding the es-\ntimates of the effective masses.\n[32] M. Bianchi, R. C. Hatch, J. Mi, B. B. Iversen, and P. Hof-\nmann, Phys. Rev. Lett. 107, 086802 (2011).Simultaneous conduction and valence band quantisation in\nultra-shallow, high density doping profiles in semiconductors\nF. Mazzola,1J. W. Wells,1A. C. Pakpour-Tabrizi,2\nR. B. Jackman,2B. Thiagarajan,3Ph. Hofmann,4and J. A. Miwa4\n1Center for Quantum Spintronics, Department of Physics,\nNorwegian University of Science and Technology, NO-7491 Trondheim, Norway\n2London Centre for Nanotechnology and Department of\nElectronic and Electrical Engineering, University College London,\n17-19 Gordon Street, London WC1H 0AH, U.K.\n3MAX-lab, PO Box 118, S-22100 Lund, Sweden\n4Department of Physics and Astronomy and\nInterdisciplinary Nanoscience Center (iNANO),\nAarhus University Ny Munkegade 120 DK-8000 Aarhus\n(Dated: November 10, 2021)\n1arXiv:1801.00373v1  [cond-mat.mtrl-sci]  31 Dec 2017Details regarding the fit analysis of the quantised valence band states, qw 1and\nqw 2, for different Si encapsulation thicknesses\nIn this section we provide details on how a combination of energy distribution curves\n(EDCs) and momentum distribution curves (MDCs) were used to determine the positions\nof the quantised valence band states, qw 1and qw 2. Fig. S1 shows the angle resolved\nphotoemission spectroscopy (ARPES) data for three δ-layer samples, each with a different\nsilicon encapsulation thickness: 2 nm, 3 nm and 4 nm. The positions of the bands were\ndetermined by extracting EDCs and MDCs. The EDCs and MDCs were fitted with a\nLorenztian convoluted with a Gaussian in order to extract the peak positions, which were\nthen marked by green crosses and overlaid on the ARPES spectra shown in Fig. S1. The\ngreen crosses were then fitted with an even sixth-order polynomial,\nE=a+bx2+cx4+dx6(1)\nin order to capture the quantised valence band positions. In this equation, ais the valence\nband maximum, and b,canddare coefficients to fit the band dispersions. The resulting\nfits to the peak positions, extracted from the EDCs and MDCs, for the qw 1(orange) and\nqw2(yellow) states are shown here in Fig. S1 and in Fig. 2 of the main paper. With regard\nto the 3 nm and 4 nm cases, the top of the qw 1band, near k /bardbl= 0, is not visible because of\na nearby surface state residing at a binding energy of ≈1 eV, while for the qw 2state the\ntop of the band is readily visible. On the other hand, the bands are sharper for the qw 1\nstate compared to those of the qw 2state for an approximate range of 0.2 <k/bardbl<0.5. The\ndifferences in broadness and intensity of the bands account for the different number and\npositions of extracted EDCs and MDCs seen in Fig. S1.\n2a)2 nm \nEF\n2EB (eV)\n0 0.4 -0.4\nk|| (Å-1)1b)3 nm\n0 0.4 -0.4\nk|| (Å-1)c)4 nm\n0 0.4 -0.4\nk|| (Å-1)FIG. S1. ARPES data for three δ-layer samples, each with a different silicon encapsulation thick-\nnesses: (a) 2 nm, (b) 3 nm and (c) 4 nm. The green crosses represent the peak maxima of the\nquantised valence bands extracted from EDCs and MDCs. The peak positions were then fitted\nwith an even sixth-order polynomial: qw 1(orange) and qw 2(yellow).\nDetermining the effective masses for the quantised valence band states, qw 1and\nqw 2, for different silicon encapsulation thicknesses\nBased on the fits presented in Fig. S1, we determined effective mass values for qw 1and\nqw2states near the band maxima, for three δ-layer samples with different silicon encapsu-\nlation thicknesses: 2 nm, 3 nm and 4 nm. From the curvature of the qw 1state at k /bardbl= 0,\nwe find effective masses of 0.82 me±0.13, 0.91me±0.08 and 0.92 me±0.08 corresponding\nto silicon encapsulation thicknesses of 2 nm, 3 nm and 4 nm. Using the same method for\nthe qw 2state, we find effective masses of 0.34 me±0.12, 0.70me±0.09 and 0.64 me±0.08\nfor respective silicon encapsulation thicknesses of 2 nm, 3 nm and 4 nm. We note that the\nuncertainties, stated along with the determined effective masses, were propagated through\nthe even sixth-order polynomial used to fit the quantised valence band states (see previous\nsection for details). The effective masses and uncertainties for the different samples are\nsummarised in Table S1.\nNote that for the 2 nm case the uncertainties associated with the effective mass values for\nqw1and qw 2are larger than the uncertainties for the thicker Si encapsulations. And, the\n3TABLE S1. Summary of effective masses and uncertainties (at the band maxima) determined for\nthe quantised valence band states for different silicon encapsulation thicknesses.\nencapsulation (nm) qw1 qw2\n2 0.82me±0.13 0.34me±0.12\n3 0.91me±0.08 0.70me±0.09\n4 0.92me±0.08 0.64me±0.08\neffective mass for qw 2for the 2 nm case differs significantly from the effective mass values\nfor qw 2determined for both the 3 nm and 4 nm cases. The larger uncertainties associated\nwith the 2 nm encapsulation thickness is hardly surprising given that the band positions\ncould not be accurately determined for quantised valence band states in the region of -0.2\n<k/bardbl<0.2; see Fig. S1(a). Additionally, we provide a summary of the estimated positions of\nthe band maxima for the quantised valence band states for the 2 nm, 3 nm and 4 nm silicon\nencapsulation thicknesses in Table S2. For the thinner Si encapsulation thicknesses (i.e 1 nm\nand 2 nm) it is likely that there is a more pronounced spatial overlap of the surface state\nat 1 eV binding energy with the quantised valence band states, compared with the thicker\nencapsulations (i.e. 3 nm and 4 nm), that can manifest as broadening and energy shifts of\nthe states. Such an effect can lead to a less accurate determination of the effective mass.\nTABLE S2. Band maxima for the quantised valence bands states for different silicon encapsulation\nthicknesses. The last column of the table gives the separation energies between qw 1and qw 2for\nthe different samples. Uncertainties are stated, except for the 2 nm case. Due to the lack of green\ncrosses near the top of the band maxima for the 2 nm sample, see Fig. S1(a), it was not possible\nto provide a reasonable uncertainty.\nencapsulation (nm) qw1 qw2 qw1−qw2\n2 1.1 eV 1.2 eV 0.1 eV\n3 1.02±0.08 eV 1.32±0.09 eV 0.30±0.17 eV\n4 1.02±0.08 eV 1.19±0.04 eV 0.17±0.12 eV\nWe expect that the different silicon encapsulation thicknesses will have a small affect on\n4the effective mass values for the quantised valence bands states, but given the magnitude\nof the associated uncertainties, the effective mass values for the qw 1state are in agreement\nwith the heavy-hole state in the bulk valence band of Si. The effective mass values for the\nqw2state are somewhat less, but also probably derived from the bulk heavy-holes since their\neffective mass values are more similar to that of the bulk heavy-hole state than the bulk\nlight-hole state.\n5"    },    {        "title": "1801.00596v1.Evaluation_of_two_photon_polarization_density_matrix_of_polarization_entangled_photon_pairs_generated_through_biexciton_resonant_hyper_parametric_scattering.pdf",        "content": "arXiv:1801.00596v1  [quant-ph]  2 Jan 2018Evaluation of two-photon polarization density matrix of po larization-entangled\nphoton-pairs generated through biexciton resonant hyper- parametric scattering\nY. Yamamoto,1G. Oohata,1and K. Mizoguchi1\n1Department of Physical Science, Graduate School of Science ,\nOsaka Prefecture University, 1-1 Gakuen-cho, Naka-ku, Sak ai 599-8531, JAPAN\n(Dated: January 3, 2018)\nWe have investigated the excitation-power dependence of po larization-entangled photon-pairs\ngenerated from a CuCl single crystal using biexciton resona nt hyper-parametric scattering. The\nmeasured two-photon polarization density matrix which cor responds to the two-photon polarization\nstate changes with increasing the excitation power. The eva luation of the tangle and the linear\nentropy obtained from the measured density matrix indicate s that the two-photon polarization\ndensity matrix can be expressed as the mixed state of an ideal state and a totally mixed state,\nand the mixture ratio of the totally mixed state increases as the excitation density increases. The\nvariations of the density matrix and the mixture ratio with t he excitation power originate from the\ngeneration of the multiple photon pairs.\nI. INTRODUCTION\nThe study for the generation of quantum entangled\nphoton pair is an important subject to progress quan-\ntum optics and quantum information1,2. The sponta-\nneous parametric down conversion (SPDC) is generally\nutilized as the generation method of quantum entangled\nphoton pairs3. In recent years, the generation methods\nbytheutilizationofthethirdordernonlinearopticalphe-\nnomena have been reported4,5. These methods have the\nfeatures as the enhancement of the effective generation\narea of photon pairs6and the generation of the photon\npairs in the wavelength range of the telecommunication\nbands7. In the third-order nonlinear phenomena, the\ngeneration method by using biexciton resonant hyper-\nparametric scattering (RHPS) which is induced by the\nresonance of the two-photon transition to a discrete en-\nergy state show the high generation efficiency of entan-\ngled photon pairs8,9. Furthermore, in RHPS, because the\ncharacteristics of the excited quantum state in materials\nreflect the generated entangled photon pair, the proper-\nties of entangled photon pairs are hardly affected by the\nconditions of excitation light. We consider that the gen-\neration method using RHPS is suitable to apply to the\nnon-classical light source with high brightness such as\nentanglement laser10. However, there are few reports on\ngeneration method using RHPS, and the quantum states\nunder the condition of the generation of multiple photon\npairsarenot discussed both theoreticallyand experimen-\ntally. Therefore,theinvestigationoftheexcitation-power\ndependence of quantum states of photon pairs is an im-\nportant subject to clarify the quantum states under the\ngeneration of the multiple photon pairs. We have been\nstudied the generation method of entangled photon pairs\nby using RHPS, and reported that the generation rates\nof photon pairs and the generation efficiency of photon\npair per unit excitation power have been estimated by\ntwo-photon time-correlation measurements11. However,\nthe quantum states of the photon pairs under various\nobservation conditions have been not discussed and theexcitation-power dependence of the quantum state has\nbeen not clarified. We believe that it is important to\nclarify the quantum state of photon pairs under the high\npower excitation.\nIn this paper, we report on the excitation-power de-\npendenceofthequantumstatesofentangledphotonpairs\nemitted from CuCl crystalsby RHPS. We havemeasured\nthe polarization projection for photon pairs generated by\nRHPS at various excitation powers. Two-photon polar-\nization density matrices have been reconstructed from\nthe experimental results. The fidelities of the ideal state\nwith the measured density matrices have been evaluated.\nFurthermore, we have estimated the linear entropy and\ntangle from the measured density matrix of entangled\nphoton pairs and clarified the relationship between the\nlinear entropy and the tangle at various excitation pow-\ners. Theobtainedrelationshipbetweenthelinearentropy\nandthetangleisconsistentwiththatintheWernerstate,\nwhich can be explained by taking account of multiple\nphoton pairs due to the increase in the generation rate\nof photon pairs.\nII. EXPERIMENT AND ANALYSIS\nA CuCl single crystal sample with a size of approx-\nimately 5 ×5×0.1 mm3grown from vapor phase was\nused to generate the entangled photon pairs by RHPS.\nThe temperature of the sample was maintained at 10 K\ninacryostat. Thepump pulseswerethesecondharmonic\nlight of a mode-locked Ti:Sapphire pulse laser at a repe-\ntition rate of 80 MHz. For the excitation condition, the\ncenter wavelength and the spectral width of the pump\npulses were selected to λ=390 nm and ∆ λ= 0.2 nm, re-\nspectively, by using a 4f optical system composed of two\nlenses, two gratings, and a slit. The center wavelength\nof the pump pulses corresponds to the two-photon reso-\nnance of the biexciton state in CuCl. The pump pulses\npassing through a neutral density filter were focused on\nthe sample, wherethe spotsize ofthe pump pulseson the2\n(a)\n0.06 W\n 0.13 W\n0.25 W\n0.63 W 0.88 W1.0 \n0.8 \n0.6 \n0.4 \n0.2 \n0.0 Fidelity F \n3 4567\n10 2 3 4567\n100 \nExcitation Power ( µW) Ideal stete \nUpper limit of separable state \nTotally-mixed state (b) \nFIG. 1. (a) The real parts of the measured density matrices\nfor the two-photon polarization states generated from RHPS\nmethod at various excitation densities. (b) Excitation-po wer\ndependence of Fidelity F. Dashed lines indicate the expected\nvalue of the fidelity for ideal state, an upper limit of separa ble\nstate, and totally mixed state.\nsamplewasapproximately100 µmφ. Theemitted lightin\nthe directions of about 45 and -45 degrees from the exci-\ntation optical axis were taken into the multi-mode fibers\nand guided to monochromators. The polarized compo-\nnents were selected through a polarization analyzer com-\nposed of quarterwaveplates, half waveplates, and polar-\nizing plates at the front of the fibers. To satisfy a phase\nmatching condition of RHPS, the detected wavelength\nof the RHPS light emitted from CuCl was selected by\nmonochromators, and the RHPS light was detected us-\ning single-photoncounting photomultipliers. The photon\ncounting signals were input to a time correlator, and the\ncoincidence counts were measured. Moreover, the po-\nlarization projection measurements were carried out at\nvarious excitation powers. According to the method re-\nported by Kwiat et al.12, the quantum state tomography\nfor two-photon polarization density matrix was obtained\nby measuring 16 kinds of polarized projection.\nIII. RESULTS AND DISCUSSION\nThe ideal state for polarization entangled photon pairs\ngenerated by RHPS is represented as8\n|ψi/an}bracketri}ht=1√\n2(|HH/an}bracketri}ht+|VV/an}bracketri}ht), (1)\nwhere H and V indicate horizontal and vertical linear\npolarization, respectively. The density matrix ρifor the\nideal state is given by\nρi=|ψi/an}bracketri}ht/an}bracketle{tψi|=\n1/2 0 0 1/2\n0 0 0 0\n0 0 0 0\n1/2 0 0 1/2\n. (2)Moreover, the fidelity Fof the ideal state with the mea-\nsured density matrix ρobtained from the quantum state\ntomography is defined as\nF=/an}bracketle{tψi|ρ|ψi/an}bracketri}ht. (3)\nTo clarify the excitation-power dependence of the fidelity\nfor the entangled photon pairs generated by RHPS, the\ntwo-photon polarization density matrices have been in-\nvestigated at various excitation powers. The two-photon\npolarization density matrices at various excitation pow-\ners, which arereconstructedfromthe resultsofthe polar-\nization projection measurements, were shown in Fig. 1\n(a). It is found that when the excitation power is the\nlowest, the measured density matrix is similar to the\nideal state. The fidelity of the ideal state with the mea-\nsured density matrix at the lowest excitation power is\n0.91 which is almost consistent with the value reported\nby Oohata et al.9. Moreover, the measured density ma-\ntrices are changed with increasing the excitation inten-\nsity. As the excitation power is increased, the non-\ndiagonal components |HH/an}bracketri}ht/an}bracketle{tVV|and|VV/an}bracketri}ht/an}bracketle{tHH|of the\nmeasured density matrices are decreased and the diago-\nnalcomponents |HV/an}bracketri}ht/an}bracketle{tHV|and|VH/an}bracketri}ht/an}bracketle{tVH|areincreased.\nIn order to quantitatively evaluate the deviation from\nthe ideal state with the increase in the excitation power,\nthe fidelity Fwas calculated at various excitation powers\n(Fig. 1 (b)). It is clear that the fidelity is monotonously\ndecreased with increasing the excitation power. The fi-\ndelity in the range larger than about 50 µW is smaller\nthan the classical upper limit of the fidelity, which indi-\ncates that the quantum entanglement between photons\nin the photon pair emitted from CuCl completely disap-\npears.\nThe tangle Tand the linear entropy SLas the phys-\nical quantities13,14are obtained to reveal the origin of\nthe excitation-power dependence of the density matrix.\nThe tangle Trepresents the degree of entanglement.The\ntangleTtakes values from 0 to 1, and indicates that the\ndegree of entanglement is higher as the tangle becomes\nclose to 1. The linear entropy SLrepresents the mixed-\nness of the density matrix. The linear entropy SLtakes\nvalues from 0 to 1. SLtakes 0 in the pure state, and\nshows a value close to 1 as many states are mixed. The\nrelationship between the tangle Tand the linear entropy\nSLobtained at various excitation powers is plotted in\nFig. 2. The dashed line indicates the maximum entan-\ngled state13, and the gray area shows the physically im-\npossible states. As the excitation power is increased, the\nmeasured density matrix changes from the state with the\nhigh tangle and low entropy to one with the low tangle\nand high entropy. From the obtained relationship be-\ntweenTandSL, we predict that the density matrix un-\nder the high excitation power constructs the mixed state\nconsisting of the ideal state and the totally mixed state,\nwhich is called Werner state15. Using an ideal density\nmatrixρiand a totally mixed state ρTM=I/4, whereI\nis the unit matrix, the Werner state ρWis defined as\nρW= (1−g)ρi+gρTM. (4)3\n1.0\n0.8\n0.6\n0.4\n0.2\n0.0Tangle T\n1.0 0.8 0.6 0.4 0.2 0.0\nLinear Entropy SL Experiments\n Werner State5 µW\n10 µW\n20 µW\n50 µW\n70 µW\nFIG. 2. Relationship between Linear entropy SLand Tangle\nT. Red circles indicate the experimental values of SLand\nTobtained from the measured density matrices at various\nexcitation powers. Solid line indicates the Werner states a nd\ngray area corresponds to the states that are not physically\nrealized.\nHere,gis the mixture ratio of the ideal state and the to-\ntally mixed state, which takes values from 0 to 1. More-\nover, it is known that the Werner state monotonically\nchanges from the state with low entropy and high tangle\nto one with high entropy and low tangle states as the\nparametergchanges from 0 to 1. The solid line in Fig. 2\nshows the relationship between TandSLin the Werner\nstate obtained by Eq. (4). The experimental results al-\nmost agree with the solid line. This agreement means\nthatthemeasureddensitymatrixobeystheWernerstate,\nwhich indicatesthat the parameter gis increasedwith in-\ncreasing the excitation power; namely, the proportion of\nthe totally mixed state increases with the increase of the\nexcitation power.\nTo understand the effect of the excitation power on\nthe measured density matrix, it is necessary to clarify\nthe origin of the totally mixed state. We have obtained\nthe parameters gat various excitation powers by fitting\nthe measured density matrix to the Werner state. The\nexcitation-powerdependence of the parameter gis shown\nin Fig. 3. The experimental results marked with black\ncircles indicate that the parameter gis monotonously in-\ncreased with the increase of the excitation power, which\nindicates that the proportion of the totally mixed state\ndepends on excitation power.\nWhen it is assumed that the originof the totally mixed\nstate is due to the photoluminescence by bound excitons\nin CuCl and the scattering light of the pump pulses9, the\ndetectionrateofthetotallymixedstateisproportionalto\nthe coincidence count rate of two photons which depends\non the square of the excitation power, because the pho-\ntoluminescence by bound excitons and the scattering of\nthe pump pulses belong to a linear optical phenomenon.\nOn the other hand, in RHPS which is related to a third-\norder nonlinear parametric process, the detection rate\nof photon pairs is proportional to the square of the ex-\ncitation power. Therefore, the ratio of the two-photon\ncoincidence count rate by RHPS to that by the photolu-\nminescence and the scattering takes constant values withthe excitation power. This property for the ratio of the\ntwo-photon coincidence count rate has been experimen-\ntally confirmed in the previous report12. Moreover, this\nresult means that the parameter gcaused by the pho-\ntoluminescence and scattering is independent on excita-\ntion power, which indicates that the obtained excitation-\npower dependence of the parameter gwill not originate\nfrom the photoluminescence by bound excitons and the\nscattering of the pump pulses. Therefore, we consider\nthat the change of the measured density matrix and the\nexcitation-power dependence of the parameter gwill be\ncaused by multiple photon pairs.\nAs the excitation power increases, the generation rate\nof photon pairs by RHPS increases, and thus the proba-\nbility of multiple photon pairs simultaneously generated\nalso increases. When the multiple photon pairs are gen-\nerated, the measured density matrix is represented by\nthe base of number states corresponding to the number\nof generated photon pairs. In this case, to completely\nclarify the measured density matrix, we need at least\ntwice as the number of detectors as the number of pho-\ntonpairs10. Inthe presentstudy, becausethe coincidence\ncount rates have been observed by using two detectors,\nthe information up to only two photons can be obtained.\nWhen three or more photons are generated, it is consid-\nered that the polarized projection which is statistically\nobtained differs from the correlationof only two photons.\nTakesue et al. have theoretically calculated the effect of\nthe generation of multiple photon pairs on polarization\ncorrelation measurements with two detectors16. When\nmultiple photon pairs are generated independently, the\ncoincidence probabilities RHH,RHV, andRHRfor the\nrespective polarization projections HH, VV, and HR are\nexpressed by\nRHH=∞/summationdisplay\nx=0f(x)Pp(x,µ)≃α2/parenleftbiggµ\n2+µ2\n4/parenrightbigg\n,(5)\nRHV=∞/summationdisplay\nx=0g(x)Pp(x,µ)≃α2µ2\n4, (6)\nRHR=∞/summationdisplay\nx=0h(x)Pp(x,µ)≃α2/parenleftbiggµ\n4+µ2\n4/parenrightbigg\n,(7)\nwherex,Pp,αandµarethe number ofphoton pairsgen-\nerated simultaneously, the Poisson distribution function,\nthe detection efficiency of photons, the average photon\npair number, respectively. f(x),g(x), andh(x) expressed\nas follows are the all coincidence probabilities of detec-\ntion of photons of the polarization projections HH, HV\nand HR, respectively, when xpairs are generated.\nf(x) =x/summationdisplay\ny=01\n2xx!\ny!(x−y)!/braceleftbig\n1−(1−α)x−y/bracerightbig2,(8)\ng(x) =x/summationdisplay\ny=01\n2xx!\ny!(x−y)!/braceleftbig\n1−(1−α)x−y/bracerightbig\n×{1−(1−α)y},(9)4\nh(x) =\nx/summationdisplay\nj=01\n2xx!\nj!(x−j)!{{1−(1−α)}j\n\n×/parenleftiggx/summationdisplay\nl=01\n2xx!\nl!(x−l)!{{1−(1−α)}l/parenrightigg\n.(10)\nThe two-photon polarization density matrix obtained\nfrom these polarized projections is expressed as12\nρp(µ)≃1\n4+4µ\n2+µ0 0 2\n0µ0 0\n0 0µ0\n2 0 0 2+ µ\n\n=1\n1+µρi+µ\n1+µρTM. (11)\nThis equation indicates that the density matrix can be\nexpressed by the mixed state of the ideal state and the\ntotally mixed state, namely, the Werner state. Moreover,\nthe parameter gis expressed by g=µ/(1 +µ), which\nis monotonously increased with increasing µ. On the\nother hand, since the simultaneous detection efficiency\nηxaffects the visibility of the time-correlation signal, it\nis expected that ηxwill havea significantinfluence on the\ncalculation of the density matrix12. Then, the detection\nprobabilities for the polarized projections, R′\nHH,R′\nHV,\nandR′\nHR, are represented as\nR′\nHH=Nmax/summationdisplay\nx=0f′(x,ηx)Pp(x,µ), (12)\nR′\nHV=Nmax/summationdisplay\nx=0g′(x,ηx)Pp(x,µ), (13)\nR′\nHR=Nmax/summationdisplay\nx=0h′(x,ηx)Pp(x,µ), (14)\nby taking account of ηxbased on Eq. (5) - (7). Here,\nto carry out an nummerical calculation to these polar-\nizxation projections, we put Nmaxas an upper limit of\nthe number of the simultaneous generated photon-pairs.\nf′(x,ηx),g′(x,ηx) andh′(x,ηx) expressed as follows are\nthe coincidence probabilities of the polarization projec-\ntions HH, HV and HR, respectively, when xpairs are\nobserved with the simultaneous detection efficiency ηx.\nf′(x,ηx) =x/summationdisplay\nk=0x−k/summationdisplay\nm=0X(x,k,m,η x)f(x,k,m) (15)\ng′(x,ηx) =x/summationdisplay\nk=0x−k/summationdisplay\nm=0X(x,k,m,η x)g(x,k,m) (16)\nh′(x,ηx) =x/summationdisplay\nk=0x−k/summationdisplay\nm=0X(x,k,m,η x)h(x,k,m) (17)\nwhere,\nX(x,k,m,η x) =ηk\nx/parenleftbigg1−ηx\n2/parenrightbiggx−kx!\nk!(x−k−m)!m!,\n(18)1.0\n0.8\n0.6\n0.4\n0.2\n0.0g\n3456789\n1023456789\n100\nExcitation Power ( µW) experiment\n ηx=0.001\n ηx=0.03\n ηx=0.20\n ηx=1.00\nFIG. 3. Excitation-power dependence of g. Black circles in-\ndicate the obtained value of gfrom the measured density ma-\ntrices at various excitation powers. Solid lines indicate c al-\nculated excitation-power dependence of gat various values of\nthe simultaneous detection efficiency ηx(0.001, 0.03, 0.20 and\n1.00).\nf(x,k,m) =k/summationdisplay\ny=0x−k−m/summationdisplay\nz=0m/summationdisplay\nw=0/parenleftbigg1\n2/parenrightbiggxk!(x−k−m)!m!\n(k−y)!y!(x−k−m−z)!z!(m−w)!w!\n×/braceleftbig\n1−(1−α)y+z/bracerightbig/braceleftbig\n1−(1−α)y+w/bracerightbig\n,\n(19)\ng(x,k,m) =k/summationdisplay\ny=0x−k−m/summationdisplay\nz=0m/summationdisplay\nw=0/parenleftbigg1\n2/parenrightbiggxk!(x−k−m)!m!\n(k−y)!y!(x−k−m−z)!z!(m−w)!w!\n×/braceleftbig\n1−(1−α)y+z/bracerightbig/braceleftbig\n1−(1−α)k−y+m−w/bracerightbig\n,\n(20)\nh(x,k,m) =/parenleftiggx−m/summationdisplay\ny=01\n2x−m(x−m)!\n(x−m−y)!y!/braceleftbig\n1−(1−α)x−m−y/bracerightbig/parenrightigg\n×/parenleftiggk+m/summationdisplay\nz=01\n2k+m(k+m)!\n(k+m−z)!z!/braceleftbig\n1−(1−α)k+m−z/bracerightbig/parenrightigg\n.\n(21)\nUsing Eq. (12) - (14), the polarization projections are\ncalculated at various values of ηx, and thus the two-\nphotonpolarizationdensitymatricesareobtainedbyper-\nforming the quantum state tomography. In this calcula-\ntion, we adopted Nmax= 15. The solid lines in Fig. 3\nindicate the calculatedparameter gplotted asfunction of\nexcitation power at various values of ηx. The excitation-\npower dependence of gdrastically changes with the value\nofηx. Atηx= 0.03, the calculated excitation-power de-\npendence of gis in almost agreement with the experi-\nmental results. From these results, we demonstrate that\nthe excitation-powerdependence ofthe measureddensity\nmatrix originates from the generation of multiple photon\npairs. Moreover, the fact that the mixture ratio of the\nideal state and the totally mixed state gdepends on ex-5\ncitation power can be explained by taking account of the\ngenerationof multiple photon pairs and the simultaneous\ndetection efficiency ηx.\nIV. CONCLUSION\nWe have investigated the polarization projections of\nthe photon pairs generated by RHPS at various excita-\ntion powers. The two-photon polarization density matri-\nces, which arereconstructedfromthe resultsofthe polar-\nization projection measurements, drastically change with\nthe excitation powers. The fidelities of the ideal statewith the measured density matrices are monotonously\ndecreasing with increasing the excitation power. The re-\nlationshipbetweenthetangleandlinearentropyobtained\nat various excitation powers indicates that the measured\ndensity matrix under the high excitation powers reflects\nthe mixed state consisting of the ideal state and the to-\ntallymixed state. Moreover,the proportionofthe totally\nmixed state increases with the increase in the excitation\npower, which originates from the generation of multiple\nphoton pairs.\nThis work was supported by Research Foundation for\nOpto-Science and Technology and Support Center for\nAdvanced Telecommunications Technology Research.\n1D. Bouwmeester, J.-w. Pan, K. Mattle, M. Eibl, H. Wein-\nfurter, and A. Zeilinger, Nature 390, 575 (1997).\n2P. Kok, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J.\nMilburn, Rev. Mod. Phys. 79, 135 (2007).\n3P. Kwiat, K. Mattle, H. Weinfurter,\nA. Zeilinger, A. Sergienko, and Y. Shih,\nPhys. Rev. Lett. 75, 4337 (1995).\n4L. J. Wang, C. K. Hong, and S. R. Friberg,\nJ. Opt. B Quantum Semiclassical Opt. 3, 346 (2001).\n5J. E. Sharping, K. F. Lee, M. A. Foster, A. C. Turner,\nB. S. Schmidt, M. Lipson, A. L. Gaeta, and P. Kumar,\nOpt. Express 14, 12388 (2006).\n6X. Li, L. Yang, X. Ma, L. Cui, Z. Y. Ou, and D. Yu,\nPhys. Rev. A 79, 033817 (2009).\n7S. Dong, Q. Zhou, W. Zhang, Y. He, W. Zhang, L. You,\nY. Huang, and J. Peng, Opt. Express 22, 359 (2014).8K. Edamatsu, G. Oohata, R. Shimizu, and T. Itoh,\nNature431, 167 (2004).\n9G. Oohata, R. Shimizu, and K. Edamatsu,\nPhys. Rev. Lett. 98, 140503 (2007),\narXiv:0607139v2 [arXiv:quant-ph].\n10C. Simon and D. Bouwmeester,\nPhys. Rev. Lett. 91, 1 (2003).\n11Y. Yamamoto, G. Oohata, and K. Mizoguchi,\nOpt. Express 24, 6034 (2016).\n12D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G.\nWhite, Phys. Rev. A 64, 052312 (2001).\n13W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).\n14S. Bose and V. Vedral, Phys. Rev. A 61, 040101 (2000).\n15W. Munro, D. James, A. White, and P. Kwiat,\nPhys. Rev. A 64, 030302 (2001).\n16H. Takesue and K. Shimizu,\nOpt. Commun. 283, 276 (2010)."    },    {        "title": "1801.01946v1.Ground_state_properties_of_light_kaonic_nuclei_signaling_symmetry_energy_at_high_densities.pdf",        "content": "arXiv:1801.01946v1  [nucl-th]  6 Jan 2018Ground-state properties of light kaonic nuclei signaling s ymmetry energy at high\ndensities\nRong-Yao Yang, Si-Na Wei, Wei-Zhou Jiang\nSchool of Physics, Southeast University, Nanjing 211189, C hina\nAsensitivecorrelation betweentheground-stateproperti esoflightkaonicnucleiandthesymmetry\nenergy at high densities is constructed under the framework of relativistic mean-field theory. Taking\noxygen isotopes as an example, we see that a high-density cor e is produced in kaonic oxygen nuclei,\ndue to the strongly attractive antikaon-nucleon interacti on. It is found that the 1 S1/2state energy\nin the high-density core of kaonic nuclei can directly probe the variation of the symmetry energy\nat supranormal nuclear density, and a sensitive correlatio n between the neutron skin thickness and\nthe symmetry energy at supranormal density is established d irectly. Meanwhile, the sensitivity of\nthe neutron skin thickness to the low-density slope of the sy mmetry energy is greatly increased in\nthe corresponding kaonic nuclei. These sensitive relation ships are established upon the fact that the\nisovector potential in the central region of kaonic nuclei b ecomes very sensitive to the variation of\nthe symmetry energy. These findings might provide another pe rspective to constrain high-density\nsymmetry energy, and await experimental verification in the future.\nPACS numbers: 21.65.Mn, 13.75.Jz, 21.10.Gv, 21.60.Gx\nI. INTRODUCTION\nThe determination of symmetry energy is of crucial importance in co ntemporary nuclear physics, since it can\nprovide vital information on understanding the basic aspects of th e strong interaction, such as three-body force [1–3],\ntensor force [4] and short range correlations [5–7]. Up to now, the density dependence of symmetry energy is still not\nwell understood, even though lots of effort has been made to extr act the symmetry energy from theory, astrophysical\nobservations and terrestrial experiments [2, 3, 8–21]. Though ne utron stars (NSs) are natural laboratories for testing\nthe properties of asymmetric nuclear matter, the symmetry ener gy constrained by NS observation has not reached\na satisfactory precision, due to limited data and indistinct observab les, e.g., the radii of NSs [20, 21]. Meanwhile,\nconstraints on the symmetry energy from heavy-ion collisions suffe r from low detection efficiency, large systematic\nerror and transport model dependence [10, 22–24]. The propert ies of finite nuclei are of great use in constraining the\nsymmetry energy near or beneath the normal density [2, 11–13, 2 5–28] owing to the high precision of the structural\ndata of finite nuclear systems. For instance, the symmetry energ y and its density slope at normal nuclear density can\nbe determined by the nucleon global optical potentials, which can be extracted from nucleon-nucleus scatterings and\nthe single-particle energy levels of bound states [12]. Additionally, a c onnection between the nuclear symmetry energy\nat subnormal density and the symmetry energy coefficients of finit e nuclei can be constructed from the measured mass\nof finite nuclei [16, 26–29] and the liquid-gas phase transition [30, 31 ]. In particular, the neutron skin thickness of\nneutron-rich nuclei has captured extensive attention, consider ing that it can serve as a bridge linking the symmetry\nenergy and the structure of NSs [16, 32–36]. After great efforts have been made to measure the root-mean-square\n(rms) radius of neutron density distribution in finite nuclei [37–41], t he symmetry energy near normal nuclear density\nhas been constrained increasingly well. Despite all this progress, inf erring the high density properties of symmetry\nenergy from finite nuclei is difficult since the density of a nucleus satu rates at normal nuclear density. If a high-\ndensity core in a finite nucleus could be created, it would be very inter esting to investigate the relationship between\nthe neutron skin thickness of such nuclei and the symmetry energ y.\nWe should look for a unique way to achieve a higher nuclear density in a fi nite nuclear system, which cannot\nbe obtained just by adding more nucleons. A possible option is to incor porate the strangeness degree of freedom,\nforming exotic nuclei, such as Λ hyperons [42–44] and kaon mesons [45–47]. Noticeably, the core density of kaonic\nnuclei could possibly be as high as two times the normal nuclear densit y due to the strongly attractive antikaon-\nnucleon interaction [46]. In fact, the properties of exotic nuclei ha ve played a special role in constraining the nuclear\nforces. Specifically, the importance of the three-body force and the role of the tensor force are exhibited in exotic\nstates, such as the halo [48–50] and Hoyle states [51, 52], the shell evolution anomaly [53, 54] and the novel magic\nnumbers far off the β-stability [55–57]. These imply that kaonic nuclei can perhaps be rega rded as candidates to\nprobe the symmetry energy at high densities. On the other hand, k aonic nuclei themselves are fascinating systems\nowing to their special features, such as the breaking of pseudosp in symmetry [58] and the halo structure of light\nkaonic nuclei [59]. Therefore, many theoretical works have focus ed on the study of kaonic nuclei [46, 47, 58–62], and\ncontinuous experimental efforts have been made to search for th em [63–71], though the existence of kaonic nuclei has\nnot yet been verified. To the best of our knowledge, the relationsh ip between kaonic nuclei and symmetry energy\nhas seldom been investigated. Hence, we will try to construct a dat a-to-data correlation between the ground-state\nproperties of kaonic nuclei and the symmetry energy in this paper u nder the framework of the relativistic mean-field2\n(RMF) theory. In the RMF theory, the interactionsaremediated b y mesons, which makesit veryeasyto incorporatea\nK−mesonself-consistently. Acharacteristicofthe RMF theoryis tha t there is alargeattractivescalarand a repulsive\nvector, which is of great importance for interpreting the pseudos pin symmetry in nuclei [72, 73]. Meanwhile, the RMF\ntheory can not only reproduce nicely the ground-state propertie s for finite nuclei in the whole nuclide table [74–77],\nbut also give a successful description of exotic nuclei [78, 79]. In th e past, plenty of research on hyper-nuclear systems\nwas also based on the RMF theory [43, 44, 80–84]. Thus, the RMF th eory should be competent in our investigation\non kaonic nuclei. It will be found that the strong antikaon-nucleon a ttraction can considerably amplify the sensitivity\nof the ground-state properties of kaonic nuclei to the symmetry energy at high densities.\nThe paper is organized as follows. In Section 2, the RMF formulas for the symmetry energy and kaonic nuclei are\ndeduced. The numerical results are presented in Section 3, followe d by a brief summary.\nII. FORMALISM\nTosimulatedifferentdensitydependenceofthesymmetryenergy, weintroduceacouplingtermbetweentheisoscalar\nvector and isovector vector potential, as done in Ref. [32]. The inte racting Lagrangian density for nucleons is given\nby\nLint=¯ψB[gσσ−gωγµωµ−gργµτ3bµ\n0−e1+τ3\n2γµAµ]ψB\n−1\n3g2σ3−1\n4g3σ4+1\n4c3(ωµωµ)2+4Λvg2\nρg2\nωωµωµb0µbµ\n0, (1)\nwhereσ, ω, b 0, Arepresent the isoscalar scalar, isoscalar vector, isovector vect or, and electromagnetic field, respec-\ntively.gi(i=σ,ω,ρ) are the corresponding coupling constants between meson fields a nd nucleons. g2andg3denote\nthe strengths of the nonlinear terms of the scalar field. c3is the parameter for the non-linear ωself-coupling term.\nΛvis the coupling between the isoscalar vector and isovector vector p otential which could be used to adjust the\nhigh-density symmetry energy.\nIn the mean-field approximation, we can derive the symmetry energ y from the Lagrangian density Eq. (1) using\nEsym(ρB) =1\n2[∂2E(ρB,δ)/∂δ2]|δ=0whereδis the isospin asymmetry. Then, we get\nEsym(ρB) =1\n2(gρ\nm∗ρ)2ρB+k2\nF\n6E∗\nF, (2)\nwhere the effective mass of the ρmesonm∗\nρ=/radicalBig\nm2ρ+8Λv(gωgρω0)2andE∗\nF=/radicalbig\nk2\nF+M∗2is the Fermi energy with\nthe effective mass of the nucleon M∗=MB−gσσ.\nAt a particular density ρ′, the symmetry energy can be approximately expanded as\nEsym(ρB) =Esym(ρ′)+L\n3ρB−ρ′\nρ′+κsym\n18(ρB−ρ′)2\nρ′2, (3)\nwhere L and κsymare the density slope and curvature of the symmetry energy at ρ′, respectively. The density slope\nL is defined as L(ρ′) = 3ρ′∂Esym\n∂ρ|ρ=ρ′. Some studies have found that L and κsymare correlated with each other [85].\nAsκsymis very poorly known, the determination of L with high precision could g ive valuable information on the\nsymmetry energy.\nTo investigate kaonic nuclei, the antikaonic sector should be incorpo rated into the interacting Lagrangian density\nfor nucleons Eq. (1). The Lagrangian density for kaons is written a s [62]\nLKN= (DµK)†(DµK)−(m2\nK−gσKmKσ)K†K, (4)\nwhere the covariant derivative is defined as Dµ≡∂µ+iVµwithVµ=gωKωµ+gρKb0µ+e1+τ3\n2Aµ, andK=/parenleftbigK+\nK0/parenrightbig\n,\nK†= (K−,¯K0). In this paper, we only consider charged kaons, since kaonic nucle i are systems where the K−meson\nis implanted into a finite nucleus, i.e., K=K+andK†=K−. From this Lagrangian, it can be inferred that the\nK−Ninteraction is mediated by σ, ω, ρmeson fields and the electromagnetic field, which are coherently att ractive,\nconsistent with experimental data. This makes the formation of a K−-nucleus bound state possible.\nWith the Lagrangian density for nucleons Eq. (1) and K−meson Eq. (4), together with the Lagrangian densities\nfor free particles, we can obtain the equations of motion using the s tandard RMF treatment [74], which can be written\nas3\nTABLE I: The parameters Λ vandgρthat simulate various density dependencies of the symmetry energy. The symmetry\nenergy at ρ0, 1.5ρ0and its density slope at 0 .7ρ0are in units of MeV. Other unlisted parameters are the same as the TM2 [104].\nΛvgρL(0.7ρ0)Esym(ρ0)Esym(1.5ρ0)\n0.0 9.357 106.53 35.90 55.19\n0.005 9.778 97.65 34.55 49.81\n0.010 10.264 88.79 33.39 46.17\n0.015 10.830 79.92 32.37 43.53\n0.020 11.502 71.05 31.47 41.54\n0.025 12.316 62.19 30.68 39.97\n0.030 13.332 53.31 29.96 38.71\n[−iα·∇+β(MB−gσσ)+gωω+gρτ3b0+e1+τ3\n2A]ψB=ǫψB, (5)\n(−∇·∇+m2\nK−E2\nK−+Π)K−= 0, (6)\n(−∇·∇+m2\nφ)φ=sφ=\n\ngσρs−g2σ2−g3σ3+gσKmKK−K+σ,\ngωρv−c3ω3−8Λvg2\nρg2\nωωb2\n0−gωKρK−ω,\ngρρ3−8Λvg2\nρg2\nωω2b0−gρKρK−, ρ,\neρp−eρK−, photon,(7)\nwhereρs, ρv, ρ3, ρp, ρK−are the scalar, vector, isovector, proton, and K−density, respectively. EK−is the energy\neigenvalue of the K−meson.sφis the so-called source term for mesons or photons (and mphoton= 0). The K−\nself-energy term is expressed as\nΠ =−gσKmKσ−2EK−V−V2, (8)\nwhereV=gωKω+gρKb0+1+τ3\n2eA. The parameters in Eq. (4) can be fitted to the depth of the K−optical potential\nat normal nuclear density, where the optical potential is defined a sUopt(p,ρ0) =ω(p,ρ0)−/radicalbig\np2+m2\nK, withω(p,ρ0)\nbeing the in-medium energy at saturation density. Strictly, one sho uld consider the effects from the K−absorption by\nnucleons. This could be done by introducing an imaginary potential int o theK−self-energyterm Π in Eq. (8) [60–62].\nWe have found that the stationary-state properties of kaonic nu clei are only slightly affected by the incorporation of\nthe imaginary part in the RMF models when the K−optical potential is around -100 MeV depth at normal nuclear\ndensity. Therefore, the imaginary part in the K−self-energy term is neglected here, as it has no impact on the\nconclusion in the paper.\nSolving these equations in an iterative procedure, one can accurat ely obtain the ground-state properties of normal\nnuclei and kaonic nuclei. In each iteration step, we use the shooting method to obtain energy eigenvalues for nucleons\nand antikaons. Specifically, the energy eigenvalues are moderately adjusted to connect the wave function smoothly\nat matching points in this boundary value problem. Note that the ant ikaon eigenvalue is not sensitive to the choice\nof the specific first-order derivative at boundaries. Readers can refer to Ref. [58] for more concrete numerical details.\nThen, we can construct the correlations between the structura l properties of these finite nuclear systems and different\ndensity dependence of the symmetry energy.\nIII. RESULTS AND DISCUSSION\nThe antikaon-nucleon interaction is demonstrated to be strongly a ttractive from the data extracted from kaonic\natoms [86–91], KN scatterings [92–97] and heavy-ion collisions [98–10 3]. The data from kaonic atoms and KN\nscatteringmainlyprovideinformationonlow-density K−Ninteractions. Oneneedstousesomespecificextrapolations\nto obtain the depth of the K−optical potential at normal nuclear density. However, these ext rapolations cause large\ndiversification, giving the depth at normal nuclear density ranging f rom -200 MeV to -50 MeV according to different\nmodels[86–97]. Heavy-ioncollisionscandirectlyprobethe properties at appropriatelyproduceddensities andproduce\nan almost consistent K−potential depth around -100 MeV at normal nuclear density [98–10 3]. Thus, we adopt -100\nMeVas the K−optical potentialdepth atsaturationdensity [100], todetermine t he valueofthe unique free parameter\ngσK, asgρKandgωKcan be determined by the SU(3) relation: 2 gωK= 2gρK=gρπ= 6.04.\nWe adopt the RMF parameter set TM2 as a starting point in this work, since TM2 was initially designed for light\nnuclei [104]. The differentdensity dependence ofthe symmetryene rgycanbe obtainedbyadjusting the parametersΛ v4\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53/s48/s50/s48/s52/s48/s54/s48/s69/s115/s121/s109/s32/s40/s77/s101/s86/s41/s32/s32/s32/s32/s32/s32/s32/s32/s76/s32/s40/s77/s101/s86/s41\n/s32/s49/s48/s54/s46/s53/s51\n/s32/s57/s55/s46/s54/s53\n/s32/s56/s56/s46/s55/s57\n/s32/s55/s57/s46/s57/s50\n/s32/s55/s49/s46/s48/s53\n/s32/s54/s50/s46/s49/s57\n/s32/s53/s51/s46/s51/s49\nFIG. 1: (Color online) The symmetry energy as a function of nu clear matter density with various slopes (L) at 0.7 ρ0obtained\nfrom adjusting parameter Λ vandgρbased on TM2, see Table I.\n/s48 /s50 /s52 /s54 /s56/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50/s49/s46/s54\n/s75/s45/s119/s105/s116/s104/s32/s75/s45\n/s32/s32/s68/s101/s110/s115/s105/s116/s105/s101/s115/s32/s40\n/s48/s41\n/s114/s32/s40/s102/s109/s41/s32/s49/s56\n/s79/s32/s40/s119/s47/s111/s32/s75/s45\n/s41\n/s32/s49/s56\n/s79/s32/s40/s119/s105/s116/s104/s32/s75/s45\n/s41\n/s32/s75/s45\n/s32/s105/s110/s32/s49/s56\n/s75/s45/s79\n/s32/s50/s50\n/s79/s32/s40/s119/s47/s111/s32/s75/s45\n/s41\n/s32/s50/s50\n/s79/s32/s40/s119/s105/s116/s104/s32/s75/s45\n/s41\n/s32/s75/s45\n/s32/s105/s110/s32/s50/s50\n/s75/s45/s79/s119/s47/s111/s32/s75/s45\nFIG. 2: (Color online) The nuclear density and K−density distribution in normal nuclei (18O,22O) and kaonic nuclei (18\nK−O,\n22\nK−O) with the parameter set TM2. The corresponding density dist ributions with various Λ vare very similar to the one with\nthe TM2, thus they are not shown here for simplicity. The “w/o K−” denotes normal nuclei without K−and the “with K−”\ndenotes the corresponding kaonic nuclei.\nandgρin Eq. (1) with a fixed a symmetry energy of 24.93MeV at ρ= 0.7ρ0(the same as TM2 and ρ0= 0.132fm−3),\nsimilarto what wasdone in Ref. [32]. This simple treatment producesa nearlyconstantbinding energyper nucleonfor\noxygenisotopeswhen Λ vischanged. However,the densitydependence ofthe symmetryen ergydiversifiesconsiderably\nat high densities due to different coupling strengths between isosca lar vector and isovector vector field, as shown in\nFig. 1. When Λ vchanges from 0 to 0.03, the symmetry energy becomes softer at h igh densities and the density slope\nof the symmetry energy at 0.7 ρ0varies from 106.53 MeV to 53.31 MeV. The specific parameters and pr operties are\nlisted in Table I.\nUsing the parameters in Table I, we calculate the ground-state pro perties of oxygen isotopes, including the density\ndistributions, nuclear potentials and the single-particle energy. In Fig. 2, we display the nuclear density and K−\ndensity distributions in18Oand22Osystems. As one can see, in the core of kaonic nuclei, the nuclear de nsity is\nlargely increased by the implanted K−meson. This is the so-called shrinkage effect, which may be construc tive for\nproducing cold high-densitymatter in the laboratory. In fact, it is t he stronglyattractiveinteractionbetween nucleons\nandK−meson that makes the compression in the core of a nucleus possible, resulting in this shrinkage effect. Indeed,\nthe increase of the nuclear density in kaonic nuclei almost coincides w ith theK−density distribution ,which is mainly\nstacked in the core region of kaonic nuclei.\nAccompanying the shrinkage effect, the inner nuclear states in kao nic nuclei are considerably affected by the\nembedding of the K−meson. Shown in Fig. 3 is the single-particle energy of the 1 S1/2state for protons and neutrons\nas a function of symmetry energy at 1.5 ρ0which can be adjusted through different Λ vin Table I. The kaonic nuclei\nhave an average core density around 1.5 ρ0, and the correlation between the 1 S1/2state energy in the core region\nand the symmetry energy at 1.5 ρ0can directly probe the variation of the symmetry energy at supran ormal nuclear\ndensity. We can see that the single-particle energies of 1 S1/2states for kaonic nuclei (solid curves) are clearly larger\nthan those for the corresponding normal nuclei (dashed curves ), i.e., the nucleons in the 1 S1/2state of kaonic nuclei5\n/s52/s48 /s52/s53 /s53/s48 /s53/s53/s52/s52/s52/s56/s53/s50/s53/s54/s54/s48/s52/s48/s52/s52/s52/s56/s53/s50/s53/s54\n/s50/s50\n/s79/s32/s50/s50\n/s75/s45/s79/s32\n/s69/s115/s121/s109/s32/s40/s77/s101/s86/s41/s49/s56\n/s79/s32/s49/s56\n/s75/s45/s79/s32/s83/s105/s110/s103/s108/s101/s45/s112/s97/s114/s116/s105/s99/s108/s101/s32/s101/s110/s101/s114/s103/s121/s32/s40/s77/s101/s86/s41/s32/s32/s80/s114/s111/s32/s119/s105/s116/s104/s32/s75/s45\n/s32 /s32/s78/s101/s117/s32/s119/s105/s116/s104/s32/s75/s45\n/s32/s80/s114/s111/s32/s119/s47/s111/s32/s75/s45\n/s32/s32 /s32/s78/s101/s117/s32/s119/s47/s111/s32/s75/s45\nFIG. 3: (Color online) The single-particle energies of the 1 S1/2state in18Oand22Osystems as a function of the symmetry\nenergy at 1.5 ρ0. The “Pro” and “Neu” labels denote the 1 S1/2state for protons and neutrons, respectively.\nare more tightly bound. This is a straightforward consequence of t he strong antikaon-nucleon attraction that would\nchange the potentials for nucleons in nuclei. Noticeably, the differen ce in the single-particle energy between protons\nand neutrons in kaonic nuclei becomes increasingly large with the incr ease of high-density symmetry energy, while it\nis much less obvious in normal nuclei. This actually establishes a direct s ignal to sensitively probe the high-density\nsymmetry energy. The rms quantities can be calculated using the de nsity distributions as weight functions. Here, we\nfocus on the neutron and proton rms radii and define the neutron skin thickness as the difference between the neutron\nand proton rms radii. The results for the neutron skin thickness in n eutron-rich18Oand22Osystems are plotted in\nFig. 4. As one can see from the left-hand panels in Fig. 4, the relation ship between the neutron skin thickness and\nthe density slope of the symmetry energy at 0 .7ρ0is nearly linear in both18Oand22Osystems. The neutron skin\nthickness in kaonic nuclei (solid lines) is larger than that in normal nuc lei (dashed lines) due to the shrinkage effect\nthat occurs in the inner states of kaonic nuclei. The gradient of the neutron skin thickness with increasing density\nslope of the symmetry energy in18\nK−Ois moderately larger than that in normal nucleus18O. Strikingly, the neutron\nskin thickness in22\nK−Ois obviously more sensitive to the density slope of the symmetry ener gy, as compared to that of\n22O. To further understand the role of the K−meson in affecting the sensitivity of the neutron skin thickness to th e\ndifference in the symmetry energy, we display in the right-hand pane ls of Fig. 4 the neutron skin thickness of18\nK−O\nand22\nK−Oas a function of the symmetry energy at 1.5 ρ0. Since this density appears to be close to the average density\nof the core of kaonic nuclei, the consequence of the symmetry ene rgy in the core region could also be reflected to the\nneutron skin. It can be clearly seen from panel (d) in Fig. 4 that the neutron skin thickness in22\nK−Ochanges rather\nrapidly with increasing symmetry energy at 1.5 ρ0. The smaller variation observed for18\nK−Oin panel (c), compared\nto that of22\nK−O, is attributed to the smaller isospin asymmetry. This can be also seen by comparing the curves for\nnormal nuclei22Oand18O, as shown in Fig. 4. Nevertheless, the sensitivities of the neutron s kin thickness to the\nisospin effect and symmetry energy are both magnified by the embed ding of the K−meson. To reveal the physics\nbehind these phenomena, we plot in Fig. 5 the isovector potential in18Oand18\nK−O,22Oand22\nK−Oas a function of\nradius. In normal nucleus18O, the isovector potential is very small, in accordance with its small iso spin asymmetry.\nThe variation of the isovector potential with different density depe ndence of the symmetry energy is insignificant\nin18O, responsible for a relatively slow change of the neutron skin thickne ss (see Fig. 4). However, the isovector\npotential in the core of18\nK−Oundergoes an obvious change for various symmetry energies, in sh arp contrast to the\ncase in18O. Thus, the embedment of K−meson makes the effect of different symmetry energy on the isovec tor\npotential more distinguishable. The situation for22Osystems in the lower panels of Fig. 5 is very similar to that of\n18Osystems. However, the isovector potential in22Ois much larger than that in18O, since22Opossesses a larger\nisospin asymmetry. Actually, the symmetry energy effect on the iso vector potentials in22\nK−Ois amplified by both the\nimplantedK−meson and the larger isospin asymmetry. Specifically, the potential in the central region of22\nK−Ohas\na 55% variation as the density slope L changes from 53.31 MeV to 106.5 3 MeV. Hence, the implantation of the K−\nmeson greatly magnifies the difference in the isovector potential th at is induced by the various density dependencies\nof the symmetry energy, responsible for the larger changes in the neutron skin thickness of kaonic nuclei. We have6\n/s48/s46/s49/s50/s48/s46/s49/s54/s48/s46/s50/s48/s48/s46/s50/s52\n/s97/s41\n/s49/s56\n/s79/s32/s82\n/s110 /s45/s32/s82\n/s112 /s32/s32/s32/s40/s102/s109 /s41\n/s53/s48 /s54/s48 /s55/s48 /s56/s48 /s57/s48 /s49/s48/s48 /s49/s49/s48/s48/s46/s51/s54/s48/s46/s52/s48/s48/s46/s52/s52/s98/s41\n/s50/s50\n/s79\n/s76/s32/s40/s77/s101/s86/s41/s49/s56\n/s75/s45/s79\n/s32\n/s32/s119/s105/s116/s104/s32/s75/s45\n/s32/s119/s47/s111/s32/s75/s45/s99/s41\n/s52/s48 /s52/s53 /s53/s48 /s53/s53/s50/s50\n/s75/s45/s79/s100/s41\n/s69/s115/s121/s109/s32/s40/s77/s101/s86/s41\nFIG. 4: (Color online) The neutron skin thickness in18Oand22Osystems as a function of the slope of symmetry energy at\n0.7ρ0(left-hand panels) and as a function of the symmetry energy a t 1.5ρ0(right-hand panels).\n/s48 /s50 /s52 /s54 /s56/s100/s41\n/s50/s50\n/s75/s45/s79\n/s114/s32/s40/s102/s109/s41/s32/s32/s32/s32/s32/s32/s32/s32/s76/s32/s40/s77/s101/s86/s41\n/s32/s49/s48/s54/s46/s53/s51\n/s32/s57/s55/s46/s54/s53\n/s32/s56/s56/s46/s55/s57\n/s32/s55/s57/s46/s57/s50\n/s32/s55/s49/s46/s48/s53\n/s32/s54/s50/s46/s49/s57\n/s32/s53/s51/s46/s51/s49\n/s48 /s50 /s52 /s54/s45/s49/s50/s45/s54/s48/s99/s41/s103 /s98\n/s48 /s32/s40/s77/s101/s86/s41\n/s114/s32/s40/s102/s109/s41/s98/s41\n/s49/s56\n/s75/s45/s79\n/s45/s49/s50/s45/s54/s48\n/s50/s50\n/s79/s49/s56\n/s79/s97/s41\nFIG. 5: (Color online) The isovector potential in18Oand18\nK−O,22Oand22\nK−Oas a function of radius. The Lvalue marked is\nobtained at 0.7 ρ0.\nalso conducted a series of investigations based on the RMF paramet er set NL-SH [105] and FSUGold [106]. It is\nfound that the conclusions are qualitatively the same. These result s indicate that light kaonic nuclei can be used\nas candidates to constrain the high-density symmetry energy eith er with appropriate theoretical approaches or by\nperforming experiments. The photo-nucleus or pion-nucleus reac tions [68, 107], e.g., γ+22O→K++22\nK−O orπ−\n+22F→K∗++22\nK−O, may be used to produce kaonic nuclei and then directly measure t he radius information via\nthe strong interaction between the nucleons and the outgoing kao ns in these reactions.\nIV. SUMMARY\nIn this paper, we have investigated the correlation between the gr ound-state properties, especially the neutron skin\nthickness, of kaonic oxygen isotopes and the symmetry energy in t he framework of the RMF theory. By introducing a\ncoupling between the isoscalar vector and isovector vector fields, we can simulate different density dependencies of the\nsymmetry energy. We have directly established a sensitive correlat ion between the ground-state properties of kaonic\noxygen isotopes and the symmetry energy at supranormal densit y. It is found that the sensitivity of the neutron skin\nthickness to the low-density slope of the symmetry energy is great ly amplified in the corresponding kaonic nuclei.\nThese sensitive relationships are established upon the fact that th e isovector potential in the central region, where a\nhigh density core is formed due to the embedding of the K−meson, becomes very sensitive to the variation of the\nsymmetry energy. 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Tellgren\nHylleraas Centre for Quantum Molecular Sciences, Department of Chemistry,\nUniversity of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo, Norway\n(Dated: November 8, 2021)\nWe show that the particle density, \u001a(r), and the paramagnetic current density, jp(r), are not\nsu\u000ecient to determine the set of degenerate ground-state wave functions. This is a general feature\nof degenerate systems where the degenerate states have di\u000berent angular momenta. We provide a\ngeneral strategy for constructing Hamiltonians that share the same ground state density, yet di\u000ber\nin degree of degeneracy. We then provide a fully analytical example for a noninteracting system\nsubject to electrostatic potentials and uniform magnetic \felds. Moreover, we prove that when ( \u001a;jp)\nis ensemble ( v;A)-representable by a mixed state formed from rdegenerate ground states, then any\nHamiltonian H(v0;A0) that shares this ground state density pair must have at least rdegenerate\nground states in common with H(v;A). Thus, any set of Hamiltonians that shares a ground-state\ndensity pair ( \u001a;jp) by necessity has at least have one joint ground state.\nA cornerstone of modern density-functional theory\n(DFT) is the Hohenberg-Kohn theorem [1], which states\nthat the ground-state particle density of a quantum-\nmechanical system determines up to an additive constant\nthe one-body potential vof the same system. The orig-\ninal argument was limited to systems that have unique\nground states. Although DFT can be formulated with-\nout recourse to the Hohenberg-Kohn theorem, using the\nconstrained-search and Lieb's convex analysis formalisms\n[2, 3], the result strengthens and adds insight to the the-\nory. Firstly, the alternative DFT formulations establish\nonly that densities determine various contributions to the\ntotal energy; in particular, the exchange-correlation en-\nergy, and properties given as functional derivatives of the\nenergy with respect to the scalar potential. The stronger\nstatement that the wave function and Hamiltonian, and\nconsequently allproperties of a system, are determined\nstill requires the Hohenberg-Kohn theorem. Secondly,\nwhereas alternatives are known for ground-state DFT,\nthe available formulation of time-dependent DFT is most\nclosely related to the Hohenberg-Kohn formulation [4].\nWhen the Hamiltonian contains a magnetic vector po-\ntential in addition to the scalar potential, the particle\ndensity alone is no longer su\u000ecient for a rigorous for-\nmulation of DFT. The most well-established extension is\ncurrent-density-functional theory (CDFT), where it has\nbeen proven that the particle density, \u001a, and the param-\nagnetic current density, jp, determine the non-degenerate\nground-state [5, 6] (see Eqs. (2) and (3) for the de\fni-\ntion of\u001aandjp, respectively). We use the term weak\nHohenberg-Kohn theorem (cf. [7], Sec. III D) for this\nresult and, following Dreizler and Gross [8], denote the\ninvertible map from non-degenerate ground states to den-\nsities byD. The reason for the term weak is that such a\nresult would be implied by the stronger but false state-\nment that ( \u001a;jp) determines ( v;A) [6, 9, 10]. Thus, the\nmap, denotedC, from (v;A) to non-degenerate ground\n\u0003andre.laestadius@kjemi.uio.nostates is not invertible. The situation can be summarized\nas\n(v;A)C! D\n\u001d(\u001a;jp): (1)\nThe fact that the map Cis not invertible does not\npreclude a density-functional formulation in terms of \u001a\nandjp. Indeed, the existence of the map D\u00001in non-\ndegenerate paramagnetic CDFT is enough to de\fne a\ncorresponding Hohenberg-Kohn functional [5]. Further-\nmore, the Hohenberg-Kohn variational principle holds for\nthe density pair ( \u001a;jp) and a theory of density function-\nals can be based on these variables [5, 6]. For further\ndiscussion on the choice of variables for current-density\nfunctionals we refer to [9, 10], see also the mathematical\nanalyses in [11{13] and the related [7]. For the status of\nthe Hohenberg-Kohn theorem for physical current den-\nsity instead of the paramagnetic current density, we refer\nto previous work showing that existing attempted proofs\nare \rawed [9, 10] and the recent progress towards a pos-\nitive result using the total current density [14, 15].\nThe aim of this work is to investigate a weak\nHohenberg-Kohn result in CDFT without the assump-\ntion of a unique ground state. Given an N-electron wave\nfunction , de\fne the particle density and the paramag-\nnetic current density according to\n\u001a (r1) =NZ\nj j2dr\u00001; (2)\njp\n (r1) =NImZ\n r1 dr\u00001; (3)\nwhereR\ndr\u00001denotes integration over all space for all but\none particle and  denotes the complex conjugate of  .\nFurthermore, given a vector potential Awe may com-\npute the total current density as the sum j=jp+\u001aA.\nFor vanishing A, the Hohenberg-Kohn theorem states\nthat if\u001a1=\u001a2, thenV1=V2+ constant, where Vk=P\njvk(rj) [1]. The proof of this result relies on the\nfact that if  is a ground state of both systems, then\n(V1\u0000V2) = constant\u0002 . If does not vanish on aarXiv:1801.09606v1  [physics.chem-ph]  29 Jan 20182\nset of positive (Lebesgue) measure, we have V1=V2+\nconstant (almost everywhere). At any rate V1=V2up to\na constant holds on the complement of N =f = 0g.\nAssuming that the measure of N is zero (i.e., assum-\ning that that the Schr odinger equation has the unique-\ncontinuation property from sets of positive measure), the\nproof can be completed by means of the variational prin-\nciple as \frst suggested in [1]. A generalization of the\noriginal Hohenberg-Kohn theorem that includes degen-\neracy was given in [16]. [See also the work of Lammert\n[17] for further analysis of the set N in connection with\nthe Hohenberg-Kohn theorem in DFT.]\nIn the presence of a magnetic \feld, a ground state  \ndoes not uniquely determine the Hamiltonian H[6, 9, 10].\nThis leads to complications in the following way: We\ndemonstrate that a given pair \u001aandjpmay arise from\ntwo di\u000berent pairs of vandAthat do not share the same\nset of ground states . This shows that the conclusion of\nTheorem 9 in [10] does not hold in general. Nonetheless,\nany set of ground-state density matrices that have the\nsame density pair ( \u001a;jp) are ground states of the same\nset of Hamiltonians (see also [18] and the discussion that\ncomes before Theorem 9 in [10]). We furthermore prove\nthat (\u001a;jp) at least determine one ground state, and un-\nder certain assumptions, the full set. This constitutes\na weak ensemble Hohenberg-Kohn result in degenerate\nCDFT.\nIn what follows, our point of departure is a quantum\nmechanical system of N(spinless) electrons subjected to\nboth a magnetic \feld and a scalar potential. The Hamil-\ntonian is\nH(v;A) =H0+NX\nj=1\u00101\n2f\u0000irj;A(rj)g+v(rj)+1\n2A(rj)2\u0011\n;\nwheref\u0001;\u0001\u0001gdenotes the anti commutator and H0is the\nuniversal part of H, independent of the external poten-\ntialsvandA. We let\nH0(\u0015) =1\n2NX\nj=1\u0000\n\u0000r2\nj+\u0015X\nj6=kr\u00001\njk\u0001\n;0\u0014\u0015\u00141;\nwhere\u0015= 1 corresponds to fully interacting electrons\nand\u0015= 0 the non-interacting case.\nWe start by demonstrating that ( \u001a;jp) does not deter-\nmine the set of possibly degenerate ground states. The\ngeneral idea is that for systems with cylindrical symmetry\nabout thez-axis, degeneration can either be introduced\nor lifted by the application of an external magnetic \feld.\nFor example, consider a cylindrically symmetric Hamil-\ntonianH(v+A2=2;0) with a ground-state degeneracy,\nwhere the ground states are distinguished by di\u000berent\neigenvalues of Lz. The Hamiltonian H(v;A) shares the\nsame eigenstates, but the eigenvalue degeneracies are now\nlifted by the orbital Zeeman e\u000bect. At least for su\u000e-\nciently weak magnetic \felds along the z-axis, the state\nwith minimal Lzis then the unique ground state. The\nidea can also be applied in the other direction. That is,suppose a magnetic \feld has been tuned so that H(v;A)\nhas a ground state degeneracy, where the ground states\nare distinguished by di\u000berent Lzvalues. The degener-\nacy is then lifted in the spectrum of the Hamiltonian\nH(v+A2=2;0).\nIn order to avoid relying on numerical results, we shall\nfocus on a two-dimensional non-interacting system of N\nelectrons subject to a magnetic \feld. De\fne rj= (xj;yj),\nv(r) =1\n2!2r2andA= (B=2)(\u0000y;x;0), whereB\u00150 is\nthe strength of a uniform magnetic \feld perpendicular to\nthe plane, i.e., B=Bez. Sincef\u0000irj;A(rj)g=BLz;j,\nthe system's Hamiltonian is given by\nH=H0(\u0015) +NX\nj=1\u0010B\n2Lz;j+hB2\n8+!2\n2i\nr2\nj\u0011\n:(4)\nLet\u0015= 0 such that H0=PN\nj=1(\u0000r2\nj=2). We write\nH=PN\nj=1hj, where (dropping the index j) the one-\nelectron operator his given by\nh=\u00001\n2r2+B\n2Lz+hB2\n8+!2\n2i\nr2:\nLet ~!=p\n(B=2)2+!2. The eigenfunctions of hin polar\ncoordinates ful\fll (see for instance [19])\n\u001en;m(r;') =Crjmjeim'Ljmj\nn(~!r2)e\u0000~!r2=2;\nwhereLjmj\nnare the associated Laguerre polynomials,\nn= 0;1;:::andm= 0;\u00061;:::The corresponding eigen-\nvalues, or orbital energies, are given by\n\"n;m= (2n+ 1 +jmj) ~!+mB\n2: (5)\nThe \frst few \"n;mare plotted in Fig. 1 for a \fxed !=!0.\nTo prove our claim, Fig. 1 shows that it is enough to\nstudy a system with N= 3 electrons (other particle num-\nbers are also possible). Let jn;mibe the abstract state\nvector corresponding to the single-particle wave function\n\u001en;m. Set!=!0,B=B0=!0=p\n2,e0= 15!0=p\n8 and\n 0\n0=j0;0i\nj0;\u00001i\nj0;1i;\n 0=j0;0i\nj0;\u00001i\nj0;\u00002i;\nwhere\ndenotes antisymmetrized tensor product. Then\nby direct computation, using B0=!0=p\n2,\nH 0\n0= (\"0;0+\"0;\u00001+\"0;1) 0\n0=e0 0\n0;\nH 0= (\"0;0+\"0;\u00001+\"0;\u00002) 0=e0 0;\nand 0\n0and 0are both degenerate ground states of H,\nwith energy e0. See also Fig. 1 where \"0;1=\"0;\u00002at the\npoint (B0;7B0=2)\u0019(0:57;1:98) for!0= 0:8. Further-\nmore,Lz 0\n0= 0 whereas Lz 06= 0.\nNext, letH0be a Hamiltonian of the form (4) for a dif-\nferent system of the same number of electrons, but with3\n0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2\nB (235 kT)00.511.522.533.54Orbital energy (hartree)\nFIG. 1. Orbital energies \"n;m(see Eq. (5)) for != 0:8\nas a function of the magnetic \feld strength B\u00150. Orbital\nenergies with m= 0 are shown as dotted black curves, m=\n\u00061 as dashed blue curves, m=\u00062 as solid red curves, m=\u00063\nas dashed dot magenta curves, and m=\u00064 as solid green\ncurves.\nB= 0 and!=p\n(B0=2)2+!2\n0. It then follows from (4)\nthat\nH=H0+B0\n2Lz:\nThus, 0\n0is the unique ground-state (up to a phase) of\nH0, with energy e0. Note that the given example can\nbe adapted to include spin. Adding the spin-Zeeman\ntermgB\u0001s=2 to the one-electron operator h, as well as\nhaving each orbital instead doubly occupied, gives a level\ncrossing at a di\u000berent Band electron number N.\nNow, if we compute \u001aandjpfrom just 0\n0, the pair\n(\u001a;jp) is (v;A)-representable from both HandH0,\n(v;A)C!f 0; 0\n0g!(\u001a;jp) f 0\n0gC (v0;0):\nThe Hamiltonians HandH0do not share the same set of\nground states and consequently, we have proved: Knowl-\nedge of\u001aandjpis not enough to determine the set of\nground states .\nIn order to obtain fully analytical results, we have fo-\ncused on a non-interacting model system, i.e., \u0015= 0.\nAnother candidate for analytical results is two-electron\nquantum dot with fully interacting ( \u0015= 1) electrons|\nwe refer to the work in [19], see also [20]|although the\nfact that exact solutions are only known for a discrete\nset of parameter values makes this case harder. Fur-\nthermore, level-crossings are ubituitous in more compli-\ncated systems as well. For example, quantum rings [21],\natomic systems [22, 23], and molecular systems [23, 24]\nall feature level crossings of the type analyzed here. Theexistence of level-crossings does not depend on the pres-\nence or absence of the spin-Zeeman term. In particu-\nlar, the lithium atom in a homogeneous magnetic \feld\nexhibits such a level crossing: In [22] that includes the\nspin-Zeeman term (see Sec. IV A and Fig. 1, Table II and\nTable III), the ground-state has Lz= 0 for \feld strengths\nup to a certain value after which a level crossing occurs\nand there are ground states with both Lz= 0 andLz6= 0.\nArguing as above, we can \fnd a system without a mag-\nnetic \feld that shares the ground-state with Lz= 0 and\nfurthermore, for this system, the ground state is unique.\nIt is interesting to note that the above situation can-\nnot arise for the hydrogen atom in a uniform magnetic\n\feld. LetH=1\n2(\u0000ir+A)2\u0000jrj\u00001, be the Hamiltonian\nthat models a hydrogen atom in a uniform magnetic \feld\ngenerated by the vector potential A=B\n2r?,B > 0 and\nr?= (\u0000y;x;0). We denote the ground-state energy e0\nand let\u0015m= infLz=mRH, whereRHis the Rayleigh-\nRitz quotient of H, i.e.,\n\u0015m= inf\nLz=mh ;H i\nh ; i:\nTheorem 4.6 in [25] states e0=\u00150, and furthermore\n\u00150< \u0015\u00001< ::: since limjrj!0+v= 0. Thus, no level\ncrossing occurs in this system.\nWe now turn to a positive result. To obtain a weak\nensemble Hohenberg-Kohn result, denote \n Hthe set of\nground states belonging to Hand letf kgm\nk=1be an or-\nthonormal basis of \n H. We here assume that m< +1,\ni.e., the multiplicity of the ground-state energy e0is \f-\nnite. For a basisf kgm\nk=1, 0\u0014\u0015k\u00141 andPm\nk=1\u0015k= 1,\nlet \u0000H(\u00151;:::;\u0015m) =Pm\nk=1\u0015k kih kbe a density ma-\ntrix ofH. A ground-state particle density \u001aand para-\nmagnetic current density jpofHare then given by \u001a=\nTr \u0000H^\u001a=Pm\nk=1\u0015k\u001a kandjp= Tr \u0000H^jp=Pm\nk=1\u0015kjp\n k.\nConversely, given a particle density \u001aand a param-\nagnetic current density jpwe say that they are ( v;A)-\nensemble-representable if there exists Hwith a \u0000Hsuch\nthat \u0000H7!(\u001a;jp). We use the standard shorthand \u0000 H7!\n(\u001a;jp) to denote \u001a=Pm\nk=1\u0015k\u001a kandjp=Pm\nk=1\u0015kjp\n k.\nHere, of course,f kgm\nk=1is a basis for \n H.\nWe have: Suppose that \u0000kis a ground-state density\nmatrix ofHkand moreover that \u0000k7!(\u001a;jp)fork= 1;2.\nThen \u00001is a ground-state density matrix for H2and vice\nversa.\nWe can prove this claim as follows. Writing Hl=\nHk+ (Hl\u0000Hk), we have for l6=k\nTr \u0000kHl=ek+Z\njp\u0001(Al\u0000Ak)dr\n+Z\n\u001a(vl\u0000vk+ (A2\nl\u0000A2\nk)=2)dr:\nConsequently Tr \u0000 1H2+ Tr \u0000 2H1=e1+e2. Moreover,\nsinceel\u0014Tr \u0000kHlit followsel= Tr \u0000kHland \u0000kis also\na ground-state density matrix of Hl. The result is illus-\ntrated in Fig. 2.\nThere are some immediate consequences of the above4\n••\u00001\u00002⌦H1⌦H2••(v1,A1)(v2,A2)•(⇢,jp)CCDDFigure 4: Illustration of the second result.\n4\nFIG. 2. Two Hamiltonians have di\u000berent sets of degener-\nate ground states (indicated by ellipses). Suppose the den-\nsity matrices \u0000 1and \u0000 2are ground states of H(v1;A1) and\nH(v2;A2), respectively. Assume further that they map to the\nsame density, \u0000 17!(\u001a;jp) and \u0000 17!(\u001a;jp). Then it follows\nthat \u0000 1is also a ground state of H(v2;A2) and that \u0000 2is\nalso a ground state of H(v1;A1). Thus, both \u0000 1and \u0000 2are\nlocated in the intersection of the two ellipses.\nfact. In particular, we stress that a Hohenberg-Kohn\nfunctional can still be constructed in the degenerate case,\nsinceFHK(\u001a;jp) = Tr \u0000H0has a unique value indepen-\ndent of which ground state \u0000 7!(\u001a;jp) that is used. Fur-\nthermore, if the ground-states of H1andH2are non-\ndegenerate, then \u001a1=\u001a2andjp\n1=jp\n2implies \n H1= \nH2.\nThis is the result of Vignale and Rasolt [5].\nReturning to the degenerate case, as demonstrated in\nthe \frst part of this work \n H1= \nH2is not true in\ngeneral even though \u0000 k7!(\u001a;jp). We next introduce\na de\fnition. Given a ( v;A)-ensemble-representable den-\nsity pair (\u001a;jp), there exists an Hwith ground state \u0000 H\nsuch that\u001a= Tr \u0000H^\u001aandjp= Tr \u0000H^jp. Letr(\u0000H)\ndenote the rank of \u0000 H, i.e., the number of nonzero eigen-\nvalues\u0015kof \u0000H. We have the following weak ensemble\nHohenberg-Kohn result:\nAssume that H1andH2have the sets of ground-\nstates \nH1with (orthonormal) basis  1; 2;:::; mand\n\nH2with (orthonormal) basis \u001e1;\u001e2;:::;\u001en. Assume\n\u000017!(\u001a1;jp\n1)and\u000027!(\u001a2;jp\n2), where \u0000kis a ground-\nstate density matrix of Hk. If\u001a1=\u001a2andjp\n1=jp\n2, it\nfollows that \nH1\\\nH26=;. Moreover, with the notation\nrk=r(\u0000k)then there are at least max(r1;r2)linearly\nindependent common ground states of the two systems\nand\ndim\nH1\\\nH2\u0015max(r1;r2):\nIf in addition r1=dim\nH1andr2=dim\nH2, then\n\nH1= \nH2.\nTo prove the above, assume that \u001a1=\u001a2=\u001aand\njp\n1=jp\n2=jp. For the \frst part, suppose \n H1\\\nH2=;\nand letf\u0015kgm\nk=1satisfy 0\u0014\u0015k\u00141 andP\nk\u0015k= 1 such\nthat\u001a=Pm\nk=1\u0015k\u001a kandjp=Pm\nk=1\u0015kjp\n k. We thenhave strict inequality\ne2<mX\nk=1\u0015kh k;H2 ki=e1\u0000Z\njp\u0001(A2\u0000A1)dr\n+Z\n\u001a(v2\u0000v1+ (A2\n2\u0000A2\n1)=2)dr: (6)\nOn the other hand, let f\u0016lgn\nl=1satisfy 0\u0014\u0016l\u00141\nandPn\nl=1\u0016l= 1 such that \u001a=Pn\nl=1\u0016l\u001a\u001elandjp=Pn\nl=1\u0016ljp\n\u001el. Again using \n H1\\\nH2=;, it holds\ne1<nX\nl=1\u0016lh\u001el;H1\u001eli=e2\u0000Z\njp\u0001(A1\u0000A2)dr\n+Z\n\u001a(v1\u0000v2+ (A2\n1\u0000A2\n2)=2)dr: (7)\nAdding (6) and (7) gives e1+e2< e1+e2, which is a\ncontradiction and \n H1\\\nH26=;.\nFor the second part, we use that \u0000 k7!(\u001a;jp) im-\nplies that \u0000 1is a ground-state density matrix of H2\n(and vice versa). To obtain a contradiction, assume\ndim \nH1\\\nH2< r 1. Without loss of generality, let\n 1; 2;:::; m02\nH2and m0+1; m0+2;:::; m=2\nH2,\nwherem0<r1\u0014m. This implies\nTr \u0000 1H2=0\n@m0X\nk=1+mX\nk=m0+11\nA\u0015kh k;H2 ki>e2\nand \u0000 1is not a ground-state density matrix of H2. By\nabove, this is a contradiction. Hence, there are at least\nr1ground states  k2\nH2.\nThe proof that there are at least r2ground states \u001ek2\n\nH1is completely analogous, and we can conclude that\nthere are at least max( r1;r2) common ground states of\ntwo systems and dim \n H1\\\nH2\u0015max(r1;r2).\nLastly, with r1=mandr2=n, we obtain from the\nprevious step\nmin(m;n)\u0015dim \nH1\\\nH2\u0015max(m;n):\nThis can only hold when m=n, and consequently\n\nH1= \nH2. This completes the proof.\nTo summarize, we have proved that a density pair\n(\u001a;jp) in general does not determine the full set of ground\nstates. The counterexample we have provided demon-\nstrates that a given ( \u001a;jp) may correspond to either a\nsystem with a unique ground state, or a system with de-\ngenerate ground states. All that is known is that any sys-\ntem that has ( \u001a;jp) as a ground-state density pair must\nat least share one ground state. While a fully analyti-\ncal proof is tractable in special cases, such as noninter-\nacting systems, the counterexample only requires that a\nlevel-crossing can be tuned by a magnetic \feld. Hence,\nthis situation is common and can be established numer-\nically in many systems, such as the lithium atom. More-\nover, we have proved a positive result. When ( \u001a;jp) is\nensemble (v;A)-representable by a mixed state formed5\nfromrdegenerate ground states, then any Hamiltonian\nH(v0;A0) that shares this ground state density pair must\nhave at least rdegenerate ground states in common with\nH(v;A). Finally, we emphasize that the complications\nin CDFT due to degeneracy does not e\u000bect the general-\nized Hohenberg-Kohn functional since any ground-state\n\u00007!(\u001a;jp) has the same expectation value Tr \u0000 H0.Acknowledgments. We acknowledge the support of the\nNorwegian Research Council through the CoE Hyller-\naas Centre for Quantum Molecular Sciences Grant\nNo. 262695. Furthermore, AL acknowledges support\nfrom ERC-STG-2014 Grant Agreement No. 639508 and\nEIT is grateful for support by the Norwegian Research\nCouncil through the Grant No. 240674. We thank A. M.\nTeale for helpful comments.\n[1] P. Hohenberg and W. Kohn, Phys. Rev. B 1964, 136,\n864.\n[2] M. Levy, Proc. Natl. Acad. Sci. U.S.A. 1979, 76, 6062.\n[3] E. H. Lieb, Int. J. Quant. Chem. 1983, 24, 243.\n[4] E. Runge and E. K. U. Gross, Phys. Rev. Lett. 1984, 52,\n997.\n[5] G. Vignale and M. Rasolt, Phys. Rev. Lett. 1987, 59,\n2360.\n[6] K. Capelle and G. Vignale, Phys. Rev. B 2002, 65,\n113106.\n[7] E. I. Tellgren, A. Laestadius, T. Helgaker, S. Kvaal and\nA. M. Teale, J. Chem. Phys. 2018, 148, 024101.\n[8] R. M. Dreizler and E. K. U. Gross, Density Functional\nTheory; An Approach to the Quantum Many-Body Prob-\nlem(Springer-Verlag 1990)\n[9] E.I. Tellgren, S. Kvaal, E. Sagvolden, U. Ekstr om, A. M.\nTeale and T. Helgaker, Phys. Rev. A 2012, 86, 062506.\n[10] A. Laestadius and M. Benedicks, Int. J. Quant. Chem.\n2014, 114, 782.\n[11] A. Laestadius, Int. J. Quant. Chem. 2014, 114, 1445.\n[12] A. Laestadius, J. Math. Chem. 2014, 52, 2581.\n[13] S. Kvaal and T. Helgaker, J. Chem. Phys. 2015, 143,\n184106.[14] E. I. Tellgren, Phys. Rev. A 2018, 97, 012504.\n[15] M. Ruggenthaler, Ground-State Quantum-Electro-\ndynamical Density-Functional Theory , 2017\narXiv:1509.01417.\n[16] H. Englisch and R. Englisch, Physica A 1983, 121, 253.\n[17] P.E. Lammert, A Gamut of Hohenberg-Kohn properties\n2016, preprint arXiv:1412.3876.\n[18] K. Capelle, C. A. Ullrich and G. Vignale, Phys. Rev. A\n2007, 76, 012508.\n[19] M. Taut, J. Phys. A: Math. Gen. 1994, 27, 1045.\n[20] M. Taut, P. Machon and H. Eschrig, Phys. Rev. A 2009,\n80, 022517.\n[21] S. Viefers, P. Koskinen, P. Singha Deo and M. Manninen,\nPhysica E 2004, 21, 1.\n[22] O.-A. Al-Hujaj and P. Schmelcher, Phys. Rev. A 2004,\n70, 033411.\n[23] S. Stopkowicz, J. Gauss, K. K. Lange, E. I. Tellgren and\nT. Helgaker, J. Chem. Phys. 2015, 143, 074110.\n[24] K. K. Lange, E. I. Tellgren, M. R. Ho\u000bmann and T. Hel-\ngaker 2012, 337327.\n[25] J. E. Avron, I. W. Herbst and B. Simon, Commun. Math.\nPhys. 1981, 79, 529."    },    {        "title": "1802.04333v1.Quantum_oscillations_in_a_biaxial_pair_density_wave_state.pdf",        "content": "Quantum oscillations in a biaxial pair density wave state\nM. R. Norman1and J. C. S\u0013 eamus Davis2, 3\n1Physical Sciences and Engineering Directorate, Argonne National Laboratory, Argonne, IL 60439\n2LASSP, Department of Physics, Cornell University, Ithaca, NY 14853\n3Condensed Matter Physics Department, Brookhaven National Laboratory, Upton, NY 11973\n(Dated: April 4, 2022)\nThere has been growing speculation that a pair density wave state is a key component of the\nphenomenology of the pseudogap phase in the cuprates. Recently, direct evidence for such a state\nhas emerged from an analysis of scanning tunneling microscopy data in halos around the vortex\ncores. By extrapolation, these vortex halos would then overlap at a magnetic \feld scale where\nquantum oscillations have been observed. Here, we show that a biaxial pair density wave state gives\na unique description of the quantum oscillation data, bolstering the case that the pseudogap phase\nin the cuprates may be a pair density wave state.\nThe discovery of charge density wave correlations in\ncuprates by neutron and x-ray scattering, scanning tun-\nneling microscopy (STM), and nuclear magnetic reso-\nnance has had a profound in\ruence on the \feld of high\ntemperature superconductivity, but a number of observa-\ntions indicate that the cuprate pseudogap phase involves\nmore than just charge ordering [1]. Evidence for pairing\ncorrelations, as well as time reversal symmetry break-\ning, is apparent depending on the particular experimental\nprobe. In an attempt to make sense of various con\rict-\ning interpretations, it was speculated that a pair density\nwave (PDW) state, evident in numerical studies of the\nt\u0000Jand Hubbard models [2, 3], could be the `mother'\nphase and also gives a natural explanation for angle re-\nsolved photoemission data [4]. More direct evidence has\nemerged from STM using a superconducting tip, where\nit was shown that the pairing order parameter was mod-\nulated in space [5]. But more telling evidence has re-\ncently come from looking at scanning tunneling data in\na magnetic \feld [6]. There, direct evidence was found for\nbiaxial order in a halo surrounding the vortex cores at a\nwave vector that was one-half that of the charge density\nwave correlations, exactly as expected based on PDW\nphenomenology [7]. This last observation leads to an\nobvious conjecture. One can estimate the \feld at which\nthese vortex halos overlap [8, 9], and this \feld is the same\nat which a long range ordered charge density wave state\nhas been seen by x-rays scattering [10]. Interestingly, this\nis virtually the same \feld at which quantum oscillations\nalso become evident [11, 12]. This implies that the small\nelectron pockets inferred from these data are due to the\nstate contained in these vortex halos.\nThe most successful model for describing quantum os-\ncillation data is that of Harrison and Sebastian [13]. By\nassuming a biaxial charge density wave state, they are\nable to form nodal pockets by folding of the Fermi arcs\nobserved by photoemission to form an electron diamond-\nshaped pocket centered on the \u0000-point side of the Fermi\narc observed by angle resolved photoemission [14]. In\ntheir scenario, as this pocket grows, eventually a Lif-\nshitz transition occurs, leading to a hole pocket centeredaround the \u0000 point itself. A central question is whether\nan alternate model could have a similar phenomenology.\nTo explore this issue, we consider a biaxial PDW state\n[15] with a wave vector of magnitude Q=\u0019=4aas ob-\nserved in the recent STM data [6]. The secular matrix\nfor such a state is of the form:\n0\nBBBBBB@\u000f~k\u0001~k+(Q\n2;0)\u0001~k\u0000(Q\n2;0)\u0001~k+(0;Q\n2)\u0001~k\u0000(0;Q\n2)\n\u0001~k+(Q\n2;0)\u0000\u000f\u0000~k\u0000(Q;0)0 0 0\n\u0001~k\u0000(Q\n2;0)0 \u0000\u000f\u0000~k+(Q;0)0 0\n\u0001~k+(0;Q\n2)0 0 \u0000\u000f\u0000~k\u0000(0;Q)0\n\u0001~k\u0000(0;Q\n2)0 0 0 \u0000\u000f\u0000~k+(0;Q)1\nCCCCCCA\nHere, we assume a d-wave form for the PDW order pa-\nrameter, \u0001 ~ q=\u00010\n2(cos(qxa)\u0000cos(qya)), with its argu-\nment,~ q=~k+~Q\n2, being the Fourier transform of the rel-\native coordinate of the pair (the center of mass Fourier\ntransform being ~Q). We also ignore all of the other o\u000b\ndiagonal components, which arise from the secondary\ncharge order, as they only lead to quantitative correc-\ntions to the results presented here. For \u000f~kwe assumed\nthe tight binding dispersion of He et al. [16] for Bi2201.\nWe do this for two reasons. First, this was the dispersion\nconsidered in previous work on PDWs [4]. Second, there\nare no complications in this dispersion associated with\nbilayer splitting.\nTo proceed, we need to de\fne the spectral function, A,\nas measured by angle resolved photoemission:\nA(!;~k) =\u0000\n\u0019ci(~k)2\n(!\u0000Ei(~k))2+ \u00002(1)\nHere,Eiis the i'th eigenvalue of the secular matrix, ci\nthe~kcomponent of the corresponding eigenvector (the\nanalogue of the particle-like Bogoliubov component), and\n\u0000 a phenomenological broadening parameter.\nIn Fig. 1, we show the spectral weight and eigenvalue\ncontours at != 0 for four values of \u0001 0. Deep in the\npseudogap phase (large \u0001 0), a small electron pocket cen-\ntered along the diagonal (0 ;0)\u0000(\u0019;\u0019) is observed whose\n\rat edge follows the spectral weight. As such, this pocketarXiv:1802.04333v1  [cond-mat.supr-con]  12 Feb 20182\n2.01.51.00.50.02.01.51.00.50.0(a)\n2.01.51.00.50.02.01.51.00.50.0(b)\n2.01.51.00.50.02.01.51.00.50.0(d)\n2.01.51.00.50.02.01.51.00.50.0(c)\nFIG. 1: Spectral weight and eigenvalue contours at != 0 for\na pair density wave state with its amplitude, \u0001 0, being (a)\n25 meV, (b) 50 meV, (c) 75 meV and (d) 100 meV (the x\nand y axes are kxandkyin units of \u0019=a). Arrows point to\nthe center of the electron pocket ((c) and (d)) and the hole\npocket ((a) and (b)). The normal state dispersion is given\nby He et al. [16]. Here, the modulus of the PDW ordering\nvector,Q, is\u0019=4a, as observed in recent STM experiments\n[6]. For the spectral weight, a phenomenological broadening\nparameter, \u0000, of 25 meV is assumed.\nshould dominate the deHaas-vanAlphen (dHvA) ampli-\ntude, unlike the other pockets which exhibit no spectral\nweight [17]. As the hole doping increases (smaller \u0001 0),\nthis pocket undergoes a Lifshitz transition, resulting in a\nlarger hole pocket also centered along the diagonal that\nresembles that obtained in the phenomenological YRZ\n(Yang-Rice-Zhang) model for the cuprates [18, 19]. Once\nthe gap collapses, then one recovers the much larger hole\npocket centered at ( \u0019=a;\u0019=a ) that is characteristic of the\noverdoped state [20]. We remark that the biaxial order\nis critical in forming these smaller pockets, though hints\nof them can be found in earlier work that assumed a uni-\naxial PDW instead [21{24] (the last two of these papers\naddressing the dHvA data).\nWe quantify this by plotting the area of the pocket\n(in the dHvA units of Tesla) along with the cyclotron\nmass as a function of \u0001 0in Fig. 2. One sees a mod-\nest dependence of the pocket area on \u0001 0except for the\npronounced jump at the Lifshitz transition, along with\nthe associated mass divergence at the Lifshitz transition.\nThese dependencies are in good accord with dHvA data\nas a function of hole doping [25], including the mass di-\nvergence, noting that quantitative details are in\ruenced\nby the dispersion and chemical potential (that is, the con-\n4005006007008009001000\n020406080100Frequency (Tesla)Δ0 (meV)hole\nelectron(a)\n0.911.11.21.31.41.51.6\n020406080100MassΔ0 (meV)(b)FIG. 2: (a) deHaas-vanAlphen (dHvA) frequency (area of\nthe pocket) and (b) cyclotron mass versus \u0001 0as derived from\nFig. 1. Note the Lifshitz transition where the small electron\npocket for large \u0001 0converts to a larger hole pocket for smaller\n\u00010. As a reference, the area of the large normal state hole\npocket centered at ( \u0019=a;\u0019=a ) for \u0001 0= 0 is 17.68 kilo-Tesla\nwith a cyclotron mass of 3.69.\nversion of the x-axis of Fig. 2 to doping is in\ruenced not\nonly by the doping dependence of \u0001 0, but also by the\ndoping dependence of the band structure and chemical\npotential). Moreover, the results presented here o\u000ber a\nprediction. That is, beyond the mass divergence (as \u0001 0\ndecreases), there should be a small doping range where\na large hole pocket of roughly twice the size of the elec-\ntron pocket occurs before the very large hole pocket in\nthe overdoped regime forms when the gap collapses. This\nprediction is supported by Hall data that shows a region\nof the phase diagram between p=0.16 andp=0.19 where\nthe Hall constant rapidly changes [26], with p=0.16 being\nwhere the mass divergence referred to above occurs, and\np=0.19 where the large Fermi surface is recovered (here,\npis the doping).\nWe feel that the scenario o\u000bered here is an attractive\nalternate to models based on a charge density wave. It\nis not only consistent with recent STM data in the vor-\ntex halos [6], but also consistent with magneto-transport\ndata that indicate the presence of pairing correlations for\nmagnetic \felds not only up to but well beyond the resis-\ntive H c2[27]. This is in line as well with previous theoret-\nical work on quantum oscillations in a d-wave vortex liq-\nuid [28]. Certainly, we hope that the model o\u000bered here\nwill lead to additional studies in high magnetic \felds to\nde\fnitively determine whether a pair density wave state\nreally exists.\nIn summary, the work presented here bolsters the case\nthat the enigmatic pseudogap phase in the cuprates is a\npair density wave state.\nThis work was supported by the Center for Emergent\nSuperconductivity, an Energy Frontier Research Center\nfunded by the US DOE, O\u000ece of Science, under Award\nNo. DE-AC0298CH1088. We thank Stephen Edkins, Mo-\nhammad Hamidian and Andrew Mackenzie for access\nto their vortex halo STM data in advance of publica-\ntion. We also acknowledge Neil Harrison, Peter John-3\nson, Marc-Henri Julien, Catherine Kallin, Steve Kivel-\nson, Patrick Lee, Brad Ramshaw, Subir Sachdev, Suchi-\ntra Sebastian, Todadri Senthil, and Louis Taillefer for\nvarious discussions.\n[1] E. Fradkin, S. A. Kivelson and J. M. Tranquada, Rev.\nMod. Phys. 87, 457 (2015).\n[2] P. Corboz, S. R. 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Randeria, Nature Com-\nmun. 4, 1700 (2013)."    },    {        "title": "1803.08992v1.Clogging_and_Depinning_of_Ballistic_Active_Matter_Systems_in_Disordered_Media.pdf",        "content": "arXiv:1803.08992v1  [cond-mat.soft]  23 Mar 2018Clogging and Depinning of Ballistic Active Matter Systems i n Disordered Media\nC. Reichhardt and C.J.O. Reichhardt\nTheoretical Division and Center for Nonlinear Studies,\nLos Alamos National Laboratory, Los Alamos, New Mexico 8754 5, USA\n(Dated: September 26, 2018)\nWe numerically examine ballistic active disks driven throu gh a random obstacle array. Formation\nof a pinned or clogged state occurs at much lower obstacle den sities for the active disks than for\npassive disks. As a function of obstacle density we identify several distinct phases including a\ndepinned fluctuating cluster state, a pinned single cluster or jammed state, a pinned multicluster\nstate, a pinned gel state, and a pinned disordered state. At l ower active disk densities, a drifting\nuniform liquid forms in the absence of obstacles, but when ev en a small number of obstacles are\nintroduced, the disks organize into a pinned phase-separat ed cluster state in which clusters nucleate\naround the obstacles, similar to a wetting phenomenon. We ex amine how the depinning threshold\nchanges as a function of disk or obstacle density, and find a cr ossover from a collectively pinned\ncluster state to a disordered plastic depinning transition as a function of increasing obstacle density.\nWe compare this tothe behavior of nonballistic active parti cles and show that as we varythe activity\nfrom completely passive to completely ballistic, a clogged phase-separated state appears in both the\nactive and passive limits, while for intermediate activity , a readily flowing liquid state appears and\nthere is an optimal activity level that maximizes the flux thr ough the sample.\nI. INTRODUCTION\nThere is a wide range of soft and hard matter sys-\ntems that can be modeled as collectively interacting par-\nticles which, when driven over quenched disorder, exhibit\npinning-depinning behavior as well as transitions be-\ntween different types of sliding regimes1,2. Such dynam-\nics occur for vortex motion in type-II superconductors3,4,\nsliding charge density waves5, depinning of classical\nWignercrystals6,7, currentdrivenmotionofskyrmionsin\nchiral magnets8,9, colloids interacting with random10–15\nor ordered substrates16, sliding in frictional systems17,\nmagnetic domain wall motion18,19, erosion20, granular\nmatter21,22, driven pattern forming systems23, geophys-\nical models of plate tectonics24, and the motion of dis-\nlocations in crystalline materials25. The substrate may\nbe random, ordered, or partially ordered, and it can be\nmodeled as localized pinning sites with a finite trap-\nping strength or as impenetrable obstacles. Under an\napplied drive, the particles exhibit a variety of pinned\nand moving order-disorder transitions that can be char-\nacterizedby the movingstructure, pattern formationfea-\ntures, changes in the velocity force curves, and fluctu-\nation phenomena1,2. In the systems listed above, the\nparticles themselves are passive or experience only ther-\nmal fluctuations, so the driving is strictly externally ap-\nplied; however, recently a growing number of studies\nhave focused on what are called active matter systems\ncontaining self-driven particles with an activity that is\noften modeled as arising from driven diffusive or run-\nand-tumble dynamics26,27. In the absence of a substrate,\nactive disks exhibit a transition from a uniform gas or\nliquid state to a phase-separated or cluster state consist-\ning of a high density solid coexisting with a low den-\nsity active gas28–34. This transition occurs for fixed ac-\ntivity as a function of increasing density or for fixed\ndensity with increasing activity level. Active mattersystems have also been been studied in the context of\nparticle shape effects27,35, active rotators36, passive and\nactive mixtures37,38, boundary effects39–43, and ratchet\neffects44–46.\nSeveral studies have examined different types of ac-\ntive matter systems coupled with ordered47–49or dis-\nordered substrates33,50–65. Numerical simulations show\nthat when a drift force is applied to run-and-tumble\ndisks moving through a random obstacle array, the flux\nthrough the system is non-monotonic as a function of ac-\ntivity level, indicating that there is an optimal run length\nor run correlation time that maximizes the flux of disks\nthrough the obstacles33. At small run lengths, the disks\nbehave thermally and easily become trapped behind the\nobstacles, giving a low disk flux. As the activity level\nor run length increases, the disks can more readily move\naround the obstacles, increasing the flux; however, when\nthe run length is too large, a self-pinning effect occurs in\nwhich the disks self-cluster around the obstacles, reduc-\ning the flux33,64. As a result, the flux is maximized when\nthe disks are active and the self-clustering is weak, while\nit is reduced when the activity becomes high enough for\nsignificant clustering or self-trapping to occur. When the\nrun lengths are very long, the average flux under an ap-\nplied drive is strongly reduced, but it never reaches zero\nsince there are still long-time dynamical rearrangements\nthat produce avalanches or intermittent flow of disks in\nthe direction of the drift force64. Analytic and theoret-\nical studies of active systems without a drift force also\nindicate that there is an optimal activity level that max-\nimizes the diffusion in the system63. If the obstacles are\nreplacedbypinning sites, the onset ofclusteringcanhave\nthe opposite effect of increasing rather than decreasing\nthe flux, since the clusters act like large rigid objects\nthat are poorly trapped by individual pinning sites59.\nWhen the activity is low, a uniform liquid state appears\nand individual disks can be trapped readily by individual2\npinning sites59. These works also showed that within the\nphase separated state, increasing the number of obsta-\ncles or pinning sites produces a disorder-induced transi-\ntionfromaphase-separatedstatetoauniformdisordered\nstate57,59.\nIn addition to the run-and-tumble models, several sim-\nulation studies of swarming or flocking active systems\nwith quenched disorder show that there is an optimal\nnoise level for the appearance of flocking50,53, and that\nincreasing the disorder strength can induce a transition\nfrom a flocking to a non-flocking state52. Experiments\non colloidal rollers that exhibit flocking behavior have\nalso revealed a transition from a drifting flocking state to\na non-flocking state as a function of increasing obstacle\ndensity, where the flow becomes increasingly filamentary\nas the disordered state is approached56.\nIn these systems, the activity is stochastic in nature.\nFor run-and-tumble particles, after each run interval the\nparticle randomly reorients and runs in a new direction,\nwhile in driven diffusive models, there is a noise term\ncontrolling the rate of rotational diffusion. The deter-\nministic limit of active matter is purely ballistic flow\nwhere the swimming direction of each particle is per-\nmanently fixed. This can be achieved in run-and-tumble\ndisk systems by setting the running time to infinity, or\nin driven diffusive systems by setting the rotational dif-\nfusion coefficient to zero51. In the deterministic regime,\nthe particles can form a cluster state even at very low\ndensities, and Bruss et al.66argue that phase separation\noccurs when the mean time between collisions is smaller\nthan the mean duration of an individual collision. Pre-\nvious studies51of an active ballistic system showed that\nthe disks can form a frozen cluster state where almost\nall the fluctuations are lost. This frozen state arises due\nto the lack of stochastic behavior in the system, and the\nresulting cluster can be regarded as an absorbed state.\nIn contrast, at long but finite run times, the cluster can\ngradually evolve over time due to the possibility of rare\nstochastic reorganizations64.\nPrevious studies of the active ballistic limit did not\ninclude an applied drift force, so no pinning-depinning\nphenomena occurred. In this work we consider active\nballistic disks driven through an obstacle array, and we\nmeasure the averagedrift mobility ∝angbracketleftV∝angbracketrightof the disks in the\ndirectionofthedrivingforce FD. Wefindthatthesystem\ncan evolve toward a completely pinned or clogged state\nwith∝angbracketleftV∝angbracketright ≈0. We describe our simulation in Sec. II. In\nSec. III wecomparethe active pinning orcloggingto that\nobservedin the zero activityor passivelimit. The critical\ndensity of obstacles needed to induce the formation of a\nclogged state is much higher in the passive system than\nin the active ballistic system, and we show that in gen-\neral the active ballistic disks are much more susceptible\nto forming clogs than the passive disks. In the active sys-\ntem, clogging is associated with the formation of a clus-\nter state. As a function of increasing obstacle density, we\nidentifyfourphases: aslidingstatewith eitherafluctuat-\ning cluster ora liquid structure; a pinned singlecluster orjammedstate consistingofalargeclusterheld in placeby\na small number of obstacles; a pinned multicluster state\ncontaining several distinct pinned clusters; and a pinned\ndisordered state in which the disk density remains spa-\ntially uniform. As the active disk density increases, we\nalso find what we term a pinned gel state where the disks\nform a percolating labyrinth structure. For low active\ndisk densities where a flowing uniform liquid appears in\nthe absence of obstacles, we find that introduction of a\nsmall number of obstacles causes a transition to a pinned\ncluster state, which we compare to the active wetting of\nclusters around an obstacle. In Sec. IV we examine the\nvelocity-force relations and the behavior of the critical\ndepinning force Fc. The depinning threshold for a clus-\nter is finite, and there is a pronounced increase in Fcat\nthe transition from the pinned single cluster state to the\npinned disordered state, reminiscent of the “peak effect”\nobserved in superconducting vortex systems at a transi-\ntion from a collectively pinned ordered or quasiordered\nvortex crystal to a vortex glass state2,3. In Sec. V we\nshow how passive clogging can be connected to the bal-\nlistic clogging limit by considering finite but increasing\nrun times for run-and-tumble active disks in obstacle ar-\nrays. In both the passive and ballistic clogged states,\nthe disks form a cluster, while between these two limits a\nflowing liquid structure appears. There is an optimal run\ntime at which the disk flux is maximized, and the flux\ndecreases with increasing run length until the disks form\na completely clogged state at infinite run times. We dis-\ncusshowtheseresultscanberelatedtogranularjamming\ntransitions67–69and jamming in systems with quenched\ndisorder70,71, where the activity or the obstacle density\nrepresent an additional set of parameters that can be\nused to induce a jammed state. In Sec. VI we summarize\nour results.\nII. SIMULATION\nWe consider a two-dimensional system with periodic\nboundary conditions in the xandy-directions containing\nNaactive or mobile disks that interact through a stiff\nrepulsive harmonic spring, Fs=k(d−2R)Θ(d−2R)ˆd,\nwheredis the distance between two disks, dis the dis-\nplacement vector, k= 100is the spring constant, and the\ndisk radius is R= 0.5. For this value of kthe disk-disk\noverlaps remain very small, allowing us to define the ac-\ntivediskdensityintermsoftheareacoveredbythe disks,\nφa=NaπR2/L2, where the system size is L= 100. In\nthe limit of no activity and no obstacles, the disks form\na hexagonal lattice at φa= 0.9. In addition to the mo-\nbile active disks, we introduce Nobsobstacles that are\nidentical to the active disks but are permanently fixed\nin place. The obstacles are initially placed in a hexago-\nnal lattice and are randomly diluted until we reach the\ndesired obstacle density of φobs=NobsπR2/L2. Placing\nthe obstacles in an initial hexagonal lattice ensures that\nthere is a fixed minimum distance between any two ob-3\nstacles, thereby avoiding the rare obstacle density fluctu-\nations such as large gaps or tight obstacle clustering that\ncan arise from a completely random obstacle placement\nand dominate the dynamics. The total disk coverage of\nboth active and fixed disks is φtot=φobs+φa. The dy-\nnamics of the active disks is obtained from the following\noverdamped equation of motion:\nηdri\ndt=Fi\ns,a+Fi\nm+Fi\ns,obs+FD. (1)\nHereη= 1.0 is the damping constant and the interac-\ntion with other active disks Fs,ahas the form of the stiff\nspring repulsion Fs. Each active disk has a motor force\nFi\nmthat is constant in magnitude and oriented in a ran-\ndomly chosen direction. In run-and-tumble systems, the\nmotor force orientation is held fixed during the run time\nτr, after which a new random orientation is chosen for\nthe next running time. In the ballistic limit, we set τr\nto infinity so that the running direction never changes.\nThe forces from the obstacles Fs,obsare also given by Fs,\nand the external driving force FD=FDˆ xis applied uni-\nformly to all active disks. We also consider the passive\nparticle limit in which Fm= 0 and the only driving force\nis the externally applied FD. To initialize the system,\nafter establishing the location of the obstacles we place\nmobile disks with artificially reduced radii in nonoverlap-\nping locations and allow the mobile disks to rearrange\nwhile gradually expanding the radii to the final value of\nR= 0.5. With this method we can reach disk densities\nup toφtot= 0.86.\nTo characterize the system we measure the average\ndrift velocity of the disks ∝angbracketleftV∝angbracketright=∝angbracketleftN−1\na/summationtextNa\ni=0vi·ˆx∝angbracketrightin\nthe direction of the drive. We wait 1 ×106simulation\ntime steps for the system to settle into a steady state and\nthen average over an additional 9 ×106simulation time\nsteps to obtain ∝angbracketleftV∝angbracketright. We have found that increasing the\nwaiting time produces negligible changes in the results.\nIn this work, unless otherwise noted we set FD= 0.05,\n|Fm|= 0.5, andτr=∞; however,wealsoconsidervaried\nFDand finite τr.\nIII. PHASES OF ACTIVE BALLISTIC DISKS\nIn Fig. 1(a) we plot the fraction Cmax/Naof active\ndisks in the largest cluster versus time in simulation time\nsteps for a system with FD= 0.05,Na= 4450, and\nNobs= 50, giving an active disk density of φa= 0.3495,\nan obstacle density of φobs= 0.00393, and an overall\ndensity of φtot= 0.3534. The largest cluster size Cmaxis\ndefined as the largest number of disks in direct contact\nwith each other as determined using the cluster identifi-\ncation algorithm described in Ref.72. Figure 1(b) shows\nthe corresponding disk velocity V=N−1\na/summationtextNa\ni=0vi·ˆ xver-\nsus time. Both Cmax/NaandVexhibit strong fluctu-\nations during the first 2 .5×106time steps, indicating\nthat the system is in a dynamic unpinned fluctuating\nstate. The disks then become trapped in a single pinned00.20.40.60.81Cmax/Na\n02×1064×1066×1068×106\ntime00.010.020.030.040.050.06\nV00.10.20.30.40.5Cmax/Na\n0.01.0×1072.0×107\ntime00.010.020.030.04\nV(a)\n(b)ab\ncd\n(c)\n(d)\nFIG.1: (a,b)Anactiveballistic systemwithanareacoverag e\nofφa= 0.3495 active disks and φobs= 0.00393 obstacles for\na totalφtot= 0.3534. An external drift force of FD= 0.05 is\napplied in the positive x-direction. (a) The fraction of disks in\nthe largest cluster, Cmax/Na, versus time in simulation time\nsteps. (b) The drift velocity Vof the active disks vs time.\nThere is a transition from a fluctuating cluster state that is\ndrifting in the direction of drive, illustrated in Fig. 2(a) , to a\npinned single cluster, shown in Fig. 2(b). At the transition ,\nCmax/Naabruptly increases to a value close to one and V\nsimultaneously drops nearly to zero. The letters aandb\nindicate the times corresponding to the images in Fig. 2(a,b ).\n(c)Cmax/Naand (d) Vvs time for the same system in the\npassive|Fm|= 0 limit at φtot= 0.3534 and φobs= 0.1178.\nForthisvalueof φtot, thesystemcanreachapinnedorclogged\nstate only when φobs≥0.098. The system evolves over time\ninto a pinned state, with a gradual drop in Vaccompanied by\na gradual increase in Cmax/Na. The initial unclogged state at\nthe time marked cis illustrated in Fig. 2(c), while the V= 0\npinned state at the time marked dis shown in Fig. 2(d).\ncluster, as shown by the sudden jump in Cmax/Nato\nCmax/Na≈1.0 which is accompanied by a drop in V\nto nearly zero. There are still some small fluctuations\nin bothCmax/NaandVdue to the presence of a small\nnumber of freely running disks that do not join the clus-\nter. In Fig. 2(a) we show a snapshot of the depinned\nfluctuating clusters at time 1 .0×106, where the active\ndisks form temporary clusters. Here V= 0.046, which\nis close to the expected obstacle-free value of V= 0.05.\nA pinned single cluster state appears at time 3 ×106,\nas illustrated in Fig. 2(b), where the active disks form a\nsingle large immobile cluster. We find that even a small\nnumber of obstacles ( Nobs/Na= 0.011) can produce a\npinned state for active ballistic disks. In contrast, for\npassive disks at the same total density of φtot= 0.3534,\nthe system does not reach a pinned or clogged state until\nNobs/Na≥0.384, indicating that nearly 35 times more\nobstacles are required to pin the passive disks compared\nto the active ballistic disks.\nIn Fig. 1(c,d) we plot Cmax/NaandVversus time\nfor a passive |Fm|= 0 system with φtot= 0.3534\nandφobs= 0.1178, where the disks reach a pinned or4\nx (a)y\nx (b)y\nx (c)y\nx (d)y\nFIG. 2: (a,b) The active ballistic disk positions (red cir-\ncles) and obstacle locations (blue circles) for the system i n\nFig. 1(a,b) with φtot= 0.3534,φobs= 0.00393, and a drift\nforceFD= 0.05 applied in the positive xdirection. (a) De-\npinned fluctuating clusters appear at time 1 ×106, marked\nain Fig. 1(a). (b) The pinned single cluster state at time\n3×106, marked bin Fig. 1(a). (c,d) Passive disk positions\n(green circles) and obstacle locations (blue circles) for t he sys-\ntem from Fig. 1(c,d) with φtot= 0.3534 and φobs= 0.1178.\n(c) The initial flowing state at the time marked cin Fig. 1(c).\n(d) The clogged or pinned state at time 3 ×106, marked din\nFig. 1(c).\nclogged state. In contrast to the active ballistic system\nin Fig. 1(a,b), the pinned state does not appear abruptly;\ninstead, the passive disks continuously evolve toward the\npinned state over time, with a growing number of pinned\nclusters gradually emerging as indicated by the steady\nincrease in Cmax/Naand the gradual decrease in V. In\nFig. 2(c) we illustrate the initial uniform spatial distri-\nbution of the passive disks, while in Fig. 2(d) we show\na snapshot of the disk positions in the clogged state at\na time of 2 ×106. Unlike the active ballistic disks in\nFig. 2(b), the passive disks in Fig. 2(d) do not form\na single clump but instead assemble into a number of\nsmaller clumps. This indicates that the passive and ac-\ntive clogged or pinned states are very different in nature.\nIn Fig. 3(a,b) we plot CL=∝angbracketleftCmax/Na∝angbracketrightand∝angbracketleftV∝angbracketrightversus\nφobsfor the active ballistic system from Fig. 1(a) with\nφtot= 0.3534 at FD= 0.05. For φobs<0.0039 the\nsystem is in a depinned fluctuating cluster state, labeled\nphase I fc, where∝angbracketleftV∝angbracketrightis finite and the long-time average\nofCmax/NaisCL≈0.5. For 0 .0039≤φobs<0.025,\nwe find a pinned single cluster state, denoted phase II,\nwithCL>0.9 and∝angbracketleftV∝angbracketright ≈0.0. In Fig. 4(a) we il-00.20.40.60.81\nCL\n00.10.20.30.40.5\nCL\n10-410-310-210-1\nφobs00.010.020.030.040.05\n<V>\n10-410-310-210-1\nφobs00.010.020.030.040.05\n<V>(a)\n(b)(c)\n(d)IIIIVIII\ndcba\nabdc\nFIG. 3: (a) CLand (b) ∝angbracketleftV∝angbracketrightvs obstacle density φobsfor\nactive ballistic disks with FD= 0.05 andφtot= 0.3534. For\nφobs<0.0039,∝angbracketleftV∝angbracketrightis finite and the system is in phase I fc, the\ndepinned fluctuating cluster state, while at the transition to\nphase II, the pinned single cluster state, ∝angbracketleftV∝angbracketrightdrops to zero.\nPhase IIextendsfrom 0 .0039≤φobs<0.025 andis illustrated\nin Fig. 4(a). For 0 .025≤φobs<0.065, the system is in\nphase III, a pinned multicluster state, illustrated in Fig. 4(b),\nwhile for φobs≥0.065 the system is in phase IV, the pinned\ndisordered phase illustrated in Fig. 4(c,d) at φobs= 0.0942\nandφobs= 0.1963. The labels atodin (a) indicate the\nvalues of φobsat which the images in Fig. 4 are obtained. (c)\nCLand (d)∝angbracketleftV∝angbracketrightvsφobsfor passive disks at the same FDand\nφtot. Here there are only two phases: a plastic flow state\nφobs<0.098, and a completely pinned or clogged state for\nφobs≥0.098. There is a peak in CLatφobs= 0.098 where\n∝angbracketleftV∝angbracketrightdrops to zero. The labels atodin (c) indicate the values\nofφobsat which the images in Fig. 5 are obtained.\nlustrate phase II at φobs= 0.01178. Phase II can be\nviewed as a pinned jammed state in which a single clump\nhas nucleated around the obstacles and acts as a rigid\nsolid. Active disks that are not adjacent to obstacles\nare pinned or prevented from moving by other active\ndisks through contact interactions, so the collective pin-\nning of the clump is dominated by disk-disk interactions\nratherthan by direct disk-obstacleinteractions. Since we\nare using monodisperse disks rather than the bidisperse\ndisk mixture commonly studied in jammed systems, the\nparticles forming the cluster have a substantial amount\nof hexagonal or crystalline ordering, whereas typical 2D\njammed systems form amorphous rather than polycrys-\ntalline packings68,70; however, in both our system and\nthe jamming systems, it is the disk-disk contact inter-\nactions that cause the system to act like a solid that\ncan be pinned by a small number of obstacles. Previ-\nous work51on active ballistic systems revealed similar\nlarge-scale frozen cluster states, but did not include an\nexternal drift force. In the present study, the formation\nof the single frozen cluster in the presence of a drift force\nresults in a pinned state.\nFor 0.025≤φobs<0.065 in Fig. 3(a,b), we observe5\nx (a)y\nx (b)y\nx (c)y\nx (d)y\nFIG. 4: The active ballistic disk positions (red circles) an d\nobstacle locations (blue circles) for the system in Fig. 3(a ,b)\nwithFD= 0.05 andφtot= 0.3534 obtained at the values\nofφobsmarked by the letters atodin Fig. 3(a). (a) The\npinned single cluster phase II at φobs= 0.01178. (b) The\npinned multicluster phase III at φobs= 0.039. (c) The pinned\ndisordered phase IV at φobs= 0.0942 consists of a group of\nsmall clusters. (d) The pinned disordered phase IV at φobs=\n0.1963 is composed of even smaller clusters.\na pinned multicluster state termed phase III in which\n∝angbracketleftV∝angbracketright= 0 where CLdecreases from CL= 0.9 toCL= 0.12\nwith increasing φobs. A snapshot of the disk positions in\nphaseIIIat φobs= 0.039appearsin Fig.4(b). For φobs≥\n0.065,thesystemisinphaseIV,apinneddisorderedstate\nwithCL<0.15 in which the disks form numerous small\nclumps that gradually decrease in size with increasing\nφobs, as illustrated in Fig. 4(c) at φobs= 0.0942 and in\nFig. 4(d) at φobs= 0.1963.\nWe plotCLand∝angbracketleftV∝angbracketrightversusφobsfor passive disks with\nφtot= 0.3534 and FD= 0.05 in Fig. 3(c,d). The pas-\nsive disks do not reach a pinned state with ∝angbracketleftV∝angbracketright ≈0 until\nφobs> φc= 0.098. Figure 3(c) shows that CLis small\nfor lowφobsand increases to a peak value of CL= 0.5\njust below φc. Forφobs> φc,CLdecreases with increas-\ning obstacle density. In Fig. 5(a) we show a snapshot of\nthe flowing state at φobs= 0.01178. Although no clus-\nters appear, the disks tend to form one-dimensional (1D)\nflowing chains. In the flowing state at φobs= 0.055,\nillustrated in Fig. 5(b), small clusters are beginning to\nappear. The pinned cluster state near the peak value\nofCLatφobs= 0.1178 is shown in Fig. 2(d). Above\nthe peak in CL, the size of the clusters decreases with\nincreasing obstacle density and a clogged state forms as\nshown in Fig. 5(c) for φobs= 0.14137 and in Fig. 5(d)x (a)y\nx (b)y\nx (c)y\nx (d)y\nFIG. 5: Passive disk positions (green circles) and obstacle\nlocations (bluecircles) for thesystem inFig. 4(c,d) with FD=\n0.05 andφtot= 0.3534 obtained at the values of φobsmarked\nby the letters atodin Fig. 3(c). (a) The flowing state at\nφobs= 0.01178. (b) At φobs= 0.055, clusters begin to form.\n(c) The clogged state at φobs= 0.14137. (d) The clogged\nstate at φobs= 0.1963.\n0.001 0.01 0.1\n|φc  - φobs|0.031250.0625\n<V>\nFIG. 6: The scaling of V∝ |φc−φobs|βfor the passive disks\nin Fig. 3(d) with φc= 0.098 and β= 0.35.\nforφobs= 0.1963, where the clusters have become quite\nsmall.\nThe peak or divergence in CLfor the passive disk sys-\ntem in Fig. 3(c) suggests that the onset of complete clog-\nging atφcoccurs at a critical point. We have tried per-\nforming a power law fit CL∝ |φc−φobs|νon either side\nof the divergence; however, we find only a limited range6\n02000400060008000\n<Cmax>\n0.0001 0.001 0.01 0.1\nφobs00.20.40.60.8\nCLab\ncd (a)\n(b)\nFIG.7: (a) ∝angbracketleftCmax∝angbracketright, theaverage numberofdisksinthelargest\ncluster, vs φobsatFD= 0.05 forφtot= 0.668 (Ntot= 8500,\nturquoise), 0.589 (7500, light green), 0.511 (6500, red), 0 .432\n(5500, dark green), 0.354 (4500, blue), 0.275 (3500, maroon ),\n0.196 (2500, violet), 0.118 (1500, orange), and 0.0786 (100 0,\nmagenta). (b) The corresponding CLvsφobscurves. The I-II\ntransition is associated with a jump or increase in ∝angbracketleftCmax∝angbracketrightand\nCL, while at large φobs, the system enters a pinned disordered\nphase as indicated by the drop in CmaxandCLto nearly zero.\nThe labels atodin (a) indicate the values of φobsat which\nthe images in Fig. 8 were obtained.\nfor the fit resulting in strong variations in the exponent.\nWe find more consistent scaling of the average drift ve-\nlocity as φcis approached, with ∝angbracketleftV∝angbracketright ∝ |φc−φobs|β, as\nshown in Fig. 6 where β= 0.35. We have studied the\ncritical clogging behavior of passive disks in more depth\nin Ref.14, where we find a robust power law divergence in\nthe transient times near φcconsistent with an absorbing\nphase transition. The focus of the present work is active\nballistic jamming and we measure the passive disks for\ncomparison. Ourresultsindicatethatthe onsetofpinned\nor clogged states for the active ballistic disks is very dif-\nferent in nature from the clogging of passive disks. In\nparticular, we find only two phases for the passive disks\nand four phases for the active ballistic disks. The I fc-II\ntransition marking the onset of a pinned state for the ac-\ntive ballistic disks produces discontinuities in both ∝angbracketleftV∝angbracketright\nandCL, consistent with a first order phase transition,\nwhile for the passive disks, the onset of pinning or clog-\nging has the character of a second order phase transition\nor a crossover phenomenon.\nIn Fig. 7(a) we plot ∝angbracketleftCmax∝angbracketright, the average number of\ndisks in the largest cluster, versus φobsfor the active bal-\nlistic system at varied φtotto highlight the evolution of\nphases I through IV. Figure 7(b) shows the collapse of\nthe curves when the same results are plotted in terms\nofCL, the average fraction of disks in the largest clus-x (a)y\nx (b)y\nx (c)y\nx (d)y\nFIG. 8: The active ballistic disk positions (red circles) an d\nobstacle locations (blue circles) for the system in Fig. 7 ob -\ntained at the values of φtotandφobsmarked by the letters a\ntodin Fig. 7. (a) The drifting liquid for φtot= 0.196 and\nφobs= 0.003926. (b) The pinned gel phase containing a large-\nscale percolating cluster at φtot= 0.589 and φobs= 0.1256.\n(c) The pinned disordered phase at φtot= 0.589 and φobs=\n0.283. (d) The high density depinned fluctuating cluster state\natφtot= 0.668 and φobs= 0.000156.\nter, versus φobs. Forφtot≤0.275, phase I is a uniform\ndrifting liquid with CL<0.04, as illustrated in Fig. 8(a)\nforφtot= 0.196 atφobs= 0.003926. As φobsincreases,\nthereis atransitionfromthe drifting liquid phaseIto the\npinned single cluster phase II, as indicated by the large\nincrease in CLto a value of CL= 0.8 or higher. The\nobstacle density φobsat which the I-II transition occurs\nshifts upward as φtotdecreases, and at the lowest values\nofφtotthat we consider, the system always remains in\nphase I, as shown for φtot= 0.0786 in Fig. 7. We note\nthat the maximum allowed value of φobsdecreases with\ndecreasing φtotsince it is bounded by the total disk den-\nsity. For φtot≥0.35, instead of the drifting liquid phase\nI, we find the depinned fluctuating cluster state I fcas\ndescribed previously, and in all cases the I-II and I fc-II\ntransitions are associated with a jump or increase in CL.\nThe II-III transition also shifts to higher values of φobs\nwith increasing φtot.\nWithin the pinned disordered phase IV in Fig. 7, an\nadditional feature emerges for φtot= 0.432 in the form\nof a peak in ∝angbracketleftCmax∝angbracketrightandCLnearφobs= 0.115. This\npeak grows in both height and extent with increasing\nφtot. At the onset of the pinned disordered phase IV,\nCLis low and the disks form a small number of isolated\nclumps. As φobsincreases, these clumps break apart and7\n0.1 0.2 0.3 0.4 0.5 0.6\nφtot10-310-210-1\nφobs\nII\nIIIIIV\nIIIpc\nIfc\nFIG. 9: The locations of the phases in Fig. 7 as a func-\ntion ofφobsvsφtot. Phase I (magenta) is the drifting liq-\nuid state; phase I fc(pink) is the depinned fluctuating clump\nstate; phaseII(yellow)isthepinnedsingleclusterstate; phase\nIII (dark green) is the pinned multicluster state; phase III pc\n(light green) is the pinned gel state; and phase IV (blue) is\nthe pinned disordered state.\nthe disks are spread more evenly over the substrate. At\nhigher overall disk densities φtot≥0.432, this produces a\npercolation transition in which the broken clumps merge\nto form a pinned gel or labyrinth state of the type il-\nlustrated in Fig. 8(b) at the peak in ∝angbracketleftCmax∝angbracketrightandCLfor\ntheφtot= 0.589 and φobs= 0.1256 system in Fig. 7.\nFor higher φobs, the active ballistic disks spread further\napartandthegeltransformstoapinneddisorderedstate,\nas shown in Fig. 8(c) for the φtot= 0.589 system from\nFig. 7 at φobs= 0.283. The emergence of the intermedi-\nate pinned gel state is responsible for the additional peak\nin∝angbracketleftCmax∝angbracketrightandCL. At the highest value φtot= 0.668 in\nFig. 7, the II-III transition is lost and the ∝angbracketleftCmax∝angbracketrightand\nCLcurves become nearly featureless below the transition\nto the pinned disordered phase IV. When the total disk\ndensityishigh,motioninthedepinnedfluctuatingcluster\nphase I fcbecomes less intermittent and the steady state\nvalue ofCLincreases to CL>0.95. The high density de-\npinned fluctuating cluster state is illustrated in Fig. 8(d)\nforφtot= 0.668 and φobs= 0.00156.\nUsing the results in Fig. 7, we can construct a phase\ndiagram showing the evolution of the different phases as\na function of φobsversusφtot, as shown in Fig. 9. The\nunpinned state is a drifting liquid (phase I) for low φtot\nand a depinned fluctuating clump state (phase I fc) for\nhighφtot. Phase II is the pinned single clump state and\nphase III is the pinned multiclump state. For large φtot\nwe find a window of phase III pc, the pinned gel state,\natφobsvalues above phase III. Phase IV is the pinned\ndisordered state. The features in the CLand∝angbracketleftV∝angbracketrightcurves\nindicate that the I-II and I fc-II transitions are first order\nin nature, while the II-III and III-IV transitions are con-\ntinuous or show crossoverbehavior. In Ref.14we perform0 0.2 0.4 0.6 0.8\nφtot00.20.40.60.81\nCL\n00.20.40.60.8Cmax/Na\n0.05.0×1061.0×1071.5×107\ntime00.020.04\nV(a)\n(b)\n(c)\nFIG. 10: (a) CLvsφtotfor active ballistic disks with FD=\n0.05. Light green circles: In the obstacle-free system φobs=\n0, clustering begins near φtot= 0.28. Dark green squares:\nA system with φobs= 0.00157, showing that inclusion of a\nsmall number of obstacles can stabilize a cluster state at mu ch\nlower densities φtot≈0.1. (c)Cmax/Naand (d) instantaneous\nvelocity Vversus time for a system with φtot= 0.196 and\nφobs= 0.00785 that remains in phase I but is near the I-\nII transition. Drops in Vindicate the temporary formation\nof a pinned cluster as shown by the corresponding jumps in\nCmax/Na.\nadetailed studyofthebehaviorofpassivedisksasafunc-\ntion ofφobsversusφtot, where we find that the pinned\nphase appears at much higher values of φobsthan in the\nactive ballistic system. We note that there may be ad-\nditional phases in the active ballistic system at values of\nφtothigher than those shown in Fig. 9, particularly upon\napproaching φtot= 0.9 where the system crystallizes into\na close-packed lattice.\nIn nonballistic active particle systems that undergo\ndriven diffusion or have run-and-tumble motion with fi-\nniteτr, a phase separated state appears in the absence of\nquenched disorder as a function of disk density φtotand\nactivity. Typically there is a density φmin\ntotbelow which\nphase separation does not occur. For the ballistic active\nmatter system we consider, in the absence of obstacles a\nphase separated state appears only for φtot≥0.35. The\nintroduction of obstacles makes it possible for a cluster\nstate to nucleate at much smaller values of φtot. For\nFD= 0.05, we find clustering at densities as small as\nφtot≈0.1. To illustrate this more clearly, in Fig. 10\nwe plotCLversusφtotfor systems with φobs= 0 and\nφobs= 0.00157. In the obstacle-free system, the cluster\nsize begins to increase near φtot= 0.28, reaching a value\nofCL= 0.8 atφtot= 0.47. In contrast, adding a small\nnumber of obstacles shifts the onset of clustering down8\n00.20.4\n<V>\n00.20.4\n<V>\n0 0.2 0.4\nFD0.40.50.60.70.80.91\nCL\n0 0.2 0.4\nFD00.020.040.060.08\nCL(a)\n(b)(c)\n(d)\nFIG. 11: (a) ∝angbracketleftV∝angbracketrightand (b)CLvsFDfor active ballistic disks\nwithφtot= 0.3534 and φobs= 0.00178. At FD= 0.05 the\nsystem is in phase II, the pinned single cluster state, and at\nFD=Fc= 0.1 it depins and enters phase I fc. (c)∝angbracketleftV∝angbracketrightand\n(d)CLvsFDfor passive ballistic disks at the same values of\nφtotandφobs. There is no depinning threshold and clustering\ndoes not occur.\ntoφtot= 0.095, and CLreaches a value of CL= 0.8\natφtot= 0.12. This indicates that the obstacles are\nresponsible for nucleating stable clusters over the range\n0.12< φtot<0.28. The disorder-induced cluster state\ncan be viewed in terms of a wetting phenomenon where\ntheactiveparticlesaccumulatenotalongwalls73butnext\nto the obstacles.\nTheabilityofaclustertoformatlowobstacledensities\nalsodependson FD. AsFDdecreases,thetransitionfrom\nthe drifting liquid phase I to the pinned single cluster\nphase II occurs at lower φobs. In general, even when FD\nis too large to stabilize phase II for a given value of φobs,\na transient pinned cluster can still form on a temporary\nbasis. An example of this appears in Fig. 10(b,c) where\nwe plot Cmax/Naand instantaneous velocity Vversus\ntime for a system with φtot= 0.196 and φobs= 0.00785,\nwhich is close to the I-II transition on the phase I side.\nThere are a series of dips in Vcorrelated with jumps\ninCmax/Nawhich arise when a pinned cluster forms,\nreducing the velocity temporarily. The cluster quickly\nbreaks apart, restoring VandCmax/Nato their steady\nstate average values.\nIV. DEPINNING AND DRIVE DEPENDENCE\nWe next consider the effect of changing FDin order\nto construct velocity-force ( v−f) curves and measure\nthe depinning threshold Fc. We compare the depinning\nof the active ballistic disks to that of passive disks. In\nFig. 11(a,b) we plot ∝angbracketleftV∝angbracketrightandCLversusFDfor an active\nballistic system with φtot= 0.3534 and φobs= 0.00178,\nwhich is in the pinned single cluster phase II at FD=\n0.05. We find a drive-induced depinning transition at00.20.40.60.81\n<V>\n0 0.2 0.4 0.6 0.8 1\nFD00.20.40.60.81\nCL00.511.5\n<V>\n0 0.5 1 1.5\nFD00.10.20.30.4\nCL(a)\n(b)(c)\n(d)\nFIG. 12: (a) ∝angbracketleftV∝angbracketrightand (b) CLvsFDfor active ballistic\ndisks with φtot= 0.3534 at φobs= 0.00235 (light blue),\n0.01178 (red), 0 .03927 (green), 0 .0628 (dark blue), 0 .0785\n(violet), 0 .0942 (brown), and 0 .1256 (magenta). (c) ∝angbracketleftV∝angbracketright\nand (d) CLvsFDfor passive disks with φtot= 0.3534 at\nφobs= 0.00235 (light blue), 0 .01178 (red), 0 .03927 (green),\n0.0628 (dark blue), 0 .094274 (violet), and 0 .1267 (brown),\nshowing that a finite depinning threshold does not appear\nuntilφobs>0.094274.\nFD=Fc= 0.1fromphaseIItophaseI fc, asshownbythe\nsharp drop in CLthat coincides with the onset of a linear\nincrease in ∝angbracketleftV∝angbracketrightwith increasing FD. In Fig. 11(c,d) we\nshow∝angbracketleftV∝angbracketrightandCLversusFDforapassivedisksystemwith\nthe same values of φtotandφobs. There is no depinning\nthreshold and CL<0.005 for all values of FD, indicating\na complete lack of clustering.\nIn Fig. 12(a,b) we plot ∝angbracketleftV∝angbracketrightandCLversusFDfor the\nactiveballistic diskswith φtot= 0.3534atφobs= 0.00235\nto 0.1256. We find that the II-I fctransition, correspond-\ning to the depinning threshold, drops to lower values of\nFDasφobsdecreases, as indicated most clearly by the\nφobs= 0.00235 curve in Fig. 12(a) which has Fc= 0.035.\nThe II-I fcdepinning transition is generally quite sharp,\nand the system goes directly from the pinned state to\na fully flowing state without passing through a regime\nin which moving and pinned active particles coexist. In\ncontrast, the depinning transition separating phases III\nand IV for φobs≥0.03927 is smooth or continuous and is\nplastic in nature, so that above depinning only a portion\nof the active disks are flowing while the other portion\nremains pinned. In studies of depinning in other sys-\ntems such as superconducting vortices or colloidal parti-\ncles moving over quenched disorder, elastic depinning is\nassociated with a sharp transition in the v−fcurve and\na scaling of V∝(FD−Fc)αwithα <1.0. Plastic de-\npinning is accompanied by an extensive nonlinear regime\nin thev−fcurves with α >1.0. In our active ballistic\ndisk system, the resolution of the v−fcurves is not high\nenough to perform a scaling analysis; however, the quali-\ntative change in the v−fcurve at the Ifc-II transition is9\n0 0.05 0.1\nφobs00.511.5\nCL, Fc 0123\nFc\n0.2 0.4 0.6 0.8\nφtot00.20.40.60.81\nCL(a)\n(b)\n(c)\nabcd\nFIG. 13: (a) Fc(red squares) and the value of CLatFD=\n0.05 (green circles) vs φobsfor the active ballistic disks at\nφtot= 0.3534. The depinning threshold remains finite down\ntoφobs= 0.0008, and there is an increase in Fcat the onset of\nphase IV. (b) Fcand (c) the value of CLatFD= 0.05 vsφtot\nforφobs= 0.09427. The behavior of Fcis nonmonotonic, and\nFcincreases with increasing φtotforφtot>0.6 whenCL= 1,\nindicating the formation of a jammed state. The labels ato\ndin (c) indicate the values of φtotat which the images in\nFig. 14 were obtained.\nconsistent with an elastic or collective depinning at low\nφobscrossing over to plastic depinning for higher φobs.\nFor the III-IV depinning transition, there is generally a\nsmall peak in CLatFcproduced when the disks start to\naccumulate behind the obstacles, and in all cases there\nis an overall drop in CLat higher values of FDin the\nmoving phase. For φobs= 0.1256, depinning does not\noccur until FD=Fc= 1.35.\nIn Fig. 12(c,d) we plot ∝angbracketleftV∝angbracketrightandCLversusFDfor the\npassive disks with φtot= 0.3534 at φobs= 0.00235 to\n0.1267. There is no finite depinning threshold for φobs≤\n0.094274. We find an extended regime of nonlinear flow\nforφobs>0.0628associatedwith a coexistence offlowing\nand clogged disks. In Fig. 12(d), CL<0.01 at all FDfor\nφobs= 0.00235 and φobs= 0.01178, while for higher φobs\nthere is generally a decrease in CLwith increasing FD\nforFD>0.3. The maximum value of CLoccurs for\nφobs= 0.094274, which is just below the obstacle density\nat whicha finite depinning threshold firstappears. These\nresults show that for varied FD, the active ballistic disks\nhave a much higher susceptibility to becoming pinned\nthan the passive disks.\nIn Fig. 13(a) we plot the evolution of the depinning\nthreshold Fcalong with the value of CLatFD= 0.05\nversusφobsfor the active ballistic disks at φtot= 0.3534.\nThere is a sharp increase from Fc= 0 at φobs= 0\ntoFc= 0.007 atφobs= 0.0008, the lowest nonzero\nobstacle density we considered, for which the ratio of\nobstacles to active particles is Na/Nobs= 440. For\n0.0045< φobs<0.039, there is a more gradual lin-\near increase of Fcwith increasing φobsover the range\nof phase II depinning through half of the phase III de-\npinning. This is followed by a regime of roughly constant\nFcfor 0.039< φobs<0.065 in the second half of phase\nIII depinning, while in phase IV for φobs>0.65, there\nis a rapid increase in Fcwhich coincides with a drop inx (a)y\nx (b)y\nx (c)y\nx (d)y\nFIG. 14: The active ballistic disk positions (red circles) a nd\nobstacle locations (blue circles) for the system in Fig. 13( b,c)\nwithφobs= 0.09427 obtained at the values of φtotmarked\nby the letters atodin Fig. 13(c). (a) The pinned liquid\natφtot= 0.1963. (b) A pinned weakly clustered state at\nφobs= 0.275 (phase IV). (c) A pinned gel state at φtot= 0.51\n(phase III pc). (d) A jammed solid state at φtot= 0.8246.\nCL. Plasticdepinning appearsin phaseIV, andthe rapid\nincrease in Fcin this regime is reminiscent of the “peak\neffect” phenomenon observed for the depinning of super-\nconducting vortices. At low disorder strength, the vor-\ntices form a crystal that that can be collectively pinned\nby a small number of pinning sites; however, when the\npinning strength is increased, the crystalline structure\nbreaks apart, the vortex structure becomes amorphous,\nand there is a pronounced increase in the depinning force\nthatismuchlargerthanwhatwouldbeexpectedfromthe\nincrease in the pinning strength2,3. In the active ballis-\ntic disk system, the clusters have considerable crystalline\norder, and when the amount of disorder is increased by\nraising the number of obstacles, the large clusters break\nup into smaller clusters that can be pinned more easily,\nleading to the increase in Fc.\nIn Fig. 13(b,c) we plot Fcand the value of CLat\nFD= 0.05 versus φtotfor the active ballistic disks at\nφobs= 0.09427. For low φtot<0.35, a disordered but\nstrongly pinned phase appears, as indicated by the large\nFcand the low CL<0.1. In Fig. 14(a) we illustrate the\ndisk configuration at φtot= 0.1963 where a pinned liquid\nwith small local disk clusters forms. As φtotincreases, Fc\ndecreases and a pinned weakly clustered state emerges,\nas shown in Fig. 14(b) at φtot= 0.275. Since the num-\nber of obstacles is fixed, as φtotincreases, each obstacle\nmust restrain a larger number of mobile disks, causing10\n00.511.5\nFc\n0123\nFc\n0 0.05 0.1\nφobs00.20.40.60.81\nCL\n0.2 0.4 0.6 0.8\nφtot00.20.40.60.81\nCL(a)\n(b)(c)\n(d)\nFIG. 15: (a) Fcand (b) the value of CLatFD= 0.05 vs\nφobsatφtot= 0.3534 for the active ballistic disks (circles)\nand passive disks (squares). (c) Fcand (d) the value of CLat\nFD= 0.05 vsφtotatφobs= 0.094 for the active ballistic disks\n(circles) and passive disks (squares). At high φtot, both the\nactive and passive disks undergo a transition into a jammed\nstate.\nx (a)y\nx (b)y\nFIG. 16: The passive disk positions (green circles), obstac le\nlocations (blue circles), and disk trajectories for the sys tem in\nFig. 15(c,d)(a) At φtot= 0.196, 1D flowchannels coexist with\npinned disks. (b) At φtot= 0.4712, collective interactions\nbetween disks cause a larger fraction of the disks to flow.\nFcto decrease with increasing φtot. In Fig. 14(c) we plot\nthe disk configuration at φtot= 0.51, where the system\nforms a pinned gel phase with low Fc. Figure 13(c,d)\nshows a local minimum in Fcnearφtot= 0.628, where\nCL= 0.92. Forφtot>0.628,Fcbegins increasing with\nincreasing φtotandCLapproaches CL= 1 as the disks\nassemble into a single jammed solid packing, as shown in\nFig. 14(d) at φtot= 0.8246.\nIn Fig. 15(a,b) we plot Fcand the value of CLat\nFD= 0.05versus φobsatφtot= 0.3534for active ballistic\ndisksandpassivedisks. Thedepinning thresholdremains\nfinite in the active system for φobs≥0.0008, whereas in\nthe passive system the depinning threshold drops to zero\nwhenφobs≤0.094. This indicates that 100 times fewer\nobstacles are needed to pin the active system compared\nto the passive system. In addition, the depinning thresh-00.10.20.30.40.5\nM\n00.10.20.30.4\nM\n10-410-2100102\nlr00.20.40.60.8\nCL\n10-410-2100102\nlr00.20.40.60.81\nCL(a)\n(b)(c)\n(d)a\nbcd\nFIG.17: (a)Thediskmobility Mand(b)CLvsrunlength lr\nfor finite run time active disks at FD= 0.05,φtot= 0.51 and\nφobs= 0.14137. The low lrbehavior is similar to that found\nin the passive disk limit while the high lrbehavior is similar\nto that found in the active ballistic limit. Between these tw o\nlimits the disks form a uniform density, highly mobile liqui d\nstate. The labels atodin (b) indicate the values of lrat\nwhich the images in Fig. 18 were obtained. (c) Mand (d)\nCLvslrfor finite run time active disks at φtot= 0.667 and\nφobs= 0.14137.\nold for the active system is always higher than that of\nthe passive system. In Fig. 15(c,d) we show Fcand the\nvalue ofCLatFD= 0.05 versus φtotatφobs= 0.094 for\nthe active ballistic and passive disks. Here the depinning\nthreshold for the passive disks does not become finite un-\ntilφtot>0.7, the density above which CLincreases to\nCL= 1.0. The high density active ballistic and passive\ndisk states are similar in nature and are dominated by\nthe formation of a jammed solid state. Even when the\ndepinning threshold in the passive disk system is zero,\nthev−fcurves can show strongly nonlinear behavior,\nand a large fraction of the passive disks remain pinned or\nimmobile for φtot<0.35, the same total disk density at\nwhich the active ballistic disks exhibit a large increase in\nFc. In Fig. 16(a) we show the passive disk locations and\ntrajectories at φtot= 0.196 and FD= 0.05. A portion\nof the disks move in 1D filamentary channels while the\nremaining disks are pinned. These filamentary channels\npersist down to FD= 0.0 in the passive disks, but are\ngenerally absent for active ballistic disks. In the passive\ndisk system, as φtotincreases, cooperative interactions\nbetween the mobile disks reduce the overall trapping and\nlead to a higher fraction of flowing disks, as illustrated\nin Fig. 16(b) at FD= 0.05 andφtot= 0.4712. As φtot\nfurther increases, the system approaches a jammed solid\nwith a finite depinning threshold.11\nx (a)y\nx (b)y\nx (c)y\nx (d)y\nFIG. 18: The finite run time active disk locations (orange\ncircles) and obstacle locations (blue circles) for a system with\nφtot= 0.51 andφobs= 0.14137 obtained at the values of\nlrmarked by the letters atodin Fig. 17(b). (a) The low\nmobility clogged state at lr= 0.001. (b) The high mobility\nuniform liquid at lr= 2.0. (c) The actively clogged state at\nlr= 10. (d) The actively clogged state at lr= 200.\nV. FINITE RUN TIME ACTIVE DISKS\nWe next show how the passive disk pinning and ac-\ntive ballistic disk pinning limits can be connected to each\nother by considering run-and-tumble particles where we\ngradually increase the running time from close to zero,\nwhich is the passive limit, to large values which approach\nthe ballistic limit. In Fig. 17(a,b) we plot the mobility\nM=∝angbracketleftV∝angbracketright/∝angbracketleftV0∝angbracketrightper disk and CLversus run length lr\nfor a finite run time active disk system at φtot= 0.51,\nφobs= 0.14137, and FD= 0.05. Here ∝angbracketleftV0∝angbracketrightis the average\ndriftvelocityofanindividualdiskintheabsenceofobsta-\ncles or other disks, so for FD= 0.05,∝angbracketleftV0∝angbracketright= 0.05, and in\nthe obstacle-free limit, M= 1.0. The disk dynamics are\nthe same as those described in Eq. 1 except the running\ntimeτrisnowfinite. Forconvenience,wecharacterizethe\nactivity level in the system at a fixed |Fm|in terms of the\nrun length lr=|Fm|τr, so that large lrcorresponds to\nlargeτr. In the passive limit, lr= 0, while in the ballistic\nlimit,lrisinfinite. Forthe parametersweconsiderin this\nsection, the system reaches a completely pinned state in\nboth the passive and ballistic limits. In Fig. 17(a) at\nlowlr,∝angbracketleftM∝angbracketrightis small, indicating that a clogged state has\nformed that is similar to the passive disk clogged state\nwhich appears at lr= 0. At the same time, CL>0.8 in\nFig. 17(b), indicating the formation of a large cluster. In00.10.20.30.40.5\nM\n00.050.10.150.2\nM\n10-410-2100102\nlr00.10.20.3\nCL\n10-410-2100102\nlr00.10.2\nCL(a)\n(b)(c)\n(d)\nFIG. 19: (a) The disk mobility Mand (b)CLvs run length\nlratFD= 0.05,φtot= 0.3543 and φobs= 0.14137 for finite\nrun time active disks. (c) Mand (d)CLvslratφtot= 0.2356\nandφobs= 0.14137 for finite run time active disks, where the\nsystem is always in the disordered regime.\nFig. 18(a) we plot the disk configurations for lr= 0.001,\nwhere the clogged state has highly heterogeneous local\ndisk density. For 0 .01< lr<5, an easily flowing liquid\nappears, as indicated by the increase in Mand the drop\ninCLtoCL<0.01. We illustrate the flowing liquid at\nlr= 2.0 in Fig. 18(b), where the disk density is uniform\nand clustering behavior is lost. As lrincreases, Mde-\ncreases once self-clustering of the disks begins to occur,\nas shown in Fig. 18(c) for lr= 10. At large lr,CLap-\nproaches CL= 0.9 andMdrops to a low value. In this\nregime, the image of the lr= 200 system in Fig. 18(d) in-\ndicates that clustering similar to that found in the active\nballistic pinned state is present. We note that M >0\nfor any finite lrsince there is always a chance that the\nactivitycanunpin afractionofthe disks evenwhen lrbe-\ncomes large. The motion in the finite but large lrregime\nbecomes highly intermittent and exhibits avalanche-like\nfluctuations, as has been described in detail elsewhere64.\nIn Fig. 17(c,d) we plot MandCLversuslrfor the run-\nand-tumble disks at a higher φtot= 0.667. We find the\nsame trend in which clustering occurs at both small and\nlargelr, with a correspondingly low value of M, while\nfor intermediate lr, the clustering disappears, CLis low,\nandMis high. The maximum value of Mis lower at\nφtot= 0.667 than at φtot= 0.51 due to the crowding\neffect that appears at higher disk densities.\nIn Fig. 19(a,b) we plot MandCLfor finite run time\nactive disks at φtot= 0.3534 and φobs= 0.1413. We\nfind the same trend as in Fig. 17 where high values of\nCLare associated with low values of M; however, at this\nlower total disk density φtot, the maximum value of CL\nis reduced. A peak in CLnearlr= 0.005 indicates the\nappearance of additional clustering just before the activ-\nity becomes large enough to liquefy the system and in-12\ncreaseM. This suggeststhat the activity becomes strong\nenough to cause untrapping of individual disks at lower\nlrbut that the untrapped disks then pool into clusters\nthat are too large to be broken apart by individual disks\nuntillrincreases, at which point the clumps break apart\nand the system flows in a liquid state. In Fig. 19(c,d)\nwe plotMandCLversuslrfor finite run length disks at\nφtot= 0.2356 and φobs= 0.1413. At this low total disk\ndensity, little clustering occurs and CLis always small.\nWe still observe a peak in Mat intermediate lrvalues;\nhowever, Misrelativelylowoverall,reachingamaximum\nvalue of only M= 0.1. This result emphasizes the role of\ncollective disk-disk interactions in liquefying the system\nand increasing the mobility for intermediate lr.\nAlthough we consider only monodisperse disks in this\nwork, the introduction of obstacles causes the clogged\nstates to exhibit a considerable amount of structural dis-\norder, similar to that found in jammed states for bidis-\nperse or completely amorphous systems67,69,70. For pas-\nsive disks at low obstacle densities, previous studies have\nshown that jamming occurs near the obstacle-free jam-\nming density of φjor point J, but that the jamming\ntransition shifts to lower disk densities as the obstacle\ndensity increases69,71. When the obstacle density is high\nenough, the behavior of the system changes and instead\nof a uniform jammed state, the passive disks assemble\ninto a clogged state with strongly heterogeneous local\ndisk density14. For active disks we find three limiting\nregimes of behavior. At high disk densities, the effect of\nthe activity becomes negligible and the behavior is sim-\nilar to that found in the passive high disk density limit,\nwhich is controlled by jamming near point J in amor-\nphous systems or crystallization at a density φtot= 0.9\nfor monodisperse disks. The second limiting regime ap-\npears for low activity and intermediate disk densities,\nwhere the active disks form a clogged configuration sim-\nilar to that found for the clogging of passive disks. The\nthird limiting regime, consisting of the actively pinned\nor clogged states that appear for active ballistic disks or\nfor disks with finite but large lr, is unique to the active\ndisks and does not appear in passive disk systems. This\nregime extends over a wide range of obstacle densities\nand can assume the form of phase II, III, or IV, as de-\nscribed in this work. Our results suggest that in addition\nto the density, temperature, and load axes on the jam-\nming phase diagram67,68, there could be two additional\naxes, the activity level and the obstacle density or disor-\nder level, that produce jammed states.\nIn future work for both the passive and active disk sys-\ntems, it would be interesting to investigate how different\nthe pinned phases are. For example, at high disk den-\nsities near the jamming transition, it is likely that the\nsystem is highly stable to perturbations since there are\nfeweravailabledegreesoffreedom. Similarly, inaclogged\nstate with high obstacle densities, a perturbed system is\nlikely to fall back into the same ora similar cloggedstate.\nOn the other hand, in phases II or III, a small perturba-\ntion could readily break up one or more of the clusters,permitting the system to flow again, so that although\nthe active systems are more susceptible to clogging, the\nclogged state they reach may be more fragile than that\nformed by passive disks.\nVI. SUMMARY\nWe have examined the pinning and clogging behav-\niors of active run-and-tumble disks in the ballistic limit\ndriven through an array of obstacles. As a function of in-\ncreasing obstacle density, we find four generic phases: an\nunpinned fluctuating cluster phase, a pinned single clus-\nter phase in which a small number of obstacles can pin a\nlarge number of active disks, a pinned multiclump phase,\nand a pinned disordered phase. We find that in contrast\nto passive disks, the active ballistic disks can reach a\npinned state at relatively low obstacle densities. Within\nthe pinned disordered phase, as the density of active bal-\nlistic disks increases, a pinned gel state or labyrinth pat-\ntern appears. By constructing velocity-force curves we\nfind that the active ballistic disks exhibit a finite depin-\nningthreshold. Asthe obstacledensity increases, thereis\na transition from collective depinning of the pinned clus-\nters to plastic depinning of the disordered pinned states\nthat is associated with a large increase in the depinning\nthreshold. As a function of total disk density, the de-\npinning threshold for the active ballistic disks is non-\nmonotonic, dropping at intermediate disk densities when\ncollective disk-disk interactionsreduce the threshold, but\nrising at high disk densities as the system approaches a\njammed orcrystallinestate. In contrast, for passivedisks\nthe depinning threshold is only finite at high disk den-\nsities, while at lower disk densities filamentary channels\nof disk flow form that are absent in the active disk sys-\ntem. For both the active and passive disks, the pinned\norcloggedstates are phase separated; however, the phase\nseparation appears at a much lower obstacle density for\nthe active disks. Finite run time active disks provide a\nconnection between the active ballistic and passive disk\nsystems, andexhibit alowmobility phaseseparatedstate\nfor short run times in the passive limit as well as for long\nrun times in the active ballistic limit. At intermediate\nrun times, the active disks form an easily flowing uni-\nform liquid with reduced clustering, and there is an opti-\nmal level of activity that maximizes the flux through the\nsystem. We describe how our results can be related to\nsystemsthatexhibitjamming. 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Soto, Wetting transitions displayed\nbypersistent active particles, Phys.Rev. Lett. 119, 078001\n(2017)."    },    {        "title": "1804.00252v1.The_impossibility_of_expanding_the_square_root_of_the_electron_density_as_a_linear_combination_of_elements_of_a_complete_set_of_basis_functions.pdf",        "content": "arXiv:1804.00252v1  [physics.chem-ph]  1 Apr 2018The impossibility of expanding the square root of the electr on\ndensity as a linear combination of elements of a complete set of\nbasis functions\nOmololu Akin-Ojo\nICTP East African Institute for Fundamental Research,\nUniversity of Rwanda, Kigali, Rwanda and\nDepartment of Physics, University of Ibadan, Ibadan, Nigeria\n(Dated: July 3, 2021)\nIn orbital-free density functional theory (OFDFT), an equa tion exists for ψ=√n,\nthe square root of the ground state electron density n. We show that ψcannot be\nexpanded as a linear combination of elements of a complete se t of basis functions\nexcept in the case of one or two electron systems. This is unli ke the case for the\nground state of a system of identical bosons in which the squa re root of the ground\nstate bosonic density can have an expansion as a linear combi nation of elements of\na complete set of basis functions.2\nI. INTRODUCTION\nThe original density functional theory (DFT), which is orbital-free DFT (OFDFT), ex-\npresses all functionals in terms of the electron ground state dens ity. This is in contrast to\nthe popular version of DFT, Kohn-Sham DFT (KSDFT), in which the kin etic energy den-\nsity functional is written in terms of one-electron orbitals. Resear ch is ongoing to discover\nan accurate kinetic energy density functional (KEDF) given only in t erms of the ground\nstate density nof the system. In this work, it is shown that another issue that sho uld be\nconsidered seriously in OFDFT research is the proper mathematical representation of the\nsquare root of the density.\nOFDFT writes the total energy of the system Eas a functional of the density n:\nE[n] =T[n]+W[n] (1)\nwhereTis the kinetic energy density functional and Wis the total potential energy of the\nelectrons. Optimization of Eq. 1 with respect to the density n, subject to the condition that\nnintegrates to the total number of electrons N, gives the Euler equation:\nδT[n]\nδn+veff[n] =µ, (2)\nwhereveff=δW/δnandµis a Lagrange multiplier, which happens to be the chemical\npotential of the system. Although Eq. 2 can be solved to obtain n, the solutions, however,\nare not guaranteed to be non-negative everywhere. In order to have non-negative solutions,\nthe optimization is often done with respect to√nand Eq. 2 is written in the form:\n/bracketleftBiggδT[n]\nδn+veff[n]/bracketrightBigg√n=µ√n, (3)\nfrom which, after solving for√n, the density n= (√n)2is easily obtained.\nIn another vein, one can simply go the route of Ref. 1 who showed th at the exact ground\nstate density satisfies the equation:\n/bracketleftBigg\n−¯h2\n2m∇2+us(r)/bracketrightBigg√n=µ√n (4)\nin whichmis the electron mass. Expressions for the effective potential usare given in the\nsame paper. Perhaps, because Eqs. 2 and 4 appear similar to single- particle Schr¨ odinger3\nequations, they are often solved by expanding ψ=√nas a linear combination of elements\nof a complete basis set of functions, {gk}, i.e.,\nψ=√n=/summationdisplay\nαdαgα(r) (5)\nwhere, the coefficients dkare simply scalars.\nBecause of the difficulty in obtaining an accurate KEDF or, equivalent ly, an accurate\neffective potential us, Kohn and Sham2replaced the kinetic energy of the many-electron\nsystem with that of a non-interacting system and wrote the latter in terms of single-particle\norbitalsφi. This led to the Kohn-Sham equations:\n/bracketleftBigg\n−¯h2\n2m∇2+veff(r)/bracketrightBigg\nφi=ǫiφi. (6)\nIn terms of the orbitals, the ground state electron density is given as:n=/summationtextocc\ni|φi|2, where\nthe sum is over occupied electron states. Since veff(r) is real, the orbitals can also be chosen\nto be real. Now, each orbital φican be written as a linear combination of elements of a\ncomplete basis set of functions, {gk}, i.e.,\nφi=/summationdisplay\nαc(i)\nαgα(r) (7)\nConsequently, the electron density nbecomes:\nn=occ/summationdisplay\ni|φi|2=occ/summationdisplay\ni/summationdisplay\nα/summationdisplay\nβc(i)\nαc(i)\nβgα(r)gβ(r) (8)\n=/summationdisplay\nα/summationdisplay\nβ/parenleftBiggocc/summationdisplay\nic(i)\nαc(i)\nβ/parenrightBigg\ngα(r)gβ(r) (9)\nAt the same time, we obtain from Eq. 5:\nn=ψ2=/summationdisplay\nα/summationdisplay\nβdαdβgα(r)gβ(r) (10)\nComparing Equations 9 and 10, one concludes that:\ndαdβ=occ/summationdisplay\nic(i)\nαc(i)\nβ (11)\nfrom which one can get:\nd2\nα= (dαdα) =occ/summationdisplay\nic(i)\nαc(i)\nα=occ/summationdisplay\ni(c(i)\nα)2(12)4\nand:\n(dαdβ)2=/parenleftBiggocc/summationdisplay\nic(i)\nαc(i)\nβ/parenrightBigg2\n(13)\nApplication of the Cauchy-Schwartz inequality gives:\n(dαdβ)2=/parenleftBiggocc/summationdisplay\nic(i)\nαc(i)\nβ/parenrightBigg2\n≤occ/summationdisplay\ni(c(i)\nα)2occ/summationdisplay\ni(c(i)\nβ)2=d2\nαd2\nβ (14)\nThe equality holds if and only if, for given αandβ,c(i)\nα= Constant ×c(i)\nβfor each orbital\nφi. This only occurs for a bosonic system (in which all the particles are in the same state).\nIt also occurs if there is only one term in the sum, i.e., for a single-elect ron system (or two-\nelectron system in which the two electrons are in the same spatial or bital state but different\nspin states – “spin up and spin down” / singlet spin state).\nExpression 14 is the main result of this work which establishes that ( dαdβ)2<d2\nαd2\nβfor all\nsystems with identical particles, except those enumerated above . This contradiction implies\nthat the square root of the electron density cannot be written as a linear combination of\nelements of a complete set of basis functions, except for one- or t wo-electron systems. This\nmay explain why attempts to solve Eq. 3 or 4 by linear expansions of ψ=√noften leads to\ndifficulties in convergence of the iterative self consistent field appro ach3,4typically used for\nDFT calculations. Attempts to minimize the energy functional Eq. 1 d irectly with respect\ntoψ, the square root of the density, are also bound to be unsuccessf ul ifψis expanded as a\nlinear combination of elements of a complete set of basis functions wh ose coefficients are the\nvariables of the optimization procedure. However, the use of othe r non-expansion methods,\nsuch as the finite-difference method can succeed. This work does n ot imply that no solution\nof Eq. 4 can be found but that such solutions will not correspond to the ground state density\n(of non-interacting electrons) in the general case if ψis expanded as a linear combination of\nelements of a complete set of basis functions.\nII. CONCLUSIONS\nThis work shows that the square root of the electron density cann ot be expanded as a\nlinear combination of elements of a complete set of basis functions ex cept in the case of one\nor two electron systems.5\n1M. Levy, J. P. Perdew, and V. Sahni, Phys. Rev. A 30, 2745 (1984).\n2W. Kohn and L. J. Sham, Phys. Rev. 140, 1133 (1965).\n3G. K.-L. Chan, A. J. Cohen, and N. C. Handy, J. Chem. Phys. 114, 1063 (2001).\n4V. Karasiev and S. Trickey, Computer Physics Communication s183, 2519 (2012)."    },    {        "title": "1804.08670v2.Head_on_collision_of_multi_state_ultralight_BEC_dark_matter_configurations.pdf",        "content": "arXiv:1804.08670v2  [gr-qc]  10 Sep 2018Head-on collision of multi-state ultralight BEC dark matte r configurations\nF. S. Guzm´ an1and Ana A. Avilez2\n1Laboratorio de Inteligencia Artificial y Superc´ omputo, In stituto de F´ ısica y Matem´ aticas,\nUniversidad Michoacana de San Nicol´ as de Hidalgo. Edificio C-3,\nCd. Universitaria, 58040 Morelia, Michoac´ an, M´ exico.\n2Departamento de F´ ısica, Centro de Investigaci´ on y de Estu dios Avanzados del IPN,\nA. P. 14–740, 07000, Ciudad de M´ exico, M´ exico.\n(Dated: July 13, 2021)\nDensity profiles of ultralight Bose-Condensate dark matter inferred from numerical simulations of\nstructure formation, ruled by the Gross-Pitaevskii-Poiss on (GPP) system of equations, have a core-\ntail structure. Multi-state equilibrium configurations of the GPP system on the other hand, have a\nsimilar core-tail density profile. We now submit these multi -state configurations to highly dynamical\nscenarios and show their potential as providers of appropri ate density profiles of structures. What\nwe do is to present the simulation of head-on collisions betw een two equilibrium configurations of\nthe GPP system of equations, including the collision of grou nd state with multi-state configurations.\nWe study the regimes of solitonic and merger behavior, and sh ow generic properties of the dynamics\nof the system, including the relaxation process and attract or density profiles. We show the merger\nof multi-state configurations have the potential to produce core-tail density profiles, with the core\ndominated by the ground state and a halo dominated by an addit ional states.\nPACS numbers: keywords: dark matter – Bose condensates\nI. INTRODUCTION\nScalar fields have been essential in modern physics, at\nfundamental level the Higgs scalar that is responsible for\nthe mass of elementary particles according to the stan-\ndard model of particles, is the most remarkable example.\nHowever, scalar fields have also been quite popular in\nmodeling new forms of matter and energy whose exis-\ntence manifests at astronomical and cosmological scales.\nParticularly, they have been used as effective field the-\nories of dark matter and dark energy. Concerning dark\nmatter in particular, a common application has been the\nmodeling of astrophysical sized objects as classical so-\nlutions of a canonical scalar field holding a wave-length\ncomparable to stellar scales or larger [1, 8, 26, 39]. In\nparticular at galactic sizes with the aim of proposing a\nmodel of dark matter understood as condensates [25, 48]\nand studied as classical scalar fields as well, with an in-\nterpretation of an effective theory [30]. From the cos-\nmological scenario it was found that the boson mass had\nto be ultralight in order to reproduce the observed dis-\ntribution of matter at very large scales[29, 40]. Similar\nproposalsrelyingon thepreviousnotionofhaloesashuge\ncondensatesor classicalscalarfield configurationswasre-\nconsidered several times and different models arose un-\nder different denominations [6, 21, 23]. In recent years,\nthe support of these models from a cosmological point of\nview has been enhanced and the interpretation as a Bose\nCondensate has become more suitable [5, 9, 11, 24, 28].\nThe dynamics of the system of equations suitable for\nscalar fields and Bose Condensates were first analyzed\nin the nineties, using the tools of numerical relativity\napplied to the solution of the evolution equations, both\nfor complex scalar fields with stationary solutions and\nreal scalar fields with non-stationary solutions [45–47].These pioneering works showed the stability properties\nand shed light on the existence and stability conditions\nof so called solitonic solutions in full General Relativ-\nity and in the linearized weak-field regime, known as the\nSchroedinger-Poisson (SP) system. A relaxation mecha-\nnism calledgravitationalcoolingwasproposed, that later\nwas applied to galactic sized structures in order to pro-\nvide a structure formation scenarioafter the turn-around\npoint [10, 17, 21]. Ever since, with the motivation of\ndark matter and a relaxation process, more modern ap-\nproaches suggested that the equations of the SP system\nare associated to a macroscopic Bose gas making up a\nBose-Einstein condensate (BEC), both fully dynamical\n[21] or in limits like the Thomas-Fermi limit [6].\nUnder this approach, after condensation, most of the\nindividual bosons lay in the ground state and the macro-\nscopic wave function scales as the occupation number of\nthe BEC. In the mean-field approximation, the macro-\nscopic wave function of such a system is described by\nthe non-linear Gross-Pitaevskii equation [16, 34, 35]. In\nthis way, the interpretation of the formerly SP system,\nbecomes the Gross-Pitaevskii equation for a Bose Con-\ndensate with a potential trap that happens to be a grav-\nitational potential sourced by the gas density itself. An\nimportant feature ofthis approachto model darkmatter,\nis that the boson requires as an ultralight mass of order\nm∼10−22eV, which suffices to suppress the formation\nof small structures, implying that a cut-off in the halo\nmass function and the matter power spectrum. Thus in\nthis type of models the missing satellites is not a problem\n[22–24, 28, 30, 42, 49]. In addition, simulations of merg-\ners of pure substructures as BECS at galactic and cluster\nscales, show that the equilibrium density profiles of this\nsort of dark matter have a core-like shape which is highly\nconvenient in order to naturally ease the core-cusp con-\ntroversy [4, 12, 14, 37, 42]. The core is compared to the2\nsolitonic solution resulting from the stationary solution\nof the GPP system [20, 21], which has a convenient set\nof properties, chief among them that these are attractor\nsolutionsforratherarbitraryinitialconfigurations[2,21].\nIn a cosmological context, the linear cosmological per-\nturbations theory along with the background cosmology\narisen from this model –plus a cosmological constant–\nreproduces the observations at very large scales such\nas the linear matter power spectrum of the distribu-\ntion of matter in the Universe and the CMB among\nothers [7, 27, 38, 50, 52]. Beyond the linear cosmo-\nlogical regime, N-body simulations in three dimensions,\nshowthat predictionsofthe matter powerspectrum from\nCDM and SFDM at cluster and supercluster scales at\nthe non-linear cosmological threshold are indistinguish-\nable. However, at galactic and intergalactic scales the\nstory changes, because the density profiles of dark mat-\nter in the two scenarios differs importantly [31, 41–43].\nIn this direction, recent simulations of mergers of soli-\ntonic solutions of the Schrodinger-Poisson system with-\nout self-interactions, indicate the final density profile\nhaveacentralsolitoniccoredensityprofilesurroundedby\na Navarro-Frenk-White(NFW)-like density tail [43, 44].\nAlthough these simulations have improved substantially\nour understanding of the structure formation in BEC\ndark matter, a long track needs to be walked in order\nto realistically simulate the real structure formation. So\nfar, in all the mentioned works, it is implicitly assumed\nthat the distribution of the BEC is described only by a\nsingle wave function and it has been shown thoroughly\nthat configurations in this state, when sufficiently iso-\nlated, they should approach a ground state configuration\nas long as the only relaxation mechanism is the gravita-\ntional cooling [21].\nExploring further out single wave function systems, it\nwas found that unlike ground state configurations, multi-\nstate configurations show the property of having a core\nand atail naturally[4, 51]. We now explorethe dynamics\nof this type of configurations in non-stationary scenarios.\nWhat we do in this paper is to simulate the head-on col-\nlision of two multi-state configurations and explore first,\nhowthestatesdeformduetotheinteractionwithanother\nstructure and second, the possibilities that a multi-state\nscenario has at explaining the profile of structures result-\ning from binary mergers. We explore for the first time\nthe head-on encounter of configurations initially laying\nnot only in ground state but in different combinations of\nmulti-states. Specifically, we simulate two types of col-\nlisions: the solitonic regime (or high velocity case) and\nthe merger regime (or low velocity case). In the first\nregime, the magnitude of the head-on momenta is large\nenough to compensate the gravitational binding energy.\nOn the contrary, systems within the merger regime are\nbounded. In the two regimes we track the evolution pro-\ncess and simulate various representative scenarios, and\nfor each one we present particular information revealing\nthe main features of the dynamics.\nFinally, westudy tworepresentativemergingcases, thecollision of a ground state with another ground state and\nthe collision of a two-states configurations with another\ntwo-states configurations. We study the relaxation pro-\ncess and the density profile of the final configurations in\norder to sketch the potential properties in a more general\nscenario of structure formation in the sense that we fit\nthe density profile of the final configuration.\nThe paper is organized as follows. In Section II we\ndescribe the GPP system of equations and the methods\nused to solvethem. In Section III we present the tests for\nthe code used and in Section IV we present the results\nof our analysis. Finally in Section V we present some\nconclusions and describe the possibilities of the multi-\nstate scenario in the frame of structure profiles.\nII. METHODS\nA. Solution of the GPP system\nThe GPP system of equations represents the descrip-\ntion of a Bose Condensate in the regime at which the GP\napproach holds. A particular sign of the system is that\nthe condensate trap potential of the GP equation corre-\nspondstoagravitationalpotentialsourcedbythedensity\nof the wave function of the condensate, interpreted as a\nmatter density [32]. The regular GP equation includes\nthe self-interaction among bosons that has effects on the\ndistribution of matter [21], nevertheless in most of cur-\nrentapplicationsto the ultralightdarkmattermodel, the\ncondensates are described by the GPP in the free field\ncase with zero self-interaction [3, 15, 41, 43, 44], and this\nis an assumption we use in this paper as well.\nThe generalization of the GPP system to multi-states\nis as follows. Considering a system described by nstates,\nandusingscaledquantitiessuchthatweremovethephys-\nical constants (see e.g. [18, 19]), the evolution of the sys-\ntem is described by the following GPP system of equa-\ntions\ni∂tΨ1=Hψ1\n...\ni∂tΨn=HΨn\n∇2V=|Ψ1|2+...+|Ψn|2, (1)\nwhere Ψ 1(t,x),...,Ψn(t,x) represent the wave functions\ncorrespondingtothe groundstate ofnode-lesswavefunc-\ntions until the most excited state with n−1 nodes and\nthe Hamiltonian operator is H=−1\n2∇2+V. Notice that\nthe different states are evolvedwith separateSchr¨ odinger\nequations, which are coupled through the gravitational\npotentialV,inturnsourcedbytheadditionofthedensity\nof each of the states. The first nequations are evolution\nequations for each of the wave functions, whereas Pois-\nsonequationactsasaconstraintofthe evolution, sincein\ngeneral the gravitational potential Vwill change in time\nas the density of each state i, specifically ρi=|Ψi|2is3\nexpected to change in general. This system of equations\ndescribes the evolution of the configurations in our anal-\nysis. The evolution of a general system then consists in\nthe solution of the system of equations (1) for consistent\ninitial conditions. This approach defines an initial value\nproblem for nwave functions Ψ 1,...,Ψnon the domain\nx∈R3andt≥0, with the consistency condition that V\nis the solution of Poisson equation in (1) at initial and\nfurther time. The initial conditions are free for the wave\nfunctions and Vis restricted by the densities ρiof the\nvarious wave functions.\nWe solve this problem numerically in 3D with ax-\nial symmetry following the recipe presented in [3, 15]\nand described as follows. Since the system of a head-\non encounter is axysymmetric we define the problem on\na two dimensional numerical domain in cylindrical co-\nordinatesr∈[0,rmax]×z∈[zmin,zmax]. We dis-\ncretize this domain with uniform resolution ri=i∆r,\nzj=zmin+j∆z,i,j∈N, where we choose the spa-\ntial resolutions ∆ r= ∆z. We set initial wave functions\nΨ1(0,x),...,Ψn(0,x)ateachgridpointofthediscretedo-\nmain. ThenwesolvePoissonequationfor Vusingbound-\nary conditions up to the quadrupole moment with a Suc-\ncessive Over Relaxation (SOR) method with optimal ac-\nceleration parameter. This sets the initial conditions to\nbe evolved. The boundary conditions provided for the\nsolution of Poisson equation is multipolar with leading\nmonopolar term given by V(rboundary) =−N/rboundary,\nwhereN=/summationtextNiis the total number of particles of all\nthe states as defined below in (3) and rboundaryis the dis-\ntance of a point belonging to the boundary to the center\nof the domain.\nThe evolution over a discrete time step ∆ tof the wave\nfunctions is carried out using the method of lines with a\nthird order Runge-Kutta integrator. At each intermedi-\nate step of the integration of the wave functions we solve\nPoisson equation, which is mandatory since Vsources\nSchr¨ odinger’sequations. We doit usingthe samemethod\nas for the initial time. Then this procedure is repeated\nover a given number of time steps with constant ∆ tuntil\nrequired.\nOnthe otherhand, particularlyinvolvedarewellposed\nboundary conditions for the Schr¨ odinger type of equa-\ntions. In practice what is best and has shown to work,\nis the implementation of a sponge. A sponge is im-\nplemented by adding an imaginary potential near the\nboundary of the numerical domain V→V+Vim. The\neffect of this imaginary potential is that produces a sink\nof particles that in the continuity equation reflects as\n∂ρ/∂t+1\n2∇ ·[Ψ∇Ψ∗−Ψ∗∇Ψ] = 2Vim|Ψ|2. We thus\nimplement this sponge outside a sphere of radius rsponge\nin a region outside of the configurations that are to be\ncollided. In order to absorb as many modes as possi-\nble, we choose Vimto have a smooth profile of the form.\nVim=−V0\n2[2 + tanh(r−rsponge)/δ−tanh(rsponge/δ)],\nwhereδis the width of a transition region between the\nphysical zone where Vim= 0 and the zone of the sponge\n(see e.g. [19]).-0.2 0 0.2 0.4 0.6 0.8 1\n 0  2  4  6  8  10Wave Functions\nrψ1ψ2\nFIG. 1. Wave functions of the equilibrium configuration with\ntwo states. Notice that ψ1(r) has no nodes whereas ψ1(r) has\none node. Should the Schr¨ odinger equation be microscopic\ninstead of macroscopic, the second state would be called the\nfirst excited state.\nB. Multi-state stationary configurations\nSince we will collide stationary configurations we pro-\nvide a description on how they are constructed. For this\nwe follow the recipe in [51]. First it is assumed that the\nwave functions in system (1) to have spherical symmetry\nand harmonic time dependence Ψ n(t,x) =ψn(R)eiωnt,\nwhereRis the spherical coordinate. This particular time\ndependenceimpliesthedensities ρnaretime-independent\nand the spherical symmetry makes Poisson equation an\nordinaryequationalong R, whichmakeseasytointegrate\nthe gravitational potential V(R).\nBy imposing boundary conditions of regularity at the\norigin for the wave functions ψn(R) and the gravita-\ntional potential V(R), and isolation conditions, namely\ndemanding ψn(R) and its derivative to be zero, and a\nmonopolar boundary condition for V(R) at a numerical\nboundary far from the origin R=Rboundary, one sets a\nStrum-Liouville problem for the wave functions and the\npotential at the same time, and whose eigenvalues are\nthe frequencies ωn. This problem is solved using a multi-\nshooting routine that pushes Rboundaryas far as possible,\nso that the boundary conditions are fulfilled [51].\nThe results are the equilibrium wave functions ψn(R)\nand the gravitational potential V(R). In this paper we\nonly consider at most three states and as an example we\nshow in Fig. 1 the solution for a two state configuration.\nNotice that these configurations are constructed using\nspherical coordinates with resolution higher than that of\nthe axisymmetric numerical domain where we perform\nthe collision. What we do with the equilibrium configu-\nration is that we interpolate the wave functions into the\naxisymmetric domain as described below, before initiat-\ning the evolutions.4\nC. Setting initial data for two configurations that\nwill collide\nEach of the configurations that will collide is an equi-\nlibrium configuration with a given number of states and\na given set of wave functions. The head-on collision will\nhappen along the z−axis starting at the same distance\nfrom the numerical origin at ( r= 0,z= 0). The evo-\nlution assumes the superposition of the wave functions\nof each of the two configurations that we use Ψ Lfor the\nconfiguration located initially at the left ( z <0) and Ψ R\nfor the configuration at the right ( z >0). If Ψ Lhasn\nstates, welabel itswavefunctions ψL1,...,ψLn, ifthe con-\nfiguration Ψ Rhasmstates, we label its wave functions\nψR1,...,ψRm. Assuming 1 < m < n , the wave function\nrepresenting the superposition of the ground state con-\nfigurations will be Ψ 1=ψL1+ψR1, the wave function\nfor the first excited state will be Ψ 2=ψL2+ψR2and so\non and so forth until Ψ m=ψLm+ψRm,...,Ψn=ψLn.\nThese superposed wave functions are those to be evolved\nusing the GPP system (1). In this paper the maximum\nvalueofmornis3. Fortheparticularcaseofonlyground\nstate configurations, a single wavefunction suffices to de-\nscribe the collision of two configurations [3, 13, 33, 44],\nas well as for the study of structure formation [41]. This\ncorresponds to the case m=n= 1 and the unique wave\nfunction is the superposition of the wave function of each\nconfiguration Ψ 1=ψL1+ψR1, and the evolution is ruled\nby only one Schr¨ odinger equation in (1). In order to put\nthis idea into practice one needs to interpolate the multi-\nstate equilibrium configurations constructed in the pre-\nvious section and center them as follows. We center Ψ L\nat the point ( r,z) = (0,−z0) and Ψ Rat (r,z) = (0,z0).\nIn order to set the configurations in the axial domain\nhere the evolution is to be carried out, we interpolate\nthe numerical solution of the wave functions of Ψ Land\nΨR. We choose z0so that the two configurations are\nfar enough from one another in order for the interference\nterm/an}b∇acketle{tψL,ψR/an}b∇acket∇i}htto be smaller than round-off error. The\ncollision will be head-on and along the zaxis, thus the\naddition of linear momentum to the initial blobs is ap-\nplied through the transformation of the wave functions\nas followsψL→eipz·xψLandψR→e−ipz·xψR.\nDiagnostics\nAssuming the wave functions to be orthogonal facili-\ntate the estimates of some physical quantities. The ex-\npectation value of an operator ˆAwould be given by\nA=/summationdisplay\nk/integraldisplay\nΨ∗\nkˆAΨkd3x. (2)\nwhere the summation runs overthe number ofwavefunc-\ntions involved. Of particular importance are the kinetic\nenergy, the gravitational energy and the number of par-\nticles associated to each wave function, that we calculate\nrespectively asKi=−1\n2/integraldisplay\nΨ∗\ni∇2Ψid3x,\nWi=1\n2/integraldisplay\nΨ∗\niVΨid3x,\nNi=/integraldisplay\nρid3x, (3)\nfor each wave function Ψ i. The integration is carried\nout over the whole numerical domain. As a particular\ncase there is the collision of two configurations, like in\n[2, 3, 41, 43, 44], where only one wave function for the\ntwo configurations was prescribed.\nOne of the exciting properties of binary GPP configu-\nrations is that they are able to form bounded final sys-\ntems depending on the value of the total energy given by\nE=K+W. Whenever E <0 it happens that the gravi-\ntational binding is sufficiently strong as to confine the in-\ndividualconfigurationsinabounded regionwhilst acom-\nmon gravitational potential well arises. Otherwise, when\nE >0, GPP solutions are dispersive because the gravi-\ntational binding is strong enough to keep individual con-\nfigurations confined in an bounded region and therefore\nthe final configuration tears apart into out-going states.\nThat the profile of the resulting states is pretty similar\nto the initial one, justifies the configuration to be some\ntimes called solitonic.\nIII. TESTS\nBefore we move to the central study of the paper, we\nshow that the code passes two basic tests. We verify that\nsome well known cases are properly reproduced, namely,\ntheevolutionofasingleconfigurationlayinginanexcited\nstateandthe collisionoftwoconfigurationsinthe ground\nstate.\nA. Tests with single configurations\nA first test consists in reproducing the evolution of\nequilibrium configurations in different multi-states. The\ndensity profile of eachconfigurationmust remain station-\nary while the wave function associated to each state os-\ncillates.\nAccording to our notation above, for a configurationin\nthegroundstatecase,asinglewavefunction {Ψ1}suffices\nto describe the evolution of the system, whereas for the\ncase with two states we use two {Ψ1,Ψ2}and three func-\ntions{Ψ1,Ψ2,Ψ3}for a configuration with three states.\nIn Fig. 2 we show snapshots of the wave functions\nfor each of the three cases and the corresponding den-\nsities defined as ρi=|Ψi|2. The plots show that each\nwave function oscillates whereas the corresponding den-\nsity remains nearly stationary with small amplitude os-\ncillations. This behavior indicates that the code solves5\n-1.5-1-0.5 0 0.5 1 1.5\n-10 -5  0  5  10Ψ1\nzΨ1\n 0 0.2 0.4 0.6 0.8 1 1.2\n-10 -5  0  5  10ρ1\nzρ1\n-1.5-1-0.5 0 0.5 1 1.5\n-10 -5  0  5  10Ψ21,Ψ22\nzΨ1Ψ2\n 0 0.2 0.4 0.6 0.8 1 1.2\n-10 -5  0  5  10ρ21,ρ22\nzρ1ρ2\n-1.5-1-0.5 0 0.5 1 1.5\n-10 -5  0  5  10Ψ31,Ψ32,Ψ33\nzΨ1Ψ2Ψ3\n 0 0.2 0.4 0.6 0.8 1 1.2\n-10 -5  0  5  10ρ31,ρ32,ρ33\nzρ1ρ2ρ3\nFIG. 2. Evolution of the wave functions and densities of con-\nfigurations 1, 2 and 3. The density of each state remains\nnearly stationary whereas the wave function is oscillating .\nThe tests run until t= 100 in code units. The evolution\nof these configurations was calculated with the axially sym-\nmetric code on the domain r∈[0,20], z∈[−20,20] with\nresolution ∆ r= ∆z= 0.2 and the configuration set at the\ncoordinate center.\nthe system of equations (1) consistently with the theory\nand with previous numerical results [51].\nB. Second test: collision of two ground state\nconfigurations\nAs an extra test for our code we choose the collision of\ntwo ground state configurations. For this we use similar\nparameters as those used in [3, 15]. This case requires\nonly one wave function Ψ 1= ΨL1+ ΨR1according to\nthe notation described above. In order to add momen-\ntum along the head-on direction, we redefine the wave\nfunction of both configurations as follows\nΨL1→eipz·zΨL1,\nΨR1→e−ipz·zΨR1,\nwith Ψ 1= ΨL1+ΨR1as the initial wave function. One\nthen solvesPoissonequation at initial time and continues\nthe evolution using the system (1) for n= 1.\nHigh initial velocity case. In order to obtain the soli-\ntonic behavior we use pz= 1.5 in code units, which cor-\nresponds to a largeinitial linear momentum giving rise to\na positive total energy E=K+W >0 and therefore the 0 0.2 0.4 0.6 0.8 1\n-30 -20 -10  0  10  20  30t=0ρ\nz 0 0.5 1 1.5 2 2.5 3 3.5\n-30 -20 -10  0  10  20  30t=9.8ρ\nz\n 0 0.2 0.4 0.6 0.8 1\n-30 -20 -10  0  10  20  30t=19.6ρ\nz-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8\n 0  5  10  15  20<pz>\nzz<0\nz>0\nFIG. 3. Solitonic case of a ground state - ground state col-\nlision. We show the density ρ1at various times along the\nz-axis, before, during and after the interaction. The arrows\nindicate the direction of motion of the two configurations. W e\nalso show the transfer of linear momentum integrated in the\nleft and right domains. For this we show the expectation\nvalue of the operator pzin the half domain z <0 and in the\nother half domain z >0. The evolution was calculated on the\nnumerical domain r∈[0,30], z∈[−30,30] with resolution\n∆r= ∆z= 0.12 and the configurations were initially located\nusingz0= 15.\nsystem turns out to be unbounded. In Fig. 3 we show\nthe density profiles at initial time and various snapshots\nduring and after the interaction between the two config-\nurations. We also show a plot of the linear momentum\ntransfer from the half domain z <0 to the half domain\nz >0 during the process and in the opposite direction.\nFor this we calculate the expectation value of the linear\nmomentum along the head-on direction ( z), in the two\nregions using /an}b∇acketle{tpz/an}b∇acket∇i}ht=/integraltext\nDΨ∗\n1(−i∂\n∂z)Ψ1d3x, where the do-\nmainDcan be the half domain z<0 or the other z>0.\nThe plot shows that the momentum transfered from the\nleft domain z <0 where initially the Lconfiguration\n(with positive momentum) is set, to the right domain\nz >0 and from the right domain (with the Rconfigura-\ntion initially with negative momentum) to the left, shows\nthe same behavior. An important observation is that the\ndensity profiles at time t= 0 andt= 19.6 is not exactly\nthe same, a comparison would reveal small differences in\nshape, at first glance final individual configurations are\nwider than the initial ones. Strictly speaking, for solitons\nthe density is the same in the asymptotic past and future\ntimes [36]. In the present study, equilibrium configura-\ntions are not solitons due to the presence of the gravita-\ntional potential involved, which influences the shape in\nthe asymptotic time, aside of the fact that the simula-\ntion is being carried out in a finite spatial and temporal\ndomain, which prevents a strict asymptotic analysis.\nLow velocity case. The scenario for the merger of the\ntwo configurations requiresthe momentum to be smaller.\nFor this we choose two values of the head-on momentum6\npz= 0.25,0.5 that help showing the behavior of the\nsystem and the relaxation process that leads to a final\nstructure in a long term evolution. A helpful quantity\nthat determines whether a system is approaching equi-\nlibrium is 2 K+W, which in theory is zero for a viri-\nalized system, and has been used in the past to moni-\ntor the Gravitational Cooling process of configurations\nruled by the GPP system [21, 45] . Based on this diag-\nnostics, we show in Fig. 4 the behavior in time of this\nquantity, together with the central value of the density\nρ1=|Ψ1|2. At the beginning 2 K+Wis positive and far\nfrom zero because the head-on momentum involved with\nthe kinetic energy of the system dominates; after the two\nconfigurations have merged, this quantity oscillates with\nadecreasingamplitude and isexpected to be zeroasymp-\ntotically, which would indicate the system tends towards\na virialized final state.\nThe central value of the density on the other hand also\nshows an asymptotic behavior, that is, it approaches a\nconstant value after oscillating with a decreasing ampli-\ntude. This is an indication that the configuration re-\nsulting from the merger tends to be time-independent\nasymptotically. Noticethat thetwovaluesofthehead-on\nmomentum ( pz= 0.25,0.5) result in two different central\ndensities of the final configuration. The reason is that\neven though the system is bounded, part ofthe density of\nthe initiallymovingconfigurationsescapesfromthe grav-\nitational potential. We use two values of pzin order to\nillustrate that the fasterthe motion ofthe configurations,\nthe smaller the final configuration. This is illustrated in\nthe bottom panel of Fig. 4, where we show N1in time.\nThe fact that this quantity decreases initially indicates\nthat a finite amount ofparticles of the system gets off the\ndomain, and also the fact that it approaches a constant\nvalue indicates that a finite number of particles remains\ntrapped in the gravitational potential. Finally in this\nFigure we also show snapshots of the density ρ1for the\ncasepz= 0.5, during the final stages of the simulation\nbetweent= 3000 and 4000 in code units. The density\noscillates as shown by the plot of its central density, with\na decreasing amplitude.\nIn theory an averagein time of this density could serve\nto fit the profile of the final configuration as done for in-\nstance in 3D simulations [41, 44]. Indeed, as a part of\nour tests, here we make such a procedure in order to ob-\ntain a final density profile for the final configuration of\nthe merger and be able to compare with profiles from\n3D structure formation simulations. By carrying out a\ntime-average of the radial density profile of the merger\nalong a period of oscillation, we compute a mean density\nprofile which is illustrated later in Fig. 11. In previ-\nous studies, specifically based on the profile of structures\nafter mergers [44], the final density profile splits into a\nsoliton-like core and power-law tail, such layout is due\nto the fact that the resulting configuration has non-zero\nangular momentum. In contrast, for the head-on case,\nwe find that the final configuration is made of a pure\nsolitonic state whose average density profile is perfectly-0.4-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8\n 1  10  100  10002K+W\ntpz = 0.5\npz = 0.25\n 0 0.5 1 1.5 2 2.5 3\n 10  100  1000ρ\ntpz = 0.5\npz = 0.25\n 1 1.5 2 2.5 3 3.5\n 0  500  1000  1500  2000  2500  3000  3500  4000N\ntpz = 0.5\npz = 0.25\n 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35\n-20 -15 -10 -5  0  5  10  15  20ρ\nzpz=0.5\nFIG. 4. Merger of ground state - ground state case. We\nshow diagnostics of configurations that merge, for initial m o-\nmentumpz= 0.5,0.25. First the quantity 2 K+W, that\napproaches asymtotically to zero in time. Second, the value\nof the density in the center of the merger, showing an oscil-\nlatory behavior with decreasing amplitude. We also calcula te\nN1=/integraltext\nρ1d3x, which is the total number of particles within\nthe whole numerical domain, indicating that part of the ma-\nterial gets off the domain and during the relaxation process\nremains nearly constant. Finally we show a sample of snap-\nshots of the density of the final configuration for the case\npz= 0.5, that shows an oscillatory behavior. The evolution\nwas calculated on the domain r∈[0,30], z∈[−30,30] with\nresolution ∆ r= ∆z= 0.12 and the configurations were ini-\ntially located with z0= 15.\ndescribed by a Gaussian fit given by:\nρ(r)\nρ0=e−(x\n6.37)2\n, (4)\nwhich is a behavior consistent with the attractor proper-\nties of ground state configurations. Fits for the final core\nresulting from 3D simulations [41, 43] is reported to obey\nρ∼(1 +0.091(r/rc))−8. In contrast, our final distribu-\ntion obeys (4) which turns out to be much more shallow\nthan the former. On the other side, while ourfinal profile\nturns out to be purely solitonic over the whole domain,\nthe fit from [41] decays more slowly at the edge follow-\ning a NFW envelope. Such an apparent discrepancy is\nexpected to happen due to the different dynamics with\nstructure formation in [41], binary mergers with orbital\nmomentum in [44] and our binary head-on case. Tan-\ngential encounters involve interference mechanisms and\ntidal effects that do not seem to match the soliton profile\nof the final state at some point in the edge of the core\ngiving rise to a core-tail profile. Nevertheless, we shall\nsee along the next section that when multi-states are in-\nvolved in the collision such a behavior of the tail of the\nfinal configuration arises.7\nIV. RESULTS\nA. Ground state vs a two-states collision\nIn this case we set the state Ψ Lto be the multi-\nstate configuration characterized by two wave functions\n{ΨL1,ΨL2}, whereas the state Ψ Ris characterized only\nby a single wave function Ψ R1. Recall these wave func-\ntions correspond to equilibrium spherically symmetric\nconfigurations,andin ordertoaddlinearmomentummo-\nmentum in the head-on direction, these wave functions\nare redefined as follows\nΨL1=eipz·zΨL1\nΨL2=eipz·zΨL2\nΨR1=e−ipz·zΨR1\nIn this case the system (1) requires the solution of two\nSchor¨ odinger equations and the gravitational potential is\nsourced with two densities ρ1=|Ψ1|2andρ2=|Ψ2|2,\nwhere Ψ 1= ΨL1+ ΨR1and Ψ 2= ΨL2. We study two\nscenarios, one exploring a high linear momentum, that\neventually could show the solitonic behavior found in the\npure ground state configurations, and a second one of\nsmall momentum that allows the merger of the two con-\nfigurations.\nHigh velocity case . For this we use the linear momen-\ntumpz= 1.5, in order for the two configurations to go\nthrough each other. The result of the interaction be-\ntween the ground state configuration and the configura-\ntion made of two first states is shown in Fig. 5. Likewise\nin the previous case of ground state configurations, we\nshow snapshots of the density and the momentum trans-\nfer during the interaction.\nDuring the stage at which individual configurations\noverlap inside a region around z= 0, an interesting be-\nhavior can be noticed. As the snapshot at t= 9.8 shows,\nonlyρ1shows an interference pattern with dominant am-\nplitude and a while after the interaction, the amplitude\nof this density decreases. On the other hand ρ2does not\nproduce such a pattern and its amplitude is enhanced.\nBesides, notice that for this multi-state, the initial and\nfinal profiles differ importantly, the initially excited state\nsolution loses its nodes after the collision.\nWe measure the momentum transference by calculat-\ning/an}b∇acketle{tpz/an}b∇acket∇i}ht1,2for the two wave functions. /an}b∇acketle{tpz/an}b∇acket∇i}ht1is non-zero\ninitially, positive in the left half-domain and negative\nin the right half-domain, and after the interaction it is\ntransferred from one half-plane to the opposite one. On\nthe other hand /an}b∇acketle{tpz/an}b∇acket∇i}ht2is non-zero only for the configu-\nration on the left, and after the interaction its value is\ntransferred to the opposite half-domain.\nLow velocity case. We produce a merger case scenario\nusingpz= 0.5 which has negative total energy and thus\ncorresponds to a bounded system. For this case, we show\nthat the quantity 2 K1+ 2K2+W1+W2in Fig. 6 os-\ncillates and approaches to zero asymptotically in time. 0 0.2 0.4 0.6 0.8 1\n-30 -20 -10  0  10  20  30t=0ρ1\nρ2ρ\nz 0 0.5 1 1.5 2 2.5 3 3.5\n-30 -20 -10  0  10  20  30t=9.8\nρ2ρ1ρ\nz\n 0 0.2 0.4 0.6 0.8 1\n-30 -20 -10  0  10  20  30t=19.6ρ1\nρ2ρ\nz-2-1.5-1-0.5 0 0.5 1 1.5 2\n 0  5  10  15  20<pz>\nzΨ1 z<0\nΨ1 z>0\nΨ2 z<0\nΨ2 z>0\n-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8\n 0  5  10  15  20<pz>\nz\nz<0\nz>0\nFIG.5. Solitoniccaseofagroundstate-two-statesencount er.\nDensity of the binary configuration at various times along th e\nz-axis, before, during and after the interaction. The unla-\nbeled arrows indicate the direction of motion of the two con-\nfigurations. We also show the transfer of linear momentum\nintegrated in the z<0 andz>0half domains, for each of the\ntwo states that are being evolved. For this we show the expec-\ntation value /angbracketleftpz/angbracketrightintegrated in the domain z <0 and in the\ndomainz >0 for both, the ground state and the excited state\nwave functions. The evolution was calculated on the domain\nr∈[0,30], z∈[−30,30] with resolution ∆ r= ∆z= 0.12 and\nthe configurations were initially located with z0= 15.\nWe also show the total energy and the integrals N1and\nN2illustrating how the system loses particles and then\napproaches a stationary regime. Nevertheless, since we\nlaunched the two configurations with the same head-on\nmomentum, and the center of mass is off the center of\ncoordinates, since there are no dissipative effects other\nthan the gravitational cooling thorough the emission of\nparticles, thecenterofmassofthe systemoscillates. This\nis illustrated by the value of /an}b∇acketle{tpz/an}b∇acket∇i}htintegrated in the two\nhalves of the domain z>0, z<0, showing that the mo-\nmentum is being transferred back and forth between the\ntwo regions. We include this scenario because it seems to\nshow the properties required for dark matter in extreme\nscenarios like the bullet cluster, where, luminous mat-\nter seems to slow down with respect to the dark matter\ndistribution. Finally we present snapshots of the gravi-\ntational potential, whose minimum oscillates during var-\nious cycles without a noticeable dissipation.\nB. Ground state vs a three-states configuration\ncollision\nLike in the previous case, we set the state Ψ Lto be\nthe multi-state configurationcharacterizedby three wave\nfunctions {ΨL1,ΨL2,ΨL3}, whereas the state Ψ Ris char-\nacterized only by one wavefunction Ψ R1. In order to add\nlinear momentum in the head-on direction, these wave\nfunctions are redefined as follows8\n 0 0.5 1 1.5 2 2.5 3 3.5\n 0  500  1000  1500  2000  2500  3000  3500  4000N\ntN1N2\n-0.5 0 0.5 1 1.5 2 2.5 3\n 1  10  100  10002K+W\nt2(K1+K2) + W1+W2\n-0.3-0.2-0.1 0 0.1 0.2 0.3\n 0  500  1000  1500  2000  2500  3000  3500  4000<pz>\nzΨ1 z<0\nΨ1 z>0\n-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8\n 0  5  10  15  20<pz>\nz-1.4-1.2-1-0.8-0.6-0.4-0.2 0\n-30 -20 -10  0  10  20  30V\nz\nFIG. 6. Merger case of a ground state - two-states encounter.\nFirst we show N1andN2ofthe system, showing it approaches\na constant number of particles during the process. Second we\nplot the quantity 2 K1+ 2K2+W1+W2, which oscillates\nwith decreasing amplitude around zero. This indicates that\nthe system tends toward a state of relaxation. In the bottom\nwe show /angbracketleftpz/angbracketrightintegrated in the two halves of the domain for\nΨ1. The oscillatory behavior indicates that even though the\nsystem as awhole shows arelaxation process, the particles a re\nbeing transferred from one half of the domain to the other.\nFinally we show snapshots of the gravitational potential V\nalong thezaxis for various times after the first merge. The\nposition of the minimum oscillates from negative to positiv e\nvalues ofz. The evolution was calculated on the domain r∈\n[0,30], z∈[−30,30] with resolution ∆ r= ∆z= 0.12 and the\nconfigurations were initially located with z0= 15.\nΨL1=eipz·zΨL1\nΨL2=eipz·zΨL2\nΨL3=eipz·zΨL3\nΨR1=e−ipz·zΨR1.\nIn this case the system (1) requires the solution of three\nSchor¨ odinger equations and the gravitational potential is\nsourced with three densities ρ1=|Ψ1|2,ρ2=|Ψ2|2and\nρ3, where Ψ 1= ΨL1+ΨR1, Ψ2= ΨL2and Ψ 3= ΨL3.\nFor ahigh velocity case we again use the momentum\npz= 1.5 and show the densities of each state in Fig.\n7. The results are similar to those in the previous case.\nThe density ρ1shows an interference pattern whereas\nthe other tow states approximately retain their profile\nduring the evolution. After the interaction it happens\nthat the first excited state is amplified, whereas the sec-\nond excited state flattens. The solitonic behavior is not\nperfect again and the densities associated to each state\nsuffer deformations due to the interaction between the\ntwo configurations through the gravitational potential.\nLow velocity case. By applying the same head-on mo-\nmentum as in the previous cases pz= 0.5, the results are\npretty similar to those of the merger of a ground state vs\na two-states configuration. In brief, the final configura-\ntion tends globally toward a virialized state, however the 0 0.2 0.4 0.6 0.8 1\n-30 -20 -10  0  10  20  30t=0ρ1\nρ3ρ2ρ\nz 0 0.5 1 1.5 2 2.5 3 3.5\n-30 -20 -10  0  10  20  30t=9.8\nρ1\nρ2\nρ3ρ\nz\n 0 0.2 0.4 0.6 0.8 1\n-30 -20 -10  0  10  20  30t=19.6ρ1\nρ2ρ3ρ\nz-2-1.5-1-0.5 0 0.5 1 1.5 2\n 0  5  10  15  20<pz>\nzΨ1 z<0\nΨ1 z>0\nΨ2 z<0\nΨ2 z>0\nΨ3 z<0\nΨ3 z>0\n-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8\n 0  5  10  15  20<pz>\nz\nz<0\nz>0\nFIG. 7. Solitonic case of a ground state - three-states con-\nfiguration encounter. Density of the binary configuration at\nvarious times along the z-axis, before, during and after the\ninteraction. The arrows indicate the direction of motion of\nthe two configurations. We also show the transfer of lin-\near momentum integrated in the left and right domains for\neach of the two states that are being evolved. For this we\nshow/angbracketleftpz/angbracketrightintegrated in the domain z <0 and in the do-\nmainz >0 for both, the ground state and the excited state\nwave functions. The evolution was calculated on the domain\nr∈[0,30], z∈[−30,30] with resolution ∆ r= ∆z= 0.12 and\nthe configurations were initially located with z0= 15.\nfinal blob oscillates in space around the center of coordi-\nnates.\nC. Two-states vs two-states configurations collision\nNow we show the result of the collision of two non-\nground state configurations. For this we choose the two\nconfigurations to have two states. In this case the con-\nfiguration at the left has two wave functions {ΨL1,ΨL2}\nand the configurationat the righthas alsotwowavefunc-\ntions{ΨR1,ΨR2}.\nThe addition of linear momentum reassigns the wave\nfunctions in the following manner\nΨL1=eipz·zΨL1,\nΨL2=eipz·zΨL2,\nΨR1=e−ipz·zΨR1,\nΨR2=e−ipz·zΨR2.\nFinally, following the description in subsection IIC, the\nwavefunction ofthe ground state is given by Ψ 1= ΨL1+\nΨR1, whereas the first excited state wave function of the\nsystem is Ψ 2= ΨL2+ΨR2.\nIn this case the system (1) requires the solution of two\nSchor¨ odinger equations and the gravitational potential is\nsourced with two densities ρ1=|Ψ1|2andρ2=|Ψ2|2.\nLike in the previous cases, in order to learn about the9\n 0 0.2 0.4 0.6 0.8 1\n-30 -20 -10  0  10  20  30t=0ρ1\nρ2ρ\nz 0 0.5 1 1.5 2 2.5 3 3.5\n-30 -20 -10  0  10  20  30t=9.8\nρ2ρ1ρ\nz\n 0 0.2 0.4 0.6 0.8 1\n-30 -20 -10  0  10  20  30t=19.6ρ1\nρ2ρ\nz-2-1.5-1-0.5 0 0.5 1 1.5 2\n 0  5  10  15  20<pz>\nzΨ1 z<0\nΨ1 z>0\nΨ2 z<0\nΨ2 z>0\nFIG. 8. Solitonic case of a two states - two states encounter.\nWe show snapshots of the interaction in the high velocity\nregime of two configurations with two states. In this case the\ntwo wave functions show an interference patter. Also shown\nare the values of /angbracketleftpz/angbracketrightcalculated with Ψ 1and Ψ 2that shows\nthe momentum is being transferred from one half domain to\nthe other one. The evolution was calculated on the domain\nr∈[0,30], z∈[−30,30] with resolution ∆ r= ∆z= 0.12 and\nthe configurations were initially located with z0= 15.\ngeneral behavior of the system we execute high velocity\nand low velocity cases.\nHigh velocity case. We again use pz= 1.5 and look for\na type of solitonic behavior of the system. The results\nare shown in Fig. 8. At the moment of collision there\nis an interaction pattern of the two densities ρ1andρ2.\nThe solitonic behavior is actually very poor, since the\ndensity of the ground state deforms significantly as seen\nby its amplitude, and the excited state density deforms\nsignificantly after the interaction as can be seen in the\nsnapshot at t= 19.6, where the nodes have been lost.\nLow velocity case. For the merger case we use pz= 0.5\nand show the results in Fig. 9. We find in the first place\nthat the quantity 2 Ki+Wicalculated for each state does\nnot approach zero if calculated independently, however\nthequantity2( K1+K2)+W1+W2approacheszero,which\nindicates the system tends toward a virialized state. In\nthe second place we notice that the central density of the\nfinalconfigurationofeachofthetwostatesoscillateswith\nno decreasing amplitude. In fact the excited state some-\ntimes has nodes and some others it does not, a behavior\nthat shows to be periodic. Instead of decreasing, the cen-\ntral value of ρ1andρ2oscillate with various frequency\nmodes and non-decreasing amplitude. This behavior is\npuzzling because on the one hand the system approaches\na virialized state whereas the densities do not seem to\napproach a stationary regime. Nevertheless, it happens\nthat the addition of the densities of the two states oscil-\nlates with a decreasing amplitude.\nWe choose this particular case to discuss some poten-\ntial interesting results to the axionic dark matter models.\nIn the context of ultralight axion dark matter, 3D sim--1-0.5 0 0.5 1 1.5 2 2.5\n 1  10  100  10002K+W\nt2K1 + W12K2+W2\n-0.5 0 0.5 1 1.5 2 2.5 3 3.5\n 1  10  100  10002K+W\nt2(K1+K2) + W1+W2\n 0 0.5 1 1.5 2 2.5\n 500  1000  1500  2000  2500  3000  3500  4000ρ\ntρ1ρ2 \n 0 0.5 1 1.5 2 2.5\n 500  1000  1500  2000  2500  3000  3500  4000ρ T\ntρ1 + ρ2\nFIG. 9. Merger case of a two states - two states encounter. In\nthe first panel we show 2 K1+W1and 2K2+W2as function\nof time. In the second panel we show the addition of the two,\nwhich indicates that the whole system is approaching a viria l-\nized state in asymptotic time. We also show the central value\nof densities ρ1andρ2of the final configuration; the behavior\nis oscillatory and interestingly does not show the tendency to\na constant time independent value., instead it oscillates w ith\nvarious modes. However the central value of ρT=ρ1+ρ2\napproaches better to an asymptotic value. The evolution was\ncalculated on the domain r∈[0,30], z∈[−30,30] with reso-\nlution ∆r= ∆z= 0.12 and the configurations were initially\nlocated with z0= 15.\nulations show that the mass function of structures have\na profile given by a core with a density profile similar to\nthat of ground state configurations and a halo surround-\ning the core [2, 3, 41, 44].\nThe idea is to extract the density profile of the final\nconfiguration in order to estimate potential implications\nto axion dark matter astrophysics. The final configu-\nrations are not stationary, and actually they have never\nbeenshowntobeeitherin2D[44]norin3D[41]analyses.\nIn our case we notice a low frequency mode that modu-\nlates the oscillation of the two densities with a period of\nnearly 500 in code units. What we do is to calculate an\naverageof the total density ρT=ρ1+ρ2and fit a density\nprofile for such average. For this we averaged snapshots\nofρTfromt=3500 to 4000, which is nearly the period\nthat modulates the oscillation of the densities.\nIn Fig. 10 we show the average of the densities of each\nof the two states and their addition. This average shows\nthe contribution of a core and a halo. This is one of the\nconvenientfeaturesofmulti-stateconfigurations[51]. We\nshow here that similar states can be formed through the\ncollision of two configurations with more than one state.\nWhat we do next is to fit the average profile of ρTas\ndone for the ground - ground merger case in section in\nIIIB and compare. We find that there is a core governed\nby a Gaussian and a power-law tail given by the profiles\nρcore∼e−(z\n1.33)2\n, z<z c\nρhalo∼z−3.1, z>z c (5)10\n 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8\n-10 -5  0  5  10Density\nzaverage of ρ1average of ρ2average of ρΤ\nFIG. 10. Average of ρ1,ρ2and the total density ρT=ρ1+ρ2\nin the time range between 3500 and 4000. This plot shows\nthe contribution of each state to the total density.\n10-1010-810-610-410-2100102\n 0.001  0.01  0.1  1  10  100Density\nzground -- ground  ρ1two states -- two states  ρT\nFIG. 11. Density profile showing a core-like slope and a power\nlaw tail. We show three cases, the result of the ground-groun d\ncollision case, the result of a ground state with a two state\nconfiguration and the result of the merger of two configura-\ntions with two states.\nwhich is illustrated in Fig. 11. We define the core-limit\nzcasthe point wherethe profile ρThas aninflexion. This\nshows the ground state vs ground state head-on mergers\ngives rise to solitonic solutions resulting profiles, whereas\nthe merger of two-states vs two-states case results in a\ncore with a tail envelope.\nV. CONCLUSIONS AND FINAL COMMENTS\nWe present the head-on collision of multi-state equi-\nlibrium configurations ruled by the GPP system. As an\nattempt to study solitonic behavior of colliding multi-\nstate configurations, we performed simulations with a\nhigh head-onmomentum, and found that there isno such\na strict solitonic behavior as given by definition in terms\nof the constancy of the density profile. Instead, the cou-\npling between the wave functions of each state throughthe common gravitational potential, deforms the initial\ndensity profiles of each state after the collision in the\nasymptotic time, in comparison with the case of the col-\nlision of two ground state configurations in [3, 13].\nWe also simulated the head-on collision with smaller\ninitial momentum in order to produce mergers. In this\ncase some interesting findings turn up. These results are\ngeneric taking into account the various cases treated in\nour analysis. In the first case we analyzed the merger\nof two ground state configurations and find that: 1) The\nresulting configuration relaxes, 2) it tends towards a sta-\ntionary regime while oscillating, 3) its density profile ap-\nproaches that of a stationary ground state configuration.\nThis is consistent with the attractor nature of equilib-\nrium configurations. From a naive standpoint, the pre-\nvious findings might be controversial with the results in\n[44], where the resulting configurations have a core-tail\nprofile. However such controversy would be apparent be-\ncause in the later case the initial configuration that even-\ntually merge, have non-vanishing orbital momentum and\ntherefore the final configuration has a non-trivial rota-\ntion velocity field, whereas our merger involves head-on\nencounters that lead to a final configuration with non-\nrotational velocity field.\nAnotherinterestingcaseis the merger of two-state con-\nfigurations , for which we find that: 1) The quantity\n2(K1+K2) +W1+W2approaches to zero asymptoti-\ncally in time, which indicates that the total system tends\nto a virialized state. 2) A long while after the collision,\neven though the densities associated with the two states\nρ1andρ2oscillatewith a modulated amplitude, the total\ndensityρT=ρ1+ρ2approaches a stationary state. This\nis an interesting hint concerning to structure formation\nsince, at least in this case when multistates merge, indi-\nvidualconfigurationsdonot virialize,howeverthesystem\ndoes as a whole. 3) We obtained an averaged profile of\nρTand found that it has a core plus a tail structure that\ncould serve to explain the results in simulations of struc-\nture formation. These results are qualitatively consistent\nwith those of [41] and [44] although having different fits\nfor the core, nonetheless the shape of the tail envelope\ncoincides in either works. This merger case of multi-\nstate configurations is the most illustrative of our analy-\nsis, showing the potential of assuming the condensate to\nbe made of multi-states.\nSo far, we have presented the results of a theoreti-\ncal study, and have shown how the densities resulting\nfrom the merger of multi-state configurations can be po-\ntentially interesting within the frame of the ultralight\nscalar field dark matter. A direct application of the re-\nsults shown in this paper within the astrophysical sce-\nnario, will have to consist of a massive set of simulations\nbetween multi-state configurations, with different linear\nmomentum and relative masses, as an attempt to define\na universal density profile resulting from mergers of dif-\nferent initial configurations, assuming the GPP system\nmodels the dynamics of a Bose-Einstein condensate of\nultralight bosons with masses of the order of 10−22eV.11\nACKNOWLEDGMENTS\nWe specially thank Argelia Bernal, for providing the\ndata for the multi-state configurations in [51]. This\nresearch is supported by grants CIC-UMSNH-4.9 and\nCONACyT 258726 (Fondo Sectorial de Investigaci´ onpara la Educaci´ on). A.A.L acknowledges financial sup-\nport from CONACyT posdoctoral fellowship. The simu-\nlations were carried out in the computer farm funded by\nCONACyT 106466 and the Big Mamma cluster at the\nIFM.\n[1] M. R. Baldeschi, R. Ruffini, and G. B. Gelmini. 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Yuan, and Liang Fu\nDepartment of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA\nWe study electronic ordering instabilities of twisted bilayer graphene with n= 2 electrons per\nsupercell, where correlated insulator state and superconductivity are recently observed. Motivated\nby the Fermi surface nesting and the proximity to Van Hove singularity, we introduce a hot-spot\nmodel to study the e\u000bect of various electron interactions systematically. Using renormalization\ngroup method, we \fnd d/p-wave superconductivity and charge/spin density wave emerge as the two\ntypes of leading instabilities driven by Coulomb repulsion. The density wave state has a gapped\nenergy spectrum at n= 2 and yields a single doubly-degenerate pocket upon doping to n>2. The\nintertwinement of density wave and superconductivity and the quasiparticle spectrum in the density\nwave state are consistent with experimental observations.\nI. INTRODUCTION\nRecently superconductivity was discovered near a cor-\nrelated insulator state in bilayer graphene with a small\ntwist angle \u0012\u00191:1\u000e[1, 2], where the moir\u0013 e pattern cre-\nates a superlattice with a periodicity of about 13 nm. A\ncorrelated insulating state is found at the \flling of n= 2\nelectrons per supercell ( n= 0 is the charge neutrality\npoint). Electron or hole doping away from n= 2 by elec-\ntrostatic gating leads to a superconducting dome, similar\nto cuprates. Insulating states are also found in trilayer\ngraphene with moir\u0013 e superlattice [3]. The nature of su-\nperconducting and insulating states in graphene super-\nlattices are now under intensive theoretical study [4{13].\nFor\u0012\u00191:1\u000e, the low-energy mini-band of twisted bi-\nlayer graphene has a narrow bandwidth of 10 meV scale\n[14{26]. However, this energy scale is still much larger\nthan the energy gaps of the superconducting and insulat-\ning states, which are on the order of 1 K. Moreover, resis-\ntivity shows metallic behavior above 4 K. This is rather\ndi\u000berent from the case of a Mott insulator in the strong\ncoupling limit, which would become insulating at much\nhigher temperature. Based on these considerations, in\nthis work, we take a weak coupling approach to study\nordered states driven by electron correlation in twisted\nbilayer graphene.\nWhile details of the band structure remain to be fully\nsorted out, a number of prominent features of the nor-\nmal state fermiology are robust and noteworthy. First, at\nsmall twist angle, the two valleys of graphene have negli-\ngibly small single-particle hybridization and give rise to\ntwo separate Fermi surfaces that intersect each other in\nthe mini Brillouin zone [26{28]. Second, as the carrier\ndensity increases, Fermi pockets associated with a given\nvalley \frst appear around the Dirac points at KandK0\nin the mini Brillouin zone, then these KandK0pockets\nmerge at a saddle point on the \u0000{ Mline to become a\nsingle pocket centered at \u0000. The saddle point associated\nwith this Lifshitz transition has a Van Hove singular-\nity (VHS) with a logarithmic divergence of the density of\nstates (DOS). Third, realistic band structure calculations\n[28] show that near the Van Hove energy the Fermi sur-\nfaces of di\u000berent valleys contain nearly parallel segments\nГK\nM\nK’ГK\nM\nK’\nQ-Q+\nQ ’\n32\n13\n21(a) (b)\n(c) (d)FIG. 1: (a), (b) Two Fermi surfaces at the Van Hove energy\nfrom di\u000berent valleys shown in red and blue, reproduced from\nMoon and Koshino's band structure calculation for twisted\nbilayer graphene with \u0012= 2\u000e[28]. Shaded areas are \flled and\nVHS appear at the points, where two Fermi surfaces encir-\nclingKandK0touch. Each Fermi surface has C3symme-\ntry about \u0000, K, andK0. The Fermi surfaces in (a) and (b)\nare related by C2rotation with respect to an in-plane axis\nalong \u0000{K. (c) Two Fermi surfaces slightly away from the\nVan Hove energy. DOS is larger near the VHS points (hot\nspots), where electron interaction predominates. We assign\npatches (circles with dashed lines) centered at hot spots. (d)\nThree inequivalent wave vectors ( Q+,Q\u0000, andQ0), along with\nsymmetry-related ones (not shown), connect various pairs of\nhot spots.\nand hence are nearly nested, see Fig. 1. Such Fermi sur-\nface nesting strongly enhances density wave \ructuations.\nWhen the Fermi energy crosses the Van Hove energy, a\nconversion between electron and hole carriers is expected\nand indeed observed from the sign change of Hall resis-\ntance as a function of doping for \u0012= 2\u000e[28] and\u0012= 1:8\u000e\n[27]. Remarkably, this sign change occurs at the \fllingarXiv:1805.06449v2  [cond-mat.str-el]  6 Jul 20182\nn= 2, indicating that the Fermi energy is very close to\nVHS. VHS in twisted graphene layers was also detected\nfrom pronounced peaks in DOS in STM measurements\n[29]. As the twist angle becomes smaller, the energy sep-\naration between VHS in conduction and valence bands\nis found to decrease rapidly from 430 meV at \u0012= 3:4\u000e\nto 82 meV at 1 :79\u000eand 12 meV at 1 :16\u000e. The strong re-\nduction of bandwidth is expected to magnify the e\u000bects\nof electron correlation. This understanding is consistent\nwith the fact that correlated insulator and superconduct-\ning states are found for \u0012= 1:1\u000e, but not for \u0012= 1:8\u000e\nand 2\u000e. In the following, we take the Fermi surface with\ngood nesting condition and in proximity to VHS as a\nstarting point and study its instabilities in the presence\nof electron interactions.\nDue to the divergent DOS at VHS, electron interac-\ntion predominates in patches of the Brillouin zone around\nsaddle points, or \\hot spots\". When multiple hot spots\nare present at a given energy, various scattering processes\namong them may interfere with each other, leading to in-\ntertwined density wave and superconducting instabilities.\nSuch hot-spot models were studied with renormalization\ngroup (RG) approach in the context of cuprates [30{34],\nand recently, by Nandkishore, Levitov, and Chubukov in\nthe context of doped monolayer graphene [35].\nIn this paper, we study interaction-driven ordering in-\nstabilities of twisted bilayer graphene around the \flling\nn= 2 using RG by patching the Brillouin zone where\nthe DOS is considerably larger than other parts. Our\nRG analysis shows how the electron interaction changes\nas the energy scale is reduced. Nontrivial RG \rows\nof the coupling constants are found as a consequence\nof the nesting of Fermi surfaces. Susceptibility calcu-\nlations reveal the possibility of various superconduct-\ning and spin/charge density-wave states at low tempera-\nture. When Coulomb repulsion is the dominant interac-\ntion,d/p-wave superconductivity and charge/spin den-\nsity wave at a particular nesting wave vector emerge as\ntwo leading instabilities. The density-wave state is found\nto have a gapped energy spectrum at n= 2 and yields a\nsingle doubly-degenerate pocket upon doping to n>2.\nII. MODEL\nWe set up a model to analyze the electron interaction\ne\u000bect in twisted bilayer graphene around the \flling n= 2.\nOur analysis of the interaction-driven instabilities focuses\non hot spots, which dominates in the DOS. The hot spots\nare patches on the Brillouin zone. The Fermi surface\nnesting in the hot spots may potentially lead to ordering\ninstabilities. In the following, we introduce a hot-spot\nmodel for twisted bilayer graphene and then the notion\nof Fermi surface nesting in hot spots.A. Hot-spot model\nTo consider the electron interaction e\u000bect and resul-\ntant ordering instabilities near the \flling n= 2, we focus\non hot spots in the Brillouin zone which possess signi\f-\ncantly larger electronic spectral weights compared to the\nother region. Such hot spots are obtained by patching the\nBrillouin zone around the saddle points. The patches are\nlabeled by \u001c=\u00061 for Fermi surfaces from the two val-\nleys,\u001bfor spins, and \u000b= 1;:::; 3 for the patches of a\ngiven valley (Fig. 1). We denote the position of a patch\ncenter by k\u000b\u001c. The three inequivalent wave vectors con-\nnecting the patch centers are de\fned by Q+=k\u000b\u001c\u0000k\f\u001c\n(intravalley), Q\u0000=k\u000b\u001c\u0000k\f\u001c0andQ0=k\u000b\u001c\u0000k\u000b\u001c0\n(intervalley) ( \u001c6=\u001c0,\u000b6=\f), see Fig. 1(d).\nElectron interaction is treated as scattering among the\npatches. By analogy with the g-ology model in one-\ndimensional physics [36{38] and Shankar's RG approach\n[39], we write down the general interaction Hamiltonian\ncompatible with lattice rotational symmetry\nHint=1\n24X\ni;j=1X\n\u000b1;:::;\u000b 4\u001c1;:::;\u001c 4X\n\u001b\u001b0gij y\n\u000b1\u001c1\u001b y\n\u000b2\u001c2\u001b0 \u000b3\u001c3\u001b0 \u000b4\u001c4\u001b:\n(1)\nHere the patch indices satisfy \u000b1=\u000b36=\u000b2=\u000b4(i= 1),\n\u000b1=\u000b46=\u000b2=\u000b3(i= 2),\u000b1=\u000b26=\u000b3=\u000b4(i= 3),\nand\u000b1=\u000b2=\u000b3=\u000b4(i= 4). The valley indices\n\u001c1;:::;\u001c 4obey the same rule, associated with j. This\nrule is diagrammatically shown in Fig. 2(a). The inter-\naction describes sixteen independent scattering processes\nwith coupling constants gijIn the following analysis, we\nconsider the momentum-conserving processes depicted in\nFig. 2(b). Scattering processes related to these ones by\nlattice symmetry are not shown. Note that gi3,g12,\ng21, andg34do not generally conserve crystal momen-\ntum since the patches are located away from the Bril-\nlouin zone boundary (see Fig. 12). Umklapp processes\nare allowed only when the hot spots are located at spe-\ncial momenta. The analysis for that case includes more\nor allgij, and is presented in Appendix F. Among the\nnine momentum-conserving terms, g11,g14,g22,g24,g44\nare associated with forward scattering processes, and g31,\ng32,g41,g42are BCS scattering processes involving two\nelectrons with opposite momenta.\nThe nine momentum-conserving terms gijin the in-\nteraction Hamiltonian Eq. (1) can be divided into three\ngroups with di\u000berent index j. Forgi4terms, all four op-\nerators belong to one valley ( \u001c1=\u001c2=\u001c3=\u001c4), thus de-\nscribing intravalley interactions. gi2terms are the prod-\nuct of two spin- and valley-conserving fermion bilinear\noperators that are associated with two di\u000berent valleys;\nwe call them intervalley density interactions. gi1terms\nare the product of two spin-conserving but valley-\ripping\nfermion bilinear operators; we call them intervalley ex-\nchange interactions.\nWe now discuss the microscopic origin of these inter-\naction terms and how the coupling constants gijshould3\ng14\ng24\ng44g22g32g42\ng11g31\ng411(a)\n(b)3 42\nFIG. 2: (a) Diagrammatic representation of scattering pro-\ncesses. The four diagrams describe the change of either sad-\ndle point or valley index, where solid and dashed lines corre-\nspond to electron propagators with di\u000berent indices. (b) Nine\nmomentum-conserving scattering processes out of sixteen dis-\ntinct scattering processes gij. Hexagons are the Brillouin zone\nboundaries and the dots are located at saddle points with two\ncolors corresponding to the two valleys. There are three types\nin the interactions (from left to right): intravalley, intervalley\ndensity, and intervalley exchange interactions.\nin principle be determined. First, when projected to the\nlowest mini-band, the long-range part of Coulomb inter-\naction generates intra- and intervalley density interaction\ngi2,gi4, while the short-range part of Coulomb interac-\ntion on graphene's lattice scale generates intervalley ex-\nchange interaction gi1involving large momentum trans-\nfer. Since Wannier functions in twisted bilayer graphene\nextend over tens of nanometers, the long-range part of\nCoulomb interaction is expected to dominate. Based on\nthis factor alone, one would expect density interactions\ngi2,gi4to be orders-of-magnitude larger than the inter-\nvalley exchange interaction gi1. On the other hand, it\nshould be noted that the long-range Coulomb interac-\ntion strength is reduced by screening from excited bands\nthat span a wide range of energies from \u001810 meV up to\nthe bandwidth of graphene layers \u001810 eV. The process\nof integrating out these excited bands as well as those\nstates of the lowest band outside the patches will sig-\nni\fcantly renormalize the coupling constants to be used\nin our patch theory. Moreover, their values are also af-\nfected by electron-phonon coupling. Since typical phonon\nenergy in graphene is much larger than the mini-band\nwidth, it is reasonable to integrate out the phonons to ob-\ntain phonon-mediated electron-electron attraction [10],\nwhich renormalizes the values of coupling constants.\nIn this work, we treat the bare values of gijas phe-\nnomenological parameters and calculate \rows of these\ncoupling constants under RG to the one-loop order.\nStrictly speaking, such a perturbative RG analysis is only\nlegitimate for weak coupling. However, instabilities to-\nward superconductivity and/or density waves are found\nwithin the weak-coupling regime (see below), which jus-ti\fes the one-loop RG analysis.\nB. Susceptibilities and Fermi surface nesting\nFermion loops in the RG calculation are associated\nwith bare susceptibilities in the particle-hole and particle-\nparticle channels:\n\u001fph(q\u0000;!) =Z\nk2patchf(\u000f\u001c\nk+k\u000b\u001c)\u0000f(\u000f\u001c0\nk+k\f\u001c0)\n!\u0000\u000f\u001c\nk+k\u000b\u001c+\u000f\u001c0\nk+k\f\u001c0;(2)\n\u001fpp(q+;!) =Z\nk2patchf(\u000f\u001c\nk+k\u000b\u001c)\u0000f(\u0000\u000f\u001c0\n\u0000k+k\f\u001c0)\n!\u0000\u000f\u001c\nk+k\u000b\u001c\u0000\u000f\u001c0\n\u0000k+k\f\u001c0;(3)\nwhere q\u0006=k\u000b\u001c\u0006k\f\u001c0,f(\u000f) is the Fermi distribution,\nand\u000f\u001c\nkis the energy dispersion of valley \u001cwith\u000f= 0 on\nthe Fermi surface.\nThere are in total eight susceptibilities in the particle-\nparticle and particle-hole channels at various wave vec-\ntors:\n\u001f0+=\u001fpp(Q0;!); \u001f 0\u0000=\u001fpp(0;!); (4)\n\u001f1s=\u001fph(Qs;!); (5)\n\u001f2+=\u001fph(0;!); \u001f 2\u0000=\u001fph(Q0;!); (6)\n\u001f3s=\u001fpp(Q\u0000s;!); (7)\nwheres= +;\u0000correspond to intra- and intervalley com-\nponents, respectively.\nAmong these susceptibilities, \u001f2+(!= 0) is the DOS\nwithin a patch at Fermi energy \u001a0.\u001f0\u0000is theQ= 0\nCooper-pair susceptibility, which involves patches at op-\nposite momenta, belonging to the di\u000berent valleys. Re-\ngardless of Fermi surface geometry, \u001f0\u0000exhibits a loga-\nrithmic divergence \u001f0\u0000(!) =\u001a0\n4ln\u0003\n!in the presence of\ntime-reversal symmetry, where \u0003 is the high-energy cut-\no\u000b and depends on the patch size.\nBesides the BCS channel, susceptibilities in other chan-\nnels may \fnd divergences when Fermi surfaces are per-\nfectly nested: \u000f\u001c\nk+k\u000b\u001c=\u0000\u000f\u001c0\nk+k\f\u001c0for the particle-hole\nchannels and \u000f\u001c\nk+k\u000b\u001c=\u000f\u001c0\n\u0000k+k\f\u001c0for the particle-particle\nchannels. In general, the interplay between BCS and\nnesting-related interactions can lead to nontrivial RG\n\rows of coupling constants [39, 40].\nAs shown in Fig. 1, the Fermi surfaces associated with\ndi\u000berent valleys have nearly parallel segments connected\nby the following nesting vectors: Q\u0000andQ0connect oc-\ncupied states of one valley and unoccupied states of an-\nother, while Q+connects occupied states around Kand\nthose around K0associated with the same valley; see\nFig. 1(d). Such Fermi surface nesting strongly enhances\nthree types of bare susceptibilities: intervalley charge or\nspin density wave susceptibility at Q\u0000andQ0(particle-\nhole channel), and intervalley pair density wave at Q+\n(particle-particle channel).\nIn the ideal case where the Fermi surface is perfectly\nnested (see Appendix A for the detailed discussion), these4\nFIG. 3: One-loop corrections to the coupling constants. The\nsolid lines represents the fermion propagators and the wavy\nlines correspond to interactions. The leftmost diagram in-\nvolves a particle-particle loop, and the other three diagrams\nhave particle-particle loops.\nsusceptibilities are logarithmically divergent like the BCS\nchannel. When the Fermi surface is nearly nested, the\nlogarithmic frequency energy dependence still holds ap-\nproximately within a range \u0003 0< !\u0014\u0003, but the diver-\ngence in the !!0 limit is cuto\u000b below a smaller energy\nscale \u0003 0associated with the deviation from perfect nest-\ning.\nFinally, when the VHS lies close to the Fermi surface,\nscatterings among states near VHS points receive partic-\nularly large RG corrections because of the large spectral\nweight. This justi\fes our use of patch RG approach.\nWhen the VHS lies exactly on the Fermi surface, the\nDOS at!= 0 is logarithmically divergent and thus leads\nto an additional log divergence in susceptibilities, see dis-\ncussion in Appendix B.\nIII. RG ANALYSIS\nLoop corrections to the coupling constants su\u000ber from\ndivergences due to Fermi surface nesting and divergent\nDOS at the Van Hove energy, which are to be cured with\nthe RG method. We consider corrections to one-loop or-\nder (Fig. 3). In the hot-spot model, each loop corrections\nis associated with the susceptibilities Eqs. (4){(7). Since\nthe Cooper-pair susceptibility \u001f0\u0000always gives the lead-\ning divergence regardless of the Fermi surface geometry,\nwe set the RG scale by y\u0011\u001f0\u0000(\u000f). In the Wilsonian\nRG procedure, we integrate out high-energy modes and\nrescale the remaining low-energy modes as we increase\nthe RG scale y, which is qualitatively similar to lowering\nthe temperature Tdown from \u0003. The other susceptibil-\nities are measured with respect to y, parameterized by\ndas(y) =d\u001fas\ndy: (8)\nBy de\fnition, d0\u0000= 1 always holds.\nA. RG equations\nThe RG equations for the coupling constants involve\nthe parameters das(y) de\fned in Eq. (8) as a function\nof the RG scale y. In general, das(y) depends on y,\nexcept when the corresponding susceptibility \u001fas(y) di-\nverges similarly to the BCS susceptibility. This occurs inthe presence of Fermi surface nesting. Then the corre-\nsponding density-wave channel has a divergent suscepti-\nbility, and hence das(y) =dasis a constant less than or\nequal to 1 [33{35].\nIn an ideal case for nesting where Fermi surface\ncomprises of corner-sharing triangles (see Fig. 7), per-\nfect nesting occurs simultaneously in three channels:\nthree susceptibilities of the intervalley type, \u001f1\u0000(Q\u0000),\n\u001f2\u0000(Q0), and\u001f3\u0000(Q+), are all logarithmically divergent\nsimilar to\u001f0\u0000, so thatd1\u0000;d2\u0000andd3\u0000are nonzero con-\nstants. In contrast, none of the intravalley susceptibili-\nties\u001fa+is divergent. Thus da+(y) decay as y\u00001(away\nfrom the Van Hove energy) or y\u00001=2(at the Van Hove\nenergy; see Appendix B), and hence become negligible at\nlargey. We neglect the subleading terms da+in the fol-\nlowing analysis. (RG equations for a generalized model\nwithda+and additional interaction terms are presented\nin Appendix F.)\nAs shown in Fig. 1, the Fermi surface of twisted bi-\nlayer graphene is nearly (but not perfectly) nested. In\nthis case, the intervalley susceptibilities \u001fa\u0000(a= 1;2;3)\nstill have a logarithmic dependence on energy from \u0003\ndown to a smaller energy \u0003 0. Equivalently, the parameter\nda\u0000(y) is approximately constant within the correspond-\ning range of the RG scale 0 \u0014y < y 0. In the following,\nwe shall analyze the RG \row within this energy range of\ninterest, where the Fermi surface is regarded as nested.\nForda+= 0, we obtain the RG equations for the nine\nmomentum-conserving coupling constants as follows:\n_g14= _g24= _g44= 0; (9)\n_g22=\u0000d3\u0000(g2\n11+g2\n22) +d1\u0000(g2\n22+g2\n32); (10)\n_g32=\u0000(g2\n31+g2\n32+ 2g31g41+ 2g32g42)\n+ 2d1\u0000g22g32;(11)\n_g42=\u0000(2g2\n31+ 2g2\n32+g2\n41+g2\n42) +d2\u0000g2\n42; (12)\n_g11=\u00002d3\u0000g11g22\n+ 2d1\u0000(g11g22\u0000g2\n11+g31g32\u0000g2\n31);(13)\n_g31=\u00002(g31g32+g31g42+g32g41)\n+ 2d1\u0000(g11g32+g22g31\u00002g11g31);(14)\n_g41=\u00002(2g31g32+g41g42) + 2d2\u0000(g41g42\u0000g2\n41):(15)\nWe use the convention _ g\u0011dg=dy .\nEquations (9){(15) show how di\u000berent coupling con-\nstants change under RG. Among them, the intervalley in-\nteractionsg22andg11involve two patches not related by\ntime-reversal symmetry, hence receive corrections solely\nfrom scattering processes related to intervalley nesting.\ng32,g42,g31, andg41involve two patches related by time-\nreversal symmetry, hence receive corrections from both\nBCS and nesting-related processes. The intravalley in-\nteractionsgi4do not \row because they do not participate\nin either process.\nDetails of the RG \row in the nine-dimensional param-\neter space are complicated and can in general be ac-\nquired numerically. (See Appendix D for the simplest5\ncase without nesting, where the analytic solution is ob-\ntained.) Nonetheless, its general feature can be under-\nstood easily: BCS corrections decrease repulsive interac-\ntions under RG, while nesting-related corrections tend to\nincrease repulsive interactions in the corresponding chan-\nnels. This important trend is a useful guideline to un-\nderstand the behavior of the RG \row, which we present\nlater.\nB. Ordering instabilities\nTo analyze various possible instabilities, we consider\nsusceptibilities associated with s- andd-wave spin-singlet\nsuperconductivity ( s-SC etc.),p- andf-wave spin-triplet\nsuperconductivity, CDW, SDW, and pair density wave\n(PDW). Both p- andd-wave pairings have two degenerate\ncomponents: ( px;py) and (dxy;dx2\u0000y2); see Appendix C\nfor detail. Three di\u000berent wave vectors, Q+,Q\u0000, and\nQ0associated with density-wave orders are indicated by\nsuperscripts +,\u0000, and0, e.g., SDW\u0000and CDW0.\nWhen only the intervalley Fermi surface nesting is con-\nsidered, the relevant instabilities are superconductivity,\nCDW/SDW at wave vectors Q\u0000andQ0, and PDW at\nQ+. An occurrence of an instability is indicated by a\ndivergence of the corresponding susceptibility. The sus-\nceptibility are obtained by an RPA-like resummation [40]\n\u001f\u0011(y) =\u001f0\n\u0011(y)\u0000\u001f0\n\u0011(y)V\u0011(y)\u001f0\n\u0011(y) +:::\n=\u001f0\n\u0011(y)\n1 +V\u0011(y)\u001f0\u0011(y); (16)\nwhere\u0011is used to label various ordering susceptibilities.\n\u001f0\n\u0011(y) is the bare susceptibility in the absence of interac-\ntion andV\u0011(y) is the e\u000bective interaction strength asso-\nciated with the ordering.\nBy a straight-forward diagrammatic calculation, we\n\fndV\u0011for various ordering channels as follows:\nVs;d-SC= 2(g42+g41\u0006g32\u0006g31) (singlet SC) ;(17)\nVp;f-SC= 2(g42\u0000g41\u0007g32\u0006g31) (triplet SC) ;(18)\nVCDW\u0000= 4(g11+g31)\u00002(g22+g32); (19)\nVCDW0= 4g41\u00002g42; (20)\nVSDW\u0000=\u00002(g22+g32); (21)\nVSDW0=\u00002g42; (22)\nVPDW+= 2(\u0000g11+g22): (23)\nDetailed description and derivation of interaction\nstrengths and susceptibilities are found in Appendix C.\nWhen the parameters dasare constant, to the leading\norder iny,\u001f\u0011(y) is written as\n\u001f\u0011(y) =dasy\n1 +V\u0011(y)dasy: (24)\nThe susceptibility diverges at 1+ V\u0011(yc)\u001f\u0011(yc) = 0, lead-\ning to an instability at yc. An instability occurs only ifthe interaction strength is attractive: V\u0011<0. In mean-\n\feld theory, the interaction strength V\u0011(y) is treated as\na constant determined by the bare values of the coupling\nconstantsgij. Then the instability temperature is given\nbyT\u0011= \u0003 exp[\u0000(cpdasjV\u0011j)\u00001=p], fory=cplnp(\u0003=\u000f)\n(cp>0) withp= 1 away from the VHS or p= 2 at\nthe VHS. Our analysis here further takes into account\nthe RG-scale dependence of coupling constants gijand\nhence the interaction strength V\u0011(y). We shall see that\nthe running coupling constants leads to results beyond\nmean-\feld theory.\nC. Intertwined superconductivity and density\nwaves\nSince the intervalley exchange interaction is likely\nsmaller than the density-density interaction, it is instruc-\ntive to \frst analyze cases only with the density-density\ninteractions. For simplicity, we set the strengths of all\ndensity-density interactions equal ( g14=g24=g44=\ng22=g32=g42>0). In the absence of the exchange in-\nteractions, some susceptibilities become degenerate as we\nsee from Eqs. (17){(23): s-SC andf-SC,d-SC andp-SC,\nand CDW and SDW at each wave vector. Such degen-\neracy results from the fact that each valley has its own\ncharge conservation and spin rotation symmetry. Similar\ndegeneracy also occurs in exciton insulators when only\nlong-range Coulomb interaction is considered [41].\nWith this choice of coupling constants, a mean-\feld\nanalysis does not \fnd any superconducting instability\nbecause the pairing interactions shown in Eqs. (17) and\n(18) are zero. In the presence of Fermi surface nesting, a\ndensity wave instability is found in mean-\feld analysis,\nwhose wave vector is Q\u0000ifd1\u0000(g22+g32)>d2\u0000g42, and\nQ0vice versa.\nOur RG analysis including the scale dependence of\nthe coupling constants gij\fnds qualitatively di\u000berent re-\nsults. Figures 4(a) and (b) are the phase diagrams from\nthe one-loop RG analysis on the ( d1\u0000;d2\u0000) plane with\nd3\u0000= 0 and the ( d1\u0000;d3\u0000) plane with d2\u0000= 0, respec-\ntively. First, in both cases, d- orp-wave superconductiv-\nity is found for small d1\u0000, which is absent in mean-\feld\ntheory. Second, the Q\u0000density-wave state is far more\ndominant than the Q0density-wave state: it already oc-\ncurs at very small nesting parameter d1\u0000, even when the\nFermi surface nesting condition is much stronger at wave\nvectorQ0.\nTo understand why the Q\u0000density wave and d/p-wave\nsuperconductivity emerge as leading instabilities, we ex-\namine the \row of intervalley density interactions g22,g32,\nandg42. Sinceg32andg42are associated with BCS scat-\ntering processes, they receive renormalization even with-\nout nesting and decrease under RG when their initial\nvalues are repulsive, as shown in Figs. 4(c), (d). In con-\ntrast, since g22is associated with a forward scattering\nprocess, it is marginal without nesting. According to\nthe RG equation Eq. (10), Fermi surface nesting in the6\n02.55.07.510.012.5>15.0\n0.0 0.1 0.2 0.3 0.4 0.50.00.10.20.30.40.5\nd1-d2-\n0.0 0.1 0.2 0.3 0.4 0.50.00.10.20.30.40.5\nd1-d3-g0cy\nCDW-/SDW-CDW-/SDW-\nd-SC/p-SCd-SC/p-SC\n0.0 0.5 1.0 1.5 2.0 2.5 3.00.10.5151050100\ng0yχη/χ0-\n0.0 0.5 1.0 1.5 2.0 2.5 3.0-0.20.00.20.40.60.8\ng0ygij/g0\nPDW+\n0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.10.5151050100\ng0yχη/χ0-\n0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-0.20.00.20.40.60.8\ng0ygij/g0p-SCf-SC\nSDW-s-SC,\nd-SC,\nCDW ,-(d) (c)(b) (a)\n(e) (f)g ,,14g24g44\ng22\ng32\ng42intravalley interactions\nintervalley density interactions\nFIG. 4: Phase diagrams with the density-density interactions. (a), (b) Phase diagrams obtained by the RG analysis in the\npresence of the density-density interactions ( g14=g24=g44=g22=g32=g42). Two parameters for nesting are varied in\neach phase diagram: (a) d1\u0000andd2\u0000(d3\u0000= 0); and (b) d1\u0000andd3\u0000(d2\u0000= 0). Colors represent critical RG scales for\ninstabilities: warm (cool) colors correspond to high (low) energy and temperature. The solid lines represent phase boundaries,\nobtained in the range of g0yc\u001415. There are CDW0/SDW0and normal regions in the vicinity of d1\u0000= 0 (not shown), but\ncritical temperatures are extremely low for those instabilities ( g0yc>15). (c), (d) Running coupling constants and (e), (f)\nsusceptibilities for possible orders. We show the results for two cases with di\u000berent strength of nesting: (c), (e) d1\u0000= 0:4 and\n(d), (f)d1\u0000=d3\u0000= 0:25. The vertical dashed lines indicate the positions where instabilities occur.\nparticle-hole channel ( d1\u0000>0) increases g22under RG.\nTherefore, in the presence of Fermi surface nesting, only\ng22grows without suppression from the BCS process and\nthus strongly enhances the Q\u0000density wave \ructuation,\nmaking it dominate over the Q0density wave.\nAlthoughg32andg42both decrease under RG, the for-\nmer decreases slower because BCS process and density\nwave nesting at wave vector Q\u0000tend to renormalize g32\nin the opposite way; see Eq. (11). Therefore, a negative\ng42\u0000g32<0 is generated and its magnitude grows un-\nder RG. As shown in Eqs. (17) and (18), this attraction\nprovides pairing interaction for both d-SC andp-SC and\nthus enhances these superconducting susceptibilities, see\nFig. 4(f). The attractive pairing interaction should bestronger than that for the Q\u0000density wave. Fermi sur-\nface nesting in the particle-particle channel ( d3\u0000>0) as-\nsists superconductivity in that it suppresses the increase\nofg22. Finite nesting in the particle-particle channel\nyields nonzero PDW susceptibility, but the interaction\nis repulsive for the PDW+\ructuation; see Eq. (23).\nWe conclude that in the presence of repulsive interval-\nley density interactions, the two leading instabilities are\ncharge/spin density wave at wave vector Q\u0000andp/d-\nwave superconductivity. When the Fermi surface nesting\nin the particle-hole channel is strong (weak), the density\nwave state (superconductivity) is favored.7\n(b)\n(e)(a)\n(d) (f)(c)\ns-SC\nd-SC\np-SC\nf-SC\nCDW-\nSDW-\ng ,,14g24g44\ng22\ng32\ng42\ng11\ng31\ng41intravalley interactions\nintervalley exchange interactionsintervalley density interactions\n0 1 2 3 4 5 6 70.10.5151050100\ng0yχη/χ0-\n0 1 2 3 4 5 6 7-0.20.00.20.40.60.8\ng0ygij/g0\n0.0 0.5 1.0 1.5 2.00.10.5151050100\ng0yχη/χ0-\n0.0 0.5 1.0 1.5 2.0-0.20.00.20.40.60.8\ng0ygij/g0\n0.0 0.5 1.0 1.50.10.5151050100\ng0yχη/χ0-\n0.0 0.5 1.0 1.5-0.20.00.20.40.60.8\ng0ygij/g0\nFIG. 5: E\u000bect of the exchange interactions. (a){(c) Running coupling constants and (d){(f) susceptibilities for possible\norderings. We choose the strengths of density-density interactions g14=g24=g44= 1 (intravalley) and g22=g32=g42= 1\n(intervalley). Fermi surface nesting is weak in (a) and (d) with d1\u0000= 0:1; and strong in (b), (c), (e), and (f) with d1\u0000= 0:4.\nFinite exchange interactions lift the degeneracies of susceptibilities; cf. Figs. 4(e), (f). We choose the exchange interactions to\nbe initially repulsive ( g11=g31=g41= 0:1) in (a), (b), (d), and (e); and attractive ( g11=g31=g41=\u00000:1) in (c) and (f).\nThe vertical dashed lines indicate the positions where instabilities occur.\nIV. ROLE OF INTERVALLEY EXCHANGE\nINTERACTION\nThe degeneracies of d/p-wave superconductivity and\nof CDW/SDW susceptibilities are lifted when intervalley\nexchange interactions g11,g31,g41are included. Their\nbare values depend on microscopic details as we have\ndiscussed in Sec. II A. For example, such interactions\ncan arise from intervalley scattering mediated by opti-\ncal phonons. Since the typical phonon frequency is much\nlarger than the mini-band width, intervalley exchange\ninteractions between low-energy electrons may be even\nattractive.\nFigure 5 shows the RG \rows including both density-\ndensity and exchange interactions. With a small d1\u0000\n[Figs. 5(a), (d)], the superconducting instabilities are\ndominant but a larger d1\u0000favors density-wave states\n[Figs. 5(b), (c), (e), (f)]. Since their initial values are\nchosen to be small, the change of exchange interactions\nunder RG is considerably smaller than that of the density\ninteractions. Nonetheless, degeneracies of susceptibilities\nare lifted by \fnite exchange interactions. We show cases\nfor repulsive interaction in Figs. 5(a), (b), (d), (e) and\nfor attractive interaction in Figs. 5(c), (f).\nRoughly speaking, repulsive interaction prefers d-SC\ntop-SC and SDW\u0000to CDW\u0000, and attractive interaction\nprefers the converse. This can be seen from our expres-\nsions for interaction strengths shown in Eq. (17), etc. De-\npending on the choice of intervalley exchange interactions\nand nesting parameter d1\u0000, any of the four orders| d-SC,\np-SC, CDW\u0000, and SDW\u0000|can be the leading instabil-\nity.While the bare values of intervalley exchange inter-\nactions are hard to obtain accurately, we now discuss\nanother important factor in selecting between d- andp-\nwave SC, and between CDW\u0000and SDW\u0000. Both SDW\nandp-wave SC (which is spin-triplet) breaks the SU(2)\nspin rotational symmetry, while the CDW and d-wave\nSC (which is spin-singlet) do not. With SU(2) symme-\ntry, it is known that thermal \ructuations associated with\nGoldstone modes prevent any true long-range spin order\nin two dimensions. This argument suggests that d-SC\nor CDW\u0000can still be realized at nonzero temperature,\neven when the leading susceptibility above the ordering\ntemperature is p-SC or SDW\u0000.\nV. ELECTRONIC STRUCTURE OF\nDENSITY-WAVE STATES\nWe now examine the electronic structure in a CDW\u0000\nstate and show that at \flling the n= 2, the CDW\u0000\nstate can be insulating. The same conclusion applies to a\ncollinear SDW\u0000because it can be mapped to the CDW\u0000\nstate by performing a sign change on electrons of one spin\npolarization in one valley.\nIn density-wave states, the Brillouin zone in momen-\ntum space is reduced because of the enlarged unit cell\nin real space. The previously distinct momentum eigen-\nstates now can hybridize, resulting in a band structure\nreconstruction. When the number of electrons in the en-\nlarged unit cell is an even integer, the resulting CDW\nstate can be a band insulator.\nThe CDW\u0000order can occur at three equivalent wave8\nKr Mr Kr' Г(b)\nM Q1Q2\nQ3K\nKr\nK’Kr’MrГ(a)\n-1-0.500.5  E/D\nFIG. 6: (a) Reduced Brillouin zone in a triple- Qdensity-wave\nstate. We assume the three ordering vectors Qj(j= 1;2;3),\nwhich are parallel to the \u0000{ Mlines and satisfyjQjj=Q\u0000=\nG=4. (b) Energy spectrum in the CDW\u0000or collinear SDW\u0000\nstate with order parameter \u0001 = 0 :16D.Dis the original\nconduction bandwidth. There are 32 bands in the reduced\nBrillouin zone and the 17th band from the bottom is colored\nin red. Filling of 16 bands corresponds to the \flling n= 2.\nvectors related by the C3rotational symmetry:\nQ1=Q\u0000n0;Q2=Q\u0000n2\u0019=3;Q3=Q\u0000n\u00002\u0019=3:\n(25)\nwithn\u001e= (cos\u001e;sin\u001e). Below we shall consider a triple-\nQCDW state, where the above three wave vectors form\nthe new reciprocal vectors, and hence de\fne the reduced\nBrillouin zone. Compared to a single- Qstate, the triple-\nQstate is expected to be energetically favorable as it\ngaps more parts of the Fermi surface especially around\nthe hot spots with large DOS.\nIn the CDW\u0000state, intervalley order parameter\nh y\nk+Qi;\u001c\u001b k\u0016\u001c\u001bi(\u0016\u001c=\u0000\u001c) becomes nonzero. The mean-\n\feld Hamiltonian in the CDW\u0000state thus includes the\nCDW potential in addition to the original electron dis-\npersion:\nHCDW =X\nk\u001c\u0014\n\u000f\u001c\nk y\nk\u001c k\u001c+ \u00013X\nj=1( y\nk+Qj;\u001c k\u0016\u001c+ H:c:)\u0015\n:\n(26)\nSince the spin structure is irrelevant, we have dropped\nthe spin index \u001b. Here we assume that the CDW order\nparameters at Q1,Q2,Q3are equal, so that the resulting\nstate is invariant under the three-fold rotation.\nFor the original Fermi surface shown in Fig. 1, the\nCDW\u0000wave vector connecting a pair of hot spots is close\nto the commensurate vector Q\u0000'j\u0000Mj=2 =G=4, where\nGis the length of the original reciprocal lattice vectors\nof twisted bilayer graphene. (The analytic expression of\nthe energy dispersion in the normal state \u000f\u001c\nkis given in\nAppendix E.) With this choice of CDW wave vector Q\u0000,\nthe reduced Brillouin zone is 4 \u00024 smaller than the orig-\ninal Brillouin zone and can be constructed as shown in\nFig. 6(a). Since there are two conduction bands (one per\nvalley) in the original Brillouin zone, there are 32 bands\nin the reduced Brillouin zone. A complete \flling of 16bands corresponds to the \flling of n= 2, where corre-\nlated insulating behavior was experimentally observed.\nWhen the CDW\u0000order parameter is small, the Fermi\nsurface at the \flling n= 2 is not fully gapped due to\nimperfect nesting. A full gap appears for \u0001 &\u0001c. For\na realistic Fermi surface with good nesting condition, we\n\fnd the critical value of the order parameter \u0001 c= 0:15D,\nwhereDis the bandwidth of the original conduction\nband. The fact \u0001 c\u001cDjusti\fes our weak coupling ap-\nproach.\nThe gapped energy spectrum in the CDW\u0000state with\n\u0001 = 0:16Dis presented in Fig. 6(b). Importantly, we\nnote that the direct gap in the CDW state is located at\n\u0000 in the reduced Brillouin zone. A single electron pocket\n(with two-fold spin degeneracy) is present above the gap,\nwhile two nearly degenerate hole pockets are present be-\nlow the gap. The hole pockets have much heavier mass\nthan the electron. These features are consistent with\nquantum oscillation measurements at densities slightly\naway fromn= 2, as we shall discuss in the next section.\nFor the commensurate CDW state with Q\u0000=G=4\nconsidered here, the scattering process labeled by g43car-\nries momentum 2 Q0= 4Q\u0000=G, and thus it is allowed.\nThis process corresponds to the intervalley exchange in-\nteraction and it is presumably smaller than intravalley\nand intervalley density interactions. We con\frm that in-\nclusion of small g43does not alter the RG \row much, and\nwe obtain qualitatively the same result [42].\nVI. DISCUSSIONS\nIn this section, we compare our results with the exper-\niments on twisted bilayer graphene [1]. We have found\nfrom RG analysis the intertwining of unconventional su-\nperconductivity and density-wave instabilities. We have\nobtained from band structure calculations the gapped\nspectrum of density-wave states at the \flling n= 2.\nOn the experiment side, the resistivity measurement at\nzero magnetic \feld near n= 2 observes a metallic behav-\nior at high temperatures, then an upturn of resistivity in\nan intermediate temperature region, before superconduc-\ntivity appears at the lowest temperature. Furthermore,\nthe in-plane upper critical \feld of the superconducting\nstate is found to be comparable to the Pauli limit, indi-\ncating spin-singlet pairing. The change from insulating\nto superconducting behaviors is consistent with the inter-\ntwined density wave and SC instabilities, shown by the\nevolution of susceptibility with decreasing energy scale\nin Figs. 4(e), (f) and also Figs. 5(e){(f). Finally, when\nsuperconductivity is destroyed by the magnetic \feld, re-\nsistivity becomes insulating at the lowest temperature.\nWe interpret this T= 0 insulating state as a\nCDW/SDW state at wave vector Q\u0000. We have analyzed\na triple-Q\u0000CDW/collinear SDW phase with 4 \u00024 pe-\nriodicity and have shown that a moderate density-wave\norder parameter fully gaps the energy spectrum at the\n\fllingn= 2, consistent with the insulating behavior of9\nresistivity at low temperature. Importantly, at densities\nslightly above n= 2 (or doping towards complete \flling\nof mini-bands), a single pocket with two-fold degeneracy\nis found in quantum oscillation measurements. This is\nconsistent with our \fnding of a single pocket with spin\ndegeneracy above the gap. On the other hand, at den-\nsities slightly below n= 2, quantum oscillations have so\nfar not been observed. This is consistent with the fact\nthat the pockets below the gap in our density-wave state\nhave heavy mass.\nAcknowledgments\nWe thank Pablo Jarillo-Herrero, Yuan Cao, Valla\nFatemi, Oskar Vafek, Leonid Levitov, Sankar Das Sarma,\nAllan MacDonald, Matthew Yankowitz, Cory Dean and\nespecially Eva Andrei for helpful discussions. This work\nis supported by the DOE O\u000ece of Basic Energy Sci-\nences, Division of Materials Sciences and Engineering un-\nder award de-sc0010526. LF is partly supported by the\nDavid and Lucile Packard Foundation.\nAppendix A: Fermi surface nesting\n1. Perfect and near nesting\nFermi surface nesting provides singularities in suscep-\ntibilities. The susceptibilities (Lindhard functions) in the\nparticle-hole and particle-particle channels are given by\n\u001fph\n\u001c\u001c0(q;!) =Z\nkf(\u000f\u001c\nk+q)\u0000f(\u000f\u001c0\nk)\n!\u0000\u000f\u001c\nk+q+\u000f\u001c0\nk; (A1)\n\u001fpp\n\u001c\u001c0(q;!) =Z\nkf(\u000f\u001c\nk+q)\u0000f(\u0000\u000f\u001c0\n\u0000k)\n!\u0000\u000f\u001c\nk+q\u0000\u000f\u001c0\n\u0000k: (A2)\n\u001cand\u001c0denote Fermi surfaces, which correspond to the\nvalley degrees of freedom for the case of twisted bilayer\ngraphene.\nThe conditions for perfect nesting is given by\n\u000f\u001c\nk+q=\u0000\u000f\u001c0\nk (particle-hole channel) ; (A3)\n\u000f\u001c\nk+q=\u000f\u001c0\n\u0000k (particle-particle channel) : (A4)\nWhen the Fermi surfaces are perfectly nested, i.e., one of\nthe above conditions holds in a certain area of the Bril-\nlouin zone, singularities in the susceptibilities are found\nin the static limit !!0. One \fnds a logarithmic di-\nvergence in a susceptibility with perfectly-nested Fermi\nsurfaces in two dimensions, so that the susceptibility has\nln(\u0003=!) dependence.\nIn order to observe logarithmic dependence in suscep-\ntibilities at !, the nesting condition should hold at the\nenergy scale determined by !. In other words, if the\nconditions are approximately met with an accuracy of\n(a) (b)\n32\n13\n21FIG. 7: Simpli\fed Fermi surface to show Fermi surface nest-\ning with di\u000berent wave vectors. (a) and (b) are Fermi surfaces\nfor di\u000berent valley degrees of freedom. Labels 1 ;2;3 are the\npatch indices, assigned at the hot spots.\naround!, we see ln(\u0003 =!) behavior at the energy scale !.\nIt allows to relax the nesting conditions at !to be\n\u000eph(!)\u0011\f\f\f\f\f\u000f\u001c\nk+q+\u000f\u001c0\nk\n!\f\f\f\f\f\u001c1; (A5)\n\u000epp(!)\u0011\f\f\f\f\f\u000f\u001c\nk+q\u0000\u000f\u001c0\n\u0000k\n!\f\f\f\f\f\u001c1; (A6)\nfor the particle-hole and particle-particle channels, re-\nspectively. When Fermi surfaces are perfectly nested, we\nhave\u000eph= 0 or\u000epp= 0, and ln(\u0003 =!) dependence in the\nsusceptibility holds down to the lowest energies. On the\nother hand, when Fermi surfaces are nearly nested with\n\u000eph/pp(!)\u001c1 for \u0003 0< !\u0014\u0003, we see a logarithmic\nenhancement with in the range \u0003 0<!\u0014\u0003, and it is cut\no\u000b by the lower bound \u0003 0.\nOur RG analysis focus on the energy range \u0003 0<!\u0014\n\u0003, where Fermi surfaces are regarded as nearly nested,\nand hence assume the parameters dasare constant within\nthe range. For energy scale below \u0003 0,dascan no longer\nbe regarded as constant, and we need to consider the y\ndependence of das.\n2. Inner and outer Fermi surfaces\nFigure 7 shows simpli\fed Fermi surfaces at the Van\nHove energy, where the Fermi surfaces are approximated\nas corner-sharing triangles, to emphasize Fermi surface\nnesting. For convenience, we regard those Fermi surfaces\nas large hole Fermi surfaces encircling the \u0000 point. At a\nsaddle point located along the \u0000{ Mline, a Fermi surface\ntouches another from an adjacent Brillouin zone. There-\nfore, there are two Fermi surfaces in one patch around\na hot spot when we consider the reduced Brillouin zone.\nFor convenience, we call the Fermi surface from the \frst\nBrillouin zone as the \\inner\" Fermi surface and the other\nfrom the second Brillouin zone as the \\outer\" Fermi sur-\nface.\nFrom the \fgure, we \fnd four distinct Fermi surface\nnestings. First, the Fermi surfaces of patches 1 and 10\n(1 and 10are from di\u000berent valleys) are perfectly nested10\nin the particle-particle channel, yielding the BCS insta-\nbility. It always holds for time-reversal-invariant sys-\ntems. There are also symmetry-related pairs: 2{20and\n3{30. Second, the inner Fermi surfaces are nested be-\ntween 1 and 10, leading to the logarithmic dependence\nin\u001fph(Q0;!) and constant d2\u0000. In addition, by look-\ning at both inner and outer Fermi surfaces, we can \fnd\nthat the Fermi surfaces from 1 and 20are nested both in\nthe particle-hole and particle-particle channels. It gives\nlogarithmic dependences in \u001fph(Q\u0000;!) and\u001fpp(Q+;!),\nwhich results in constant d1\u0000andd3\u0000, respectively.\nThe discussion can be done in parallel near the Van\nHove energy. When the Fermi energy is slightly above\nthe Van Hove energy, we can regard the Fermi surfaces as\nhole pockets around the \u0000 point similarly as above. If we\nconsider the \flling slightly below the Van Hove energy, we\nshould instead look at electron pockets, which surround\ntheKandK0points; cf. Fig. 1(c). In this case, a hot\nspot involves two Fermi surfaces from the two electron\npockets. Still, a hot spot contains two Fermi surfaces,\nand Fermi surface nesting for both of them should be\nconsidered.\nAppendix B: Susceptibilities near the Van Hove\nenergy\nThe DOS logarithmically diverges at a saddle point\nin two dimensions because of the Van Hove singularity.\nWhen we approximate the energy dispersion near a sad-\ndle point as \u000f(\u000ek) =A\u000ek2\nx\u0000B\u000ek2\ny, we obtain the DOS\n(in the vicinity of the saddle point) as\n\u001a0(\u000f) =1\n2\u00192p\nABln\u0003\n\u000f; (B1)\nwhere \u0003 is the high-energy cuto\u000b, which corresponds to\nthe patch size in the present case.\nWe have de\fned the eight susceptibilities \u001fasin the\nparticle-hole and particle-particle channels with four dif-\nferent wave vectors in Eqs. (4){(7). Among the eight\nsusceptibilities, \u001f2+corresponds to the DOS; \u001f2+(!) =\n\u001a0(!). The divergence in the limit !!0 is relaxed by\n\fnite chemical potential or away from the Van Hove en-\nergy:\n\u001f2+(!) =1\n2\u00192p\nABln\u0003\nmax(!;\u0016): (B2)\nThe susceptibility in the BCS channel \u001f0\u0000su\u000bers from\nthe Cooper instability in the presence of time-reversal\nsymmetry. An alternative way to state this is that the\nFermi surfaces are perfectly nested in this channel. We\nnote that\u001f0\u0000is an intervalley susceptibility since the\nsaddle points at k\u000b\u001cand\u0000k\u000b\u001c(=k\u000b;\u001c06=\u001c) belong to dif-\nferent valleys. Calculating \u001f0\u0000(!), we obtain\n\u001f0\u0000(!) =\u001a0(!)\n4ln\u0003\n!=1\n8\u00192p\nABln\u0003\nmax(!;\u0016)ln\u0003\n!:\n(B3)For\u0016= 0 (at the Van Hove energy), \u001f0\u0000has a double\nlog singularity. It is the leading divergence in the eight\nsusceptibilities and it sets the RG scale y. It is important\nto note that the two logarithms have di\u000berent origins:\none logarithmic singularity [ln(\u0003 =maxf!;\u0016g)] is from the\nVan Hove singularity and the other [ln(\u0003 =!)] from the\nFermi surface nesting. The logarithmic divergence from\nFermi surface nesting has already been discussed. The\ndivergence of the DOS is suppressed away from the Van\nHove energy. However, if \u0016\u001c\u0003 is satis\fed, there still\nare large spectral weights in the patches around the hot\nspots, which justi\fes the use of patch RG.\nWhen Fermi surfaces are nested for a susceptibility in\na certain channel and a wave vector (say, \u001fas), a loga-\nrithmic dependence from Fermi surface nesting is present.\n(One can \fnd an explicit calculation for the case of mono-\nlayer graphene in e.g. Ref. [43].) Then, \u001fashas the\nsame singularity as \u001f0\u0000and the corresponding param-\neterdas(y) =d\u001fas=dybecomes constant as we de\fne the\nRG scale by y\u0011\u001fpp(0;\u000f). In contrast, Fermi surface\nnesting is absent for \u001fas, the corresponding parameter\ndas(y) decays as y\u00001=2fory\u001d1 [33, 35].\nAppendix C: Susceptibilities for ordering\ninstabilities\n1. Interaction strengths for instabilities\nNow we consider an interaction strength that may trig-\nger an ordering instability. We write two-particle inter-\nactions in the particle-particle and particle-hole channels\nas\n^Vph=Vph y\n1 2 y\n3 4; (C1)\n^Vpp=Vpp y\n1 y\n2 3 4: (C2)\nOne can perform a mean-\feld analysis by considering\na mean \feld composed of the \frst or last two fermion\noperators on the right-hand sides. Those two types of\ninteractions are diagrammatically depicted in Fig. 8(a).\nIn the present model, there are 16 distinct scattering\nprocesses, and to the lowest order, interaction strengths\nfor ordering instabilities are written as linear combina-\ntions of those coupling constants [Fig. 8(b)]. Various in-\nteraction strengths V\u0011corresponding to ordering \u0011with\nall 16 coupling constants are given as follows:\nVs-SC= 2(g42+g41+g32+g31); (C3)\nVp-SC= 2(g42\u0000g41\u0000g32+g31); (C4)\nVd-SC= 2(g42+g41\u0000g32\u0000g31); (C5)\nVf-SC= 2(g42\u0000g41+g32\u0000g31); (C6)\nVCDW+=4(g12+g14+g32+g34)\n\u00002(g21+g24+g31+g34);(C7)\nVCDW\u0000=4(g11+g13+g31+g33)\n\u00002(g22+g23+g32+g33);(C8)11\n(a)\n(c)(b)\nSC, PDW\nCDW SDW\nFIG. 8: Diagrammatic representation of interactions and summation for the susceptibility. (a) Diagrams for interactions in\nthe particle-particle channel Vppand in the particle-hole channel Vph. The two-particle interactions are drawn by four-leg\nladders. (b) Interactions for various orderings to the lowest order. (c) RPA-like resummation for susceptibilities.\nΔ Δ\nΔ ΔΔ\nΔΔ\nΔ Δ\nΔ ΔΔ\nΔ ΔΔ Δ\n-Δ-Δ\n-Δ-Δ\n-Δ -Δ\n-Δ -ΔΔ/2\nΔ/2-Δ/2\n-Δ/2\n-Δ/2-Δ/2\n-Δ/2-Δ/2\n0s fx(x-3y 2) dxy px\ndx2-y2 py0 0\n0\nFIG. 9: SC pairing symmetries and order parameters at hot spots. Note that the order parameters are not normalized for\neach pairing.\nVCDW0=2(2g41\u0000g42+g43)\n\u00002(g12+g13) + 4(g21+g23);(C9)\nVSDW+=\u00002(g21+g24+g31+g34); (C10)\nVSDW\u0000=\u00002(g22+g23+g32+g33); (C11)\nVSDW0=\u00002(g12+g13+g42+g43); (C12)\nVPDW+= 2(\u0000g11\u0000g12+g21+g22); (C13)\nVPDW\u0000= 2(\u0000g13\u0000g14+g23+g24); (C14)\nVPDW0= 2(g33+g34+g43+g44); (C15)\nVc=\u0000(g11+g14) + 2(g22+g24)\u0000g41+ 2g42+g44;\n(C16)\nVs=\u0000(g11+g14+g41+g44): (C17)\nIn addition to instabilities for superconductivity and den-\nsity waves, we also write down the interaction strengths\nfor the charge compressibility Vcand the uniform spin\nsusceptibility Vs.\nAs for superconductivity, we can consider six di\u000berent\npairing symmetries because of the six patches. The six\npairing symmetries are distinct in regard to the super-\nconducting order parameters \u0001 at the patches (Fig. 9).Eachp- andd-wave pairing has two di\u000berent pairings,\nbut those two give the same interaction strength V\u0011. We\nassume that the system respects the point group D3,\nwhich has the representations A1,A2, andE. From\nthis viewpoint, s- andf-wave pairings belong to the one-\ndimensional representations A1andA2, respectively, and\neachp- andd-wave pairing belong to the two-dimensional\nrepresentation E. For the other interaction strengths, we\nassume the A1representation.\n2. Derivation of susceptibilities for ordering\ninstabilities\nWe calculate the susceptibilities for ordering \u0011 \u001f\u0011by\nsumming up RPA-like diagrams, shown in Fig. 8(c); see\nalso Ref. [40]. It yields the susceptibility \u001f\u0011in the form\nof\n\u001f\u0011=\u001f0\n\u0011\u0000\u001f0\n\u0011V\u0011\u001f0\n\u0011+\u001f0\n\u0011V\u0011\u001f0\n\u0011V\u0011\u001f0\n\u0011\u0000:::=\u001f0\n\u0011\n1 +V\u0011\u001f0\u0011;\n(C18)12\n-1.0 -0.5 0.0 0.5 1.0-1.0-0.50.00.51.0\ng4+g3+\nFIG. 10: RG \rows on the ( g4+;g3+) plane without Fermi\nsurface nesting other than the BCS channel. Black solid lines\nare the separatrix lines of the \rows. One \fnds the same RG\n\row forg4\u0000andg3\u0000.\nwhere\u001f0\n\u0011is the susceptibility calculated without inter-\naction.\u001f0\n\u0011are equal to \u001faswithfasgdepending on\n\u0011: 0+ (PDW0), 0\u0000(all SC), 1+ (CDW+, SDW+), 1\u0000\n(CDW\u0000, SDW\u0000), 2\u0000(CDW0, SDW0), 3+ (PDW\u0000), and\n3\u0000(PDW+).\nAppendix D: Analysis of the RG equations without\nnesting\nThe RG equations (9){(15) involve many variables and\nparameters, and a full analysis of all nine running cou-\npling constants is rather complicated. Here we analyze\nthe simplest case where there is no Fermi surface nesting\nother than the BCS channel ( das= 0, except for d1\u0000= 1\nby de\fnition). Then, only g31,g32,g41, andg42receive\none-loop corrections since they can be regarded as BCS\ninteractions. The RG equations for the four coupling\nconstants are written as\ndg3\u0006\ndy=\u0000g2\n3\u0006\u00002g3\u0006g4\u0006; (D1)\ndg4\u0006\ndy=\u00002g2\n3\u0006\u0000g2\n4\u0006; (D2)\nwhere we de\fne\nga\u0006=ga2\u0006ga1(a= 3;4): (D3)\nWe note that ga+andga\u0000characterize spin-singlet and\nspin-triplet SC, respectively. The RG \row for g4+and\ng3+is shown in Fig. 10. The coupled RG equations can\nbe rewritten as\nd(g4\u0006+ 2g3\u0006)\ndy=\u0000(g4\u0006+ 2g3\u0006)2; (D4)\nd(g4\u0006\u0000g3\u0006)\ndy=\u0000(g4\u0006\u0000g3\u0006)2; (D5)\nand thus are readily solved.\n02468\nE/t1(a) (b)\nK\nГ M\nK’K\nГ M\nK’FIG. 11: Energy contour plots of the energy bands \u000f\u001c\nk,\nEq. (E3). (a) and (b) correspond to the energy dispersions\nof the two di\u000berent valleys. The parameters are chosen as\nt0= 5:4,t1= 1,t2=\u00001,t0\n2= 0:3, andt3= 0:6 The Lifshitz\ntransition and accompanying VHS are found at E=t 1\u00196:1.\nThe bandwidth is D= 7:2.\nWhen we neglect the exchange interactions, g4\u0006and\ng3\u0006are reduced to be g42andg32, respectively. If we fur-\nther assume that those two intervalley density-density in-\nteractions have the same amplitude g42=g32, the equal-\nity holds under the RG \row from Eq. (D5). Each cou-\npling constant obeys the RG equation _ g=\u00003g2, which is\nin accordance with Shankar's result for BCS interactions\nwithout nesting [39].\nAppendix E: Energy dispersion of a normal state\nWe construct an analytic form of the energy dispersion\nwith two valley degrees of freedom, belonging to the point\ngroupD3and satisfying a \flling condition. For the latter,\nwe require that the energy dispersion have VHS points\nat the \flling n= 2. The symmetry conditions are given\nas follows: The C3rotational symmetry is kept within\na valley, and hence the energy dispersion satis\fes the\nrelation\n\u000f\u001c\nC3k=\u000f\u001c\nk: (E1)\nThere also exist in-plane C2rotations about the \u0000{ K\nlines, where one of them is taken along the kyaxis. The\nin-planeC2rotation interchanges the two valley and \rips\nkx, thus requiring\n\u000f\u001c\nkx;ky=\u000f\u0016\u001c\n\u0000kx;ky: (E2)\nA function that obeys the symmetry conditions\nEqs. (E1) and (E2) can be represented as a Fourier series\nbecause of the underlying lattice periodicity. Up to the\nthird order, the symmetry-allowed terms are given by\n\u000f\u001c\nk=2X\ni=0\u0002\nt0+t1cos(ki\ny) +t2cos(p\n3ki\nx) +\u001ct0\n2sin(p\n3ki\nx)\n+t3cos(2ki\ny)\u0003\n; (E3)\nwhere we de\fne ki\ny=Ci\n3kyandki\nx=Ci\n3kx. The param-\neterst1,t2,t0\n2, andt3are real. In addition to the D313\ng12\ng13 g21g23g33\ng34g43\nFIG. 12: Seven momentum-nonconserving scattering pro-\ncesses.\nsymmetry, we further impose the \flling condition, which\nis ful\flled by tuning the parameters t1,t2,t0\n2, andt3. By\nchoosing e.g. t1= 1,t2=\u00001,t0\n2= 0:3, andt3= 0:6,\nwe \fnd saddle points of the energy dispersion, i.e., VHS\npoints, at the \flling n= 2, as shown in Fig. 11. The en-\nergy bands extend in the range of 0 \u0014E=t1\u0014D= 7:2.\nAppendix F: Generalized model and RG equations\n1. Generalized model\nWe generalize the model for twisted bilayer graphene,\nconsisting of two valleys with SU(2) spin, to a ( Nv\u0002Ns)-\nband model, consisting of Nvvalleys with SU( Ns) spins.\nWe assume that the valley degrees of freedom are non-\ndegenerate in the kinetic part, and as a result, there are\nNvseparate bands each with Ns-fold degeneracy from\nspin; i.e., the kinetic component has U(1) \u0002SU(Ns) sym-metry. Each Fermi surface is assumed to possess np\nhot spots. Such a model is applicable for example to\ncuprates, monolayer graphene, and graphene superlat-\ntices. Our results for graphene superlattices are obtained\nby setting Nv=Ns= 2 andnp= 3. A model for\ncuprates corresponds to Nc= 1,Ns= 2, andnp= 2 [30{\n34], and one for monolayer graphene to Nc= 1,Ns= 2,\nandnp= 3 [35]. All those cases are in two dimensions; a\ngeneralization to other dimensions is straightforward.\nThe orbital index takes \u001c= 1;:::;Nv, the spin index\n\u001b= 1;:::;Ns, and the saddle point index \u000b= 1;:::;np.\nRede\fning the range of the indices, we use the same inter-\naction as Eq. (1). As long as we have the same set of cou-\npling constants, the structure of the RG equations does\nnot depend either on the position of the saddle points in\nthe Brillouin zone or the dimensionality of the system,\nwhich merely changes the parameters das. Note that the\ndimensionality constrains geometrically possible Nvand\nnsif we consider the same set of coupling constants. Also,\nnote that umklapp processes in the valley space ( j= 3)\nexplicitly violate the valley U(1) symmetry.\n2. RG equations for the coupling constants\nFor the generalized ( Nv\u0002Ns)-band model where each\nFermi surface has npsaddle points, we obtain the RG\nequations for the sixteen coupling constants gijto one-\nloop level (Fig. 3):\ndg11\ndy=\u0000Nvd3\u0000(g11g22+g12g21) +Nvd2+(g11g44+g14g41)\n+ 2d1\u0000[g11g22+g31g32+ (Nv\u00001)(g13g23+g2\n33)]\u0000Nsd1\u0000[g2\n13+g2\n33+ (Nv\u00001)(g2\n11+g2\n31)];(F1)\ndg12\ndy=\u00002d3\u0000[g12g22+ (Nv\u00001)g11g21] + 2d2\u0000[g12g42+ (Nv\u00001)g13g43]\n+ 2d1+[g12g24+g32g34+ (Nv\u00001)(g14g21+g31g34)]\u0000NvNsd1+(g12g14+g32g34);(F2)\ndg13\ndy=\u0000Nvd3+(g13g24+g14g23) +Nvd2\u0000(g12g43+g13g42)\n+ 2d1\u0000[g13g22+g32g33+ (Nv\u00001)(g11g23+g31g33)]\u0000NvNsd1\u0000(g11g13+g31g33);(F3)\ndg14\ndy=\u00002d3+[g14g24+ (Nv\u00001)g13g23] + 2d2+[g14g44+ (Nv\u00001)g11g41]\n+ 2d1+[g14g24+g2\n34+ (Nv\u00001)(g12g21+g31g32)]\u0000Nsd1+[g2\n14+g2\n34+ (Nv\u00001)(g2\n12+g2\n32)];(F4)\ndg21\ndy=\u0000Nvd3\u0000(g11g12+g21g22) +Nvd1+(g21g24+g31g34) +d2\u0000[g12g41+g21g42+ (Nv\u00001)(g13g43+g23g43)]\n\u00002Nsd2\u0000[g23g43+ (Nv\u00001)g21g41];\n(F5)\ndg22\ndy=\u0000d3\u0000[g2\n12+g2\n22+ (Nv\u00001)(g2\n11+g2\n21)] +d1\u0000[g2\n22+g2\n32+ (Nv\u00001)(g2\n23+g2\n33)]\n+ 2d2+[g14g42+g22g44+ (Nv\u00001)(g11g44+g24g41)]\u0000NvNsd2+(g22g44+g24g42);(F6)14\ndg23\ndy=\u0000Nvd3+(g13g14+g23g24) +Nvd1\u0000(g22g23+g32g33)\n+ 2d2\u0000[g12g43+g23g42+ (Nv\u00001)(g13g41+g21g43)]\u0000NvNsd2\u0000(g21g43+g23g41);(F7)\ndg24\ndy=\u0000d3+[g2\n14+g2\n24+ (Nv\u00001)(g2\n13+g2\n23)] +d1+[g2\n24+g2\n34+ (Nv\u00001)(g2\n21+g2\n31)]\n+ 2d2+[g14g44+g24g44+ (Nv\u00001)(g11g42+g22g41)]\u00002Nsd2+[g24g44+ (Nv\u00001)g22g42];(F8)\ndg31\ndy=\u0000Nv(g31g42+g32g41)\u0000Nv(np\u00002)g31g32+Nvd1+(g21g34+g24g31)\n+ 2d1\u0000[g11g32+g22g31+ (Nv\u00001)(g13g33+g23g33)]\u00002Nsd1\u0000[g13g33+ (Nv\u00001)g11g31];(F9)\ndg32\ndy=\u00002[g32g42+ (Nv\u00001)g31g41]\u0000(np\u00002)[g2\n32+ (Nv\u00001)g2\n31] + 2d1\u0000[g22g32+ (Nv\u00001)g23g33]\n+ 2d1+[g12g34+g24g32+ (Nv\u00001)(g14g31+g21g34)]\u0000NvNsd1+(g12g34+g14g32);(F10)\ndg33\ndy=\u0000Nvd0+(g33g44+g34g43)\u0000Nv(np\u00002)d0+g33g34+Nvd1\u0000(g22g33+g23g32)\n+ 2d1\u0000[g13g32+g22g33+ (Nv\u00001)(g11g33+g23g31)]\u0000NvNsd1\u0000(g11g33+g13g31);(F11)\ndg34\ndy=\u00002d0+[g34g44+ (Nv\u00001)g33g43]\u0000(np\u00002)d0+[g2\n34+ (Nv\u00001)g2\n33] + 2d1+[g24g34+ (Nv\u00001)g21g31]\n+ 2d1+[g14g34+g24g34+ (Nv\u00001)(g12g31+g21g32)]\u00002Nsd1+[g14g34+ (Nv\u00001)g12g32];(F12)\ndg41\ndy=\u0000Nvg41g42\u0000Nv(np\u00001)g31g32+Nvd2+g41g44+Nv(np\u00001)d2+g11g14\n+ 2d2\u0000[g41g42+ (Nv\u00001)g2\n43] + 2(np\u00001)d2\u0000[g12g21+ (Nv\u00001)g13g23]\n\u0000Nsd2\u0000[g2\n43+ (Nv\u00001)g2\n41]\u0000Ns(np\u00001)d2\u0000[g2\n23+ (Nv\u00001)g2\n21];(F13)\ndg42\ndy=\u0000[g2\n42+ (Nv\u00001)g2\n41]\u0000(np\u00001)[g2\n32+ (Nv\u00001)g2\n31]\n+d2\u0000[g2\n42+ (Nv\u00001)g2\n43] + (np\u00001)d2\u0000[g2\n12+ (Nv\u00001)g2\n13]\n+ 2d2+[g42g44+ (Nv\u00001)g41g44] + 2(np\u00001)d2+[g14g22+ (Nv\u00001)g11g24]\n\u0000NvNsd2+g42g44\u0000NvNs(np\u00001)d2+g22g24;(F14)\ndg43\ndy=\u0000Nvd0+g43g44\u0000Nv(np\u00001)d0+g33g34+Nvd2\u0000g42g43+Nv(np\u00001)d2\u0000g12g13\n+ 2d2\u0000[g42g43+ (Nv\u00001)g41g43] + 2(np\u00001)d2\u0000[g12g23+ (Nv\u00001)g13g21]\n\u0000NvNsd2\u0000g41g43\u0000NvNs(np\u00001)d2\u0000g21g23;(F15)\ndg44\ndy=\u0000d0+[g2\n44+ (Nv\u00001)g2\n43]\u0000(np\u00001)d0+[g2\n34+ (Nv\u00001)g2\n33] +d2+[3g2\n44+ (Nv\u00001)g2\n41+ 2(Nv\u00001)g41g42]\n+ (np\u00001)d2+[g2\n14+ 2g14g24+ (Nv\u00001)(g2\n11+ 2g11g22)]\n\u0000Nsd2+[g2\n44+ (Nv\u00001)g2\n42]\u0000Ns(np\u00001)d2+[g2\n24+ (Nv\u00001)g2\n22]:\n(F16)\nThe RG equations for the case of twisted bilayer\ngraphene are obtained with Nv=Ns= 2 andnp= 3. 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We con\frm numeri-\ncally the qualitatively same result even when both d1\u0000\nandd2\u0000are nonzero, considering that g43is an inter-\nvalley exchange interaction and hence it is likely to be\nrather smaller than the intravalley interactions and the\nintervalley density interactions.\n[43] J. Gonz\u0013 alez, Phys. Rev. B 78, 205431 (2008)."    },    {        "title": "1806.10885v1.Non_collinear_Magnetism_and_Phonon_Dispersion_Relation_in_Vacancy_Induced_Phosphorene_Monolayer.pdf",        "content": "arXiv:1806.10885v1  [cond-mat.mtrl-sci]  28 Jun 2018Non collinear Magnetism and Phonon Dispersion Relation in V acancy Induced\nPhosphorene Monolayer\nSushant Kumar Behera and Pritam Deb∗\nAdvanced Functional Material Laboratory, Department of Ph ysics,\nTezpur University (Central University), Tezpur-784028, I ndia.\n(Dated: April 24, 2022)\nWe have studied the electronic, magnetic and linear phonon d ispersion behavior of Phosphorene\nmonolayer using rst principle based ab initio method. Phosp horene monolayer is a semiconducting\nsystem with a dimensional dependent variable range of band g ap. Vacancy has been done to study\nthe geometry and physical behavior of the monolayer system. Pristine, vacancy induced mono-\nlayer and vacancy induced doped monolayer are included in th e calculation. Dopant concentration\nhas been well checked via optimization algorithm to maintai n the dilute magnetic semiconduct-\ning behavior of the monolayer system. Density of states and p artial density of state indicates the\ncontribution of individual orbitals in the system. Band clo sing nature in observed in vacancy and\ndoped vacancy states indicating closed dense states and met allic behavior of the perturbed phases.\nBoth antiferromagnetic and ferromagnetic ordering is incl uded in our calculation to get a charm of\nboth ordering in the physical properties of the system. Land au energy level distribution is mapped\nvia Fermi surface with linear dispersion relation in terms o f phonon vibrational density of states\nand linear dispersion relations. The results of linear phon on density of states corroborating with\nelectronic density of states.\nkeywords: Phosphorene, Phonon Dispersion relation, Non-collinear M agnetism, Landau Energy Level, Mapping\nI. INTRODUCTION\nTwo dimensional(2D) atomic crystalsare stable mate-\nrials carryingmany different interesting properties unlike\ntheir 3D counterparts with potential applications[1, 2].\nElectronic properties in 2D materials vary from semi-\nconducting to metallic depending on the composition\nand thickness. Among the 2D materials, Phosphorene,\nthe most stable allotrope, is a single or few layer of\nblack phosphorus with a stacked puckered structure\nwhere individual layers held together by van der Waals\nforces. Structural anisotropy is supported via orthogonal\nunit cell with armchair and zigzag configurations along\nthe x and y direction, respectively[3]. As a result, it\npossess a unique structural characteristics along the\narmchair direction and a bilayer configuration along the\nzigzag direction. This unique structural arrangement\nis the key factor of the novel anisotropic physical (i.e.\nelectronic structure, magnetic moment development,\nphonon dispersion, etc.) properties of phosphorene\nmonolayer. Vacancy and doping in pristine monolayer\nhelp in modulating the physical phenomena of the\npristine sheet. As a result, the overall electronic, mag-\nnetic and mapping of Landau energy levels in terms of\nFermi energyaremodulated forenhanced characteristics.\nDiluted magnetic semiconductors (DMS) are com-\npounds with semiconducting nature where magnetic\natoms are used to replace their constituent atoms[4]. It\nis our current interest to gain moment at room temper-\nature in a few classes of layered DMS materials. Still,\n∗Corresponding Author: pdeb@tezu.ernet.ininduced magnetism in 2D semiconducting materials has\nremained an interesting and unexplored field of current\nresearch.\nRecently, first-principles based theoretical calculations\nare followed to understand the magnetic phenomena\nand phonon dispersion frequency of layered DMS mate-\nrials. Moreover, low symmetry and high anisotropy of\nphosphorous atoms show interesting platform to study\nmagnetic behavior and their phonon frequency in a form\nof puckered honeycomb lattice in phosphorene mono-\nlayer. These reports stimulate lots of research efforts\nin evaluating the magnetic moment development and\nphonon dispersion relation in monolayer phosphorene\nvia doping or vacancy for potential application purposes.\nIn this present work, we use density functional theory\ncombined with PHONON codes to calculate electronic,\nmagnetic and phonon dispersion properties. Moreover,\nwe study the electronic structure, magnetic behavior\nand phonon dispersion in phosphorene monolayer with\nvacancy and doping iron atom.\nII. METHODOLOGY\nThe density functional theory calculations were done\nby plane-wave based Quantum Espresso (QE) code with\nPAW potential and PBE-GGA exchange correlations[5].\nWe maintain 20 ˚Aof vacuum to minimize closest atomic\ninteractions in the monolayer in supercell form. We\nuse 48 eV as cutoff energy for geometry optimization\nand 9×9×1 grid for sampling the Brillouin zone with\nMonkhorst-Pack scheme in a 3 ×3×1 supercell. For\nthe projection of the density of states (PDOS), we use a2\nFIG. 1. The optimized geometry of phosphorene monolayer in ( a) pristine phosphorene, (b) one in the top is the zigzag dire ction\nof pure phosphorene and the bottom one is the armchair direct ion of pure phosphorene with 36 atoms, (c) phosphorene with\nmonovacancy, (d) phosphorene with divacancy having a total of 34 atoms, (e) doped monovacancy phosphorene and (f) doped\ndivacancy phosphorene. The red circle denotes the vacancy c reated and the green circle denotes the dopant i.e. the Fe ato m.\nFIG. 2. Projected density of states of (a) MV, (b) DV, (c) DMV a nd (d) DDV. The dotted black line presents the Fermi energy\n(EF) level.\n27×27×1grid for Brillouin-zone integrations. The band\nstructure is plotted along the high symmetric points of\nΓ, M, K and Γ in reciprocal space. Then, we could get\nthe phonon dispersion the systems using the PHONON\npackage integrated in the Quantum Espresso package.\nWe take five systems during our simulation, such as\npristine phosphorene monolayer (Pristine), monovacancy\nin monolayer (MV), divacancy in monolayer (DV), iron\ndoped in MV named as doped monovacancy in mono-\nlayer (DMV) and iron doped in DV named as doped\ndivacancy in monolayer (DDV). The five systems are\nsimulated under three different magnetic orientations,\ne.g. nonmagnetic (NM), antiferromagnetic (AFM) and\nferromagnetic (FM) ordering.III. RESULTS AND DISCUSSIONS\nThe geometry optimized structures of the phospho-\nrene monolayer (shown in Figure 1) are obtained using\ngeometry optimization with the quasi-Newtonian BFGS\nalgorithm. The mapping of Landau energy level is found\nfor lowest iteration steps to gain stability of the surface\natomic configuration. The constituent atoms are mobile\nduring calculation to fix the minimum energy based\nstable configurations.\nAtomic orbitals are projected (Figure 2) to suitable\nenergy level to identify near Fermi level activity of the\nvacancy induced and doped vacancy based monolayer\nsystems. Transition from semiconducting to metallic3\nFIG. 3. Band structure of pristine Phosphorene in different\norientations of (a) NM, (b) AFM and (c) FM. The dotted ver-\ntical lines presents the symmetric points in reciprocal spa ce.\nThe high symmetric points are Γ-M-K-Γ in the calculation.\nThe dotted red line presents the Fermi energy ( EF) level.\nphase is found from the bandgap closing behaviour\n(Figure 3) of the systems apart from pristine one.\nThe defect states develops finite nonzero magnetic\nmoments due to imbalanced surface spins in the pristine\nmonolayer. Doped Fe atom enhances the moment value\nshowing suitability towards room temperature dilute\nferromagnetic semiconductor. The phonon frequency\ngap (94 c m−1for pristine, 90 c m−1for MV and 66.7\ncm−1for DV) goes on decreasing as we go on creating\nvacancy in the pristine phosphorene to obtain rst\nmonovacancy and then divacancy.IV. CONCLUSIONS\nIn summary, we sum up the results of electronic\nand magnetic properties of the systems under various\nmagnetic ordering. Overlapping projection states of the\natomic orbitals near Fermi level proves the evidence of\nmetallic nature in the monolayer systems. Moreover, the\nbandgap closing behavior of the vacancy induced and\ndoped systems compared to pristine system indicates the\nphase transformation from semiconducting to metallic\nphase which is suitable for magnetic property study.\nPhonon frequency gap and lattice dynamics in terms of\nLandau energy mapping can be successfully tuned by\ncreating defects in the system in the form of vacancy and\ndoping. Thus phosphorene has a bright future as a room\ntemperature DMS layered material and Phononic crystal\nin radio-frequency communications and ultrasound\nimaging for medicine and nondestructive testing.\nACKNOWLEDGMENTS\nSKB acknowledges DST, Govt of India for providing\nINSPIRE Fellowship. The authors would like to thank\nTezpur University for providing HPCC facility.\n[1] S. K. Behera and P. Deb, RSC Advances 7, 31393 (2017).\n[2] P. D. S. K. Behera and A. Ghosh, Chemistry Select 2,\n3657 (2017).\n[3] F. Hao and X. Chen, J. Appl. Phys. 2016, 165104 (2006).[4] K. S. et al., Rev. Mod. Phys. 82, 1633 (2010).\n[5] P. G. et al., J. Phys. Condens. Matter. 21, 395502 (2009)."    },    {        "title": "1807.00296v2.Hidden_pair_density_wave_order_in_cuprate_superconductors.pdf",        "content": "Hidden pair-density-wave order in cuprate superconductors\nShiping Feng\u0003, Deheng Gao, Yiqun Liu, Yingping Mou, and Shuning Tan\nDepartment of Physics, Beijing Normal University, Beijing 100875, China\nWhen the Mott insulating state is suppressed by charge carrier doping, the pseudogap phenomenon\nemerges, where at the low-temperature limit, superconductivity coexists with some ordered elec-\ntronic states. Within the framework of the kinetic-energy-driven superconductivity, the nature of\nthe pair-density-wave order in cuprate superconductors is studied by taking into account the pseu-\ndogap e\u000bect. It is shown that the onset of the pair-density-wave order does not produce an ordered\ngap, but rather a novel hidden order as a result of the interplay of the charge-density-wave order\nwith superconductivity. As a consequence, this novel hidden pair-density-wave order as a subsidiary\norder parameter coexists with the charge-density-wave order in the superconducting-state, and is\nabsent from the normal-state.\nPACS numbers: 74.72.Kf, 74.25.Jb, 74.20.Mn,71.45.Lr, 71.18.+y\nKeywords: Charge-density-wave order; Pair-density-wave order; Cuprate superconductors\nOne of the most remarkable features in cuprate su-\nperconductors is the presence of the pseudogap in the\ntemperature range above the superconducting (SC) tran-\nsition temperature Tcbut below the pseudogap crossover\ntemperature T\u00031{3. In particular, the experimental re-\nsults from the angle-resolved photoemission spectroscopy\n(ARPES) observations indicated that this pseudogap can\nbe identi\fed as being a region of the self-energy e\u000bect\nin which it suppresses the spectral weight of the low-\nenergy quasiparticle excitation spectrum4{10, where the\nlow-energy spectral weight of the quasiparticle excitation\nspectrum on the electron Fermi surface (EFS) around the\nantinodal region is gapped out by the pseudogap, leav-\ning behind the low-energy spectral weight of the quasi-\nparticle excitation spectrum only located at the discon-\nnected segments around the nodal region to form the\nFermi pockets. This Fermi pocket consists of the Fermi\narc and back side of Fermi pocket. However, the ex-\nperimental results observed recently from di\u000berent mea-\nsurement techniques11{22demonstrated that the charge-\ndensity-wave (CDW) order exists within the pseudogap\nphase, appearing below a temperature TCDW well above\nTcin the underdoped regime, and coexists with supercon-\nductivity below Tc.TCDW is the temperature where the\nCDW order develops, and is of the order of T\u0003. Further-\nmore, the CDW vector QCDW was reported to connect\ntwo tips of the straight Fermi arcs11{22. These experi-\nmental results show that the CDW order plays a very\nimportant role in the physics of the pseodogap phase of\ncuprates superconductors11{22. On the other hand, the\nmeasurements of the Hall coe\u000ecient in high magnetic\n\felds indicated that the EFS reconstruction by the CDW\norder ends at a critical doping that is lower than the\npseudogap critical point23, which therefore show that the\npseudogap and CDW order are separate phenomena23.\nDespite the general consensus about the existence\nof the CDW order in cuprate superconductors11{23, its\n\u0003Electronic address: spfeng@bnu.edu.cnphysical origin is still controversial. Theoretically, two\nmain scenarios are disputing the explanation of the na-\nture of the CDW order. In one of the scenarios24{30,\nthe CDW vectors spanning the tips of the Fermi arcs\nare a manifestation of the pseudogap formation due to\nthe CDW order, while in the other31{33, the pesudo-\ngap phase is attributed to the pair-density-wave (PDW)\nstate, where electrons on the same side of EFS are paired,\nand therefore is di\u000berent from the CDW state. In this\nscenario, the CDW order only appears in the pesudo-\ngap phase as a subsidiary order parameter, and the tips\nof the Fermi arcs themselves result from an EFS insta-\nbility around the antinodal region that is distinct from\nthe CDW order. In this case, some natural questions are\nraised: (i) what type ordered-state is the crucial ordered-\nstate for the competition with superconductivity? (ii)\ndoes the PDW order coexist with the CDW order and su-\nperconductivity? In our recent studies28{30, the physical\norigin of the CDW order and of its interplay with super-\nconductivity in cuprate superconductors have been stud-\nied within the framework of the kinetic-energy-driven SC\nmechanism by taking into account the pseudogap e\u000bect,\nwhere the main features of the CDW order in cuprate su-\nperconductors are qualitatively reproduced11{22, includ-\ning the doping dependence of the CDW vector. In partic-\nular, we show that the CDW order both in the SC- and\nnormal-states is driven by the pseudogap-induced EFS\ninstability, with the CDW vector that is well consistent\nwith the wave vector connecting the straight tips of the\nFermi arcs. However, although the CDW order coexists\nwith superconductivity below Tc, this CDW order an-\ntagonizes superconductivity30. In this paper, we try to\ndiscuss the physical origin of the PDW order in cuprate\nsuperconductors and of its connection with the interplay\nof the CDW order and superconductivity along with this\nline. We show that the PDW order is generated by the\ninterplay between the CDW order and superconductivity,\ntherefore this PDW order appears automatically in the\nSC-state as a subsidiary order parameter, and is absent\nfrom the normal-state. Moreover, we demonstrate that\nthe onset of the PDW order does not produce an orderedarXiv:1807.00296v2  [cond-mat.supr-con]  26 Nov 20182\ngap, and in this sense, the PDW order is a novel hidden\norder.\nOur present study of the physical origin of the\nhidden PDW order in cuprate superconductors is\nbased on the framework of the kinetic-energy-driven\nsuperconductivity34,35, which has been employed pre-\nviously to study the unusual behavior of the SC-state\nquasiparticle excitations in cuprate superconductors36,37\nand the related interplay of the CDW order andsuperconductivity30. Most importantly, this kinetic-\nenergy-driven superconductivity is associated with the\nsingle-particle diagonal and o\u000b-diagonal Green's func-\ntionsG(k;\u001c\u0000\u001c0) =\u0000< T\u001cCk\u001b(\u001c)Cy\nk\u001b(\u001c0)>and\n=y(k;\u001c\u0000\u001c0) =<T\u001cCy\nk\"(\u001c)Cy\n\u0000k#(\u001c0)>of thet-Jmodel in\nthe charge-spin separation fermion-spin representation,\nwhich have been derived within the framework of the full\ncharge-spin recombination as36,\nG(k;!) =1\n!\u0000\"k\u0000\u00061(k;!)\u0000[\u00062(k;!)]2=[!+\"k+ \u0006 1(k;\u0000!)]; (1a)\n=y(k;!) =\u0000\u00062(k;!)\n[!\u0000\"k\u0000\u00061(k;!)][!+\"k+ \u0006 1(k;\u0000!)]\u0000[\u00062(k;!)]2; (1b)\nwhere the bare electron excitation spectrum \"k=\n\u0000Zt\rk+Zt0\r0\nk+\u0016, with\rk= (coskx+ cosky)=2,\r0\nk=\ncoskxcosky,tandt0are the nearest-neighbor (NN) and\nnext NN electron hopping amplitudes in the t-Jmodel,\nrespectively, Zis the number of the NN or next NN\nsites on a square lattice, and \u0016is the chemical potential,\nwhile the electron self-energies \u0006 1(k;!) in the particle-\nhole channel and \u0006 2(k;!) in the particle-particle channel\noriginated from the interaction of electrons with spin ex-\ncitations have been evaluated in terms of the full charge-\nspin recombination, and are given explicitly in Ref. 36.\nIn the framework of the kinetic-energy-driven super-\nconductivity, the electron self-energy \u0006 2(k;!) describes\na coupling of the electron pair interaction strength and\nelectron pair order parameter, and in the static limit, it\nthus is de\fned as the momentum dependence of the SC\ngap30,36 \u0016\u0001s(k) = \u0006 2(k;!= 0). On the other hand, the\nelectron self-energy \u0006 1(k;!) describes the single-particle\ncoherence, and therefore is closely related to the pseudo-\ngap as30,36,\n\u00061(k;!)\u0019[\u0016\u0001PG(k)]2\n!+\"0k; (2)\nwhere the energy spectrum \"0kand the momentum de-\npendence of the pseudogap \u0016\u0001PG(k) have been obtained\nexplicitly in Ref. 36.\nWith the help of the above single-particle diagonal\nGreen's function (1a), we can obtain the quasiparti-\ncle excitation spectrum I(k;!)/nF(!)A(k;!), with\nthe fermion distribution nF(!) and the electron spectral\nfunctionA(k;!) =\u00002ImG(k;!). This quasiparticle ex-\ncitation spectrum therefore describes the energy and mo-\nmentum dependence of the ARPES spectrum38{40. On\nthe experimental hand, the combination of the resonant\nX-ray scattering (RXS) data and EFS measured results\nusing ARPES have revealed a quantitative link between\nthe CDW vector QCDW and the momentum vector con-\nnecting the tips of the straight Fermi arcs11,15,16, whichin this case coincide with the hot spots on EFS. However,\non the theoretical hand, we28{30have studied the quanti-\ntative connection between the collective response of the\nelectron density and the low-energy electronic structure\nwithin the framework of the kinetic-energy-driven super-\nconductivity, and the obtained results are well consistent\nwith these RXS and ARPES experimental data11,15,16.\nIn Fig. 1, we plot the quasiparticle excitation spectrum\nI(k;0) as a function of the momentum in (a) the SC-\nstate and (b) the normal-state at doping \u000e= 0:09 with\ntemperature T= 0:002Jfor parameters t=J= 2:5 and\nt0=t= 0:3. Obviously, the results in Fig. 1 indicates that\nthere are two continuous contours in momentum space,\nwhich are labeled as kFandkBS, respectively. How-\never, some striking features appear: (A) The low-energy\nspectral weight on the constant energy contours kFand\nkBSaround the antinodal region has been gapped out\nby the pseudogap, and then the low-energy quasiparticle\nexcitations occupy disconnected segments located at the\ncontours kFandkBSaround the nodal region; (B) How-\never, the highest peak heights are located at the tips of\nthe disconnected segments, where the most quasiparti-\ncles are accommodated; (C) The tips of the disconnected\nsegments on the contours kFandkBSconverge at the\nhot spots to form a closed Fermi pocket8{10,41, with the\ndisconnected segment at the \frst contour kFis called the\nFermi arc, while the other at the second contour kBSis\nassociated with the back side of the Fermi pocket; (D)\nThese Fermi pockets appear both in the SC- and normal-\nstates, and are not symmetrically located in the Brillouin\nzone, i.e., they are not centered around [ \u0006\u0019=2;\u0006\u0019=2]; (E)\nThe quasiparticle scattering wave vector between the tips\nof the straight Fermi arcs both in the SC- and normal-\nstates shown in Fig. 1 (red lines) at the underdoping\n\u000e= 0:09 isQHS= 0:280 (hereafter we use the reciprocal\nunits), which is in good agreement with the experimental\naverage value of the CDW vector QCDW\u00190:29 observed\nboth in the SC- and normal-states of the underdoped3\n\tB\n \tC\nFIG. 1: (Color online) The intensity map of the quasiparticle excitation spectrum I(k;0) in the [kx;ky] plane in (a) the SC-state\nand (b) the normal-state at \u000e= 0:09 withT= 0:002Jfort=J= 2:5 andt0=t= 0:3. The pairing of electrons and holes at kand\nk+Q[red lines in (a) and (b)] drives the CDW order formation, and the electron pairing at kand\u0000k\u0000Qstates [dashed-red\nlines in (a)] governs the hidden PDW order, while the electron pairing at kand\u0000kstates [yellow lines in (a)] is responsible\nfor superconductivity.\ncuprate superconductors11{22, indicating that the CDW\norder both in the SC- and normal-states is driven by the\nEFS instability. However, it should be emphasized that\nthe momentum dependence of the CDW gap at the tips\nof the Fermi arcs is very small ( \u00180)11{22, which directly\ncontradicts the standard CDW picture where an energy\ngap is expected at precisely that points. Moreover, we\nhave also shown that this CDW vector is doping depen-\ndent, with the magnitude of the CDW vector QCDW that\ndecreases with the increase of doping, also in good agree-\nment with experimental results11,15,16.\nWe are now ready to discuss the physical origin of the\nPDW order in cuprate superconductors and of its con-\nnection with the interplay between the CDW order and\nsuperconductivity. In the above discussions, the quasi-\nparticle excitation spectrum I(k;!) is obtained in terms\nof the electron spectral function, and therefore the essen-\ntial behavior of the quasiparticle excitations in cuprate\nsuperconductors is completely determined by the elec-\ntron spectral function [then the single-particle diagonal\nGreen's function (1a) and the related electron self-energy\n(2)]. However, we \fnd that these single-particle diago-\nnal and o\u000b-diagonal Green's functions in Eq. (1) and\nthe related electron self-energy in Eq. (2) can be also\nreproduced exactly by a phenomenological Hamiltonian,\nH=X\nk\u001b\"kCy\nk\u001bCk\u001b\u0000X\nk\u001b\"0kCy\nk+Q\u001bCk+Q\u001b\n+X\nk\u001b\u0016\u0001PG(k)(Cy\nk+Q\u001bCk\u001b+Cy\nk\u001bCk+Q\u001b)\n\u0000X\nk\u0016\u0001s(k)(Cy\nk\"Cy\n\u0000k#+C\u0000k#Ck\"); (3)\nwhere the dispersions of \"kand\"0k, the SC gap \u0016\u0001s(k),and the pseudogap \u0016\u0001PG(k) are given explicitly in Eqs.\n(1) and (2). In particular, the pseudogap \u0016\u0001PG(k) has\nbeen identi\fed as the momentum dependence of the\nCDW gap. In our previous discussions, we29have shown\nthat this pseudogap \u0016\u0001PG(k) has a strong angular de-\npendence, where \u0016\u0001PG(kF) exhibits the largest value\naround the antinodes, however, the actual minimum of\n\u0016\u0001PG(kF)\u00180 does not appear around the nodes, but lo-\ncates exactly at the tips of the Fermi arcs, which is well\nconsistent with the experimental observations11{22. This\nphenomenological Hamiltonian (3) consists of two parts,\nthe CDW part with the momentum dependence of the\nCDW gap \u0016\u0001PG(k), and the SC part with the momen-\ntum dependence of the SC gap \u0016\u0001s(k). In this case, two\nbasic low-energy excitations for the SC quasiparticle and\nCDW quasiparticle, respectively, should emerge as the\npropagating modes, with the scattering of the SC quasi-\nparticles that mainly are responsible for superconductiv-\nity, while the scattering of the CDW quasiparticles domi-\nnates the CDW \ructuation. It should be emphasized that\nthis type Hamiltonian (3) has been usually employed to\nphenomenologically discuss the physical behavior of the\nCDW order and of its interplay with superconductivity\nin cuprate superconductors24{27.\nIn the framework of the equation of motion, the time-\nFourier transform of the single-particle diagonal and o\u000b-\ndiagonal Green's functions G(k;!) and=y(k;!) of the\nHamiltonian (3) satis\fes the following equations42,\n(!\u0000\"k)G(k;!) +\u0016\u0001s(k)=y(k;!)\n+\u0016\u0001PG(k)<T\u001cCk+Q\u001b(\u001c)Cy\nk\u001b(\u001c0)>!= 1; (4a)\n(!+\"k)=y(k;!) +\u0016\u0001s(k)G(k;!)\n+\u0016\u0001PG(k)<T\u001cCy\nk+Q\"(\u001c)Cy\n\u0000k#(\u001c0)>!= 0:(4b)4\nHowever, it is clear from these equations that for obtain-\ning the single-particle diagonal and o\u000b-diagonal Green's\nfunctionsG(k;!) and=y(k;!), we need to introduce an-\nother two Green's functions,\nGCDW(k;\u001c\u0000\u001c0) =\u0000<T\u001cCk+Q\u001b(\u001c)Cy\nk\u001b(\u001c0)>;(5a)\n=y\nPDW(k;\u001c\u0000\u001c0) =<T\u001cCy\nk+Q\"(\u001c)Cy\n\u0000k#(\u001c0)>; (5b)\nwhich therefore describe the CDW and PDW states, re-\nspectively. After a straightforward calculation, we \fnd\nthat the time-Fourier transform of the CDW and PDW\nGreen's functions GCDW(k;!) and=y\nPDW(k;!) satis\fes\nthe following equations42,\nGCDW(k;!) =\u0016\u0001PG(k)\n!+\"0kG(k;!); (6a)\n=y\nPDW(k;!) =\u0000\u0016\u0001PG(k)\n!\u0000\"0k=y(k;!): (6b)\nSubstituting these CDW and PDW Green's functions in\nEq. (6) into Eq. (4), the single-particle diagonal and\no\u000b-diagonal Green's functions G(k;!) and=y(k;!) of\nthe Hamiltonian (3) are obtained explicitly, and then\nthese obtained single-particle diagonal and o\u000b-diagonal\nGreen's functions G(k;!) and=y(k;!) are the exactly\nsame as quoted in Eq. (1) and the related the electron\nself-energy in Eq. (2).\nFrom the above discussions, we therefore con\frm\nthat the single-particle diagonal and o\u000b-diagonal Green's\nfunctionsG(k;!) and=y(k;!) in Eq. (1) and the re-\nlated the electron self-energy in Eq. (2) obtained based\non the kinetic-energy-driven superconductivity can be re-\nproduced exactly from the phenomenological Hamilto-\nnian (3) in terms of the CDW and PDW Green's func-\ntionsGCDW(k;!) and=y\nPDW(k;!), which therefore also\nreveal clearly the secrets of the PDW order: (i) As in\nthe case of the CDW order, the appearance of the PDW\nGreen's function =y\nPDW(k;!) in the calculation of the\nsingle-particle diagonal and o\u000b-diagonal Green's func-\ntionsG(k;!) and=y(k;!) also means the existence of\nthe PDW order as shown in Fig. 1a (dashed-red lines)\nand its coexistence with the CDW order and supercon-\nductivity in the SC-state. However, since the Hamilto-\nnian (3) describes obviously the interplay between the\nCDW order and superconductivity only, the automatical\nappearance of the PDW order is therefore generated by\nthe interplay of the CDW order with superconductivity,\nand can be thought to be the subsidiary order parame-\nter; (ii) However, in a clear contrast to the case of the\nCDW order, the appearance of the PDW order does not\nneed to introduce an obvious PDW order parameter in\nthe Hamiltonian (3), in other words, the appearance of\nthe PDW order does not produce an ordered gap in the\nquasiparticle excitation spectrum, and in this sense, this\nPDW order is a novel hidden order.\nIn the normal-state, the SC gap \u0016\u0001s(k) = 0, and thenthe phenomenological Hamiltonian (3) is reduced as,\nH=X\nk\u001b\"kCy\nk\u001bCk\u001b\u0000X\nk\u001b\"0kCy\nk+Q\u001bCk+Q\u001b\n+X\nk\u001b\u0016\u0001PG(k)(Cy\nk+Q\u001bCk\u001b+Cy\nk\u001bCk+Q\u001b); (7)\nwhere the Hamiltonian (7) consists of the CDW part\nonly with the momentum dependence of the CDW gap\n\u0016\u0001PG(k). In this case, only one basic low-energy exci-\ntation for the CDW quasiparticle emerges as the prop-\nagating mode. In particular, the single-particle Green's\nfunctionG(k;!) of the Hamiltonian (7) is evaluated in\nterms of the CDW Green's function GCDW(k;!) only,\nand then the obtained single-particle Green's function is\nalso the exactly same as the single-particle Green's func-\ntion obtained based on the kinetic-energy-driven super-\nconductivity and the related the electron self-energy in\nthe normal-state28,29,36. In this normal-state case, the\ninterplay of the CDW order with superconductivity dis-\nappears, leading to that the novel hidden PDW order is\ntherefore absent in the normal-state.\nFinally, we emphasize that since the CDW gap in the\nHamiltonians (3) and (7) has been identi\fed as the pseu-\ndogap \u0016\u0001PG, this is also why the CDW order exists within\nthe pseudogap phase, appearing below a temperature of\nthe order of T\u0003well aboveTcin the underdoped regime,\nand coexists with the hidden PDW order and supercon-\nductivity below Tc11{23.\nIn conclusion, within the framework of the kinetic-\nenergy-driven superconductivity, we have studied the\nphysical origin of the PDW order in cuprate supercon-\nductors and of its connection with the interplay of the\nCDW order with superconductivity by taking into ac-\ncount the pseudogap e\u000bect. Our results show that the\nPDW order is generated by the interplay of the CDW or-\nder with superconductivity, and therefore automatically\nemerges as a subsidiary order parameter in the SC-state.\nHowever, the onset of the PDW order does not produce\nan ordered gap in the quasiparticle excitation spectrum,\nand in this sense, this PDW order is a novel hidden or-\nder. The theory also predicts that this novel hidden PDW\norder appeared automatically as a subsidiary order pa-\nrameter in the SC-state is absent from the normal-state,\nwhich should be veri\fed by future experiments.\nAcknowledgements\nThe authors would like to thank Professor Yongjun\nWang for helpful discussions. This work was supported\nby the National Key Research and Development Program\nof China under Grant No. 2016YFA0300304, and the\nNational Natural Science Foundation of China (NSFC)\nunder Grant Nos. 11574032 and 11734002.5\n1See, e.g., H ufner, S. et al., Rep. Prog. 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Magn. 29, 3027 (2016).\n30Gao, D. et al., Physica C 551, 72 (2018).\n31Lee, P. A., Phys. Rev. X 4, 031017 (2014).\n32See, e.g., Fradkin, E. et al ., Rev. Mod. Phys. 87, 457\n(2015).\n33Wang, Y. et al., Phys. Rev. B 91, 115103 (2015).\n34Feng, S., Phys. Rev. B 68, 184501 (2003); Feng, S. et al.,\nPhysica C 436, 14 (2006); Feng, S. et al., Phys. Rev. B.\n85, 054509 (2012).\n35See, e.g., the review, Feng, S. et al., Int. J. Mod. Phys. B\n29, 1530009 (2015).\n36Feng, S. et al., Physica C 517, 5 (2015).\n37Gao, D. et al., J. Low Temp. Phys. 192, 19 (2018).\n38See,e.g., the review, Damascelli, A. et al., Rev. Mod. Phys.\n75, 473 (2003).\n39See, e.g., the review, Campuzano, J. C. et al., In: Ben-\nnemann, K. H. and Ketterson, J. B. (eds.) Physics of Su-\nperconductors, vol. II, p. 167. Springer, Berlin Heidelberg\nNew York (2004).\n40See,e.g., the review, Zhou, X. J. et al., In: Schrie\u000ber, J. R.\n(eds.) Handbook of High-Temperature Superconductivity:\nTheory and Experiment, p. 87. Springer, New York (2007).\n41Zhao, H. et al., Physica C 534, 1 (2017).\n42See, e.g., Mahan, G. D., Many-Particle Physics. Plenum\nPress, New York (1981)."    },    {        "title": "1808.04994v2.Density_driven_correlations_in_many_electron_ensembles__theory_and_application_for_excited_states.pdf",        "content": "Density-driven correlations in many-electron ensembles:\ntheory and application for excited states\nTim Gould\nQld Micro- and Nanotechnology Centre, Griffith University, Nathan, Qld 4111, Australia\nStefano Pittalis\nCNR-Istituto Nanoscienze, Via Campi 213A, I-41125 Modena, Italy\nDensity functional theory can be extended to excited states by means of a unified variational\napproach for passive state ensembles. This extension overcomes the restriction of the typical density\nfunctional approach to ground states, and offers useful formal and demonstrated practical benefits.\nThe correlation energy functional in the generalized case acquires higher complexity than its ground\nstate counterpart, however. Little is known about its internal structure nor how to effectively\napproximate it in general. Here we show that such a functional can be broken down into natural\ncomponents, including what we call “state-” and “density-driven” correlations, with the former\namenable to conventional approximations, and the latter being a unique feature of ensembles. Such\na decomposition, summarised in eq. (6), provides us with a pathway to general approximations that\nare able to routinely handle low-lying excited states. The importance of density-driven correlations\nis demonstrated, an approximation for them is introduced and shown to be useful.\nElectronic structure theory has transformed the study\nof chemistry, materials science and condensed matter\nphysics, by enabling quantitative predictions using com-\nputers. But a general solution to the many-electron prob-\nlem remains elusive, because the electron-electron inter-\nactions imply highly non-trivial correlations among the\nrelevant degrees of freedoms. Out of the numerous elec-\ntronic structure methodologies, density functional theory\n[1–3] (DFT) has become the dominant approach thanks\nto its balance between accuracy and speed, achieved by\nusing the electron density as the basic variable, then map-\nping the original interacting problem onto an auxiliary\nnon-interacting problem.\nDFT gives access to ground states, but not excited\nstates, meaning alternatives must be used for impor-\ntant processes like photochemistry or exciton physics\n[4]. Its time-dependent extension (TDDFT) does offer\naccess to excited states at reasonable cost [5, 6], and\nis thus commonly employed for this purpose. Routine\napplications of TDDFT reuse ground-state approxima-\ntions by evaluating them on the instantaneous density,\nthe so-called adiabatic approximation. This approach\nfails badly, however, when many-body correlations defy\na time-dependent mean-field picture, including for im-\nportant charge transfer excitations [7, 8].\nOne highly promising alternative involves tackling\nboth ground and excited eigenstates by means of one and\nthe same density functional approach [9–12], using en-\nsemble DFT (EDFT). EDFT is appealing because it can\nautomatically deal with otherwise difficult orthogonal-\nity conditions and can potentially tap into more than 30\nyears of density functional approximation development.\nEDFT has been shown to solve problems that are difficult\nfor TDDFT, such as charge transfers, double excitations,\nand conical intersections [13–23].\nConsolidating the preliminary success of EDFT intouseful approximations requires further understanding of\nhow many-body correlations get encoded in EDFT and\nhow they can be approximated generally. The correlation\nenergy of many-electron ground states is traditionally di-\nvided into dynamical (weak) and static (strong) correla-\ntions. This decomposition is by no means unambiguous,\nyet is very useful both for designing, and understanding\nthe limitations of, approximations [24]. Both static and\ndynamic correlations are also present in ensembles. But\nthe internal structure of the correlation energy functional\nfor ensembles is, by necessity, more complex. Little is\nknown about its specific properties and quirks.\nIn this Letter, we reveal a decomposition of the en-\nsemble correlation energy that lends itself both to an ex-\nact evaluation andto a universal approximation scheme.\nOur decomposition uncovers components of the correla-\ntion energy in multi-state ensembles, that will be missed\nby direct reuse of existing density functional approxima-\ntions on pure-state contributions. We show that the ad-\nditional components are unique features of EDFT and\ncan lead to significant errors, if ignored. We thus point\nout a crucial missing step on the path to upgrade existing\napproximations for correlations.\nThe components revealed through our decomposition\n–density-driven correlations – have so far gone unno-\nticed, and are similar to, but not the same as density-\ndriven errors of approximations [25]. Ultimately, these\ncomponents appear because the Kohn-Sham scheme in\nEDFT provides the exact overall ensemble particle den-\nsity, but not the density of each state in the ensemble.\nOur approach makes use of recent results on the Hartree-\nexchange component of the ensemble energy [26] and in-\ntroduces a generalization of the Kohn-Sham machinery.\nWe shall describe our construction first formally and then\nalso by means of direct applications. The relevance of the\ndensity-driven correlation is thus established unambigu-arXiv:1808.04994v2  [physics.chem-ph]  17 Apr 20192\nously for prototypical cases.\nA primer on EDFT: For a given electron-electron inter-\naction strength λ, external potential v, and set of weights\nWone can find[10] an ensemble density matrix,\nˆΓλ[v;W] =/summationdisplay\nwκ|κλ/angbracketright/angbracketleftκλ|≡arg min\nˆΓ→WTr/bracketleftBig\nˆΓˆHλ[v]/bracketrightBig\n,(1)\nso thatEλ[v;W] = Tr[ ˆΓλˆHλ[v]] =/summationtext\nκwκEλ\nκis the en-\nergy of the ensemble system. Here W={wκ}describes\na set of non-negative weights that obey/summationtext\nκwκ= 1.\nA consequence of (1) is that |κλ/angbracketrightare eigenfunctions of\nˆHλ[v] =ˆT+λˆW+/integraltext\nˆn(r)v(r)drsorted so that wκ≤wκ/prime\nfor eigenvalues Eλ\nκ> Eλ\nκ/primewhereEλ\nκ=/angbracketleftκ|ˆH|κ/angbracketright, making\nthe ensemble a passive state from which no work can be\nextracted[27]. We can, without loss of generality, assign\nequal weights whenever interacting states are degenerate.\nExcitation energies can be found via derivatives or differ-\nences ofE1with respect to relevant excited state weights\nwκ>0[9, 11, 22, 28].\nBy the Gross-Oliveira-Kohn (GOK) theorems [10–12]\nand the usual assumption that all densities of inter-\nest are ensemble v-representable, there exists a poten-\ntial,vλ[n;W]≡arg maxu{Eλ[u;W]−/integraltext\nnudr},that is a\nunique functional of nandW. Notice here we allow λto\nvary while keeping nconstant to connect “adiabatically”\nthe non-interacting ( λ= 0,v0≡vs) with the fully inter-\nacting limits ( λ= 1,v1≡v). To simplify discussion, we\nfurther restrict to the “strong adiabatic” case that the\nordering of occupied states ( wκ>0) asλ→0+is the\nsame as at λ= 1, i.e. that the energy ordering of low-\nlying states is adiabatically preserved. This is true in the\ncases considered here and the majority of cases amenable\nto EDFT – exceptions, we suspect, may include magnetic\nstates such as those with relevant orbital degeneracies in\ncombination with strong and spin-orbit interactions. Our\nconsequent discussion should be extended to cover such\nexceptions.\nSincevλ→nandn→vλare unique mappings at all\nrelevantλ, for weightsW, we can define the universal\nensemble density functional\nFλ[n]≡/summationdisplay\nκwκ/angbracketleftκλ|ˆT+λˆW|κλ/angbracketright≡Tr[ˆΓλ(ˆT+λˆW)] (2)\nwhere|κλ/angbracketrightare eigenstates of [ ˆT+λˆW+ˆvλ]|κλ/angbracketright=Eλ\nκ|κλ/angbracketright,\nˆΓλ=/summationtextwκ|κλ/angbracketright/angbracketleftκλ|and Tr[ ˆΓλˆn] =/summationtextwκ/angbracketleftκλ|ˆn|κλ/angbracketright=n.\nFor brevity, we now drop explicit references to W.\nMaking use of the Kohn-Sham (KS) ensemble, the in-\nteracting universal functional at λ= 1 (F[n]≡F1[n])\ncan be decomposed as F[n] =Ts[n] +EHx[n] +Ec[n]\nwhereTs[n],EHx[n] andEc[n] are the ensemble KS ki-\nnetic, Hartree-exchange (Hx) energy, and correlation en-\nergy functionals. We shall focus on cases involving de-\ngeneracies for different spin states but no ambiguities for\nthe spatial degree-of-freedom – this is sufficient for eluci-\ndating the main points of this work. Thus, the KS kineticand Hx energy are given, respectively, by\nTs[n]≡F0[n] =/summationdisplay\nwκTs,κ[n], (3)\nEHx[n]≡lim\nλ→0+Fλ[n]−F0[n]\nλ=/summationdisplay\nκwκΛHx,κ[n],(4)\nwhereTs,κ=/angbracketleftκ0+|ˆT|κ0+/angbracketright, ΛHx,κ=/angbracketleftκ0+|ˆW|κ0+/angbracketright.|κ0+/angbracketright\nare orthogonal (formally non-interacting) eigenstates as\nwell as proper spin eigenstates – they thus may be lin-\near combinations of Slater determinants which “opti-\nmize”EHx[26]. Of relevance to our discussion are the\nfollowing three facts: (1) TsandEHxare functionals\nof a shared set of occupied one-body orbitals φi[n](r)\nobeying [ ˆt+vs[n]]φi[n](r) =/epsilon1i[n]φi[n](r); (2) Some\nstates (e.g. singlet/triplet) can have the same KS den-\nsity and kinetic energy, but different KS-pair densities\nand Hx energies; (3) KS density and kinetic terms may\nbe expressed as ns,κ=/angbracketleftκ0+|ˆn|κ0+/angbracketright=/summationtext\niθκ\ni|φi|2and\nTs,κ=/summationtext\niθκ\niti, whereθκ\ni∈{0,1,2}are occupation fac-\ntors for spin-orbital i. By contrast, Hartree-exchange\nterms Λ Hx,κ[{φi}] =1\n2/integraltext\ndrdr/primeW(r,r/prime)n2Hx,κ(r,r/prime) must\nbe expressed via the KS-pair densities n2Hx,κ(r,r/prime) =\n/angbracketleftκ0+|ˆn(r)ˆn(r/prime)−ˆn(r)δ(r−r/prime)|κ0+/angbracketright.\nApart from the stated restrictions, so far no approx-\nimations have been made. Thus, we can complete the\npicture by defining the correlation energy functional\nEc[n] :=F[n]−FEXX[n], (5)\nas the difference between the unknown Fand the exact\nexchange (EXX) functional FEXX≡Ts+EHx. While\nformally correct, the above expression has limited effec-\ntiveness in practice. In what follows, we shall introduce\nwhat we argue is a more useful expression for Ec[n] , due\nto its ability to distinguish pure-state correlations from\nthose introduced by ensembles.\nMoving toward this objective, it is important to note\nthat the KS densities ns,κare not the same as the densi-\nties of interacting states nκ. As an example, consider the\nlowest lying triplet (ts) and singlet (ss) excited states in\nH2. The KS densities of the singlet and triplet excitation\nare equal to each other while the interacting ones are not,\ni.e.ns,ts=ns,ss=|φ0|2+|φ1|2(note, spatial orbitals are\nthe same for spin either up or down) and nts/negationslash=nss[22].\nThe same overall ensemble density is, by construction,\nobtained from the KS and the real ensemble. This fact\nis not specific to H 2, and its implications for the correla-\ntion energy of ensembles forms the bulk of the remainder\nof this letter. We shall first proceed formally, and then\nreview and test key results in concrete cases.\nState- and density-driven ensemble correlations: First,\nit is useful to recall that the energy components can be\nrestated from functionals of ninto functionals of the\n(ensemble) KS potential. As mentioned above, Λ Hx,κ\ndepends on the same set of single-particle orbitals as\nTs,κandns,κ. Thus, they can all be transformed into3\na functional of a potential, by replacing φi[n] byψi[vs]≡\nφi[n[vs]], where [ ˆt+vs]ψi[vs] =εi[vs]ψi[vs]. Therefore,\nany functional of the single-particle orbitals can be read-\nily expressed as a functional of the KS potential; e.g.,\nns,κ[vs]≡/summationtext\niθκ\ni|ψi[vs]|2,Ts,κ[vs] and Λ Hx,κ[vs].\nAs a second and crucial step, we seek to general-\nize the KS procedure by finding, for each state |κ/angbracketright,\na unique and state-dependent KS-like system with ef-\nfective potential vκ\nssuch that ns,κ[vs→vκ\ns] =nκ\nis the resulting density – note, ns,κ=/summationtext\niθκ\ni|ψi[vs]|2\nandnκ=/summationtext\niθκ\ni|ψi[vκ\ns]|2use the same set of occupa-\ntion factors. Finding the corresponding effective poten-\ntial relies on two conditions being satisfied: (i) that at\nleast onevκ\nsexists; (ii) that multiple valid potentials\n(i.e.,vκ\ns,1,vκ\ns,2→nκ) can be distinguished through a bi-\nfunctionalvκ\ns[nκ,n]≡arg minvκs→nκ/bardblvs[n],vκ\ns/bardblnthat se-\nlectsvκ\nsas the potential yielding nκthat is closest to\nthe true KS potential vsyieldingn, according to some\nmeasure/bardblv1,v2/bardblnthat can depend explicitly on n– one\nexample is:/bardblv1,v2/bardbln=/integraltext\nn(r)|v1(r)−v2(r)|dr.\nRegarding (i), the two-electron states considered here\n(see later discussion) can be mapped to KS ground-states\nwith well-defined and unique potentials. KS-like equa-\ntions for specific eigenstates have also been introduced\nto retrieve excitations of Coulomb systems [29, 30]. Ad-\nditional details and discussion appears in the supplemen-\ntary material. Regarding (ii), more than one metric may\nwork for the purpose. This implies some arbitrariness for\nintermediate quantities [eqs (8) and (9), below], yet no\ndifference for their sum [eq. (6)].\nOncevκ\nsis determined, we introduce ¯Ts,κ[nκ,n]≡\nTs,κ[vs→vκ\ns[nκ,n]] and ¯ΛHx,κ[nκ,n]≡ΛHx,κ[vs→\nvκ\ns[nκ,n]], where the original functionals are transformed\nby replacing the KS orbitals ψi[vs]→ψi[vκ\ns] in the orbital\nfunctionals, to give energy bifunctionals of the specific\ndensitynκand the total ensemble density n. We thus\nextend all key functionals to be specified for ensemble\ndensity components, as well as globally. For the special\ncasenκ=ns,κwe are guaranteed to find vκ\ns[ns,κ,n] =\nvsby construction. It then follows that Ts[n] =/summationtext\nκwκ¯Ts,κ[ns,κ,n],EHx[n] =/summationtext\nκwκ¯ΛHx,κ[ns,κ,n].\nFinally, we can express the correlation energy as:\nEc[n] =ESD\nc[n] +EDD\nc[n], (6)\nwhere\nESD/DD\nc [n]≡/summationdisplay\nκwκ¯ESD/DD\nc,κ [nκ,n]. (7)\nHere, the “pure” state-driven (SD),\n¯ESD\nc,κ[nκ,n] :=¯Fκ[nκ,n]−¯FEXX\nκ[nκ,n], (8)\nand “ensemble” density-driven (DD),\n¯EDD\nc,κ[nκ,n] :=¯FEXX\nκ[nκ,n]−FEXX\nκ[n] (9)terms are defined using ¯Fκ[nκ,n] :=Eκ[n]−/integraltext\ndrnκ(r)v[n](r), ¯FEXX\nκ[nκ,n] := ¯Ts,κ[nκ,n] +\n¯ΛHx,κ[nκ,n], andFEXX\nκ[n] :=Ts,κ[n] + Λ Hx,κ[n]≡\n¯FEXX\nκ[ns,κ,n] (sincens,κdepend onvs[n]).\nEq. (6) is the key result of the present work. It ex-\npresses the correlation energy of GOK ensembles in terms\nof: (a) state-driven correlations [eq. (8)] which are like\nthe usual pure state correlation energy, but involve bi-\nfunctionals of [ nκ,n];and(b) density-driven correlations\n[eq. (9)], which resemble difference between exact ex-\nchange energies at different pure state densities. The\nlabelling of SD terms as “pure” and DD as “ensemble”\ncan now be explained. In a pure state, ns,gs=ngs=n\nand thusEDD\nc= 0, as expected. Moreover, in anyensem-\nble, the ground-state term ¯ESD\nc,gsdepends only on ngs, and\nnot onn(sincevgs\nsis unique). By contrast, ¯EDD\nc,gsalways\ndepends on both nandngs, so varies with the overall\nchoice of ensemble. Density-driven correlations are con-\nsequently a unique, yet unavoidable, feature of EDFT\n– they appear because the KS system cannot simultane-\nously reproduce the densities of all ensemble components.\nImplications: First of all, our decomposition need not\nhandle problematic self- or ghost- interactions [31–33].\nBecause, our correlation functional is defined on top of an\nensemble Hartree-exchange which is already maximally\nfree from such spurious interactions. Any spurious inter-\nactions present must thus be the result of approximation.\nOur decomposition, of course, is not meant to tame un-\navoidable strong correlations in the SD terms.\nWe now turn to how our scheme can help in the de-\nvelopment of new approximations. Inspired by the prin-\nciple of minimal effort, one might seek to replace the\nentire correlation energy with the SD terms, eq. (8), by\nreusing any standard DFT approximation (DFA), i.e. set\nESD\nc,κ[nκ,n]→EDFA\nc[ns,κ]. The idea of reusing standard\nDFAs in ensembles is not new in EDFT, and with appro-\npriate care has been shown to give good results in excited\nstate and related non-integer ensembles [14, 31, 34]. In\nthe present context [see eq. (6) and eq. (7)], however,\nwe can appreciate that such a procedure: (a) replaces\nthe interacting densities of the SD terms by their non-\ninteracting counterparts, to make use of ingredients that\nare available in a typical calculations; (b) disregards the\nadditional functional dependence of the SD terms on n;\nand (c) misses the DD terms entirely.\nNext, we show that the contribution of the DD terms\nare indeed of relevant magnitude, when all the exact\nquantities are evaluated numerically. Then, we shall dis-\ncuss approximations.\nApplications: Having established the basic theory, let\nus now study the role of density-driven correlations in two\nelectron soft-Coulomb molecules. These tunable (via pa-\nrameterµ) one-dimensional molecules can exhibit chemi-\ncally interesting properties such as charge transfer excita-\ntions (µ= 2) or strong correlations ( µ= 0) [22] and thus\nallow important physics to be analyzed with full control.4\nFIG. 1. Decomposition of the correlation energy of the charge\ntransfer (top) and strongly-correlated (bottom) cases. The\nshaded regions show the relative significance of density-driven\nand state-driven correlations, with the former contributing\napproximately one quarter of the total correlation energy\nin the charge transfer case. The inset of the bottom panel\nillustrates the unzoomed plot. Here we set a mixture of\n60/30/10% respectively for the three lowest energy states.\nDetails are in the in the Supplementary Material.\nWe restrict ourselves to ensembles involving the\nground- (gs), triplet-excited (ts) and singlet-excited (ss)\nstates only. We perform our calculations in three steps:\nStep 1: Solve the two electron Hamiltonian ˆHwith one-\nand two-body interactions terms to obtain interacting\nstate-specific terms Eκ,|κ/angbracketright,nκ,F1\nκ=/angbracketleftκ|ˆT+ˆW|κ/angbracketright=\nEκ−/integraltext\ndxnκ(x)v(x), for the three states κ∈{gs,ts,ss},\nand ensemble averages therefrom, e.g., n=/summationtext\nκwκnκand\nF1=/summationtext\nκwκF1\nκ.\nStep 2: Invert[35] the density using the single-particle\norbital Hamiltonian ˆh=−1\n2∂2\nx+v(x) to findv(x) =\nvs(x)→n(x) and real-valued orbitals φ0andφ1that\nare required for the KS eigenstates. Here, vsdepends\non the density nand groundstate weight wgsonly, as\nn= (1 +wgs)φ2\n0+ (1−wgs)φ2\n1. From these terms, cal-\nculatens,κ,n2Hx,κTs,κand Λ Hx,κ, and ensemble aver-\nages, again for κ∈{gs,ts,ss}. Here,Ts,ts=Ts,ssand\nns,ts=ns,ssbut Λ Hx,ts/negationslash= Λ Hx,ssandn2Hx,ts/negationslash=n2Hx,ss.\nStep 3: Carry out separate inversions using ngs=\n2ψ0[vgs\ns]2,nts=ψ0[vts\ns]2+ψ1[vts\ns]2andnss=ψ0[vss\ns]2+\nψ1[vss\ns]2/negationslash=ntsto obtain the three unique potentials vκ\ns.\nThen use the resulting orbitals ψ0[vκ\ns] andψ1[vκ\ns] to cal-\nculate ¯Ts,κ[nκ,n] and ¯ΛHx,κ[nκ,n] on the interacting den-\nsities of the three states, and thus obtain the final ingre-\ndients for eqs (6)–(9).\nIn Figure 1 we show the correlation energy for two\nexamples of bond breaking (which occurs at R≈3), re-\nsolved into total, DD and SD components. One example\nexhibits charge transfer excitations (top, µ= 2), and the\nother involves strong correlations (bottom, µ= 0). We\nchoose an ensemble with 60% groundstate, 30% triplet\nstate and 10% singlet state (60/30/10%).\nThe first thing to notice is that in the “typical” charge\ntransfer case, the DD correlations form a substantial por-\ntion of the total correlation energy, about 25% on av-\nerage. This highlights the importance of capturing, or\n-0.4-0.20.00.20.4Error [eV]\nR [au]Charge transfer molecule µ=2 @ 60/40/0%\n0 1 2 3 4-0.4-0.20.00.20.4Error [eV]Charge transfer molecule µ=2 @ 60/30/10%\nSDA\n+DDAFIG. 2. The error Err( R) = ∆Eapprox\nHxc−∆Eexact\nHxc in excitation\nenergies ∆E=E−Egs, shown relative to the dissociation\nlimit, Err( R= 4). Shown are 60/40/0% (top) and 60/30/10%\n(bottom) mixtures for the charge transfer case. We report\nboth the pure state-driven only approximation (SDA) and the\nSD term plus the DD approximation (+DDA). Shaded regions\nindicate where including the DD approximation improves on\nthe SD-only case (orange), or worsens it (blue). The diamond\nindicates the equilibrium interatomic distance.\napproximating it somehow: a raw application of even a\nnearly perfect approximation to the SD correlations will\nmiss around one quarter of the correlation energy. The\nstrongly correlated case has a similar breakdown for small\nR, but becomes dominated by the SD correlations for\nlargeR. This is not surprising, as the SD term captures\nthe multi-reference physics that gives rise to most of the\ncorrelation energy, whereas the DD term contains only\nweaker dynamic correlations. The various densities that\ngive rise to the DD correlations are shown and discussed\nin the Supplementary Material.\nOf final note, close inspection of the strongly correlated\ncase reveals a subtle point: for R≥3, the DD correla-\ntion energy is positive . At first glance this might seem\nto be impossible – correlation energies should always be\nnegative. However, it reflects the fact that the DD corre-\nlation energy is defined via an energy difference between\ntwo states which come from different many-body prob-\nlems with different densities. Thus, the negative sign is\nnot guaranteed by any minimization principle.\nSo far we have been concerned with exact quantities.\nBut for applications, it is essential to derive approxima-\ntions. For a proof-of-principle demonstration, let us focus\non charge transfers in 1D molecules. We approximate the\nSD terms using available ingredients for our 1D model –\nworking in 3D would let us generate a variety of forms\nby tapping into the existing DFT zoo. The reported ap-\nproximations use numerically exact KS densities ns,κ.\nWe generate a SDA by combining the ensemble ex-\nact Hx results with a local spin density approximation\n(LSDA) for correlation, parametrised for the 1D soft-\nCoulomb potential [36–38]. But we adapt the LSDA ac-\ncording to the formalism laid out by Becke, Savin and\nStoll [39] – which is useful for dealing with multiplets.\nFull details are provided in the Supplementary material.\nThe key point to be addressed here is the approx-5\nimation for the DD terms (DDA). As far as charge\ntransfer are concerned, intuition suggests that an elec-\ntrostatic model may work well for a first DDA. Thus,\nwe proposeEDDA\nc =/summationtext\nκwκ/braceleftbig\nEH[nκ→˜nκ]−EH[ns,κ]/bracerightbig\n.\nThis expression involves the KS densities ns,κand ˜nκ=\nSκns,κ[1 +a∆ns,κ+b∆n2\ns,κ] which accounts for the fact\nthat in real situations we may not access the exact nκ.\nHere,Sκis chosen to ensure the correct number of elec-\ntrons, and the term ∆ ns,κ=ns,κ−n(i.e., the deviation\nof the state density ns,κfrom the full ensemble density\nn), ensures that the correction is zero in the case of a\npure state. Parameters a=−0.28 andb= 0.12 are\nfound via optimization. Additional information on our\nDDA, including comparisons with the exact DD term,\nare provided in the Supplementary material.\nFigure 2 shows errors in our approximations for the\n60/30/10% case from earlier, and a 60/40/0% case with-\nout singlet excitations. Although the proposed approxi-\nmation neglects both kinetic and x-like contributions [see\neq. (9)], its performance is remarkably good. Including\nthe DDA improves results for almost all chemically rele-\nvantR(see orange shading).\nSummary and outlook : Correlations in ensemble den-\nsity functional theory (EDFT) are more than the simple\nsum of their parts. They naturally divide into state-\ndriven (SD) and density-driven (DD) contributions, the\nformer being amenable to direct translation of existing\nDFT approximations, and the latter being a unique prop-\nerty of ensembles. In prototypical ensembles of excited\nstates, DD correlations account for up to 30% of the over-\nall correlation energy. Therefore, accurate approximation\nof the correlation energy requires simultaneous consider-\nation of the SD and DD components.\nA simple approximation to the DD correlations was\ndevised and evaluated in model situations. Thus, ac-\ncounting for both SD and DD correlations was shown\nto be both feasible and promising to prompt progress in\nEDFT. Development of general approximations, exten-\nsion to deal with systems that may challenge our simpli-\nfying “strong adiabatic” assumption, and generalization\nof key concepts and procedures presented here to other\nensembles [28, 40–42] are being pursued.\n[1] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864\n(1964).\n[2] W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).\n[3] R. O. Jones, Rev. Mod. Phys. 87, 897 (2015).\n[4] S. Matsika and A. I. Krylov, Chem. Rev. 118, 6925\n(2018).\n[5] E. Runge and E. K. Gross, Phys. Rev. Lett. 52, 997\n(1984).\n[6] M. E. Casida and M. Huix-Rotllant, Annu. Rev. Phys.\nChem. 63, 287 (2012).\n[7] C. A. Ullrich and I. V. Tokatly, Phys. Rev. B 73, 235102(2006).\n[8] N. T. Maitra, J. Phys.: Cond. Matter 29, 423001 (2017).\n[9] A. K. Theophilou, Journal of Physics C: Solid State\nPhysics 12, 5419 (1979).\n[10] E. K. U. Gross, L. N. Oliveira, and W. Kohn, Phys. Rev.\nA37, 2805 (1988).\n[11] E. K. U. Gross, L. N. Oliveira, and W. Kohn, Phys. Rev.\nA37, 2809 (1988).\n[12] L. N. Oliveira, E. K. U. Gross, and W. Kohn, Phys. Rev.\nA37, 2821 (1988).\n[13] M. Filatov and S. Shaik, Chem. Phys. Lett. 304, 429\n(1999).\n[14] M. Filatov, M. Huix-Rotllant, and I. Burghardt, J.\nChem. Phys. 142, 184104 (2015).\n[15] M. Filatov, WIREs Comput. Mol. Sci. 5, 146 (2015).\n[16] M. Filatov, “Ensemble DFT approach to excited\nstates of strongly correlated molecular systems,” in\nDensity-Functional Methods for Excited States , edited by\nN. Ferr´ e, M. Filatov, and M. Huix-Rotllant (Springer In-\nternational Publishing, Cham, 2016) pp. 97–124.\n[17] O. Franck and E. Fromager, Mol. Phys. 112, 1684 (2014).\n[18] K. Deur, L. Mazouin, and E. Fromager, Phys. Rev. B\n95, 035120 (2017).\n[19] A. Pribram-Jones, Z.-h. Yang, J. R. Trail, K. Burke, R. J.\nNeeds, and C. A. Ullrich, J. Chem. Phys. 140(2014).\n[20] Z.-h. Yang, J. R. Trail, A. Pribram-Jones, K. Burke,\nR. J. Needs, and C. A. Ullrich, Phys. Rev. A 90, 042501\n(2014).\n[21] Z.-h. Yang, A. Pribram-Jones, K. Burke, and C. A. Ull-\nrich, Phys. Rev. Lett. 119, 033003 (2017).\n[22] T. Gould, L. Kronik, and S. Pittalis, J. Chem. Phys.\n148, 174101 (2018).\n[23] F. Sagredo and K. Burke, J. Chem. Phys. 149, 134103\n(201).\n[24] S. Ghosh, P. Verma, C. J. Cramer, L. Gagliardi, and\nD. G. Truhlar, Chem. Rev. (2018).\n[25] M.-C. Kim, E. Sim, and K. Burke, Phys. Rev. Lett. 111,\n073003 (2013).\n[26] T. Gould and S. Pittalis, Phys. Rev. Lett. 119, 243001\n(2017).\n[27] M. Perarnau-Llobet, K. V. Hovhannisyan, M. Huber,\nP. Skrzypczyk, N. Brunner, and A. Ac´ ın, Phys. Rev.\nX5, 041011 (2015).\n[28] K. Deur and E. Fromager, J. Chem. Phys 150, 094106\n(2019).\n[29] M. Levy and A. Nagy, Phys. Rev. Lett. 83, 4361 (1999).\n[30] P. W. Ayers, M. Levy, and A. Nagy, The Journal of\nChemical Physics 143, 191101 (2015).\n[31] E. Pastorczak and K. Pernal, J. Chem. Phys. 140,\n18A514 (2014).\n[32] A. Pribram-Jones, Z.-h. Yang, J. R. Trail, K. Burke, R. J.\nNeeds, and C. A. Ullrich, J. Chem. Phys. 140, 18A541\n(2014).\n[33] N. I. Gidopoulos, P. G. Papaconstantinou, and E. K. U.\nGross, Phys. Rev. Lett. 88, 033003 (2002).\n[34] E. Kraisler and L. Kronik, Phys. Rev. Lett. 110, 126403\n(2013).\n[35] T. Gould and J. Toulouse, Phys. Rev. A 90(2014).\n[36] N. Helbig, J. I. Fuks, M. Casula, M. J. Verstraete, M. A.\nMarques, I. Tokatly, and A. Rubio, Phys. Rev. A 83,\n032503 (2011).\n[37] L. O. Wagner, E. Stoudenmire, K. Burke, and S. R.\nWhite, Phys. Chem. Chem. Phys. 14, 8581 (2012).\n[38] (2017), private communication from Michele Casula.6\n[39] A. Becke, A. Savin, and H. Stoll, Theor. Chim. Acta 91,\n147 (1995).\n[40] J. P. Perdew, R. G. Parr, M. Levy, and J. L. Balduz,\nPhys. Rev. Lett. 49, 1691 (1982).[41] T. Gould and J. F. Dobson, J. Chem. Phys. 138, 014103\n(2013).\n[42] B. Senjean and E. Fromager, Phys. Rev. A 98, 022513\n(2018).Supplementary Material for\n“Density-driven correlations in many-electron ensembles:\ntheory and application for excited states”\nTim Gould\nQld Micro- and Nanotechnology Centre, Griffith University, Nathan, Qld 4111, Australia\nStefano Pittalis\nCNR-Istituto Nanoscienze, Via Campi 213A, I-41125 Modena, Italy\nWe provide further information on: (i) auxiliary (KS-like) non-interacting systems for excited\npure-state interacting densities; (ii) The 1D model system used in numerical examples; (iii) KS and\nexact densities for pure states in ensembles; (iv) the approximations introduced in the manuscript.\nI. ON THE AUXILIARY NON-INTERACTING\nSYSTEMS FOR EXCITED PURE-STATE\nINTERACTING DENSITIES\nIn the paper, we briefly discuss circumstances in which\neffective KS-like potentials can be shown to exist for\nexcited pure-state interacting densities. Here, we ex-\npand on this discussion. Since most systems of physical\nor chemical interest exhibit “nice” particle densities, we\nshall not go into the discussion of peculiar or subtle cases.\nNote, this ‘machinery’ is introduced in our work to de-\ncompose the energy correlation into the state-driven and\ndensity-driven energy correlations. The orbitals obtained\nfrom the aforementioned KS-like potentials are required\nto calculate ¯FEXX[nκ,n]. Other terms in the decomposi-\ntion can be either regarded as functionals of the densities\nalone or can be computed by using standard (ensemble)\nKS orbitals. All the required definitions are given within\nthe paragraph stating equation (8) in the main text.\nFirstly, in the case of excited states of Coulomb sys-\ntems we can use the work in Ref. 1 and 2 to pro-\nvide a general proof. We can follow the arguments of\nRef. 2, to obtain Ts,κ[nκ] from their equation (8). Then,\nsince, equation (12) provides the potential vκ\nsfornκwe\ncan obtain the orbitals ψi[vκ\ns] for use in ¯FEXX\nκ[nκ,n] =\nTs,κ[nκ] + Λ Hx,κ[{ψi[vκ\ns]}].\nSecondly, the cases in the manuscript are for soft-\nCoulomb systems and are not amenable to the above\ntreatment. However, they involves an ensemble com-\nposed only of a doubly-occupied singlet ground-state,\nand the first-lying triplet and singlet states formed by\npromotion of a single orbital. For these states, we show\nhere, that the particle densities may be retrieved by us-\ning single-particle orbitals of non-interacting ground-state\nproblems – in the sense to be specified below.\nThe case of the actual ground states are obvious and,\nthus, do not need to be discussed further. For the lowest-\nlying triplets, we only need to observe that they behave\neffectively as ground states in the minimizations of the\nenergy within the given (triplet) multiplicity. This is,\nin fact, an old trick used in the KS literature to access\nlowest-lying states of any prescribed multiplicity.\nThis trick, however, does not apply to the first-lyingexcited singlets (in our examples, the actual ground\nstates are singlets). But at the level of non-interacting\nstates, singlet and triplet lead to particle densities of the\nsame form [see, for example, eq. (2) below]. Thus, we\nmay retrieve the particle density as well as the single-\nparticle orbitals for our excited interacting singlet (which\ndiffers from the particle density of our excited interact-\ningtriplet) from the lowest-lying triplet of a different\nsystem. Hence, the corresponding single-particle orbitals\nfor all three states may be determined uniquely by usual\nKS inversion routines[3]. The orbitals thus obtained can\nthen be used to calculate ¯FEXX[nκ,n].\nFinally, there is yet another way to rely on standard\nKS inversion procedures. Our specific case is for two\ninteracting electrons. For it, we may search for the\nauxiliary local potential that has a closed-shell ground\nstate for four non -interacting electrons having double\nthe prescribed particle density. This state has same or-\nbitals as the desired two-electron non-interacting excited\ntriplet/singlet but with doubled occupations [to see this\nmathematically, double the second expression in eq. (2),\nbelow, to get n4-el≡2nts/ß= 2|φ0|2+ 2|φ1|2, which is\nthe usual four-electron KS density expression]. Thus, we\ncan eventually obtain ¯FEXX[nκ,n].\nTherefore, the existence (and, indeed, uniqueness) of\nthe potentials is guaranteed in most cases of interest.\nWe also note, in passing, that the ability of the various\nCoulomb functionals of Refs 1 and 2 to deal with specific\nexcited states might also offer a route for bypassing the\nstrong adiabatic assumption which is so far required in\nour work.\nII. 1D SOFT-COULOMB MOLECULES\nThe 1D soft-Coulomb molecules we study, both in the\nmanuscript and here, involve a Hamiltonian\nˆH=−1\n2[∂2\nx1+∂2\nx2] +W(x1−x2) +v(x1) +v(x2),\n(1)\nin one spatial dimension. Here, the electron-electron in-\nteraction term is W(x) = (1\n4+x2)−1\n2. The parametrisedarXiv:1808.04994v2  [physics.chem-ph]  17 Apr 20192\nexternal potential is v(x) =−W(x+R/2)−W(x−R/2)−\nµe−(x−R/2)2, for nuclear distance Rand adjustable well-\ndepthµon the right atom.\nThis model is reasonably straightforward to solve nu-\nmerically. More importantly, by varying the parameter\nµ(the effective well-depth on the right atom) we can ex-\nplore its behaviour from strongly correlated (using µ= 0)\nto charge transfer (using µ= 2) states.\nIII. KS AND EXACT DENSITIES FOR PURE\nSTATES IN ENSEMBLES\nIn the manuscript we highlight the effect of density-\ndriven correlations on energies, both in exact and approx-\nimate cases. Here we analyze the various particle densi-\nties for the two cases shown in Figure 1 of the manuscript.\nNote, in this section ‘KS’ strictly refers to its original\nmeaning as intended in EDFT. Therefore, the KS system\nwhich yields a prescribed ensemble particle density does\nnot need to reproduce the particle densities of each pure\nstate in the same ensemble. The two-electron systems\nwe consider have degenerate (for the triplet/singlet) KS\ndensities of the form\nns,κ(x) =/braceleftBigg\n2|φ0(x)|2, κ = gs,\n|φ0(x)|2+|φ1(x)|2, κ = ts/ss.(2)\nwhereφ0(x) andφ1(x) are the required single-particle\norbitals.\nThus, Supplementary Figure 1 shows density differ-\nences ∆nκ=ns,κ−nκbetween KS and true states in\nthe left and middle panels, and the true densities at the\nright. It also shows the weighted mean absolute density\ndifference ∆|n|=/summationtext\nκwκ|∆nκ|, to visually summarise\nthe density difference that may affect the energy, keep-\ning in mind that/summationtext\nκwκ∆nκ= 0. In all cases it is clear\nthat the density differences are substantial.\nSpecifically, Supplementary Figure 1 shows results for\nthe caseR= 4 withµ= 0 (strong correlations) and\nµ= 2 (charge transfer). It includes the 60/30/10% case\n(middle) analysed in the main manuscript, but also a\n60/40/0% case (left) without any singlet contribution.\nThe singlet-free case lets us explore a subtle point on\nhow the weights affect the densities.\nOne particularly interesting feature is that the strongly\ncorrelated case (bottom) shows fundamentally different\ndeviations when the singlet is neglected or included, re-\nflecting the large errors ns,ts−ntsandns,ß−nßin si-\nmultaneously trying to represent the triplet and singlet\nstates using ns,ts=ns,ß=|φ0|2+|φ1|2. Any calculation\nof ensembles involving the three lowest energy configura-\ntions will need to handle such a difficult case via a direct\ndensity-driven correlation energy approximation.IV. DEFINITION OF APPROXIMATE\nCORRELATION FORMS\nA. State-driven contributions\nWe set out to approximate the state-driven correlations\nusing the exact ensemble Hartree-exchange (HX) plus a\nmodification of the correlation energy as defined in local\nspin density approximation (LSDA) [see eq. (7) below].\nSince we have discuss the Hx expression in detail in the\nmanuscript, let us first focus on correlation.\nThe correlation energy per electron in the LSDA for\n1D soft-Coulomb systems takes the form [4, 5]\n/epsilon1c(rs)≈−1\n2rs+Er2\ns\nA+Brs+Cr2s+Dr3slog(1 +αrs+βrm\ns).\n(3)\nwherers= 1/(2n). Variations due non-vanishing polar-\nization,ζ= (n↑−n↓)/n, are accounted for with\n/epsilon1c(rs,ζ) =(1−ζ2)/epsilon1c(rs,ζ= 0) +ζ2/epsilon1c(rs,ζ= 1).(4)\nThe results reported in [4] and [5] are for W/prime= (1 +\nX2)−1\n2, whereas we use W= (1\n4+X2)−1\n2. Unpub-\nlished parameters for /epsilon1cfor our case were obtained by\nprivate communication[6]: A= 7.4070,B= 1.11663,\nC= 1.8923,D= 0.0960119,E= 0.024884,α= 2.43332,\nβ= 0.0142507 and m= 2.9198 (forζ= 0); andA1=\n5.248B1= 0,C1= 1.568,D1= 0.1286,E1= 0.00321,\nα1= 0.0539,β1= 0.0000156, and m1= 2.959 (for\nζ= 1).\nWe can adapt the LSDA such to use ζto distinguish\nthe various states yet preserving their multiplet struc-\nture. This may be implemented with the replacement\nζ(x)→ζκ(x) =/radicalBigg\nmax/bracketleftbig\n0,1−2n2Hx,κ(x,x)\nnκ(x)nκ(x)/bracketrightbig\n,(5)\nwheren2Hx,κ(x,x) stands for the pair density of non-\ninteracting KS states (see in the main manuscript). In\ndoing this, we are borrowing aides from the work of\nBecke, Savin and Stoll [7] and make the additional re-\nquest that imaginary values are avoided by means of the\nmax-function – note, this expression provides the exact\npolarization for single Slater determinants.\nLet us see in detail how we deal with the states ana-\nlyzed in the previous section. Readily, we find\nn2Hx,κ(x,x/prime) =\n\n2|φ0(x)|2|φ0(x/prime)|2, κ = gs,\n|φ0(x)φ1(x/prime)−φ1(x)φ0(x/prime)|2, κ = ts,\n|φ0(x)φ1(x/prime) +φ1(x)φ0(x/prime)|2, κ = ss.\n(6)\nThrough eqs (2), (5) and (6), we obtain ζgs= 0,ζts= 1,\nandζß=/radicalbig\nmax[0,(n0−n1)2−4n0n1]/(n0+n1).3\nStrongly correlated molecule μ=0\nCharge transfer molecule μ=2\nSupplementary Figure 1: Density differences for charge transfer (top) and strongly-correlated (bottom) cases,\nand with only triplet states (left) or with singlet states as well (centre). The line plots show 5 ×the density\ndifference for each state (navy, teal and orange dashed lines), and the cream shaded area shows −5×the weighted\naverage absolute density difference ∆ |n|. The final panel (right) shows the true densities of the three states, for\nvisual comparison.\nHence, our state-driven approximation is defined by\nthe expression\nESDA\nHxc=/summationdisplay\nκwκ/braceleftbigg\nΛHx,κ+/integraldisplay\ndxns,κ(x)/epsilon1c(rs,κ(x),ζκ(x))/bracerightbigg\n,\n(7)\nwhere Λ Hx,κis given in the main manuscript. Because\neq. (7) is the only state-driven approximation we use in\nthe main text, no risk of confusion may emerge by de-\nnoting its expression compactly and simply as SDA.\nB. Density-driven contributions\nIn introducing a first approximation for the density-\ndriven correlations (DDA), we allow ourselves a “min-\nimalistic” approach. As discussed in the manuscript, a\ndensity driven term essentially involves the difference be-\ntween ¯Ts+¯EHxcalculated at interacting density nκand\nKohn-Sham density ns,κ. It thus involves a complicated\norbital dependence. To avoid this, we assume for simplic-\nity that the difference in the kinetic energy and exchange\ncontributions may be small, relative to the simpler elec-\ntrostatic term, so that\nEDDA\nc =/summationdisplay\nκwκ/braceleftbig\nEH[nκ]−EH[ns,κ]/bracerightbig\n. (8)\nBut in a typical calculation we would not have access to\nthe exactnκ. Thus we replace it with\nnκ→˜nκ=Sκns,κ/bracketleftbig\n1 +a∆ns,κ+b∆n2\ns,κ/bracketrightbig\n.(9)\nThe term Sκ=Ne/(Ne+aMκ,1+bMκ,2), where\nNe=/integraltext\nn(x)dx, ∆ns,κ=ns,κ−n, andMκ,p=/integraltext\nns,κ(x)∆np\ns,κ(x)dx, is chosen to ensure the correct\nnumber of electrons. ∆ ns,κ=ns,κ−nis employed to\nguarantee no contribution in the case of a pure state,\nwhere it must be zero since nκ=0=ns,κ=0=n. Note,\nour goal is not to find accurate densities, but to find an\naccurate approximation for the density-driven correlation\nenergyEDD\nc.\nFinally, the two parameters a=−0.28 and\nb= 0.12 are found by minimizing the error\nminC/summationtext\nR|EDDA\nc(R)−EDD\nc(R)−C|for allRover 10\ndissociation curves (90 calculations in total for 9 values\nofRin each). Here, we allow for a systematic deviation\n(via constant C) with the aim of yielding a good ap-\nproximation for the more-important energy differences,\ne.g. the approximate dissociation curves. The bench-\nmarks cover all R∈(0,1/2,1,...4),µ∈(0,2), andW∈\n(90/10/0%,80/20/0%,70/30/0%,60/40/0%,60/30/10%)\n– we thus seek to reduce the risk of overfitting for our\ntest cases.\nHere Supplementary Figure 2 shows results from the\nDDA and from the exact calculations, shown relative to\ntheir values in the dissociation limit, i.e. R= 4 for all\npractical purposes. The approximation is almost perfect\nin the 60/40/0% case, but less good in the 60/30/10%\ncase, see discussion below. Nonetheless, it gives values\nwithin 0.2 eV of the exact results even in its worst cases.\nThe 60/30/10% case identifies a major limitation of\nour approximation: the density-driven correction for the\nsinglet and triplet excitations are necessarily the same,\nas their KS densities are the same. This means the\napproximation gives essentially the same results for the\n60/30/10% and 60/40/0% cases, despite the two having\ndifferent exactEDD\nc(the small difference is caused by the\ndifferent state and overall densities). Future approxima-\ntions might use the pair-density n2Hx,κof the states or\npossiblyζκ, as above in (5), to improve their flexibility.\n[1] M. Levy and A. Nagy, Phys. Rev. Lett. 83, 4361 (1999). [2] P. W. Ayers, M. Levy, and A. Nagy, The Journal of Chem-4\n-0.4-0.20.00.20.4Ec(R)−Ec(4) [eV]\nR [au]Charge transfer molecule µ=2 @ 60/40/0%\n0 1 2 3 4-0.4-0.20.00.20.4Ec(R)−Ec(4) [eV] Charge transfer molecule µ=2 @ 60/30/10%\nEDDA\nc\nEc\n∆EDDA\nc\nSupplementary Figure 2: Deviations\nEc(R)−Ec(R= 4) of the density-driven approximation\nEDDA\nc compared to the exact results EDD\nc. Also shown\nare the errors ∆EDDA\nc =EDDA\nc−EDD\ncof the\napproximation, which are less than 0.1 eV in all cases.ical Physics 143, 191101 (2015).\n[3] T. Gould and J. Toulouse, Phys. Rev. A 90(2014).\n[4] N. Helbig, J. I. Fuks, M. Casula, M. J. Verstraete, M. A.\nMarques, I. Tokatly, and A. Rubio, Phys. Rev. A 83,\n032503 (2011).\n[5] L. O. Wagner, E. Stoudenmire, K. Burke, and S. R.\nWhite, Phys. Chem. Chem. Phys. 14, 8581 (2012).\n[6] (2017), private communication from Michele Casula.\n[7] A. Becke, A. Savin, and H. Stoll, Theor. Chim. Acta 91,\n147 (1995)."    },    {        "title": "1808.10537v2.Nuclear_kinetic_density_from_ab_initio_theory.pdf",        "content": "Nuclear kinetic density from ab initio theory\nMichael Gennari\u0003\nUniversity of Waterloo, 200 University Avenue West, Waterloo, Ontario N2L 3G1, Canada\nTRIUMF, 4004 Wesbrook Mall, Vancouver, British Columbia V6T 2A3, Canada\nPetr Navr\u0013 atily\nTRIUMF, 4004 Wesbrook Mall, Vancouver, British Columbia V6T 2A3, Canada\n(Dated: June 6, 2022)\nBackground: The nuclear kinetic density is one of many fundamental, non-observable quantities in\ndensity functional theory (DFT) dependent on the nonlocal nuclear density. Often, approximations\nmay be made when computing the density that may result in spurious contributions in other DFT\nquantities. With the ability to compute the nonlocal nuclear density from ab initio wave functions,\nit is now possible to estimate e\u000bects of such spurious contributions.\nPurpose: We derive the kinetic density using ab initio nonlocal scalar one-body nuclear densities\ncomputed within the no-core shell model (NCSM) approach, utilizing two- and three-nucleon chiral\ninteractions as the sole input. The ability to compute translationally invariant nonlocal densities\nallows us to gauge the impact of the spurious center-of-mass (COM) contributions in DFT quantities,\nsuch as the kinetic density, and provide ab initio insight into re\fning energy density functionals.\nMethods: The nonlocal nuclear densities are derived from the NCSM one-body densities calculated\nin second quantization. We present a review of COM contaminated and translationally invariant\nnuclear densities. We then derive an analytic expression for the kinetic density using these nonlocal\ndensities, producing an ab initio kinetic density.\nResults: The ground state nonlocal densities of4,6,8He,12C, and16O are used to compute the\nkinetic densities of the aforementioned nuclei. The impact of COM removal techniques in the\ndensity are discussed and compared to a procedure applied in DFT. The results of this work can be\nextended to other fundamental quantities in DFT.\nConclusions: The use of a general nonlocal density allows for the calculation of fundamental quan-\ntities taken as input in theories such as DFT. This allows benchmarking COM removal procedures\nand provides a bridge for comparison between ab initio and DFT many-body techniques.\nI. INTRODUCTION\nIn this paper, we derive an analytic expression for the\nkinetic density of a nuclear system from the nonlocal\nscalar one-body nuclear densities calculated in the no-\ncore shell model (NCSM) [1] approach. In particular,\nwe use the method introduced in Ref. [2] to construct\nthe microscopic nonlocal one-body density. The ab initio\nNCSM is a rigorous many-body technique which treats\nallAnucleons as active degrees of freedom and takes re-\nalistic two- and three-nucleon forces as the sole input.\nThe model is suited for the description of light nuclei\n(A.16) as it is able to account for many-nucleon corre-\nlations producing high-quality wave functions.\nWith the NCSM nuclear densities it is possible to com-\npute quantities fundamental to density functional theory\n(DFT), such as the kinetic density, using ab initio wave\nfunctions. In systems such as16O, this can allow for\ndirect comparison between COM removal procedures in\ndi\u000berent many-body techniques, such as DFT. DFT is\na many-body method for calculating nuclear properties\nacross the nuclear chart, which has been practiced in nu-\nclear physics for approximately 40 years [3{9]. It involves\n\u0003mgennari5216@gmail.com\nynavratil@triumf.cathe minimization of an energy functional with respect to\nseveral system densities (kinetic, spin, isospin, etc.).\nDFT has had great success and has made signi\fcant\nprogress in the description of medium- to heavy-mass nu-\nclei [10{14]. In these types of systems, contributions\nfrom the center-of-mass (COM) and the nonlocality in\nthe density are reduced in e\u000bect. However, if DFT is to\nextend its reach to more systems, the size of these two\ne\u000bects must be reviewed to ensure they are under control.\nRecently, there has been a signi\fcant e\u000bort to con-\nstruct a bridge between DFT and ab initio approaches\nto the nuclear many-body problem [15{18], with the ul-\ntimate goal of reducing the phenomenological nature of\nDFT and creating direct connections to the underlying\nquantum chromodyanimcs (QCD). Motivated by this ef-\nfort, we attempt to contribute to the construction of such\na bridge by computing a fundamental input of energy\ndensity functionals, the nuclear kinetic density, from ab\ninitio many-body wave functions with chiral potentials\nas the sole input. In this work, we study16O and other\nlight nuclei using the NCSM many-body method to ex-\nplore the e\u000bects of nonlocality and COM contamination\nin the nuclear kinetic density.\nThe paper is organized as follows: in Sec. II, divided\ninto three subsections, we focus on the theoretical con-\nstruction of the kinetic density. The NCSM formalism\nis discussed in Sec. II A, the derivation of the nonlocal\ntranslationally invariant density is reviewed in Sec. II B,arXiv:1808.10537v2  [nucl-th]  25 Jan 20192\nand \fnally the construction of the kinetic density from\nthe NCSM nonlocal densities in Sec. II C. In Sec. III we\ndiscuss the nonlocal density results, we present results for\nthe kinetic density, and we compare the contributions of\nthe spurious center of mass (COM) and e\u000bects of the\nremoval. In Sec. IV we draw our conclusions. In the ap-\npendix, Sec. V, we present the derivations necessary to\nthis work.\nII. THEORETICAL FRAMEWORK\nA. NCSM\nEvaluating the kinetic density in the most consistent\nmanner from ab initio wave functions requires knowledge\nof the translationally invariant nonlocal density. The\nA-nucleon eigenstates required to calculate the nonlocal\none-body density are computed according to the ab ini-\ntioNCSM approach [1]. Within this many-body method,\nnuclei are considered to be systems of Anonrelativistic\npoint-like nucleons interacting via high-quality, realistic\ntwo- and three-body inter-nucleon interactions. Each in-\ndividual nucleon is treated as an active degree of freedom\nand the translational invariance of observables, the an-\ngular momentum, and the parity of the nucleus under\nconsideration are conserved. The many-body wave func-\ntion is expanded over a basis of antisymmetric A-nucleon\nharmonic oscillator (HO) states. The basis contains up\ntoNmaxHO excitations above the lowest possible Pauli\ncon\fguration. The basis is characterized by an additional\nparameter \n, the frequency of the HO well. Additional\ninformation on the many-body method and the trunca-\ntion scheme can be found in Ref. [1].\nNCSM wave functions are computed through the di-\nagonalization of the translationally invariant nuclear\nHamiltonian, which includes both two- and three-body\n(NN+3N) forces:\n^HjA\u0015J\u0019Ti=EJ\u0019T\n\u0015jA\u0015J\u0019Ti; (1)\nwith\u0015distinguishing eigenstates with identical J\u0019T.\nIn general, we can accelerate convergence of the HO ex-\npansion by applying a similarity renormalization group\n(SRG) transformation on the NN and 3Ninterac-\ntions [19{23]. While large gains in convergence can be\nachieved by performing the SRG evolution, it induces\nhigher body terms and can introduce a further depen-\ndence on the momentum-decoupling scale \u0015SRG if the\nunitarity of the SRG transformation is violated.\nIn the present work, we used the NN chiral potential\nat N4LO with a cuto\u000b \u0003 = 500 MeV, recently developed\nby Entem, Machleidt, and Nosyk [24, 25]. This interac-\ntion will be denoted as NN-N4LO(500). We also included\nthe three-nucleon potential at the next-to next-to leading\norder (N2LO) with simultaneous local [26] and nonlocal\nregularization, a more complete description of which can\nbe found in Ref. [2]. The 3N component will be denotedas 3Nlnl, making the notation for the total interaction\nutilized for the densities NN-N4LO(500)+3Nlnl.\nIn calculations with the NN-N4LO(500)+3Nlnl inter-\naction, we have worked with ~\n = 20 MeV and \u0015SRG\nranges of 1.6 to 2.0 fm\u00001, both previously determined to\nbe optimal and suitable for the given s- andp-shell nuclei\nunder consideration [2]. Note that the Nmax= 8 calcu-\nlations for12C and16O were obtained using importance-\ntruncated NCSM basis [27, 28].\nB. Nonlocal density\nIn the current work we use the generalized COM re-\nmoval method of Ref. [2], which extended the results of\nlocal densities in Ref. [29] to generate nonlocal one-body\ndensity matrices. We note the di\u000berences between this\napproach and the alternate approaches of Ref. [30, 31]\nreside in the method of COM removal, which in this\nand previously cited works is done directly in coordi-\nnate space, while in Ref. [30, 31] the COM removal from\nnonlocal and local densities is performed in momentum\nspace.\nIn coordinate representation, the nonlocal form of the\nnuclear density operator is de\fned as\n\u001aop(~ r;~ r0) =AX\ni=1\u0000\nj~ rih~ r0j\u0001i=AX\ni=1\u000e(~ r\u0000~ ri)\u000e(~ r0\u0000~ r0\ni):(2)\nThe matrix element of this operator between a general\ninitial and \fnal state obtained in the Cartesian coordi-\nnate single-particle Slater determinant (SD) basis is writ-\nten as\nSDhA\u0015jJjMjj\u001aop(~ r;~ r0)jA\u0015iJiMiiSD\n=X1\n^Jf(JiMiKkjJfMf)\u0012\nY\u0003\nl1(^r)Y\u0003\nl2(^r0)\u0013(K)\nk\n\u0002Rn1;l1\u0000\nj~ rj\u0001\nRn2;l2\u0000\nj~ r0j\u0001\n\u0002(\u00001)l1+l2+K+j2+1\n2^j1^j2^K\u001a\nj2l21\n2\nl1j1K\u001b\n\u0002(\u00001)\n^KSDhA\u0015fJfjj(ay\nn1;l1;j1~an2;l2;j2)(K)jjA\u0015iJiiSD:\n(3)\nWe suppress the isospin and parity quantum num-\nbers for simplicity. In Eq. (3), the NCSM eigen-\nstates (1) have the subscripts SD denoting that we\nused Slater determinant HO basis that include COM\ndegrees of freedom as opposed to the translationally\ninvariant Jacobi coordinate HO basis [32]. Further,\n^\u0011=p2\u0011+ 1 andRn;l(j~ rj) is the radial HO wave func-\ntion with the oscillator length parameter b=q\n~\nm\n,\nwheremis the nucleon mass. The one-body density\nmatrix elements are introduced in second-quantization,\nSDhA\u0015fJfjj(ay\nn1;l1;j1~an2;l2;j2)(K)jjA\u0015iJiiSD. Both~ rand\n~ r0are measured from the center of the HO potential well.3\nAs a result of this construction, the density contains a\nspurious COM component.\nWe require the removal of the COM component from\nthe nonlocal density if we are to compute a consistent\nkinetic density. This is enabled by the factorization of\nthe Slater determinant and Jacobi eigenstates,\nh~ r1:::~ rA~ \u001b1:::~ \u001bA~ \u001c1:::~ \u001cAjA\u0015JMiSD=\nh~\u00181:::~\u0018A\u00001~ \u001b1:::~ \u001bA~ \u001c1:::~ \u001cAjA\u0015JMi\u001e000(~\u00180);(4)\nwith the ground state COM component, labeled in\nEq. (4) as\u001e000(~\u00180). This is given as the N= 0 HO state\nwith~\u00180proportional to the A-nucleon COM coordinate.\nThe matrix element of the translationally invariant op-\nerator as given in Ref. [2], \u001atrinv\nop(~ r\u0000~R;~ r0\u0000~R), between\ngeneral initial and \fnal states is then given by (compare\nto Eq. (3))\nhA\u0015jJjMjj\u001atrinv\nop(~ r\u0000~R;~ r0\u0000~R)jA\u0015iJiMii\n=\u0010A\nA\u00001\u00113\n2X1\n^Jf(JiMiKkjJfMf)\n\u0002\u0000\nMK\u0001\u00001\nnln0l0;n1l1n2l2\u0012\nY\u0003\nl(\\~ r\u0000~R)Y\u0003\nl0(\\~ r0\u0000~R)\u0013(K)\nk\n\u0002Rn;l\u0010r\nA\nA\u00001j~ r\u0000~Rj\u0011\nRn0;l0\u0010r\nA\nA\u00001j~ r0\u0000~Rj\u0011\n\u0002(\u00001)l1+l2+K+j2\u00001\n2^j1^j2\u001aj1j2K\nl2l11\n2\u001b\n\u0002SDhA\u0015fJfjj(ay\nn1;l1;j1~an2;l2;j2)(K)jjA\u0015iJiiSD\n(5)\nwhere\n\u0000\nMK\u0001\nnln0l0;n1l1n2l2\n=X\nN1;L1(\u00001)l+l0+K+L1\u001a\nl1L1l\nl0K l 2\u001b\n^l^l0\n\u0002hnl00ljN1L1n1l1li1\nA\u00001hn0l000l0jN1L1n2l2l0i1\nA\u00001:\n(6)\nIn Eq. (5), the Rn;l\u0010q\nA\nA\u00001j~ r\u0000~Rj\u0011\nis the radial harmonic\noscillator wave function in terms of a relative Jacobi coor-\ndinate,~\u0018=\u0000q\nA\nA\u00001(~ r\u0000~R). The\u0000\nMK\u0001\nnln0l0;n1l1n2l2ma-\ntrix (6) introduced in Ref. [29] includes generalized har-\nmonic oscillator brackets of the form hnl00ljN1L1n1l1lid\ncorresponding to a two particle system with a mass ratio\nofd, as outlined in Ref. [33].\nThe nonlocal density expressions presented here can\nbe related to the local densities in Ref. [29] by restricting\nthe coordinates such that ~ r=~ r0, or\n\u001a(~ r) =\u001a(~ r;~ r0)j~ r=~ r0=\u001a(~ r;~ r): (7)\nThe normalization of the nonlocal density is consistent\nwith Ref. [29] such that the integral of the local form\nZ\nd~ rhA\u0015JMj\u001aop(~ r;~ r)jA\u0015JMi=A (8)returns the number of nucleons for both (3) and (5).\nFinally, make note that the proton and neutron densi-\nties are obtained separately by introducing (1\n2\u0006tzi) fac-\ntors, respectively, in Eq. (2). This results in the inclusion\nof a proton or neutron index in the creation and anihi-\nlation operators, as the COM operators commute with\nisospin operators. The normalization (8) then becomes\nZorNfor the proton and neutron density respectively.\nC. Kinetic density\nIn DFT, the kinetic density is just one of several sys-\ntem densities which contribute to the local energy den-\nsityH(~ r). The kinetic density is not itself an observable,\nhowever when combined with the potential interaction\nterms, the resultant local energy density His an observ-\nable from which nuclear properties can be computed [34].\nThe kinetic term in H(~ r) is given by\nHkinetic (~ r) =~2\n2m\u001c0(~ r); (9)\nwheremis the nucleon mass and \u001c0=\u001cp+\u001cnis the total\nkinetic density [35].\nWith the nonlocal nuclear densities constructed, it is\nnow possible to compute the kinetic density of a given\nnuclear system from ab initio theory. We act upon the\nnonlocal density by a Laplacian-like operator according\nto the following relation described in Ref. [36],\n\u001cN(~ r) =\u0014\n~r\u0001~r0\u001aN(~ r;~ r0)\u0015\n~ r=~ r0; (10)\nwhere Ndenotes the nucleon type for protons ( p) and\nneutrons (n). In order to derive a computable expres-\nsion for this quantity, we require several relations. It is\nuseful to begin by writing the kinetic density in spherical\ncomponent form as\nrur0\n\u0000u\u001a(~ r;~ r0) =\nX\nn;l;n0;l0;K;k;ml;ml0\u000bK;i;f\nn;l;n0;l0(lmll0ml0jLM)\n\u0002\u0014\nruRn;l(r)Y\u0003\nl;ml(^r)\u0015\u0014\nr0\n\u0000uRn0;l0(r0)Y\u0003\nl0;ml0(^r0)\u0015\n;\n(11)\nwhereu= 0;\u00061 and\u000bK;i;f\nn;l;n0;l0is de\fned for the transla-\ntionally invariant density as\n\u000bK;i;f\nn;l;n0;l0=X\nn1;l1;j1;n2;l2;j2\u0012A\nA\u00001\u00133=2\n\u00021\n^Jf(JiMiKkjJfMf)\u0000\nMK\u0001\u00001\nn;l;n0;l0;n1;l1;n2;l2\n\u0002(\u00001)l1+l2+K+j2\u00001\n2^j1^j2\u001aj1j2K\nl2l11\n2\u001b\n\u0002SDhA\u0015fJfjj(ay\nn1;l1;j1~an2;l2;j2)(K)jjA\u0015iJiiSD:\n(12)4\nWe note that \u000bK;i;f\nn;l;n0;l0is di\u000berent for the COM contami-\nnated density. We now discuss several relations necessary\nfor the derivation of the kinetic density, explicitly shown\nin the appendix. The \frst set of relations are analytic\nexpressions for the spherical components of ~rf(~ r)Ym\nl(^r),\nwhich can be found in section 5.8.3 of Ref. [37]. In these\nrelations, we see explicit dependence on the derivative\nof the RHO function. In order to remove a direct de-\npendence on a \frst order di\u000berential, we introduce the\nfollowing relation for the derivative of the RHO,\ndRnl\ndr=l\nrRnl\u00001\nb\u0012r\nn+l+3\n2Rn;l+1(r)+pnRn\u00001;l+1(r)\u0013\n:\n(13)\nFor the derivation of the Eq. (13), see the appendix,\nSec. V A. Using these relations, along with additional\nangular momentum algebra, we may now evaluate the\nexpression for the kinetic density in terms of spherical\ncomponents, which takes the form\n\u001cN(~ r) =\u0014\nr0r0\n0\u001aN(~ r;~ r0)\u0000r +1r0\n\u00001\u001aN(~ r;~ r0)\n\u0000r\u00001r0\n+1\u001aN(~ r;~ r0)\u0015\n~ r=~ r0:(14)\nThe derivation of the r0r0\n0\u001aNcomponent is shown in\nSec. V B. The outlined procedure can be followed exactly\nfor ther+1r0\n\u00001\u001aNandr\u00001r0\n+1\u001aNcomponents, with\nonly minor di\u000berences in the angular momentum algebra.\nIII. RESULTS\nA. Nonlocal density\nIn this section, as in Ref. [2], we discuss results for the\nnonlocal densities obtained from the NCSM wave func-\ntions using the approach described in Sec. II B.\nTo highlight the signi\fcance of COM removal in lighter\nsystems, we considered the4;8He and16O systems.\nWe computed the translationally invariant and COM\ncontaminated nonlocal densities, given by Eq. (5) and\nEq. (3), respectively. Note that all \fgure plots of the\nCOM contaminated density are labeled wiCOM while\nthe translationally invariant density plots are labeled\ntrinv. The ground state densities of the nuclei are shown\nwith all angular dependence factorized out for plotting.\nProton densities are shown in blue, neutron densities are\nshown in red, and total nuclear densities are shown in\nblack.\nIn Fig. 1, Fig. 2, and Fig. 3 we show com-\nparisons between calculations of nonlocal and local\n4He densities with the bare NN-N4LO(500) interaction\nand with the previously described SRG-evolved NN-\nN4LO(500)+3Nlnl interaction. The SRG-evolved inter-\naction is computed at the two- plus three-body level,\nwith all higher body SRG induced terms neglected. An\nNmax= 18 basis space was used for the bare interaction,\nFigure 1. Ground state4He nonlocal proton and neutron\ndensities calculated using the bare NN-N4LO(500) interaction\nwith anNmax= 18 basis space. An oscillator frequency of\n~\n = 20:0 MeV was used for this calculation.\nand anNmax= 14 basis space was used for the SRG-\nevolved interaction. Comparing the nonlocal densities\nbetween the bare and SRG-evolved interactions, di\u000ber-\nences in the predicted structure are evident, such as the\nreduction of the peak in the bare calculation. However,\nan arguably more important feature of the bare density\nis that it tends to have an initial plateau extending to\none fermi, where it begins a rapid fall o\u000b towards zero\ndensity. The rate of the declination is di\u000berent from the\nSRG-evolved interaction, which produces a density with\na smoother transition between the peak value and the fall\nof towards zero. These alterations are more noticeable in\nFig. 3. In the top panel, we compare the local densi-\nties of the SRG-evolved two-body interaction with and\nwithout the chiral three-nucleon interaction. NN+3Nind\nSRG labels the two-body SRG-evolved interaction with\ninduced three-body terms, and NN+3N SRG labels the\ntwo-body SRG-evolved interaction with the induced and\nchiral three-body interaction terms (full three-body in-\nteraction). Notice the di\u000berences in the density when\nutilizing the SRG-evolved interaction with and without\nthe chiral three-body interaction. For all intents and pur-\nposes, we will now treat the SRG-evolved interaction as\na separate, physically realistic interaction independent of\nthe bare interaction.\nIn Fig. 2 and Fig. 4, we show results for the\nCOM contaminated and translationally invariant nonlo-\ncal proton and neutron density of4;8He using the NN-\nN4LO(500)+3Nlnl interaction. Nmax= 14 andNmax=\n10 basis spaces were used, respectively, with a \row pa-\nrameter\u0015SRG= 2:0 fm-1and an oscillator frequency of\n~\n = 20:0 MeV. To appreciate the magnitude of spuri-\nous COM contamination in light nuclei, notice the signif-\nicant di\u000berences in the predicted structure of the4;8He5\nFigure 2. Ground state4He nonlocal proton and\nneutron densities calculated using the SRG-evolved NN-\nN4LO(500)+3Nlnl interaction with an Nmax= 14 basis space.\nAn oscillator frequency of ~\n = 20:0 MeV and a \row param-\neter of\u0015SRG= 2:0 fm-1were used for the calculations.\nFigure 3. Ground state trinv local density comparison\nfor4He. Panel a: We show calculations with two-body\n(NN+3Nind SRG) and two- plus three-body (NN+3N SRG)\nSRG-evolved interactions. Panel b: We show calculations\nwith the bare two-body NN-N4LO(500) interaction.\nsystems between the wiCOM andtrinv densities. The\nstructure di\u000berences are particularly noticeable at small\nrandr0. The trinv has noticeably sharper features and\nthe edges of the density tend to fall o\u000b more rapidly than\nin the case of the wiCOM density. Clearly there is sub-\nstantial suppression of the density at small distances and\na smoothing of the density over large distances coming\nfrom the COM contamination. It is also important to\nnote the di\u000berences in density for protons and neutrons\nFigure 4. Ground state8He nonlocal proton and neutron\ndensities calculated using the NN-N4LO(500)+3Nlnl interac-\ntion with an Nmax= 10 basis space. An oscillator frequency\nof~\n = 20:0 MeV and a \row parameter of \u0015SRG= 2:0 fm-1\nwere used for the calculations.\nin a system such as8He, where we see a visible smoothing\nof the neutron density over greater distances due to the\nexotic structure of the nucleus.\nFor comparison, in Fig. 5, we present a similar calcu-\nlation for the proton and neutron densities of16O. There\nare noticeably smaller e\u000bects from the COM removal in\ncomparison to the very light systems,4;8He. A notable\nfeature of the COM contamination is that it diminishes\nwith increasing A, and so reduced e\u000bects are expected in\nlarger systems. While the peaks of the trinv density are\nnot quite as pronounced, the smoothing e\u000bect appears to\npresent in this larger system as the edges still fall to zero\nslightly more rapidly than the wiCOM density. Never-\ntheless, while the COM removal e\u000bects are reduced, ob-\njects or observables highly sensitive to the structure of\nthe density will still be impacted by these di\u000berences, as\nshown in Ref. [2]. One would then expect that an object\nsuch as the kinetic density, a term dependent upon a gra-\ndient on each coordinate, Eq. (10), would experience an\nampli\fcation of these structure di\u000berences.\nWe now present the local proton and neutron densities,\n\u001aN(r) =\u001aN(r;r), for4;8He and16O for further analysis.\nReferring to Fig. 6 and Fig. 7 for the local densities of\nlight nuclei, there are notably drastic e\u000bects resulting\nfrom the COM removal procedure. If accurate nuclear\nstructure calculations are to be performed for lighter sys-\ntems, one must properly treat the COM contamination\nin these systems. Additionally, in studying the local den-\nsities of16O in Fig. 8, one can see structural di\u000berences\npresent in the larger system which were not so easily ob-\nserved in the nonlocal density \fgures. From the local\ndensities we observe that these structure di\u000berences are\napparent and still relevant in the larger systems, even6\nFigure 5. Ground state16O nonlocal proton and neutron\ndensities calculated using the NN-N4LO(500)+3Nlnl interac-\ntion with an Nmax= 8 importance truncated basis space. An\noscillator frequency of ~\n = 20:0 MeV and a \row parameter\nof\u0015SRG= 2:0 fm-1were used for the calculations.\nFigure 6. Ground state4He local proton ( panel a ), neutron\n(panel b ), and total densities ( panel c ) computed using the\nNN-N4LO(500)+3Nlnl interaction with an Nmax= 14 basis\nspace, an oscillator frequency of ~\n = 20:0 MeV, and a \row\nparameter of \u0015SRG= 2:0 fm-1.\nthough the COM contribution diminishes with increas-\ningA-nucleon number. As a result, we expect that the\nCOM removal process will produce noticeable changes in\nthe kinetic densities for16O.\nFigure 7. Ground state8He local proton ( panel a ), neutron\n(panel b ), and total densities ( panel c ) computed using the\nNN-N4LO(500)+3Nlnl interaction with an Nmax= 10 basis\nspace, an oscillator frequency of ~\n = 20:0 MeV, and a \row\nparameter of \u0015SRG= 2:0 fm-1.\nFigure 8. Ground state16O local proton ( panel a ), neu-\ntron ( panel b ), and total densities ( panel c ) computed us-\ning the NN-N4LO(500)+3Nlnl interaction with an Nmax= 8\nimportance truncated basis space, an oscillator frequency of\n~\n = 20:0 MeV, and a \row parameter of \u0015SRG= 2:0 fm-1.\nB. Kinetic density\nIn the following section we present the main result\nof this work; kinetic densities computed from ab initio\nNCSM nonlocal densities using the method outlined in\nSec. II C. For completeness, we present results ranging\nfrom4He to16O, though we emphasize that any reason-\nable comparison with DFT can only be done with the\nlatter.\nAs for the densities, we present results for4He using7\nFigure 9. Ground state trinv kinetic density comparison\nfor4He. Panel a: We show calculations with two-body\n(NN+3Nind SRG) and two- plus three-body (NN+3N SRG)\nSRG-evolved interactions. Panel b: We show calculations\nwith the bare two-body NN-N4LO(500) interaction. Nonlo-\ncal densities were computed as previously described in Fig. 1\nand Fig. 2, respectively.\nSRG-evolved chiral two-body (NN+3Nind SRG) and chi-\nral two- plus three-body (NN+3N SRG) interactions in\nthe top panel, as well as the bare NN-N4LO(500) inter-\naction kinetic densities in the bottom panel of Fig. 9. As\npreviously discussed, we see signi\fcant di\u000berences with\nthe inclusion of the chiral three-body interaction terms\nwhen using the SRG-evolved interaction to compute the\nkinetic density. The most signi\fcant di\u000berences in the\npredicted structure of the SRG-evolved and bare inter-\nactions occur at ranges of less than one fermi. To re-\niterate, moving forward we will treat the SRG-evolved\nNN-N4LO(500)+3Nlnl interaction as a di\u000berent, physi-\ncally realistic interaction.\nLet us now consider the lighter systems to gauge the\nsigni\fcance of the COM removal process, and to under-\nstand how the COM contamination may impact objects\ndependent on the nonlocal density. In Figs. 10, 11 and\n12, we present results for the kinetic density of4;6;8He,\nrespectively, with the nonlocal proton and neutron densi-\nties computed as previously described in Sec. III A. As ex-\npected, for small A-nucleon systems we observe tremen-\ndous di\u000berences in the trinv and wiCOM kinetic den-\nsities, most signi\fcantly in the case of4He. The am-\npli\fcation of the density structure di\u000berences is quite\npronounced, and further we see the suppression previ-\nously attributed to the COM contamination appearing in\nthe kinetic densities. We \fnd the maximum suppression\noccurring in the short range distances, while the COM\ncontamination tends to spread the kinetic densities over\nlarger distances, as was observed for the nonlocal densi-\nties. Notice that not only do we see signi\fcant di\u000berences\nwith smallr, but we see fairly pronounced changes in the\nFigure 10. Ground state4He comparisons between the trinv\nand wiCOM kinetic densities. Proton ( panel a ), neutron\n(panel b ), and total kinetic densities ( panel c ) are shown.\nThe nonlocal density was computed as previously described\nin Sec. III A. The expectation value of the intrinsic kinetic\nenergy for4He is 51:91 MeV.\nFigure 11. Ground state6He comparisons between the trinv\nand wiCOM kinetic densities. Proton ( panel a ), neutron\n(panel b ), and total kinetic densities ( panel c ) are shown.\nThe nonlocal density was computed as previously described\nin Sec. III A. The expectation value of the intrinsic kinetic\nenergy for6He is 78:26 MeV.\nlong range behavior of the kinetic density in nuclei like\n6;8He.\nLet us now consider heavier A-nucleon systems, which\ncan provide a method of directly gauging the impact of\nCOM contamination in energy density functionals. In\nFig. 13 and Fig. 14, we present the results for the kinetic\ndensity of12C, with the nonlocal densities computed as\npreviously described in Sec. III A. As expected from pre-\nvious results, the wiCOM nonlocal densities suppress the8\nFigure 12. Ground state8He comparisons between the trinv\nand wiCOM kinetic densities. Proton ( panel a ), neutron\n(panel b ), and total kinetic densities ( panel c ) are shown.\nThe nonlocal density was computed as previously described\nin Sec. III A. The expectation value of the intrinsic kinetic\nenergy for8He is 116:30 MeV.\nkinetic density for small r, however the e\u000bect is not as\npronounced as in the lighter systems of4;6;8He.\nIn both the12C and16O systems, we see signi\fcantly\nreduced e\u000bects during the COM removal process. This\nmay be in part due to the highly spherical shape of a sys-\ntem such as16O, though this requires further inspection.\nWhile reduced, the trinv andwiCOM nonlocal densities\nmaintain a non-negligible di\u000berence which may provide\nsome corrections if used as an input for energy density\nfunctionals.\nLet us now turn to a discussion on the integration of\nthe kinetic density operator. Note that in this work we\nconsider solely the J= 0 ground states. In the case of\nthe translationally invariant kinetic density, upon inte-\ngration over the spatial coordinates, we exactly repro-\nduce the expectation value of the ground state intrinsic\nkinetic energy of the nucleus, which can be independently\ncalculated from two-body densities introduced in second\nquantization. The expectation value is given by Eq. (15),\nhTinti=1\n4X\nabcdhabjTintjcdi\n\u0002SDhA\u0015JTjay\naay\nbadacjA\u0015JTiSD:(15)\nWhen considering the COM contaminated kinetic den-\nsity, one recovers the expectation value of the intrinsic\nkinetic energy plus the expectation value of the kinetic\nenergy of the COM. The results for the hTintiare sum-\nmarized in Table I. The recovery of the intrinsic kinetic\nenergy after COM removal is direct con\frmation of suc-\ncess of the procedure, and can be summarized by the\nFigure 13. Ground state12C comparisons between the trinv\nand wiCOM kinetic densities. Proton ( panel a ), neutron\n(panel b ), and kinetic total densities ( panel c ) are shown.\nThe nonlocal density was computed as previously described\nin Sec. III A. The expectation value of the intrinsic kinetic\nenergy for12C is 219:84 MeV.\nFigure 14. Ground state16O comparisons between the trinv\nand wiCOM kinetic densities. Proton ( panel a ), neutron\n(panel b ), and total kinetic densities ( panel c ) are shown.\nThe nonlocal density was computed as previously described\nin Sec. III A. The expectation value of the intrinsic kinetic\nenergy for16O is 301:69 MeV.\nfollowing set of relations, Eq. (16) and Eq. (17),\nhTinti=SDhA\u0015JTj\u0012~2\n2m\u001ctrinv\n0\u0013\njA\u0015JTiSD\n=~2\n2mZ1\n0r2\u001ctrinv\n0(r)dr;(16)9\nFigure 15. Ground state Nmaxconvergence results for4He\ntrinv kinetic neutron density. The nonlocal density was com-\nputed as previously described in Sec. III A.\nFigure 16. Ground state Nmaxconvergence results for16O\ntrinv kinetic neutron density. The nonlocal density was com-\nputed as previously described in Sec. III A.\nhTwiCOMi=SDhA\u0015JTj\u0012~2\n2m\u001cwiCOM\n0\u0013\njA\u0015JTiSD\n=~2\n2mZ1\n0r2\u001cwiCOM\n0 (r)dr\n=SDhA\u0015JTj~2\n2m\u0012\n\u001cint\n0+\u001cCOM\n0\u0013\njA\u0015JTiSD\n=hTinti+3\n4~\n;\n(17)\nwheremis the nucleon mass and \u001c0is the total kinetic\ndensity. Note that these relations are always true in the\nNCSM, whereas in other methods are only true if con-\nvergence to an exact many-body solution is achieved. InNucleus NmaxhTinti Error (\u0006)\n4He (bare) 18 62.73\u00060.01 %\n4He 14 51.91\u00060.01 %\n6He 12 78.27\u00061.4 %\n8He 10 116.30\u00063.1 %\n12C 8 IT 219.84\u00061.2 %\n16O 8 IT 301.69\u00060.8 %\nTable I. Ground state mean intrinsic kinetic energy values\nand percent errors for all aforementioned nuclei calculated us-\ning the NN-N4LO(500)+3Nlnl interaction (except4He-bare,\nwhich are the results for the bare NN-N4LO(500) interaction).\nAllhTintivalues are in MeV. Note ITrefers to an importance\ntruncated basis space. Percent errors are calculated using the\ndi\u000berence between the maximal Nmaxvalue and the previous\nvalue.\nFig. 15 and Fig. 16, we present ground state Nmaxcon-\nvergence plots for the kinetic density of the nuclei4He\nand16O. We achieve rapid convergence in4He when ap-\nplying the NN-N4LO(500)+3Nlnl interaction at a basis\nsize ofNmax= 10, as the \fnal three Nmaxcalculations\n(Nmax= 10;12;14) overlap completely. Similarly, we are\nable to see good convergence trends in16O at an impor-\ntance truncated basis size of Nmax= 8, as this calcula-\ntion is only mildly di\u000berent from the the Nmax= 6 basis\nspace calculation. Let it be noted that given our use of\nthe harmonic oscillator basis, all densities - and density\ndependent quantities - have \\unphysical\" asymptotic be-\nhaviour due to the Gaussian tail resulting from the basis\nexpansion.\nC. Comparison to basic COM treatment in DFT\nLet us now revisit the form of Eq. (9). This Hkinetic\nterm has no additional treatment for the COM contam-\nination. However, a basic COM treatment can be intro-\nduced in DFT [38{40]. In Eq. (18), a term inversely pro-\nportional to the number of nucleons is subtracted from\nthe standard Hkinetic to treat the COM contamination:\nHkinetic (~ r) =~2\n2m\u0012\n1\u00001\nA\u0013\n\u001c0(~ r); (18)\nwhere\u001c0would be\u001cwiCOM in our calculations. In Fig. 17,\nwe show trinv,wiCOM , and DFT calculations of the ki-\nnetic density for4;8He,12C, and16O, obtained using the\nNN-N4LO(500)+3Nlnl interaction. The DFT curve is10\nFigure 17. Ground state total kinetic density results for4He (panel a ),8He (panel b ),12C (panel c ), and16O (panel\nd) calculated with the NN-N4LO(500)+3Nlnl interaction. The nonlocal densities for the nuclei were computed as previously\ndescribed in Sec. III A. The DFT kinetic density was obtained by using Eq. (19), \u001cDFT(r) = (1\u00001\nA)\u001cwiCOM (r).\nobtained by application of Eq. (18), so\n\u001cDFT(~ r) =\u0012\n1\u00001\nA\u0013\n\u001cwiCOM (~ r): (19)\nThe most important item to note about the plots is the\ndi\u000berence in the kinetic density pro\fle when comparing\ntheab initio calculation to the mock DFT calculation,\nwhich includes the aforementioned COM treatment. The\ndi\u000berences between the predicted kinetic density struc-\nture are easier seen in the lighter nuclei, where the e\u000bects\nof COM removal are more drastic. Nevertheless these\ne\u000bects are still appreciable in the larger systems under\nconsideration. The DFT calculation including the COM\ntreatment has reduced the overall size of the wiCOM ki-\nnetic density signi\fcantly. In particular, the inclusion\nof this1\nAterm pushes the short range segments of the\nDFT curve further from the ab initio translationally in-\nvariant kinetic density, whereas the long range portions\nare pushed closer. As expected, with increasing nucleon\nnumber the total change from the wiCOM kinetic den-\nsity is reduced, yet still non-negligible in a system such\nas16O. This comparison has shown that perhaps this\n\frst order COM correction employed in DFT is an in-\naccurate method of treating contamination in low-mass\nnuclei, pushing the structure of the kinetic density fur-\nther from the ab initio prediction. While this was noted\nas a potential source of error of the technique, inducingslightly unphysical trends with respect to the nucleon\nnumber [39, 40], there have been few e\u000borts to quantify\nthe magnitude of the e\u000bect with respect to ab initio cal-\nculations. As was the purpose of this work, we may now\ndirectly compare NCSM and DFT kinetic density results\nand, as it is not itself an observable, we may compare\nthe e\u000bects that these structural changes will have on the\nminimization of the energy density functional. We antic-\nipate that these COM corrections in the kinetic density\nwill produce \fne structure corrections to ground state\nobservables.\nIn Table II, we present the mean kinetic energy values\nfor the trinv,wiCOM , and DFT calculations. Compar-\ning thehTintiandhTDFTicolumns, one can see that the\nmean values agree well across both COM removal tech-\nniques, with the hTDFTiconsistently slightly underesti-\nmating the true value of the mean. Notably, the increas-\ning nucleus size increases the di\u000berence between the ab\ninitio and DFT mean intrinsic kinetic energy values, indi-\ncating some form of unphysical trend with respect to the\nA-nucleon number in the DFT calculation. The inclu-\nsion of this1\nAterm in the DFT calculation does appear\nto reduce the integral of the kinetic density appropriately,\ne\u000bectively removing spurious COM contamination from\nthe mean value intrinsic kinetic energy, albeit with a very\ndi\u000berent structural prediction for the kinetic density.11\nNucleus NmaxhTintihTwiCOMihTDFTi\n4He 14 51.91 66.91 50.18\n6He 12 78.26 93.26 77.72\n8He 10 116.30 131.30 114.89\n12C 8 IT 219.84 234.84 215.27\n16O 8 IT 301.69 316.69 296.90\nTable II. Ground state mean intrinsic kinetic energy values\nusing trinv,wiCOM , and DFT kinetic densities for all afore-\nmentioned nuclei, calculated with the NN-N4LO(500)+3Nlnl\ninteraction. AllhTiivalues are in MeV. Note ITrefers to an\nimportance truncated basis space. The hTDFTiis calculated\nby using Eq. (19), hTDFTi= (1\u00001\nA)hTwiCOMi. The values\nofhTintiandhTwiCOMidi\u000ber just by3\n4~\n, see Eq. (17).\nIV. CONCLUSIONS\nMotivated by the recent e\u000borts to connect DFT and ab\ninitio approaches to the nuclear many-body problem, the\npurpose of this work was to provide ab initio predictions\nfor the nuclear kinetic density, a fundamental input of en-\nergy density functionals in DFT, such that comparisons\ncan then be produced for both the many-body meth-\nods and the COM removal techniques. We used the ap-\nproach of Ref. [2] to construct both COM contaminated\nand translationally invariant nonlocal one-body densities.\nThe kinetic densities were then computed following the\nprocedure outlined in Sec. II C, which provided an ana-\nlytic expression in terms of the one-body density matrix\nelements that was then evaluated numerically. The nu-\nclear density and kinetic density results were obtained\nusing the SRG-evolved NN-N4LO(500)+3Nlnl chiral in-teraction [2, 25, 26].\nThe calculation of the one-body density matrix ele-\nments and nonlocal densities requires the knowledge of\nthe many-body nuclear wave functions, which in this\nwork were computed from the ab initio NCSM approach.\nIn Sec. III A, we showed results with and without the\nground state COM contamination for the densities of\n4;8He and16O, obtained from the NCSM wave functions.\nAs observed in the Sec. III B, the COM removal pro-\ncess produces non-negligible structure changes in both\nthe nonlocal densities and, further, in the kinetic densi-\nties of4;6;8He,12C and16O. In Sec. III C, we performed\na comparison of the trinv kinetic density to a basic COM\nremoval technique used in DFT [38{40]. While the COM\ntreatment provided good agreement for the mean value\nintrinsic kinetic energy of the nuclei, the DFT kinetic\ndensity was shown to be structurally di\u000berent from the ab\ninitio calculations, forcing long-range behavior closer to\nand short-range behavior further from the NCSM result.\nComparisons such as this provide insight into re\fning en-\nergy density functionals, perhaps providing \fne structure\ncorrections to ground state observables in DFT. It should\nbe noted that given the latest DFT developments, e.g.,\nin Ref. [15], where it is attempted to derive the energy\ndensity functionals from chiral forces, a direct compar-\nison to our calculations will be possible as exactly the\nsame chiral forces can be used as input in both types of\ncalculations.\nIn conclusion, the development of a general nonlocal\ndensity allows for the calculation of fundamental quanti-\nties taken as input in theories such as DFT. This provides\nthe communities with a means to better gauge the di\u000ber-\nences in many-body techniques and procedures for COM\nremoval. Although the COM removal e\u000bect is reduced\nin largerA-nucleon systems, it is still non-negligible and\ncan motivate the need to include a procedural technique\nfor removing the COM or motivate a check against the\nexisting techniques of COM removal.\nV. APPENDIX\nA. Derivative of radial harmonic oscillator function\nTo begin, we introduce existing derivative and recurrence relations for Laguerre polynomials:\nd\ndrLl\nn(r) =\u0000Ll+1\nn\u00001(r) (20)\nLl\nn(r) +Ll+1\nn\u00001(r) =Ll+1\nn(r) (21)\nRecall that the radial harmonic oscillator (RHO) function is given by\nRn;l(r) =s\n2\u0000(n+ 1)\n(b2)l+3\n2\u0000(n+l+3\n2)rlexp\u0010\n\u0000r2\n2b2\u0011\nLl+1\n2n\u0012r2\nb2\u0013\n; (22)12\nwherebis the harmonic oscillator length and \u0000 is the gamma function. We now de\fne\n\rn;l;b=s\n2\u0000(n+ 1)\n(b2)l+3\n2\u0000(n+l+3\n2)(23)\nfor simplicity. Taking the radial derivative and using (20), we have\ndRn;l\ndr=\rn;l;bh\nlrl\u00001exp\u0010\n\u0000r2\n2b2\u0011\nLl+1\n2n\u0012r2\nb2\u0013\n\u0000rl+1\nb2exp\u0010\n\u0000r2\n2b2\u0011\nLl+1\n2n\u0012r2\nb2\u0013\n\u00002rl+1\nb2exp\u0010\n\u0000r2\n2b2\u0011\nLl+3\n2\nn\u00001\u0012r2\nb2\u0013i\n:(24)\nNow making use of (21) and rewriting in terms of RHO functions,\ndRn;l\ndr=l\nrRn;l(r)\u0000\rn;l;brl+1\nb2exp\u0010\n\u0000r2\n2b2\u0011h\nLl+1\n2n\u0012r2\nb2\u0013\n+ 2Ll+3\n2\nn\u00001\u0012r2\nb2\u0013i\ndRn;l\ndr=l\nrRn;l(r)\u0000\rn;l;brl+1\nb2exp\u0010\n\u0000r2\n2b2\u0011h\nLl+3\n2n\u0012r2\nb2\u0013\n+Ll+3\n2\nn\u00001\u0012r2\nb2\u0013i\ndRn;l\ndr=l\nrRn;l(r)\u00001\nb2\rn;l;b\"\nRl+1\nn(r)\n\rn;l+1;b+Rl+1\nn\u00001(r)\n\rn\u00001;l+1;b#(25)\nwe can derive our \fnal result,\ndRn;l\ndr=l\nrRn;l\u00001\nb\u0014r\nn+l+3\n2Rn;l+1(r) +pnRn\u00001;l+1(r)\u0015\n(26)\nB. Derivation of kinetic density\nWe begin from the expression of the kinetic density in terms of the nonlocal density, Eq. (10), and expand using\nthe relation ~r\u0001~r0=P\n\u0016=\u00061;0(\u00001)\u0016r\u0016r0\n\u0000\u0016. We may write the kinetic density in component form as\n\u001cN(~ r) =\u0014\nr0r0\n0\u001aN(~ r;~ r0)\u0000r +1r0\n\u00001\u001aN(~ r;~ r0)\u0000r\u00001r0\n+1\u001aN(~ r;~ r0)\u0015\n~ r=~ r0: (27)\nLet us consider solely the contribution of the r0r0\n0\u001aN(~ r;~ r0)j~ r=~ r0component since the procedure is identical for all\nthree components. We suppress the Nisospin label. It is convenient to rewrite this expression as follows,\nr0r0\n0\u001a(~ r;~ r0)j~ r=~ r0=X\nn;l;n0;l0;K;k;ml;ml0\u000bK;i;f\nn;l;n0;l0(lmll0ml0jLM)\u0014\nr0Rn;l(r)Y\u0003\nl;ml(^r)\u0015\u0014\nr0\n0Rn0;l0(r0)Y\u0003\nl0;ml0(^r0)\u0015\n;(28)\nwhere\u000bK;i;f\nn;l;n0;l0is de\fned in Eq. (12). Using the following relation from Ref. [37],\nr0Rn;l(r)Y\u0003\nl;ml(^r) =s\n(l+ 1)2\u0000m2\nl\n(2l+ 1)(2l+ 3)\u0012dRn;l(r)\ndr\u0000l\nrRn;l(r)\u0013\nY\u0003\nl+1;ml(^r)\n+s\nl2\u0000m2\nl\n(2l\u00001)(2l+ 1)\u0012dRn;l(r)\ndr+l+ 1\nrRn;l(r)\u0013\nY\u0003\nl\u00001;ml(^r);(29)\nfor both ther0andr0\n0terms, we then expand and evaluate our result at ~ r=~ r0to arrive at the formula shown\nin Eq. (30). Note that we group all spherical harmonics under the same collective index Linstead of having four13\nseparate angular momentum indices. We have\nr0r0\n0\u001a(~ r;~ r0)j~ r=~ r0\n=X\nn;l;n0;l0;K;k;ml;ml0\u000bK\nn;l;n0;l0(lmll0ml0jLM)\n\u0002\"\nc00\u0012dRn;l(r)\ndr\u0000l\nrRn;l(r)\u0013\u0012dRn0;l0(r)\ndr\u0000l0\nrRn0;l0(r)\u0013\n+c01\u0012dRn;l(r)\ndr\u0000l\nrRn;l(r)\u0013\u0012dRn0;l0(r)\ndr+l0+ 1\nrRn0;l0(r)\u0013\n+c02\u0012dRn;l(r)\ndr+l+ 1\nrRn;l(r)\u0013\u0012dRn0;l0(r)\ndr\u0000l0\nrRn0;l0(r)\u0013\n+c03\u0012dRn;l(r)\ndr+l+ 1\nrRn;l(r)\u0013\u0012dRn0;l0(r)\ndr+l0+ 1\nrRn0;l0(r)\u0013#\nY\u0003\nLM(^r);(30)\nwhere thec0jcoe\u000ecients are complicated angular momentum factors. As an example, the c00factor is provided below\nin Eq. (31).\nc00=s\n(l+ 1)2\u0000m2\nl\n(2l+ 1)(2l+ 3)s\n(l0+ 1)2\u0000m2\nl0\n(2l0+ 1)(2l0+ 3)s\n(2l+ 3)(2l0+ 3)\n4\u0019(2L+ 1)\n\u0002(l+ 1mll0+ 1ml0jLM) (l+ 1 0l0+ 1 0jL0)(31)\nA similar procedure can be performed for the r+1r0\n\u00001\u001aNandr\u00001r0\n+1\u001aNcomponents of the kinetic density. 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Maruhn,\nThe European Physical Journal A-Hadrons and Nuclei\n7, 467 (2000)."    },    {        "title": "1809.07496v1.Optimal_mass_transport_and_kernel_density_estimation_for_state_dependent_networked_dynamic_systems.pdf",        "content": "Optimal Mass Transport and Kernel Density Estimation\nfor State-Dependent Networked Dynamic Systems\nMathias Hudoba de Badyn, Utku Eren, Behc ¸et Ac ¸ıkmes ¸e and Mehran Mesbahi\nAbstract — State-dependent networked dynamic systems are\nones where the interconnections between agents change as a\nfunction of the states of the agents. Such systems are highly\nnonlinear, and a cohesive strategy for their control is lacking\nin the literature. In this paper, we present two techniques\npertaining to the density control of such systems. Agent states\nare initially distributed according to some density, and a\nfeedback law is designed to move the agents to a target density\nprofile. We use optimal mass transport to design a feedforward\ncontrol law propelling the agents towards this target density.\nKernel density estimation, with constraints imposed by the\nstate-dependent dynamics, is then used to allow each agent\nto estimate the local density of the agents.\nI. INTRODUCTION\nNetworked dynamic systems arise in many synthetic and\nnatural systems in science and engineering [1]; in particular\nmulti-agent systems offer an interesting control paradigm.\nEach agent augments the system with an additional (local)\ncomputational resource, motivating the concept of distributed\ncontrollers & estimators [2], [3], [4], leading to a notion of\nlocal control versus global control. The latter seeks to design\ncontrol laws to guide groups of agents to a desired objective,\nand the former seeks to design on-board controllers for each\nagent to facilitate their role in the global control law.\nA class of the more widely used local control laws is\ncalled consensus , where each agent averages data from\ntheir neighbours to compute a parameter related to their\nobjective – for example, heading, position or a formation\ncenter [1], [5], [6]. The attractive feature of consensus is\nhow the interconnections between agents – the so-called\nnetwork orgraph topology – affect the performance and\nagreement characteristics of the algorithm [7], [8]. Graph-\ntheoretic characteristics, such as symmetry, structural balance\nand graph spectra provide additional insights into the control-\ntheoretic behaviour of consensus [9], [10], [11], [12], [13].\nGlobal controllers for multi-agent systems take many\nforms [14], eg., potential field approaches [15], smoothed\nparticle hydrodynamics [16] and density control [17]. In [18],\n[19], density control with only relative measurements of\nposition between agents is considered, and mean-field control\nis used to tackle multi-agent interactions by considering a\nmass flow in the large- Nlimit [20], [21], [22].\nThe main interest area of the current paper is net-\nworked dynamic systems in which the underlying network\nis time-varying. Examples of such systems are switching\nand proximity-based consensus [1], [6], [23] and the Vicsek\nflocking model [24]. State-dependency refers to networks in\nThe research of the authors was supported by NSERC un-\nder grant number CGSD2-502554-2017, the U.S. ARL and the\nU.S. ARO under contract number W911NF-13-1-0340, and the U.S.\nAFOSR under grant number FA9550-16-1-0022. The authors are\nwith the William E. Boeing Department of Aeronautics & As-\ntronautics, University of Washington, Seattle WA, USA. Emails:\nfhudomath,ue,behcet,mesbahi g@uw.edu ©IEEE 2018which the underlying graph varies based on the state of the\nnodes. State-dependent networks have been considered in [7],\n[20], [22], [25], [26], [27], [28], but few underlying princi-\nples for designing controllers to account for this difficult\nnonlinearity has been proposed. A wide range of real-life\nnetworked systems are state-dependent.\nTo tackle this problem, we propose a twofold extension\nof the work in [17]. First, we consider state-dependent\nnetworked dynamic systems, instead of single-integrator\ndynamics. Second, we propose a control method for these\nsystems by using a feedback-based density control law that\nutilizes optimal mass transport (OMT). OMT was considered\nfor linear systems in [29], non-linear systems in [30] and\nfor density tracking of non-interacting agents in [31]. Our\ncontribution considers OMT for multi-agent systems, in\nparticular ones with state-dependent dynamics.\nIn the OMT problem, the initial and final densities \u001a0&\u001a1\nof the agents are specified. The solution to the problem yields\na time-dependent density profile with boundary conditions\nimposed by \u001a0;\u001a1, and a velocity field that together satisfy\nthe continuity equation.\nWe aim to use this velocity field as a feedforward control\ninput to the state-dependent multi-agent system, coupled with\na density control feedback law. Using a modified form of\nkernel density estimation (KDE) that takes into account the\nstate-dependent dynamics, we will show that the combination\nof the two control techniques will allow us to propose\na physically realizable control strategy for state-dependent\nnetworked dynamic systems.\nThis paper is organized as follows. The mathematical\npreliminaries, including notation, optimal mass transport, and\ndensity control with the KDE procedure are reviewed in §II.\nThe problem statement and paper contributions are outlined\nin §III. We present the feedforward controller based on OMT\nin §IV, and the state-dependent KDE in §V. Examples are\nprovided in §VI, and the paper is concluded in §VII.\nII. MATHEMATICAL PRELIMINARIES AND\nBACKGROUND\nA. Mathematical Notation\nWe follow the standard graph theory notation listed in [1].\nA measure space (\u0006;A;\u0016)is a triple containing a sample\nspace \u0006\u001aRn, a\u001b-algebra of subsets of \u0006, and a measure\n\u0016that assigns the ‘size’ \u0016(A)2R+to a setA2A. The\nBorel\u001b-algebraBis generated from the countable unions,\nintersections and complements of open subsets of Rn. The\nLebesgue measure \u0015assigns to a closed interval [a;b]\u001aR\nthe ‘size’b\u0000a; this can be extended to Rnby considering\nproducts of measures. A measure \u0016is called absolutely\ncontinuous with respect to a measure \u0017if\u0017(A) = 0 =)\n\u0016(A) = 0 forA2 A ;\u0016is called Lebesgue absolutely\ncontinuous if\u0017=\u0015. This is denoted \u0016\u001c\u0017. If\u0016\u001c\u0017,arXiv:1809.07496v1  [math.OC]  20 Sep 2018there exists a function f:R!R+called a Radon-Nikodym\nderivative , ordensity , of\u0016with respect to \u0017. It is denoted\nf:=d\u0016\nd\u0017, and satisfies the property that for all A2 A ,\n\u0016(A) =R\nAfd\u0017. Let\u0019(x;y)be a joint measure on X\u0002Y .\nWe denote the set of such joint measures \u0019byP(X;Y). The\nmarginal\u0019xof\u0019onXis defined as the push-forward under\nthe projection map XonX:\u0019x=X#\u0019;whereX(x;y) =x.\nSimilarly, the marginal \u0019yof\u0019onYis given by\u0019y=Y#\u0019;\nwhereY(x;y) =y. We denote the convolution of two\nfunctionsf;g asf ?g , or the convolution of a function f\nand measure \u0016asf ?\u0016 .\nB. Optimal Mass Transport\nInformally speaking, the optimal mass transport problem\nis to find a mapping between two densities that minimizes\nsome cost. Formally speaking, we consider two measures1\n\u00160;\u00161onRnwith equal mass:R\nRnd\u00160=R\nRnd\u00161:The\noptimal mass transport problem [32], [33] is to find a\nmeasurable map T:Rn!Rntaking\u00160to\u00161via the\nfollowing optimization problem:\nminimizeR\nRnc(x;T(x))d\u00160(x)\nsubject toR\nx2Ad\u00161(x) =R\nT(x)2Ad\u00160(x);8A\u001aRn;(1)\nwherecis a cost function depending on the initial and\ntransported masses. The constraint in Problem (1) means\nthat\u00161is the push-forward measure of \u00160under the map\nT, in that for each Borel set B2B:=\u001b(Rn), we have that\n\u00161(B) =\u00160\u0000\nT\u00001(B)\u0001\n. This is denoted as T#\u00160=\u00161.\nA generalization of Problem (1) by Kantorovich is able to\npick out the optimal map T, if it exists, for a certain class\nof costscunder the assumption of absolute continuity of the\nmeasures [34]. Here, we consider a joint distribution \u0019(x;y)\nonRn\u0002Rnand solve for the optimal admissible measure\n\u0019given some cost c(x;y).\nThe set of admissible measures \u0019are those whose\nmarginals are \u00160and\u00161:X#\u0019=\u00160; Y#\u0019=\u00161:This\nis equivalent to requiring\n\u0019(A\u0002Rn) =\u00160(A); \u0019(Rn\u0002B) =\u00161(B) (2)\nfor all measurable A\u001aRnandB\u001aRn. The Kantorovich\nrelaxed optimal mass transport problem [34] is given by\nminimizeR\nRn\u0002Rnc(x;y)d\u0019(x;y)\nsubject to \u00192f\u00192P(Rn;Rn) s:t:(2) holdsg:(3)\nProposition 1 ([32], [33]): For quadratic costs c(x;y) =\nkx\u0000yk2, the support of the optimal joint measure \u0019\u0003(x;y)\nof Problem (3) is exactly the graph of the optimal map T\u0003(x)\nminimizing Problem (1).\nFor quadratic costs, Benamou and Brenier formulated an\nequivalent problem in terms of a constrained fluid mechanics\nmodel.\nProposition 2 ([35]): Given Lebesgue absolutely contin-\nuous\u00160;\u00161with Radon-Nikodym densities \u001a0;\u001a1respec-\ntively, Problem (3) with quadratic costs is equivalent to\ninf\u001a;vR\nRnR1\n01\n2kv(t;x)k2\u001a(t;x)dtdx\nsubject to@\u001a\n@t+r\u0001(v\u001a) = 0\n\u001a(0;x) =\u001a0(x); \u001a(1;y) =\u001a1(y):(4)\n1If the measures are Lebesgue absolutely continuous, one can equivalently\nconsider densities \u001a0;\u001a1.Furthermore, the solution to Problem (4) is of the form\nv(t;x) =r'(t;x);where'(t;x)is the Lagrange multiplier\nof the constraints and the solution to the Hamilton-Jacobi\nequation@t\u001e+1\n2jr\u001ej2= 0.\nRemark 1: The optimal map T\u0003of Problem (3) in the case\nof quadratic costs can be reconstructed from the variable\nv(t;x)from the solution of Problem (4). This formally\nestablishes the equivalence stated in Proposition 2 [35].\nC. Density Control and Kernel Density Estimation\nIn [17], a feedback control law to drive a group of\nsingle-integrator agents to a desired density profile \u001a1(x;t)\nwas analyzed. The following feedback law is proposed to\ncompute the velocity field as a function of the error in density\n\b(x;t) :=\u001a(x;t)\u0000\u001a1(x)as\nv(x;t) =\u0000\u000br\b(x;t)\n\u001a(x;t): (5)\nDensity control of multi-agent systems is impacted by\nthe ability of individual agents to discern the local density\nprofile from measurements of their neighbours. Since the\nnumber of agents is finite, the local density profile must be\napproximated from finitely many samples ri(t).\nThis can be accomplished using kernel density estima-\ntion [36], [37]. The kernel density estimate ^\u001a(t;x)at any\npointx2Rnand timet2R+is given by\n^\u001a(t;x) =Z\nRn\"dY\nk=11\nhkK\u0012x[k]\u0000\u0018[k]\nhk\u0013#\ndPN(t;\u0018):(6)\nHere,K:R!Ris called the smoothing kernel ,hkis called\nthesmoothing parameter , anddPN(t;\u0018)is a sum of Dirac\nmeasures at sample points:\ndPN(t;\u0018) =1\nNX\nr(t)2S(t)\u000e(\u0018\u0000r(t))d\u0018:\nSince\u000e(\u0001)is the Dirac delta functional, Equation (6) can be\nwritten as\n^\u001a(t;x) =1\nNNX\ni=1\"dY\nk=11\nhkK \nx[k]\u0000r[k]\ni\nhk!#\n:\nThe control law (5) then uses the estimate ^\u001a(x;t)in place\nof knowledge of the true density \u001a(x;t), where the sample\npointsr(t)are taken to be the agent states:\nv(x;t) =\u0000\u000br(^\u001a(x;t)\u0000\u001a1(x))\n^\u001a(x;t):\nIn [17], Gaussian kernels were used – this induces an\nall-to-all communication; every agent is able to sample\nthe position every other agent. The control law (5) has a\nconvergence guarantee listed in Theorem 6 of [17].\nIII. PROBLEM STATEMENT AND\nCONTRIBUTIONS\nIn this paper, we consider state-dependent networked dy-\nnamic systems with Nagents on a bounded region R\u001aRn,\nwhere theith agent’s state evolves according to the dynamics\n_xi=f(G(x);x) +Bui(x;t);x:= (x1;:::;xn):OMT Swarm\u001a(x;t)\nKDEv(x;t) x\n^\u001a(x;t)\nFig. 1: Block diagram of density control scheme\nA prototypical example of such a system is state-dependent\nconsensus\n_xi=X\nj6=iA(xi;xj)\u0001(xi\u0000xj) +ui; (7)\nwhere the edge weight wij:=A(xi;xj)changes depending\non the state of the agent iand its neighbour j. In general,\none can consider an interaction kernel H(x)that generates\na consensus-like dynamics by convolution with the Dirac\nmeasure supported at agent states [20], [22]:\n\u0016N(x) =1\nNNX\nj=11fxjg(x) =1\nNNX\nj=1\u000e(x\u0000xj):(8)\nUsing Equation (8), we can write a general multi-agent\nsystem as\n_xi= (H?\u0016N) (xi) +ui:\nA simple example motivated by robotics is proximity-based\nedge switching, where A(xi;xj) = 1 ifkxi\u0000xjk\u0014rand\n0 otherwise, where ris some communication radius. This\ncorresponds to an interaction kernel H(x) =x1kxk\u0014r(x):\nAs the number of agents Ngrows sufficiently large, one\ncan consider the time-dependent density of agents\u001a(x;t)\nover a region of the state space. In the context of mean-\nfield control, the formal large- Nlimit of the dynamics (7)\nproduces the mean field dynamics [20], [22],\n@\u001a\n@t+rx\u0001[(P(\u001a(x;t);t) +u)\u001a] = 0\nP(\u001a(x;t);x) =R\nA(x;y)(y\u0000x)\u001a(y;t)dy:(9)\nWe now state the contributions of this paper, namely the\nfeedforward OMT scheme with kernel density estimation\nshown in Figure 1.\nContribution 1: Feed-Forward Density Control With Opti-\nmal Mass Transport.\nWe consider a generalization of the Brenier-Benamou\nOMT problem (4) with the continuity equation constraint\nreplaced by the mean-field dynamics (9):\ninf\u001a;vRR1\n01\n2kv(t;x)k2\u001a(t;x)dtdx\nsubject to@t\u001a+rx\u0001[(P(\u001a(x;t);x) +v)\u001a] = 0\nP(\u001a(x;t);x) =R\nA(x;y)(y\u0000x)\u001a(y;t)dy\n\u001a(0;x) =\u001a0(x); \u001a(1;y) =\u001a1(y):(10)\nThe solution to Problem (10) yields two variables with\nimportant physical interpretations. The time-varying density\n\u001a(t;x)represents the mass of agents with dynamics (7)\nwith inputs ui(x;t) :=v(xi;t); the velocity field v(x;t)is\nprecisely the input uigiven to agent iat positionxat time\nt.\nProblem (10) assumes that the initial and final masses of\nagents are distributed in the mean-field limit according todensities\u001a0and\u001a1, respectively. When considering finitely\nmany agents, any initial density will take the form of\nEquation (8) - namely, it will be a Dirac measure supported\nat the agent states, also called the empirical density .\nHence, in general, the boundary conditions on the density\nin Problem (10) can be either deterministic (in the case of\nDirac measures supported at the agent states), or probabal-\nistic (in the sense that the initial/final agent states xi(0)and\nxi(1)are randomly distributed according to the densities \u001a0\nand\u001a1). In the latter case, as N!1 , the Dirac measure\nsupported at the agent states at time tconverges in a formal\nsense to the density \u001a(\u0001;t)[20], [22].\nIn either case, we consider the velocity field v(x;t)as a\nfeed-forward input to the dynamics (7). Since the number\nof agents is finite, the empirical density at time twill\nonly approximate the density \u001a(x;t)from the solution of\nProblem (10).\nContribution 2: Feedback Density Control with Kernel\nDensity Estimation and State-Dependent Constraints.\nIn a state-dependent networked dynamic system, eg.,\nEquation (7), the existence of an edge indicates some notion\nof information transfer between agents. Hence, a physical\nestimation scheme and density control law can only allow i\nto sample those agents jsuch thatA(xi;xj)6= 0. The second\ncontribution of this paper is to extend the KDE procedure\nin [17] by solving a quadratic program for an optimal kernel\nthat takes into account the state-dependent communication\nconstraints.\nIV. FEEDBACK CONTROL OF STATE-DEPENDENT\nNETWORKED DYNAMIC SYSTEMS\nConsider the state-dependent consensus dynamics (7).\nLetv1(x;t)denote the velocity field from the solution to\nProblem (10), and let v2(x;t)denote the velocity field from\nthe control law (5). Our proposed control law is then given\nby the velocity field (with \u000b>0),\nu(x;t) =8\n<\n:v1(x;t)\u0000\u000br(\u001a(x;t)\u0000\u001a1(x))\n\u001a(x;t)0\u0014t\u00141\nv2(x;t)\u0000P(\u0016(x;t);x) t\u00151(11)\nP(\u0016(x;t);x) =Z\nA(x;y)(y\u0000x)\u0016(y;t)dy:\nOf course, the switch at t= 1 is completely arbitrary, and\ncan be altered by changing the time horizon of Problem (10).\nThe main result of this section is the following theorem,\nan extension of Theorem 6 in [17]. Informally, it states that\nas the number of agents in the system tends as N!1 ,\nthe velocities of the agents performing the state-dependent\ncontrol law (11) will vanish asymptotically.\nTheorem 1: Consider a system of NagentsS(t)on a\nbounded regionR \u001a Rnwith individual dynamics given\nby(7)and with control law (11). Further suppose that\nthe initial swarm density \u001a(x;t)and target density \u001a1(x)\nsatisfy the boundary condition r\b(x;t) = 0 on@R. As\nt!1 , for sufficiently large Nthe error density \b(x;t)\nconverges to zero: limt!1\b(x;t) = 0;forx2R and so\n^\u001a(x;t)!\u001a1(x). Furthermore, the velocities of all agents\nvanish asymptotically: limt!1_x(t) = 0;forx2R:\nThe proof is discussed in the Appendix.Fig. 2: Illustration of (proximity-based) state-dependent con-\nstraints on the KDE procedure. Dotted lines indicate samples\nthe center agent cannot measure.\nV. DENSITY ESTIMATION FOR KERNELS\nWITH COMPACT SUPPORT\nConsider a state-dependent consensus dynamics as in\nEquation (7) where A(xi;xj)is a state-dependent edge\nweight. In order to implement the density control law, each\nagent must be able to estimate the density of nearby agents to\ngenerate the correct velocity field. In this section, we discuss\noptimal kernels designed to achieve this task that are subject\nto the state-dependent constraints imposed by A(xi;xj).\nThe state-dependent constraints in some (informal) sense\ndenote ‘information transfer’ between agents iandj. If\nAij= 0, then agents iandjcannot detect each other, and the\nKDE procedure should reflect this. To illustrate this notion,\nconsider Figure 2.\nIn 1D kernel density estimation, there are two parameters\nselected a priori that influence the quality of the estimated\nprobability density function, namely the kernel Kand the\nsmoothing parameter h. We consider an optimal selection\nofKsubject to the state-dependent constraints; we leave\nthe task of selecting hfor future work. For now, we just\nneed the following assumption on has a function of the\nnumber of samples: limN!1Nh(N) =1:The standard\nmetric for measuring the quality of the estimated probability\ndensity function is given by the mean integrated square error\nEMISE:=E\u0002R\n(^\u001a(x)\u0000\u001a(x))dx\u0003\n:\nBy extracting out the dependence on the number of sam-\nplesN, and the choice of smoothing parameter h, one can\nobtain the asymptotic mean integrated square error (AMISE)\n[36]:EMISE:=EAMISE +o\u0000\n(hn)\u00001+h4\u0001\n:One can factor\nthe AMISE into a product of two terms, one depending on\nhand one depending on K:EAMISE =C1(K)C2(h), where\nC1(K) :=\"\u0012Z\nK(x)2dx\u00134\u0012Z\nx2K(x)dx\u00132#1=5\n:\nHence, by fixing a:=R\nx2K(x)dxdepending on the\nlength of the boundary of our estimation horizon, the\nonly parameter left to optimize is the roughness ofK(x):R\nK(x)2dx. We can write an optimization problem as fol-\nlows:\nminimizeR\nK(x)2dx\nsubject toR\nK(x) = 1;R\nxK(x) = 0R\nx2K(x) =a2<1; K(x)\u00150:(12)\nIn one dimension, the solution to Problem (12) is given by\n[36],\nKa(x) =3\n41\nap\n5\"\n1\u0000\u0012x\nap\n5\u00132#\n1fjxj\u0014ap\n5g:(13)\n-2 -1.5 -1 -0.5 0 0.5 1 1.5 200.20.40.60.811.2Fig. 3: Optimal kernels with unconstrained support, and sup-\nport constrained to [\u00002;2]n[1=4;3=4], with second moment\na= 5\u00001=2. The unconstrained kernel solution is exactly\ngiven by Equation (13).\nWe wish to find the solution to a modified version of this\nproblem where we enforce a compact support constraint\nof the formfK(x) = 0; x2 A;Accompactg. To this\nend, notice that Problem (12) can be numerically solved by\ndiscretizing it as follows.\nDenote the region of the problem as X=fx:jxj\u0014Bg.\nDiscretizeXintoNpoints spaced dxapart. Let the vector\nx:=fxigN\ni=12[\u0000B;B]Nconsist of these points, and let\nk:=fkigN\ni=1be the vector of the kernel Kevaluated at these\npoints, i.e.ki=K(xi). Discretizing the integrals yields a\nquadratic program of the form:\nminimize kTk\nsubject toPN\ni=1kidx= 1;PN\ni=1xiki= 0PN\ni=1x2\nikidx=a2; ki\u00150;1\u0014i\u0014N:(14)\nAs discussed before, an agent’s state-dependent density\nestimate will depend on sampling points from agents that\nhave an edge between them. Hence, in the density estimate\nfor agenti, the kernel Kwill only depend upon the state of\nagentiand its neighbours Ni. The density estimate of agent\niis written as\n^\u001ai(t;x(t)) =1\nNhdX\nj2Ni\"dY\nk=1Ki\u0012xi(t)\u0000xj(t)\nh;xj\u0013#\nwhere the support of the kernel is restricted to the support\nof the state-dependent edge weight A(xi;xj):\nA(xi;xj) = 0 =)K(h\u00001(xi\u0000xj);xj) = 0:\nTo extend Problem (14) to multi-dimensional systems, we\nconsider multiplicative kernels forxi2Rn, where each di-\nmension is estimated independently: K(x) =QN\nk=1Kk(xk):\nThis yields the final optimal kernel problem with compact\nsupport constraint. For brevity, we show the explicit form for\nthe 2D problem, as it is clear (yet notationally cumbersome)\nhow to write the general ND problem:\nminimizePNx\niPNy\njk2\nij\nsubject toPNx\ni=1PNy\nj=1kijdxdy = 1; kij\u00150;8i;jPNx\ni=1xikij= 0;PNx\ni=1x2\nikijdx=a2;8jPNy\nj=1yjkij= 0;PNy\nj=1y2\njkijdx=a2;8i\nkij= 0 if(xi;yj)2A;Accompact:\nIt is important to note that removing a compact inter-\nval from the kernel may bias the density estimate - thisFig. 4: Left: Initial density \u001a0. Right: Target density \u001a1.\nFig. 5: Optimal density profiles over time, and superimposed\nagent states using the feedback density control law.\nis unavoidable. The compact support constraint defines a\nselection-biased distribution (SBD) ; each agent samples the\ndistribution g(x) =w(x)\u001a(x)=\u0016, withw(x) =1Ac(x);\u0016=R\nw(x)\u001a(x)dx. The standard unbiased kernel density esti-\nmate of a SBD involves multiplying the kernel by a factor\nof\u0016=w(x)[38], [39], which is unbounded for our choice of\nw(x). Techniques for unbiasing ^\u001a(x)will be left for future\nwork.\nVI. EXAMPLES\nWe numerically simulate N= 200 agents with interaction\nkernelH(x) =x1kxk\u00140:01(x). The velocity field P[\u0016](x)is\nevaluated with the Dirac measure (8), effectively yielding N\nsingle-integrator agents that are able to only sample agents\na short distance away from each other.\nThe optimal mass transport problem was solved using\nopen-source code, utilizing a primal-dual algorithm [40],\n[41]. The density profile and velocity field was calculated\nover a 100\u0002100\u0002100 grid inx;y andtspace. The\ninitial density \u001a0(x)was a 2D Gaussian at the center of\na[0;1]2grid, and the target density was a ring of 2D\nGaussians, as shown in Figure 4. The superimposed optimal\ndensity profile over time, and the states of the agents over\ntime integrating the feedback and feedforward control law\nare shown in Figure 5. As one can see, the agents are\nmore organized around the final density distributions when\nusing the feedback law as opposed to just integrating the\nfeedforward law, as shown in Figure 6.\nVII. CONCLUSION\nIn this paper, we examined density control of state-\ndependent networked dynamic systems. We utilized the opti-\nmal mass transport problem to design a feed-forward velocity\nfield propelling agents with initial conditions sampled from\na density profile \u001a0to some target density \u001a1. We then\nFig. 6: Optimal density profiles over time, and superimposed\nagent states with only the feedforward control.\ntackled the problem of using a density feedback control law\nwith sparse measurements dictated by the state-dependent\nedge switching constraints of the agents. We utilized kernel\ndensity estimation to convert measurements of neighboring\nagents into a local estimate of the swarm density, which\nwas then used to calculate a feedback density control law.\nIn particular, a quadratic program was designed to find the\noptimal kernel subject to the state-dependent edge switching.\nThere are many open problems remaining, here we discuss\nseveral. First, the selection of an optimal interaction distance\nr=hfor proximity-based edge switching. This will depend\non, for example, \u001a0; \u001a1andN. Ifhis large, this will\nrequire more on-board computation and sensing capability; if\nhis small, agents will be isolated. Second, one can consider\nthe task of determining a state-dependent kernel yielding an\nunbiased estimate of \u001a.\nAPPENDIX\nWe now state the proof of Theorem 1.\nProof: First, recall the following theorem about con-\nsistency of the estimate ^\u001a(x;t).\nTheorem 2 ([42]): Consider a kernel density estimation\nscheme for the target density \u001a(x). Suppose the smoothing\nparameterhis chosen as a function of the number of samples\nN:limN!1Nh(N) =1:Then, at each point of continuity\nxof\u001a, the estimator ^\u001aN(x)is weakly consistent in that for\nall\u000f>0,limN!1P(j\u001aN(x)\u0000\u001a(x)j>\u000f) = 0:\nBy Theorem 2, under the assumption on the smoothing\nparameterh, we have that as N!1 ,^\u001a(t;x)!\u001a(t;x)\nwith probability 1 for any finite t.\nConsider the following Lyapunov function:\nV(t) =Z\nR\u0012\u001a(x;t)\n\u000b\u00132\n_xT_x dx: (15)\nAsN! 1 , the velocity field _xapproaches the mean-\nfield limit [20]: _x=P(\u0016;t) +u(x;t);whereP(\u0016;x) =R\nA(x;y)(y\u0000x)\u0016(y;t)dy;and\u0016:=\u0016(y;t)is the measure\nsatisfying@t+r\u0001((P(\u0016;t) +u(x;t))\u0016) = 0:Under the\ncontrol law (11), it follows that for sufficiently large t, the\nLyapunov function (15) can be written as\nV(t) =Z\nRr\b(x;t)Tr\b(x;t)dx:The time derivative of V(t)is then given by\n_V(t) =\u000bZ\nR\u0018(x;t)T\u0001\u0018(x;t)dx;\nwhere\u0018(x;t) :=r\b(x;t). Sincer\b(x;t) = 0 on@R, we\nhave that\u0018(x;t) = 0 on@Rwhich is a Dirichlet boundary\ncondition. It follows that _V(t)<0since the Dirichlet\nproblem for the Laplace operator has strictly negative eigen-\nvalues [43].\nTherefore, by LaSalle’s Invariance Principle, we can con-\nclude that limt!1r\b(x;t) = 0 , and so limt!1\b(x;t) =\nconstant. However, since r\b(x;t) = 0 on@R,\nthe massR\nR\b(x;t)dxis conserved for all t > 0\n(in thatR\nR\b(x;t)dx = 0 ) and so we have thatR\nR^\u001a(x;t)dx=R\nR\u001a(x;t)dx: Consequently, it follows\nthatlimt!1\b(x;t) = 0 , and hence limt!1\u001a(x;t) =\nlimt!1^\u001a(x;t)which completes the proof.\nREFERENCES\n[1] M. Mesbahi and M. Egerstedt, Graph-Theoretic Methods in Multiagent\nNetworks . Princeton University Press, 2010.\n[2] B. Ac ¸ıkmes ¸e, M. Mandi ´c, and J. L. Speyer, “Decentralized observers\nwith consensus filters for distributed discrete-time linear systems,”\nAutomatica , vol. 50, no. 4, pp. 1037–1052, 2014.\n[3] M. 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The associated density-wave\ngap functions become nodal, and the net nodal vorticity is determined by the monopole charge of\nthe pairing Berry phase. The gap function nodes become zero-energy Weyl nodes of the bulk spectra\nof quasi-particle excitations. These states can occur in doped Weyl semimetals with nested electron\nand hole Fermi surfaces enclosing Weyl nodes of the same chirality in the weak coupling regime.\nTopologically non-trivial low-energy Fermi arc surface states appear in the density-wave ordering\nstate as a consequence of the emergent zero-energy Weyl nodes.\nIntroduction. { Charge density wave (CDW) order-\ning, the spontaneous ordering of electron density or bond\nstrength, is an important phenomenon in correlated elec-\ntron systems [1, 2]. The broken translational symmetry\nof CDW ordering often arises from a Peierls instability,\nwhich is driven by electron-phonon interactions between\nnested Fermi surfaces that lead to the softening of phonon\nmodes and accompanying periodic lattice distortions [3].\nNovel topological electron excitations can exist at defects\nin the CDW order, such as half-fermion modes localized\naround the domain walls of the Peierls distortion in the\none-dimensional polyacetylene chain [2, 4]. The CDW\ninstability may also be driven by electron-electron inter-\nactions, as studied in the context of high-T ccuprates\n[5{7]. Analogously to unconventional superconductivity,\nCDW order may also possess unconventional symmetries,\nforming a non-trivial representation of the lattice sym-\nmetry group. For example, CDW order with a d-wave\nform factor was proposed to compete and coexist with\nsuperconductivity [5, 6].\nStudy of the Berry phase of Bloch wave states in lat-\ntice systems has led to the discovery of a plethora of\ntopological states, such as quantum anomalous Hall in-\nsulators [8, 9] and topological insulators [10{13]. Fur-\nthermore, the discovery of Weyl semimetals [14{47] has\nopened up a new avenue for studying topological phases\nin semi-metallic systems. As in quantum anomalous Hall\ninsulators where a Chern number structure arises from\nquantized Berry \rux over the two-dimensional Brillouin\nzone, in three-dimensional semimetals the Fermi surfaces\nhave a Chern number structure due to the Weyl points\nacting as sources or sinks of Berry \rux.\nAfter doping, magnetic Weyl semimetals can host\nmonopole harmonic superconductivity, a novel class of\ntopological states. As opposed to typical unconventional\nsuperconductors, such as d-wave high Tccuprates, and\np-wave super\ruid3He, in monopole harmonic supercon-\nductors the gap function \u0001( k) cannot be described by\nspherical harmonics and their lattice counterparts [47].Instead, these systems carry \\pairing monopole charge\",\na generalization of Berry phase from single-particle states\nto a two-particle order parameter. When the pairing\noccurs between two Fermi surfaces with opposite Chern\nnumbers, which can be the case when the enclosed Weyl\npoints have opposite chiralities, Cooper pairs acquire\nnon-trivial Berry phase structure. As a result, the gap\nfunction cannot be well de\fned over the entire Fermi\nsurface. Consequently, the Fermi surface becomes nodal\nwith total vorticity determined by the pairing monopole\ncharge associated with the two-particle Berry phase.\nIn this article, we study non-trivial Berry phase struc-\nture for a class of order parameter gap functions lying\nin the particle-hole channel. As an example that can\nbe realized in a doped Weyl semimetal, CDW order-\ning will be considered. When two nested Fermi sur-\nfaces, one electron-like and one hole-like, carry di\u000berent\nChern numbers, the CDW order formed between these\ntwo Fermi surfaces inherits non-trivial band structure\ntopology that can be seen in its gap function \u001a(k). As\nwith the gap function of monopole harmonic supercon-\nductivity,\u001a(k) cannot be globally well de\fned in momen-\ntum space and becomes nodal. The nontrivial Berry \rux\nenforces a nonzero total vorticity of \u001a(k) determined by\nthe di\u000berence in Chern number between the two nested\nFermi surfaces that is independent of the concrete mech-\nanism for CDW ordering. The nodes of the CDW gap\nfunction emerge as new Weyl nodes in the low-energy\nquasi-particle spectra that are distinct from the original\nband structure Weyl points. The chiralities of the emer-\ngent quasi-particle Weyl nodes are determined by the\nband structure in which the single-particle Weyl points\nhave been shifted away from the Fermi surfaces after dop-\ning.\nGap function Berry \rux and nodes for CDW order-\ning. { We begin with a minimal description of a pair\nof electron-like and hole-like Fermi surfaces, which carry\nopposite Chern numbers and are well nested. Such Fermi\nsurfaces can be realized in a 3D Weyl semimetal systemarXiv:1810.08715v2  [cond-mat.str-el]  19 Sep 20192\naround two Weyl points of the same chirality. Consider\ntwo Weyl points of positive chirality located at K+\neand\nK+\nhwith energies\u0000E0andE0respectively and Fermi en-\nergy at\u0016= 0. A hole-like Fermi surface denoted FS h;C\nis centered around K+\nhwith Chern number Cor, equiv-\nalently, monopole charge q=1\n2C. Similarly, an electron-\nlike Fermi surface denoted FS e;\u0000Cis centered around K+\ne\nwith Chern number \u0000C. An example of a system with\nthis Fermi surface structure is considered in Eq. (3) be-\nlow.\nSince FSh;Cand FSe;\u0000Care well nested, they favor a\nCDW instability, inter-Fermi surface particle-hole pair-\ning, under repulsive interactions. The two-particle CDW\norder parameter exhibits a non-trivial Berry \rux quan-\ntization, which can be seen as follows. After project-\ning to the low-energy Fermi surfaces, the electron cre-\nation operators on FS h;Cand FSe;\u0000Ccan be de\fned as\n\u000by\n\u0006(p) =P\na=\";#\u0018\u0006;a(p)cy\na(K+\nh(e)+p), where pis the mo-\nmentum relative to the Weyl node at K+\nh(e),arefers to the\nspin or pseudospin degrees of freedom, and \u0018\u0006(p) is the\nspinor eigenfunction carrying monopole charge \u0006q. Here\nplies on the surface Sthat results from shifting FS e;\u0000C\nby\u0000K+\netowards the origin. We de\fne the particle-hole\nchannel pairing operator\nP\u0000+(p) =\u000by\n\u0000(p)\u000b+(p); (1)\nwhich creates an electron on FS e;\u0000Cand a hole on\nFSh;C. The single-particle Berry connection is A\u0006(p) =P\nai\u0018\u0003\n\u0006;a(p)rp\u0018\u0006;a(p), and the Berry \rux penetrating\nFS\u0006is given by\r\nSdp\u0001rp\u0002A\u0006(p) =\u00064\u0019q. It can\nbe shown that the pairing Berry connection associated\nwithP\u0000+(p) isA\u0000+(p) =A\u0000(p)\u0000A+(p), and the net\npairing Berry \rux through Sis\r\nSdp\u0001rp\u0002A\u0000+(p) =\n4\u0019qCDW , whereqCDW =\u00002q.\nThe non-zero Berry \rux through Sleads to a non-\ntrivial vortex structure for the CDW gap function. The\nCDW interaction Hamiltonian after mean-\feld decompo-\nsition is expressed as\nH\u001a=X\np\u001a\u0000+(p)P\u0000+(p) +\u001a+\u0000(p)P+\u0000(p);(2)\nwhereP+\u0000=Py\n\u0000+. The gap function \u001a\u0000+is conjugate to\nthe CDW operator P\u0000+(p), and\u001a+\u0000=\u001a\u0003\n\u0000+. Because of\nthe non-trivial gauge \feld A\u0000+,\u001a\u0000+(p) cannot be glob-\nally well de\fned on S. This follows from examining the\ngauge invariant circulation \feld v\u0000+(p) =r\u001e\u0000+(p)\u0000\nA\u0000+(p), where\u001e\u0000+is the phase of \u001a\u0000+.v\u0000+is well\nde\fned except at the nodes of \u001a\u0000+(p), and each node\nhas an integer-valued vorticity gi=1\n2\u0019\u000b\nCidp\u0001v\u0000+(p).\nCiis an in\fnitesimal loop around the node pi, with the\npositive loop direction de\fned with respect to the lo-\ncal normal vector. The total vorticity of \u001a\u0000+overSisP\nigi= 2qCDW , where the sum is over all the nodes\nonS. As a consequence, the enclosure of a non-zero\nnet monopole charge gives rise to nodes of \u001a\u0000+(p) on\nkz\nFSe,1 FSe,-1 FSh,1 FSh,-1\nkzky(a)\n (b)\nFIG. 1. (a) The four Fermi surfaces of the band Hamiltonian\nin Eq. (4) with V0=t= 0:49. They are divided into two\nwell-nested pairs: FS h;1and FSe;\u00001which enclose the Weyl\npoints K+\nh;ewith positive chirality, and FS h;\u00001and FSe;1,\nwhich enclose the Weyl points K\u0000\ne;hwith negative chirality.\n(b) Energy spectrum as a function of kzin the absence of the\nCDW ordering and V0=t= 0:2. The surface states localized\non they= 0 boundary are plotted in red connecting K\u0000\neand\nK+\nh, and K\u0000\nhandK+\ne.\nS. The non-trivial nodal structure necessitates the use\nof the monopole harmonic functions [48], as opposed to\nthe usual spherical harmonics to describe the order pa-\nrameter.\nDoped Weyl Semimetal with Nested Fermi Surfaces. {\nTo demonstrate the above topological nodal structure,\nwe employ the band Hamiltonian\nH=X\nk;\na;bcy\na(k)[h(k) +V(k)]abcb(k) +H\u001a; (3)\nwherea;brefers to the pseudospin degree of freedom,\ntypically realized by AandBsublattices; H\u001ais the mean-\n\feld Hamiltonian for CDW ordering speci\fed below; and\nwe assume the chemical potential \u0016= 0. The matrix\nkernelh(k) of the band Hamiltonian is\nh(k) =tz\u0012\n2\u0000coskx\u0000cosky\u0000\r+ cos2kz\u0013\n\u001cz\n+txsinkx\u001cx+tysinky\u001cy;(4)\nwith Pauli matrices \u001cx;y;z de\fned in the A;B basis and\npseudospin-dependent hopping amplitudes tx;y;z. Here\n\r= 1=2 controls the location of the Weyl points along kz.\nFor simplicity, we choose tx;y;z =tin this paper. The cor-\nresponding lattice model giving rise to h(k) is presented\nin Supplemental Materials (S.M.) I. The momentum-\ndependent potential V(k) takes the form\nV(k) =V0coskzI; (5)\nwithIthe 2\u00022 identity matrix. V0plays a role similar to\na chemical potential by controlling the size of the Fermi\nsurface.\nWithout loss of generality, we assume V0>0. This\nmodel possesses four Weyl points, all located on the kz\naxis, at K\u0000\ne= (0;0;\u00003\u0019\n4),K+\nh= (0;0;\u0000\u0019\n4),K\u0000\nh=\u0000K+\nh,\nandK+\ne=\u0000K\u0000\ne, where the upper indices \u0006refer to the3\nchiralities of the Weyl points and the lower indices eand\nhrefer to whether the Fermi surface associated with the\nWeyl point is electron-like or hole-like. The potential\nV(k) shifts the points K\u0000\nhandK+\nhup in energy, forming\nthe respective hole-like Fermi surfaces FS h;1and FSh;\u00001,\nas shown in Fig. 1 ( a). Similarly, the points K+\neandK\u0000\ne\nare shifted down in energy, forming the electron pockets\nFSe;\u00001and FSe;+1. This model is a modi\fcation of the\nmodels in Refs. 17 and 49 to allow four Weyl points with\nnesting. The electron and hole Fermi pockets enclosing\nthe Weyl points with the same chirality are nested with\nthe commensurate wavevector Q= (0;0;\u0019). This nesting\ncondition is satis\fed so that portions of the Fermi surface\nseparated by Qhave the same shape. Under an open\nboundary condition along the y-direction and periodic\nboundary conditions along xandz, the energy spectrum\nin the absence of the CDW ordering is plotted in Fig. 1\n(b) as a function of kzalong thekx= 0 cut. The surface\nFermi arc states are shown in red.\nCDW ordering is imposed through the mean-\feld\nHamiltonian\nH\u001a=X\nk;\na;b=A;Bcy\na(k+Q)\u001aab(k)cb(k) +h:c:; (6)\nwhere we take \u001a(k) =\u001a\u001czand\u001ais the magnitude of the\nCDW ordering. \u001a(k) is diagonal in the sublattice Aand\nBbasis, which describes two sublattices with di\u000berent\ncharge densities. Below we will see that this CDW or-\ndering does not open a full gap over the Fermi surface\nbut instead becomes nodal with a non-trivial vorticity.\nTopological Nodal CDW. { We \frst consider the CDW\ngap function connecting FS h;1and FSe;\u00001which en-\nclose the Weyl nodes K+\nhandK+\ne, respectively. For\nsmallV0=tand\u001a=t, the Fermi surfaces are close to\nthe Weyl nodes and the single-particle states corre-\nspond to the helicity eigenstates satisfying ^p\u0001\u001c\u0018\u0006=\n\u0007\u0018\u0006, where\u0018\u0006corresponds to Berry \rux monopole\ncharges of q=\u00061\n2. Explicitly, \u0018\u0006can be repre-\nsented as\u0018+(^p) = (\u0000sin\u0012p\n2e\u0000i\u001ep;cos\u0012p\n2)Tand\u0018\u0000(^p) =\n(cos\u0012p\n2;sin\u0012p\n2ei\u001ep)T, where\u0012pand\u001epare the polar and\nazimuthal angles of ^pand a gauge convention has been\nchosen. The electron creation operator for the eigen-\nstate on the helical Fermi surface FS h;1is\u000by\n+;h(p) =P\na\u0018+;a(^p)cy\na(K+\nh+p), and that for FS e;\u00001is\u000by\n\u0000;e(p) =P\na\u0018\u0000;a(^p)cy\na(K+\ne+p).\nThe gap function \u001a\u001czin Eq. (6) can now\nbe projected onto the helical Fermi surfaces FS h;1\nand FSe;\u00001, where the projected gap function con-\njugate to P\u0000+(p) =\u000by\n\u0000;e(p)\u000b+;h(p) is\u001a\u0000+(p) =\n\u0000\u001ap\n8\u0019=3Yq=\u00001;l=1;m=0(\u0012p;\u001ep), in terms of monopole\nharmonicsYqlm[48]. For pnear the north pole of FSe;\u00001,\nwhere\u0012p= 0, the projected gap function is \u001aN\n\u0000+(p) =\n\u0000\u001asin\u0012pe\u0000i\u001ep. By applying a gauge transformation, the\nprojected gap function near the south pole, where \u0012p, canbe similarly shown to be \u001aS\n\u0000+(p) =\u0000\u001asin\u0012pei\u001ep. The\nprojected gap function has nodes at the poles, where\nsin\u0012p= 0. After taking into account the contribution\nof the Berry connection A\u0000+(p), the circulation \feld of\nthe gap function is v\u0000+(p) =\u0000cot\u0012p^\u001ep. Integrating v\naround in\fnitesimal loops near \u0012p= 0 and\u0019reveals a\ngap function vorticity of \u00001 near both poles, hence the\ntotal vorticity is \u00002 on the Fermi surface surrounding\nFSe;\u00001, consistent with qCDW =\u00001.\nThe CDW gap function nodes are actually low-energy\nWeyl points generated by interactions for the mean-\feld\nHamiltonian. Around the nested Fermi surfaces FS e;\u00001\nand FSh;1, the low-energy two-band Hamiltonian is\nH2band=X\np y(p)n\n(tjpj\u0000\u0016)\u001bz\u0000\u001asin\u0012p(e\u0000i\u001ep\u001b+\n+ei\u001ep\u001b\u0000)o\n (p); (7)\nwhere (p) = (\u000b\u0000;e(p);\u000b+;h(p))Tand\u0016=\u0000V(K+\ne+p).\nThe interaction-induced Weyl node at the north pole,\ndenoted K+\nn, has positive chirality as can be shown by\nexpandingH2bandabout the north pole, where the helical\nbasis is regular. The south pole, denoted K+\ns, is the site\nof a singularity in the helical basis and thus needs to\nbe treated more carefully. Taking into account the 4 \u0019-\n\rux from the Dirac string penetrating the south pole,\nor equivalently changing the gauge choice to place the\nsingularity at the North pole, this Weyl node can also be\nshown to possess positive chirality as well.\nThe positive chiralities of K+\nn;sare in fact determined\nby the chiralities of the original band structure Weyl\nnodes K+\ne;h, which are away from the chemical potential\nand hence lie in the high energy sector. Nevertheless,\nthey still determine the chirality of the low-energy Weyl\nnodes, independent of the details of the mechanism of the\nCDW ordering. Typically, the low-energy physics is not\nsensitive to the details at high energy, but the topologi-\ncal structure at low-energy in our case is indeed inherited\nfrom the topology at high energy, and thus the emer-\ngence of the low-energy Weyl fermions are topologically\nprotected.\nSimilar analysis can also be performed in parallel for\nthe CDW ordering connecting the nested Fermi surfaces\nFSh;\u00001and FSe;1surrounding K\u0000\nh;e, respectively. The\ntwo low-energy Weyl nodes denoted K\u0000\nn;son the nested\nFSh;\u00001and FSe;1have negative chirality, which is again\ndetermined by the Weyl nodes K\u0000\nh;eat high energies. In\ntotal, the sum of chiralities of all the Weyl nodes, includ-\ning the original band structure ones and the interaction-\ninduced ones, remain zero as required by the Nielsen-\nNinomiya theorem [50, 51].\nThe emergent Weyl nodes in the monopole CDW gap\nfunction as well as novel topological surface states are\ndemonstrated in the quasi-particle energy spectra in Fig.\n2, where we take open boundary conditions along y-\ndirection and periodic boundary conditions along xand4\n(a)\n (b)\n(c)\n (d)\n (e)\nFIG. 2. The bulk and surface spectra for the topological\nCDW ordering with emergent Weyl nodes of Eq. (6). The\nopen boundaries are perpendicular to the y-axis, and only\nsurface states localized at the y= 0 boundary are shown. Pa-\nrameter values are \u001a=t= 0:1 andV0=t= 0:2. a) The disper-\nsions along the cut at kx= 0 with varying kzin the reduced\nBZ with 0 \u0014kz\u0014\u0019.K\u0000\neandK+\nhatkz<0 are folded into\nthe reduced BZ. Surface state spectra are plotted in red. The\ndispersions for varying kxare shown for constant kzcuts at\nb)kz=\u0019\n4,c)kz=3\u0019\n8,d)kz=5\u0019\n8, ande)kz=3\u0019\n4. Green\nand magenta respectively indicate surface states with major-\nity weight in the 0 \u0014kz\u0014\u0019and\u0000\u0019\u0014kz\u00140 components.\nzdirections. Because of the nesting vector Q= (0;0;\u0019),\nthe reduced Brillouin zone (BZ) with 0 \u0014kz\u0014\u0019is con-\nsidered. As shown in Fig. 2 ( a), two pairs of emergent\nzero-energy Weyl nodes K\u0000\ns;nandK+\ns;nappear at the\nkz-axis near K\u0000\nhandK+\nein the bulk quasi-particle en-\nergy spectrum. The surface states localized at the y= 0\nboundary are shown in color. Away from the Fermi sur-\nface, there are two branches of chiral surface states for\n(K\u0000\nn)z< kz<(K+\ns)z, due to BZ folding of the original\nFermi arcs in Fig. 1 (b). The number of branches of sur-\nface states changes as kzmoves across K\u0000\ns,K\u0000\nn,K+\nsand\nK+\nn, as shown in Fig. 2 ( b)\u0018(e). The surface states in-\nside the CDW gaps with ( K\u0006\ns)z<kz<(K\u0006\nn)zare quasi-\nparticles that are superpositions of electron states at k\nandk+Qdue to the particle-hole pairing near the Fermi\nsurface. The component with majority weight changes\nwhen these surface states cross zero energy in Fig. 2 ( b)\nand (e), in contrast to the surface states shown in ( c) and\n(d) where each branch of surface states is dominated by\na single component. Thus, the CDW gaps admit novel\nzero-energy quasi-particles as equal superpositions of k\nandk+Qstates, that are a CDW analog of Majorana\nmodes. These novel zero-energy modes are symmetric\nunder the k$k+Qsymmetry of the Hamiltonian H,\nanalogous to the charge conjugation symmetry of Majo-\nrana modes.\n(a)\n (b)\nFIG. 3. Evolution of the low-energy Weyl nodes as the or-\ndering strength \u001ais increased with \fxed V= 0:2t. a) At\n\u001a\u00190:458t, a gap atkz= 0 starts to open as K\u0000\nsmerges with\nK+\nn. b) By\u001a= 0:6t, there is a gap at all kzafter K\u0000\nnhas\nmerged with K\u0000\nsatkz=\u0019\n2.\nA natural question is whether the projection to the\nlow-energy helical Fermi surface remains valid as the\nCDW ordering strength \u001abecomes large. We \fnd that\nthe zero-energy Weyl nodes remain robust until \u001areaches\na critical value \u001ac. As\u001ais increased, the low-energy Weyl\nnodes K\u0000\nnandK+\nsare pushed closer to each other, as are\nK\u0000\nsandK+\nn. As shown in Fig. 3 ( a),K\u0000\nsandK+\nn\frst\nmerge atkz= 0, opening a gap, and K\u0000\nnandK+\nsremain\nseparated. Further enlarging \u001a,K\u0000\nnandK+\nsmerge next\natkz=\u0019\n2after which the gap opens. After both pairs\nof zero-energy Weyl nodes merge, the system becomes\nfully gapped around zero energy, as shown in Fig. 3( b).\nThe surface states in this case cross the bulk gap as kx\nis varied.\nIncreasing\u001abeyond the point at which the system \frst\nbecomes gapped, there must eventually be an additional\ngap closing followed by a gap opening. To see this, con-\nsider that in the \u001a!1 limit, the spectrum must be\nconcentrated close to \u0006\u001a, the eigenvalues of H\u001a. In this\nlarge\u001alimit, there can then be no surface states cross-\ning the gap, and the surface states shown in Fig. 3( b)\nmust eventually disappear following a second transition\nas\u001aincreases beyond \u001a= 0:6t. A more detailed explana-\ntion is presented in S.M. II. This process is similar to the\ntransition between weak- and strong-coupling topological\nsuperconductivity, in which the strong-coupling state is\ntopologically trivial.\nConnection to experiments. { The general require-\nment for material realizations of monopole CDW states\nis the presence of nested Fermi surfaces enclosing Weyl\npoints of the same chirality, which can occur in either an\ninversion-breaking or, as in this work, a time-reversal-\nbreaking Weyl semimetal. In addition to the intrinsic\nordering mechanism, CDW states can also be driven by\nan ultra-fast laser pulse, as demonstrated in a layered\nchalcogenide system [52]. If the CDW state instead oc-\ncurs between Fermi surfaces enclosing Weyl points of op-\nposite chiralities, the gap function has zero Berry phase\nand can either be fully gapped or develop nodes. In such5\na case, the symmetry of the gap function is represented by\nconventional spherical harmonics or their lattice counter-\npart and is determined energetically by speci\fc ordering\nmechanisms.\nTo experimentally characterize the monopole CDW,\nphase sensitive bulk detection, which can be provided by\nResonant Inelastic X-ray Scattering (RIXS), is desired to\nextract the non-trivial total vorticity of phase winding in\nthe gap function. It has been shown that, with appropri-\nate choices of polarization, the RIXS intensity exhibits\nnodal lines in a momentum transfer q-plane [53] when\nthe phase winding of a single-particle Weyl Hamiltonian\nin the same momentum plane is nontrivial. This result\nis insensitive to the energy resolution within an energy\nwindow such that the Weyl bands around di\u000berent Weyl\npoints are well separated in momentum space. Hence,\nRIXS is expected to probe the emergent zero-energy Weyl\nnodes and their phase windings in the monopole CDW\nordering. Furthermore, it can map out the non-trivial\ntotal vorticity of the gap function, which is absent in the\nusual gapped or even nodal CDW states. Detailed anal-\nysis will be deferred to a future publication.\nDiscussion. { It would be interesting to investigate\nother orderings besides the monopole CDW in this sys-\ntem. Since most Weyl semimetals are not strongly cor-\nrelated, we focus on Fermi surface instability, which can\ngive rise to charge or spin density wave states and su-\nperconductivity in the particle-hole and particle-particle\nchannels respectively. In this model, the superconduc-\ntivity arising from zero-momentum pairing occurs be-\ntween Fermi surfaces with opposite Chern numbers and\nis thus an example of the recently proposed monopole\nsuperconductivity [47]. This superconducting gap func-\ntion, like the density wave gap functions, is nodal and\ncharacterized by monopole harmonics. It is possible\nto only have the CDW instability without supercon-\nductivity or, if their energy scales are comparable, to\nhave both instabilities coexist. Analogous to the coex-\nistence of the CDW and superconductivity seen in the\ntransition-metal dichalcogenide NbSe 2[54{58], competi-\ntion between monopole harmonic orders is possible, and\nthis interesting problem will be studied in future work.\nConclusions. { We have studied the inter-Fermi-\nsurface particle-hole pairing between well-nested Fermi\nsurfaces enclosing Weyl nodes of the same chirality. The\nCDW ordering in this case gives rise to gap functions\npossessing topologically protected nodes independent of\nthe details of ordering interactions. 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After a Fourier transformation, the band\nHamiltonian becomes\nH=X\nr;\na;b=A;B(\u0014\ncy\na(r+^ ex)\u0012\n\u0000tx\n2i\u001cx\u0000tz\n2\u001cz\u0013\nabcb(r) +h:c:\u0015\n+\u0014\ncy\na(r+^ ey)\u0012\n\u0000ty\n2i\u001cy\u0000tz\n2\u001cz\u0013\nabcb(r) +h:c:\u0015\n+\u0014\ncy\na(r+^ ez)\u0012V0\n2I\u0013\nabcb(r) +h:c:\u0015\n+\u0014\ncy\na(r+ 2^ ez)\u0012tz\n4\u001cz\u0013\nabcb(r) +h:c:\u0015\n+\u0002\ncy\na(r) (tx(5=2\u0000\r)\u001cz)abcb(r)\u0003)\n;\nwherecy\na(r) andca(r) are respectively the electron cre-\nation and annihilation operators on site rwith sublattice\na=A;B.^ ei(i=x;y;z ) are the unit lattice vectors\npointing in the idirection and \u001ciare Pauli matrices in\nthe sublattice basis. The \frst three terms describe near-\nest neighbor hopping in the x-,y-, andz- directions with\nsublattice dependent hopping amplitudes tiandV0. The\nfourth is a next nearest neighbor hopping term in the\nz-direction. The last term describes a sublattice modu-\nlated on-site potential with parameter \r, which controls\nthe position of the Weyl points along the kzaxis.\nII. STRONG-COUPLING CDW\nAs discussed in the main text, there can be no surface\nstates crossing the gap once \u001ais su\u000eciently large. This\ncan be seen explicitly by considering the Hamiltonian in\na BdG-like form,\nhBdG(k) =V0coskz\u001bz\nI+ sinkxI\n\u001cx+ sinkyI\n\u001cy\n+\u00123\n2\u0000coskx\u0000cosky+ cos2kz\u0013\nI\n\u001cz\n+\u001a\u001bx\n\u001cz:\n(S1)\nAs in the main text, tx;y;z =tandV0and\u001aare ex-\npressed in units of t. The\u001cmatrices are pseudospin Pauli\nmatrices corresponding to the A;B sublattice degrees of\nfreedom, while the \u001bPauli matrices correspond to the k\nandk+Qparts of the BdG basis [ cA(k);cB(k);cA(k+\nQ);cB(k+Q)]T. By explicitly diagonalizing the BdG\nHamiltonian, it can be shown that the gap can close only\nwhenkx;ky= 0 or\u0019, where the energy spectrum can be\nFIG. S1. Curves of the critical coupling \u001acat which the gap\ncloses as a function of kzwithV0= 0:2. The top curve de-\nscribes the gap closing at kx=\u0019, while the bottom curve\ndescribeskx= 0. Along the middle curve, there are simulta-\nneous gap closings at both kx= 0 andkx=\u0019. The gap does\nnot close for positive \u001aat any other values of kx.\nwritten\nE(k)\u0006;\u0006=\u0006(~\rk+ cos2kz)\u0006q\n\u001a2+V2\n0cos2kz(S2)\nwith\n~\rk=8\n><\n>:7\n2(kx;ky) = (\u0019;\u0019)\n3\n2(kx;ky) = (0;\u0019);(\u0019;0)\n\u00001\n2(kx;ky) = (0;0):(S3)\nCritical values of \u001afor which the gap closes somewhere\nin the BZ are given by\n\u001ac(k) =q\n(~\rk+ cos2kz)2\u0000V2\n0cos2kz: (S4)\nFor open boundary conditions in the ydirection and peri-\nodic boundary conditions in kxandkz, a gap closing will\nbe seen in the spectrum at ( kx;kz) if there is a positive\ncritical\u001ac(k) at (kx;kz) for anyky.\nThe critical values \u001acas a function of kzare shown\nin Fig. S1 for V0= 0:2. There are only three critical\ncurves, as there can only be gap closings at kx= 0 or\u0019,\ngiving only three possible values for ~ \rk. The middle curve\ncontains information about both kx= 0 andkx=\u0019,\nthough these transitions are qualitatively di\u000berent. The\n\frst transition mentioned in the main text corresponds\nto the bottom curve as \u001ais increased from 0 to 0 :5. For\n\u001abetween the bottom and middle critical curves, there\nis an edge state extending across all kzin thekx= 0 cut\nas shown in Fig. 3.\nOnce\u001abegins to cross the middle curve, at \u001ac= 1:5,\nthe gap closes at kz=\u0019=2 for both kx= 0 andkx=\n\u0019. Increasing \u001athrough the middle curve, the surface\nstates atkx= 0 begin to disappear as a gap opens from\nkz=\u0019=2 outwards until the gap nodes are pushed toS-2\nkz=\u0019for\u001ac\u00192:49. Increasing \u001afurther leaves the\nsystem gapped at all points along kx= 0, and no surface\nstates will cross the gap for larger \u001a. Atkx=\u0019instead,\ncrossing the middle curve introduces surface states as the\ngap opens from kz=\u0019=2 outwards. These surface states\ndisappear after the \fnal transition as \u001ais increased past\nthe top curve, and a fully gapped system remains with no\nsurface states crossing the gap once \u001aexceeds\u001ac\u00194:5.\nThe gap closings at kx=\u0019are shown for \u001acrossing\nthe middle and top critical curves in Fig. S2. The tran-\nsition that occurs at kx=\u0019as\u001acrosses the top curve\nis qualitatively similar to the transition at kx= 0 as\u001a\ncrosses the middle curve.\n(a)\n (b)\n(c)\n (d)\nFIG. S2. Gap closings as \u001ais varied at kx=\u0019. a) At\u001a= 1:5,\nthe gap closes at kz=\u0019=2. b)\u001a= 1:8. As\u001ais increased\npast the initial gap closing, a gap with surface states opens\nand the gap nodes are pushed outwards. c) \u001a= 2:8. Once\nthe gap nodes reach kz=\u0019, the gap opens leaving a surface\nstate. d)\u001a= 3:8. As\u001aincreases further, the gap closes at\nkz=\u0019=2 again and reopens, pushing the gap nodes towards\nkz=\u0019and eliminating the surface states."    },    {        "title": "1811.02805v3.PaDNet__Pan_Density_Crowd_Counting.pdf",        "content": "1\nPaDNet: Pan-Density Crowd Counting\nYukun Tian, Yiming Lei, Junping Zhang, Member, IEEE, and James Z. Wang\nAbstract —Crowd counting is a highly challenging problem in\ncomputer vision and machine learning. Most previous methods\nhave focused on consistent density crowds, i.e., either a sparse\nor a dense crowd, meaning they performed well in global\nestimation while neglecting local accuracy. To make crowd\ncounting more useful in the real world, we propose a new\nperspective, named pan-density crowd counting, which aims to\ncount people in varying density crowds. Specifically, we propose\nthe Pan-Density Network (PaDNet) which is composed of the\nfollowing critical components. First, the Density-Aware Network\n(DAN) contains multiple subnetworks pretrained on scenarios\nwith different densities. This module is capable of capturing pan-\ndensity information. Second, the Feature Enhancement Layer\n(FEL) effectively captures the global and local contextual features\nand generates a weight for each density-specific feature. Third,\nthe Feature Fusion Network (FFN) embeds spatial context and\nfuses these density-specific features. Further, the metrics Patch\nMAE (PMAE) and Patch RMSE (PRMSE) are proposed to better\nevaluate the performance on the global and local estimations. Ex-\ntensive experiments on four crowd counting benchmark datasets,\nthe ShanghaiTech, the UCF CC 50, the UCSD, and the UCF-\nQNRF, indicate that PaDNet achieves state-of-the-art recognition\nperformance and high robustness in pan-density crowd counting.\nIndex Terms —Crowd Counting, Density Level Analysis, Pan-\ndensity Evaluation, Convolutional Neural Networks.\nI.INTRODUCTION\nCROWD counting has broad applications in public safety,\nemergency evacuation, smart city planning, and news\nreporting [1]. However, due to perspective distortions, severe\nocclusions, high-variant densities, and other problems, crowd\ncounting has been a persistent challenge in computer-vision\nand machine-learning domains. Existing work has largely\nfocused on consistent density crowds, i.e., either a sparse or a\ndense crowd. Yet in the real world, an image of a crowd may\nhave areas of inconsistent densities due to camera perspective\nas well as naturally varying distribution of people in the\ncrowd. Accurately counting people in crowds, thus, entails\nconsideration of all density variations. To emphasize this\ndensity-inclusive focus, we name our approach pan-density\ncrowd counting .\nThe needs for pan-density crowd counting is evident in the\ncrowd image examples shown in Figure 1. The images convey\ntwo properties: (i) different crowds have varying densities and\ndistributions, and more importantly, (ii) the densities of local\nY . Tian, Y . Lei and J. Zhang are with the Shanghai Key Labora-\ntory of Intelligent Information Processing, School of Computer Science,\nFudan University, Shanghai 200433, China (e-mails: fyktian17, ymlei17,\njpzhang g@fudan.edu.cn).\nJ. Z. Wang is with the College of Information Sciences and Technology,\nThe Pennsylvania State University, University Park, PA 16802, USA (e-mail:\njwang@ist.psu.edu).\nManuscript received February 27, 2019; revised September 20, 2019;\naccepted October 22, 2019 (Corresponding author: Junping Zhang.)\nFig. 1. Example crowd images. The last image is from the Shang-\nhaiTech Part B dataset [10] and the other images are from the UCF-QNRF\ndataset [11]. These images show crowds of diverse densities and distributions.\nFurthermore, the densities of local regions could be inconsistent in the same\nscene.\nregions can be inconsistent even in the same scene. Because\nprior methods focus on the global evaluation in varying-\ndensity scenes, they cannot sufficiently capture pan-density\ninformation to achieve good performance in terms of local\naccuracy. Their recognition accuracy is therefore limited in\ndealing with pan-density crowd counting.\nSome earlier methods count sparse pedestrians by us-\ning a sliding window detector [2, 3]. Regression-based ap-\nproaches [4, 5] utilize hand-crafted features extracted from\nlocal image patches to count sparse crowds. Due to severe\nocclusions, these methods have limited performance in dense\ncrowd counting. Inspired by the success of convolutional neu-\nral network (CNN) [6, 7, 8, 9], researchers employed CNN-\nbased methods to predict a density map which includes impor-\ntant spatial distribution information for dense crowd counting.\nSingle-column CNNs [12, 13] adopt multiple convolutional\nlayers to extract features, and these features are fed into a\nfully connected layer to count people in dense scenes. While\nthese single-column-based methods are suitable for single-\ndensity crowd counting, they cannot fully capture pan-density\ninformation.\nIn order to count crowds with varying densities, multi-\ncolumn-based network methods have been developed [10, 14,\n15, 16]. These methods contain several columns of CNNs\nthat have different-sized filters to capture multi-scale infor-\nmation. For instance, filters with larger receptive fields are\nmore useful for modeling the density maps corresponding to\nlarger head regions. However, these multi-scale-based methods\nhave relatively low efficiency because they cannot accurately\nrecognize a specific density crowd or reasonably utilize thearXiv:1811.02805v3  [cs.CV]  9 Jan 20202\n(a) Original\n (b) GT Count: 553\n (c) MCNN Count: 557.2\nFig. 2. The original image is from the ShanghaiTech Part A dataset [10]. The ground truth is shown in (b). The density map generated by MCNN in shown\nin (c). The global estimation of MCNN is close to the ground truth, but the local estimation is biased. Accurate global estimation, however, is because the\nunderestimation of region 2 offsets the overestimation of regions 3 and 4.\nfeatures learned by networks across columns.\nLiet al. [17] shows that because these methods cannot\naccurately learn different features for each column, they result\nin some ineffective and redundant branches. To address this\nissue, Sam et al. [15] proposed a Switch-CNN through training\nthe switch classifier to select the optimal regressor for one\ninput patch. Yet, Switch-CNN is limited because it chooses\none of the results of different subnetworks rather than fusing\nthem. High-variant density exists not only at the whole image\nlevel, but also at the image patch level. Each subnetwork of\nSwitch-CNN is trained on a specific density subdataset and\nthus cannot utilize the whole dataset. Therefore, the single\nsubnetwork has limited recognition performance and cannot\novercome the feature covariate shift problem.\nMost of these multi-column-based models fuse the feature\nmaps generated by different columns via a 1\u00021convolutional\nlayer. As a result, the operation suffers from multi-scale model\ncompetition. A more reasonable way of fusing feature maps\nis to assign different weights for the subnetworks. Sindagi et\nal.[16] proposed a Contextual Pyramid CNN (CP-CNN) to\nincorporate contextual information for achieving low counting\nerror and high-quality density maps. This approach, however,\nhas a high computation complexity in predicting the global and\nlocal contexts. Further, predicting the local context is a difficult\ntask, and once the prediction is biased, overall performance is\nseverely limited.\nIn addition, although they are accurate in estimating the\nglobal count in the scene, the fatal flaw of most crowd-\ncounting methods is the bias of their local estimations.\nAn example in Figure 2 shows that the estimation (=557.2)\ngenerated by Multi-Column CNN (MCNN) [10], which is\na representative crowd-counting algorithm, is close to the\nground truth of 553. However, the local estimations are quite\ninaccurate. For example, the estimation of MCNN in the region\n2 is 143.5, while the ground truth is 192.7. There also exists\nrelatively large biases in the regions 3 and 4. By observing the\nbiased local estimation, it appears that the high accuracy of\nglobal estimation stems from a fact that the underestimation\nof region 2 offsets the overestimation of regions 3 and 4. It\ncan also be seen that two general evaluation metrics of crowd\ncounting, MAE and RMSE, prefer estimating global accuracy\nand robustness over estimating local ones.\nIn order to tackle the aforementioned problems, we proposea novel model, called PaDNet, for pan-density crowd counting.\nPaDNet is composed of three critical components. First, the\nDensity-Aware Network (DAN) contains multiple subnetworks\npretrained on scenarios with different densities. This module\nis capable of capturing the pan-density information. Second,\nthe Feature Enhancement Layer (FEL) effectively captures the\nglobal and local contextual features and generates a weight\nfor each density-specific feature. Third, the Feature Fusion\nNetwork (FFN) embeds spatial context and fuses these density-\nspecific features instead of choosing one.\nOur main contributions are summarized below.\n\u000fWe propose a novel end-to-end architecture named PaD-\nNet for pan-density crowd counting. Further, we explore\nthe impact of density-level division on estimation perfor-\nmance. Through extensive experiments on four bench-\nmark crowd datasets, PaDNet obtains the best perfor-\nmance and high robustness in pan-density crowd counting\ncompared with state-of-the-art algorithms.\n\u000fIn order to evaluate both local accuracy and robustness,\nwe propose two new evaluation measures, i.e., Patch\nMAE (PMAE) and Patch RMSE (PRMSE). They con-\nsider both global accuracy and robustness as well as local\nones.\nThe remainder of the paper is organized as follows. Sec-\ntion II introduces related works in crowd counting. In Sec-\ntion III, we present the details of our method and then we\npresent and analyze the experimental results in Section IV.\nWe offer final thoughts in Section V.\nII.RELATED WORKS\nExisting crowd counting algorithms can be roughly catego-\nrized into detection-based methods, regression-based methods,\nand CNN-based methods. Below we give a brief survey of\nthese three categories.\nA.Detection-based Methods\nDetection-based methods of crowd counting utilize a\nmoving-window detector to identify pedestrians and count the\nnumber of people in an image [18]. Some researchers have\nproposed extracting common features from appearance-based\ncrowd images in order to count crowds [19, 20, 21], but these\napproaches have obtained limited recognition performance3\nFig. 3. The PaDNet consists of Feature Extraction Network (FEN), Density-Aware Network (DAN), Feature Enhancement Layer (FEL), and Feature Fusion\nNetwork (FFN). FEN extracts low-level feature of images. DAN employs multiple subnetworks to recognize different density levels in crowds and to generate\nthe feature map ( ^Yi). FEL captures the global and local features and learns an enhancement rate to boost the feature map ^Yiand generates ^Y0\ni. FFN fuses\n^Y0\niand generates the final density map for counting.\nimprovement when dealing with dense crowd counting. To\novercome this issue, researchers used part-based methods to\ndetect the specific body parts such as the head or the shoulder\nto count pedestrians [22, 23]. However, these detection-based\nmethods are only suitable for counting sparse crowds because\nthey are affected by severe occlusions.\nB.Regression-based Methods\nTo address the problem of occlusion, regression-based meth-\nods have been introduced for crowd counting. The main idea of\nregression-based methods is to learn a mapping from low-level\nfeatures extracted from local image patches to the count [4, 5].\nThese extracted features include foreground features, edge\nfeatures, textures, and gradient features such as local binary\npattern (LBP), and histogram oriented gradients (HOG). The\nregression approaches include linear regression [24], piece-\nwise linear regression [25], ridge regression [26], and Gaussian\nprocess regression. Although these methods refine the previous\ndetection-based ones, they ignore spatial distribution infor-\nmation of crowds. To utilize spatial distribution information,\nthe method proposed by Lempitsky et al. [27] regresses a\ndensity map rather than the crowd count. The method learns a\nlinear mapping between local patch features and density maps,\nthen estimates the total number of pedestrians via integrating\nover the whole density map. The method proposed by Pham\net al. [28] learns a non-linear mapping between local patch\nfeatures and density maps by using random forest. Most recent\nregression-based methods are based on the density map.\nC.CNN-based Methods\nBenefit from CNN’s strong ability to learn representations, a\nvariety of CNN-based methods have recently been introduced\nin crowd counting. As a pioneering work for crowd counting\nwith CNN, the method proposed by Wang et al. [12] adopted\nmultiple convolutional layers to extract features and sentthese features into a fully connected layer to predict numbers\nin extremely dense crowds. Another work [29] pretrained a\nnetwork for certain scenes and selected similar training data\nto fine-tune the pretrained network based on the perspective\ninformation. The main drawback is that the approach requires\nperspective information which is not always available.\nObserving that the densities and appearances of image\npatches are of large variations, Zhang et al. [10] further\nproposed MCNN architecture for estimating the density map.\nIn their work, different columns are explicitly designed for\nlearning density variations across different feature resolutions.\nDespite different sizes of filters, it is difficult for different\ncolumns to recognize varying density crowds, and this lack\nof recognition results in some ineffective branches. Sindagi et\nal.[30] proposed a multi-task framework to simultaneously\npredict density classification and generate the density map\nbased on high-level prior information. They further proposed a\nfive-branch contextual pyramid CNNs, short for CP-CNN [16],\nto incorporate contextual information of the crowd for achiev-\ning lower counting errors and high-quality density maps.\nHowever, CP-CNN has high computational complexity and\ncannot be applied in real-time scene analysis. Inspired by\nMCNN, the work proposed by Sam et al. [15] includes a\nSwitch-CNN, where the switch classifier is trained to select\nthe optimal regressor for one input patch. In the prediction\nphase, Switch-CNN can only use a column network that is\nconsistent with the classification result of that patch, without\nincorporating all trained subnetworks. High-variation densities\nnot only exist at the whole image level, but also at the image\npatch level. Therefore, the single subnetwork has limited\nrecognition performance and cannot overcome the feature\ncovariate shift problem. Kang et al. [31] proposed a method of\nfusing multi-scale density predictions of corresponding multi-\nscale inputs, while Deb et al. [32] designed an aggregated\nmulti-column dilated convolution network for perspective-\nfree counting. However, none of these works consider local4\n(a)C= 50 ,D= 24:1\n (b)C= 50;D= 55:0\n(c)C= 78;D= 20:8\n (d)C= 241;D= 20:8\nFig. 4. The average distance between adjacent people is more reasonable for\nrepresenting the dense degree of crowd compared with the number of people.\nSeveral instances of the SHA dataset [10] are shown. Cis the number of\npeople.Dis the dense degree of the crowd calculated by Eq. (2) (smaller is\ndenser). The number of people is the same in figures (a) and (b), but (a) is\ndenser than (b) and Din (a) is smaller than that in (b). On the other hand,\nDis the same in figures (c) and (d), but the number of people in (c) is far\nless than that in (d).\ninformation.\nTo avoid the issues of ineffective branches and expensive\ncomputation in previous multi-column networks, Li et al. [17]\nintroduced a deeper single-column-based dilated convolutional\nnetwork called CSRNet. Cao et al. [33] developed an encoder-\ndecoder-based scale aggregation network for crowd counting.\nObserving the importance of temporal information in counting\ncrowds, a bi-directional ConvLSTM-based [34] spatiotemporal\nmodel was proposed for video crowd counting [35].\nMost of the CNN-based methods count crowds by predict-\ning a density map based on l2regression loss. However, l2\nis sensitive to outliers and blurs the density map. Shen et\nal.[36] thus proposed a GANs-based method to generate high-\nquality density maps, and a strong regularization constraint\nwas conducted on cross-scale crowd density estimation. In\naddition, Liu et al. [37] combined the detection-based and\nthe regression-based method for dealing with density variation\nfor crowd coutning. Shi et al. [38] proposed a framework\nwhich produced generalizable features by using deep negative\ncorrelation learning (NCL). Liu et al. [39] leveraged unlabeled\ndata to enhance the feature representation capability of the\nnetwork. Inspired by image generation, the method proposed\nby Ranjan et al. [40] is an iterative crowd counting framework\nwhich generates a low-quality density map first and then\ngradually evolves it to a high-quality density map. Note that\nthese methods are inferior in achieving robust recognition\nperformance in pan-density crowd counting, which is the\nproblem we are tackling.III. OURAPPROACH\nOur framework is illustrated in Figure 3. The proposed\nPaDNet consists of four components: Feature Extraction\nNetwork (FEN), Density-Aware Network (DAN), Feature En-\nhancement Layer (FEL), and Feature Fusion Network (FFN).\nFEN extracts the low-level features of images. DAN contains\nmultiple subnetworks pretrained on scenarios with different\ndensities and is used to capture pan-density information. FEL\ncaptures the global and local features to learn an enhancement\nrate or a weight for each density-specific feature map. Finally,\nFFN aggregates all of the refined features to generate the final\ndensity map for counting the crowd. Below, we will describe\nour proposed PaDNet in details.\nA.Feature Extraction Network (FEN)\nA main difficulty in crowd counting is that the background\nand the density level have drastical variations in real-world\nscenes. To apply deep learning for such a situation, a suffi-\nciently large training dataset is required. However, the largest\nexisting training dataset only contains 1,201 images. As was\ndone in many deep learning models [17, 16, 38, 36, 39], we use\npretrained models to avoid overfitting. Note that because most\nof the popular backbones such as VGG-16 [41], ResNet [42],\nand GoogLeNet [43] are trained on the ImageNet [44], which\nis a classification task, while crowd counting is a regression\ntask, these backbones cannot be directly inserted into our\nmodel. Thus, we have to develop an alternative process. The\nwork of Yosinski et al. [45] considers that the front-end of\nthe network learns task-independent general features which\nare similar to Gabor filters and color blobs, and the back-\nend of the network learns task-specific features. Based on this\nconsideration, we choose the first ten convolutional layers of a\npretrained VGG-16 with Batch Normalization [46] and ReLU\nas FEN. Formally, the feature extraction process is as follows,\n^Yfeat=FFEN(X); (1)\nwhereXis an input image, ^Yfeatis the base feature and has\n512 channels, and FFEN(\u0001)is the FEN.\nB.Density-Aware Network (DAN)\nThe goal of DAN is to capture pan-density information.\nTherefore, each subnetwork in DAN is pretrained on scenarios\nwith specific density so that it can recognize specific density\ncrowds. However, determining the ground truth for an image’s\ndensity level depends on human experiences. A straightfor-\nward way is to focus on the number of people in the image.\nDue to differences in crowd distributions, there exists scenarios\nwhere the number of people is the same while the density of\nthe crowd is different. To address this issue, the work proposed\nby Sam et al. [15] suggests that using the average distance\nbetween adjacent heads is more effective than mere head count\nto represent the dense degree of the crowd. Therefore, we\ncalculate the dense degree for an image patch as follows,\nD=1\nPPX\ni=1QX\nj=1dij; (2)5\nAlgorithm 1 Training Phase\nInput: input crowd image patches dataset S\nOutput: output the parameters \u0002PaDNet\nInit: Dividing the whole image patches SintoNclusters\nS1;S2:::SNvia K-means clustering algorithm.\nfori= 1 toepoch 1do\nforj= 1 toNdo\nTrainingjth subnetwork with Sjupdate \u0002j\nLmse=1\nkSjkPkSjk\ni=1k^Yi\u0000ZGT\nik2\n2\nSaving the best state \u0002jofjth subnetwork\nend for\nend for\nLoading the bestf\u0002jgN\n1forPaDNet\nfori= 1 toepoch 2do\nTraining PaDNet withSupdate \u0002PaDNet and don’t\nfreeze the \u0002j.\nL=1\nkSkPkSk\ni=1kFPaDNet (Xi)\u0000ZGT\nik2\n2+\u0015Lce\nend for\nreturn \u0002PaDNet\nAdam is applied with learning rate at 10\u00005and weight\ndecay at 10\u00004\nwherePis the number of people in an image patch, dij\nrepresents the distance between the ith subject and its jth\nnearest neighbor, and Qis the calculated maximum number of\nnearest neighbors. Intuitively, the smaller the value of D, the\ndenser the crowd. Examples are shown in Figure 4 indicates\nthat the average distance is more reasonable to represent the\ndense degree of the crowd.\nTABLE I\nDAN CONSISTS OF MULTIPLE SUBNETWORKS . THE NUMBER OF\nSUBNETWORK IS RELATED TO CLASSES OF DENSITY LEVEL . THE\nCONVOLUTIONAL LAYERS ’PARAMETERS ARE DENOTED AS\n“CONV (KERNEL SIZE ,OUTPUT CHANNELS ).” E VERY CONVOLUTIONAL\nLAYER IS FOLLOWED BY BATCH NORMALIZATION [46] AND RELU.\nLevel-1 Level-2 Level-3 Level-4\nConv(9, 384) Conv(7, 256) Conv(5, 128) Conv(5, 128)\nConv(9, 256) Conv(7, 128) Conv(5, 64) Conv(5, 64)\nConv(7, 128) Conv(5, 64) Conv(3, 32) Conv(3, 32)\nConv(5, 64) Conv(3, 32) Conv(3, 16) Conv(3, 16)\nConv(1, 1) Conv(1, 1) Conv(1, 1) Conv(1, 1)\nIn DAN, the number of subnetworks is the same as the\nnumber of density categories. We design different network\nconfigurations from level-1 to level-4 subnetworks. The con-\nfigurations are shown in Table I. The lower-level networks are\nused to recognize sparse crowds; the higher-level networks\nare used to recognize dense crowds. For sparse crowds, the\ndistances between adjacent people are larger and the head sizes\nare typically larger than they are in dense crowds. Therefore,\nwe use lower-level subnetworks with larger filters to recognize\nthe sparser crowds and higher-level subnetworks with smaller\nfilters to recognize denser crowds. As the density level in-creases, the sizes of the filters gradually become smaller. The\nfilters of each subnetwork are pyramidal and become smaller\nfor enhancing the multi-scale ability of the subnetwork. In\naddition, the lower subnetworks include more filters than the\nhigher subnetworks do in each layer because the distribution of\na dense scene is more uniform than it is in a sparse scene. The\nwork of Li et al. [17] suggests that too many pooling layers\ncan reduce the spatial information of feature map. Therefore,\nthere is no pooling layer in DAN. Formally, given the base\nfeature ^Yfeatas input for DAN, each subnetwork generates a\ndensity-specific feature as follows,\n^Yi=Fsub i(^Yfeat); (3)\nwhere ^Yiis the density-specific feature generated by the ith\nsubnetwork, and Fsub i(\u0001)denotes theith subnetwork of DAN.\nC.Feature Enhancement Layer (FEL)\nAlthough each subnetwork of DAN can recognize a specific\ndensity crowd, feature maps at varying levels of density\nshould be weighted with different importance because of the\nnonuniform distributions of crowds. Therefore, we design a\nFeature Enhancement Layer (FEL) to assign different weights\nfor varying feature maps. Specifically, these subnetworks of\nDAN generate density-specific feature maps, ^Y1,^Y2,:::,^Yn.\nWe concatenate them as input for FEL. FEL consists of a\nSpatial Pyramid Pooling (SPP) [47] and a Fully Connected\n(FC) layer. SPP performs three scales pooling operations for\neach ^Yi. Theith operation divides feature map into i\u0002i\nregions with pooling to capture the global and local contextual\nfeatures. Then the FC layer analyzes the contextual features\nto weight the importance for each ^Yi. Formally, given several\ndensity-specific feature maps ^Y1,^Y2,:::,^Ynas input for FEL\nas follows,\nv=FFEL(^Y1;^Y2;:::;^Yn); (4)\nwhereFFEL is FEL and vis the vector ( v1,v2,:::,vN)\ngenerated by FEL, we weight the importance for each density-\nspecific feature ^Yias follows,\n^Y0\ni=^Yi\u0002(1 +wi); (5)\nwi=exp(vi)PN\nj=1exp(vj); (6)\nwhere the number 1denotes that retaining the original feature\nof theith subnetwork, and widenotes the importance for this\nfeature map. The cross-entropy loss is used to train FEL.\nD.Feature Fusion Network (FFN)\nIn order to fuse the refined feature maps^Y0\nis, we propose a\nsophisticated network, named Feature Fusion Network (FFN),\nto embed spatial context effectively and combine all feature\nmaps for generating the final density map. FFN consists\nof Conv(7, 64), Conv(5, 32), Conv(3, 32), and Conv(1, 1).\nEvery convolutional layer is followed by Batch Normalization\nand ReLU. Inspired by U-Net [48] and DenseNet [49], skip\nconnections can make up for the lost information and improve6\nthe performance. Before Conv(1, 1) layer, we further add a\nskip connection to concatenate ^Yis.\nThe detail of the training procedure is shown in Algorithm 1.\nThe loss function for training the PaDNet is given as follows,\nL=Lmse+\u0015Lce; (7)\nLmse=1\nMMX\ni=1kFPaDNet (Xi)\u0000ZGT\nik2\n2; (8)\nwhereMis the number of the training samples and \u0015is the\nweight factor ofLcewith the settings listed in Table II. The\ndenser the crowd, the larger the \u0015. In a sparse crowd, the value\nof training lossLmseis very small. Therefore, we set a small\nvalue for\u0015.FPaDNet (Xi)is the density map estimated by the\nPaDNet.ZGT\niis theith ground truth.\nTABLE II\nTHE PARAMETER SETTINGS OF \u0015FOR DIFFERENT DATASETS .\nDataset \u0015\nShanghaiTech A[10]\n1 UCF CC50[50]\nUCF QNRF[11]\nShanghaiTech B[10] 0:1\nUCSD[25] 0:01\nIV.EXPERIMENTS\nWe now evaluate PaDNet on four crowd counting bench-\nmark datasets: the ShanghaiTech [10], the UCSD [25], the\nUCF CC50 [50], and the UCF-QNRF [11]. We compare\nPaDNet with five state-of-the-art algorithms including D-\nConvNet [38], ACSCP [36], ic-CNN [40], SANet [33], and\nCSRNet [17]. Further, we conduct extensive ablation experi-\nments to analyze the effect of different components in PaDNet.\nWe detail experimental settings and results below.\nA.Data Preparation\nWe resize the training images to 720 \u0002720, and crop nine\npatches from each image. Four of them contain four quarters\nof the image without overlapping. The remaining five patches\nare randomly cropped from the image. By using horizontal flip\nfor these patches, we can get 18 patches from each image.\nWe calculate the dense degree Dusing Eq. (2) for every\npatch. Then the K-means algorithm is performed to cluster\nall image patches into Csubsets with varying density level.\nTo avoid sample imbalance, we continue to randomly crop the\npatches from the original images to balance each subset so that\neach category will have an equivalent number of patches. The\nsetups of different datasets are listed in Table III. Note that\nthe UCSD [25] is a sparse dataset, hence we set Qto2.\nThe ground truth is generated by blurring the head an-\nnotations with a normalized Gaussian kernel (sum to one).\nGeometry-adaptive kernel used for generating the density map,\nas in [10], is defined as:\nF(x) =NX\ni=1\u000e(x\u0000xi)\u0002G\u001bi(x);with\u001bi=\fdi;(9)TABLE III\nQNEAREST NEIGHBORS ARE CALCULATED FOR DIFFERENT DATASETS .\nDataset QNearest Neighbors\nShanghaiTech[10]\n5 UCF CC50[50]\nUCF QNRF[11]\nUCSD[25] 2\nwherexiis the position of ith head in the ground truth \u000e\nanddiis the average distance of Knearest neighbors. We\nconvolve\u000e(x\u0000xi)with a Gaussian kernel with parameter\n\u001bi. For the ShanghaiTech [10], the UCF CC50 [50], and\nthe UCF-QNRF [11] datasets, we set \fto0:3. The UCSD\ndataset [25] does not satisfy the assumptions that the crowd\nis evenly distributed, so we set \u001bof the density map to 3.\nB.Evaluation Metrics\nThe general evaluation metrics of crowd counting are mean\nabsolute error (MAE) and root mean squared error (RMSE).\nHere MAE is defined as\nMAE =1\nMMX\ni=1jCXi\u0000CGT\nXij; (10)\nand RMSE is defined as\nRMSE =vuut1\nMMX\ni=1\u0000\nCXi\u0000CGT\nXi\u00012; (11)\nwhereMis the number of test samples, CXiandCGT\nXiare\nthe estimated number of people and the ground truth for the\nith image, respectively. Moreover, the MAE and the RMSE\nreflect the algorithm’s global accuracy and robustness.\nHowever, MAE and RMSE cannot be used to evaluate\nlocal performance. The GAME metric [51] that has been used\nin vehicle counting to evaluate local estimation has some\nsimilarity. But it just covers a limited range of scales and does\nnot adequately reflect the robustness for pan-density crowd\ncounting. Therefore, we expand MSE and RMSE to patch\nmean absolute error (PMAE) and patch root mean squared\nerror (PRMSE), respectively to accommodate our needs.\nPMAE =1\nn\u0002Mn\u0002MX\ni=1jCXi\u0000CGT\nXij; (12)\nPRMSE =vuut1\nn\u0002Mn\u0002MX\ni=1\u0000\nCXi\u0000CGT\nXi\u00012: (13)\nSpecifically, we split each image into npatches of same size\nwithout overlapping and calculate the MAE and RMSE of the\npatches. That is, PMAE and PRMSE are able to completely\nreflect the algorithm’s local accuracy and robustness. Note that\nwhennequals to 1, PMAE and PRMSE will degenerate into\nMAE and RMSE, respectively.7\n71.765.1\n59.267.5108.6104.5\n98.1107.6\n5060708090100110120\nPaDNet-1 PaDNet-2 PaDNet-3 PaDNet-4SHA Dataset\nMAE RMSE12.4\n9.1\n8.110.522.3\n15.5\n12.215.7\n510152025\nPaDNet-1 PaDNet-2 PaDNet-3 PaDNet-4SHB Dataset\nMAE RMSE281.8\n185.8228267.8387.7\n278.3298.7373.3\n150200250300350400\nPaDNet-1 PaDNet-2 PaDNet-3 PaDNet-4UCF_CC_50 Dataset\nMAE RMSE\n118.3108.4\n101.896.5207.1\n194.7\n180.8\n170.2\n90110130150170190210\nPaDNet-1 PaDNet-2 PaDNet-3 PaDNet-4UCF-QNRF Dataset\nMAE RMSE1.17\n0.850.91.001.48\n1.061.131.26\n0.811.21.4\nPaDNet-1 PaDNet-2 PaDNet-3 PaDNet-4UCSD Dataset\nMAE RMSE\nFig. 5. We conduct experiments on all datasets to analyze the effect of density level division. PaDNet-N indicates that we divide the dataset into Nclasses, and\nPaDNet hasNsubnetworks. PaDNet-2 achieves the best recognition performance on the UCSD and UCF CC50 datasets. PaDNet-3 has superior recognition\nperformance on the ShanghaiTech dataset. PaDNet-4 performs the best on the UCF-QNRF dataset.\nTABLE IV\nCOMPARISON ON THE SHANGHAI TECH DATASET\nPart A Part B\nMethod MAE RMSE MAE RMSE\nZhang et al. [29] 181.8 277.7 32.0 49.8\nMCNN [10] 110.2 173.2 26.4 41.3\nSwitch-CNN [15] 90.4 135.0 21.6 33.4\nCP-CNN [16] 73.6 106.4 20.1 30.1\nLiu et al. [39] 73.6 112.0 13.7 21.4\nIG-CNN [52] 72.5 118.2 13.6 21.1\nD-ConvNet [38] 73.5 112.3 18.7 26.0\nACSCP [36] 75.7 102.7 17.2 27.4\nic-CNN [40] 68.5 116.2 10.7 16.0\nCSRNet [17] 68.2 115.0 10.6 16.0\nSANet [33] 67.0 104.5 8.4 13.6\nOurs 59.2 98.1 8.1 12.2\nC.Datasets and Comparisons\n1)The ShanghaiTech dataset :This dataset contains 1,198\nannotated images from a total of 330,165 people, each of\nwhich is annotated at the center of the head. The dataset is\ndivided into Part A and Part B. Part A contains 482 images\nrandomly crawled from the Internet. The training set has 300\nimages and the testing set has 182 images. Part B contains 716\nimages taken from the busy streets of the metropolitan areas in\nShanghai. The training set has 400 images, and the testing set\nhas 316 images. The density of Part A is higher than Part B,\nand the density varies significantly. We test the performance of\nPaDNet on Part A and Part B as the other approaches did and\nreport the best performance in Table IV. PaDNet achieves the\nbest performance among all approaches. Specifically, it has an\n11.6% MAE and a 6.1% RMSE improvement for the Part ATABLE V\nCOMPARISON ON THE UCF CC 50DATASET\nMethod MAE RMSE\nZhang et al. [29] 467.0 498.5\nMCNN [10] 377.6 509.1\nSwitch-CNN [15] 318.1 439.2\nCP-CNN [16] 295.8 320.9\nLiu et al. [39] 337.6 434.3\nIG-CNN [52] 291.4 349.4\nD-ConvNet [38] 288.4 404.7\nACSCP [36] 291.0 404.6\nic-CNN [40] 260.9 365.5\nCSRNet [17] 266.1 397.5\nSANet [33] 258.4 334.9\nOurs 185.8 278.3\ndataset compared with the second-best approach, SANet [33].\n2)The UCF CC50 dataset :This is an extremely dense\ncrowd dataset. It contains 50 images of different resolutions\nwith counts ranging from 94 to 4,543 with an average of\n1,280 individuals in each image. The training set only has\n40 images and the testing set only has 10 images. To more\naccurately verify the performance of PaDNet, we adopt a\n5-fold cross-validation following the standard setting in [50].\nExperiments shown in Table V indicate that PaDNet achieves a\n28.1% MAE improvement compared with SANet, and a 13.3%\nRMSE improvement compared with CP-CNN. These results\nindicate that PaDNet is suitable for extremely dense scenes.\n3)The UCSD dataset :The UCSD dataset [25] is a sparse\ndensity dataset that is a 2,000-frame video dataset chosen from\none surveillance camera on the UCSD campus. The ROI and\nthe perspective map are provided in the dataset. The resolution\nof each image is 238 \u0002158, and the crowd count in each8\n(a) Original\n (b) Ground Truth Count: 162\n (c) PaDNet-1 Count: 74.8\n(d) PaDNet-2 Count: 127.2\n (e) PaDNet-3 Count: 136.5\n (f) PaDNet-4 Count: 75.3\nFig. 6. An example result of the SHA dataset. (b) shows the ground truth density map. (c)-(f) are density maps generated by PaDNet-1, PaDNet-2, PaDNet-3\nand PaDNet-4, respectively.\nTABLE VI\nCOMPARISON ON THE UCSD DATASET\nMethod MAE RMSE\nZhang et al. [29] 1.60 3.31\nMCNN [10] 1.07 1.35\nSwitch-CNN [15] 1.62 2.10\nACSCP [36] 1.04 1.35\nHuang et al. [53] 1.00 1.40\nCSRNet [17] 1.16 1.47\nSANet [33] 1.02 1.29\nOurs 0.85 1.06\nTABLE VII\nCOMPARISON ON THE UCF-QNRF DATASET\nMethod MAE RMSE\nIdrees et al. [50] 315.0 508.0\nEncoder-Decoder [54] 270.0 478.0\nCMTL [30] 252.0 514.0\nResNet101 [42] 190.0 277.0\nDenseNet201 [49] 163.0 226.0\nMCNN [10] 277.0 426.0\nSwitch-CNN [15] 228.0 445.0\nIdrees et al. [11] 132.0 191.0\nOurs 96.5 170.2\nimage varies from 11 to 46. As Chan et al. did, we use frames\nfrom 601 to 1,400 as the training set and the remaining frames\nfor testing. All the frames and density maps are masked with\nROI. The results are listed in Table VI. Our method achieves\nsuperior performance both on a highly dense crowd dataset and\na sparse crowd dataset. Our method has a 15.0% MAE and a\n17.8% RMSE improvement for the UCSD dataset comparedwith the second-best approach, SANet and Huang et al. ’s\nmethod [53].\n4)The UCF-QNRF dataset :We further evaluate the recog-\nnition performance of PaDNet on the UCF-QNRF dataset [11],\nwhich is the newest and largest crowd dataset. The UCF-\nQNRF contains 1.25 million humans marked with dot annota-\ntions and consists of 1,535 crowd images with wider variety of\nscenes containing the most diverse set of viewpoints, densities,\nand lighting variations. The minimum and the maximum\ncounts are 49 and 12,865, respectively. Meanwhile, the median\nand the mean counts are 425 and 815.4, respectively. We use\n1,201 images for training set and 334 images for testing,\nfollowing [11]. The results are listed in Table VII. PaDNet\nobtains the lowest MAE performance, and a 26.9% MAE\nimprovement compared with the second-best approach, i.e.,\nIdrees et al. [11].\nD.Algorithmic Studies\nWe explore PaDNet from three aspects: 1) the effect of\ndensity level, 2) the effects of different components in PaDNet,\nand 3) the performance of PaDNet in pan-density crowd\ncounting.\n1)The effect of density level :The setting of the density\nlevel affects data preprocessing and the number of subnet-\nworks, and this setting relates to experimental results. The\nexperimental results are shown in Figure 5, where Nin the\nname PaDNet- Nindicates that we divide the dataset into N\nclasses and PaDNet has Nsubnetworks. Specifically, when\nNequals to 1, PaDNet does not have FEL or FFN. As seen\nin Figure 5, PaDNet-2 achieves the best recognition perfor-\nmance on the UCSD and UCF CC50 datasets. PaDNet-3 has\nsuperior performance on the ShanghaiTech dataset. PaDNet-\n4 performs the best on the UCF-QNRF dataset. Intuitively,9\n(a) Original\n (b) GT Count: 86\n (c) Est Count: 107.6\n (d) Est Count: 80.2\n (e) Est Count: 83.0\nFig. 7. An example result of the SHA dataset [10]. The density maps are generated by different configuration of PaDNet. (b) shows the ground truth. (c) is\nthe result of the PaDNet without FEL and Skip Connection (SC). (d) is the result of PaDNet only without SC. (e) is the result of PaDNet.\nTABLE VIII\nTHEPMAE AND PRMSE OFPADN ET COMPARE WITH CSRN ET AND MCNN.\nMethodsn = 1 n = 4 n = 9 n = 16\nPMAE PRMSE PMAE PRMSE PMAE PRMSE PMAE PRMSE\nMCNN [10] 112.8 173.0 34.6 58.4 17.1 30.3 10.1 19.1\nCSRNet [17] 68.8 107.8 19.8 37.3 9.6 19.9 5.7 13.2\nPaDNet w/o FEL&SC 65.0 103.2 20.6 38.5 10.6 21.6 6.3 14.1\nPaDNet w/o SC 60.4 100.8 18.3 35.8 9.1 19.3 5.5 12.7\nPaDNet 59.2 98.1 17.9 35.4 8.8 19.1 5.3 12.7\nTABLE IX\nTHE EFFECTS OF DIFFERENT COMPONENTS IN PADN ET ON THE SHA\nDATASET .\nMethod MAE RMSE\nPaDNet w/o FEL&SC 65.0 103.2\nPaDNet w/o SC 60.4 100.8\nPaDNet 59.2 98.1\ndifferent datasets should have been adapted to different number\nof subnetworks. On the other hand, the number of subnetworks\nshould fit the distribution of the dataset. For examples, the\ncrowd count in each image varies from 11 to 46 in the UCSD\ndataset, and the UCF CC50 dataset has an average of 1,280\npersons in each image. Based on our experiments, we notice\nthat when the division of density level is 2, the corresponding\nperformance is the best because of the micro-variation density\nin the UCSD and UCF CC50 datasets.\nFor the ShanghaiTech dataset, PaDNet-3 performs better\nthan PaDNet-2 because of the high-variation density in the\nShanghaiTech dataset. Note that PaDNet-4 performs worse\nthan PaDNet-3 on this dataset. In general, we argue that the\nmore abundant the training data for each subnetwork, the\nstronger the generalization ability of the subnetwork. Since\nPaDNet-1 only has one subnetwork, it is difficult to cover\nall the distributions of crowd. As the number of subnetworks\nincreases, the amount of training data for each subnetwork\ndecreases. While each subnetwork makes it easier to cover\nthe distribution, it also reduces generalization ability to some\nextent. For the UCF-QNRF dataset which has a greater density\nvariation, the minimum and the maximum counts are 49 and\n12,865, respectively. It has 1,201 original images for training.\nThus, we divide the UCF-QNRF dataset into 4 levels and\nPaDNet-4 achieves the best recognition performance.\nIn order to comprehend this reason intuitively, the den-\nsity maps generated by PaDNet-1, PaDNet-2. PaDNet-3 andPaDNet-4 are shown in Figure 6. The density map generated\nby PaDNet-1 is slightly blurred and PaDNet-1 cannot recog-\nnize different density crowds. As the increase of subnetworks,\nthe recognition abilities of PaDNet-2 and PaDNet-3 become\ngradually stronger. The generated density map achieves higher\nquality, and the estimated count is more precise. For PaDNet-\n4, although it has a clear density map, the estimated count\nis biased. As the generalization ability of each subnetwork\nweakens, which is caused by overfitting, PaDNet-4 cannot\naccurately recognize the bottom-left corner of the image.\n2)Effects of different components :We analyze the effects\nof different components of PaDNet in three aspects: (i) the\neffects of FEL and Skip Connection, (ii) the effect of DAN,\nand (iii) the effect of weighting strategy for feature maps.\nThe effects of FEL and Skip Connection. We conduct\nthe experiments on the ShanghaiTech Part A (SHA) dataset\nto analyze the effects of FEL and Skip Connection (SC) in\nPaDNet-3. The results are listed in Table IX.\nThe first method is the baseline of PaDNet-3 and does\nnot have FEL or SC. In the second method, only FEL is\nintroduced to analyze the effect of FEL. In the third method,\nFEL and SC are incorporated. The baseline method uses the\nsame weights to fuse the feature maps generated by DAN,\nand the corresponding MAE is just 65.0. Specially, when\nFEL is introduced into the framework, the MAE is improved\nto 60.4. Thus, this approach is reasonable for fusing feature\nmaps with learned weights. After SC is introduced into the\nframework, the MAE is improved to 59.2 justifying that SC\nis also an effective trick. The generated density maps are\nshown in Figure 7. By comparing these density maps, we can\nfurther analyze the effects of different components in PaDNet.\nThe density map generated by the baseline method is slightly\nblurred because it overestimates the count. Concretely, the\nbottom of the density map is biased. When FEL is employed\nto adjust the weights of feature maps, the generated density\nmap becomes accurate and the overestimation at the bottom10\nTABLE X\nTHEPMAE AND PRMSE OFPADN ET-NCOMPARE WITH CSRN ET AND MCNN ON MORE DATASETS .\nDatasets Methodsn = 1 n = 4 n = 9 n = 16\nPMAE PRMSE PMAE PRMSE PMAE PRMSE PMAE PRMSE\nSHA [10]MCNN [10] 112.8 173.0 34.6 58.4 17.1 30.3 10.1 19.1\nCSRNet [17] 68.8 107.8 19.8 37.3 9.6 19.9 5.7 13.2\nPaDNet-1 71.1 108.6 20.7 36.3 10.1 20.0 6.1 12.8\nPaDNet-2 65.1 104.5 19.7 36.7 9.8 20.0 5.8 12.8\nPaDNet-3 59.2 98.1 17.9 35.4 8.8 19.1 5.3 12.7\nPaDNet-4 67.5 107.6 19.4 37.1 9.4 20.2 5.6 12.9\nSHB [10]MCNN 26.6 43.3 9.1 17.5 4.8 10.1 3.1 7.1\nCSRNet 9.8 16.1 3.3 7.1 1.8 4.4 1.2 3.2\nPaDNet-1 12.4 22.3 4.0 9.4 2.1 5.5 1.3 4.0\nPaDNet-2 9.1 15.5 3.0 6.7 1.6 4.1 1.1 3.0\nPaDNet-3 8.1 12.2 3.0 5.7 1.6 3.4 1.1 3.0\nPaDNet-4 10.5 15.8 3.6 7.0 1.9 4.2 1.3 3.1\nUCF CC50 [50]MCNN 378.1 504.3 114.1 179.4 53.6 84.8 32.3 54.0\nCSRNet 256.4 355.2 73.7 110.7 36.5 60.3 21.6 36.1\nPaDNet-1 281.8 387.7 82.3 121.9 39.6 64.7 23.7 39.4\nPaDNet-2 185.8 278.3 65.9 93.3 33.4 53.6 20.2 32.2\nPaDNet-3 228.0 298.7 74.2 104.2 35.4 60.4 21.6 35.6\nPaDNet-4 267.8 373.3 78.4 121.8 36.6 61.4 22.2 38.0\nUCF-QNRF [11]MCNN 273.3 408.0 78.3 134.1 37.1 68.4 22.0 43.0\nCSRNet 114.6 208.4 32.0 69.1 15.7 37.9 9.3 23.6\nPaDNet-1 118.3 207.1 33.3 70.4 15.9 38.6 9.6 24.6\nPaDNet-2 108.4 194.7 31.6 71.7 15.6 39.8 9.5 25.1\nPaDNet-3 101.8 180.8 30.4 65.4 15.4 37.7 9.3 23.7\nPaDNet-4 96.5 170.2 28.7 62.8 14.5 35.7 8.9 22.7\nof the image is eliminated. However, the density map loses\na little information in the middle of the image. When SC is\nemployed, the lost information is supplemented.\nTABLE XI\nTHE EFFECTS OF DIFFERENT CHANNEL CONFIGURATIONS OF DAN ON\nTHE UCF-QNRF DATASET .\nMethod MAE RMSE\nPaDNet-3- ascend 106.7 184.0\nPaDNet-3- equal 103.5 179.7\nPaDNet-3- descend 101.8 180.8\nPaDNet-3- double 100.5 174.4\nPaDNet-4 96.5 170.2\nThe effect of DAN. We design DAN with pyramidal filters\nto enhance the ability of capturing pan-density information. In\nparticular, the lower-density subnetworks have relatively larger\nfilters that is similar to many multi-scale methods [10, 15,\n16]. Hence, we mainly explore the effects of different channel\nconfigurations of DAN, and then conduct the studies on the\nUCF-QNRF dataset. The results are listed in Table XI. We\nhave three channel configurations for DAN of PaDNet-3, i.e.,\nascend ,descend andequal . The descend listed in Table I is\nour final design. The ascend means the lower subnetworks has\nfewer channels. The equal means all the subnetworks have the\nsame number of channels as level-2 subnetwork (Table I).\nNote that PaDNet-3- descend obtains the best recognition\nperformance compared with PaDNet-3- ascend and PaDNet-3-\nequal . It confirms our speculation that lower-level subnetworks\nshould equip with more filters than higher-level subnetworksin each layer because sparse scenes have more variations\nof crowds and environmental interference. When compared\nwith sparse scenes, the distribution of dense crowds is closer\nto the uniform distribution. PaDNet-3- ascend has the fewest\nchannels for level-1 subnetwork (Table I), hence it obtains\nthe worst results. Meanwhile, in order to exploit whether the\nnumber of parameters of DAN could greatly affect the final\nperformance, we double the channels of level-3 subnetwork\n(Table I), and named as PaDNet-3- double , to make PadNet-3\nand PaDNet-4 have same number of parameters. The MAE\nof PaDNet-3- double is 100.5 which is better than PaDNet-3\nbecause of the added parameters, but PaDNet-3- double still\ncannot achieve the competitive performance compared with\nPaDNet-4. Therefore, the number of subnetworks of DAN has\na greater influence on final performance than does the number\nof parameters of DAN.\nThe effect of weighting strategy for feature maps. A\nnormal idea is that a network learns a wito weight the\nith feature map, but such a weighting strategy can only\nobtain limited performance improvement. Thus, we weight the\nimportance for each feature map using Eq. (5). In order to\nverify the effectiveness of our strategy, we conduct the ablation\nstudies on the SHA dataset with PaDNet-3. The results are\nlisted in Table XII. We discover that using 1+wias the weight\nhas a better performance than using wias the weight. We\nargue that these feature maps generated by subnetworks are\nhigh-level features and quite close to the final density map.\nHere,wiis not the pixel-level weight and the value of wi\nranging in (0;1). Usingwias the weight for the whole feature\nmap could lose the local features of crowd. However, the form11\nMCNN Count:  1273.5 : CSRNet Count: 985.0 : PaDNet Count: 1046.0 : GT Count: 1068 :\nGT Count: 61 :\nGT Count: 48 :MCNN Count: 22.9 : CSRNet Count: 58.4 : PaDNet Count: 60.6 :\nPaDNet Count : 48.7 : MCNN Count: 59.0 : CSRNet Count: 43.9 :\nGT Count: 700 :\nGT Count: 2364 : MCNN Count:  2481.4 :CSRNet Count: 940.7 :\nCSRNet Count: 2121.3 :PaDNet Count: 610.6 :\nPaDNet Count: 2317.5 :\nMCNN Count:  588.9\nFig. 8. Example experimental results. The images in each row are original crowd image, the ground truth, the result generated by MCNN, the result generated\nby CSRNet (the state-of-the-art method based on single-column dilated convolutional network), and the result generated by our PaDNet, respectively. The\nimages of the first two rows are in the SHB dataset. The images of the third row are in the SHA dataset. The remaining images are in the UCF CC50\ndataset.\nTABLE XII\nTHE EFFECTS OF DIFFERENT WEIGHT STRATEGIES FOR FEATURE MAPS .\nMethod MAE RMSE\nPaDNet(wias the\nweight)62.4 104.7\nPaDNet( 1 +wias the\nweight)59.2 98.1\nof1 +wiexhibits the combination of the original and the\nimportant features. Therefore, this strategy accurately weights\nthe importance of feature maps at different levels based on\nwis and will not lose the original features.\n3)Performance in pan-density crowd counting :We eval-\nuate the performance of our method in pan-density crowd\ncounting from two aspects: (i) the performance in different\ndensity scenes, and (ii) the performance at local regions ofthe same scene. We conduct the experiments on SHA with\nPaDNet-3. Meanwhile, we compare PaDNet with MCNN and\nCSRNet1. In order to evaluate the performance in different\ndensity scenes, we divide the SHA dataset into five groups\naccording to increasing density level. The results are shown in\nFigure 9. PaDNet achieves the best recognition performance\non the density levels 2, 3, and 5. CSRNet obtains the best\nperformance on density level 1 and MCNN on level 4. How-\never, PaDNet achieves competitive performance as CSRNet\nand MCNN on levels 1 and 4. Thus, PaDNet demonstrates\nboth better accuracy and higher robustness in different density\nscenes.\nAs mentioned above, most existing methods performed\nwell in global estimation while neglecting local accuracy.\nWe evaluate the local accuracy and robustness for PaDNet\n1We implemented MCNN and CSRNet algorithms and obtained almost the\nsame results.12\n1342213134271038\n+12+5+1+19-60\n+5-15-25-22-108\n+58+26-9-8-111\n10020030040050060070080090010001100\n1 2 3 4 5Average Count\nDensity LevelGT PaDNet CSRNet MCNN\nFig. 9. Histogram of average crowd numbers estimated by different methods\non five groups splited from the SHA dataset according to increasing density\nlevel.\naccording to proposed PMAE and PRMSE. We calculate\nPMAE and PRMSE when n(Eq. 12 and 13.) is set to 1, 4, 9\nand 16. The results are listed in Table VIII. The performance\nof PaDNet is better than MCNN and CSRNet under various\nconditions. This suggests that regardless of global or local\ncounts are examined, PaDNet achieves highly accurate and\nrobust estimation in pan-density crowd counting.\nMeanwhile, we calculate PMAE and PRMSE of the ablated\nPaDNet. By comparing the last three rows of Table VIII,\nboth FEL and SC are justified to be effective. Especially, for\nPaDNet without FEL and SC, the global MAE and RMSE are\nbetter than CSRNet. But the PMAE and PRMSE are worse\nthan CSRNet. When FEL is introduced into the framework,\nthe local evaluation is improved and the results are better\nthan CSRNet. The experiments show that FEL is beneficial\nfor improving the global and local recognition performance.\nIn order to evaluate the effectiveness of PMAE and PRMSE\nand to make it convenient for other researchers to follow pan-\ndensity crowd counting, we calculate proposed PMAE and\nPRMSE for MCNN, CSRNet, PaDNet-1, PaDNet-2, PaDNet-\n3, and PaDNet-4 on the SHA, SHB, UCF CC50, and UCF-\nQNRF datasets. The results are listed in Table X. It shows\nthat PMAE and PRMSE are effective and robust metrics for\nevaluating the global and local accuracy and robustness. For\nexample, in the SHA and UCF-QNRF datasets, the MAE of\nCSRNet is worse than PaDNet-2, while the PMAE of CSRNet\nis better than PaDNet-2. These results demonstrate that PMAE\nand PRMSE can effectively evaluate both global and local\nperformances.\nFigure 8 shows some density maps predicted by MCNN,\nCSRNet, and PaDNet. The density maps generated by MCNN\nare a little blurred, and the estimated counts are quite biased.\nSimilarly, the density maps generated by CSRNet are also\nblurred in extremely dense scenes. In contrast, the density\nmaps yielded by PaDNet indicate that not only is the local\ntexture fine-grained but also the global one has high quality.\nConsequently, the counts of PaDNet are the closest to the\nground truth.\nNote that the trade-off for better performance is that datapreprocessing is more complex because we must use different\ndensity datasets to pretrain the corresponding subnetworks.\nFurthermore, it takes about five hours to train the PaDNet on\nthe ShanghaiTech Part A dataset using four NVIDIA GTX\n1080Ti GPUs. In the prediction phase, the computation only\ncosts 0.11 seconds on average for an image with one such\nGPU. Therefore, PaDNet can be readily deployed in real-time\nscene crowd counting.\nV. C ONCLUSIONS\nWe propose a novel end-to-end deep learning framework\nnamed PaDNet for pan-density crowd counting. PaDNet can\nfully leverage pan-density information. Specifically, the com-\nponent DAN can effectively recognize different density crowds\nwhile the component FEL improves both global and local\nrecognition performances. Meanwhile, the new evaluation\nmetrics PMAE and PRMSE, which are extended from MAE\nand RMSE, not only evaluate the global accuracy and robust-\nness but also the local ones. Extensive experiments on four\nbenchmark datasets indicated that PaDNet attained the lowest\npredictive errors and higher robustness in pan-density crowd\ncounting when compared with state-of-the-art algorithms. In\nthe future, we will explore how to simplify network architec-\nture for pan-density crowd counting.\nACKNOWLEDGMENTS\nY . Tian, Y . Lei, and J. Zhang were supported in part\nby the National Key R & D Program of China (No.\n2018YFB1305104), the National Natural Science Foundation\nof China (NSFC 61673118), Shanghai Municipal Science\nand Technology Major Project (No. 2018SHZDZX01) and\nZJLab. J. Z. Wang was supported by The Pennsylvania State\nUniversity. The authors would like to thank Hanqing Chao, Qi\nZhou, Yuan Cao, and Haiping Zhu for their assistance. They\nare grateful to the reviewers and the Associate Editor for their\nconstructive comments.\nREFERENCES\n[1] M. Ford, “Trumps press secretary falsely claims: ‘largest\naudience ever to witness an inauguration, period’,”\nThe Atlantic , January 21, 2017. [Online]. Avail-\nable: https://www.theatlantic.com/politics/archive/2017/\n01/npenalty-n@Minauguration-crowd-size/514058/\n[2] B. Wu and R. Nevatia, “Detection of multiple, partially\noccluded humans in a single image by Bayesian com-\nbination of edgelet part detectors,” in Proceedings of\nthe IEEE International Conference on Computer Vision ,\nvol. 1, 2005, pp. 90–97.\n[3] M. Wang and X. 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Brox, “U-Net: Convo-\nlutional networks for biomedical image segmentation,” in\nProceedings of the International Conference on Medical\nImage Computing and Computer-Assisted Intervention ,\n2015, pp. 234–241.\n[49] G. Huang, Z. Liu, L. v. d. Maaten, and K. Q. Wein-\nberger, “Densely connected convolutional networks,” in\nProceedings of the IEEE Conference on Computer Vision\nand Pattern Recognition , 2017, pp. 2261–2269.\n[50] H. Idrees, I. Saleemi, C. Seibert, and M. Shah, “Multi-\nsource multi-scale counting in extremely dense crowd\nimages,” in Proceedings of the IEEE Conference on\nComputer Vision and Pattern Recognition , 2013, pp.\n2547–2554.\n[51] R. Guerrero-Gmez-Olmedo, B. Torre-Jimnez, R. Lpez-\nSastre, S. Maldonado-Bascn, and D. Ooro-Rubio, “Ex-\ntremely overlapping vehicle counting,” in Iberian Confer-\nence on Pattern Recognition and Image Analysis , 2015.\n[52] D. Babu Sam, N. N. Sajjan, R. Venkatesh Babu, and\nM. 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Popescu1,2\n1School of Physics, University of Bristol, Tyndall Avenue, B ristol BS8 1TL, U.K. and\n2Institute for Theoretical Studies, ETH Zurich\n(Dated:)\nIn this note I will present a subtle interplay between densit y matrices and the knowledge about\ntheir preparation, and I will argue that there is a need to con sider a new type of quantum state, in\nbetween pure states and density matrices.\nQuantum mechanics presents many crucial differences\nfrom the classical world that we know in our daily life.\nOne of the most dramatic of them is, arguably, the one\nrelated to density matrices: In quantum mechanics it is\npossible to have completely different physical situations\nwhich nevertheless cannot be distinguished from one an-\nother by any observation. This is the case of various\nensembles of systems prepared in ways corresponding to\nthe same density matrix. When information is also given\nabout how these different situations were prepared, we\ncan tell them apart, but as long as we only have the sys-\ntems themselves, no observation can differentiate them.\nIn this short note I will present a new and subtle inter-\nplaybetween densitymatricesandknowledgeabouttheir\npreparation.\nSuppose Alice prepares a large ensemble of spin 1 /2 par-\nticles by taking each particle separately, tossing an unbi-\nassed coin, and then preparing the particle either “up”\nor “down” along the z-axis. The particles are numbered,\nand Alice notes their preparation in a notebook; she\ntherefore knows the state (“up- z” or “down- z”) for each\nparticle.\nThis ensemble of particles corresponds to the identity\ndensity matrix.\nSuppose now Alice were to give this ensemble to Bob,\nwithout telling him anything about the preparation. As\nitiswell-known,Bobhasabsolutelynowaytodistinguish\nthis ensemble from any other ensemble that corresponds\nto the same density matrix (identity density matrix in\nthis case), such as an ensemble of particles prepared by\ntaking each particle separately, tossing an unbiased coin,\nand then preparing the particle either “up” or “down”\nalong the x axis. All he can do is to is to certify that\nthe ensemble corresponds to the identity density ma-\ntrix, but he can learn nothing about which particular\nensemble corresponding to that density matrix was pre-\npared. The “ x-ensemble” is physically different from the\n“z-ensemble” yet Bob has no way to distinguish them.\nAlice however, knowing what she did, can do much more\nthan what Bob can: she can predict what results she will\nget if she were to measure again the particles along thez-axis. So, in particular, she could check if her ensemble\nwas substituted by another ensemble corresponding to\nthe identity density matrix.\nAll of the above is well known.\nSuppose now that Alice loses her notebook. She still re-\nmembers that she prepared the particles polarised along\nthe z axis, but not in which direction, “up- z” or “down-\nz” each particle is. Apparently now she is in no better\nposition than Bob. Indeed, exactly like Bob, she can no\nlonger predict the outcomes of the measurements along\nany axis, including the z-axis. Moreover, exactly like\nBob, she no longer has any way to determine whether or\nnot her ensemble was substituted by another ensemble\ncorresponding to the identity density matrix.\nAsshe nolongercandistinguish theensemble shecreated\nfrom any other ensemble corresponding to the same den-\nsity matrix, we could be tempted to conclude that now,\nfor all practical purposes, all she has is a density matrix,\nnot a specific ensemble, exactly as Bob.\nYet, Alice still has some information about the original\nensemble, namely that it is a “z-ensemble”, i.e. that each\nparticle is polarised either “up-z” or “down-z”. That\nrepresents a certain knowledge about the physical made-\nup of the system, and it is more than what Bob knows.\nBut it seems that makes no difference.\nSurprisingly however, as I will show now, there is a dif-\nference.\nConsider the following scenario. After Alice prepared\nthe ensemble, as described above, she put it in a secure\nplace where nobody, including her, had access to it for\na given time period; the particles are well isolated and\nremain undisturbed during this entire time. Meanwhile\nshe “loses” her notebook. Actually the notebook is not\nlost but stolen by Charles. Charles then goes to a judge\nand claims that he is the one that preparedthe ensemble,\nand that Alice had nothing to do with it. Hence the\nensemble, when released from its secure place, should be\ngiven to him.\nCharles feels certain to win since, using the stolen note-\nbook, he could prove to the judge that he knows what2\nthe ensemble is by correctly predicting the outcomes of\nz-spin measurements. He also knows that Alice has no\nway that she could prove to the judge that she knows\nwhat the ensemble is, since she cannot predict correctly\nthe outcomes of any spin measurements on this ensem-\nble. Alice, however, can disprove Charles’ claim that she\nhas no knowledge whatsoever what the ensemble is:\nHearing Charles’ claim that she doesn’t know what the\nensemble is, Alice asks the judge not to revealto her any-\nthing about what ensemble Charles claims to have pre-\npared. She then tells the judge that this is a z-ensemble.\nThen she asks the judge to request Charles to disclose\nwhat the ensemble is, and to prove it by predicting the\nresults of the spin measurements. The judge should then\nperform the measurements and check Charles’ predic-\ntions. Of course, Charles has no option but to reveal\nthat this is a z-ensemble, since if he would say anything\nelse, he could not predict the result of the measurements.\nNow, the probabilitythatAlice, just bychance, indicated\nthe correct spin polarisation is vanishingly small, so, by\nthe above procedure, she clearly proves she knows what\nthe ensemble is.\nAn even more sophisticated scenario is also possible.\nSuppose Charles convinces the judge that the polarisa-\ntion direction he prepared is an important commercial\nsecret, and cannot be disclosed. Alice can still win. She\nasks for the spins to be measured along a direction of her\nchoosing, and agrees to lose if Charles is able to predict\nthe results with probability only slightly better than 1/2.\nThe judge agrees to this procedure since (a) Alice can\nonly succeed if she is able to choose a direction perpen-\ndicular to the direction of polarisation; if she doesn’t\nknow the polarisation, she has vanishingly small prob-\nability of doing this, and (b) if Alice doesn’t know thedirection and chooses a direction at random, Charles can\nwin without disclosing too much about the actual polar-\nisation direction (even if he can predict the results with\ngreater probability, he only needs to do it slightly better\nthan 1/2, which gives him plenty of room to disguise the\nactual polarisation direction).\nObviously,sinceAliceknowsthatthespinswereprepared\npolarised “up” or “down” along the z-axis, she requires\nCharles to measure along, say, the x-axis, where he can-\nnot guess the results better than random and she wins.\nTo conclude, Alices knowledge is meaningful: from her\npoint of view the ensemble is more than the density ma-\ntrix to which it corresponds, although, by herself, she\ncannot make any predictions better that Bob, who has\nno knowledge at all, and for whom the state is just a\ndensity matrix. This shows the existence of a new type\nof physical “state”, which sits somewhere in between the\npure state that describes the situation for Charles, who\nhas full knowledge of the preparation, and the density\nmatrix that describes the situation for Bob who has no\nknowledge at all. It is a new mathematical object that,\nI believe, has to be introduced in quantum mechanics.\nThe above examples are, of course, very artificial and\nwith no immediate application; they are however most\nprobably just the tip of an iceberg, and I feel they have\ndeepimplicationsonourunderstandingofwhatquantum\nstates are, what is physical about them, and what role\nour knowledge plays.\nAcknowledgements: I thank Tony Short for simpli-\nfying my original example, as well as discussions with\nRalph Silva, Emmanuel Zambrini and Paul Skrzypczyk.\nI also thank the Institute for Theoretical Studies, ETH\nZurich for its support during this research."    },    {        "title": "1811.10754v1.Phenomenological_level_density_model_with_hybrid_parameterization_of_deformed_and_spherical_state_densities.pdf",        "content": "arXiv:1811.10754v1  [nucl-th]  27 Nov 2018ARTICLE Typeset with jnst.cls <ver.2.1>\nPhenomenological level density model with hybrid paramete rization of\ndeformed and spherical state densities\nNaoya Furutachi∗ †, Futoshi Minato and Osamu Iwamoto\nNuclar Data Center, Japan Atomic Energy Agency, Tokai-mura , Naka-gun, Ibaraki 319-1195, Japan\nA phenomenological level density model that has different level den sity parameter sets\nfor the state densities of the deformed and the spherical states , and the optimization of\nthe parameters using experimental data of the average s-wave n eutron resonance spacing\nare presented. The transition to the spherical state from the de formed one is described\nusing the parameters derived from a microscopic nuclear structur e calculation. The nu-\nclear reaction calculation has been performed by the statistical mo del using the present\nlevel density. Resulting cross sections for various reactions with t he spherical, deformed\nand transitional target nuclei show a fair agreement with the expe rimental data, which\nindicates the effectiveness of the present model. The role of the ro tational collective en-\nhancement in the calculations of those cross sections is also discuss ed.\nKeywords: nuclear level density; nuclear data; statistica l model; neutron spectrum\n∗Corresponding author. Email: naoya.furutachi@riken.jp\n†Present address: RIKEN Nishina center, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan\n1J. Nucl. Sci. & Technol. Article\n1. Introduction\nThe level density (LD) is a key ingredient in the nuclear reaction calcu lation using\nthe statistical model. The accuracy of the calculated nuclear reac tion observables for\nvarious reaction channels relies on the LD, and therefore a number of theoretical works\nemploying phenomenological[1,2,3,4] or microscopic models[5,6,7,8 ,9,10,11] have been\ndevoted to achieve a reliable LD. While the microscopic models are basic ally free from\nadjustable parameters and suitable to predict LDs of nuclei away f rom the stability line,\nthe phenomenological models that have analytical formula and adju stable parameters are\nstillusefultocalculateLDsofnuclei aroundthestabilitylineforthe practicalapplications.\nGenerally, the reliability of the phenomenological models isensured wit h theexperimental\ninformation of excitation energies and spin-parity of the low-lying dis crete states, and the\naverage of the s-wave neutron resonance spacing D0.\nOne of the key effects for LD is the enhancement due to the collectiv e nuclear ex-\ncitations. It is theoretically indicated that the collective rotational excitation brings an\nextremely large enhancement on the LD, which amounts to 10 ∼100 magnitude at the\nneutron threshold energy of stable nuclei[12,4]. In spite of its huge effect, the phenomeno-\nlogical LD models without the explicit treatment of the collective enha ncement have\nbeen successfully applied to the nuclear reaction calculations for pr actical uses, for exam-\nple, the LD model of Gilbert and Cameron[1] without the collective enh ancement[2] has\nbeen mainly used in the statistical model calculation of the neutron in duced reaction un-\nder 20 MeV for the nuclear data evaluation of Japanese Evaluated N uclear Data Library\n(JENDL)[13].Thereasonwhy such aLDmodel doesnotcause seriou sproblems innuclear\nreaction calculations is conjectured that the collective enhanceme nt is effectively taken\ninto account in LD parameters, if they are optimized using the exper imentalD0[2,3].\nActually, such an effective LD model works well for the optimization o f the asymptotic\nlevel density parameter to reproduce D0. Koning et al.[3] have derived the global LD\n2J. Nucl. Sci. & Technol. Article\nparameter systematics for the several LD models with and without explicit treatment of\nthe collective enhancement. As for the Fermi-Gas based models, bo th the collective and\nthe effective LDs have a similar precision for the reproduction of D0to each other.\nIt is noted that, besides the phenomenological models discussed he re, the importance\nof the explicit treatment of the collective excitation is rather obviou s in the microscopic\nLD calculations using Hartree-Fock plus Bardeen-Cooper-Schrieff er (HF+BCS) theory\nwith the partition function method[6,7], and Hartree-Fock-Bogiliub ov (HFB) theory with\nthe combinatorial method[8,9,10,11]. All these studies treatedt he collective excitation ex-\nplicitly, andfounda fair agreement between the calculated D0andthe experiments. These\nresults indicate that if the intrinsic state densities are calculated wit hout the parametriza-\ntion, the collective enhancements are naturally required.\nThe role of the explicit treatment of the collective enhancement in ph enomenological\nLD models can be discussed from nuclear reaction calculations. Konin g et al.[3] have\napplied the the effective and the collective LD models to systematic ca lculations of the\nnuclear reactions. The calculated cross sections are systematica lly different between them\nfor various reactionchannels. The difference is expected to be mor e significant ina nuclear\nreaction at a higher incident energy, because the asymptotic beha viors of LD models with\nand without the collective enhancement are quite different. Actually , the important role of\nthecollective enhancement inthe cross section calculationfor thep rojectile fragmentation\nwith a relativistic incident energy have been reported[14].\nHowever, there remains problems in the description of the collective enhancement in\nphenomenological models. One is the fading of the collective enhance ment as a function\nof the excitation energy. Although there are some theoretical inv estigations about the\nfading of the collective enhancement[15], it is difficult to confirm their v alidity directly\nfrom the experiments, because it is expected that there is a finite m ean deformation even\nwith the excitation energy of several tens of MeV[15] for well-def ormed nuclei. In addition\n3J. Nucl. Sci. & Technol. Article\nto that, it is also difficult to describe the rotational collective enhanc ement for nuclei in\nthe transitional region, because the interaction between the sing le-particle states and the\ncollective states plays a significant role in this case.\nOur aim in this paper is to present a reliable LD necessary for the prec ise calculation\nof nuclear reaction observables using the statistical model. For th is purpose, we propose\na new phenomenological model based on the LD model of Gilbert and C ameron[1], in\nwhich the state densities of the deformed and spherical states ha ve different level density\nparameters. The optimizationof the parameters areperformed b y fitting theexperimental\nD0with distinction between deformed and spherical nuclei. The LDs of t he deformed and\nthe spherical states are smoothly connected by the damping func tion, in analogy with the\nway used in the microscopic calculations based on HF+BCS and HFB[6, 7,8,9,10,11]. The\nfading of the rotational collective enhancement is effectively descr ibed in this way. Since\nthere is no direct experimental information about the fading of the rotational collectivity,\nwe utilized the microscopic nuclear structure calculation to determin e the parameters\nin the damping function. By the composition of the deformed and sph erical states, the\ntransitional state may be also effectively taken into account.\nIn this study, much attention is paid on the effectiveness of the pre sent LD model for\nthe actual nuclear reaction calculations. We use CCONE code[16] to calculate the cross\nsections, whicharecomparedwiththeexperimental data.Atthes ametime,weinvestigate\nthe role of the explicit treatment of the collective enhancement in nu clear reactions.\nThis paper is organized as follows. In Sec. 2, the formulation of the p resent LD model,\nthe optimization procedure of the level density parameters, and t he microscopic nuclear\nstructure calculation are presented. In Sec. 3, first the charac teristics of the present LD\nis discussed, then the results of the nuclear reaction calculations a re shown. Sec. 4 sum-\nmarizes this work.\n4J. Nucl. Sci. & Technol. Article\n2. Formulation\nWe present a new phenomenological LD model that is described with t he LDs of\nthe deformed and the spherical states connected by the damping function in a similar\nway to that used in the microscopic calculations based on HF+BCS and HFB[6,7,8,9,\n10,11]. By optimizing the level density parameters for the deforme d and the spherical\nstates separately, reliable LDs for both deformed and spherical n uclei are expected to be\nachieved. We call the present model as the hybrid model to distingu ish from the existing\nphenomenological collective models.\n2.1. hybrid level density model\nThe LD of the present hybrid model ρhis described with the LD of the spherical state\nρsph, and that of the deformed state ρdef,\nρh(U,J) =\n\n(1−fdam(Ex))ρsph(U−Edef,J)+fdam(Ex)ρdef(U,J) (Edef≥Ecut)\nρsph(U,J) (Edef< Ecut),(1)\nwhich are smoothly connected by the damping function fdam,\nfdam(Ex) =1\n1+e(Ex−Ets)/de, de=CEts. (2)\nHereEx,UandJaretheexcitationenergy,thepairingcorrectedexcitationenerg yandthe\ntotal angular momentum of the nucleus. In this formulation, the fa ding of the rotational\ncollectivity is phenomenologically expressed by the transition from ρdeftoρsph. Since\nexperimental information about the fading of the rotational collec tivity is limited, we\nderived the parameters EdefandEtsthat control this transition from the microscopic\nnuclear structurecalculation, which isexplained inSec. 2.2..Thepara meterEdefisdefined\nas the energy difference between the deformed ground state and the minimum energy of\nthe spherical state. If Edefis smaller than Ecut, the level density is approximated with\nρsph. The parameter Ecutis arbitrary fixed at 0.3 MeV. The parameter Etsis the central\nenergy of the transition, which is estimated utilizing information of th e deformation at a\n5J. Nucl. Sci. & Technol. Article\nfinitetemperature.Thewidthparameter deofthedampingfunctionisphenomenologically\ndetermined supposing a linear dependence on Etswith the adjustable parameter C. The\ndetaileddiscussion fortheparameter CisgiveninSec. 2.4..The pairingcorrectedeffective\nexcitation energy Uis,\nU=Ex−2∆ for even-even nuclei\n=Ex−∆ for odd nuclei\n=Exfor odd-odd nuclei\n∆ = 11 /√\nA. (3)\nThe functions ρsphandρdefare described by the phenomenological Fermi-gas model\nwith the level density parameters asandad, respectively,\nρsph(U,J) =Rs(U,J)ωs(U)√\n2πσs,\nρdef(U,J) =KrotRd(U,J)ωd(U)√\n2πσd, (4)\nωs,d(U) =√π\n12exp(2/radicalbig\nas,dU)\na1/4\ns,dU5/4, (5)\nhereRs,d(U,J) are the spin distribution functions, and ωsandωdare the state densities\nforρsphandρdef, respectively. The rotational collective enhancement is explicitly tr eated\ninρdefby applying the enhancement factor Krot. Contrary to the rotational collective\nenhancement, vibrational one is not explicitly treated in our formula tion. We expect that\nitisimplicitly takenintoaccountthroughtheoptimizationofthelevel d ensityparameters.\nThe level density parameters as,dare given as,\nas,d(U) =as,d(∗)/bracketleftbigg\n1+Esh\nU(1−e−γU)/bracketrightbigg\n, (6)\nhereas,d(∗) are the asymptotic level density parameters described by the sy stematics,\nas,d(∗) =αs,dA(1−βs,dA−1/3). (7)\nThe parameters αs,d,βs,d, andγare optimized using the experimental D0, as explained\n6J. Nucl. Sci. & Technol. Article\nin the next subsection. The shell correction energy Eshis defied as,\nEsh=Mexp−MLDM, (8)\nhere the mass formula of Myers and Swiatecki are[17] used for MLDM. It is noted that the\npairing energy systematics in Eq. 3 is consistent with the one used in t he calculation of\nMLDM.\nThe spin distribution function Rs,d(U,J) are\nRs,d(U,J) =2J+1\n2σ2\ns,dexp/bracketleftBigg\n−(J+1/2)2\n2σ2\ns,d/bracketrightBigg\n, (9)\nhere we employ the shell-corrected spin dispersion function of Mugh abghab and Dun-\nford[18],\nσ2\ns,d=I0/radicalbig\nas,dU\nas,d(∗), (10)\nI0=2\n5m0R2A\n(/planckover2pi1c)2= 0.01389A5/3MeV−1. (11)\nThe rotational enhancement factor Krotis written as,\nKrot=σ2\n⊥, (12)\nσ2\n⊥=I0(1+β2\n3)/radicalbigg\nU\nad. (13)\nIn the present model, the composite formula of Gilbert and Cameron [1] is used. The\nlow excitation energy region below the matching energy Emis described by the constant\ntemperature part ρCT(Ex,J),\nρGC(Ex,J) =Rh(U,J)ρCT(Ex) (Ex< Em),\nρGC(Ex,J) =ρh(Ex,J) (Ex≥Em). (14)\nHere the spin distribution function Rh(U,J) is calculated by,\nRh(U,J) =ρh(U,J)/ρtot\nh(U), ρtot\nh(U) =/summationdisplay\nJρh(U,J), (15)\n7J. Nucl. Sci. & Technol. Article\nwhereρCTis given by,\nρCT(Ex) =1\nTexp/parenleftbiggEx−E0\nT/parenrightbigg\n, (16)\nhereE0andTaredetermined fromthe usual matching condition[1]. Thepairing co rrected\nmatching energy Um=Ex−2∆ (even-even), Ex−∆ (odd), Ex(odd-odd) are given by\nthe simple systematics,\nUsys\nm=pAx, (17)\nwhere the mass dependence of the systematics is introduced to fit theUmdetermined to\nreproduce the experimental discrete level numbers. The optimiza tion procedures for the\nparameters p,xare explained later.\nIf the pairing corrected energy Uis smaller than 0, the spin distribution function\nRh(U,J) cannot be calculated by Eq. 15. To avoid this, we simply extrapolate Rh(U,J)\natU= 1 MeV to U <1 MeV region.\nFinally, we assume the equal parity distribution function, namely\nρGC(Ex,J,Π) =1\n2ρGC(Ex,J). (18)\n2.2. microscopic nuclear structure calculation\nIn thepresent model, results ofthemicroscopic nuclear structur e calculationisutilized\nto determine the transition from the deformed LD to the spherical LD. We performed\nthe nuclear structure calculation using FTHFB theory, and derived the most probable\ndeformation β2as a function of the excitation energy. The excitation energy is calc ulated\nusing the energy expectation values of the system with the temper atureT,\nEx=E(T)−E(T= 0). (19)\nThe calculation was executed with HFBTHO code[19], where the energ y density func-\ntional of SkM*[20] was used. We employed the surface-volume mixed type pairing in-\n8J. Nucl. Sci. & Technol. Article\nteraction with the pairing cutoff energy ǫcut= 60 MeV. The neutron and the proton\npairing strengths are determined to reproduce the experimental pairing gap derived from\nthe three-point mass difference for120Sn and138Ba, which have the proton and neutron\nclosed shells of Z=50 and N=82, respectively.\n[Figure 1 about here.]\nInFig.1,themostprobable β2asafunctionoftheexcitationenergyisshown.Basically\nthe most probable β2decreases as the excitation energy increases, but its behavior is\ndifferent for each nucleus. For example, while80Se has a larger β2than133Cs at the\nground state, the most probable β2decreases more rapidly and becomes 0 at slightly\nsmaller energy than133Cs. We define Etsas the energy where the most probable β2\nvalue becomes 0, because it can be a indicative of the loosing of the ro tational collective\nenhancement, and derived it systematically for stable nuclei. The ob tainedEtsare shown\nin Fig. 2. We found that the most of the deformed nuclei of A <150 have Etsof 10∼20\nMeV. This means that the disappearance of the deformation may aff ect nuclear reactions\nwith incident beam energy even below 20 MeV, which are often calculat ed using the\nstatistical model for nuclear data libraries. For deformed nuclei in A >150 region, the\nmost of them have large Etswhich are well above the maximum excitation energy of the\ncompound nucleus formed with 20 MeV incident nucleon.\nIn the present model, we suppose that the spherical states appe ar in the excited state\nafter exhausting the deformation energy that is defined as the en ergy difference between\nthe spherical and the deformed ground state energies,\nEdef=Eβ2=0\nconst.(T= 0)−E(T= 0). (20)\nThis energy is subtracted from the excitation energy of ρsph(U,J), as described by Eq. 1.\n[Figure 2 about here.]\n9J. Nucl. Sci. & Technol. Article\n2.3. effective and collective level density models\nFor comparison, we also derive the LDs using the effective and collect ive models. The\neffective model is defined with ρsph(U,J) used in the present hybrid model,\nρeff(U,J) =ρsph(U,J), (21)\nand the collective model is defined as,\nρcol(U,J) = max([ Krot−1]f(Ex)+1,1)Rd(U,J)ωd(U)√\n2πσd,\nfdam(Ex) =1\n1+e(Ex−Ecol)/dcol, (22)\nhereEcolanddcolare fixed at 30 MeV and 5 MeV, which are the values used by Koning\net al.[3]. For both ρeff(U,J) andρcol(U,J), the constant temperature part are combined\nin the same way as the hybrid model.\n2.4. optimization procedure\nBasically the optimization of the systematics for the asymptotic leve l density param-\neter was performed in a similar way to Mengoni and Nakajima[2]. It is no ted that the\nconstant temperature model is not used in the optimization proced ure for the asymptotic\nlevel density parameters for simplicity.\nThe parameters to be optimized using the experimental values of th e average s-wave\nneutron resonance spacing D0areαs,d,βs,dandγin Eq. 6 and 7. We determine αs,dand\nβs,dto minimize χ2\nadefined as,\nχ2\na= Σi(alocal\ni(∗)−asys\ni(∗))2\nasys\ni(∗), (23)\nherealocal\ni(∗) is the asymptotic level density parameter derived to reproduce t he experi-\nmentalD0for each nucleus, and asys\ni(∗) is that calculated by Eq. 7. Here iis the index\nto specify nucleus. The experimental D0values for 300 nuclei are taken from RIPL-3\n10J. Nucl. Sci. & Technol. Article\ndatabase[21]. Once αs,dandβs,dare determined, we calculate fD0rmsdefined as,\nfD0\nrms= exp/bracketleftBigg\n1\nNmaxNmax/summationdisplay\ni=1ln2D0(cal.)\nD0(exp.)/bracketrightBigg1/2\n, (24)\nwhereD0(cal.) are calculated using asys(∗). The above procedure is performed using var-\niousγparameters, and finally the set of αs,d,βs,dandγthat gives the minimum value of\nfD0rmsis determined. Obtained parameters and fD0rmsare listed in Table 1.\n[Table 1 about here.]\nIn more detail, the procedure to determine αs,dandβs,dis divided into two steps. First\nwe determine αsandβs. For the spherical nuclei that have the condition Edef< Ecut,D0\nis calculated only from ρsph. Therefore, alocal\nscan be determined independently from ad. In\nthe left top panel of Fig. 3, alocal\ns(∗) of 108 nuclei with Edef< Ecutare shown by the open\nsquares, and asys\ns(∗) determined by minimizing χ2\nawith these alocal\ns(∗) is shown by the solid\nline. Secondly, αdandβdare determined. To calculate D0for nuclei with Edef≥Ecut,\nbothas(∗) andad(∗) are necessary. We calculate as(∗) usingasys\ns(∗) determined from the\nabove procedure, and derive alocal\nd(∗) to reproduce the experimental D0for 182 nuclei\nwithEdef≥Ecut. The obtained alocal\nd(∗) andasys\nd(∗) are shown by the open circles and\nthe broken line in the left top panel of Fig. 3, respectively. It is clear ly seen that smaller\nad(∗) values are required compared to as(∗), which indicates that the spherical and the\ndeformed intrinsic states should have different state densities. It is noted that we excluded\n10 nuclei with small deformations of Ecut< Edef<0.5 MeV, in which ρhis dominated by\nρsph. In such a case, extremely large or small values of alocal\nd(∗) appears to reproduce D0,\nand it is unfavorable for the optimization of asys\nd(∗).\n[Figure 3 about here.]\nThe hybrid model has an additional parameter Cthat adjusts the width parameter\ndeoffdam. While we use the theoretical values for the central energy Etsoffdam, the\nwidth parameter dethat express a smoothness of the transition is quite phenomenolog ical.\nTherefore, we investigated the dependence on Cin the calculation of D0. In Fig. 4, fD0rmsas\n11J. Nucl. Sci. & Technol. Article\nafunctionof Cisshown. Whileitisclear thatasmall Cisnotpreferable, Cdependence of\nfD0rmsis so moderate in lager Cregion, which means that D0cannot be an strong constraint\nonC. Basically we used C= 0.35 that is smaller than the optimal value for D0that is\naround 0.70, since a better agreement between calculations and ex perimental data of the\nnuclear reaction cross sections was obtained with C= 0.35, in the case of (n,2n) reactions\nfor Se isotopes discussed in the next section.\n[Figure 4 about here.]\nWe also optimized the parameters for the effective and the collective LD models. For\nthese models, all the experimental D0values for 300 nuclei are used for the optimization\nofasys(∗). The obtained alocal(∗) andasys(∗) for the effective and the collective models are\nshown in the middle and the bottom panels of Fig. 3, respectively, and the parameters\ninasys(∗) andfD0rmscalculated using the optimized asys(∗) are listed in Table 1. Although\nsignificantly different parameters are required for asys\ns(∗) andasys\nd(∗), the resulting fD0rmsare\nsimilar among the effective, collective and hybrid models. As already me ntioned in the\nintroduction, the essentiality of the explicit treatment of the collec tive enhancement is\nhardly seen in the calculation of D0, if the phenomenological LD models optimized using\nthe experimental D0are used.\nFinally, the parameters in the constant temperature part of LD ar e optimized. The\nparameters to be optimized are pandxin Eq. 17 to calculate Usys\nm. They are determined\nto minimize χ2calculated as same as Eq. 23 using Usys\nmandUlocal\nm, andUlocal\nmis determined\nto minimize\nflev\nrms= exp/bracketleftBigg\n1\nNmaxNmax/summationdisplay\ni=1ln2LEi(i)(cal.)\nLEi(i)(exp.)/bracketrightBigg1/2\n, (25)\nhereLEi(i) is the cumulative number of the discrete levels at the excitation ene rgyEiof\nthe experimentally observed i-th level, and Nmaxis the number of levels to be compared.\nThe experimental data of the discrete levels are taken from RIPL- 3 database[21]. Since\nthere may be discrete levels that have not been observed, the cum ulative number of the\n12J. Nucl. Sci. & Technol. Article\nobserved levels is expected to deviate from the reality with increase of the excitation\nenergy. We assume that the deviation is small if the cumulative numbe r of the observed\nlevels is much smaller than the maximum number of the observed levels, and arbitrary\ntake 70% of the maximum number as Nmax. Nuclei with more than 100 observed levels\nare used to determine the parameters of Usys\nm. In Fig. 5, the obtained Ulocal\nmandUsys\nmare\nshown by the symbols and the solid line, respectively. It is seen that Ulocal\nmare roughly\nreproduced by the mass dependence of Usys\nm, except for the values around A∼200. We\ntake priority to achieve better precision for UminA <200 region, which are relevant to\nthe nuclear reaction calculations in the next section, andexcluded Ulocal\nminA >200region\nfrom the fitting for this preference. In the final results present ed in the next section, the\noptimized Usys\nmis used to calculate LD.\n[Figure 5 about here.]\n2.5. Nuclear reaction models\nThe nuclear reaction calculations have been executed using CCONE c ode[16]. The\ncode composed of the optical model, two-component exciton mode l, distorted-wave Born\napproximationandHauser-Feshbach statistical model. Asforthe opticalmodel, theglobal\noptical potential parameters of Koning and Delaroche[22] was use d. LDs of the hybrid, ef-\nfective and collective models are adopted to Hauser-Feshbach sta tistical model in CCONE\ncode by using the tabulated numerical data of RIPL-3 format[21].\n3. results\n3.1. Total level densities\nBefore showing the results of the nuclear reaction calculations, th e characteristics of\nthe hybrid model are discussed from the total LDs in comparison wit h the effective and\ncollective models. InFig.6,the totalLDsof82Se,90Zr,169Tmand197Auinwide excitation\n13J. Nucl. Sci. & Technol. Article\nenergy range, and those magnified around the neutron threshold are show in the left and\nright panels, respectively. The parameters relevant to the defor mation that determine the\ncharacteristic of the present hybrid model are summarized in Table 2. As described by Eq.\n1 and 2, the transition to ρsphfromρdefis made by these parameters. Hereafter, we denote\nthe LDs of the hybrid, effective and collective models as ρh,ρeffandρcol, respectively.\n[Figure 6 about here.]\n[Table 2 about here.]\nFirst of all, for the spherical90Zr case,ρhis close to ρeffin the entire region, while ρcol\nis significantly different from them, because there is the rotational collective enhancement\neven in the spherical nuclei with the fixed Ecolof 30 MeV. In addition to that, because of\nthe difference in the asymptotic level density parameters, the incr ease rate of ρcolabove\n30 MeV is also different from ρhandρeff. As for169Tm that has a developed deformation\nwithβ2= 0.32,ρhshows a similar behavior to ρcolbelow about 30 MeV. They deviates\nfrom each other above 30 MeV, because the rotational collective e nhancement fades in\nρcolaround this energy, but does not in ρh. As for82Se that has a moderately developed\ndeformation of β2= 0.16, the component of ρdefinρhis decreasing around Ex∼Ets=7.5\nMeV. In Ex>20 MeV, ρhcomes closer to ρeff, because the component of ρsphdominates\nin this region. The difference between ρhandρeffin the asymptotic region is characterized\nwith the energy shift by Edef.197Au has a smaller β2of 0.13 but has a larger Edefthan\n82Se. The increment of ρhsignificantly reduces around Ex∼Ets=13 MeV because the\ndifference between the spherical LD shifted by Edefand the deformed LD is large. Above\n20 MeV, the increase rate of ρhcomes closer to ρeff, and deviates from ρcol.\nThe LDs around the neutron threshold Snare shown in the right panel of Fig. 6 as a\nfunction of Ex−Sn. Since the asymptotic LD parameters are optimized for all of ρh,ρeff\nandρdefusing the experimental D0, they are close to each other at Sn. However, there is\na difference in the increase rate of these LDs. In any case, ρeffhas a larger increase rate\n14J. Nucl. Sci. & Technol. Article\nthanρcol. Whether the increase rate of ρhis similar to that of ρefforρcolis determined\nby the deformation. It is close to ρefffor the spherical90Zr, andρcolfor the deformed\n169Tm and197Au. As for82Se,ρhhas even smaller increase rate than ρcol, because the\ncomponent of ρdefdisappears just around Sninthiscase. The increase ratesof LDsaround\nSnhave remarkable influences on the nuclear reaction calculations exp lained in the next\nsubsection.\n3.2. cross sections of (n,xn) and (p,xn) reactions\nInthissection,wetesttheeffectivenessofLDsandalsodiscussth eroleoftherotational\ncollective enhancement from the calculations of (n,xn) and (p,xn) re actions. The experi-\nmental data of the cross sections to be compared are taken from EXFOR[23] throughout\nthis section.\nTo illustrate the role of the rotational collective enhancement, the (n,2n) and (n,3n)\nreactions with90Zr and169Tm targets that are spherical and deformed, respectively, are\ncalculated. In addition to that, these nuclei have a plenty of (n,2n) experimental data to\nbecompared. There arealso (n,3n) experimental data for169Tm, but not for90Zr.Instead,\nthe (n,3n) cross sections of89Y are calculated.\n[Figure 7 about here.]\nThe results are shown in Fig. 7. As discussed in the previous subsect ion,ρhis similar\ntoρeffif the nucleus is spherical. Therefore, for the90Zr target, the (n,2n) cross sections\ncalculated using ρhandρeffare also similar, and they show good agreement with the\nexperimental data. However, ρcolis different from them even for the spherical90Zr, and\ncannot reproduce the experimental data. On the other hand, fo r the deformed169Tm\ntarget, the cross sections calculated with ρhare similar to those with ρcol. Compared to\ntheresults with ρeff, the(n,2n) and (n,3n) cross sections aresuppressed below 12 MeV and\n25 MeV, respectively. The (n,3n) cross sections and the competing (n,2n) cross sections\n15J. Nucl. Sci. & Technol. Article\nabove 15 MeV show good agreement with the experimental data. Th e difference in the\ncalculated (n,2n) cross sections mainly come from the difference in th e LDs of the target\nnuclei. In the (n,2n) reaction, first the N+1 compound nucleus is formed, then it emits\none neutron. If LD of the target nucleus has smaller increase rate around the neutron\nthreshold, the emitted neutron brings more energy, which results in the increase of the\ncompetitive inelastic channel cross section, and decrease of the ( n,2n) cross section. Later\nthe difference in the neutron emission spectrum is discussed in detail.\n[Figure 8 about here.]\nNext we discuss the (n,2n) cross sections of Se isotopes shown in Fig . 8. If the target\nnucleus have a moderate deformation with Etsclose toSn, the (n,2n) cross section cal-\nculated with ρhshows non negligible dependence on de, which is the width parameter of\nfdam.76Se,78Se,80Se and82Se haveEts=12.2, 11.1, 10.1 and 7.5 MeV, and Sn=11.1, 10.5,\n9.9 and 9.3 MeV, respectively. The (n,2n) cross sections calculated w ithρhandρcolshow\nsuppression from those with ρeff, as in the cases of90Zr and169Tm. As for the results with\nρh, the degrees of the suppression depend on de. The results calculated using C= 0.35\nand 0.70 are also compared in Fig. 8. If deis smaller, a decrease of the component of ρdef\ninρhis more rapid, which results in a smaller increase rate of LD. Therefor e, the (n,2n)\ncross sections calculated with C= 0.35 tend to be suppressed compared to those with\nC= 0.70. While this effect is not significant for76Se,78Se and80Se cases, a noticeable\ndifference is foundfor82Se, because82SehasEtsjust below Sn. Inthis case, thecomponent\nofρdefbecomes 0 just around SnifC= 0.35 is used, which results in the significantly\nsmall increase rate of LD around Snas shown in Fig. 6. As for82Se, the (n,2n) cross\nsections calculated with C= 0.35 are even smaller than those calculated with ρcol.\nThese results indicate that the effect of the fading of the rotation al collective enhance-\nment around Sncan be seen in the (n,2n) cross section. The validity of this effect sho uld\nbe studied using as many experimental data as possible, but not so m any (n,2n) exper-\n16J. Nucl. Sci. & Technol. Article\nimental data are available for nuclei that have Etsclose toSn. Although the number of\nexperiments is limited, Se isotopes have the systematic experimenta l data of Frehaut et\nal.[24]. The calculated results with C= 0.35 well agree with those data renormalized by\nthe factor of 1.08, which is derived by Vonach et al.[25].\n[Figure 9 about here.]\nAnother nucleus that has a plenty of experimental data and a mode rate deformation\nis197Au. The calculated results of197Au(n,xn) cross sections are shown in Fig. 9. As\ndiscussed in the case of Se isotopes, the values of EtsandSnare important to understand\nthe characteristics of the cross section calculated with ρh.Etsis 13 MeV for197Au, while\nSnandS2nare 6.6 MeV and 15.0 MeV, respectively. Since Etsis much larger than Snand\njust below S2n, both (n,2n) and (n,3n) cross sections show suppression from the results\nwithρeffbelow 14 MeV and 25 MeV, respectively. However, the (n,2n) and (n,3 n) cross\nsections in 15 MeV < En<25 MeV, which are competing, show a disagreement with the\nexperimental data. To investigate how the calculated cross sectio ns depend on the degrees\nof the deformation, a modified ρhfor197Au that has arbitrary chosen EtsandEdefvalues\nof 8 MeV and 1 MeV is used to calculate the cross sections. The result s are also shown in\nFig. 9. Since Ets= 8 MeV is well under S2n, the suppression of the (n,3n) cross sections\nbelow 25 MeV is small. As a consequence, this results with the modified ρhshow a better\nagreement with the experimental data in 15 MeV < En<25 MeV. As for the (n,4n)\nand (n,5n) cross sections, the results with both ρhofEts= 8 and 13 MeV are similar,\nbecause the incident energies are higher enough from Etsfor these channels, which means\nthe complete disappearance of the component of ρdef. The results with ρhsignificantly\ndeviate from those with ρcolin the higher incident energy region due to the difference\nof LDs in the asymptotic region. Several experimental data above 40 MeV support the\nresults with ρh.\n[Figure 10 about here.]\n17J. Nucl. Sci. & Technol. Article\nThe suppression of (n,xn) cross sections calculated with ρhandρcolfrom those with\nρeffis related to the difference in the evaporated neutron emission spec trum. To show\nthis, the neutron emission spectrum ofnatSe(n,xn)natZr(n,xn) and197Au(p,xn) reactions\nare calculated. The results are shown in the left panel of Fig. 10. Th e neutron emission\nspectrum ofnatZr(n,xn) reaction at 14.1 MeV calculated with ρcolshows a noticeable\nenhancement around 5 MeV from those calculated with ρhandρeffand a disagreement\nfrom the experimental data. It is consistent with the (n,2n) cross section calculated with\nρcol, which significantly deviates from the experimental data. Since ρcolhas a smaller\nincrease rate at a excitation energy close to the incident nucleon en ergy, the evaporated\nneutrons fromthe compound nucleus tend to bring larger energies compared to the results\nwithρhandρeff. In most cases, ρcolhas a smaller increase rate than ρeff, even for spherical\nnuclei. InnatSe(n,xn) case, the calculated result with ρhis similar to ρcol, which show\nenhancement from the result with ρeffaround 5 MeV.\nIn the right panel of Fig. 10, the neutron emission spectrum of105,106,108,110Pd(p,xn)\nreactions at Ep= 26.1 MeV are shown. For105Pd,106Pd,108Pd and110Pd,Etsare calcu-\nlated to be 10.0, 11.0, 14.3 18.0 and 20.3 MeV, respectively. While all of f our Pd isotopes\nhave moderate deformations around β2∼0.2, the difference in Etsresults in the signifi-\ncant difference in the evaporated neutron emission spectrum. Sinc e110Pd has the largest\nEtsthat is close to Ep, the component of ρdefinρhaffects the neutron emission from the\ncompound nucleus. In this case, the neutron emission spectrum ca lculated with ρhis close\ntoρcol, and deviates from that with ρeff. IfEtsis much smaller than Ep, the component\nofρdefhas a small influence on the neutron emission from the compound nuc leus. There-\nfore, the neutron emission spectrum calculated with ρhare similar to those with ρeffin\n105Pd(p,xn) and106Pd(p,xn) cases. This result illustrates the characteristic of the pr esent\nLD model, and at the same time, the role of the collective enhancemen t in the evaporated\nneutron emission spectrum.\n18J. Nucl. Sci. & Technol. Article\n4. Summary\nTo construct a new phenomenological LD model for a better precis ion of the nuclear\nreaction calculation, and to investigate the role of the rotational c ollective enhancement\nin the nuclear reaction at the same time, we proposed the hybrid mod el in which the LDs\nof the deformed and the spherical states described by Fermi-Gas model are connected by\nthe damping function. We optimized the asymptotic level density par ameter systematics\nfor the LDs of the deformed and the spherical states separately using the experimental D0\nof deformed and spherical nuclei, respectively. The information of the nuclear deformation\nderived from the FTHFB calculation was utilized. The obtained LD was in troduced in the\nnuclear reaction calculation using the statistical model, and the cro ss sections of (n,xn)\nand (p,xn) reactions were discussed.\nWe found that the LD with the rotational collective enhancement te nds to have a\nsmaller increase rate compared to that with no explicit collective enha ncement, which\nresults in a higher energy neutron emission from the compound nucle us. The (n,xn) cross\nsections with incident neutron energies just above the threshold a re suppressed because\nof this mechanism. In many cases, cross sections calculated with th e transitional model\nwere similar to those with the effective model and the collective model for the nuclear re-\nactions for the spherical and the deformed targets, respective ly. We showed the calculated\nexamples for the spherical90Zr and the deformed169Tm targets, both of which agree with\nthe experiments.\nDepending on the incident nucleon energy and the degree of the def ormation of the\ntarget nucleus, the cross sections have sensitivity to a certain en ergy range of LD where\nthe component of the deformed state is decreasing. In76,78,80,82Se(n,2n) reactions, the\ndecreasing component of the deformed state results in a good agr eement between the cal-\nculated and the experimental cross sections. In197Au(n,xn) reactions, how cross sections\ndepend on the degrees of the deformation was shown. These resu lts indicate that a more\n19J. Nucl. Sci. & Technol. Article\nreliable prediction of deformations in excited states may lead to a mor e precise calculation\nof cross sections.\nThese results indicate that the present model is effective for prec ise calculations of\nnuclear reactions for both the spherical and deformed targets. Since the calculated cross\nsection depends on the predicted deformations, a more precise cr oss section calculation\ncan be achieved with a more reliable nuclear structure calculation in fu ture. This model\nalso can be a tool to investigate the fading of the rotational collect ive enhancement in\nnuclear excited states through the nuclear reaction calculation.\nAcknowledgement\nThis work was funded by ImPACT Program of Council for Science, Te chnology and\nInnovation (Cabinet Office, Government of Japan).\nReferences\n[1] GilbertA,CameronAGW.Acompositenuclearleveldensityformula withshellcorrections.Canadian\nJ Phys. 1965; 43:1446.\n[2] Mengoni A, Nakajima Y. Fermi-Gas Model Parametrization of Nuc lear Level Density. J Nucl Sci\nTechnol. 1994; 31:151-162.\n[3] Koning AJ, Hilaire S, Goriely S.Global and local level density models. Nucl Phys A.2008;810:13-76.\n[4] Ignatyuk AV, Istekov KK, Smirenkin GN.Role of collective effects in the systematics of nuclear level\ndensities. Sov J Nucl Phys. 1979; 29:450.\n[5] Goriely S. A new nuclear level density formula including shell and pair ing correction in the light of\na microscopic model calculation. Nucl Phys A. 1996; 605:28-60.\n[6] Demetriou P, Goriely S. Microscopic nuclear level densities for pra ctical applications. Nucl Phys A.\n2001; 695:95-108.\n[7] Minato F. J Nucl Sci Technol. 2011; 48:984-992.\n20J. Nucl. Sci. & Technol. Article\n[8] Hilaire S, Delaroche JP, Girod M.Combinatorialnuclear level densit ies based on the Gogny nucleon-\nnucleon effective interaction. Eur Phys J A. 2001; 12:169-184.\n[9] Hilaire S, Goriely S. Global microscopic nuclear level densities within t he HFB plus combinatorial\nmethod for practical applications. Nucl Phys A. 2006; 779:63-81.\n[10] Goriely S, Hilaire S, Koning AJ. Improved microscopic nuclear level densities within the Hartree-\nFock-Bogoliubov plus combinatorial method. Phys Rev C. 2008; 78:0 64307.\n[11] Hilaire S, Girod M, Goriely S, et al. Temperature-dependent comb inatorial level densities with the\nD1M Gogny force. Phys Rev C. 2012; 86:064317.\n[12] Bour A, Mottelson BR. Nuclear Structure Vols I and II. New Yor k: W. A. Benjamin Inc; 1969 and\n1975.\n[13] Shibata K, Osamu I, Tsuneo N, et al.JENDL-4.0: A new library for nuclear science and engineering.\nJ Nucl Sci Technol. 2011; 48:1.\n[14] Junghans AR, De jong M, Clerc H-G, et al. Projectile-fragment yields as a probe for the collective\nenhancement in the nuclear level density. Nucl Phys A. 1998; 629:6 35.\n[15] Hansen G, Jensen AS. Energy dependence of the rotational e nhancement factor in the level density.\nNucl Phys. 1983; 406:236.\n[16] Iwamoto O, Iwamoto N, Kunieda S, et al.The CCONE Code System and its Application to Nuclear\nData Evaluation for Fission and Other Reactions. Nucl Data Sheets . 2016; 131:259-288.\n[17] Myers WD, Swiatecki WJ. Nucl Phys. 1966; 81:1.\n[18] Mughabghab SF, Dunford C.Nuclear Level Density and the Effe ctive Nucleon Mass.Phys Rev Lett.\n1998; 81:4083.\n[19] Stoitsov MV, Schunck N, Kortelainen M, et al. Axially deformed so lution of the Skyrme-Hartree-\nFock-Bogoliubov equations using the transformed harmonic oscillat or basis (II) HF BTHO v2.00d:\nA new version of the program. Comp Phys Communications. 2013; 18 4:1592-1604.\n[20] Bartel J, Quentin P.Towardsa better parametrisationof Sky rme-likeeffective forces: a critical study\nof the SkM force. Nucl Phys A. 1982; 386:79-100.\n21J. Nucl. Sci. & Technol. Article\n[21] Capote R, Herman M, Obloˇ zinsk´ y, et al. RIPL - Reference Inp ut Library for calculation of nuclear\nreactions and nuclear data evaluations. Nucl Data Sheets. 2009; 110:3107-3214.\n[22] Koning AJ, Delaroche JP. Local and global nucleon optical mode ls from 1 keV to 200 MeV. Nucl.\nPhys. A. 2003; 713: 231-310.\n[23] Otuka N, Dupont E, Semkova V, et al. Towards a more complete a nd accurate experimental nuclear\nreaction data library (EXFOR): international collaboration betwee n nuclear reaction data centres\n(NRDC). Nucl Data Sheets. 2014;120:272-276.\n[24] FrehautJ,BertinA,BoisR, etal.Statusof(n, 2n)crosssect ionmeasurementsatBruyeres-le-Chatel.\nIn: Bhat MR, Pearlstein S, editors. Proceedings of Symposium on Ne utron Cross-sections from 10\nto 50 MeV; 1980 May 12-14. Upton (NY): Brookhaven National Lab oratory; 1980. p. 399-411.\n[25] Vonach H, Pavlik A, Strohmaier B.Accurate determination of (n , 2n) cross sections for heavy nuclei\nfrom neutron production spectra. Nucl Sci Eng. 1990;106:409-4 14.\n22J. Nucl. Sci. & Technol. Article\nTable 1 Parameters of the hybrid, effective and collective models, and calcu latedfD0rms.\nhybrid effective collective\nfD0rms 1.66 1.74 1.66\nαs[MeV−1] 0.07110 0.06573\nαd[MeV−1] 0.01291 0.03960\nβs-3.608 -4.385\nβd-30.54 -5.708\nγ[MeV−1] 0.072 0.073 0.098\np[MeV] 547 55 76\nx -1.10 -0.54 -0.74\nEcut[MeV] 0.30\nC 0.35\n23J. Nucl. Sci. & Technol. Article\nTable 2 Calculated β2,Edef,EtsandEmof82Se,90Zr,169Tm and197Au. The experimental values of\nthe one neutron separation energies are also shown.\nβ2Edef(MeV) Ets(MeV) EmSn(MeV)\n82Se 0.16 1.31 7.5 6.7 9.3\n90Zr 0 0 0 6.2 12.0\n169Tm 0.32 19.2 90.5 2.8 8.0\n197Au -0.13 3.1 13.0 2.4 6.9\n24J. Nucl. Sci. & Technol. Article\nFigure Captions\nFigure 1 Most probable deformation β2as a function of the excitation energy\ncalculated by FTHFB.\nFigure 2 Parameter Etsderived from FTHFB calculation.\nFigure 3 Calculateda(∗)(leftpanel)and D0(rightpanel)forthehybrid, effective\nand collective models. The a(∗) determined to reproduce D0of each\nnucleus and calculated from the systematics are shown by the symb ols\nand lines, respectively.\nFigure 4 Dependence of fD0rmson the additional parameter Cfor the hybrid model.\nFigure 5 Pairing corrected matching energy Ulocal\nmobtained by minimizing flev\nrms\nof each nucleus and Usys\nmcalculated by Eq. 23 are shown by the symbols\nand the solid line, respectively. The red symbols are results for even -\neven nuclei, and the green ones for odd and odd-odd nuclei.\nFigure 6 Total level densities of the hybrid (solid line), effective (dashed line)\nand collective (dotted line) LD models for82Se,90Zr,169Tm and197Au\nas a function of Ex(left panel) and Ex−Sn(right panel).\nFigure 7 Cross sections of (n,2n) reactions for90Zr and169Tm, and (n,3n) reac-\ntions for89Y and169Tm. Calculated results using ρh,ρeffandρcolare\nshown by solid, dashed and dotted lines, respectively. They are com -\npared with the experimental data taken from EXFOR shown by symb ols.\n25J. Nucl. Sci. & Technol. Article\nFigure 8 Cross sections of (n,2n) reactions for Se isotopes. Calculated res ults\nare same as in Fig. 7 except for the result using ρhwithC=0.70\nshown by dash-dotted line. The experimental data of Frehaut et a l. are\nrenormalized by a factor of 1.08[25] (circle).\nFigure 9 Cross sections of (n,xn) reactions for197Au. Calculated results are same\nas in Fig. 7 except for the result using ρhwithEts= 8 MeV shown by\ndash-dotted line.\nFigure 10 Neutron emission cross sections of (n,xn) and (p,xn) reactions. Ca lcu-\nlated results are same as in Fig. 7. The experimental data are taken\nfrom EXFOR.\n26J. Nucl. Sci. & Technol. Article\n 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35\n 0  5  10  15  20|β2|\nexcitation energy [MeV]80Se133Cs184W\nFigure 1 Most probable deformation β2as a function of the excitation energy calculated by FTHFB.\n27J. Nucl. Sci. & Technol. Article\n 0 20 40 60 80 100\n 0  25  50  75  100  125  150  175  200Ets [MeV]\nmass number\nFigure 2 Parameter Etsderived from FTHFB calculation.\n28J. Nucl. Sci. & Technol. Article\n 0 5 10 15 20 25 30\n 0  50  100  150  200  250a(*) [MeV-1]\nmass number 0 5 10 15 20 25 30a(*) [MeV-1] 0 5 10 15 20 25 30a(*) [MeV-1]as(*) local\nad(*) local\nas(*) sys\nad(*) sys\nhybrid\neffective\ncollectivehybrid\neffective\ncollectivefrms=1.66\nfrms=1.74\nfrms=1.66\n10-210-1100101\n0 50 100 150 200 250D0(cal.)/D0(exp.)\nmass number10-210-1100101D0(cal.)/D0(exp.)10-210-1100101D0(cal.)/D0(exp.)\nFigure 3 Calculateda(∗) (left panel) and D0(rightpanel) forthe hybrid,effectiveandcollectivemodels.\nThe a(∗) determined to reproduce D0of each nucleus and calculated from the systematics are shown\nby the symbols and lines, respectively.\n29J. Nucl. Sci. & Technol. Article\n 1.64 1.66 1.68 1.70 1.72 1.74 1.76 1.78 1.80 1.82 1.84\n 0 0.2  0.4  0.6  0.8  1 1.2  1.4  1.6frms\nC\nFigure 4 Dependence of fD0rmson the additional parameter Cfor the hybrid model.\n30J. Nucl. Sci. & Technol. Article\n 0 2 4 6 8 10 12 14\n 50  100  150  200  250Um [MeV]\nmass numbersystematics\neven\nodd\nFigure 5 Pairing corrected matching energy Ulocal\nmobtained by minimizing flev\nrmsof each nucleus and\nUsys\nmcalculated by Eq. 23 are shown by the symbols and the solid line, respe ctively. The red symbols\nare results for even-even nuclei, and the green ones for odd and o dd-odd nuclei.\n31J. Nucl. Sci. & Technol. Article\n10010510101015102010251030\n 0 10 20 30 40 50 60 70 80\nEx [MeV]197Au10510101015102010251030total level density [MeV-1]\n169Tm105101010151020\n90Zr105101010151020\n82Se\nρhρeffρcol\n105106107\n-3 -2 -1 0 1 2 3\nEx-Sn [MeV]197Au106107108total level density [MeV-1]\n169Tm105106 90Zr104105 82Se\nρhρeffρcol\nFigure 6 Total level densities of the hybrid (solid line), effective (dashed line) and collective (dotted\nline) LD models for82Se,90Zr,169Tm and197Au as a function of Ex(left panel) and Ex−Sn(right\npanel).\n32J. Nucl. Sci. & Technol. Article\n0200400600800100012001400\n 12  14  16  1890Zr(n,2n)89Y(n,3n)cross section [mb]\nneutron energy [MeV] 22  24  26  28  30  32  34\nneutron energy [MeV]EXFOR\nρhρeffρcol\n05001000150020002500\n 8  10  12  14  16  18169Tm(n,2n)169Tm(n,3n)cross section [mb]\nneutron energy [MeV] 16  18  20  22  24  26  28  30\nneutron energy [MeV]EXFOR\nρhρeffρcol\nFigure 7 Cross sections of (n,2n) reactions for90Zr and169Tm, and (n,3n) reactions for89Y and169Tm.\nCalculated results using ρh,ρeffandρcolare shown by solid, dashed and dotted lines, respectively.\nThey are compared with the experimental data taken from EXFOR s hown by symbols.\n33J. Nucl. Sci. & Technol. Article\n02004006008001000120014001600\n 11  12  13  14  15  16  1680Se(n,2n)82Se(n,2n)76Se(n,2n)78Se(n,2n)cross section [mb]\nneutron energy [MeV] 10  11  12  13  14  15\nneutron energy [MeV]0200400600800100012001400cross section [mb]EXFOR\nFrehaut+(80)*1.08ρh C=0.35\nρh C=0.70\nρeffρcol\nFigure 8 Cross sections of (n,2n) reactions for Se isotopes. Calculated res ults are same as in Fig. 7\nexcept for the result using ρhwithC=0.70 shown by dash-dotted line. The experimental data of\nFrehaut et al. are renormalized by a factor of 1.08[25] (circle).\n34J. Nucl. Sci. & Technol. Article\n05001000150020002500\n 8  10  12  14  16  18197Au(n,2n)197Au(n,3n)cross section [mb]\nneutron energy [MeV] 16 18 20 22 24 26 28 30 32 34 36\nneutron energy [MeV]EXFOR\nρh Ets=13 MeV\nρh Ets8 MeV\nρeff\nρcol\n0500100015002000\n 25 30 35 40 45 50 55 60 65cross section [mb]\nneutron energy [MeV] 25 30 35 40 45 50 55 60 65197Au(n,4n)197Au(n,5n)\nneutron energy [MeV]EXFOR\nρh Ets=13 MeV\nρh Ets8 MeV\nρeff\nρcol\nFigure 9 Cross sections of (n,xn) reactions for197Au. Calculated results are same as in Fig. 7 except\nfor the result using ρhwithEts= 8 MeV shown by dash-dotted line.\n35J. Nucl. Sci. & Technol. Article\n10-1100101102103104\n 0 1 2 3 4 5 6 7 8dσ/dE [mb/MeV]\nneutron energy [MeV]197Au(p,xn)*10-1\nEp=11.2 MeVnatZr(n,xn)*10-1\nEn=14.1 MeVnatSe(n,xn)\nEn=14.6 MeVρh\nρeff\nρcol\nEXFOR\n10-210-1100101102103\n 3 4 5 6 7 8 9 10 11dσ/dE [mb/MeV]\nneutron energy [MeV]105Pd(p,xn)*10-3106Pd(p,xn)*10-2108Pd(p,xn)*10-1110Pd(p,xn)Ep=26.1 MeVρh\nρeff\nρcol\nEXFOR\nFigure 10 Neutron emission cross sections of (n,xn) and (p,xn) reactions. Ca lculated results are same\nas in Fig. 7. The experimental data are taken from EXFOR.\n36"    },    {        "title": "1812.01966v2.Managing_uncertainty_in_data_derived_densities_to_accelerate_density_functional_theory.pdf",        "content": "Managing uncertainty in data-derived densities to accelerate\ndensity functional theory\nAndrew T. Fowler1, Chris J. Pickard1,2, and James A. Elliott1\n1Department of Materials Science and Metallurgy, University of Cambridge, 27 Charles\nBabbage Road, Cambridge, CB3 0FS, United Kingdom\n2Advanced Institute for Materials Research, Tohuku University, 2-1 1 Katahira, Aoba,\nSendai, 980-8577, Japan\nAbstract\nFaithful representations of atomic environments and\ngeneral models for regression can be harnessed to\nlearn electron densities that are close to the ground\nstate. One of the applications of data-derived elec-\ntron densities is to orbital-free density functional the-\nory. However, extrapolations of densities learned\nfrom a training set to dissimilar structures could\nresult in inaccurate results, which would limit the\napplicability of the method. Here, we show that\na non-Bayesian approach can produce estimates of\nuncertainty which can successfully distinguish accu-\nrate from inaccurate predictions of electron density.\nWe apply our approach to density functional the-\nory where we initialise calculations with data-derived\ndensities only when we are con\fdent about their\nquality. This results in a guaranteed acceleration\nto self-consistency for con\fgurations that are simi-\nlar to those seen during training and could be useful\nfor sampling based methods, where previous ground\nstate densities cannot be used to initialise subsequent\ncalculations.\n1 Introduction\nDensity functional theory (DFT) has seen widespread\nadoption in many areas of research spanning the nat-\nural sciences due to its high predictive capability atmodest computational cost and transferability across\ndi\u000berent systems [1]. The staggering number of appli-\ncations and papers that exploit DFT are a testament\nto its value in Materials Science [2].\nThe foundations of DFT are the Hohenberg-Kohn\ntheorems [3]. The \frst of these expresses the total en-\nergy of a many-electron system as a functional, F[n],\nof the ground state electron density, n(x), where x\ndenotes a location in real space. The second theorem\ntells us the ground state density is found by minimis-\ningF[n] with respect to n(x). Although an exact\nform forF[n] has not been established, the unknown\ncomponents can be separated into a kinetic energy\ncontribution, T[n] and a term called the exchange\ncorrelation functional, Exc[n] [4]. The magnitude\nof contributions from Exc[n] to the total energy are\nknown to be relatively small and so the exchange cor-\nrelation term can be approximated to some extent by\napproaches like the local density approximation and\nthe generalised gradient approximation [5, 6]. The\nkinetic energy term cannot however be so well ap-\nproximated and a universally applicable functional is\nstill unknown [7]. This forces many applications to\nan alternative paradigm, Kohn-Sham (KS) DFT [5].\nHere,T[n] is replaced by an expectation over inde-\npendant electron wave functions. In many cases, this\nvastly improves the accuracy of the kinetic energy\ncontribution to F[n] but it introduces a signi\fcant\nincrease in the computational expense [8].\n1arXiv:1812.01966v2  [cond-mat.mtrl-sci]  28 Feb 2019With the recent renewed interest in machine learn-\ning, theoretical attempts to learn T[n] have been\nsupplemented with data-driven inferences [9]. These\nare hampered by di\u000eculties in approximating gradi-\nents@T[n]=@n(x), an evaluation which is necessary\nin \fnding the ground state density [10]. Recently, an\napproach to circumvent this issue was proposed, stim-\nulating a new wave of interest in data-driven orbital\nfree (OF) DFT [11, 12, 13]. The alternative route to\nevaluating data-derived OF functionals on the ground\nstate density is to empirically infer the ground state\ndensity itself, removing the variational optimisation\nofF[n] completely. Two possible issues with this ap-\nproach ultimately stem from the availability of data.\nWhileT[n] andn(x) may be very accurate for struc-\ntures similar to those seen during training, when ex-\ntrapolating for unfamiliar structures, either T[n] or\nn(x) may give predictions that are far from the true\nvalues.\nThe key contribution that we make in this work is\nto show that predictive uncertainty can be harnessed\nto prevent poor extrapolations of n(x) for structures\nthat are dissimilar to those seen during training. We\nillustrate how such a measure of con\fdence can be\napplied to accelerate KS DFT by initialising calcula-\ntions with a data-driven contribution only when we\nare con\fdent about its quality. We note that such an\napplication is most suited to sampling methods such\nas nested sampling, where subsequent structures are\nnot guaranteed to be similar [14]. To the best of\nour knowledge, ab initio. nested sampling has yet to\nbe realised due to the prohibitive computational re-\nquirements of standard KS DFT. This work may con-\ntribute, in some part, to realising such calculations.\nFor other applications like molecular dynamics or ge-\nometry optimisation, a temporary history of ground\nstate densities can be applied to subsequent con\fg-\nurations in the calculation. This results in succes-\nsive calculations being initialised fairly close to their\nground state, rendering any improvements made from\na data-derived density to be much less signi\fcant.2 Quantifying uncertainty\nEvaluating an error or measure of con\fdence in a\ndata-driven prediction like n(x) is a well studied\nproblem [15, 16]. Applications of uncertainty quan-\nti\fcation have recently begun appearing in Materials\nScience, with some even in DFT, such as the linear\nmodel exchange correlation functional of Aldegunde\net al. [17, 18, 19, 20, 21]. In this work, we show\nthat useful applications of a predictive uncertainty\ninn(x) can be realised for just one of many possible\napproaches. By illustrating a proof-of-concept appli-\ncation to accelerating KS DFT, we hope to encourage\na greater awareness of the advantages of quantifying\nuncertainty and to stimulate interest in alternative\nmethods and applications such as in OF DFT.\n2.1 Non-Bayesian regression\nIn the following we adopt the notation that nand\nxrefer to a known ideal model contribution to elec-\ntron density and a corresponding representation for\nthe environment of that density point, respectively.\nSpeci\fcally, we adopt the bispectrum representation\nforx= (xlocal;xglobal), which is a concatenation of\nlocal and global contributions [22, 23]. We refer the\nreader to section A of the Appendix for further de-\ntail and also note that explicit dependence of nupon\nxhas been dropped in this section to improve clar-\nity. In this work, we use a non-Bayesian approach to\nquantify uncertainty. Although a Bayesian method to\nparametric regression will give a more reliable mea-\nsure of uncertainty, evaluating uncertainty from the\npredictive distribution for non-linear models is not a\nsimple task and often sampling is involved which can\nincur signi\fcant computational overhead [15].\nWe propose a model in which observations of the\ntrue ground state density nare prone to random er-\nror which is distributed normally about the model\npredictions \u0016(x;w):\np(njx;w) =N(nj\u0016(x;w);\u001b(x;w)2): (1)\nWe also introduce a dependency of the variance\nof this random error, \u001b(x;w)2, on the environment\nx, which is known as a heteroskedastic model for\n2noise [24]. We use a fully connected feed-forward\nneural network with hidden network weights w, to\ncalculate\u0016(x;w) and\u001b(x;w)2. Observations of\nnare treated as independent and identically dis-\ntributed random variables and to infer w, we calcu-\nlate the maximum likelihood estimate by maximising\nthe productQ\nn;xp(njx;w) over all observations in\nthe training set.\nTo quantify error in the predictions of ngiven a\nnewx, we adopt an ensemble of Nensneural net-\nworks, each with network weights wi. Adopting a\nuniformly weighted Gaussian mixture, the likelihood\nof the ensemble is then:\np(njx;W) =1\nNensNensX\ni=1N\u0000\nnj\u0016(x;wi);\u001b(x;wi)2\u0001\n=N\u0000\nnjnML(x;W);\u001bML(x;W)2\u0001\n(2)\nwhere W= (w1;:::;wNens) andp(njx;W) is also\na normal distribution [25]. Uncertainty in our pre-\ndiction ofnis given by the variance of p(njx;W),\n\u001bML(x;W)2and can be evaluated as:\n\u001bML(x)2=1\nNensNensX\ni=1\u0016(x;wi)2\u0000nML(x)2\n+1\nNensNensX\ni=1\u001b(x;wi)2\nnML(x) =1\nNensNensX\ni=1\u0016(x;wi):(3)\n2.2 Doing no harm\nTo apply our model for prediction uncertainty\n\u001bML(x)2in (3) to accelerate KS DFT, we need to\nevaluate a global measure of uncertainty for an en-\ntire structure. We call this measure H[p\u001bML], where\nan unknown dependency on the empirical prior distri-\nbutionp\u001bMLof\u001bMLis shown explicitly. In this work,\nwe adopt the very simple measure that:H[p\u001bML] =Ep\u001bML[ln(\u001bML)]\n=1\nNNX\ni=1ln(\u001bML(xi))(4)\nforNdensities in a crystal. We now abbreviate\nH[p\u001bML] =Hand introduce a tapering function\n\u0000(H), which is essentially a step function with a con-\ntrollable transition point and length scale. For details\nof the speci\fc form of \u0000 used in this work, we refer\nthe reader to section C of the Appendix. With \u0000( H),\nwe can control the empirical contribution nML(x) to\nan initial density estimate:\nn(x) =n0(x) + \u0000(H)nML(x): (5)\nn0(x) represents any standard initialisation technique\nfor the density in DFT but typically, this is a com-\nbination of the radial components of electron density\nfor atoms assumed to be in vacuum. The ideal model\ncontribution nfrom section 2.1 is the di\u000berence of\nthe true ground state density and the standard ini-\ntial contribution, n(x)\u0000n0(x) from (5).\nWe note that an alternative strategy could be to ta-\nper empirical contributions locally at each grid point,\nbut we choose a global approach to discourage spuri-\nous non-smoothness in nML(x)\u0000(\u001bML(x)) that might\noccur ifj\u001bML(x+\u000ex)\u0000\u001bML(x)j>>0, for a small per-\nturbation in environment \u000ex. We also note that the\ne\u000bects of any random error in \u001bML(x) are signi\fcantly\nreduced by considering distribution averages. While\nwe found that the simple choice of Hused in (4)\nworked very well at identifying uncertain predictions\nfor the applications in this work, a more informative\nmeasure of the distribution p\u001bMLmay improve this\ndistinction further. Higher order moments of p\u001bML\nsuch as the distribution variance for example could\nbe utilised, in addition to knowledge about the dis-\ntribution mean.\n3 Results\nIn this section we illustrate how the non-Bayesian ap-\nproach to uncertainty quanti\fcation adopted in this\n30 5 10 15\nx/(\u0017A)51015y/(\u0017A)\n147\n\u001bML(x)\nFigure 1: A model trained on pristine graphene iden-\nti\fes a large degree of prediction uncertainty, \u001bML(x)\ndenoted by greyscale shading, in the area surround-\ning a 7-5 pair defect. We note that \u001bML(x) is given\nin units of 10\u00002e\u0017A\u00003.\nwork can qualitatively distinguish accurate from inac-\ncurate values of the data-derived contribution nML(r).\nWe also show how the number of self-consistent \feld\niterations needed to reach self-consistency in a KS\nDFT calculation can be reduced as the initial den-\nsity tends to the exact ground state density.\nFor environments dissimilar to those seen during\ntraining, we expect a larger predictive uncertainty.\n7-5 defect in graphene Figure 1 shows \u001bML(x)\nfor a single layer of graphene with a 7-5 pair (Dienes)\ntopological defect [26]. Only densities from a single\npristine layer of graphene were used during training.\nIn the area immediately surrounding the defect, pre-\ndictive uncertainties increase (denoted by dark shad-\ning), identifying this region as an environment dis-\nsimilar to the defect-free layer.\nIn-plane strain in graphite In Figure 2, we com-\npare the prediction uncertainty of graphite with 0%\nand 5% in-plane strain. Speci\fcally, we show the\n[100] lattice vector contour and \fnd that predictions\nare signi\fcantly more certain for the 0% contour0:0 0:2 0:4 0:6 0:8 1:0\n[100]0:00:10:20:3nML(x)\n0%strain\n5%strain\nFigure 2: A model trained only on primitive graphite\nwith 0% in-plane lattice strain identi\fes a region of\nhigh degree of uncertainty when making predictions\nalong the [100] contour of a primitive graphite crystal\nwith 5% in-plane strain. The shaded regions show the\nintervalnML(r)\u00063\u001bML(x) and the dashed lines show\nthe true ground state density. We note that charge\ndensities are given in units of e\u0017A\u00003.\nwhich was seen during training, than the 5% con-\ntour that was not. Further details of the bispectrum\nand KS DFT calculations for Figures 1 and 2 can be\nfound in the Appendix, section C.\n3.1 Accurate initial densities\nTo motivate our application of uncertainty quanti\f-\ncation to KS DFT, we examine the convergence of\nsingle point KS DFT calculations to self-consistency,\nas we perturb initial densities away from the exact\nground state via perturbations to the ideal model\ncontribution, n(x)\u0000n0(x) in (5).\nWe study a non-metallic crystal, graphite, and cal-\nculate the ground state density for several hundred\nprimitive cell con\fgurations sampled from a NPT\nmolecular dynamics trajectory. The components of\nthe discrete Fourier transform of the ideal model con-\ntribution are perturbed by additive Gaussian noise.\nBy taking the inverse transform, we have a contin-\n410\u0000310\u0000210\u00001 1\nRMSE / (e\u0017A\u00003)\u00008\u0000404dSCF\nPulay\nBroyden\nFigure 3: As the initial KS DFT densities are de-\nformed away from the true ground state, the num-\nber of iterations necessary to reach self-consistency\nincreases and the improvement from standard DFT\ninitialisation, dSCF, decreases. Pulay and Broyden\ndensity mixing schemes result in a very similar con-\nvergence for all calculations.\nuously deformed version of the true density. We\nmeasure deformation by the root-mean-squared er-\nror (RMSE) of the perturbed and true ground state\ndensity.\nIn Figure 3, the self-consistent calculation is ini-\ntialised with charge densities which are increasingly\ndeformed versions of the ground state. We see that\nas the magnitude in deformation from the ground\nstate, the RMSE, increases, so too does the number\nof iterations needed to reach self-consistency. The\nquantity displayed on the abscissa, dSCF, is the im-\nprovement, in the number of iterations, relative to a\ncalculation with the standard initial density. As de-\nformations increase, the improvement decreases. The\nhashed and shaded areas in Figure 3 represent con-\n\fdence intervals of 67%, showing that the relation\nbetween RMSE and convergence to self-consistency\nis stochastic to some degree.\nTo ensure that any empirical method for initialising\nKS DFT densities does not negatively a\u000bect conver-\ngence to self-consistency in regions where the empiri-cal densities extrapolate poorly, uncertainty quanti\f-\ncation is clearly needed. For further details of the\nDFT calculations in Figure 3, see section C of the\nAppendix.\n4 Applying global uncertainty\nAs we saw by the calculations in Figure 3, a mea-\nsure of con\fdence in density is necessary if we are to\nuse empirical densities in DFT in a \\safe\" manner.\nNot wanting to leave things worse than how we found\nthem, we hope to ensure that every calculation ini-\ntialised by a data-driven density does no worse than\nits ordinary counterpart.\nTo illustrate that a global measure of con\fdence in\npredictions, Hfrom (4), can be applied to acceler-\nate KS DFT, we \frst consider using empirical den-\nsities without using knowledge of their uncertainty.\nAfter training on 5 primitive cell graphite con\fgura-\ntions from a NVT molecular dynamics simulation at\nT= 300 K, we predict densities for all 300 crystals in\nour data set. Details of the empirical model and KS\nDFT calculations can be found in the Appendix, sec-\ntion C. Without applying any information about un-\ncertainty, we blindly initialise Broyden density mix-\ning (DM) DFT calculations and record the reduction\nin the number of iterations to self-consistency, dSCF,\nrelative to a calculation with a standard initial den-\nsity. Next, we calculate the global con\fdence measure\nH=E[ln(\u001bML)] for each crystal and categorise crys-\ntals into discrete sets according to their dSCF score.\nWe show the corresponding empirical joint distribu-\ntionp(E[ln(\u001bML)];dSCF) in Figure 4.\nWe can expand the empirical joint distribution\np(H;dSCF) =p(dSCFjH)p(H) (6)\nin terms of the unknown conditional distribution\np(dSCFjH) from which we want to decide if a given\nprediction, His good enough to initialise a KS DFT\ncalculation. Taking the prior p(H) as constant,\np(H;dSCF)/p(dSCFjH) and we look for a gap in H\nbetweenp(H;dSCF\u00150) andp(H;dSCF<0). It is\nhere that a transition point can be set in the taper-\ning function \u0000( H), to reduce uncertain predictions\nto zero. This can be visualised by comparing the\n56\nE[ln(\u001bML)]-6.2\n-6 dSCF\n4320-1p(E[ln(\u001bML)];dSCF)\nFigure 4: The empirical joint distribution\np(H;dSCF) evaluated for a set of primitive graphite\ncon\fgurations shows separable categorisations of\nperformance into good (dSCF >0) and poor\n(dSCF<0) predictions.peak at dSCF =\u00001 with that at dSCF=0. Although\nthere is some overlap between these two peaks, there\nis almost zero overlap between dSCF=-1 and all other\npeaks. This means that Hcan be used to identify the\nquality of density predictions before any DFT calcu-\nlation is made. We note that the joint distribution\nshown in Figure 4 is a smoothed approximation of the\ntrue empirical distribution but that important prop-\nerties such the width of each conditional distribution\np(dSCFjH), are preserved.\nIn fact we see that for the small study here, the\nexpectation of Hconditioned on dSCF,\nEp(HjdSCF) [H] =Z\ndHp(HjdSCF)H (7)\nwhich is the dashed line in Figure 4, follows a\nmonotonic relation with dSCF. This shows that pre-\ndicted uncertainty really does correspond in a mono-\ntonic way to actual error. Using the distribution of\npredicted uncertainties over a crystal, we can iden-\ntify model predictions which are poor and will harm\nconverge to self-consistency. By e\u000bectively turning o\u000b\npoor predictions using \u0000( H), empirical corrections to\nthe initial KS density can be applied only for crystals\nwhich are similar to those seen during training.\n4.1 Accelerating self-consistency\nNow that we have established a mechanism to detect\nglobal uncertainty in density, we can apply this to\nsingle point KS DFT calculations to accelerate con-\nvergence to self-consistency. In Figure 5 we compare\nthe empirical distributions p(dSCF) for Broyden DM\nDFT calculations performed using data-derived den-\nsities with and without tapering. The upper half of\nFigure 5 shows a number of extrapolations where\npoor predictions of density have a negative e\u000bect\nupon convergence (dSCF <0). In the lower half, un-\ncertain predictions have been identi\fed and reduced\nto zero, increasing the peak at dSCF = 0.\nCrucially, the computation time required to evalu-\nate our data-derived density estimate is just less than\nthe time taken to evaluate a single SCF iteration. For\nfurther details of the calculations in Figure 5 involv-\ning KS DFT parameters, see section C of the Ap-\npendix. Despite our model being trained only on 57\n2 0 2 400.20.4untapered\n-2 0 2 4\ndSCF00.20.4taperedp(dSCF)\nFigure 5: When data-derived densities are used in\nKS DFT without using uncertainty prediction (un-\ntapered contributions), there is a non-zero chance\np(dSCF) that inaccurate predictions can harm con-\nvergence to self-consistency (dSCF <0). When pre-\ndiction uncertainties are applied to identify (taper)\nunfamiliar crystals, only a neutral or positive speed\nup is seen.con\fgurations, a large proportion of crystals exhibit\na speed up in converging to self-consistency. Such an\ne\u000bect could also arise by poorly choosing a test set of\ncrystals, whereby all atom positions remain in almost\nidentical positions. A trivial approach of applying the\nground state density from a random crystal, or an av-\nerage of ground state densities over all crystals, would\ntherefore achieve similar, or better results. However,\nin fact this is not the case. When such simulations\nwere run, we found that almost all ( \u001895%) of pre-\ndictions obtained dSCF = 0. Our test set is in fact a\nrather dissimilar collection of con\fgurations, most of\nwhich involve signi\fcant shifts in registry across the\nbasal plane, as con\fgurations jump from one AB-\nstacked state to another. We attribute the ability\nof our model to infer useful predictions from such\nan incredibly small number of con\fgurations to the\nfact that each crystal in the training set contributes\nO(104) grid points. Simply put, more data leads to\na better inference, even when a large number of data\npoints come from the same crystal.\n4.2 Wider applicability\nTo this point, all calculations in this work have been\nmade to illustrate that data-derived densities can\nbe applied to a single system to improve the stan-\ndard analytical initial densities that are used in KS\nDFT. We consider the wider implications of this work\nbeyond graphite by comparing the dissimilarity of\nground state and standard initial densities for a col-\nlection of 29 metals and 37 non-metals under both\nlow and high pressure. We \fnd that all of the met-\nals we consider have initial densities that are much\ncloser to the ground state density than with graphite\nwhile the converse is true for approximately half of\nthe non-metals studied here. We use the RMSE of all\ndensity grid points within a crystal as a measure of\ndissimilarity between these two densities. To classify\nmetals and non-metals we use the density of states at\nthe Fermi level. We use a value of 0 :2e(eV)\u00001which\nis just above the density of states for the metalloid\nAs to classify the two classes.\nWe show in Figure 6 a smoothed approximation\nof the conditional distribution p( log10(RMSE)j\u0011) for\nmetal or non-metals \u0011along with a dashed vertical8\n10\u0000310\u0000210\u00001 1\nRMSE=(e\u0017A\u00003)0.000.250.500.751.001.251.50p( log10(RMSE)j\u0011)metal\nnon-metal\ngraphite\nFigure 6: The RMSE between the standard ini-\ntial and ground state densities is much smaller for\nthe metals studied here then with graphite and ap-\nproximately half of the non-metals. Four-component\nGaussian mixture models approximate the condi-\ntional distributions p(log10(RMSE)j\u0011) of the RMSE\ngiven the characterisation \u0011of the system. The ver-\ntical dashed line shows the RMSE of graphite.line showing the RMSE between the standard ini-\ntial and ground state densities for graphite. The\nlogarithm of the RMSE illustrates that the RMSE\ndi\u000bers by almost two orders of magnitude between\ngraphite and some of the metals. We note that the\napproximate representation of p( log10(RMSE)j\u0011) is\na four-component Gaussian mixture model of the true\ndata [15]. Based upon the systems studied here, we\nsummarise that data-derived densities may in general\nbe more suitably applied to non-metals than metals.\nDetails of these systems as well as and the KS DFT\ncalculations that were used to calculate the RMSE in\nFigure 6 can be found in section C of the Appendix.\n5 Discussion\nWe have shown that uncertainty quanti\fcation can\nbe applied to accelerate KS DFT for sampling meth-\nods like nested sampling, attaining a maximum speed\nup of 57%1for the systems studied in this work. We\nview the approach taken in this work as more a proof\nof concept than a \fnal solution, con\fdent that excit-\ning developments and insights are accessible to future\nwork. To this end, we note that the approach taken\nin this work is just one of many possible methods.\nWe use this section to discuss what we believe to be\nthe most prominent disadvantages of this approach\nand outline a few ideas that could address these short\ncomings.\nWhile our parametric approach leads to a compu-\ntation time for data-derived densities that is smaller\nthan a single self-consistent \feld cycle, the time\nneeded to train densities from a single crystal is or-\nders of magnitude larger than a full DFT calcula-\ntion. Although sampling methods require thousands\nof crystal con\fgurations, the time to train or re\fne\nan data-derived density should ideally be as close to a\nsingle KS DFT calculation as possible. Some heuris-\ntic techniques, such as maximising the sample vari-\nance of observations in a smaller training subset, may\ngive some reduction in this, but a more promising av-\nenue could be to use approximate Bayesian inference,\nsuch as deterministic variational inference [27]. A\n1See section D of the Appendix for the de\fnition of speed\nup that we adopt here.Bayesian approach, even when the posterior distribu-\ntion is approximate, could also lead to more reliable\nuncertainty estimates. The non-Bayesian approach\nadopted in this work does not guarantee that \\false\npositives\" cannot occur when determining if con\f-\ndence should be placed in a data-derived density or\nnot. In addition, Bayesian online learning could allow\nfor an incremental approach to learning densities such\nthat re\fnements are made during sampling, only to\ncrystals which are dissimilar to all of those that were\npreviously seen during training [28].\nThe approach that we use in this work to make de-\ncisions about con\fdence in the density, does not take\naccount of the type of crystal. For applications like\nnested sampling where several di\u000berent phases are\nsampled from, it may become essential for our deci-\nsion process to include knowledge about the global\nenvironment, such as from a global bispectrum rep-\nresentation of the crystal. An unsupervised method\nsuch as a Gaussian mixture model may be necessary\nto associate crystals with nearby clusters and to ap-\nply decisions using a predetermined set of distinct\ncon\fdence thresholds for each cluster.\nAn aspect that we havent considered in this work\nis the question of which method of minimising the KS\nHamiltonian, given an initial density, gives the low-\nest computation time. Although this is a well studied\nproblem, perhaps new insights are possible when an\nestimate of con\fdence is available in the initial den-\nsity [29].\nWe note that our discussion of KS DFT and the\napplication of data-derived densities to accelerate\nconvergence to self-consistency in this work has so far\nignored spin. For many systems and processes such\nas radicals, transition metal complexes or homolytic\nbond breaking, the spatial wave functions of opposing\nspin states are not equal (  \u000b(r)6= \f(r)) [30, 31, 32].\nSpin-unrestricted KS DFT is a generalisation of\nthe spin-restricted form, where  \u000b(r)6= \f(r)\nis possible and the variational minimisation of\ntotal energy E[n;Q] is performed with respect\nto both the total electron density n(r) and the\nspin density Q(r) =P\nij \u000b(r)j2\u0000P\nij \f(r)j2[33].\nInitial densities for unrestricted spin therefore\nrequireQ(r) in addition to n(r). A general-\nisation of the data-derived densities used inthis work to unrestricted spin DFT could be\nrealised by adopting p(tjx;w) =N(tj\u0016;\u0003\u00001)\nfort= (n(r);Q(r)). A parametric model\nwould then represent x!\u0000\n\u0016;\u0003\u00001\u0001\nrather then\nx!(\u0016;\u001b2) as in (1). The generalisation of (2)\nleads to p(tjx;W) =N(tj\u0016ML;(\u0003ML)\u00001where\n\u0016ML= (nML;QML) and the covariance matrix\n(\u0003ML)\u00001represents uncertainty in the initial data-\nderived total ( nML) and spin ( QML) densities. The\nsimplest way to apply \u0003MLto identify uncertain pre-\ndictions might be to sum the diagonal components\nof (\u0003ML)\u00001to de\fne a scalar measure analogous to\n\u001bMLin (4). We also note that E[n;Q] is well known\nto exhibit a number of stationary points and in the\nabsence of any knowledge about the ground state\nofQ, some form of approximate global optimisation\nmust be utilised. If the data-derived densities\nare su\u000eciently accurate then global optimisation\nfor spin-unrestricted DFT could be abandoned\naltogether, providing signi\fcant reductions to the\ncomputation required.\n6 Conclusions\nWe have shown that a non-Bayesian treatment of pre-\ndictive uncertainty can be applied to electron density\nregression to identify crystals that are dissimilar to\nthose seen during training. We have applied this ap-\nproach to KS DFT where we have been able to iden-\ntify and prevent unfamiliar crystals from negatively\ne\u000becting convergence to self-consistency. For the sys-\ntems studied in this work, where con\fdent predic-\ntions were made we saw a maximum speed up in con-\nvergence to self-consistency of 57%1and cautiously\nnote that further improvements could be made with\na more in depth study of the approach to minimise\nthe KS Hamiltonian. Crucially, our predictions can\nbe evaluated in less time than a single self-consistent\n\feld iteration for a primitive crystal, meaning that\nour application to KS DFT could be useful for meth-\nods like nested sampling.\nWe view this work as a proof of concept. Quan-\ntifying uncertainty in predicted densities is shown\nto be a fruitful endeavour and we hope our work\nwill encourage further applications and alternative\n9approaches, for example in orbital free DFT. More\ngenerally, this work motivates more sophisticated\ntreatments of interpolation, or caching, which are\ncurrently treated deterministically to accelerate high\nperformance plane wave DFT codes [34, 35]. We an-\nticipate that a paradigm shift towards \\probabilistic\ncaching\", or regression, will lead to the e\u000ecient use\nof previously computed data.\nAcknowledgements\nThe authors would like to thank Nick Woods for\nreading an earlier version of this manuscript and for\nsharing a number of insightful observations regarding\nspin-polarised DFT that motivate our discussion of\nextending data-derived densities to spin-unrestricted\nDFT. We also thank Georg Schusteritsch for many\nhelpful discussions regarding KS DFT and the self-\nconsistent \feld procedure as well as our referees for\nproviding constructive comments that has helped to\nimprove this manuscript. Additionally, we thank the\nUKCP consortium, grant number EP/P022596/1 and\nthe Royal Society through a Royal Society Wolfson\nResearch Merit award, on behalf of C.J.P. . 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C.\nPayne, \\First principles methods using\nCASTEP,\" Zeitschrift fur Kristallographie 220\nno. 5-6, (Jan, 2005) 567{570.\n[35] J. Hafner, \\Ab-initio simulations of materials\nusing VASP: Density-functional theory and\nbeyond,\" Journal of Computational Chemistry\n29no. 13, (Oct, 2008) 2044{2078.\n[36] S. Gra\u0014 zulis, A. Da\u0014 skevi\u0014 c, A. Merkys,\nD. Chateigner, L. Lutterotti, M. Quir\u0013 os, N. R.\nSerebryanaya, P. Moeck, R. T. Downs, and\nA. Le Bail, \\Crystallography Open Database\n(COD): an open-access collection of crystal\nstructures and platform for world-wide\ncollaboration,\" Nucleic Acids Research 40\nno. D1, (Jan, 2012) D420{D427.[37] M. Hellenbrandt, \\The Inorganic Crystal\nStructure Database (ICSD)Present and\nFuture,\" Crystallography Reviews 10no. 1,\n(Jan, 2004) 17{22.\n12Appendix\nA\nIn the bispectrum approximation, elements of local\nand global contributions to the representation of en-\nvironment, x(r), are determined by the projections\ncnlmof local and global environment into radial ( n)\nand spherical harmonic ( lm) bases.\nclocal\nnlm(r) =X\ni2\nrgn(dri)Ylm(d\u0012i;d\u001ei)\ncglobal\nnlm=1\nNNX\ni=1X\nj2\nrign(drij)Ylm(d\u0012ij;d\u001eij):(8)\n\nris a spherical volume of \fnite radius surrounding\na point in real space r. Indicesiandjdenote pairs\nof theNatoms contained in the primitive cell of a\ncrystal. dri=jri\u0000rj, drij=jrj\u0000rijand d\u0012and d\u001e\nare the polar and azimuth projections respectively of\nri\u0000randrj\u0000ri.\nB\nThere are many equally adequate tapering functions\nwhich could be chosen. We used:\n\u0000(\u001b(r);\u001b\u0003) =(\n~\u001b4\n1+~\u001b4;~\u001b>0\n0;~\u001b<0\n~\u001b=\u001b\u0003\u0000\u001b(r)\n\u001bscale(9)\nTable 1: Calculation parameters.\nbispectrum Neural network KS DFT\nCalculation rcut(\u0017A)nmaxlmaxNens nodesEcut(eV)k-point grid\nFigure 1 4 3 3 5 2\u0002100 400 [36 36 1]\nFigure 2 6 10\u00038\u000310 2\u000250 400 [36 36 6]\nFigure 3 - - - - - 800 [36 36 6]\nFigure 4 6 6 6 10 2\u0002200 300 [18 18 4]\nFigure 5 4 4 4 5 2\u0002150 300 [18 18 4]\nFigure 6 - - - - - 800 (0.1,0.1,0.1) \u0017A\u00001simply because it has property that every nth\nderivative@(n)\u0000=@\u001b(r)(n)remains continuous.\nC\nFor all of the KS DFT calculations in Table 1, a\nFermi surface smearing width of 250 K, an energy\ntolerance of 10\u00006eV/atom, the PBE exchange\ncorrelation functional and Broyden density mixing\nwere used, except for the calculations in Figure 3\nwhich used both Broyden and Pulay density mixing\nand the calculations in Figure 6 which used a\nsmearing width of 300 K. All graphite and graphene\ncon\fgurations except for the NPT calculations of\n\fgure 3 had an in-plane C-C spacing of 1.42 \u0017A\nand an inter-layer spacing of 3.34 \u0017A. In addition,\na vacuum corresponding to a unit cell of 20 \u0017A in\nthe plane-normal axis was adopted for the graphene\nlayer in Figure 1 and in Figure 5 a tapering function\nof the form in (9) was used with the threshold and\nscaling factor ( \u001b\u0003;\u001bscale) = (\u00006:83;10\u00003).\u0003denotes\nuse of the power spectrum rather than bispectrum\nrepresentation for the calculations in Figure 2. The\nnotation adopted in 1 regarding the number of nodes\nused in each neural network, is that x\u0002ydenotes a\nneural network of xnode layers, each containing y\nnodes. Table 2 lists the database, unique identifying\nnumber and characterisation of each system used to\ngenerate the calculations in Figure 6. We supply\ninput \fles for all data sets within this work at\nhttps://github.com/andrew31416/densityregression/\ntree/master/data sets.\n13Table 2: The collection of metals and non-metals from Figure 6 were taken from the\nCrystallography Open Database (COD) [36] and the Inorganic Crystal Structure Database\n(ICSD) [37]. A unique identifying number is given for each crystal along with its character-\nisation as discussed in section 4.2\nDatabase Identi\fer Characterisation Database Identi\fer Characterisation\nCOD 9008572 non-metal COD 9008531 metal\nCOD 9008594 non-metal COD 9008468 metal\nCOD 9008595 non-metal COD 9008544 metal\nCOD 9008569 non-metal COD 9008482 metal\nCOD 9008577 non-metal COD 9008478 metal\nCOD 9008561 non-metal COD 9008485 metal\nCOD 1011098 non-metal COD 9008463 metal\nCOD 9008568 non-metal COD 9008458 metal\nICSD 193853 non-metal COD 9008552 metal\nICSD 26158 non-metal COD 9008501 metal\nICSD 9863 non-metal COD 9008490 metal\nICSD 18154 non-metal COD 9008522 metal\nICSD 16516 non-metal COD 9008470 metal\nICSD 15598 non-metal COD 9008549 metal\nICSD 41440 non-metal COD 9008513 metal\nICSD 2130 non-metal COD 9008536 metal\nICSD 411857 non-metal COD 9008514 metal\nICSD 27249 non-metal COD 9008546 metal\nICSD 20904 non-metal COD 9008584 metal\nICSD 22156 non-metal COD 9008570 metal\nICSD 84461 non-metal COD 9008543 metal\nICSD 15390 non-metal COD 9008525 metal\nICSD 39566 non-metal COD 9008512 metal\nICSD 27798 non-metal COD 9008557 metal\nICSD 40914 non-metal COD 9008477 metal\nICSD 16428 non-metal COD 9008558 metal\nICSD 165592 non-metal ICSD 15535 metal\nICSD 22157 non-metal ICSD 63670 metal\nICSD 19079 non-metal ICSD 653014 metal\nICSD 29128 non-metal\nICSD 107946 non-metal\nICSD 60559 non-metal\nICSD 16262 non-metal\nICSD 187642 non-metal\nICSD 22158 non-metal\nICSD 18012 non-metal\nICSD 77378 non-metalD\nWe adopt the convention that the speed up\n\u001c=\u0012Nold\u0000Nnew\nNnew\u0013\n\u0002100% (10)\nfor data-derived initial densities that require\nNnewself-consistent \feld iterations to reach self-\nconsistency. Noldis the number of iterations re-\nquired for a standard calculation that uses a non\ndata-derived initial density. \u001cis de\fned such that a\ndata-derived density that halves then required num-\nber of self-consistent \feld iterations corresponds to a\n100% speed up.\n15"    },    {        "title": "1812.11462v3.Entanglement_of_multiphoton_polarization_Fock_states_and_their_superpositions.pdf",        "content": "arXiv:1812.11462v3  [quant-ph]  6 May 2019Entanglement of multiphoton two-mode polarization Fock st ates and of their\nsuperpositions.\nS. V. Vintskevich1,3, D. A. Grigoriev1,3, N.I. Miklin4, M. V. Fedorov1,2\n1A.M. Prokhorov General Physics Institute, Russian Academy of Sciences, 38 Vavilov st., Moscow, 119991, Russia\n2National Research University Higher School of Economics,\n20 Myasnitskaya Ulitsa, Moscow, 101000, Russia\n3Moscow Institute of Physics and Technology, Dolgoprudny, M oscow Region, Russia and\n4Institute of Theoretical Physics and Astrophysics,\nNational Quantum Information Center, Faculty of Mathemati cs,\nPhysics and Informatics, 80-308, Gdansk, Poland\n(Dated: May 8, 2019)\nDensity matrices of pure multiphoton Fock polarization sta tes and of arising from them reduced\ndensity matrices of mixed states are expressed in similar wa ys in terms of matrices of correlators\ndefined as averaged products of equal numbers of creation and annihilation operators. Degree of\nentanglement of considered states is evaluated for various combinations of parameters of states and\ncharacter of their reduction.\n1. INTRODUCTION\nThere is a growing interest in the science of quantum\ninformationtomultiparticleentanglement. It findsappli-\ncationsinquantumcomputingandquantumerrorcorrec-\ntion [1],[2], as well as in quantum networks [3]. The lat-\nter includes, in particular, communication among many\nparties that is enhanced by shared multiparticle entan-\nglement. The most promising recourse for establishing\nthis type of entanglement are, of course, multi-photon\nsystems. Thus, there is a natural interest in studying en-\ntanglement properties of states of many photons. Recent\nexperiments also shown that entanglement of up to ten\nphotons can be observed in a lab [4].\nIn this work we consider pure multiphoton two-mode\npolarization Fock states and their superpositions. We\ngive a general definition of the density matrices of such\nstates as well as of the density matrices of mixed states\narising from pure Fock states after their partial reduc-\ntion over a series of photon variables. Elements of such\ndensity matrices are expressed in terms of correlators de-\nfined as averaged products of equal numbers of creation\nand annihilation operators with different distributions of\noperators over two polarization modes. we will calculate\nparameters characterizing the degree of entanglement in\nsuch states and investigate their dependence on features\nof the original pure states and on the ways of their re-\nduction.\nNote that for biphoton states the method of density\nmatrices of the described type was suggested by D.N.\nKlyshkoin 1997[5] and somewhatlaterused in the works\n[6, 7]. More recently there was a series of works on some\naspects of entanglement in multipohoton states [8–12].\nBut as far as we know, there were no works where the\nKlyshkomethodofdensitymatriceswouldbe generalized\nfor multiphoton states with numbers of photons higher\nthan 3. Such generalization is one of the main goals of\nthis work. The second goal is characterizing entangle-\nment of multiphoton states in terms of Schmidt decom-\npositions and their parameters, which will be new too.Note also that, though the Schmidt decomposition was\nknown in mathematics since 1906 [13], in the fields of\nmodern quantum optics and quantum information it was\nintroduced by J.H. Eberly and coworkers at first in 1994\n[14] and then in 2004 [15]. A much more general and de-\ntailed description of the Schmidt decomposition, as well\nas its applications, were given in the review paper [16].\n2. DENSITY MATRICES\nLet us consider an arbitrary pure state |Ψ(n)/an}bracketri}htofn\nphotons having identical frequencies and identical given\npropagation directions but distributed arbitrarily be-\ntween two polarization modes, horizontal and vertical\nones,HandV. Two-modepolarization basic Fock states\nare states with given numbers of horizontally and verti-\ncally polarized photons nHandnVsuch thatnH+nV=\nn\n|ΨnH,nV/an}bracketri}ht=|nH,nV/an}bracketri}ht=a†nH\nHa†nV\nV√nH!nV!|0/an}bracketri}ht.(1)\nMore general n-photon polarization states to be consid-\nered are superpositions of basic Fock states\n|Ψ(n)/an}bracketri}ht=n/summationdisplay\nnH=0CnH|nH,nV/an}bracketri}ht|nV=n−nH(2)\nwith/summationtextn\nnH=0|CnH|2= 1. The wave functions of all n-\nphoton states |Ψ(n)/an}bracketri}htdepend on nsingle-photon vari-\nablesσi, Ψ(n)({σi}) =/an}bracketle{t{σi}|Ψ(n)/an}bracketri}htand, explicitly, they\nare given by symmetrized products of nsingle-photon\nwave functions [17]. In the case of polarization modes\nthe single-photon wave functions in these products are\nψH(σi) =δσi,HandψV(σj) =δσj,V. In the matrix rep-\nresentation ψH(σi) =/parenleftbig1\n0/parenrightbig\niandψV(σj) =/parenleftbig0\n1/parenrightbig\nj[18].\nNote that sometimes it’s possible to meet in litera-\nturementionsaboutparticle-ormode- entanglementand\naboutdifferencesorsimilaritiesbetweenthem. Wedonot2\nuse such concepts here because in our opinion the type of\nentanglement to be studied can be much more correctly\ninterpreted as related to uncertainty of distributions of\nparticle variables between modes or, shortly, as the vari-\nableentanglement. For two-modepolarizationstatesthis\nmeans an uncertainty of attachment of polarization vari-\nablesσitoH- orV-modes.\nDirect products of ntwo-line columns/parenleftbig1\n0/parenrightbig\niand/parenleftbig0\n1/parenrightbig\nj\nform a basis of columns with 2nelements (“rows”)\nand with different locations of a single unit in one\nof these “rows”. Written down in this basis ex-\nplicitly, the multiphoton wave function can be used\nfor constructing the density matrix ρ(n)({σi},{σ′\ni}) =\nΨ(n)({σi})Ψ(n)†({σ′\nj}). However at high values of the\nphoton numbers nthis procedure is rather cumbersome\nto be reproduced explicitly. Fortunately, there is a much\nmore compact algorithm for constructing multiphoton\ndensity matrices to be described and discussed below.\nBut of course, at any given ncorrectness of the used be-\nlow matrix representationscan be checked and confirmed\ndirectly by the described derivations based on the use of\nthe multiphoton wave functions Ψ(n)({σi}).\nThus, for any pure two-mode multiphoton state |Ψ(n)/an}bracketri}ht\nits 2n×2ndensity matrix can be presented symbolically\nin the following form\nρ(n)=1\nn!/parenleftig/braceleftig\n/an}bracketle{t(a†\nH)n−k2(a†\nV)k2an−k1\nHak1\nV/an}bracketri}ht/bracerightig/parenrightig\n(3)\nwith averaging understood as /an}bracketle{t.../an}bracketri}ht=/an}bracketle{tΨ(n)|...|Ψ(n)/an}bracketri}ht.\nSuch mean products of the creation and annihilation op-\nerators can be referred to as correlators. The integers k1\nandk2(both≥0 and≤n) in Eq. (3) numerate, cor-\nrespondingly, groups of columns and rows in the matrix.\nAt anygivenvaluesof k1andk2columnsand rowsrepeat\nthemselves Ck1,2ntimes, where Ck\nn=n!/k!(n−k)! are the\nbinomial coefficients. Note also that the total powers of\ncreation operators and total powers of annihilation oper-\nators in all elements are the same: ( n−k2)+k2=nand\n(n−k1)+k1=n. But proportionsbetween powers of the\ncreation operators in the H- andV-modes change from\none line of the matrix to another and they are controlled\nby the integer k2. Similarly, proportions between pow-\ners of the annihilation operators in the H- andV-modes\nchange from one column of the matrix to another and\nthey are controlled by the integer k1.\nThe simplest examples arethe density matricesof pure\none-photon and two-photon polarization states\nk1= 0 1 k2\nρ(1)=/parenleftbigg\n/an}bracketle{ta†\nHaH/an}bracketri}ht /an}bracketle{ta†\nHaV/an}bracketri}ht\n/an}bracketle{ta†\nVaH/an}bracketri}ht /an}bracketle{ta†\nVaV/an}bracketri}ht/parenrightbigg\n0\n1(4)\nandρ(2)=1\n2×\nk1= 0 1 1 2 k2\n×\n/angbracketlefta†2\nHa2\nH/angbracketright /angbracketleft a†2\nHaHaV/angbracketright /angbracketleft a†2\nHaHaV/angbracketright /angbracketleft a†2\nHa2\nV/angbracketright\n/angbracketlefta†\nHa†\nVa2\nH/angbracketright /angbracketlefta†\nHa†\nVaHaV/angbracketright /angbracketlefta†\nHa†\nVaHaV/angbracketright /angbracketlefta†\nHa†\nVa2\nV/angbracketright\n/angbracketlefta†\nHa†\nVa2\nH/angbracketright /angbracketlefta†\nHa†\nVaHaV/angbracketright /angbracketlefta†\nHa†\nVaHaV/angbracketright /angbracketlefta†\nHa†\nVa2\nV/angbracketright\n/angbracketlefta†2\nVa2\nH/angbracketright /angbracketleft a†2\nVaHaV/angbracketright /angbracketleft a2\nVaHaV/angbracketright /angbracketleft a†2\nVa2\nV/angbracketright\n0\n1\n1\n2(5)\nand so on.\nAs mentioned above, the biphoton density matrix ρ(2)\n(5) was written down by Klyshko [5] and used in Refs.\n[5–7]. Note however that the next step used for working\nwith the density matrix (5) consisted in simple crossing\nout one of two coinciding rows and one of two coincid-\ning columns. This reduces the 4th-order matrix to the\n3-dimensional one, but it changes significantly features\nof the arising matrix. In particular, its trace becomes\ndifferent from one in contrast to the density matrix (5).\nAlso it does not provide a correct transition to the so\ncalled coherence matrix of biphoton qutrits [18]. Indeed,\nthe most general polarization biphoton state is qutrit,\nthe sate vector of which is\n|Ψ(2)/an}bracketri}ht=C1|2H/an}bracketri}ht+C2|1H,1V/an}bracketri}ht+C3|2V/an}bracketri}ht(6)\nwithCibeing arbitrary complex constants obeying the\nnormalization condition/summationtext\ni|Ci|2= 1.The natural co-\nherence matrix of this state is\nρ(2)\ncoh=/parenleftigg|C1|2C∗\n1C2C∗\n1C3\nC∗\n2C1|C2|2C∗\n2C3\nC∗\n3C1C∗\n3C2|C3|2/parenrightigg\n=\n\n/angbracketlefta†2\nHa2\nH/angbracketright\n2/angbracketlefta†2\nHaHaV/angbracketright√\n2/angbracketlefta†2\nHa2\nV/angbracketright\n2\n/angbracketlefta†\nHa†\nVa2\nH/angbracketright√\n2/angbracketlefta†\nHa†\nVaHaV/angbracketright/angbracketlefta†\nHa†\nVa2\nV/angbracketright√\n2\n/angbracketlefta†2\nVa2\nH/angbracketright\n2/angbracketlefta†2\nVaHaV/angbracketright√\n2/angbracketlefta†2\nVa2\nV/angbracketright\n2\n.(7)\nEvidently, the last expression (7) does not coincide with\nthat of Eq. (5) with, e.g., deleted the third column and\nthird row. So, the procedure of crossing out repeated\ncolumns and rows can not be considered as mathemati-\ncally correct. To make it correct, one has to make first\nthe unitary transformation of the matrix (5) [18], after\nwhich all elements in one of rows and one of columns\nin the 4 ×4 matrix turn zero. For the matrix (5) the\nrequired unitary transformation has the form\nρ(2)→/tildewideρ(2)=Uρ(2)U†\nwith\nU=\n1 0 0 0\n0 1/√\n2 1/√\n2 0\n0 1/√\n2−1/√\n2 0\n0 0 0 1\n.\nOnly after this transformation the arising single line\nand single column with zero elements can be safely re-\nmoved without changing general features of the origi-\nnal density matrix and providing the correct expression3\nfor the coherence matrix (7) [18]. In principle, similar\ntransformations can be found also for density matrices\nof higher-order states, n >2. But in the following dis-\ncussion we will not use such transformations by keeping\nall the full 2ndimensionality of the density matrices un-\nchanged, with repeating identical columns and rows of\nthe density matrix completely conserved. Actually this\nrepetitionofcolumnsandrowsisrelateddirectlywiththe\nsymmetry features of multi-boson wave functions. This\nsymmetry is not seen explicitly in the multiphoton state\nvectors of the type (1), but in the wave function of po-\nlarization variables they are present in the form of terms\ndiffering only by transposition of variables [17–19]. Such\nterms in the wave functions are responsible directly for\nappearance of repeated columns and rows in the density\nmatrices.\n3. REDUCED DENSITY MATRICES\nAs known the degree of entanglement of pure quan-\ntum states is related directly to the degree of mixing of\nreduced state. The concept of reduced states arises when\none represents a complicated pure states as if consisting\nof two parts. And then reduction is averaging over one\nof these two parts giving rise to possibly mixed state of\nthe other part. In the simplest cases of n= 2 andn= 3\ndefinitions of two parts are evident: these parts consist\nof two single-photon states in the case of biphotons, and\nthey consist of a single-photon and two-photon states in\nthe case of a pure three-photon original states. In the\ncases of states with large numbers of photons, n≥4,\nthere are more than one ways of imagining how the orig-\ninaln-photon state can be divided for two parts. E.g.\nforn= 4 there are are two ways of the gedanken split-\nting this state for two parts: 4 = 2 + 2 and 4 = 3 + 1\n[12]. Thus, in these cases one can speak about different\ndegrees of entanglement corresponding to different ways\nof splitting the original state for two parts.\nMathematically, a standard way of reducing den-\nsity matrices of pure states consists in using their\nwave-function representation ρ(n)({σi},{σ′\ni}) =\nΨ(n)({σi})Ψ(n)†({σ′\ni}), equalizing one or several\nvariablesσi=σ′\niand summing the product Ψ(n)Ψ(n)†\nover the variable(s) σm. But the procedure is rather\ncumbersome for states with many photons and with\nall symmetry requirements to the multi-boson wave\nfunctions completely taken into account. Fortunately,\nthe result of such calculations can be presented in a\nrelatively simple form with elements of the reduced\ndensity matrices expressed in terms of correlatorssimilar\nto those arising in the described above density matrices\nof pure states. By assuming that for an n-photon state\nwe reduce the density matrix ρ(n)with respect to n−m\nvariables, we can write the following general expressionfor the resulting reduced 2m-order density matrix\nρ(m;n)\nr=(n−m)!\nn!/parenleftig/braceleftig\n/an}bracketle{t(a†\nH)m−k2(a†\nV)k2am−k1\nHak1\nV/an}bracketri}ht/bracerightig/parenrightig\n(8)\nwith the previous definition of averaging in correlators\n/an}bracketle{t.../an}bracketri}ht=/an}bracketle{tΨ(n)|...|Ψ(n)/an}bracketri}htand with the previous meaning of\nthe integers k2andk1(m≥k1,2≥0) numerating groups\nof columns and rows, at given k1andk2repeatedCk1,2m\ntimes. Below are some examples of the reduced matrices.\nThe single-photon reduced density matrices of arbi-\ntrary puren-photon states |Ψ(n)/an}bracketri}htarising atm= 1 have\nthe form\nρ(1;n)\nr=1\nn/parenleftbigg\n/an}bracketle{ta†\nHaH/an}bracketri}ht /an}bracketle{ta†\nHaV/an}bracketri}ht\n/an}bracketle{ta†\nVaH/an}bracketri}ht /an}bracketle{ta†\nVaV/an}bracketri}ht/parenrightbigg\n. (9)\nFor basic Fock states |ΨnH,nV/an}bracketri}ht(1) these matrices are\nvery simple\nρ(1;n)\nr=1\nnH+nV/parenleftbigg\nnH0\n0nV/parenrightbigg\n, (10)\nand they correspond to the Schmidt entanglement pa-\nrameter\nK(nH, nV) =1\nTr[(ρ(1;n)\nr)2]=(nH+nV)2\nn2\nH+n2\nV.(11)\nIn the case of even total numbers of photon n=nH+nV,\nas a function of nH, the Schmidt parameter Kachieves\nmaximum at nH=nV=n/2 andKmax=2. At other\nrelations between of nHandnVthe Schmidt parameter\nKis smaller than Kmax. In the cases of odd numbers of\nphotonsnthe maximal values of the Schmidt parameter\nare achieved at nH= [n/2] andnH= [n/2]+ 1, where\nthe symbol [ x] denotes in this case the integer closest\nto but smaller than x. Maximal values of the Schmidt\nparameter in these cases are somewhat smaller than 2.\nThe simplest example of the basic Fock state with odd n\nis that of three-photon states |1H,2V/an}bracketri}htand|2H,1V/an}bracketri}ht. In\nboth cases Equation (11) gives K= 9/5 in agreement\nwith the results of the work [12]. The main conclusion\nfrom this brief analysisconcerns achievableentanglement\nofn-photon basic Fock states with respect to division\nfor subsystems of a single-photon and an ( n−1)-photon\nstates: entanglement of such states with respect to such\ndivision for subsystems does not exceed that occurring in\nthe case of biphoton states, and the maximal entangle-\nment with K= 2 or close to 2 is achieved in the states\nwith maximally close numbers of horizontally and verti-\ncally polarized photons, nHandnV.\nThe two-photon reduced density matrices of arbitrary\npuren-photon states Ψ(n)arise in the cases of m= 2 and\ntheir general form is given by\nρ(2;n)\nr=1\nn(n−1)×\n\n/angbracketlefta†2\nHa2\nH/angbracketright /angbracketleft a†2\nHaHaV/angbracketright /angbracketleft a†2\nHaHaV/angbracketright /angbracketleft a†2\nHa2\nV/angbracketright\n/angbracketlefta†\nHa†\nVa2\nH/angbracketright /angbracketlefta†\nHa†\nVaHaV/angbracketright /angbracketlefta†\nHa†\nVaHaV/angbracketright /angbracketlefta†\nHa†\nVa2\nV/angbracketright\n/angbracketlefta†\nHa†\nVa2\nH/angbracketright /angbracketlefta†\nHa†\nVaHaV/angbracketright /angbracketlefta†\nHa†\nVaHaV/angbracketright /angbracketlefta†\nHa†\nVa2\nV/angbracketright\n/angbracketlefta†2\nVa2\nH/angbracketright /angbracketleft a†2\nVaHaV/angbracketright /angbracketleft a†2\nVaHaV/angbracketright /angbracketleft a†2\nVa2\nV/angbracketright\n.(12)4\nFormally, this density matrix looks identical to that of\nEquation (5), though normalization factors in these two\nmatrices are different. But even more important differ-\nence concerns the meaning of averaging in correlators in\nthese matrices. If the case of the density matrix of a\npure two-photon states ρ(2)(5) averaging is defined as\n/an}bracketle{tΨ(2)|...|Ψ(2)/an}bracketri}ht. Incontrast,inthecaseofthesecond-order\nreduced density matrix (12) correlatorsin this matrixare\ndefined as /an}bracketle{tΨ(n)|...|Ψ(n)/an}bracketri}ht, wheren >2. Note also that\nall described matrices, both of pure states (3)-(5) and of\nmixed states (8)-(12), obey the same important feature:\ntheir traces are equal to one.\nFor evaluatingthe degreeof entanglementofmultipho-\nton states |Ψ(n)/an}bracketri}httheir reduced density matrices have to\nbe diagonalized numerically after which the found eigen-\nvaluesλ(m;n)\nican be used for finding the Schmidt entan-\nglement parameter or the entropy of the reduced density\nmatrices\nK=1/summationtext\niλ2\niandSr=−/summationdisplay\niλilog2λi.(13)\nBefore presenting specific results of calculations, it’s\nworth making a note concerning features ofthe described\nabove density matrices and differences between their fea-\ntures in the cases of basic Fock states (1) and their su-\nperpositions (2). In the case of single basic Fock states\ntheir pure-state and reduced density matrices have many\nzeros. In fact, averaging over basic Fock zeroes all cor-\nrelators containing products of creation and annihilation\noperators in one of two modes in different powers, e.g.,\nsuchas(a†\nH)paq\nHwithp/ne}ationslash=qandthesameforthevertical-\npolarizationmode. Owingto this, the densitymatricesof\nsingle Fock statesturn out havinga diagonal-blockstruc-\nture. The following Equation represents an example of\nsuch a diagonal-block second-order reduced density ma-\ntrixρ(2;4)\nr(12)forthestate |2H,2V/an}bracketri}htreducedwith respect\nto two variables ( m= 2):\nρ(2;4)\nr=\n1/6 0 0 0\n0 1/3 1/3 0\n0 1/3 1/3 0\n0 0 0 1 /6\n (14)\nIn this matrix three diagonal blocks are located (a) at\nthe crossing of the first line and first column, (b) at the\ncrossing of the 2nd and 3rd lines with the 2nd and 3rd\ncolumns and (c) at the crossing of the 4th line and 4th\ncolumn. Each block gives only one non-zero eigenvalue,\nandtheyareequalto, correspondingly,1/6,2/3,and1/6,\nwhich gives K= 2 in accordance with the result shown\nin Figure 1.\nInageneralcaseofthereduceddensitymatrices ρ(m;n)\nr\n(8) corresponding to the original states Ψ nH,nV(1) the\nnon-zero square blocks arise at crossings of the lines and\ncolumns with equal numbers of integers k1andk2,k1=\nk2≡kwith 0≤k≤m, and the dimensionality of each\nsuch blocks is Ck\nm. The number of bloks equals to m+1.\nAll elements inside each block are equal to each other.Owing to equality of elements inside a block, each block\nhas only one nonzero eigenvalue, and eigenvalues of the\nreduced density matrix can be expressed via these non-\nzero eigenvalues of blocks. Explicitly they are given by\nλk=(n−m)!\nn!Ck\nm/an}bracketle{tΨnH,nV|(a†\nHaH)m−k(a†\nVaV)k|ΨnH,nV/an}bracketri}ht\n=(n−m)!\nn!m!\nk!(m−k)!nH!\n(nH−m+k)!nV!\n(nV−k)!,(15)\nwith additional limitations\nk≤min{nV,m}and\nk≥max{m−nH,0} ≡max{nV−(n−m),0}.(16)\nNotice that at m=n=nH+nVthe reduced matrix\nρ(m;n)\nrturns into the density matrix of a pure state ρ(n).\nIn this case the limitations (16) take the form k≤nV\nandk≥nV, and they are compatible with each other\nonly atk=nV. This means that at a given value of\nnVthe density matrix ρ(n)has only one nonzero block\ncharacterized by k=nV. A simple algebra shows that\nin this case Equation (15) yields λk= 1 as it has to be\nfor a pure state.\nThe described features of the reduced density matrices\ncorresponding to the basic two-mode Fock states Ψ nH,nV\n(1) simplify significantly diagonalization of these matri-\nces and their Schmidt-mode analysis. The situation ap-\npears to be absolutely different in the case of superposi-\ntions of basic states Ψ(n)(2). In this case the diagonal-\nblock structure of matrices does not exist anymore and\nthe reduced density matrices have to be diagonalized\nwithout any helping simplifications.\n4. RESULTS\nThe results of calculations are presented in a series of\npictures of Figures 1-6. The first of these pictures (Fig-\nure 1) corresponds to multiphoton states |Ψ(n)/an}bracketri}htwith the\ntotal number of photons n, wherenis taken even, and\nwith equal numbers of photons with horizontaland verti-\ncal polarizations, nH=nV=n/2. The state is assumed\nto be imagined consisting of two parts with the same\nnumbers of photons in each, n/2. The reduced density\nmatrix of such subsystem is ρ(n\n2,n)(m=n−m=n/2\nin notations of Equation (8)). Its eigenvalues are λiand\nthe Schmidt entanglement parameter is determined by\nthe first expression in Equation (13). In Figure 1 the\nSchmidt parameter is shown in its dependence on the\ntotal number of photons in the state |Ψ(n)/an}bracketri}ht. As seen\nfrom the picture of Figure 1, in the considered case the\nSchmidt entanglement parameter and, hence, the degree\nof entanglement are monotononically growing function\nof the number of photons. In other words, multiphoton\nFock states can have much higher resource of entangle-\nment than usually considered biphoton states.5\nFigure 1: The calculated Schmidt entanglement parameter\nK(n) for states |Ψ(n)/angbracketright=|nH,nV/angbracketrightwith even n, equal numbers\nof horizontally and vertically polarized photons, nH=nV=\nn/2, and with the gedanken splitting of the states for two m-\nphoton states with equal numbers of photons, m= (n−m) =\nn/2; the dotted line corresponds to Kapprof Equation (17)\nNote that the curve in Figure 1 can be perfectly ap-\nproximated by the analytical expression\nKappr≈0.62+n0.54. (17)\nCoincidence ofthis model curvewith the numericallycal-\nculated one is so perfect that in the picture of Figure\n1 they look indistinguishable, except for a small region\nn <4. The main qualitative conclusion from this com-\nparison is that as a function of the total number of pho-\ntonsn, the Schmidt entanglement parameter K(n) grows\nroughly as the root square of n.\nSimilar conclusions can be deduced from calculations\nof the entropy of reduced state Srdefined by the second\nexpressioninEquations(13). Forthesamestateasinthe\nprevious calculations the function Sr(n) plotted in Fig-\nure 2 is seen to be monotonically growing and being very\nsimilar to the curve of Figure 1. This confirms the con-\nclusion about growing degree of entanglement with the\ngrowingnumberofphotonsandconfirmscompatibilityof\nthe entropy and Schmidt parameter for characterization\nn0Sr\n11.522.533.5\n10 50 40 20 30 \nFigure 2: The same as in Figure 1 but for the entropy of\nthe reduced state rather than for the Schmidt entanglement\nparameterof the degree of entanglement.\nThe picture of Figure 3 describes dependencies of the\nSchmidt entanglement parameter Kon the relation be-\ntweenhorizontallyandverticallypolarizedphotonsinthe\nFock states with given total numbers of photons n: if the\nnumberofverticallypolarizedphotonsis nV=k≤n,the\nnumber of horizontally polarized photons is nH=n−k,\nand the number kvaries along the horizontal axis in the\npicture of Figure 3. In this series of calculations the\ndegree of reduction is taken to be as high as possible,\nm= 1, i.e., the reduced state is a single-photon one and\nits reduced density matrix is ρ(1;n)\nrof Equation (9). As\nseen well from the pictures at all values of nthe Schmidt\nnumberKand the degree of entanglement are maximal\nwhen the numbers of vertically and horizontally polar-\nized photons in the state |Ψ(n)/an}bracketri}htare equal (k=n/2) or\nmaximally close to each other (in the case of odd n).\nThe picture of Figure 4 shows the dependence of the\nSchmidt entanglementparameterofthe Fockstate |Ψ(n)/an}bracketri}ht\nonm/n, i.e., on the ratio of number mof variables re-\nmaining in the state after its reduction to the total num-\nber of photons (or their variables) nin the original pure\nstate. The picture shows clearly that entanglement of\nthe state |Ψ(n)/an}bracketri}htis maximal when it is considered as split\nfor two parts with equal number of photons in each parts\n(m/n= 0.5).\nThe picture ofFigure 5 shows the dependence of eigen-\nvaluesλkon their numbers kfor the reduced density ma-\ntricesρ(m;n)\nrof the state with the total number of pho-\ntonsn= 120,nH=nVand differentdegreesofreduction\nn−m.\nThe results shown in Figure 5 show that in spite of a\ngrowing degree of entanglement in strongly multiphoton\nstates, eigenvalues of all reduced density matrices remain\nconcentrated in a restricted region of not too high val-\nues. This means that the effective dimensionality of the\ncorresponding Hilbert spaces remains not too high. This\nconclusion is important for approximatenumerical calcu-\nlations because it opens a possibility of performing these\nFigure 3: The Schmidt entanglement parameter Kas a func-\ntion of the number nVof vertically polarized photons in the\nstates|Ψ(n)/angbracketright; total numbers of photons nare shown near the\ncurves; the assumed division for subsystems is n→1+(n−1)6\nm/n0 0.2 0.4 0.6 0.8 1 K\n123423\n1\nFigure 4: The Schmidt entanglement parameter Kof the\nstates|ΨnH,nV/angbracketright(1) vs. the ratio “the number mof variables\nin the reduced density matrix divided by the total number of\npolarization variables or the total number of photons n”; the\ncurves correspond to n= 6(1),8(2)and24(3)\nk10 00.10.20.3\n246 8kλ\nFigure 5: Arranged in descending order, eigenvalues λkof the\nreduced density matrices ρ(m,n)\nr(8) of the state |ΨnH,nV/angbracketright(1)\nwithnH=nV= 60 and n=nH+nV= 120 and different\ndegrees of reduction: m= 50 (solid line), 30 (dashed line)\nand 10 (dash-dotted line)\ncalculations in smaller- dimensionality matrices forming\nthe main cores for finding relatively large eigenvalues λk.\nLet us consider now an example of states more compli-\ncated than a single basic Fock state. Let the state under\nconsideration be given by\n|Ψ/an}bracketri}ht=n/summationdisplay\nm=1Cm|(n−m)H,mV/an}bracketri}ht. (18)\nLet us take the coefficients Cmin the Gaussian form\nCm=Nexp/parenleftbigg\n−(m−m0)2\n2σ2/parenrightbigg\n. (19)\nwith the normalization factor Ngiven by\nN=/bracketleftiggn/summationdisplay\nm=0exp/parenleftbigg\n−(m−m0)2\nσ2/parenrightbigg/bracketrightigg−1/2\n(20)\nandm0is that value of mat which the squared co-\nefficients |Cm|2are maximal. As mentioned above inthis case diagonalization of the reduced density matrix\nis more complicated because this matrix does not have\nanymore a diagonal-block structure, and it has to be di-\nagonalized as a whole, without any simplifications. Nev-\nertheless, the results of such calculations are presented in\nFigure 6 for three different values of the parameter m0\nin the Gausssian distribution of Equation (19).\n0 2 4 6 8 10 K\n0.511.522.5\nσ\nFigure 6: The Schmidt entanglement parameter Kfor the\nstate (18) with n= 6 and m0= 3,2,1,0 (from top to down\nat small values of σ)\nOne of the most interesting features of the curves in\nFigure 6 concerns disappearance of entanglement ( K=\n1) at some definite point σ0. In principle, this does not\ncontradict, e.g., to the known features of the simplest su-\nperposition of Fock states - biphoton polarization qutrit\n(6) characterized by three constants C1,C2,C3. As\nknown [19], its degree of entanglement can be charac-\nterized either by the Schmidt entanglement parameter\nKor by the so called concurrence C=|2C1C3−C2\n2|\n[20], which are related to each other by a simple formula\nC=/radicalbig\n2(1−K−1). It’s known also that entanglement\nof qutrit disappears when C= 0 or 2C1C3=C2\n2. This\neffect of disappearing entanglement at some specific re-\nlation between the qutrit’s parameter seems to be anal-\nogous to the effect of missing entanglement of the state\n(18) atσ=σ0\n5. CONCLUSION\nThus, in this paper the density-matrix approach used\nearlier for biphoton states is generalized for the case of\nmultiphoton two-mode polarization states. Both pure\ntwo-modeFockstatesandtheirsuperpositionswithgiven\ntotal numbers of photons are considered. In this method\nelements of density matrices are expressed in terms of\nmean values of products of photon creation and annihi-\nlation operators.Structures of the arising density matri-\nces reduced with a part of polarization variables is dis-\ncussed. Eigenvalues λkof the reduced density matrices\nare found analytically for Fock states and numerically\nfor their superpositions.These results are used for find-\ningthedegreeofentanglementofmultiphotonstateswith\nrespect to their division for pairs of states with smaller7\nnumbers of photons. The degree of entanglement is es-\ntimated either by the Schmidt entanglement parameter\nK= 1//summationtext\nkλ2\nkor by the entropy of the reduced states\nS=−/summationtext\nkλklog2λk. The main qualitative conclusion\nis that the degree of entanglement is maximal if num-\nbers of photon in two modes, nHandnV, are maximally\nclose to each other and if multiphoton states are consid-\nered as consisting to two parts with approximately (or\nexactly) equal numbers of photons in each of two parts.\nThe maximal degree of entanglement is found to be agrowing function of the number of photons as shown in\nFigures 1 and 2.\nAcknowledgement\nThe work is supported by the Russian Foundation for\nBasic Research, grant 18-02-00634.\n[1] P.B. Shor. PRA, 52:R2493(R), 1995.\n[2] R. Raussendorf and H. J. Briegel. PRL, 86:5188, 2001.\n[3] H. J. Kimble. Nature, 453:1023, 2008.\n[4] X-L. Wang et al. PRL, 117:210502, 2016.\n[5] D.N. Klyshko. JETP, 84:1065, 1997.\n[6] A.V. Burlakov and M.V. Chekhova. JETP Letters ,\n75:432, 2002.\n[7] L.A. Krivitskii, S.P. Kulik, G.A. Maslennikov, and M.V.\nChekhova. Quantum Electronics , 35:69, 2005.\n[8] Jun-Yi Wu and H.F. Hofmann. New Journal of Physics ,\n19:103032, 2017.\n[9] B.J. Dalton, J. Goold, B.M. Gaqrraway, and M.D. Reid.\nPhysica Scripta , 92:023005, 2017.\n[10] K. Wang, S.V. Sucjkov, J.G. Titchener, A. Szameit, and\nA.A. Sukhorukov. ArXiv, quant-ph:1808.05038v1, 2018.\n[11] Y. Lu and Q. Zhao. ArXiv, quant-ph:1403.0673v2, 2005.[12] M.V. Fedorov and N.I. Miklin. Laser Physics , 25:035204,\n2015.\n[13] E. Schmidt. Math. Ann. , 63:433, 1906.\n[14] R Grobe, K Rzazewski, and J H Eberly. J. Phys. B: At.\nMol. OpL Phys. , 27:L503, 1994.\n[15] C.K. Law and J.H. Eberly. Phys. Rev. Lett. , 92:127903.\n[16] M.V. Fedorov and N.I. Miklin. Contemporary Physics ,\n55:94, 2014.\n[17] S.S. Schweber. An Introcuction to Relativistic Quantum\nField Theory . Harper and Row, New York, USA, 1961.\n[18] M.V. Fedorov, P.A. Volkov, and Yu.M. Mikhailova.\nJETP, 35:115, 2012.\n[19] M V Fedorov, P A Volkov, J M Mikhailova, S S Straupe,\nand S P Kulik. New Joural of Physics , 13:083004, 2011.\n[20] R.W. Wootters. Phys. Rev. Lett. , 80:2245, 1998."    },    {        "title": "1812.11572v1.Van_der_Waals_equation_of_state_and_PVT_properties_of_real_fluid.pdf",        "content": "Van der Waals equation of state and PVT – properties of real fluid  \n \nUmirzakov I. H.  \nInstitute of Thermophysics, Pr. Lavren teva St., 1, Novosibirsk, Russia, 630090  \ne-mail: umirzakov@itp.nsc.ru  \n \nAbstract  \n   It is shown that: in the case when two parameters of the Van der Waals equation of state are \ndefined from the critical temperature and pressure  the exact parametrical solution of the \nequations of the liquid -vapor phase equilibrium of the Van der Waals fluid  quantitatively \ndescribes the experimental dependencies of the saturated pressure of argon on the temperature \nand reduced vapor density ; it can describe qualitatively the temperature dependencies of the \nreduced vapor and liquid densities of argon ;  and it gives the quantitative descri ption of the \ntemperature dependencies of the reduced densities near critical point.  When the parameters are \ndefined from the critical pressure and density  the parametric solution describes quantitatively the \nexperimental dependencies of the saturated pressure of argon on the density and reduced \ntemperature , it can describe qualitatively the dependencies of the vapor and liquid densities on \nthe reduced temperature , and it gives the quantitative description of the dependencies of the \ndensities on the reduced temperature  near critical point.  If the parameters are defined from the \ncritical temperature and density  then the exact solution describe s quantitatively the experimental \ndependencies of the reduced saturated pressure of argon on the density and temperature, it \ndescribes qualitatively the temperature dependenc ies of the vapor and liquid densities of argon , \nand it gives the quantitative description of the temperature dependenc ies of the vapor and liquid \ndensities  near critical point.   \n    It is also shown that the Van der Waals equation of state describes quantitatively the reference \nexperimental PVT - data for the gas and supercritical fluid states for the under -critical densities of \nargon, the dependencies of the saturation pressure on the temperature and vapor density , and the \ndependence of the vapor density of argon on temperature if the parameters are defined from the \ncritical pressure and temperature.  \n \nKeywords  Argon, L iquid -Vapor Phase Equilibrium, F luid, Critical Point , Coexistence  \n \nIntroduction  \n \n       As known the similarity laws allow one to set a correspondence between different \nthermodynamic values without explicit use of the equation of state (EOS) [1].  Some of \nconsequences of the van der Waals equation (Vd W-EOS) are valid for a great number of various \nmodels and real systems described by completely different equations of state [1]. The well -\nknown examples of such similarities are the principle of corresponding states and law of \nrectilinear diameter [2]. Accor ding to V dW-EOS an ideal line for  pressure on temperature -\ndensity plane along which the compressibility factor is equal to unity is straight linear [3].  There \nis considerable experimental evidence, confirming the linearity of this line for many other \nsubstances and models [1,  2, 4-7]. This line is also named as “Zeno line” (obtained from “Z = \none”) [2]. This regularity appears to have much wider area of applicability than the original \nVDW equation [1]. Besides, the Zeno -line is used to check both the refe rence [8] and \nsemiempirical [9] equations of states.      VdW-EOS is also used to generate new similarity relations [1,  10, 11]. Predictions of VdW-\nEOS for lines along which an enthalpy, free energy and chemical potential of van der Waals \nfluid coincides w ith their values for the ideal gas describe many substances and model systems \n[1, 6, 12, 13]. For more details see [1] and references here in.  \n    There are two alternative theories of the critical region of the liquid -vapor first -order phase \ntransitions. The first theory is the traditional theory of the region near single critical point based \nupon Ising -like scaling theory with crossover to classical equations of state [1 4-20]. VdW -EOS \n[18] and the fundamental equations of state [2 0], which are based upon the concept of a single \ngas-liquid critical point, are representations of these classical equations of state.  \n   The second theory is the “meso -phase” hypothesis of Woodcock [26]. According to the “meso -\nphase” hypothesis : at criti cal and supercritical temperatures on the thermodynamic (density, \npressure) -plane exists  a region where the pure substance is in  the “meso -phase”, which consists \nof small clusters that are gas like and clusters of macroscopic size that are liquid like ; there is \nexist a line of critical points over a finite range of densities at critical temperature and pressure \ninstead of single critical point ; and the pressure in the “meso -phase”  is linear function of \ndensity. This hypothesis is reminiscent of an old co ncept of the supercritical fluid as a mixture of \n“gasons” and “liquidons” that has turned out to be inconsistent with the experimental evidence \n[16,17]. \n    Some predictions of the “meso -phase” hypothesis were criticized by Sengers and Anisimov \n[16] and Um irzakov [27]. According to [ 16] in contrast to the conjecture of Woodcock, there is \nno reliable experimental evidence to doubt the existence of a single critical point in the \nthermodynamic limit and of the validity of the scaling theory for critical thermo dynamic \nbehavior.  \n     According to [26] the Van der Waals critical point does not comply with the Gibbs phase rule \nand its existence is based upon a hypothesis rather than a thermodynamic definition. The paper \n[27] mathematically demonstrates that a crit ical point is not only based on a hypothesis that is \nused to define values of two parameters of VdW -EOS. Instead, the author of [27] argue d that a \ncritical point is a direct consequence of the thermodynamic phase equilibrium conditions \nresulting in a singl e critical point. It was shown that the thermodynamic conditions result in the \nfirst and second partial derivatives of pressure with respect to volume at constant temperature at \na critical point equal to zero which are usual conditions of an existence of a  critical point  [27]. \n    The papers [28] and [29] were the responses to the critique of some predictions of the \nhypothesis in [4] and [27], respectively.  \n    The paper [28] was criticized in [30]. It was shown [30] that: (1) the expressions for the \nisoch oric  and isobaric (\nPC ) heat capacities of liquid and gas, coexisting in phase equilibrium, \nthe heat capacities at saturation of liquid and gas (\nC ) and the heat capacity (\nC ) used in \nWoodcock’s article [28] are incorrect; (2) the conclusions of the article based on the comparison \nof the incorrect \nVC , \nPC , \nC and \nC  with experimental data are also incorrect; (3) the lever rule \nused in [28] cannot be used to define \nVC  and \nPC  in the two -phase coexistence region; and (4) \nthe correct expression for the isochoric heat capacity describes the experimental data well .  \n   The Van der Waals equation of state was criticized in [29].     According to [29] “state functions \nof van der Waals’ equation fail to describe the thermodynamic properties of gases and gas–\nliquid coexistence”, “the van der Waals equation fails to describe even qualitatively the \nthermodynamic properties of gas –liquid coexistence in the critical region, and “the liquid –gas \ncritical point is not a property that the van der Waals equation ca n make any statements about”.  \n    We show in this paper that  above statements of [29] are incorrect.  We show that in the case \nwhen two parameters of Van der Waals equation of state are defined from the critical \ntemperature and pressure  the exact parametri cal solution of the equations of the liquid -vapor \nphase equilibrium of Van der Waals fluid  (VdW -fluid) , which is described by VdW -EOS, \nquantitatively describes the experimental dependencies of the saturated pressure of argon on the temperature and reduced vapor density ; it can describe qualitatively the temperature \ndependencies of the reduced vapor and liquid densities of argon ;  and it gives the quantitative \ndescri ption of the temperature dependencies of the reduced densities near critical point.  W hen \nthe parameters are defined from the critical pressure and density , the parametric solution \ndescribes quantitatively the experimental dependencies of the saturat ion pressure of argon on the \ndensity and reduced temperature , it can describe qualitatively the dependencies of the vapor and \nliquid densities on the reduced temperature , and it gives the quantitative description of the \ndependencies of the densities on the reduced temperature  near critical point. If the parameters are \ndefined from the critical temper ature and density  then the exact solution describe s quantitatively \nthe experimental dependencies of the reduced saturated pressure of argon on density and \ntemperature, it describes qualitatively the temperature dependenc ies of the vapor and liquid \ndensitie s of argon , and it gives the quantitative description of the temperature dependenc ies of \nthe vapor and liquid densities  near critical point.   It is also shown that the Van der Waals \nequation of state describes quantitatively the experimental dependencies of the saturation \npressure on the temperature and vapor density  of argon [22] , and the experimental dependence of \nthe vapor density of argon on temperature if the parameters are defined from the cri tical pressure \nand temperature, and VDW -EOS describes quan titatively the reference experimental PVT - data \n[21] for gas state and supercritical fluid states for under -critical densities of argon.  \n \nComparison of p arametric solution of the coexistence equations of Van der Waals fluid \nwith  liquid -vapor equilibrium of real fluid  \n \n     The Van der Waals’ equation of state (VdW -EOS ) [18] is the relation  \n2) 1/( ),( an bn nkTTnp \n,                                                                                                        (1) \nwhere \np is the pressure,  \nT is the temperature, \nn  is the number density  of the  particle s  (atom s \nor molecule s), \nk is the Boltzmann constant, \na  and \nb  are the positive constants.  The values of \nthe number density, pressure and temperature at the critical point of liquid -vapor first -order \nphase transition  are equal to \nb nc 3/1 ,  \n227/b apc  and \nkb a Tc 27/8 , respectively [ 18, 23-\n25]. The mass density \n  is equal to  \nn , where \n  is the mass of the particle.  \n    It is easy to see that VdW -EOS defines the exact position of the critical point on the \nthermodynamic (\np ,\nT)- ,  (\nn ,\np)- and  (\nT ,\nn)- planes and, hence, it describes the density -\ntemperature -pressure relation and phase equilibrium line near the critical point (see Figs. 1 -3) if \nthe coefficients \na  and \nb  of VdW -EOS are defined from  (\ncp,\ncT),   (\ncp,\ncn) and (\ncT ,\ncn) using \nthe relations  \nc c p Tk a 64/ 2722\n,     \nc cp kTb 8/ ,                                                                                             (2) \n2/3c cnp a\n,              \ncn b 3/1 ,                                                                                                   (3) \nc cn kTa 8/ 9\n,           \ncn b 3/1  ,                                                                                                  (4) \n respectively. Here \ncp , \ncT and \ncn  are the values of the pressure, temperature and density of the \nreal fluid at the critical point.       According to the parametric solution [ 23-25, 27] of the equations corresponding to the liquid -\nvapor phase equilibrium of VdW -fluid the saturation pressure \n)(Tpe  and the densities of liquid \n)(TnL\n and vapor \n)(TnV  of VdW -fluid are defined  \n)(2 22 4 4 2 22 4 2 2 2\n)1 4 2 2 2)(1 ()1 4)(1 (2))(( /\nTyyy y y y yy y y y\ney ye e ey ee ye ye eyTyFa bkT\n  \n,                                  (5) \n)(2 22 2\n2 2 )1 2(12))(( )(\nTyyy yy y\nL Ly e e yye eyTyF Tbn\n \n,                                                            (6) \n)(2 42 2\n1 2)2 2(12))(( )(\nTyyy yy y\nV Vy y e eye eyTyF Tbn\n \n,                                                              (7) \n \n)(2 2)( )]( 1/[)()( ))(( /)(\nTyyL L L P e yFyF yFyF TyFabTp\n  \n.                                                (8)  \n    The temperature dependencies of the saturation pressure \n)(Tpe  and the densities of liquid \n)(TnL\n and vapor \n)(TnV  of VdW -fluid are defined from Eqs. 6-8 using the temperature \ndependence of the parameter \n)(Ty   defined from Eq. 5.      \n     We have from Eqs. 5 -8 using Eq. 2  \n))((8/ 27 TyF T Tc\n,                                                                                                                   (9) \n))(( 8 /)( TyFz nTnLc c L\n,                                                                                                            (10) \n))(( 8 /)( TyFz nTnVc c V\n,                                                                                                           (11) \n))(( 27)( TyFp TpPc e\n.                                                                                                              (12)  \nwhere \nc c c c kTnpz /   is the value of the compressibility factor of the real fluid at the critical \npoint. The dependencies \n)(Tpe , \n)(TnL  and \n)(TnV  are defined from Eqs. 10-12 using the \ntemperature dependence of the parameter \n)(Ty   defined from Eq. 9.      \n    Fig. 1 demonstrates that Eqs. 9-12 - the exact parametrical solution of the equations of the \nliquid -vapor phase equilibrium of  VdW -fluid - quantitatively describe the experimental \ndependencies [22] of the saturated pressure of argon on the temperature and reduced (to critical) \nvapor densi ty near critical point, and they can describe qualitatively the reduced vapor and liquid \ndensities of argon on the temperature in the case when the parameters \na   and \nb   are defined \nfrom \n) ,(c cpT  using Eq. 2  and \nK Tc  567.150 , \n1 039948.0  molkg  , \n3 6.535  mkgc   \nand \nMPa pc  863.4  for argon [21,22] . \n     VdW -EOS predicts in this case  for the critical mass density \n3  418mkg  which is \nconsiderabl y less than \n3  6.535  mkgc  of argon [22] . Therefore VdW -EOS describes the \nsaturation pressure as the function of the reduced vapor density  (Fig. 1b) and the reduced liquid \nand vapor densities as the function of temperature  (Fig. 1c) .   \n    We have from Eqs. 5-8 using Eq. 3  \n))(( 9/1 / TyFz TTc c\n,                                                                                                             (13) \n))(( 3)( TyFn TnLc L\n,                                                                                                                 (14) \n))(( 3)( TyFn TnVc V\n,                                                                                                                 (15) ))(( 27)( TyFp TpPc e.                                                                                                              (16)  \nThe dependencies \n)(Tpe , \n)(TnL  and \n)(TnV  are defined from Eqs. 14-16 using the temperature \ndependence of the parameter \n)(Ty   defined from Eq. 13.      \n \n \n \n  \nFig. 1. Fig. 1a - temperature dependence of saturation \npressure \nep  of argon: blue circles correspond to \nexperimental data [22] and solid red line – to Eqs. 9 \nand 12. Fig. 1b – dependence of saturation pressure \nep\n of argon on reduced vapor density \nc Vnn/ : blue \ncircles correspond to [22] and solid red line – to Eqs. \n9, 11 and 12. Fig. 1c -  temperature dependencies of \nreduced coexistence liquid \nc Lnn/  and vapor \nc Vnn/\n densities of argon: blue open circles  \ncorrespond to [22] and solid red circles - to Eqs.  9-11.  \n    Fig. 2  shows that  Eqs. 13-16 - the exact parametrical solution of the equations of the liquid -\nvapor phase equilibrium of  VdW -fluid describes quantitatively the experimental dependencies  \n[22] of the saturated pressure of argon on the density and reduced (to critical) temperature nea r \ncritical point, and it can describe qualitatively  the vapor and liquid densities of argon on the \nreduced temperature when the parameters \na   and \nb   are defined from \n) ,(c cpn  using Eq. 3 . \nVdW -EOS predicts in this case for the critical temperature \nK 350  which is considerabl y greater \nthan \nK Tc  687.150  of argon [22] . Therefore VdW -EOS describes the saturation pressure  (Fig. \n2a) and the  liquid   and vapor densities (Fig. 2c) as the functions of the reduced temperature.   \n    We have from Eqs. 5 -8 using Eq. 4  \n))((8/ 27 TyF T Tc\n,                                                                                                                 (17) \n))(( 3)( TyFn TnLc L,                                                                                                                 (18) \n))(( 3)( TyFn TnVc V\n,                                                                                                                 (19) \n))(( 8/81 /)( TyFz pTpP c c e \n.                                                                                                  (20)  \n \n \n  \n \n  \nFig. 2.  Fig. 2a - dependence of saturation pressure \nep\n of argon on reduced temperature \ncTT/ : blue \ncircles correspond to experimental data [22] and solid \nred line – to Eqs. 13 and 16. Fig. 2 b – dependence of \nsaturation pressure of argon on vapor density \nVn :  \nblue circles correspond to [22] and solid red line – to \nEqs. 13, 15 and 16. Fig. 2 c -  dependencies of \ncoexistence liquid \nLn  and vapor \nVn  densities of \nargon  on reduced temperature \ncTT/ : blue open \ncircles  correspond to [22] and solid red circles - to \nEqs. 13-15.  \n \nThe dependencies \n)(Tpe , \n)(TnL  and \n)(TnV  are defined from Eqs. 18-20 using the temperature \ndependence of the parameter \n)(Ty   defined from Eq. 17.  \n   Fig. 3  shows that Eqs. 17 -20 describe quantitatively the experimental dependencies [22] of the \nreduced (to critical) saturated pressure of argon on density and  temperature near critical point, \nand it describes qualitatively  the vapor and liquid densities of argon on temperature when  the \nparameters \na   and \nb  are defined from \n) ,(c cTn  using Eq. 4 . VdW -EOS predicts in this case for \nthe critical pressure \nMPa 27.6  which is considerable greater than \nMPa pc  863.4  of argon [22]. \nTherefore VdW -EOS describes the reduced saturation pressure as the functions of the \ntemperature (Fig. 3a) and vapor pressure (Fig. 3b).   \n    One can see from Eqs. 5 -6 [27] and Fig. 1 [27] that the saturated vapor density of the VdW -\nfluid is non -negative for any value of the temperature. Figs. 1c, 2c and 3c show the same.  So, in   \n \n \n \n Fig. 3. Fig. 3a - temperature dependence of reduced \nsaturation pressure \nc epp/  of argon: blue circles \ncorrespond to experimental data [22] and solid red \nline – to Eqs. 17 and 20. Fig. 3b –  dependence of  \nreduced saturation pressure  \nc epp/  of argon on  \nvapor density \nVn :  blue circles correspond to [22] \nand solid red line  – to Eqs. 17, 19 and 20. Fig. 3 c -  \ntemperature dependencies of coexistence liquid \nLn  \nand vapor \nVn  densities of argon: blue open circles  \ncorrespond to [22] and solid red circles - to Eqs.    17-\n19. \n \ncontrast to the Capture of Fig. 4 [29], VdW -EOS is not absurd for temperatures below 110 K. \n \nVan der Waals equation of state and  liquid -vapor equilibrium of real fluid  \n \n    We obtain from Eq. 1 using Eqs. 2 -4 \ncc\nc cc\npnTk\nnkTpnkTpTnp6427\n88),(222\n\n,                                                                                             (21) \n223\n/ 33),(\ncc\nc nnp\nnnnkTTnp \n,                                                                                                       (22) \ncc\nc nnkT\nnnnkTTnp89\n/ 33),(2\n\n.                                                                                                     (23) \n \n    It is easy to see from Eqs. 21 -23 that one can define the temperature as the functio n of the \npressure and density  by  use of the following equations  \n \n \n \n \n \n \nFig. 4. Blue circles correspond to experimental data [22] . Figs. 4a, 4b - temperature dependence of saturation \npressure \nep  of argon: solid red line – to Eq. 21, dotted  brown  line – to Eq. 22 and dashed black line to Eq. 23. \nFigs.4c, 4d  –  dependence of saturation pressure  \nep  of argon on vapor  mass density  \nV :   blue circles correspond \nto [22] , solid red line – to Eq. 21, dotted  brown  line – to Eq. 22 and dashed black line to Eq. 23. Fig. 4e, 4f  -   \ndependenc e of temperature on saturated vapor  mass density  of argon\nV  : solid red line – to Eq. 24, dotted  brown  \nline – to Eq. 25 and dashed black line to Eq. 26.  \n \n\n\n\n\n\n\n\ncc\ncc\npnkT\npnTkpnkpnT816427 1),(222\n,                                                                                (24) \n\n\n\n\n\n\n\nc ccnn\nnnppnkpnT 33\n31),(22\n,                                                                                       (25) \n\n\n\n\n\n\n\nc cc\nnn\nnnkTpnkpnT 389\n31),(2\n.                                                                                       (26) \n    The dependence of the saturated vapor pressure of argon  on the temperature is shown  on Figs. \n4a and 4 b. As one can see Eq. 2 1 describes quantitatively the experimental data [22] in the \ninterval from the triple point temperature to the critical one, and Eqs. 2 2-23 describe \nquantitatively the data near triple point temperature  only. Fig. 4 b shows that the Eq. 2 1 gives the \nbest description of the data [22] on the saturated vapor pressure  near triple point temperature.  \n    The dependence of the saturated vapor pressure of argon  on the mass density of the saturated \nvapor is presented on Figs. 4 c and 4 d. As one can see Eq. 2 1 describes quantitatively the \nexperimental data [22] in the interval from the triple point temperature to the critical one, and \nEqs. 2 2-23 describe quantitati vely the data near triple point temperature. Fig. 4 d demons trates  \nthat Eq. 2 1 gives the best description of the data [22] near triple point.  \n   The dependence of the temperature on the saturated vapor mass density of argon  \nV Vn  is \npresented on Figs. 4e and 4f. As evident Eq. 24 describes quantitatively the experimental data \n[22] in the interval from the triple point temperature to the critical one, and Eqs. 25 -26 describe \nquantitatively the data near triple point temperature.  Fig. 4f shows that the Eq. 24 gives the best \ndescription of the data [22] near triple point temperature.  \n \nVan der Waals equation of state and  PVT -properties  of real fluid  \n \n    The comparison of the Van der Waals equation of state (Eq. 21) with the reference \nexperimental PVT –data [21] is shown on Figs. 5. The data for the vapor density on the \ncoexistence line [22] are also included to the comparison. The values of temperature s \ncorresponding to isotherms of [21 , 22] (column 1), the phase state of argon (column 2,  where \n“scf” denotes the supercritical state ), the density region of the experimental data [21] described \nby VdW -EOS (column 3), number of experimental points of  [21, 22] described by VdW -EOS \n(column 4), the full number of experimental points at the isotherm (column 5), and the value of \nthe quadratic root of the mean of the sum of the squares of the displacements (square root of the \nstandard deviation)  \n%1006427\n88\n112/1\n1222\n\n\n\n\n\n\n\n  \nN\nii\ncic\nci ciicppnTk\nkTnpkTnp\nN\n \nof the predictions of VdW -EOS from the data [21,22] (column 6) for the isotherm are presented \nin Table 1 . The abbreviation “oe” (“ue”) in the column 7 of Table 1 denotes that VdW -EOS \noverestimat es (underestimates) the data on the isotherm.  \n    Table 1 and Fig. 5 show that Eq. 21 describes nine experimental isotherms of the gas state of \nargon and its eighteen  supercritical experimental isotherms [21 , 22] for the under -critical \ndensities with the good accuracy. As one can see from the last column of Table 1  VDW -EOS \n(Eq. 21) overestimates all nine isotherms  for the temperatures that are less er or equal to the \ncritical one , and  it underestimates ten isotherms from the temperature interval \nK TK  340  190\n. One can conclude using the data from the last row of Table 1 that Eq. 21 \ndescribes with a good accuracy 78% of the experimental data [21 , 22] for the gas and \nsupercritical states of argon for the densities less than the critical one.  The full number of \nexperimental PVT -data is equal to 638 [21] therefore Eq. 21 describes quantitatively 66% of \nthem. The columns 4 and 5 show that Eq. 21 describes quantitatively all experimental PVT -data \nof gas at 8 isotherms from the temperature interva l \nK TK  148  110  and all data at 10 \nsupercritical isotherms from the temperature interval \nK TK  340  190 . \n  \n \n \n \n \n \n \nFig. 5. Isotherms of argon: red symbols correspond to experimental data [2 1] and lines – to VDW -EOS Eq. 21. a) \nBlack solid line and squares correspond to \nK 110 , blue dotted -dashed line and diamonds – to \nK 120 , blue dashed \nline and circles – to \nK 130 , and blue solid line and triangles – to \nK 135 . b) Black solid line and squares \ncorrespond to \nK 140 , blue dotted -dashed line and diamonds – to \nK 143 , blue dashed line and circles – to \nK 146 , \nand blue solid line and triangles – to \nK 148 . c) Black solid line and squares correspond to \nK 7.150 , blue dotted -\ndashed line and diamonds – to \nK 153 , blue dashed line and circles – to \nK 155 , and blue solid line and triangles – \nto \nK 157 . d) Black solid line and squares correspond to \nK 160 , blue dotted -dashed line and diamonds – to \nK 165\n, blue dashed line and circles – to \nK 170 , and blue solid line and triangles – to \nK 175 . e) Black solid line \nand squares correspond to \nK 180 , blue dotted -dashed line and diamonds – to \nK 190 , blue dashed line and circles \n– to \nK 200 , and blue solid line and triangles – to \nK 220 . f) Black solid line and squares correspond to \nK 250 , \nblue dotted -dashed line and diamonds – to \nK 265 , blue dashed line and circles – to \nK 280 , and blue solid line \nand triangles – to \nK 295 . g) Black solid line and squares correspond to \nK 310 , blue dotted -dashed line and \ndiamonds – to \nK 325 , and blue dashed line and circles – to \nK 340 . \n \nTable 1. Description of experimental isotherms of argon [2 1] presented on Fig. 5.  \n1 2 3 4 5 6 7 \nKT  ,\n phase state of \nargon  density \nregion  number of described \npoints   full number of \nexp.points  \n%  ,  oe - overestimates  \nue - underestimates  \n110 gas \n)(TV 6 6 3.2 oe \n120 gas \n)(TV 6 6 3.1 oe \n130 gas \n)(TV 5 5 4.7 oe \n135 gas \n)(TV 7 7 4.0 oe \n140 gas \n)(TV 9 9 4.3 oe \n143 gas \n)(TV 9 9 3.7 oe \n146 gas \n)(TV 10 10 3.7 oe \n148 gas \n)(TV 13 13 2.8 oe \n150.7  gas \nс 16 33 2.2 oe \n153 scf \nC 17 41 1.9   \n155 scf \nC 17 34 1.3   \n157 scf \nC 16 31 1.3   \n160 scf \nC 19 33 1.2   \n165 scf \nC 19 31 1.2   \n170 scf \nC 19 28 1.3   \n175 scf \nC 22 28 1.6   \n180 scf \nC 23 27 1.9   \n190 scf \nC 27 27 2.3 ue \n200 scf \nC 17 17 2.7 ue \n220 scf \nC 17 17 2.8 ue \n250 scf \nC 18 18 2.7 ue \n265 scf \nC 17 17 2.5 ue \n280 scf \nC 20 20 2.5 ue \n295 scf \nC 17 17 2.3 ue \n310 scf \nC 17 17 2.2 ue \n325 scf \nC 20 20 1.9 ue \n340 scf \nC 17 17 2.0 ue \n       \n420  \n538      \n  \n \n Conclusion\n We show ed that:  \n1) in the case when two parameters of Van der Waals equation of state are defined from the \ncritical temperature and pressure the exact parametrical solution of the equations of the \nliquid -vapor phase equilibrium of Van der Waals fluid quantitatively describes  the \nexperimental dependencies of the saturated pressure of argon on the temperature and reduced \nvapor density; it can describe qualitatively the temperature dependencies of the reduced \nvapor and liquid densities of argon;  and it gives the quantitative de scription of the \ntemperature dependencies of the reduced densities near critical point;  \n2) when the parameters are defined from the critical pressure and density , the parametric \nsolution describes quantitatively the experimental dependencies of the saturated pressure of \nargon on the density and reduced temperature, it can describe qualitatively the dependencies \nof the vapor and liquid densities on the reduced temp erature, and it gives the quantitative \ndescription of the dependencies of the densities on the reduced temperature near critical \npoint ; \n3) if the parameters are defined from the critical temperature and density  then the exact solution \ndescribes quantitatively  the experimental dependencies of the reduced saturated pressure of \nargon on the density and temperature, it describes qualitatively the temperature dependenc ies \nof the vapor and liquid densities of argon, and it gives the quantitative description of the \ntemperature dependenc ies of the vapor and liquid densities near critical point ; \n4) the Van der Waals equation of state describes quantitatively the experimental dependencies \nof the saturation pressure on the temperature and vapor density  [22], and the experime ntal \ndependence of the vapor density of argon on the temperature if the parameters are defined \nfrom the cr itical pressure and temperature;  \n5) the Van der Waals equation of state describes quantitatively the reference experimental  \nPVT - data [21] for gas and supercritical fluid states for the  under -critical densities of argon  \nif the parameters are defined from the cr itical pressure and temperature . \nReferences  \n1. 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Woodcock, Failures of van der Waals’ Equation at the Gas –Liquid Critical Point, Int. J. \nThermophys.  2018 , 39, 120 . https://doi.org/10.1007/s10765 -018-2444 -6 \n30. I. H. Umirzakov,  On the Interpretation of Near -Critical Gas –Liquid Heat Capacities by L. V. \nWoodcock, Int. J. Thermophys. (2017) 38, 139: Comments, Int. J. Thermophys . 2018 , 39, 139 .  \nhttps://doi.org/10.1007/s10765 -018-2414 -z. "    },    {        "title": "1901.01739v1.High_Density_Reflection_Spectroscopy_I__A_case_study_of_GX_339_4.pdf",        "content": "MNRAS 000, 1{12 (2018) Preprint 8 January 2019 Compiled using MNRAS L ATEX style \fle v3.0\nHigh Density Re\rection Spectroscopy I. A case study of\nGX 339-4\nJiachen Jiang,1?Andrew C. Fabian,1Jingyi Wang,2Dominic J. Walton,1\nJavier A. Garc\u0013 \u0010a,3;4Michael L. Parker5, James F. Steiner,2and John A. Tomsick6\n1Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA\n2MIT Kavli Institute for Astrophysics and Space Research, MIT, 70 Vassar Street, Cambridge, MA 02139, USA\n3Cahill Center for Astronomy and Astrophysics, California Institute of Technology, Pasadena, CA 91125, USA\n4Dr. Karl Remeis-Observatory and Erlangen Centre for Astroparticle Physics, Sternwartstr. 7, D-96049 Bamberg, Germany\n5European Space Agency (ESA), European Space Astronomy Centre (ESAC), E-28691 Villanueva de la Ca~ nada, Spain\n6Space Sciences Laboratory, 7 Gauss Way, University of California, Berkeley, CA 94720-7450, USA\nAccepted XXX. Received YYY; in original form ZZZ\nABSTRACT\nWe present a broad band spectral analysis of the black hole binary GX 339-4 with\nNuSTAR andSwift using high density re\rection model. The observations were taken\nwhen the source was in low \rux hard states (LF) during the outbursts in 2013 and 2015,\nand in a very high \rux soft state (HF) in 2015. The high density re\rection model can\nexplain its LF spectra with no requirement for an additional low temperature thermal\ncomponent. This model enables us to constrain the density in the disc surface of\nGX 339-4 in di\u000berent \rux states. The disc density in the LF state is log¹necm\u00003º\u001921,\n100 times higher than the density in the HF state ( log¹necm\u00003º=18:93+0:12\n\u00000:16). A\nclose-to-solar iron abundance is obtained by modelling the LF and HF broad band\nspectra with variable density re\rection model ( ZFe=1:50+0:12\n\u00000:04Z\fandZFe=1:05+0:17\n\u00000:15Z\f\nrespectively).\nKey words: accretion, accretion discs - X-rays: binaries - X-rays: individual (GX 339-\n4)\n1 INTRODUCTION\nThe primary X-ray spectra from black holes (BHs) can be\ndescribed by a power-law continuum, which originates from\na high temperature compact structure external to the black\nhole accretion disc. This high temperature compact struc-\nture is called the corona. The interaction between the pri-\nmary power-law photons and the disc top layer can pro-\nduce both emission, including \ruorescence lines and re-\ncombination continuum, and absorption edges. These fea-\ntures are referred to as the disc re\rection spectrum (e.g.\nGeorge & Fabian 1991; Garc\u0013 \u0010a & Kallman 2010). The disc\nre\rection spectrum is highly a\u000bected by relativistic e\u000bects,\nsuch as Doppler e\u000bect and gravitational redshift, due to the\nstrong gravitational \feld in the vicinity of black holes (e.g.\nReynolds & Nowak 2003). For example, relativistic blurred\nFe K\u000bemission line features have been detected in re\rection\nspectra of both Active Galactic Nuclei (AGN, e.g. MCG-\n6\u000030\u000015, Tanaka et al. 1995) and Galactic BH X-ray Bi-\n?E-mail: jj447@cam.ac.uknary sources (XRB, e.g. Cyg X-1, Barr et al. 1985). Rel-\nativistic re\rection spectra can provide information on the\ndisc-corona geometry, such as the coronal region size and the\ndisc inner radius. By comparing relativistic re\rection spec-\ntra in di\u000berent observations, we can investigate changes of\nthe inner accretion processes through the evolution of the X-\nray \rux states in both highly variable AGNs (e.g. Mrk 335,\nIRAS 13224\u00003809, Parker et al. 2014; Jiang et al. 2018) and\nXRBs that show di\u000berent \rux states (e.g. XTE J1650-500,\nReis et al. 2013).\nThe existence of two di\u000berent \rux states in XRB was\n\frst realized in the X-ray emission of the XRB Cyg X-1\n(Oda et al. 1971). Its X-ray spectrum can change from a\nsoft spectrum featured by a strong thermal component to a\nhard spectrum featured by a strong disc re\rection compo-\nnent. The soft state, which is also characterised by no radio\ndetection, is identi\fed as the `high' \rux (HF) state and the\nhard state with associated radio detection, is identi\fed as\nthe `low' \rux (LF) state, due to the large \rux variation dur-\ning the transition. Measurements in the HF soft states of BH\nXRB o\u000ber good evidence that the accretion disc is extended\n©2018 The AuthorsarXiv:1901.01739v1  [astro-ph.HE]  7 Jan 20192 J. Jiang et al.\nto the innermost stable circular orbit (ISCO, e.g. LMC X-3,\nSteiner et al. 2010). Most of the spin measurements of soft\nstates are based on the assumption that the inner radius is\nlocated at ISCO (e.g. Gou et al. 2014; Walton et al. 2016).\nIn the LF hard state, the disc is predicted to be truncated\nat a large radius and replaced by an advective \row at small\nradii (Esin et al. 1997; Narayan 2005). Although there is ev-\nidence that the disc is truncated as measured by re\rection\nspectroscopy at X-ray luminosities LX\u00190:1%LEdd(Tomsick\net al. 2009; Narayan & McClintock 2008), there is a substan-\ntial debate whether the disc is truncated in the intermediate\n\rux hard state due to di\u000berent spectral modelling or instru-\nmental pile-up e\u000bects (see the discussion in Wang-Ji et al.\n2018).\nA common result obtained by re\rection modelling of\nblack hole X-ray spectra is high iron abundance compared\nto solar. For example, Walton et al. (2016) found a value\nofZFe\u00194Z\fin Cyg X-1 and Parker et al. (2015) obtained\nZFe\u00194:7Z\fin the same source. Similarly, an iron abun-\ndance of ZFe\u00192\u00005Z\fis required for another BH XRB\nV404 Cyg (Walton et al. 2017). Such a high iron abundance\nhas been commonly seen in AGNs as well (e.g. Chiang et al.\n2015; Parker et al. 2018). Wang et al. (2012) found that\nthe metallicty of the out\rows in di\u000berent quasars can vary\nbetween 1.7{6.9 Z\f. Reynolds et al. (2012) suggested that\nthe radiation-pressure dominance of the inner disc may en-\nhance the iron abundances. However radiative levitation ef-\nfects make predictions for a change of the inner disc iron\nabundance, which is di\u000ecult to be observed in AGNs due to\ntheir longer dynamical timescales.\nAnother possible explanation for the high iron abun-\ndances is high density re\rection. Most versions of avail-\nable disc re\rection models assume a constant electron den-\nsity ne=1015cm\u00003for the top layer of the BH accretion\ndisc, which is appropriate for very massive supermassive\nblack holes in AGNs (e.g. MBH>108M\f). For example,\nan upper limit of ne<1015:3cm\u00003is obtained in Seyfert\n1 galaxy 1H0419 \u0000577 ( MBH\u00191:3\u0002108M\f, Grupe et al.\n2010) by \ftting its XMM-Newton spectra with variable den-\nsity re\rection model (Jiang submitted). At higher electron\ndensity, the free-free process becomes more important in\nconstraining low energy photons, increasing the tempera-\nture of the top layer of the disc, and thus turning the\nre\rected emissions below 1 keV into a blackbody shaped\nspectrum (Ross & Fabian 2007; Garc\u0013 \u0010a et al. 2016). Such a\nmodel can potentially relieve the very high iron abundance\nrequired in previous re\rection spectral modelling. For in-\nstance, Tomsick et al. (2018) obtained an electron density\nofne\u00193\u00021020cm\u00003by \ftting the Cyg X-1 intermediate\n\rux state spectra with the high electron density re\rection\nmodel. Although the iron abundance was \fxed at the solar\nvalue during the spectral \ftting, the model successfully ex-\nplains the spectra. Jiang et al. (2018) \ftted the narrow line\nSeyfert 1 galaxy IRAS 13224 \u00003809 spectra and obtained an\nelectron density of ne>1019:7cm\u00003with an iron abundance\nofZFe\u00195Z\f, which is signi\fcantly lower than the previ-\nous results ZFe\u001920Z\fand closer to the iron abundance\nmeasured in the ultra-fast out\row of the same source.\nHigher densities may also potentially explain the weak\nlow temperature thermal component found in the LF state\nof the XRBs (e.g. Reis et al. 2008; Wang-Ji et al. 2018) and\nat least some of the soft excess commonly seen in Seyfertgalaxies (e.g. Fabian et al. 2009; Chiang et al. 2015; Jiang\net al. 2018). The inclusion of the high electron density e\u000bects\nsigni\fcantly decreases the \rux of the best-\ft blackbody com-\nponent in IRAS 13224 \u00003809 required for the spectral \ftting\npurpose (Jiang et al. 2018). It is also interesting to note that\nthe best-\ft \rux and temperature of the blackbody compo-\nnent that accounts for the soft excess in IRAS 13224 \u00003809\nshow a F/T4relation, indicating a constant area origin\nof the soft excess emission (Chiang et al. 2015; Jiang et al.\n2018).\nGX 339-4 is a low mass X-ray binary (LMXB) and\nshows activity in a wide range of wavelength from optical\nto X-ray. The mass of the central black hole still remains\nuncertain. For example, Heida et al. (2017) obtained a black\nhole mass of 2\u000010M\fby studying its near infrared spec-\ntrum and a mass of >5M\fis obtained previously by Hynes\net al. (2003a,b); Mu~ noz-Darias et al. (2008). The distance\nhas been estimated to be \u00197kpc (Zdziarski et al. 2004).\nGX 339\u00004has shown frequent outbursts and multiple X-\nray observations have been taken during di\u000berent spectral\nstates of GX 339-4. In its hard state, its X-ray spectrum\nshows a broad iron emission line and a power-law contin-\nuum with a photon index varying between \u0000\u00191:5\u00002:5\nacross di\u000berent \rux levels (Miller et al. 2004, 2006, 2008).\nReis et al. (2008) presented a systematic study of its high\nand low hard state XMM-Newton and RXTE spectra by\ntaking the blackbody radiation from the disc into the top\nlayer as well as the Comptonization e\u000bects into modelling,\nand obtained a black hole spin of a\u0003=0:94\u00060:02. More re-\ncently, Parker et al. (2016) obtained a disc iron abundance of\nZFe\u00196:6Z\ffor the HF soft state NuSTAR andSwift spectra\nof GX 339-4. In this study, the disc inner radius is assumed\nto be located at ISCO and a black hole spin of a\u0003>0:95\nis obtained by combining disc thermal spectral and re\rec-\ntion spectral modelling. Later, Wang-Ji et al. (2018) found\nZFe\u00198Z\ffor the LF state of the same source observed\nby the same instruments. Similar conclusions were found by\nanalysing its stacked RXTE spectra at the LF states (Gar-\nc\u0013 \u0010a et al. 2015) and NuSTAR spectra during the outburst of\n2013 (F urst et al. 2015).\nIn this paper, we present a high density re\rection in-\nterpretation of both LF and HF state spectra of GX 339-4.\nThe same NuSTAR and Swift spectra as in Parker et al.\n(2016); Wang-Ji et al. (2018) are considered. In Section 2,\nwe introduce the data reduction process; in Section 3, we\nintroduce the details of high density re\rection modelling of\nthe LF and HF spectra of GX 339-4; in Section 4, we present\nand discuss the \fnal spectral \ftting results. The high den-\nsity re\rection modelling of AGN spectra are presented in a\ncompanion paper (Jiang in prep).\n2 OBSERVATIONS AND DATA REDUCTION\nThe weekly MAXI hardness-intensity diagram (HID) for\nthe 2009-2018 period of GX 339-4 (Matsuoka et al. 2009)\nis shown in Fig. 1, showing a standard `q-shaped' behaviour\nduring the outbursts. GX 339-4 went through two outbursts\neach in 2013 and 2015. 11 NuSTAR observations in total,\neach with a corresponding Swift snapshot, were triggered\nduring these two outbursts, shown by the arrow in Fig. 1.\nTheNuSTAR LF observations in 2015 were taken only dur-\nMNRAS 000, 1{12 (2018)High Density Re\rection I. 3\nTable 1. NuSTAR andSwift observations of GX 339-4 in 2013 and 2015. WT: window timing mode; PC: photon counting mode.\nObs NuSTAR obsID Date exp.(ks) Swift obsID Date exp.(ks) Mode\nHF 80001015003 2015-03-04 30.9 00081429002 2015-03-04 1.9 WT\nLF1 80102011002 2015-08-28 21.6 00032898124 2015-08-29 1.7 WT\nLF2 80102011004 2015-09-02 18.3 00032898126 2015-09-03 2.3 WT\nLF3 80102011006 2015-09-07 19.8 00032898130 2015-09-07 2.8 WT\nLF4 80102011008 2015-09-12 21.5 00081534001 2015-09-12 2.0 PC\nLF5 80102011010 2015-09-17 38.5 00032898138 2015-09-17 2.3 WT\nLF6 80102011012 2015-09-30 41.3 00081534005 2015-09-30 2.0 PC\nLF7 80001013002 2013-08-11 42.3 00032490015 2013-08-12 1.1 WT\nLF8 80001013004 2013-08-16 47.4 00080180001 2013-08-16 1.9 WT\nLF9 80001013006 2013-08-24 43.4 00080180002 2013-08-24 1.6 WT\nLF10 80001013008 2013-09-03 61.9 00032898013 2013-09-02 2.0 WT\nLF11 80001013010 2013-10-16 98.2 00032988001 2013-10-17 9.6 WT\n2015\tobservations\t(HF)\n2015\tobservations\t(LF1-6)\n2013\tobservations\t(LF7-11)Count\tRate\t(2-20keV,\tcts\ts -1 \tcm -2 )\n0.1\n1\nHardness\t(F\n4-10\tkeV\n/F\n2-4\tkeV\n)\n0.1\n1\nFigure 1. Weekly MAXI hardness-intensity diagram for the\n2009-2018 period of GX 339-4. The black square and the arrows\ncorrespond to the dates of the HF observations and the LF(1-\n11) observations analysed in this work. The arrows show the \rux\nchange during the NuSTAR monitoring of the outbursts. NuS-\nTAR observations were taken during the rise and decay of the\noutburst in 2013, and only during the decay of the outburst in\n2015.\ning the decay of the outburst. In this work, we consider all of\ntheNuSTAR observations taken during these two outbursts.\nIn March 2015, GX 339-4 was detected with strong thermal\nand power-law components by Swift , suggesting strong ev-\nidence of a HF state with a combination of disc thermal\ncomponent and re\rection component. One NuSTAR target\nof opportunity observation was triggered with a simultane-\nousSwift snapshot. See the black square in Fig. 1 for the \rux\nand hardness state of the source during its HF observations.\nA full list of observations are shown in Table 1.2.1 NuSTAR Data Reduction\nThe standard pipeline NUPIPELINE V0.4.6, part of HEA-\nSOFT V6.23 package, is used to reduce the NuSTAR data.\nThe NuSTAR calibration version V20171002 is used. We\nextract source spectra from circular regions with radii of\n100 arcsec, and the background spectra from nearby circular\nregions on the same chip. The task NUPRODUCTS is used\nfor this purpose. The 3-78 keV band is considered for both\nFPMA and FPMB spectra. The spectra are grouped to have\na minimum signal-to-noise (S/N) of 6 and to oversample by\na factor of 3.\n2.2 Swift Data Reduction\nThe Swift observations are processed using the standard\npipeline XRTPIPELINE V0.13.3. The calibration \fle ver-\nsion used is x20171113. The LF observations taken in the\nWT mode are not a\u000bected by the pile-up e\u000bects. The source\nspectra are extracted from a circular region with a radius\nof 20 pixels1and the background spectrum spectra are ex-\ntracted from an annular region with an inner radius of 90\npixels and an outer radius of 100 pixels. The LF observations\ntaken in the PC mode are a\u000bected by the pile-up e\u000bects. By\nfollowing Wang-Ji et al. (2018) where they estimated the\nPSF \fle, a circular region with a radius of 5 pixels is ex-\ncluded in the center of the source region. The 0.6{6 keV\nband of all the LF Swift XRT spectra are considered. The\nHF observation was taken in the WT mode and was a\u000bected\nby pile-up e\u000bects. By following Parker et al. (2016), a circu-\nlar radius of 10 pixels is excluded in the center of the source\nregion. The 0.6{1 keV of the HF Swift XRT spectrum at a\nvery high \rux state is ignored due to known issues of the\nRMF redistribution issues in the WT mode2. The Swift\nXRT spectra are grouped to have a minimum S/N of 6 and\nto oversample by a factor of 3.\n11 pixel\u00192:3600\n2See following website for more XRT WT mode calibration in-\nformation. http://www.swift.ac.uk/analysis/xrt/digest cal.php\nMNRAS 000, 1{12 (2018)4 J. Jiang et al.\n3 SPECTRAL ANALYSIS\nXSPEC V12.10.0.C (Arnaud 1996) is used for spectral anal-\nysis, and C-stat is considered in this work. The Galactic\ncolumn density towards GX 339-4 remains uncertain. The\nvalue of combined NHIandNH2obtained by Willingale et al.\n(2013) is 5:18\u00021021cm\u00002. However, Kalberla et al. (2005)\nreported a column density of 3:74\u00021021cm\u00002in the Lei-\nden/Argentine/Bonn survey. The Galactic column density\nvalues measured by di\u000berent sets of broad band X-ray spec-\ntra are di\u000berent too. For example, Wang-Ji et al. (2018)\nobtained\u00194\u00021021cm\u00002while Parker et al. (2016) obtained\na higher value of 7:7\u00060:2\u00021021cm\u00002. We therefore \fxed the\nGalactic column density at 3:74\u00021021cm\u00002in the beginning\nof our analysis and allow it to vary to obtain the best-\ft\nvalue for each set of spectra. For local Galactic absorption,\nthetbabs model is used. The solar abundances of Wilms\net al. (2000) are used in tbabs . An additional constant model\nconstant has been applied to vary normalizations between\nthe simultaneous spectra obtained by di\u000berent instruments\nto account for calibration uncertainties.\n3.1 Low Flux State (LF) Spectral Modelling\nWe analyze all the LF NuSTAR observations publicly avail-\nable prior to 2018 and they have discussed in F urst et al.\n(2015); Wang-Ji et al. (2018). Fig. 2 shows the ratio plots of\nLF1-11 spectra \ftted with a Galactic absorbed power-law\nmodel obtained by \ftting only the corresponding NuSTAR\nspectra. All the LF spectra show a broad emission line fea-\nture around 6.4 keV with a Compton hump above 20 keV.\nThey provide a strong evidence of a relativistic disc re\rec-\ntion component. By following Garc\u0013 \u0010a et al. (2015); Wang-Ji\net al. (2018), we model the features with a combination of\nrelativistic disc re\rection and a distant re\rector for the nar-\nrow emission line component. A more developed version of\nreflionx (Ross & Fabian 2005) is used to model the rest-\nframe disc re\rection spectrum. The reflionx grid allows the\nfollowing free parameters: disc iron abundance ( ZFe), disc\nionization log¹\u0018º, disc electron density ne, high energy cuto\u000b\n(Ecut), and photon index ( \u0000). All the other element abun-\ndances are \fxed at the solar value (Morrison & McCammon\n1983). The ionization parameter is de\fned as \u0018=4\u0019Fn,\nwhere Fis the total illuminating \rux and nis the hydrogen\nnumber density. The photon index \u0000and high energy cut-\no\u000bEcutare linked to the corresponding parameters of the\ncoronal emission modelled by cutoffpl in XSPEC. A con-\nvolution model relconv (Dauser et al. 2013) is applied to\nthe rest frame ionized disc re\rection model reflionx to ap-\nply relativistic e\u000bects. A simple power-law shaped emissivity\npro\fle is assumed ( \u000f/r\u0000q) and the emissvity index qis al-\nlowed to vary during the \ft. Other free parameters are the\ndisc viewing angle iand the disc inner radius rin/ISCO. The\nionization of the distant re\rector is \fxed at the minimum\nvalue\u0018=10. The other parameters of the distant re\rector\nare linked to the corresponding parameters in the disc re-\n\rection component. The BH spin parameter a\u0003is \fxed at\nits maximum value 0:998(Kerr 1963) to fully explore the\nrinparameter. We use cflux , a simple convolution model\nin XSPEC, to calculate the 1{10 keV \rux of each model\ncomponent. For future reference and simplicity, we de\fne\nan empirical re\rection fraction as frefl=FrefFplin the 1{10 keV band, where Frefand Fplare the \rux of the disc\nre\rection component and the coronal emission calculated\nbycflux . Note that this is not the same as the physically\nde\fned re\rection fraction discussed by Dauser et al. (2016).\nThe \fnal model is tbabs * ( cflux*(relconv*reflionx)\n+ cflux*reflionx + cflux*cutoffpl) in XSPEC format.\nThis model can \ft all LF spectra successfully with no obvi-\nous residuals. For example, it o\u000bers a good \ft for the LF1\nspectra with C-stat/ \u0017= 1043.52/948. A ratio plot of LF1\nspectra \ftted with this model is shown in the top panel\nof Fig. 3. The best-\ft values of some key parameters that\na\u000bect the spectral modelling below 3 keV are following:\nNH=3:4+0:2\n\u00000:1\u00021021cm\u00002,log¹\u0018/erg cm s\u00001º=3:18+0:07\n\u00000:06, and\nlog¹ne/cm\u00003º=20:6\u00060:3. Our best-\ft column density is\nconsistent with the Galactic column density measured in\nKalberla et al. (2005).\nWe notice that previously the spectral modelling re-\nquires a low temperature multicolour disc thermal compo-\nnent diskbb (kT=0:46keV) when using the model with the\ndisc electron density ne\fxed at log¹ne/cm\u00003º=15for LF1\nobservation (Wang-Ji et al. 2018). However the normaliza-\ntion of this component is very low and weakly constrained.\nSimilarly, a weak thermal component is also required in the\nanalysis of its XMM-Newton hard state observations (Reis\net al. 2008) and other earlier NuSTAR observations (Reis\net al. 2013). The di\u000berence in spectral modelling may re-\nsult from the following two reasons: one is the high den-\nsity re\rection model, where a blackbody-shaped emission\narises in the soft band when the disc electron density ne\nbecomes higher than 1015cm\u00003; the other is the uncertain\nneutral absorber column density, which was measured to\nbeNH=4:12+0:08\n\u00000:12\u00021021cm\u00002in Wang-Ji et al. (2018) and\nhigher than our best-\ft value for the LF1 spectra.\nIn order to test for an additional diskbb component,\nwe \frst \ft the spectra with NH\fxed at the higher Galac-\ntic column density NH=5:18\u00021021cm\u00002obtained by Will-\ningale et al. (2013) rather than the value from Kalberla et al.\n(2005). An additional diskbb component improves the \ft by\nonly\u0001C-stat=1.1. See the middle panel of Fig. 3 for the cor-\nresponding ratio plot. Only an upper limit of the diskbb\nnormalization is of Ndiskbb<1:5\u0002105found. Compared with\nthe result in Wang-Ji et al. (2018), a lower disc inner temper-\nature of kT=0:24+0:08\n\u00000:10keV is required in this \ft. Second,\nwe \ft LF1 spectra with the absorber column density as a\nfree parameter (bottom panel of Fig 3). A contour plot of\nC-stat distribution on the NHvs.Ndiskbb parameter plane is\ncalculated by STEPPAR function in XSPEC and shown in\nFig. 4. It clearly shows a strong degeneracy between the ab-\nsorber column density and the normalization of the diskbb\ncomponent. The \ft is only improved by \u0001C-stat=3 with 2\nmore free parameters after including this diskbb component.\nSee Fig. 3 for ratio plots against di\u000berent continuum mod-\nels. By varying the Galactic column density, it only slightly\nchanges the \ft of the Swift XRT spectrum. Therefore, we\nconclude that an additional diskbb component is not nec-\nessary for LF1 spectral modelling when the disc density\nparameter neis allowed to vary. In order to visualize the\nspectral di\u000berence with di\u000berent ne, we show the best-\ft re-\n\rection model component for LF1 in Fig. 5 in comparison\nwith the best-\ft model for the same observation assuming\nlog¹necm\u00003º=15cm\u00003in Wang-Ji et al. (2018). With a disc\nMNRAS 000, 1{12 (2018)High Density Re\rection I. 5\nHF\n0.4\n0.6\n0.8\n1\n1.2\nEnergy\t(keV)\n1\n3\n10\n30\nLF11\n0.8\n1\n1.2\nLF10\n0.8\n1\n1.2\nLF9\n0.8\n1\n1.2\nLF8\n0.8\n1\n1.2\nLF7\n0.8\n1\n1.2\nLF6\n0.8\n1\n1.2\n1.4\nEnergy\t(keV)\n1\n3\n10\n30\nLF5\n0.8\n1\n1.2\n1.4\nLF4Ratio\n0.8\n1\n1.2\n1.4\nLF3\n0.8\n1\n1.2\n1.4\nLF2\n0.8\n1\n1.2\n1.4\nLF1\n0.8\n1\n1.2\n1.4\nFigure 2. First 11 panels: ratio plots of GX 339-4 FPM (blue crosses: FPMA; green crosses: FPMB) and XRT (red circles) spectra\n\ftted with a Galactic absorbed power law for LF observations in 2015 (LF1-6) and 2013 (LF7-11). Last panel: ratio plot of HF spectra\n\ftted with a Galactic absorbed power law plus a simple blackbody component for the very high \rux soft state observation in 2015. All\nthe spectra show a broad line emission feature around 6.4 keV and a strong Compton hump above 10 keV.\ndensity as high as log¹necm\u00003º=20:6, a quasi-blackbody\nemission arises in the soft band and accounts for the excess\nemission below 2 keV. Similar conclusions are found for the\nother sets of LF spectra. Future pile-up free high S/N obser-\nvation below 2 keV, such as from NICER , may help constrain\nmore detailed spectral shape of LF states of GX 339-4.\nSo far we have achieved the best-\ft model for the LF\nspectra individually. We also undertake a multi-epoch spec-\ntral analysis with disc iron abundance ZFeand disc viewing\nangle ilinked between LF1-11 spectra. All the other param-\neters are allowed to vary during the \ft. A table of all the\nbest-\ft parameters are shown in Table. 2. The best-\ft mod-\nels and corresponding ratio plots are shown in Fig. 6. We\nallow the column density of the neutral absorber to vary indi\u000berent epochs to investigate any variance. A slightly higher\ncolumn density ( NH\u00194:1\u00021021cm\u00002) is required for LF6,7.\nThe emissivity index of the relativistic re\rection spectrum\nis weakly constrained in LF3-6 observations but largely con-\nsistent with the Newtonian value q=3, except for the LF1\nobservation. We can also con\frm that the disc is not trun-\ncated at a signi\fcantly large radius, such as r=100rg(Plant\net al. 2015). A slight iron over abundance compared to so-\nlar is required ( ZFe=1:5+0:12\n\u00000:04) for the spectral \ftting. The\npower-law continuum is softer in the \frst two observations\ntaken at higher \rux levels but remains consistent at 90%\ncon\fdence during the rest of the decay in 2015. The photon\nindex in LF7-10 during the outburst in 2013 is consistent\nat 90% con\fdence as well. The re\rection fraction decreases\nMNRAS 000, 1{12 (2018)6 J. Jiang et al.\nFree\tN\nH\n\tw/o\tDISKBB\t\nC-stat/\nν=1043.52/948\n0.9\n1\n1.1\nFree\tN\nH\n\tw\tDISKBB\t\nC-stat/\nν=1040.12/946\n0.9\n1\n1.1\nEnergy\t(keV)\n1\n30\nFreeze\tN\nH\n\tw\tDISKBB\t\nC-stat/\nν=1042.43/947Ratio\n0.9\n1\n1.1\nFigure 3. Ratio plots for LF1 spectra against di\u000berent continuum\nmodels. Red circles: XRT; blue crosses: FPMA; green crosses:\nFPMB. See text for more details.\nlog\t(DISKBB\tNormalization)\n1\n2\n3\n4\n5\nN\nH\n\t(10\n22\ncm\n-2\n)\n0.3\n0.35\n0.4\n0.45\n0.5\nFigure 4. A contour plot of C-stat distribution on the\nGalactic absorption column density NHvs. diskbb model\nnormalization parameter plane for LF1 spectra when \ftted\nwith tbabs*(diskbb+cutoffpl+relconv*reflionx+reflionx) . It\nshows a clear degeneracy between two parameters. The lines show\nthe 1\u001b(red solid line), 2 \u001b(blue dashed line), and 3 \u001bcontours\n(green dash-dot line).\nLF1\nlog(n\ne\n)=20.6\nlog(n\ne\n)=15\t(Wang+18)E2f(E)\n0.01\n0.1\nEnergy\t(keV)\n1\n10Figure 5. The best-\ft relativistic high density re\rection model\nfor LF1 spectra (solid line) and the best-\ft relativistic re\rection\nmodel obtained by Wang-Ji et al. (2018) assuming log¹necm\u00003º=\n15(dotted line). The plot is only shown for comparison of spec-\ntral shape. An additional diskbb component is required to \ft the\nbroad band spectra in Wang-Ji et al. (2018).\nwith the decreasing total \rux during the \rux decay in 2015.\nThis is likely caused by a receding inner disc radius at the\nvery low \rux states or a change of the coronal geometry\n(e.g. its height above the disc). Moreover, the multi-epoch\nspectral analysis of all LF observations shows tentative ev-\nidence for an anti-correlation between disc density and X-\nray band \rux. For example, the disc density increases from\nlog¹necm\u00003º=20:60+0:23\n\u00000:12in the highest \rux state (LF1) to\nlog¹necm\u00003º=21:45+0:06\n\u00000:13in the lowest \rux state (LF6). The\n\rux level of the cold re\rection component remains consis-\ntent, indicating that this emission arises from stable material\nat a large radius from the central black hole. We will discuss\nother \ftting results, such as the electron density measure-\nments, in Section 4.\n3.2 High Flux State (HF) Spectral Modelling\nThe same NuSTAR andSwift observations of GX 339-4 in\na HF state analysed in Parker et al. (2016) are considered\nhere. A ratio plot of the HF spectra \ftted with a Galactic ab-\nsorbed power-law model and a simple disc blackbody compo-\nnent diskbb is shown in the bottom panel of Fig. 2. The HF\nspectra show a broad emission line feature at the iron band\nand a Compton hump above 10 keV, indicating existence of\na relativistic disc re\rection component similar with all the\nLF spectra. A multicolour disc blackbody component diskbb\nis used to account for the strong disc thermal component.\nThe full model is tbabs * ( cflux*(relconv*reflionx) +\ncflux*reflionx + cflux*cutoffpl + diskbb) in XSPEC\nformat. This model provides a good \ft with C-\nstat/\u0017=1048.68/971. The best-\ft model is shown in the last\npanel of Fig. 6 and the best-\ft parameters are shown in the\nlast column of Table 2. A disc density of log¹necm\u00003º=\nMNRAS 000, 1{12 (2018)High Density Re\rection I. 7\nFigure 6. Top: the \frst 11 panels show the best-\ft models obtained by \ftting LF1-11 spectra simultaneously. The last panel shows the\nbest-\ft model obtained by \ftting only HF spectra. Red solid lines: total model; blue dotted lines: relativistic re\rection model; purple\ndashed lines: coronal emission; green dash-dot lines: distant re\rection component; orange dash-dot-dot lines: disc thermal spectrum (only\nneeded in the HF spectral modelling). Bottom: ratio plots against the corresponding best-\ft models shown in the upper panels. Red\ncircles: XRT; blue crosses: FPMA; green crosses: FPMB.\n18:93+0:12\n\u00000:16is found in HF observations which is 100 times\nlower than the best-\ft value in LF observations.\nSo far we have obtained a good \ft for the HF spectrum\nof GX 339-4. A higher column density is required for the neu-\ntral absorber ( NH=6:2\u00060:2\u00021021cm\u00002) compared to the LF\nobservations ( NH\u00193:4\u00021021cm\u00002). Parker et al. (2016) ob-\ntained a higher column density of NH=7:2\u00060:2\u00021021cm\u00002\nfor the same observation assuming ne=1015cm\u00003for the\ndisc. Both our result and Parker et al. (2016) are higher than\nthe Galactic absorption column density estimated at other\nwavelengths (e.g. Kalberla et al. 2005), indicating a possi-\nble extra variable neutral absorber along the line of sight.\nOnly an upper limit of the \rux of the distant cold re\rec-\ntor is achieved ( log¹Fdisº<\u000010:89). The 1{10 keV band \rux\nvalues of the disc re\rection component and the coronal emis-\nsion increase by a factor of 13 and 6 respectively comparedto LF1. The best-\ft broad band model shows the highest\nre\rection fraction among all the observations considered in\nthis work, indicating a geometry change of the disc corona\nsystem such as a small inner disc radius. A solar iron abun-\ndance ( 1:05+0:17\n\u00000:15) is required for the HF spectra, which is\nlower than the value obtained by Parker et al. (2016), where\na disc density of ne=1015cm\u00003is assumed.\n4 RESULTS AND DISCUSSION\nWe have obtained a good \ft for the LF and the HF pectra\nof GX 339-4. The LF spectral modelling requires a high disc\ndensity of log¹necm\u00003º\u001921with no additional low temper-\nature thermal component. The HF spectral modelling re-\nquires a 100 times lower density ( log¹necm\u00003º=18:93+0:12\n\u00000:16)\nMNRAS 000, 1{12 (2018)8 J. Jiang et al.\nTable 2. The best-\ft parameters obtained by \ftting 1) LF1-11 spectra simultaneously 2) only HF spectra. u: unconstrained; l: linked;\nf: \fxed. Freflis the \rux of the relativistic disc re\rection component measured between 1{10 keV; Fplis the \rux of the coronal emission\nmeasured at the same energy band; Fdisis for the distant cold re\rector. The re\rection fraction freflis de\fned as FreflFplfor simplicity.\nL0:1\u0000100keV is the 0.1{100 keV band Galactic-absorption corrected luminosity calculated using the best-\ft model. A black hole mass\nMBH=10M\fand a distance d=10kpc are assumed.\nParameter Unit LF1 LF2 LF3 LF4 LF5 LF6\nNH 1021cm\u000023:22+0:14\n\u00000:083:20+0:10\n\u00000:123:1+0:3\n\u00000:23:4+0:5\n\u00000:53:2+0:2\n\u00000:34:1+0:8\n\u00000:7\nq - 6+3\n\u000022:5+3:6\n\u00000:4<7 5(u) 6(u) 4(u)\nrin ISCO <4.7 <8 <11 11+4\n\u00007<32 21+14\n\u000012\nlog¹\u0018º erg cm s\u000013:18+0:06\n\u00000:393:13+0:12\n\u00000:083:12+0:09\n\u00000:173:10+0:17\n\u00000:043:12+0:11\n\u00000:083:08+0:05\n\u00000:02\nlog¹neº cm\u0000320:60+0:23\n\u00000:1220:64+0:16\n\u00000:1321:1+0:4\n\u00000:221:49+0:14\n\u00000:1320:82+0:31\n\u00000:1521:45+0:06\n\u00000:13\na\u0003 - 0.998(f) (l) (l) (l) (l) (l)\ni deg 34\u00062 (l) (l) (l) (l) (l)\nZFe Z\f 1:50+0:12\n\u00000:04(l) (l) (l) (l) (l)\nlog¹Freflº erg cm\u00002s\u00001\u00009:42+0:04\n\u00000:09\u00009:72+0:07\n\u00000:03\u00009:91+0:13\n\u00000:05\u000010:13+0:11\n\u00000:09\u000010:21+0:03\n\u00000:06\u000010:53+0:10\n\u00000:11\nEcut keV 350+92\n\u0000124>287 >255 >290 >350 >420\n\u0000 - 1:594+0:004\n\u00000:0101:530+0:018\n\u00000:0451:49+0:02\n\u00000:051:517+0:010\n\u00000:0301:485+0:007\n\u00000:0291:49+0:03\n\u00000:02\nlog¹Fplº erg cm\u00002s\u00001\u00009:22+0:05\n\u00000:03\u00009:25\u00060:03\u00009:32+0:02\n\u00000:03\u00009:420+0:011\n\u00000:028\u00009:601+0:009\n\u00000:013\u00009:82\u00060:02\nkT keV - - - - - -\nNdiskbb - - - - - - -\nfrefl - 0:631+0:010\n\u00000:0160:3388+0:0041\n\u00000:00100:224+0:006\n\u00000:0020:195+0:003\n\u00000:0020:246+0:003\n\u00000:0020:194+0:003\n\u00000:002\nlog¹Fdisº erg cm\u00002s\u00001\u000011:01+0:14\n\u00000:20\u000011:58+0:16\n\u00000:31\u000011:40+0:19\n\u00000:33\u000011:4+0:2\n\u00000:3\u000011:16+0:15\n\u00000:20<-11.69\nC-stat/\u0017 11870.25/11305\nL0:1\u0000100keVLEdd % 2.7 2.5 2.2 1.7 1.2 0.6\nContinued\nParameter Unit LF7 LF8 LF9 LF10 LF11 HF\nNH 1021cm\u000024:1+0:3\n\u00000:23:84+0:18\n\u00000:163:41+0:18\n\u00000:173:71+0:16\n\u00000:233:75+0:12\n\u00000:146:2\u00060:2\nq - >0:5 >2 >3 >2.7 4(u) 5:88+1:01\n\u00000:77\nrin ISCO <25 <17 <10 <1.51 <25 1(f)\nlog¹\u0018º erg cm s\u000013:29+0:04\n\u00000:033:26\u00060:03 3:23+0:016\n\u00000:0203:23+0:02\n\u00000:053:26+0:04\n\u00000:033:88+0:08\n\u00000:12\nlog¹neº cm\u0000321:0\u00060:2 21 :25+0:23\n\u00000:1721:15\u00060:15 20:93+0:12\n\u00000:0821:57+0:20\n\u00000:1718:93+0:12\n\u00000:16\na\u0003 - (l) (l) (l) (l) (l) >0.93\ni deg (l) (l) (l) (l) (l) 35:9+1:6\n\u00002:0\nZFe Z\f (l) (l) (l) (l) (l) 1:05+0:17\n\u00000:15\nlog¹Freflº erg cm\u00002s\u00001\u000010:071+0:009\n\u00000:010\u00009:93\u00060:04\u00009:71+0:03\n\u00000:02\u00009:52+0:03\n\u00000:04\u000010:41+0:02\n\u00000:07\u00008:30+0:03\n\u00000:02\nEcut keV >380 >320 >430 >330 >410 500(f)\n\u0000 - 1:427+0:065\n\u00000:0161:419+0:012\n\u00000:0101:421+0:009\n\u00000:0151:42\u00060:02 1:478+0:011\n\u00000:0182:357+0:019\n\u00000:018\nlog¹Fplº erg cm\u00002s\u00001\u00009:467+0:007\n\u00000:008\u00009:278+0:011\n\u00000:013\u00009:068+0:010\n\u00000:021\u00008:94+0:009\n\u00000:008\u00009:701+0:015\n\u00000:014\u00008:46+0:06\n\u00000:05\nkT keV - - - - - 0:831+0:03\n\u00000:05\nNdiskbb - - - - - - 1649+76\n\u000049\nfrefl - 0:249+0:005\n\u00000:0070:22+0:02\n\u00000:020:228+0:017\n\u00000:0110:26\u00060:02 0:195+0:012\n\u00000:0301:45+0:06\n\u00000:03\nlog¹Fdisº erg cm\u00002s\u00001\u000011:00+0:10\n\u00000:12\u000010:73+0:02\n\u00000:13\u000010:62\u00060:08\u000010:50+0:09\n\u00000:08\u00001:06+0:08\n\u00000:13<-10.89\nC-stat/\u0017 1048.68/971\nL0:1\u0000100keVLEdd % 1.2 2.0 3.1 4.0 0.7 26.7\ncompared to LF observations. In this section, we discuss\nthe spectral \ftting results by comparing with previous data\nanalysis and accretion disc theories.\n4.1 Comparison with previous results\nFirst, the most signi\fcant di\u000berence from previous results is\nthe close-to-solar disc iron abundance. Previously, Parkeret al. (2016); Wang-Ji et al. (2018) obtained a disc iron\nabundance of ZFe=6:6+0:5\n\u00000:6Z\fand ZFe\u00198Z\frespectively\nby analysing the same spectra considered in this work. Sim-\nilar result was achieved by F urst et al. (2015). A high iron\nabundance of ZFe=5\u00061Z\fwas also found by analysing\nstacked RXTE spectra (Garc\u0013 \u0010a et al. 2015). All of their\nwork was based on the assumption for a \fxed disc density\nne=1015cm\u00003. By allowing the disc density neto vary as a\nMNRAS 000, 1{12 (2018)High Density Re\rection I. 9\nLF1-11\nHFΔ\tC-stat\n0\n5\n10\n15\n20\ni\t/\tdeg\n25\n30\n35\n40\nHF\n0\n5\n10\n15\n20\na\n*\n0.9\n0.925\n0.95\n0.975\n1\nLF1-11\nHFΔ\tC-stat\n0\n10\n20\n30\nZ\nFe\n/Z\n⊙\n1\n1.5\n2\nFigure 7. C-stat contour plots for the disc iron abundance and the relativistic re\rection parameters obtained by \ftting LF 1-11 spectra\nsimultaneously (solid lines) and only the HF spectra (dashed line). The solid lines show the 1 \u001b, 2\u001b, and 3\u001branges.\nfree parameter during spectral \ftting, we obtained a close-\nto-solar disc iron abundance ( ZFe=1:50+0:12\n\u00000:04Z\ffor LF ob-\nservations and ZFe=1:05+0:17\n\u00000:15Z\ffor HF observations). The\nbest-\ft disc iron abundance for the LF spectra is slightly\nhigher than the value for the HF spectra at 90% con\fdence.\nHowever they are consistent within 2 \u001bcon\fdence range. See\nthe left panel of Fig. 7 for the constraints on the disc iron\nabundance parameter. A similar conclusion was achieved by\nanalysing the intermediate \rux state spectra of Cyg X-1\n(Tomsick et al. 2018) with variable density re\rection model.\nHowever a \fxed solar iron abundance was assumed in their\nmodelling.\nSecond, the best-\ft disc viewing angle measured for\nGX 339-4 is di\u000berent in di\u000berent works. The middle panel of\nFig. 7 shows the constraint of the disc viewing angle given\nby multi-epoch LF spectral analysis and HF spectral anal-\nysis. The two measurements are consistent at the 90% con-\n\fdence level ( i=34\u00062deg for the LF observations and\ni=35:9+1:6\n\u00002:0deg for the HF observations). Although our best-\n\ft value is lower compared with the measurement in Wang-\nJi et al. (2018) ( i=40\u000e) and higher than the measurement\nin Parker et al. (2016) ( i=30\u000e), all the previous re\rection\nbased measurements are consistent with our results at 2 \u001b\nlevel. Similarly Tomsick et al. (2018) measured a di\u000berent\nviewing angle for Cyg X-1 compared with previous works.\nIt indicates that a slightly di\u000berent viewing angle measure-\nment might be common when allowing the disc density ne\nto vary as a free parameter during the spectral \ftting.\nThird, a high black hole spin ( a\u0003>0:93) is given by the\ndisc re\rection modelling in the HF spectral analysis. Due\nto the lack of precise measurement of the source distance\nand the central black hole mass, we can only give a rough\nestimation of the inner radius through the normalization of\nthediskbb component in the HF observations. The normal-\nization parameter is de\fned as Ndiskbb =¹rinD10kpcº2cosi,\nwhere the D10kpc is the source distance in units of 10 kpc\nandiis the disc viewing angle. The best-\ft value is Ndiskbb =\n1649+76\n\u000049, corresponding to an inner radius of rin\u001945km\u00193rg\nassuming MBH=10M\fand D=10kpc. We also \ftted the\nthermal component with kerrbb model (Li et al. 2005) as\nin Parker et al. (2016). kerrbb is a multi-colour blackbody\nmodel for a thin disc around a Kerr black hole, which in-cludes the relativistic e\u000bects of spinning black hole. The\nBH spin and the disc viewing angle are linked to the cor-\nresponding parameters in relconv . However we found the\nsource distance and the central black hole mass parameters\ninkerrbb are not constrained during the spectral \ftting.\nkerrbb model gives a slightly worse \ft with \u0001C-stat\u00197and\n2 more free parameters compared to the diskbb model. Since\nthe black hole mass and distance measurement is beyond\nour purpose, we decide to \ft the thermal spectrum in the\nHF observation with diskbb for simplicity. See Parker et al.\n(2016) for more discussion concerning the black hole mass\nand the source distance measurements obtained by \ftting\nwith kerrbb . In conclusion, the high spin result in this work\nis obtained by modelling the relativistic disc re\rection com-\nponent in the HF state of GX 339-4 and consistent with\nprevious re\rection-based spectral analysis (e.g. Plant et al.\n2015; Garc\u0013 \u0010a et al. 2015; Parker et al. 2016; Wang-Ji et al.\n2018). Kolehmainen & Done (2010) found an upper limit of\na\u0003<0:9by analysing RXTE spectra. However they assumed\nthe disc viewing angle is the same as the binary orbital incli-\nnation, which is not necessarily the case (e.g. Tomsick et al.\n2014; Walton et al. 2016).\nFourth, there is a debate whether the disc is truncated\nat a signi\fcant radius in the brighter phases of the hard\nstate. Compared with the results obtained by modelling the\nsame spectra with ne=1015cm\u00003and an additional diskbb\ncomponent (Wang-Ji et al. 2018), we \fnd larger upper limit\nof the inner radius in the LF2-5 observations. For example,\nan upper limit of rin<8RISCO is obtained for the LF2 ob-\nservation, larger than rin=1:8+3:0\n\u00000:6RISCO found by Wang-Ji\net al. (2018). Such di\u000berence could be due to di\u000berent mod-\nelling of the disc re\rection component. The constraints on\nthe inner radius rinare shown in the top right panel of Fig. 8.\nNevertheless, we con\frm that the inner radius is not as large\nasrin\u0019100rgas proposed by previous analysis (e.g. Plant\net al. 2015).\n4.2 High density disc re\rection\nThe LF and HF NuSTAR andSwift spectra of GX 339-4\ncan be successfully explained by high density disc re\rec-\ntion model with a close-to-solar iron abundance for the disc.\nMNRAS 000, 1{12 (2018)10 J. Jiang et al.\nLF1-6\t(2015)\nLF7-11\t(2013)Rin/ISCO\n0\n10\n20\n30\n40\nL\nx\n/L\nEdd\n\t(%)\n0\n1\n2\n3\n4\nLF1\nLF2\nLF3\nLF4\nLF5\nLF6\nLF7\nLF8\nLF9\nLF10\nLF11Δ\tC-stat\n2\n4\n6\n10\nR\nin\n/ISCO\n0\n10\n20\n30\n40\nFigure 8. Left: the best-\ft inner radius of the disc vs. the X-ray luminosities for LF observations. Red points represent LF1-6 observations\nand blue points represent LF7-11 observations. The X-ray luminosity LXis the 0.1{100 keV band Galactic-absorption corrected luminosity\ncalculated using the best-\ft model. A black hole mass MBH=10M\fand a distance d=10 kpc are assumed. The error bars show the 90%\ncon\fdence ranges of the measurements. See Table 2 for values. Right: the constraints on the inner disc radius for each LF spectra shown\nin di\u000berent colours. The solid lines show the 1 \u001b, 2\u001b, and 3\u001branges.\nIn the low \rux hard states, no additional low-temperature\ndiskbb component is required in our modelling. Instead, a\nquasi-blackbody emission in the soft band of the disc re-\n\rection model \fts the excess below 2 keV. At higher disc\ndensity, the free-free process becomes more important in\nconstraining low energy photons, increasing the disc sur-\nface temperature, and thus turning the re\rected emission\nin the soft band into a quasi-blackbody spectrum. See Fig. 5\nfor a comparison between the best-\ft high density re\rection\nmodel for LF1 observation and a constant disc density model\n(ne=1015cm\u00003).\nIn LF states of GX 339-4, a disc density of ne\u0019\n1021cm\u00003is required for the spectral \ftting. Our multi-\nepoch spectral analysis shows tentative evidence that the\ndisc density increases from log¹necm\u00003º=20:60+0:23\n\u00000:12in the\nhighest \rux state (LF1) to log¹necm\u00003º=21:45+0:06\n\u00000:13in\nthe lowest \rux state (LF6) during the decay of the out-\nburst in 2015, except for LF5 observation. See Table 2 for\nnemeasurements. Similar pattern can be found in LF7-\n10 observations. In HF state of GX 339-4, we measure a\ndisc density of ne\u00191019cm\u00003by \ftting the broad band\nspectra with a combination of high density disc model and\na multi-colour disc blackbody model. The disc density in\nHF state is 100 times lower than that in LF states. The\n0.1{100 keV band luminosity of GX 339-4 in HF state\n(LX\u00190:28LEdd) is 10 times the same band luminosity in\nLF states ( LX=0:01\u00000:03LEdd). While the accretion rate\nis rather small, the anti-correlation between the disc density\nand the X-ray luminosity log¹ne1ne2º/\u0000 2 log¹LX1LX2ºis\nfound to agree with the expected behaviour of a standard\nradiation-pressure dominated disc (e.g. Shakura & Sunyaev\n1973; Svensson & Zdziarski 1994). See Section 4.3 for more\ndetailed comparison between the measurements of the disc\ndensity and the predictions of the standard disc model.\n4.3 Accretion rate and disc density\nSvensson & Zdziarski (1994) (hereafter SZ94) reconsideredthe standard accretion disc model (Shakura & Sunyaev 1973,\nhereafter SS73) by adding one more parameter to the disc\nenergy balance condition - a fraction of the power associated\nwith the angular momentum transport is released from the\ndisc to the corona, denoted as f. Only 1\u0000fof the released\naccretion power is dissipated in the colder disc itself.\nBy following SZ94, we can obtain a relation between ne\nand ffor a radiation-pressure dominated disc:\nne=1\n\u001bTRS256p\n2\n27\u000b\u00001R32Ûm\u00002»1\u0000¹RinRº12¼\u00002»\u0018¹1\u0000fº¼\u00003;(1)\nwhere\u001bT=6:64\u00021025cm2is the Thomson cross section; RS\nis the Schwarzschild radius; Ris in the units of RS;Ûmis de-\n\fned asÛm=ÛMc2LEdd=LBol\u000fLEdd;LEdd=4\u0019GMm pc\u001bT=\n2\u0019¹mpmeº¹mec3\u001bTºRSis the Eddington luminosity; \u0018is the\nconversion factor in the radiative di\u000busion equation and cho-\nsen to be 1, 2 or 2/3 by di\u000berent authors (SZ94). For a\nradiation-pressure dominated disc, the disc density nede-\ncreases with increasing accretion rate Ûm. In Fig. 9, we plot\nthe radiation-pressure dominated disc solutions for \u0018=1in\ngreen lines and a solution for \u0018=2in blue.\nWhenÛm<0:1and fapproaches unity, the radiation-\npressure dominated radius disappears and gas pressure\nstarts dominates the disc. The relation between neand f\nfor a gas-pressure dominated disc is\nne=1\n\u001bTRSK\u000b\u0000710R\u00003320Ûm25»1\u0000¹RinRº12¼25»\u0018¹1\u0000fº¼\u0000310;\n(2)\nwhere K=2\u000072¹512p\n2\u00193\n405º310¹\u000bfmp\nmeº910¹RS\nreº310.\u000bfis the\n\fne-structure constant; mpis the proton mass; meis the\nelectron mass; reis the classical electron radius. An example\nof a gas-pressure dominated disc solution for R=2RSand\nf=0:01is shown by the black line in Fig. 9 for comparison.\nFor a gas-pressure dominated disc, the disc density increases\nwith increasing accretion rate.\nThe best-\ft disc density and accretion rate values ob-\ntained by \ftting GX 339-4 LF1-11 and HF spectra with high\nMNRAS 000, 1{12 (2018)High Density Re\rection I. 11\nξ\t=1\nξ\t=2\nGas\tDominated\nRadiation\tDominated\nm\n>\nm\nEdd\nf=0.8\nf=0.4\nf=0.01Density\tlog(n e )\n18\n20\n22\n24\n26\nAccretion\tRate\tlog(\nm\n)\n−2\n−1\n0\n1\nFigure 9. Disc density log¹necm\u00003ºvs. accretion rate log¹Ûmº\nbased on the radiation-pressure dominated (green) and gas-\npressure dominated (black) disc solutions in SZ94, assuming\nMBH=10M\fand\u000b=0:1. The solid green lines show the solutions\nwith di\u000berent coronal power fraction fatR=2RS. The dashed\nand dotted green lines show the radiation pressure-dominated\nsolution with f=0:01atR=6;8RSrespectively. The black\ncircular points show the surface disc density and the mass ac-\ncretion rate measurements of GX 339-4 in LF and HF states\nin 2015 and the black squares show the measurements for ob-\nservations in 2013. The mass accretion rate is estimated using\nÛm=LBol\u000fLEdd\u0019L0:1\u0000100keV\u000fLEddin this work. A Novikov-\nThorne accretion e\u000eciency \u000f=0:2(Novikov & Thorne 1973; Agol\n& Krolik 2000) and an inner disc radius of Rin=1RSis assumed\nfor a spinning black hole with a\u0003=0:95. The black vertical line\nshows the Eddington accretion limit ÛmEdd=1\u000f\u00195.\u0018=1is\nassumed during the calculation as in SZ94. The blue solid line\nis shown for the radiation-pressure dominated disc solution with\nR=2RSandf=0:01, assuming \u0018=2.\ndensity re\rection model are shown by black points in Fig. 9.\nThe accretion rate is calculated using Ûm=LBol\u000fLEdd\u0019\nL0:1\u0000100keV\u000fLEdd, where\u000fis the accretion e\u000eciency and\nL0:1\u0000100keV is the 0.1{100keV band absorption corrected lu-\nminosity calculated using the best-\ft model. According to\nNovikov & Thorne (1973); Agol & Krolik (2000), an accre-\ntion e\u000eciency of \u000f=20% is assumed for a spinning black\nhole with a\u0003=0:95measured in Section 3.2. A black hole\nmass of 10 M\fand a source distance of 10 kpc are assumed.\nThe 0.1{100 keV band luminosity in the HF state of\nGX 339-4 is approximately 10 times the same band lumi-\nnosity in the LF states. The disc density in the HF state is\n2 orders of magnitude lower than the density in the LF1-\n6 states. The anti-correlation between its density and ac-\ncretion rate is expected according to the radiation-pressure\ndominated disc solution in SZ94 ( log¹neº/\u0000 2 log¹Ûmº, as in\nEq. 1). However the disc density measurements for GX 339-4\nare lower than the predicted values for corresponding accre-\ntion rates. See Fig. 9 for comparison between measurementsand theoretical predictions in SZ94. Following are possible\nexplanations for the mismatch: 1. the disc density shown in\nEq. 1 is assumed to be constant throughout the vertical pro-\n\fle of the disc (SS73). The neparameter we measure using\nre\rection spectroscopy is however the surface disc density\nwithin a small optical depth (Ross & Fabian 2007). For ex-\nample, three-dimensional MHD simulations show that the\nvertical structure of radiation-pressure dominated disc den-\nsity is centrally concentrated (e.g. Turner 2004; Hirose et al.\n2006). 2. the accretion rate might be underestimated by as-\nsuming LBol\u0019L0:1\u0000100keV , although we do not expect other\nbands of GX 339-4 to make a large contribution to its bolo-\nmetric luminosity; 3. there is a large uncertainty on the black\nhole mass, the disc accretion e\u000eciency, and the source dis-\ntance measurements for GX 339-4. For example, the most\nrecent near-infrared study shows that the central black hole\nmass in GX 339-4 could be within 2\u000010M\f(Heida et al.\n2017). Nevertheless, the use of the high density re\rection\nmodel enables us to estimate the density of the disc surface\nin di\u000berent \rux states of an XRB and an anti-correlation\nbetween neand LXhas been found in GX 339-4.\n4.4 Future work\nIn our work, we conclude that the high density re\rection\nmodel can explain both the LF and HF spectra of GX 339-4\nwith a close to solar iron abundance. No additional black-\nbody component is statically required during the spectral\n\ftting of the LF states. On one hand, the strong degeneracy\nbetween the diskbb component and the absorber column\ndensity is due to the low S/N of the Swift XRT observa-\ntions. More pile-up free soft band spectra are required to\nobtain a more detailed spectral shape at the extremely LF\nstate of GX 339-4, such as from NICER . On the other hand,\nmore detailed spectral modelling is required. For example,\na more physics model, such as Comptonization model, is re-\nquired for the coronal emission modelling in the broad band\nspectral analysis. The disc thermal photons from the disc to\nthe re\rection layer need to be taken into account in future\nre\rection modelling, especially in the XRB soft states where\na strong thermal spectrum is shown.\nACKNOWLEDGEMENTS\nJ.J. acknowledges support by the Cambridge Trust and the\nChinese Scholarship Council Joint Scholarship Programme\n(201604100032). D.J.W. acknowledges support from an\nSTFC Ernest Rutherford Fellowship. A.C.F. acknowledges\nsupport by the ERC Advanced Grant 340442. M.L.P. is sup-\nported by European Space Agency (ESA) Research Fellow-\nships. J.F.S. has been supported by NASA Einstein Fellow-\nship grant No. PF5-160144. J.A.G. acknowledges support\nfrom the Alexander von Humboldt Foundation. 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In a simulation study, we compare the performance of\ndifferent versions of the smoother.\nIndex Terms —Extended object tracking, smoothing, random\nmatrix, Gaussian, Wishart, inverse Wishart\nI. I NTRODUCTION\nMultiple Object Tracking ( MOT) denotes the process of\nsuccessively determining the number and states of multi-\nple dynamic objects based on noisy sensor measurements.\nTracking is a key technology for many technical applications\nin areas such as robotics, surveillance, autonomous driving,\nautomation, medicine, and sensor networks. Extended object\ntracking is defined as MOT where each object generates\nmultiple measurements per time step and the measurements\nare spatially structured on the object, see [1].\nExtended object tracking is applicable in many different\nscenarios, e.g., environment perception for autonomous vehi-\ncles using camera, lidar and automotive radar. The multiple\nmeasurements per object and time step create a possiblity\nto estimate the object extent, in addition to the position and\nthe kinematic properties such as velocity and heading. This\nestimation requires an object state space model, including\nmodelling of the object dynamics and the measurement pro-\ncess. Extended obejct models include the Random Matrix\nmodel [2], [3], the Random Hypersurface model [4], and\nGaussian Process models [5]. A comprehensive overview of\nextended object tracking can be found in [1].\nIn this paper we focus on the Random Matrix model, also\nknown as the Gaussian inverse Wishart ( GIW) model. The\nrandom matrix model was originally proposed by Koch [2],\nand is an example of a spatial model. In this model the shape\nof the object is assumed to be elliptic. The ellipse shape is\nsimple but still versatile, and the random matrix model has\nbeen integrated into many different multiple extended object\nKarl Granstr ¨om is with the Department of Electrical Engineer-\ning, Chalmers University of Technology, Gothenburg, Sweden. E-mail:\nkarl.granstrom@chalmers.se .\nJakob Bramst ˚ang did his part of this work as a Master’s Student at the\nDepartment of Electrical Engineering, Chalmers University of Technology,\nGothenburg, Sweden. He is currently with Knightec AB, Stockholm, Sweden.\nE-mail: jakob.bramstang@knightec.se .TABLE I\nNOTATION\n\u000fRn: space of vectors of dimension n\n\u000fIn\u0002n: space of non-singular n\u0002nmatrices\n\u000fSd\n+: space of positive semi-definite d\u0002dmatrices\n\u000fSd\n++: space of positive definite d\u0002dmatrices\n\u000fId: unit matrix of size d\u0002d\n\u000f0m\u0002n: all-zerom\u0002nmatrix\n\u000f\n: Kronecker product\n\u000fj\u0001j: set cardinality\n\u000fdiag (\u0001): diagonal matrix\n\u000fE[\u0001]: expected value\n\u000fN(x;m;P ): Gaussian pdf for random vector x2Rnxwith mean\nvectorm2Rnxand covariance matrix P2Snx\n+\n\u000fIWd(X;v;V): inverse Wishart pdf for random matrix X2Sd\n++\nwith degrees of freedom v>2dand parameter matrix V2Sd\n++, see,\ne.g., [23, Def. 3.4.1]\n\u000fWd(X;v;V): Wishart pdf for random matrix X2Sd\n++with degrees\nof freedomv\u0015dand parameter matrix V2Sd\n++, see, e.g., [23, Def.\n3.2.1]\n\u000fGBII\nd(X;a; b; \n;\t): Generalized matrix variate beta type II pdf\nfor random matrix X2Sd\n++with degrees of freedom a >d\u00001\n2,\nb >d\u00001\n2, parameter matrix \t2Sd\n+, and parameter matrix \nsuch\nthat(\n\u0000\t)2Sd\n++, see, e.g., [23, Def. 5.2.4]\ntracking frameworks [6]–[15]. Indeed, the random matrix\nmodel is applicable in many real scenarios, e.g., pedestrian\ntracking using video [6]–[8] or lidar [9] and tracking of boats\nand ships using marine radar [14]–[20].\nThe focus of this paper is on Bayesian smoothing for the\nrandom matrix model. A preliminary version of this work was\npresented in [21]. This paper is a significant extension of [21],\nand presents the following contributions:\n\u000fClosed form smoothing expressions for the conditional\nGIW model from [2].\n\u000fClosed from smoothing expressions for the factorized\nGIW model from [3].\n\u000fA simulation study that compares the derived smoothers\nto both prediction and filtering.\nAs a minor contribution, a closed form expression for the\nrandom matrix prediction from [22] is presented.\nThe rest of the paper is organized as follows. A problem\nformulation is given in the next section. In Section III a\nreview of the random matrix model is given. Smoothing for the\nconditional GIW model is presented in Section IV; smoothing\nfor the factorised GIW model is presented in Section V.\nResults from a simulation study are presented in Section VI.\nConcluding remarks are given in Section VII.arXiv:1901.05301v1  [eess.SP]  11 Jan 20192\nII. P ROBLEM FORMULATION\nLet\u0018kdenote the extended object state at time k, letZk\ndenote the set of measurements at time step k, and let Z1:k\ndenote the sets of measurements from time 1up to, and\nincluding, time k. Bayesian extended object filtering builds\nupon two steps, the Chapman-Kolmogorov prediction\np(\u0018k+1jZ1:k) =Z\np(\u0018k+1j\u0018k)p(\u0018kjZ1:k)d\u0018k (1)\nwherep(\u0018k+1j\u0018k)is the transition density, and the Bayes\nupdate\np(\u0018k+1jZ1:k+1) =p(Zk+1j\u0018k+1)p(\u0018k+1jZ1:k)R\np(Zk+1j\u0018k+1)p(\u0018k+1jZ1:k)d\u0018k+1(2)\nwherep(Z1:k+1j\u0018k+1)is the measurement likelihood. The\nfocus of this paper is on Bayesian extended object smoothing,\np(\u0018kjZ1:K) =p(\u0018kjZ1:k)Zp(\u0018k+1j\u0018k)p(\u0018k+1jZ1:K)\np(\u0018k+1jZ1:k)d\u0018k+1;\n(3)\nwhereKis the final time step. For the random matrix model,\nthe Chapman-Kolmogorov prediction and Bayes update have\nbeen covered extensively in previous litterature, see, e.g., [2],\n[3], [22], [24] for the prediction, and, e.g., [2], [3], [24]–[27]\nfor the update. In this paper, we focus on Bayesian extended\nobject smoothing. In previous literature, smoothing is only\ndiscussed briefly in [2, Sec. 3.F], and complete details are\nnot given.\nBayesian filtering and smoothing for the random matrix\nmodel is an example of assumed density filtering : the func-\ntional form of the state density is to be preserved in the pre-\ndiction and the update. It is therefore necessary that Bayesian\nsmoothing also preserves the functional form of the extended\nobject state density. Two different assumed state densities can\nbe found in the literature: the conditional Gaussian inverse\nWishart [2], and the factorized Gaussian inverse Wishart [3].\nThe problem considered in this paper is to use the Bayesian\nsmoothing equation (3) to compute the smoothing GIW param-\neters for both the conditional model and the factorized model.\nIII. R EVIEW OF RANDOM MATRIX MODEL\nIn this section we give a brief review of the random matrix\nmodel; a longer review can be found in [1, Sec. 3.A].\nIn the random matrix model [2], [3], the extended object\nstate is a tuple \u0018k= (xk;Xk)2Rnx\u0002Sd\n++. The vector xk2\nRnxrepresents the object’s position and its motion properties,\nsuch as velocity, acceleration, and turn-rate. The matrix Xk2\nSd\n++represents the object’s extent, where dis the dimension\nof the object; d= 2 for tracking with 2D position and d= 3\nfor tracking with 3D position. The matrix Xkis modelled as\nbeing symmetric and positive definite, which means that the\nobject shape is approximated by an ellipse.\nIn the literature, there are two alternative models for the ex-\ntended object state density, the conditional and the factorised.In the conditional model, first presented in [2], the following\nstate density is used,\np(\u0018kjZ1:`) =p(xkjXk;Z1:`)p(XkjZ1:`) (4a)\n=N\u0000\nxk;mkj`;Pkj`\nXk\u0001\n\u0002IWd\u0000\nXk;vkj`;Vkj`\u0001\n; (4b)\nwheremkj`2Rnx,Pkj`2Ss\n+,vkj`>2d,Vkj`2Sd\n++,\nands=nx\nd. In this model, the random vector xconsists\nof ad-dimensional spatial component (the position) and its\nderivatives (velocity, acceleration, etc.), see [2, Sec. 3]. Thus,\ns\u00001 =nx\nd\u00001describes up to which derivative the kinematics\nare described, see [2, Sec. 3].\nIn the factorised model, first presented in [3], the following\nstate density is used,\np(\u0018kjZ1:`) =p(xkjZ1:`)p(XkjZ1:`) (5a)\n=N\u0000\nxk;mkj`;Pkj`\u0001\nIWd\u0000\nXk;vkj`;Vkj`\u0001\n;\n(5b)\nwheremkj`2Rnx,Pkj`2Snx\n+,vkj`>2d, andVkj`2Sd\n++.\nIn this model, the random vector xconsists of a d-dimensional\nspatial component (the position) and additional motion param-\neters; note that, in contrast to the conditional model, here the\nmotion parameters are not restricted to being derivatives of the\nspatial component, and non-linear dynamics can be modelled,\nsee further in [3].\nThe random matrix transition density can expressed as\np(\u0018k+1j\u0018k) =p(xk+1;Xk+1jxk;Xk) (6a)\n=p(xk+1jXk+1;xk;Xk)p(Xk+1jxk;Xk)(6b)\n=p(xk+1jXk+1;xk)p(Xk+1jxk;Xk) (6c)\nwhere the last equality follows from a Markov assumption,\nsee [2]. The random matrix measurement likelihood can be\nexpressed on a general form as\np(Zkj\u0018k)/Y\nz2Zkp(zjxk;Xk) (7)\nNote that the modelling of the extended object measurement\nset cardinality is outside the scope of this work, see [1, Sec.\n2.C] for an overview of different models for the number of\nmeasurements.\nIV. C ONDITIONAL MODEL SMOOTHING\nIn the conditional model, we have conditional Gaussian\ninverse Wishart densities, cf. (4b), and under assumed density\nfiltering we seek a smoothed density of the same form, i.e.,\np(\u0018kjZ1:K) =p(xkjXk;Z1:K)p(XkjZ1:K) (8a)\n=N\u0000\nxk;mkjK;PkjK\nXk\u0001\n\u0002IWd\u0000\nXk;vkjK;VkjK\u0001\n: (8b)3\nTABLE II\nCONDITIONAL MODEL :PREDICTION\nmk+1jk= (Fk\nId)mkjk\nPk+1jk=FkPkjkFT\nk+D\nvk+1jk=d+ 1 +\u0012\n1 +vkjk\u00002d\u00002\nn\u0013\u00001\n(vkjk\u0000d\u00001)\nVk+1jk=\u0012\n1 +vkjk\u0000d\u00001\nn\u0000d\u00001\u0013\u00001\nAVkjkAT\nA. Assumptions and modelling\nThe following assumptions are made for the conditional\nGIW model, see [2, Sec. 2].\nAssumption 1: The time evolution of the extent state is\nassumed independent of the kinematic state,\np(Xk+1jxk;Xk) =p(Xk+1jXk): (9)\n\u0003\nAssumption 2: The extent changes slowly with time,\nXk+1\u0019Xk, such that for the kinematic state, conditioned\non the extent state, the following holds,\np(xkjXk)\u0019p(xkjXk+1); (10)\np(xk+1jXk+1)\u0019p(xk+1jXk); (11)\np(xk+1jXk+1;xk)\u0019p(xk+1jXk;xk): (12)\n\u0003\nThe validity of Assumptions 1 and 2 is discussed in [2].\nIn the conditional random matrix model, the transition\ndensity (6c) is Gaussian-Wishart, see [2, Sec. 3.A/B],\np(\u0018k+1j\u0018k)\u0019p(xk+1jXk+1;xk)p(Xk+1jXk) (13a)\n=N(xk+1; (Fk\nId)xk;Dk\nXk+1)(13b)\n\u0002Wd\u0012\nXk+1;nk;Xk\nnk\u0013\nwhere thes\u0002smatrixFkis the motion model, the s\u0002smatrix\nDkis the process noise, and the degrees of freedom nk\u0015d\ngovern the uncertainty of the time evolution of the extent.\nThis transition density was generalised by [24] by introducing\nad\u0002dparameter matrix Afor the extent transition,\np(\u0018k+1j\u0018k)\u0019N(xk+1; (Fk\nId)xk;Dk\nXk+1)(13c)\n\u0002Wd\u0012\nXk+1;nk;AXkAT\nnk\u0013\nIn the remainder of the paper, we consider this generalised\ntransition density. The measurement model is [2, Sec. 3.D]\np(zjxk;Xk) =N(z; (Hk\nId)xk;Xk); (14)\nwhere the 1\u0002smatrixHkis the measurement model.\nB. Prediction, update, and smoothing\nThe prediction and the update for the conditional model\nare reproduced in Table II and in Table III, respectively. The\nsmoothing is given in the following theorem.\nTheorem 1: Let the densities p(\u0018kjZ1:k),p(\u0018k+1jZ1:K)\nandp(\u0018k+1jZ1:k)be conditional Gaussian inverse WishartTABLE III\nCONDITIONAL MODEL :UPDATE\nmkjk=mkjk\u00001+ (K\nId)\"\nPkjk=Pkjk\u00001\u0000KSKT\nvkjk=vkjk\u00001+jZkj\nVkjk=Vkjk\u00001+N+Z\n\"=\u0016z\u0000(H\nId)mkjk\u00001\n\u0016z=1\njZkjP\nz2Zkz\nZ=P\nz2Zk(z\u0000\u0016z) (z\u0000\u0016z)T\nS=HPkjk\u00001HT+1\njZkj\nK =Pkjk\u00001HTS\u00001\nN =S\u00001\"\"T\nTABLE IV\nCONDITIONAL MODEL :SMOOTHING\nmkjK=mkjk+ (G\nId)\u0000\nmk+1jK\u0000mk+1jk\u0001\nPkjK=Pkjk\u0000G\u0000\nPk+1jk\u0000Pk+1jK\u0001\nGT\nvkjK=vkjk+\u0011\u00001\u0010\nvk+1jK\u0000vk+1jk\u00002(d+1)2\nn\u0011\nVkjK=Vkjk+\u0011\u00001A\u00001\u0000\nVk+1jK\u0000Vkjk\u0001\n(A\u00001)T\nG=PkjkFT\nkP\u00001\nk+1jk\n\u0011= 1 +vk+1jK\u0000vkjk\u00003(d+1)\nn\n(4b), and let the transition density be Gaussian Wishart\n(13c). The smoothed density p(\u0018kjZ1:K), see (3), is con-\nditional Gaussian inverse Wishart, see with parameters\n(mkjK;PkjK;vkjK;VkjK)given in Table IV. \u0003\nThe proof of Theorem 1 is given in Appendix B.\nV. F ACTORIZED MODEL\nFor the random matrix model in [3], we have factorised\nGaussian inverse Wishart densities, cf. (5), and under assumed\ndensity filtering we seek a smoothed density of the same form,\ni.e.,\np(\u0018kjZ1:K) =p(xkjZ1:K)p(XkjZ1:K) (15a)\n=N\u0000\nxk;mkjK;PkjK\u0001\n\u0002IWd\u0000\nXk;vkjK;VkjK\u0001\n: (15b)\nA. Assumptions, approximations\nThe following assumption is made for the factorized GIW\nmodel, see [3].\nAssumption 3: The time evolution of the kinematic state is\nindependent of the extent state,\np(xk+1jXk+1;xk) =p(xk+1jxk) (16)\n\u0003\nThe validity of this assumption is discussed in [3], [22]. The\ntransition density is Gaussian Wishart [22],\np(\u0018k+1j\u0018k) =N(xk+1;fk(xk);Qk) (17)\n\u0002Wd\u0012\nXk+1;nk;M(xk)XkMT(xk)\nnk\u0013\nwhere the function fk(\u0001) :Rnx!Rnxis the motion model,\nthenx\u0002nxmatrixQkis the process noise covariance, the4\nTABLE V\nFACTORIZED MODEL :PREDICTION\nIfMx=A, whereAis ad\u0002dinvertible matrix,\nmk+1jk=fk(mkjk)\nPk+1jk= ~FkPkjk~FT\nk+Q\nvk+1jk=d+ 1 +\u0010\n1 +vkjk\u00002d\u00002\nn\u0011\u00001\n(vkjk\u0000d\u00001)\nVk+1jk=\u0010\n1 +vkjk\u0000d\u00001\nn\u0000d\u00001\u0011\u00001\nAVkjkAT\n~Fk=rxfk(x)jx=mkjk\nelse,\nmk+1jk=fk(mkjk)\nPk+1jk= ~FkPkjk~FT\nk+Q\nvk+1jk=d+ 1 +\u0011\u00001(vkjk\u0000d\u00001)\nVk+1jk=\u0011\u00001\u0010\n1\u0000d+1\ns\u0011\u0010\n1\u0000d+1\nn\u0011\nC2\n\u0011= 1 + (vkjk\u00002d\u00002)\u0010\n1\ns+1\nn\u0000d+1\nns\u0011\ns=d+1\ndTrn\nC1C2(C1C2\u0000Id)\u00001o\n~Fk=rxfk(x)jx=mkjk\nC1=Ekjkh\u0000\nMxVkjkMT\nx\u0001\u00001i\nC2=Ekjk\u0002\nMxVkjkMT\nx\u0003\ndegrees of freedom nk\u0015dgovern the uncertainty of the time\nevolution of the extent, and the function M(\u0001) :Rnx!Id\u0002d\ndescribes how the extent changes over time due to the object\nmotion. For example, M(\u0001)can be a rotation matrix. In what\nfollows, we write Mx=M(x)for brevity.\nThe measurement model is\np(zjxk;Xk) =N\u0010\nz;~Hkxk;\u001aXk+Rk\u0011\n; (18)\nwhere thed\u0002nxmatrix ~Hkis the measurement model, \u001a>0\nis a scaling factor, and Rk2Sd\n+is the measurement noise\ncovariance. The scaling factor \u001aand the noise covariance\nRkwere added to better model scenarios where the sensor\nnoise is large in relation to the size of the extended object,\nsee discussion in [3, Sec. 3]. In this paper, to enable a\nstraightforward comparison to the conditional model, which\nassumes that the sensor noise is small in comparison to the\nsize of the extended object, we focus on the case \u001a= 1 and\nRk=0d\u0002d.\nB. Prediction, update, and smoothing\nThe prediction and the update for the conditional model\nare reproduced in Table V and in Table VI, respectively. The\nsmoothing is given in the following theorem.\nTheorem 2: Let the densities p(\u0018kjZ1:k),p(\u0018k+1jZ1:K)\nandp(\u0018k+1jZ1:k)be factorised Gaussian inverse Wishart\n(5), and let the transition density be Gaussian Wishart\n(17). The smoothed density p(\u0018kjZ1:K), see (3), is fac-\ntorised Gaussian inverse Wishart, see with parameters\n(mkjK;PkjK;vkjK;VkjK)given in Table VII. \u0003\nThe proof of Theorem 2 is given in Appendix C.\nC. Expected value approximation\nNote that both the prediction and the smoothing require\nexpected values, see C1andC2in Table V, and C3andC4inTABLE VI\nFACTORIZED MODEL :UPDATE\nmkjk=mkjk\u00001+K\"\nPkjk=Pkjk\u00001\u0000KSKT\nvkjk=vkjk\u00001+jZkj\nVkjk=Vkjk\u00001+^N+^Z\n\"=\u0016z\u0000~Hmkjk\u00001\n\u0016z=1\njZkjP\nzi2Zkzi\nZ=P\nzi\nk2Zk\u0000\nzi\u0000\u0016z\u0001\u0000\nzi\u0000\u0016z\u0001T\n^X=Vkjk\u00001\u0000\nvkjk\u00001\u00002d\u00002\u0001\u00001\nY=\u001a^X+R\nS= ~HPkjk\u00001~HT+Y\njZkj\nK =Pkjk\u00001~HTS\u00001\n^N= ^X1\n2S\u00001\n2\"\"TS\u0000T\n2^XT\n2\n^Z= ^X1\n2Y\u00001\n2ZY\u0000T\n2^XT\n2\nTABLE VII\nFACTORIZED MODEL :SMOOTHING\nIfMx=A, whereAis ad\u0002dinvertible matrix,\nmkjK=mkjk+Gk\u0000\nmk+1jK\u0000mk+1jk\u0001\nPkjK=Pkjk\u0000G\u0000\nPk+1jk\u0000Pk+1jK\u0001\nGT\nvkjK=vkjk+\u0011\u00001\u0010\nvk+1jK\u0000vk+1jk\u00002(d+1)2\nn\u0011\nVkjK=Vkjk+\u0011\u00001A\u00001\u0000\nVk+1jK\u0000Vk+1jk\u0001\n(A\u00001)T\nG=Pkjk~FT\nkP\u00001\nk+1jk\n\u0011= 1 +vk+1jK\u0000vk+1jk\u00003(d+1)\nn\nelse\nmkjK=mkjk+Gk\u0000\nmk+1jK\u0000mk+1jk\u0001\nPkjK=Pkjk\u0000G\u0000\nPk+1jk\u0000Pk+1jK\u0001\nGT\nvkjK=vkjk+\u0011\u00001\n2\u0010\ng\u00002(d+1)2\nh+d+1\u0011\nVkjK=Vkjk+\u0011\u00001\n3C4\nG=Pkjk~FT\nkP\u00001\nk+1jk\nW =Vk+1jK\u0000Vk+1jk\nw=vk+1jK\u0000vk+1jk\ng=\u0011\u00001\n1\u0010\nw\u00002(d+1)2\nn\u0011\nh=d+1\ndTrn\nC3C4(C3C4\u0000Id)\u00001o\n\u00111= 1 +w\u00003(d+1)\nn\n\u00112= 1 +g\u00003d\u00003\nh+d+1\n\u00113= 1 +g\u0000d\u00001\nh\u0000d\u00001\nC3=EkjK\u0014\u0010\nM\u00001\nxW(M\u00001\nx)T\u0011\u00001\u0015\n=EkjK\u0002\nMT\nxW\u00001Mx\u0003\nC4=EkjKh\nM\u00001\nxW(M\u00001\nx)Ti\n=EkjKh\u0000\nMT\nxW\u00001Mx\u0001\u00001i\nTable VII. For a Gaussian distributed vector x\u0018N (m;P ),\nthe expected value of MxVMT\nxcan be approximated using\nthird order Taylor expansion,\nC1\u0019(M(m)VM(m)T)\u00001\n+nxX\ni=1nxX\nj=1d2(MxVMT\nx)\u00001\ndx[i]dx[j]\f\f\f\f\f\nx=mP[i;j](19)5\nwhere x[i]is theith element of x,P[i;j]is thei;jth element\nofP. The necessary differentiations are\ndMxVMT\nx\ndx[j]=dMx\ndx[j]VMT\nx+MxVdMT\nx\ndx[j](20a)\nd2MxVMT\nx\ndx[i]dx[j]=d2Mx\ndx[i]dx[j]VMT\nx+dMx\ndx[j]VdMT\nx\ndx[i]\n+dMx\ndx[i]VdMT\nx\ndx[j]+MxVd2MT\nx\ndx[i]dx[j](20b)\nand, for any function Nx=N(x),\nd2N\u00001\nx\ndx[i]dx[j]=N\u00001\nxdNx\ndx[j]N\u00001\nxdNx\ndx[i]N\u00001\nx\u0000N\u00001\nxd2Nx\ndx[i]dx[j]N\u00001\nx\n+N\u00001\nxdNx\ndx[i]N\u00001\nxdNx\ndx[j]N\u00001\nx (20c)\nThe expected values C2,C3andC4can be approximated\nanalogously.\nVI. S IMULATION STUDY\nIn this section we present the results of a simulation study.\nIn all simulations, the dimension of the extent is d= 2.\nA. Implemented smoothers\nThree different smoothers were implemented.\n1) Conditional GIW model with constant velocity motion\nmodel (CCV): The state vector contains 2D Cartesian position\nand velocity, xk= [px\nk; py\nk; vx\nk; vy\nk]T,nx= 4ands= 2. The\nfollowing models are used,\nFk=\u00141T\n0 1\u0015\n; (21a)\nDk=\u001b2\na\"\nT4\n4T3\n2\nT3\n2T2#\n; (21b)\nHk=\u00021 0\u0003\n: (21c)\nA=Id, andnk= 100 , whereTis the sampling time.\n2) Factorized GIW model with constant velocity motion\nmodel (FCV): The state vector contains 2D Cartesian position\nand velocity, xk= [px\nk; py\nk; vx\nk; vy\nk]T, andnx= 4. The\nfollowing models are used,\nfk(x) =\u0014I2TI2\n02\u00022I2\u0015\nx; (22a)\nQk=\u001b2\na\"\nT4\n4I2T3\n2I2\nT3\n2I2T2I2#\n(22b)\n~Hk=\u0002I202\u00022\u0003\n(22c)\nnk= 100 , andMx=Id.\n3) Factorized GIW model with coordinated turn motion\nmodel (FCT): The state vector contains 2D Cartesian positionand velocity, as well as turn-rate, xk= [px\nk; py\nk; vx\nk; vy\nk; !k]T,\nandnx= 5. The following models are used,\nfk(xk) =2\n6641 0sin(T!k)\n!k\u00001\u0000cos(T!k)\n!k0\n0 11\u0000cos(T!k)\n!ksin(T!k)\n!k0\n0 0 cos( T!k)\u0000sin(T!k) 0\n0 0 sin( T!k) cos(T!k) 0\n0 0 0 0 13\n775xk;(23a)\nQk=Gdiag\u0000\n[\u001b2\na; \u001b2\na; \u001b2\n!]\u0001\nGT; (23b)\nG=2\n4T2\n2I202\u00021\nTI202\u00021\n01\u00022 13\n5 (23c)\nMx=\u0014cos(T!)\u0000sin(T!)\nsin(T!) cos(T!)\u0015\n; (23d)\n~Hk=\u0002I202\u00023\u0003\n(23e)\nandnk=1. For the matrix transformation function Mxwe\nhave the following,\nM\u00001\nx=MT\nx (24a)\ndMx\ndx[i]=8\n<\n:Th\n\u0000sin(T!)\u0000cos(T!)\ncos(T!)\u0000sin(T!)i\ni= 5\n02\u00022 i6= 5(24b)\nd2Mx\ndx[i]dx[j]=8\n<\n:T2h\n\u0000cos(T!) sin(T!)\n\u0000sin(T!)\u0000cos(T!)i\ni;j= 5\n02\u00022 i;j6= 5(24c)\nB. Simulated scenarios\nWe focused on two types of scenarios: in the first the true\ntracks were generated by a constant velocity model; in the\nsecond the true tracks were generated by a coordinated turn\nmodel. This allows us to test the different smoothers both\nwhen their respective motion models match the true model,\nand when there is motion model mis-match.\nThe CV tracks were generated using the CV model (21a)\nand (21b) with \u001ba= 1; the extent’s major axis was simulated\nto be aligned with the velocity vector of the extended object.\nThe CT tracks were generated using the CT model (23a) and\n(23b) with\u001ba= 1 and\u001b!=\u0019=180; the extent’s major axis\nwas simulated to be aligned with the velocity vector of the\nextended object.\nFor both motion models, in each time step k, a detection\nprocess was simulated by first sampling a probability of detec-\ntionpD, and, if the object is detected, sampling Nzdetections\nusing a Gaussian likelihood. We simulated two combinations:\n(pD; Nz) = (0:25;10)and(pD; Nz) = (0:75;10).\nC. Performance evaluation\nFor performance evaluation of extended object estimates\nwith ellipsoidal extents, a comparison study has shown that\namong six compared performance measures, the Gaussian\nWassterstein Distance ( GWD ) metric is the best choice [28].\nThe GWD is defined as [29]\n\u0001kj`=kpk\u0000^pkj`k2(25)\n+ Tr\u0012\nXk+^Xkj`\u00002\u0010\nX1\n2\nk^Xkj`X1\n2\nk\u00111\n2\u0013\n;6\nwhere the expected extended object state is\n^\u0018kj`=Ep(\u0018kjZ1:`)[\u0018k] =\u0010\n^xkj`;^Xkj`\u0011\n(26)\n=\u0012\nmkj`;Vkj`\nvkj`\u00002d\u00002\u0013\n(27)\nandpkis the position part of the extended object state vector\nxk.\nD. Results\nWe show results for estimates ^\u0018kj`for`2fk\u00001; k; Kg,\ni.e., prediction, filtering and smoothing. Results for true tracks\ngenerated by a CV model are shown in Figure 1; results for\ntrue tracks generated by a CT model are shown in Figure 2.\nWe see that in all cases, as expected, the smoothing errors\nare smaller than the filtering errors, which are smaller than\nthe prediction errors. This confirms that the derived smoothers\nwork as they should. It is also in accordance with expectations\nthat the performance is worse when the probability of detection\nis lower. Perhaps counter-intuitive is that for CV true motion,\nthe FCT smoother performs best despite the modelling error in\nthe motion model. We believe that this is due, at least in part,\nto the standard assumption that the orientation of the extent\nellipse is aligned with the velocity vector. The motion noice on\nthe velocity vector introduces rotations on the extent ellipse,\nand the motion model used in FCT captures these rotations\nbetter.\nVII. C ONCLUSIONS AND FUTURE WORK\nThis paper presented Bayesian smoothing for the random\nmatrix model used in extended object tracking. Two variants of\nGaussian inverse Wishart state densities exist in the literature,\na conditional and a factorised; closed form Bayesian smooth-\ning was derived for both of them. The derived smoothers were\nimplemented and tested in a simulation scenario. In future\nwork the smoothers will be used with real data, e.g., data\nfrom camera, lidar or radar.\nAPPENDIX\nA. Preliminary results\nIn this appendix we present some preliminary results that\nare used in the proofs of Theorems 1 and 2. The first two\nLemmas regard products, and ratios, of inverse Wishart pdfs,\nrespectively.\nLemma 1: The product of two inverse Wishart pdfs is\nproportional to an inverse Wishart pdf,\nIWd(X;a;A)IWd(X;b;B)\n/IWd(X;a+b;A+B) (28)\n\u0003\nProof: This follows from the definition of the inverse Wishart\npdf.\nLemma 2: The fraction of two inverse Wishart pdfs is\nproportional to an inverse Wishart pdf,\nIWd(X;a;A)\nIWd(X;b;B)/IWd(X;a\u0000b;A\u0000B) (29)\u0003\nProof: This follows from the definition of the inverse Wishart\npdf.\nThe following two Lemmas are related to the use Wishart\ntransition densities and inverse Wishart state densities for the\nextent matrix.\nLemma 3: For Wishart and inverse Wishart pdfs, the fol-\nlowing holds,\nWd\u0012\nY;n;MXMT\nn\u0013\n/IWd\u0010\nX;n;nM\u00001Y\u0000\nM\u00001\u0001T\u0011\n: (30)\n\u0003\nProof: This follows from the definitions of the Wishart pdf\nand the inverse Wishart pdf.\nLemma 4:Z\nWd(X;v;V)IWd(V;w;W ) dV\n=GBII\nd\u0012\nX;v\n2;w\u0000d\u00001\n2;W;0d\u0002d\u0013\n(31)\nProof: see [23, Prob. 5.33].\nLemma 5:Z\nIWd(X;v;V)Wd(V;w;W ) dV\n=GBII\nd\u0012\nX;w\n2;v\u0000d\u00001\n2;W;0d\u0002d\u0013\n(32)\nProof: see [22, Thm. 3].\nFor approximations of densities, the Kullback-Leibler di-\nvergence ( KL-div) is often minimised to find the optimal\napproximation. The following Lemma is about approximation\nby a factorised density.\nLemma 6: For two random variables xandX, with joint\ndensityp(x;X), the factorised density q?(x)q?(X)that min-\nimises the KL-div top(x;X),\nq?(x)q?(X) = arg min\nq(x)q(X)KL (p(x;X)jjq(x)q(X))(33)\nis given by the marginals,\nq?(x) =Z\np(x;X)dX (34)\nq?(X) =Z\np(x;X)dx (35)\n\u0003\nProof: This is a previously know result that follows from the\ndefinition of the Kullback-Leibler divergence.\nApproximation of matrix valued densities as Wishart, or\ninverse Wishart, densities, by minimisation of the KL-div, is\npresented in [22, Thm. 1] and [30, Thms. 3 & 4]. The KL-\ndiv minimisation leads to matching of the expected logarithm\nof the determinant of the extent matrix, as well as either the\nexpected extent matrix, or the expected inverse extent matrix.\nA simpler closed form approximation is obtained if instead the\nexpected random matrix and the expected inverse are match,\ni.e., not matching the expected log-determinant; this is shown\nin the following four Lemmas.7\nTime0 10 20 30 40 50 60 70 80 90 100∆k|k−1\n81012Prediction errors\nTime0 10 20 30 40 50 60 70 80 90 100∆k|k\n6789Filtering errors\nTime0 10 20 30 40 50 60 70 80 90 100∆k|K\n345Smoothing errors\nTime0 10 20 30 40 50 60 70 80 90 100∆k|k−1\n46810Prediction errors\nTime0 10 20 30 40 50 60 70 80 90 100∆k|k\n33.544.5Filtering errors\nTime0 10 20 30 40 50 60 70 80 90 100∆k|K\n1.522.53Smoothing errors\nFig. 1. Results for object tracks generated by a CV model, with probability of detection pD= 0:25(left) andpD= 0:75(right) and, if detected, Nz= 10\nobject measurements (both). CCV is shown in blue, FCV is shown in red, and FCT is shown in orange. Each line is the median from 1000 Monte Carlo\nsimulations.\nTime0 10 20 30 40 50 60 70 80 90 100∆k|k−1\n1015202530Prediction errors\nTime0 10 20 30 40 50 60 70 80 90 100∆k|k\n810121416Filtering errors\nTime0 10 20 30 40 50 60 70 80 90 100∆k|K\n4567Smoothing errors\nTime0 10 20 30 40 50 60 70 80 90 100∆k|k−1\n4681012Prediction errors\nTime0 10 20 30 40 50 60 70 80 90 100∆k|k\n33.544.55Filtering errors\nTime0 10 20 30 40 50 60 70 80 90 100∆k|K\n234Smoothing errors\nFig. 2. Results for object tracks generated by a CT model, with probability of detection pD= 0:25(left) andpD= 0:75(right) and, if detected, Nz= 10\nobject measurements (both). CCV is shown in blue, FCV is shown in red, and FCT is shown in orange. Each line is the median from 1000 Monte Carlo\nsimulations.\nLemma 7: By matching the expected values E[X]and\nE[X\u00001], the inverse Wishart density IWd(X;v;V)can\nbe approximated by a Wishart density Wd(X;w;W )with\nparameters\nw=v\u0000d\u00001 (36)\nW=V\n(v\u00002d\u00002)(v\u0000d\u00001)(37)\n\u0003\nProof: this follows from the definitions of the expected values,\nsee [23].\nLemma 8: By matching the expected values E[X]and\nE[X\u00001], the Wishart density IWd(X;w;W )can be ap-\nproximated by an inverse Wishart density Wd(X;v;V)withparameters\nv=w+d+ 1 (38)\nV=Ww(w\u0000d\u00001) (39)\n\u0003\nProof: this follows from the definitions of the expected values,\nsee [23].\nLemma 9: By matching the expected values E[X]\nandE[X\u00001], the generalized Beta type 2 density\nGBII\nd\u0000\nX;a\n2;b\n2;A;0d\u0002d\u0001\ncan be approximated by a Wishart8\ndensityWd(X;w;W )with parameters\nw=ab\na+b\u0000d\u00001(40)\nW=(a+b\u0000d\u00001)A\nb(b\u0000d\u00001)(41)\n\u0003\nProof: this follows from the definitions of the expected values,\nsee [23].\nLemma 10: By matching the expected values E[X]\nandE[X\u00001], the generalized Beta type 2 density\nGBII\nd\u0000\nX;a\n2;b\n2;A;0d\u0002d\u0001\ncan be approximated by an\ninverse Wishart density Wd(X;v;V)with parameters\nv=ab\na+b\u0000d\u00001+d+ 1\n=(a+d+ 1)(b+d+ 1)\u00002(d+ 1)2\na+b\u0000d\u00001(42)\nV=a(a\u0000d\u00001)\na+b\u0000d\u00001A (43)\n\u0003\nProof: this follows from the definitions of the expected values,\nsee [23].\nB. Conditional model smoothing\nFor conditional densities (4a) and the transition density\n(13a), under Assumptions 1 and 2, the Bayesian smoothing\n(3) leads to a conditional smoothed density\np(\u0018kjZ1:K) =p(xkjXk;Z1:K)p(XkjZ1:K) (44a)\nwhere\np(xkjXk;Z1:K) =p(xkjXk;Z1:k)\n\u0002Zp(xk+1jxk;Xk)p(xk+1jXk;Z1:K)\np(xk+1jXk;Z1:k)dxk+1 (44b)\np(XkjZ1:K) =p(XkjZ1:k)\n\u0002Zp(Xk+1jXk)p(Xk+1jZ1:K)\np(Xk+1jZ1:k)dXk+1 (44c)\nThe proof of (44) is given in (45). We get the following\nsmoothed conditional GIW density\np(xkjXk;Z1:K) =N\u0000\nxk;mkjK;PkjK\nXk\u0001\n(46a)\np(XkjZ1:K) =IWd\u0000\nXk;vkjK;VkjK\u0001\n(46b)\nwith the parameters given in Table IV. The proof of (46a) is\nsimple; the details follow the proof of the RTS-smoother, see,\ne.g., [31, Thm. 8.2]. The proof of (46b) is given in (47).\nC. Factorized model smoothing\nIn the factorized case, there is no known analytical solution\nthat gives a smoothed density of the desired factorized form;\ntherefore approximations are necessary. Factorized density\napproximations are common in so called variational infer-\nence, and the factors are typically found by minimising the\nKullback-Leibler divergence, see, e.g., [32, Ch. 10]. To find\na factorised smoothed density, we apply Lemma 6 to thesmoothed joint density p(\u0018kjZ1:K), given in (3), and obtain\nthe following two smoothing equations,\np(xkjZ1:K) =Z\np(\u0018kjZ1:K)dXk (48a)\n=p(xkjZ1:k)Zp(xk+1jxk)p(xk+1jZ1:K)\np(xk+1j;Z1:k)dxk+1(48b)\nand\np(XkjZ1:K) =Z\np(\u0018kjZ1:K)dxk (49a)\n=p(XkjZ1:k)\u0002 (49b)ZZp(Xk+1jxk;Xk)p(Xk+1jZ1:K)\np(Xk+1jZ1:k)p(xkjZ1:K) dXk+1dxk\nThe proof of the marginalisation (48) is given in (50), and\nthe proof of the marginalisation (49) is given in (51).\nFor the kinematic vector, we have that for Gaussian densities\np(xk+1jZ1:K)andp(xk+1jZ1:k), see (5), and a Gaussian tran-\nsition density p(xk+1jxk), see (17), the smoothed kinematic\nstate density is Gaussian with parameters given by the standard\nRTS-smoothing backwards step, given in, e.g., [31, Thm. 8.2].\nWe get the result in Table VII.\nFor the extent matrix, the smoothing (49) does not have\nan analytical solution, and approximations are necessary. The\nderivation of the result in Table VII is given in (52).\nREFERENCES\n[1] K. Granstr ¨om, M. Baum, and S. Reuter, “Extended Object Tracking:\nIntroduction, Overview and Applications,” Journal of Advances in\nInformation Fusion , vol. 12, no. 2, pp. 139–174, Dec. 2017.\n[2] W. Koch, “Bayesian approach to extended object and cluster tracking\nusing random matrices,” IEEE Transactions on Aerospace and Electronic\nSystems , vol. 44, no. 3, pp. 1042–1059, Jul. 2008.\n[3] M. Feldmann, D. Fr ¨anken, and J. W. Koch, “Tracking of extended\nobjects and group targets using random matrices,” IEEE Transactions\non Signal Processing , vol. 59, no. 4, pp. 1409–1420, Apr. 2011.\n[4] M. Baum and U. Hanebeck, “Extended object tracking with random\nhypersurface models,” IEEE Transactions on Aerospace and Electronic\nSystems , vol. 50, no. 1, pp. 149–159, Jan. 2013.\n[5] N. 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Svensson, “Gamma Gaussian inverse-\nWishart Poisson multi-Bernoulli Filter for Extended Target Tracking,”\ninProceedings of the International Conference on Information Fusion ,\nHeidelberg, Germany, Jul. 2016.9\np(\u0018kjZ1:K) =p(\u0018kjZ1:k)Zp(\u0018k+1j\u0018k)p(\u0018k+1jZ1:K)\np(\u0018k+1jZ1:k)d\u0018k+1 (45a)\nA1\u0019p(\u0018kjZ1:k)Zp(xk+1jxk;Xk+1)p(Xk+1jXk)p(xk+1jXk+1;Z1:K)p(Xk+1jZ1:K)\np(xk+1jXk+1;Z1:k)p(Xk+1jZ1:k)d\u0018k+1 (45b)\nA2\u0019p(\u0018kjZ1:k)Zp(xk+1jxk;Xk)p(Xk+1jXk)p(xk+1jXk;Z1:K)p(Xk+1jZ1:K)\np(xk+1jXk;Z1:k)p(Xk+1jZ1:k)d\u0018k+1 (45c)\n=p(xkjXk;Z1:k)p(XkjZ1:k)Zp(xk+1jxk;Xk)p(xk+1jXk;Z1:K)\np(xk+1jXk;Z1:k)dxk+1Zp(Xk+1jXk)p(Xk+1jZ1:K)\np(Xk+1jZ1:k)dXk+1(45d)\n=p(xkjXk;Z1:k)Zp(xk+1jxk;Xk)p(xk+1jXk;Z1:K)\np(xk+1jXk;Z1:k)dxk+1p(XkjZ1:k)Zp(Xk+1jXk)p(Xk+1jZ1:K)\np(Xk+1jZ1:k)dXk+1 (45e)\n=p(xkjXk;Z1:K)p(XkjZ1:K) (45f)\np(XkjZ1:K) =IWd\u0000\nXk;vkjk;Vkjk\u0001ZWd\u0010\nXk+1;nk;Xk\nnk\u0011\nIWd\u0000\nXk+1;vk+1jK;Vk+1jK\u0001\nIWd\u0000\nXk+1;vk+1jk;Vk+1jk\u0001 dXk+1 (47a)\nL2/IWd\u0000\nXk;vkjk;Vkjk\u0001Z\nWd\u0012\nXk+1;nk;Xk\nnk\u0013\nIWd\u0010\nXk+1; ~vk+1;~Vk+1\u0011\ndXk+1 (47b)\nL3=IWd\u0000\nXk;vkjk;Vkjk\u0001Z\nIWd(Xk;nk;Xk+1nk)IWd\u0010\nXk+1; ~vk+1;~Vk+1\u0011\ndXk+1 (47c)\nL7\u0019IWd\u0000\nXk;vkjk;Vkjk\u0001Z\nWd\u0012\nXk;nk\u0000d\u00001;Xk+1nk\n(n\u00002d\u00002)(n\u0000d\u00001)\u0013\nIWd\u0010\nXk+1; ~vk+1;~Vk+1\u0011\ndXk+1 (47d)\nL4=IWd\u0000\nXk;vkjk;Vkjk\u0001\nGBII\nd \nXk;nk\u0000d\u00001\n2;~vk+1\u0000d\u00001\n2;nk~Vk+1\n(n\u00002d\u00002)(n\u0000d\u00001);0d\u0002d!\n(47e)\nL10\u0019IWd\u0000\nXk;vkjk;Vkjk\u0001\nIWd \nXk;~vk+1nk\u00002(d+ 1)2\n~vk+1+nk\u00003d\u00003;nk~Vk+1\n~vk+1+nk\u00003d\u00003!\n(47f)\nL1/IWd0\n@Xk;vkjk+(vk+1jK\u0000vk+1jk)\u00002(d+1)2\nnk\n1 +vk+1jK\u0000vk+1jk\u00003(d+1)\nnk;Vkjk+Vk+1jK\u0000Vk+1jk\n1 +vk+1jK\u0000vk+1jk\u00003d\u00003\nnk1\nA (47g)\n[13] M. 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Available: http://arxiv.org/10\np(xkjZ1:K) =Z\np(\u0018kjZ1:K)dXk (50a)\n=Z\np(\u0018kjZ1:k)Zp(\u0018k+1j\u0018k)p(\u0018k+1jZ1:K)\np(\u0018k+1jZ1:k)d\u0018k+1dXk (50b)\nA3=Z\np(xkjZ1:k)p(XkjZ1:k)ZZp(xk+1jxk)p(Xk+1jxk;Xk)p(xk+1jZ1:K)p(Xk+1jZ1:K)\np(xk+1j;Z1:k)p(Xk+1jZ1:k)dxk+1dXk+1dXk (50c)\n=p(xkjZ1:k)Zp(xk+1jxk)p(xk+1jZ1:K)\np(xk+1j;Z1:k)dxk+1Z\np(XkjZ1:k)Zp(Xk+1jxk;Xk)p(Xk+1jZ1:K)\np(Xk+1jZ1:k)dXk+1dXk(50d)\n=p(xkjZ1:k)Zp(xk+1jxk)p(xk+1jZ1:K)\np(xk+1j;Z1:k)dxk+1 (50e)\np(XkjZ1:K) =Z\np(\u0018kjZ1:K)dxk (51a)\n=Z\np(\u0018kjZ1:k)Zp(\u0018k+1j\u0018k)p(\u0018k+1jZ1:K)\np(\u0018k+1jZ1:k)d\u0018k+1dxk (51b)\nA3=Z\np(xkjZ1:k)p(XkjZ1:k)ZZp(xk+1jxk)p(Xk+1jxk;Xk)p(xk+1jZ1:K)p(Xk+1jZ1:K)\np(xk+1j;Z1:k)p(Xk+1jZ1:k)dxk+1dXk+1dxk (51c)\n=p(XkjZ1:k)ZZp(Xk+1jxk;Xk)p(Xk+1jZ1:K)\np(Xk+1jZ1:k)p(xkjZ1:k)Zp(xk+1jxk)p(xk+1jZ1:K)\np(xk+1j;Z1:k)dxk+1dXk+1dxk (51d)\n=p(XkjZ1:k)ZZp(Xk+1jxk;Xk)p(Xk+1jZ1:K)\np(Xk+1jZ1:k)p(xkjZ1:K) dXk+1dxk (51e)\nabs/1510.01225\n[27] E. Saritas and U. Orguner, “A random matrix measurement update\nusing taylor-series approximations,” in Proceedings of the International\nConference on Information Fusion , Cambridge, UK, Jul. 2018, pp.\n1756–1763.\n[28] S. Yang, M. Baum, and K. Granstr ¨om, “Metric for performance evalua-\ntion of elliptic extended object tracking methods,” in IEEE International\nConference on Multisensor Fusion and Integration for Intelligent Sys-\ntems, Baden-Baden, Germany, Sep. 2016.\n[29] C. R. Givens and R. M. Shortt, “A class of Wasserstein metrics for\nprobability distributions.” The Michigan Mathematical Journal , vol. 31,\nno. 2, pp. 231–240, 1984.\n[30] K. Granstr ¨om and U. Orguner, “On Spawning and Combination of\nExtended/Group Targets Modeled with Random Matrices,” IEEE Trans-\nactions on Signal Processing , vol. 61, no. 3, pp. 678–692, Feb. 2013.\n[31] S. S ¨arkk¨a,Bayesian Filtering and Smoothing . Cambridge University\nPress, 2013.\n[32] C. M. Bishop, Pattern recognition and machine learning . New York,\nUSA: Springer, 2006.11\np(XkjZ1:K) =p(XkjZ1:k)ZZp(Xk+1jxk;Xk)p(Xk+1jZ1:K)\np(Xk+1jZ1:k)p(xkjZ1:K) dXk+1dxk (52a)\n=p(XkjZ1:k)ZZWd\u0010\nXk+1;nk;M(xk)XkMT(xk)\nnk\u0011\nIWd\u0000\nXk+1;vk+1jK;Vk+1jK\u0001\nIWd\u0000\nXk+1;vk+1jk;Vk+1jk\u0001 dXk+1p(xkjZ1:K) dxk (52b)\nL2/p(XkjZ1:k) (52c)\n\u0002ZZ\nWd\u0012\nXk+1;nk;M(xk)XkMT(xk)\nnk\u0013\nIWd\u0000\nXk+1;vk+1jK\u0000vk+1jk;Vk+1jK\u0000Vk+1jk\u0001\ndXk+1p(xkjZ1:K) dxk\nL3=p(XkjZ1:k)ZZ\nIWd\u0000\nXk;nk;nkM\u00001(xk)Xk+1M\u0000T(xk)\u0001\nIWd(Xk+1;w;W ) dXk+1p(xkjZ1:K) dxk (52d)\nL7\u0019p(XkjZ1:k)\n\u0002ZZ\nWd\u0012\nXk;nk\u0000d\u00001;nkM\u00001(xk)Xk+1M\u0000T(xk)\n(nk\u0000d\u00001)(nk\u00002d\u00002)\u0013\nIWd(Xk+1;w;W ) dXk+1p(xkjZ1:K) dxk (52e)\nL4=p(XkjZ1:k)Z\nGBII\nd\u0012\nXk;nk\u0000d\u00001\n2;w\u0000d\u00001\n2;nkM\u00001(xk)WM\u0000T(xk)\n(nk\u0000d\u00001)(nk\u00002d\u00002);0d\u0002d\u0013\np(xkjZ1:K) dxk (52f)\nL10\u0019p(XkjZ1:k)Z\nIWd\u0012\nXk;wnk\u00002(d+ 1)2\nw+nk\u00003d\u00003;nkM\u00001(xk)WM\u0000T(xk)\nw+nk\u00003d\u00003\u0013\nN\u0000\nxk;mkjK;PkjK\u0001\ndxk (52g)\n[22, Thm. 2]\u0019p(XkjZ1:k)Z\nIWd\u0012\nXk;wnk\u00002(d+ 1)2\nw+nk\u00003d\u00003;nkVxk\nw+nk\u00003d\u00003\u0013\nWd\u0000\nVxk;h;h\u00001C4\u0001\ndVxk (52h)\nL5\u0019p(XkjZ1:k)GBII\nd\u0012\nXk;h\n2;1\n2(w\u0000d\u00001)(nk\u0000d\u00001)\nw+nk\u00003d\u00003;nkh\u00001C4\nw+nk\u00003d\u00003;0\u0013\n(52i)\nL10\u0019IWd\u0000\nXk;vkjk;Vkjk\u0001\nIWd\u0012\nXk;\u0011\u00001\n2\u0012\ng\u00002(d+ 1)2\nh+d+ 1\u0013\n;\u0011\u00001\n3C4\u0013\n(52j)\nL1/IWd\u0012\nXk;vkjk+\u0011\u00001\n2\u0012\ng\u00002(d+ 1)2\nh+d+ 1\u0013\n;Vkjk+\u0011\u00001\n3C4\u0013\n(52k)"    },    {        "title": "1902.07469v1.On_the_Probability_Density_of_the_Nuclei_in_a_Vibrationally_Excited_Molecule.pdf",        "content": "On the Probability Density of the Nuclei in a Vibrationally Excited\nMolecule\nAxel Schild, Laboratory for Physical Chemistry, ETH Z urich, Switzerland\nApril 19, 2022\nAbstract\nFor localized and oriented vibrationally excited molecules, the one-body probability density of the nuclei (one-\nnucleus density) is studied. Like the familiar and widely used one-electron density that represents the probability of\n\fnding an electron at a given location in space, the one-nucleus density represents the probability of \fnding a nucleus\nat a given position in space independent of the location of the other nuclei. In contrast to the full many-dimensional\nnuclear probability density, the one-nucleus density contains less information and may thus be better accessible\nby experiment, especially for large molecules. It also provides a quantum-mechanical view of molecular vibrations\nthat can easily be visualized. We study how the nodal structure of the wavefunctions of vibrationally excited states\ntranslates to the one-nucleus density. It is found that nodes are not necessarily visible: Already for relatively\nsmall molecules, only certain vibrational excitations change the one-nucleus density qualitatively compared to the\nground state. It turns out that there are some simple rules for predicting the shape of the one-nucleus density\nfrom the normal mode coordinates, and thus for predicting if a vibrational excitation is visible in a corresponding\nexperiment.\nQuantum-mechanically, the state of an approximately isolated molecule is described by a wavefunction that depends on\nthe location of all nuclei and electrons. The corresponding probability density, which represents the distribution of the\nparticles in the high-dimensional con\fguration space of their coordinates, is thus di\u000ecult to visualize, to comprehend,\nand also to measure. Notwithstanding, there is an intuitive semi-classical picture of a molecule in chemistry. The\nemergence of this picture is a non-trivial problem[1, 2] and looking at static states[3, 4, 5] as well as the quantum\ndynamics[6, 7, 8, 9] of small isolated molecules can lead to surprising insights. An important role for understanding\nthe chemical picture of a molecule is certainly played by the large di\u000berence of electronic and nuclear masses. This\nmass di\u000berence results in a strong spatial localization of the nuclei compared to the electrons, which in turn motivates\na separation of the molecular wavefunction into a (marginal) nuclear wavefunction and an electronic wavefunction that\nconditionally depends on the location of the nuclei. While such a separation is exact[10], its practical application is\nusually in terms of the Born-Oppenheimer approximation [11] where the e\u000bect of the nuclear motion on the electronic\nwavefunction is neglected and only the chosen position of the nuclei is relevant [12].\nIn the semi-classical picture of a molecule, the theoretical treatment of nuclei and electrons is di\u000berent: The delocalized\nelectrons are considered to be quantum particles, while the comparably localized nuclei are often approximated as\nclassical particles. However, nuclei are also quantum particles and for small molecules the nuclear (many-body)\nprobability densities have often been calculated and analyzed.[13, 14, 15, 16] Also, there has been recent interest in\nmeasuring e.g. the nuclear probability density [17, 18, 19, 20, 21, 22, 23] or the nuclear \rux (current) density.[24, 25, 26]\nAn intriguing example of the quantum nature of the nuclei is the measurement of the nuclear density of a vibrationally\nexcited H+\n2-molecules by means of Coulomb explosion imaging.[21] The measured nuclear density, which depends only\non the relative distance of the two nuclei, shows the expected nodal pattern of vibrationally excited states.\nSimilar measurements can be made for the electronic probability density, as exempli\fed by the measurement of the\ncorrelated two-electron probability density of the H 2molecule.[27] For more than two electrons, however, there is a\ndimensionality problem because multiple electrons have to be measured in coincidence and their probability density is\ndi\u000ecult to grasp for the human intuition which is based on a three-dimensional experience of the world. What can be\ndone instead is to consider the electronic one-body probability density, also known as the one-electron density. The\none-electron density is the marginal density of \fnding one electron at a given location in space independent of where\nthe other electrons are. It is the central quantity of Density Functional Theory [28], it is easily visualized, and it is\ndirectly accessible to experiment if the molecule is localized: For example, an electron scanning tunneling microscope\ndoes essentially measure the one-electron density of a localized molecule and can provide intuitive images of molecules\non surfaces [29]. However, while some information about the electronic state can be extracted from the one-electron\ndensity,[30] this task is in general di\u000ecult: Although the many-electron density is qualitatively di\u000berent for di\u000berent\nexcited states due to the appearance of nodes in the wavefunction, the corresponding one-electron densities may be\nvery similar.\nInspired by the experimental imaging of the one-electron density, the question arises if one-nucleus densities can be\nmeasured and what information about the nuclear state they contain. To measure a one-nucleus density, only the\nposition of one nucleus relative to the lab frame needs to be measured, hence visualization and interpretation of\n1arXiv:1902.07469v1  [physics.chem-ph]  20 Feb 2019Axel Schild\nexperimental data can be more direct than in coincidence measurements of the relative position of multiple nuclei.\nSimilar to the one-electron density, the one-nucleus density can also represent a three-dimensional picture of all nuclei\nif the experiment is insensitive to the type of the nuclei or if the data of di\u000berent types of nuclei are combined. It\ncan provide a straightforward visualization of the quantum state of the nuclei in a molecule and can in this way\ncomplement our understanding of molecular behavior. However, to obtain the one-nucleus density the molecule needs\nto be localized, e.g. on a surface or with the help of a trap[31]. This author is not aware of any experiments that\nprovide the one-nucleus density of a molecular system with an accuracy that can e.g. resolve vibrational excitations,\nbut in principle such experiments are possible.\nClearly, a measurement of the one-nucleus density would provide a picture of the quantum nature of nuclei that is\ndirectly accessible to our spatial conception. But would it provide information about the vibrational state of the\nmolecule, i.e., would the nodal structure of wavefunctions in excited states be visible in the one-nucleus density?\nThis question is investigated in the following with the help of the nuclear wavefunction obtained from a normal\nmode analysis, i.e., obtained from the local harmonic approximation of the potential energy surface for the nuclear\ncon\fguration of lowest energy[32]. The nuclear wavefunction  nucis then a product of a translational, a rotational,\nand a vibrational part,\n nuc= trans\nnuc\u0002 rot\nnuc\u0002 vib\nnuc: (1)\nThe translational part  trans\nnuc represents translation of the whole molecule in space and can always be factored exactly,\nwhile the separation of  rot\nnuc(which describes rotations of the whole molecule) from  vib\nnucis only valid for small dis-\nplacements from the equilibrium con\fguration[33, 34, 35, 36]. Typically, a normal mode analysis aims at computing\nthe vibrational frequencies (and maybe analyzing the normal mode coordinates) while  nucis of little interest. How-\never, if those frequencies are in good agreement with measured frequencies, the function  nucis likely also a good\napproximation to the exact nuclear wavefunction. Then, statements about how qualitative features of  nuctranslate\nto the approximate one-nucleus density can be expected to be also true for the exact one-nucleus density.\nIn the following, the one-nucleus density of the nuclei for di\u000berent states of  vib\nnucis studied and it is investigated how\nthe nodes of the wavefunction in excited states manifest in the one-nucleus density. As shown below, the nuclei are\nrather localized. In analogy to the classical representation of the Nnuclei as Npoints in a three-dimensional space,\nthe sum of the one-nucleus densities for all individual (types of) nuclei yields a density in three-dimensional space\nwhere the probability distribution of each nucleus is clearly visible. For brevity, hereafter this sum is simply called the\none-nucleus density (in analogy to the one-electron density) and the nuclear many-body probability density in the N-\ndimensional con\fguration space is called the N-nucleus density. A detailed description of how the one-nucleus densities\nare obtained is given in the Supporting Information. Here, only the approximations and assumptions are discussed to\npoint out when the approach is applicable. The aim is to determine the one-nucleus density \u001a(R) from the approximate\nnuclear wavefunction  nuc(X), where R= (R1;R2;R3) is a three-component vector and where X= (X1;:::;XN) stands\nfor theNthree-component position vectors Xjof the nuclei. The one-nucleus densities for each nucleus are obtained\nby integrating the N-nucleus densityj nuc(X)j2over all but the coordinates of the selected nucleus,\n\u001aj(R) =Z\n\u0001\u0001\u0001Z\nj nuc(X)j2dXf1\u0001\u0001\u0001Nngnj\f\f\f\f\nXj=R(2)\nfordXf1\u0001\u0001\u0001Nngnj=dX1:::dXj\u00001dXj+1:::dXNn. The one-nucleus density of the molecule is the sum of the one-nucleus\ndensities for all nuclei,\n\u001a(R) =NX\nj=1\u001aj(R): (3)\nThe one-nucleus density gives the probability to \fnd any nucleus in a given region of space, but from the relative\nspatial location it is immediately clear if the nucleus is e.g. an oxygen or a hydrogen nucleus.\nThe nuclear wavefunction is obtained as follows: 1) The Born-Oppenheimer approximation is made to obtain a\nSchr odinger equation for the nuclear wavefunction alone, with a potential energy surface V(X). Only the electronic\nground state is considered. 2) A standard normal mode analysis[32] at one of the minima of Vis made. From\nthis calculation Nnormal mode coordinates qjand the frequencies of their harmonic oscillator (HO) potentials are\nobtained. The nuclear wavefunction is a product of HO wavefunctions in each normal mode coordinate, and coupling\nof rotational and vibrational degrees of freedom is neglected.[34, 35] 3) There are six (\fve for linear molecules) normal\nmodes for which the corresponding HO frequency is zero, representing translation and rotation of the molecule. The\nnuclear wavefunction has the form of (1). It is assumed that  trans\nnuc and rot\nnucare normalized Gaussian functions with\na very small width corresponding to a frequency of 0.5 Eh=~. This acts like a constraint on the coordinates and has\nthe e\u000bect of localizing and orienting the molecule. The resulting probability density can be interpreted either as a\ncut through the full density or as the nuclear density given the molecule is at a certain position and oriented in a\ncertain way. A practical advantage of this choice for  trans\nnuc and rot\nnucis that the wavefunction becomes a product of\n2Axel Schild\nHO eigenfunctions in all modes. The nuclear wavefunction is\n nuc(X) =3NY\nj=1\u001ej\u0000\nqj(X);mj\u0001\n; (4)\nwhere\u001ej(qj;mj) is a HO wavefunction with quantum number mj, and where mj=0for the normal mode coordinates\nfor translation and rotation of the whole molecule. In the Supporting Information, it is described how the high-\ndimensional integral of (2) with wavefunction (4) can be done analytically. All quantum chemical calculations are\nmade with the program Psi4[37] using 3rd order M\u001cller-Plesset pertubation theory and a cc-pVTZ basis set[38].\nR3/a0\n1.5\n0.0 1.5\nR2/a00.01.5R3/a0R3/a0\n1.5\n0.0 1.5\nR2/a00.01.5R3/a0\n1.5\n0.0 1.5\nR2/a00.01.5R3/a0(010)\nυ2\n1.5\n0.0 1.5\nR2/a01.5\n1.5\n0.0 1.5\nR2/a0\n1.5\n0.0 1.5\nR2/a00.01.5R3/a0(001)\nυ2υ1\nυ1υ3\nυ3\nFigure 1: Normal modes of a water molecule (arrows showing the extent and directionality (color) of the nuclear\ndisplacement along the mode): symmetric stretch \u00171, bending mode \u00172, and antisymmetric stretch \u00173. The top-row\nshows sketches of the harmonic oscillator densities along the modes for the \frst excitation of the symmetric stretch\n(010), the bottom row for the \frst excitation of the antisymmetric stretch (001). A light \flling of these densities\nmeans that locally other modes point in the same direction, while a dark \flling means that this is not the case.\nThe \frst example is the one-nucleus density of the water molecule. Figure 1 shows its familiar vibrations as arrows\nindicating the motion of classical nuclei. There are three vibrational normal mode coordinates which are labeled\naccording to the usual spectroscopic notation[39]: the symmetric stretch \u00171, the bending mode (scissoring) \u00172, and\nthe antisymmetric stretch \u00173. The one-nucleus densities of the water molecule for some excitations of these modes are\nshown in Figure 2 as contour plots in the molecular plane, labeled as ( m1;m2;m3), wheremjis the quantum number of\nmode\u0017j. Insets magnify the details of the nuclear density around the oxygen nucleus, as the density there is much\nmore localized compared to the hydrogen nuclei due to the mass di\u000berence.\nThe one-nucleus density of the vibrational ground state for each nucleus looks like a product of Gaussian functions\noriented along the directions of the normal modes (not shown). The one-nucleus densities of the \frst excitation in\neach mode, (100), (010), and (001), show how a node in the \frst excited state of the HO along the corresponding\nnormal modes translates to the one-nucleus density. From the pictures, it seems that the node in the wavefunction\nof one of these modes leads to a depletion of the one-nucleus density along this normal mode coordinate, but not to\nexact nodes or nodal planes in the one-nucleus density. There are two reasons for the absence of exact nodes: First,\na Gaussian distribution for the translational and rotational modes is assumed. Depending on the width of the these\ndistributions, the resulting one-nucleus densities become broader and loose their structure. A very narrow Gaussian\ndistribution is chosen, hence this reason is of minor importance. The main reasons for the absence of exact nodes in\nthe one-nucleus density is that those nodes only exists in con\fguration space, while the reduction of the N-nucleus\ndensity to the one-nucleus density as given in (2) in general does not yield zero anywhere in space.\nTo understand the one-nucleus density in Figure 2, the analytic form of the N-nucleus density needs to be investigated.\nIt is a product of HO densities in all normal modes, because the wavefunction (4) is a product of HO wavefunctions.\nThese 1d-HO densities can be visualized by functions centered at the equilibrium position of the three nuclei, with\nextent and direction as given by the arrows. In Figure 1, the idea is illustrated for the \frst excited state of the bending\nmode, (010), and of the antisymmetric stretch, ( 001).\nSome predictions can be made about the qualitative features of the one-nucleus densities at the nuclei by means of a set\nof simple rules. These rules are called the LOcal COmparison (LOCO) rules, because they are based on a comparison\nof the normal mode coordinates at the location of each nucleus separately. The LOCO rules are as follows: For a\nnucleus, the magnitudes and directions of the displacements along the normal modes are compared. (a) If only one\nnormal mode displaces the nucleus in a certain direction or if there is one normal mode that displaces the nucleus in\na certain direction much stronger than the other normal modes, the nodes of the wavefunction due to an excitation of\n3Axel Schild\n1.5 0.0 1.5\nR2/a00.01.5R3/a0\n1.5 0.0 1.5\nR2/a00.01.5R3/a0\n1.5 0.0 1.50.01.5R3/a01.5 0.0 1.5\nR2/a00.01.5R3/a0\n1.5 0.0 1.5\nR2/a00.01.5R3/a0\n1.5 0.0 1.50.01.5R3/a01.5 0.0 1.5\nR2/a00.01.5\n1.5 0.0 1.5\nR2/a00.01.5(010) (100) (001)\n(002) (101) (111)\n1.5 0.0 1.50.01.5\n(003) (102) (201)\nR2/a0 R2/a0 R2/a0R3/a0R3/a0R3/a0\nFigure 2: Contour plots of the one-nucleus density of a localized and oriented water molecule in the molecular plane\nfor di\u000berent vibrationally excited states. State labels ( m1;m2;m3) indicate the number of quanta in normal modes \u00171,\n\u00172,\u00173, respectively.\nthis normal mode are clearly visible as depletions in the one-nucleus density. (b) If several normal modes displace the\nnucleus in the same direction by similar magnitude, an excitation of one of these modes is not necessarily visible in\nthe one-nucleus density. In general, the more such modes exist, the less likely it is that an excitation in one of these\ncan be recognized in the one-nucleus density. It follows that typically, only the normal modes that displace a nucleus\nthe most in a given direction can have a strong in\ruence on the qualitative shape of the one-nucleus density. (c) If\nthere are two (or more) modes that displace the nucleus in the same direction, simultaneous excitation of these modes\nmay show combination features, as exempli\fed below.\nFor example, the one-nucleus density of states (100), (010) and (001) can be understood from the LOCO rules (a) and\n(b) as follows: In Figure 1, sketches of the harmonic oscillator wavefunctions in the modes are shown. At the hydrogen\nnuclei,\u00171and\u00173point in a similar direction, while \u00172is perpendicular. Thus, the excitation of \u00172(state (010)) leads\nalmost to a nodal plane in the one-nucleus density at the hydrogen nuclei, cf. Figure 2. In contrast, excitation of \u00171\n(state (100)) or \u00173(state (001)) lead to a signi\fcantly less pronounced depletion of the one-nucleus density at the\nequilibrium position of hydrogen. For the oxygen nucleus, the situation is reversed, as \u00171and\u00172point in the same\ndirection and \u00173is perpendicular. Consequently, the one-nucleus density of state (001) shows a depletion at the oxygen\nequilibrium position, while no depletion is seen in state (100). As \u00172displaces the oxygen nucleus stronger than \u00171\n(which is, however, hardly visible on the scale of Figure 1), a depletion due to the node in \u00172is visible in state (010).\nA situation similar to that of state (001) is found for states (002) and (003), i.e. when the HO function of \u00173is in\nits second or third excited state. For (002), there is the expected triple-maximum structure at the oxygen nucleus\n(with a lower central maximum), while the central maximum at the hydrogen nuclei is not visible. For state (003)\nfour maxima are found in the one-nucleus density at the oxygen nucleus (although the two central ones are too weak\nto be clearly seen in Figure 2), while at the hydrogen nuclei still only two maxima can be found.\nAn example for LOCO rule (c) is found if two normal modes are excited that locally have similar magnitude and\ndirection. For state (101) three maxima appear at the hydrogen nuclei, similar to a second excited state of the HO,\nbut the central maximum is strongest while the outer maxima are weaker. At the oxygen nucleus only two maxima\nthat point along the direction of normal \u00173are found, i.e. a combination of the features of the one-nucleus densities\nfor states (100) (only one maximum in the direction of \u00171) and (001) (two maxima in the direction of \u00173). For states\n(102) and (201) the qualitative features of the one-nucleus densities can be explained analogously and in accord with\nthe LOCO rules, especially the four maxima at the hydrogen nuclei and a triple maximum at the oxygen nucleus for\nstate (102), but only a double maximum at the state (201).\nLast, for state (111) the resulting one-nucleus density is a simple combination of features of states (010) and (101),\nbecause\u00172is locally perpendicular to the other two vibrational modes at the hydrogen nuclei. At the oxygen nucleus,\nthe e\u000bect of exciting two modes with similar direction and magnitude, LOCO rule (c), is that three maxima are found\n4Axel Schild\nR2/a00.01.5R3/a0\nOD H\n0.01.5R3/a0\n1.5 0.0 1.5\nR2/a01.5 0.0 1.5\nR2/a01.5 0.0 1.5\nR2/a00.01.5R3/a0υ30.01.5R3/a0\nR2/a0OD H\n0.0R3/a0υ2\n(010)R2/a0OD H\n0.0R3/a0υ1\n(100) (001)\nFigure 3: Top: Normal mode coordinates of the mono-deuterated water molecule: O-D stretch \u00171, bending mode \u00172,\nO-H stretch \u00173. Bottom: Contour plots of the one-nucleus density of a localized and oriented mono-deuterated water\nmolecule in the molecular plane for vibrational states corresponding to the \frst excitations of the normal modes shown\nabove.\nalong coordinate R3upon closer inspection, although the central one is hardly visible.\nThe normal modes of the water molecule have a symmetry with respect to the hydrogen nuclei. If the symmetry of the\nnuclear structure is broken by replacing one hydrogen nucleus with a deuterium nucleus, a very di\u000berent picture for\nthe one-nucleus densities is obtained. The normal mode coordinates for a mono-deuterated water molecule are given\nin Figure 3. The bending mode \u00172is similar to the bending mode of water, but \u00171is now the O-D stretch mode that\nonly displaces the deuterium nucleus strongly, while \u00173is the O-H stretch mode that almost exclusively displaces the\nhydrogen nucleus. Thus, according to the LOCO rules it is expected that any excitation of \u00172is clearly visible in the\none-nucleus density at the hydrogen and deuterium nucleus, that an excitation of \u00171is only visible at the deuterium\nnucleus, and that and excitation of \u00173is only visible at the hydrogen nucleus. The prediction of the LOCO rules is\naccurate, as the one-nucleus densities of the \frst excited states of each of these modes shown in Figure 3 illustrate.\nA consequence of the LOCO rules is that for molecules with many nuclei, only certain (of the lowest) excited states\nare qualitative visible in the one-nucleus density, i.e. those that are the ones displacing a nucleus the most in a certain\ndirection of space. This is indeed the case. For example, for the benzene molecule the one-nucleus density of none\nof the \frst excited states of any normal mode in the molecular plane is qualitatively di\u000berent from the vibrational\nground state. The situation changes if one hydrogen nucleus is exchanged by a deuterium nucleus. In the Supporting\nInformation, the one-nucleus densities for the only two normal mode coordinates that strongly displace the deuterium\nnucleus are shown. Further examples given in the Supporting Information are an analysis of ethene and mono-\ndeuterated ethene as molecules that contain less nuclei than benzene but where similar e\u000bects are found, and methane\nas an example of a non-planar molecule. The TOC graphic shows the one-nucleus density of one of the vibrational\nstates responsible for the blue color of water.[40]\nIn conclusion, it is found that similar to how a one-electron density provides understanding of the behavior of electrons,\nthe one-nucleus densities of molecules can provide an interesting opportunity to measure and to visualize the quantum\nnature of the nuclei: Compared to the N-nucleus density, where the location of all Nnuclei of a molecule needs to be\nknown, the one-nucleus density provides an accessible representation of the nuclear structure that complements the\nclassical picture of the nuclei. While a measurement of the one-nucleus density is undoubtedly di\u000ecult because the\nnuclei are localized in a relatively small region, it is in principle possible, especially for light \\quantum\" nuclei like\nhydrogen or deuterium that are comparably delocalized.\nImportantly, in contrast to the one-electron density where the electronic state is in general not qualitatively visible\n(or where, to this authors knowledge, there are no rules available to predict if this is the case), the vibrational state\nof a molecule may be directly visible in the one-nucleus density. The presented LOCO rules allow to predict, from a\nnormal mode analysis of the nuclear wavefunction, without much e\u000bort which vibrational excitations would be visible\nin the one-nucleus density and thus which molecules are rewarding targets for experimental investigations.\nAcknowledgement\nThe author is grateful to J. Manz (Freie Universit at Berlin) for stimulating and supporting this work and to E.K.U.\nGross (MPI \u0016\bHalle) for extensive discussions about the topic.\nSupporting Information Available: Details of the computation and further examples of one-nucleus densities.\n5Axel Schild\nSupporting Information for \\On the Probability Density of the Nuclei in a\nVibrationally Excited Molecule\": Computational Details\n1 The local harmonic approximation\nThe local harmonic approximation is brie\ry reviewed. For a detailed description, see [32]. In the Born-Oppenheimer\napproximation, the nuclear wavefunction is obtained from\n \n\u0000NX\nj=1~2@2\nXj\n2Mj+V(X)!\n\t(X) =E\t(X) (5)\nwith masses Mjfor the coordinates Xj. Around a minimum XeqofV, the potential is expanded in a Taylor series to\nsecond order in terms of the displacement coordinates x=X\u0000XeqandV(Xeq) is set to zero. Additionally, mass-weighted\ndisplacement coordinates ~ xj=pMjxjare introduced. Then (5) becomes\n \n\u0000NX\nj=1~2@2\n~ xj\n2+1\n2NX\nj=1NX\nk=1Ujk~ xj~ xk!\n\t(X(~ x)) =E\t(X(~ x)) (6)\nwith\nUjk=@Xj@XkV(X)\f\f\nX=XeqpMjMk(7)\nThe matrix Uis diagonalized by the matrix of eigenvectors Q,\nQT\u0001U\u0001Q=\n (8)\nwhereQTQ=QQT=diag (1) and\n=diag (!2) is the diagonal matrix of the eigenvalues !2\nj. This yields\nNX\nj=1NX\nk=1Ujk~ xj~ xk=~ xT\u0001U\u0001~ x=~ xT\u0001Q\u0001QTU\u0001Q\u0001QT\u0001~ x=NX\nj=1!2\nj~ q2\nj (9)\nwith normal mode coordinates\n~ qj=QT~ x=NX\nk=1Qkj~ xk=NX\nk=1pMkQkjxk: (10)\nIn terms of the normal mode coordinates, (5) becomes\n NX\nj=1 \n\u0000~2@2\n~ qj\n2+!2\nj\n2~ q2\nj!!\n\t(X(~ q)) =E\t(X(~ q)) (11)\nWith the coordinate scaling\nqj=r!j\n~~ qj=NX\nk=1r\n!jMk\n~Qkjxk (12)\nthe wavefunction becomes a product\n\t(X(q)) := (q) =NY\nj=1\u001ej(qj;mj) (13)\nwith the eigenfunctions of the quantum harmonic oscillator with unit mass and unit frequency[41, 42]\n\u001ej(qj;mj) =p\n2mjmj!\u00121\n\u0019\u00131\n4\ne\u0000q2\nj\n2bmj\n2cX\nk=0(\u00001)k\n4kk!(mj\u00002k)!qmj\u00002k\nj; (14)\nwhere the \roor function brcgives the largest integer smaller than or equal to r. However, there is one catch: Six\n(or \fve, for linear molecules) of the frequencies !jare zero, as they belong to translations and rotations of the full\nsystem. There are no external potentials that break the translational and rotational invariance of the nuclear system,\nhence it is broken arti\fcially by setting a non-zero value for these frequencies !j. The quantum numbers mjfor these\ndegrees of freedom are set to zero so that the density in these modes is a Gaussian function with a width determined\nby the chosen frequency. In the limit !j!0, a\u000e-distribution is obtained for j\u001ej(qj;0)j2.\n6Axel Schild\n2 The density\nIn the main article, the nuclear one-nucleus density is denoted as \u001a(R) and it is de\fned as sum of the individual one-\nnucleus densities \u001aj(R). In this notes it is described how to obtain \u001aj(R), but the notation is slightly di\u000berent. We\nstart from the density \u001a[N](x1;:::;xN) in theN-dimensional con\fguration space in terms of displacement coordinates\nxj, and we aim at computing the one-nucleus density \u001a[3](x1;x2;x3) of the particle with displacement coordinates\nx1;x2;x3, by integrating over x4;:::;xN. To obtain\u001a(R), all we need to do is to shift \u001a[3](x1;x2;x3) to the equilibrium\nposition of the considered nucleus, to repeat the procedure for the coordinates of all other nuclei, and to add all those\ndensities. We use the notation \u001a[N]because we give the solution of the integrals iteratively, by computing \u001a[N\u00001],\u001a[N\u00002],\netc., with each of the densities in this series depending on one coordinate less compared to the previous density.\nThe explicit expression for the Nn-body or N-coordinate density \u001a[N](x) can now be given. We require that\nZ\n\u0001\u0001\u0001Z\nj\t(X)j2dX1:::dXN=Z\n\u0001\u0001\u0001Z\n\u001a[N](x)dX1:::dXN!=1 (15)\nand have\nZ\nj\u001ej(qj;mj)j2dqj=1; (16)\nwhereR\nrepresents the de\fnite integralR1\n\u00001throughout this text. Thus, the density is\n\u001a[N](x) =JqxNY\nj=1\u0000\n\u001ej(qj;mj)\u00012(17)\nwith the Jacobian determinant Jqxfor the coordinate transformation (12) from xtoqthat ensures that \u001a[N](x) is\nnormalized to one when integrating over all x. Explicitly, with the transformation matrix\nTjk=r\n!jMk\n~Qkj (18)\nwe have\nJqx=jdet(Tjk)j (19)\nThen\n\u001a[N](x) =JqxNY\nj=12mjmj!p\u0019e\u0000q2\njbmj\n2cX\nk=0bmj\n2cX\nl=0(\u00001)k+l\n4k+lk!l!(mj\u00002k)!(mj\u00002l)!q2(mj\u0000(k+l))(20)\n=Jqx\u0000[N](x)NY\nj=1P(2mj)\nj(x) (21)\nwith the Gaussian function\u0003\n\u0000[N](x) =exp \n\u0000NX\nj=1qj(x)2!\n(22)\nand with the polynomial of order 2mj\nP(2mj)\nj(x) =2mjmj!p\u0019bmj\n2cX\nk=0bmj\n2cX\nl=0(\u00001)k+l\n4k+lk!l!(mj\u00002k)!(mj\u00002l)!q(x)2(mj\u0000(k+l))(23)\nBefore inserting the explicit de\fnition of qj(x), we rewrite this polynomial by changing the summation variables to\nk+l!k, (k\u0000l)=2!l, so that\nP(2mj)\nj(x) =2bmj\n2cX\nk=0cmj;kq(x)2(mj\u0000k)=2bmj\n2cX\nk=0p(2(mj\u0000k))\nj (24)\nwith coe\u000ecients\ncmj;k=2mj\u00002kmj!p\u0019LkX\nl=\u0000Lk(\u00001)k\n\u0000k\n2+l\u0001\n!\u0000k\n2\u0000l\u0001\n!\u0000\nmj\u00002\u0000k\n2+l\u0001\u0001\n!\u0000\nmj\u00002\u0000k\n2\u0000l\u0001\u0001\n!(25)\n\u0003A factor\u0019N=2could be added to have \u0000[N](x) properly normalized. Instead, it is included in the de\fnition of the polynomial coe\u000ecients.\n7Axel Schild\nthat are obtained with summation boundaries\nLk=1\n2\u0010jmj\n2k\n\u0000\f\f\fk\u0000jmj\n2k\f\f\f\u0011\n(26)\nAn alternative that is more practical for numerical implementations is to change the summation variables to k+l!k, (k\u0000l)=2!l, so\nthat the coe\u000ecients become\ncmj;k=2mj\u00002kmj!p\u0019LkX\nl=\u0000Lk\n2jl(\u00001)k\n\u0000k+l\n2\u0001\n!\u0000k\u0000l\n2\u0001\n! (mj\u0000(k+l))! (mj\u0000(k\u0000l))!(27)\nLk=jmj\n2k\n\u0000\f\f\fk\u0000jmj\n2k\f\f\f (28)\nwhere the sum over lnow has increments \u0001l=2.\n3 Initial parameters\nThe general form of the N-nucleus density (21) is\n\u001a[N](x) =\u0000[N](x) \nA[N](0)+NX\nk1;k2=1A[N](2)\nk1k2xk1xk2+\u0001\u0001\u0001+NX\nk1;:::;k2M=1A[N](2M)\nk1:::k2Mxk1:::xk2M!\n(29)\nwith the Gaussian function (22) given as\n\u0000[N](x) =exp \n\u0000NX\nj=1NX\nk=1S[N]\njkxjxk!\n(30)\nIt is a multivariate Gaussian function multiplied by a polynomial composed of monomials of degree 0;2;:::;2M, where\nM=NX\nj=1mj (31)\nis the sum of quantum numbers. The superscript [ N] of the coe\u000ecients for the monomials A[N](2\u000b)andS[N]\njkindicates\nthat those are the parameters of the N-nucleus density. Below, we see that when we integrate over coordinate xNwe\nobtain an N\u00001-nucleus density \u001a[N\u00001]that looks like (29), except that the sums terminate at N\u00001and the coe\u000ecients\nchanged. By determining the new coe\u000ecients, we can perform all integrals that are necessary to derive the one-nucleus\ndensity iteratively. Note that the Jacobian determinant is included in the initial coe\u000ecients, cf. (36).\nHowever, \frst we have to determine the initial values of the coe\u000ecients. Inserting the transformation equation of the\ncoordinates (12),(18) into the de\fnition of the Gaussian function (22) yields\nS[N]\njk=NX\nl=1TljTlk (32)\nSimilarly, inserting (12),(18) into the de\fnition of the polynomials (24) for each quantum number jshows that these\npolynomials are a sum (over kj) of monomials\np2(mj\u0000kj)\nj =JqxNX\nl1=1\u0001\u0001\u0001NX\nl2(mj\u0000kj)=1A(2(mj\u0000kj));j\nl1:::l2(mj\u0000kj)xl1:::xl2(mj\u0000kj) (33)\nwith coe\u000ecientsy\nA(2(mj\u0000kj));j\nl1:::l2(mj\u0000kj)=cmj;kjTjl1:::Tjl2(mj\u0000kj) (34)\nThe polynomial occurring in the de\fnition of the density (21) is\nNY\nj=1P(2mj)\nj=2bm1\n2cX\nk1=0\u0001\u0001\u00012bmN\n2cX\nkN=0p2(m1\u0000k1)\n1:::p2(mN\u0000kN)\nN (35)\nyThese coe\u000ecients are symmetric w.r.t. exchange of any two indices.\n8Axel Schild\nBy comparing the de\fnition of the coe\u000ecients in the density (29) with the from of the polynomial (35) we see how\nto obtain the coe\u000ecients A[N](2\u000b): First, we compute all monomial coe\u000ecients (34). Then, we take the tensor product\nalongj,z\nJqx\u0002A(2(m1\u0000k1));1\nl1:::l2(m1\u0000k1)\nA(2(m2\u0000k2));2\nl1:::l2(m2\u0000k2)\n\u0001\u0001\u0001\nA(2(mN\u0000kN));N\nl1:::l2(mN\u0000kN)(36)\nfor all possible combinations of the kj-index. The number of indices of the resulting object is the sum of the number of\nindices of the individual coe\u000ecients and corresponds to the order of the polynomial ( 2\u000b) to which it belongs. Adding\nall the results of the same order yields the initial coe\u000ecients A[N](2\u000b)of theN-nucleus density.\n4 Integration\nNext, we need to integrate over one variable, say, xN. For this purpose, we need the integralx[42]\nZ\ne\u0000ax2+bx+cdx=r\n\u0019\naeb2\n4a+c=I0 (37)\nTaking the derivative of (37) w.r.t. band comparing with the de\fnition of the Hermite polynomials yields\nZ\nxne\u0000ax2+bx+cdx=bn\n2cX\nm=0n!\n2nm!(n\u00002m)!bn\u00002m\nan\u0000mI0 (38)\nIntegrating the density (29) over xMyields\n\u001a[N\u00001](x) =Z\n\u001a[N](x) =MX\ni=02iX\nj=0N\u00001X\nk1=1\u0001\u0001\u0001N\u00001X\nki\u0000j=1^P(2i)\njA[N](2i)\nk1:::k2i\u0000jN:::xk1:::xk2i\u0000jI[N]\nj (39)\nHere,A[N](2i)\nk1:::k2i\u0000jN:::represents the coe\u000ecient for the monomial of order 2iwith2i\u0000jindices that run from 1toN\u00001,\nand the remaining jindices set to N. The operator ^P(2i)\njconstructs the sum of all\u00002i\nj\u0001\npermutations of the indices set\ntoNwith those that run from 1toN\u00001.\nTo perform the integral of the density over the last coordinate xN, we \frst have to group the monomials in (29) for each order according\nto the exponent of xN(which corresponds to the index jofIj) to \fnd\nZ\n\u001a[N](x) =A[N](0)I0+N\u00001X\nk1=1N\u00001X\nk2=1A[N](2)\nk1k2xk1xk2I0+N\u00001X\nk1=1\u0010\nA[N](2)\nk1N+A[N](2)\nNk1\u0011\nxk1I1+A[N](2)\nNNI2\n+N\u00001X\nk1=1N\u00001X\nk2=1N\u00001X\nk3=1N\u00001X\nk4=1A[N](4)\nk1k2k3k4xk1xk2xk3xk4I0\n+N\u00001X\nk1=1N\u00001X\nk2=1N\u00001X\nk3=1\u0010\nA[N](4)\nk1k2k3N+A[N](4)\nk1k2Nk3+A[N](4)\nk1Nk2k3+A[N](4)\nNk1k2k3\u0011\nxk1xk2xk3I1\n+N\u00001X\nk1=1N\u00001X\nk2=1\u0010\nA[N](4)\nk1k2NN+A[N](4)\nk1Nk2N+A[N](4)\nNk1k2N+A[N](4)\nk1NNk2+A[N](4)\nNk1Nk2+A[N](4)\nNNk1k2\u0011\nxk1xk2I2\n+N\u00001X\nk1=1\u0010\nA[N](4)\nk1NNN+A[N](4)\nNk1NN+A[N](4)\nNNk1N+A[N](4)\nNNNk1\u0011\nxk1I3+A[N](4)\nNNNNI4+: : : (40)\nAt each order 2i, the coe\u000ecients of each integral Ijare obtained as sum of the\u00002i\nj\u0001\npossible permutations of the index Noccurring jtimes\nin the coe\u000ecients A[N](2i). Unfortunately, in general A[N](2i)is not symmetric when exchanging two indices. However, it is constructed as\ntensor product of arrays that have this property, hence this symmetry may be exploited to some extent, if desired.\nWe now assume (correctly) that \u001a[N\u00001]has the same functional form as \u001a[N], but with new coe\u000ecients A[N\u00001]andS[N\u00001],\n\u001a[N\u00001](x)!=exp \n\u0000N\u00001X\nj=1N\u00001X\nk=1S[N\u00001]\njkxjxk! \nA[N\u00001](0)+N\u00001X\nk1;k2=1A[N\u00001](2)\nk1k2xk1xk2+\u0001\u0001\u0001+N\u00001X\nk1;:::;k2M=1A[N\u00001](2M)\nk1:::k2Mxk1:::xk2M!\n(41)\nThe integrals I[N]\njthat occur here are of the form (38) and are given by\nI[N]\n0=s\n\u0019\nS[N]\nNNexp \n\u0000N\u00001X\nj=1N\u00001X\nk=1 \nS[N]\njk\u0000S[N]\njNS[N]\nkN\nS[N]\nNN!\nxjxk!\n(42)\nzThese coe\u000ecients are not symmetric w.r.t. exchange of any two indices anymore, but only within certain blocks. This makes the\nequations later a little bit more elaborate.\nxR\nis still the de\fnite integral from x=\u00001 tox=1.\n9Axel Schild\nand\nI[N]\nn=bn\n2cX\nm=0(\u00001)nn!\n4mm!(n\u00002m)!I[N]\n0\u0010\nS[N]\nNN\u0011n\u0000mN\u00001X\nk1=1\u0001\u0001\u0001N\u00001X\nkn\u00002m=1S[N]\nk1N:::S[N]\nkn\u00002mNxk1:::xkn\u00002m (43)\nrespectively.\nEquations (42) and (43) can be derived from (37) and (38) by making the identi\fcations\na=S[N]\nNN b=\u00002N\u00001X\nj=1S[N]\njNxj c=\u0000N\u00001X\nj=1N\u00001X\nk=1S[N]\njkxjxk (44)\nWe see from (42) that because I[N]\n0occurs in all integrals I[N]\nn, the new coe\u000ecients for the exponential are\nS[N\u00001]\njk =S[N]\njk\u0000S[N]\njNS[N]\nkN\nS[N]\nNN(45)\nThe new coe\u000ecients for the polynomial are\nA[N\u00001](2\u000b)\nk1:::k2\u000b=s\n\u0019\nS[N]\nNNMX\ni=\u000b2iX\nj=2(i\u0000\u000b)(\u00001)jj!\n4i\u0000\u000b(i\u0000\u000b)!(j\u00002(i\u0000\u000b))!^P(2i)\njA[N](2i)\nk1:::k2i\u0000jN:::S[N]\nk2i\u0000j+1N:::S[N]\nk2\u000bN\u0010\nS[N]\nNN\u0011j+\u000b\u0000i(46)\nSome remarks are in order: First, we note that the indices k1;::: now only have N\u00001entries. Second, the term\nA[N](2i)\nk1:::k2i\u0000jN:::S[N]\nk2i\u0000j+1N:::S[N]\nk2\u000bNof the last equation has to be read as follows: We take A[N](2i)\nk1:::k2iand set the last jindices\nequal toN. Then, we make a tensor multiplication with so many vectors S[N]\nkjNthat the resulting object has 2\u000bindices.\nThe second step of the integration of \u001a[N](x) overxNis to group the resulting polynomial according to the orders of the monomials and add\nthe respective contributions. The structure of \u001a[N\u00001](x) can be visualized as follows:\n\u00100\n0\u0011\n[0]\b[00] (i=0)\n\u00102\n0\u0011\n[2]\b[00] +\u00102\n1\u0011\n[1]\b[10] +\u00102\n2\u0011\n[0]\b[01;20] (i=1)\n\u00104\n0\u0011\n[4]\b[00] +\u00104\n1\u0011\n[3]\b[10] +\u00104\n2\u0011\n[2]\b[01;20] +\u00104\n3\u0011\n[1]\b[11;30] +\u00104\n4\u0011\n[0]\b[02;21;40] (i=2)\n\u00106\n0\u0011\n[6]\b[00] +\u00106\n1\u0011\n[5]\b[10] +\u00106\n2\u0011\n[4]\b[01;20] +\u00106\n3\u0011\n[3]\b[11;30] +\u00106\n4\u0011\n[2]\b[22;31;40] +\u00106\n5\u0011\n[1]\b[12;31;50] +\u00106\n6\u0011\n[0]\b[03;22;41;60]\n(i=3)\nThe binomial coe\u000ecients are a reminder of the permutations induced by the permutation operator ^P(2i)\njacting on A[N](2i). The number\n[i\u0000j] left of\bis the degree of the polynomial that is not included in the integration (as it does not contain the integration variable),\nand has the coe\u000ecient A[N](2i). The numbers [ a;b; : : :] are the degrees of the polynomials coming from the integral I[N]\n(j). The\bmeans that\nthe orders have to be added, i.e. [ 1]\b[1;3] represents one monomial of order 2and one monomial of order 4in the \fnal expression. The\ncolors of the numbers right of \bindicate the same order of the resulting monomial and the subscript indicates the number mof (43) from\nwhich the monomial is obtained. We note that all contributions to A[N\u00001](0)come from the terms for m=i, all contributions to A[N\u00001](2)\ncome from the terms for m=i\u00001, etc. From this structure, (46) can be obtained as follows: First, we change the labels of the summation\nvariables of the coordinates in (43) from k1; : : : ;kj\u00002mtok2i\u0000j+1; : : : ;k2(i\u0000m),\nI[N]\nj=jj\n2k\nX\nm=0(\u00001)jj!\n4mm!(j\u00002m)!I[N]\n0\u0010\nS[N]\nNN\u0011j\u0000mN\u00001X\nki\u0000j+1=1\u0001\u0001\u0001N\u00001X\nki\u00002m=1S[N]\nk2i\u0000j+1N: : :S[N]\nk2(i\u0000m)Nxk2i\u0000j+1: : :xk2(i\u0000m)(47)\nso that we can insert this formula directly into the equation for the density after \frst integration (39),\n\u001a[N\u00001](x) =MX\ni=02iX\nj=0jj\n2k\nX\nm=0(\u00001)jj!\n4mm!(j\u00002m)!s\u0019\nS[N]\nNN\u0000[N\u00001]\n\u0010\nS[N]\nNN\u0011j\u0000mN\u00001X\nk1=1\u0001\u0001\u0001N\u00001X\nki\u00002m=1^P(2i)\njA[N](2i)\nk1:::k2i\u0000jN:::S[N]\nk2i\u0000j+1N: : :S[N]\nk2(i\u0000m)Nxk1: : :xk2(i\u0000m)(48)\n!=\u0000[N\u00001]0\n@A[N\u00001](0)+N\u00001X\nk1;k2=1A[N\u00001](2)\nk1k2xk1xk2+\u0001\u0001\u0001+N\u00001X\nk1;:::;k2M=1A[N\u00001](2M)\nk1:::k2Mxk1: : :xk2M1\nA (49)\nNow we have to identify A[N\u00001](2\u000b)of (49) in (48) by setting m=i\u0000\u000band by ensuring that the limits of the sums over iandjare adjusted\naccordingly.\nTo obtain the one-nucleus density, equations (45) and (46) need to be iterated until only three indices are left. Those\nbelong to the displacement coordinates x1;x2;x3. In order to obtain the one-nucleus density for the other nuclei, the\nprocedure is repeated after appropriate permutation of the columns of transformation matrix Tjk.\nLast, we note that that we could ignore all factors 1=p\u0019in the original and updated coe\u000ecients because each integration\ncancels one of the of the initially Nfactors. Then we need to multiply the \fnal one-nucleus density with \u0019\u00003=2to obey\nthe normalization condition.\n10Axel Schild\n5 Note on the computational implementation\nThe approximations that are used for the nuclear wavefunction allow to compute the vibrational one-nucleus densities\nfor molecules with a relatively large number of nuclei Nn. However, the resulting polynomial is of order 2M, where\nMis the sum of the vibrational quanta in the system. The current numerical implementation stores arrays of the\npolynomial coe\u000ecients that are of dimension N2M\nn, hence with the number of quanta Mthe memory limit is quickly\nreached, so that in practice only computations for M\u00144are possible on a modern workstation. With a di\u000berent\nnumerical implementation this problem can possibly be avoided, but for large Mthe local harmonic approximation\nthat is made in the normal mode analysis is questionable anyway, hence this is not practical restriction.\n11Axel Schild\nSupporting Information for \\On the Probability Density of the Nuclei in a\nVibrationally Excited Molecule\": Additional Examples\nA note on the labeling of the normal modes: In contrast to the main article, molecules with more that three\nnuclei are discussed in the following, hence there are more normal modes. To have a uniform and simple notation for\nall molecules, the following convention is used to label the normal modes: The normal modes are numbered according\nto the frequency of their corresponding harmonic oscillators in ascending order. Modes 1 to 6 are those of translation\nand rotation of the molecule. The symmetry of the mode is ignored in the numbering.\nA note on the labeling of the states: In these document, only one-nucleus densities for a single excitation of\none mode or, for methane, for two singly excited modes or one double excited mode are shown. The state is labeled by\nthe number(s) of the excited modes.\nIn the following, present one-nucleus densities for selected states of water, mono-deterated benzene, ethene, mono-\ndeuterated ethene (D-ethene), and methane are presented and it is discussed how their qualitative shape can be\npredicted by using the LOCO rules and the normal mode coordinates. All computations were performed as described\nin the main article, hence the Born-Oppenheimer approximation is used in a local harmonic approximation at the\nnuclear con\fguration of minimum energy (the equilibrium con\fguration), and it is assumed that the molecule is\nlocalized and oriented. The three-dimensional reference space has coordinates R1;R2;R3. For each molecule, \frst the\nnormal mode coordinates are discussed and thereafter selected densities are presented.\nAs mentioned in the main article, for the benzene molecule the \frst excitations of any of the normal mode coordinates\ndo not lead to a qualitative change of the one-nucleus density compared to the ground state because the hydrogen\nand oxygen nuclei are displaced in the same direction by multiple normal modes. In contrast, for mono-deterated\nbenzene there are only two normal modes that signi\fcantly displaced the deuterium nucleus. These coordinates are\nalso perpendicular with respect to each other, hence their excitations are clearly visible in the one-nucleus density.\nThis is shown in \fgure 4.\nThe equilibrium con\fguration of ethene is planar, hence the normal mode coordinates correspond to displacements of\nthe nuclei that are either completely in the molecular plane, or perpendicular. Hence, the discussion is restricted to\nthe normal modes in the molecular plane, which is de\fned as the R2-R2plane. For ethene, \fgure 5 shows the selected\nnormal mode coordinates. The modes are numbered by increasing frequency of the corresponding harmonic oscillator,\nand modes 1-6 are those of translation and rotation of the whole system.\nFrom the \fgure, it can be seen that at the hydrogen nuclei there are many normal modes that correspond to dis-\nplacements in a similar spatial direction, which also have similar magnitude. For example, modes 7, 11, 12, and 13\nall displace the hydrogen nuclei to a similar extend in a similar direction, while modes 15, 16, 17, and 18 do the same\nin an almost perpendicular direction. Hence, from the LOCO rules it can be concluded that an excitation of just one\nof these modes does not yield any notable qualitative changes of the one-nucleus density, i.e. there are no new clear\nminima appearing that correspond to the nodes of the wavefunction in such an excited state. This is indeed the case,\nand all one-nucleus densities for an excitation along one of these modes is qualitatively similar to the ground state.\nThe situation at the carbon nuclei, however, is somewhat di\u000berent. Although there are again many modes that displace\nthese nuclei in the same direction, there are two modes that displace them stronger than all other modes: Mode 11\ndisplaces the oxygen nuclei comparably strongly along R2, while mode 14 displaces the oxygen nuclei comparably\nstrongly along R3. This di\u000berence can be seen in the one-nucleus densities for the respective excited states. Figure 6\nshows contour plots of the one-nucleus densities for the \frst excited states of mode 11 and mode 14, and insets show a\nmagni\fcation of the regions around the carbon nuclei. Several contour maps with di\u000berent line spacing are used to be\nable to see details of the density of the hydrogen nuclei and of the carbon nuclei in the same picture. The density at\nthose nuclei has two local maxima and a depletion in the region of the equilibrium position. No other of the excited\nstates corresponding to the \frst excitation any of the normal modes shows this qualitative features, neither at the\ncarbon nuclei nor at the hydrogen nuclei.\nThe situation is altered if the symmetry is broken by isotope substitution. The normal mode coordinates for D-ethene\nare given in \fgure 7. As is clear from the \fgure, there are several normal modes that should, if excited, according\nto the LOCO rules yield a clear qualitative imprint in the one-nucleus density, because they are the only ones that\ndisplace the considered nucleus signi\fcantly in a certain direction. To illustrate, normal modes 7, 12, 15, and 17\nare used. Normal modes 7 and 15 are the ones that displace the deuterium nucleus the strongest, and in almost\nperpendicular directions. Modes 12 and 17 displace the top-right hydrogen nucleus the strongest, and also in mutually\nalmost perpendicular directions. Consequently, excitations of those modes can be expected to have the strongest\nqualitative impact on the one-nucleus density at the given nucleus.\nFigure 8 shows the one-nucleus density for the excites states represented by \frst excitation of modes 7, 12, 15, or\n17. It can clearly be seen that excitations of mode 7 and 15 as well as 12 and 17 show two maxima and a depletion\ncorresponding to the node in the wavefunction of the excited state, at the deuterium nucleus and the top-right hydrogen\nnucleus, respectively. There are more examples of this behavior at the other nuclei, but none of these is as pronounced\nas the ones depicted in \fgure 8.\n12Axel Schild\nLast, it should be illustrated that the LOCO rules also hold for non-planar molecules. For this purpose, methane is\nconsidered. In its equilibrium con\fguration, this molecule has four hydrogen nuclei at the vertices of a tetrahedron,\nand a carbon nucleus at its center. As it is hard to draw the one-nucleus density of the nuclei in a picture, only one of\nthe four equivalent hydrogen nuclei is considered. For the carbon nucleus being at the origin, this nucleus is located\nat ca. (0;2;0)a0. All normal mode coordinates at this nucleus are shown in \fgure 9 as arrows in the R1-R2-,R1-R3-,\nandR2-R3-plane.\nFrom the \fgure, it can be seen that at this nucleus, in each direction there are only two relevant normal modes: mode\n8 and 10 along R1, mode 12 and 15 along R2, and mode 9 and 11 along R3. Mode 8, 15, and 9 are those displacing\nthe nucleus the strongest, although only mode 15 has a clear margin with respect to mode 12, while the displacement\nalong modes 10 and 11 are very close to those of modes 8 and 9, respectively\nIn this example, the excited states corresponding to two quanta in these modes are considered. Contour plots of the\none-nucleus densities are given in \fgure 10 for excitations of mode 8 and 10, in \fgure 11 for excitations of mode 12\nand 15, and in \fgure 12 for excitations of mode 9 and 11.\nIt is found that a double excitation of mode 15 has a clear triple-maximum structure reminiscent of the density of\nthe harmonic oscillator wavefunction in this node. According to the LOCO rules this is expected, because mode 15\ndisplaces the considered nucleus the most. Also double excitations of modes 8 or 9 yield triple-maximum structures,\nalthough the central maximum is almost invisible. Again, this can be rationalized because the wavefunction is in its\nground state along modes 10 and 11, which have similar e\u000bects on the nucleus.\nThe combined e\u000bect of exciting two locally similar modes can also be observed. In \fgures 10, 11, and 12, the one-\nnucleus density of the excited state corresponding to an excitation of mode 8 and 10, mode 12 and 15, and mode 9 and\n11 are also shown. A triple-maximum structure is found, which is most pronounced when the two modes are locally\nsimilar, i.e. for excitation of mode 8 and 10 as well as mode 9 and 11.\n13Axel Schild\n42024\nR2/a042024R3/a0\n42024\nR2/a042024R3/a042024\nR2/a042024R3/a0\nCCC\nC\nC\nC DHH\nH\nH\nH\n4 2 0 2 4\nR2/a042024R3/a0\nCCC\nC\nC\nC DHH\nH\nH\nH14\n31\nFigure 4: Left: Normal mode coordinates of the mono-deuterated benzene molecule. Right: Contour plots of the one-\nnucleus densities of a localized and oriented mono-deuterated benzene molecule in the molecular plane for vibrational\nstates corresponding to the \frst excitations of the normal modes shown to the left.\n14Axel Schild\n2 0 2\nR2/a0202R3/a0\nCCH H\nH H\n2 0 2\nR2/a0202R3/a0\nCCH H\nH H\n2 0 2\nR2/a0202R3/a0\nCCH H\nH H\n2 0 2\nR2/a0202R3/a0\nCCH H\nH H2 0 2\nR2/a0202R3/a0\nCCH H\nH H\n2 0 2\nR2/a0202R3/a0\nCCH H\nH H\n2 0 2\nR2/a0202R3/a0\nCCH H\nH H\n2 0 2\nR2/a0202R3/a0\nCCH H\nH H\n2 0 2\nR2/a0202R3/a0\nCCH H\nH H7 11 12\n13 14 15\n16 17 18\nFigure 5: Normal modes of ethene that are con\fned to the molecular plane. The modes are labeled according to\nfrequency, with modes 1-6 representing translation and rotation of the whole molecule. The arrows show the extent\n(length) of the displacement of the nuclei along the mode and the directionality of the displacement (color).\n15Axel Schild\n20 2\nR2/a0202R3/a0\n20 2\nR2/a0202R3/a011 14\nFigure 6: Contour plots of the one-nucleus densities of localized and oriented ethene in the molecular plane for the\nvibrational states corresponding to the \frst excitation along normal modes 11 and 14. Insets at the top and bottom\nshow a magni\fed view of the region around the oxygen nuclei.\n16Axel Schild\n2 0 2\nR2/a0202R3/a0\nCCD H\nH H\n2 0 2\nR2/a0202R3/a0\nCCD H\nH H\n2 0 2\nR2/a0202R3/a0\nCCD H\nH H7 11 12\n2 0 2\nR2/a0202R3/a0\nCCD H\nH H\n2 0 2\nR2/a0202R3/a0\nCCD H\nH H\n2 0 2\nR2/a0202R3/a0\nCCD H\nH H\n2 0 2\nR2/a0202R3/a0\nCCD H\nH H\n2 0 2\nR2/a0202R3/a0\nCCD H\nH H\n2 0 2\nR2/a0202R3/a0\nCCD H\nH H13 14 15\n16 17 18\nFigure 7: Normal modes of mono-deuterated ethene that are con\fned to the molecular plane. The modes are labeled\naccording to frequency, with modes 1-6 representing translation and rotation of the whole molecule.\n17Axel Schild\n20 2\nR2/a0202R3/a0\n20 2\nR2/a0202R3/a020 2\nR2/a0202R3/a0\n20 2\nR2/a0202R3/a07 12\n15 17\nFigure 8: Contour plots of the one-nucleus densities of localized and oriented mono-deuterated ethene in the molecular\nplane for the vibrational states corresponding to the \frst excitation along normal modes 7, 12, 15, and 17. Insets at\nthe top and bottom show a magni\fed view of the region around the oxygen nuclei.\n18Axel Schild\n12\n158\n108\n10 1215\n911 911\nFigure 9: Normal modes of one of the hydrogen nuclei of methane in the R1-R2-,R1-R3-, andR2-R3-plane (the center of\nmass of the molecule is at the origin). For those corresponding to the largest displacements in a given direction, their\nnumbers (ordered according to increasing frequency of the normal modes, with modes 1-6 corresponding to translation\nand rotation of the molecule) are given.\n19Axel Schild\n0.4 0.20.0 0.2 0.4\nR1/a00.40.20.00.20.4R2/a0\n0.4 0.20.0 0.2 0.4\nR1/a00.40.20.00.20.4R3/a0\n0.4 0.20.0 0.2 0.4\nR2/a00.40.20.00.20.4R3/a0\n0.4 0.20.0 0.2 0.4\nR1/a00.40.20.00.20.4R2/a0\n0.4 0.20.0 0.2 0.4\nR1/a00.40.20.00.20.4R3/a0\n0.4 0.20.0 0.2 0.4\nR2/a00.40.20.00.20.4R3/a0\n0.4 0.20.0 0.2 0.4\nR1/a00.40.20.00.20.4R2/a0\n0.4 0.20.0 0.2 0.4\nR1/a00.40.20.00.20.4R3/a0\n0.4 0.20.0 0.2 0.4\nR2/a00.40.20.00.20.4R3/a08,8 8,8 8,8\n10,10 10,10 10,10\n8,10 8,10 8,10\nFigure 10: Contour plots of the one-nucleus densities of the hydrogen nucleus of methane of \fgure 9 for a state where\ntwo quanta are distributed in the normal modes. Left column: R1-R2-plane. Middle column: R1-R3-plane. Right\ncolumn: R2-R3-plane. Top row: One-nucleus density for the state corresponding to the second excitation of mode 8.\nMiddle row: One-nucleus density for the state corresponding to the second excitation of mode 10. Bottom row: One-\nnucleus density for the state corresponding to the \frst excitation of both mode 8 and mode 10. Note that compared\nto the normal modes in \fgure 9, the hydrogen nucleus was shifted to the origin in R2-direction.\n20Axel Schild\n0.4 0.20.0 0.2 0.4\nR1/a00.40.20.00.20.4R2/a0\n0.4 0.20.0 0.2 0.4\nR1/a00.40.20.00.20.4R3/a0\n0.4 0.20.0 0.2 0.4\nR2/a00.40.20.00.20.4R3/a0\n0.4 0.20.0 0.2 0.4\nR1/a00.40.20.00.20.4R2/a0\n0.4 0.20.0 0.2 0.4\nR1/a00.40.20.00.20.4R3/a0\n0.4 0.20.0 0.2 0.4\nR2/a00.40.20.00.20.4R3/a0\n0.4 0.20.0 0.2 0.4\nR1/a00.40.20.00.20.4R2/a0\n0.4 0.20.0 0.2 0.4\nR1/a00.40.20.00.20.4R3/a0\n0.4 0.20.0 0.2 0.4\nR2/a00.40.20.00.20.4R3/a012,12 12,12 12,12\n15,15 15,15 15,15\n12,15 12,15 12,15\nFigure 11: Contour plots of the one-nucleus densities of the hydrogen nucleus of methane of \fgure 9 for a state where\ntwo quanta are distributed in the normal modes. Left column: R1-R2-plane. Middle column: R1-R3-plane. Right\ncolumn: R2-R3-plane. Top row: One-nucleus density for the state corresponding to the second excitation of mode\n12. Middle row: One-nucleus density for the state corresponding to the second excitation of mode 15. Bottom row:\nOne-nucleus density for the state corresponding to the \frst excitation of both mode 12 and mode 15. Note that\ncompared to the normal modes in \fgure 9, the hydrogen nucleus was shifted to the origin in R2-direction.\n21Axel Schild\n0.4 0.20.0 0.2 0.4\nR1/a00.40.20.00.20.4R2/a0\n0.4 0.20.0 0.2 0.4\nR1/a00.40.20.00.20.4R3/a0\n0.4 0.20.0 0.2 0.4\nR2/a00.40.20.00.20.4R3/a0\n0.4 0.20.0 0.2 0.4\nR1/a00.40.20.00.20.4R2/a0\n0.4 0.20.0 0.2 0.4\nR1/a00.40.20.00.20.4R3/a0\n0.4 0.20.0 0.2 0.4\nR2/a00.40.20.00.20.4R3/a0\n0.4 0.20.0 0.2 0.4\nR1/a00.40.20.00.20.4R2/a0\n0.4 0.20.0 0.2 0.4\nR1/a00.40.20.00.20.4R3/a0\n0.4 0.20.0 0.2 0.4\nR2/a00.40.20.00.20.4R3/a09,9 9,9 9,9\n11,11 11,11 11,11\n9,11 9,11 9,11\nFigure 12: Contour plots of the one-nucleus densities of the hydrogen nucleus of methane of \fgure 9 for a state where\ntwo quanta are distributed in the normal modes. 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Elsevier, 2008.\n24"    },    {        "title": "1902.09621v1.Targeting_Multiple_States_in_the_Density_Matrix_Renormalization_Group_with_The_Singular_Value_Decomposition.pdf",        "content": "Targeting Multiple States in the Density Matrix Renormalization Group with The\nSingular Value Decomposition\nE. F. D'Azevedo,1W. R. Elwasif,1N. D. Patel,2and G. Alvarez3\n1Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831 , USA\n2Department of Physics, The Ohio State University, Columbus, Ohio 43210 , USA\n3Computational Sciences and Engineering Division and Center for Nanophase Materials Sciences,\nOak Ridge National Laboratory, Oak Ridge, Tennessee 37831 , USA\n(Dated: February 27, 2019)\nPACS numbers: 71.10.Fd, 71.27.+a, 02.70.Hm, 79.60.-i\nI. ABSTRACT\nIn the Density Matrix Renormalization Group (DMRG),\nmultiple states must be included in the density matrix\nwhen properties beyond ground state are needed, in-\ncluding temperature dependence, time evolution, and\nfrequency-resolved response functions. How to include\nthese states in the density matrix has been shown in the\npast. But it is advantageous to replace the density matrix\nby a singular value decomposition (SVD) instead, because\nof improved performance, and because it enables multiple\ntargeting in the matrix product state description of the\nDMRG. This paper shows how to target multiple states\nusing the SVD; it analyzes the implication of local sym-\nmetries, and discusses typical performance improvements\nusing the example of the Hubbard model's photoemission\nspectra on a ladder geometry.\nII. INTRODUCTION\nThe density matrix renormalization group (DMRG)\n[1,2] \fnds the ground state of local Hamiltonians, such as\nthose that appear in condensed matter theory. The com-\nputational complexity of DMRG grows as a polynomial\nin the number of lattice sites, if the local Hamiltonian\nis formulated in a one dimensional chain. Moreover, for\nquasi one dimensional systems, the computational com-\nplexity grows as a polynomial in the length of the long\ndimension, such that the DMRG can e\u000eciently truncate\nall local Hamiltonians on these lattices. The error made\nby the DMRG can be estimated, and it converges to the\nexact result as more and more states are kept. To perform\nthe truncation the lattice is divided in two disjoint parts:\na system and an environment.\nIn conventional DMRG, the reduced density matrix \u001aS\nof the system with respect to the environment is com-\nputed using only the ground state vector, and then \u001aSis\ndiagonalized. The eigenvectors Wof the reduced density\nmatrix are used to e\u000bect a change of basis. In the new\nbasis, the eigenvalues d\u000bof the reduced density matrix\ndetermine the importance of each state, and states below\ncertain importance are discarded. If Aoperates only onthe system and \u0016A=WyAW in the new basis, then\nhAi= Tr(A\u001aS) =X\n\u000b\u0016A\u000b;\u000bd\u000b=X\n\u000b<m\u0016A\u000b;\u000bd\u000b+EA(m);\nwhereEA(m) =X\n\u000b>=m\u0016A\u000b;\u000bd\u000b\u0014\u0016AmaxX\n\u000b>=md\u000b\nis the DMRG error. Instead of changing basis using\nthe eigenvectors of the reduced density matrix, one may\ninstead perform a singular value decomposition (SVD)\nof the ground state vector. The square of the singular\nvalues then correspond to the eigenvalues of the reduced\ndensity matrix. This SVD approach, though algebraically\nequivalent, is faster, because the state does not need to be\n\\squared\" to create a density matrix. Furthermore, the\nSVD approach blends well in the matrix product state\n(MPS) formulation of the DMRG [3, 4].\nIn applications of the DMRG to frequency depen-\ndent observables [ 5{7], or to time-dependent Hamilto-\nnians [ 8,9], or to \fnite temperature properties [ 10], one\nneeds multiple states instead of just a single state like in\nground state DMRG. For example, in the correction vector\nmethod for frequency dependent DMRG [ 6], three states\nare needed to obtain the angle resolved photo-emission\nspectra and the Green functions: the ground state vector,\nthe vector resulting from the application of the creation\nor destruction operator to the ground state, and the cor-\nrection vector. To obtain the dynamical spin structure\nfactor the vectors resulting from the application of spin\noperators to the ground state are needed instead.\nYet the DMRG algorithm is not immediately applica-\nble to arbitrary states, and was originally developed to\ncompute the ground state of the Hamiltonian instead.\nThis di\u000eculty has been successfully overcome (see [ 11]\nand references therein) by rede\fning the reduced density\nmatrix of the \\system\" or left block Las:\n\u001aS\n\u000b;\u000b0=X\n\f2RX\nfwf fy\n\u000b;\f f\n\u000b0;\f; (1)\nwhere\u000band\u000b0label states in the left block L,\fthose\nof the right block R, andf fgfis the set of states of the\nsuperblockL[R that are needed for observations. The\nweightswfare chosen to be non-zero. The states  fare\nsaid to be targeted by the DMRG algorithm. Because of\ntheir inclusion in the reduced density matrix, these statesarXiv:1902.09621v1  [cond-mat.str-el]  25 Feb 20192\ncan be obtained with a precision that scales similarly to\nthat of the ground state in the static formulation of the\nDMRG.\nBut it is advantageous to replace the density matrix\nconstruction by a singular value decomposition (SVD),\nas explained in the single state case. In particular, it\nenables multiple state targeting in the MPS formulation,\nand thus addition of MPSs is not needed, neither is their\nconsequent compression. The SVD for multiple states is\nless trivial to perform: The use of symmetries is far from\nobvious in the SVD approach with multiple states, and\nrequires careful thinking of the equations. Parallelization\nover symmetry patches speeds up the simulation further.\nSubsection III A explains the singular value decomposi-\ntion for multiple states, and III B describes the acceler-\nation brought about by symmetry patches. Subsection\nIII E overviews the related matrix-vector multiply, which\nis the most time consuming sub-algorithm of the DMRG.\nSubsection IV A applies these algorithms to a Hubbard\nmodel formulated on a two-leg ladder, and describes the\nspectral function obtained. Subsection IV B analyses the\nperformance pro\fle of the sub-algorithms, emphasizing\nthe improvements brought about by the replacement of\nthe reduced density matrix calculation by the SVD decom-\nposition. The conclusions provide an overview of impli-\ncations for the MPS formulation of the DMRG; pointers\nto the free and open source computer programs used are\nalso given.\nIII. THEORY\nA. Singular Value Decomposition\nWe \frst brie\ry summarize the conventional and SVD\napproaches with one state, often the ground state. We\nconsider a lattice partitioned into a left part or \\system\"\nwithnsstates, and a right part or \\environment\" with\nnestates. We consider the full lattice or superblock states\nlabeled by \u000bon the left, and \fon the right. If there\nis only one vector  to target then the reduced density\nmatrix of the left part is \u001a\u000b0;\u000b=P\n\f \u000b0;\f \u0003\n\u000b;\f, which we\nshall represent as \u001a=  y. (The reduced density matrix\nof the right part is similar, and will not be considered in\nwhat follows.) One then fully diagonalizes \u001a=WyDW,\nwhereWare the eigenvectors of \u001aandDis the diagonal\nmatrix of its eigenvalues. It is known that we can replace\nthe diagonalization of \u001aby the singular value decompo-\nsition (SVD) of  =USVy, where it can be proved by\nsubstitution that Ucoincides with the eigenvectors of\n\u001a, and that the square of Scontains in its diagonal the\neigenvalues of \u001a[4].\nWe now extend the single state approach to the tar-\ngeting of multiple vectors  f, 0\u0014f <F . The de\fnition\nof the reduced density matrix is then \u001a=P\nfwf f fy,\nwhere the weights wf>0, and the sum over fruns over\nall theFvectors that need to be targeted. We de\fne the\nvectorXf;\u000b;\f=pwf f\n\u000b;\fsuch thatXcan be thought ofas a matrix of F\u0002nsrows andnecolumns;\u001a=XXy\nand the replacement for the full diagonalization of \u001ais\nthen the SVD of X:\nXf;\u000b;\f=X\nk;k0Uf;\u000b;kSk;k0Vy\nk0;\f: (2)\nOne can then see by substitution into the equality\n\u001a=XXythatUcontains the eigenvectors of \u001aand\nS2its eigenvalues. We must now carefully develop these\nformulas to take into account symmetry patches by classi-\nfying the states by their local symmetry properties. This\nis needed to preserve symmetries and helpful to speed\nup the computational simulation. Patching formulas are\ndivided in two: (i) organizing  fin patches to be ready\nfor SVD, and (ii) performing the actual SVD in patches.\nB. Symmetry Patches\nThe DMRG algorithm represents the full Hamiltonian\nHas sum of Kronecker product of operators\nH=HL\nI+I\nHR+X\nkCk\nL\nCk\nR; (3)\nwhereCLare matrices on the left block; these matrices\nvary from model to model: for the Hubbard model they\nare the electron creation matrices, for the Heisenberg\nmodel, these are the z-component of the spin Sz, and the\nmatricesS+that increase the z-component of the spin,\nand similarly for other models. Likewise, the counterpart\nmatrices on the right block are denoted by CR.\nThe states in each vector  can be reorganized by\ngrouping them according to the left or right quantum\nnumber. For example, if there are ns= 1000 states on left\nor system and ne= 4000 states on right or environment,\nthen the vector can be reshaped as a 1000 \u00024000 matrix.\nThe non-zero entries in  can be grouped as rectangular\n\\patches\", such that  \u000b;\f= 0 unless q\u000b+q\f=qtarget ,\nwhereq\u000b= (ne(\u000b);Sz(\u000b)) withne(\u000b) the number of\nelectrons of state \u000b, andSz(\u000b) the spin component z\nof state\u000b. This symmetry constraint would then lead\nto a patched matrix  \u000b;\fthat is not necessarily block\ndiagonal. Figure 1 shows the case where the basis has\nbeen reordered such that  \u000b;\fbecomes block diagonal,\nwhich is always possible but not necessary. Whether block\ndiagonal or not, there is only one patch for each block\nrow or block column, and the SVD of  \u000b;\fcan then be\ncomputed independently as the SVD of each individual\nrectangular patch.\nC. SVD in Patches\nThe DMRG algorithm described previously reshapes\nthe lowest eigenvector  produced by the Lanczos algo-\nrithm into a L\u0002Rrectangular matrix to form the density\nmatrix\u001a= y whereLis the number of basis vectors3\n0 1,000 2,000 3,000 4,00002004006008001,000\nαβ\nFIG. 1: Example of a full vector  \u000b;\fin the superblock, when\nreshaped as 1000 \u00024000 matrix to highlight non-overlapping\nrectangular patches. The basis has been reordered such that\n \u000b;\fbecomes block diagonal, which is always possible but not\nnecessary.\nin \\left\" and Rthe number of basis vectors in \\right\".\nThe eigen decomposition of \u001ais then used in a truncation\nprocess to select the largest dominant eigen-pairs to be\nused in the next step.\nThe eigen decomposition can be more e\u000eciently ob-\ntained by performing singular value decomposition (SVD)\nof rectangular matrices over the patches in  . Let us \frst\nobserve that the matrix \u001ais block diagonal. The number\nof diagonal blocks is the number of symmetry patches in\nthe eigen-vector  . If iis the rectangular matrix for the\nithpatch, then the ithdiagonal block of \u001ais y\ni i. (Do\nnot confuse  ia symmetry patch of the single vector  ,\nwith fthef-th vector in the multiple target case.) Note\nthat the economy version of SVD of a m\u0002nmatrixA\ngivesA=U\u0003S\u0003Vywherek\u0014min(m;n) is the rank of\nmatrixA; matrixUism\u0002kand has orthogonal columns\nandUy\u0003U=Ik; matrixSis ak\u0002kdiagonal matrix\nthat contains the singular values; matrix Visn\u0002kand\nhas orthogonal columns Vy\u0003V=IkwithIkthe identity\nmatrix of size k\u0002k. The right singular vectors are then\nthe eigen vectors of AyAand the singular values are the\npositive square roots of the eigen-values,\nAyA= (USVy)y(USVy) =V(SyS)Vy: (4)\nThus the truncation process can proceed by independently\ncomputing the SVD of rectangular matrices over each\npatch of concurrently. The right singular vectors are\nthe eigen-vectors; the eigen-values are the squares of\nsingular values. The eigen-pairs of the density matrix can\nthus be obtained but without explicitly forming \u001a.D. Multiple Targets\nStates included in the reduced density matrix of ei-\nther system or environment are said to be targeted. In\nground state calculations, the ground state is often (but\nnot always [ 12]) the only state targeted. In order to apply\nDMRG to \fnite time or to \fnite temperature or to \fnite\nfrequency problems, other states need to be targeted. We\nnow give a brief summary of these applications. In each ap-\nplication or variant, states other than the ones mentioned\nbelow may be included in the reduced density matrix or\ntargeted. For \fnite time, the time evolved states of a wave-\npacket need to be targeted. For \fnite temperature, the\nin\fnite temperature state and the inverse-temperature-\nevolved state need to be targeted. For \fnite frequency,\nthe correction vectors need to be targeted.\nWithout loss of generality, we consider F= 3 states\nincluded in the reduced density matrix\n\u001a=Ay\n0A0+Ay\n1A1+Ay\n2A2; (5)\nwhere we have absorbed the weights wfinto the matrices\nsuch thatAf\u0011pwfBf. Each matrix Afis the same\nshapembyn;and\u001aisnbyn:With\u001aso de\fned, the\nDMRG proceeds with its truncation procedure by using\nthe eigenvalues and eigenvectors of \u001a.\nThe SVD version stacks the matrices into a bigger\nmatrix 3mbyn;\nAmat=\u0012\nA0\nA1\nA2\u0013\n(6)\nand uses the SVD decomposition of Amatto truncate\nthe DMRG states. It can then be veri\fed that this pro-\ncess is equivalent to the traditional process of using the\neigenvalues and eigenvectors of \u001agiven by Eq. (5). First,\nAy\nmatAmatisn\u00023mby 3m\u0002nand the result is nbyn\nand equal to Ay\n0A0+Ay\n1A1+Ay\n2A2;which is\u001a:Next, the\neconomy SVD version of Amat;yields at most min(3 m;n)\nnon-zero singular values. If Amat=USVy, then\n\u001a=Ay\nmatAmat=V(SyS)Vy; (7)\nyields the information related to eigen decomposition of\n\u001a;Sis diagonal and so is SyS.\nE. Matrix-Vector Product\nThe matrix-vector multiply of Hwith a vector in the\nsuperblock is the most time-consuming sub-algorithm of\nthe DMRG. Although not directly related to the density\nmatrix or the SVD of the targeted states, we brie\ry review\nit here and in the supplemental [13] for completeness.\nAs mentioned in III B, the reorganization or grouping\nof states by quantum numbers to identity the rectangu-\nlar patches o\u000bers a signi\fcant computational advantage\nwhere the target Hamiltonian can be further expressed\nas sum of Kronecker products of small matrices. The4\nevaluation of the matrix-vector multiplication H can\ntake advantage of the properties of Kronecker products\n[13] and be e\u000eciently evaluated as many independent\nmatrix-matrix multiplications.\nGiven a targeted quantum number, there are only a\n\fnite number of admissible combinations of left and right\nquantum numbers that forms the upper bound on the\nnumber of patches Np. This reorganization or group-\ning corresponds to a matrix block partitioning of the\nleft and right operator matrices. The evaluation of the\nmatrix-vector multiplication H can be viewed as ma-\ntrix operations Y=CXwhereCis aNpbyNpblock\npartitioned matrix. Each submatrix C[I;J] in the block\npartition can be expressed as a sum of Kronecker prod-\nucts,C[I;J] =P\nkA(k)\nIJ\nB(k)\nIJ. EachA(k)\nIJ( orB(k)\nIJ)\ncorresponds to a left (or right) operator in Eq. (3). There\nis a corresponding matching partition in vectors Xand\nYasY[I] =P\nJC[I;J]X[J] for row partition index I.\nThe expression C[I;J]X[J] =P\nk(A(k)\nIJ\nB(k)\nIJ)X[J] can\nbe evaluated as matrix-matrix multiplications [14]\nC[I;J]X[J] = KX\nkA(k)\nIJ\nB(k)\nIJ!\nX[J]\n=KX\nk(B(k)\nIJX[J](A(k)\nIJ)t) =KX\nk(W(k)\nIJ(A(k)\nIJ)t)\nwhereW(k)\nIJ=B(k)\nIJX[J], the segment X[I] is reshaped\nas a matrix of appropriate shape, andtindicates matrix\ntranspose. Note that depending on the speci\fc model\nor geometry and number of operators, some of the A(k)\nIJ\norB(k)\nIJmatrices may be the identity matrix or even\nbe zero, i.e., there may be di\u000berent number (perhaps\neven zero) of Kronecker matrices in C[I;J] block. Each\nevaluation of Y[I] can proceed independently. Moreover,\ninC[I;J] there may be multiple independent matrix-\nmatrix multiplication operations in computing W(k)\nIJ=\nB(k)\nIJX[J] over index k. Thus the reorganization or group\ninto patches exposes many independent matrix-matrix\nmultiplication operations that can be evaluated in parallel.\nIV. CASE STUDY: SPECTRAL FUNCTION OF\nHUBBARD LADDERS\nA. The Model and Dynamic Observables\nTo study the SVD of multiple states, we consider the\nparadigm of a Mott insulator simulated with DMRG using\nthe Hubbard Hamiltonian [15, 16]\nH=X\ni;j;\u001btijcy\ni\u001bcj\u001b+UX\nini\"ni#: (8)\nThe Hilbert space where Hacts is the tensor product of\nHilbert spaces on each site of the lattice. Each one-siteHilbert space has four states in its computational basis:\nempty, occupied with electron up, occupied with electron\ndown, and occupied with two electrons of di\u000berent spin.\nThe operator cy\ni\u001bcreates an electron on site iwith spin\n\u001b, andni\u001bis a diagonal operator that multiplies the\nstate by 1 if there is an electron with spin \u001bat that\nsite, or yields the zero of the Hilbert space if not. The\nhopping matrix tcorresponds to a two-leg ladder, that\nis,ti;j=tx=tyif sitesiandjare neighbors on a two-\nleg ladder and 0 otherwise. We take tx=ty= 1 as\nthe unit of energy. Ladders have been investigated as\nthe next step beyond chains and as a proxy for the fully\ntwo-dimensional lattice; ladders are also of interest in the\nsimulation of high temperature superconductors.\nTo obtain the angle-resolved photoemission spectra\nA(q;!) at momentum qand frequency !, the states  fin\nEq. (1) must include [ 6] the ground state (which ought to\nbe computed \frst), crjgs> (the state resulting from the\napplication of the destruction operator crto the ground\nstate), and the real and imaginary parts of the correction\nvectorjcv(r;!;\u0011 )i\u0011(z\u0000H)\u00001crjgsi, wherez=!+i\u0011,\nandris a given site. (We do not use arrows to denote sites,\neven though sites belong to a ladder and are thus two\ndimensional.) One then observescy\nr0so thatA+(r;r0;!) =\nhgsjcy\nr0jcvi, and one also must add [ 17] the counterpart\nA\u0000to obtainA(r;r0;!).\nIf the lattice were periodic and A(r;r0;!) depended\nonly on the Euclidean distance between sites randr0,\nthen we could \fx (say) rand varyr0. DMRG simulations\nare usually done with open boundary conditions in the\nlong dimension, and ours is no exception. We use never-\ntheless an approximation that we shall name \\center site\napproximation\" [ 18], where we take one central site r=c\n\fxed, and vary the other site r0. This approximation\naccelerates our simulation because in one single DMRG\nrun we can obtain values A(c;r0;!) for allr0at a \fxed!.\nIn parallel, we carry out the needed simulations to obtain\nall frequencies !of interest. We then perform a Fourier\ntransform from space to momentum qto \fnally compute\nA(q;!) of physical and experimental signi\fcance. Figure\n2 shows the result for both qy= 0 andqy=\u0019on a 32\u00022\nladder atU= 4, with 4 holes for an electronic density\nofn= 0:9375. Due to the \\center site approximation\"\nand the open boundary condition in the long direction of\nthe ladder, the imaginary part of the spectral function is\nslightly negative at frequencies where we would expect it\nto be zero, a problem that we have not tried to hide but\nshow in the \fgure for explication purposes. If needed, such\nproblems can be controlled either by \fnite-size scaling or\nby using frequency \flters.\nFigure 2 reproduces results presented in [ 19], now in\na larger lattice at high accuracy. The weights for the\nstates included in the SVD is 0 :3 for each correction\nvector, 0:3 for thecrjgsivector, and 0 :1 for the ground\nstate vectors. (Other weight values could have been used;\nsee the discussion in [ 6].) The chemical potential is at\n\u0016= 1:20\u00060:05. Theqy= 0 bonding band shows high\nintensity at an energy near the chemical potential and at5\n−3−2−1 0 1 2 3−4−20246\nqxEnergy\n0.02.04.0\n−3−2−1 0 1 2 3−4−20246\nqxEnergy\n0.01.02.03.04.05.0\nFIG. 2: (a) A(qx;qy= 0;!) of a 32 \u00022 Hubbard ladder with\ntx=ty= 1 as unit of energy, U= 4, and electronic n= 0:9375\nyielding a chemical potential equal to 1 :20\u00060:05. (b) Same\nas (a) forqy=\u0019\na momentum near qx=\u00062. A double feature appears\naround momentum qx= 0, and toward qx=\u0006\u0019the\nspectral intensity appears at the high energy end of the\nspectrum (!= 3:5). Theqy= 0 anti-bonding band has\nintensity at the chemical potential and around momentum\nqx= 0, but most intensity is at higher energies; the\nHubbard band appears as expected at an energy U= 4\nabove.\nB. Performance Pro\fle\nThe formation of the density matrix takes a part of\nthe performance pro\fle of a typical non-ground state\nDMRG run: It accounts for approximately 40% of the\ntotal run time. (For ground state runs the density-matrix\nconstruction takes less percentage-wise [ 20].) When re-\nplaced by the SVD algorithm, this percentage reduces\nto only 5% of the total run time. Figure 3 shows a bar-\nchart of the density matrix (DM) subalgorithm and its\nreplacement by SVD. The \fgure indicates small varia-\ntions in sub-algorithms not directly related to the SVD\nor DM, because the states that are discarded vary even\nin identical runs due to the degeneracy of the eigenval-\nues of the density matrix, at least to machine precision.\n0 200 400 600 800 1,0001,2001,400fullRunSVDfullRunDM\nseconds\nMVP SVD/DM RestFIG. 3: Wall times in seconds spent by the matrix-vector\nmultiplication, the SVD or the DM depending on the type of\nsub-algorithm used, and the rest of the run, for both the run\ndone with SVD (fullRunSVD) and for the run done with the\nconventional density-matrix construction (fullRunDM). The\nrun for frequency !=\u00002 was chosen; other frequencies are\nsimilar.\nNevertheless, \fgure 3 illustrates the substantial gain in\nCPU performance that the use of SVD with symmetry\npatches brings about. The rest of time is spent in other\nsub-algorithms, of which the construction of Hamiltonian\nconnections between \\system\" and \\environment\" at over\n50% of run time, is the most time consuming, as we now\nexplain.\nIn ground state DMRG, the most computational ex-\npensive kernel is the on-the-\ry computation of the Hamil-\ntonian connections between system and environment in\norder to perform Lanczos decomposition to obtain the\nground state. In the case of frequency dependent DMRG,\nthe most computational expensive kernel is the Lanczos\ntridiagonalization of the Hamiltonian, which again re-\nduces to the on-the-\ry computation of the Hamiltonian\nconnections between system and environment. As ex-\nplained in reference [ 21], the correction vectors [ 6] are\nobtained in Krylov-space considering that\n(z\u0000H)\u00001cjgsi= (z\u0000VyTV)\u00001cjgsi; (9)\nwherez=!+i\u0011with\u0011 > 0,Vthe Lanczos vectors;\nthere arensLanczos vectors, each of size nb.Tofns\nrows andnscolumns is the tridiagonal decomposition of\nH.Hofnbrows andnbcolumns is too large to store in\nmemory, and needs to be built on-the-\ry. We have found\nthatns= 400 Lanczos vectors are enough to study a\n32\u00022 ladder. We have kept at most m= 2000 states for\nthe DMRG, such that nb\u001442m2is the dimension of the\ntruncated Hilbert space; the factor 4 arises because the\none-site Hilbert space of the Hubbard model is composed\nof 4 states.\nWe have analyzed the performance of the model just\ndescribed in the realistic case of running the computer\nprogram in high performance computer systems, and in-\ncluding the use of the GPU for the matrix-vector product\nalgorithm. Figure 4 shows the total walltimes in seconds\nfor a simulation using the traditional DMRG approach\nof building the reduced density matrix, and using the\nSVD approach instead, for each frequency of interest. For\nall frequencies, the SVD approach decreases total run-\ntime by a factor of approximately 1.25; larger frequencies6\n−4−2 0 2 4 6 81,0001,2001,4001,600\nωWalltime (seconds)\nNOSVD\nSVD\nFIG. 4: Walltimes in seconds for each frequency run !using\nconventional DMRG (dashed line) and using SVD (solid line).\nGPU CPU\n!fullRunSVD fullRunDM fullRunSVD fullRunDM\nGS 191 231 193 241\n\u00002 1073 1406 1690 1969\n0 1043 1380 1710 1947\n2 1048 1402 1757 2014\nTABLE I: Wall times in seconds of typical runs depending\non the use of the SVD algorithm or the density matrix (DM)\nalgorithm, for runs done using the GPU or the CPU for the\nmatrix-vector product algorithm. Note that the SVD or the\ndensity matrix diagonalization was always done on CPU.\ntake longer to converge in this Krylov-space algorithm as\ndetailed in ref. [21].\nTable I compares the same run when done on GPU\nand when done on CPU; only the matrix-vector-product\nalgorithm was run on GPU when indicated. The SVD\nalgorithm is already at 5% of run time and it might not be\nbene\fcial to run it on the GPU due to memory overhead.\nThe results shown in the table should not be taken to infer\nthat the GPU is linearly faster than the CPU, because\nonly the most computationally expensive sub-algorithm\nwas done on the GPU: the matrix-vector product. Other\nsub-algorithms, the SVD of the vectors, the summation\nof sparse matrices, and the wave-function-transformation\n[22] were done on the CPU. Data movement between the\nCPU and the GPU is another well-known limiting factor\nthat must be taken into account.\nV. CONCLUSION\nWhether the MPS formulation or the conventional for-\nmulation of the DMRG is used, multiple states must betargeted for observations beyond ground state. The sin-\ngular value decomposition helps both formulations. In\nthe MPS formulation, targeting multiple states replaces\nthe addition of MPSs and their subsequent compression\n[4] at the expense of the maximum bond dimension m. In\nthe conventional formulation the SVD replaces the more\nexpensive density matrix sub-algorithm, substantially re-\nducing the time to solution.\nFuture work will apply the computational insights and\ntheory explained in this paper to the simulation of mod-\nels on more than two leg ladders, of interest as a proxy\nto the fully two-dimensional lattice. The real frequency\nspectral functions in these models, of interest due to the\nexistence of angle-resolved photo-emission spectroscopy\nand neutron scattering data, has mostly been inacces-\nsible theoretically and is only now been computed on\nlarge enough lattices and with enough precision to help\nexplain the transitions and interactions that cause the\nexperimentally measured spectra.\nAll the theory and computation presented for real fre-\nquency applies straightforwardly to \fnite temperature T\nby replacing the initial state with the in\fnite temperature\nstate obtained from the ground state of entangler Hamil-\ntonians [ 10,23] and the Lehman formulation in Eq. (9)\nby thermal evolution at \\imaginary time\" \f= 1=T. Like-\nwise, real time evolution (when using Krylov-space de-\ncomposition) will show maximal computational cost in\nthe matrix-vector product [ 24]. The computer programs\nused (including DMRG++ [25]) are described in the\nsupplemental [ 13], where details to enable reproduction\nof the results presented here can also be found.\nAcknowledgments\nThe performance and algorithmic improvements have\nbeen supported by the scienti\fc Discovery through Ad-\nvanced Computing (SciDAC) program funded by U.S.\nDepartment of Energy, O\u000ece of Science, Advanced Sci-\nenti\fc Computing Research and Basic Energy Sciences,\nDivision of Materials Sciences and Engineering. The\ncase study was supported by the Laboratory Directed\nResearch and Development Program of ORNL. Software\ndevelopment has been partially supported by the Center\nfor Nanophase Materials Sciences, which is a DOE O\u000ece\nof Science User Facility.7\n[1]S. R. White, Phys. Rev. Lett. 69, 2863 (1992), URL\nhttps://link.aps.org/doi/10.1103/PhysRevLett.69.\n2863.\n[2]S. R. White, Phys. Rev. B 48, 10345 (1993), URL https:\n//link.aps.org/doi/10.1103/PhysRevB.48.10345 .\n[3]S. Rommer and S. Ostlund, Phys. Rev. B 55, 2164 (1997).\n[4] U. Schollw ock, Annals of Physics 96, 326 (2010).\n[5] K. Hallberg, Phys. Rev. B 52, 9827 (1995).\n[6] T. K uhner and S. White, Phys. Rev. B 60, 335 (1999).\n[7] E. Jeckelmann, Phys. Rev. B 66, 045114 (2002).\n[8]S. R. White and A. E. Feiguin, Phys. Rev. Lett. 93,\n076401 (2004).\n[9]S. R. Manmana, A. Muramatsu, and R. M. Noack, in AIP\nConf. Proc. , edited by A. Avella and F. Mancini (2005),\nvol. 789, pp. 269{278, also in http://arxiv.org/abs/cond-\nmat/0502396v1.\n[10]A. R. Feiguin and S. R. White, Phys. Rev. B 72, 020404\n(2005).\n[11] U. Schollw ock, Rev. Mod. Phys. 77, 259 (2005).\n[12]S. R. White, Phys. Rev. B 72, 180403 (2005), URL https:\n//link.aps.org/doi/10.1103/PhysRevB.72.180403 .\n[13]See Supplemental Material at https://g1257.github.\nio/papers/84/ for a description and usage of the com-\nputer code.\n[14]C. F. Van Loan, Journal of Computational and Applied\nMathematics 123, 85 (2000).\n[15]J. Hubbard, Proc. R. Soc. London Ser. A 276, 238 (1963).\n[16]J. Hubbard, Proc. R. Soc. London Ser. A 281, 401 (1964).[17]P. Dargel, A. Honecker, R. Peters, R. M. Noack, and\nT. Pruschke, Phys. Rev. B 83, 161104(R) (2011).\n[18]R. M. Noack, S. R. White, and D. J. Scalapino, EPL (Eu-\nrophysics Letters) 30, 163 (1995), URL http://stacks.\niop.org/0295-5075/30/i=3/a=007 .\n[19]J. Riera, D. Poilblanc, and E. Dagotto, The European\nPhysical Journal B - Condensed Matter and Complex\nSystems 7, 53 (1999), ISSN 1434-6036, URL https://\ndoi.org/10.1007/s100510050588 .\n[20]C. Nemes, G. Barcza, Z. Nagy, rs Legeza, and P. Szolgay,\nComputer Physics Communications 185, 1570 (2014),\nISSN 0010-4655, URL http://www.sciencedirect.com/\nscience/article/pii/S0010465514000654 .\n[21]A. Nocera and G. Alvarez, Phys. Rev. E 94,\n053308 (2016), URL https://link.aps.org/doi/10.\n1103/PhysRevE.94.053308 .\n[22]S. R. White, Phys. Rev. Lett. 77, 3633 (1996), URL\nhttps://link.aps.org/doi/10.1103/PhysRevLett.77.\n3633.\n[23]A. Nocera and G. Alvarez, Phys. Rev. B 93,\n045137 (2016), URL https://link.aps.org/doi/10.\n1103/PhysRevB.93.045137 .\n[24]G. Alvarez, L. G. G. V. D. da Silva, E. Ponce, and\nE. Dagotto, Phys. Rev. E 84, 056706 (2011).\n[25]G. Alvarez, Computer Physics Communications 180, 1572\n(2009)."    },    {        "title": "1903.04046v2.The_Local_Density_Approximation_in_Density_Functional_Theory.pdf",        "content": "arXiv:1903.04046v2  [math-ph]  28 Oct 2019THE LOCAL DENSITY APPROXIMATION IN DENSITY\nFUNCTIONAL THEORY\nMATHIEU LEWIN, ELLIOTT H. LIEB, AND ROBERT SEIRINGER\nAbstract. We give the first mathematically rigorous justification of\nthe Local Density Approximation in Density Functional Theo ry. We\nprovide a quantitative estimate on the difference between th e grand-\ncanonical Levy-Liebenergy ofagivendensity(thelowest po ssible energy\nof all quantum states having this density) and the integral o ver the\nUniform Electron Gas energy of this density. The error invol ves gradient\nterms and justifies the use of the Local Density Approximatio n in the\nsituation where the density is very flat on sufficiently large r egions in\nspace.\nc/circlecopyrt2019 by the authors. This paper may be reproduced, in its enti rety,\nfor non-commercial purposes. Final version to appear in Pure and Ap-\nplied Analysis .\nContents\n1. Introduction 2\n2. Main result 3\n2.1. The grand-canonical Levy-Lieb functional 3\n2.2. The Uniform Electron Gas 5\n2.3. The Local Density Approximation 6\n3. A priori estimates on T(ρ) andE(ρ) 8\n3.1. Known lower bounds 8\n3.2. Upper bound on the best kinetic energy 10\n3.3. Upper bound on E(ρ) 14\n4. Proof of Theorems 1 and 2 14\n4.1. Upper bound in terms of local densities 15\n4.2. Lower bound in terms of local densities 22\n4.3. A convergence rate for tetrahedra 24\n4.4. Proof of Theorem 1 26\n4.5. Replacing the local density by a constant density 27\n4.6. Lipschitz regularity of eUEG 30\n4.7. Proof of Theorem 2 31\nAppendix A. Classical case 36\nReferences 39\nDate: October 29, 2019.\n12 M. LEWIN, E.H. LIEB, AND R. SEIRINGER\n1.Introduction\nDensity Functional Theory (DFT) [10, 44, 11, 4, 47] is the mos t efficient\napproximation of the many-body Schr¨ odinger equation for e lectrons. It is\nused in several areas of physics and chemistry and its succes s in predict-\ning the electronic properties of atoms, molecules and mater ials is unprece-\ndented. Among the many functionals that have been developed over the\nyears [42], the Local Density Approximation (LDA) is the standard and sim-\nplest scheme [18, 22, 10, 44, 45]. It is not as accurate as its s uccessors\ninvolving gradient corrections, but it is considered as “the mother of all ap-\nproximations” [46] and it is still one of the methods of choice in solid state\nphysics.\nIn the orbital-free formulation of Density Functional Theo ry [25, 35], the\nLocal Density Approximation consists in replacing the full ground state\nenergy by a local functional as follows:\nFLL(ρ)≈1\n2ˆ\nR3ˆ\nR3ρ(x)ρ(y)\n|x−y|dxdy+ˆ\nR3eUEG/parenleftbig\nρ(x)/parenrightbig\ndx. (1)\nHereρis the given one-particle density of the system and FLL(ρ) is theLevy-\nLieb functional [25,35], themainobjectofinterestinDFT.Thisisthelowes t\npossible Schr¨ odinger energy of all quantum states having t he prescribed\ndensityρ. The first term on the right side is called the directorHartree\nterm. It is the classical electrostatic interaction energy of th e density ρand\nit is the only nonlocal term in the LDA. The second term is the e nergy of the\nUniform Electron Gas (UEG) [10, 44, 13, 30], containing all of the kinetic\nenergy and the exchange-correlation energy in our conventi on. That is,\neUEG(ρ0) is the ground state energy per unit volume of the infinite ele ctron\ngas withtheprescribedconstant density ρ0over thewholespace(fromwhich\nthe direct term has been dropped). The rationale for the appr oximation (1)\nis to assume that the density is almost constant locally (in l ittle boxes of\nvolumedx), and to replace the local energy per unit volume by that of th e\ninfinite gas at that density ρ(x).1\nOur goal in this paper is to justify the approximation (1) in t he appropri-\nate regime where ρis flat in sufficiently large regions of R3. We will prove\nthe following quantitative estimate\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleFLL(ρ)−1\n2ˆ\nR3ˆ\nR3ρ(x)ρ(y)\n|x−y|dxdy−ˆ\nR3eUEG/parenleftbig\nρ(x)/parenrightbig\ndx/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n/lessorequalslantεˆ\nR3/parenleftbig\nρ(x)+ρ(x)2/parenrightbig\ndx+C(1+ε)\nεˆ\nR3|∇√ρ(x)|2dx\n+C\nε4p−1ˆ\nR3|∇ρθ(x)|pdx(2)\nfor allε >0, where FLL(ρ) is the grand canonical version of the Levy-Lieb\nfunctional. The parameters p >3 and 0 < θ <1 should satisfy some\nconditions which will be explained below. For instance, p= 4 and θ= 1/2\n1It is often more convenient to fix the densities ρ↑(x) andρ↓(x) of, respectively, spin-up\nand spin-down electrons instead of the total density ρ(x) =ρ↑(x)+ρ↓(x). All our results\napply similarly to this situation, as explained below in Rem ark 4.THE LOCAL DENSITY APPROXIMATION IN DENSITY FUNCTIONAL THEO RY 3\nis allowed. After optimizing over ε, this justifies the LDA when the two\ngradient terms are much smaller than the local term\n\n\nˆ\nR3|∇√ρ(x)|2dx≪ˆ\nR3/parenleftbig\nρ(x)+ρ(x)2/parenrightbig\ndx,\nˆ\nR3|∇ρθ(x)|pdx≪ˆ\nR3/parenleftbig\nρ(x)+ρ(x)2/parenrightbig\ndx.\nFor instance for a rescaled density in the form\nρN(x) :=ρ/parenleftbig\nN−1/3x/parenrightbig\nwith´\nR3ρ= 1, we obtain after taking ε=N−1/12\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleFLL(ρN)−N5\n3\n2ˆ\nR3ˆ\nR3ρ(x)ρ(y)\n|x−y|dxdy−Nˆ\nR3eUEG/parenleftbig\nρ(x)/parenrightbig\ndx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslantCN11\n12.\nThe bound (2) is, to our knowledge, the first estimate of this k ind on\nthe fundamental functional FLL. Although it should be possible to extract\na definite value of the constant Cfrom our proof, it is probably very large\nand we have not tried to do it. The factor 1 /ε4p−1is also quite large and\nit is an open problem to improve it. We hope that our work will s timulate\nmore results on the functional FLLin the regime of slowly varying densities.\nIn physics and chemistry, the exchange-correlation energy is defined by\nsubtracting a kinetic energy term T(ρ) fromFLL(ρ). In this paper we also\nderive a bound on T(ρ) which, when combined with (2), provides a bound\non the exchange-correlation energy similar to (2). This is e xplained below\nin Remark 3.\nIn the next section we provide the precise mathematical defin ition ofFLL\nandeUEG, and we state our main theorem containing the estimate (2). I n\nSection 3 we review some known a prioriestimates on FLLand prove a new\nupper bound on the kinetic energy. Section 4 contains the pro of of our main\nresults. Finally, in Appendix A we discuss a similar bound in the classical\ncase where the kinetic energy is dropped, extending thereby our previous\nresult in [30].\nAcknowledgments. The authors thank the Institut Henri Poincar´ e for its\nhospitality. This project has received funding from the Eur opean Research\nCouncil (ERC) under the European Union’s Horizon 2020 resea rch and in-\nnovation programme (grant agreements AQUAMS No 694227 of R. S. and\nMDFT No 725528 of M.L.).\n2.Main result\n2.1.The grand-canonical Levy-Lieb functional. Letusconsideraden-\nsityρ∈L1(R3,R+) such that√ρ∈H1(R3). Naturally we should assume\nin addition that´\nR3ρ=Nis an integer, but here we will work in the grand\ncanonical ensemble where this is not needed. The grand-canonical Levy-Lieb4 M. LEWIN, E.H. LIEB, AND R. SEIRINGER\nfunctional [25, 35, 30] is defined by\nFLL(ρ) :=\ninf\nΓn=Γ∗\nn/greaterorequalslant0/summationtext∞\nn=0Tr(Γn)=1/summationtext∞\nn=1ρΓn=ρ/braceleftigg∞/summationdisplay\nn=1TrHn/parenleftigg\n−n/summationdisplay\nj=1∆xj+/summationdisplay\n1/lessorequalslantj<k/lessorequalslantn1\n|xj−xk|/parenrightigg\nΓn/bracerightigg\n.(3)\nHere\nHn:=L2\na((R3×{1,...,q})n,C)\nis then-particle space of antisymmetric square-integrable funct ions on (R3×\n{1,...,q})n, withqspin states (for electrons q= 2). The family of opera-\ntors Γ = {Γn}n/greaterorequalslant0forms a grand-canonical mixed quantum state, that is, a\nstate over the fermionic Fock space (commuting with the part icle number\noperator). The density of each Γ nis defined by\nρΓn(x) =\nn/summationdisplay\nσ1,...,σn\n∈{1,...,q}ˆ\nR3(n−1)Γn(x,σ1,x2,...,xn,σn;x,σ1,x2,...,xn,σn)dx2···dxn\nwhere Γ n(x1,...,σn;x′\n1,...,σ′\nn) is the kernel of the trace-class operator Γ n.\nThis kernel is such that\nΓn(xτ(1),στ(1),...,xτ(N),στ(N);x′\n1,σ′\n1,...,x′\nN,σ′\nN)\n= Γn(x1,σ1,...,xN,σN;x′\nτ(1),σ′\nτ(1)...,x′\nτ(N),σ′\nτ(N))\n=ε(τ) Γn(x1,σ1,...,xN,σN;x′\n1,σ′\n1...,x′\nN,σ′\nN)\nfor every permutation τ∈SNwith signature ε(τ)∈ {±1}.\nIf´\nR3ρ=N∈Nand we restrict ourselves to mixed states Γ where\nonly Γ Nis non-zero, we obtain Lieb’s functional [35]. If we further assume\nthat Γ N=|Ψ/a\\}b∇acket∇i}ht/a\\}b∇acketle{tΨ|is a rank-one projection, then we find the original Levy\nor Hohenberg-Kohn functional [18, 25]. It is well known [35] that working\nwith mixed states has several advantages, in particular we o btain a convex\nfunction of ρ.\nThe grand-canonical version (3) is less popular but still im portant phys-\nically.2It is also a convex function of ρ. The fact that we can appeal\nto states with an arbitrary number nof particles (but still a fixed average\nnumber´\nR3ρ) will considerably simplify several technical parts of our study.\nWe expect that our main result (Theorem 2 below) holds the sam e for the\ncanonical functionals, for which the energy is minimized ov er mixed or pure\nstates with Nparticles.\nIt is useful to subtract the direct term from FLL, hence to consider the\nenergy\nE(ρ) :=FLL(ρ)−1\n2ˆ\nR3ˆ\nR3ρ(x)ρ(y)\n|x−y|dxdy. (4)\n2The grand canonical functional FLLis the weak- ∗lower semi-continuous closure of\nthe canonical Lieb functional [27]. Hence it appears natura lly in situations where some\nparticles can be lost, e.g. in scattering processes.THE LOCAL DENSITY APPROXIMATION IN DENSITY FUNCTIONAL THEO RY 5\nThe (grand-canonical) exchange-correlation energy is defined by Exc(ρ) :=\nE(ρ)−T(ρ) where\nT(ρ) := inf\nΓn=Γ∗\nn/greaterorequalslant0/summationtext∞\nn=0Tr(Γn)=1/summationtext∞\nn=1ρΓn=ρ/braceleftigg∞/summationdisplay\nn=1TrHn/parenleftigg\n−n/summationdisplay\nj=1∆xj/parenrightigg\nΓn/bracerightigg\n= inf\n0/lessorequalslantγ=γ∗/lessorequalslant1\nργ=ρTr(−∆)γ\n(5)\nis the lowest possible kinetic energy. We will study the func tionalT(ρ) in\nSection 3.2 below.\n2.2.The Uniform Electron Gas. In [30, Section 5], we have defined the\nuniform electron gas, which is obtained in the limit when ρapproaches a\nconstant function in the whole space. This is believed to be t he same as the\nground state energy of Jellium, where the density is not nece ssarily constant\nbut the electrons instead evolve in a constant background [3 6]. This has\nrecently been proved in the classical case in [31, 7] and the s ame is expected\nin the quantum case.\nThe following result is a slight improvement of [30, Thm. 5.1 ].\nTheorem 1 (Quantum Uniform Electron Gas) .Letρ0>0. Let{ΩN} ⊂R3\nbe a sequence of bounded connected domains with |ΩN| → ∞, such that ΩN\nhas a uniformly regular boundary in the sense that\n|∂ΩN+Br|/lessorequalslantCr|ΩN|2/3,for allr/lessorequalslant|ΩN|1/3/C,\nfor some constant C >0. LetδN>0be any sequence such that δN/|ΩN|1/3→\n0andδN|ΩN|1/3→ ∞. Letχ∈L1(R3)be a radial non-negative function\nof compact support such that´\nR3χ= 1and´\nR3|∇√χ|2<∞. Denote\nχδ(x) =δ−3χ(x/δ). Then the following thermodynamic limit exists\nlim\nN→∞E/parenleftbig\nρ01ΩN∗χδN/parenrightbig\n|ΩN|=eUEG/parenleftbig\nρ0/parenrightbig\n(6)\nwhere the function eUEGis independent of the sequence {ΩN}, ofδNand\nofχ.\nFor more properties of the UEG energy eUEGwe refer to [30] and the\nreferences therein. In [30, Thm. 5.1] we rather optimized ov er the values of\nρin the transition region around ∂ΩN. We were able to prove the simple\nlimit (6) only when Ω Nis a tetrahedron. Using an upper bound on E(ρ)\nthat will be derived later in Proposition 1, we are now able to treat more\nreasonable limits in the form of (6). The proof of Theorem 1 is provided in\nSection 4.4 below.\nThe function χδNis used to regularize the function ρ01ΩNwhich cannot\nbethe density of a quantum state, since its squareroot is not inH1(R3) [35].\nThe first condition δN/|ΩN|1/3→0 implies that the smearing happens in a\nneighborhoodof the boundary ∂ΩNwhich has a negligible volume compared\nto|ΩN|. The second condition δN|ΩN|1/3→ ∞ensures that the kinetic\nenergy in the transition region stays negligible in the ther modynamic limit.\nRemark 1. The same result holds under the weaker condition that Ω Nhas\nanη–regular boundary, which means that |∂ΩN+Br|/lessorequalslantC|ΩN|η/parenleftbig\nr|ΩN|−1/3/parenrightbig6 M. LEWIN, E.H. LIEB, AND R. SEIRINGER\nfor allr/lessorequalslant|ΩN|1/3/C, withη(t)→0 when t→0+. The condition\nδN|ΩN|1/3→ ∞is then replaced by δ−2\nNη(δN|ΩN|−1/3)→0.\n2.3.The Local Density Approximation. We are now able to state our\nmain result.\nTheorem 2 (Local Density Approximation) .Letp >3and0< θ <1such\nthat\n2/lessorequalslantpθ/lessorequalslant1+p\n2. (7)\nThere exists a constant C=C(p,θ,q)such that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleE(ρ)−ˆ\nR3eUEG/parenleftbig\nρ(x)/parenrightbig\ndx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslantεˆ\nR3/parenleftbig\nρ(x)+ρ(x)2/parenrightbig\ndx\n+C(1+ε)\nεˆ\nR3|∇√ρ(x)|2dx+C\nε4p−1ˆ\nR3|∇ρθ(x)|pdx(8)\nfor every ε >0and every non-negative density ρ∈L1(R3)∩L2(R3)such\nthat∇√ρ∈L2(R3)and∇ρθ∈Lp(R3).\nThe constant C=C(p,θ,q) in our estimate (8) depends on the number\nof spin states q(q= 2 for electrons), in addition to the parameters pand\nθ. It diverges when p→3+. Ifp→3+then we can take θ→5/6−.\nOur estimate therefore applies to densities ρwith compact support, which\nvanish at the boundary of their support like δ(x)awitha >4/5, where\nδ(x) = d(x,∂ρ−1({0})). In particular, densities which vanish linearly are\nallowed. Our proof allows one to consider more singular dens ities, that is,\ntorelaxtheconstraintthat θp/lessorequalslant1+p/2, butthenthepowerof εdeteriorates.\nOur estimate (8) is certainly not optimal and it is an interes ting challenge\nto improve it. We conjecture that a similar inequality holds withρ+ρ2\nreplaced by ρ4/3+ρ5/3which have the scaling of the Coulomb and kinetic\nenergies, respectively. The higher power ρ2arises from the trial state used\nin our upper bound (Proposition 1) and it is used to control so me errors\nappearing when merging quantum systems with overlapping su pports. This\nis explained in Section 4.1 below. Finally, the last gradien t term in (8)\nis used to control local variations of ρinL∞. One could expect gradient\nerrors involving only´\nR3|∇√ρ|2and´\nR3|∇ρ1/3|2which are believed to arise\nin the gradient expansion of the uniform electron gas for, re spectively, the\nCoulomb and kinetic energies.\nOne interesting case is when the density is given by a fixed fun ctionρ\nwith´\nR3ρ= 1, which is rescaled in the manner\nρN(x) =ρ(N−1/3x).\nAfter taking ε=N−1/12in (8) we obtain the following simple bound\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleE(ρN)−Nˆ\nR3eUEG/parenleftbig\nρ(x)/parenrightbig\ndx/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n/lessorequalslantCN5\n12ˆ\nR3|∇√ρ(x)|2dx+CN11\n12ˆ\nR3/parenleftig\nρ(x)+ρ(x)2+|∇ρθ(x)|p/parenrightig\ndx.\n(9)THE LOCAL DENSITY APPROXIMATION IN DENSITY FUNCTIONAL THEO RY 7\nIt is conjectured [21, 24, 23, 26, 11] that the next order in th e expansion of\nE(ρN) should involve the gradient correction to the kinetic ener gy\nˆ\nR3|∇√ρN(x)|2dx=N1\n3ˆ\nR3|∇√ρ(x)|2dx\nand the gradient correction to the Coulomb energy\nˆ\nR3/vextendsingle/vextendsingle/vextendsingle∇ρ1/3\nN(x)/vextendsingle/vextendsingle/vextendsingle2\ndx=N1\n3ˆ\nR3/vextendsingle/vextendsingle/vextendsingle∇ρ1/3(x)/vextendsingle/vextendsingle/vextendsingle2\ndx.\nIn particular, the next order should be proportional to N1/3. It remains an\nopen problem to establish this rigorously.\nIn the classical case where the kinetic energy is neglected, the limit of\nEcl(ρN)/Nwas found in our previous work [30], but without a quantitati ve\nestimate on the remainder. We can give an estimate similar to (8) in the\nclassical case, with a lower power of εin front of the gradient term. This is\njust a slight adaptation of the proof in [30], which is much ea sier than the\nquantum case. The argument is explained for completeness in Appendix A.\nIn the classical case the limit for Ecl(ρN)/Nwas later extended to Riesz\ninteractions |x|−sand other dimensions d/greaterorequalslant1 in [8]. Although our result (8)\nin thequantum case can probablybeextended to other Riesz in teractions by\nusing ideas from [12, 19, 15, 8], we only consider here the phy sically relevant\n3D Coulomb case for shortness.\nRemark 2 (Canonical case) .We expect an inequality similar to (8) for the\n(mixed) canonical version of E(ρ) where´\nR3ρ=N∈Nand Γn= 0 for\nn/\\e}atio\\slash=N. However our proof does not adapt in an obvious way to this cas e.\nRemark 3 (Exchange-correlation energy) .In physics and chemistry, the\nLDA is usually expressed in terms of the exchange-correlation energy . In\nthe grand-canonical setting it is defined by\nExc(ρ) =E(ρ)−T(ρ),\nwithT(ρ) the lowest possible kinetic energy (5). The functional T(ρ) is\nstudied in Section 3 below, where it is proved that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleT(ρ)−q−2\n3cTFˆ\nR3ρ(x)5\n3dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslantεq−2\n3ˆ\nR3ρ(x)5\n3dx+C\nε13\n3ˆ\nR3|∇√ρ(x)|2dx\n(10)\nwithcTF= 35/341/3π4/3/5 the Thomas-Fermi constant. The lower bound\nwas derived by Nam [43] and we prove the missing upper bound (w ith a\nbetter power of ε) in Theorem 3 below. Actually, by following our proof of\nTheorem 2 (simply discarding the Coulomb interaction) we ca n also prove a\nlower bound on T(ρ), with an error similar to the right side of (8) but with\na smaller power of εin front of |∇ρθ|p. This provides the following estimate8 M. LEWIN, E.H. LIEB, AND R. SEIRINGER\non the exchange-correlation energy\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleExc(ρ)−ˆ\nR3eUEG/parenleftbig\nρ(x)/parenrightbig\ndx+q−2\n3cTFˆ\nR3ρ(x)5\n3dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n/lessorequalslantεˆ\nR3/parenleftbig\nρ(x)+ρ(x)2/parenrightbig\ndx+C(1+ε)\nεˆ\nR3|∇√ρ(x)|2dx\n+C\nε4p−1ˆ\nR3|∇ρθ(x)|pdx.(11)\nFor a rescaled density ρN(x) =ρ(x/N1/3) we obtain the same rate of con-\nvergence N11/12as in (9).\nRemark 4 (Local Spin Density Approximation) .In practice, it is often\nconvenient to not fix the total density but, rather, the densi ty of each spin\ncomponent\nρσ(x) =\n/summationdisplay\nn/greaterorequalslant1n/summationdisplay\nσ2,...,σn\n∈{1,...,q}ˆ\nR3(n−1)Γn(x,σ,x2,...,xn,σn;x,σ,x2,...,xn,σn)dx2···dxn\nforσ∈ {1,...,q}. SimilarlyasinTheorem1onecandefinethecorresponding\nspin-polarizedUEGenergy eUEG(ρ1,...,ρq)oftheuniformelectrongaswhere\nthe electrons of spin σare assumed to have the constant density ρσ. By\nfollowing the arguments in this paper, one can then prove the estimate\nsimilar to (8)\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleE(ρ1,...,ρq)−ˆ\nR3eUEG/parenleftbig\nρ1(x),...,ρq(x)/parenrightbig\ndx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslantεˆ\nR3/parenleftbig\nρ(x)+ρ(x)2/parenrightbig\ndx\n+C(1+ε)\nεˆ\nR3|∇√ρ(x)|2dx+C\nε4p−1ˆ\nR3|∇ρθ(x)|pdx.(12)\nIt is only for simplicity of notation that we work with the tot al density\nρ=/summationtextq\nσ=1ρσ.\n3.A priori estimates on T(ρ)andE(ρ)\nLower bounds on E(ρ) in (4) are well known and will be recalled below.\nUpper bounds are somewhat difficult to derive due to the constr aint that\nthe quantum states considered need to have the exact given de nsityρ. In\nthis section we prove an upper bound on the best kinetic energ y and use it\nto derive an upper bound on E(ρ). Because our bounds are of independent\ninterest we work in this section in any dimension d/greaterorequalslant1. First we quickly\nrecall the known lower bounds.\n3.1.Known lower bounds. We recall that the Lieb-Thirring inequal-\nity [39, 40, 38] states that there exists a positive constant cLT=cLT(d)>0\nsuch that\nTr(−∆)γ/greaterorequalslantq−2\ndcLTˆ\nRdργ(x)1+2\nddx (13)THE LOCAL DENSITY APPROXIMATION IN DENSITY FUNCTIONAL THEO RY 9\nforevery self-adjoint operator γonL2(Rd,Cq)suchthat0 /lessorequalslantγ/lessorequalslant1. Thebest\nconstant cLTis unknown but has been conjectured to be the semi-classical\nconstant\ncTF=4π2d\n(d+2)/parenleftbiggd\n|Sd−1|/parenrightbigg2\nd\n(14)\nin dimension d/greaterorequalslant3. In [43], Nam has proved that\nTr(−∆)γ/greaterorequalslantq−2\ndcTF(1−ε)ˆ\nRdργ(x)1+2\nddx−κ\nε3+4\ndˆ\nRd|∇√ργ(x)|2dx(15)\nfor every ε >0 and some constant κ=κ(d), in all space dimensions d/greaterorequalslant1.\nWe also recall the Hoffmann-Ostenhof inequality [17] which st ates that\nTr(−∆)γ/greaterorequalslantˆ\nRd/vextendsingle/vextendsingle∇√ργ(x)/vextendsingle/vextendsingle2dx (16)\nand always imposes that√ρ∈H1(Rd). Theinequality (16) does not require\nthe fermionic constraint 0 /lessorequalslantγ/lessorequalslant1.\nThe Lieb-Oxford inequality [32, 37, 20, 38] states that the t otal Coulomb\nenergy is bounded from below by\n∞/summationdisplay\nn=1TrHn/parenleftigg/summationdisplay\n1/lessorequalslantj<k/lessorequalslantn1\n|xj−xk|/parenrightigg\nΓn/bracerightigg\n/greaterorequalslant1\n2ˆ\nR3ˆ\nR3ρ(x)ρ(y)\n|x−y|dxdy−1.64ˆ\nR3ρ(x)4\n3dx(17)\nwhereΓ = {Γn}is agrand-canonical quantumstate satisfyingtheconditio ns\nin (4). Inspired by [2], this bound was recently generalized in [29] to\n∞/summationdisplay\nn=1TrHn/parenleftigg/summationdisplay\n1/lessorequalslantj<k/lessorequalslantn1\n|xj−xk|/parenrightigg\nΓn/bracerightigg\n/greaterorequalslant1\n2ˆ\nR3ˆ\nR3ρ(x)ρ(y)\n|x−y|dxdy−/parenleftigg\n3\n5/parenleftbigg9π\n2/parenrightbigg1\n3\n+ε/parenrightiggˆ\nR3ρ(x)4\n3dx\n−0.001206\nε3ˆ\nR3|∇ρ(x)|dx(18)\nbut the constant (3 /5)(9π/2)1/3≃1.4508 is not expected to be the optimal\nLieb-Oxford constant.\nUsing (13) together with (17), we obtain the following.\nCorollary 1 (Lower bound on E(ρ)).We have\nE(ρ)/greaterorequalslantq−2\n3cLTˆ\nR3ρ(x)5\n3dx−1.64ˆ\nR3ρ(x)4\n3dx (19)\nfor every ρ/greaterorequalslant0such that√ρ∈H1(Rd).\nThe constants can be improved at the expense of adding gradie nt correc-\ntions, using (15) and (18).10 M. LEWIN, E.H. LIEB, AND R. SEIRINGER\n3.2.Upper bound on the best kinetic energy. Let us recall that the\nlowest possible kinetic energy of a fixed density ρ∈L1(Rd,R+) with√ρ∈\nH1(Rd) reads\nT(ρ) := min\n0/lessorequalslantγ=γ∗/lessorequalslant1\nργ=ρTr(−∆)γ.\nIn [30], we have shown that for ρ/lessorequalslant1,\nT(ρ)/lessorequalslantˆ\nRd|∇√ρ|2+q−2\ndcTFˆ\nRdρ\nby using the trial state\nγ=/radicalbig\nρ(x)1/parenleftbigg\n−∆/lessorequalslantd+2\ndcTFq−2\nd/parenrightbigg/radicalbig\nρ(x).\nOur goal in this section is to prove a similar bound without th e assumption\nthatρ/lessorequalslant1. Coherent states [33] can usually give good boundson the ki netic\nenergy but they do not preserve the density. The main difficult y here is to\nconstruct a state having the exact given density ρ. The next result says that\nthe semi-classical approximation to the kinetic energy is a n upper bound to\nthe exact T(ρ), up to some gradient corrections.\nTheorem 3 (Upper bound on the best kinetic energy) .There are two con-\nstantsκ1,κ2>0depending only on the space dimension dsuch that\nT(ρ)/lessorequalslantq−2\ndcTF(1+κ1ε)ˆ\nRdρ(x)1+2\nddx+κ2(1+√ε)2\nεˆ\nRd|∇√ρ(x)|2dx,\n(20)\nfor every ρ∈L1(Rd,R+)with√ρ∈H1(Rd)and every ε >0, wherecTFis\nthe Thomas-Fermi constant (14).\nNote that the gradient correction in (20) has a better behavi or inεthan\nin Nam’s lower bound (15).\nIn dimension d= 1, March and Young have given the proof of a better\nestimate without the parameter ε, in [41, Eq. (9)]:\nT(ρ)/lessorequalslantq−2cTFˆ\nRρ(x)3dx+ˆ\nR/vextendsingle/vextendsingle/vextendsingle/parenleftbig√ρ/parenrightbig′(x)/vextendsingle/vextendsingle/vextendsingle2\ndx.\nIn the same paper they also state a result in 3D (for a constant c > cTF)\nbut the proof has a mistake. This was mentioned as a conjectur e in [35, Sec.\n5.B]. Our result (20) can therefore be seen as a solution to th e March-Young\nproblem. We conjecture that a similar bound holds without th e parameter\nεin dimension d= 2,3 as well.\nRemark 5 (Explicit constants in 3D) .In dimension d= 3 one can take\nκ1= 1, κ 2= 48 in (20).\nTheseconstantsarenotoptimalandtheyareonlydisplayedf orconcreteness.\nOurproofallowstoslightlyimprovetheconstantsunderthe assumptionthat\nεis small enough. For instance for ε/lessorequalslant1, we have the better inequality\nT(ρ)/lessorequalslantq−2\n3cTF/parenleftig\n1+ε\n15/parenrightigˆ\nR3ρ(x)5\n3dx+19\nεˆ\nR3|∇√ρ(x)|2dx.(21)THE LOCAL DENSITY APPROXIMATION IN DENSITY FUNCTIONAL THEO RY 11\nProof of Theorem 3. For simplicity we only write the proof in the no-spin\ncaseq= 1. Recall that the free Fermi sea\nPr=1/parenleftbigg\n−∆/lessorequalslantd+2\ndcTFr2/d/parenrightbigg\nhas the constant density ρPr=rand the constant kinetic energy density\ncTFr1+2/d. In particular, if we take\nγ=/radicalbig\nf(x)1/parenleftbigg\n−∆/lessorequalslantd+2\ndcTFr2/d/parenrightbigg/radicalbig\nf(x)\nfor some f/greaterorequalslant0, we obtain\nργ=rf(x),Tr(−∆)γ=rˆ\nRd|∇/radicalbig\nf(x)|2dx+cTFr1+2\ndˆ\nRdf(x)dx.\n(22)\nSee for instance [30, Sec. 5] for details. In addition, we hav e in the sense of\noperators 0 /lessorequalslantγ/lessorequalslantf(x), hence γis a fermionic one-particle density matrix\nunder the additional condition that f/lessorequalslant1.\nLet now η/greaterorequalslant0 be a smooth non-negative function such that\nˆ∞\n0η(t)dt= 1,ˆ∞\n0η(t)dt\nt/lessorequalslant1. (23)\nUsing the smooth layer cake principle\nρ(x) =ˆ∞\n0η/parenleftbiggt\nρ(x)/parenrightbigg\ndt\nwe introduce the trial state\nγ=ˆ∞\n0/radicaligg\nη/parenleftbiggt\nρ(x)/parenrightbigg\n1/parenleftbigg\n−∆/lessorequalslantd+2\ndcTFt2/d/parenrightbigg/radicaligg\nη/parenleftbiggt\nρ(x)/parenrightbiggdt\nt.\nIn the sense of operators, we have\n0/lessorequalslantγ/lessorequalslantˆ∞\n0η/parenleftbiggt\nρ(x)/parenrightbiggdt\nt=ˆ∞\n0η(t)dt\nt/lessorequalslant1.\nIn addition, γhas the required density\nργ(x) =ˆ∞\n0η/parenleftbiggt\nρ(x)/parenrightbigg\ndt=ρ(x).\nHenceγis admissible and it can be used to get an upper bound on T(ρ).\nFrom (22), its kinetic energy is\nTr(−∆)γ=ˆ\nRddxˆ∞\n0dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∇x/radicaligg\nη/parenleftbiggt\nρ(x)/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n+cTFˆ\nRdρ(x)1+2\nddxˆ∞\n0η(t)t2\nddt.12 M. LEWIN, E.H. LIEB, AND R. SEIRINGER\nNote that\nˆ\nRddxˆ∞\n0dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∇x/radicaligg\nη/parenleftbiggt\nρ(x)/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n=ˆ\nRddxˆ∞\n0dt/vextendsingle/vextendsingle/vextendsingle∇xη/parenleftig\nt\nρ(x)/parenrightig/vextendsingle/vextendsingle/vextendsingle2\n4η/parenleftig\nt\nρ(x)/parenrightig\n=ˆ\nRddx|∇ρ(x)|2\n4ρ(x)4ˆ∞\n0t2dt/vextendsingle/vextendsingle/vextendsingleη′/parenleftig\nt\nρ(x)/parenrightig/vextendsingle/vextendsingle/vextendsingle2\nη/parenleftig\nt\nρ(x)/parenrightig\n=ˆ\nRd|∇√ρ(x)|2dxˆ∞\n0t2η′(t)2\nη(t)dt.\nHence we have proved that\nTr(−∆)γ=ˆ\nRd|∇√ρ(x)|2dxˆ∞\n0t2η′(t)2\nη(t)dt\n+cTFˆ\nRdρ(x)1+2\nddxˆ∞\n0η(t)t2\nddt\nfor allη/greaterorequalslant0 satisfying the two constraints (23). The smallest constan t we\ncan get in front of the ρ1+2/dterm iscTF, by concentrating ηat the point\nt= 1, but this makes the other term blow up. If we fix\nˆ∞\n0η(t)t2\nddt= 1+ε\nthen the best constant we can get in front of the gradient term is given by\nthe variational problem\nC(ε) := inf\nη/greaterorequalslant0´∞\n0η=1´∞\n0η/t/lessorequalslant1´∞\n0t2/dη/lessorequalslant1+εˆ∞\n0t2η′(t)2\nη(t)dt.\nWe claim that C(ε)/lessorequalslantconst.(1 +ε−1) forεsmall enough, which we prove\nby an appropriate choice of η.\nLet us first take, for instance,\nηε(t) =3\n2ε3(t−1)21(1/lessorequalslantt/lessorequalslant1+ε)+3\n2ε3(1+2ε−t)21(1+ε/lessorequalslantt/lessorequalslant1+2ε).\n(24)\nThen´\nηε= 1 and´\nηε(t)/tdt/lessorequalslant1 sinceηεis supported on [1 ,∞). Using\nthe simple boundsˆ∞\n0η(t)t2\nddt/lessorequalslant(1+2ε)2/d\nandˆ∞\n0t2η′(t)2\nη(t)dt/lessorequalslant(1+2ε)2ˆ∞\n0η′(t)2\nη(t)dt=12(1+2ε)2\nε2,\nwe obtain (after changing εintoε/2)\nT(ρ)/lessorequalslantcTF(1+ε)2\ndˆ\nRdρ(x)1+2\nddx+48(1+ε)2\nε2ˆ\nRd|∇√ρ(x)|2dx,∀ε >0.\n(25)THE LOCAL DENSITY APPROXIMATION IN DENSITY FUNCTIONAL THEO RY 13\nThe behavior of the correction in front of the semi-classica l termcTFis\nnot optimal for small ε. It can be replaced by 1+ κ1ε2, forε/lessorequalslant1. To see\nthis we slightly translate the function (24) to the left by an amount−εband\nintroduce\nηε,b(t) =3\n2ε3(t−1+εb)21(1−εb/lessorequalslantt/lessorequalslant1+(1−b)ε)\n+3\n2ε3(1+(2−b)ε−t)21(1−εb+ε/lessorequalslantt/lessorequalslant1+(2−b)ε).(26)\nThen we have\nˆ∞\n0ηε,b(t)dt\nt= 1+(b−1)ε+/parenleftbigg11\n10−2b+b2/parenrightbigg\nε2\n+/parenleftbigg\n−13\n10+33\n10ε−3b2+b3/parenrightbigg\nε3+O(ε4)\nand\nˆ∞\n0ηε,b(t)t2\nddt= 1−2\nd(b−1)ε\n+1\n10d2/parenleftbig\n22−40b+20b2−11d+20bd−10b2d/parenrightbig\nε2+O(ε3).\nThe unique bεsuch that´∞\n0ηε,bε(t)dt\nt= 1 satisfies\nbε= 1−ε\n10−3\n350ε3+O(ε4)\nand for this bεwe have\nˆ∞\n0ηε,bε(t)t2\nddt= 1+2+d\n10d2ε2+O(ε4),\nˆ∞\n0t2η′\nε,bε(t)2\nηε,bε(t)dt=12\nε2+O(1).\nThis is how we can get (21) for εsmall enough (after replacing ε2byε)./square\nRemark 6. In the 3D case we can take for instance b= 1−ε/10−4ε3/350.\nOne can then verify that\nˆ∞\n0ηε,b(t)dt\nt/lessorequalslant1,ˆ∞\n0ηε,b(t)t2\n3dt/lessorequalslant1+ε2\n15\nandˆ∞\n0t2η′\nε,bε(t)2\nηε,bε(t)dt/lessorequalslant19\nε2\nfor allε/lessorequalslant1. Hence\nT(ρ)/lessorequalslantcTF/parenleftbigg\n1+ε2\n15/parenrightbiggˆ\nR3ρ(x)5\n3dx+19\nε2ˆ\nR3|∇√ρ(x)|2dx,∀ε/lessorequalslant1.(27)\nCombining with (25), we find the estimate (20) for κ1= 1 and κ2= 48.14 M. LEWIN, E.H. LIEB, AND R. SEIRINGER\n3.3.Upper bound on E(ρ).It is well known that any fermionic one-\nparticle density matrix γ(i.e., an operator satisfying 0 /lessorequalslantγ=γ∗/lessorequalslant1) is\nrepresentable by a quasi-free state Γ γin Fock space [1]. The two-particle\ndensity matrix of such a state is given by Wick’s formula\nΓ(2)\nγ(x1,σ1,x2,σ2;y1,σ′\n1,y2,σ′\n2)\n=γ(x1,σ1;y1,σ′\n1)γ(x2,σ2;y2,σ′\n2)−γ(x1,σ1;x2,σ2)γ(y1,σ′\n1;y2,σ′\n2).(28)\nIn particular, the corresponding interaction energy with p air potential wis\n1\n2¨\nRd×Rdw(x−y)\nρ(x)ρ(y)−q/summationdisplay\nσ,σ′=1|γ(x,σ;y,σ′)|2\ndxdy.\nFrom this we immediately obtain the following.\nCorollary 2 (Upper bound on E(ρ) in dimension d/greaterorequalslant1).We have\nE(ρ)/lessorequalslantq−2\ndcTF(1+κ1ε)ˆ\nRdρ(x)1+2\nddx+κ2(1+√ε)2\nεˆ\nRd|∇√ρ(x)|2dx\n(29)\nfor every ρ/greaterorequalslant0such that√ρ∈H1(Rd).\nThis is for the grand-canonical version (4) of the Levy-Lieb functional\nwhich is the object of concern in this paper. It was proved in [ 34] that any\nfermionic γwith integer trace N= Tr(γ) is also the one-particle density\nmatrix of an N-particle mixed state Γ on the fermionic space HN, such that\nthe corresponding two-particle density matrix satisfies\nΓ(2)/lessorequalslantΓ(2)\nγ\nin the sense of operators. From the positivity of the Coulomb potential we\ndeduce immediately the following result for mixed canonica l states.\nCorollary 3 (Upperboundinthemixedcanonicalcase) .Letρ∈L1(Rd,R+)\nbe such that´\nRdρ=N∈N, and√ρ∈H1(Rd). Then there exists a mixed\nstateΓon the fermionic space/logicalandtextN\n1L2(Rd×{1,...,q})such that ρΓ=ρand\nTr\nN/summationdisplay\nj=1−∆xj+/summationdisplay\n1/lessorequalslantj<k/lessorequalslantN1\n|xj−xk|\nΓ\n/lessorequalslantq−2\ndcLT(1+κ1ε)ˆ\nRdρ(x)1+2\nddx+κ2(1+√ε)2\nεˆ\nRd|∇√ρ(x)|2dx.\n4.Proof of Theorems 1 and 2\nOur proof is divided into several steps. The first is to show th at the\nenergy is essentially local, that is, to prove that\nE(ρ)≈/summationdisplay\nkE/parenleftbig\nρχk/parenrightbig\n(30)\nwhere{χk}is a smooth partition of unity,/summationtext\nkχk= 1. The precise state-\nment of (30) will involve upper and lower bounds, as well as an average over\nthe translations, rotations and dilations of the partition itself. The lowerTHE LOCAL DENSITY APPROXIMATION IN DENSITY FUNCTIONAL THEO RY 15\nbound was indeed already shown in [30] using the Graf-Schenk er inequal-\nity [14]. The upper bound is the main new ingredient of our pro of. The two\nbounds are derived in Sections 4.1 and 4.2. This will allow us to provide a\nrather simple proof of Theorem 1 in Section 4.4, using the con vergence for\ntetrahedra which will be studied in Section 4.3.\nInSection4.5wewillestimate thedeviation oftheenergywh enwereplace\nρχkby a constant function, say ρ(xk)χkfor some xkin the support of χk. If\nρis essentially constant in the corresponding region, the er ror will be small,\nbut ifρis not constant we bound the energy using some gradient terms ,\nutilizing our upper bound (29).\nAfter showing the Lipschitz regularity of eUEGin Section 4.6, we will\nbe able to conclude the proof of Theorem 2 in Section 4.7. We re place\nE(ρ(xk)χk) by´\nR3eUEG(ρ(x))χk(x)dxwhenρis large enough on supp( χk),\nusing some quantitative estimates derived for tetrahedra i n Section 4.3.\nIn the rest of the paper we call Ca generic constant which can some-\ntimes change from line to line, but which only depends on q(the number\nof spin states) and p,θ, the two parameters appearing in the statement of\nTheorem 2.\n4.1.Upper bound in terms of local densities. Our main goal here is\nto give an upper bound on the energy E(ρ) by splitting ρinto a sum of local\ndensities. In the classical case, we have the exact subaddit ivity property\n(see [30, Lem. 2.5])\nEcl(ρ1+ρ2)/lessorequalslantEcl(ρ1)+Ecl(ρ2)\nwhich considerably simplifies the analysis and was one of the main tools of\nour previous work [30]. In particular we immediately find an u pper bound\nin the form\nEcl(ρ)/lessorequalslant/summationdisplay\nkEcl(ρχk)\nfor a partition of unity χk. In the quantum case this is not as easy. The first\ndifficulty is that we cannot cut sharply and have to use a smooth partition\nof unity. This has the consequence that neighboring local de nsities overlap.\nBut then, for two densities ρ1andρ2with overlapping support, it is not\nobvious how to relate E(ρ1+ρ2) withE(ρ1) andE(ρ2). This is due to the\nfermionic nature of the electrons which puts a very strong co nstraint on trial\nstates. If we take two trial quantum states for ρ1andρ2, we cannot simply\ntake their tensor product and use it as a trial state for ρ1+ρ2. The tensor\nproduct does not have the fermionic symmetry, and if we anti- symmetrize\nit the density is not equal to ρ1+ρ2anymore.\nFor this reason, we will use an incomplete partition of unity with holes, in\norder to make sure that the local quantum states are not overl apping. Since\nwe want to get the exact density, holes are however in princip le not allowed.\nInstead of filling the holes with electrons, our idea is to rat her average over\nall the possible rotations, translations and dilations of t he partition of unity,\nwhich will make the holes disappear in average. All the argum ents of this\nsection apply the same to a tiling made of cubes, but for a bett er matching\nwith the lower boundwe will consider a tiling made of tetrahe dra. Ourlower16 M. LEWIN, E.H. LIEB, AND R. SEIRINGER\nbound relies on the Graf-Schenker inequality [14] which req uires the use of\ntetrahedra.\nLet us consider the unit cube C1= (−1/2,1/2)3, which is the union of 24\ndisjoint identical tetrahedra ∆1,...,∆24, all of volume 1 /24. Since the cube\ncan be repeated in the whole space, we obtain a tiling of R3with tetrahedra:\nR3=/uniondisplay\nz∈Z324/uniondisplay\nj=1(∆j+z) (31)\nand the corresponding partition of unity\n/summationdisplay\nz∈Z324/summationdisplay\nj=11ℓ∆j(x−ℓz)≡1,for a.e.x∈R3,\nfor any fixed size ℓ >0 of the tiles. Any ∆jcan be written as ∆j=µj∆\nwhere∆is a reference tetrahedron with 0 as its center of mass. Here\nµj= (zj,Rj)∈C1×SO(3) is an appropriate translation and rotation,\nwhich acts as µjx=Rjx−zj, hence∆j=Rj∆−zj. In each tetrahedron\nwe now place the regularized characteristic function\nχℓ,δ,j:=1\n(1−δ/ℓ)31ℓµj(1−δ/ℓ)∆∗ηδ. (32)\nHereηδ(x) = (10/δ)3η1(10x/δ) whereη1is a fixed C∞\ncnon-negative radial\nfunction with support in the unit ball and such that´\nR3η1= 1. Assuming\nthatδ/lessorequalslantℓ/2, thefunction χℓ,δ,jhas its supportwell inside ℓ∆j, at a distance\nproportionalto δfromitsboundary. Theprefactorhasbeenchosentoensure\nthatˆ\nR3χℓ,δ,j=ℓ3\n24=|ℓ∆|.\nThe function\n/summationdisplay\nz∈Z324/summationdisplay\nj=1χℓ,δ,j(x−ℓz)\nis equal to (1 −δ/ℓ)−3>1 inside the tiles but vanishes in a neighborhood of\nthe boundary of the tiles. It is the incomplete partition of u nity which we\nhave mentioned above. We obtain a partition of unity after av eraging over\nthe translations of the tiling:\n1\nℓ3ˆ\nCℓ/summationdisplay\nz∈Z324/summationdisplay\nj=1χℓ,δ,j(x−ℓz−τ)dτ= 1,for a.e.x∈R3.(33)\nHereCℓ= (−ℓ/2,ℓ/2)3=ℓC1is the cube of side length ℓ. This is because\nfor anyf∈L1(R3),\nˆ\nCℓ/summationdisplay\nz∈Z3f(x−ℓz−τ)dτ=ˆ\nCℓ/summationdisplay\nz∈ℓZ3f(x−z−τ)dτ\n=ˆ\nR3f(x−τ)dτ=ˆ\nR3f(τ)dτ.\nThe main result of this section is the following upper bound.THE LOCAL DENSITY APPROXIMATION IN DENSITY FUNCTIONAL THEO RY 17\nProposition 1 (Upper bound in terms of local densities) .There exists a\nuniversal constant Csuch that for any√ρ∈H1(R3), any0< δ < ℓ/ 2and\nany0< α <1/2,\nE(ρ)/lessorequalslant/parenleftbiggˆ1+α\n1−αds\ns4/parenrightbigg−1ˆ1+α\n1−αdt\nt4ˆ\nSO(3)dRˆ\nCtℓdτ\n(tℓ)3×\n×/summationdisplay\nz∈Z324/summationdisplay\nj=1E/parenleftig\nχtℓ,tδ,j(R· −tℓz−τ)ρ/parenrightig\n+Cδ2log(α−1)ˆ\nR3ρ2.(34)\nIn particular, we can find ℓ′∈(ℓ(1−α),ℓ(1+α)),δ′∈(δ(1−α),δ(1+α))\nand an isometry (τ,R)∈R3×SO(3)such that\nE(ρ)/lessorequalslant/summationdisplay\nz∈Z324/summationdisplay\nj=1E/parenleftig\nχℓ′,δ′,j(R· −ℓ′z−τ)ρ/parenrightig\n+Cδ2log(α−1)ˆ\nR3ρ2.(35)\nTherightsideof (34)involves ourincompletepartitionofu nitywithholes,\nwhich is rotated (with the rotation R), translated (with the translation τ)\nand dilated (with the dilation parameter t). The error is small only when\nδ(the size of the holes) is small. However we cannot take δ= 0 since that\nwould make the gradient of the densities χtℓ,tδ,j(R· −tℓz−τ)ρblow up.\nNevertheless, the statement is that the energy decouples an d the holes can\nbe neglected, at the expense of an error of the order δ2´\nR3ρ2. In (34) we use\ndilations for purely technical reasons, in order to better c ontrol error terms.\nProof of Proposition 1. Using (33), we write our density ρas follows\nρ(x) =1\nℓ3ˆ\nCℓ/summationdisplay\nz∈Z324/summationdisplay\nj=1χℓ,δ,j(x−ℓz−τ)ρ(x)dτ. (36)\nFor every fixed τ∈Cℓ, we can construct a grand canonical trial state Γ τ\nhaving the density\nρΓτ(x) =/summationdisplay\nz∈Z324/summationdisplay\nj=1χℓ,δ,j(x−ℓz−τ)ρ(x).\nFor this we pick Γ τ=/circlemultiplytext24\nj=1/circlemultiplytext\nz∈Z3Γτ,z,jwhere each Γ τ,z,jhas the density\nρΓτ,z,j(x) =χℓ,δ,j(x−ℓz−τ)ρ(x)\nand minimizes the corresponding energy E(χℓ,δ,j(· −ℓz−τ)ρ). Since the\nquantum states Γ τ,z,jhave disjoint supports, we can anti-symmetrize the\nstate Γ τin the standard manner. We denote by Γ τ,athe anti-symmetrized\nstate. The energy of Γ τ,ais equal to that of Γ τand so is its density ρΓτ,a=\nρΓτ. Finally we take as trial state\nΓ =1\nℓ3ˆ\nCℓΓτ,adτ18 M. LEWIN, E.H. LIEB, AND R. SEIRINGER\nwhich satisfies by construction that ρΓ=ρ. We find the upper bound\nE(ρ)/lessorequalslant1\nℓ3ˆ\nCℓE(ρΓτ)dτ+1\nℓ3ˆ\nCℓD(ρΓτ)dτ−D(ρ)\n=1\nℓ3ˆ\nCℓ/summationdisplay\nz∈Z324/summationdisplay\nj=1E/parenleftbig\nχℓ,δ,j(·−ℓz−τ)ρ/parenrightbig\ndτ+1\nℓ3ˆ\nCℓD(ρΓτ−ρ)dτ.\n(37)\nHere we employ the usual notation\nD(ρ) :=1\n2ˆ\nR3ˆ\nR3ρ(x)ρ(y)\n|x−y|dxdy. (38)\nIn the second line of (37) we have used that the energy of a tens or product\nof states of disjoint supports is the sum of the energies of th e pieces [30].\nThis is because the cross terms in the direct energy exactly c ancel with the\nmany-particle interactions of different states in the tensor product.\nThe error term in (37) is solely due to the nonlinearity of the direct term\nand it may be rewritten as\n1\nℓ3ˆ\nCℓD(ρΓτ−ρ)dτ=1\nℓ3ˆ\nCℓD\n/summationdisplay\nz∈Z324/summationdisplay\nj=1(1ℓ∆j−χℓ,δ,j)(·−ℓz−τ)ρ\ndτ.\nFor every real-valued ( ℓZ3)–periodic function f, we have\n1\nℓ3ˆ\nCℓD/parenleftig\nf(·−τ)ρ/parenrightig\ndτ\n=1\n2/summationdisplay\nk∈(2π/ℓ)Z3/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nℓ3ˆ\nCℓf(z)e−ik·zdz/vextendsingle/vextendsingle/vextendsingle/vextendsingle2¨\nR3×R3eik·(x−y)\n|x−y|ρ(x)ρ(y)dxdy\n= 2π/summationdisplay\nk∈(2π/ℓ)Z3/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nℓ3ˆ\nCℓf(z)e−ik·zdz/vextendsingle/vextendsingle/vextendsingle/vextendsingle2ˆ\nR3|/hatwideρ(p)|2\n|p−k|2dp.\nHence we obtain\n1\nℓ3ˆ\nCℓD(ρΓτ−ρ)dτ= 2π/summationdisplay\nk∈2πZ3/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\nC1fδ/ℓ(z)e−ik·zdz/vextendsingle/vextendsingle/vextendsingle/vextendsingle2ˆ\nR3|/hatwideρ(p)|2\n|p−k/ℓ|2dp,\nwith\nfε(x) =24/summationdisplay\nj=1/parenleftbigg\n1µj∆−1\n(1−ε)31µj(1−ε)∆∗ηε/parenrightbigg\n(39)\nforǫ=δ/ℓ. We haveˆ\nC1fδ/ℓ(z)dz= 0.\nSince all the functions appearing in the sum on the right side of (39) are\nsupported in the unit cube, we also obtain for k∈2πZ3\\{0},\nˆ\nC1fδ/ℓ(z)e−ik·zdz=−(2π)3\n(1−ε)324/summationdisplay\nj=1/hatwider1µj(1−ε)∆(k)/hatwideη1(εk).(40)THE LOCAL DENSITY APPROXIMATION IN DENSITY FUNCTIONAL THEO RY 19\nThis results in the final formula for the error term\n1\nℓ3ˆ\nCℓD(ρΓτ−ρ)dτ\n= (2π)7/summationdisplay\nk∈2πZ3\nk/ne}ationslash=0|/hatwideη1(εk)|2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n(1−ε)324/summationdisplay\nj=1/hatwider1µj(1−ε)∆(k)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2ˆ\nR3|/hatwideρ(p)|2\n|p−k/ℓ|2dp.\n(41)\nIn order to control the denominator |p−k/ℓ|2, we are going to average our\ncalculation over all the rotations of the tiling. We also rep laceℓandδby,\nrespectively, tℓandtδand we average over t∈(1−α,1+α) with a weight\nt−4. Rotating the tiling is the same as rotating ρ. In addition, ǫ=δ/ℓis\nindependent of t. Hence we are left with estimating\n/parenleftbiggˆ1+α\n1−αdt\nt4/parenrightbigg−1ˆ1+α\n1−αdt\nt4ˆ\nSO(3)dR1\n|p−Rk/(tℓ)|2\n=ℓ2\n4π|k|23(1−α2)3\n2α(3+α2)ˆ1\n1−α\n1\n1+αr2drˆ\nS2dω1\n|p′−ωr|2\n=ℓ2\n4π|k|23(1−α2)3\n2α(3+α2)1Aα∗1\n|·|2(p′)\nwithp′=pℓ/|k|and where Aαis the annulus\nAα=/braceleftbigg1\n1+α/lessorequalslant|x|/lessorequalslant1\n1−α/bracerightbigg\n.\nWe will use the following estimate\nLemma 1. We have\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble1Aα∗1\n|·|2/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL∞/lessorequalslantCαlog(α−1) (42)\nfor allα/lessorequalslant1/2.\nThe proof of (42) is a simple computation which is provided at the very\nend of the proof. Using (42) we obtain\n/parenleftbiggˆ1+α\n1−αdt\nt4/parenrightbigg−1ˆ1+α\n1−αdt\nt4ˆ\nSO(3)dR1\n(tℓ)3ˆ\nCtℓD(ρΓτ,R−ρ)dτ\n/lessorequalslantClog(α−1)\n/summationdisplay\nk∈2πZ3\nk/ne}ationslash=0ℓ2\n|k|2|/hatwideη1(εk)|2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n(1−ε)324/summationdisplay\nj=1/hatwider1µj(1−ε)∆(k)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nˆ\nR3ρ2\n(43)\nand it remains to estimate the sum in the parenthesis. For thi s we have to\nbound/summationtext24\nj=1/hatwider1µj(1−ε)∆(k).20 M. LEWIN, E.H. LIEB, AND R. SEIRINGER\nLemma 2 (Fourier transform of the reduced tetrahedra) .We have\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n(1−ε)324/summationdisplay\nj=1/hatwider1µj(1−ε)∆(k)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n/lessorequalslantC\nε4+ε2|k|2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\nC1/parenleftbigg\nx−24/summationdisplay\nj=1zj1∆j/parenrightbigg\ne−ik·xdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n(44)\nfor all0< ε <1/2and allk∈2πZ3\\{0}.\nProof.We recall that ∆j=Rj∆1−zjwithµj= (zj,Rj)∈C1×SO(3).\nWe have\n(1−ε)−3/hatwider1µj(1−ε)∆(k) = (2π)−3/2(1−ε)−3ˆ\nµj(1−ε)∆e−ik·xdx\n= (2π)−3/2ˆ\n∆e−ik·(Rj(1−ε)x−zj)dx\n= (2π)−3/2ˆ\nµj∆e−ik·x+iεk·(x−zj)dx.\nSincek∈2πZ3\\{0}, the integral vanishes at ε= 0 after summing over j.\nInserting the derivative at ε= 0 yields\n(1−ε)−324/summationdisplay\nj=1/hatwider1µj(1−ε)∆(k) =i(2π)−3/2εk·ˆ\nC1/parenleftbigg\nx−24/summationdisplay\nj=1zj1∆j/parenrightbigg\ne−ik·xdx\n−ε224/summationdisplay\nj=1ˆ1\n0(1−s)dsˆ\n∆j/parenleftbig\nk·(x−zj)/parenrightbig2e−ik·x+iεsk·(x−zj)dx.\nWe claim that the second term is uniformly bounded with respe ct tok.\nIndeed, one integration by parts gives\nˆ1\n0(1−s)dsˆ\n∆j/parenleftbig\nk·(x−zj)/parenrightbig2e−ik·x+iεsk·(x−zj)dx\n=iˆ1\n01−s\n1−εsdsˆ\n∆jk·(x−zj) (x−zj)·∇xe−ik·x+iεsk·(x−zj)dx\n=iˆ1\n01−s\n1−εsdsˆ\n∂∆jk·(x−zj) (x−zj)·nj(x)e−ik·x+iεsk·(x−zj)dx\n−4iˆ1\n01−s\n1−εsdsˆ\n∆jk·(x−zj)e−ik·x+iεsk·(x−zj)dx. (45)\nHerenj(x)isthenormalizedvectorperpendicularto ∂∆jpointingoutwards.\nIntegrating once more in the same manner (involving the edge s of the faces\nof∂∆jfor the first term), we see that (45) is bounded uniformly in k, hence\nwe obtain (44). /squareTHE LOCAL DENSITY APPROXIMATION IN DENSITY FUNCTIONAL THEO RY 21\nInserting (44) in (43), we obtain the two error terms\nε2/summationdisplay\nk∈2πZ3\nk/ne}ationslash=0|/hatwideη1(εk)|2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆ\nC1/parenleftbigg\nx−24/summationdisplay\nj=1zj1∆j/parenrightbigg\ne−ik·xdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n/lessorequalslantε2(2π)3/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublex−24/summationdisplay\nj=1zj1∆j/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(C1)=Cε2\n(using here that /ba∇dbl/hatwideη1/ba∇dblL∞/lessorequalslant(2π)3/2) and\nε4/summationdisplay\nk∈2πZ3\nk/ne}ationslash=0|/hatwideη1(εk)|2\n|k|2∼\nε→0ε3ˆ\nR3|/hatwideη1(k)|2\n|k|2dk.\nRecalling that ε=δ/ℓ, our final estimate on the averaged error is propor-\ntional to\nδ2log(α−1)/parenleftbigg\n1+δ\nℓ/parenrightbiggˆ\nR3ρ2.\nIn order to conclude the proof of Proposition 1, it remains to provide the\nProof of Lemma 1. We have\n1Aα∗1\n|·|2(x) = 2πˆ1\n1−α\n1\n1+αr2drˆπ\n0sin(ϕ)dϕ1\nr2+|x|2−2r|x|cosϕ\n=π\n|x|ˆ1\n1−α\n1\n1+αlog/parenleftigg\nr+|x|/vextendsingle/vextendsingler−|x|/vextendsingle/vextendsingle/parenrightigg\nrdr\n=π|x|ˆ 1\n|x|(1−α)\n1\n|x|(1+α)log/parenleftigg\nr+1/vextendsingle/vextendsingler−1/vextendsingle/vextendsingle/parenrightigg\nrdr.\nFor|x|/lessorequalslant1/5 and 0< α <1/2, the integrand is bounded on the correspond-\ning interval and we obtain\n1Aα∗1\n|·|2(x)/lessorequalslantCα.\nSimilarly, for |x|/greaterorequalslant4 and 0< α <1/2 the integrand can be estimated by\nr2, which gives again\n1Aα∗1\n|·|2(x)/lessorequalslantCα\n|x|2/lessorequalslantCα.\nFinally, for 1 /5/lessorequalslant|x|/lessorequalslant4, we have\n1Aα∗1\n|·|2(x)/lessorequalslantCα+Cˆ 1\n|x|(1−α)\n1\n|x|(1+α)/vextendsingle/vextendsinglelog/vextendsingle/vextendsingler−1/vextendsingle/vextendsingle/vextendsingle/vextendsingledr.\nThe last integral is over an interval of length\n1\n|x|(1−α)−1\n|x|(1+α)=2α\n|x|(1−α2)/lessorequalslant40\n3α.22 M. LEWIN, E.H. LIEB, AND R. SEIRINGER\nThe integral is maximum when the interval is placed at the div ergence point\nr= 1. So we have\nˆ 1\n|x|(1−α)\n1\n|x|(1+α)/vextendsingle/vextendsinglelog/vextendsingle/vextendsingler−1/vextendsingle/vextendsingle/vextendsingle/vextendsingledr/lessorequalslantˆ1+40\n3α\nmax(0,1−40\n3α)/vextendsingle/vextendsinglelog/vextendsingle/vextendsingler−1/vextendsingle/vextendsingle/vextendsingle/vextendsingledr/lessorequalslantCαlog(α−1).\n/square\nThis concludes the proof of Proposition 1. /square\n4.2.Lower bound in terms of local densities. Next we turn to the\nlower bound. We are going to use the same tiling made of tetrah edra, with\nthe difference that we do not insert any hole. Similarly to (32) , we introduce\nξℓ,δ,j:=1ℓµj∆∗ηδ (46)\nwhich forms a smooth partition of unity, without holes,\n/summationdisplay\nz∈Z324/summationdisplay\nj=1ξℓ,δ,j(x−ℓz) = 1.\nProposition 2 (Lower bound in terms of local densities) .There exists a\nuniversal constant Csuch that for any√ρ∈H1(R3)and any δ >0with\n0< δ/ℓ < 1/C, we have\nE(ρ)/greaterorequalslant1−Cδ/ℓ\nℓ3/summationdisplay\nz∈Z324/summationdisplay\nj=1ˆ\nSO(3)ˆ\nCℓE/parenleftig\nξℓ,δ,j(R· −ℓz−τ)ρ/parenrightig\ndRdτ\n−C\nℓˆ\nR3/parenleftig/parenleftbig\n1+δ−1/parenrightbig\nρ+δ3ρ2/parenrightig\n.(47)\nIn particular, we can find an isometry (τ,R)∈R3×SO(3)such that\nE(ρ)/greaterorequalslant/parenleftbigg\n1−Cδ\nℓ/parenrightbigg/summationdisplay\nz∈Z324/summationdisplay\nj=1E/parenleftig\nξℓ,δ,j(R· −ℓz−τ)ρ/parenrightig\n−C\nℓˆ\nR3/parenleftig/parenleftbig\n1+δ−1/parenrightbig\nρ+δ3ρ2/parenrightig\n.(48)\nProof.For a state Γ =/circleplustext\nn/greaterorequalslant0Γnon Fock space(commuting with theparticle\nnumberoperator) andaninteraction potential w, weintroducethesimplified\nnotation\nCw(Γ) :=/summationdisplay\nn/greaterorequalslant2TrHn\n/summationdisplay\n1/lessorequalslantj<k/lessorequalslantnw(xj−xk)\nΓn (49)\nand\nDw(ρ) :=1\n2ˆ\nR3ˆ\nR3w(x−y)ρ(x)ρ(y)dxdy. (50)\nFor the Coulomb potential w(x) =|x|−1we simply use the notation C(Γ)\nandD(ρ). For the kinetic energy, we write\nT(Γ) :=/summationdisplay\nn/greaterorequalslant1TrHn/parenleftigg\n−n/summationdisplay\nj=1∆xj/parenrightigg\nΓn (51)THE LOCAL DENSITY APPROXIMATION IN DENSITY FUNCTIONAL THEO RY 23\nand finally denote by\nE(Γ) :=T(Γ)+C(Γ)−D(ρΓ).\nthe total energy, with the direct term subtracted.\nThe proof uses the well known fact that, for any interaction p otentialw\nand any state Γ on Fock space, we have\nCw(ρ)−Dw(ρ)/greaterorequalslant\n\n−w(0)\n2ˆ\nR3ρΓwhen/hatwidew∈L1(Rd) and/hatwidew/greaterorequalslant0,\n−´\nR3w\n2ˆ\nR3(ρΓ)2whenw∈L1(Rd) andw/greaterorequalslant0.(52)\nFor the first bound see for instance [30, Eq. (4.8)]. The secon d bound uses\nonly that Cw(Γ)/greaterorequalslant0 andDw(ρ)/lessorequalslant/ba∇dblw/ba∇dblL1/ba∇dblρ/ba∇dbl2\nL2/2, by Young’s inequality.\nWe are now ready to prove (48). The smeared Graf-Schenker ine quality\nfrom [14, Lemma 6] states that the potential\n/tildewidewℓ(x) =1\n|x|−/parenleftbigg\n1−κδ\nℓ/parenrightbigg/tildewidehℓ,δ(x)\n|x|−κδ2\nℓ/tildewidehℓ,δ(0)\n|x|(δ+|x|)\nhas a positive Fourier transform for all ℓ > κδ, with/tildewidewℓ(0) =−κℓ−1/tildewidehℓ,δ(0),\nwhere\n/tildewidehℓ,δ(x−y) =1\n|ℓ∆|ˆ\nSO(3)/parenleftbig\n1ℓ∆∗ηδ/parenrightbig\n∗/parenleftbig\n1−ℓ∆∗ηδ/parenrightbig\n(Rx−Ry)dR\n=1\n|ℓ∆|ˆ\nSO(3)ˆ\nR3/parenleftbig\n1R−1ℓ∆+z∗ηδ/parenrightbig\n(y)/parenleftbig\n1R−1ℓ∆+z∗ηδ/parenrightbig\n(x)dzdR.\nHere∆is a tetrahedron and κ >0 is a large enough constant. In addition,\nwe have from [30, Proof of Lem. 5.5] that the potential\n/tildewiderWδ(x) =1\n|x|(δ+|x|)−e−√\n2\nδ|x|\nδ|x|\nis positive and has positive Fourier transform, with /tildewiderWδ(0) =√\n2δ−2.\nArguing exactly as in [30, Lem. 5.5] using (52), we find that fo r any\nfermionic grand-canonical mixed state Γ = ⊕n/greaterorequalslant0Γnwith density ρΓ, we\nhave\nC(Γ)−D(ρΓ)/greaterorequalslant1−κδ/ℓ\nℓ3/summationdisplay\nz∈Z324/summationdisplay\nj=1ˆ\nSO(3)ˆ\nCℓ/braceleftbigg\nC/parenleftig\nΓ|√\nξℓ,δ,j(R·−ℓz−τ)/parenrightig\n−D/parenleftig\nξℓ,δ,j(R· −ℓz−τ)ρΓ/parenrightig/bracerightbigg\ndRdτ\n−C\nℓˆ\nR3ρΓ−Cδ3\nℓˆ\nR3(ρΓ)2. (53)\nHere Γ |fis the geometrically f–localized state on Fock space [9, 16, 27],\nthat is, the unique state which has the k-particle reduced density matrices\nf⊗kΓ(k)f⊗k. The last term proportional to δ3/ℓcomes from the L1norm of\nℓ−1δe−√\n2\nδ|x||x|−1.24 M. LEWIN, E.H. LIEB, AND R. SEIRINGER\nFor the kinetic energy we use the IMS formula as in [14, 16] and [30,\nLem. 5.6], which yields an error in the form\nN\nℓ3ˆ\nR3|∇/radicalbig\nξℓ,δ,j|2=O/parenleftbiggN\nℓδ/parenrightbigg\n,\nwhereN=´\nR3ρΓ. For the total energy we obtain\nE(Γ)/greaterorequalslant1−κδ/ℓ\nℓ3/summationdisplay\nz∈Z324/summationdisplay\nj=1ˆ\nSO(3)ˆ\nCℓE/parenleftig\nΓ|√\nξℓ,δ,j(R·−ℓz−τ)/parenrightig\ndRdτ\n−C\nℓ/parenleftbigg\n1+1\nδ/parenrightbiggˆ\nR3ρΓ−Cδ3\nℓˆ\nR3(ρΓ)2,(54)\nwhich yields the result. /square\n4.3.A convergence rate for tetrahedra. In this section we study the\nconvergence of the energy per unit volume for tetrahedra and find a conver-\ngence rate. We introduce the energy per unit volume of a tetra hedron at\nconstant density ρ0>0\ne∆(ρ0,ℓ,δ) :=|ℓ∆|−1E/parenleftig\nρ01ℓ∆∗ηδ/parenrightig\n(55)\nwhereηδ(x) = (10/δ)3η1(10x/δ) withη1a fixedC∞\ncnon-negative radial\nfunction with support in the unit ball and such that´\nR3η1= 1. We prove\nthe following\nProposition 3 (Thermodynamiclimit for tetrahedra) .For every fixed ρ0>\n0, we have\nlim\nδ/ℓ→0\nδ3/ℓ→0\nℓδ→∞e∆(ρ0,ℓ,δ) =eUEG(ρ0). (56)\nForδ/lessorequalslantℓ/Cand0< α <1/2, we have the upper bound\ne∆(ρ0,ℓ,δ)/lessorequalslanteUEG(ρ0)+Cρ0\nℓ/parenleftig\n1+δ−1+δ3ρ0+δρ2/3\n0/parenrightig\n,(57)\nand the averaged lower bound\n/parenleftbiggˆ1+α\n1−αds\ns4/parenrightbigg−1ˆ1+α\n1−αe∆(ρ0,tℓ,tδ)dt\nt4/greaterorequalslanteUEG(ρ0)−Cδ2ρ2\n0log(α−1).(58)\nIf in addition ρ1/3\n0ℓ/greaterorequalslantC, we have the pointwise lower bound\ne∆(ρ0,ℓ,δ)/greaterorequalslanteUEG(ρ0)−Cδρ5/3\n0+ρ4/3\n0\nℓ−Cρ23/15\n0+ρ18/15\n0\nℓ2/5.(59)\nThe constant Conly depends on the chosen regularizing function η1. It is\nindependent of ρ0,ℓ,δ,α.\nWe will later see that the condition δ3/ℓ→0 is actually not needed in\nthe limit (56). It is an interesting problem to replace the er ror term in the\nlower bound (59) by an error similar to the upper bound (57). N ote that\nthe error term in (58) goes to zero only when δ→0 whereas (59) does not\nrequireδ→0.THE LOCAL DENSITY APPROXIMATION IN DENSITY FUNCTIONAL THEO RY 25\nProof.For fixed ρ0>0 andδ >0, the existence of the limit (56) for ℓ→ ∞\nwas proved in [30], using a lower bound similar to (48).\nWe consider a large tetrahedron ℓ′∆, smeared at a scale δ′and a tiling\nof smaller tetrahedra of size ℓ≪ℓ′, smeared at scale δ. Applying our lower\nbound (47), we find\ne∆(ρ0,ℓ′,δ′)\n/greaterorequalslant1−Cδ\nℓ\n|ℓ′∆|ℓ3/summationdisplay\nz∈Z324/summationdisplay\nj=1ˆ\nCℓˆ\nSO(3)E/parenleftig\nρ0(1R(ℓµj∆−ℓz−τ)∗ηδ)(1ℓ′∆∗ηδ′)/parenrightig\ndRdτ\n−Cρ0\nℓ/parenleftig\n1+δ−1+δ3ρ0/parenrightig\nwhere for the error term we have used thatˆ\nR3(ρ01ℓ′∆∗ηδ′)2=ρ2\n0ˆ\nR3(1ℓ′∆∗ηδ′)2/lessorequalslantρ2\n0ˆ\nR31ℓ′∆∗ηδ′=ρ2\n0|ℓ′∆|.\nFor all the tetrahedra such that R(ℓµj∆−ℓz−τ)+Bδ/10⊂(ℓ′−δ′)∆, we\nobtain exactly |ℓ∆|e∆(ρ0,ℓ,δ) in the integral. The other tetrahedra are at\na distance proportional to ℓ+δ+δ′from the boundary of ℓ′∆. Hence, using\nour lower bound (19) on the energy, they give rise to an error t erm of the\norderρ4/3\n0(ℓ+δ+δ′)/ℓ′. We obtain\ne∆(ρ0,ℓ′,δ′)/greaterorequalslant/parenleftbigg\n1−Cσδ\nℓ−Cσℓ+δ+δ′\nℓ′/parenrightbigg\ne∆(ρ0,ℓ,δ)\n−Cρ4/3\n0ℓ+δ+δ′\nℓ′−Cρ0\nℓ/parenleftig\n1+δ−1+δ3ρ0/parenrightig\n.(60)\nHereσ= 1 ife∆(ρ0,ℓ,δ)/greaterorequalslant0 andσ= 0 otherwise.\nAfter taking the limit ℓ′→ ∞at fixedℓ,δ,δ′,ρ0, we obtain\neUEG(ρ0)/greaterorequalslant/parenleftbigg\n1−Cσδ\nℓ/parenrightbigg\ne∆(ρ0,ℓ,δ)−Cρ0\nℓ/parenleftig\n1+δ−1+δ3ρ0/parenrightig\n.\nIt follows from our upper bound (29) (see also [30, Rmk. 5.4]) that\neUEG(ρ0)/lessorequalslantq−2/3cTFρ5/3\n0\nfor allρ0>0. Hence after dividing by 1 −Cσδ/ℓwe have shown the claimed\nupper bound\ne∆(ρ0,ℓ,δ)/lessorequalslanteUEG(ρ0)+Cρ0\nℓ/parenleftig\n1+δ−1+δ3ρ0+δσρ2/3\n0/parenrightig\n.\nWe may use exactly the same argument using our upper bound (34 ) in place\nof (48) and we obtain the lower bound (58).\nNext we replace ℓbytℓandδbytδin our lower bound (60) on the energy\nof the large simplex of size ℓ′, and average over t∈(1/2,3/2) with the\nmeasure t−4. We then insert our lower bound (58) and, after collecting th e\ndifferent error terms, we obtain\ne∆(ρ0,ℓ′,δ′)/greaterorequalslanteUEG(ρ0)−Cℓ+δ+δ′\nℓ′(ρ5/3\n0+ρ4/3\n0)\n−Cρ0\nℓ/parenleftig\n1+δ−1+δ3ρ0+δρ2/3\n0/parenrightig\n−Cδ2ρ2\n0.26 M. LEWIN, E.H. LIEB, AND R. SEIRINGER\nIt is natural to choose δ=ℓ−1/3(ρ0)−4/9which provides the estimate\ne∆(ρ0,ℓ′,δ′)/greaterorequalslanteUEG(ρ0)−Cℓ+ℓ−1/3(ρ0)−4/9+δ′\nℓ′(ρ5/3\n0+ρ4/3\n0)\n−Cρ13/9\n0+ρ10/9\n0\nℓ2/3−Cρ11/9\n0\nℓ4/3−Cρ2/3\n0\nℓ2\nand then ℓ= (ℓ′)3/5ρ−2/15\n0which gives\ne∆(ρ0,ℓ′,δ′)/greaterorequalslanteUEG(ρ0)−Cδ′\nℓ′(ρ5/3\n0+ρ4/3\n0)−Cρ23/15\n0+ρ18/15\n0\n(ℓ′)2/5\nunder the assumption that ℓ′(ρ0)1/3/greaterorequalslantC. This is exactly (59). The two\nbounds (57) and (59) give the limit (56). /square\n4.4.Proof of Theorem 1. Let ΩNbe a sequence of domains as in the\nstatement, that is, such that |ΩN| → ∞and|∂ΩN+Br|/lessorequalslantCr|ΩN|2/3for\nallr/lessorequalslant|ΩN|1/3/C. Assume also that δN|ΩN|−1/3→0.\nBy following the proof of (60) we see that a similar inequalit y holds with\nthe large tetrahedron ℓ′∆replaced by Ω N. This gives\nE/parenleftbig\nρ01ΩN∗ηδN/parenrightbig\n|ΩN|/greaterorequalslant/parenleftbigg\n1−Cσδ\nℓ−Cσℓ+δ+δN\n|ΩN|1/3/parenrightbigg\ne∆(ρ0,ℓ,δ)\n−Cρ4/3\n0ℓ+δ+δN\n|ΩN|1/3−Cρ0\nℓ/parenleftig\n1+δ−1+δ3ρ0/parenrightig\n.\nUnder the sole condition that δN|ΩN|−1/3→0, the right side tends to\neUEG(ρ0) if we take for instance δfixed and ℓ=|ΩN|1/6.\nWe then use the upper bound (34) with α= 1/2 as well as the fact that\nE/parenleftig\nρ0χtℓ,tδ,j(R· −tℓz−τ)1ΩN∗ηδN/parenrightig\n/lessorequalslantCρ0ℓ3/parenleftig\nρ2/3\n0+(ℓδ)−1/parenrightig\n+Cρ0ˆ\nR3χtℓ,tδ,j(R· −tℓz−τ)/vextendsingle/vextendsingle∇/radicalbig\n1ΩN∗ηδN/vextendsingle/vextendsingle2\nby (29), for the tetrahedra close to the boundary. We find\nE/parenleftbig\nρ01ΩN∗ηδN/parenrightbig\n|ΩN|\n/lessorequalslant/parenleftbigg\n1+Cσℓ+δ+δN\n|ΩN|1/3/parenrightbigg/parenleftiggˆ3/2\n1/2ds\ns4/parenrightigg−1ˆ3/2\n1/2e∆(ρ0,tℓ,tδ)dt\nt4\n+Cℓ+δ+δN\n|ΩN|1/3ρ0/parenleftig\nρ2/3\n0+(ℓδ)−1/parenrightig\n+Cρ0\nδN|ΩN|1/3+Cρ2\n0δ2.(61)\nWe have used here that\n1\n|ΩN|ˆ\nR3/vextendsingle/vextendsingle∇/radicalbig\n1ΩN∗ηδN/vextendsingle/vextendsingle2/lessorequalslantC\nδN|ΩN|1/3.\nUnder the additional assumption that δN|ΩN|1/3→ ∞, we may choose for\ninstance ℓ=|ΩN|1/6andδ=|ΩN|−1/12, which yields the result. /squareTHE LOCAL DENSITY APPROXIMATION IN DENSITY FUNCTIONAL THEO RY 27\n4.5.Replacing the local density by a constant density. The goal\nof this section is to provide estimates on the variation of th e energy in a\n(smeared) tetrahedron, when we replace the local density by a constant,\nchosen to be either the minimum or the maximum of the density i n the\ntetrahedron.\nProposition 4 (Replacing ρby aconstant locally) .Letp >3and0< θ <1\nsuch that\n2/lessorequalslantpθ/lessorequalslant1+p\n2. (62)\nThere exists a constant C=C(p,θ,q)such that, for ℓ/greaterorequalslantCandδ/lessorequalslantℓ/C, we\nhave\nE/parenleftbig\nρ(1ℓ∆∗ηδ)/parenrightbig\n/lessorequalslantE/parenleftbig\nρ(1ℓ∆∗ηδ)/parenrightbig\n+Cεˆ\nR3/parenleftbig\nρ+ρ2/parenrightbig\n(1ℓ∆∗ηδ)\n+Cˆ\nR3ρ/vextendsingle/vextendsingle/vextendsingle∇/radicalbig\n1ℓ∆∗ηδ/vextendsingle/vextendsingle/vextendsingle2\n+C\nεˆ\nR3|∇√ρ|2(1ℓ∆∗ηδ)\n+C/parenleftbiggℓ2p\nεp−1+ℓp\nε5\n4p−1/parenrightbiggˆ\nℓ∆+Bδ|∇ρθ|p(63)\nand\nE/parenleftbig\nρ(1ℓ∆∗ηδ)/parenrightbig\n/greaterorequalslantE(ρ(1ℓ∆∗ηδ))−Cεℓ3/parenleftbig\nρ+ρ2/parenrightbig\n−Cℓ2\nδρ\n−C\nεˆ\nR3|∇√ρ|2(1ℓ∆∗ηδ)\n−C/parenleftbiggℓ2p\nεp−1+ℓp\nε5\n4p−1/parenrightbiggˆ\nℓ∆+Bδ|∇ρθ|p(64)\nfor all0< ε/lessorequalslant1/2, where\nρ= min\nx∈supp(1ℓ∆∗ηδ)ρ(x),ρ= max\nx∈supp(1ℓ∆∗ηδ)ρ(x)\nare respectively the minimum and maximum value of ρon the support of\n1ℓ∆∗ηδ.\nUnder the assumption that´\nℓ∆+Bδ|∇ρθ|pis finite, the density ρis con-\ntinuous on ℓ∆+Bδ, so that ρandρare well defined.\nWe have already discussed in the beginning of Section 4.1 the difficulty\nof deriving a subadditivity-type estimate relating E(ρ1+ρ2) toE(ρ1) and\nE(ρ2). The following lemma provides a rather rough inequality, w hich how-\never will be sufficient for our purposes.\nLemma 3 (Rough subadditivity estimate) .Letρ1,ρ2∈L1(R3,R+)be two\ndensities such that√ρ1,√ρ2∈H1(R3). Then\nE(ρ1+ρ2)/lessorequalslantE(ρ1)+Cεˆ\nR3/parenleftig\nρ5/3\n1+ρ4/3\n1/parenrightig\n+Cε−2/3ˆ\nR3ρ5/3\n2\n+Cˆ\nR3|∇√ρ2+ερ1|2+1−ε\nεD(ρ2) (65)\nfor all0< ε/lessorequalslant1.28 M. LEWIN, E.H. LIEB, AND R. SEIRINGER\nHere we have in mind that ρ2is small compared to ρ1and we estimate\nE(ρ1+ρ2) in terms of E(ρ1) plus some error terms. The worse error in\nthe estimate (65) is D(ρ2)/ε, because it grows much faster than the volume.\nLater we will only use (65) locally and this bad term will not b e too large.\nBut it will be responsible for the large power of εin front of the gradient\ncorrectioninourmainestimate(8). Weconjecturethatther eisaninequality\nsimilar to (65) without the term D(ρ2)/ε.\nNote that we can estimateˆ\nR3|∇√ρ2+ερ1|2/lessorequalslantˆ\nR3|∇√ρ2|2+εˆ\nR3|∇√ρ1|2\nby the convexity of ρ/ma√sto→ |∇√ρ|2.\nProof of Lemma 3. Fix anε∈(0,1] and consider two optimal states Γ 1and\nΓ2in Fock space, for ρ1andρ2/ε+ρ1, respectively. Then\nΓ := (1−ε)Γ1+εΓ2\nis a proper quantum state which has the density\nρΓ= (1−ε)ρ1+ε/parenleftigρ2\nε+ρ1/parenrightig\n=ρ1+ρ2.\nInserting this trial state and using (29) for E(ρ2/ε+ρ1), we deduce that\nE(ρ1+ρ2)/lessorequalslant(1−ε)E(ρ1)+C\nε2/3ˆ\nR3ρ5/3\n2+Cˆ\nR3|∇√ρ2+ερ1|2\n+Cεˆ\nR3ρ5/3\n1−D(ρ1+ρ2)+(1−ε)D(ρ1)+εD(ρ1+ρ2/ε).\nWe have\n−D(ρ1+ρ2)+(1−ε)D(ρ1)+εD(ρ1+ρ2/ε) =1−ε\nεD(ρ2).\nBy the Lieb-Oxford inequality E(ρ1)/greaterorequalslant−C´\nR3ρ4/3\n1, and the result follows.\n/square\nWe are now able to provide the\nProof of Proposition 4. We write ρ=ρ+(ρ−ρ) and apply (65). We obtain\nE/parenleftbig\nρ(1ℓ∆∗ηδ)/parenrightbig\n/lessorequalslantE/parenleftig\nρ(1ℓ∆∗ηδ)/parenrightig\n+Cεˆ\nR3/parenleftig\nρ5/3+ρ4/3/parenrightig\n(1ℓ∆∗ηδ)\n+C\nε2/3ˆ\nR3(ρ−ρ)5/3(1ℓ∆∗ηδ)+1\nεD/parenleftig\n(ρ−ρ)(1ℓ∆∗ηδ)/parenrightig\n+Cˆ\nR3/vextendsingle/vextendsingle/vextendsingle∇/radicalig\n(1ℓ∆∗ηδ)(ρ−(1−ε)ρ)/vextendsingle/vextendsingle/vextendsingle2\n.\nIn the first line we have used that ρ/lessorequalslantρon the support of 1ℓ∆∗ηδand that\n1ℓ∆∗ηδ/lessorequalslant1. First we can bound ρ4/3+ρ5/3byρ+ρ2. Next, using\n|∇/radicalbig\nfg|2=|∇(fg)|2\n4fg/lessorequalslantf|∇g|2\n2g+g|∇f|2\n2f,THE LOCAL DENSITY APPROXIMATION IN DENSITY FUNCTIONAL THEO RY 29\nand∇(ρ−(1−ε)ρ) =∇ρ= 2√ρ∇√ρ, we can bound the gradient term\npointwise by\n/vextendsingle/vextendsingle/vextendsingle∇/radicalig\n(1ℓ∆∗ηδ)(ρ−(1−ε)ρ)/vextendsingle/vextendsingle/vextendsingle2\n/lessorequalslant(1ℓ∆∗ηδ)2ρ/vextendsingle/vextendsingle∇√ρ/vextendsingle/vextendsingle2\nρ−(1−ε)ρ+2ρ/vextendsingle/vextendsingle/vextendsingle∇/radicalbig\n1ℓ∆∗ηδ/vextendsingle/vextendsingle/vextendsingle2\n.\nSinceρ/greaterorequalslantρ, we have ερ/lessorequalslantρ−(1−ε)ρand hence\nρ\nρ−(1−ε)ρ/lessorequalslant1\nε.\nThis gives the estimate on the gradient term\nˆ\nR3/vextendsingle/vextendsingle/vextendsingle∇/radicalig\n(1ℓ∆∗ηδ)(ρ−(1−ε)ρ)/vextendsingle/vextendsingle/vextendsingle2\n/lessorequalslant2\nεˆ\nR3|∇√ρ|2(1ℓ∆∗ηδ)+2ˆ\nR3ρ/vextendsingle/vextendsingle/vextendsingle∇/radicalbig\n1ℓ∆∗ηδ/vextendsingle/vextendsingle/vextendsingle2\n.\nNext we estimate the terms involving ρ−ρin terms of the gradient of ρθ.\nWe use the Sobolev inequality in the bounded set ℓ∆+Bδ\n/ba∇dblu/ba∇dblp\nL∞(ℓ∆+Bδ)/lessorequalslantCℓp−3ˆ\nℓ∆+Bδ|∇u(x)|pdx (66)\nforp >3 and every continuous uwhich vanishes at least at one point in\nℓ∆+Bδ(we always assume δ/lessorequalslantℓ/Cso thatℓ∆+Bδis included in a ball\nof radius proportional to ℓ). By the Hardy-Littlewood-Sobolev inequality,\nthis gives\nD/parenleftig\n(ρ−ρ)(1ℓ∆∗ηδ)/parenrightig\n/lessorequalslantC/vextenddouble/vextenddouble(ρ−ρ)(1ℓ∆∗ηδ)/vextenddouble/vextenddouble2\nL6/5\n/lessorequalslantC/vextenddouble/vextenddouble/vextenddoubleρθ−ρθ/vextenddouble/vextenddouble/vextenddouble2\nL∞(ℓ∆+Bδ)/parenleftbiggˆ\nR3ρ6\n5(1−θ)(1ℓ∆∗ηδ)/parenrightbigg5\n3\n/lessorequalslantC/parenleftbigg\nℓ2pˆ\nℓ∆+Bδ|∇ρθ|p/parenrightbigg2\np/parenleftbiggˆ\nR3ρ2p\np−2(1−θ)(1ℓ∆∗ηδ)/parenrightbigg1−2\np\n/lessorequalslantCε/parenleftbiggℓ2p\nεp−1ˆ\nℓ∆+Bδ|∇ρθ|p+εˆ\nR3ρ2p\np−2(1−θ)(1ℓ∆∗ηδ)/parenrightbigg\n.(67)\nIn the second estimate we have used that\nρ−ρ/lessorequalslantC(ρθ−ρθ)ρ1−θ\nsinceθ/lessorequalslant1. In the third estimate we have used H¨ older’s inequality to obtain\nan integral to the power 1 −2/p. This yields some power of ℓwhich has\nbeen taken into account in the first factor. In order to bound ρ2p\np−2(1−θ)by\nρ+ρ2we need that\n1/lessorequalslant2p\np−2(1−θ)/lessorequalslant2\nwhich is equivalent to our assumption (62).30 M. LEWIN, E.H. LIEB, AND R. SEIRINGER\nSimilarly, we can bound the other error term as followsˆ\nR3(ρ−ρ)5\n3(1ℓ∆∗ηδ)\n/lessorequalslantC/vextenddouble/vextenddouble/vextenddoubleρθ−ρθ/vextenddouble/vextenddouble/vextenddouble5\n3a\nL∞(ℓ∆+Bδ)ˆ\nR3ρ5\n3(1−θa)(1ℓ∆∗ηδ)\n/lessorequalslantC/parenleftbigg\nℓpˆ\nℓ∆+Bδ|∇ρθ|p/parenrightbigg5a\n3p/parenleftbiggˆ\nR3ρ5p\n3p−5a(1−θa)(1ℓ∆∗ηδ)/parenrightbigg1−5a\n3p\n/lessorequalslantCε2\n3/parenleftbiggℓp\nεp\na−1ˆ\nℓ∆+Bδ|∇ρθ|p+εˆ\nR3ρ5p\n3p−5a(1−θa)(1ℓ∆∗ηδ)/parenrightbigg\n(68)\nwhere 0 < a/lessorequalslant1 is a parameter to be chosen. As before we need the\ncondition\n1/lessorequalslant5p\n3p−5a(1−θa)/lessorequalslant2\nin order to bound the last term by ρ+ρ2. This is equivalent to\n2−p\n5a/lessorequalslantθp/lessorequalslant1+2p\n5a\nwhere the left inequality is always satisfied under our assum ption (62). If\nwe choose a= 1 then the upper bound on pθis stronger than (62). Hence\nwe rather choose a= 4/5 and obtain (63).\nThe argument for (64) is similar. This time we write ρ=ρ+(ρ−ρ) and\nobtain from (65)\nE/parenleftbig\nρ(1ℓ∆∗ηδ)/parenrightbig\n/lessorequalslantE/parenleftig\nρ(1ℓ∆∗ηδ)/parenrightig\n+Cεˆ\nR3/parenleftig\nρ5/3+ρ4/3/parenrightig\n(1ℓ∆∗ηδ)\n+C\nε2/3ˆ\nR3(ρ−ρ)5/3(1ℓ∆∗ηδ)+1\nεD/parenleftig\n(ρ−ρ)(1ℓ∆∗ηδ)/parenrightig\n+Cˆ\nR3/vextendsingle/vextendsingle/vextendsingle∇/radicalbig\n(1ℓ∆∗ηδ)(ρ−(1−ε)ρ)/vextendsingle/vextendsingle/vextendsingle2\n.\nThe gradient term can be bounded above byˆ\nR3/vextendsingle/vextendsingle/vextendsingle∇/radicalbig\n(1ℓ∆∗ηδ)(ρ−(1−ε)ρ)/vextendsingle/vextendsingle/vextendsingle2\n/lessorequalslant2\nεˆ\nR3|∇√ρ|2(1ℓ∆∗ηδ)+2ρˆ\nR3/vextendsingle/vextendsingle/vextendsingle∇/radicalbig\n1ℓ∆∗ηδ/vextendsingle/vextendsingle/vextendsingle2\n/lessorequalslant2\nεˆ\nR3|∇√ρ|2(1ℓ∆∗ηδ)+Cℓ2\nδρ.\nThe other terms are estimated as before, using that ρ−ρ/lessorequalslantρ. /square\n4.6.Lipschitz regularity of eUEG.In this section we prove that the UEG\nenergyeUEGis locally Lipschitz. The main result is the following.\nProposition 5 (Lipschitz regularity of eUEG).There exists a universal con-\nstantCso that\neUEG(ρ)−C/parenleftbig\nρ1\n3+ρ2\n3/parenrightbig\nρ′/lessorequalslanteUEG(ρ−ρ′)/lessorequalslanteUEG(ρ)+Cρ′ρ1\n3(69)\nfor every 0/lessorequalslantρ′/lessorequalslantρ. In particular, we have\n|eUEG(ρ1)−eUEG(ρ2)|/lessorequalslantC/parenleftig\nmax(ρ1,ρ2)1\n3+max(ρ1,ρ2)2\n3/parenrightig\n|ρ1−ρ2|.(70)THE LOCAL DENSITY APPROXIMATION IN DENSITY FUNCTIONAL THEO RY 31\nProof.By scaling we have\nE/parenleftbig\nαρ(α1\n3·)/parenrightbig\n= min\nΓ/braceleftig\nα2\n3T(Γ)+α1\n3(C(Γ)−D(ρΓ))/bracerightig\n/lessorequalslantα1\n3E(ρ)\nforα/lessorequalslant1. This proves that\nE/parenleftbig\nαρ01ℓ/α1\n3∆∗χδ/α1\n3/parenrightbig\nℓ3|∆|=e∆(αρ0,ℓ/α1\n3,δ/α1\n3)\nα/lessorequalslantα1\n3e∆(ρ0,ℓ,δ).\nPassing to the limit using Proposition 3, we find\neUEG(αρ0)/lessorequalslantα4\n3eUEG(ρ0)\nfor every 0 /lessorequalslantα= 1−ε/lessorequalslant1, hence\neUEG/parenleftbig\n(1−ε)ρ0/parenrightbig\n/lessorequalslant(1−ε)4\n3eUEG(ρ0)/lessorequalslanteUEG(ρ0)+CεeUEG(ρ0)−.\nHere we have used the notation x−= max(−x,0) for the negative part.\nUsing that eUEG(ρ0)/greaterorequalslant−cLOρ4/3\n0by the Lieb-Oxford inequality (with cLO/lessorequalslant\n1.64), we obtain\neUEG/parenleftbig\n(1−ε)ρ0/parenrightbig\n/lessorequalslanteUEG(ρ0)+Cερ4\n3\n0\nfor all 0/lessorequalslantε/lessorequalslant1. This proves the upper bound in (69).\nSimilarly, we can write (still for 0 /lessorequalslantα/lessorequalslant1)\nE/parenleftbig\nαρ(α1\n3·)/parenrightbig\n+cLOα1\n3ˆ\nR3ρ4\n3\n= min\nΓ/braceleftbigg\nα2\n3T(Γ)+α1\n3/parenleftbigg\nC(Γ)−D(ρΓ)+cLOˆ\nR3ρ4\n3/parenrightbigg/bracerightbigg\n/greaterorequalslantα2\n3/parenleftbigg\nE(ρ)+cLOˆ\nR3ρ4\n3/parenrightbigg\n.\nThis gives as before\neUEG(αρ0)+α4\n3cLOρ4\n3\n0/greaterorequalslantα5\n3/parenleftbigg\neUEG(ρ0)+cLOρ4\n3\n0/parenrightbigg\n.\nUsing this time eUEG(ρ0)/lessorequalslantCρ5/3\n0, we obtain\neUEG/parenleftbig\n(1−ε)ρ0/parenrightbig\n/greaterorequalslanteUEG(ρ0)−Cε/parenleftbig\nρ4\n3\n0+ρ5\n3\n0/parenrightbig\nfor all 0/lessorequalslantε/lessorequalslant1. /square\n4.7.Proof of Theorem 2. We have derived all the estimates we need to\nprove the main inequality (8) in Theorem 2.\nLetρ∈L1(R3,R+)∩L2(R3,R+) be any density so that ∇√ρ∈L2(R3)\nand∇ρθ∈Lp(R3). First we recall from our upper bound (29) and the lower\nbound (19) that\n|E(ρ)|/lessorequalslantcTFq−2/3(1+ε)ˆ\nR3ρ5/3+cLOˆ\nR3ρ4/3+C(1+ε)\nεˆ\nR3|∇√ρ|2.\nSimilarly, we have\n|eUEG(ρ)|/lessorequalslantcTFq−2/3ρ5/3+cLOρ4/3. (71)\nIn particular, the inequality (8) is obvious for large εand we only have to\nconsider small ε.32 M. LEWIN, E.H. LIEB, AND R. SEIRINGER\nIn our upper bound (34) and our lower bound (47), the worse coe fficient\ninvolving ℓandδin front of ρ+ρ2isδ2+1/(ℓδ). This suggests to take\nδ=√ε, ℓ=ε−3\n2 (72)\nwhich we do for the rest of the proof. In fact, in our proof we wi ll replace ℓ\nandδbytℓandtδand average over t∈[1/2,3/2].\nStep 1. Upper bound. Let usfirsttake 1 /4/lessorequalslantε3/2ℓ/lessorequalslant2and1/4/lessorequalslantδε−1/2/lessorequalslant2\nand derive an upper bound on E(ρ(1ℓ∆∗ηδ)). We recall that ∆is a\ntetrahedron of volume 1 /24 as described in Section 4.1 and that ηδ(x) =\n(10/δ)3η1(10x/δ) withη1a fixedC∞\ncnon-negative radial function with sup-\nport in the unit ball and such that´\nR3η1= 1. We denote by\nρ:= min\nsupp(1ℓ∆∗ηδ)ρ,ρ:= min\nsupp(1ℓ∆∗ηδ)ρ\nthe maximal and minimal values of ρon the support of 1ℓ∆∗ηδ, as in\nProposition 4. We use the upper bound (63) from Proposition 4 which\nquantifies the error made when replacing E(ρ(1ℓ∆∗ηδ)) byE(ρ(1ℓ∆∗ηδ)).\nFor the latter we then use our estimate (57) in Proposition 3 o n the energy\nof a smeared tetrahedron. With our choice (72) of ℓandδin terms of ε, this\nleads to\nE/parenleftbig\nρ(1ℓ∆∗ηδ)/parenrightbig\n/lessorequalslanteUEG(ρ)ˆ\nR31ℓ∆∗ηδ+Cεˆ\nR3/parenleftbig\nρ+ρ2/parenrightbig\n(1ℓ∆∗ηδ)\n+Cˆ\nR3ρ/vextendsingle/vextendsingle/vextendsingle∇/radicalbig\n1ℓ∆∗ηδ/vextendsingle/vextendsingle/vextendsingle2\n+C\nεˆ\nR3|∇√ρ|2(1ℓ∆∗ηδ)+C\nε4p−1ˆ\nℓ∆+Bδ|∇ρθ|p.(73)\nIn the first line we have bounded the error terms in (57) by\nε3\n2ρ/parenleftig\n1+ε−1\n2+ε3\n2ρ+√ερ2/3/parenrightig\n/lessorequalslantCε(ρ+ρ2)\nin order to to simplify our final bound. In the support of 1ℓ∆∗ηδwe have\nby (69)\neUEG(ρ)/lessorequalslanteUEG(ρ(x))+C(ρ(x)−ρ)ρ(x)1\n3\nhence\neUEG(ρ)ˆ\nR31ℓ∆∗ηδ/lessorequalslantˆ\nR3eUEG(ρ)(1ℓ∆∗ηδ)+Cˆ\nR3(ρ−ρ)ρ1\n3(1ℓ∆∗ηδ).\nSimilarly as we did for (68), we can bound for 0 < a/lessorequalslant1\nˆ\nR3(ρ−ρ)ρ1\n3(1ℓ∆∗ηδ)\n/lessorequalslantC/vextenddouble/vextenddouble/vextenddoubleρθ−ρθ/vextenddouble/vextenddouble/vextenddoublea\nL∞(ℓ∆+Bδ)ˆ\nR3ρ4\n3−θa(1ℓ∆∗ηδ)\n/lessorequalslantC/parenleftbigg\nε−3\n2pˆ\nℓ∆+Bδ|∇ρθ|p/parenrightbigga\np/parenleftbiggˆ\nR3ρ4−3θa\n3p\np−a(1ℓ∆∗ηδ)/parenrightbigg1−a\np\n/lessorequalslantC/parenleftbigg1\nε3\n2p+p\na−1ˆ\nℓ∆+Bδ|∇ρθ|p+εˆ\nR3ρ4−3θa\n3p\np−a(1ℓ∆∗ηδ)/parenrightbigg\n.(74)THE LOCAL DENSITY APPROXIMATION IN DENSITY FUNCTIONAL THEO RY 33\nAgain we need\n1/lessorequalslant4−3θa\n3p\np−a/lessorequalslant2\nwhich is equivalent to\n2−2p\n3a/lessorequalslantθp/lessorequalslant1+p\n3a\nwhere the left side is automatically satisfied under our main assumption (7).\nIn order to get an error controlled by the other gradient term s, we need\n3\n2p+p\na−1/lessorequalslant4p−1\nwhich requires a/greaterorequalslant2/5. Taking a= 2/3 provides the smallest power of ε.\nCollecting our estimates, we have proved the following uppe r bound on the\nenergy in a tetrahedron\nE/parenleftig\nρ(1ℓ∆∗ηδ)/parenrightig\n/lessorequalslantˆ\nR3eUEG/parenleftbig\nρ(x)/parenrightbig\n(1ℓ∆∗ηδ)dx+Cεˆ\nR3(1ℓ∆∗ηδ)/parenleftbig\nρ+ρ2/parenrightbig\n+C\nε4p−1ˆ\nℓ∆+Bδ|∇ρθ|p+C\nεˆ\nR3(1ℓ∆∗ηδ)|∇√ρ|2\n+Cˆ\nR3ρ/vextendsingle/vextendsingle/vextendsingle∇/radicalbig\n1ℓ∆∗ηδ/vextendsingle/vextendsingle/vextendsingle2\n. (75)\nHere we have considered a tetrahedron placed at the origin fo r simplicity,\nbut we of course get a similar inequality for any tetrahedron , by translating\nand rotating ρ.\nNext we recall our upper bound (34) on the total energy E(ρ)\nE(ρ)/lessorequalslant/parenleftiggˆ3/2\n1/2ds\ns4/parenrightigg−1ˆ3/2\n1/2dt\nt4ˆ\nSO(3)dRˆ\nCtℓdτ\n(tℓ)3×\n×/summationdisplay\nz∈Z324/summationdisplay\nj=1E/parenleftig\nχtℓ,tδ,j(R· −tℓz−τ)ρ/parenrightig\n+Cεˆ\nR3ρ2,(76)\nwithχℓ,δ,j:= (1−ε2)−31ℓµj(1−ε2)∆∗ηδ. We also recall from Section 4.1\nthatδ/ℓ= (tδ)/(tℓ) =ε2. Inserting (75) into (76) and using the fact (33)\nthatχtℓ,tδ,jforms a partition of unity after averaging over translation s and\nrotations, we obtain\nE(ρ)/lessorequalslant(1−ε2)3ˆ\nR3eUEG/parenleftig\n(1−ε2)−3ρ(x)/parenrightig\ndx+Cεˆ\nR3/parenleftbig\nρ+ρ2/parenrightbig\n+C\nεˆ\nR3|∇√ρ|2+C\nε4p−1ˆ\nR3|∇ρθ|p.(77)\nNote that when we sum over the tiling, the sets tℓµj∆+Btδhave finitely\nmany intersections, which just results in a bigger constant in front of |∇ρθ|p.34 M. LEWIN, E.H. LIEB, AND R. SEIRINGER\nWe have also used that\nˆ\nSO(3)dRˆ\nCtℓdτ\n(tℓ)3/summationdisplay\nz∈Z324/summationdisplay\nj=1ˆ\nR3ρ/vextendsingle/vextendsingle/vextendsingle/vextendsingle∇/radicalig\nχtℓ,tδ,j(R· −ℓz−τ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n=24\n(tℓ)3ˆ\nR3ρˆ\nR3/vextendsingle/vextendsingle/vextendsingle/vextendsingle∇/radicalig\nχtℓ,tδ,j(R· −ℓz−τ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n/lessorequalslantC\nℓδˆ\nR3ρ=Cεˆ\nR3ρ.\nFrom Proposition 5 and (71), we have\n(1−ε2)3ˆ\nR3eUEG/parenleftig\n(1−ε2)−3ρ/parenrightig\n/lessorequalslantˆ\nR3eUEG/parenleftbig\nρ/parenrightbig\n+Cε2ˆ\nR3/parenleftig\nρ4\n3+ρ5\n3/parenrightig\nhence we obtain the desired upper bound\nE(ρ)/lessorequalslantˆ\nR3eUEG/parenleftbig\nρ/parenrightbig\n+εˆ\nR3/parenleftbig\nρ+ρ2/parenrightbig\n+C\nεˆ\nR3|∇√ρ|2+C\nε4p−1ˆ\nR3|∇ρθ|p(78)\nforεsmall enough.\nStep 2. Lower bound. The lower bound is slightly more tedious since all our\nlower estimates involve ρwhich can in general not be bounded by ρ. We\nshall argue as follows. First we average our lower bound (47) overt. This\ngives\nE(ρ)/greaterorequalslant/parenleftiggˆ3\n2\n1\n2ds\ns4/parenrightigg−1ˆ3\n2\n1\n2dt\nt41−Cε\n(tℓ)3×\n×/summationdisplay\nz∈Z324/summationdisplay\nj=1ˆ\nSO(3)ˆ\nCtℓE/parenleftig\nξtℓ,tδ,j(R· −tℓz−τ)ρ/parenrightig\ndRdτ\n−Cεˆ\nR3/parenleftbig\nρ+ε2ρ2/parenrightbig\n.(79)\nWe recall that here ξℓ,δ,j=1ℓµj∆∗ηδ, see Section 4.2. In order to use the\nsame argument as for the upper bound, we are going to prove the estimate\n/parenleftiggˆ3\n2\n1\n2ds\ns4/parenrightigg−1ˆ3\n2\n1\n2dt\nt4(tℓ)3/braceleftbigg\nE/parenleftig\nρ(1tℓ∆∗ηtδ)/parenrightig\n−ˆ\nR3eUEG/parenleftbig\nρ(x)/parenrightbig\n(1tℓ∆∗ηtδ)dx\n+Cˆ\nR3ρ|∇/radicalbig\n1tℓ∆∗ηtδ|2+Cεˆ\nR3/parenleftbig\nρ+ρ2/parenrightbig\n(1tℓ∆∗ηtδ)\n+C\nεˆ\nR3|∇√ρ|2(1tℓ∆∗ηtδ)/bracerightbigg\n/greaterorequalslant−C\nε4p−1ˆ\n2ℓ∆+B2δ|∇ρθ|p.(80)\nThat the last integral is over the larger set 2 ℓ∆+B2δwill only affect the\nmultiplicative constant C. Inserting (80) into (79) gives a bound as in (78)\nbut in the opposite direction. This concludes the proof of th e theorem and\nit therefore only remains to prove (80).\nWith an abuse of notation we consider the minimal and maximal values\nover the larger set 2( ℓ∆+Bδ),\nρ:= min\n2ℓ∆+B2δρ,ρ:= min\n2ℓ∆+B2δρ (81)THE LOCAL DENSITY APPROXIMATION IN DENSITY FUNCTIONAL THEO RY 35\ninstead of the corresponding definitions on the smaller set ℓ∆+Bδ. First\nwe again recall that, by (29) and (19), we have\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleE/parenleftig\nρ(1tℓ∆∗ηtδ)/parenrightig\n−ˆ\nR3(1tℓ∆∗ηtδ)eUEG/parenleftbig\nρ(x)/parenrightbig\ndx/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n/lessorequalslantCˆ\nR3(1tℓ∆∗ηtδ)/parenleftbig\nρ4/3+ρ5/3/parenrightbig\n+Cˆ\nR3(1tℓ∆∗ηtδ)|∇√ρ|2+Cˆ\nR3ρ/vextendsingle/vextendsingle/vextendsingle∇/radicalbig\n1tℓ∆∗ηtδ/vextendsingle/vextendsingle/vextendsingle2\n.\nHence there is nothing to prove when\nˆ\nR3/parenleftbig\nρ4/3+ρ5/3/parenrightbig\n(1tℓ∆∗ηtδ)/lessorequalslantCεˆ\nR3/parenleftbig\nρ+ρ2/parenrightbig\n(1tℓ∆∗ηtδ)+1\nε4p−1ˆ\n2ℓ∆+B2δ|∇ρθ|p.\nThis is the case if ρ1/3/lessorequalslantCε, for instance. Hence we may assume in the\nfollowing that ρ/greaterorequalslantCε3and that\nˆ\nR3/parenleftbig\nρ4/3+ρ5/3/parenrightbig\n(1tℓ∆∗ηtδ)/greaterorequalslant1\nε4p−1ˆ\n2ℓ∆+B2δ|∇ρθ|p.\nBy (66), this implies\nℓ3\nε3pθ−4ρpθ/greaterorequalslantC\nℓp−3ε4p−1/parenleftig\nρθ−ρθ/parenrightigp\n,\nthat is,\nρθ−ρθ/lessorequalslantCε3\np(1+5p\n6−θp)ρθ.\nUnder our assumption (7) on pandθthe exponent is positive, hence we\ndeduce that for εsmall enough\nρ/lessorequalslantCρ/lessorequalslantCρ(x)\non 2ℓ∆+B2δ. With this additional information we can use our previous\nestimates.\nBy arguing exactly as in the proof of Proposition 4 with ρthe maximum\nover 2ℓ∆+B2δinstead of the support of 1tℓ∆∗ηtδ, we get the estimate\nsimilar to (64)\nE/parenleftbig\nρ(1tℓ∆∗ηtδ)/parenrightbig\n/greaterorequalslantE(ρ(1tℓ∆∗ηtδ))−Cεℓ3/parenleftbig\nρ+ρ2/parenrightbig\n−C\nεˆ\nR3|∇√ρ|2(1tℓ∆∗ηtδ)−C\nε4p−1ˆ\n2ℓ∆+B2δ|∇ρθ|p.(82)\nFrom thefactthat ρ/lessorequalslantCρ, thesecondterm ontheright sidecan bebounded\nby\nCεˆ\nR3/parenleftbig\nρ+ρ2/parenrightbig\n(1tℓ∆∗ηtδ).\nThen we average over tand use our lower estimate (58) on the averaged\nenergy of a tetrahedron. This gives\n/parenleftiggˆ3\n2\n1\n2ds\ns4/parenrightigg−1ˆ3\n2\n1\n2dt\nt4E(ρ(1tℓ∆∗ηtδ))\n(tℓ)3|∆|/greaterorequalslanteUEG(ρ)−Cερ2.36 M. LEWIN, E.H. LIEB, AND R. SEIRINGER\nThe last term can again be bounded by\nε(tℓ)−3ˆ\nR3ρ2(1tℓ∆∗ηtδ)\nand included into the average over t. Finally, using (69) and ρ/lessorequalslantCρ, we\ninfer that\neUEG(ρ)/greaterorequalslanteUEG(ρ(x))−C(ρ−ρ(x))ρ(x)1\n3\non the support of 1tℓ∆∗ηtδ. To conclude the proof of (80) we can proceed\nin the same way as for the upper bound (75). This concludes the proof of\nTheorem 2. /square\nAppendix A.Classical case\nIn the classical case where the kinetic energy is neglected, the grand\ncanonical energy functional is defined [30] by\nEcl(ρ) := inf/summationtext∞\nn=0Pn(R3n)=1/summationtext∞\nn=1ρPn=ρ∞/summationdisplay\nn=1ˆ\n(R3)n/summationdisplay\n1/lessorequalslantj<k/lessorequalslantn1\n|xj−xk|dPn(x1,...,xn)\n−1\n2ˆ\nR3ˆ\nR3ρ(x)ρ(y)\n|x−y|dxdy(83)\nwhere each Pnis a symmetric probability measure on ( R3)nwith density\nρPn(x) =nˆ\nR3(n−1)dPn(x,x2...,xn).\nThis classical energy (83) is obtained from the quantum ener gy in the limit\nlim\nα→0α−4\n3E(α3ρ(α·)/parenrightbig\n=Ecl(ρ)\nsee [5, 3, 28, 6]. When´\nR3ρ=N∈N, the canonical version of Eclreads\nEcan\ncl(ρ) := inf\nρP=ρˆ\n(R3)N/summationdisplay\n1/lessorequalslantj<k/lessorequalslantN1\n|xj−xk|dP(x1,...,xN)\n−1\n2ˆ\nR3ˆ\nR3ρ(x)ρ(y)\n|x−y|dxdy.(84)\nIn [30] we have shown that for ρN(x) =ρ1(N−1/3x) with´\nR3ρ1= 1,\nlim\nN→∞Ecan\ncl(ρN)\nN= lim\nN→∞Ecl(ρN)\nN=cUEGˆ\nR3ρ(x)4\n3dx (85)\nwhere\ncUEG= lim\nρ→0+eUEG(ρ)\nρ4/3<0\nis the energy per unit volume of the classical uniform electr on gas at den-\nsity 1. In this appendix we quickly explain how to derive the f ollowing\nquantitative estimate on the convergence rate in (85).THE LOCAL DENSITY APPROXIMATION IN DENSITY FUNCTIONAL THEO RY 37\nTheorem 4 (Estimate in the (grand-canonical) classical case) .Letp >3\nand0< θ <1such that θp/greaterorequalslant4/3. There exists a universal constant\nC=C(p,θ)such that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleEcl(ρ)−cUEGˆ\nR3ρ(x)4\n3dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslantεˆ\nR3/parenleftbig\nρ(x)+ρ(x)4\n3/parenrightbig\ndx\n+C\nεbˆ\nR3|∇ρθ(x)|pdx(86)\nwith\nb= max/braceleftbig\n2p−1,(1+3θ)p−4/bracerightbig\n,\nfor every ε >0and every non-negative density ρ∈L1(R3)∩L4/3(R3)such\nthat∇ρθ∈Lp(R3).\nUnder the condition that θp/lessorequalslant1+p/3, which is slightly more restrictive\nthan in the quantum case (7), we get the much smaller power 2 p−1 ofεin\nfront of the gradient term. Then, after optimizing (86) in ε, we obtain the\nquantitative estimate\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleEcl(ρN)−N cUEGˆ\nR3ρ(x)4\n3dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n/lessorequalslantCN5\n6/parenleftbiggˆ\nR3/parenleftbig\nρ(x)+ρ(x)4\n3/parenrightbig\ndx/parenrightbigg1−1\n2p/parenleftbiggˆ\nR3|∇ρθ(x)|pdx/parenrightbigg1\n2p\n(87)\nfor every ρN(x) =ρ1(N−1/3x) and for 4 /3/lessorequalslantθp/lessorequalslant1+p/3. The rate N5/6is\nbetter than the N11/12obtained in the quantum case, but still far from the\nexpected rate N1/3.\nProof.Theestimate (86)follows fromtheLieb-Oxfordinequality ( 17)(with-\nout the gradient term) when εis large, so we only have to consider the case\nwhereεis small.\nWe use again the tiling (31) and, as in the proof in [30], the up per bound\nEcl(ρ)/lessorequalslant/summationdisplay\nz∈Z324/summationdisplay\nj=1Ecl(1ℓµj∆+ℓzρ)/lessorequalslant/summationdisplay\nz∈Z324/summationdisplay\nj=1Ecl(1ℓµj∆+ℓzρ) (88)\nwhereρ= min ℓµj∆+ℓzρ. The inequality (88) is a consequence of the sub-\nadditivity and the negativity of Ecl. Now it follows from [30, Cor. 3.4] and\nfrom the Graf-Schenker inequality as in Section 4.3, that in a tetrahedron\ncUEGρ4/3\n0/lessorequalslantEcl(ρ01ℓ∆)\nℓ3|∆|/lessorequalslantcUEGρ4/3\n0+Cρ0\nℓ.\nThis provides the upper bound\nEcl(ρ)/lessorequalslantcUEGˆ\nR3ρ4/3+|cUEG|/summationdisplay\nz∈Z324/summationdisplay\nj=1ˆ\nℓµj∆+ℓz/parenleftig\nρ4/3−ρ4/3/parenrightig\n+C\nℓˆ\nR3ρ.38 M. LEWIN, E.H. LIEB, AND R. SEIRINGER\nFor the rest of the argument we use the notation ε= 1/ℓ. In each tetrahe-\ndron we can follow the argument in (74) and estimate\nˆ\nR3(ρ4\n3−ρ4\n3)(1ℓ∆∗ηδ)\n/lessorequalslantC/vextenddouble/vextenddouble/vextenddoubleρθ−ρθ/vextenddouble/vextenddouble/vextenddoublea\nL∞(ℓ∆+Bδ)ˆ\nR3ρ4\n3−θa(1ℓ∆∗ηδ)\n/lessorequalslantC/parenleftbigg1\nεpˆ\nℓ∆+Bδ|∇ρθ|p/parenrightbigga\np/parenleftbiggˆ\nR3ρ4−3θa\n3p\np−a(1ℓ∆∗ηδ)/parenrightbigg1−a\np\n/lessorequalslantC/parenleftbigg1\nεp+p\na−1ˆ\nℓ∆+Bδ|∇ρθ|p+εˆ\nR3ρ4−3θa\n3p\np−a(1ℓ∆∗ηδ)/parenrightbigg\nwith 0< a/lessorequalslant1. In order to estimate the second term by ρ+ρ4/3, we need\nthat\n1/lessorequalslant4−3θa\n3p\np−a/lessorequalslant4\n3\nwhich is equivalent to\n4\n3/lessorequalslantθp/lessorequalslant1+p\n3a\nand which we assume for the rest of the proof.\nWe finally turn to the lower bound. Up to an appropriate rotati on and\ntranslationofthetiling(or, equivalently, ofthedensity ρ), theGraf-Schenker\ninequality gives the following lower bound [30, p. 100]\nEcl(ρ)/greaterorequalslant/summationdisplay\nz∈Z324/summationdisplay\nj=1Ecl(1ℓµj∆+ℓzρ). (89)\nWe recall that\nEcl(1ℓµj∆+ℓzρ)/greaterorequalslant−cLOˆ\nℓµj∆+ℓzρ4\n3\nby the Lieb-Oxford inequality (17). We have nothing to prove in any tetra-\nhedronℓµj∆+ℓzsuch that\ncLOˆ\nℓµj∆+ℓzρ4\n3/lessorequalslantεˆ\nℓµj∆+ℓz/parenleftbig\nρ+ρ4\n3/parenrightbig\n+A\nεp+p\na−1ˆ\nℓµj∆+ℓz|∇ρθ|p.\nHereAis a large constant to be chosen later. This is in particular t he case\nwhenρ1/3= max ℓµj∆+ℓzρ/lessorequalslantε/cLO. So we may assume that\ncLOˆ\nℓµj∆+ℓzρ4\n3> εˆ\nℓµj∆+ℓz/parenleftbig\nρ+ρ4\n3/parenrightbig\n+A\nεp+p\na−1ˆ\nℓµj∆+ℓz|∇ρθ|p\nand that ρ >(ε/cLO)3. This implies\nρθ−ρθ/lessorequalslantCε1\na−1\np\nA1\npρ4\n3p/lessorequalslantCε3\np(1+p\n3a−θp)\nA1\npρθ.\nThe power of εis non-negative when, again,\nθp/lessorequalslant1+p\n3a.THE LOCAL DENSITY APPROXIMATION IN DENSITY FUNCTIONAL THEO RY 39\nForAlarge enough (or εsmall enough) this gives ρ/lessorequalslantCρ/lessorequalslantCρin the\nsimplexℓµj∆+ℓz. The rest of the argument is then exactly the same as\nfor the upper bound.\nAs a conclusion we obtain the bound (86) with the error term\nC\nεp+p\na−1ˆ\nℓµj∆+ℓz|∇ρθ|p\nand the restrictions that p >3, 0< θ/lessorequalslant1, 0< a/lessorequalslant1 and\n4\n3/lessorequalslantθp/lessorequalslant1+p\n3a.\nIn order to minimize the power of εwe want to take aas large as possible,\nthat is,\na= min/parenleftbigg\n1,p\n3(θp−1)/parenrightbigg\n.\nForθp/lessorequalslant1 +p/3 we take a= 1 whereas for θp >1 +p/3 we choose the\nother value and get the stated inequality (86). /square\nRemark 7 (Canonical case) .Aswas mentioned in(85), in[30]wecould also\nhandle the canonical case. We would easily obtain a quantita tive estimate\non the canonical energy Ecan\ncl(ρ) if we knew the speed of convergence of\nℓ−3Ecan\ncl(ρ01ℓ∆) to its limit cUEG|∆|. 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Lieb ,A lower bound for Coulomb energies , Phys. Lett. A, 70 (1979), pp. 444–\n446.\n[33] ,Thomas-Fermi and related theories of atoms and molecules , Rev. Mod. Phys.,\n53 (1981), pp. 603–641.\n[34] ,Variational principle for many-fermion systems , Phys. Rev. Lett., 46 (1981),\npp. 457–459.\n[35] ,Density functionals for Coulomb systems , Int. J. Quantum Chem., 24 (1983),\npp. 243–277.\n[36]E. H. Lieb and H. Narnhofer ,The thermodynamic limit for jellium , J. Stat. Phys.,\n12 (1975), pp. 291–310.\n[37]E. H. Lieb and S. Oxford ,Improved lower bound on the indirect Coulomb energy ,\nInt. J. Quantum Chem., 19 (1980), pp. 427–439.\n[38]E. H. Lieb and R. Seiringer ,The Stability of Matter in Quantum Mechanics ,\nCambridge Univ. Press, 2010.THE LOCAL DENSITY APPROXIMATION IN DENSITY FUNCTIONAL THEO RY 41\n[39]E. H. Lieb and W. E. Thirring ,Bound on kinetic energy of fermions which proves\nstability of matter , Phys. Rev. Lett., 35 (1975), pp. 687–689.\n[40] ,Inequalities for the moments of the eigenvalues of the Schr¨ odinger hamiltonian\nand their relation to Sobolev inequalities , Studies in Mathematical Physics, Princeton\nUniversity Press, 1976, pp. 269–303.\n[41]N. H. March and W. H. Young ,Variational methods based on the density matrix ,\nProc. Phys. Soc., 72 (1958), p. 182.\n[42]N. Mardirossian and M. Head-Gordon ,Thirty years of density functional theory\nin computational chemistry: an overview and extensive asse ssment of 200 density\nfunctionals , Molecular Physics, 115 (2017), pp. 2315–2372.\n[43]P. T. Nam ,Lieb-Thirring inequality with semiclassical constant and gradient error\nterm, J. Funct. Anal., 274 (2018), pp. 1739–1746.\n[44]R. Parr and W. Yang ,Density-Functional Theory of Atoms and Molecules , Inter-\nnational Series of Monographs on Chemistry, Oxford Univers ity Press, USA, 1994.\n[45]J. P. Perdew and S. Kurth ,Density Functionals for Non-relativistic Coulomb\nSystems in the New Century , Springer Berlin Heidelberg, Berlin, Heidelberg, 2003,\npp. 1–55.\n[46]J. P. Perdew and K. Schmidt ,Jacob’s ladder of density functional approximations\nfor the exchange-correlation energy , AIP Conference Proceedings, 577 (2001), pp. 1–\n20.\n[47]A. Pribram-Jones, D. A. Gross, and K. Burke ,DFT: A theory full of holes? ,\nAnnu. Rev. Phys. Chem., 66 (2015), pp. 283–304.\nCNRS & CEREMADE, Universit ´e Paris-Dauphine, PSL University, 75016\nParis, France\nE-mail address :mathieu.lewin@math.cnrs.fr\nDepartments of Mathematics and Physics, Jadwin Hall, Princ eton Univer-\nsity, Washington Rd., Princeton, NJ 08544, USA\nE-mail address :lieb@princeton.edu\nIST Austria (Institute of Science and Technology Austria), Am Campus 1,\n3400 Klosterneuburg, Austria\nE-mail address :robert.seiringer@ist.ac.at"    },    {        "title": "1903.04894v5.Observing_evolution_from_steady_state.pdf",        "content": "arXiv:1903.04894v5  [physics.gen-ph]  11 May 2023Observing evolution from steady state\nHerman Telkamp\nJan van Beverwijckstraat 104, 5017JA Tilburg, The Netherla nds∗\nWe study cosmology in the time-translation symmetry of the c onformal FLRW frame ¯g=a−2g,\nwhere constant matter density induces constant curvature R−2\n0and where evolution on the common\nlight cone is observed from a steady state. Equipartition of recessional and peculiar components of\nkinetic energy of the gravitational field add to a total curva ture24R−2\n0of twice the scalar curvature\nof de Sitter universe, predicting a matter density Ωm=1/24, orh≈0.73. Projecting the equilibrium\nstate on the present state in ΛCDM returns ˆh≈0.68and densities within confidence limits of Planck\n2018 results.\nInvariance of de Sitter universe with respect to expan-\nsion of the scale factor is physically attributed to un-\nknown vacuum energy. However, in an essay of 1906 on\nthe relativity of space, Poincarï¿œ imagined invariance\nof an expanding universe as arising from uniform spatial\nexpansion; that is, expansion without exception of gravi-\ntationally bound objects, so that everything expands, in-\ncluding photon wavelength and ruler, but nothing seems\nto change [1]. The uniformly expanding universe thus\nshows the time-translation symmetry of de Sitter uni-\nverse, while constant matter density can be related to\nconstant curvature R−2\n0. The expansion of physical units\nof length makes space in the frame of the comoving ob-\nserver static and intrinsically stable. In this frame the\nuniverse must be infinitely old and in thermodynamic\nequilibrium, meeting the perfect cosmological principle.\nRepeated observation of cosmological parameters, like\ndensity or temperature, will show no change, hence zero\nredshift drift, as evidence of a stationary state in a static\nspace. However, in terms of the present unit of length,\nthe universe continues to expand from past into future, as\nrevealed by cosmological redshift on the past light cone.\nThe same universe then appears evolving and of finite\nage, like in Big Bang cosmology. This presents the dual-\nity of Poincarï¿œ’s uniformly expanding universe.\nUniform expansion can be represented by the confor-\nmal metric ¯g=a−2g, wheregis the FLRW metric. In\nstandard coordinates, the line element of ¯gis\nd¯s2=a−2dt2−dr2\n1−kr2/r2\n0−r2dΩ2, k=−1,0,1.(1)\nProperties of ¯ghave been studied before, e.g., by Deruelle\nand Sasaki [2], implications of which we discuss. Since\nconformal frames have the light cone in common, cos-\nmological observation in the FLRW frame can be rein-\nterpreted in terms of gravitational time dilation in the\nstatic space of the conformal frame ¯g, which is our main\nsubject.\nOne expects a cosmology of gravitational time dilation\nin the thermostatic equilibrium of the conformal frame\nto deviate considerably from Big Bang cosmology in the\nFLRW frame. Still, the two frames remain each other’s\nconformal dual, so the two cosmologies must be consis-\ntent. The conformal frame has the distinct property oftime-translation symmetry, i.e., is maximally symmet-\nric, and clearly this must have distinct implications to\ncosmology in the FLRW frame. This becomes transpar-\nent when expressing the line element in conformal time\ndη=a−1dt, so that the metric assumes a static form,\nd¯s2=dη2−dr2\n1−kr2/r2\n0−r2dΩ2. (2)\nFlorides showed that the existence of a static represen-\ntation constrains solution space in the FLRW frame to\nspacetimes of constant curvature [3]. In particular, if the\nconstant density is positive, this regards de Sitter space-\ntime. Constant energy density of the total particle mat-\nter fluid in the conformal frame, ¯ρm=ρm0, can then be\nconsidered to act as cosmological constant, i.e., (in units\nwhere8\n3πG=c= 1),\nR−2\n0= ¯ρm=ρm0. (3)\nFrom this assumption we derive properties and obser-\nvational viability of a nonempty de Sitter cosmology in\nthe conformal frame. The Friedmann equation is ob-\ntained as null solution of the gravitational field at the\nde Sitter horizon, that is, from the FLRW null met-\nric in isotropic coordinates. Conformal invariance of\nthe null solution means that the Friedmann equation is\nidentical in both frames, yet is to be interpreted differ-\nently. For example, in the expanding frame an energy\ndensity evolves with time, ρ(t) =ρ0a(t)−n, and is ho-\nmogeneous, i.e., constant in space, while in the confor-\nmal frame the same energy density evolves spatially, i.e.,\n¯ρ(r) =ρ0(1 +z(r))n, yet is constant in time. The two\nmeet on the light cone (’then=there’). The present state\nin the expanding frame therefore coincides with the local\nstationary state in the conformal frame, hence is con-\nstant,ρ(t0) =ρ0= ¯ρ(0). Evolution in the uniformly\nexpanding universe thus seems to regard past and fu-\nture only. This means that in the dual universe density\nparameters (e.g., of the ΛCDM model) in fact are fixed\nand must somehow relate to steady state parameters, as\nshown. For simplicity we will refer to ’recessional’ and\n’peculiar’ energy densities in both frames, even while re-\ncessional energy density is transformed into curvature of\ntime in the conformal frame.2\nProperties of ¯g.—In the static frame of Eq.(2), particle\nrest mass ¯mis constant, i.e., d¯m/dη= 0. Sincead¯m/dt=\nd¯m/dη= 0, mass¯mmust be constant in the conformal\nframe too. Indeed, the conformal factor a−1restores the\ntime-translation symmetry that is lost in the expanding\nuniverse. Deruelle and Sasaki [2] show that mass min\nthe FLRW frame transforms in the conformal frame into\n¯m=am, (4)\nand point out that CMB temperature ¯T=aT=const.\nThis suggests that presumed constant particle mass min\nthe expanding frame (as implied by ρba3=const) would\nconflict with constant particle mass ¯m=amin the time-\ntranslation symmetry of the conformal frame. Turning\nthe argument around: ¯m=const in the conformal frame\nimpliesm∝a−1in the expanding frame, like cosmic\ntemperature T∝a−1. Then, on the light cone of the\nuniformly expanding universe, baryon density evolves as\nρb(a) =ρb0a−4=ρb0(1+z)4= ¯ρb(z), (5)\nthat is, as pressureless matter density in terms of evolving\nrest mass m∝a−1. On the other hand, Eq.(5) seems to\nrecognize the inherent relativistic aspect of baryons, bot h\nat the subatomic level and in the interaction with the\ncosmic gravitational field. For our purposes, the main\nthing is consistency of the two interpretations, so that in\neither view the total matter content of baryons, photons\nand neutrinos appears as a single radiation density\nρm=ρb+ργ+ρν=ρm0a−4. (6)\nA uniform equation of state decouples evolution of the\nscale factor from particle fluid composition (as one ex-\npects in de Sitter universe). Hereafter, we will not as-\nsumeρm∝a−4, instead independently retrieve this rela-\ntion further on, as part of the Friedmann equation.\nCurvature of time in flat space. —Spatial flatness of de\nSitter universe follows directly from the well known differ-\nential equation of the event horizon radius, ˙Re=HRe−1\n[4], where constant Re=R0impliesH=R−1\n0, i.e., flat de\nSitter expansion at deceleration q=−1. Intrinsic spatial\nflatness of de Sitter universe is appealing since it avoids\nthe unstable equilibrium of flat space in standard cos-\nmology. Also, it may explain the remarkable flat volume\nof Misner-Sharp mass, regardless of k[5]. These proper-\nties suggest that, physically, curvature energy density in\nde Sitter universe can only be associated with curvature\nof time. Since q≡ −¨aa/˙a2is dimensionless, curvature\nenergy density indeed is free to curve time in spatially\nflat de Sitter universe. The FLRW metric in isotropic\ncoordinates allows relocation of all curvature to the time\ndimension by the conformal factor a−1(1 +kr2/r2\n0), so\nthat space is intrinsically flat, static and stable, and the\nline element of the conformal metric ˜greads\nd˜s2=a−2(1+kr2/r2\n0)2dt2−dr2−r2dΩ2.(7)In the following we relate k=−1,+1to, respectively, the\nin- and outgoing peculiar component of the gravitational\nfield at the horizon. Then the evolving curvature den-\nsity represents the peculiar kinetic energy density of the\nfield, while the constant ’vacuum’ density ρm0=R−2\n0\nrepresents the recessional kinetic energy density of the\nfield. One expects the evolving curvature density to cause\nevolving gravitational time dilation. This, in turn, affect s\nboth vacuum and curvature density, and so on, until some\npoint of equilibrium. To account for this interaction of\npeculiar and recessional kinetic energy we shall rely on\nthe metric.\nMetric analysis. —The Friedmann equation of de Sitter\nuniverse can be derived from evaluation of the in- and\noutgoing null solutions at the event horizon, where the\ninward directed stationary null front satisfies ar=R0.\nSubstitution and differentiation of r=R0/ain the radial\nnull metric ( dΩ = 0 ) returns the Friedmann equation of\nthe recessional kinetic energy density, i.e.,\n˙a2\na2=1\nR2\n0(1+kr2/r2\n0)2=1\nR2\n0(1+ka−2)2.(8)\nThis is mirrored by substitution of a=R0/rin the met-\nric, returning an identical peculiar kinetic energy densit y\n˙r2\nr2=1\nR2\n0(1+kr2/r2\n0)2=1\nR2\n0(1+ka−2)2=˙a2\na2.(9)\nThis equipartition of kinetic energy shows the duality of\npermanent equilibrium, ˙a2/a2= ˙r2/r2, across evolution\ninto past and future, relative to a constant present state.\nAs suggested, we associate k=−1,1with, respectively,\nthe ingoing and outgoing gravitational field at the hori-\nzon. Since these fields coexist, the corresponding kinetic\nenergy densities in Eq.(9) coexist too. Hence, the energy\ndensity of the ensemble is given by the sum over k=±1,\nwhere the alternating cross term vanishes, i.e.,\n˙a2\na2=2\nR2\n0(1+k2a−4) =˙r2\nr2. (10)\nThis is where we see curvature energy density appear-\ning in the form of gravitational radiation density, i.e.,\nas a uniform total matter density, in agreement with\nρm∝a−4in Eq.(6). Accounting for equipartition of ki-\nnetic energy in 3-dimensional space, the recessional and\npeculiar energy densities amount to\nH2≡˙a2\na2=6\nR2\n0(1+a−4) =˙r2\nr2. (11)\nwhereH≡˙a/ais the Hubble parameter, expressing ex-\npansion rate. Since (dR/R)2= (da/a+dr/r)2, the re-\ncessional and peculiar kinetic energy densities combine\ninto a total kinetic energy density (recessional and pecu-\nliar motion can be seen to occupy independent degrees of\nfreedom [6]). Like before, the cross term vanishes in the3\nensemble, hence the Friedmann equation of total density\nin a nonempty de Sitter universe is\n˜H2≡˙R2\nR2=˙a2\na2+˙r2\nr2=12\nR2\n0(1+a−4) = 2H2.(12)\nThe constant ˜H2\n0= 24R−2\n0= 24ρm0= 2H2\n0equals twice\nthe scalar curvature of de Sitter spacetime. This predicts\na density of local energy of the matter\nΩm=1\n24= 0.04166.... (13)\nSinceΩm≈Ωb, this can be validated using the baryon\ndensity estimate Ωbh2= 0.0224±0.0001(Planck 2018\n[7]). Without introducing dark components, this predicts\na Hubble constant,\nh=/radicalbig\n24Ωbh2= 0.7332±0.0016, (14)\nwithin the range of some of the most accurate distance\nladder estimates of the Hubble constant, e.g., 0.732±\n0.013by Riess et al. [8], or0.733±0.018by Wong et\nal.[9]. The slightly lower, but nearly as accurate BBN\nestimate of baryon density Ωbh2= 0.02166±0.00026 by\nCooke et al. [10] predicts h= 0.7210±0.0043, still within\nrange of the above distance ladder estimates.\nThe Friedmann equation in Eq.(12) shows some inter-\nesting properties. The constant radius of curvature R0\nimplies constant present densities, while the total matter\ndensity is seen to evolve into past and future, i.e., past\ndivergence of ρm∝a−4still refers to a hot past. At the\nsame time, curvature of time prevents a past singularity\nto actually occur in the local frame of the observer. The\nsolution thus unifies favorable aspects of Big Bang and\nsteady state scenarios [11, 12].\nA curious implication of constant densities in the con-\nformal frame is that even the ’age’ parameter t0is a con-\nstant. The Friedmann equation (11) has solution\na(t) =A1/4sinh(2√\n6t/R0)1/2, (15)\nwhere the density ratio A≡Ωm/ΩΛ= 1. Solving for\na(t0) = 1 gives a constant ’age’\nt0=1\n2√\n6R0asinh(1) (16)\nas ultimate consequence of time-translation symmetry in\nthe conformal frame. The constant age parameter obvi-\nously does not represent elapsed clock time. This seem-\ningly problematic notion of age is still sensible, provided\nthat clock rate slows down, so that age becomes a con-\nstantt0in terms of the continuously stretching units of\npresent time (as Eq.(16) shows, in fact representing the\nconstant curvature of de Sitter universe). Deceleration\nof clock rate is seen indeed in the conformal metrics in\nEqs.(1) and (7), i.e., locally ( r= 0) proper time of theobserver dilates as conformal time dη=a−1dt, there-\nfore diverges into the past of an infinitely old stationary\nuniverse, while t0is fixed. So in the past, at arbitrary\nredshifts, the age of the universe was t0, like today. This\nwould make the recently reported existence of very mas-\nsive galaxies at high redshifts conceivable [13].\nTransformation to ΛCDM.—Temperature anisotropy\nof CMB radiation is due to baryon density fluctuations,\nwhich in stationary state are characterized by harmonic\nacoustic oscillations. The angular power spectrum repre-\nsents the peculiar kinetic energy densities of the harmonic\ncomponents, the sum of which is taken as a measure of\ncosmic energy density ρ0, therefore of the recessional ki-\nnetic energy density represented by the Hubble constant\nH2\n0= ˙a(t0)2/a(t0)2. This same relationship is seen in\nthe vacuum/radiation model of Eq.(11). In fact, this\nmodel can be transformed into the basic ΛCDM model\nof vacuum and pressureless matter densities, so that the\ncalculated ΛCDM parameters can be compared with con-\ncordance model estimates. This is rather straightforward\nsince the two models are related by transformation of the\nscale factor ˆa=a1/βalong with scaling of the time coor-\ndinatet→ˆt=t/β. Choosing β=3/4converts radiation\ndensity∝a−4into the form of pressureless matter den-\nsity∝ˆa−3. The transformation brings the solution of\nthe scale factor in Eq.(15) into the form\nˆa(ˆt) =ˆA1/4βsinh(2√\n6ˆt/R0)1/2β. (17)\nForˆa(ˆt0) = 1 andˆt0=t0/β=4\n3t0, the density ratio is\nˆA≡ˆΩm/ˆΩΛ=sinh/parenleftbig4\n3asinh(1)/parenrightbig−2= 0.4659.... Thus,\nˆΩm= 1−ˆΩΛ=ˆA(1+ˆA)−1= 0.3179... . (18)\nThis matches the Planck 2018 estimate ΩPlanck\nm= 0.315±\n0.007[7]. The Friedmann equation in terms of ˆa(ˆt)fol-\nlows from differentiation of the solution in Eq.(17) and\nsome algebra. Finally, relabeling ˆt→t, the equation of\nthe basic ΛCDM model in the frame of the observer is\nˆH(t)2≡˙ˆa2\nˆa2=1\nβ212\nR2\n0/parenleftbigˆΩΛ+ˆΩmˆa(t)−3/parenrightbig\n.(19)\nBy our initial assumption, R−2\n0=ρm0, the Hubble con-\nstant of the ΛCDM model can be related to the total\nmatter density, i.e.,\nˆH2\n0=1\nβ2·12R−2\n0=16\n9·12R−2\n0= 211\n3ρm0.(20)\nWithΩb≈Ωm=1\n24, the BBN estimate of baryon den-\nsityΩbh2= 0.02166±0.00026 by Cooke et al. [10] gives\nan estimate of the Hubble constant ˆh=/radicalBig\n8\n9·24Ωbh2=\n0.6798±0.0041for theΛCDM model. This is in agree-\nment with the Planck 2018 estimate ˆhPlanck= 0.674±\n0.005. Yet, the higher Planck 2018 estimate of the4\nbaryon density Ωbh2= 0.0224±0.0001 [7]) predicts\nˆh= 0.691±0.002, at some distance from ˆhPlanck.\nTheΛCDM expression ˆH2\n0=8\n9·24R−2\n0shows a\nsubstantially lower value of the Hubble constant than\nthe total density expression ˜H2\n0= 24R−2\n0of the vac-\nuum/radiation model in Eq.(12), i.e., ˜H0/ˆH0=/radicalbig\n9/8≈\n1.061. This may seem to explain most of the distance be-\ntween CMB and distance ladder estimates of the Hubble\nconstant. It is not evident, however, that distance ladder\nestimates exactly express total density, as represented by\n˜H0, even though one expects both recessional and pecu-\nliar kinetic energy density to contribute to cosmological\nredshift. Nonetheless, the present analysis indicates the\npossibility of alternative interpretations of the Hubble\nconstant and confirms the strong model dependence of\nparameter estimates, as noted in [7].\n∗herman_telkamp@hotmail.com\n[1] H. Poincare, Science and method; The Relativity of Space\n(T. Nelson London, 1914).[2] N. Deruelle and M. Sasaki, in Cosmology, Quantum Vac-\nuum and Zeta Functions (Springer Berlin Heidelberg,\nBerlin, Heidelberg, 2011), pp. 247–260, ISBN 978-3-642-\n19760-4.\n[3] P. S. Florides, General Relativity and Gravitation 12,\n563 (1980).\n[4] V. Faraoni, Phys. Rev. D 84, 024003 (2011).\n[5] P. Binetruy and A. Helou, Class. Quant. Grav. 32,\n205006 (2015).\n[6] H. Telkamp, Phys. Rev. D 98, 063507 (2018).\n[7] N. Aghanim et al. (Planck collaboration), Astron. Astro -\nphys.641, A6 (2020).\n[8] A. G. Riess et al., Astrophys. J. Letters 908, L6 (2021).\n[9] K. C. Wong et al., Mon. Not. Roy. Astron. Soc. 498, 1420\n(2019).\n[10] R. J. Cooke, M. Pettini, and C. C. Steidel, The Astro-\nphysical Journal 855, 102 (2018).\n[11] F. Hoyle, Mon. Not. Roy. Astron. Soc. 108, 372 (1948).\n[12] H. Bondi and T. Gold, Mon. Not. Roy. Astron. Soc. 108,\n252 (1948).\n[13] I. Labbé, P. van Dokkum, E. Nelson, R. Bezanson, K. A.\nSuess, J. Leja, G. Brammer, K. Whitaker, E. Mathews,\nM. Stefanon, et al., Nature (London) 616, 266 (2023),\n2207.12446."    },    {        "title": "1904.03036v2.Probability_representation_of_quantum_channels.pdf",        "content": "arXiv:1904.03036v2  [quant-ph]  8 Apr 2019Probability representation of quantum channels\nAshot Avanesov1,2,∗and Vladimir I. Man’ko1,2,3,†\n1Department of General and Applied Physics,\nMoscow Institute of Physics and Technology (State Universi ty) Institutskii per. 9,\nDolgoprudnyi, Moscow Region 141700, Russia\n2Lebedev Physical Institute, Russian Academy of Sciences\nLeninskii Prospect 53, Moscow 119991, Russia\n3Tomsk State University, Department of Physics,\nLenin Avenue 36, Tomsk 634050, Russia\n(Dated: April 9, 2019)\nAbstract\nUsing the known possibility to associate the completely pos itive maps with density matrices and\nrecent results on expressing the density matrices with sets of classical probability distributions of\ndichotomic random variables we construct the probability r epresentation of the completely positive\nmaps. In this representation, any completely positive map o f qudit state density matrix is identified\nwith the set of classical coin probability distributions. E xamples of the maps of qubit states are\nstudied in detail. The evolution equation of quantum states is written in the form of the classical-\nlike kinetic equation for probability distributions ident ified with qudit state.\n∗avanesov@phystech.edu\n†manko@lebedev.ru\n1I. INTRODUCTION\nIt is known that any quantum operation of quantum states defined in ad-dimensional\nHilbert space Hdcorresponds to a nonnegative hermitian operator acting on the te nsor prod-\nuct of Hilbert spaces Hd⊗Hd[1]. Channel-state duality or Choi-Jamio/suppress lkowski isomorphism\n[2, 3] has a fundamental physical meaning and vastly applicated in th e field of quantum\ninformation theory.\nSince the origin of quantum mechanics, the pure states of quantum systems are described\nby wave function or wave vector that is an element of a Hilbert space [4, 5]. The quantum\nstate also can be determined by the density matrix [6, 7] that is a no nnegative hermitian\noperator acting on the Hibert space. Moreover, it is impossible to pr esent the state of\na quantum system in the presence of thermal fluctuation in the for m of a wave function.\nNaturally, the density matrix describes these states called mixed st ates.\nThroughout the whole period of the development of quantum mecha nics and its applica-\ntions, the alternative approaches to describe quantum states we re being suggested. Some of\nthem initially were developed in attempts to construct the formalism o f quantum mechanics\nthat is similar to classical statistical mechanics one [8–12]. Authors o f these works identified\nquantum states with different kinds of the quasiprobability distribut ions which are functions\nof position and momentum.\nAlso, there were attemts to construct the representation of qu antum mechanics based\nonly on fair probability distribution [13–16]. Some approaches propo sed the usage of non-\nKolmogorov probability theories [17].\nIn [18] it was introduced the tomographic probability representatio n of states of the\nquantum system with continuous variables, where the quantum sta te is associated with fair\nprobability distributions. This representation relates to the notion of optical tomogram\nfunction which was introduced in [19, 20] where authors proposed t o measure tomogram\nprobabilities in purpose to reconstruct the Wigner function of a pho ton quantum state. The\napproach of [18] was expanded for the case of systems with discre te variables and notion\nof spin-tomogram was introduced [21, 22]. For further information on the tomographic\nprobability representation of quantum mechanics, we refer to rev iew [23].\nThe evolution of closed quantum systems is described by the Schr¨ o dinger equation [24].\nHence, the state of the closed system undergoes the unitary tra nsformation. It is also\n2important to consider the problem of the evolution of open quantum systems. The general\nnonunitary transformation of the quantum state is determined by completely positive map\nof the density matrix and can be presented in the form [25]\nˆρ→/summationdisplay\nkˆAkˆρˆA†\nk, (1)\nwhere ˆρis the initial density matrix of the state and ˆAis an arbitrary matrix called Kraus\noperator [26]. The main aspects of the map (1) are widely discussed in the literature, and\nit is also known as quantum channel, e. g. see [27]. The notion of dynam ical maps were\nintroduced in [28, 29]. The evolution equation of open quantum syste ms was derived in\n[30, 31].\nIn the case of qudit systems, the state can be reconstructed fr om the finite number of\nvalues of the tomographic function. For instance, we need only thr ee real parameters to\ndetermine the state of qubit [32]. Hence, we can use three probabilit ies to describe the qubit\nstate. Recently, quantum suprematism or probability representa tion of quantum mechanics\nof discrete variables systems was suggested [33–36]. Initially, it was proposed for the case\nof qubit systems. Here, the state associated with three probabilit y distributions based on\ntomographic probabilities, the quantum observable is treated as th e set of classical-like\nvariables, though the dependence between statistics of quantum observable and its classical-\nlike variables is not straightforward [37]. Generalization to the descr iption of qudit states in\nterms of the probabilities also was proposed [38].\nThe goal of our approach is to present the formulation of quantum states and the state\nevolution equations in terms of probability distributions and stochas tic (classical-like) ki-\nnetic equations. In the present paper, we are aimed to utilize the ch annel-state duality in\nthe framework of the developed probability representation and th us to express the quan-\ntum operation as the set of probability distributions. Then, it is also p ossible to derive\nthe stochastic equation for these probabilities that correspond t o the evolution equation\nof quantum systems. In the present paper, we consider the unita ry evolution of the qubit\nsystem.\n3II. QUDIT STATES IN PROBABILITY REPRESENTATION\nLet us give a brief review of the main aspects of probability represen tation of the states of\nthe finite-level quantum (qudit) systems. This approach is directly derived from the notion\nof spin tomographic function that is introduced for spin systems. I n the case of spin jit\nhas the expression w(m, /vector n ) =/angbracketleftm|ˆU(/vector n)ˆρˆU†(/vector n)|m/angbracketright, wherem=−j,−j+ 1,..., j is spin\nprojection onto direction determined by the vector /vector n,|m/angbracketrightis eigenvector in computational\nbasis (z-basis). The unitary matrix ˆU(/vector n) is chosen by the way that ˆU†(/vector n)|m/angbracketrightis eigenvector of\noperator of spin projection onto direction /vector n, that is expressed in the form nxˆσx+nyˆσy+nzˆσz.\nHere ˆσx, ˆσy, ˆσzare Pauli matrices. Thus, the value w(m, /vector n ) is the probability of the outcome\nmthe measurement of spin projection onto direction of vector /vector n.\nThe knowledge of tomographic function allows us to reconstruct th e density matrix of the\nstate of the quantum system. Moreover, we need only a finite numb er of its values. To be\nprecisely in the case of the n-level quantum system its density matrices determined by n2−1\nreal parameters. Finally, we can present these parameters as th e tomographic probabilities.\nFor example, in the case of qubit systems, we need only three tomog raphic probabilities\nto describe the state p1=/angbracketleft+|ˆUxˆρˆU†\nx|+/angbracketright, p2=/angbracketleft+|ˆUyˆρˆU†\ny|+/angbracketright, p3=/angbracketleft+|ˆρ|+/angbracketright, where ˆUx=\n1√\n2\n1 1\n1−1\n,ˆUy=1√\n2\n1i\ni1\nand|+/angbracketright=\n1\n0\n. In other words p1=w(+, x),p2=w(+, y),\nandp3=w(+, z). Then, the density matrix of the qubit state can be presented in t he form\nˆρ=\np1/parenleftbig\np2−1\n2/parenrightbig\n−i/parenleftbig\np3−1\n2/parenrightbig\n/parenleftbig\np2−1\n2/parenrightbig\n+i/parenleftbig\np3−1\n2/parenrightbig\n1−p1\n. (2)\nThese probability parameters form three dichotomic probability dist ributions. However,\nthey must satisfy the restriction/summationtext3\ni=1/parenleftbig\npi−1\n2/parenrightbig2≤1\n4.\nIn essence, in the probability representation of the qudit systems states are described\nby the finite set of probability distributions. In the case of qubit sys tems, we use the three\ndistributions corresponding to the measurements of spin project ion onto three perpendicular\ndirections x,yandz.\nIn the case of ququart systems, the probability parametrization o f the density matrix can\nbe presented in the form\n4ˆρ=\np1+p2+p3−2/parenleftbig\np4−1\n2/parenrightbig\n−i/parenleftbig\np5−1\n2/parenrightbig /parenleftbig\np6−1\n2/parenrightbig\n−i/parenleftbig\np7−1\n2/parenrightbig /parenleftbig\np8−1\n2/parenrightbig\n−i/parenleftbig\np9−1\n2/parenrightbig\n/parenleftbig\np4−1\n2/parenrightbig\n+i/parenleftbig\np5−1\n2/parenrightbig\n1−p1/parenleftbig\np10−1\n2/parenrightbig\n−i/parenleftbig\np11−1\n2/parenrightbig /parenleftbig\np12−1\n2/parenrightbig\n−i/parenleftbig\np13−1\n2/parenrightbig\n/parenleftbig\np6−1\n2/parenrightbig\n+i/parenleftbig\np7−1\n2/parenrightbig /parenleftbig\np10−1\n2/parenrightbig\n+i/parenleftbig\np11−1\n2/parenrightbig\n1−p2/parenleftbig\np14−1\n2/parenrightbig\n−i/parenleftbig\np15−1\n2/parenrightbig\n/parenleftbig\np8−1\n2/parenrightbig\n+i/parenleftbig\np9−1\n2/parenrightbig /parenleftbig\np12−1\n2/parenrightbig\n+i/parenleftbig\np13−1\n2/parenrightbig /parenleftbig\np14−1\n2/parenrightbig\n+i/parenleftbig\np15−1\n2/parenrightbig\n1−p3\n\n(3)\nThus, we can determine the state of the quantum system as the se t of 12 dichotomic prob-\nability distributions and one of the size 4. In other words, instead of density matrix we de-\nscribe the ququart state as a set of probability distributions Ξ(4)=/braceleftBig\n/vectorP,/vectorPi, i= 4,...,15/bracerightBig\n,\nwhere\n/vectorP=\np1+p2+p3\n1−p1\n1−p2\n1−p3\n,/vectorPi=\npi\n1−pi\n. (4)\nAs we see a bit further this representaion of ququart states will be useful in the construction\nof probability representation of completely positive maps of qubit sy stems.\nIII. COMPLETELY POSITIVE MAPS AND CHOI-JAMIO/suppress LKOWSKI ISOMOR-\nPHISM\nThe general linear transformation Fof the quantum system states can be presented in\nthe form\nF[ˆρ]ij=d/summationdisplay\ni0=1d/summationdisplay\nj0=1Dii0, jj0ρi0j0, (5)\nwhereρi0j0denote the elements of the density matrix ˆ ρ.\nIf operation Fpreserves the hermiticity and nonnegativity of eigenvalues of the m atrix\ntransforming matrix ˆ ρ, then the map Fis called positive. We also can add the requirements\nof trace-preserving, so the matrix F[ˆρ] would be the density one.\nBy using the aforementioned requirements of preserving hermiticit y and trace we can\nconclude that Dii0, jj0=D∗\njj0, ii0and/summationtextd\ni=1Dii0, ij0=δi0j0.\n5Finally, let us recall the definition of completely positive maps. Here, in addition to\nthe examined d-level quantum system HA, we introduce the n-level environment HEand\nconsider the maps of the whole system HA⊗ HE. In particular, we are interested in the\nmaps of the form In⊗F, whereInis the identical operator which acts in the space of the\nstates of the system HEandFis the transformation of the states of the system HA. If for\neverynthe map In⊗Fis positive, then the map Fis a completely positive one.\nLet us introduce the matrix of the form\nˆD=d/summationdisplay\nk,l,i,j=1Dki, lj|k/angbracketright/angbracketleftl|⊗|i/angbracketright/angbracketleftj|. (6)\nThe matrix ˆDaccordingly to [28] is called dynamical. Due to the properties of positiv e\nmaps, it is hermitian one that has the trace equal to d.\nFinally, the Choi theorem tells us that the dynamical matrix of the co mpletely positive\nmap can only have nonnegative eigenvalues.\nThus, we briefly reviewed the main aspects of the description of com pletely positive\nmaps. The corresponding to the transformation of d-level system dynamical matrix ˆDis\nthe Hermitian, positive-semidefinite matrix with the fixed trace (act ually Tr ˆD=d). Hence,\nthe matrix1\ndˆDsatisfies all the requirements of the density matrix. The Choi-Jamio /suppress lkowski\nisomorphism makes the correspondence between the completely po sitive maps of d-level\nsystems and the density matrices of d2-level systems.\nIV. PROBABILITY REPRESENTATION OF COMPLETELY POSITIVE MAPS\nAs it was shown, the set of d2−1 probability parameters (that was contained in one\nvector/vectorP) determined the state of d-level quantum system. From these probabilities we can\nconstruct N−1 =d2−ddichotomic distributions and one distribution of size that equal to\nd. We denote the set of these distributions as Ξ(d).\nWe remind that there is the isomorphism between the completely posit ive maps of the\nstates of d-level systems and the states of d2-level systems. Therefore, we can use the\nintroduced probability parametrization of the normalized dynamical matrix1\ndˆD. By ac-\ncomplishing this procedure, we come to probability representation o f the transformation of\nquantum states.\n6Let us demonstrate the possibility of probability description of comp letely positive maps\non the example of qubit systems.\nThe dynamical matrix of the completely positive map of qubit state is h ermitian and its\ntrace is equal to 2. We can utilize the parametrization of ququart st ate (3). We also should\nnote that the probability parameters that determine the consider ed map must satisfy the\nfollowing relations p1+p3=3\n2p4+p14= 1p5+p15= 1.\nHence, the completely positive map of qubit state is described by the vector/vectorPof 15\nprobability parameters or by the set of probability distributions Ξ(4). The vector /vectorPcan be\nobtained from the vectorized matrix ˆD. By vectorizing we mean the procedure when we\ntake raws of the initial matrix of size n×nand put them in one raw accordingly to their\nposition in the matrix, then by transposition operation obtain the ve ctor of size n2. In other\nwords the product of vectorization of matrix ˆDis vector /vectorDof the form\n/vectorD=d/summationdisplay\nk,l,i,j=1Dki, lj|k/angbracketright|i/angbracketright|l/angbracketright|j/angbracketright. (7)\nThe probability vector /vectorPis a linear transform of vectorized dynamical matrix\n/vectorP=ˆA·/vectorD+/vectorb (8)\nwhere ˆAis a matrix of size 15 ×16. Nonzero elements of the matrix are A1,6=A2,11=\nA3,16=−1\n2,A4,2=A4,5=A6,3=A6,9=A8,4=A8,13=A10,7=A10,10=A12,8=\nA12,14=A14,12=A14,15=1\n4andA5,2=−A5,5=A7,3=−A7,9=A9,4=−A9,13=\nA11,7=−A11,10=A13,8=−A13,14=A15,12=−A15,15=i\n4. All elements of vector /vectorb\nexceptb1,b2andb3are equal to1\n4. For the first three entries of vector /vectorbwe haveb1=b2=\nb3=1\n2. Note, that we have the map R16→R15, that is why it is possible to use other\ntransformation matrix ˆAand vector /vectorb.\nWe are also able to reconstruct the dynamic matrix ˆDfrom the given vector of probability\nparameters /vectorP. Here, it is also presented as the linear map\n/vectorD=ˆB·/vectorP+/vector c (9)\nNonzero elements of matrix ˆBareB1,1=B1,2=B1,3=B2,4=B3,6=B4,8=B5,4=\n−B6,1=B7,10=B8,12=B9,6=B10,10=−B11,2=B12,14=B13,8=B14,12=B15,14=\n−B16,3= 2 and B2,5=B3,7=B4,9=−B5,5=B7,11=B8,13=−B9,7=−B10,11=\n7B12,15=−B13,9=−B14,13=B15,15=−2i. For elements of vector /vector cwe have c1=−4,\nc6=c11=c16= 2,c2=c3=c4=c7=c8=c12=−1 +iandc5=c9=c10=c13=c14=\nc15=−1−i.\nV. STOCHASTIC EQUATIONS\nThe evolution process can be described by the completely positive ma p that depends on\nthe time parameter. We have shown that it is possible to use the prob ability distributions\nto describe the transformations of quantum states. In order to develop our approach, we\ndecide to find the equation for the vector of probability parameter s/vectorP. In the present paper,\nit is considered a unitary evolution. In the case the dynamical matrix can be expressed in\nthe form ˆD=/vectorU·/vectorU†, where/vectorUis vectorized unitary matrix ˆUthat obeys to Shr¨ odinger\nequation. Then, the evolution equation for the dynamical matrix ˆDtakes the form\nidˆD\ndt=/bracketleftBig\nˆH⊗ˆId,ˆD/bracketrightBig\n, (10)\nwhere [ ˆA,ˆB] is commutator of operators ˆAand ˆBand ˆHis Hamiltonian of the system. Thus,\nthe dynamical matrix obeys the von Neumann equation for the d2-level system described by\nthe Hamiltonian ˆH⊗ˆId. Thus, we can obtain the evolution equation for vector /vectorD\nid/vectorD\ndt=ˆQ·/vectorD (11)\nwhere ˆQ=ˆH⊗ˆI8−ˆI8⊗ˆH.\nFinally, we come to the description of the evolution process in terms o f the probability\nrepresentation. The equation for the vector of probability param eters/vectorPhas the form\nid/vectorP\ndt=ˆAˆQˆB·/vectorP+ˆAˆQ·/vector c (12)\nWe should add the initial condition to the last equation so that the pro blem would have\na unique solution. In the time moment, t= 0 the transformation matrix ˆMis the unity\nmatrix. Therefore, the corresponding dynamic matrix ˆDafter normalization is the density\nmatrix of maximally entangled state ˆD(0)= (|0/angbracketright|0/angbracketright+|1/angbracketright|1/angbracketright)(/angbracketleft0|/angbracketleft0|+/angbracketleft1|/angbracketleft1|). Thus, we\nfinally derive that all initial probability parameters are equal to1\n2except the following ones\np(0)\n1=p(0)\n2=p(0)\n8= 1, p(0)\n3=1\n2.\n8VI. SUMMARY\nTo conclude we point out the main results of the work. We demonstra ted on an example\nof qudits that the quantum states can be identified with probability d istributions. A new\naspect of this work is that we expressed the matrix elements of the evolution operator of\nthe qudit system with given Hamiltonian in terms of probabilities and the time evolution\nequation for the unitary evolution of the system is presented in the form of classical-like\nkinetic equation (12) for these probability distributions.\nThe solutions of the kinetic equation are shown to provide the proba bility representation\nform of the completely positive quantum channels.\nThe approach developed in the paper can be extended to open syst em evolution and it\nwill be done in a future publication.\n[1] I. Bengtsson and K. 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Man’ko, J Russ Lase r Res 38, 416 (2017).\n10"    },    {        "title": "1904.06411v2.The_Quantum_Cocktail_Party_Problem.pdf",        "content": "The Quantum Cocktail Party Problem\nXiao Liang,1, 2Yadong Wu,2and Hui Zhai2,\u0003\n1Laboratory of Quantum Information, University of Science and Technology of China, Hefei, 230026, China\n2Institute for Advanced Study, Tsinghua University, Beijing, 100084, China\nThe cocktail party problem refers to the famous selective attention problem of how to \fnd out\nthe signal of each individual sources from signals of a number of detectors. In the classical cocktail\nparty problem, the signal of each source is a sequence of data such as the voice from a speaker, and\neach detector detects signal as a linear combination of all sources. This problem can be solved by\na unsupervised machine learning algorithm known as the independent component analysis. In this\nwork we propose a quantum analog of the cocktail party problem. Here each source is a density\nmatrix of a pure state and each detector detects a density matrix as a linear combination of all\npure state density matrix. The quantum cocktail party problem is to recover the pure state density\nmatrix from a number observed mixed state density matrices. We propose the physical realization of\nthis problem, and how to solve this problem through either classical Newton's optimization method\nor by mapping the problem to the ground state of an Ising type of spin Hamiltonian.\nIntroducation.\nThe cocktail party problem refers to the phenomenon\nthat the brain of a listener can focus on a single voice\nwhile \fltering out a range of other voices in a multi-\ntalker situation, say, in a cocktail party [1]. This selec-\ntive attention problem is \frst de\fned as the \\cocktail\nparty problem\" (CPP) by C. Cherry in 1953 [2]. For\nseveral decades, it is an important research subject for\nboth neuroscience to understand how human or animals\nsolve this problem and computer science to design al-\ngorithms to solve this problem. During recent years,\nmachine-learning based approach to solve the CPP is\nessential for many industrial applications such as auto-\nmated speech recognization. The independent compo-\nnent analysis (ICA) is such an algorithm particularly\nsuitable for the CPP. The CPP can also found its ap-\nplication in physical science such as astrophysics data\nanalysis [3, 4]. Recently, it has also been proposed to use\nthe spirt of CPP and ICA method to extract the eigen\nfrequency of a quantum system from a dynamical probe\n[5].\nLet us \frst brie\ry review the classical CPP (c-CPP).\nConsidering N-independent speakers in a room, they\nspeak simultaneously and the voice of each speaker is a\nsource denoted by a sequence si(t) (i= 1;:::;N ). There\nare alsoMdetectors in the room. Each detector detects\na signalxj(t) (j= 1;:::;M ) that is considered to be a\nlinear combination of all sources si(t), as schematically\nshown in Fig. 1(a). That is to say, we have a M\u0002N-\ndimensional matrix Aand\nxj(t) =X\njiAjisi(t): (1)\nTo concentrate on one of the speakers, it means that we\nshould \fnd out A\u00001such that we can determine si(t)\nassi(t) =P\nj(A\u00001)ijxj(t) from the signals of all detec-\ntors. For human, we only have two ears which means\nthe number of detectors is two. But for computer algo-\nrithm problem, we make the situation simpler by consid-\nFIG. 1. Schematic of the cocktail party problem. (a) Classical\ncase. The voices emitted from di\u000berent source are mixed and\ndetected by detectors. (b) Quantum case. The wave functions\nfrom di\u000berent sources mix spatially and the density matrix at\ndi\u000berent places are detected by detectors.\nering that there are more detectors than sources, that is,\nM >N . Even though, this is still an ill-de\fned problem\nif no information of the source is known. In practices, we\nutilize the information that being voice of an individual\nspeaker, each source si(t) displays certain feature and is\nmore regular than a mix of several voices. By performing\nstatistics over tfor each sequence si(t), we can determine\nthe entropy of the sequence and we use the criterion that\nthe entropy of each sequence should be minimized to de-\ntermine each si(t). This is how we solve the CPP with\nthe ICA method [6].\nIn this work we will propose a quantum analogy of\nthe CPP, termed as the quantum cocktail party problem\n(q-CPP). We will discuss how to solve the q-CPP with\nan analogy of the ICA method. We should also present\na mathematical statement that can help us to map the\nloss function to a Hamiltonian of Ising spins. Though\nby classical Monte Carlo, we show that the ground state\nspin con\fguration can solve the q-CPP, we point out that\nthis spin Hamiltonian can be solved more e\u000eciently by\nquantum methods, for example, by quantum simulation\nand quantum annealing.\nResults.arXiv:1904.06411v2  [quant-ph]  16 Apr 20192\nQuantum CPP. Here we \frst propose the q-CPP. We\nconsiderNdi\u000berent sources, and each source sito be\na density matrix of a pure state as j\u001eiih\u001eij, wherej\u001eii\nis a normalized quantum state in a Hilbert space with\ndimensiond. There are Mnumber of detectors, and the\nsignalxjdetected by each detector is a density matrix of\na mixed state denoted by \u001ajas\n\u001aj=X\njiAjij\u001eiih\u001eij: (2)\nWe also normalize \u001ajto be trace unity, which requireP\niAji= 1 for all j. The q-CPP is de\fned as that,\nsuppose that we know su\u000ecient number of \u001aj, whether\none can \fnd out Ajito recover eachj\u001eiih\u001eij.\nHere we will brie\ry discuss the uniqueness of the solu-\ntion. First of all, we should emphasize that in order to\nensure the solution is unique, it is important to require\nthat di\u000berentj\u001eiiarenotorthogonal to each other. Sec-\nondly, when the number of detector increases by one, the\nconstraints increases by d2and the free parameters in-\ncreases byN\u00001, so we will consider the situation that\nd2>N. Lastly, it is always good to have su\u000ecient num-\nber of detectors, normally we consider M > N . We\ndo not rigorously prove the uniqueness of the solution,\nbut we \fnd that in practices, generically we always \fnd\nunique solution when these conditions are satis\fed.\nA physical realization of the q-CPP can be proposed\nas follows. Let us consider a particle whose internal\nHilbert space is a product of two degrees of freedom as\nH=HA\u0002HB. The dimensionality of HAisdand the di-\nmensionality ofHBisN, andfjsiig(i= 1;:::;N ) forms\na complete set of bases in HB. A wave function j\bican\nbe generally expanded in these bases as\nj\bi=NX\ni=1'i(r)j\u001eiijsii: (3)\nFor a more physical picture, one can consider Eq. 3\nas particle emitted from Ndi\u000berent sources, and the\nwave function inHBisjsiifor particle emitted from the\nsource-i, as schematically shown in Fig. 1(b). If we place\nMdetectors in di\u000berent places riand the quantum mea-\nsurement traces out the Hilbert space HB, it results in\nMdi\u000berent density matrix as Eq. 2 with Aji=j'i(rj)j2,\nthus, it is also natural to require Ajito be positive num-\nbers. In practices, these density matrices can be con-\nstructed by the quantum state tomography. We consider\nthe situation that both the wave function '(r) andj\u001eii\nare unknown. The q-CPP is to determine them from \u001aj\n(j= 1;:::;M ).\nTo \fnd out the pure state, the most important infor-\nmation we use here is that the density matrix of a pure\nstate has the property that \u001a2=\u001a. Thus, the scheme is\nto \fnd out a proper combination of \u001ajas\u001a=PM\nj=1wj\u001aj\nwith the normalization conditionP\njwj= 1 that canClassical CPP Quantum CPP\nSource Voice of each speaker Each pure state\nDetector Mixed voices Mixed state\nLoss function Minimizing entropy Minimizingj\u001a2\u0000\u001aj\nTABLE I. A comparison between the classical and the quan-\ntum cocktail party problem in term of di\u000berent de\fnition of\nsource, di\u000berent role of detector and di\u000berent loss functions.\nminimizej\u001a2\u0000\u001aj. This is equivalent to say, we de\fne the\nloss function as\nF=X\nmnj(\u001a2\u0000\u001a)mnj2\n=X\nmnjMX\nj=1(\u001a2\njw2\nj\u0000\u001ajwj)mn+MX\ni6=j=1(\u001ai\u001aj)mnwiwjj2:\n(4)\nA comparison between c-CPP and q-CPP is summarized\nin the Table I.\nOptimization with Newton's Method. We \frstly use the\nclassical Newton's method to optimize the loss function\nF, and the update rule of wis:\nw(t+\u0001t) =w(t)\u0000F0[w(t)]\nF00[w(t)]; (5)\nwhere w=fw1;:::;wMgis aM-dimensional vector.\nHereF0andF00are respectively the \frst order and the\nsecond order gradient of the loss function F. When the\nloss function reaches the minimum, it should yield F= 0,\nthus the optimization is completed. This process does not\nrequire any information of Aijandj\u001eiias a prior.\nTo test this algorithm, we \frst randomly generate a\nset of pure state in the form of\nj\u001ei=1pP\nkjckj2X\nkckjki; (6)\nwherefjkigis a complete set of basis in the Hilbert space\nHAwith dimension chosen as dA= 8, andckis a real co-\ne\u000ecient uniformly sampled in the range of [ \u00005;5]. To\ngenerate\u001ai, we randomly sample Aijin the range of\n[0;1], then normalized under the constraintP\njAi;j= 1.\nFor the example shown in Fig. 2, we choose three\npure states \u001ai=j\u001eiih\u001eijand the \fdelities between the\nthree pure states are F(\u001a1;\u001a2):= 0:56,F(\u001a1;\u001a3):= 0:54\nandF(\u001a2;\u001a3):= 0:88, where the \fdelity is de\fned as\nF(\u001aa;\u001ab) = Trpp\u001aa\u001abp\u001aa.\nWe then use the Newton's method to solve the q-CPP.\nwis initialized in such a way that wi(i= 1;:::;M\u00001)\nare uniformly sampled in the range of [ \u00002;2] andwM\nis determined by the constraintP\niwi= 1. Then, we\ncan reach a convergent solution following Eq. 5. In\nthe example of Fig.2, three di\u000berent \u001afcan be found3\n3579\nDetectornumber0.50.81.0\n(a)F(f,1)\nF(f,2)\nF(f,3)\n3579\nDetectornumber0.50.81.0\n(b)F(f,2)\nF(f,1)\nF(f,3)\n3579\nDetectornumber0.50.81.0\n(c)F(f,3)\nF(f,1)\nF(f,2)\n19232731\nQubitnumber0.20.61.0\n(d)F(f,1)\nF(f,2)\nF(f,3)\n19232731\nQubitnumber0.20.61.0\n(e)F(f,2)\nF(f,1)\nF(f,3)\n19232731\nQubitnumber0.20.61.0\n(f)F(f,3)\nF(f,1)\nF(f,2)\nFIG. 2. (Color online)(a-c) The \fdelities versus number of detectors for three di\u000berent solutions found by the Newton's method.\n(d-f) The \fdelities versus number of spins found by looking for the ground state of the Ising Hamiltonian. Here the number of\nspins means the number of detectors. In all cases, the solid dots denote the \fdelities between the output density matrix \u001afand\nthree input pure state density matrices, and the dashed lines denote the \fdelities between the three input pure state density\nmatrices.\nby the Newton's method depending on di\u000berent initial-\nization, and their \fdelities with \u001ai(i= 1;2;3) are shown\nin Fig. 2(a-c). One can see that there is always one\n\fdelity equalling unity. For instance, for the case Fig.\n2(a),F(\u001af;\u001a1):= 1, andF(\u001af;\u001a2),F(\u001af;\u001a3) are con-\nsistent with F(\u001a1;\u001a2) andF(\u001a1;\u001a3), respectively. This\nmeans that the resulting \u001afrecovers\u001a1. Similarly, in the\ncases of Fig. 2(b) and (c), the resulting \u001afrecovers\u001a2\nand\u001a3, respectively. We have also tried di\u000berent num-\nber of detectors. For the case with three sources, we \fnd\nthat the performance is good as long as the number of\ndetectors is equal or greater than three.\nMapping to a Hamiltonian Problem. Since Eq. 4 is a\nfunction offwjg, and if we restrict the value of all wjto\nbe\u00061, minimizing Eq. 4 can be regarded as \fnding the\nground state of a Hamiltonian of the Ising spins. If we\nreplacewjas\u001bz\nj, Eq. 4 can be written into a Hamiltonian\nform as\n^H=X\nijklAijkl\u001bz\ni\u001bz\nj\u001bz\nk\u001bz\nl+X\nijkBijk\u001bz\ni\u001bz\nj\u001bz\nk+X\nijCij\u001bz\ni\u001bz\nj;\n(7)\nwhere\nAijkl= Tr(\u001ai\u001aj)(\u001ak\u001al); (8)\nBijk=\u0000Tr(\u001ai\u001aj)\u001ak\u0000Tr\u001ai(\u001aj\u001ak); (9)\nCij= Tr\u001ai\u001aj; (10)\nHere we set the energy unit of the Hamiltonian as unity.In order to satisfy the constraintP\njwj= 1, we require\nthe number of spin to be odd and the total magnetization\nto be unity. Note that this Hamiltonian contains four,\nthree and two-body interactions. In this Hamiltonian,\nthe number of sites are equal to the number of detectors.\nHere it is worth emphasizing that only computing the\ncoe\u000ecients listed in Eq. 8-10 depends on the pure state\nHilbert dimension dof the original quantum problem,\nand the complicity of the Hamiltonian Eq. 7 itself will\nnot increase as dincreases. Given that in the previous\ndiscussion of uniqueness of the solution, we prefer to have\na larged, this is a great advantage of this approach.\nNow the question is whether we can restrict all wjto\nbe\u00061. Here we make the following statement:\nStatement: We consider each source siis a vector, and\nM-number of signal xj(j= 1;:::;M ) as a mixing of N-\nnumber of sources siwritten as xj=P\njiAjisi, where\nallAjiare positive numbers ranging between zero and\nunity without any other restrictions. We construct y=PM\nj=1wjsjwherewjcan only take\u00061. For each given\nMand for a speci\fed target sk, we optimizefwjg(j=\n1;:::;M ) to minimizejy\u0000skjand the minimized value\nis denoted byjy\u0000skjM\nmin. We state that\nlim\nM!1jy\u0000skjM\nmin!0: (11)\nThe meaning of this statement is that, as long as the\nnumber of the detector is su\u000ecient, we can always restrict\nwjto be\u00061.4\n1/5 1/111/151/25\n1/M101\n102\n103\n104\n|ysk|M\nmin\nk=1\nk=2\nFIG. 3. Numerical veri\fcation of the Statement . Here we\nconsider \fve sources and Mis the number of detectors.\nWe have veri\fed this statement with numerical simu-\nlations. As an example, we consider \fve sources and each\nof them is an eight-dimensional vector. As shown in Fig.\n3, we plotjy\u0000skjM\nminas a function of 1 =Mand \fnd that\nit does converge to zero as Mincreases.\nNow we show the ground state spin con\fguration of\nthis Hamiltonian can determine the solution of the q-\nCPP. AsMbecomes large, the Hilbert space dimension\nof the Hamiltonian increases and it is hard to solve the\nground state by the exact diagnolization. Hence, here we\nuse the classical Monte-Carlo method to \fnd the ground\nstate approximately. In our simulations, the initial tem-\nperature is unity which is the same as the energy unit of\nthe Hamiltonian. During the annealing process, the tem-\nperature is reduced epoch by epoch, and in each epoch\nthe temperature is reduced by1\nn(n+1), wherenis the\nepoch number. In each epoch, we randomly \rip the spin\nfor 12000 times with the acceptance probability Paccept\ngiven by\nPaccept =(\n1Et0<Et\ne\u0000\f(Et0\u0000Et)Et0\u0015Et; (12)\nwhereEt0andEtare the eigenenergies after and before\nrandomly \ripping the spins, respectively. \f= 1=(kbT)\nis the inverse temperature. In the simulation, it su\u000bers\nfrom the problem of trapped into local minimum. If so,\nwhen we regard \u001bz\njaswjand reconstruct density matrix\n\u001af=P\njwj\u001aj,\u001afmay not be a positive de\fnite ma-\ntrix. Hence, during the annealing process, we simultane-\nously run two criterions. We require the von Neumann\nentropy of the reconstructed density matrix, de\fned as\n\u0000Tr\u001aflog\u001afto be close to zero, and we also require the\nreconstructed density matrix to be positive de\fnite. We\nstop the annealing process at a temperature when these\ntwo criterions are well satis\fed.\nThe results of the Monte Carlo calculation are pre-\nsented in Fig. 2(d-f). Here we also choose three dif-\nferent input density matrices with \fdelity mutually as\nF(\u001a1;\u001a2):= 0:13,F(\u001a1;\u001a3):= 0:70 andF(\u001a2;\u001a3):= 0:45.Similar as the results from the Newton's method, depend-\ning on di\u000berent initialization, we can \fnd di\u000berent spin\ncon\fgurations that can construct three di\u000berent density\nmatrices\u001af. For instance, for the case shown in Fig.\n2(d), one can \fnd the \fdelity F(\u001af;\u001a1) is very close to\nunity, and F(\u001af;\u001a2) andF(\u001af;\u001a3) are consistent with\nF(\u001a1;\u001a2) andF(\u001a1;\u001a3), respectively. We can also see\nthat the results get improved as the number of detectors\nincreases. For instance, in Fig. 2(d), for M= 19, 23, 27,\nF(\u001af;\u001a1) are 0:991, 0:998 and 0:999, which gradually ap-\nproaches unity, and F(\u001af;\u001a2) are 0:187, 0:143 and 0:136,\nwhich gradually approaches the input value F(\u001a1;\u001a2).\nFinally, we should remark that both exact diagnoaliza-\ntion and classical Monte Carlo have limitations for solv-\ning the Hamiltonian like Eq. 7. The most e\u000ecient way\nto \fnd out its ground state is through quantum method,\neither by quantum simulation or by quantum annealing\n[7].\nConclusion and Outlook.\nIn summary, we have proposed a quantum analogy of\nthe cocktail party problem. The essential point is to re-\nplace the classical signal from each source with quan-\ntum data. In the classical problem, it is the independent\nclassical signals that are emitted from di\u000berent sources,\nwhich are mixed and detected by the detectors. In the\nquantum problem, instead, it is the quantum wave func-\ntions that are emitted from di\u000berent sources, and here\nthe \\independent\" means that the wave functions are or-\nthogonal in part of the Hilbert space. The wave functions\ninterfere and the reduced density matrix are observed by\ndetectors. In both cases, the goal is to recover the indi-\nvidual sources from the information of detectors only.\nWe also show that solving this problem can be mapped\nto \fnding the ground state of an Ising type spin Hamilto-\nnian, and we propose to solve this Hamiltonian by quan-\ntum simulator or quantum annealing. We note that,\nto ensure the uniqueness of the solution, it is preferred\nto keep both the Hilbert space dimension dof the pure\nstate wave function and the number of detector Mlarge\nenough. Under this situation, this quantum approach\nhas advantage that on one hand, the complicity of the\nHamiltonian does not increase with the increasing of d;\nand on the other hand, the number of Ising spins equal to\nMand the quantum approach can exhibit its advantage\nwhenMis large.\nWe envision that one can generalize this quantum\nversion to more complicated situation, such as time-\ndependent wave function. Similar as that the classical\nCPP is very useful in classical information processing,\nwe believe the quantum version of CPP can also be use-\nful in quantum information processing.\nCompeting Interests\nThe authors declare that there are no competing inter-\nests.\nAuthor Contribution\nAll authors contribute extensively to the entire project.5\nData Availability\nData sets were generated or analyzed during the cur-\nrent study.\nAcknowledgment. This work is supported MOST un-\nder Grant No. 2016YFA0301600 and NSFC Grant No.\n11734010.\n\u0003hzhai@tsinghua.edu.cn\n[1] M. A. Bee and C. Micheyl, The cocktail party problem:\nwhat is it? How can it be solved? And why should ani-\nmal behaviorists study it? , J. Comp. Psychol. 122235-251\n(2008).[2] E. C. Cherry, Some experiments on the recognition of\nspeech, with one and with two ears , J. Acoust. Cos. Am.\n25975-979 (1953).\n[3] I. P. Waldmann, Of Cocktail Parties and Exoplanets , As-\ntrophys. J. 74712 (2012).\n[4] J. Crowder and N. J. Cornish, Solution to the galactic fore-\nground problem for LISA , Phys. Rev. D 75043008 (2007).\n[5] Y. Wu and H. Zhai, Generalized Independent Component\nAnalysis for Extracting Eigen-Modes of a Quantum Sys-\ntem, arXiv: 1904.05067\n[6] A. Hyv arinen and E. Oja, Independent component analy-\nsis: algorithms and applications , Neural Netw. 13411-430\n(2000).\n[7] S. E. V.-Andraca, W. C.-Santos, C. McGeoch and M.\nLanzagorta, A cross-disciplinary introduction to quantum\nannealing-based algorithms , Contemp. Phys, 59174-197\n(2018)."    },    {        "title": "1904.06780v1.Truncation_scheme_of_time_dependent_density_matrix_approach_III.pdf",        "content": "arXiv:1904.06780v1  [nucl-th]  14 Apr 2019EPJ manuscript No.\n(will be inserted by the editor)\nTruncation scheme of time-dependent density-matrix appro ach\nIII\nMitsuru Tohyama1and Peter Schuck2,3\n1Kyorin University School of Medicine, Mitaka, Tokyo 181-86 11, Japan\n2Institut de Physique Nucl´ eaire, IN2P3-CNRS, Universit´ e Paris-Sud, F-91406 Orsay Cedex, France\n3Laboratoire de Physique et de Mod´ elisation des Milieux Con dens´ es, CNRS et, Universit´ e Joseph Fourier, 25 Av. des Mar tyrs,\nBP 166, F-38042 Grenoble Cedex 9, France\nReceived: date / Revised version: date\nAbstract. The time-dependent density-matrix theory (TDDM) where the Bogoliubov-Born-Green-\nKirkwood-Yvon hierarchy for reduced density matrices is tr uncated by approximating a three-body density\nmatrix with one-body and two-body density matrices is appli ed to the Lipkin model. It is shown that in\nthe large Nlimit the ground state in TDDM approaches the exact solution . Various truncation schemes\nfor the three-body density matrix are also tested for an exte nded three-level Lipkin model.\nPACS.21.60.Jz Nuclear Density Functional Theory and extensions (includes Hartree-Fock and random-\nphase approximations)\n1 Introduction\nTheequationsofmotionforreduceddensitymatriceshave\na coupling scheme known as the Bogoliubov-Born-Green-\nKirkwood-Yvon (BBGKY) hierarchy [1] where an n-body\ndensity matrix couples to n-body and n+1-body density\nmatrices. To solve the equations of motion for the one-\nbody and two-body density matrices, we need to truncate\nthe BBGKY hierarchy at a two-body level. The simplest\ntruncation scheme is to approximate a three-body den-\nsity matrix with the antisymmetrized products of the one-\nbody and two-body density matrices neglecting the cor-\nrelated part of the three-body density matrix (the three-\nbody correlation matrix) [2,3]. We refer to this trunca-\ntion scheme as the time-dependent density-matrix theory\n(TDDM). In some cases TDDM overestimates ground-\nstate correlations [4], gives unphysical results [5,6] and\ncausesinstabilitiesoftheobtainedsolutions[7]forstrongly\ninteracting cases. Obviously the problems originate in the\ntruncation scheme where the three-body correlation ma-\ntrix is completely neglected. This is related to the loss of\nN-representability [8] which is preserved only in the case\nof the untruncated BBGKY hierarchy. To remedy to such\ndifficulties of the naive truncation scheme, better approx-\nimations for the three-body correlation matrix have been\nproposed. One is to approximate the three-body corre-\nlation matrix with the products of the correlated part of\nthe two-body correlationmatricesbasedon a perturbative\nconsideration [9], which corresponds to the cumulant ex-\npansion [10]: The three-body correlation matrix may also\nbe called the three-body cumulant. We refer to this trun-\ncationschemeasTDDM1. Itwasfoundin theapplicationsof TDDM1 to the Lipkin model [11] that TDDM1 can\nremedy the overestimation of ground-state correlations in\nTDDM. It was also found that TDDM1 underestimates\nground-state correlations in strongly interacting regions\nwhen the number of particles Nbecomes large. This sug-\ngests that higher-order effects which effectively reduce the\ncontribution of the three-body correlation matrix should\nbe taken into account. We proposed another truncation\nscheme [12] which includes a normalization factor in the\nthree-body correlation matrix in TDDM1. We refer to it\nas TDDM2. In this work we pursue our investigation of\nthe cumulant approximation to the correlated part of the\nthree-body density matrix which enters the equation of\nmotion for the two-body density matrix. In particular we\nwill continue in this sense our investigation of the two-\nlevel Lipkin model where we will find that TDDM solves\nthis model exactly in the thermodynamic limit staying in\nthe symmetry unbroken (spherical) basis. Additionally we\nthen also study the three-level Lipkin model [13] which\nwill turn out to have a quite different behavior with re-\nspect to the three-body correlations. We will find a par-\nticular approximationscheme for the cumulant expression\nof the three-body correlation matrix which also solves the\nthree-level Lipkin model almost exactly for the ground\nstate energy in the large Nlimit staying in the symme-\ntry unbroken description. However, this scheme will not\nbe able to reproduce the rotational spectrum. A detailed\ndiscussion of this behavior will be given and the particu-\nlarities of the three-level Lipkin model in this respect will\nbe discussed. At the end, we will try to give some general\nconclusions concerning the performance of the cumulant2 Mitsuru Tohyama, Peter Schuck: Truncation scheme of time- dependent density-matrix approach III\napproximationto the three-body correlationmatrix in the\nTDDM approach.\nThe paper is organized as follows. The three trunca-\ntion schemes TDDM, TDDM1 and TDDM2 are explained\nin sect. 2. The results for the two and three-level Lipkin\nmodels are presented in sect. 4 and sect. 4 is devoted to\nsummary.\n2 Formulation\nWe consider a system of Nfermions and assume that the\nHamiltonian Hconsisting of a one-body part and a two-\nbody interaction\nH=/summationdisplay\nαα′∝an}bracketle{tα|t|α′∝an}bracketri}hta†\nαaα′+1\n2/summationdisplay\nαβα′β′∝an}bracketle{tαβ|v|α′β′∝an}bracketri}hta†\nαa†\nβaβ′aα′,\n(1)\nwherea†\nαandaαare the creation and annihilation opera-\ntors of a particle at a single-particle state α.\n2.1 Time-dependent density-matrix theories\nThe TDDM approaches give the coupled equations of mo-\ntion for the one-body density matrix (the occupation ma-\ntrix)nαα′and the correlated part of the two-body density\nmatrixCαβα′β′. These matrices are defined as\nnαα′(t) =∝an}bracketle{tΦ(t)|a†\nα′aα|Φ(t)∝an}bracketri}ht, (2)\nCαβα′β′(t) =∝an}bracketle{tΦ(t)|a†\nα′a†\nβ′aβaα|Φ(t)∝an}bracketri}ht\n−(nαα′(t)nββ′(t)−nαβ′(t)nβα′(t)),(3)\nwhere|Φ(t)∝an}bracketri}htisthetime-dependenttotalwavefunction |Φ(t)∝an}bracketri}ht=\nexp[−iHt]|Φ(t= 0)∝an}bracketri}ht. We use units such that ¯ h= 1. The\nequations of motion for nαα′andCαβα′β′are written as\ni˙nαα′=/summationdisplay\nλ(ǫαλnλα′−nαλǫλα′)\n+/summationdisplay\nλ1λ2λ3[∝an}bracketle{tαλ1|v|λ2λ3∝an}bracketri}htCλ2λ3α′λ1\n−Cαλ1λ2λ3∝an}bracketle{tλ2λ3|v|α′λ1∝an}bracketri}ht], (4)\ni˙Cαβα′β′=/summationdisplay\nλ(ǫαλCλβα′β′+ǫβλCαλα′β′\n−ǫλα′Cαβλβ′−ǫλβ′Cαβα′λ)\n+Bαβα′β′+Pαβα′β′+Hαβα′β′\n+/summationdisplay\nλ1λ2λ3[∝an}bracketle{tαλ1|v|λ2λ3∝an}bracketri}htCλ2λ3βα′λ1β′\n+∝an}bracketle{tλ1β|v|λ2λ3∝an}bracketri}htCλ2λ3αα′λ1β′\n− ∝an}bracketle{tλ1λ2|v|α′λ3∝an}bracketri}htCαλ3βλ1λ2β′\n− ∝an}bracketle{tλ1λ2|v|λ3β′∝an}bracketri}htCαλ3βλ1λ2α′], (5)whereCαβγα′β′γ′isthethree-bodycorrelationmatrixwhich\nisneglected inthe originalversionofTDDM [2,3]. The en-\nergy matrix ǫαα′is given by\nǫαα′=∝an}bracketle{tα|t|α′∝an}bracketri}ht+/summationdisplay\nλ1λ2∝an}bracketle{tαλ1|v|α′λ2∝an}bracketri}htAnλ2λ1,(6)\nwhere the subscript Ameans that the corresponding ma-\ntrix is antisymmetrized. The matrix Bαβα′β′in eq. (5)\ndoes not contain Cαβα′β′and describes 2p–2h and 2h–\n2p excitations, where p and h refer to particle and hole\nstates, respectively. The terms Pαβα′β′andHαβα′β′con-\ntainCαβα′β′and describe p–p (and h–h) and p–h corre-\nlations to infinite order, respectively [3]. These matrices\nare given in ref. [3] but listed again in Appendix A for\ncompleteness. Equations (4) and (5) satisfy the conser-\nvation laws of the total energy and the total number of\nparticles [2,3]. As mentioned above, Cαβγα′β′γ′in eq. (5)\nis neglected in TDDM [2,3]. In order to remedy difficul-\nties of TDDM, we have proposed a truncation scheme for\nCαβγα′β′γ′[9] such that\nCp1p2h1p3p4h2=/summationdisplay\nhChh1p3p4Cp1p2h2h,(7)\nCp1h1h2p2h3h4=/summationdisplay\npCh1h2p2pCp1ph3h4.(8)\nThese expressions were derived from perturbative con-\nsideration [9] using the following CCD (Coupled-Cluster-\nDoubles)-like ground state wavefunction |Z∝an}bracketri}ht[14]\n|Z∝an}bracketri}ht=eZ|HF∝an}bracketri}ht (9)\nwith\nZ=1\n4/summationdisplay\npp′hh′zpp′hh′a†\npa†\np′ah′ah, (10)\nwhere|HF∝an}bracketri}htis the Hartree-Fock (HF) ground state and\nzpp′hh′is antisymmetric under the exchanges of p ↔p′\nand h↔h′. We refer to the truncation scheme given\nby eqs. (7) and (8) as TDDM1. In the applications of\nTDDM1 to the Lipkin model it was found that although\nTDDM1 effectively suppresses excess two-body correla-\ntions in small Nsystems, it overestimates the contribu-\ntions of the three-body correlation matrix in large Nsys-\ntems. To make TDDM1 applicable to large Nsystems, we\nmodified the truncation scheme [12] to\nCp1p2h1p3p4h2=1\nN/summationdisplay\nhChh1p3p4Cp1p2h2h,(11)\nCp1h1h2p2h3h4=1\nN/summationdisplay\npCh1h2p2pCp1ph3h4.(12)\nThe factor Nis given by\nN= 1+1\n4/summationdisplay\npp′hh′Cpp′hh′Chh′pp′, (13)Mitsuru Tohyama, Peter Schuck: Truncation scheme of time-d ependent density-matrix approach III 3\nwhich plays a role in reducing the three-body correlation\nmatrix in stronglyinteractingregionsofthe Lipkin model.\nWe refer to the truncation scheme given by eqs. (11) and\n(12) as TDDM2. The conservation of the total energy and\ntotal particle number is not affected by the above approx-\nimations for the three-body correlation matrix because its\nanti-symmetry properties under the exchange of single-\nparticle indices is respected.\n3 Results\n3.1 Two-level Lipkin model\nFirst we consider the usual two-level Lipkin model. The\nLipkin model [11] describes an N-fermions system with\ntwoN-fold degenerate levels with energies ǫ/2 and−ǫ/2,\nrespectively. The upper and lower levels are labeled by\nquantumnumber pand−p,respectively,with p= 1,2,...,N.\nWe consider the standard Hamiltonian\nH=ǫJz+V\n2(J2\n++J2\n−), (14)\nwhere the operators are given as\nJz=1\n2N/summationdisplay\np=1(c†\npcp−c−p†c−p), (15)\nJ+=J†\n−=N/summationdisplay\np=1c†\npc−p. (16)\nTheHFgroundstatewherethelowestsingle-particlestates\ngiven by the operators {a−p}(p= 1,2,· · ·N) are fully\noccupied\n|HF∝an}bracketri}ht=N/productdisplay\np=1a†\n−p|0∝an}bracketri}ht (17)\nis obtained by the transformation\n/parenleftbigg\na†\n−p\na†\np/parenrightbigg\n=/parenleftbigg\ncosαsinα\n−sinαcosα/parenrightbigg/parenleftbigg\nc†\n−p\nc†\np/parenrightbigg\n.\n(18)\nThe HF energy is given by\nE(α) =−Nǫ\n2+Nǫsin2α\n+VN(N−1)sin2αcos2α. (19)\nUsingχ=|V|(N−1)/ǫ, we can express EHFwhich mini-\nmizesE(α) as\nEHF=/braceleftbigg\n−Nǫ\n2(χ≤1)\nNǫ\n4(−χ−1/χ) (χ >1),(20)\nwhere cos2 α= 1/χis satisfied.SincethesolutioninTDDMiscloselyrelatedtothatin\nthe deformed Hartree-Focktheory(DHF) as shownbelow,\nwe discuss some properties of the DHF solution. In DHF\nthe occupation matrix is given by\nn−p(n−p−p) = cos2α, (21)\nnp(npp) = 1−n−p= sin2α, (22)\nnp−p= cosαsinα. (23)\nIn DHF higher reduced density matrices have no corre-\nlated parts. For example the 2p–2h element of the two-\nbody density matrix is expressed with the occupation ma-\ntrices as\nρpp′−p−p′=∝an}bracketle{tDHF|c†\n−pc†\n−p′cp′cp|DHF∝an}bracketri}ht\n=np−pnp′−p′= cos2αsin2α.(24)\nSimilarly, the ph–ph element is given by\nρp−p′−pp′(p∝ne}ationslash=p′) =∝an}bracketle{tDHF|c†\n−pc†\np′c−p′cp|DHF∝an}bracketri}ht\n=np−pn−p′p′= cos2αsin2α.(25)\nThe three-body density matrix is also expressedas the an-\ntisymmetrized products of three occupation matrices such\nthat\nρp−p′p′′pp′−p′′= cos2αsin4α, (26)\nwhich is the same as nppρ−p′p′′p′−p′′. This means that the\ncorrelatedpartofthethree-bodydensitymatrix Cp−p′p′′pp′−p′′\nvanishes in DHF.\nIn TDDM (also in TDDM1 and TDDM2) the ground\nstate is obtained using the adiabatic method [18,19] start-\ning from the non-interacting ”spherical” HF state where\nthe occupation matrix nαα′has no off-diagonal elements.\nSince the TDDM equations eqs. (4) and (5) maintain the\nsymmetry, nαα′always has no off-diagonal elements. Sim-\nilarly,ρpp′−p−p′andρp−p′−pp′(p∝ne}ationslash=p′) have no uncorre-\nlated parts in TDDM. As shown below, there are strong\nsimilarities between the TDDM and DHF solutions such\nthatCpp′−p−p′andCp−p′−pp′correspond to eqs. (24) and\n(25), respectively. The diagonal elements eqs. (21) and\n(22) in DHF also correspond to those in TDDM. This\nsuggests the following relation for the expectation value\nof an operator ˆQ\n∝an}bracketle{tˆQ∝an}bracketri}htTDDM≈ ∝an}bracketle{tΨDHF|ˆQ|ΨDHF∝an}bracketri}ht\n≈1\n2(∝an}bracketle{tDHF(α)|ˆQ|DHF(α)∝an}bracketri}ht\n+∝an}bracketle{tDHF(−α)|ˆQ|DHF(−α)∝an}bracketri}ht),(27)\nwhere|ΨDHF∝an}bracketri}htis a ’spherical’ wavefunction given by\n|ΨDHF∝an}bracketri}ht=|DHF(α)∝an}bracketri}ht+|DHF(−α)∝an}bracketri}ht√\n2.(28)\nHere|DHF(α)∝an}bracketri}htmeans the deformed HF ground state with\nα. In fact the following relations hold\n∝an}bracketle{tDHF(α)|DHF(−α)∝an}bracketri}ht= (cos2α−sin2α)N4 Mitsuru Tohyama, Peter Schuck: Truncation scheme of time- dependent density-matrix approach III\n/s48 /s49 /s50 /s51 /s52 /s53/s45/s49/s56/s45/s49/s54/s45/s49/s52/s45/s49/s50/s45/s49/s48/s45/s56/s45/s54/s69/s48/s47/s32/s84/s68/s68/s77\n/s32/s84/s68/s68/s77/s50\n/s32/s72/s70/s44/s32/s68/s72/s70\n/s32/s69/s120/s97/s99/s116\n/s32/s84/s68/s68/s77/s49\n/s32/s68/s84/s68/s68/s77/s78/s61/s49/s50\nFig. 1.Ground-state energy in TDDM (solid line) as a func-\ntion ofχ=|V|(N−1)/ǫforN= 12. The results in TDDM1\nwhere the three-body correlation matrix is given by eqs. (7)\nand (8) are shown with the dashed line. The green (gray) line\ndepicts the results in TDDM2 where the three-body correla-\ntion matrix is given by eqs. (11) and (12). The results in HF\nand DHF( χ >1) are shown with the dotted line. The open\ncircles (DTDDM) indicate the results in TDDM calculated us-\ning the ’deformed’ single-particle states and the DHF groun d\nstate. The exact values are given by the dot-dashed line.\n(29)\n∝an}bracketle{tDHF(α)|c†\nαcβ|DHF(−α)∝an}bracketri}ht ∝(cos2α−sin2α)N−1\n(30)\n∝an}bracketle{tDHF(α)|c†\nαc†\nβcβ′cα′|DHF(−α)∝an}bracketri}ht ∝(cos2α−sin2α)N−2.\n(31)\nSince 0<(cos2α−sin2α)<1, the above expectation\nvalues between |DHF(α)∝an}bracketri}htand|DHF(−α)∝an}bracketri}htbecome quite\nsmall for large N, justifying eq. (27).\nTo illustrate how TDDM, TDDM1 and TDDM2 be-\nhave in small Nregions, we first present the results for\nN= 12. The ground-stateenergy E0calculated in TDDM\n(solid line) is shown in fig. 1 as a function of χ. The re-\nsults in TDDM1 where the three-body correlation matrix\nis givenby eqs.(7) and (8) areshownwith the dashedline.\nThe results in TDDM2 where the three-body correlation\nmatrix is given by eqs. (11) and (12) are shown with the\ngreen (gray) line. The results in HF and DHF ( χ >1) and\ntheexactvaluesareshownwiththedottedanddot-dashed\nlines, respectively. The open circles indicate the results in\nTDDM calculated using the ’deformed’ (symmetry bro-\nken) single-particle states and the DHF ground state as\nthe starting ground state. We refer to this scheme as the\ndeformed TDDM (DTDDM). When the ’deformed’ basis\nis used, the two-body correlation matrix is so small that\nboth DTDDM and the deformed TDDM1 give similar re-\nsults. The exact values are given by the dot-dashed line.\nIn this small Nsystem TDDM overestimates two-body\ncorrelations [9,12] and TDDM1 underestimates them in/s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s78/s61/s49/s50/s110/s112\n/s32/s84/s68/s68/s77\n/s32/s84/s68/s68/s77/s50\n/s32/s72/s70/s44/s32/s68/s72/s70\n/s32/s69/s120/s97/s99/s116\n/s32/s84/s68/s68/s77/s49\n/s32/s68/s84/s68/s68/s77\nFig. 2.Same as fig. 1 but for the occupation probability npof\nthe upper state.\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s78/s61/s49/s50/s67/s112/s112/s39/s45/s112/s45/s112/s39\n/s32/s84/s68/s68/s77\n/s32/s84/s68/s68/s77/s50\n/s32/s72/s70/s44/s32/s68/s72/s70\n/s32/s69/s120/s97/s99/s116\n/s32/s84/s68/s68/s77/s49\n/s32/s68/s84/s68/s68/s77\nFig. 3. Same as fig. 1 but for the 2p–2h element Cpp′−p−p′\nof the two-body correlation matrix. The results in DHF show\nn2\np−p.\nstrongly interacting regions. TDDM2 cures this problem\nand the agreement with the exact solutions is much im-\nproved. DTDDM also provides the additional correlation\nenergy missing in DHF. The occupation probability npof\nthe upper state and the 2p–2h element Cpp′−p−p′of the\ncorrelation matrix calculated in TDDM are shown in figs.\n2 and 3, respectively. In DHF n2\np−pis shown as a quan-\ntitycorrespondingto Cpp′−p−p′. TDDM1 andTDDM2 de-\nscribe well E0,npandCpp′−p−p′in the transition region\nχ≈1, where DHF fails to reproduce the exact values.\nFigures 2 and 3 indicate that the improvement in E0from\nTDDM to TDDM2 is due to the appropriate reduction\nofCpp′−p−p′. Figures 1–3 also show that DHF becomes\ngood approximation with increasing interaction strength\nand that the improvement in E0from DHF to DTDDM is\nbrought by that in np. The deviation of npandCpp′−p−p′Mitsuru Tohyama, Peter Schuck: Truncation scheme of time-d ependent density-matrix approach III 5\n/s48 /s49 /s50 /s51 /s52 /s53/s45/s55/s48/s45/s54/s48/s45/s53/s48/s45/s52/s48/s45/s51/s48/s45/s50/s48/s69/s48/s47/s32/s84/s68/s68/s77\n/s32/s72/s70/s44/s32/s68/s72/s70\n/s32/s69/s120/s97/s99/s116\n/s32/s84/s68/s68/s77/s49/s78/s61/s53/s48\nFig. 4.Ground-state energy in TDDM (solid line) as a func-\ntion ofχforN= 50. The exact values are given by the dot-\ndashed line. The dashed line depicts the results in TDDM1.\nThe results in TDDM2 lie between the TDDM results and the\nexact values and are not shown here. The results in HF and\nDHF(χ >1) are shown with the dotted line but cannot be\ndistinguished from the exact values in the scale of the figure\nexcept for the region χ≈1.\n/s48 /s49 /s50 /s51/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s78/s61/s53/s48/s110/s112/s32/s84/s68/s68/s77\n/s32/s72/s70/s44/s32/s68/s72/s70\n/s32/s69/s120/s97/s99/s116\nFig. 5.Same as fig. 4 but for the occupation probability npof\nthe upper state.\nin DTDDM at χ= 1.5isdue to the factthat the deformed\nstate becomes unstable in DTDDM near χ= 1.\nNow we present the TDDM results for N= 50. The\nground-stateenergy E0calculated in TDDM (solid line) is\nshown in fig. 4 as a function of χ. The results in TDDM1\nare given by the dashed line. The dotted and dot-dashed\nlines depict the results in HF and DHF ( χ >1) and the\nexact values, respectively. The results in HF and DHF\ncannot be distinguished from the exact values in the scale\nof the figure except for the region χ≈1. The results\nin TDDM2 lie between the TDDM results and the exact\nvalues and are not shown here. In this large Nsystem,/s48 /s49 /s50 /s51/s48/s46/s48/s48/s46/s49/s48/s46/s50/s78/s61/s53/s48/s67/s112/s112/s39/s45/s112/s45/s112/s39 /s32/s84/s68/s68/s77\n/s32/s72/s70/s44/s32/s68/s72/s70\n/s32/s69/s120/s97/s99/s116\nFig. 6. Same as fig. 4 but for the 2p–2h element Cpp′−p−p′\nof the two-body correlation matrix. The results in DHF show\nn2\np−p.\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s40/s69/s49/s45/s69/s48/s41/s47/s32/s84/s68/s68/s77\n/s32/s82/s80/s65\n/s32/s69/s120/s97/s99/s116/s78/s61/s50/s48/s48\nFig. 7.Excitation energy of the first excited state calculated\nin TDDM (dots) a function of χforN= 200. The exact values\nare shown with the dot-dashedline. The dotted line depicts t he\nresults in RPA.\nthe ground-state energies in TDDM and TDDM2 agree\nwell with the exact solutions and the DHF energies also\nbecome close to the exact values including the transition\nregion. This is also seen in npandCpp′−p−p′shown in\nfigs. 5 and 6, respectively. The factor Nin eqs. (11) and\n(12) plays a role in drastically reducing the three-body\ncorrelation matrix, which makes TDDM2 almost equiva-\nlentto TDDM [12]. In DHF the three-bodydensitymatrix\nhas no correlated parts. The fact that DHF becomes good\napproximation for the ground state of the Lipkin model\nwith increasing number of particles explains that TDDM\nwhich has no three-body correlationmatrix becomes good\napproximation for large N. We checked that TDDM gives\nthe almost exact ground state for much larger N.6 Mitsuru Tohyama, Peter Schuck: Truncation scheme of time- dependent density-matrix approach III\n/s48 /s49 /s50 /s51/s48/s49/s50/s51/s52/s40/s69/s50/s45/s69/s48/s41/s47/s78/s61/s50/s48/s48\n/s32/s69/s120/s97/s99/s116\n/s32/s84/s68/s68/s77\n/s32/s82/s80/s65\nFig. 8. Same as fig. 7 but for the second excited state. The\ndotted line depicts the RPA results for the first excited stat e.\nFinally we present the excitation energies of the first\nand second excited states calculated in TDDM for N=\n200 where TDDM gives the almost exact ground-state.\nThe energy of the first excited state is calculated using\nthe TDDM equations eqs. (4) and (5) and assuming np−p,\nCpp′−pp′andC−p−p′−pp′are small. The second excited\nstate which has the same symmetry as the ground state is\ncalculated in a different manner. In the adiabatic method\nusedforthe ground-statecalculationsthereisasmallmix-\ning of excited states which have the same symmetry as the\nground state. The excitation energy of the second excited\nstate can be estimated from the frequency of the small\noscillation of nppin the adiabatic method. The obtained\nresultsforthe firstandsecondexcitedstatesarecompared\nwith the exact solutions (dot-dashed line) in figs. 7 and 8,\nrespectively. The dots in figs. 7 and 8 depict the TDDM\nresults. The results in RPA for the first excited state are\nshown in figs. 7 and 8 with the dotted line. In contrast to\nRPA TDDM describes the decreasing excitation energy of\nthe first excited state with increasing interaction strength\nbeyondχ= 1. For χ >1.073 TDDM does not give an os-\ncillating solution to the first excited state. This indicates\nthat the three-body correlation matrix plays some role in\nthe first excited state. With increasing χthe RPA results\nforthe firstexcitedstateapproachtheexactresultsforthe\nsecond excited states as shown in fig. 8. The agreement of\nthe TDDM results with the exact values is good for both\nthe first and second excited states. This is related to the\nfact that TDDM becomes exact with increasing number\nof particles.\nIn this subsection we found out that in the two-level\nLipkin model the three-body correlation matrix vanishes\nin the large N(thermodynamic) limit and that TDDM\nsolves the ground state of this model exactly staying in\nthe symmetry unbroken description. How far our findings\ncan be transposed to other models of this type of a one\nsite spin model is an open question but our studies mayhelp to analyze the situation also in other cases. We now\nwill turn to the case of the three-level Lipkin model.\n3.2 Three-level Lipkin model\nNext we consider the three-level Lipkin model [13] with\nthe single-particlelevels labeled 0, 1 and 2. Since the num-\nberofunoccupiedstatesislargerthanthatoftheoccupied\nstates, the three-level Lipkin model may be more realistic\nthanthe two-levelLipkin modelandit hasoftenbeen used\nto test extended mean-field theories [15,16,17]. We choose\nthe Hamiltonian which is invariant under the exchange of\n1 and 2:\nH=ǫ(ˆn1+ ˆn2)+V\n2/parenleftBig\nK2\n1+K2\n2+(K†\n1)2+(K†\n2)2/parenrightBig\n,\n(32)\nwhere\nˆnα=N/summationdisplay\ni=1c†\nαicαi α= 0,1,2,(33)\nKα=N/summationdisplay\ni=1c†\nαic0iα= 1,2.(34)\nTheHFgroundstatewherethelowestsingle-particlestates\ngiven by the operators {a0i}(i= 1,2,···N) are fully oc-\ncupied\n|HF∝an}bracketri}ht=N/productdisplay\ni=1a†\n0i|0∝an}bracketri}ht (35)\nis obtained by the transformation\n\na†\n0i\na†\n1i\na†\n2i\n=\ncosαcosβsinαsinβsinα\n−sinαcosβcosαsinβcosα\n0−sinβcosβ\n\nc†\n0i\nc†\n1i\nc†\n2i\n.\n(36)\nThe HF energy is independent of βand given by\nE(α,β) =Nǫsin2α+VN(N−1)sin2αcos2α.(37)\nUsingχ=|V|(N−1)/ǫ, we can express EHFwhich mini-\nmizesE(α,β) as\nEHF=/braceleftbigg0 (χ≤1)\nNǫ\n4(2−χ−1/χ) (χ >1).(38)\nThe ground-state wavefunction which has the symmetry\nunder the exchange of 1 and 2 may be written as\n|Ψ∝an}bracketri}ht=1\nπ/integraldisplayπ/2\n−π/2|DHF(α,β)∝an}bracketri}htdβ, (39)\nwhere|DHF(α,β)∝an}bracketri}htis the DHF ground state with αand\nβ. In the following we show that eq. (39) gives a non-\nvanishing three-body correlation matrix. To evaluate anMitsuru Tohyama, Peter Schuck: Truncation scheme of time-d ependent density-matrix approach III 7\nexpectation value of an operator ˆQusing eq. (39), we need\nthe overlap ∝an}bracketle{tDHF(α,β′)|ˆQ|DHF(α,β)∝an}bracketri}ht. We showsome ex-\namplesoftheoverlapsusing X= cos2α+cos(β′−β)sin2α\n[15]:\n∝an}bracketle{tDHF (α,β′)|DHF(α,β)∝an}bracketri}ht=XN, (40)\n∝an}bracketle{tDHF (α,β′)|c†\n11c11|DHF(α,β)∝an}bracketri}ht\n= cosβ′cosβsin2αX(N−1), (41)\n∝an}bracketle{tDHF (α,β′)|c†\n01c†\n02c12c11|DHF(α,β)∝an}bracketri}ht\n= cos2βcos2αsin2αX(N−2), (42)\n∝an}bracketle{tDHF (α,β′)|c†\n01c†\n12c02c11|DHF(α,β)∝an}bracketri}ht\n= cosβ′cosβcos2αsin2αX(N−2),(43)\n∝an}bracketle{tDHF (α,β′)|c†\n11c†\n12c†\n03c13c02c11|DHF(α,β)∝an}bracketri}ht\n= cos2β′cos2βcos2αsin4αX(N−3),(44)\nwhere 01 ,11,···mean the quantum number ( α,i) for the\nsingle-particle state given in eq. (33). All components of\nthe density matrices do not depend on the quantum num-\nberi.SinceXNforlarge Nissharplypeakedat β′−β≈0,\nthe diagonal assumption β′=βis justified for the over-\nlaps. Then the density matrices obtained with eq. (39) are\ngiven as follows after the βintegration (we choose mag-\nneticquantumnumberssothattheexpressionsaregeneral\nbut in the end there will, of course, be no dependence on\nmagnetic states):\nnpp=∝an}bracketle{tΨ|c†\n11c11|Ψ∝an}bracketri}ht=1\n2sin2α,(45)\nCpp′hh′=∝an}bracketle{tΨ|c†\n01c†\n02c12c11|Ψ∝an}bracketri}ht\n=1\n2cos2αsin2α, (46)\nCph′hp′=∝an}bracketle{tΨ|c†01c†12c02c11|Ψ∝an}bracketri}ht\n=1\n2cos2αsin2α, (47)\nρph′p′′pp′h′′=∝an}bracketle{tΨ|c†\n11c†\n12c†\n03c13c02c11|Ψ∝an}bracketri}ht\n=3\n8cos2αsin4α. (48)\nSinceρph′p′′pp′h′′contains four particle-state indices, the\nβintegration makes it impossible to express ρph′p′′pp′h′′\nby the product of nppandCph′hp′which respectively have\ntwo particle-state indices. From the above expressions we\nobtain the correlated part of the three-body density ma-\ntrix as\nCph′p′′pp′h′′=C11,02,13,11,12,03\n=∝an}bracketle{tΨ|c†\n11c†\n12c†\n03c13c02c11|Ψ∝an}bracketri}ht\n− ∝an}bracketle{tΨ|c†\n11c11|Ψ∝an}bracketri}ht∝an}bracketle{tΨ|c†\n12c†\n03c13c02|Ψ∝an}bracketri}ht\n=1\n8cos2αsin4α. (49)\nSimilarly,thethree-bodycorrelationmatrix Cph′h′′p′hh′′=\nC11,02,03,12,01,03corresponding to eq. (8) is evaluated us-\ning the DHF wavefunction eq. (39) and it is found that/s48 /s49 /s50 /s51 /s52 /s53/s45/s50/s53/s45/s50/s48/s45/s49/s53/s45/s49/s48/s45/s53/s48/s69/s48/s47/s32/s84/s68/s68/s77/s49/s45/s98\n/s32/s69/s120/s97/s99/s116\n/s32/s68/s72/s70\n/s32/s84/s68/s68/s77\n/s32/s84/s68/s68/s77/s50\n/s32/s84/s68/s68/s77/s49/s78/s61/s50/s48\nFig. 9. Ground-state energy of the three-level Lipkin model\ncalculated in TDDM1-b (solid line) as a function of χforN=\n20. The dashed, dot-dashed and two-dot-dashed lines depict\nthe results in TDDM, TDDM1 and TDDM2, respectively. The\nresults in HF and DHF( χ >1) are shown with the dotted line.\nThe exact values are given by the dots.\nC11,02,03,12,01,03= 0 as is the case of the two-level Lip-\nkin model. This is because ρph′h′′p′hh′′contains only two\nparticle-state indices, which allows us to express it as\nnhhCph′hp′. On the other hand, the three-body correlation\nmatrix of the form Cpp′p′′hp′h′′, which enters the equation\nofmotion for Cph′p′h, is also non-vanishing because it con-\ntains four particle-state indices. Using eq. (39) it is eval-\nuated to be the same as eq. (49) that is (cos2αsin4α/8).\nThe above analysis indicates that the three-body correla-\ntion matrix in the strongly interacting region ( χ >1) of\nthe three-level Lipkin model is quite different from that\nof the two-level Lipkin model and that it should be prop-\nerly treated. However,extending the truncation scheme to\ninclude the equation of motion for the three-body corre-\nlation matrix is a difficult task and is not pursued in this\nwork. In the following we just point out that there exists\na simple truncation scheme for the three-body correlation\nmatrix which gives a good description of the ground state\nfor the whole range of the interaction strength. In this ap-\nproach the quadratic approximation for the pph–pph type\nthree-body correlation matrix given by eq. (7) is used and\nthe phh–phh type given by eq. (8) is omitted consider-\ning the above analysis based on the DHF wavefunction.\nWe refer to this truncation scheme as TDDM1-b. We first\npresent the results in TDDM1-b and then discuss why\nTDDM1-b works.\nTheground-stateenergycalculatedinTDDM1-b(solid\nline)isshowninfigs.9and10asafunctionof χforN= 20\nand 200, respectively. The exact values are given by the\ndots. The dashed and dot-dashed lines depict the results\nin TDDM and TDDM1, respectively. The results in HF\nand DHF( χ >1) are shown with the dotted line but in\nthe case of N= 200 they cannot be easily distinguished\nfrom the TDDM1-b results in the scale of the figure. The8 Mitsuru Tohyama, Peter Schuck: Truncation scheme of time- dependent density-matrix approach III\n/s48 /s49 /s50 /s51 /s52 /s53/s45/s50/s53/s48/s45/s50/s48/s48/s45/s49/s53/s48/s45/s49/s48/s48/s45/s53/s48/s48/s69/s48/s47/s32/s72/s70/s44/s32/s68/s72/s70\n/s32/s84/s68/s68/s77/s49/s45/s98\n/s32/s69/s120/s97/s99/s116\n/s32/s84/s68/s68/s77\n/s32/s84/s68/s68/s77/s49/s78/s61/s50/s48/s48\nFig. 10. Ground-state energy of the three-level Lipkin model\ncalculated in TDDM1-b (solid line) as a function of χforN=\n200. The exact values are given by the dots. The dashed and\ndot-dashed lines depict the results in TDDM and TDDM1,\nrespectively. The results in HF and DHF( χ >1) are shown\nwith the dotted line but cannot be easily distinguished from\nthe TDDM1-b results in the scale of the figure.\n/s48 /s49 /s50 /s51/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53/s48/s46/s50/s48/s110/s49/s78/s61/s50/s48\n/s32/s84/s68/s68/s77/s49\n/s32/s84/s68/s68/s77/s49/s45/s98\n/s32/s69/s120/s97/s99/s116\n/s32/s72/s70/s44/s32/s68/s72/s70\nFig. 11. Occupationprobabilityofthestate1ofthethree-level\nLipkin model calculated in TDDM1-b (solid line) as a functio n\nofχforN= 20. The dot-dashed line depicts the results in\nTDDM1. The results in HF and DHF( χ >1) given by eq. (45)\nare shown with the dotted line. The exact values are given by\nthe dots.\nresults in TDDM2 are given with the two-dot-dashed line\nin fig. 9. In the case of N= 200 they are close to those\nin TDDM as in the case of the two-level Lipkin model\nand are not shown in fig. 10. In contrast to the case of\nthe two-level Lipkin model TDDM strongly overestimates\ntwo-bodycorrelationsin the three-levelLipkin model even\nforN= 200. This is due to the fact that the three-body\ncorrelation matrix cannot be neglected in the three-level\nLipkin model as discussed above. DHF gives a good de-/s48 /s49 /s50 /s51/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53 /s78/s61/s50/s48/s67/s112/s112/s39/s104/s104/s39\n/s32/s84/s68/s68/s77/s49\n/s32/s84/s68/s68/s77/s49/s45/s98\n/s32/s69/s120/s97/s99/s116\n/s32/s72/s70/s44/s32/s68/s72/s70\nFig. 12. Same as fig. 11 but for Cpp′hh′. The results in HF\nand DHF are given by eq. (46).\n/s48 /s49 /s50 /s51/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53/s48/s46/s50/s48/s110/s49/s32/s84/s68/s68/s77/s49/s45/s98\n/s32/s72/s70/s44/s32/s68/s72/s70\n/s32/s69/s120/s97/s99/s116\n/s32/s84/s68/s68/s77/s49/s78/s61/s50/s48/s48\nFig. 13. Same as fig. 11 but for N= 200.\n/s48 /s49 /s50 /s51/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53/s67/s112/s112/s39/s104/s104/s39/s78/s61/s50/s48/s48\n/s32/s84/s68/s68/s77/s49/s45/s98\n/s32/s72/s70/s44/s32/s68/s72/s70\n/s32/s69/s120/s97/s99/s116\n/s32/s84/s68/s68/s77/s49\nFig. 14. Same as fig. 12 but for N= 200.Mitsuru Tohyama, Peter Schuck: Truncation scheme of time-d ependent density-matrix approach III 9\nscription of the ground-state energy for N= 200 as in the\ncaseofthetwo-levelLipkinmodel.TheresultsinTDDM1-\nb agree well with the exact solutions both in N= 20\nand 200. TDDM1 underestimates the correlation effects\nfor large χbecause the phh–phh component of the three-\nbody correlation matrix given by eq. (8) is non-vanishing.\nThe occupation probability n1of the single-particle level\nlabeled 1 and the 2p–2h element of the two-body corre-\nlation matrix calculated in TDDM1-b are also in good\nagreement with the exact values as shown in figs. 11–14.\nWe have checked that the agreement of the TDDM1-b re-\nsults with the exact solutions extends at least to χ= 10\nin the case of N= 200. The results in TDDM1 are not so\ngoodasthoseinTDDM1-bbuttheagreementwiththeex-\nact values are reasonable in the transition region ( χ≈1).\nIn the case of the two-level Lipkin model the ground state\nenergies calculated in TDDM1-b approximately come in\nthe middle between the results in TDDM and TDDM1.\nIn the following we try to explain why TDDM1-b gives\ngood results for the three-level Lipkin model. In the large\nNcase of the three-level Lipkin model (and also the two-\nlevel Lipkin model) the coupling of Cpp′hh′andChh′pp′\ntoCph′p′hdescribed by the Hαβα′β′term in eq. (5) is\ndominant (see eq. (53)). The three-body correlation ma-\ntrix of the form Cpp′p′′hp′h′′, which enters the equation of\nmotion for Cph′p′h, is neglected in TDDM1-b(and also in\nTDDM1). This is because such an element of the three-\nbody correlation matrix is of higher order than eq. (7) in\nthe perturbative regime. In the strongly interactingregion\nof the three-level Lipkin model Cpp′p′′hp′h′′becomes non-\nvanishing as discussed above. As a consequence of the ne-\nglect ofCpp′p′′hp′h′′,Cph′p′his overestimated in TDDM1-\nb. IfCpp′p′′hp′h′′is assumed to be np′p′Cpp′′hh′′/2 using\neqs. (45) and (46), the overestimationin eq. (5) for Cph′p′h\ncan be evaluated to be n1Cpp′hh′/(n0−2n1), where n0is\nthe occupation probability of the single-particle level la-\nbeled 0. It is impossible to analyticallycalculateits contri-\nbution to Cph′p′hbut comparison of Cph′p′hin TDDM1-b\nand that in DHF which is given by eq. (47) shows that\nthe deviation of Cph′p′his well expressed as\nC2\npp′hh′\n2(n0−n1). (50)\nThe quadratic form is understandable because n1is deter-\nmined by Cpp′hh′in eq. (4). In the equation of motion for\nCpp′hh′in TDDM1-b one ofthe four termsin eq. (5) which\ninclude the three-body correlationmatrix ( C3) cancels the\noverestimated part of Cph′p′h: eq. (50) is multiplied by\n−2(n0−n1) due to the occupation factors in eq. (53).\nConsequently, the three remaining C3terms in TDDM1-b\nplay a role as the effective C3terms. In fig. 15 the three\nquarters of eq. (7) calculated in TDDM1-b (solid line) are\ncompared with the exact values of C3(dots) and the DHF\nvalues (dotted line) given by eq. (49). The TDDM1-b val-\nues well simulates the χdependence of the exact C3. This\nis the reasonbehind why the simple truncation scheme eq.\n(7) gives good results.\nOur study of the three-level Lipkin model has shown\nthat contrary to the two-level case the three-body corre-/s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s48/s48/s46/s48/s48/s53/s48/s46/s48/s49/s48/s48/s46/s48/s49/s53/s67/s51/s78/s61/s50/s48/s48\n/s32/s68/s72/s70\n/s32/s84/s68/s68/s77/s49/s45/s98/s40/s51/s47/s52/s41\n/s32/s69/s120/s97/s99/s116\nFig. 15. Three-body correlation matrix C11,02,13,11,12,03(C3)\nused in TDDM1-b (solid line) multiplied by a factor 3 /4 as a\nfunction of χforN= 200. The results in HF and DHF( χ >1)\nare shown with the dotted line. The exact values are given by\nthe dots.\nlation matrix cannot be neglected in the large N(thermo-\ndynamic) limit. Working in the symmetry unbroken basis,\nthe cumulant approximation to the correlated part of the\nthree body density matrix works very well for the ground\nstateenergyifonlythe pph–pph termis retained.We gave\narguments why the phh–phh particle contribution must\nbe discarded. However, the validity of TDDM1-b remains\nrestricted to the particularities of the model and conclu-\nsions for the general case cannot be made. On the other\nhand, staying in the symmetry unbroken description, one\ncannot describe the rotational spectrum which emerges\nin the three-level Lipkin model [13] if the two upper lev-\nels become degenerate as is the case of eq. (32). For such\ncases with a spontaneously broken symmetry it is in gen-\neral better to start with a symmetry broken basis and to\nrecover good symmetry with projection techniques. DT-\nDDM which consists of the simplest truncation scheme\nforC3and the symmetry broken single-particle basis may\nbe applicable for such cases. In the past we applied our\ncumulant approximation eqs. (7) and (8) to the more re-\nalistic case of16O with promising results [20]. In these\nexamples the symmetry unbroken (spherical) description\nis certainly the choice to be used.\nThe excited states of the three-level Lipkin model can\nbe calculated in the same manners as those used for the\ntwo-level Lipkin model. The results obtained from the\nTDDM1-b based approaches are shown in figs. 16 and 17.\nThe exact eigenstates are labeled by the quantum num-\nber∝an}bracketle{tL2\n0∝an}bracketri}ht= 0,1,2,· · ·whereL0=i(K12−K21) [17]\nwithK12(=K†\n21) =/summationtext\nic†\n1ic2i, and the ground state has\n∝an}bracketle{tL2\n0∝an}bracketri}ht= 0. In fig. 17 the exact values for the lowest ex-\ncited state with the same quantum number as the ground\nstate are shown. Both the first and second excited states\nin TDDM1-b have χdependence which is similar to the\ncase of the two-level Lipkin model. The first excited state10 Mitsuru Tohyama, Peter Schuck: Truncation scheme of time -dependent density-matrix approach III\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s69/s49/s47/s78/s61/s50/s48/s48\n/s32/s69/s120/s97/s99/s116\n/s32/s84/s68/s68/s77/s49/s45/s98\n/s32/s82/s80/s65\nFig. 16. Excitation energy of the first excited state in\nTDDM1-b (open squares) as a function of χforN= 200.\nThe results in RPA are shown with the dotted line and the\nexact values are given by the dots.\n/s48 /s49 /s50 /s51/s48/s49/s50/s51/s52/s69/s50/s47/s78/s61/s50/s48/s48\n/s32/s69/s120/s97/s99/s116\n/s32/s84/s68/s68/s77/s49/s45/s98\nFig. 17. Excitation energy of the second excited state in\nTDDM1-b (open squares) as a function of χforN= 200.\nThe results in RPA for the first excited states are shown with\nthe dotted line. The dots depict the exact values for the lowe st\nexcited state with /angbracketleftL2\n0/angbracketright= 0.\nin TDDM1-b collapses beyond χ= 1.045. A more accu-\nrate treatment of the three-body correlation matrix than\nTDDM1-b is required.\n4 Summary\nWe studied the time-dependent density-matrix theory (\nTDDM) in the large Nand strong coupling limits of the\nLipkin models. It was found that in the two-level Lip-\nkin model the ground state calculated using the original\ntruncation scheme of TDDM where the three-body corre-\nlation matrix is completely neglected approaches the ex-act solution with increasing number of particles. It was\npointed out that this is related to the fact that the de-\nformed Hartree-Fock approximation (DHF) gives the ex-\nactground-stateenergiesinthelarge Nlimit.Therelation\nbetween the occupation matrix in DHF and the correla-\ntion matrix in TDDM was also discussed. The small am-\nplitude limit of TDDM was also found to describe the ex-\ncited states of the two-level Lipkin model very well in the\n’spherical’region.Inthe ’deformed’regionthefirstexcited\nstatefallstozeroasitshould.However,itfallstozerowith\na finite slope contrary to the exact solution which goes to\nzero asymptotically. It indicates that the three-body cor-\nrelation matrix neglected in TDDM still plays some role\nin excited states.\nIt was shown that in the three-level Lipkin model a\nsimple truncation scheme where the three-body correla-\ntionmatrixisapproximatedbythe squareofthe two-body\ncorrelation matrix gives good results for the ground-state\nproperties. Also the first two excited states are quite well\nreproduced far into the strongly correlated (deformed) re-\ngion working in the symmetry unbroken basis.\nWe pointed to the fact that the influence of the three-\nbody correlationmatrix in the two- and three-level Lipkin\nmodels is quite different. While in the former model the\ngenuine three-body correlations totally disappear in the\nlargeNlimit, they partially vanish in the latter model.\nWe gave reasons for this different behavior and suggested\nthat for other spin models similar studies could eventually\nalso give insight into the role of three-body correlations\nthere.\nThe fact that three-body correlation matrix in the\nstrong coupling (’deformed’) regions is apparently quite\nmodel dependent is perturbing and sheds some doubt of\nthe ’blind’ applicability of the cumulant approximation\n(TDDM1) to the three-body correlation matrix in ’de-\nformed’ regions while working in the spherical basis. On\nthe other hand more realistic applications to16O [20] and\nother models [9] where the three-body correlation matrix\ndoes not disappear have shown in the past that the cumu-\nlant approximation is quite powerful for symmetry unbro-\nken systems. It was also shown that TDDM1 gives reason-\nable results for the three-level Lipkin model as long as the\ninteraction is not so strong. Therefore we think that the\ncumulant approximation is applicable to realistic systems.\nA Terms in eq. (5)\nThe terms in eq. (5) are given below. Bαβα′β′describes\nthe 2p–2h and 2h–2p excitations.\nBαβα′β′=/summationdisplay\nλ1λ2λ3λ4∝an}bracketle{tλ1λ2|v|λ3λ4∝an}bracketri}htA\n×[(δαλ1−nαλ1)(δβλ2−nβλ2)nλ3α′nλ4β′\n−nαλ1nβλ2(δλ3α′−nλ3α′)(δλ4β′−nλ4β′)].\n(51)Mitsuru Tohyama, Peter Schuck: Truncation scheme of time-d ependent density-matrix approach III 11\nParticle – particle and h–h correlations are described by\nPαβα′β′\nPαβα′β′=/summationdisplay\nλ1λ2λ3λ4∝an}bracketle{tλ1λ2|v|λ3λ4∝an}bracketri}ht\n×[(δαλ1δβλ2−δαλ1nβλ2−nαλ1δβλ2)Cλ3λ4α′β′\n−(δλ3α′δλ4β′−δλ3α′nλ4β′−nλ3α′δλ4β′)Cαβλ1λ2].\n(52)\nHαβα′β′describes p–h correlations.\nHαβα′β′=/summationdisplay\nλ1λ2λ3λ4∝an}bracketle{tλ1λ2|v|λ3λ4∝an}bracketri}htA\n×[δαλ1(nλ3α′Cλ4βλ2β′−nλ3β′Cλ4βλ2α′)\n+δβλ2(nλ4β′Cλ3αλ1α′−nλ4α′Cλ3αλ1β′)\n− −δα′λ3(nαλ1Cλ4βλ2β′−nβλ1Cλ4αλ2β′)\n−δβ′λ4(nβλ2Cλ3αλ1α′−nαλ2Cλ3βλ1α′)].\n(53)\nReferences\n1. M. Bonitz, Quantum kinetic theory, second edition\n(Springer, 2015).\n2. S. J. Wang and W. Cassing, Ann. Phys. 159, 328 (1985).\n3. M. Gong and M. Tohyama, Z. Phys. A 335, 153 (1990).\n4. S. Takahara, M. Tohyama and P. Schuck, Phys. Rev. C 70,\n057307 (2004).\n5. K.-J. Schmitt, P. -G. Reinhard and C. Toepffer, Z. Phys.\nA336, 123 (1990).\n6. T. Gherega, R. Krieg, P. -G. Reinhard and C. Toepffer,\nNucl. Phys. A 560, 166 (1993).\n7. M. Tohyama, J. Phys. Soc. Jpn. 81, 054707 (2012).\n8. A.J.Coleman andV.I.Yukalov, Reduced density matrices:\nCoulson’s challenge (Springer, New York, 2000).\n9. M. Tohyama and P. Schuck, Eur. Phys. J. A 50, 7 (2014).\n10. D. A. Mazziotti, Phys. Rev. A 60, 3618 (1999).\n11. H. J. Lipkin, N. Meshkov and A. J. Glick, Nucl. Phys. 62,\n188 (1965).\n12. M.TohyamaandP.Schuck,Eur.Phys.J.A 53,186(2017).\n13. S. Y. Li, A. Klein and R. M. Dreizler, J. Math. Phys. 11,\n975 (1970).\n14. I. Shavitt and R. J. Bartlett, Many-body methods in chem-\nistry and physics (Cambridge, 2009).\n15. G. Holzwarth and T. Yukawa, Nucl. Phys. A 219, 125\n(1974).\n16. K. Hagino and G. F. Bertsch, Phys. Rev. C 61, 024307\n(2000).\n17. D. S. Delion, P. Schuck and J. Dukelsky, Phys. Rev. C 72,\n064305 (2005).\n18. M. Tohyama, Phys. Rev. A 71(2005) 043613.\n19. D. A. Mazziotti, Phys. Rev. Lett. 97, 143002 (2006).\n20. M. Tohyama, Phys. Rev. C 91, 017301 (2015)."    },    {        "title": "1906.06043v1.Friedel_oscillations_in_2D_electron_gas_from_spin_orbit_interaction_in_a_parallel_magnetic_field.pdf",        "content": "Friedel oscillations in 2D electron gas from spin-orbit interaction in a parallel\nmagnetic field\nI. V. Kozlov and Yu. A. Kolesnichenkoa)\nB.I. Verkin Institute for Low Temperature Physics and Engineering,  National Academy of Sciences of Ukraine, \npr. Nauki, 47, Khar ’kov, 61103,  Ukraine\nEffects associated with the interference of electron waves around a magnetic point defect  in\ntwo-dimensional electron gas with  combined  Rashba-Dresselhaus  spin-orbit interaction in the \npresence  of a parallel magnetic field are theoretically investigated. The effect of a magnetic field \non the anisotropic  spatial distribution of the local  density  of states and the local density of magneti-\nzation  is analyzed. The existence of oscillations of the density of magnetization with scattering by a \nnon-magnetic defect  and the contribution of magnetic scattering (accompanied  by spin- flip) in the \nlocal  density of electron  states are  predicted.\n1. Introduction\nInterest in research on quasi-two-dimensional [hereinafter\nfor brevity, two-dimensional (2D)] conductive systems results\nfrom new capabilities for studying various quantum effectsthat are absent or are very small in mass conductors. Among\nsuch effects are spatial oscillations of the density of electrons\nn(ϵ\nF,r) theoretically predicted by Friedel,1depending on the\ndistance rfrom a point defect or conductor edge ( ϵFis Fermi\nenergy). Friedel oscillations (FO) are associated with interfer-ence of incident and re flected electron waves and in an isotro-\npic conductor have a period Δr=πħ/p\nF(where pFis the\nFermi impulse). Direct observation of FO became possiblewith the development of scanning tunnel microscopy (STM).\n2\nThus, when studying in Refs. 3and4the surface (111) of\nnoble metals by means of STM, FO of the local density ofstates (LDS) ρ(ϵ,r ) = dn/dϵ|\nϵ=ϵFwere detected, arising as\nthe result of the scattering of 2D surface states on individual\nadsorbed atoms. It was further shown that the study of the\nperiods and contours of the constant phase of FO makes itpossible to obtain new information on the local characteris-\ntics of the spectrum of charge carriers and on the process of\ntheir scattering by an individual point defect of known nature\n(see reviews\n5–7and the references quoted in them).\nDuring electron scattering on a magnetic impurity at tem-\nperature Tbelow the Kondo temperature TK,8there appears in\nthe FO of the electron density n(ϵF,r) an additional shift of the\nphase of the oscillations depending on r/rK(rK=ħνF/TKis the\nKondo length) which, however, is absent in oscillations ρ(ϵF,\nr)o ft h eL D Sa sm e a s u r e db ym e a n so fS T M .9The magnetic\ndefect, along with oscillations of the electron density, results inoscillations of the local density of magnetization (LDM) ofelectron gas m(ϵ\nF,r), which is a magnetic analogue of the\nFO10(hereinafter we will use for brevity the term FO of the\nLDM). Spatial oscillations of the LDM are investigated bymeans of a spin-polarized scanning tunnel microscope having\na contact (tip) with a magnetic covering.\n11\nThe asymmetrical con finement electric potential (con-\nfinement potential) that limits the motion of electrons along\nthe normal to the edge (for surface states) or to the border of\nthe heterotransition with a quantum well, as well as theabsence of a center of inversion of a bulk crystal, result inspin-orbit interaction (SOI) (see the monograph,\n12which sig-\nnificantly affects the thermodynamic and kinetic characteris-\ntics of such 2D systems.13The FO near an individual point\ndefect in 2D electron gas with Rashba SOI18,19were experi-\nmentally observed in Refs. 14–17, and in this case a rather\nlarge number of theoretical works20–27were devoted to the\nanalysis of the oscillatory dependence of the LDS. The LDM\nwas theoretically studied in Ref. 28for electron scattering of\nsurface states of Au(111) on an adsorbed Co atom, taking\ninto account the Rashba SOI and the Kondo effect.\nIn a number of 2D systems, combined Rashba19and\nDresselhaus31SOI (R-D SOI) is produced, based on semicon-\nductors such as zincblende (III –V zincblende and wurtzite)\nand SiGe (see Refs. 29and30). With R-D SOI, the scattering\nlaw of charge carriers is anisotropic, and this results in signi fi-\ncant modi fication of the FO. Thus, FO beats were predicted in\nRef.32, and it is shown in Ref. 33that the FO are also essen-\ntially anisotropic, and contain more than two harmonics for acertain relationship of the constants of SOI.\nAs a consequence of the dependence on the wave vector\nof the direction of electron spin in 2D systems with SOI, theparallel magnetic field not only results in Zeeman splitting,\nbut also signi ficantly changes the energy spectrum (see for\nexample Refs. 34and35). By changing the amplitude and\ndirection of the magnetic field, it is possible by a change in\nthe spectrum to control all electron characteristics of a 2Dconductor with R-D SOI.\nIn this article we studied FO oscillations around the\npoint of a magnetic defect in 2D electron gas with R-D SOI\nlocated in a parallel magnetic field. General expressions for\nthe LDS and LDM and their asymptotic expressions at large\ndistances from the defect were obtained in the Born approxi-\nmation. The dependences of the FO of the LDS and LDM onthe amplitude and direction of the magnetic field were ana-\nlyzed. The effect of the occurrence of oscillations of thedensity of magnetization during scattering by a non-magneticdefect and the dependence of the oscillations of the density\nof states on the magnitude of the magnetic moment of the\ndefect were predicted.2. Statement of the problem\n2.1. The Hamiltonian\nUsing the calibration A= (0, 0, Bxy–Byx), we write the\nHamiltonian of the 2D electron gas with R-D SOI in a paral-\nlel magnetic fieldB=(Bx,By,0) in the absence of defects, as\na linear approximation on the wave vector operator ^k¼\n/C0irr(see, for example, Refs. 34and35):\n^H0¼/C22h2(^k2\nxþ^k2\ny)\n2mσ0þα(σx^ky/C0σy^kx)þβ(σx^kx/C0σy^ky)\nþg/C3\n2μB(BxσxþByσy), (1)\nwhere mis the effective mass of an electron, B=(Bx,By,0) is\nthe induction of the magnetic field,σx,y,zare Pauli matrices,\n^σ0is the 2 × 2 identity matrix, αandβare constants of the\nRashba ( α) and Dresselhaus ( β) SOI, μBis the Bohr magne-\nton, and g* is the effective g-factor of the 2D system, which\ncan signi ficantly differ from the value g0= 2 for free elec-\ntrons.12For de finiteness we will assume that the spin-orbit\ninteraction constants are positive.\nThe eigenvalues and eigenfunctions of the Hamiltonian\nin(1)are written as\nϵ1,2(k)¼/C22h2k2\n2m+A(kx,ky),\nA¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi\n(hxþαkyþβkx)2þ(hyþαkxþβky)2q\n, (2)\nψ1,2(r)¼1\n2πffiffiffi\n2peikr 1\neiθ1,2/C18/C19\n;1\n2πeikrf(θ1,2), (3)\nwhere the f(θ1, 2) are the spin parts of the wave functions of (3),\nsinθ1(k)¼hy/C0αkx/C0βky\nA,c o s θ1(k)\n¼hx/C0αky/C0βkx\nA,θ2¼θ1þπ: (4)\nIn order to reduce writing in later formulas, we introduce the\ndesignation\nh¼g/C3\n2μBB: (5)\nThe spin orientation for each of the branches of the spectrum\nin(2)is defined by the mean\ns1,2(θ)¼fy(θ1,2)σf(θ1,2)¼(cosθ1,2, sinθ1,2, 0), (6)\nand, in accordance with the formulas in (4), this depends not\nonly on the direction of the wave vector, but also on itsvalue, for combined R-D SOI in the presence of a parallel\nmagnetic field.\n2.2. Scattering by a defect\nWe will model the interaction of electrons with a point\nmagnetic defect at the point r= 0 using the two-dimensional\nδ-potential, often used in the examination of various physicalproblems:36\nD(r)¼γσ0þ1\n2Jσ/C20/C21\nδ(r), (7)\nwhere γ> 0 is the constant of potential interaction of elec-\ntrons with the defect; ^σ=(^σx,^σy,^σz) is the Pauli vector; and J\nis the effective magnetic moment of the defect which differs\nfrom its true value S(S≥1) due to the Kondo effect, resulting\nin partial screening of the spin Sby conductivity electrons.\nWe consider the direction of the vector Jto be fixed, and will\nnot consider the processes of revolution and precession of thedefect spin.\nTemperature Tis assumed equal to zero. Such an\napproach is quite justi fiable since the quantum interferen-\ntial phenomena to which FO pertain are usually observedat low temperatures, when scattering of electrons on\nphonons is suf ficiently small. For T=0 , t h e L D S ρ(ϵ\nF,r)\nand LDM m(ϵF,r) can be calculated with the aid of the\nretarded Green function ^GR(E,r1,r2) in the coordinate rep-\nresentation\nρ(ϵF,r)¼/C01\nπIm Sph\n^GR(ϵF,r,r)i\n, (8)\nm(ϵF,r)¼/C01πIm Sph\n^σ^G\nR(ϵF,r,r)i\n: (9)\nWe will take account of the effect of electron scattering at\na defect in the Born approximation (see, for example, Ref.\n37from the potential of scattering (7), after presenting the\nGreen function in the form of a decomposition\n^GR(ϵ,r1,r2)/C25^GR\n0(ϵ,r1/C0r2)þ^GR0(ϵ,r1)D(r)^GR0(ϵ,/C0r2),\n(10)\nin which the retarded Green function in the absence of\ndefects, ^G0R(ϵF,r1–r2), depends only on the difference of\ncoordinates r=r1–r2:\n^GR\n0(ϵ,r)¼1\n(2π)2ð1\n/C01d2keikr\nϵ/C0^H0þi0;ϵ[R: (11)\nThe Hamiltonian ^H0is defined by expression (1). Naturally,\nformula (10)can be used to describe the FO only at suf ficiently\nlarge distances from the defect r>rDwhen the term associated\nwith scattering is small. The value of rDwith respect to the\norder of magnitude of size can be estimated as the Fermi wave-\nlength rD∼λF∼ħ/pFfor potential scattering, and as the\nKondo length rD–rK=ħvF/TKfor magnetic scattering.\n3. The Green function\n3.1. General ratios\nWe derived in Ref. 38exact analytical expressions for\nthe equilibrium Green function at temperature zero for a 2D\nsystem of electrons with R-D SOI, and their asymptotes forlarge values of the spatial variable. Here we will provide\ncertain relationships based on earlier-obtained results\n38\nwhich will be necessary for further calculations\nThe equilibrium retarded Green function (11) can be\npresented in the form of a decomposition on the Paulimatrices\n^GR\n0(ϵ,r)¼g0(ϵþi0,r)^σ0þgx(ϵþi0,r)^σx\nþgy(ϵþi0,r)^σy,ϵ,[R, (12)\nwhere\ng0(ϵ,r)¼1\n2(2π)2X\nj¼1,2ð\nd2keikr 1\nϵ/C0ϵj(k), (13)\ngx(ϵ,r)\ngy(ϵ,r)/C26/C27\n¼1\n2(2π)2X\nj¼1,2ð\nd2keikrcosθj(k)\nsinθj(k)/C26/C271\nϵ/C0ϵj(k),\n(14)\nθjis the angle de fining the direction of spin of an electron\n(6). Its dependence on the pulse is de fined by formula (3).\nEach of the terms with j= 1, 2 contributes to the Green func-\ntion of one of the branches of the energy spectrum ϵ1,2(2).\nThe components g0,x,yof the Green function (12) satisfy the\nsymmetry relationships\ng0(r,h)¼g0(/C0r,/C0h),gx,y(r,h)¼/C0gx,y(/C0r,/C0h):\n(15)\nFurther, we will assume α≠βin all calculations, except for\nspecial cases when the equality of the constants of SOI is\nstipulated separately. For α≠β, it is convenient to transition\nto new variables of integration ^kandf.\nkx¼kx0þ~kcosf,ky¼ky0þ~ksinf, (16)\nwhere\nkx0¼hαsinwhþβcoswh\nα2/C0β2;ky0¼/C0hαcoswhþβsinwh\nα2/C0β2\n(17)\nare the coordinates of the point of contact of the branches ofthe spectrum, in which the energy is\nϵ1,2(k0)¼/C22h2(k2\n0xþk2\n0y)\n2m¼h2/C22h2α2þβ2þ2αβsin2wh\n2m(α2/C0β2)2\n¼ϵ0.0, (18)\nand the angle whspecifies the direction of the magnetic field\nh=h(coswh, sin wh, 0). The wave vector k0=(kx0,ky0),\ncorresponding to a point with energy ϵ0, is determined from\nthe condition A(kx0,ky0)= 0 in the expressions in (2).\nIn the variables in (16), the dependence of energy ϵ1, 2\non~kandfis written as\nϵ1,2(~k,f)¼/C22h2~k2\n2m/C0/C22h2~k\n2λ1,2(f)þϵ0, (19)\nwhere\nλ(1,2)(f)¼hαsin (f/C0wh)/C0βcos ( fþwh)\nα2/C0β2\n+m\n/C22h2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi\nα2þβ2þ2αβsin (2 f):q\n(20)\nIn the new coordinates of (16), the point of contact of the\nbranches of the spectrum corresponds to ^k=0 .\nThe angles de fining the direction of spin, θ1, 2(f),\ndepend only on the direction of the wave vector (angle f) and\nthe constants of SOI, after the replacement in (16):\nsinθ1,2(f)¼+αcosfþβsin fffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi\nα2þβ2þ2αβsin 2 fp ,\ncosθ1,2(f)¼+αsinfþβcos fffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiα2þβ2þ2αβsin 2 fp : (21)\nHence, for each branch of the spectrum, the directions of elec-\ntron spin s1,2(f) are antisymmetric relative to the point ^k=0 .\nFor each angle f, the directions of spin of electron belonging to\ndifferent branches of the spectrum are strictly opposite.\nSubstituting expressions (16) and (21) into formulas\n(13)and(14), after integration on ^kwe derive\nwhere ^k=k±(1, 2)(ϵ) are the roots of the equations\nϵ1,2(~k,f)¼ϵ, (24)\nk(1,2)\n+¼λ(1,2)+ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi\n(λ(1,2))2þ2m(ϵ/C0ϵ0)\n/C22h2r\n, (25)F(k,r)¼ð1\n0d~k\n~k/C0kei~kr¼eikr/C0Ci(/C0kjrj)þiSi(kr)þiπ\n2sgnr/C20/C21\n,\nr[R,k[C,\n(26)\nthe angle wrin formulas (22) and(23) determines the direc-\ntion of the vector r=r(coswr, sinwr, 0).g0(ϵ,r)¼/C0m\n(2π/C22h)2exp [ i(kx0coswrþky0sinwr)þ]X\nj¼1,2þ\ndfX\n+k(j)\n+\nk(j)\n+/C0k(j)\n+Fk(j)\n+,rcos ( f/C0wr)/C16/C17\n,ϵ[C, (22)\ngx(ϵ,r)\ngy(ϵ,r)/C26/C27\n¼/C0m\n(2π/C22h)2exp [ i(kx0coswrþky0sinwr)r]X\nj¼1,2þ\ndfcosθj(f)\nsinθj(f)/C26/C27X\n+k(j)\n+\nk(j)\n+/C0k(j)\n+Fk(j)\n+,rcos ( f/C0wr)/C16/C17\n,ϵ[C, (23)In the speci fic case of equality of the constants of SOI\nα=βand the directions of the magnetic field along the axis\ny=–xandh=h/√2(–1, 1, 0), the integrals in formulas (13)\nand(14)may be expressed through the Bessel functions:\ng0(ϵþi0,r)¼1\n2(Gþ(ϵ,r)þG(ϵ,r)),ϵ[R, (27)\ngx(ϵþi0,r)\ngy(ϵþi0,r)/C26/C27\n¼+1\n2ffiffiffi\n2p(Gþ(ϵ,r)/C0G(ϵ,r)),ϵ[R, (28)\nwhere\nG+(ϵ,r)¼exp+iffiffiffiffiffiffiffiffiffi\n2mαp\n/C22h2(xþy)/C18/C19\nGR\n2Dϵþffiffiffiffiffiffiffiffiffi\n2mαp\n/C22h2+h,r/C18/C19\n,\n(29)\nandGR\n2D(ϵ,r) is the retarded Green function of free 2D elec-\ntrons,\nGR\n2D(ϵ,r)¼/C0m\n2/C22h2iH(1)\n0ffiffiffiffiffiffiffiffiffi\n2mϵp\njrj=/C22h/C0/C1\n; ϵ.0\n2\nπK0ffiffiffiffiffiffiffiffiffiffiffi\n2mjϵjp\njrj=/C22h/C16/C17\n;ϵ,08\n<\n:, (30)H0(n)(x) is the Hankel function, and K0(x) is the modi fied\nBessel function of the second kind.\n3.2. Asymptotic formulas\nFor large r,the stationary phase method39makes it pos-\nsible to derive very simple asymptotic expressions of formu-\nlas(22) and(23). Then the stationary phase f=fst(j)should be\ndetermined from the equation\nd\ndfkj\n+cos ( f/C0wr)/C0/C1\njf¼f(j)\nst¼0, (31)\nin which the k±(j)are the positive real solutions (25) of\nEq.(24)forϵ∈R.\nAs a result of standard calculations,38we obtain the fol-\nlowing asymptotes of the components of the Green function\nin (12)\ng0(ϵþi0,r)≃/C0i\n2ffiffiffiffiffi\n2πpX\nj¼1,2X\ns1\n/C22hv(j)ffiffiffiffiffiffiffi\njKjjp\nr\n/C2exp iSjr/C0iπ\n4sgnKj/C20/C21\njf¼f(j)\nst,ϵ[R, (32)\nSj(f,wr)¼k(j)\n+(f) cos ( f/C0wr), (34)\nwhich are valid for Sjr≫1. All functions must be calculated\nat the points of the stationary phase f=fst(j). The summation\nonstakes account of the possibility of the existence, on a\nnon-convex isoenergy contour belonging to a branch of the\nspectrum ϵ2(k), of several points of the stationary phase cor-\nresponding to the given direction of the vector r. In formulas\n(32) and (33), K1,2(f)≠0 is the curvature of the isoenergy\ncurve ϵ1, 2(f)=ϵ.\nKj(f)¼k(j)\n+(f)2þ2k(j)\n+(f)2/C0k(j)\n+(f)k(j)\n+(f)\n(k(j)\n+(f)þk(j)\n+(f)2)3=2, (35)\nandvj≠0 is the absolute value of the velocity of an electron\nv(j)=∇kϵj/ħ. Hereinafter, a dot over a function signi fies dif-\nferentiation on the angle fof the direction of a wave vector in\nthe displaced coordinates (16). Solutions of Eq. (31)satisfying\nthe inequality Sj(f) [see formula (34)] correspond to the condi-\ntion of parallelism of vectors randv(j)(see also Ref. 43)\nrn(j)\nvjf¼f(j)\nst¼r;n(j)\nv(f)¼v(j)\njv(j)j\n¼+/C0k(j)\n+sinfþk(j)\n+cosfffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi\nk(j)2\n+þk(j)2\n+q ,k(j)\n+cosf/C0k(j)\n+sinfffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik\n(j)2\n+þk(j)2\n+q0\nB@1\nCA:\n(36)The values of the phase krof rapidly oscillating functions as\nr→∞in the integrals in (13)and(14)can be interpreted43in\nterms of the support function40of the isoenergy contour ϵ1, 2\n(k)=ϵ.\nSj(k)¼kn(j)\nv(k);k[ϵi(k)¼ϵ,( 3 7 )\nknowing which, it is possible to restore the contour and find its\ncurvature at any point.\n4. Friedel oscillations\n4.1. Fermi contours\nIt was shown in Refs. 41–43that the geometry of the\nconstant phase lines of the FO oscillations of the LDS and\ntheir period depend on the local geometry of the Fermisurface. Therefore, in this section we will introduce some\ninformation that will be needed later regarding the energy\nspectrum of the system being studied.\nIn the case of 2D electron gas with R-D SOI placed in a\nparallel magnetic field, the energy spectrum contains two\nbranches ϵ\n1, 2(k)≠ϵ1, 2(–k)(2), not possessing central\nsymmetry. The surface ϵ=ϵ1(k) is always convex, at the\nsame time that the surface ϵ=ϵ2(k) for a de fined region of\nvalues of SOI constants and magnetic field contains saddle\npoints and areas of negative Gaussian curvature (see for\nexample Refs. 34and35. There exists a critical value of thegx(ϵþi0,r)\ngy(ϵþi0,r)/C26/C27\n≃/C0i\n2ffiffiffiffiffi\n2πpX\nj¼1,2X\nscosθj\nsinθj/C26/C271\n/C22hv(j)ffiffiffiffiffiffiffiffiffi\njKjjrp exp iSjr/C0iπ\n4sign K j/C20/C21\njf¼f(j)\nst,ϵ[R, (33)magnetic field\nhc¼(α2/C0β2)2\nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi\nα4þ6α2β2þβ4þ4αβ(α2þβ2) sin 2whp , (38)\nfor values less than which, for h<hc2the function ϵ=ϵ1(k)\nhas no extrema and takes the smallest value ϵ=ϵ0(18)at the\npoint of contact of the branches of the spectrum, and for h>\nhc,ϵ=ϵ1(k) has an absolute minimum.\nIn the coordinate system of (16), the equations of Fermi\ncontours k(ϵF,f)=k±(1,2)are de fined by the formulas (25) for\nϵ=ϵFin the field of values of parameters for which k±(1, 2)≥0.\nIn the case when the inequality ϵF>ϵis satis fied, the func-\ntions k+(1, 2) (ϵF,f) > 0 for any values of fwhile at the same\ntime the roots k–(1, 2) (ϵF,f) < 0 are always negative. To each\nbranch of the spectrum there corresponds one Fermi contour\nk(ϵF,f)=k+(1, 2).F o r ϵF<ϵ0the real roots of Eq. (24) exist\nwhen the inequality is satis fied\n2m(ϵ0/C0ϵF)\n/C22h2/C20(λ(j))2,\nand both roots k±(j)take positive values in some interval of\nangles for which λj≥0. In this case the Fermi contours do\nnot cover the coordinate origin ( kx0,ky0) (17), and the two\npoints k=k±(j)on the same Fermi contour correspond to the\nsame direction of the angle f. Due to the dependence of the\nenergy at the point of contact of the branches of the spec-trum ϵ0(18) and the functions λ(j)(f)( 2 0 )o n h, it is possi-\nble by varying the magnitude and direction of the magnetic\nfield to change smoothly the energy spectrum and therefore\nthe geometry of isoenergy contours.\n4.2. LDS oscillations\nUsing expressions (8)and(10), we will present LDS in\nthe form of the sum of two components:\nρ(ϵF,r)¼ρ0(ϵF)þΔρ(ϵF,r), (39)\nwhere ρ0(ϵF) is the density of states of two-dimensional\ndegenerate gas from R-D SOI in the absence of defects:34,38\nρ0(ϵF)¼m\nπ/C22h2;ϵF/C21ϵ0\nm\n2π2/C22h2X\nj¼1,2þ\ndfλ(j)\nffiffiffiffiffiffi\nξ(j)p Θ(λ(j))Θ(ξ(j));ϵF/C20ϵ0,8\n>><\n>>:\n(40)\nξ(1,2)¼(λ(1,2))2þ2m(ϵF/C0ϵ0)\n/C22h2: (41)\nThe part of LDS Δρ(ϵF,r) that depends on scattering\ndescribes the FO. Using decomposition (12) of the Green\nfunction over Pauli matrices, we derive\nThe lack of a center of inversion of the electron scattering law (2) results in the appearance in Δρin(42) of a component pro-\nportional to the components Jx, y, z of the magnetic moment of the defect, which goes to zero for h=0 .\nUsing asymptotic expressions (32) and (33) for the Green functions at large distances from the defect, we derive\nQik¼1\n2π2/C22hv(i)/C22v(j)jKi/C22Kjjr;fij¼/C0π\n4(sign Kiþsign /C22Kj):\n(44)\nThe angles wJandθJspecify the direction of the vector of\nthe magnetic moment of the defect.\nJ¼J(coswJsinθJ, sinwJsinθJ, cos θJ): (45)\nThe line over a function signi fies that its value is taken at the\npoint k=^kst(1, 2)for which the velocity v(1, 2)of an electron isdirected opposite to the direction of the vector r,andnv(1,2)\n(^kst(1,2))=–nv(1,2)(kst(1,2)). Hence, each of the components in\nthe sum (43) takes account of the contribution to LDS of\nelectron back-scatter with a transition between two points\nwith opposite direction of the velocity on the same ( i=j)o r\ndifferent ( i≠j) Fermi contours.\nThe asymptotic formula in (43) makes it possible to\neasily interpret the reason for the occurrence of a magnetic\ncontribution to the density of states: due to the connection\nbetween the directions of spin and the wave vector, the mag-\nnetic scattering that results in spin flip at some angle Δθi=θiΔρ(r)¼/C02\nπIm{γ[g0(r)g0(/C0r)þgx(r)gx(/C0r)þgy(r)gy(/C0r)]þJx[g0(r)gx(/C0r)þgx(r)g0(/C0r)]\nþJy[g0(r)gy(/C0r)þgy(r)g0(/C0r)]þiJz[gy(r)gx(/C0r)/C0gx(r)gy(/C0r)]}: (42)\nΔρ(r)¼/C0X\ni,j¼1,2X\nsQij2γcos2θi/C0θj\n2/C20/C21\nþJsinθJ/C16\ncos/C0\nwJ/C0wi/C1\nþcos/C0\nwJ/C0θj/C1/C17/C18/C19/C26\n/C2sin/C16/C0\nSiþSj/C1\nrþfij/C17\n/C0JcosθJsin/C0\nθi/C0θJ/C1\ncos/C16/C0\nSiþSj/C1\nrþfij/C17\n}, (43)–/C22θj≠0, ±πand that corresponds to the change of velocity\nof an electron to its opposite, changes the flux of electrons\npropagated in the opposite direction.\n4.3. LDM oscillations\nWe present an expression for LDM m(r) in the form of\na sum not dependent on the coordinates of the componentm0and the oscillatory additive Δm(r)\nm(r)¼m0þΔm(r): (46)\nIt is clear that due to the common relationships (12) and (9),\nthe component of the density of magnetization m0z=0 .\nHowever, the components m0x, ymay be distinct from zero at\nthe Fermi energy ϵF≤ϵ0:\nUsing a representation of the Green function in the form of decomposition on Pauli matrices (12), we will write expressions for\nthe component of the vector Δm(r) in the following form:\nUsing the derived formulas (48) –(50) and asymptotic expressions (32) –(33) for the component of the Green function (12)\nfor large r,w efi nd the LDM components that are oscillatory with distance from the defect:\nThe derived results (51)–(53)define the dependence of the FO\nof the LDM on an external magnetic field. As in the case of\nLDS, the non-magnetic contribution to LDM can be easilyinterpreted on the basis of the asymptotic formulas (51)–(53):back-scatter by a non-magnetic defect is accompanied by a\ntransition to a state with the spin direction turned relative to the\ninitial at an angle Δθ\ni=θi–/C22θj≠0, ±π , which results in a\nchange of the local density of magnetization around the defect.mx,y¼0, ϵF/C21ϵ0,\nm\n2π2/C22h2X\nj¼1,2þ\ndfcosθi\nsinθi/C26/C27λ(j)\nffiffiffiffiffiffi\nξ(j)p Θ(λ(j))Θ(ξ(j)),ϵF/C20ϵ0:8\n><\n>:(47)\nΔmx(r)¼/C02\nπIm{Jx[g0(r)g0(/C0r)þgx(r)gx(/C0r)/C0gy(r)gy(/C0r)]þJy[gx(r)gy(/C0r)þgy(r)gx(/C0r)]\n/C0iJz[g0(r)gy(/C0r)þgy(r)g0(/C0r)]þγ[g0(r)gx(/C0r)þgx(r)g0(/C0r)]}, (48)\nΔmy(r)¼/C02πIm{J\ny[g0(r)g0(/C0r)þgx(r)gx(/C0r)/C0gy(r)gy(/C0r)]þJx[gy(r)gx(/C0r)þgx(r)gy(/C0r)]\n/C0iJz[g0(r)gx(/C0r)þgx(r)g0(/C0r)]þγ[g0(r)gy(/C0r)þgy(r)g0(/C0r)]}, (49)\nΔmy(r)¼/C02πIm{J\nz[g0(r)g0(/C0r)þgx(r)gx(/C0r)/C0gy(r)gy(/C0r)]þiJx[g0(r)gy(/C0r)þgy(r)g0(/C0r)]\n/C0iJy[g0(r)gx(/C0r)/C0gx(r)g0(/C0r)]þiγ[gx(r)gy(/C0r)/C0gy(r)gx(/C0r)]}: (50)\nΔmx(r)¼/C0X\ni,j¼1,2X\nsQijJsinθjcoswJcos2θiþ/C22θj\n2/C18/C19\nþsinwJsin (θiþ/C22θj)/C18 /C20/C21/C26\nþJcosθJ(sinθi/C0sin/C22θj)\n/C2cos (( Siþ/C22Sj)rþfij)þγ(cos θiþsinθiþcos/C22θjþsin/C22θj)]/C2sin (( Siþ/C22Sj)rþfij)}, (51)\nΔmy(r)¼/C0X\ni,j¼1,2X\nsQijJsinθj2 sinwJsin2θiþ/C22θj\n2/C18/C19\nþcoswJsin (θiþθj)/C18 /C20/C21/C26\n/C0JcosθJ(sinθi/C0sin/C22θj)\n/C2cos (( Siþ/C22Sj)rþfij)þγ(cos θiþsinθiþcos/C22θjþsin/C22θj)] sin (( Siþ/C22Sj)rþfij)}, (52)\nΔmz(r)¼/C0X\ni,j¼1,2X\nsQij2Jcosθjsin2θiþ/C22θj\n2/C18/C19\nþsin (( Siþ/C22Sj)rþfij)/C26\n/C0[JsinθJ(sinwJ(sinθi/C0sin/C22θj)\n/C0coswJ(cos θi/C0cos/C22θj))þγsin (θi/C0/C22θj)] cos (( Siþ/C22Sj)rþfij)}: (53)4.4. The special case α=βand h x=–hy. Analytical solution\nIn the speci fic case that α=βandh=h√2(–1, 1, 0),\ntwo branches of the spectrum are crossed along the parabola\nϵ¼/C22h2k2\ny1\n2mþ/C22h2h2\n8mα2,kx1¼kxþkyffiffiffi\n2p¼h\n2α,ky1¼kx/C0kyffiffiffi2p :\n(54)\nForϵ\nF>ħ2h2/8mα2, the Fermi contours have two common\npoints and each of them consists of two arcs of circles of\nradius k(±)=√(2mϵ±)/ħ(see Fig. 5):\nk(+)¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi\n2mjϵ+jp\n=/C22h,ϵ+¼ϵFþ2mα2\n/C22h2+h: (55)\nThe spin directions on each of the arcs that comprise one\ncontour are opposite: θ+=3π/4 or θ_=–π/4, and the arcs\nwith the identical spin directions θ±form complete circles\n[see Fig. 5(a)].\nUsing the expressions for the Green functions in (27)and\n(28), we derive the following expression for the oscillatorypart of the LDS:\nΔρ(r)¼γþJy/C0Jx\n2ffiffiffi\n2p/C18/C19\nRþþγ/C0Jy/C0Jx\n2ffiffiffi2p/C18/C19\nR\n/C0,( 5 6 )\nwhere\nR+(r)¼m2\n4π/C22h4J0(k(+)r)Y0(k(+)r)Θ(ϵþ), (57)\nwhere J0(x)a n d Y0(x) are the Bessel functions of the first and\nsecond kind, respectively. For large values of the arguments\nk(±)r≫1, we have\nR+(r)≃/C0m2\n4π2/C22h4k(+)rcos (2k(+)r): (58)\nThe components of the oscillatory part of the LDM, Δm(r),\nare de fined by the following expressions:\nΔmx(r)¼γffiffiffi\n2pRþþγffiffiffi2p/C0Jy/C0Jx\n2/C18/C19\nR/C0Jz\n2R,( 5 9 )\nΔmy(r)¼γffiffiffi\n2pþJy/C0Jx\n2/C18/C19\nRþ/C0γffiffiffi2pR\n/C0/C0Jz\n2R,( 6 0 )\nFig. 1. ( a) Typical form of Fermi con-\ntours for ϵF>ϵ0in a parallel magnetic\nfield. ( b) Oscillatory part of LDS\nΔ/C22ρ(/C22x,/C22y) for scattering by a non-\nmagnetic defect ( J= 0). The following\nvalues of parameters are used: /C22α= 0.7,\n/C22β= 0.3, /C22h= 0.6, wh= 2.0.\nFig. 2. FO of the LDM for scattering by a a non-magnetic defect with ϵF>ϵ0.(a) Distribution of the component Δ/C22mznormal to the plane. ( b) Distribution of\nthe absolute valueffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi\nΔ/C22m2\nxþΔ/C22m2yq\nof the LDM component in the plane. Arrows indicate the direction of the vector Δ/C22m¼(Δ/C22mx,Δ/C22my). Values of the parameters\ncoincide with those given in Fig. 1.Δmz(r)¼Jz\n2(Rþ/C0R/C0)Jy/C0Jx\n2ffiffiffi\n2p R,( 6 1 )where the designations R±(r)a r ed e fined by expression (57)\nand\nIn the case being studied, the FO of the LDS (56) are isotro-\npic, and at large distances from the defect contain two har-\nmonics with periods Δr=π/k(±), associated with back-scatter\nbetween states belonging to different Fermi contours, while\nat the same time in LDM oscillation (59) –(61), a contribution\nis made by the transitions between states of the same Fermicontour that result in the appearance of harmonics of the FO,with periods\nΔr¼\n2π\nk(þ)þk(/C0)+4mα\n/C22h2sinwrþπ\n4/C16/C17 , (64)depending on the direction wrin the coordinate space. It\nfollows from formula (63)that the lines of constant phase for\noscillations of the LDM with the period in (64) are ellipses\n(for k(+)+k(–)>4mα/ħ2) or hyperbolas (for k(+)+k(–)\n<4mα/ħ2), and the point r= 0 coincides with one of the foci.\n5. Discussion of results\nThe effect of a parallel magnetic field the FO of a 2D\nelectron gas with Rashba-Dresselhaus SOI results from two\nmain causes. First, the magnetic fieldBbreaks the central\nsymmetry of the Fermi contours and changes their local\nFig. 3. ( a) Fermi contours for ϵ0=ϵF\n(18), h>hc(38),wh=3π/4. The black\npoint shows k0, the point of contact of\nthe branch of the range of (17).(b)\nLDS for scattering by magnetic defect\nJ=(J, 0, 0) and γ=0 (7). The\nfollowing values of parameters areused: /C22α= 0.5, /C22β= 0.2, /C22h= 1.4.\nFig. 4. FO of the LDS Δ/C22ρ(/C22r) with\nscattering by a magnetic defect, J= (1,\n–1, 0) J0/√2 and γ= 0, near the value\nof the magnetic field /C22hmin1= 1.36,\nwhere the Fermi level passes through\nthe point of the minimum of the\nbranch of the range ϵ1.(a)/C22hmin1</C22h=\n1.4. ( b)/C22hmin1>/C22h= 1.28. The following\nvalues of parameters are used: /C22α= 0.5,\n/C22β= 0.1, wh=3π/4.R(r)¼m2\n2π/C22h4sin2ffiffiffi\n2p\nmα\n/C22h2(xþy)/C20/C21 /C26\nJ0(k(þ)r)Y0(k(/C0)r)þJ0(k(/C0)r)Y0(k(þ)r)/C27\nΘ(ϵþ)Θ(ϵ/C0), (62)\nR(r)≃/C04m2\nπ2/C22h4rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi\nk(þ)k(/C0)p sin k(þ)þk(/C0)þ4mα\n/C22h2sinwrþπ\n4/C16/C17/C18/C19\nr/C20/C21/C26\n/C0sin k(þ)þk(/C0)þ4mα\n/C22h2sinwrþπ\n4/C16/C17/C18/C19\nr/C20/C21 /C27\n;k(+)r/C291:(63)geometry. The back-scatter of electrons, which provides the\nchief contribution to the FO, corresponds to transitions\nbetween states with opposite direction of velocity which in\nturn depends on the fieldB. As a result, a change to the mag-\nnitude and direction of the vector Bchanges both the period\nof the FO [as a consequence of the change to the magnitudeof the wave vector corresponding to a point of a stationaryphase, see (34)] and their amplitude [as a consequence of the\nchange of the curvature of the Fermi contour, see (35)]. The\nsecond main circumstance is the change of electron spinunder the effect of the direction of the field [see (6)]. Since\nwith SOI the spin of an electron and its wave vector are inter-connected, the matrix elements of transitions between twoquantum states with back-scatter depend on the direction\nof the spins before and after scattering. For B=0 ,r e p l a c e -\nment of the direction of a wave vector by its o pposite\nresults in a spin- flip for states on the same Fermi contour,\nand to preserving its direction upon transition to another\nFermi contour. As a result, due to the selection rules forspin, there are no components in the FO of the LDS that\ndepend on the magnetic defect moment, and there are nat-\nurally no FO of the LDM during scattering by a non-magnetic defect (see Refs. 23,26,a n d33). In a parallel\nmagnetic field during back-scatter, states corresponding to\na spin- flip to some angle depending on the fieldBare per-\nmissible. As a result, new harmonics appear in the FO of\nthe LDS, whose periods depend only on the characteristics\nof one of the Fermi contours, as well as components pro-portional to the magnetic defect moment J.F o rt h es a m e\nreason, the FO of the LDM cont ain harmonics propor-\ntional to the constant γof the potential interaction of an\nelectron with a defect. Note that the listed conclusions\nremain valid in the presence of only one SOI type (Rashba\nor Dresselhaus), and all our analytical results make it pos-sible to set one of the constants of SOI equal to zero.\nAsymptotic formulas (43) and (51) –(53) provide a com-\nplete qualitative description of all harmonics of the FO of\nthe LDM and LDS, and the dependences of their periods\nand amplitudes on the fieldB.\nThe spatial distributions of local densities of states\nand magnetization in Figs. 1–5, obtained by means of the\ngeneral expressions (42) and (48) –(50), are only several\nspeci fic examples illustrating the variety of the nature of\nthe anisotropy of Friedel oscillations in the system understudy. We use the following dimensionless values in thecreation of the diagrams:\n/C22α¼\nmα\n/C22h2kF,/C22β¼mβ\n/C22h2kF,/C22h¼h\nϵF¼2mh\n/C22h2k2\nF,\n/C22k¼k\nkF,/C22r¼kFr,/C22ϵ¼ϵ\nϵF,( 65)\nΔ/C22ρ(/C22r)¼π2/C22h4\nm2(γþJ)/C18/C19\nΔρ(r);Δ/C22m(/C22r)¼π2/C22h4\nm2(γþJ)/C18/C19\nΔm(r),\n(66)\nwhere kF=√(2mϵF)/ħ. The arrows placed on the dia-\ngrams on the Fermi contours show the direction of the\nvector of velocity (arrows directed perpendicular to the line\nof the Fermi contour) and the direction of electron spin.\nFig. 1(a) demonstrates a violation of the central sym-\nmetry of the Fermi contours and a change in direction ofthe spins under the effect of a parallel magnetic field\nunder conditions of R-D SOI. With scattering by a non-\nmagnetic defect, the FO of the LDS Δ/C22ρ(/C22r)[ F i g . 1(b)] pre-\nserve the central symmetry, but no longer have symmetry\nrelative to axes x=yand x=–y,w h i c he x i s t si nt h e\nabsence of fieldB=0 .\n33The dashed lines in Fig. 1(b) show\nthe directions of the maximum amplitude of the FO thatcoincide with the direction of velocity at the in flection\npoints of Fermi contour belonging to branch ϵ\n2.E a c ht w o\nlines (with identical length of a stroke) limit the “fan”of the\ndirections, in which the FO have more than two harmonics.\nIn the zero field, both “fans ”coincide.\nFig. 2illustrates the effect of the emergence of a non-\nuniform distribution of the density of magnetization existingonly in a magnetic field with scattering by a non-magnetic\ndefect. It is interesting to note that potential scattering leads\nto nonzero density of magnetization not only in the plane 2D\nof electrons [Fig. 1, Eq. (6)] in which the vector Blies, but\nalso in the direction perpendicular to this plane [Fig. 1(a)].\nIn Fig. 3, the FO of the LDS caused by magnetic scatter-\ning for the special case when the Fermi energy coincides\nwith the energy ϵ\n0at the point of a contact of the branches\nof the range in (18), and a magnetic field greater than the\ncritical hc(38), i.e., a branch of the range ϵ=ϵ1(k) are given\nhas an absolute minimum with energy ϵ1min<ϵ0.\nFigure 4visually demonstrates the essential change in\nthe nature of the FO of the LDS associated with magnetic\nFig. 5. FO of the LDS /C22ρin the case of\nequality of constants of R-D SOI for\nscattering by a non-magnetic defect,\nγ≠0 and J=0 . ( a) Fermi contours,\n(b) oscillations of the z-component of\nthe LDM, /C22mz, for scattering by a mag-\nnetic defect, γ= 0 and J= (0, 0, J). The\nfollowing values of parameters were\nused in constructing the graphs: /C22α=/C22β=\n0.3,/C22h=1 ,wh=3π/4.scattering in a narrow interval of magnetic fields near the\nvalue\nh¼h(1)\nmin¼m\n2/C22h2(αþβ)2þϵF,wh¼3π\n4, (67)\ncorresponding to the decreases in the minimum of the branch\nof range ϵ1with Fermi level ϵF.F o rh >hmin(1)there is only\none Fermi contour belonging to branch ϵ2[inset in Fig. 4(a)]\nand de fining the unique harmonic of the FO in Fig. 4(a).F o r\nmagnetic fields lower than the minimum fieldh<h min(1),a\nsecond Fermi contour occurs [inset in Fig. 4(b)], belonging\nto branch ϵ1(a 2D analog of the topological Lipshitz transi-\ntion with the appearance of a new cavity of the Fermi\nsurface), which results in the emergence of a second har-\nmonic of FO with greater period and amplitude, as in\nFig. 4(b). The distributions of the LDS are constructed for\nthe case where h>hc(38)andϵF<ϵ0(18).\nFigure 5pertains to the special case α=βandh=h/\n√2(–1, 1, 0), examined in section 5.3. For the chosen\nvalues of parameters, the Fermi contours have two common\npoints, i.e., ϵF>ħ2h2/8mα2[Fig. 5(a)]. According to the\nasymptotic formula (63), there are visible in Fig. 5ellipses of\nlines of constant phase in the spatial distribution of the LDScomponent /C22m\nzaround the magnetic defect.\n6. Conclusion\nWe investigated the effect of a parallel magnetic field on\nFriedel oscillations of the local densities of states and of\nmagnetization in a 2D electron gas with R-D SOI that are\nassociated with scattering by a magnetic defect. It is shown\nthat a magnetic field breaking the central symmetry of the\nlaw of scattering results in the appearance in the FO of\nthe LDS of harmonics caused by magnetic scattering, and the\nnon-magnetic impurity generates FO of the LDM. 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B 74, 085411 (2006).\n42Ye. S. Avotina, Yu. A. Kolesnichenko, S. B. Roobol, and J. M. Ruitenbeek,Fiz. Nizk. Temp. 34, 268 (2008) [Low Temp. Phys. 34, 207 (2008)].\n43N. V. Khotkevych-Sanina, Yu. A. Kolesnichenko, and J. M. van\nRuitenbeek, New J. Phys. 15, 123013 (2013)."    },    {        "title": "1907.04697v2.Gapless_regime_in_the_charge_density_wave_phase_of_the_finite_dimensional_Falicov_Kimball_model.pdf",        "content": "arXiv:1907.04697v2  [cond-mat.str-el]  19 Aug 2019Gapless regime in the charge density wave phase of the finite d imensional\nFalicov-Kimball model\nMartin Žonda, Junichi Okamoto and Michael Thoss\nInstitute of Physics, Albert-Ludwig University of Freibur g,\nHermann-Herder-Strasse 3, 791 04 Freiburg, Germany\nThe ground-state density of states of the half-filled Falico v-Kimball model contains a charge-\ndensity-wave gap. At finite temperature, this gap is not imme diately closed, but is rather filled in\nby subgap states. For a specific combination of parameters, t his leads to a stable phase where the\nsystem is in an ordered charge-density-wave phase, but ther e is high density of states at the Fermi\nlevel. We show that this property can be, in finite dimensions , traced to a crossing of sharp states\nresulting from the single particle excitations of the local ized subsystem. The analysis of the inverse\nparticipation ratio points to a strong localization in the d iscussed regime. However, the pronounced\nsubgap density of states can still lead to a notable increase of charge transport through a finite size\nsystem. We show this by focusing on the transmission in heter ostructures where a Falicov-Kimball\nsystem is sandwiched between two metallic leads.\nI. INTRODUCTION\nThe Falicov-Kimball model (FKM)1is one of the\nsimplest models for the description of correlated elec-\ntrons on a lattice. Over time, it has become a stan-\ndard tool for the investigation of various phenomena\nincluding crystallization2–4, metal-insulator and valence\ntransitions5–16, inhomogeneous charge and spin orderings\n17–26, nonlocal correlations27–30, ferroelectricity31–35,\nmixtures of heavy and light cold atoms in optical\nlattices36–39, transport through layered systems40–45or\ndifferent nonequilibrium phenomena46–55.\nIts biggest advantage over the paradigmatic Hubbard\nmodel56lies in the fact that it is accessible by exact\nmethods. It is exactly solvable in the limit of infinite\ndimensions (infinite coordination number) by means of\ndynamical mean field theory (DMFT)57–60and it can be\naddressed by an exact, sign-problem-free Monte Carlo\n(MC) method24,61–63in finite dimensions. Both meth-\nods take advantage of the fact that the FKM combines\nquantum and classical degrees of freedom.\nDespite the simplicity and accessibility of this model,\nits research continues to offer new and often surprising re-\nsults. This is true even for its simplest spin-less version a t\nthe particle-hole symmetric point. For example, nonequi-\nlibrium DMFT and cluster approximation method stud-\nies showed that its quantum subsystem does not ther-\nmalize after an interaction quench48,50,55; simple gener-\nalizations of the FKM can be used to study the inter-\nplay of topology and interactions at finite temperatures64\nor fractionalization of particles into charge and spin\nobjects65; it was used to derive universal features of the\ncritical metal-insulator transition that are transferabl e\nto other Hubbard-like models13,66, utilized in studies of\ndifferent quasiparticles67,68; and, as discussed below, the\nphase diagram of the model is still in question as well.\nThe Hamiltonian of the spinless FKM at half fillingreads\nHFK=−t/summationdisplay\n/an}bracketle{tl,l′/an}bracketri}ht/parenleftig\nd†\nldl′+d†\nl′dl/parenrightig\n+U/summationdisplay\nl/parenleftbigg\nf†\nlfl−1\n2/parenrightbigg/parenleftbigg\nd†\nldl−1\n2/parenrightbigg\n(1)\nwhere the first term describes nearest-neighbor hopping\nof spinless electrons on a lattice. The second term rep-\nresents a Coulomb-like local interaction between the lo-\ncalizedfparticle and itinerant delectron on the same\nlattice site. The terms with factor −1\n2secure the half-\nfilling conditions Nf+Nd=L/2for chemical potential\nµ= 0. HereNf(d)is the total number of f(d) particles\nandLis the total number of lattice points.\nThe phase diagram of this model in finite as well as in-\nfinite dimensions contains three main phases (see Fig. 1):\nan ordered charge density wave (CDW) phase (OP) that\nexists at low temperatures57,58,62,69,70, a gapless disor-\ndered phase for weak interaction Uand high tempera-\ntures (DPw), and a gapped disordered phase for strong\ninteraction Uand high temperatures (DPs)60,71.\nHowever, this is not a complete picture. A recent study\nof the model on a two-dimensional ( D= 2) lattice72\nshowed that in the thermodynamic limit DPw exhibits\nAnderson localization which destabilizes the metallic-li ke\nphase reported in older works focused on relatively small\nlattice sizes61,62. Therefore, all three phases are insulat-\ning in the thermodynamics limit. Nevertheless, for any\nfinite system size, there is a crossover from an Ander-\nson localized insulating phase at intermediate Uthrough\na weakly localized regime, with the above mentioned\nmetallic-like character, to a Fermi gas at U= 0. In addi-\ntion, a series of papers proved that in infinite dimensions\n(D→ ∞) there is a stable ordered CDW phase without\na gap at the Fermi level73–77.\nThe gapless CDW phase in the infinite dimensional\nFKM is related to the existence of narrow bands in the\ndensity of states (DOS) that are formed inside the CDW\ngap at finite temperatures. These subgap bands come\nin pairs placed symmetrically around the center of the2\n 0 0.1 0.2 0.3\n 0 2 4 6 8 10 12 14 16 18T/t\nU/tOPDPsDPw 0 0.2\n-4-2 0 2 4U = 2t T = 1tDOS\nE/t 0 0.4\n-10 -5 0 5 10U = 10t\nT = 1tDOS\nE/t\n 0 3\n-8-6-4-2 0 2 4 6 8U = 10t\nT = 0.06tDOS\nE/t\nFigure 1. Simplified phase diagram of the spinless FKM on\na square 2D lattice with ordered CDW phase (OP) and dis-\nordered phases in weak (DPw) and strong (DPs) interaction\nregimes. The insets illustrate typical d-electron DOS in the\nrespective phases.\ngap and merge for a finite range of interaction strengths\nUand temperatures. The resulting merged single sub-\ngap band is centered around the Fermi level and, con-\nsequently, there is no gap at the Fermi level. Lemański\nargued that this merging is related to the inversion of the\nsubgap bands belonging to two sublattices of a bipartite\nlattice at critical interaction Uc76. Here a sublattice A\nconstitutes such lattice points that all their nearest neig h-\nbors belong to the sublattice B and vice versa (they are\nalternating). The DOS calculated for each sublattice sep-\narately contains both subgap bands placed symmetrically\naround the Fermi energy. However, they differ in width\nand height. This property is related to the CDW ordering\nbecause the average f-electron occupancy differs for the\nsublattices as was discussed in various studies73–75. What\nis interesting is that at some critical Ucthe qualitatively\ndifferent subgap bands belonging to different sublattices\nflip positions.\nThe open question is if there is an analog of such a\nband crossing in finite dimensions as well. The subgap\nstates, respective bands, had already been discussed in\nbothD= 2andD= 322,61,62. However, a systematic\nstudy focused on the region around Ucis missing. The\npresent paper has the aim to fill this gap. We show that\nthere is indeed a clear crossing of distinct subgap states\nin finite dimensions. Moreover, we reveal the underlying\nmechanism of the crossing by focusing on the single par-\nticle excitations from the CDW ground state. We show\nthat these excitations significantly influence the density\nof states up to surprisingly high temperatures approach-\ningTcof the order-disorder transition. We also demon-\nstrate that, despite the presence of strong localization,\nthe crossing can support charge transmission through a\nfinite-sized system in the gapped regime. This is done\nby addressing a heterostructure where the finite system\nmodeled by the FKM is sandwiched between two metal-\nlic leads. We mostly focus on the D= 2case, butD= 1\nandD= 3are addressed as well.\nThe rest of the paper is organized as follows. In Sec. IIwe outline the methodology for addressing the thermo-\ndynamic properties of the FKM and introduce the model\nof the heterostructure as well as the method for studying\ncharge transport in it. The main results are presented in\nSec.III, where we first analyze the origin and properties\nof the sharp features of the subgap DOS in Sec. III A\nand then show in Sec. III Bhow these affect the charge\ntransport through finite FKM coupled to metallic leads.\nSection IVconcludes with a summary. In the Appendices\nwe show some analytical results on the existence of the\ngap and the positions of sharp subgap states.\nII. METHODS\nA. Thermodynamic properties\nThefparticles in the FKM represent classical degrees\nof freedom. This can be seen from the fact that the\nf-particle occupation numbers f†\nlflcommute with the\nentire Hamiltonian in Eq. ( 1). This allows us to replace\nany operator f†\nlflby its eigenvalues wl= 0orwl= 1\nand write a partial Hamiltonian for a particular classical\nconfiguration w. After neglecting the constant energy\nshift−U/4, the Hamiltonian in Eq. ( 1) reads for a chosen\nconfiguration w\nHw=/summationdisplay\nl,l′hll′d†\nldl′−U\n2Nf=/summationdisplay\nαεα˜d†\nα˜dα−U\n2Nf.(2)\nThereby,εjare the eigenvalues of the matrix with ele-\nmentshll′=Uwlδll′−tδ/an}bracketle{tl,l′/an}bracketri}ht, whereδ/an}bracketle{tl,l′/an}bracketri}ht= 1when the\nlattice positions represented by vectors landl′are the\nnearest neighbors and zero otherwise. The ground-state\nconfiguration for any bipartite lattice at the particle-\nhole symmetric point is the checkerboard ordering of f\nparticles2,71. The corresponding configuration wcan be\nwritten aswl= (1 +eiπl)/2for any hypercubic lattice.\nIt is easy to show (see Appendix A) that such an effec-\ntive potential opens a gap of the width Uin the band\nstructure which is centered around the Fermi energy.\nAn advantage of the Falicov-Kimball model is that the\nmean values of any d-electron operator ˆOcan be written\nin the form\n/angbracketleftig\nˆO/angbracketrightig\n= Trw/angbracketleftig\nˆO/angbracketrightig\nd≡1\nZ/summationdisplay\nwe−βF(w)/angbracketleftig\nˆO/angbracketrightig\nd,(3)\nwhere\nF(w) =−1\nβ/summationdisplay\nαln/bracketleftbig\n1+e−βεα/bracketrightbig\n−U\n2Nf, (4)\nwithZ=/summationtext\nwe−βF(w)being the partition function (we\nassumeµ= 0). Here/an}bracketle{t./an}bracketri}htdis the trace over the d-electron\nsubsystem for fixed w61. As this is a single-particle prob-\nlem, the trace can be calculated effectively using exact\nnumerical diagonalization. The sum over configurations3\nwcan then be calculated using a Metropolis algorithm\nbased Monte Carlo method24,61–63,78.\nThe quantities that we are mostly interested in are the\nnormalized DOS defined as\nDOS(ε) =1\nLTrw/summationdisplay\nαδ(ε−εα) (5)\nwhereTrw≡1\nZ/summationtext\nwe−βF(w)and averaged inverse partic-\nipation ratio (IPR)\nIPR(ε) = Tr w/summationtext\niDOSi(ε,w)2\nDOS(ε,w)2, (6)\nwhereDOSi(ε,w) =/summationtext\nαδ(ε−εα)UiαU†\nαi/Lis the local\nDOS and the matrix Uconsists of the eigenvectors be-\nlonging to eigenvalues εαof the matrix hfrom Eq. ( 2)\ncalculated for a particular configuration wand arranged\nin columns. The matrix Ucan be evaluated numerically\nfor a finite system and it can be chosen to be real.\nThe finite size scaling of the IPR can be used for\nstudying localization of the itinerant electrons in the\nsystem79–81. TheIPRscales as 1/Lfor a completely itin-\nerant system states and converges to a finite value with\nincreasingLfor strongly localized states. In the case of\nperfect spacial localization to a single lattice point, the\nIPR converges to one. Note, that because of the finite\nsize of our lattices, we use a Gaussian broadening of the\nδ-functions, δ(ε−εα)≈exp[−(ε−εα)2/(2b2)]/(b√\n2π).\nIn most of presented cases, we set the broadening con-\nstant tob= 0.002t. This small value is a compromise\nbetween the effort to suppress the influence of the arti-\nficial broadening on our results (especially IPR) and the\npreservation of the stability of the calculations of the IPR\nfor a broad range of temperatures and lattice sizes. We\ndiscuss the influence of the Gaussian broadening on our\nresults in detail below.\nB. Heterostructure\nBesides studying an isolated FKM, we also address a\nheterostructure Hh=HFK+/summationtext\nl=L,RHl\nlead+Hl\nhyb, where\na two dimensional FK system is sandwiched between two\nmetallic leads as illustrated in Fig. 2. The central sys-\ntemHFKis finite in xbut in principle infinite in the\nydirection (modeled by periodic boundary conditions).\nThe leads and hybridization terms read\nHl\nlead=−tl/summationdisplay\n/an}bracketle{tm,n/an}bracketri}ht/parenleftig\nc†\nl,mcl,n+cl,nc†\nl,m/parenrightig\n+ǫl/summationdisplay\nnc†\nl,ncl,n,\nHl\nhyb=−γl/summationdisplay\n/an}bracketle{ti,n/an}bracketri}ht/parenleftig\nc†\nl,ndi+d†\nicl,n/parenrightig\n, (7)\nwheretlis the hopping for lead l=L,R,ǫrepresents\nan energy shift of the lead, and γlis hopping parameter\nbetween the system and the lead l.\nWe have two main motivations for addressing this more\ncomplex setup. First, the broadening of the system statest tLγL tRγRμL\nμR\nxy V\n2B\nFigure 2. Schematic picture of the heterostructure. The bla ck\npart represents the two-dimensional FKM system with near-\nest neighbor hopping t. The red parts are noninteracting leads\nwith hopping tL,Rand the hybridization interaction with hop-\npingγL,Ris indicated in blue. The leads are characterized by\nelliptic surface DOSs. The voltage drop Vis introduced by\nmutual shift of ǫL,Rwhere the condition µL,R=ǫL,Ris used\nto keep the bands half filled at any applied voltage.\nis in the case of the heterostructure provided naturally by\nthe coupling γlto the semi-infinite leads, which allows to\ntest the results obtained for isolated system potentially\ninfluenced by an artificial broadening of the δfunctions.\nSecond, we want to reveal how the existence of the finite\nDOS in the CDW gap influences the transport properties\nof the model.\nThe properties of the heterostructure can be effec-\ntively addressed by a combination of a sign-problem free\nMonte-Carlo with the nonequilibrium Green’s function\ntechnique approach45. We assume in our analysis that\nthe central FK system was in the distant past decoupled\nfrom the leads and that both system and leads had been\nin thermal equilibrium. The occupation numbers of the\nfelectrons are integrals of motion, therefore, their distri -\nbution can be calculated for the isolated system as it will\nnot change after coupling to the leads. Here, in contrast\nto the previous subsection, we assume open boundary\nconditions of the central system at the system-lead in-\nterface. Test calculations show, that if system is large\nenough (Lx>10) the influence of the boundary condi-\ntions on the f-electron distribution is negligible. We fur-\nther assume, that the semi-infinite leads are unaffected\nby the system and are modeled by parallel chains cou-\npled to the central system as shown Fig. 2. Therefore,\nthe leads can be characterized by their surface density of\nstates\nρl(ε) =2\nπB2/radicalig\nB2−(ε−ǫl)2,\nwith band half-width B= 2tlcentered around the band\nenergy shift ǫlfrom Eqs. ( 7). This allow us to write an\nexact formal solution for the Green’s function of the het-\nerostructure for a particular configuration w(for details,\nsee Refs.45,82):\nGr,a(ε) =gr,a(ε)+gr,a(ε)Σr,a(ε)Gr,a(ε),(8)\nG<(ε) =Gr(ε)Σ<(ε)Ga(ε). (9)4\nHere,Gr(a)is the retarded (advanced) Green’s function\nof the coupled system, G<is the lesser Green’s func-\ntion of the coupled system, and gr(a)(ε)is the retarded\n(advance) Green’s function of the bare system with com-\nponents:\ngr,a\nαβ(ε) =δαβ\nε−εα±i0. (10)\nThe total tunneling self-energies Σr,a,<=Σr,a,<\nL+Σr,a,<\nR\nhave the components\nΣr,a\nl,αβ(ε) = Λl,αβ(ε)±i\n2Γl,αβ(ε),\nΣ<\nl,αβ(ε) =iΓl,αβ(ε)fl(ε−µl),\nΓl,αβ(ε) = 2πγ2U{sl}\nαβρl(ε), (11)\nΛl,αβ(ε) =\n\n2γ2\nB2U{sl}\nαβ(ε−ǫl)for|ε−ǫl|<B\n2γ2\nB2U{sl}\nαβ/bracketleftbigg\n(ε−ǫl)∓/radicalig\n(ε−ǫl)2−B2/bracketrightbigg\nfor(ε−ǫl)≷±B\nU{sl}\nαβ=/summationdisplay\ni∈ {sl}U†\nβiUiα,\nwhere{sL,R}are the sets of system lattice positions at\nthe left and right interfaces, fl(ε)is the Fermi function,\nandµl=L,Ris the chemical potential of the leads.\nThe transmission function, which contains the most de-\ntailed information on charge transport, has for a specific\nwa compact form83,\nΘw(ε) = Tr d[ΓL(ε)Gr(ε)ΓR(ε)Ga(ε)],(12)\nwhere the trace goes over the d-electron subsystem. Its\ntotal mean value is obtained by a trace over the f-\nelectron subsystem which is done by the MC method.\nSimilarly, the local density of states of a coupled system\n(heterostructure) for a specific wcan be calculated as\nLDOSh i(ε,w) =i\n2πL/summationdisplay\nα,βUiαU†\nβiGr\nαβ(ε)(13)\n−/summationdisplay\nα,βU†\nαiUiβGa\nαβ(ε),\nand it allows us to define an averaged generalized inverse\nparticipation ratio (gIPR)\ngIPR(E) = Tr w/summationtext\niLDOSh2\ni(ε,w)\nDOSh2(ε,w), (14)\nwhereDOSh(ε,w) = Tr di[Gr(ε)−Ga(ε)]/2πL.\nIn the following, we set µL=ǫLandµR=ǫR, which\ncorresponds to half-filled lead bands and we introduce a\nfinite voltage drop as V=µL−µRwith antisymmetric\nconditionµL=−µR. In equilibrium, we set µL=µR=\n0and assume that the electrostatic potential (set to zero)\nof the central system is uninfluenced by the leads. This\nfixes the half-filling condition for the central system. We\nfocus on equivalent leads γ=γL=γRwith a broad band\nhalf-widthB= 10t.III. RESULTS\nA. Subgap density of states\nThe typical subgap bands, calculated for D= 2\nat temperatures in the vicinity of the CDW phase\ntransition61,62,72, resemble the exact DMFT results cal-\nculated for infinite dimensions73–76. However, in finite\ndimensions the subgap bands reduce with the decreasing\ntemperature into sharp features pinned mostly to few\ndistinct energies. This is illustrated in Fig. 3, where we\nshow the subgap ( |ε|< U/2) low-temperature DOS cal-\nculated for L= 24×24, three values of U, and various\ntemperatures below Tc.\nThe character of the subgap DOS changes with U. For\nU= 1t, there are two pronounced maxima placed sym-\nmetrically at ε∼ ±0.25U; forU= 2.5t, a sharp max-\nimum is centered around ε= 0and is accompanied by\ntwo sharp features of approximately a third of its height,\nwhich are placed at ε∼ ±0.15U; forU= 4t, four sharp\nlocal maxima of comparable heights exist at ε∼ ±0.13U\nandε∼ ±0.21U. Considering their positions, the sub-\ngap maxima have the same qualitative dependence on U\nthat was described for the subgap bands for D→ ∞73–76.\nNamely, the maxima approach each other with increasing\nUuntil they merge at some critical Ucand then, above\nit, draw apart.\nThe position of these most pronounced distinct local\nmaxima does not depend on the temperature and, in con-\ntrast to other states, their weight is not completely neg-\nligible even for a very low temperatures.\nBecause the d-electron DOS is dictated by the distribu-\ntion offparticles, one can expect that the origin of these\ndistinct maxima must be in some states reflecting the low\nenergy disruptions of the ground-state checkerboard or-\ndering. To analyze this conjecture we focus on three typ-\nical singlef-electron excitations from the checkerboard\nordering illustrated in the top panels of Fig. 4. They\nrepresent an addition of one felectron to the otherwise\nperfect checkerboard ordering; a removal of one felec-\ntron and, finally, a displacement of a single felectron\nto the nearest unoccupied lattice point. The actual spa-\ncial position of these three defects does not play a role,\nas we are assuming periodic boundary conditions. Note\nthat because felectrons can be also interpreted as ions,\nthe studied defects can be seen as an equivalent of typi-\ncal lattice defects such as vacancies (Schottky defects) or\ninterstitial defects (Frenkel defects).\nBecause the above disruptions of the checkerboard pat-\ntern play a role of classical single impurities, they lead\nto sharp bound states inside of the CDW gap84. We plot\nthe normalized position of these states (i.e., their calcu-\nlated eigenenergies) in Fig. 4as a function of Uin all\nthree realistic dimensions (for details and analytical re-\nsults, see Appendix B). The positions of the the main\npeaks in the finite temperature subgap DOS plotted in\nFig.3coincide with the eigenenergies at the same Uand\nD= 2shown in Fig. 4(b) (they are also signaled by the5\n— 0 0.1 0.2\n-0.5 -0.25  0.25  0.5a\nU = 1tDOS\nε/UT=0.04t\nT=0.03t\nT=0.02t\n 0 0.2 0.4\n-0.25  0  0.25  0.5b\nU = 2.5tDOS\nε/UT=0.10t\nT=0.08t\nT=0.06t\n 0 0.1 0.2 0.3\n-0.5 -0.25  0.25  0.5c\nU = 4tDOS\nε/UT=0.13t\nT=0.12t\nT=0.10t\nT=0.08t10-210-1100\n-0.015  0.015T=0.095tDOS\nε/UL=24×24\nL=30×30\nL=40×40\nFigure 3. Details of the subgap density of states ( |ε|< U/2)\nfor three values of of Uand various values of temperature.\nAll data have been calculated on a two-dimensional square\nlattice with size L= 24×24and periodic boundary conditions.\nThe sharp states, calculated for particular configurations w\nin the sampling process, have been broadened by a Gaussian\nbroadening (see text) with b= 0.002t. The arrows signal the\npositions of the bound states belonging to single f-electron\nexcitations discussed in the text. Their color coding is the\nsame as in Fig. 4(b). The inset in panel (b) is the detail of\nthe DOS close to the Fermi level calculated for L= 24×24,\n30×30andb= 0.002tatT= 0.095t.\narrows in Fig. 3). The highest peaks in Fig. 3reflect an\naddition or removal of an felectron and the second in\norder reflect a single f-electron displacement. Therefore\nwe can conclude that these simple f-electron excitations\nare the main underlaying mechanism for the stable finite-\ntemperature subgap anomalies.\nThe results show that displacing an felectron leads to\ntwo subgap bound states placed symmetrically around\nε= 0. Adding or removing an felectron leads to a\nsingle subgap state. These are related by εadd=−εrem\nand cross each other at critical Ucwhich depends on the\ndimension. For the one-dimensional case we get Uc=\n4/√\n3, forD= 2it isUc∼=2.5, andD= 3leads toUc∼=\n3.18t(see Appendix B)85. This means that similarly to\nthe infinite dimensional case76, the main subgap features\nexchange roles at Uc.\nAnother qualitative change that takes place at Uccan-0.5-0.25 0 0.25 0.5\n 0  5  10  15  20a\nD = 1εsg /U\nU/tadd\nrem\ndis\n-0.5-0.25 0 0.25 0.5\n 0  5  10  15  20b\nD = 2εsg /U\nU/tadd\nrem\ndis\n-0.5-0.25 0 0.25 0.5\n 0  5  10  15  20c\nD = 3εsg /U\nU/tadd\nrem\ndis\n 0 0.2 0.4 0.6 0.8\n 0 1 2 3d\nD=2(Ee-Eg)/t\nU/t 0  2  4  6e\nD=1\nU/tdis\nadd, rem\nNd = L - Nfdisplaced added removed\nFigure 4. a-c) Position of the subgap states for disrupted\ncheckerboard ordering of felectrons in different dimensions.\nThree single f-electron excitations are considered. Namely,\nadding one additional felectron to an unoccupied lattice\npoint (add), removing one felectron (rem) and displacing one\nelectron from its position to a nearest neighboring unoccup ied\npoint (dis) as illustrated for D= 2case in the top three pan-\nels. The results have been obtained using the analytical and\nsemi-analytical formulas from Appendix B and represent the r-\nmodynamic limit solutions ( L→ ∞). d-e) Difference between\nthe minimal energy of the disrupted and perfect checkerboar d\nordering of felectrons. The energy differences for add and\nrem cases are identical. The dotted line represents the ener gy\nof the system where the half-filling Nf+Nd=Lis enforced.\nSolid red line breaks this condition below Ucby one particle\nNf+Nd=L±1.\nbe seen from the difference between the minimal energy\nof our disrupted configurations ( Ee) and the real ground-\nstate energy of the checkerboard ordering ( Eg) plotted in\nFig.4(d). There is a kink in the Ee−Egdependence on U\n(solid red line) exactly at Ucfor and added as well as re-\nmovedfelectron. This is because below Ucthe minimal\nenergy requirement always sets Nd=L/2, which leads to\nNd+Nf=L+1for a configuration with an additional\nfelectron and Nd+Nf=L−1for a removed one. This\nmay seem strange, considering that µ= 0should ensure\nthe half-filling, but it is a straightforward consequence of\nthe finite lattice size. The number of eigenenergies εα\nequalsL. These are, for a perfect checkerboard pattern,\ndivided equally into two bands (Appendix A). For a dis-\nrupted configuration, the subgap state εaddis pulled out\nfrom the lower band. Consequently, as it is energetically\nadvantageous to occupy this state if εadd<0, which is\nthe case below Uc, this leads to Nd=L/2. AboveUc,\nwe haveεadd>0, which leaves the state unoccupied and\nthereforeNd=L/2−1. The opposite is true for εrem.\nTherefore, the half-filling for any single of these excita-\ntions is restored only above Ucand we need a combination\nof them to fulfill this requirement below Uc. The energy6\n 0 0.1 0.2 0.3 0.4 0.5\n 1  2  3  4  5  6aDOS(ε=0)\nU/tT=1.00t\nT=0.12t\nT=0.10t\nT=0.08t\nT=0.06t\nT=0.04t\n10-410-310-210-1\n 2.4  2.5\nU/t0.00.20.4\n 200  400  600  800  1000U = 2.5t, b = 0.002bDOS(ε=0)\nLT = 0.07t  \nT = 0.095t\nT = 1.0t    \n0.00.30.6\n10-210-1L = 20×20\nU = 2.5tcDOS(ε=0)\nbT=0.095t\nT=0.5t\nFigure 5. (a) Density of states at the Fermi level as a func-\ntion ofUfor a square lattice L= 20×20. The inset shows\ndetails of the peak at Uc= 2.5twhere the bound states from\nFig. 4(b) cross each other. (b) Finite size scaling of the DOS\nat the Fermi level for various temperatures calculated with\nbroadening b= 0.002t. (c) Dependence of the DOS at the\nFermi level on the used artificial broadening.\nprofile for forced condition Nf+Nd=Lis plotted in\nFigs.4(d,e) using a dotted line. Note that the situation\nforD= 1is somewhat more complicated as here the\ndisplacement of a single fparticle can have a lower en-\nergy than adding or removing a localized particle at both\nweak and strong interaction limits [see Fig. 4(e)].\nThe above discussed single f-electron excitations have\na profound effect on the DOS even at surprisingly high\ntemperatures. This is already illustrated by the sharp\nsubgap features in the finite temperature DOS plotted in\nFig.3. Nevertheless, we are especially interested in the\nordered phase with high DOS at the Fermi level analo-\ngous to the one observed in infinite dimensions75. There-\nfore, we show in Fig. 5(a) the dependence of DOS (ε= 0)\nonUat various temperatures. The highest temperature\nT= 1trepresents the disordered phase, T= 0.12tis just\nabove the critical temperature for U= 2.5tand the rest\nis below it. Figure 5(a) is a direct D= 2analog of the\ninfinite dimensional case shown in Fig. 7 of the work by\nLemański and Ziegler75. Although both cases share some\nqualitative characteristics, like the increase of the DOS\naroundUcfor small temperatures, there is one distinct\ndifference. The increase of the DOS around Ucis not\nonly much sharper, but for T/lessorsimilarTcit also leads to val-\nues of DOS which are higher than the high temperature\nlimit where the gap is completely closed. This is again\nillustrated in Fig. 6(a), where we show the temperature\ndependence of the DOS (ε= 0) forD= 2. The posi-\ntion of the maximum of DOS ( Tm∼0.095t) is clearly\nbelow the critical temperature ( Tc∼0.12t). Moreover,\nthe DOS profile is stable with respect to the lattice size\nas it is illustrated in Fig. 6(a), Fig. 5(b), and in the inset\nof Fig. 3(b), where we show the detail of the DOS around\nthe Fermi level.\nThe above-discussed DOS were calculated with an\nartificial Gaussian broadening of the δ-functions with\nb= 0.002t. The effect of the Gaussian broadening on\nthe DOS (ε= 0) is completely negligible in the disor-dered phase as illustrated in Fig. 5(c) by the red circles.\nThe situation in the ordered CDW phase is more com-\nplicated, especially for critical Ucand low temperatures.\nFigure 5(c) shows that the DOS (ε= 0) calculated at\nT= 0.095t(black circles) increases with the decreasing\nbroadening. Also in Fig. 6(a), one can see by follow-\ning the dashed line, which represents the DOS (ε= 0)\ncalculated for L= 20×20and broader broadening of\nb= 0.004t, that bigger artificial broadening leads to a\ndecrease of the calculated DOS (ε= 0)maximum.\nThe reason is twofold. For the D= 2, the band around\nε= 0has a fine structure, as is shown in the inset of\nFig.3(b), and has a clear maximum at ε= 0. A wide\nartificial broadening smooths this structures, which sig-\nnificantly lowers the DOS at the Fermi level. From our\ndata it is not clear if this fine structure will disappear in\nthe thermodynamic limit. It seems not to be the case,\nas the detail of the DOS shown in the inset of Fig. 3(b)\ndepends only weakly on the lattice size, but a study on\nmuch bigger lattices is required to confirm this.\nEven more important is that despite the broadening\ninto a band provided by multi-particle excitations, the\nstates at the Fermi level stay very sharp even for large\nlattices, as is shown in the inset of Fig. 6(b). This ef-\nfect becomes even more crucial with decreasing temper-\nature as here the configurations with just one fparticle\nadded or removed from a perfect checkerboard ordering\ncan have a very high weight. This is shown in Fig. 6(b),\nwhere we plotted the total MC weight of these config-\nurationw∼/parenleftbig\ne−βF(wadd)+e−βF(wrem)/parenrightbig\n/Zas a function\nof temperature for various lattice sizes (solid lines) and\ncompare it with the analogous weight for checkerboard\nordering (dashed line). There is a clear maximum at\nwhich the combined weight of the above excited states is\nalmost one-third of the total one. This can explain the\nextremely sharp subgap states for low temperatures in\nthe finite system shown in Fig. 386.\nThe enhanced DOS at the Fermi level caused by the\ncrossing of the sharp maxima might raise a question if the\nphase has still an insulating character. However, know-\ning that the prevailing mechanism behind this effect is\nthe impurity like single-particle excitation of the local-\nized subsystem, one can expect a strong localization of\nthedelectrons. We show by studying the scaling of the\naveraged IPR( ε= 0) depicted in Fig. 7(a) that this is\nreally the case. For localized states, the IPR should sat-\nurate with increasing lattice size to a finite value. Note\nthat IPR →1would point to a complete localization of\nthedelectrons to a single lattice point. Therefore, the\nsaturation of IPR to ∼0.16forT= 0.095tshown in\nFig.7(a) (red circles) still points to a strong localization.\nOn the other hand, the slow decline of the IPR for T= 1t\n(blue circles) points to a weak localization as expected in\nthis regime for a finite size system45,72.\nAs shown in Fig. 7(b), the IPR is, in contrast to\nthe DOS results discussed above, relatively stable for a\nbroad range of broadening parameter b(note the logarith-\nmic scale) even below the critical temperature. There-7\n 0 0.2 0.4 0.6\n 0  0.1  0.2  0.3  0.4  0.5ab=0.002tDOS(ε=0)\nT/tL=20×20\nL=24×24\nL=30×30\nb=0.004t\n 0 0.2 0.4 0.6 0.8 1\n 0  0.1  0.2  0.3  0.4  0.5bw\nT/t10-210-1100\n-0.015  0.015L=24×24T=0.095tDOS\nε/Ub=0.0002\nb=0.002\nb=0.02\nFigure 6. (a) Density of states at the Fermi level as a func-\ntion of temperature for D= 2atU= 2.5tcalculated with\nbroadening parameter b= 0.002t(solid lines) for various lat-\ntice sizes and with b= 0.004tforL= 24×24(dashed line).\nThe vertical black dotted line signals the critical tempera ture\nof the order-disorder phase transition. (b) Dependence of t he\nstatistical weight of the configurations with perfect CDW or -\ndering (green dashed line calculated for L= 20×20) and\nconfigurations with a single fparticle added or removed from\nCDW.\nfore, the conclusions of strong localization of the crossed\nstates is not affected by the artificial broadening of the\nδ-functions.\nTo further support these findings, we provide an al-\nternative test of the above conclusions in the next sub-\nsection. Instead of artificial broadening, we consider a\nheterostructure where the system is coupled to two semi-\ninfinite metallic leads. These provide a different kind\nof broadening of the system states. It can be argued\nthat this broadening is more natural, but it is also un-\neven, because the broadening of the LDOS decreases with\nincreasing distance from the system-leads interface41,45.\nIn addition, this setup allows us to study the transport\nthought a finite system.\nB. Transport properties of a heterostructure\nWe have shown recently45that the typical phases of the\nFKM have different influence on the transport properties\nof a heterostructure. However, that study did not focus\non the particular case where the system is in CDW phase\nbut has a large density of states at the Fermi level as is\nthe case in Fig. 6(a). Here we study the effect of the finite\nDOS in this regime on the transport properties for a finite 0.01 0.1\n 200  400  600  800  1000a\nU = 2.5tIPR(ε=0)\nLT = 0.095t\nT = 1.0t    \ngIPR, T = 0.095t, γ = 2t\ngIPR, T = 1.0t, γ = 2t\n 0.08 0.12 0.16\n10-210-1100L = 20×20×10\nbIPR(ε=0)\nbT=0.095t\nT=0.5t\nFigure 7. (a) System size scaling of the averaged inverse par -\nticipation ratio for U= 2.5tatT= 0.095t(red circles) and\nT= 1t(blue circles) and system size scaling of the generalized\naveraged inverse participation ratio for the same paramete rs\nandγ= 2t(black and green squares). (b) Dependence of the\ninverse participation on the artificial broadening of the de lta\nfunctions. The IPR for T= 0.5tis multiplied by factor of 10.\n 0 0.4 0.8 1.2 1.6\n 0  0.1  0.2  0.3  0.4  0.5  0.6U = 2.5t\nL=20×20DOSh(ε=0)\nT/tγ = 0.5t\nγ = 1t\nγ = 2t\n 0 1 2\n 50 300  600  900γ = 2tDOSh(ε=0)\nLT = 0.095t\nT = 1.0t\nFigure 8. Averaged DOS of the coupled system calculated\nforU= 2.5t,L= 20×20and three values of γ. The posi-\ntion of the maximum T∼0.095tcoincides with the position\nof the maxima in Fig. 6(a). The inset represents the finite\nsize scaling of DOS at T= 0.095t(maximum) and 1t(high\ntemperature).\ndimensional central system. Because of that, we first\nhave to readdress the problem of DOS and localization\nfor the heterostructure.\nIn contrast to the isolated FKM studied in the pre-\nvious section, in the heterostructure the broadening of\nthe system states results naturally from the coupling to\nthe semi-infinite leads. In Fig. 8, we show the averaged\nDOSh(ε= 0)forL= 20×20and three values of γ. The8\n 0 0.04 0.08 0.12 0.16 0.2\n 0.025  0.1 0.2  0.5  1 2 4L = 20×20×10gIPR(ε=0)\nγT=0.095t\nT=1.0t\nFigure 9. gIPR as a function of γforT= 0.095tandT= 1t\nand central system size L= 20×20. The gIPR for T= 1tis\nmultiplied by factor of 10for the sake of clarity.\nDOSh(ε= 0)above the critical temperature, signaled by\nvertical dotted line, is identical to the one in Fig. 6(a)\nand does not depend on the lattice size as can be seen\nin the inset of Fig. 8(blue line). The positions of the\nmaxima are in compliance as well ( T∼0.095t) and the\nmaximum of DOSh (ε= 0)is well above its value in the\ndisordered phase. In addition, the DOSh (ε= 0)shown\nin Fig. 8depends only weakly on the chosen values of\nγ’s. This supports the conclusion that the crossing of\nthe major subgap bands at critical Ucan lead to a DOS\nat the Fermi level, which exceeds its values in the gapless\ndisordered phase. On the other hand, the DOSh (ε= 0)\ncalculated for T= 0.095tdepends much stronger on the\nlattice size (red circles in the inset of Fig. 8) than the\nequivalent DOS calculated for isolated FKM. This, as\nwell as the increasing error bars, is a direct consequence\nof the fact that the broadening coming from the leads is\nnot homogeneous in the central system as its effect on\nthe LDOSh decreases with the distance from the system-\nleads interfaces45. Consequently, the broadening in the\ncentral part of the system coming from the leads might\nvanish fast with the increased lattice size.\nThe qualitative differences between the natural and ar-\ntificial broadening allow us to perform an alternative in-\nvestigation of the localization based on the gIPR defined\nin Eq. ( 14). A direct comparison of the finite size scaling\nof the gIPR calculated for γ= 2twith the IPR is shown\nFig.7(a). The gIPR for T= 0.095t(black squares) has\nthe same profile as IPR and it convergences to a similar\nfinite value, which confirms strong localization even for\nthe coupled system. Similarly, the scaling of the gIPR\nin the disordered phase represented by T= 1t(green\nsquares) points to a weakly localized central system at\nbest (see also Ref.45). The comparison with the IPR also\nreveals that the coupling to the leads can suppress the\nlocalization in the finite system in this regime.\nWe analyze the effect of coupling to the leads on the\nlocalization in the finite system ( L= 20×20) in more de-\ntail in Fig. 9. Both gIPR curves calculated at T= 0.095t\nandT= 1tare saturated at low values of the coupling γ.\nThe saturated values are in good agreement with the ones\ncalculated for the isolated system [Fig. 7(b)]. This can 0 0.1 0.2 0.3 0.4 0.5\n 0.01  0.1  1aU = 2.5tΘ(ε=0)\nT/tγ=2t\nγ=1t\nγ=0.5t 0 6\n-1.5  1.5T = 0.095t\n×100Θ\nε/U\n 0 5 10 15 20\n 200  400  600  800  1000cγ = 2t\n10×Θ(ε=0)×Lx\nLT = 0.095t\nT = 1.0t 0 0.05\n 0.25  1  4bΘ(ε=0)\nV/Uγ = 2t\nγ = 1t\nFigure 10. (a) Dependence of the equilibrium transmission\nfunction on temperature calculated at Fermi level for a het-\nerostructure with U= 2.5tand the system size L= 20×20\ncoupled to two semi-infinite noninteracting leads with semi -\nelliptical surface density of states and band half-width B=\n20t. The position of the humps signalled by an arrow coin-\ncides with the maxima in Fig. 6. The inset is an example of\nthe full transmission function for γ= 1tandT= 0.095t. The\nsubgap region is multiplied by 100for the sake of visibility.\n(b) Nonequilibrium transmission function for ε= 0as a func-\ntion of voltage drop rescaled by U= 2.5t. (c) Dependence\nof the equilibrium transmission function at the Fermi level\nmultiplied by the linear size of the system on the total latti ce\nsize. The values for T= 0.095twere scaled by factor 10.\nalso be interpreted as evidence that the coupling to the\nleads provides a good independent method for studying\nthe problem of the localization.\nHowever, as we increase γ(note the logarithmic scale\nin Fig. 9) the gIPR significantly decreases for γ/apprge0.3tat\nT= 1tandγ/apprge1tforT= 0.095t. This effect is stronger\nin the disordered phase, where at strong coupling the\nalready small gIPR drops to half its weak coupling value.\nNevertheless, the localization is weakened in the ordered\nphase as well. It is therefore worth it to examine how the\ncoupling to the leads affects the transport properties in\na finite system.\nWe focus on the transmission function as this provides9\nthe most detail information on the charge transport. Fig-\nure10(a) shows the transmission function at ε= 0as a\nfunction of temperature for U= 2.5t, central system size\nL= 20×20and for three values of system lead hopping γ.\nWe focus on the equilibrium situation ( V= 0) because a\nvoltage that is smaller than the CDW gap ( V <U/2) has\nonly a small effect on the transmission function at ε= 0.\nThis is shown in Fig. 10(b) forU= 2.5tandT= 0.095t,\nwhere voltage values are spread on a logarithmic scale.\nThe transmission function in Fig. 10(a) is negligible for\nlow temperatures, but a clear “hump” for γ= 2tand local\nmaximum for γ <2t(signaled by an arrow) are present\nclose to the temperature where the DOS of the central\nsystem has its maximum [see Fig. 8]. The transmission\nfunction at Fermi level is still several magnitudes smaller\nthan its values outside the CDW gap, as seen in the inset\nof Fig. 10(a), and it is negligible for any energy within\nthe gap with the exception of the close vicinity of ε= 0.\nNevertheless, its clear increase close to T∼0.1tshows\nthat, despite the relatively strong localization, the cros s-\ning of the subgap states can influence the charge trans-\nport through a finite system. Interestingly, it actually\nenables a finite charge transmission otherwise blocked by\nthe CDW gap. Still, the transmission function signif-\nicantly increases above the critical temperature of the\norder-disorder transition and, despite much lower DOS\nin this phase, the transmission function can be several\ntimes higher than at T∼0.1t.\nThis clearly reflects the difference of the localization\nin the ordered and disordered phases already shown in\nFig.7(a). This effect can also be seen in the system size\nscaling of the transmission functions plotted in Fig. 10(c).\nTo highlight the difference, we scaled the transmission\nfunction by the linear size of the system. The weak\nlocalization in the disordered phase ( T= 1t), further\nlowered by the coupling to the leads, results in a sit-\nuation where the scaled transmission function increases\nwithL. On the other hand, the scaled transmission for\nT= 0.095t, which even for small system lattices is ap-\nproximately ten times smaller than for T= 1t, rapidly\ndecreases with growing lattice size. The strong localiza-\ntion in this regime clearly overrules even the increasing\nDOSh(ε= 0) shown in the inset of Fig. 8.\nWe can therefore conclude that the high density of\nstates in the CDW phase observed in the vicinity of Uc\ncan lead to a notable increase of charge transport through\na finite system, but this effect rapidly vanishes with in-\ncreasing system size. The reason is the relatively strong\nlocalization of the states at the Fermi level.\nIV. CONCLUSIONS\nWe have studied in realistic dimensions the structure\nof the subgap states of the FKM in the ordered CDW\nphase. We have shown that, similar to exact results in\ninfinite dimensions, there are pronounced maxima in the\nsubgap DOS placed symmetrically around the Fermi levelwhich merge around some critical value of Uand ex-\nchange their positions above it. The position of these\nmaxima does not depend on the temperature, because\nthey mainly reflect the underlying sharp states resulting\nfrom single-particle excitations of the localized f-electron\nsubsystem. The most pronounced of these states belong\nto an addition and removal of a single felectron from a\nperfect checkerboard ordering.\nThe crossing of the most pronounced subgap maxima\nleads to a rise of the DOS at the Fermi level which can\nexceed even the DOS in the disordered gapless phase.\nHowever, the states in this regime are strongly localized.\nWe have confirmed this by studying DOS and IPR for two\ndifferent setups. First, we addressed an isolated FKM,\nwhere we used an artificial broadening of the states. Sec-\nond, we studied a heterostructure, where the states had\nbeen broadened by coupling the system to simple semi-\ninfinite leads. Although the coupling to the leads can\nlower the localization in the disordered phase, both stud-\nies have lead to the same qualitative conclusions.\nIn the case of the heterostructure, we have also shown\nthat the significant increase of the DOS at critical Ucan\nboost the charge transport trough a small finite system\nin the ordered phase. It increases the charge transmis-\nsion which is otherwise suppressed in the whole range of\nenergies within the CDW gap. Here, the strength of the\ncoupling to the leads plays an important role. Never-\ntheless, with increasing lattice size this effect is quickly\nsuppressed by the strong localization.\nACKNOWLEDGMENTS\nThe authors acknowledge support by the state of\nBaden-Württemberg through bwHPC and the German\nResearch Foundation (DFG) through Grant No. INST\n40/467-1 FUGG. J. O. acknowledges support from Georg\nH. Endress Foundation. We thank Romuald Lemański\nand James K. Freericks for introducing us to the prob-\nlem of the crossing of the subgap states in FKM and\nTomáš Novotný for very helpful discussions.\nAPPENDIX A: CDW GAP\nHere we present the ground-state solution of the spin-\nless FKM on a hypercubic lattice. We show that the\ncheckerboard ordering of the felectrons opens a gap in\nthe DOS and derive general formulas for eigenvalues and\neigenvectors.\nThe Hamiltonian Eq. ( 1) for the CDW ordering of f\nelectrons in hyper cubic lattice reads\nHCDW\nFK=−t/summationdisplay\nl,δ/parenleftig\nd†\nldl+δ+h.c./parenrightig\n+U\n2/summationdisplay\nleiπld†\nldl,(15)\nwhereδindices’s the relevant nearest neighbors (e.g.,10\n1ix+0iy+0iz). Fourier transformation, given by\nd†\nk=1√\nL/summationdisplay\nleik·ld†\nl, (16)\nleads to\nHCDW\nFK=/summationdisplay\nkǫkd†\nkdk+U\n2/summationdisplay\nkd†\nkdk+π,(17)\nwhere\nǫk=−2tD/summationdisplay\ni=1cos(ki).\nBy replacing of the summation by integration and care-\nful reshaping of the Brillouin zone, the Hamiltonian in\nEq. (17) can be written in a matrix form\nHCDW\nFK=D/productdisplay\ni=2πˆ\n−πdki\n2ππ\n2ˆ\n−π\n2dk1\n2π(18)\n/parenleftig\nd†\nkd†\nk+π/parenrightig/parenleftbigg\nǫkU\n2U\n2ǫk+π/parenrightbigg/parenleftbiggdk\ndk+π/parenrightbigg\n.\nTherefore, the states are divided into two bands sepa-\nrated by CDW gap with width U\nε±\nk=±/radicaligg\nǫ2\nk+/parenleftbiggU\n2/parenrightbigg2\n.\nThe related eigenvectors stored as a columns in matrix\nQkread\nQk=\nǫk+ε+\nk /radicalBig\n(U\n2)2+(ǫk+ε+\nk)2,ǫk+ε−\nk /radicalBig\n(U\n2)2+(ǫk+ε−\nk)2\nU/2/radicalBig\n(U\n2)2+(ǫk+ε+\nk)2,U/2/radicalBig\n(U\n2)2+(ǫk+ε−\nk)2\n.(19)\nAPPENDIX B: SUBGAP STATES\nIn this Appendix, some analytical results for the posi-\ntions of the subgap states resulting from single f-electron\nexcitations are derived.\nLet us excite the groundstate Hamiltonian in Eq. ( 17)\nby adding or removing a single f-electron on position m\nHex=−Ueiπmd†\nmdm.\nOur aim here is to find the position of the sharp states\nin the spectra resulting from this single impurity. The\nHamiltonian ( 15) can be formally diagonalized by unitary\ntransformation,\n/tildewided†\nα=/summationdisplay\nlψαld†\nl,\n/tildewidedα=/summationdisplay\nldlψ†\nlα,after which the total system Hamiltonian including Hex\nreads\nH=/summationdisplay\nαεα/tildewided†\nα/tildewidedα−Ueiπm/summationdisplay\nα,βψαmψ†\nmβ/tildewided†\nα/tildewidedβ.\nWe define the Green’s function,\nGα,β(τ) =−iΘ(τ)/angbracketleftbigg/bracketleftig\n/tildewidedα(τ),/tildewided†\nβ(0)/bracketrightig\n+/angbracketrightbigg\nand use the equation of motion technique to get\n/parenleftbigg\nid\ndτ−εα/parenrightbigg\nGα,β(τ) =δα,βδ(τ)\n−Ueiπm/summationdisplay\nα′ψαmψ†\nmα′Gα′,β(τ).\nAfter the transformation into energy domain this reads\n(ε−εα+i0)Gα,β(ε) =δα,β−Ueiπm/summationdisplay\nα′ψαmψ†\nmα′Gα′,β(ε),\nor written in matrix form\n/parenleftbig\nE+Ueiπmψm/parenrightbig\nG(ε) =1,\nwhereEis diagonal matrix with elements Eα,α=ε−\nεα+i0. We are primarily interested in the position\nof the bound states inside the CDW gap which can\nbe calculated from the zeros of the determinant of the\ninverse Green’s function. Because Eis diagonal and\nψm\nαα′=ψαmψ†\nmα′we can use Sylvester’s determinant\ntheorem [ det(X+cr) = det(X)(1+rX−1c)]:\ndet/parenleftbig\nG−1(ε)/parenrightbig\n=/parenleftigg\n1+Ueiπm/summationdisplay\nαψ†\nmαψαm\nε−εα+i0/parenrightigg\n×/productdisplay\nα(ε−εα+i0). (20)\nThe position of the sharp states in the CDW gap can\ntherefore be obtained by solving\n1+eiπmU/summationdisplay\nαψ†\nmαψαm\nε−εα= 0. (21)\nWe evaluate this equation by using the transformation\nfrom Eq. ( 16) together with the decomposition M=\nQΛQ−1, whereMis the inner matrix from Eq. ( 18) and\nelements of the diagonal matrix Λareε+\nkandε−\nk. The\nproduct in Eq. ( 21) then reads\nψ†\nmαψαm=1\nL/summationdisplay\np=±/parenleftigg\n1+Ue−iπmεp\nk+ǫk/parenleftbigU\n2/parenrightbig2+(εp\nk+ǫk)2/parenrightigg\n,\nand the sum over index αchanges to\n/summationdisplay\nα→D/productdisplay\ni=2πˆ\n−πdki\n2ππ\n2ˆ\n−π\n2dk1\n2π≡/summationdisplay\nK.11\nUsing the new notation, Eq. ( 21) can be rewritten to a\nsimple form\n1 =eiπmU\nL/summationdisplay\nKUe−iπm+2ε\nε2\nk−ε2. (22)\nInD= 1this equation can be easily evaluated and reads\n2/parenleftbig\nU+2εeiπm/parenrightbig\nU√\nU2−4ε2√\n16t2+U2−4ε2= 1.\nTherefore, the solution for Uas a function of the energy\nof bound states is\nU= 4/radicaligg\n±1−2˜ε\n(±3−2˜ε)(1±2˜ε)2,\nwhere˜ε=ε/U. At the crossing of the states, we haveε= 0and therefore the critical interaction in D= 1is\nUc= 4/√\n3.\nThe evaluation of Eq. ( 22) inD= 2leads to\n1 =U2(1±2˜ε)\nπu/radicalbiggu\n16+uK/parenleftbigg16\n16+u/parenrightbigg\n,\nwhereu=U2(0.25−˜ε2)andK(x)is the elliptic integral\nof the first kind. The critical interaction in D= 2is\nUc.= 2.5t(with rounding at the third decimal place).\nTheD= 3case can be evaluated numerically and has\nthe critical interaction Uc∼=3.18t.\nThe displacement of the felectron to a neighboring\nposition can be seen as adding and removing an felec-\ntron at adjoining positions and can be represented by an\nadditional term,\nHex=−Ueiπm(d†\nmdm−d†\nm+1indm+1in),\nwhereinchoses the direction of the f-electron shift. 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Meden, Journal of Physics 31, 163001 (2019) ."    },    {        "title": "1907.04767v1.The_role_of_topological_defects_in_the_two_stage_melting_and_elastic_behavior_of_active_Brownian_particles.pdf",        "content": "The role of topological defects in the two-stage melting and\nelastic behavior of active Brownian particles\nSiddharth Paliwal\u0003and Marjolein Dijkstray\nSoft Condensed Matter, Debye Institute for Nanomaterials Science,\nUtrecht University, Princetonplein 1, 3584 CC Utrecht, The Netherlands\n(Dated: July 11, 2019)\nWe \fnd that crystalline states of repulsive active Brownian particles at high activity melt into\na hexatic phase but this transition is not driven by an unbinding of bound dislocation pairs as\nsuggested by the Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) theory. Upon reducing the\ndensity, the crystalline state melts into a high-density hexatic state devoid of any defects. Decreasing\nthe density further, the dislocations proliferate and introduce plasticity in the system, nevertheless\nmaintaining the hexatic state, but eventually melting into a \ruid state. Remarkably, the elastic\nconstants of active solids are equal to those of their passive counterparts, as the swim contribution\nto the stress tensor is negligible in the solid state. The sole e\u000bect of activity is that the stable solid\nregime shifts to higher densities. Furthermore, discontinuities in the elastic constants as a function of\ndensity correspond to changes in the defect concentrations rather than to the solid-hexatic transition.\nAccording to the Kosterlitz-Thouless-Halperin-Nelson-\nYoung (KTHNY) theory, a two-dimensional solid of pas-\nsive particles melts via a continuous transition into an\nintermediate hexatic state of quasi-long-ranged bond ori-\nentational order, and melts subsequently via a second\ncontinuous transition into a \ruid state [1{3]. These tran-\nsitions are triggered by the unbinding of topological de-\nfects, which are particles with a non-conforming number\nof neighbors with respect to that of the crystal lattice.\nThe coordination number is 6 for an ideal triangular lat-\ntice. Hence, particles with a number of neighbors Nbthat\ndeviates from 6 are classi\fed as defects. One can distin-\nguish defects that either exist freely as 5-fold and 7-fold\ndisclinations or cluster into dislocations (5-7 pairs), dis-\nlocation pairs (5-7-5-7 quartets) or higher-order clusters.\nThe debate on the melting behavior of an equilibrium sys-\ntem of hard disks as well as short-ranged repulsive disks\nwas only settled a few years ago, both in simulations [4{\n8] and experiments [9], showing a two-stage melting sce-\nnario, that deviates from the KTHNY scenario, of a con-\ntinuous solid-hexatic transition driven by an unbinding of\ndislocation pairs and a \frst-order hexatic-\ruid transition\ndue to a proliferation of grain boundaries.\nRemarkably, non-equilibrium systems of self-propelled\nparticles [10{13], which constantly convert energy from\nthe environment into persistent motion, have recently\nalso been shown to follow such a two-stage melting be-\nhaviour [14, 15]. It was found that the \frst-order nature\nof the liquid-hexatic transition persists upto a small de-\ngree of activity. The transition then becomes continuous\nuntil it reappears as a coexistence of dilute and dense\nstates at high activity [14, 15]. The solid state was found\nto melt into a hexatic state via a continuous transition. In\nthis work, we further explore this hexatic-solid transition\nwith respect to the role of topological defects discussed in\nthe KTHNY theory and investigate whether the melting\nis driven by transitions in defect concentrations.\nWe numerically simulate a system of Nactive Brown-\n0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6\nρσ20.00.81.62.44.87.212244872144Pe =v0τ/σ\n0.900.920.940.960.00.81.62.4FIG. 1. State diagram in the Pe- \u001a\u001b2representation exhibit-\ning \ruid (circle, red), \ruid-hexatic coexistence (diamonds,\nyellow), hexatic (triangle, green) and crystal (square, blue)\nstates. The symbols denote state points used in the simu-\nlations, the background colors denote the boundaries of the\nlabeled regions, and the black dashed line indicates the den-\nsities beyond which the concentration of topological defects\nvanishes in simulations of N= 72\u0002103particles. The inset\nis a magni\fcation of the low-Pe regime.\nian particles exhibiting overdamped Langevin dynamics:\n\r_ri=\u0000X\nj6=iriU(rij) +\rv0ei+p\n2\rkBT\u0003t\ni;\nwherev0eiis the self-propulsion speed, \ris the damp-\ning coe\u000ecient, kBthe Boltzmann constant, and Tthe\ntemperature of the solvent. The particle orientation\nei= (cos\u0012;sin\u0012) undergoes free rotational di\u000busion\n_\u0012i=p2Dr\u0003r\ni, whereDris the rotational di\u000busion co-\ne\u000ecient. The quantities \u0003t\niand \u0003r\niare unit-variance\nGaussian noise terms with zero mean. The particles in-\nteract with a pairwise repulsive WCA potential U( r) =\n4\"h\u0000\u001b\nr\u000112\u0000\u0000\u001b\nr\u00016i\n+\", and we set kBT=\"= 1. We \frst de-\ntermine the phase boundaries by measuring the equationarXiv:1907.04767v1  [cond-mat.soft]  10 Jul 20192\nFIG. 2. Typical sections of active Brownian particle con\fgurations (80 \u001b\u000280\u001b) for Pe = 0 :0;4:8 and 71:5 at labeled states,\nshowing 5-fold (blue) and 7-fold (red) defects, other defects (black), and particles with Nb= 6 (grey). Some vacancies are\nindicated by arrows in the hexatic and solid states. The con\fgurations at Pe = 4 :8 are similar to the corresponding passive\nstates. For Pe = 71 :5 we \fnd that with increasing density the hexatic states show a decrease in the number fraction of defects,\nwith a complete absence of defects at a density \u001a\u001b2&1:560. The positional correlations become quasi-long-ranged around a\ndensity\u001a\u001b2'1:950 for Pe = 71 :5 as shown in Fig. 1. The insets in the upper right and lower right show the same con\fgurations\nas the main panels but colored according to the hexatic and positional order parameters  6and T, respectively, following the\ncolor mapping shown at the top.\nof state (pressure-density curves), density histograms,\nand the decay of the orientational and positional corre-\nlation functions. We present the state diagram in Fig. 1\nin the activity-density (Pe- \u001a) representation, where the\nP\u0013 eclet number is de\fned by Pe = v0\u001c=\u001b. The state dia-\ngram shows that the continuous hexatic-solid transition\npersists upon introducing activity, but shifts to higher\ndensities due to the softness of the interaction potential.\nWe hereby assume that the solid state transforms into\na hexatic phase when the positional correlations decay\nwith a power law exponent \u0011T\u00141=3 as described by the\nequilibrium KTHNY theory.\nTopological defects: We identify the topological defects\nby performing a Voronoi construction and calculating the\nnumber of neighbors Nbfor each particle. We classify\nthem asNb-fold defects and account for 5-fold and 7-fold\ndefects. In Fig. 2 we show typical con\fgurations, high-\nlighting these defects in the \ruid, hexatic and crystalline\nstate at Pe = 0 ;4:8 and 71:5. At low activity Pe = 4 :8,\nsimilar defect con\fgurations are observed as for the pas-sive hard-disk systems [6]. A \fnite number fraction of\ndislocation pairs can be identi\fed in the solid state which\ncan exist due to thermal \ructuations without disturbing\nthe positional order. In the hexatic state, the presence of\nunpaired dislocation defects causes the positional order\nto decay exponentially but the orientational order decays\nalgebraically. Finally the \ruid state comprises of many\ndefect clusters and 5-fold and 7-fold disclinations, both\ndestroying the local bond orientational order. However,\nthe hexatic states at Pe = 71 :5, are signi\fcantly di\u000berent\nin defect con\fgurations. The low-density hexatic state\nat\u001a\u001b2= 1:420 shows a high number fraction of disloca-\ntions with almost no dislocation pairs, whereas the high-\ndensity hexatic state ( \u001a\u001b2= 1:600) at the same Pe shows\na complete absence of any defects.\nWe further quantify this observation by measuring the\nnumber fractions of the di\u000berent types of defects, which\nare plotted in Fig. 3(a) as a function of density for various\nPe. We clearly observe that the overall number fraction\nof defectsNtotal=Nincreases upon decreasing the den-3\nsity for all Pe. At low Pe \u00142:4, we \fnd that the solid\nphase contains mainly bound dislocation pairs, which\nmove freely in the solid phase. The number fraction\nof bound dislocation pairs Nquart=Nincreases slightly\nupon approaching the hexatic-solid transition, whereas\nthe number fraction of dislocations Npair=Nincreases\nmore rapidly. At the hexatic-solid transition, Nquart=N\nis still higher than Npair=Nbut at lower density this sce-\nnario is reversed. The high fraction of Npair=Nimplies\nthat the quasi-long-range positional order of the system\nis destroyed, suggesting that the solid-hexatic transition\nis induced by the unbinding of dislocation pairs. Upon\nreducing the density further, we \fnd that the fraction\nNfree=Nof 5- and 7-fold defects also starts to increase\ndue to the unbinding of dislocations into disclinations\nand the bond orientational order decays exponentially\nin the stable liquid phase. To summarize, we \fnd that\nthe two-step melting scenario consisting of a solid-hexatic\ntransition driven by the unbinding of dislocation pairs\nand a hexatic-\ruid transition caused by defect clusters\nas observed for the 2D passive systems of short-range\nrepulsive particles, persists at low activity.\nFor higher activity Pe \u00157:2, the behavior of disloca-\ntions is di\u000berent than for passive systems as shown in\nFig. 2. For Pe\u00157:2 the system seems to support a\nhigher fraction of dislocations as compared to disloca-\ntion pairs for all densities, as observed in Fig. 3(a) where\nNquart never exceeds Npair. This observation is more\nclearly visible in Fig. 3(b) where we plot the di\u000berences\n(Npair\u0000Nquart)=Nand (Npair\u0000Nfree)=N. We \fnd that\nfor Pe\u00142:4 and for high densities ( Npair\u0000Nquart)=N\nis negative whereas for Pe \u00157:2 this di\u000berence is always\npositive, thereby demonstrating that a higher fraction of\nisolated dislocations over bound ones is favored at higher\nactivity.\nIn Fig. 3(a) and (b) we also mark the hexatic-solid\ntransition densities as identi\fed from the spatial decay of\npositional correlations [16]. These densities agree closely\nwith the minimum in ( Npair\u0000Nquart)=Nwhich corre-\nsponds to a reversal in the trend of defect concentrations\nand con\frms our previous observation that for Pe \u00142:4\nthe hexatic-solid transition is driven by the unbinding of\ndislocation pairs. The di\u000berence ( Npair\u0000Nfree)=N, how-\never, is always positive for all densities and show similar\ntrends for all values of Pe considered.\nIn Fig. 3(a) we also locate the densities above which\nwe \fnd a complete absence of defects in the system for\nall values of Pe, marked by a cross on the x-axis and the\ncorresponding density range by cross-hatching. As we in-\ncrease the activity, this point crosses over from the dense\ncrystal state to the hexatic state. For Pe = 23 :8;47:7\nand 71:5 we clearly see that a large region correspond-\ning to the dense hexatic states is free from any kinds of\ntopological defects in contrast to the low activity sys-\ntems. This dense hexatic region, devoid of any defects,\nstill has an exponentially decaying positional order as\n0.90 0.92 0.9410−1\n10−3\n10−5Ndefect/N\nPe=0.0\nρσ2(a)Nquart/N\nNpair/NNfree/N\nNvac/NNtotal/N\n0.920.940.960.98\nPe=2.4\n1.05 1.10 1.15\nPe=7.2\n1.25 1.5010−1\n10−3\n10−5Ndefect/N\nPe=23.8\nρσ21.50 1.75\nPe=47.7\n1.5 2.0\nPe=71.5\n0.9 1.0 1.1 1.2 1.3−0.2−0.10.00.10.2∆(Ndefect/N)\n(Npair−Nquart)/N (Npair−Nfree)/N\nρσ2(b)\nPe=0.0\n2.4\n4.8\n7.2\n11.9\n23.8FIG. 3. (a) Number fraction of speci\fc defects as a function of\ndensity\u001a\u001b2for various Pe. The background colors mark the\nboundaries as indicated in the state diagram (Fig. 1) and the\nregion devoid of any defects is cross-hatched. The symbols\non the axis indicate the densities where the defects become\nabsent (\u0002), and the hexatic-solid transition densities ( ?). (b)\nThe di\u000berence in the number fraction of unpaired dislocations\nand bound dislocation pairs ( Npair\u0000Nquart)=N(squares), and\nthe di\u000berence in the number fraction of unpaired dislocations\nand free disclinations ( Npair\u0000Nfree)=N(circles) as a function\nof\u001a\u001b2for various Pe as labeled in the legend.\nindicated by the background colors corresponding to the\nstate diagram (Fig. 1). The activity-induced \ructuations\ndecorrelate the particle positions at long range and are\nresponsible for a faster decay of positional order in this\ndensity regime. The solid-hexatic transition in active sys-\ntems is thus driven by a striking non-equilibrium feature.\nWe now test the correspondence of changes in defect con-\ncentrations and the elastic response of the system with\nrespect to the predictions of the KTHNY theory.\nElastic Moduli: The KTHNY theory for the melting of\n2D equilibrium solids is based on the linear elastic prop-\nerties of a continuum. Although the equilibrium theory\nrelies on the elastic deformation energies to resolve the\ntransition to a \ruid state with a vanishing shear and\nrenormalized Young's modulus, we can extend the no-\ntion of mechanical stress, which is well-de\fned for an\nisotropic active \ruid, to describe the elastic behaviour\nin terms of the response to an externally imposed lin-4\n1.0 1.2 1.4 1.6 1.8101102103104βKσ2\n0.07.223.847.771.5\nρσ2(a)\nPe=0.0\n7.2\n23.8\n47.7\n71.5\nfit\n0.92 0.94 0.96 0.981.01.5βKσ2/16π\nρσ2(b)\nPe=0.0\n0.8\n1.6\n2.4KR\nFIG. 4. (a) Young's modulus Kas a function of density \u001a\u001b2\ncollapse onto a single master curve K/exp(a\u001a3+b\u001a2+c\u001a+\nd) for 0:0\u0014Pe\u001471:5. The plus (+) markers denote the\ndiscontinuous jump in Kfor the corresponding activity, cross\n(\u0002) markers show the densities where the defects disappear\nfor a large system size of N= 72\u0002103particles, and star\n(?) markers denote the transition densities from the decay of\n T, all o\u000bset vertically for clarity. (b) Bare Young's modulus\nK(open circles) obtained directly from the Lam\u0013 e coe\u000ecients\nand the corresponding renormalized values KR(triangles) for\nPe = 0:0;0:8;1:6 and 2:4. The vertical lines mark the hexatic-\nsolid transitions identi\fed from the decay of  TforN= 72\u0002\n103particles.\near strain on the simulation box. In our simulations, we\nstart from a perfect hexagonal initial con\fguration with\nN= 2:8\u0002103particles and measure the full stress ten-\nsorP\u000b\f, comprising of the ideal, virial and swim compo-\nnents [17], in the deformed box due to a \fxed small linear\nstrain\u000fxx2[\u00000:01;0:01]. We calculate the Lam\u0013 e elastic\ncoe\u000ecients \u0015and\u0016from the e\u000bective sti\u000bness tensor B\nobtained from the slope of a linear \ft to the stress versus\nstrain curves [18{20] (see Ref. [16] for details).\nIn Fig. 4(a) we plot the Young's modulus K= 4\u0016(\u0015+\n\u0016)=(\u0015+ 2\u0016) obtained for 0 :0\u0014Pe\u001471:5 and identify\nthe densities where Kshows a discontinuous transition.\nInterestingly, we observe that Kcollapses onto a single\nmaster curve for all Pe. This remarkable result can be\nexplained from the fact that the swim contribution to the\nstress tensor is negligible in the solid, and hence the elas-tic moduli for active solids equals the ones of their passive\ncounterparts. Activity only shifts the stability region of\nthe solid to higher densities. Additionally, in the bot-\ntom part of the \fgure we mark the densities where the\ndefects disappear for larger systems of N= 72\u0002103par-\nticles as a cross (\u0002) as well as the hexatic-solid transition\ndensities obtained from the decay of positional correla-\ntions as a star ( ?) similar to the ones marked in Fig. 3.\nWe \fnd that in the passive case the discontinuity in K\nagrees well with the hexatic-solid transition. However,\nfor Pe\u00157:2 the discontinuity in Kagrees well with the\n(dis)appearance of defects.\nIn equilibrium systems, the KTHNY theory suggests a\ncritical value of 16 \u0019for the renormalized Young's mod-\nulus\fKR\u001b2below which the solid is unstable to shear.\nThe renormalization procedure corrects for the interac-\ntions of defects at \fnite temperature. We apply the\nrenormalization procedure to the elastic constants up to\nPe\u00142:4 for which there is a \fnite fraction of defects at\nthe hexatic-solid transition.\nTo evaluate the renormalized Young's modulus we \frst\nexplicitly measure the probability of dislocation pairs pd\nin simulations and calculate the core energy Ecof de-\nfects using pd= exp(\u00002\fEc)Z(K) whereZ(K) is the\n`internal partition function' of a dislocation [21, 22]. In\nequilibrium, due to thermal \ructuations there is a \f-\nnite probability for the formation of dislocation pairs and\nthe dislocation energy Ecnear the melting transition is\nsmall but \fnite. As we increase the activity the con-\ncentration of dislocation defects near the melting tran-\nsition reduces (see Fig. 3). This observation hints that\nthe energy needed to create a dislocation pair becomes\nhigher as we increase activity. Conversely, we can inter-\npret that the unbinding energy reduces with increasing\nactivity which eases the dissociation of dislocation pairs\ninto dislocations. We then apply the recursion relations\nof the KTHNY theory [21, 22] to obtain the renormalized\nYoung's modulus KR, shown in Fig. 4(b) as a function\nof density for Pe=0, 0.8, 1.6 and 2.4. In the same plot\nwe also show the `bare' values Kas in Fig. 4(a). For\nPe = 0, we \fnd that the renormalized KRdi\u000bers sig-\nni\fcantly from the `bare' value. The density at which\nKRdrops to zero agrees well with our estimate of the\nhexatic-solid transition. However, as we increase the ac-\ntivity upto Pe = 2 :4 we \fnd that this is no longer valid.\nHence, even for a small Pe we can already see that the\npredictions of KTHNY theory based on the elastic con-\nstants deviate signi\fcantly from the transitions as ob-\ntained from the positional correlations of the particles.\nAt higher activity there is a complete absence of defects\nat the hexatic-solid transition point and the transition is\nentirely driven by activity.\nConclusions: We found that at high activity the 2D\nmelting of active Brownian particle solids into a hexatic\nstate is not driven by an unbinding of dislocation pairs\nin contrast to passive systems. The hexatic state at high5\ndensities is completely devoid of any defects, and this\ndefect-free region widens with activity. The solid-hexatic\ntransition might be driven by a growing length scale of\nregions of cooperative motion, but this requires further\ninvestigation. Interestingly, we observed that the elastic\nconstants of active solids are equal to those of the passive\ncounterparts, as the swim contribution to the stress ten-\nsor is negligible in the solid state. The activity only shifts\nthe stability regime of the solid state to higher densities.\nWe thank Berend van der Meer, Laura Filion and\nFrank Smallenburg for many useful discussions. S.P.\nand M.D. acknowledge funding from the Industrial Part-\nnership Programme \\Computational Sciences for Energy\nResearch\" (Grant No.14CSER020) of the Foundation for\nFundamental Research on Matter (FOM), which is part\nof the Netherlands Organization for Scienti\fc Research\n(NWO). This research programme is co-\fnanced by Shell\nGlobal Solutions International B.V.\n\u0003s.paliwal@uu.nl\nym.dijkstra@uu.nl\n[1] J. M. Kosterlitz and D. J. Thouless, J. Phys. C Solid\nState Phys. 6, 1181 (1973).\n[2] D. R. Nelson and B. I. Halperin, Phys. Rev. B 19, 2457\n(1979).\n[3] S. T. Chui, Phys. Rev. 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Commun. 192, 97 (2015).\n[25] L. F. Cugliandolo, P. Digregorio, G. Gonnella, and\nA. Suma, Phys. Rev. Lett. 119, 268002 (2017).\n[26] G. Clavier, N. Desbiens, E. Bourasseau, V. Lachet,\nN. Brusselle-Dupend, and B. Rousseau, Mol. Simul. 43,\n1413 (2017).\n[27] K. J. Strandburg, Rev. Mod. Phys. 60, 161 (1988).1\nSupplementary Information:\nThe role of topological defects in the two-stage melting and\nelastic behavior of active Brownian particles\nMODEL\nWe consider a two-dimensional system of isotropic Brownian particles that exhibit a self-propulsion speed v0which\nis directed along the orientation vector ei= (cos\u0012i;sin\u0012i) assigned to particle i. To describe the translational and\nrotational motion of the individual colloidal particle i= 1;:::;N we employ the overdamped Langevin dynamics:\n\r_ri=\u0000X\nj6=iriU(rij) +\rv0ei+p\n2\rkBT\u0003t\ni;\n_\u0012i=p\n2Dr\u0003r\ni; (S1)\nwhere\ris the damping coe\u000ecient due to the drag forces from the implicit solvent, kBis the Boltzmann constant,\nandTis the bath temperature. Dris the rotational di\u000busion coe\u000ecient. The quantities \u0003t\niand \u0003r\niare unit-variance\nGaussian noise terms with zero mean:\n\n\u0003t\ni(t)\u000b\n= 0;h\u0003r\ni(t)i= 0;\n\n\u0003t\ni(t)\u0003t\nj(t0)\u000b\n=I2\u000eij\u000e(t\u0000t0)\n\n\u0003r\ni(t)\u0003r\nj(t0)\u000b\n=\u000eij\u000e(t\u0000t0); (S2)\nwhere I2is the 2\u00022 identity matrix. The angular brackets h\u0001\u0001\u0001i denote an average over di\u000berent realizations of the\nnoise. The particles interact with a short-range repulsive Weeks-Chandler-Andersen (WCA) potential given by:\nU(r) = 4\"\u0014\u0010\u001b\nr\u001112\n\u0000\u0010\u001b\nr\u00116\u0015\n+\"; r\u001421=6\u001b\n= 0 r >21=6\u001b (S3)\nwhere r =jrijjis the distance between the centers of particle iandj,\u001bis the particle diameter pertaining to the\nlength scale in WCA potential, and \"is the strength of the particle interactions.\nWe set the system temperature kBT=\"= 1, following Ref. 11, and the damping coe\u000ecient \r\u001b2=\"= 1 \fxing the\ntranslational di\u000busion coe\u000ecient to correspond to the free di\u000busion of particles given by the Stokes-Einstein relation\nDt=\r\u00001kBT. This sets our time scale as \u001c=\r\u001b2=kBT. The rotational di\u000busion coe\u000ecient is set to Dr\u001c= 3 and\nwe use a time step size dt= 10\u00005\u001cfor numerically integrating the equations of motion. We de\fne a non-dimensional\nP\u0013 eclet number Pe = v0\u001c=\u001b as the ratio of the persistence length of motion to the particle diameter and perform\nsimulations using the HOOMD-Blue [23, 24] package in the range 0 \u0014Pe\u0014150. We used N= 72\u0002103particles in an\napproximately square 2D periodic simulation box with dimensions Lx;Ly\u0019250\u001bfor identifying the \ruid-hexatic-solid\ntransitions and N= 2:8\u0002103for calculating the elastic moduli. We used a regular hexagonally packed arrangement\nof particles at the overall system density as our initial con\fguration and we collect snapshots for 200 \u001c\u0000500\u001cat an\ninterval of\u001cfor analysis after allowing the system to achieve a stationary state for about 200 \u001c.\nAs mentioned in the main text, the state diagram shown in Fig. 1 was obtained by measuring the equation of state\n(pressure-density curves), the density histograms and the decay of the orientational and the positional correlation\nfunctions to locate the boundaries as precisely as possible. Speci\fcally, to locate the coexistence, we identi\fed negative\nslope regions in the pressure-density curves as well as double-peaked structure of the density histograms similar to\nthe analysis presented in Ref. 14 and Ref. 25. We describe below the orientational and positional order parameters\nused for identifying the hexatic and solid phases.\nOrientational and Positional order\nWe measure the local 6-fold orientational symmetry around particle iusing the hexatic order parameter  6(ri)\ngiven by:\n 6(ri) =1\nNbX\nj2Nbexp(\u00136\u0012ij); (S4)2\nwhereNbdenotes the number of nearest neighbors of particle iand the bond angle \u0012ijis measured as a deviation of\nthe orientation of the vector rijfrom the reference global system orientation measured from \t 6(L) averaged over all\nthe particles. We identify the nearest neighbors Nbof the particle by using a Voronoi construction.\nTo investigate the decay of positional order, we measure the positional correlation function\ngT(r) =h \u0003\nT(r0+r) T(r0)i; (S5)\nwhere T(ri) is the positional order parameter expressed as:\n T(ri) = exp(\u0013k0\u0001ri): (S6)\nHerek0is the vector in reciprocal space denoting one of the \frst Bragg peaks in the 2D structure factor S(k). The\nmagnitude of this vector is equal to that of the reciprocal lattice vector i.e. k0= (0;4\u0019=ap\n3) wherea= (2=\u001ap\n3)1=2\u001b\nis the lattice spacing in a regular hexagonal packing at a number density \u001ain a 2D geometry. According to the\nKTHNY theory, the positional order of a two-dimensional solid decays algebraically as gT(r)/r\u0000\u0011Twith an exponent\n0\u0014\u0011T\u00141=3. Upon melting, the decay of the positional correlations becomes exponential i.e. gT(r)/exp(\u0000r=\u0018T)\nwith a correlation length \u0018T, which decreases with decreasing density. We show the positional correlation functions\ngT(r) as a function of particle separation rin Fig. S1 and extract the correlation lengths \u0018Tin the case of an exponential\ndecay or the exponent \u0011Tin the case of an algebraic decay. We identify the hexatic-solid transition by locating the\ndensity at which the exponent \u0011Tbecomes smaller than 1 =3. For Pe = 0, we \fnd that the decay of gT(r) becomes\nalgebraic with \u0011T\u00191=3 at\u001a\u001b2= 0:926, marking the hexatic-solid phase transition. We locate the transition densities\nfor higher Pe in a similar manner.\n0.20.40.60.81gT(r)Pe=0.0\n0.920\n0.926\n0.930Pe=2.4\n0.960\n0.970\n0.980Pe=7.2\n1.080\n1.100\n1.120\n10 1000.20.40.60.81gT(r)Pe=23.8\n1.360\n1.400\n1.440\n10 100\nr/σPe=47.7\n1.600\n1.650\n1.700\n10 100Pe=71.5\n1.700\n1.800\n1.900\nFIG. S1. Positional correlation function gT(r) for 0\u0014Pe\u001471:5 at the labeled densities. The decay is exponential for a\nhexatic state with the correlation length diverging upon increasing the density and the decay becomes quasi-long ranged for\na crystalline state. The dashed grey line indicates algebraic decay with exponent \u0011T= 1=3 and the dotted grey lines indicate\nexponential decay with correlation lengths 50 \u001band 100\u001b.3\nSystem-size dependence\nTo check the \fnite-size e\u000bects on the decay of the positional order we also simulate a few cases with N= 288\u0002103\nparticles. We show gT(r) for the two di\u000berent system sizes at Pe = 2 :4;7:2 and 71:5 and with densities near the\nrespective hexatic-solid transition in Fig. S2. The values for the exponents \u0011T;1and\u0011T;2of the power-law decay, for\nthe small and large systems, respectively, are also listed in Fig. S2. For smaller system sizes, we observe clearly that\n\u0011T;1<1=3 for all three Pe values, which corresponds to the solid phase. For N= 288\u0002103, we \fnd that the exponent\n\u0011T;2is still close to \u0011T;1for Pe = 2:4, but for higher Pe the exponent \u0011T;2is signi\fcantly larger than \u0011T;1. Despite\nthese di\u000berences in the decay of the positional correlations, the location of the density where the defects disappear\ndoes not change and the observed hexatic region devoid of defects is robust over these investigated system sizes.\n1 10 1000.20.40.60.81.0gT(r)\nηT,2= 0.18ηT,1= 0.21Pe=2.4,ρσ2= 0.980\n1 10 100ηT,2= 0.38ηT,1= 0.26Pe=7.2,ρσ2= 1.180\n1 10 100ηT,2= 0.49ηT,1= 0.25Pe=71.5,ρσ2= 1.960\nr/σ\nFIG. S2. Comparison of the decay of positional correlations gT(r) on a log\u0000log scale for system sizes of N= 72\u0002103(orange\nlines) and 288\u0002103(blue lines) for Pe=2 :4;7:2 and 71:5 at varying densities near the hexatic-solid transition. The exponents\n\u0011T;1and\u0011T;2, obtained by \ftting gT(r)/r\u0000\u0011Tin the range 30 \u001b\u000080\u001band 50\u001b\u0000150\u001b, for a small and large system, respectively,\nare also quoted in the \fgure. The grey dotted line indicates an exponential decay with a correlation length \u0018T= 100\u001b, and the\ngrey dashed line indicates a power-law decay with exponent \u0011T= 1=3.\nELASTIC MODULI\nStress tensor\nIn order to measure the bulk pressure Pin our system consisting of Nactive Brownian particles, we employ the\nexpressions as introduced by Winkler et al.[17], but modi\fed them to the 2D case. Speci\fcally, the pressure for\nisotropic active particles in a periodic box with lateral dimensions LxandLyand 2D `volume' V=LxLyis calculated\nusingP= Tr(P) where the full stress tensor Pis given by:\nP\u000b\f=Pvir\n\u000b\f+\u000e\u000b\f(Pid\n\u000b\f+Pswim\n\u000b\f): (S7)\nHerePidis the ideal gas pressure given by Pid=\u001akBTwith\u001a=N=V the number density of the particles. The virial\ncontribution Pviris obtained using the standard virial expression\nPvir\n\u000b\f=\u00001\n4V*NX\niNX\nj6=i@ri;\fU(rij)\u0001(ri;\f\u0000rj;\f)+\n: (S8)\nThe swim pressure contribution Pswim due to the self-propulsion is given by:\nPswim=\r\u001av2\n0\n2Dr\u0000\rv0\n4VDr*NX\ni=1NX\nj6=iriU(rij)\u0001ei+\n: (S9)4\nFIG. S3. Typical particle con\fgurations with the top row showing a magni\fcation of the highlighted region shown in the bottom\nrow by a black square for selected densities belonging to a pure hexatic phase ( \u001a\u001b2\u00141:94) and a solid phase ( \u001a\u001b2\u00151:95) for\nPe = 71:5. The color coding of the particles is according to arg(  T) after subtracting the mean orientation, as shown in the\ncolor wheel on the right. We observe topological defects (colored black) in the con\fgurations for \u001a\u001b2= 1:42 and 1:55 but not\nfor\u001a\u001b2= 1:60 and 1:95 withN= 72\u0002103particles.\nSti\u000bness tensor and Lam\u0013 e elastic coe\u000ecients\nIn the linear elastic theory of isotropic solids, the elastic moduli relate the stress response of a system to an applied\nstrain. In equilibrium, the elastic moduli are related to the free energy change due to such deformations [18]. Instead,\nfor non-equilibrium systems we directly assume Hooke's law which linearly relates the mechanical stress P\u000b\fwith the\napplied strain \u000f\r\u000ethrough a symmetric sti\u000bness tensor Cgiven by:\nC=2\n664C11C120 0\nC220 0\n0 0\nC443\n775=2\n664\u0015+ 2\u0016 \u0015 0 0\n\u0015+ 2\u00160 0\n0 0\n\u00163\n775\nwhere\u0015and\u0016are the Lam\u0013 e coe\u000ecients in equilibrium systems [18]. For conciseness, we follow the Voigt notation\nabove for indexing C\u000b\f\r\u000e withxx=1,yy=2, andxy=4. If a system is under a uniform isotropic pressure, the sti\u000bness\ntensor Ccan be rewritten in terms of an e\u000bective sti\u000bness tensor Bas [19, 20]:\nB\u000b\f\r\u000e =C\u000b\f\r\u000e\u0000P(\u000e\u000b\r\u000e\f\u000e+\u000e\u000b\u000e\u000e\f\r\u0000\u000e\u000b\f\u000e\r\u000e) (S10)\nB11=C11\u0000P; B 22=C22\u0000P;\nB12=C12+P; B 44=C44\u0000P;\nwhereP= (Pxx+Pyy)=2 is the uniform pressure. The bulk modulus E, the shear modulus Gand the Young's\nmodulusKare related to the Lam\u0013 e coe\u000ecients in 2D as:\nE=\u0015+\u0016; G =\u0016; K =4\u0016(\u0015+\u0016)\n\u0015+ 2\u0016=4EG\nE+G: (S11)\nFurthermore, from equilibrium statistical thermodynamics the isothermal compressibility \u0014= 1=E, whereEis the\nbulk modulus, is expressed as:\n1\nE=\u0014=\u00001\nV\u0012@V\n@P\u0013\nT=1\n\u001a\u0012@\u001a\n@P\u0013\nT; (S12)5\nwhich can also be measured directly from the slope of P\u0000\u001acurves. Once the uniform pressure of the system and\nthe sti\u000bness tensor (or the e\u000bective sti\u000bness tensor B) are known, we obtain the elastic moduli from \u0015and\u0016using\nEq. S11. In our simulations we apply the method of box deformations to numerically evaluate the sti\u000bness tensor for\na system of interacting particles in an NVT ensemble [20, 26].\nWe extract the four non-zero elements of the sti\u000bness tensor Cby performing three kinds of deformations of the\nsimulation box following Ref. 26. In the \frst kind of deformation, the box is elongated or compressed along the\nx-direction by a small factor \u000fxxsuch that the particle coordinates in the x-direction become x0=x(1 +\u000fxx) and the\nbox length also becomes L0\nx=Lx(1 +\u000fxx). Similarly, the box can be elongated or compressed along the y-direction\ncorresponding to imposing a small linear strain \u000fyy. Both these deformations correspond to a change in the overall\ndensity of the system but the magnitude is kept small in order to stay in the linear response regime. The third\ndeformation is of a shearing type in which we change the shape of the box by keeping the volume constant. The angle\nbetween the xandydimension box vectors, aandbrespectively, is changed from \u0019=2 to\u0019=2\u0000tan\u00001(\u000fxy). The\nparticle positions are then transformed as ( x;y)!(x+y\u000fxy;y).\n−0.01 0.00 0.01\n/epsilon1xx25303540βσ2Pαα\n(a)ρσ2=1.010\n1.0301.050\n1.0701.090\n1.110Pxx\nPyyPxy\n−0.01 0.00 0.01\n/epsilon1xy−1.0−0.50.00.5βσ2Pαβ\nPe=7.2 (b)\nFIG. S4. (a) Diagonal Pxx(\u000e);Pyy(\u0003) and (b) o\u000b-diagonal Pxy(4) components of the full pressure tensor (Eq. S7) obtained\nin a deformed simulation box with N= 2:8\u0002103particles as a function of linear tensile and shearing strains, \u000fxxand\u000fxy,\nrespectively, for various state points \u001a\u001b2as labeled in the legend for Pe = 7 :2. The stress response is linear in this regime\nof small strain magnitudes. We obtain the elements of the e\u000bective sti\u000bness tensor Bfrom the slope of a linear \ft (solid and\ndashed lines) to the data points (symbols). The errorbars in the measurements are smaller than the symbol sizes.\nIn our simulations, we start from a perfect hexagonal initial con\fguration with N= 2:8\u0002103particles and deform\nthe box corresponding to the applied strain. We then measure the full stress tensor P\u000b\fafter a su\u000eciently long\nequilibration time that allows the system to reach a steady state. We perform the measurements by applying \fxed\nlinear strain \u000fxx2[\u00000:01;0:01] in intervals of 0.004. For an isotropic solid only the \frst two elements C11andC12\nare su\u000ecient to obtain the Lam\u0013 e coe\u000ecients \u0015and\u0016, which can be measured just by applying a longitudinal strain\n\u000fxx. However, for some cases we also measure the values of \u0016obtained by imposing a shearing strain \u000fxyand con\frm\nthat the two independent measurements agree. The e\u000bective sti\u000bness tensor Bis directly obtained from the slope of\na linear \ft to the stress vs. strain curves, as shown for B11;B12andB44in Fig. S4, using\nB11=@Pxx\n@\u000fxx; B 22=@Pyy\n@\u000fyy; B 12=@Pyy\n@\u000fxx; B 44=@Pxy\n@\u000fxy;\nBulk and Shear elastic moduli\nIn Fig. S5(a) and S5(b) we plot the bulk modulus Eand the shear modulus G, respectively, as a function of density\nfor various Pe obtained using the method described above. For Pe = 0 (magni\fed in the inset) we \fnd that there is\na distinct jump in both EandG, as indicated in the \fgure by a blue arrow, at a density of \u001a\u001b2= 0:926. This jump\nis indicative of the second order nature of the transition. Upon increasing Pe, we observe a similar jump appearing6\nin bothEandGat higher densities marked by arrows in the \fgure. The bulk modulus Eshows only a discontinuity\nfor higher Pe but the shear modulus Gshows a sharp drop to very small values at this transition upon reducing the\ndensity. Such a small value of the shear modulus Gindicates that the system is not a solid anymore and undergoes\nplastic deformation upon shearing. Furthermore, in the same plots we also indicate the densities where we observe\na \fnite number of defects in the simulations with N= 2:8\u0002103particles by a plus marker (+) as in the main text\nFig. 4(a). These points were determined by analyzing the sampled snapshots within our simulated time which show\na complete absence of defects at densities higher than the marked points (+).\nFor Pe = 0 the defects disappear at a density of \u001a\u001b2= 0:950 which is much higher than the point \u001a\u001b2= 0:926 at\nwhich we observe the jump in the elastic moduli. For Pe \u00157:2 we \fnd that the two transition points agree extremely\nwell. This indicates that for active cases the system becomes plastic as soon as a \fnite number of defects, mainly\ndislocations, appear in the system. On the other hand, the elastic moduli of the active solid states as a function of\ndensity collapse onto a single master curve independent of Pe. This remarkable result can be explained by the fact\nthat the swim contribution to the stress tensor is zero or negligible in the solid phase, and hence, the elastic constants\nof active solids become equal to those of passive solids at the same density. The sole e\u000bect of activity is that the\nstable solid regime shifts to higher densities with activity. A numerical \ft of the form E;G/exp(a\u001a3+b\u001a2+c\u001a+d)\nis shown as a black solid line in both Fig. S5(a) and Fig. S5(b), and agrees very well with the measurements.\nRENORMALIZATION PROCEDURE FROM KTHNY THEORY\nIn equilibrium systems, the KTHNY theory suggests that the melting transition is accompanied by a lowering\nof the Young's modulus \fKbelow a critical value of 16 \u0019. To correct for the interactions of defects present at a\n\fnite temperature a renormalization group analysis is applied to obtain the renormalized value KRof the Young's\nmodulus which can then be compared against the numerical value of 16 \u0019to identify the melting transition. The\ntheory describes the dislocation defects in 2D systems associated with a `core energy' Ec[1, 2]. The probability of\n\fnding a bound pair of such dislocation defects is given by [21, 22]:\npd= exp(\u00002\fEc)Z(K)\n= exp\u0012\n\u00002Ec\nkBT\u00132\u0019p\n3\n\fK=8\u0019\u00001I0\u0012\fK\n8\u0019\u0013\nexp\u0012\fK\n8\u0019\u0013\n(S13)\nwhereZ(K) is the internal partition function of a dislocation, and I0is a modi\fed Bessel function. The theory\nsuggests a continuous transition from the solid to the hexatic state for large core energies Ec\u00152:8kBTand a weakly\nto strongly \frst-order transition as Ecapproaches and becomes lower than a value of 2 :8kBT[27]. Typically, for\nsystems with hard-core interactions the value of Ecnear the solid-hexatic transition is \u00186kBTas found in Ref. [6]\nfor monolayers of hard spheres.\nThe renormalization group recursion relations for the Young's modulus Kare expressed as [1, 2, 22]:\n@\n@l\u00128\u0019\n\fK(l)\u0013\n= 24\u00192y2exp\u0012\fK\n8\u0019\u0013\u0014\n0:5I0\u0012\fK\n8\u0019\u0013\n\u00000:25I1\u0012\fK\n8\u0019\u0013\u0015\n(S14)\n@y(l)\n@l=\u0012\n2\u0000\fK\n8\u0019\u0013\ny+ 2\u0019y2exp\u0012\fK\n16\u0019\u0013\nI0\u0012\fK\n8\u0019\u0013\n: (S15)\nwhere the fugacity yof the dislocation-pair \ruid is obtained from an estimate of the core energy Ecas:\ny= exp\u0012\n\u0000Ec\nkBT\u0013\n: (S16)\nThe di\u000berential equations Eq. S14-S15 can be solved recursively for l= 0:::1by using the unrenormalized (`bare')\nvaluesK(0) =Kandy(0) = exp(\u0000Ec(K(0))) as the initial guesses for l= 0 and utilizing a trapezoidal (or higher\norder scheme) for performing the integration. The renormalized values are obtained from the renormalization-\row\ndiagram of y-vs-1=K(Fig. 1 in Ref. [22]) for the separatrix and KR=K(1) wheny(1) = 0. Exactly at the\ntransition, the renormalization-\row follows the separatrix and above ( T >Tm) and below ( T <Tm) the melting point\ngoes to the end points 1and 0, respectively.7\n1.0 1.2 1.4 1.6 1.8\nρσ2101102103104βEσ2\n0.07.223.847.771.5(a)\nPe =v0τ/σ\n0.0\n7.2\n23.8\n47.7\n71.5\nfit\n0.925 0.95Pe=0.0\n1.0 1.2 1.4 1.6 1.8\nρσ2101102103104βGσ2\n0.07.223.847.771.5(b)\nPe =v0τ/σ\n0.0\n7.2\n23.8\n47.7\n71.5\nfit\nFIG. S5. (a) Bulk modulus Eand (b) shear modulus G, obtained by explicitly straining the simulation box with N= 2:8\u0002103\nparticles, as a function of density \u001a\u001b2for various Pe as labeled in the legend. Both the bulk and the shear moduli collapse\nonto a single master curve indicated by the black lines which are \fts of the form E;G/exp(a\u001a3+b\u001a2+c\u001a+d). Upon\nlowering the density, the shear modulus Gdrops sharply to zero at the critical density where the defects start to appear for\nthe corresponding activity, and the bulk modulus Eshows a transition to a lower stable curve. The transition points at which\nthe defect concentration becomes zero obtained from visual inspection in a system of N= 2:8\u0002103particles are marked with\na plus (+).8\n\u0003s.paliwal@uu.nl\nym.dijkstra@uu.nl\n[1] J. M. Kosterlitz and D. J. Thouless, J. Phys. C Solid State Phys. 6, 1181 (1973).\n[2] D. R. Nelson and B. I. Halperin, Phys. Rev. B 19, 2457 (1979).\n[3] S. T. Chui, Phys. Rev. Lett. 48, 933 (1982).\n[4] E. P. Bernard and W. Krauth, Phys. Rev. Lett. 107, 155704 (2011).\n[5] M. Engel, J. A. Anderson, S. C. Glotzer, M. Isobe, E. P. Bernard, and W. Krauth, Physical Review E 87, 042134 (2013).\n[6] W. Qi, A. P. Gantapara, and M. Dijkstra, Soft Matter 10, 5449 (2014).\n[7] S. C. Kapfer and W. Krauth, Phys. Rev. Lett. 114, 035702 (2015).\n[8] J. A. Anderson, J. Antonaglia, J. A. Millan, M. Engel, and S. C. Glotzer, Phys. Rev. X 7, 021001 (2017).\n[9] A. L. Thorneywork, J. L. Abbott, D. G. A. L. Aarts, and R. P. A. Dullens, Phys. Rev. Lett. 118, 158001 (2017).\n[10] Y. Fily and M. C. Marchetti, Phys. Rev. Lett. 108, 235702 (2012).\n[11] G. S. Redner, M. F. Hagan, and A. Baskaran, Phys. Rev. Lett. 110, 55701 (2013).\n[12] J. Stenhammar, D. Marenduzzo, R. J. Allen, and M. E. Cates, Soft Matter 10, 1489 (2014).\n[13] J. T. Siebert, F. Dittrich, F. Schmid, K. Binder, T. Speck, and P. Virnau, Phys. Rev. E 98, 030601(R) (2018).\n[14] P. Digregorio, D. Levis, A. Suma, L. F. Cugliandolo, G. Gonnella, and I. Pagonabarraga, Phys. Rev. Lett. 121, 098003\n(2018).\n[15] J. U. Klamser, S. C. Kapfer, and W. Krauth, Nat. Comm. 9, 5045 (2018).\n[16] See Supplemental Material.\n[17] R. G. Winkler, A. Wysocki, and G. Gompper, Soft Matter 11, 6680 (2015).\n[18] L. Landau, E. Lifshitz, and J. Sykes, Theory of Elasticity , Course of theoretical physics (Pergamon Press, 1989).\n[19] J. R. Ray, Comput. Phys. Rep. 8, 109 (1988).\n[20] D. Frenkel and B. Smit, Understanding molecular simulation: from algorithms to applications , Vol. 1 (Elsevier, 2001).\n[21] D. S. Fisher, B. I. Halperin, and R. Morf, Phys. Rev. B 20, 4692 (1979).\n[22] S. Sengupta, P. Nielaba, and K. Binder, Phys. Rev. E 61, 6294 (2000).\n[23] J. A. Anderson, C. D. Lorenz, and A. Travesset, J. Comput. Phys. 227, 5342 (2008).\n[24] J. Glaser, T. D. Nguyen, J. A. Anderson, P. Liu, F. Spiga, J. A. Millan, D. C. Morse, and S. C. Glotzer, Comput. Phys.\nCommun. 192, 97 (2015).\n[25] L. F. Cugliandolo, P. Digregorio, G. Gonnella, and A. Suma, Phys. Rev. Lett. 119, 268002 (2017).\n[26] G. Clavier, N. Desbiens, E. Bourasseau, V. Lachet, N. Brusselle-Dupend, and B. Rousseau, Mol. Simul. 43, 1413 (2017).\n[27] K. J. Strandburg, Rev. Mod. Phys. 60, 161 (1988)."    },    {        "title": "1907.11300v1.Comparison_Between_the_f_Electron_and_Conduction_Electron_Density_of_States_in_the_Falicov_Kimball_Model_at_Low_Temperature.pdf",        "content": "Noname manuscript No.\n(will be inserted by the editor)\nComparison Between the f-Electron and Conduction-Electron Density\nof States in the Falicov-Kimball Model at Low Temperature\nR. D. Nesselrodt\u0001J. Canfield\u0001J. K. Freericks\nReceived: date / Accepted: date\nAbstract The spinless Falicov-Kimball model is one of the\nsimplest models of many-body physics. While the conduction-\nelectron density of states is temperature independent in the\nnormal state, the f-electron density of states is strongly tem-\nperature dependent—it has an orthogonality catastrophe sin-\ngularity in the metallic phase and is gapped in the insulating\nphase. The question we address here is whether the spectral\ngap is the same for both electron species as T!0. We find\nstrong evidence to indicate that the answer is affirmative.\nKeywords Mott transition\u0001Falicov-Kimball model \u0001\ndensity of states\u0001orthogonality catastrophe\n1 Introduction\nThe Falicov-Kimball model [1] is perhaps the simplest solid-\nstate model for describing strongly correlated electron sys-\ntems. The model possesses a Mott-insulator transition, a charge-\ndensity-wave ordered phase and can be solved exactly in the\nlimit of infinite dimensions [2]. Initially the model was em-\nployed to understand phase transitions in transition-metal\noxides [1], and has since been used to study a number of\nstrongly correlated systems. Its solution via dynamical mean-\nfield theory has also been reviewed [3].\nRecently, the Falicov-Kimball model has been used to in-\nvestigate core-level X-ray photoemission spectroscopy [4]\n[5]. It has been known for some time that the localized f-\nelectron Green’s function in the metallic phase of this model\nis related to the x-ray edge singularity problem [6,7] (we\nonly reference papers immediately relevant to this work here;\nR. D. Nesselrodt, J. Canfield and J. K. Freericks\nDepartment of Physics, Georgetown University\n37th and O Sts. NW, Washington, D.C. 20057 USA\nTel.: 202-687-5984\nE-mail: rdn11@georgetown.edua more complete history of X-ray edge and the Falicov-\nKimball model appears elsewhere [5]). In particular, at low\ntemperature, the density of states in the metal displays a\npower-law divergence with an interaction-dependent power\nlaw.\nThe f-electrons, unlike the conduction electrons, possess\na non-trivial and highly temperature dependent spectral func-\ntion. This has been studied previously using the numerical\nrenormalization group [8] illustrating the power law diver-\ngence of the f-electron spectrum at w=0 in the metal-\nlic phase. Here we seek to understand the gapped behav-\nior of the f-electron spectral function in the Mott-insulating\nregime, where NRG approaches are known to have accuracy\nissues. Particularly, we ask if the f-electron gap approaches\nthe (temperature-independent) conduction-electron gap in the\nlimit T!0 in the insulating phase. This question may seem\nlike it should have an obvious answer, because the f-electron\ndynamics are inherited through their interaction with the\nconduction electrons. But at nonzero temperature, we clearly\nseef-electron spectral weight within the gap, so the answer\nto the question is far from obvious.\n2 Formalism\nThe spinless Falicov-Kimball model describes two electron\nspecies, conduction ( c) and localized ( f) electrons, which\ninteract via a local Couloumb repulsion Uwhen occupying\nthe same lattice site i. We represent the creation (destruction)\nof a conduction electron at the site iby the second quantized\noperator c†\ni(ci) and of a localized electron by f†\ni(fi). As-\nsuming a common chemical potential mfor the two species,\nthe spinless Falicov-Kimball Hamiltonian is given by\nHFK=å\ni j(\u0000ti j\u0000mdi j)c†\nicj+Efå\nif†\nifi+Uå\nif†\nific†\nici\n(1)arXiv:1907.11300v1  [cond-mat.str-el]  25 Jul 20192 R. D. Nesselrodt et al.\nwhere Efis the f-electron site energy, and ti jis the hop-\nping matrix, with ti j=t=tjifor nearest-neighbor sites i\nandjwith the hopping integral ta real constant. We work\nat half filling ( m=U=2 and Ef=\u0000U=2) on the infinite-\ncoordination Bethe lattice in equilibrium.\nScaling the hopping energy twith coordination number Z\nast=t\u0003=p\nZthen in the limit Z!¥we obtain the non-\ninteracting density of states on the Bethe lattice, given by\nrZ!¥(e) =p\n4t\u00032\u0000e2\n2pt\u00032(2)\nwith t\u0003=1 our energy unit and we restrict to jej\u00142t\u0003.\nThe dynamical mean-field theory (DMFT) approach solves\nthe many-body problem by mapping the lattice problem onto\nan impurity problem because the self-energy has no momen-\ntum dependence. A self-consistent iterative approach is em-\nployed to determine the Green’s functions, with details given\nin [3]. We do not need the full conduction-electron solution\nhere, instead, we require only the bare time-ordered Green’s\nfunction for the conduction electrons G0(t)given by\nG0(t) =\u0000i\npZ¥\n\u0000¥dwIm[G0(w)]e\u0000iwt[f(w)\u0000q(t)] (3)\nwith f(w)the Fermi-Dirac distribution,\nG0(w) =1\nw+id+m\u0000l(w+id)(4)\nthe effective medium, and the dynamical mean-field l(w)\nobtained from the DMFT algorithm [3]. Here q(t)is the\nHeaviside unit-step function.\nThe conduction electrons behave like noninteracting par-\nticles, in that they have a temperature-independent density\nof states, but also like interacting electrons, since they have\na Mott metal-insulator transition at U=2.\nFig. 1: The Kadanoff-Baym-Keldysh contour, which starts\nat time 0, moves along the real time axis to time t, back along\nthe real axis to time 0, then proceeds down the imaginary\ntime axis a distance \u0000b=\u00001=T.\nWe define the greater and lesser f-electron Green’s func-\ntions by\nG>\nf(t;t0) =\u0000ihf(t)f†(t0)i; (5)\nG<\nf(t;t0) =ihf†(t0)f(t)i; (6)where the angle brackets denote the thermal average, h:::i=\nTr(exp(\u0000bHFK)(:::))=Trexp (\u0000bHFK)The creation and an-\nnihilation operators are in the Heisenberg representation. These\nGreen’s functions can be determined by selecting t,t0on cer-\ntain branches of the Kadanoff-Baym-Keldysh contour (Fig.\n1) and using the contour-ordered Green’s function,\nGc\nf(tc;t0\nc) =\u0000ihTc\u0000\nf(tc)f†(t0\nc)\u0001\ni (7)\nwhere tc,t0\ncare two times on the contour (Fig. 1). Tcis the\ncontour time-ordering operator, arranging the operators such\nthatt0\nclies before tcon the contour. For example, if we pick\nt0\ncon the upper real-time branch of the contour and tcon\nthe lower real-time branch, we recover G>\nf(t;t0)from the\ncontour-ordered Green’s function.\nIt has been shown in Ref. [9], that in equilibrium the greater\nGreen’s functions for the f-electrons take the form of a Toeplitz\ndeterminant of a continuous matrix operator over only the\npositive time branch of the contour\nG>\nf(t) =\u0000iw0e\u0000i(Ef\u0000m)tDet[0;t]\f\fd(t1\u0000t2)\u0000UG 0(t1\u0000t2)\f\f\n(8)\nwhere w0is the average density of sites without an f-electron\n(w0=1\n2at half filling), and G0(t)is the bare time-ordered\nGreen’s function determined from the dynamical mean-field\nl(w). The symbols t1andt2denote the matrix indices of the\ncontinuous matrix operator for which we evaluate the deter-\nminant; note that both times must fall within the interval\n[0;t]. (There is a similar expression for the lesser Green’s\nfunction.) To approximate this continuous matrix operator,\nwe discretize it to a conventional matrix and calculate the\ndeterminant for three different discretization time steps Dt.\nWe then perform a second-order Lagrange interpolation to\nextrapolate to the limit Dt!0. These numerical results are\nchecked for accuracy against known spectral moments of the\nGreen’s functions [10].\nIn the limit of large times, an exact analytic formula for\nthef-electron Green’s function in equilibrium can be ob-\ntained using a factorization technique from complex analy-\nsis described by McCoy and Wu [11] and called the Wiener-\nHopf sum approach. It relies on a result for infinite-sized\ndeterminants of Toeplitz matrices called Szego’s theorem.\nFurther finite-time approximations can be made to improve\nthe short-time agreement of this asymptotically exact re-\nsult with the determinant calculation given by Eq. (8) [9],\nwith slightly different formulas for the case when the inter-\naction energy Ulies above or below Uc, a critical interac-\ntion strength Uc=p\n2 on the Bethe lattice where the com-\nplex function C(w) =1\u0000UG 0(w)goes from no winding\naround the origin for U<Ucto winding once in the clock-\nwise direction (Ind C=-1) for U\u0015Uc; note that this critical\nUcis smaller than the critical Ufor the Mott transition. InComparison Between the f-Electron and Conduction-Electron Density of States in the Falicov-Kimball Model at Low Temperature 3\nthe regime with U>Uc, the analytic result for the Toeplitz\ndeterminant in Eq. (8) is\nDet[0;t]=exp\u0014t\n2pZ¥\n\u0000¥dwln¯C(w)+Z¥\n0dt0t0¯g(t0)¯g(\u0000t0)\u0015\n\u0002Dt\n2pZp\nDt\n\u0000p\nDtdw0eiw0t¯P(\u0000w0)\n¯Q(w0)(9)\nwhere ¯C(w) =exp[iwDt][1\u0000UG 0(w)], ¯g(t) =R¥\n\u0000¥dw\n\u0002exp[\u0000iwt]ln(¯C(w))=2p, and ¯P(¯Q)are integrals over all\npositive (negative) time of ¯ g(t)and satisfy\n¯C(w) =1\n¯P(w)¯Q(\u0000w): (10)\nThis expression represents a significant reduction in compu-\ntational complexity, and allows us to probe a much wider pa-\nrameter space when directly calculating the discretized de-\nterminant is not possible. Our approach will be to calculate\nthe determinant directly for short times and use the analytic\nexpression for long times, allowing us to obtain the spec-\ntral function of the f-electrons down to temperatures signif-\nicantly lower than previous calculations. We need to patch\nthe two solutions together smoothly, as described below. We\nexamine the behavior of the f-electrons as they approach\ntheir T=0 limit in both the metallic phase near the Mott-\ninsulator transition U=2 and in the Mott insulating regime.\nFinally, we define the local density of states of the f-\nelectrons, Af(w). At half-filling, there exists a particle-hole\nsymmetry in our system [ Af(w) =Af(\u0000w)], and conse-\nquently the full f-electron density of states can be expressed\nas a Fourier transform of Im G>(t)alone [12],\nAf(w) =\u00002\npZ¥\n0dtcos(wt)ImG>(t): (11)\n3 Numerics\nWe examine the temperature-dependent dynamics of the f-\nelectron spectral function in the metallic regime near the\nMott transition as well as in weakly and strongly correlated\ninsulating phases above the Mott transition, which occurs\natU=2. The spectral function is known to have a power-\nlaw like divergence in the metallic phase that disappears as\nwe move into the Mott phase [8]. In the Mott insulator, the\nf-electron density of states develops a gap with decreasing\ntemperature [13]. We explore the character of this gap to see\nif it is the same as in the temperature-independent conduc-\ntion electron density of states.\nTo utilize the Weiner-Hopf technique in Eq. (9), we per-\nform the discretized matrix calculation of Eq. (8) out to the\nlongest time computationally feasible for three different time\nstepsDtin the ratio 1 : 2 : 4. Next we employ a second-order\nLagrange extrapolation to take the limit Dt!0. We useEq. (9) to calculate the determinant out to even longer times,\nand use a weighted blending of the two functions over a\ntime range where the analytic approximation is roughly par-\nallel to the discretized determinant results. This procedure\nworks exceptionally well for high temperature, as shown\nin Fig. 4, but as we decrease the temperature, the determi-\nnant calculation requires significantly smaller time steps and\ntakes longer to reach the ”long-time” regime where our an-\nalytic result holds (see Fig. 5). This effect limits our low-\ntemperature calculations, and along with truncating the cal-\nculation at a finite time, leads to numerical artifacts near\nw=0 (see Figs. 7, 8, 9).\nFig. 2: G>(t)vs.tforU=1:5. Here we have no zero cross-\ning of the time axis, corresponding to the metallic phase of\nthe model. The inset shows the shorter-time behavior.\n4 Results\nIn the metallic regime, for U=1:5 (Fig. 3) we see evidence\nof a power-law divergence as the temperature approaches\nzero in agreement with [8]. In the time domain, we observe\na delayed decay towards zero with decreasing temperature.\nNote there is no zero crossing in Im G>(t)(Fig. 2), indicat-\ning we are still in the metallic phase.\nForU=1:9, closer to the Mott transition, we still see the\npower-law like increase in the spectral function (Fig. 6), but\nwith the development of a kink in the center of our density of\nstates, representing a precursor to the insulating Mott phase.\nAt the Mott transition, U=2 (Fig. 7), the f-electrons\npartially fill the insulating gap at high temperature, with a\nkinked density of states that does not reach zero at the min-\nimum temperature we were able to reach ( T=0:001). We\nexpect that the density of states should touch zero precisely\natT=0. This is in contrast with the conduction electrons,4 R. D. Nesselrodt et al.\nFig. 3: Spectral function Af(w)vs.w. Here we see the well\ndocumented [8] power law divergence of the orthogonality\ncatastrophe set in as T!0.\nFig. 4: Time-domain plot of Im G>(t)for high temperature\n(T=1) in the insulating phase U\u00152. Here the analytic\nformula works well for all times, and the blending between\nthe two approaches is smooth.\nwhose density of states touches zero at U=2 for all temper-\natures.\nForU=2:5 (Fig. 7), we see a similar transfer of weight\nfrom high to low frequencies as described in [13]. Here we\nsee the spectral function becomes gapped around T=0:1,\nbut continues to change shape down to T=0:001, below\nwhich the gap is frozen into place. Notice in the right inset of\nFig. 8 the small region where the density of states becomes\nslightly negative. This is an artifact of not properly capturing\nthe long-time behavior, likely due to the finite time trunca-\ntion of some long period oscillations in G>(t)and possibly\nthe blending of the numerical and asymptotic results.\nFig. 5: Time-domain plot of Im G>(t)for low tempera-\ntureT<0:05 in the insulating phase U\u00152. Here the di-\nrect determinant calculation has not reached its asymptotic\nlimit at the maximum time allowed by our computational\nresources. Note that both Green’s functions oscillate with\nthe same frequency, but these oscillations are damped much\nmore quickly when using Eq. (9). The appearance of high-\nfrequency oscillations at lower temperatures are characteris-\ntic of the f-electron’s low-temperature dynamics.\nFig. 6: Af(w)vs.wforU=1:9. We see power-law-like\nbehavior, with a central kink that sharpens with decreasing\ntemperature.\nFinally we examine the strongly correlated insulator with\nU=3 (Fig. 9). We see a similar temperature evolution to\nU=2 and U=2:5. In this case, the system becomes gapped\naround T=0:25, higher than U=2:5. Below this temper-\nature Af(w)changes more slowly as the low-temperature\nbehavior is frozen in. We observe a general trend of theComparison Between the f-Electron and Conduction-Electron Density of States in the Falicov-Kimball Model at Low Temperature 5\nFig. 7: Af(w)vswat the Mott transition, U=2. We see\na gradual evolution whereby spectral weight is transferred\nfrom higher frequency states to states near w=0.\nFig. 8: Temperature evolution of Af(w)vs.watU=2:5\nand (inset) the conduction electron density of states over two\ndifferent ranges of frequency. We see that the gap in the low-\nest temperature f-electron density of states seems to be ap-\nproaching the gap width of the conduction electron density\nof states.\ngap appearing and freezing into place at a higher temper-\nature for more strongly correlated materials. Again we see\nthe irregular behavior near w=0 from the finite time trun-\ncation. More interesting, however, is the comparison with\nthe conduction-electron density of states. We see from the\nleft inset of Fig. 9 that at low temperature the width of the\ngap is nearly indistinguishable between the two different\nelectrons—this strongly suggests that the T=0 gap of the\nf-electrons is the same as the conduction-electron gap.\nFig. 9: Temperature evolution of Af(w)vs.watU=3; (in-\nset) the conduction-electron density of states over two differ-\nent frequency ranges. We again see that the lowest tempera-\nturef-electron density of states seems to be approaching the\nwidth of the conduction electron density of states.\n5 Conclusion\nWe extensively studied the properties of the f-electron spec-\ntra of the Falicov-Kimball model in a number of interesting\ncases: just below the Mott transition, in the strongly cor-\nrelated regime, and at temperatures approaching the T=0\nlimit. By using the Weiner-Hopf technique, we were able to\nexamine these more complex situations by obtaining an an-\nalytic expression for the long-time behavior which is com-\nputationally much more efficient than the direct determinant\ncalculation, because the matrices grow in size with increas-\ning time. Due to the rich temperature-dependent dynamics\nof the f-electrons, we asked if the spectral gap in the f-\nspectrum approaches that of the (temperature-independent)\nconduction electrons at T=0. We found strong evidence\nthat this is the case. This technique could be pushed further\nby making higher-order finite-time corrections to the asymp-\ntotic formulas given in [9] and by carrying the calculations\nout to longer times.\nAcknowledgements We would like to thank A. M. Shvaika for valu-\nable conversations and suggestions. This work was supported by the\nDepartment of Energy, Office of Basic Energy Sciences, Division of\nMaterials Sciences and Engineering under Contract No. de-sc0019126.\nJ. K. F. was also supported by the McDevitt bequest at Georgetown.\nConflict of interest\nThe authors declare that they have no conflict of interest.6 R. D. Nesselrodt et al.\nReferences\n1. L.M. Falicov, J. Kimball, Phys. Rev. Lett. 22, 997 (1969). DOI\n10.1103/PhysRevLett.22.997\n2. U. Brandt, C. Mielsch, Z. Phy. B: Condes. Matt. Phys. 75, 365\n(1989). DOI 10.1007/BF01321824\n3. J.K. Freericks, V . Zlati ´c, Rev. Mod. Phys. 75, 1333 (2003). DOI\n10.1103/RevModPhys.75.1333\n4. A.M. Shvaika, J.K. Freericks, Condens. Matt. Phys. 15, 1 (2012).\nDOI 10.5488/CMP.15.43701\n5. N. Pakhira, A.M. Shvaika, J.K. Freericks, Phys. Rev. B. 99(2019).\nDOI 10.1103/PhysRevB.99.125137\n6. G. M ¨oller, A.E. Ruckenstein, S. Schmitt-Rink, Phys. Rev. B 46,\n7427 (1992). DOI 10.1103/PhysRevB.46.7427\n7. Q. Si, G. Kotliar, A. Georges, Phys. Rev. B 46, 1261 (1992). DOI\n10.1103/PhysRevB.46.1261\n8. F.B. Anders, G. Czycholl, Phys. Rev. B 71, 125101 (2005). DOI\n10.1103/PhysRevB.71.125101\n9. A.M. Shvaika, J.K. Freericks, Cond. Mat. Phys. 11, 425 (2008).\nDOI 10.5488/CMP.11.3.425\n10. S.R. White, Phys. Rev. B 44, 4670 (1991). DOI 10.1103/\nPhysRevB.44.4670\n11. B. McCoy, T. Wu, The Two-Dimensional Ising Model (Harvard\nUniversity Press, 1973)\n12. U. Brandt, M.P. Urbanek, Z. Phys. B: Conds. Matt. Phys. 89(3),\n297 (1992). DOI 10.1007/BF01318160\n13. J.K. Freericks, V .M. Turkowski, V . Zlati ´c, Phys. Rev. B. 71(2005).\nDOI 10.1103/PhysRevB.71.115111"    },    {        "title": "1907.12428v2.Learn_to_Scale__Generating_Multipolar_Normalized_Density_Maps_for_Crowd_Counting.pdf",        "content": "Learn to Scale: Generating Multipolar Normalized Density Maps for Crowd\nCounting\nChenfeng Xu1\u0003, Kai Qiu2, Jianlong Fu2, Song Bai3, Yongchao Xu1y, Xiang Bai1\n1Huazhong University of Science and Technology,2Microsoft Research Asia,3University of Oxford\nfxuchenfeng, yongchaoxu, xbai g@hust.edu.cn, fkaqiu, jianfg@microsoft.com, songbai.site@gmail.com\nAbstract\nDense crowd counting aims to predict thousands of hu-\nman instances from an image, by calculating integrals of a\ndensity map over image pixels. Existing approaches mainly\nsuffer from the extreme density variances. Such density pat-\ntern shift poses challenges even for multi-scale model en-\nsembling. In this paper, we propose a simple yet effective\napproach to tackle this problem. First, a patch-level den-\nsity map is extracted by a density estimation model and\nfurther grouped into several density levels which are de-\ntermined over full datasets. Second, each patch density\nmap is automatically normalized by an online center learn-\ning strategy with a multipolar center loss. Such a design\ncan significantly condense the density distribution into sev-\neral clusters, and enable that the density variance can be\nlearned by a single model. Extensive experiments demon-\nstrate the superiority of the proposed method. Our work\noutperforms the state-of-the-art by 4.2%, 14.3%, 27.1% and\n20.1% in MAE, on ShanghaiTech Part A, ShanghaiTech Part\nB, UCF CC50 and UCF-QNRF datasets, respectively.\n1. Introduction\nA robust crowd counting system is of significantly value\nin many real-world applications such as video surveillance,\nsecurity alerting, event planning, etc. In recent years, the\ndeep learning based approaches have been the mainstream\nof crowd counting, due to the powerful representation learn-\ning ability of convolutional neural networks (CNNs). To\nestimate the count, predominant approaches generate a den-\nsity map by CNN, from which the count of instances can be\nintegrated over image pixels.\nAlthough crowd counting has been extensively studied\nby previous methods, handling the large density variances\nwhich cause huge density pattern shift in crowd images is\n\u0003This work was done when Chenfeng Xu was a research intern at Mi-\ncrosoft Research Asia.\nyCorresponding author\nSparse Medium Dense(a)\nShanghai-B Shanghai-A UCF_CC_50 UCF-QNRF\nDataset0.000.050.100.150.200.250.300.35Mean Relative ErrorMRE Overview\nOurs\nMCNN [33]\nSwitch-CNN [23]\nCMTL [27]\nACSCP [24]\nSANet [3]CSRNet [15]\nCP-CNN [28]\nL2R [18]\nD-ConvNet-v1 [25]\nic-CNN [22]\n(b)\nFigure 1. (a) Three examples from ShanghaiTech Part A dataset,\nwhich show extreme density variances. (b) Comparison of Mean\nRelative Error on four crowd counting datasets (the scale variances\nget larger from left to right) of different approaches. Results show\nthe robustness of the proposed approach to extreme scale vari-\nances. [best viewed in color].\nstill an open issue. As illustrated in Fig. 1(a), the densi-\nties of crowd image patches can vary significantly, which\nchange from a bit sparse ( e.g., ShanghaiTech Part B) to ex-\ntremely dense ( e.g., UCF-QNRF). Such large density pat-\ntern shifts usually bring grand challenges to density pre-\ndiction by a single CNN model, due to its fixed sizes of\nreceptive fields. Remarkable progress has been achieved\nby learning a density map through designing multi-scale ar-\nchitectures [23] or aggregating multi-scale features [3, 33],\nwhich indicate that the ability to cope with density varia-\ntion is crucial for crowd counting methods. Although den-\nsity maps with multiple scales can be generated and aggre-\ngated, it is still hard to ensure robustness when the density\n1arXiv:1907.12428v2  [cs.CV]  8 Aug 2019variances get increased a lot. As shown in Fig. 1(b), most\nrecent works obtain a higher MRE1on datasets with larger\ndensity variances, which indicates that the extreme density\nvariance and pattern shift in crowd counting remains a huge\nchallenge.\nIn this paper, we propose a simple yet effective method to\nmitigate the problem caused by extreme density variances.\nThe core idea is learning to scale image patches and to facil-\nitate the density distribution condensing to several clusters,\nthus the density variance can be reduced. The scale factor\nof each image patch can be automatically learned during\ntraining, with the supervision of a novel multipolar center\nloss (MPCL). More specifically, all the patches at each den-\nsity level are optimized to approach a density center, which\ncan be updated by online calculating a mean value for each\ndensity level.\nIn particular, the proposed framework consists of two\nclosely-related steps. First, given an image, an initial den-\nsity map is generated by our designed Scale Preserving Net-\nwork (SPN). After that, each density map is divided into\nK\u0002Kpatches, and all the patch-level density maps are fur-\nther evenly divided into Ggroups, according to their density\nlevels. Second, each patch is scaled by a learned scale fac-\ntor, thus the density of this patch can converge to a center of\nits density level. The final density map for the input image\ncan be obtained by concatenating the K\u0002Kpatch-level\ndensity maps.\nExperiments are conducted on several popular\nbenchmark datasets, including ShanghaiTech [33],\nUCF CC50 [9], and UCF-QNRF [11]. Extensive evalua-\ntions demonstrate superior performance over the prior arts.\nMoreover, the cross dataset validation on these datasets fur-\nther demonstrates that the proposed method has a powerful\ntransferability. In summary, the main contributions in this\npaper are two-fold:\n- We propose a Learning to Scale Module (L2SM) to\nsolve the density variation issue in crowd counting.\nWith L2SM, different regions can be automatically\nscaled so that they have similar densities, while the\nquality of the density maps is significantly improved.\nL2SM is end-to-end trainable when adding it into a\nCNN model for density estimation.\n- The proposed L2SM added into SPN significantly\noutperforms state-of-the-art methods on three widely-\nadopted challenging datasets, demonstrating its effec-\ntiveness in handling density variation. Furthermore,\nL2SM also has a good transferability under cross\ndataset validation on different datasets, showing the\ngeneralizability of the proposed method.\n1MRE is calculated by MAE/P, where MAE denotes the standard\nMean Average Error and P is the average count of a dataset2. Related Work\nCrowd counting has attracted much attention in com-\nputer vision. Early methods frame the counting problem\nas a detection task [7, 29] that explicitly detects individual\nheads, which has major difficulty in occlusion and dense ar-\neas. The regression-based methods [4, 6, 8, 10] greatly im-\nprove the counting performance on dense areas via different\nregression functions such as Gaussian process, ridge regres-\nsion, and random forest regression. Recently, with the de-\nvelopment of deep learning, the mainstream crowd counting\nmethods switch to CNN-based methods [21, 32, 2, 33, 31, 5,\n18]. These CNN-based methods address the crowd counting\nvia regressing density map representations [14], and achieve\nhigher accuracy and transferability than the classical meth-\nods. Recent methods mainly focus on two challenging as-\npects faced by current CNN-based methods: huge scale and\ndensity variance and severe over-fitting.\nMethods addressing huge scale and density variance.\nMulti-scale is a challenging problem for many vision tasks\nincluding crowd counting. It is difficult to count the small\nheads in dense areas accurately. There are many methods at-\ntempting to handle huge scale variance. The existing meth-\nods can be roughly divided into two categories: methods\nthat explicitly rely on scale information and methods that\nimplicitly cope with multi-scale.\n1) Some methods explicitly make use of scale informa-\ntion for crowd counting. For instance, Zhang et al. [32] and\nOnoro et al. [19] adopt CNNs with provided geometric or\nperspective information. Yet, this scale related information\nis not always readily available. Sindagi et al. [28] use net-\nworks to estimate the density degree for the corresponding\nwhole and partial region based on manually setting scale de-\ngrees and fuse them as context information. Sam et al. [23]\nleverage the scale information to design different networks\nfor dividing and counting. To overcome the difficulty in\nmanually setting the scale degree, Sam et al. [1] design an\nincrementally growing CNN to deal with areas of different\ndensity degrees without involving any handcraft steps.\n2) Some other works aim to implicitly cope with the\nmulti-scale problem. Zhang et al. [33] and Cao et al. [3]\npropose to build a multi-column CNN to extract multi-scale\nfeatures and fuse them together for density map estimation.\nDifferent from multi-scale feature fusion, Liu et al. [17] at-\ntempt to encode the scale of the contextual information re-\nquired to predict crowd density accurately. In [15], Li et\nal.propose to increment the receptive field size in CNN to\nbetter leverage multi-scale information. In addition to these\nspecific network designs for implicitly handling the multi-\nscale problem, Shen et al. [24] introduce an ad hoc term in\nthe training loss function in order to pursue the cross-scale\nconsistency. In [11], Idrees et al. propose to adopt vari-\nant ground-truth density map representation with Gaussian\nkernels of different sizes to better deal with density map es-\n2Figure 2. A rational human behavior. For a given image, we are\nprone to first count in the regions of large heads ( e.g., region on the\nbottom of image), then zoom in the regions of dense small heads\nfor precise counting (see for example the region in the middle and\nits zoomed version on top right).\ntimation in areas of different density levels.\nMethods alleviating severe over-fitting. It is well-\nknown that deep CNNs [13, 26] usually struggle with the\nover-fitting problem on small datasets. Current CNN-based\ncrowd counting methods also face this challenge due to the\nsmall size and limited variety of existing datasets, lead-\ning to weak performance and transferability. To over-\ncome the over-fitting, Liu et al. [18] propose a learning-to-\nrank framework to leverage abundantly available unlabeled\ncrowd images and a self-learning strategy. Shi et al. [25]\nbuild a set of decorrelated regressors with reasonable gen-\neralization capabilities through managing their intrinsic di-\nversities to avoid severe over-fitting.\nThough many methods have been proposed to tackle the\nlarge scale and density variation issue, this problem still\nremains challenging for crowd counting. Different from\nprevious methods [33, 23, 27, 1, 3, 16], we mimic a ratio-\nnal human behavior in crowd counting through learning to\nscale dense region counting. We compute the scale ratios\nwith a novel use of multipolar center loss [30] to explicitly\nbring all the regions of significantly varied density to mul-\ntiple similar density levels. This results in a robust density\nestimation on dense regions and appealing transferability.\n3. Method\n3.1. Overview\nThe mainstream crowd counting methods model the\nproblem as density map regression using CNNs. For a given\nimage, the ground-truth density map Dis given by spread-\ning binary head locations to nearby regions with Gaussian\nkernels. For sparse regions, the ground-truth density only\ndepends on a specific person, resulting in regular Gaus-\nsian blobs. For dense regions, multiple crowded heads may\nspread to the same nearby pixel, yielding high ground-truth\ndensities with very different density patterns compared with\nsparse regions. These density pattern shifts make it difficult\nto accurately predict the density maps for both dense and\nsparse regions in the same way.\nTo improve the counting accuracy, we aim to tackle theproblem of pattern shift caused by large density variations\nand refine the prediction for highly dense regions. Specifi-\ncally, the proposed method mimics a rational behavior when\nhumans count crowds. For a given crowd image, we are\nprone to begin with dividing the image into partitions of dif-\nferent crowding levels before attempting to count the peo-\nple. For sparse regions of large heads, it is easy to count the\npeople on the original region directly. Whereas, for dense\nregions composed of crowded small heads, we need to zoom\nin the region for more accurate counting. An example of\nthis counting behavior is depicted in Fig. 2.\nWe propose a network to mimic such human behavior\nfor crowd counting. The overall pipeline is depicted in\nFig. 3, consisting of two modules: 1) Scale preserving net-\nwork (SPN) presented in Sec. 3.2. We leverage multi-scale\nfeature fusion to generate an initial density map prediction,\nwhich provides an accurate prediction on sparse regions and\nindicates the density distribution over the image; 2) Learn-\ning to scale module (L2SM) detailed in Sec. 3.3. We divide\nthe image into K\u0002Knon-overlapping regions, and select\nsome dense regions (based on the initial density estimation)\nto re-predict the density map. Specifically, we leverage SPN\nto compute a scaling factor for each selected dense region,\nand scale the ground-truth density map by changing the dis-\ntance between blobs and keeping the same peaks. The den-\nsity re-prediction for the selected regions is then performed\non the scaled features. The key to this re-prediction pro-\ncess lies in computing appropriate scaling factors. For that,\nwe adopt the center loss to centralize the density distribu-\ntions into multipolar centers, alleviating the density pattern\nshift issue and thus improving the prediction accuracy. The\nwhole network is end-to-end trainable and the training ob-\njective is presented in Sec. 3.4.\n3.2. Scale Preserving Network\nWe follow the mainstream crowd counting methods by\nregressing density maps. Precisely, we use geometry-\nadaptive kernels to generate ground-truth density maps in\nhighly congested scenes. For a given image containing P\nperson, the ground-truth annotation can be represented via\na delta function on each pixel p:H(p) =PP\ni=1\u000e(p\u0000pi),\nwherepiis the annotated location of i-th person. The den-\nsity mapDon each pixel pis then generated by convolving\nH(p)with a Gaussian kernel G:D(p) =PP\ni=1\u000e(p\u0000pi)\u0003\nG\u001bi, where the Gaussian kernel \u001biis a spread parameter.\nWe develop a CNN to regress the density map D. For a\nfair comparison with most methods, we adopt VGG16 [26]\nas the backbone network. We discard the pooling layer be-\ntween stage4 andstage5 , as well as the last pooling layer\nand the fully connected layers that follow to preserve ac-\ncurate spatial information. It is well-known that deep lay-\ners in CNN encode more semantic and high-level informa-\ntion, and shallow layers provide more precise localization\n316×168×84×42×21×1\nInitial density mapCONV+POOL\n𝐻\n8∗𝑊\n8\nFeature Pyramid𝑓𝑏\n𝑓𝑠𝑓𝑏\n𝐷L2SRe-predicted density map\nSamplerScale factor mapCNN\nSelected Features Scaled features…\nΤ𝑫□r𝟏𝟐\nΤ𝑫□r𝟐𝟐\nΤ𝑫□r𝒊𝟐\n…MPCL\nCONV…\n……..\n.K\nK…\nRe-prediction…𝐷′\n𝑓𝑏\nK x K\nUP/DOWN SAMPLINGFigure 3. Overall pipeline of the proposed method with two modules: 1) Scale Preserving Network (SPN) to generate an initial density\nmap^Dfrom stacked feature fs, and 2) Learning to Scale Module (L2SM) that computes the scale ratios rfor dense regions selected (based\non^D) from K\u0002Knon-overlapping divisions of image domain, and then re-predicts the density map ^D0for selected dense regions from\nscaled feature fb. We adopt multipolar center loss (MPCL) on relative density level reflected by ^Di=r2\nifor each region Rito explicitly\ncentralize all the selected dense regions into multiple similar density levels. This alleviates the density pattern shift issue caused by the\nlarge density variation between sparse and dense regions.\ninformation. We extract features from different stages by\napplying 3\u00023convolutions on the last layer of each stage.\nThen we pool these features extracted from stage1 tostage5\ninto16\u000216,8\u00028,4\u00024,2\u00022, and 1\u00021, respectively.\nThis results in a pyramid structure. Each spatial unit in the\npooled feature indicates the density level, hence it maps to\nthe scale information of the underlying image. These scale\npreserving features are then upsampled to the size of conv5\nby bilinear interpolation and stacked together with features\ninconv5fb. We then feed the stacked feature fsto three\nsuccessive convolutions and one deconvolution layer for re-\ngressing the density map ^D.\n3.3. Learning to Scale Module\nThe initial density prediction is accurate on sparse re-\ngions thanks to the regular individual Gaussian blobs, but\nthe prediction is less accurate on dense regions composed of\ncrowded heads lying very close to each other. As indicated\nin Sec. 3.1, this triggers the pattern shift on the target den-\nsity map. Following the rational human behavior in crowd\ncounting, we zoom in the dense regions for better count-\ning accuracy. In fact, on the zoomed version, the distance\nbetween nearby heads is enlarged, which results in regular\nindividual Gaussian blobs of target density map, alleviating\nthe density pattern shift. Such density pattern modulating\nfacilitates the prediction. Inspired by this, we first evenly\ndivide the image domain into K\u0002K(e.g.,K= 4) non-\noverlapping regions. We then select the dense regions based\non the average initial density Di=P\np2Ri^D(p)=jRijof\neach regionRi, wherejRijdenotes the area of region Ri.\nWe achieve this by learning to scale the selected dense\nregions.We first leverage the scale preserving pyramid features\ndescribed in Sec. 3.2 to compute the scaling ratio rifor each\nselected region Ri. Precisely, we downsample/upsample\nthe pooled features described in Sec. 3.2 to K\u0002K, and\nconcatenate them together. This is followed by a 1\u00021con-\nvolution to produce the scale factor map r. Each value in\nthisK\u0002Kmaprrepresents the scaling ratio for the under-\nlying region.\nOnce having the scale factor map r, we scale the feature\nfbon the selected regions accordingly through bilinear up-\nsampling. Based on the scaled feature map corresponding\nto each selected region Ri, we apply five successive convo-\nlutions to re-predict the density map for scaled Ri. We then\nresize the re-predicted density map to the original size of\nRiand multiply the density on each pixel by r2\nito preserve\nthe same counting result. The initial prediction on selected\nregions is replaced by the re-prediction of resized density\nmap.\nTo guide the density map re-prediction on the selected\nregions, we also adjust the ground-truth density map for\neach region accordingly. For each selected region Ri, in-\nstead of directly scaling the ground-truth density map in the\nsame way as feature map scaling, we first scale the binary\nhead location map, and then recompute the ground-truth\ndensity map D0\niforRibyD0\ni(p) =PPi\nm=1\u000e(p\u0000ri\u0003pm)\u0003\nG\u001bm(p), wherePiis the number of people in Ri. As\nshown in Fig. 4, such ground-truth transformation for den-\nsity map re-computation reduces the density pattern gap be-\ntween sparse regions and dense regions, facilitating the den-\nsity map re-prediction.\nThe main issue of this density map re-prediction by\nlearning to scale dense regions is to compute appropriate\n42019/3/3 pdfresizer .com-pdf-crop.pdf\nchrome-extension://oemmndcbldboiebfnladdacbdfmadadm/https://pdfresizer .com/download/361325fecb09.pdf 1 / 1\nFigure 4. An example of ground-truth transformation for density\nmap re-computation by enlarging the distance between blobs while\nkeeping the original peaks, alleviating the density pattern shift be-\ntween sparse and dense regions.\nscale ratios for the selected dense regions. Yet, there is no\nexplicit target scale suggesting how much region Rishould\nbe zoomed ideally. We would like to have the estimated\naverage density Diapproaching the ground-truth average\ndensity on the i-th region. The relative density degree of re-\ngionRicould be well reflected by di=Di=r2\ni:Assuming\nthat we make the value of difor each region close to one of\nthe multiple learnable centers, then we centralize all the se-\nlected regions to multiple similar density levels, alleviating\nthe large density pattern shift and thus improving the pre-\ndiction accuracy. This motivates us to resort to center loss\nondiwith multipolar centers. Put it simply, we attempt to\ncentralize all the selected regions into Ccenters following\ntheir average density Dacting as the unsupervised cluster-\ning.\nSpecifically, we extend the center loss to a multipolor\ncenter loss (MPCL) to handle different density levels. We\nfirst initialize the Ccenters with increasing random values\nfor more and more dense regions. Then for each center dc,\nwe follow the standard process of using center loss and up-\ndate the center for (t+ 1) -th iteration as\n\u0001dct=Pnc\ni=1(dct\u0000Dc\ni\nrc\ni\u0002rc\ni)\n1 +nc;dct+1=dct\u0000\u000b\u0001\u0001dct;(1)\nwherencrefers to the number of regions, Dc\nirefers to av-\nerage density map, rc\nirefers to scaling ratio for i-th region,\nand\u000bdenotes the learning rate for updating each center,\nrespectively. The Dc\niwill be centralized to the c-th cen-\nter in an image. During each iteration, we use the selected\nN=PC\nc=1ncdense regions to compute the center loss\nLcwith multiple centers and update network parameters as\nwell as the centers. The supervision on rusing multipo-\nlar center loss is the key to bring all the selected regions\nto multiple similar density levels, leading to robust density\nestimations.\n3.4. Training objective\nThe whole network is end-to-end trainable, which in-\nvolves three loss functions: 1) L2 loss for initial predictionof density map LDgiven byLD=\r\r\rD\u0000^D\r\r\r\n2; 2) L2 loss\nfor density map re-prediction on N=PC\nc=1ncselected\nregionsLrgiven byLr=PN\ni=1\r\r\rD0\ni\u0000^D0i\r\r\r\n2, where ^D0\ni\ndenotes the re-predicted density map on the scaled selected\nregionRi; 3) Multipolar center loss at relative density level\ndfor the selected regions Lccomputed by\nLc=CX\nc=1ncX\ni=1\r\r\r\r\rDc\ni\nrc\ni\u0002rc\ni\u0000dc\r\r\r\r\r\n2: (2)\nThe final loss function Lfor the whole network is the com-\nbination of the above three losses given by\nL=LD+\u00151\u0002Lr+\u00152\u0002Lc; (3)\nwhere\u00151and\u00152are two hyperparameters. Note that we\noptimize the loss function Lin Eq. (3) to update not only\nthe overall network parameters but also the centers fdcg.\n4. Experiments\n4.1. Datasets and Evaluation Metrics\nWe conduct experiments on three widely adopted\nbenchmark datasets including ShanghaiTech [33],\nUCF CC50 [9], and UCF-QNRF [11] to demonstrate\nthe effectiveness of the proposed method. These three\ndatasets and the adopted evaluation metrics are shortly\ndescribed in the following.\nShanghaiTech Dataset. The ShanghaiTech crowd count-\ning dataset [33] consists of 1198 annotated images divided\ninto two parts. Part A contains 482 images which are ran-\ndomly crawled from the Internet. Part B includes 716 im-\nages which are taken from the busy streets of metropolitan\narea in Shanghai city.\nUCF CC50 Dataset. This dataset is a collection of 50 im-\nages of very crowd scenes [9]. There the number of peo-\nple varies from 94 to 4543 in images. Following classical\nbenchmarks on this dataset, we use 5-fold cross-validation\nto evaluate the performance of our method.\nUCF-QNRF. UCF-QNRF dataset is the recent dataset [11]\ncontaining 1535 images. The number of people in an image\nvaries from 49 to 12865, making this dataset feature huge\ndensity variance. Furthermore, the images in this dataset\nalso have very huge resolution variance ( e.g., ranging from\n400\u0002300to9000\u00026000 ).\nEvaluation metrics. We employ two standard met-\nrics, i.e., Mean Absolute Error (MAE) and Mean Squared\nError (MSE). MAE and MSE are defined as\nMAE =1\nMMX\ni=1jci\u0000^cij;MSE =vuut1\nMMX\ni=1(ci\u0000^ci)2;\n(4)\n5MethodShanghaiTech Part A ShanghaiTech Part B UCF CC50 UCF-QNRF\nMAE MSE MAE MSE MAE MSE MAE MSE\nMCNN [33] 110.2 173.2 26.4 41.3 377.6 509.1 277 -\nCMTL [27] 101.3 152.4 20.0 31.1 322.8 397.9 252 514\nSwitch-CNN [23] 90.4 135.0 21.6 33.4 318.1 439.2 228 445\nCP-CNN [28] 73.6 112.0 20.1 30.1 298.8 320.9 - -\nACSCP [24] 75.7 102.7 17.2 27.4 291.0 404.6 - -\nL2R [18] 73.6 112.0 13.7 21.4 279.6 388.9 - -\nD-ConvNet-v1 [25] 73.5 112.3 18.7 26.0 288.4 404.7 - -\nCSRNet [15] 68.2 115.0 10.6 16.0 266.1 397.5 - -\nic-CNN [22] 69.8 117.3 10.7 16.0 260.9 365.5 - -\nSANet [3] 67.0 104.5 8.4 13.6 258.4 334.9 - -\nCL [11] - - - - - - 132 191\nVGG16 ( ours ) 72.9 114.5 12.1 20.5 225.4 372.5 120.6 205.2\nSPN ( ours ) 70.0 106.3 9.1 14.6 204.7 340.4 110.3 184.6\nSPN+L2SM ( ours ) 64.2 98.4 7.2 11.1 188.4 315.3 104.7 173.6\nTable 1. Quantitative comparison of the proposed method with state-of-the-art methods on three benchmark0 datasets.\nMethod SPNL2SM (G=3) L2SM (G=4) L2SM/S2AD (G=5)\nC= 2 C= 2 C= 1 C= 2 C= 3 C= 4 C= 5\nMAE 70.0 65.1 66.1 67.2/68.9 65.4/68.1 64.2/67.0 67.1/69.2 69.8/73.6\nMSE 106.3 100.4 103.5 102.3/110.3 100.7/107.3 98.4/105.4 101.6/108.7 104.5/113.5\nCost time (s) 0.524 0.576 0.569 0.539/0.540 0.550/0.551 0.565/0.563 0.583/0.580 0.592/0.587\nTable 2. Ablation study on different settings of dense region selection, number of centers C, and different ways of learning to scale. L2SM\ndenotes the proposed learning to scale module and S2AD denotes that we directly scale the selected regions to the average density.\nK\u0002Ksetting MAE MSE\n2\u00022 68.0 107.1\n4\u00024 67.2 106.3\n6\u00026 67.9 106.9\n8\u00028 68.5 109.1\nTable 3. Ablation study on K\u0002Kimage domain divisions for\nselecting dense region to re-predict under one center setting.\nwhereci(resp. ^ci) represents the ground-truth ( resp. esti-\nmated) number of pedestrians in the i-th image, and Mis\nthe total number of testing images.\n4.2. Implementation Details\nWe follow the setting in [15] to generate the ground-truth\ndensity map. For a given dataset, we first evenly divide all\nthe images in a dataset into Ggroups of regions with in-\ncreasing densities, and then attempt to centralize the top C\ndensest groups of regions to Csimilar density levels ( i.e.,\nCcenters involved in the center loss), respectively. In the\nfollowing, without explicitly specifying, Gis set to 5, and\nCis set to 3 for all the used datasets except for UCF CC50\ndataset. Since images from UCF CC50 dataset consist of\ncrowded people over the whole image domain, we central-\nize all regions to C= 5 similar density levels. Without\nexplicitly specified, the hyperparameter Kinvolved in di-\nviding each image into K\u0002Kregions is set to 4.\nThe loss function described in Eq. (3) is used for the\nmodel training. We set \u00151to 1 and discuss the impact of\u00152in Eq. (3) in the following. We use Adam [12] op-\ntimizer to optimize the whole architecture with the learn-\ning rate initialized to 1e-4. When training on the UCF-\nQNRF dataset containing images of very high resolutions\n(e.g.,9000\u00026000 ), we first down-sample the image of\nwhich resolution is larger than 1080p to 1920\u00021080 . Then\nwe divide each image into 2\u00022and combine them into a\ntensor with batch size equal to 4. When training on the other\ndatasets, we directly input the whole image to our network.\nDuring inference, we first generate an initial density map\n^Dfor the whole input image, and then select dense regions\nfromK\u0002Kdivisions based on the average initial density\nDion each region Ri. IfDiis larger than a predefined\nvalue for selecting the top Cgroups of regions in training,\nwe replace the initial density map prediction with scaled re-\nprediction for each selected dense region Ri.\nThe proposed method is implemented in Pytorch [20].\nAll experiments are carried out on a workstation with an In-\ntel Xeon 16-core CPU (3.5GHz), 64GB RAM, and a single\nTitan Xp GPU.\n4.3. Experimental Comparisons\nThe proposed method outperforms all the other com-\npeting methods on all the benchmarks. The quantitative\ncomparison with the state-of-the-art methods on these three\ndatasets is presented in Table 1.\nShanghaiTech. Our work outperforms SANet [3], the state-\nof-the-art method, by 2.8 in MAE and 6.1 in MSE on\n6MethodPart A!Part B Part B!Part A Part A!UCF CC50 UCF-QNRF!Part A Part A!UCF-QNRF\nMAE MSE MAE MSE MAE MSE MAE MSE MAE MSE\nMCNN [33] 85.2 142.3 221.4 357.8 397.7 624.1 - - - -\nD-ConvNet-v1 [25] 49.1 99.2 140.4 226.1 364 545.8 - - - -\nL2R [18] - - - - 337.6 434.3 - - - -\nSPN ( ours ) 23.8 44.2 131.2 219.3 368.3 588.4 87.9 126.3 236.3 428.4\nSPN+L2SM ( ours ) 21.2 38.7 126.8 203.9 332.4 425.0 73.4 119.4 227.2 405.2\nTable 4. Cross dataset experiments on ShanghaiTech, UCF CC50, and UCF-QNRF dataset for assessing the transferability of different\nmethods.\n0 0.05 0.1 0.2 0.5 1\nThe weight of MPCL60708090100110120 Error of PredictionMAE W/O TransedGT\nMSE W/O TransedGT\nMAE W/ TransedGT\nMSE W/ TransedGT\nFigure 5. Ablation study on the effect of weight of the center loss\nunder one center and on whether using ground-truth transforma-\ntion when scaling for re-prediction. W/ TransedGT means ground-\ntruth transformation is used while W/O TransedGT means it is not\nused.\nShanghaiTech Part A and 1.2 in MAE and 2.5 in MSE on\nShanghaiTech Part B. It is shown in Table 1 that L2SM im-\nproves the performance of our SPN baseline by 5.8 in MAE\nand 7.9 in MSE on ShanghaiTech Part A, and 1.9 in MAE\nand 3.5 in MSE on ShanghaiTech Part B. In fact, Shang-\nhaiTech Part A contains images that are more crowded than\nShanghaiTech Part B, and the density distribution of Shang-\nhaiTech Part A varies more significantly than that of Shang-\nhaiTech Part B. This may explain that the improvement of\nthe proposed L2SM on ShanghaiTech Part A is more signif-\nicant than that on ShanghaiTech Part B.\nUCF CC50.We then compare the proposed method with\nother related methods on UCF CC50 dataset. To the best of\nour knowledge, UCF CC50 dataset is currently the densest\ndataset publicly available for crowd counting. The proposed\nmethod achieves significant improvement over state-of-the-\nart methods. Precisely, the proposed method decreases the\nMAE from 258.4 to 188.4, and MAE from 334.9 to 315.3\nfor SANet [3].\nUCF-QNRF. We also conduct experiments on UCF-QNRF\ndataset containing images of significantly mulitiple den-\nsity distributions and resolutions. By limiting the maximal\nimage size to 1920\u00021080 , our VGG16 baseline alreadyachieves state-of-the-art performance. The proposed SPN\nbrings an improvement of 10.3 in MAE and 20.6 in MSE\ncompared with VGG16 baseline. The proposed L2SM fur-\nther boosts the performance by 5.6 in MAE and 11.0 in\nMSE.\n4.4. Ablation Study\nThe ablation studies are mainly conducted on the Shang-\nhaiTech part A dataset, as it is a moderate dataset, neither\ntoo dense nor too sparse, and covers a diverse number of\npeople heads.\nEffectiveness of different learning to scale settings. For\nthe learning to scale process, we first evenly divide the im-\nages in a whole dataset into Ggroups of regions with in-\ncreasing density, and then attempt to centralize the densest\nCgroups of regions to Csimilar density levels. As shown in\nTable 2, the number of groups Gand the number of centers\nCare important for accurate counting. For a fixed num-\nber of groups ( e.g.,G= 5), centralizing more and more\nregions leads to slightly improved counting results. Yet,\nwhen we attempt to centralize every image region, we also\nre-predict the density map for very sparse or background\nregions, bringing more background noise and thus yield-\ning slightly decreased performance. A relative finer group\ndivisions with a proper number of centers performs slightly\nbetter. As shown in Table 2, the proposed L2SM with multi-\npolar center loss performs much better than directly scaling\nthe regions to the average density (S2AD) in each group.\nTime overhead. To analyze the time overhead of the pro-\nposed L2SM, we conduct experiments under seven differ-\nent settings (see Table 2). The time overhead analysis is\nachieved by calculating the average inference time on the\nwhole ShanghaiTech Part A test set. The batch size is set\nto 1 and only 1 Titan-X GPU is used during inference. The\naverage time overhead of SPN is about 0.524s per image.\nWhen we increase the number of centers and the number\nof regions to be re-predicted, the runtime slightly increases.\nWhen using 5 centers and re-predict all the K\u0002Kregions,\nthe proposed L2SM increases the runtime by 0.068s per im-\nage, which is negligible compared with the whole runtime.\nEffectiveness of the weight of MPCL. We study the effec-\ntiveness of center loss on ShanghaiTech Part A using one\n7SPN: 1489 GT: 1265\nSPN: 1716 GT: 2256L2SM: 1299\nL2SM: 2011Figure 6. Qualitative visualization of predicted density map on two examples. From left to right: original image, prediction given by SPN,\nre-predicted density map with L2SM on selected regions (englobed by black boxes), and ground-truth density map.\ncenter by changing its weight \u00152in Eq. (3). Note that when\nthe weight\u00152is set to 0, the center loss is not used, which\nmeans that the scale ratio ris learned automatically with-\nout any specific supervision. As shown in Fig. 5, The use\nof center loss which brings regions of significantly multiple\ndensity distributions to similar density levels plays an im-\nportant role in improving the counting accuracy. It is also\nnoteworthy that the performance improvement is rather sta-\nble for a wide range of weight of the center loss.\nEffectiveness of the ground-truth transformation. We\nalso study the effect of ground-truth transformation in-\nvolved in scale to re-predict process. As shown in Fig. 5,\nthe ground-truth transformation by enlarging the distance\nbetween crowded heads is more accurate than straightfor-\nwardly scale the ground-truth density map. It is not sur-\nprised to understand that enlarging the distance between\ncrowded heads results in regular Gaussian density blobs for\ndense regions, which reduces the density pattern shift thus\nfacilitates the density map prediction.\nEffectiveness of the division. We also conduct experiments\nby varying the K\u0002Kimage domain divisions. As shown\nin Table 3. The performance is rather stable across different\nimage domain division.\n4.5. Evaluation of Transferability\nTo demonstrate the transferability of the proposed\nmethod across datasets, we conduct experiments under\ncross dataset settings, where the model is trained on the\nsource domain and tested on the target domain.\nThe cross dataset experimental results are presented in\nTable 4. We can observe that the proposed method gener-\nalizes well to unseen datasets. In particular, the proposed\nmethod consistently outperforms D-ConvNet-v1 [25] and\nMCNN [33] by a large margin. The proposed method also\nperforms slightly better than L2R [18] in transferring mod-els trained on ShanghaiTech Part A to UCF CC50. Yet,\nthe improvement is not as significant as the comparison\nwith [33, 25] on transferring between ShanghaiTech Part A\nand Part B. This is probably because L2R [18] also relies on\nextra data which may somehow help to reduce the gap be-\ntween the two datasets. As shown in Table 4, the proposed\nL2SM plays an important role in ensuring the transferabil-\nity of the proposed method. Furthermore, as shown in Ta-\nble 1 and Table 4, the proposed method under cross-dataset\nsettings performs competitively or even outperforms some\nmethods [23, 28, 27, 33] using the proper training set. This\nalso confirms the generalizability of the proposed method.\n5. Conclusion\nIn this paper, we propose a Learning to Scale Module\n(L2SM) to tackle the problem of large density variation for\ncrowd counting. We achieve density centralization by a\nnovel use of multipolar center loss. The L2SM can effec-\ntively learn to scale significantly multiple dense regions to\nmultiple similar density levels, making the density estima-\ntion on dense regions more robust. Extensive experiments\non three challenging datasets demonstrate that the proposed\nmethod achieves consistent and significant improvements\nover the state-of-the-art methods. L2SM also shows the\nnoteworthy generalization ability to unseen datasets with\nsignificantly different density distributions, demonstrating\nthe effectiveness of L2SM in real applications.\nAcknowledgement\nThis work was supported in part by the National Key\nResearch and Development Program of China under Grant\n2018YFB1004600, in part by NSFC 61703171, and in\npart by NSF of Hubei Province of China under Grant\n2018CFB199, to Dr. Yongchao Xu by the Young Elite Sci-\nentists Sponsorship Program by CAST.\n8References\n[1] D. Babu Sam, N. N. Sajjan, R. Venkatesh Babu, and\nM. Srinivasan. Divide and grow: Capturing huge diversity\nin crowd images with incrementally growing cnn. In CVPR ,\npages 3618–3626, 2018. 2, 3\n[2] L. Boominathan, S. S. Kruthiventi, and R. V . Babu. Crowd-\nnet: A deep convolutional network for dense crowd counting.\nInACM-MM , pages 640–644, 2016. 2\n[3] X. Cao, Z. Wang, Y . Zhao, and F. Su. Scale aggregation\nnetwork for accurate and efficient crowd counting. 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Detecting humans in\ndense crowds using locally-consistent scale prior and global\nocclusion reasoning. TPAMI , 37(10):1986–1998, 2015. 2\n[11] H. Idrees, M. Tayyab, K. Athrey, D. Zhang, S. Al-Maadeed,\nN. Rajpoot, and M. Shah. Composition loss for counting,\ndensity map estimation and localization in dense crowds. In\nECCV , 2018. 2, 5, 6\n[12] D. P. Kingma and J. Ba. Adam: A method for stochastic\noptimization. In ICLR , 2014. 6\n[13] Y . LeCun, Y . Bengio, and G. Hinton. Deep learning. Nature ,\n521(7553):436–444, 2015. 3\n[14] V . Lempitsky and A. Zisserman. Learning to count objects\nin images. In NIPS , pages 1324–1332, 2010. 2\n[15] Y . Li, X. Zhang, and D. Chen. Csrnet: Dilated convo-\nlutional neural networks for understanding the highly con-\ngested scenes. In CVPR , pages 1091–1100, 2018. 2, 6\n[16] L. Liu, H. Wang, G. Li, W. Ouyang, and L. Lin. Crowd\ncounting using deep recurrent spatial-aware network. IJCAI ,\n2018. 3\n[17] W. Liu, M. Salzmann, and P. Fua. Context-aware crowd\ncounting. In CVPR , pages 5099–5108, 2019. 2\n[18] X. Liu, J. van de Weijer, and A. D. Bagdanov. Leveraging\nunlabeled data for crowd counting by learning to rank. In\nCVPR , 2018. 2, 3, 6, 7, 8\n[19] D. Onoro-Rubio and R. J. L ´opez-Sastre. Towards\nperspective-free object counting with deep learning. In\nECCV , pages 615–629, 2016. 2[20] A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. De-\nVito, Z. Lin, A. Desmaison, L. Antiga, and A. Lerer. Auto-\nmatic differentiation in pytorch. 2017. 6\n[21] V .-Q. Pham, T. Kozakaya, O. Yamaguchi, and R. Okada.\nCount forest: Co-voting uncertain number of targets using\nrandom forest for crowd density estimation. In ICCV , pages\n3253–3261, 2015. 2\n[22] V . Ranjan, H. Le, and M. Hoai. Iterative crowd counting. In\nECCV , 2018. 6\n[23] D. B. Sam, S. Surya, and R. V . Babu. Switching convolu-\ntional neural network for crowd counting. In CVPR , vol-\nume 1, page 6, 2017. 1, 2, 3, 6, 8\n[24] Z. Shen, Y . Xu, B. Ni, M. Wang, J. Hu, and X. Yang. Crowd\ncounting via adversarial cross-scale consistency pursuit. In\nCVPR , pages 5245–5254, 2018. 2, 6\n[25] Z. Shi, L. Zhang, Y . Liu, X. Cao, Y . Ye, M.-M. Cheng, and\nG. Zheng. Crowd counting with deep negative correlation\nlearning. In CVPR , pages 5382–5390, 2018. 3, 6, 7, 8\n[26] K. Simonyan and A. Zisserman. Very deep convolutional\nnetworks for large-scale image recognition. In ICLR , 2015.\n3\n[27] V . A. Sindagi and V . M. Patel. Cnn-based cascaded multi-\ntask learning of high-level prior and density estimation for\ncrowd counting. In AVSS , pages 1–6, 2017. 3, 6, 8\n[28] V . A. Sindagi and V . M. Patel. Generating high-quality crowd\ndensity maps using contextual pyramid cnns. In ICCV , 2017.\n2, 6, 8\n[29] P. Viola, M. J. Jones, and D. Snow. Detecting pedestrians\nusing patterns of motion and appearance. IJCV , 63(2):153–\n161, 2005. 2\n[30] Y . Wen, K. Zhang, Z. Li, and Y . Qiao. A discriminative fea-\nture learning approach for deep face recognition. In ECCV ,\npages 499–515, 2016. 3\n[31] F. Xiong, X. Shi, and D.-Y . Yeung. Spatiotemporal modeling\nfor crowd counting in videos. In ICCV , pages 5161–5169,\n2017. 2\n[32] C. Zhang, H. Li, X. Wang, and X. Yang. Cross-scene crowd\ncounting via deep convolutional neural networks. In CVPR ,\npages 833–841, 2015. 2\n[33] Y . Zhang, D. Zhou, S. Chen, S. Gao, and Y . Ma. Single-\nimage crowd counting via multi-column convolutional neu-\nral network. In CVPR , pages 589–597, 2016. 1, 2, 3, 5, 6, 7,\n8\n9"    },    {        "title": "1908.10501v2.Infinite_invariant_density_in_a_semi_Markov_process_with_continuous_state_variables.pdf",        "content": "arXiv:1908.10501v2  [cond-mat.stat-mech]  11 Jul 2020Infinite invariant density in a semi-Markov process with con tinuous state variables\nTakuma Akimoto,1,∗Eli Barkai,2and G¨ unter Radons3\n1Department of Physics, Tokyo University of Science, Noda, C hiba 278-8510, Japan\n2Department of Physics, Bar-Ilan University, Ramat-Gan\n3Institute of Physics, Chemnitz University of Technology, 0 9107 Chemnitz, Germany\n(Dated: July 14, 2020)\nWe report on a fundamental role of a non-normalized formal steady state, i.e., an infinite invariant\ndensity, in a semi-Markov process where the state is determi ned by the inter-event time of successive\nrenewals .The state describes certain observables found in models of a nomalous diffusion, e.g., the\nvelocity in the generalized L´ evy walk model and the energy o f a particle in the trap model. In\nour model, the inter-event-time distribution follows a fat -tailed distribution, which makes the state\nvalue more likely to be zero because long inter-event times i mply small state values. We find two\nscaling laws describing the density for the state value, whi ch accumulates in the vicinity of zero in\nthe long-time limit. These laws provide universal behavior s in the accumulation process and give\nthe exact expression of the infinite invariant density. Moreover, we provide two distributional limit\ntheorems for time-averaged observables in these non-stati onary processes. We show that the infinite\ninvariant density plays an important role in determining th e distribution of time averages.\nI. INTRODUCTION\nThere is a growing number of studies on applications\nof infinite invariant densities in physical literature, rang-\ningfromdeterministicdynamicsdescribingintermittency\n[1–5], models of laser cooling [6–9], anomalous diffusion\n[10–14], fractal-time renewal processes [15], and non-\nnormalized Boltzmann states [16]. Infinite invariant den-\nsities are non-normalized formal steady states of systems\nand were studied in dynamical systems exhibiting in-\ntermittency [17–23]. The corresponding ergodic theory\nis known as infinite ergodic theory, which is based on\nMarkovian stochastic processes [24, 25], and states that\ntime averages of some observables do not converge to\nthe correspondingensembleaveragesbutbecomerandom\nvariables in the long-time limit [21, 26–31]. Thus, time\naverages cannot be replaced by ensemble averages even\nin the long-time limit. This striking feature is differ-\nent from usual ergodic systems. Therefore, finding unex-\npected links between infinite ergodictheory and nonequi-\nlibrium phenomena attracts a significant interest in sta-\ntistical physics [1–3, 9–12, 16, 32–35].\nIn equilibrium systems, time averages of an observ-\nable converge to a constant, which is given by the en-\nsemble average with respect to the invariant probabil-\nity measure, i.e., the equilibrium distribution. However,\nin nonequilibrium processes, this ergodic property some-\ntimes does not hold. In particular, distributional be-\nhaviors of time-averaged observables have been experi-\nmentally unveiled. Examples are the intensity of fluo-\nrescence in quantum dots [36, 37], diffusion coefficients\nof a diffusing biomolecule in living cells [38–41], and in-\nterface fluctuations in Kardar-Parisi-Zhang universality\nclass [42], where time averagesof an observable, obtained\nfrom trajectories under the same experimental setup, do\n∗takuma@rs.tus.ac.jpnot converge to a constant but remain random. These\ndistributional behaviors of time averages of some observ-\nables have been investigated by several stochastic models\ndescribing anomalous diffusion processes [12, 43–53].\nWhile several works have considered applications of\ninfinite ergodic theory to anomalous dynamics, one can-\nnot apply infinite ergodic theory straightforwardly to\nstochastic processes. Therefore, our goal is to provide\na deeper understanding of infinite ergodic theory in non-\nstationarystochastic processes. Tothis end, wederivean\nexact form of the infinite invariant density and expose\nthe role of the non-normalized steady state in a mini-\nmal model for nonequilibrium non-stationary processes.\nIn particular, we unravel how the infinite invariant den-\nsity plays a vital role in a semi-Markov process (SMP),\nwhich characterizes the velocity of the generalized L´ evy\nwalk (GLW) [53, 54].\nOur work addresses three issues. Firstly, what is the\npropagator of the state variable? In particular, we will\nshow its relation to the mean number of renewals in the\nstate variable. Secondly, we derive the exact form of the\ninfinite invariantdensity, which is obtained froma formal\nsteady state of the propagator found in the first part.\nFinally, we investigate distributional limit theorems of\nsome time-averaged observables and discuss the role of\nthe infinite invariant density. We end the paper with a\nsummary.\nII. INFINITE ERGODIC THEORY IN\nBROWNIAN MOTION\nBefore describing our stochastic model, we provide the\ninfinite invariant density and its role in one of the sim-\nplest models of diffusion, i.e., Brownian motion. Statis-\ntical properties of equilibrium systems or nonequilibrium\nsystems with steady states are described by a normal-\nized density describing the steady state. On the other\nhand, a formalsteadystate sometimes cannotbe normal-2\nized in nonequilibrium processes, where non-stationarity\nis essential [6–12, 16]. Let us consider a free 1D Brow-\nnian motion in infinite space. The formal steady state\nis a uniform distribution, which cannot be normalized in\ninfinite space. To see this, consider the diffusion equa-\ntion,∂tP(x,t) =D∂2\nxP(x,t), whereP(x,t) is the den-\nsity. Then, setting the left hand side to zero yields a\nformal steady state, i.e., the uniform distribution. This\nis the simplest example of an infinite invariant density\nin nonequilibrium stochastic processes, where the system\nnever reaches the equilibrium. Although the propagator\nof Brownian motion is known exactly, the role of the in-\nfinite invariant density is not so wel-known. Here, we\nwill demonstrate its use. Later, we will see parallels and\ndifferences to the results for our SMP.\nFirst, we consider the occupation time statistics. The\nclassical arcsine law states that the ratio between the\noccupation time that a 1D Brownian particle spends on\nthe positive side and the total measurement time follows\nthe arcsine distribution [55], which means that the ra-\ntio does not converge to a constant even in the long-time\nlimit and remainsa randomvariable. Moreover,the ratio\nbetween the occupation time that a 1D Brownian parti-\ncle spends on a region with a finite length and the total\nmeasurement time does not converge to a constant. In-\nstead, the normalized ratio exhibits intrinsic trajectory-\nto-trajectory fluctuations and the distribution function\nfollows a half-Gaussian, which is a special case of the\nMittag-Leffler distribution known from the occupation\ntime distribution for Markov chains [24].\nThese two laws are distributional limit theorems for\ntime-averaged observables because the occupation time\ncan be represented by a sum of indicator functions.\nTo see this, consider the Heaviside step function, i.e.,\nθ(x) = 1 ifx >0, otherwise zero. The occupation time\non the positive side can be represented by/integraltextt\n0θ(Bt′)dt′,\nwhereBtis a trajectory of a Brownian motion. The inte-\ngral ofθ(x) with respect to the infinite invariant density,\ni.e.,/integraltext∞\n−∞θ(x)dx, is clearly diverging. On the other hand,\nif we consider f(x) =θ(x−xa)θ(xb−x), i.e.; it is one\nforxa<x<x b, and zero otherwise, the integral of f(x)\nwith respect to the infinite invariant density remains fi-\nnite. Therefore, the observable for the arcsine law is not\nintegrable with respect to the infinite density while that\nforthe lattercase is integrable. Therefore, the integrabil-\nity of the observable discriminates the two distributional\nlimit theorems in occupation time statistics.\nFor non-stationary processes, the propagator never\nreaches a steady state, i.e., equilibrium state. However, a\nformal steady state exists and is described by the infinite\ninvariant density for many cases. This infinite invari-\nant density will characterize distributional behaviors of\ntime-averaged observables. For Brownian motion, this\nsteady state is trivial (uniform) and in some sense non-\ninteresting. However, we will show that this integrability\ncondition is rather general as in infinite ergodic theory of\ndynamical systems.III. SEMI-MARKOV PROCESS\nHere, we introduce a semi-Markov process (SMP),\nwhichcouples a renewal process to an observable. A\nrenewal process is a point process where an inter-event\ntime of two successive renewal points is an independent\nandidenticallydistributed (IID) randomvariable[56]. In\nSMPs, a state changes at renewal points. More precisely,\na state remains constant in between successive renewals.\nIn what follows, we consider continuous state variables.\nIn particular, the state is characterized by a continuous\nscalar variable and the scalar value is determined by the\ninter-event time. In this sense, the continuous-time ran-\ndom walk and a dichotomous process are SMPs [43, 57] .\nMoreover, time series of magnitudes/distances of earth-\nquakes can be described by an SMP because there is a\ncorrelation between the magnitude and the inter-event\ntime [58]. In the trap model [59] a random walker is\ntrapped in random energy landscapes. Because escape\ntimes from a trap are IID random variables depending\non the trap and itsmean escape time is given by the\nenergydepth of the trap , the value of the energy depth\nis also described by an SMP. Therefore, a state variable,\nin a different context, can have manymeanings (see also\nRef. [60])\nAs a typical physical example of this process, we con-\nsiderthe GLW [54]. This system can be applied to many\nphysical systems such as turbulence dynamics and sub-\nrecoil laser cooling [6–8, 53, 61–64], where the state is\nconsidered to be velocity or momentum. In the GLW a\nwalker moves with constant velocities vnover time seg-\nments of lengths τnbetween turning points occurring at\ntimestn, i.e,τn=tn−tn−1, where flight durations τn\nare IID random variables. Thus, the displacement Xn\nin time segment [ tn−1,tn] is given by Xn=vnτn. A\ncoupling between vnandτnis given by joint probability\ndensity function (PDF) ψ(v,τ). As a specific coupling\nwhich we consider in this paper, the absolute values |vn|\nof the velocities and flight durations τnin elementary\nflight events are coupled deterministically via\n|vn|=τν−1\nn, (1)\nor equivalently via\nτn=|vn|1\nν−1. (2)\nThe quantity ν >0 is an important parameter charac-\nterizing a given GLW. This nonlinear coupling was also\nconsidered in Ref. [35, 63, 65]. The standard L´ evy walk\ncorresponds to case ν= 1, implying that the velocity\ndoes not depend on the flight duration. In what follows,\nwe focus on case 0 < ν <1.Importantly, if τ→ ∞, in\nthis regime |vn| →0. Thus, we will find accumulation of\ndensity in the vicinity of v= 0. This is because we as-\nsume a power-law distribution for flight durations, that\nfavors long flight durations. Some investigations such\nas Refs. [53, 54] concentrated on the behavior in coordi-\nnate space, where a trajectory x(t) is a piecewise linear\nfunction of time t.3\nIn the following, we denote the state variable as ve-\nlocity and investigate the velocity distribution at time t,\nwhere a trajectory of velocity v(t) is a piecewise constant\nfunction of t. An SMP consists of a sequence {E1,E2,...}\nof elementary flight events En= (vn,τn). We note that\nthis sequence En(n= 1,···) is an IID random vector\nvariable. Thus, the velocity process of a GLW is charac-\nterized by the joint PDF of velocity vand flight duration\nτin an elementary flight event:\nφ(v,τ) =∝an}bracketle{tδ(v−vi)δ(τ−τi)∝an}bracketri}ht (3)\nThe symbol δ(.) denotes the Dirac delta function. PDF\nψ(τ) of the flight durations is defined through the\nmarginal density of the joint PDF ψ(v,τ):\nψ(τ) =/integraldisplay+∞\n−∞φ(v,τ)dv=∝an}bracketle{tδ(τ−τi)∝an}bracketri}ht.(4)\nSimilarly one can get PDF χ(v) for the velocities of an\nelementary event as\nχ(v) =/integraldisplay+∞\n0φ(v,τ)dτ=∝an}bracketle{tδ(v−vi)∝an}bracketri}ht.(5)\nIn L´ evywalktreatmentsusually ψ(τ) is prescribedand\nchosen as a slowly decaying function with a power-law\ntail:\nψ(τ)∼c\n|Γ(−γ)|τ−1−γ(τ→ ∞) (6)\nwith the parameter γ >0 characterizing the algebraic\ndecay andcbeing a scale parameter. A pair of param-\netersνandγdetermines the essential properties of the\nGLW and the asymptotic behavior in the velocity space.\nOf special interest is the regime 0 < γ <1. There the\nsequence of renewal points {tn,n= 0,1,2,...}, at which\nvelocityv(t) changes, i.e.,\ntn=n/summationdisplay\ni=1τi (7)\nwitht0= 0, is a non-stationary process in the sense\nthat the rate of change is not constant but varies with\ntime [6, 43]. This is because the mean flight duration\ndiverges, i.e., ∝an}bracketle{tτi∝an}bracketri}ht=/integraltext∞\n0τ ψ(τ)dτ=∞. To determine\nthelastvelocity v(t)attimet, oneneedstoknowthetime\ninterval straddling t, which is defined as τ≡tn+1−tn\nwithtn<t<t n+1and wasdiscussed in [49, 66]. In other\nwords, to determine the distribution of the velocity at\ntimet, one needs to know the time when the first renewal\noccursafter time tandthe time for the last renewalevent\nbeforet.\nIV. GENERAL EXPRESSION FOR THE\nPROPAGATOR\nA. Standard derivation\nWe are interested in the propagator p(v,t), which is\nthe PDF of finding a velocity vat timet, given that theprocess started at t= 0 withv= 0. To derive an expres-\nsion forp(v,t), we note that at every renewal time tn−1\nthe process starts anew with velocity vnuntilt<tn. So\none needs the probability R(t)dtof finding some renewal\nevent in [t,t+dt).This quantity is called sprinkling den-\nsity in the literature [6] and it isclosely related to the\nrenewal density in renewal theory [56].\nIt is obtained from a recursion relation for the PDF\nRn(t) =∝an}bracketle{tδ(t−tn)∝an}bracketri}htthat then-th renewal point tnoc-\ncurs exactly at time t. Using the PDF ψ(τ), we get the\niteration rule\nRn+1(t) =/integraldisplayt\n0dt′ψ(t−t′)Rn(t′) (8)\nwith the initial condition R0(t) =δ(t), which means that\nwe assume a renewal occurs at t= 0, i.e., ordinary re-\nnewal process [56]. Summing both sides from n= 0 to\ninfinity, one gets the equation of R(t)≡/summationtext∞\nn=0Rn(t) for\nt>0, i.e.,\nR(t) =/integraldisplayt\n0dt′ψ(t−t′)R(t′)+R0(t).(9)\nEq. (9) is known as the renewal equation. The solution\nof this equation is easily obtained in Laplace space as\n/tildewideR(s) =1\n1−/tildewideψ(s), (10)\nwhere/tildewideR(s) =/integraltext∞\n0R(t)exp(−st)dt. The integral of\nR(t) is related to the expected number of renewal events\n∝an}bracketle{tN(t)∝an}bracketri}htoccurring up to time t, i.e.,\n∝an}bracketle{tN(t)∝an}bracketri}ht=/integraldisplayt\n0R(t′)dt′. (11)\nNote that here the event at t= 0 is also counted while\nthe event at t= 0 is often excluded in renewal theory.\nWith knowledge of R(t), which in principle can be ob-\ntainedbyLaplaceinversionofEq.(10), onecanformulate\nthe solution of the propagator as\np(v,t) =/integraldisplayt\n0dt′W(v,t−t′)R(t′),(12)\nwhereW(v,t−t′) takesinto account the last incompleted\nflight event, starting at the last renewal time t′, provided\nthat the flight duration is longer than t−t′with velocity\nv. Thus,W(v,t) is given by\nW(v,t)≡/integraldisplay∞\ntdτφ(v,τ). (13)\nIntegrating this over all velocities leads to the survival\nprobability Ψ( t) of the sojourn time, i.e., the probability\nthat an event lasts longer than a given time t\nΨ(t) =/integraldisplay+∞\n−∞W(v,t)dv=/integraldisplay∞\ntdτ ψ(τ).(14)4\nUsingEqs. (5), (10), and (13) one can write down the\npropagator in Laplace space\n/tildewidep(v,s) =/tildewiderW(v,s)/tildewideR(s) =1\nsχ(v)−/tildewideφ(v,s)\n1−/tildewideψ(s).(15)\nThisisageneralexpressionofthepropagatorandanana-\nlogue of the Montroll-Weiss equation of the continuous-\ntime random walk [67]. Recalling/integraltext\ndv/bracketleftBig\nχ(v)−/tildewideφ(v,s)/bracketrightBig\n=\n1−/tildewideψ(s) gives/integraltext\ndv/tildewidep(v,s) = 1/s, implying that propaga-\ntorp(v,t) in the form of Eq. (15) is correctly normalized/integraltext\ndvp(v,t) = 1.\nInwhatfollows,asaspecificexample,weconsiderade-\nterministic coupling between τiand|vi|. The joint PDF\nφ(v,τ) is specified as follows: flight duration τiis chosen\nrandomly from the PDF ψ(τ), and the correspondingab-\nsolute value of the velocity |vi|is deterministically given\nby|vi|=τν−1\ni. Finally, the sign of viis determined with\nequal probability, implying that\nφ(v,τ) =1\n2/bracketleftbig\nδ/parenleftbig\nv−τν−1/parenrightbig\n+δ/parenleftbig\nv+τν−1/parenrightbig/bracketrightbig\nψ(τ) (16)\nwithφ(v,τ) =φ(−v,τ). Alternatively, one can specify\nthe velocity first using the PDF χ(v) =χ(−v). Then,\none can express the joint PDF φ(v,τ) also as\nφ(v,τ) =δ/parenleftBig\nτ−|v|1\nν−1/parenrightBig\nχ(v). (17)\nAlthough Eqs. (16) and (17) are equivalent, the latter\nsuggests a different interpretation of selecting an elemen-\ntaryevent, e.g. thevelocityisselectedfrom χ(v) firstand\nthen this velocity state lasts for duration τi=|vi|1\nν−1.\nObviously prescribing ψ(τ) determines χ(v) [via Eqs.(5)\nand (16)] and vice versa [via Eqs.(4) and (17)]. From\nEq. (13), one gets\nW(v,t) =χ(v)θ(|v|1\nν−1−t), (18)\nwhereθ(x) is the Heaviside step function.\nBefore deriving our main results, we give the equilib-\nrium distribution of the propagator for γ >1. Although\nwe assumed γ <1 in Eq. (6), the general expression for\nthe propagator, Eq. (15), is exact also for γ >1. For\nγ >1, the mean flight duration ∝an}bracketle{tτ∝an}bracketri}htis finite and we have\npeq(v) = lim\ns→0s/tildewidep(v,s) =/integraltext∞\n0φ(v,τ)τdτ\n∝an}bracketle{tτ∝an}bracketri}ht.(19)\nTherefore, for γ >1, as expected, the equilibrium distri-\nbution exists; i.e., the propagator reaches a steady state:\np(v,t)→peq(v) =χ(v)|v|1\nν−1\n∝an}bracketle{tτ∝an}bracketri}ht(20)\nfort→ ∞. Here, we note that the equilibrium distri-\nbution has a different form for the decoupled case, i.e.,\nφ(v,τ) =χ(v)ψ(τ). In this case, it is easily obtained as\npeq(v) =χ(v).Because the integration of R(t) gives∝an}bracketle{tN(t)∝an}bracketri}ht, we get an\nexact expression for the propagator\np(v,t) =χ(v) [∝an}bracketle{tN(t)∝an}bracketri}ht−∝an}bracketle{tN(t−tc(v))∝an}bracketri}ht],(21)\nwheretc(v)≡ |v|1\nν−1. We note that ∝an}bracketle{tN(t)∝an}bracketri}ht= 0 when\nt<0. In particular, one can express p(v,t) as\np(v,t) =χ(v)·\n\n∝an}bracketle{tN(t)∝an}bracketri}ht\n∝an}bracketle{tN(t)∝an}bracketri}ht−∝an}bracketle{tN(t−tc(v))∝an}bracketri}htfor\nfort<tc(v)\nt>tc(v).\n(22)\nSince we have madeno approximation, the solution is\nformally exact, while the remaining difficulty is to obtain\n∝an}bracketle{tN(t)∝an}bracketri}ht.This is the central result of this section.\nThe mean number ∝an}bracketle{tN(t)∝an}bracketri}htof renewals up to time\ntincreases monotonically from ∝an}bracketle{tN(t→0)∝an}bracketri}ht= 1 be-\ncause the first jump is at t0= 0+, which implies that\nlimt→0p(v,t) =χ(v), which is the velocity distribution\nof the elementary event as given by Eq. (5). For a given\nvelocityvsatisfyingt < tc(v), the function p(v,t) in-\ncreases until treachestc(v) because ∝an}bracketle{tN(t)∝an}bracketri}htis a monoton-\nically increasing function. Thereafter p(v,t) stays con-\nstant or decreases because ∝an}bracketle{tN(t)∝an}bracketri}ht−∝an}bracketle{tN(t−tc(v))∝an}bracketri}htstays\nconstant or decreases depending on whether the renewal\nsequences {tn,n= 0,1,2,...}are equilibrium sequences\nor not [56, 68–70]. This in turn depends on the shape of\nψ(τ), more precisely on the decay of ψ(τ) for largeτ, as\ndetailed below.\nFor a discussion of the velocity profile p(v,t) for a fixed\ntimet, it is more convenient to rewrite Eq. (21) for v>0\nas\np(v,t) =χ(v) [∝an}bracketle{tN(t)∝an}bracketri}ht−∝an}bracketle{tN(t−tc(v))∝an}bracketri}htθ(v−vc(t))]\n(23)\nwhere we introduced the critical velocity vc(t) =tν−1\nandvc(t) is monotonically decreasing as function of tbe-\ncause 0< ν <1. For negative v,p(v,t) follows from\nthe symmetry p(v,t) =p(−v,t). Thus, for a fixed t\nand|v|< vc(t),p(v,t) is the same as χ(v),enlarged\nby the velocity-independent factor ∝an}bracketle{tN(t)∝an}bracketri}ht, whereas for\n|v|>vc(t) it has a non-trivial v-dependence due to the\nv-dependence of ∝an}bracketle{tN(t−tc(v))∝an}bracketri}ht. Note that at velocity\nv=vc(t) the profile of vjumps by the value\nδp= lim\nε→0[p(vc(t)−ε,t)−p(vc(t)+ε,t)] =χ(vc(t)) (24)\nat the critical velocity v=vc(t) because we assume\n∝an}bracketle{tN(0)∝an}bracketri}ht= 1.\nB. Another derivation of Eq. (21)\nHere, we give another derivation of the propagator,\ni.e., Eq. (21). The joint PDF of v(t) withtsatisfying\ntn<t<t n+1, denoted by pn(v,t),can be written as\npn(v,t) =∝an}bracketle{tδ(τn+1−|v|1\nν−1)I(tn<t<t n+1)∝an}bracketri}ht,(25)5\n10 010 110 210 310 410 5\n10 -5 10 -4 10 -3 10 -2 10 -1 l(v) \nvt = 10 7\nt = 10 6\nt = 10 5\nt = 10 4\nt = 10 3p(v,t)\nFIG. 1. Time evolution of the propagator for different times\n(γ= 0.5 andν= 0.2). Symbols with lines are the results\nof numerical simulations. Dashed and dotted lines are the\ntheories, i.e., Eqs. (37) and (34), respectively. Infinite d ensity\ncan be observed for v >vc(t) while the propagator follows a\ndifferent scaling, i.e., Eq. (34), for v < v c(t). We used the\nPDFψ(τ) =γτ−1−γforτ≥1andψ(τ) = 0 forτ <1as the\nflight-duration PDF.\nwhereI(·) = 1 if the condition in the bracket is satisfied,\nand 0 otherwise. It follows that the propagator can be\nobtained as a sum over the number of renewals n:\np(v,t) =∞/summationdisplay\nn=0pn(v,t). (26)\nUsingχ(v) andtn+1=tn+τn+1, we have\np(v,t) =χ(v)∞/summationdisplay\nn=0∝an}bracketle{tI(tn<t<t n+|v|1\nν−1)∝an}bracketri}ht,(27)\nwhere we note that ∝an}bracketle{tI(tn<t)∝an}bracketri}htgives a probability:\n∝an}bracketle{tI(tn<t)∝an}bracketri}ht= Pr(N(t)>n). (28)\nThe mean of N(t) can be written as\n∝an}bracketle{tN(t)∝an}bracketri}ht=∞/summationdisplay\nn=1nPr(N(t) =n) =∞/summationdisplay\nn=0Pr(N(t)>n),(29)\nwhere we used identity Pr( N(t) =n) = Pr(N(t)> n−\n1)−Pr(N(t)>n). Therefore, we have Eq. (21).\nV. INFINITE INVARIANT DENSITY IN A\nSEMI-MARKOV PROCESS\nA. Infinite invariant density\nTo proceed with the discussion of Eq. (23), we use\nEq. (6) for the flight-duration PDF and consider γ <1.\nThe PDF of velocities χ(v) in an elementary event can\nbe obtained by Eqs. (5) and (16):\nχ(v) =1\n2ψ(|v|1\nν−1)|v|−1−1\n1−ν\n1−ν. (30)For the specific choice for ψ(τ) given in Eq. (6) the\nasymptotic form with 0 <ν <1 yields\nχ(v)∼c\n2(1−ν)|Γ(−γ)||v|−1+γ\n1−νforv→0.(31)\nFirst, we give the asymptotic behavior of ∝an}bracketle{tN(t)∝an}bracketri}htfor\nt→ ∞. Because the Laplace transform of ψ(τ) is given\nby/tildewideψ(s) = 1−csγ+o(sγ) fors→0, Eqs. (10) and (11)\nyieldsthe well-known result:\n∝an}bracketle{tN(t)∝an}bracketri}ht ∼1\ncΓ(1+γ)tγfort→ ∞.(32)\nHowever, for our purposes, we need to go beyond this\nlimit as shown below. The renewal function gives the\nexact form of the propagator [see Eq. (21)]. There are\ntwo regimes in the propagator as seen in Eq. (22). For\nt < tc(v), or equivalently v < v c(t), the propagator is\ngiven by\np(v,t) =∝an}bracketle{tN(t)∝an}bracketri}htχ(v). (33)\nIn this regime the propagator is an increasing function of\ntbecause ∝an}bracketle{tN(t)∝an}bracketri}htis a monotonically increasing function\nwhose asymptotic behavior is given by Eq. (32), whereas\nthe support ( −vc(t),vc(t)) will shrink because vc(t) =\nt−(1−ν)→0ast→ ∞.Fort≫1andv<vc(t), implying\nv<vc(t)≪1, the propagator becomes\np(v,t)∼sin(πγ)tγ\n2π(1−ν)|v|−1+γ\n1−ν. (34)\nThis is a universal law in the sense that the asymptotic\nform does not depend on the detailed form of the flight-\nduration PDF such as scale parameter c.\nFort>tc(v), or equivalently v>vc(t), the propagator\nis given through ∝an}bracketle{tN(t)∝an}bracketri}ht−∝an}bracketle{tN(t−tc(v))∝an}bracketri}ht.Fort≫1 and\nvc(v)<v≪1,\n∝an}bracketle{tN(t)∝an}bracketri}ht−∝an}bracketle{tN(t−tc(v))∝an}bracketri}ht∼=tγ−(t−tc(v))γ\ncΓ(γ+1)(35)\n∼tγ−1tc(v)\ncΓ(γ). (36)\nTherefore, the asymptotic behaviorof the propagatorbe-\ncomes\np(v,t)∼χ(v)|v|1\nν−1tγ−1\ncΓ(γ)fort→ ∞,(37)\nwhich can also be obtained simply by changing ∝an}bracketle{tτ∝an}bracketri}htin\nEq. (20) into/integraltextt\n0τψ(τ)dτexcept for the proportional con-\nstant. Here, one can define a formal steady state I∞(v)\nusing Eq. (37) as follows:\nI∞(v)≡lim\nt→∞t1−γp(v,t) =χ(v)|v|1\nν−1\ncΓ(γ).(38)\nThis does not depend on tin the long-time limit and is\na natural extension of the steady state for γ >1, i.e.,6\n10 -5 10 -4 10 -3 10 -2 10 -1 10 010 1\n10 -2 10 -1 10 010 110 2l(v) \nt = 10 6\nt = 10 7\nt = 10 8\n10 -5 10 -4 10 -3 10 -2 10 -1 10 010 1\n10 -2 10 -1 10 010 110 2l(v) \nt = 10 4\nt = 10 5\nt = 10 6\n10 -5 10 -4 10 -3 10 -2 10 -1 10 010 1\n10 -2 10 -1 10 010 110 2l(v) \nt = 10 4\nt = 10 5\nt = 10 6p(v′\u0000)\n10 −210 −510 −310 −110 1\np(v′\u0000)\n10 −510 −310 −110 0\np(v′\u0000)\n10 −510 −310 −110 1\n10 010 2\nv′\u000010 −210 010 2\nv′\u000010 −210 010 2\nv′\u000010 -5 10 -4 10 -3 10 -2 10 -1 10 \n10 -2 10 -1 10 10 10 l(v) \nt = 10 4\nt = 10 5\nt = 10 6\n10 -5 10 -4 10 -3 10 -2 10 -1 10 \n10 -2 10 -1 10 10 10 l(v) \nt = 10 4\nt = 10 5\nt = 10 6\n10 -5 10 -4 10 -3 10 -2 10 -1 10 10 \n10 -2 10 -1 10 10 10 l(v) \nt = 10 6\nt = 10 7\nt = 10 8(a) (b) (c)\nFIG. 2. Rescaled propagators for (a) ν= 0.2, (b)ν= 0.5 and (c)ν= 0.8 (γ= 0.5). Symbols with lines are the results of\nnumerical simulations for different times t. Dashed lines represent the scaling functions, i.e., Eq. (4 0).The flight-duration PDF\nis the same as that in Fig. 1.\nEq. (20). In this sense, I∞(x) is a formal steady state of\nthe system. However, I∞(v) is not normalizableand thus\nit is sometimes called infinite invariant density .Using\nEq. (31), the asymptotic form of the infinite density for\nv≪1 becomes\nI∞(v)∼γsin(πγ)\n2π(1−ν)|v|−1−1−γ\n1−ν. (39)We note that the infinite density describes the propaga-\ntor only for v >vc(t). Whilevc(t)→0 in the long-time\nlimit, the propagatorfor v≪1 is composed of two parts,\ni.e., Eqs.(34) and (37). Thesebehaviorsareillustratedin\nFig. 1, where the support of the propagator is restricted\nto|v|<1. In particular, the accumulation at zero ve-\nlocity forv<vc(t) and a trace of the infinite density for\nv > vc(t) are clearly shown. In general, the propagator\nforv≫1 is described by the small- τbehavior of the\nflight-duration PDF through Eq. (22).\nB. Scaling function\nRescalingvbyv′=t1−νvin the propagator, we find a scaling function. In particular, the res caled propagator does\nnot depend on time tand approaches the scaling function denoted by ρ(v′) in the long-time limit ( t→ ∞):\npres(v′,t)≡p(v′/t1−ν,t)/vextendsingle/vextendsingle/vextendsingle/vextendsingledv\ndv′/vextendsingle/vextendsingle/vextendsingle/vextendsingle→ρ(v′)≡\n\nsin(πγ)\n2π(1−ν)|v′|−1+γ\n1−ν (v′<1)\nsin(πγ){1−(1−v′1\nν−1)γ}\n2π(1−ν)|v′|−1+γ\n1−ν(v′≥1),(40)\nwhere we used Eq. (35) and note that v′\nc≡t1−νvc(t) = 1.In the scaling function, the long-time limit is taken in\nadvance. Thus, the scaling function describes only small- vbehaviors of p(v,t). In other words, large- vbehaviors of\nρ(v) are not matched with those of I∞(v) while large- vbehaviors of ρ(v) are matched with small- vbehaviors of I∞(v).\nThe scaling function is normalized and continuous at v′= 1 whereas p(v,t) is not continuous at v=vc(t)for finitet\nbecause the jump in the propagator at v=vcis given by Eq. (24) and χ(vc(t))→0 fort→ ∞.As shown in Fig. 2,\nrescaled propagators at different times tcoincide with the scaling function for t≫1.Note that the scaling function\ndescribes the behavior of p(v,t) forv≪1 in the long-time limit, which does not capture the behavior of p(v,t) for\nv >1. Large-vbehaviors of p(v,t) can be described by I∞(v). Although I∞(v) depends on the details of χ(v), the\nscaling function is not sensitive to all the details except for γ. In this sense, it is a general result.\nC. Ensemble averages\nThetheoryofinfiniteergodictheoryisatheoryofobservables. Th ismeansthatwemustclassifydifferentobservables\nand define the limiting laws with which their respective ensemble averag es are obtained in the long time limit. We\nwill soon consider also time averages. Consider the observable f(v). The corresponding ensemble average is given by\n∝an}bracketle{tf(v(t))∝an}bracketri}ht ≡/integraldisplay∞\n−∞f(v)p(v,t)dv=/integraldisplayvc(t)\n−vc(t)p(v,t)f(v)dv+/integraldisplay∞\nvc(t)p(v,t)f(v)dv+/integraldisplay−vc(t)\n−∞p(v,t)f(v)dv.(41)7\nIf we take the time t→ ∞, we have\n∝an}bracketle{tf(v(t))∝an}bracketri}ht∼=/integraldisplay1\n−1ρ(v′)f(v′/t1−ν)dv′+tγ−1/integraldisplay∞\nvc(t)I∞(v)f(v)dv+tγ−1/integraldisplay−vc(t)\n−∞I∞(v)f(v)dv, (42)\nwhere we performed a change of variable and used the scaling funct ion in the first term, and we also used p(v,t)∼=\ntγ−1I∞(v) for|v|> vc(t) in the second and third terms. Moreover, we assume that the sec ond and third term in\nEq. (41) does not diverge. In what follows, we consider f(v) =|v|α. Whenf(v) is integrable with respect to ρ(v),\ni.e.,/integraltext∞\n−∞ρ(v)f(v)dv <∞,αsatisfies the following inequality:\n−γ\n1−ν<α<1−γ\n1−ν. (43)\nIn this case, the leading term of the asymptotic behavior of the ens emble average is given by the first term:\n∝an}bracketle{tf(v(t))∝an}bracketri}ht ∼t−α(1−ν)/integraldisplay1\n−1ρ(v)f(v)dv(t→ ∞), (44)\nwhere we used Eq. (40):\n/integraldisplay1\n−1ρ(v)f(v/t1−ν)dv∼t−α(1−ν)/integraldisplay1\n−1ρ(v)|v|αdv(t→ ∞). (45)\nThus, the ensemble average goes to zero and infinity in the long-time limit forα>0 andα<0, respectively. On the\nother hand, when f(v) is integrable with respect to I∞(v), i.e.,/integraltext∞\n−∞I∞(v)f(v)dv<∞, wheref(v) satisfiesf(v)∼vα\nwithα>1−γ\n1−ν>0 forv→0, the second and third terms becomes\ntγ−1/integraldisplay∞\nvc(t)I∞(v)f(v)dv+tγ−1/integraldisplay−vc(t)\n−∞I∞(v)f(v)dv=tγ−1/integraldisplay∞\n−∞I∞(v)f(v)dv, (46)\nBecause the relation between α,νandγsatisfiesα(1−ν)>1−γ, the asymptotic behavior of the ensemble average\nis given by\n∝an}bracketle{tf(v(t))∝an}bracketri}ht ∼tγ−1/integraldisplay∞\n−∞I∞(v)f(v)dv(t→ ∞). (47)\nA structure of Eqs. (44) and (47) is very similar to an ordinary equilib rium averaging in the sense that there is a\ntime-independent average with respect to ρ(v) orI∞(v) on the right hand side, where the choice of ρ(v) orI∞(v)\ndepends on whether the observable is integrable with respect to ρ(v) orI∞(v). The beauty of infinite ergodic theory\nis that this can be extended to time averages, which as mentioned will be discussed below.\nIn the long-time limit, p(v,t) behaves like a delta distribution in the following sense:\n/integraldisplay∞\n−∞lim\nt→∞p(v,t)f(v)dv=/integraldisplay∞\n−∞ρ(x) lim\nt→∞f(x/t1−ν)dx=f(0). (48)\nEq. (48) is clearly obtained when f(v) is integrable with respect to ρ(v), i.e.,/integraltext∞\n−∞ρ(v)f(v)dv <∞. Even when\nf(v) is not integrable with respect to ρ(v), Eq. (48) is valid if f(v) is integrable with respect to I∞(v). In fact, the\nasymptotic behavior of the ensemble average ∝an}bracketle{tf(v(t))∝an}bracketri}htbecomes ∝an}bracketle{tf(v(t))∝an}bracketri}ht →0 =f(0) fort→ ∞, as shown above.\nTherefore, Eq. (48) is valid in this case. When both integrals diverge , Eq. (48) is no longer valid. However, if there\nexists a positive constant εsuch thatψ(τ) = 0 forτ <ε, Eq. (48) is always valid. In the long-time limit, the ensemble\naverage is trivial in the sense that it simply gives the value of the obse rvable atv= 0. At this stage, there is no\nreplacement of a “steady state” concept. However, in general, s caling function ρ(v) describes the propagator near\nv= 0 while infinite invariant density I∞(v) describes the propagator for v>0 including large- vbehaviors. Therefore,\nas shown in Eqs. (44) and (47), both the scaling function and the infi nite invariant density play an important role for\nthe evaluation of certain ensemble averages at time t.\nVI. DISTRIBUTIONAL LIMIT THEOREMS\nWhen the system is stationary, a time average ap-\nproaches a constant in the long-time limit, which impliesergodicityof the system. However, time averagesof some8X(t)(a)\nX(t)(b)-1 -0.5 0  0.5 1 \n 0  2000  4000  6000  8000  10000  12000  14000  16000 0  200 400 600 800 1000 1200 1400v(t) \nX(t) \ntf(v)=|v| \n-1 -0.5  0  1 \n 0  0 v(t) \nX(t) f(v)=|v| f(v) =|v|\n-1 -0.5 0  0.5 1 \n 0  2000  4000  6000  8000  10000  12000  14000  16000 0  500 1000 1500 2000 2500 3000 3500 4000v(t) \nX(t) \ntf(v)=|v| \n-1 -0.5  0  1 \n 0  0 v(t) \nX(t) f(v)=|v| f(v) =|v|\nFIG. 3. Trajectories of velocity v(t) and the integral of a\nfunctionf(v) =|v|, i.e.,X(t) =/integraltextt\n0|v(t′)|dt′. Because velocity\nv(t) is a piece-wise constant function, integral X(t) is a piece-\nwise linear function of t. Parameter sets ( γ,ν) are (0.8,0.2)\nand (0.5,0.8) for (a) and (b), respectively.\nobservables may not converge to a constant but properly\nscaled time averages converge in distribution when the\nsystem is non-stationary as it is case for γ <1.While we\nfocus on regime 0 < ν <1, the following theorems can\nbe extended to regime ν >1.\nTo obtain the distribution of these time averages, we\nconsider the propagator of the integrals of these observ-\nables along a trajectory from 0 to t, denoted by X(t),\nwhich are piece-wise-linear functions of tand can be de-\nscribed by a continuous accumulation process (see Fig. 3)\n[31]. Time average of function f(v) is defined by\nf(t)≡1\nt/integraldisplayt\n0f(v(t′))dt′=X(t)\nt. (49)\nAs specific examples, we will consider time averages ofthe absolute value of the velocity and the squared veloc-\nity, i.e.,f(v) =|v|orf(v) =v2. Integrated value X(t)\ncan be represented by\nX(t) =N(t)−1/summationdisplay\nn=1f(vn)τn+f(vN(t))(t−tN(t)−1).(50)\nThe stochastic process of X(t)can be characterized by a\nrecursion relation, which is the same as in the derivation\nof the velocity propagator. Let Rf(x,t) be the PDF of\nx=X(t) when a renewal occurs exactly at time t, then\nwe have\nRf(x,t) =/integraldisplayx\n0dx′/integraldisplayt\n0dt′φf(x′,t′)Rf(x−x′,t−t′)+R0\nf(x,t),\n(51)\nwhereφf(x,τ) =δ/parenleftbig\nx−f(τν−1)τ/parenrightbig\nψ(τ) andR0\nf(x,t) =\nδ(x)δ(t). Here, we assume that function f(v) is an even\nfunction. We note that we use a deterministic coupling\nbetweenτandv, i.e., Eq. (1). The PDF of X(t) at time\ntis given by\nPf(x,t) =/integraldisplayx\n0dx′/integraldisplayt\n0dt′Φf(x′,t′)Rf(x−x′,t−t′),(52)\nwhere\nΦf(x,t) =/integraldisplay∞\ntdτψ(τ)δ(x−f(τν−1)t).(53)\nThe double-Laplace transform with respect to xandt\nyields\n/tildewidePf(k,s) =/tildewideΦf(k,s)\n1−/tildewideφf(k,s), (54)\nwhere/tildewideφf(k,s) and/tildewideΦf(k,s) arethe double-Laplacetrans-\nforms ofφf(x,τ) andΦf(x,t) given by\n/tildewideφf(k,s) =/integraldisplay∞\n0dτe−sτ−kf(τν−1)τψ(τ) (55)\nand\n/tildewideΦf(k,s) =/integraldisplay∞\n0dte−st/integraldisplay∞\ntdτe−kf(τν−1)tψ(τ),(56)\nrespectively. Eq. (54) is the exact form of the PDF of\nX(t) in Laplace space.\nBefore considering a specific form of f(v), we show that there are two different classes of distributional limit\ntheorems of time averages. Expanding e−kf(τν−1)τin Eq. (55), we have\n/tildewideφf(k,s)∼=/tildewideψ(s)−k/integraldisplay∞\n0dτf(τν−1)τψ(τ)e−sτ+O(k2). (57)9\nUsing Eq. (30), one can write the second term with s→0as\n/integraldisplay∞\n0dτf(τν−1)τψ(τ) =1\n1−ν/integraldisplay∞\n0dvf(v)v3−ν\nν−1ψ(v1\nν−1) = 2/integraldisplay∞\n0f(v)v1\nν−1χ(v)dv= 2cΓ(γ)/integraldisplay∞\n0f(v)I∞(v)dv.(58)\nWhenf(v) is integrable with respect to the infinite invariant density, i.e.,/integraltext∞\n0f(v)I∞(v)dv <∞, the second term is\nstill finite for s→0. As shown below, we will see that the integrability gives a condition th at determines the shape\nof the distribution function for the normalized time average, i.e., f(t)/∝an}bracketle{tf(t)∝an}bracketri}ht.\nA. Time average of the absolute value of v\nIn this section, we show that there are two phases for\ndistributional behaviors of time averages. The phase line\nisdeterminedbyarelationbetween γandν. Asaspecific\nchoice of function f(v), we consider the absolute value of\nthe velocity, i.e., f(v) =|v|. Thus,X(t) is given by\nX(t) =N(t)−1/summationdisplay\nn=1τν\nn+τν−1\nN(t)(t−tN(t)−1),(59)\nForν < γ, the moment ∝an}bracketle{tτν∝an}bracketri}htis finite, i.e., ∝an}bracketle{tτν∝an}bracketri}ht<∞.\nThis condition is equivalent to the following condition\nrepresented by the infinite density:\n∝an}bracketle{tf(v)∝an}bracketri}htinf=/integraldisplay∞\n0f(v)I∞(v)dv<∞.(60)\nThe double Laplace transform /tildewideP|v|(k,s) is calculated in\nAppendix C (see Eq. (C4)). For s→0, the leading term\nof−∂/tildewideP|v|(k,s)\n∂k/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nk=0becomes\n−∂/tildewideP|v|(k,s)\n∂k/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nk=0∼∝an}bracketle{tτν∝an}bracketri}ht\ncs1+γ. (61)\nIt follows that the mean of X(t) fort→ ∞becomes\n∝an}bracketle{tX(t)∝an}bracketri}ht ∼∝an}bracketle{tτν∝an}bracketri}ht\ncΓ(1+γ)tγ. (62)\nSince the mean of X(t) increases with tγ, we consider\na situation where k∼sγfor smallk,s≪1 in the\ndouble-Laplace space. Thus, all the term k/sν(≪1)\nandO(k2/sγ) in Eq. (C4) can be ignored. It follows that\nthe asymptotic form of /tildewideP(k,s) is given by\n/tildewideP|v|(k,s) =csγ−1/∝an}bracketle{tτν∝an}bracketri}ht\nk+csγ/∝an}bracketle{tτν∝an}bracketri}ht. (63)\nThisisthedouble LaplacetransformofPDF G′\nt(∝an}bracketle{tτν∝an}bracketri}htx/c)\n[71], where\nGt(x) = 1−Lγ(t/x1/γ) (64)\nandLγ(x) is a one sided L´ evy distribution; i.e., the\nLaplace transform of PDF lγ(x)≡L′\nγ(x) is given bye−kγ. By a straightforward calculation one obtain the\nasymptotic behavior of the second moment as follows:\n∝an}bracketle{tX(t)2∝an}bracketri}ht ∼2∝an}bracketle{tτν∝an}bracketri}ht2t2γ\nc2Γ(1+2γ). (65)\nFurthermore, the nth moment can be represented by\n∝an}bracketle{tX(t)n∝an}bracketri}ht ∼n!Γ(1+γ)n\nΓ(1+nγ)∝an}bracketle{tX(t)∝an}bracketri}htn(66)\nfort→ ∞. It follows that random variable X(t)/∝an}bracketle{tX(t)∝an}bracketri}ht\nconvergesin distribution to a random variable Mγwhose\nPDF follows the Mittag-Leffler distribution oforderγ,\nwhere\n∝an}bracketle{te−zMγ∝an}bracketri}ht ∼∞/summationdisplay\nn=0Γ(1+γ)n\nΓ(1+nγ)(−z)n. (67)\nIn other words, the normalized time averages defined by\n∝an}bracketle{tτν∝an}bracketri}htX(t)/(ctγ) do not converge to a constant but the\nPDF converge to a non-trivial distribution (the Mittag-\nLeffler distribution ). In particular, the PDF can be rep-\nresented through the L´ evy distribution:\nG′\n1(x) =1\nγx−1\nγ−1lγ(x−1/γ) (68)\nTo quantify trajectory-to-trajectory fluctuations of the\ntime averages, we consider the ergodicity breaking (EB)\nparameter [44] defined by\nEB(t)≡∝an}bracketle{tf(t)2∝an}bracketri}ht−∝an}bracketle{tf(t)∝an}bracketri}ht2\n∝an}bracketle{tf(t)∝an}bracketri}ht2, (69)\nwhere∝an}bracketle{t·∝an}bracketri}htimplies the average with respect to the initial\ncondition. When the system is ergodic, it goes to zero as\nt→ ∞. On the other hand, it converges to a non-zero\nconstant when the trajectory-to-trajectory fluctuations\nare intrinsic. For ν <γ < 1, the EB parameter becomes\nEB(t)→ML(γ)≡2Γ(1+γ)2\nΓ(1+2γ)−1 (t→ ∞),(70)\nwhich means that time averages do not converge to a\nconstant but they become a random variable with a non-\nzerovariance. For γ >1, the EB parameteractually goes\nto zero in the long-time limit. Moreover, it also goes to\nzero asγ→1 in Eq. (70). We note that the condition\n(60) is general in a sense that the distribution of time10\naverages of function f(v) satisfying the condition (60)\nfollows the Mittag-Leffler distribution, which is the same\ncondition as in infinite ergodic theory [21].\nForν > γ,∝an}bracketle{tτν∝an}bracketri}htdiverges and equivalently ∝an}bracketle{tf(v)∝an}bracketri}htinf=\n∞, which results in a distinct behavior of the time aver-\nages. Using Eq. (C6), we have\n−∂/tildewideP|v|(k,s)\n∂k/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nk=0∼γΓ(ν−γ)\n(1+γ−ν)Γ(1−γ)1\ns1+ν(71)fors→0. The inverse Laplace transform gives\n∝an}bracketle{tX(t)∝an}bracketri}ht ∼γ|Γ(ν−γ−1)|\nΓ(1−γ)Γ(1+ν)tν(72)\nfort→ ∞. Therefore, X(t) scales as tν, which means\nthat all the terms of k/sνin Eq. (C5) cannot be ignored.\nThese terms give the higher order moments. Performing\nthe inverse Laplace transform of terms proportional to\n1/s1+νgives\n∝an}bracketle{tX(t)n∝an}bracketri}ht ∝tnν(73)\nfort→ ∞. By Eq. (C8), the EB parameter becomes\nEB(t)→A(γ,ν)≡2(1+γ−ν)Γ(1+ν)2\nΓ(1+2ν)/bracketleftbigg(1+γ−ν)Γ(2ν−γ)Γ(1−γ)\nγ(2−2ν+γ)Γ(ν−γ)2+1/bracketrightbigg\n−1 (t→ ∞).(74)\nThis EB parameter depends on γas well as ν(> γ)\nand was found also in Ref. [52]. We note that A(γ,ν)\nis a decreasing function of ν. Therefore, trajectory-to-\ntrajectory fluctuations of the time averages becomes in-\nsignificant for large ν. In particular, A(γ,ν) converges to\nML(γ) and 0 for ν→γ+0 andν→1−0, respectively.\nIn other words, the system becomes ergodic in the sense\nthat the time averagesconvergeto a constant in the limit\nofγ→1 (andν→1).\nB. Time average of the squared velocity\nForf(v) =v2,X(t) can be represented by\nX(t) =N(t)−1/summationdisplay\nn=1τ2ν−1\nn+τ2ν−2\nN(t)(t−tN(t)−1).(75)\nBy the same calculation as in the previous case, us-\ningφv2(z,τ) =δ/parenleftbig\nz−τ2ν−1/parenrightbig\nψ(τ) andΦv2(z,t) =/integraltext∞\ntdτψ(τ)δ(z−τ2ν−2t), one can express the double\nLaplace transform of P(z,t) as\n/tildewidePv2(k,s) =/tildewideΦv2(k,s)\n1−/tildewideψv2(k,s). (76)\nTherefore, the limit distribution of X(t)/∝an}bracketle{tX(t)∝an}bracketri}htcan be\nobtained in the same way as for the previous observable.\nIn particular, the Mittag-Leffler distribution is a univer-\nsal distribution of the normalized time average of v2if\n2ν−1< γ, i.e.,f(v) =v2is integrable with respect\nto the infinite invariant density. On the other hand, the\ndistribution ofnormalized time averages X(t)/∝an}bracketle{tX(t)∝an}bracketri}htbe-\ncomes another distribution for t→ ∞if 2ν−1>γ(see\nAppendix. C). It follows that ∝an}bracketle{tX(t)∝an}bracketri}ht ∝t2ν−1fort→ ∞\nand the EB parameter becomes\nEB(t)→A(γ,2ν−1). (77)(ν= 0.4)\n(ν= 0.4)\n(ν= 0.8)\n(ν= 0.8)\n 0  0.2 0.4 0.6 0.8 1  1.2 1.4\n10 310 410 510 610 710 810 910 10 EB parameter \ntEq. (52) \nEq. (56) \nEq. (59) \nf(v)=|v| \nf(v)=v 2\nf(v)=v 2\nf(v)=|v| \nf(v)=v 2\n 0  1 \n10 10 10 10 10 10 10 10 10 EB parameter Eq. (52) \nEq. (56) \nEq. (59) \nf(v)=|v| \nf(v)=v \nf(v)=v \nf(v)=|v| \nf(v)=v (ν= 0.5)Eq. (71) \nEq. (75) \nEq. (78) 10 \n!\"# !$# ) = ) = \n!\"#$%$&'&()*+,%-.\", &/0&/&(,*+,%-.\", \n11111!\"#$%$&'&()*+,%-.\", &/0&/&(,*+,%-.\", \n11111\nFIG. 5. Phase diagram of the parameter space ( ) for (a) f(v) = |v|and (b) ) = . The solid line = 1 describes the \nboundary of the infinite measure. The dotted line represents the b oundary that the average of the observable ) with respect \nto the infinite/probability measure diverges. For eq dv < and 1 (region III), the time average converges to a \nconstant, implying the EB parameter goes to zero. For dv < and 1 (region II), the EB parameter becomes \na non-zero constant given by Eq. (62), implying the time averag es remain random variables. For dv and \n1 (region I), the EB parameter becomes a non-zero constant given by Eq. (66) or (69), implying the time averages remain \nrandom variables and it depends on as well as , which is di erent from case and 1. \nACKNOWLEDGEMENT \nThis work was supported by JSPS KAKENHI Grant \nNumber 16KT0021, 18K03468 (TA). EB thanks the Is- rael Science Foundation and Humboldt Foundation for \nsupport. \nAppendix A: Exact form of the propagator outside \n,v )] \nHere, we consider the Mittag-Le er function as the \nflight-duration PDF to obtain the exact form of the prop- \nor outside [ ,v )]. The Mittag-Le er func- \ntion with parameter is defined as [70] \n=0 +1) (A1) \nTo obtain the exact form of the propagator, one needs a \nspecific form the flight-duration PDF. Here, we assume \nthat the flight-duration PDF can be written through the \nMittag-Le er function: \n) = dt ) = \n=0 1) \n(A2) \nIn fact, the asymptotic behavior is given by a power law \n[70], i.e., \n+1)sin( γπ → ∞ (A3) \nMoreover, it is known that the Laplace transform of \nis given by \n) = 1+ (A4) Therefore, the Laplace transform of becomes \n(1 )) 1+ (A5) \nand its inverse Laplace transform yields \n(1+ +1 (A6) \nfor any t > 0. For 1, ) is given by \n2(1 1+ (A7) \nIt follows that the propagator outside [ ,v )] be- \ncomes \nv,t 2(1 )sin( γπ 1+ (A8) \nfor 1 and v > t . As shown in Fig. 6, the prop- \nor outside [ ,v )] is described by Eq. (A8, \nwhereas we did not use Eq. (A2). We expect that this \nexact form is universal like the infinite density. 10 \n!\"# !$# ) = ) = \n!\"#$%$&'&()*+,%-.\", &/0&/&(,*+,%-.\", \n11111!\"#$%$&'&()*+,%-.\", &/0&/&(,*+,%-.\", \n11111\nFIG. 5. Phase diagram of the parameter space ( ) for (a) f(v) = |v|and (b) ) = . The solid line = 1 describes the \nboundary of the infinite measure. The dotted line represents the b oundary that the average of the observable ) with respect \nto the infinite/probability measure diverges. For eq dv < and 1 (region III), the time average converges to a \nconstant, implying the EB parameter goes to zero. For dv < and 1 (region II), the EB parameter becomes \na non-zero constant given by Eq. (62), implying the time averag es remain random variables. For dv and \n1 (region I), the EB parameter becomes a non-zero constant given by Eq. (66) or (69), implying the time averages remain \nrandom variables and it depends on as well as , which is di erent from case and 1. \nACKNOWLEDGEMENT \nThis work was supported by JSPS KAKENHI Grant \nNumber 16KT0021, 18K03468 (TA). EB thanks the Is- rael Science Foundation and Humboldt Foundation for \nsupport. \nAppendix A: Exact form of the propagator outside \n,v )] \nHere, we consider the Mittag-Le er function as the \nflight-duration PDF to obtain the exact form of the prop- \nor outside [ ,v )]. The Mittag-Le er func- \ntion with parameter is defined as [70] \n=0 +1) (A1) \nTo obtain the exact form of the propagator, one needs a \nspecific form the flight-duration PDF. Here, we assume \nthat the flight-duration PDF can be written through the \nMittag-Le er function: \n) = dt ) = \n=0 1) \n(A2) \nIn fact, the asymptotic behavior is given by a power law \n[70], i.e., \n+1)sin( γπ → ∞ (A3) \nMoreover, it is known that the Laplace transform of \nis given by \n) = 1+ (A4) Therefore, the Laplace transform of becomes \n(1 )) 1+ (A5) \nand its inverse Laplace transform yields \n(1+ +1 (A6) \nfor any t > 0. For 1, ) is given by \n2(1 1+ (A7) \nIt follows that the propagator outside [ ,v )] be- \ncomes \nv,t 2(1 )sin( γπ 1+ (A8) \nfor 1 and v > t . As shown in Fig. 6, the prop- \nor outside [ ,v )] is described by Eq. (A8, \nwhereas we did not use Eq. (A2). We expect that this \nexact form is universal like the infinite density. 10 \n!\"# !$# ) = ) = \n!\"#$%$&'&()*+,%-.\", &/0&/&(,*+,%-.\", \n11111!\"#$%$&'&()*+,%-.\", &/0&/&(,*+,%-.\", \n11111\nFIG. 5. Phase diagram of the parameter space ( ) for (a) ) = and (b) f(v) = v2. The solid line = 1 describes the \nboundary of the infinite measure. The dotted line represents the b oundary that the average of the observable ) with respect \nto the infinite/probability measure diverges. For eq dv < and 1 (region III), the time average converges to a \nconstant, implying the EB parameter goes to zero. For dv < and 1 (region II), the EB parameter becomes \na non-zero constant given by Eq. (62), implying the time averag es remain random variables. For dv and \n1 (region I), the EB parameter becomes a non-zero constant given by Eq. (66) or (69), implying the time averages remain \nrandom variables and it depends on as well as , which is di erent from case and 1. \nACKNOWLEDGEMENT \nThis work was supported by JSPS KAKENHI Grant \nNumber 16KT0021, 18K03468 (TA). EB thanks the Is- rael Science Foundation and Humboldt Foundation for \nsupport. \nAppendix A: Exact form of the propagator outside \n,v )] \nHere, we consider the Mittag-Le er function as the \nflight-duration PDF to obtain the exact form of the prop- \nor outside [ ,v )]. The Mittag-Le er func- \ntion with parameter is defined as [70] \n=0 +1) (A1) \nTo obtain the exact form of the propagator, one needs a \nspecific form the flight-duration PDF. Here, we assume \nthat the flight-duration PDF can be written through the \nMittag-Le er function: \n) = dt ) = \n=0 1) \n(A2) \nIn fact, the asymptotic behavior is given by a power law \n[70], i.e., \n+1)sin( γπ → ∞ (A3) \nMoreover, it is known that the Laplace transform of \nis given by \n) = 1+ (A4) Therefore, the Laplace transform of becomes \n(1 )) 1+ (A5) \nand its inverse Laplace transform yields \n(1+ +1 (A6) \nfor any t > 0. For 1, ) is given by \n2(1 1+ (A7) \nIt follows that the propagator outside [ ,v )] be- \ncomes \nv,t 2(1 )sin( γπ 1+ (A8) \nfor 1 and v > t . As shown in Fig. 6, the prop- \nor outside [ ,v )] is described by Eq. (A8, \nwhereas we did not use Eq. (A2). We expect that this \nexact form is universal like the infinite density. 10 \n!\"# !$# ) = ) = \n!\"#$%$&'&()*+,%-.\", &/0&/&(,*+,%-.\", \n11111!\"#$%$&'&()*+,%-.\", &/0&/&(,*+,%-.\", \n11111\nFIG. 5. Phase diagram of the parameter space ( ) for (a) ) = and (b) f(v) = v2. The solid line = 1 describes the \nboundary of the infinite measure. The dotted line represents the b oundary that the average of the observable ) with respect \nto the infinite/probability measure diverges. For eq dv < and 1 (region III), the time average converges to a \nconstant, implying the EB parameter goes to zero. For dv < and 1 (region II), the EB parameter becomes \na non-zero constant given by Eq. (62), implying the time averag es remain random variables. For dv and \n1 (region I), the EB parameter becomes a non-zero constant given by Eq. (66) or (69), implying the time averages remain \nrandom variables and it depends on as well as , which is di erent from case and 1. \nACKNOWLEDGEMENT \nThis work was supported by JSPS KAKENHI Grant \nNumber 16KT0021, 18K03468 (TA). EB thanks the Is- rael Science Foundation and Humboldt Foundation for \nsupport. \nAppendix A: Exact form of the propagator outside \n,v )] \nHere, we consider the Mittag-Le er function as the \nflight-duration PDF to obtain the exact form of the prop- \nor outside [ ,v )]. The Mittag-Le er func- \ntion with parameter is defined as [70] \n=0 +1) (A1) \nTo obtain the exact form of the propagator, one needs a \nspecific form the flight-duration PDF. Here, we assume \nthat the flight-duration PDF can be written through the \nMittag-Le er function: \n) = dt ) = \n=0 1) \n(A2) \nIn fact, the asymptotic behavior is given by a power law \n[70], i.e., \n+1)sin( γπ → ∞ (A3) \nMoreover, it is known that the Laplace transform of \nis given by \n) = 1+ (A4) Therefore, the Laplace transform of becomes \n(1 )) 1+ (A5) \nand its inverse Laplace transform yields \n(1+ +1 (A6) \nfor any t > 0. For 1, ) is given by \n2(1 1+ (A7) \nIt follows that the propagator outside [ ,v )] be- \ncomes \nv,t 2(1 )sin( γπ 1+ (A8) \nfor 1 and v > t . As shown in Fig. 6, the prop- \nor outside [ ,v )] is described by Eq. (A8, \nwhereas we did not use Eq. (A2). We expect that this \nexact form is universal like the infinite density. 10 \n!\"# !$# ) = ) = \n!\"#$%$&'&()*+,%-.\", &/0&/&(,*+,%-.\", \n11111!\"#$%$&'&()*+,%-.\", &/0&/&(,*+,%-.\", \n11111\nFIG. 5. Phase diagram of the parameter space ( ) for (a) ) = and (b) f(v) = v2. The solid line = 1 describes the \nboundary of the infinite measure. The dotted line represents the b oundary that the average of the observable ) with respect \nto the infinite/probability measure diverges. For eq dv < and 1 (region III), the time average converges to a \nconstant, implying the EB parameter goes to zero. For dv < and 1 (region II), the EB parameter becomes \na non-zero constant given by Eq. (62), implying the time averag es remain random variables. For dv and \n1 (region I), the EB parameter becomes a non-zero constant given by Eq. (66) or (69), implying the time averages remain \nrandom variables and it depends on as well as , which is di erent from case and 1. \nACKNOWLEDGEMENT \nThis work was supported by JSPS KAKENHI Grant \nNumber 16KT0021, 18K03468 (TA). EB thanks the Is- rael Science Foundation and Humboldt Foundation for \nsupport. \nAppendix A: Exact form of the propagator outside \n,v )] \nHere, we consider the Mittag-Le er function as the \nflight-duration PDF to obtain the exact form of the prop- \nor outside [ ,v )]. The Mittag-Le er func- \ntion with parameter is defined as [70] \n=0 +1) (A1) \nTo obtain the exact form of the propagator, one needs a \nspecific form the flight-duration PDF. Here, we assume \nthat the flight-duration PDF can be written through the \nMittag-Le er function: \n) = dt ) = \n=0 1) \n(A2) \nIn fact, the asymptotic behavior is given by a power law \n[70], i.e., \n+1)sin( γπ → ∞ (A3) \nMoreover, it is known that the Laplace transform of \nis given by \n) = 1+ (A4) Therefore, the Laplace transform of becomes \n(1 )) 1+ (A5) \nand its inverse Laplace transform yields \n(1+ +1 (A6) \nfor any t > 0. For 1, ) is given by \n2(1 1+ (A7) \nIt follows that the propagator outside [ ,v )] be- \ncomes \nv,t 2(1 )sin( γπ 1+ (A8) \nfor 1 and v > t . As shown in Fig. 6, the prop- \nor outside [ ,v )] is described by Eq. (A8, \nwhereas we did not use Eq. (A2). We expect that this \nexact form is universal like the infinite density. FIG. 4. Ergodicity breaking parameter as a function of the\nmeasurement time for ν= 0.4, 0.5, and ν= 0.8 (γ= 0.3).\nWe note that ν= 0.4 and 0.5 satisfy 2 ν−1<γwhileν= 0.8\nsatisfies 2ν−1>γ. Symbolsrepresenttheresultsofnumerical\nsimulations. Thelong-dashed linerepresentsEq. (70), the two\ndotted lines represent Eq. (74), and the dotted line represe nt\nEq. (77). The flight-duration PDF is the same as that in\nFig. 1.\nThis expression was also obtained in Ref. [52]. The expo-\nnent 2ν−1 in Eq. (77) is different from that found in the\nEB parameter for f(v) =|v|withν >γ.Therefore, our\ndistributional limit theorem is not universal but depends\non the observable. On the other hand, the exponent γin\nthe EB parameter for 2 ν−1<γis the same as that for\nf(v) =|v|withν <γ.\nFigure4showsthatourtheoryworksverywellforboth\nobservables. For f(v) =v2withν= 0.4 andν= 0.5\n(γ= 0.3), both of which satisfy 2 ν−1< γ, the EB\nparameters do not depend on ν. Moreover, Fig. 4 shows\nthatthe EB parameter given by A(γ,ν) is a decreasing\nfunction of νforγ <ν.11\n((\u0000 ( \u0001 \u0002 f(v) =|v| f(v) =v2(a) \n(b) γν\n1\n0probability measure infinite measure \n1I\nII III probability measure infinite measure \nγν\n1\n0 1I\nII III \nFIG. 5. Phase diagram of the parameter space ( γ,ν) for (a)f(v) =|v|and (b)f(v) =v2. The solid line γ= 1 describes the\nboundary of the infinite measure. The dotted line represents the boundary that the average of the observable f(v) with respect\nto the infinite/probability measure diverges. For/integraltext\nf(v)peq(v)dv <∞andγ >1 (region III), the time average converges to a\nconstant, implying the EB parameter goes to zero. For/integraltext\nf(v)I∞(v)dv<∞andγ <1 (region II), the EB parameter becomes\na non-zero constant given by Eq. (70), implying the time aver ages remain random variables. For/integraltext\nf(v)I∞(v)dv=∞and\nγ <1 (region I), the EB parameter becomes a non-zero constant gi ven by Eq. (74) or (77), implying the time averages remain\nrandom variables and it depends on γas well asν, which is different from case/integraltext\nf(v)I∞(v)<∞andγ <1.\nVII. CONCLUSION\nWe investigated the propagator in an SMP and pro-\nvided its exact form, which is described by the mean\nnumber of renewals [see Eq. (23)]. We assumed that\nχ(v) =χ(−v) and that this function has support on zero\nvelocity. More specifically, the relation v=τν−1implies\nthatlongflightdurationsfavorvelocityclosetozerosince\n0< ν <1 and this is the reason for an accumulation of\nprobability in the vicinity of zero velocity in this model.\nWe prove that the propagator accumulates in the vicin-\nity of zero velocity in the long-time limit when the mean\nflight-duration diverges (γ <1) and the coupling param-\neter fulfills ν <1. Taking a closer look at the vicinity\nofv= 0, we found universal behaviors in the asymptotic\nforms of the propagator. In particular the asymptotic\nbehavior of the propagator for v≪1 follows two scaling\nlaws, i.e., the infinite invariant density Eq. (38) and the\nscaling function Eq. (40). The scaling function describes\na detailed structure of the propagator near v= 0 includ-\ning zero velocity while the infinite invariant density de-\nscribes the propagator outside vc=tν−1. Clearlyvc→0\nwhent→ ∞, and interestingly the asymptotic form out-\nsidevcbecomes a universal form that is unbounded at\nthe origin and cannot be normalized, i.e., an infinite in-\nvariant density . One advantage of considering the topic\nwith an SMP is that we can attain an explicit expression\nfor the infinite invariant density Eq. (38). In contrast\nin general it is hard to find exact infinite invariant mea-\nsures in deterministic dynamical systems, for example in\nthe context of the Pomeau-Manniville map [17].\nFurther, while the Mittag-Leffler distribution describ-\ning the distribution of time averages of integrable ob-servables is well known, from the Aaronson-Darling-Kac\ntheorem, we considered here also another distributional\nlimit theorem[see Eqs.(74) and (77)] which describesthe\ndistributionoftimeaveragesofcertainnon-integrableob-\nservables. Therefore, the integrability of the observable\nwith respect to the infinite invariant density establishes\na criterion on the type of distributional limit law, which\nis similar to findings in infinite ergodic theory. These\nresults will pave the way for constructing physics of non-\nstationary processes. Finally, we summarize our results\nby the phase diagram shown in Fig. 5. The infinite in-\nvariant density is always observed for γ <1. On the\nother hand, the boundary of the regions I and II depends\non the observation function f(v).\nACKNOWLEDGEMENT\nThis work was supported by JSPS KAKENHI Grant\nNumber 16KT0021, 18K03468 (TA). EB thanks the Is-\nrael Science Foundation and Humboldt Foundation for\nsupport.\nAppendix A: Exact form of the propagator outside\n[−vc(t),vc(t)]\nHere, weconsideraspecific formforthe flight-duration\nPDF to obtain the exact form of the propagator outside\n[−vc(t),vc(t)]. As a specific form, we use\nψ(τ) =−d\ndtEγ(−tγ) =1\nτ1−γ∞/summationdisplay\nn=0(−1)nτnγ\nΓ(γn+γ),(A1)12p(v,t)\n10 010 110 210 310 410 5\n10 -5 10 -4 10 -3 10 -2 10 -1 l(v) \nvt = 10 7\nt = 10 6\nt = 10 5\nt = 10 4\nt = 10 3\nFIG. 6. Time evolution of the propagator for different times\n(γ= 0.5 andν= 0.2). Symbols with lines are the results of\nnumerical simulations, which is the same as those in Fig. 1.\nDashed and solid lines are the theories, i.e., Eqs. (37) and\n(A8), respectively. The flight-duration PDF is the same as\nthat in Fig. 1.\nwhereEγ(z) is the Mittag-Leffler function with parame-\nterγdefined as [72]\nEγ(z)≡∞/summationdisplay\nn=0zn\nΓ(γn+1). (A2)\nIn fact, the asymptotic behavior is given by a power law\n[72], i.e.,\nψ(τ)∼Γ(γ+1)sin(γπ)\nπτ−1−γ(τ→ ∞).(A3)Moreover,it is known that the Laplace transform of ψ(τ)\nis given by\n/tildewideψ(s) =1\n1+sγ. (A4)\nTherefore, the Laplace transform of ∝an}bracketle{tN(t)∝an}bracketri}htbecomes\n1\ns(1−/tildewideψ(s))=1\ns1+γ+1\ns, (A5)\nand its inverse Laplace transform yields\n∝an}bracketle{tN(t)∝an}bracketri}ht=1\nΓ(1+γ)tγ+1 (A6)\nfor anyt>0. Forv≪1,χ(v) is given by\nχ(v)∼1\n2(1−ν)|Γ(−γ)||v|−1+γ\n1−ν.(A7)\nIt follows that the propagator outside [ −vc(t),vc(t)] be-\ncomes\np(v,t)∼tγ−(t−v1\nν−1)γ\n2(1−ν)sin(γπ)π|v|−1+γ\n1−ν.(A8)\nfort≫1 andv > tν−1. As shown in Fig. 6, the prop-\nagator outside [ −vc(t),vc(t)] is described by Eq. (A8),\nwhereas we did not use Eq. (A1).\nAppendix B: another proof of the asymptotic behavior of the p ropagator of v\nTo obtain the propagator, i.e., the PDF of velocity vat timet, it is almost equivalent to have the PDF ψt(τ) of\ntime interval straddling t, i.e.,τN(t)−1, whereN(t)−1is the number of renewals until t(not counting the one at\nt0= 0). In ordinary renewal processes, the double Laplace transfo rm of the PDF with respect to τandtis given by\n[66]\n/tildewideφ(k,s) =/tildewideψ(k)−/tildewideψ(k+s)\ns[1−/tildewideψ(s)]. (B1)\nForγ <1, the asymptoticbehaviorofthisinverseLaplacetransformcanb ecalculatedusingatechnique fromRef. [43].\nFortandτ≫1,\nψt(τ)∼\n\nsinπγ\nπtγ\nτ1+γ/bracketleftBig\n1−/parenleftBig\n1−τ\nt/parenrightBigγ/bracketrightBig\n(τ<t)\nsinπγ\nπtγ\nτ1+γ(τ>t).(B2)\nThis is the asymptotic result, which does not depend on the details of theflight-duration PDF, i.e. different flight-\nduration PDFs give the same result if the power-law exponent γis the same. On the other hand, detail forms of\nψt(τ), e.g., the behavior for small tandτ, depend on details of the flight-duration PDF [15].13\nHere, we consider a situation where the relation between the velocit y and the flight duration is given by |v|=τν−1.\nThe PDF of velocity vat timet, i.e., the propagator, can be represented through the PDF ψt(τ):\np(v,t) =1\n2|ν−1||v|1\nν−1−1ψt(|v|1\nν−1). (B3)\nNote thatp(v,t) is symmetric with respect to v= 0. Using Eq. (B2) yields\np(v,t)∼\n\nsinπγ\n2π|1−ν|tγ|v|−1+γ\n1−ν/bracketleftBigg\n1−/parenleftBigg\n1−|v|1\nν−1\nt/parenrightBiggγ/bracketrightBigg\n(|v|>tν−1)\nsinπγ\n2π|1−ν|tγ|v|−1+γ\n1−ν (|v|<tν−1).(B4)\nThe asymptotic form for ν <1 becomes\np(v,t)∼\n\nsinπγ\n2π|1−ν|tγ−1|v|−1+γ\n1−ν(|v| ≪tν−1)\nγsinπγ\n2π(ν−1)tγ|v|−1+1−γ\nν−1(tν−1≪ |v|)(B5)\nfort→ ∞. Therefore, this is consistent with the propagator we obtained in t his paper, Eqs. (34) and (37).\nForγ >1, the PDF ψt(τ) has an equilibrium distribution, i.e., for t→ ∞the PDFψt(τ) is given by\nψt(τ)∼τψ(τ)\n∝an}bracketle{tτ∝an}bracketri}ht, (B6)\nwhere∝an}bracketle{tτ∝an}bracketri}htis the mean flight duration [49].\nAppendix C: the double Laplace transform /tildewideP(k,s)and the exact form of the second moment of X(t)forν >γ\nHere, we represent the double Laplace transform /tildewideP(k,s) as an infinite series expansion. Expanding e−kτνin\nEqs. (55) and (56), we have\n/tildewideφ|v|(k,s)∼=/tildewideψ(s)−∝an}bracketle{tτν∝an}bracketri}htk+O(k2) (C1)\nand\n/tildewideφ|v|(k,s)∼=/tildewideψ(s)+csγ\n|Γ(−γ)|∞/summationdisplay\nn=1(−1)n\nn!Γ(nν−γ)/parenleftbiggk\nsν/parenrightbiggn\n(C2)\nforν <γandγ <ν, respectively, where ∝an}bracketle{tτν∝an}bracketri}ht ≡/integraltext∞\n0τνψ(τ)dτ. Moreover, we have\n/tildewideΦ|v|(k,s)∼=1−/tildewideψ(s)\ns+c\n|Γ(−γ)|s1−γ∞/summationdisplay\nn=1(−1)n\nn!Γ(nν−γ+1)\nγ+(1−ν)n/parenleftbiggk\nsν/parenrightbiggn\n(C3)\nforγ <1. Using Eq. (54), we have\n/tildewideP|v|(k,s) =csγ\ns/bracketleftBigg\n1+1\n|Γ(−γ)|∞/summationdisplay\nn=11\nn!Γ(nν−γ+1)\nγ+(1−ν)n/parenleftbigg\n−k\nsν/parenrightbiggn/bracketrightBigg\n1\ncsγ+∝an}bracketle{tτν∝an}bracketri}htk+O(k2)\n=1\ns/bracketleftBigg\n1+∞/summationdisplay\nn=11\nn!γΓ(nν−γ+1)\nΓ(1−γ){γ+(1−ν)n}/parenleftbigg\n−k\nsν/parenrightbiggn/bracketrightBigg/bracketleftbigg\n1+∝an}bracketle{tτν∝an}bracketri}ht\nck\nsγ+O(k2/sγ)/bracketrightbigg−1\n(C4)14\nforν <γand\n/tildewideP|v|(k,s) =csγ\ns/bracketleftBigg\n1+1\n|Γ(−γ)|∞/summationdisplay\nn=11\nn!Γ(nν−γ+1)\nγ+(1−ν)n/parenleftbigg\n−k\nsν/parenrightbiggn/bracketrightBigg/bracketleftBigg\ncsγ−csγ\n|Γ(−γ)|∞/summationdisplay\nn=1Γ(nν−γ)\nn!/parenleftbigg\n−k\nsν/parenrightbiggn/bracketrightBigg−1\n=1\ns/bracketleftBigg\n1+∞/summationdisplay\nn=11\nn!γΓ(nν−γ+1)\nΓ(1−γ){γ+(1−ν)n}/parenleftbigg\n−k\nsν/parenrightbiggn/bracketrightBigg/bracketleftBigg\n1−∞/summationdisplay\nn=1γΓ(nν−γ)\nn!Γ(1−γ)/parenleftbigg\n−k\nsν/parenrightbiggn/bracketrightBigg−1\n=1\ns/bracketleftBigg\n1+∞/summationdisplay\nn=11\nn!γΓ(nν−γ+1)\nΓ(1−γ){γ+(1−ν)n}/parenleftbigg\n−k\nsν/parenrightbiggn/bracketrightBigg/bracketleftBigg\n1+∞/summationdisplay\nm=1/braceleftBigg∞/summationdisplay\nn=1γΓ(nν−γ)\nn!Γ(1−γ)/parenleftbigg\n−k\nsν/parenrightbiggn/bracerightBiggm/bracketrightBigg\n(C5)\nforν >γ.\nThe coefficient of the term proportional to kin Eq. (C5) is\n−1\ns1+ν/bracketleftbiggγΓ(ν−γ+1)\n(γ+1−ν)Γ(1−γ)+γΓ(ν−γ)\nΓ(1−γ)/bracketrightbigg\n. (C6)\nMoreover, by considering the coefficient of the term proportional tok2in Eq. (C5), the leading term of the second\nmoment of X(t) in the Laplace space ( s→0) can be represented as\n∂2/tildewideP|v|(k,s)\n∂k2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nk=0∼M2(ν,γ)\ns1+2ν, (C7)\nwhere\nM2(ν,γ) =2γΓ(2ν−γ)\n(2−2ν+γ)Γ(1−γ)+2γ2Γ(ν−γ)2\n(1+γ−ν)Γ(1−γ)2. (C8)\nIt follows that the asymptotic behavior of ∝an}bracketle{tX(t)2∝an}bracketri}htis given by Eq. (73) with n= 2.\nSince∝an}bracketle{tX(t)n∝an}bracketri}htgrows as ∝an}bracketle{tX(t)n∝an}bracketri}ht ∝tnν, one can define Mn(ν,γ) as\n∝an}bracketle{tX(t)n∝an}bracketri}ht ∼Γ(1+ν)nMn(ν,γ)\nΓ(1+nν)M1(ν,γ)n∝an}bracketle{tX(t)∝an}bracketri}htn. (C9)\nIt follows that the random variable X(t)/∝an}bracketle{tX(t)∝an}bracketri}htconverges in distribution to a random variable Mν,γwhich depends\non bothνandγ. More precisely, one obtains\n∝an}bracketle{te−zMν,γ∝an}bracketri}ht ∼∞/summationdisplay\nn=0Γ(1+ν)nMn(ν,γ)\nn!Γ(1+nν)M1(ν,γ)n(−z)n. (C10)\nTherefore, the PDF of the normalized time average defined by X(t)/tνconverges to a non-trivial distribution that is\ndifferent from the Mittag-Leffler distribution (see Fig. 7).\n[1] T. Akimoto and Y. Aizawa, J Korean Phys. Soc. 50, 254\n(2007).\n[2] N. Korabel and E. Barkai, Phys. Rev. Lett. 102, 050601\n(2009).\n[3] T. Akimoto and T. Miyaguchi, Phys. Rev. E 82,\n030102(R) (2010).\n[4] T. Akimoto and E. Barkai,\nPhys. Rev. E 87, 032915 (2013).\n[5] P. Meyer and H. Kantz, Phys. Rev. E 96, 022217 (2017).\n[6] F. Bardou, J.-P. Bouchaud, A. Aspect, and C. Cohen-\nTannoudji, Levy statistics and laser cooling: how rareevents bring atoms to rest (Cambridge University Press,\n2002).\n[7] F. Bardou, J. P. Bouchaud, O. Emile, A. Aspect, and\nC. Cohen-Tannoudji, Phys. Rev. Lett. 72, 203 (1994).\n[8] E. Bertin and F. Bardou, Am. J. Phys. , 630 (2008).\n[9] E. 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Kehrein1\n1Institute for Theoretical Physics, University of G¨ otting en, Germany\n* manuel.kreye@theorie.physik.uni-goettingen.de\nAugust 29, 2019\nAbstract\nIn weakly perturbed systems that are close to integrability, thermalization can be\ndelayed by the formation of prethermalization plateaus. We study the build-up\nof density-density correlations after a weak interaction quench in the Hubbard\nmodel in d > 1dimensions using unitary perturbation theory. Starting from a\npre-quench state at temperature T, we show that the prethermalization values of\nthe post-quench correlations are equal to the equilibrium values of the interacting\nmodel at the same temperature T. This is explained by the local character of\ndensity-density correlations.\nContents\n1 Introduction 2\n1.1 Motivation 2\n1.2 Model 3\n2 Real-time evolution of the annihilation operator 4\n2.1 Unitary perturbation theory 4\n2.2 Transformation of the Hamiltonian 6\n2.3 Transformation of the annihilation operator 7\n2.4 Time evolution and backward transformation 9\n3 Equal-time connected density-density correlation function 10\n3.1 Correlations between antiparallel spins 11\n3.2 Correlations between parallel spins 12\n3.3 Small q-limit of parallel-spin correlations in infinite spatial di mensions 15\n3.4 Relation to prethermalization 17\n4 Conclusion 18\nA Consistency checks 19\nA.1 Preservation of the canonical anti-commutation relati on 19\nA.2 Total spin-up particle number 22\nA.3 Variance of the total spin-up particle number 23\n1SciPost Physics Submission\nB Calculation of correlation functions 25\nB.1 Antiparallel-spin correlations 26\nB.2 Parallel-spin correlations 27\nB.3 Limit of infinite spatial dimensions 29\nReferences 32\n1 Introduction\n1.1 Motivation\nSeminal experiments with ultracold atoms have made it possi ble to study the thermaliza-\ntion dynamics of isolated quantum many-body systems out of e quilibrium [1–4]. The high\ncontrollability of cold atoms in optical lattices allows fo r the simulation of artificial models\nand the implementation of quantum quenches, where in the sub sequent nonequilibrium dy-\nnamics individual atoms can be tracked site- and time-resol ved. For example, Kinoshita et\nal. demonstrated that a one-dimensional Bose gas brought ou t of equilibrium remains in a\nnonthermal steady state because of the integrability of the underlying model [2].\nThese experiments have stimulated theoretical research on the question how isolated quan-\ntum systems thermalize [5]. A pure state in an isolated syste m can be described by a density\noperatorρwith Tr[ρ2] = 1. But, as it is subject to unitary time evolution, it can ne ver evolve\ninto a mixed thermal state with Tr[ ρ2]<1. However, for certain subsets of observables, a\ntime-evolved pure state can become indistinguishable from a thermal state. The general view\nis that for local observables the environment acts as a therm al bath.\nWhile after a quantum quench we expect generic nonintegrabl e systems to thermalize [3, 6],\nintegrable systems, like in the experiment by Kinoshita et a l. [2], usually do not thermalize\nbecause the set of conserved quantities strongly restricts the dynamics. However, the non-\nthermal steady states of integrable systems can be describe d by a generalized Gibbs ensemble\n(GGE) [7]. The natural question arises what happens to weakl y perturbed systems, i.e., to\nnonintegrable systems that come close to integrability. He re, one often faces the phenomenon\nof prethermalization.\nPrethermalization was discussed by Berges, Bors´ anyi and W etterich in the context of heavy-ion\ncollisions [8]. They argued that in far-from-equilibrium s ettings there can be an intermediate\ntime scale where bulk quantities, like the equation of state for hydrodynamical considerations,\nhave already reached their equilibrium value while momentu m-dependent mode quantities\nare still far from thermalization. In condensed matter phys ics, this concept was captured by\nMoeckel and Kehrein, who studied the thermalization dynami cs of the momentum distribution\nfunction after a weak interaction quench in the Hubbard mode l using unitary perturbation\ntheory [9]. They identified a prethermalization plateau whe re the momentum distribution\nbecomes quasi-stationary but still differs from equilibrium . Their result was verified by nu-\nmerical calculations in dynamical mean-field theory (DMFT) [10].\nThe relation between the prethermalization time scale and t he perturbation strength in the\nHubbard model led to the picture of near-integrability indu ced bottlenecks in the thermaliza-\n2SciPost Physics Submission\ntion dynamics emphasized by Kollar, Wolf and Eckstein [11]. They argued that prethermal-\nization plateaus can also be predicted by generalized Gibbs ensembles and that nonthermal\nsteady states in integrable systems can be interpreted as in finitely delayed prethermalization\nplateaus.\nMeanwhile, prethermalization has become a topic of vast res earch interest. It has been stud-\nied e.g. in the Hubbard model [12–14], the three-dimensiona l Heisenberg model [15], one-\ndimensional spin chains [16,17], the Luttinger model [18,1 9], as well as in models with long-\nrange interactions [20] and periodic driving [21]. All thes e works hint at the universality of\nprethermalization in perturbed systems [22,23]. Experime ntally, prethermalization has been\nobserved in cold atom systems, e.g. in one-dimensional Bose gases [24, 25] and long-range\ninteracting spin chains [26]. Furthermore, prethermaliza tion has also been discussed in the\ncontext of quantum information [27], Anderson localizatio n [28], many-body localization [29],\nquantum time crystals [30,31] and the preheating of the earl y universe [32,33].\nIn this paper, we extend the work of Moeckel and Kehrein and st udy a local quantity, more\nprecisely the equal-time density-density correlation fun ction, in the nonequilibrium Hubbard\nmodel ind >1 dimensions. We consider a weak interaction quench, allowi ng us to use uni-\ntary perturbation theory, a method that avoids secular term s [34] and is especially suited to\ndirectly compare prethermalization to equilibrium values [9].\nThe pre-quench state has a temperature Tand reaches a prethermalized post-quench state\nwhere the correlation functions are equal to the equilibriu m values of the interacting model\nat the same temperature T. Heating effects from the quench that would increase the tempe r-\nature of the post-quench state will only show on a much longer time scale that is not covered\nby our approach.\n1.2 Model\nWe study the real-time evolution in the Fermi-Hubbard model [35] ind>1 dimensions,\nH=/summationdisplay\nk,σǫk:c†\nkσckσ: +U\nΩ/summationdisplay\nk′\ni,ki:c†\nk′\n1↑ck1↑c†\nk′\n2↓ck2↓:δk′\n1+k′\n2,k1+k2, (1)\nwith a general dispersion relation ǫkand where ǫF= 0 is the Fermi energy. Udenotes the\ninteraction strength, Ω the number of lattice sites, σ∈ {↑,↓}the spins and k∈[−π,π]dthe\nmomenta corresponding to reciprocal lattice vectors. For t echnical reasons, we use normal-\nordering : ·: with respect to the Gibbs state of the non-interacting Hami ltonianH0=/summationtext\nk,σǫkc†\nkσckσ.\nWe implement a weak interaction quench by preparing the syst em in the ground state |ψ0∝an}b∇acket∇i}htof\nH0and switching on the interaction to some finite value of Uat timet= 0. As the interaction\nUis considered weak, we can treat the real-time evolution pro blem perturbatively.\nThe described quench setup and its perturbative treatment w as studied by Moeckel and\nKehrein, who calculated the time evolution of the momentum d istribution function [9]. We\nbuild up our considerations from their work and expand it to i nclude the real-time dynamics\nof the equal-time connected density-density correlation f unction\nCσ′σ\nx′,x(t) =∝an}b∇acketle{tnx′,σ′(t)nx,σ(t)∝an}b∇acket∇i}ht−∝an}b∇acketle{tnx′,σ′(t)∝an}b∇acket∇i}ht∝an}b∇acketle{tnx,σ(t)∝an}b∇acket∇i}ht, (2)\nwherenx,σ(t) = Ω−1/summationtext\nk′,kei(k′−k)xc†\nk′σ(t)ckσ(t) is the local density operator for spin- σparti-\ncles at lattice site x.\n3SciPost Physics Submission\n2 Real-time evolution of the annihilation operator\nThe general idea is to solve the Heisenberg equation of motio n for the annihilation operator\nck↑(t) in the Hubbard model using unitary perturbation theory. We will calculate the pertur-\nbative expansion of ck↑(t) up to second order in U. This result can in principle be used for the\nconstruction of a wide class of observables. As an example, w e will calculate density-density\ncorrelations, which can be evaluated for different initial st ates.\n2.1 Unitary perturbation theory\nA problem that often occurs in naive perturbative treatment s of the Heisenberg equations of\nmotion is the appearance of secular terms that grow with some power law in time. These\nsecular terms emerge when the expansion in the small paramet er indirectly includes an ex-\npansion in time.\nIn classical mechanics, one can avoid this problem by using c anonical transformations that\nbring the Hamiltonian to normal-form, before one deals with the time evolution. Hackl and\nKehrein extended this idea to the realm of quantum mechanics , where the canonical transfor-\nmations must be replaced by unitary transformations [34].\nThe general scheme is depicted in Fig. 1: By (continuous) uni tary transformations Uone ap-\nproximately diagonalizes the Hamiltonian Hand transforms the observables Oaccordingly.\nIn the energy-diagonal basis (denoted by a tilde), the Heise nberg equations of motion for the\nobservables can be solved without the appearance of secular terms. After a backward transfor-\nmationU†of the time-evolved observable to the original basis, one ca n calculate expectation\nvalues with respect to a given state |ψ∝an}b∇acket∇i}ht.\nO(t),|ψ∝an}b∇acket∇i}htH,O,|ψ∝an}b∇acket∇i}ht\n˜O(t) =ei˜Ht˜Oe−i˜Ht,˜|ψ∝an}b∇acket∇i}ht˜H,˜O,˜|ψ∝an}b∇acket∇i}ht\nU†U\ntime evolution\nFigure 1: Illustration of the unitary perturbation theory s cheme\nJust like in the classical analogue, this forward-backward scheme can also be carried out\nperturbatively and still no secular terms will appear.\nThe flow equation method\nA method for approximately diagonalizing many-body Hamilt onians has been proposed by\nWegner [36] and independently in the context of high-energy physics by G/suppress lazek and Wil-\nson [37]. One applies a sequence of continuous unitary trans formations defined by the flow\n4SciPost Physics Submission\nequation\ndH(B)\ndB= [η(B),H(B)]−, (3)\nwhereH(B= 0) is the initial interacting Hamiltonian. Wegner showed t hat under rather\ngeneral conditions the canonical generator\nηcan.(B)def= [H0(B),Hint(B)]−, (4)\nwill effectively diagonalize the Hamiltonian in the limit B→ ∞ , apart from degeneracies.\nHere,H0(B) is the diagonal part of the Hamiltonian and Hint(B) the interaction part. In\nthe one-dimensional Hubbard model, H0andHintwould already commute at B= 0 and the\nflow would become featureless. Therefore, we have to assume d>1.\nThe coupling between eq. (3) and (4) usually leads to the gene ration of an infinite series of\nhigher-order interaction terms. This problem can be avoide d by systematic expansions in the\ncoupling parameter.\nWhile the Hamiltonian will have a simple structure in the ene rgy-diagonal basis, the com-\nplicated dynamics of the interacting system is shifted to th e observables, which transform\nunder\ndO(B)\ndB= [η(B),O(B)]− (5)\nand will hence become more intricate.\nCalculations in equilibrium\nThe forward-backward scheme in Fig. 1 is especially suited f or our quench setup, because\nthe initial state |ψ0∝an}b∇acket∇i}htis very simple and evaluations of expectation values after t he backward\ntransformation can be done by utilizing\n∝an}b∇acketle{tψ0|c†\nkσckσ|ψ0∝an}b∇acket∇i}ht=nk≡Θ(−ǫk) (forσ=↑,↓), (6)\nwhere the Fermi-Dirac distribution at zero temperature is j ust the Heaviside step function\nΘ(−ǫk).\nIf we want to calculate equilibrium quantities of the intera cting system, this can better be\ndone in the energy-diagonal basis at B=∞. This is because the ground state of an interacting\nHamiltonian is more complicated, but will show the simple fe ature of eq. (6) in a basis, where\nthe Hamiltonian is diagonal.\nFor our quench setup, we will use the flow equation results for ˜OandO(t) to directly compare\nthe equilibrium to the nonequilibrium setting.\n5SciPost Physics Submission\n2.2 Transformation of the Hamiltonian\nAs a second-order ansatz for the flowing Hamiltonian we choos e\nH(B) =/summationdisplay\nk,σǫk(B) :c†\nkσckσ: +1\nΩ/summationdisplay\nk′\ni,kiUk′\n1,k1,k′\n2,k2(B) :c†\nk′\n1↑ck1↑c†\nk′\n2↓ck2↓:δk′\n1+k′\n2,k1+k2\n+1\nΩ/summationdisplay\nk′\ni,ki/summationdisplay\nσVk′\n1,k1,k′\n2,k2(B) :c†\nk′\n1σck1σc†\nk′\n2σck2σ:δk′\n1+k′\n2,k1+k2\n+ higher-order interaction terms\n+O(U3), (7)\nwith initial values ǫk(B= 0) =ǫk,Uk′\n1,k1,k′\n2,k2(B= 0) =UandVk′\n1,k1,k′\n2,k2(B= 0) = 0.\nThe higher-order interaction terms are not relevant for our calculation. Hence, the canonical\ngenerator from eq. (4) becomes\nη(B) =1\nΩ/summationdisplay\nk′\ni,ki∆ǫk′\n1,k1,k′\n2,k2(B)Uk′\n1,k1,k′\n2,k2(B) :c†\nk′\n1↑ck1↑c†\nk′\n2↓ck2↓:δk′\n1+k′\n2,k1+k2\n+1\nΩ/summationdisplay\nk′\ni,ki/summationdisplay\nσ∆ǫk′\n1,k1,k′\n2,k2(B)Vk′\n1,k1,k′\n2,k2(B) :c†\nk′\n1σck1σc†\nk′\n2σck2σ:δk′\n1+k′\n2,k1+k2\n+ higher-order interaction terms\n+O(U3), (8)\nwith ∆ǫk′\n1,k1,k′\n2,k2def=ǫk′\n1−ǫk1+ǫk′\n2−ǫk2. With this generator, the flow equation for the\nHamiltonian, given in eq. (3), yields\nǫk(B) =ǫk+U2\nΩ2/summationdisplay\nk1,k′\n2,k21−e−2(∆ǫk,k1,k′\n2,k2)2B\n∆ǫk,k1,k′\n2,k2/parenleftbig\n(1−nk1)nk′\n2(1−nk2) +nk1(1−nk′\n2)nk2/parenrightbig\n+O(U3), (9)\nUk′\n1,k1,k′\n2,k2(B) =Ue−(∆ǫk′\n1,k1,k′\n2,k2)2B\n+O(U2), (10)\nVk′\n1,k1,k′\n2,k2(B) =−U2\nΩ/summationdisplay\nk′\n3,k3∆ǫk′\n1,k1,k′\n3,k3e−(∆ǫk′\n1,k1,k′\n3,k3)2Be−(∆ǫk′\n2,k2,k3,k′\n3)2B−e−(∆ǫk′\n1,k1,k′\n2,k2)2B\n(∆ǫk′\n1,k1,k′\n3,k3)2+ (∆ǫk′\n2,k2,k3,k′\n3)2−(∆ǫk′\n1,k1,k′\n2,k2)2\n×(nk′\n3−nk3)δk′\n3+k2,k3+k′\n2\n+O(U3). (11)\nClearly, the off-diagonal terms are exponentially surpresse d throughout the flow. At B=∞,\nonly elastic collision terms with ∆ ǫk′\n1,k1,k′\n2,k2= 0 survive. In this basis, the Hamiltonian takes\non the form\n˜H=/summationdisplay\nk,σ˜ǫk:c†\nkσckσ: +U\nΩ/summationdisplay\nk′\ni,ki:c†\nk′\n1↑ck1↑c†\nk′\n2↓ck2↓:δǫk′\n1+ǫk′\n2,ǫk1+ǫk2δk′\n1+k′\n2,k1+k2+O(U2),(12)\n6SciPost Physics Submission\nwith a renormalized one-particle energy\n˜ǫk=ǫk+U2\nΩ2/summationdisplay\nk1,k′\n2,k21\n∆ǫk,k1,k′\n2,k2/parenleftbig\n(1−nk1)nk′\n2(1−nk2) +nk1(1−nk′\n2)nk2/parenrightbig\n+O(U3).(13)\nThe elastic collision terms in eq. (12) are exactly the contr ibutions that appear in the quantum\nBoltzmann equation, from which we know to become relevant at time scales t∼ρ−3\nFU−4[38].\nOur calculation, as we will show in sec. 2.4, is only stable fo r time scales up to and including\nt∼ρ−1\nFU−2and hence we neglect the elastic collisions.\n2.3 Transformation of the annihilation operator\nThe annihilation operator is transformed under the flow equa tion from eq. (5),\ndck↑(B)\ndB= [η(B),ck↑(B)]−. (14)\nThe generator from eq. (8) causes a second-order flow to the fo llowing structure,\nck↑(B) =hk(B) :ck↑:\n+/summationdisplay\nk′\ni,kiFk,k1,k′\n2,k2(B) :ck1↑c†\nk′\n2↓ck2↓:δk+k′\n2,k1+k2\n+/summationdisplay\nk′\ni,kiGk,k1,k′\n2,k2(B) :ck1↑c†\nk′\n2↑ck2↑:δk+k′\n2,k1+k2\n+ higher-order interaction terms\n+O(U3). (15)\nWe will see in a moment that Fk,k1,k′\n2,k2(B) only contributes to the correlation functions with\nits first-order correction. Hence, we only consider the foll owing effective flow equations for\nthe coefficients,\ndhk(B)\ndB=U\nΩ/summationdisplay\nk1,k′\n2,k2∆ǫk,k1,k′\n2,k2e−(∆ǫk,k1,k′\n2,k2)2BFk,k1,k′\n2,k2(B)\n×/parenleftbig\n(1−nk1)nk′\n2(1−nk2) +nk1(1−nk′\n2)nk2/parenrightbig\nδk+k′\n2,k1+k2\n+O(U3), (16)\ndFk,k1,k′\n2,k2(B)\ndB=−U\nΩ∆ǫk,k1,k′\n2,k2e−(∆ǫk,k1,k′\n2,k2)2Bhk(B)\n+O(U2), (17)\n7SciPost Physics Submission\ndGk,k1,k′\n2,k2(B)\ndB=U\nΩ/summationdisplay\nk′\n3,k3∆ǫk′\n3,k3,k′\n2,k2e−(∆ǫk′\n3,k3,k′\n2,k2)2BFk,k1,k3,k′\n3(B)\n×(nk′\n3−nk3)δk′\n2+k′\n3,k2+k3\n+U2\nΩ2/summationdisplay\nk′\n3,k3(∆ǫk,k1,k′\n2,k2)(∆ǫk,k1,k′\n3,k3−∆ǫk′\n2,k2,k3,k′\n3)hk(B)\n×e−(∆ǫk,k1,k′\n3,k3)2Be−(∆ǫk′\n2,k2,k3,k′\n3)2B−e−(∆ǫk,k1,k′\n2,k2)2B\n(∆ǫk,k1,k′\n3,k3)2+ (∆ǫk′\n2,k2,k3,k′\n3)2−(∆ǫk,k1,k′\n2,k2)2\n×(nk′\n3−nk3)δk′\n3+k2,k3+k′\n2\n+O(U3). (18)\nThe perturbative solutions with the initial condition ck↑(B= 0) =:ck↑: are easily found,\nhk(B) = 1−U2\n2Ω2/summationdisplay\nk1,k′\n2,k2/parenleftBigg\n1−e−(∆ǫk,k1,k′\n2,k2)2B\n∆ǫk,k1,k′\n2,k2/parenrightBigg2\n×/parenleftbig\n(1−nk1)nk′\n2(1−nk2) +nk1(1−nk′\n2)nk2/parenrightbig\nδk+k′\n2,k1+k2\n+O(U3), (19)\nFk,k1,k′\n2,k2(B) =−U\nΩ1−e−(∆ǫk,k1,k′\n2,k2)2B\n∆ǫk,k1,k′\n2,k2\n+O(U2), (20)\nGk,k1,k′\n2,k2(B) =U2\nΩ2/summationdisplay\nk′\n3,k3∆ǫk′\n3,k3,k′\n2,k2\n∆ǫk,k1,k3,k′\n3\n×/parenleftBigg\n1−e−(∆ǫk′\n3,k3,k′\n2,k2)2Be−(∆ǫk,k1,k3,k′\n3)2B\n(∆ǫk′\n3,k3,k′\n2,k2)2+ (∆ǫk,k1,k3,k′\n3)2−1−e−(∆ǫk′\n3,k3,k′\n2,k2)2B\n(∆ǫk′\n3,k3,k′\n2,k2)2/parenrightBigg\n×(nk′\n3−nk3)δk′\n2+k′\n3,k2+k3\n+U2\nΩ2/summationdisplay\nk′\n3,k3(∆ǫk,k1,k′\n2,k2)(∆ǫk,k1,k′\n3,k3−∆ǫk′\n2,k2,k3,k′\n3)\n(∆ǫk,k1,k′\n3,k3)2+ (∆ǫk′\n2,k2,k3,k′\n3)2−(∆ǫk,k1,k′\n2,k2)2\n×/parenleftBigg\n1−e−(∆ǫk,k1,k′\n3,k3)2Be−(∆ǫk′\n2,k2,k3,k′\n3)2B\n(∆ǫk,k1,k′\n3,k3)2+ (∆ǫk′\n2,k2,k3,k′\n3)2−1−e−(∆ǫk,k1,k′\n2,k2)2B\n(∆ǫk,k1,k′\n2,k2)2/parenrightBigg\n×(nk′\n3−nk3)δk′\n3+k2,k3+k′\n2\n+O(U3). (21)\nTaking the limit B→ ∞ , we have a solution for ˜ck↑in the energy-diagonal basis that we\ncan use for equilibrium considerations. For the nonequilib rium quench setup, we now need\nto time-evolve the annihilation operator and then transfor m it back to the original basis at\nB= 0.\n8SciPost Physics Submission\n2.4 Time evolution and backward transformation\nAs pointed out in sec. (2.2), at B=∞the time evolution up to and including t∼ρ−1\nFU−2\nis simply governed by the quadratic Hamiltonian ˜H=/summationtext\nk,σ˜ǫk:c†\nkσckσ:. For the coefficients\nof the annihilation operator, we get\n˜hk(t) =e−i˜ǫkt˜hk, (22)\n˜Fk,k1,k′\n2,k2(t) =e−i(˜ǫk1−˜ǫk′\n2+˜ǫk2)t˜Fk,k1,k′\n2,k2, (23)\n˜Gk,k1,k′\n2,k2(t) =e−i(˜ǫk1−˜ǫk′\n2+˜ǫk2)t˜Gk,k1,k′\n2,k2. (24)\nThese time-evolved functions are now used as initial condit ions for the backward transforma-\ntion toB= 0 that is also given by eqs. (16) - (18). Integrating the flow e quations backwards\nyields\nhk(t) =e−i˜ǫkt\n−U2\nΩ2e−iǫkt/summationdisplay\nk1,k′\n2,k21−ei(∆ǫk,k1,k′\n2,k2)t\n(∆ǫk,k1,k′\n2,k2)2\n×/parenleftbig\n(1−nk1)nk′\n2(1−nk2) +nk1(1−nk′\n2)nk2/parenrightbig\nδk+k′\n2,k1+k2\n+O(U3), (25)\nFk,k1,k′\n2,k2(t) =U\nΩe−iǫkt1−ei(∆ǫk,k1,k′\n2,k2)t\n∆ǫk,k1,k′\n2,k2\n+O(U2), (26)\nGk,k1,k′\n2,k2(t) =−U2\nΩ2e−iǫkt/summationdisplay\nk′\n3,k3∆ǫk′\n3,k3,k′\n2,k2\n∆ǫk,k1,k3,k′\n3\n×/parenleftBigg\n1−ei(∆ǫk,k1,k′\n2,k2)t\n(∆ǫk′\n3,k3,k′\n2,k2)2+ (∆ǫk,k1,k3,k′\n3)2−ei(∆ǫk,k1,k3,k′\n3)t−ei(∆ǫk,k1,k′\n2,k2)t\n(∆ǫk′\n3,k3,k′\n2,k2)2/parenrightBigg\n×(nk′\n3−nk3)δk′\n2+k′\n3,k2+k3\n+U2\nΩ2e−iǫkt/summationdisplay\nk′\n3,k3(∆ǫk,k1,k′\n2,k2)(∆ǫk,k1,k′\n3,k3−∆ǫk′\n2,k2,k3,k′\n3)\n(∆ǫk,k1,k′\n3,k3)2+ (∆ǫk′\n2,k2,k3,k′\n3)2−(∆ǫk,k1,k′\n2,k2)2\n×/parenleftBigg\nei(∆ǫk,k1,k′\n2,k2)t−1\n(∆ǫk,k1,k′\n3,k3)2+ (∆ǫk′\n2,k2,k3,k′\n3)2−ei(∆ǫk,k1,k′\n2,k2)t−1\n(∆ǫk,k1,k′\n2,k2)2/parenrightBigg\n×(nk′\n3−nk3)δk′\n3+k2,k3+k′\n2\n+O(U3). (27)\nTime scales: Moeckel and Kehrein argued that the perturbative solution i s stable up to\nand including time scales t∼ρ−1\nFU−2, whereρFis the density of states at the Fermi edge [9].\nIn order to see this, we evaluate the expression from eq. (25) introducing energy integrals,\nhk(t) =e−iǫkt−U2e−iǫkt/integraldisplay∞\n−∞dE1−ei(ǫk−E)t\n(ǫk−E)2Ik(E) +O(U3). (28)\n9SciPost Physics Submission\nAt temperature T, the phase space factor Ik(E) is∝ρ3\nFmax{E2,T2}. For zero temperature,\nthe integral in eq. (28) converges for all times at the Fermi s urface, where ǫk=ǫF= 0.\nAway from the Fermi surface, the integral diverges as ∼ǫ2\nktforǫk/greaterorsimilarTand as∼T2tfor\nǫk/lessorsimilarT. Therefore, the second order correction of hk(t) becomes comparable to 1 for times\nt∼ρ−3\nFU−2min{ǫ−2\nk,T−2}. This implies that the perturbative nature of our approach i s\nvalid until times t/lessorsimilarρ−1\nFU−2for a worst case estimate where ǫkis of order the bandwidth.\nHowever, one often considers only dynamical contributions in the vicinity of the Fermi edge,\nǫk≈0, and at low temperature, which much improves the stability of the time evolution.\nTogether with the general structure of the annihilation ope rator from eq. (15), we have reached\na perturbative solution of the Heisenberg equation of motio n for this operator that can be\nused to construct a wide class of observables. Before we use t his for the evaluation of density-\ndensity correlations, we check our result for consistency.\nConsistency check 1: preservation of canonical anticommutation relation\nAs the sequence of forward transformation, time evolution a nd backward transformation is\ncompletely unitary, the canonical anticommutation relati on\n/bracketleftBig\nck↑(t),c†\nk′↑(t)/bracketrightBig\n+!=δk,k′+O(U3) (29)\nshould be preserved, at least in a perturbative sense. This c ondition leads to a relation\nbetweenhk(t),Fk,k1,k′\n2,k2(t) andGk,k1,k′\n2,k2(t) that is indeed fulfilled by our solutions from\neqs. (25) - (27), see App. A.\nConsistency check 2: total spin-up particle number\nFrom our perturbative solution for ck↑(t), we can easily calculate the operator for the to-\ntal spin-up particle number,\nN↑(t)def=/summationdisplay\nkc†\nk↑(t)ck↑(t), (30)\nwhich must be conserved, because it commutes with the Hamilt onian. We can show that the\nabove solutions are also consistent with this condition, se e App. A.\n3 Equal-time connected density-density correlation funct ion\nNow, we are able to evaluate expectation values of time-evol ved observables with respect to\nthe initial state |ψ0∝an}b∇acket∇i}ht, which corresponds to the nonequilibrium quench setup. If w e calculate\nexpectation values of observables in the basis at B=∞with respect to the same state |ψ0∝an}b∇acket∇i}ht,\nthis will correspond to the interacting Hubbard model in equ ilibrium.\nThe quantity of interest is the equal-time connected densit y-density correlation function from\n10SciPost Physics Submission\neq. (2),\nCσ′σ\nx′,x(t) =1\nΩ2/summationdisplay\nk′,k,q′,qei(k′−k)x′ei(q′−q)x∝an}b∇acketle{tc†\nk′σ′(t)ckσ′(t)c†\nq′σ(t)cqσ(t)∝an}b∇acket∇i}ht\n−1\nΩ2/summationdisplay\nk′,k,q′,qei(k′−k)x′ei(q′−q)x∝an}b∇acketle{tc†\nk′σ′(t)ckσ′(t)∝an}b∇acket∇i}ht∝an}b∇acketle{tc†\nq′σ(t)cqσ(t)∝an}b∇acket∇i}ht. (31)\nWe will distinguish to two cases of antiparallel-spin and pa rallel-spin correlations, where we\nhaveC↑↓\nx′,x(t)≡C↓↑\nx′,x(t) andC↑↑\nx′,x(t)≡C↓↓\nx′,x(t) due to the spin-symmetry of the Hubbard\nmodel. Details of the calculation can be found in App. B.\n3.1 Correlations between antiparallel spins\nFor the case of antiparallel spins, the correlation functio n has a leading order contribution that\nis of first order in U. In the nonequilibrium quench scenario, we get the followin g correlation\nfunction,\nC↑↓\nx′,x(t) =2U\nΩ3/summationdisplay\nk′,kei(k′−k)(x′−x)(nk′−nk)/summationdisplay\nq′,q1−cos/parenleftbig\n(∆ǫk′,k,q′,q)t/parenrightbig\n∆ǫk′,k,q′,q(1−nq′)nqδk′+q′,k+q\n+O(U2), (32)\nwhile for the interacting Hubbard model in equilibrium, we g et\nCeq.↑↓\nx′,x=2U\nΩ3/summationdisplay\nk′,kei(k′−k)(x′−x)(nk′−nk)/summationdisplay\nq′,q1\n∆ǫk′,k,q′,q(1−nq′)nqδk′+q′,k+q\n+O(U2). (33)\nNow, we calculate the time average of the nonequilibrium cor relation function,\nC↑↓\nx′,x(t)def= lim\nt→∞1\nt/integraldisplayt\n0dt′C↑↓\nx′,x(t′). (34)\nAs our perturbative ansatz only covers time scales up to and i ncluding the prethermalization\nregime, this time average equals the prethermalization val ue of the correlation function though\nwe integrate to t=∞. The integration to t=∞also makes the initial transient of the\ncorrelation function play no role for the time average.\nThe time average of eq. (32), where only the cos-function dro ps out, is equal to the equilibrium\nresult, at least in leading order. Hence, the prethermaliza tion value is\nCpre.↑↓\nx′,x≡C↑↓\nx′,x(t)\n=Ceq.↑↓\nx′,x+O(U2). (35)\n11SciPost Physics Submission\n3.2 Correlations between parallel spins\nFor parallel spins, the correlation function is of second or der inUand therefore more intricate.\nIn the nonequilibrium setting, we find\nC↑↑\nx′,x(t) =1\nΩ2/summationdisplay\nk′,kei(k′−k)(x′−x)nk′(1−nk)\n−4U2\nΩ4/summationdisplay\nk′,kei(k′−k)(x′−x)nk′(1−nk)\n×/summationdisplay\nk1,k′\n2,k21−cos/parenleftbig\n(∆ǫk,k1,k′\n2,k2)t/parenrightbig\n(∆ǫk,k1,k′\n2,k2)2\n×/parenleftbig\n(1−nk1)nk′\n2(1−nk2) +nk1(1−nk′\n2)nk2/parenrightbig\nδk+k′\n2,k1+k2\n+2U2\nΩ4/summationdisplay\nk′,kei(k′−k)(x′−x)nk′(1−nk)\n×/summationdisplay\nk′\ni,ki1−cos/parenleftbig\n(∆ǫk′,k′\n1,k′\n2,k2)t/parenrightbig\n−cos/parenleftbig\n(∆ǫk,k1,k′\n2,k2)t/parenrightbig\n+ cos/parenleftbig\n(∆ǫk′,k,k1,k′\n1)t/parenrightbig\n(∆ǫk′,k′\n1,k′\n2,k2)(∆ǫk,k1,k′\n2,k2)\n×nk′\n2(1−nk2)δk+k′\n2,k1+k2δk′+k1,k+k′\n1\n−2U2\nΩ4/summationdisplay\nk′,kei(k′−k)(x′−x)nk′(1−nk)\n×/summationdisplay\nk′\ni,ki∆ǫk′,k,k′\n2,k2\n∆ǫk′\n1,k1,k2,k′\n2/parenleftBigg\n1−cos/parenleftbig\n(∆ǫk′,k,k′\n1,k1)t/parenrightbig\n(∆ǫk′,k,k′\n2,k2)2+ (∆ǫk′\n1,k1,k2,k′\n2)2−1−cos/parenleftbig\n(∆ǫk′,k,k′\n2,k2)t/parenrightbig\n(∆ǫk′,k,k′\n2,k2)2/parenrightBigg\n×(nk′\n1−nk1)(nk′\n2−nk2)δk′+k′\n1,k+k1δk′+k′\n2,k+k2\n+2U2\nΩ4/summationdisplay\nk′,kei(k′−k)(x′−x)/parenleftbig\nnk′(1−nk) + (1−nk′)nk/parenrightbig\n×/summationdisplay\nk′\ni,ki∆ǫk,k1,k′\n2,k2\n∆ǫk′,k′\n1,k′\n2,k2/parenleftBigg\n1−cos/parenleftbig\n(∆ǫk′,k,k1,k′\n1)t/parenrightbig\n(∆ǫk′,k′\n1,k′\n2,k2)2+ (∆ǫk,k1,k′\n2,k2)2−1−cos/parenleftbig\n(∆ǫk,k1,k′\n2,k2)t/parenrightbig\n(∆ǫk,k1,k′\n2,k2)2/parenrightBigg\n×nk1(nk′\n2−nk2)δk+k′\n2,k1+k2δk′+k1,k+k′\n1\n+2U2\nΩ4/summationdisplay\nk′,kei(k′−k)(x′−x)nk′(1−nk)\n×/summationdisplay\nk′\ni,ki(∆ǫk′\n1,k1,k2,k′\n2) + (∆ǫk′,k,k2,k′\n2)\n(∆ǫk′,k,k1,k′\n1)/parenleftBigg\n1−cos/parenleftbig\n(∆ǫk′,k,k1,k′\n1)t/parenrightbig\n(∆ǫk′\n1,k1,k2,k′\n2)2+ (∆ǫk′,k,k2,k′\n2)2/parenrightBigg\n×(nk′\n1−nk1)(nk′\n2−nk2)δk′+k1,k+k′\n1δk′+k2,k+k′\n2\n12SciPost Physics Submission\n+2U2\nΩ4/summationdisplay\nk′,kei(k′−k)(x′−x)/parenleftbig\nnk′(1−nk) + (1−nk′)nk/parenrightbig\n×/summationdisplay\nk′\ni,ki(∆ǫk′,k′\n1,k′\n2,k2) + (∆ǫk,k1,k′\n2,k2)\n(∆ǫk′,k,k1,k′\n1)/parenleftBigg\n1−cos/parenleftbig\n(∆ǫk′,k,k1,k′\n1)t/parenrightbig\n(∆ǫk′,k′\n1,k′\n2,k2)2+ (∆ǫk,k1,k′\n2,k2)2/parenrightBigg\n×nk′\n1(nk′\n2−nk2)δk+k′\n2,k1+k2δk′+k1,k+k′\n1\n+U2\nΩ4/summationdisplay\nk′,kei(k′−k)(x′−x)nk′(1−nk)\n×/summationdisplay\nk′\ni,ki1−cos/parenleftbig\n(∆ǫk′,k,k′\n1,k1)t/parenrightbig\n−cos/parenleftbig\n(∆ǫk′,k,k′\n2,k2)t/parenrightbig\n+ cos/parenleftbig\n(∆ǫk′\n1,k1,k2,k′\n2)t/parenrightbig\n(∆ǫk′,k,k′\n1,k1)(∆ǫk′,k,k′\n2,k2)\n×(nk′\n1−nk1)(nk′\n2−nk2)δk′+k′\n1,k+k1δk′+k′\n2,k+k2\n−2U2\nΩ4/summationdisplay\nk′,kei(k′−k)(x′−x)nk′\n×/summationdisplay\nk′\ni,ki1−cos/parenleftbig\n(∆ǫk′,k′\n1,k′\n2,k2)t/parenrightbig\n−cos/parenleftbig\n(∆ǫk,k1,k′\n2,k2)t/parenrightbig\n+ cos/parenleftbig\n(∆ǫk′,k,k1,k′\n1)t/parenrightbig\n(∆ǫk′,k′\n1,k′\n2,k2)(∆ǫk,k1,k′\n2,k2)\n×(1−nk1)nk′\n2(1−nk2)δk+k′\n2,k1+k2δk′+k1,k+k′\n1\n+4U2\nΩ4/summationdisplay\nk′,kei(k′−k)(x′−x)nk′\n×/summationdisplay\nk1,k′\n2,k21−cos/parenleftbig\n(∆ǫk,k1,k′\n2,k2)t/parenrightbig\n(∆ǫk,k1,k′\n2,k2)2(1−nk1)nk′\n2(1−nk2)δk+k′\n2,k1+k2\n+O(U3). (36)\nFor this solution, a further consistency check is sensible.\nConsistency check 3: variance of total spin-up particle number\nAs the total spin-up particle number is conserved, we expect this also for its variance. The\nvariance is obtained by a lattice summation over the paralle l-spin correlation function,\n/summationdisplay\nx′,xC↑↑\nx′,x(t) =∝an}b∇acketle{t(N↑(t))2∝an}b∇acket∇i}ht−(∝an}b∇acketle{tN↑(t)∝an}b∇acket∇i}ht)2. (37)\nIn App. A, we show that the lattice summation over our result f rom eq. (36) indeed yields\nthe time-independent solution\n/summationdisplay\nx′,xC↑↑\nx′,x(t) =/summationdisplay\nknk(1−nk) +O(U3). (38)\n13SciPost Physics Submission\nFor the interacting Hubbard model in equilibrium, the paral lel-spin correlation function is\nCeq.↑↑\nx′,x=1\nΩ2/summationdisplay\nk′,kei(k′−k)(x′−x)nk′(1−nk)\n−2U2\nΩ4/summationdisplay\nk′,kei(k′−k)(x′−x)nk′(1−nk)\n×/summationdisplay\nk1,k′\n2,k21\n(∆ǫk,k1,k′\n2,k2)2/parenleftbig\n(1−nk1)nk′\n2(1−nk2) +nk1(1−nk′\n2)nk2/parenrightbig\nδk+k′\n2,k1+k2\n+2U2\nΩ4/summationdisplay\nk′,kei(k′−k)(x′−x)nk′(1−nk)\n×/summationdisplay\nk′\ni,ki1\n(∆ǫk′,k′\n1,k′\n2,k2)(∆ǫk,k1,k′\n2,k2)nk′\n2(1−nk2)δk+k′\n2,k1+k2δk′+k1,k+k′\n1\n−2U2\nΩ4/summationdisplay\nk′,kei(k′−k)(x′−x)nk′(1−nk)\n×/summationdisplay\nk′\ni,ki∆ǫk′,k,k′\n2,k2\n∆ǫk′\n1,k1,k2,k′\n2/parenleftBigg\n1\n(∆ǫk′,k,k′\n2,k2)2+ (∆ǫk′\n1,k1,k2,k′\n2)2−1\n(∆ǫk′,k,k′\n2,k2)2/parenrightBigg\n×(nk′\n1−nk1)(nk′\n2−nk2)δk′+k′\n1,k+k1δk′+k′\n2,k+k2\n+2U2\nΩ4/summationdisplay\nk′,kei(k′−k)(x′−x)/parenleftbig\nnk′(1−nk) + (1−nk′)nk/parenrightbig\n×/summationdisplay\nk′\ni,ki∆ǫk,k1,k′\n2,k2\n∆ǫk′,k′\n1,k′\n2,k2/parenleftBigg\n1\n(∆ǫk′,k′\n1,k′\n2,k2)2+ (∆ǫk,k1,k′\n2,k2)2−1\n(∆ǫk,k1,k′\n2,k2)2/parenrightBigg\n×nk1(nk′\n2−nk2)δk+k′\n2,k1+k2δk′+k1,k+k′\n1\n+2U2\nΩ4/summationdisplay\nk′,kei(k′−k)(x′−x)nk′(1−nk)\n×/summationdisplay\nk′\ni,ki(∆ǫk′\n1,k1,k2,k′\n2) + (∆ǫk′,k,k2,k′\n2)\n(∆ǫk′,k,k1,k′\n1)/parenleftBigg\n1\n(∆ǫk′\n1,k1,k2,k′\n2)2+ (∆ǫk′,k,k2,k′\n2)2/parenrightBigg\n×(nk′\n1−nk1)(nk′\n2−nk2)δk′+k1,k+k′\n1δk′+k2,k+k′\n2\n+2U2\nΩ4/summationdisplay\nk′,kei(k′−k)(x′−x)/parenleftbig\nnk′(1−nk) + (1−nk′)nk/parenrightbig\n×/summationdisplay\nk′\ni,ki(∆ǫk′,k′\n1,k′\n2,k2) + (∆ǫk,k1,k′\n2,k2)\n(∆ǫk′,k,k1,k′\n1)/parenleftBigg\n1\n(∆ǫk′,k′\n1,k′\n2,k2)2+ (∆ǫk,k1,k′\n2,k2)2/parenrightBigg\n×nk′\n1(nk′\n2−nk2)δk+k′\n2,k1+k2δk′+k1,k+k′\n1\n+U2\nΩ4/summationdisplay\nk′,kei(k′−k)(x′−x)nk′(1−nk)\n×/summationdisplay\nk′\ni,ki1\n(∆ǫk′,k,k′\n1,k1)(∆ǫk′,k,k′\n2,k2)(nk′\n1−nk1)(nk′\n2−nk2)δk′+k′\n1,k+k1δk′+k′\n2,k+k2\n14SciPost Physics Submission\n−2U2\nΩ4/summationdisplay\nk′,kei(k′−k)(x′−x)nk′\n×/summationdisplay\nk′\ni,ki1\n(∆ǫk′,k′\n1,k′\n2,k2)(∆ǫk,k1,k′\n2,k2)(1−nk1)nk′\n2(1−nk2)δk+k′\n2,k1+k2δk′+k1,k+k′\n1\n+2U2\nΩ4/summationdisplay\nk′,kei(k′−k)(x′−x)nk′\n×/summationdisplay\nk1,k′\n2,k21\n(∆ǫk,k1,k′\n2,k2)2(1−nk1)nk′\n2(1−nk2)δk+k′\n2,k1+k2\n+O(U3). (39)\nComparing this to the nonequilibrium result, we realize tha t there are different prefactors in\ndifferent terms. This hampers a direct relation of the prether malization value of the post-\nquench state to the equilibrium value of the interacting mod el. We remark that the difference\nis due to the second-order corrections of hk(t). These have also been responsible for the\ndeviation of the nonequilibrium momentum distribution fun ction from the equilibrium value\nin the calculation by Moeckel and Kehrein [9].\nIn the following, we will show that the prethermalization va lue of the parallel-spin correlation\nfunction is equal to the equilibrium value at least for small momentum transfer, i.e., up to\nlinear order in q, whereqis the momentum in Fourier space associated with the distanc e\nx′−xin real space.\nIn order to perform a small momentum expansion in Fourier spa ce, we need do apply the\nlimit of infinite spatial dimensions.\n3.3 Small q-limit of parallel-spin correlations in infinite spatial di mensions\nWe Fourier transform the parallel-spin correlation functi on to momentum space, ˆC↑↑\nq(t) =\n1\nΩ/summationtext\nx′−xe−iq(x′−x)C↑↑\nx′,x(t), take the limit of an infinite dimensional lattice [39], whi ch allows\nus to replace sums over momenta by energy integrals, and expa nd the correlation function for\nsmall momentum q, which corresponds to the long-range behavior in real space .\n15SciPost Physics Submission\nFor the nonequilibrium function, we find\nˆC↑↑\nq(t) =1\nΩ2/summationdisplay\nknk+q(1−nk)\n−2U2\nΩ2/summationdisplay\nknk+q(1−nk)/integraldisplay\ndǫ1′/integraldisplay\ndǫ1/integraldisplay\ndǫ2′/integraldisplay\ndǫ2D(ǫ1′)D(ǫ1)D(ǫ2′)D(ǫ2)\n×∆ǫ2′,2\n∆ǫ1′,1,2,2′/parenleftBigg\n1−cos/parenleftbig\n(∆ǫ1′,1)t/parenrightbig\n(∆ǫ2′,2)2+ (∆ǫ1′,1,2,2′)2−1−cos/parenleftbig\n(∆ǫ2′,2)t/parenrightbig\n(∆ǫ2′,2)2/parenrightBigg\n(n1′−n1)(n2′−n2)\n+2U2\nΩ2/summationdisplay\nknk+q(1−nk)/integraldisplay\ndǫ1′/integraldisplay\ndǫ1/integraldisplay\ndǫ2′/integraldisplay\ndǫ2D(ǫ1′)D(ǫ1)D(ǫ2′)D(ǫ2)\n×(∆ǫ1′,1,2,2′) + (∆ǫ2,2′)\n(∆ǫ1,1′)/parenleftBigg\n1−cos/parenleftbig\n(∆ǫ1,1′)t/parenrightbig\n(∆ǫ1′,1,2,2′)2+ (∆ǫ2,2′)2/parenrightBigg\n(nk′\n1−nk1)(nk′\n2−nk2)\n+U2\nΩ2/summationdisplay\nknk+q(1−nk)/integraldisplay\ndǫ1′/integraldisplay\ndǫ1/integraldisplay\ndǫ2′/integraldisplay\ndǫ2D(ǫ1′)D(ǫ1)D(ǫ2′)D(ǫ2)\n×1−cos/parenleftbig\n(∆ǫ1′,1)t/parenrightbig\n−cos/parenleftbig\n(∆ǫ2′,2)t/parenrightbig\n+ cos/parenleftbig\n(∆ǫ1′,1,2,2′)t/parenrightbig\n(∆ǫ1′,1)(∆ǫ2′,2)(n1′−n1)(n2′−n2)\n+O(q2) +O(U3), (40)\nwith ∆ǫ1′,1def=ǫ1′−ǫ1. For the correlation function in equilibrium, we get\nˆCeq.↑↑\nq =1\nΩ2/summationdisplay\nknk+q(1−nk)\n−2U2\nΩ2/summationdisplay\nknk+q(1−nk)/integraldisplay\ndǫ1′/integraldisplay\ndǫ1/integraldisplay\ndǫ2′/integraldisplay\ndǫ2D(ǫ1′)D(ǫ1)D(ǫ2′)D(ǫ2)\n×∆ǫ2′,2\n∆ǫ1′,1,2,2′/parenleftbigg1\n(∆ǫ2′,2)2+ (∆ǫ1′,1,2,2′)2−1\n(∆ǫ2′,2)2/parenrightbigg\n(n1′−n1)(n2′−n2)\n+2U2\nΩ2/summationdisplay\nknk+q(1−nk)/integraldisplay\ndǫ1′/integraldisplay\ndǫ1/integraldisplay\ndǫ2′/integraldisplay\ndǫ2D(ǫ1′)D(ǫ1)D(ǫ2′)D(ǫ2)\n×(∆ǫ1′,1,2,2′) + (∆ǫ2,2′)\n(∆ǫ1,1′)/parenleftbigg1\n(∆ǫ1′,1,2,2′)2+ (∆ǫ2,2′)2/parenrightbigg\n(nk′\n1−nk1)(nk′\n2−nk2)\n+U2\nΩ2/summationdisplay\nknk+q(1−nk)/integraldisplay\ndǫ1′/integraldisplay\ndǫ1/integraldisplay\ndǫ2′/integraldisplay\ndǫ2D(ǫ1′)D(ǫ1)D(ǫ2′)D(ǫ2)\n×1\n(∆ǫ1′,1)(∆ǫ2′,2)(n1′−n1)(n2′−n2)\n+O(q2) +O(U3). (41)\nAt zero temperature, the function/summationtext\nknk+q(1−nk) can be geometrically estimated to be\nproportional to |q|for small momentum. Hence, the above solutions represent th e algebraically\ndecaying parts proportional to |x′−x|−2in real space.\nNow, we can calculate the time average of the nonequilibrium correlation function and find\n16SciPost Physics Submission\nthe prethermalization value\nˆCpre.↑↑\nq≡ˆC↑↑\nq(t)\n=ˆCeq.↑↑\nq +O(q2) +O(U3). (42)\n3.4 Relation to prethermalization\nIn generic non-integrable systems – like the Hubbard model i n higher dimensions ( d>1) – we\nexpect thermalization after the system has been brought out of equilibrium [3,6]. However, a\nnumber of systems with a two-stage course of thermalization have been found, where an inter-\nmediate prethermalization regime can be identified before t he entire system reaches thermal\nequilibrium [8–10, 40]. Berges, Bors´ anyi and Wetterich po inted out that it will be sufficient\nto look at the prethermalization regime if the quantity of in terest (for example, the equation\nof state in hydrodynamical considerations) has already obt ained an equilibrium value [8].\nConsidering heavy-ion collisions, they argued that mode qu antities like the momentum dis-\ntribution function memorize the initial conditions in the p rethermalization regime and only\ndecay in the long-time limit when thermalization commences to their thermal values. This\nleads to the formation of characteristic plateaus for momen tum-dependent quantities. In con-\ntrast, local quantities that are not explicitly momentum-d ependent quickly lose information\non the initial state and already equilibrate in the pretherm alization regime.\nIn the context of condensed matter systems, Moeckel and Kehr ein calculated the momen-\ntum distribution function Nk(t) =nk+ ∆Nk(t) for the Hubbard quench setup considered in\nthis paper and found that it reaches a prethermalization val ue\n∆Npre.\nk= 2∆Neq.\nk+O(U3) (43)\nafter the interaction quench [9]. They assumed zero tempera ture for the pre-quench state,\nwhile the equilibrium value ∆ Neq.\nkwas defined with respect to the interacting model at zero\ntemperature. The leading order contribution of the prether malization value ∆ Npre.\nkdiffers\nfrom the equilibrium function at zero temperature by a facto r 2. In contrast, the interaction\nenergy, which is a sum of local terms, already reaches the equ ilibrium value (associated with\nzero temperature) in the prethermalized regime, as Moeckel and Kehrein concluded from the\nFeynman-Hellman theorem. Eckstein, Kollar and Werner confi rmed the result from eq. (43)\nby numerical calculations in dynamical mean-field theory (D MFT) [10].\nAnother route in understanding prethermalization behavio r is picturing it as near-integrability\ninduced bottlenecks in the thermalization dynamics, as don e by Kollar, Wolf and Eckstein [11].\nThey emphasized that thermalization in nearly integrable s ystems can be massively delayed\nthe closer the system comes to integrability by the formatio n of prethermalization plateaus.\nWhile integrable systems usually relax to a nonthermal stat e that can be described by a\ngeneralized Gibbs ensemble (GGE) [7], Kollar, Wolf and Ecks tein considered an interaction\nquench in a system with a weakly perturbed Hamiltonian H=H0+gHint(starting from the\nintegrable point H0) and showed that for conserved quantities of H0the prethermalization\nvalue can also be predicted by a GGE. The approximate constan ts of motion in the perturbed\nsystem defer thermalization to time scales t≫ O (g−2) [9,11].\n17SciPost Physics Submission\nWe now turn our attention towards the prethermalized values of nonequilibrium correlation\nfunctions calculated in this paper.\nWe emphasize that the pre-quench state is a thermal state of t he noninteracting Hamiltonian\nat temperature T. The heating effect of the quench will increase the temperatur e, but only\non a time scale much longer than the time scale covered in our c alculation. The prethermal-\nization regime corresponds to times before these heating effe cts set in, that is t/lessorsimilarρ−1\nFU−2.\nThe equilibrium values Ceq.are defined with respect to equilibrium states of the interac ting\nHamiltonian at temperature T, which is equal to the temperature of the pre-quench state.\nThe density-density correlation function for antiparalle l spins reaches a prethermalization\nvalue given by eq. (35),\nCpre.↑↓\nx′,x=Ceq.↑↓\nx′,x+O(U2), (44)\nwhich, in leading order, is the equilibrium value of the inte racting model.\nWe cannot find such a relation for the prethermalization valu e of the parallel-spin correlation\nfunction from eq. (36). But we can at least make a statement fo r the long-range part, which\nis associated with the small momentum expansion of its Fouri er transform. The linear order\nexpansion for small momentum transfer from eq. (42) yields t he relation\nˆCpre.↑↑\nq =ˆCeq.↑↑\nq +O(q2) +O(U3). (45)\nHence, the long-range correlations between parallel spins show prethermal behavior equal to\nthe equilibrium behavior.\nWe conclude that our findings are close to the original pictur e of prethermalization by Berges,\nBors´ anyi and Wetterich. While in the prethermalization re gime momentum-dependent quan-\ntities like the distribution from eq. (43) differ from equilib rium, local quantities like the\ninteraction energy or the density-density correlation fun ctions from eqs. (44) and (45) already\nprethermalize to equilibrium values that are associated wi th the interacting model at a tem-\nperature equal to the pre-quench temperature. In the long-t ime behavior where the system\nthermalizes and that is not covered by our approach, we expec t the heating effect of the\nquench to further increase the temperature.\n4 Conclusion\nIn this paper, we calculated the time evolution of the annihi lation operator in the Fermi-\nHubbard model in d > 1 dimensions in a perturbative manner for weak interaction U. In\nparticular, no secular terms appear, so that the perturbati ve expansion covers time scales up\nto and including the prethermalization regime.\nWe used our result to construct equal-time density-density correlation functions for antipar-\nallel and parallel spins in a leading-order expansion. Here , we could write down the functions\nfor both the nonequilibrium case – generated by a quench star ting from the eigenstate of\nthe noninteracting model at temperature Tto a weak interaction – and the equilibrium case\ndefined by the weakly interacting model at the same temperatu reT.\nFor correlations between antiparallel spins, we calculate d the time average of the post-quench\nscenario, which corresponds to the prethermalization valu e. We demonstrated that the\nprethermalization value equals the equilibrium value at te mperature Tand explained this\n18SciPost Physics Submission\nresult by the notion that local quantities already equilibr ate in the prethermalization regime.\nFor correlations between parallel spins, we also gave close d expressions for nonequilibrium and\nequilibrium, where the leading-order contributions were o f second order in U. For a direct\ncomparison, we needed the further approximation of an infini te dimensional lattice, where\nwe could show that at least the long-range part of the correla tion function also reaches the\nequilibrium value in the prethermalization regime.\nOur approach is valid on time scales up to and including t/lessorsimilarρ−1\nFU−2and does not include the\nheating effect of the quench, which would further increase the temperature and only shows\non a much longer time scale where thermalization takes place .\nWe point out that our solutions from eqs. (25) - (27) can be use d for a second order expan-\nsion of expectation values of any observable that is compose d of at most four annihilation and\ncreation operators. Therefore, one can use the perturbativ e expansion also for other purposes.\nAcknowledgements\nWe are grateful for discussions with M. Kastner and L. Cevola ni.\nFunding information This work was supported through SFB 1073 (project B03) of the\nDeutsche Forschungsgemeinschaft (DFG).\nM.K. is financially supported by the Bisch¨ ofliche Studienf¨ orderung Cusanuswerk.\nA Consistency checks\nA.1 Preservation of the canonical anti-commutation relati on\nThe application of the forward-backward scheme depicted in Fig. 1 is a sequence of unitary\ntransformations on the annihilation and creation operator s. Hence, we expect the canonical\nanticommutation relation\n/bracketleftBig\nck↑(t),c†\nk′↑(t)/bracketrightBig\n+!=δk,k′+O(U3) (46)\nto be preserved, at least up to second order in U. This motivates a consistency check for the\ntime-evolved solutions from eqs. (25) - (27) after the unita ry perturbation theory scheme has\n19SciPost Physics Submission\nbeen applied to the annihilation operator. With its general form from eq. (15), we get\n/bracketleftBig\nck↑(t),c†\nk′↑(t)/bracketrightBig\n+=hk(t)h∗\nk′(t)δk,k′\n+/summationdisplay\nk1,k′\n2,k2Fk,k1,k′\n2,k2(t)F∗\nk′,k1,k′\n2,k2(t)/parenleftbig\n(1−nk1)nk′\n2(1−nk2) +nk1(1−nk′\n2)nk2/parenrightbig\n×δk+k′\n2,k1+k2δk,k′\n−/summationdisplay\nk′\ni,kiFk,k1,k′\n2,k2(t)F∗\nk′,k′\n1,k′\n2,k2(t)(nk′\n2−nk2) :c†\nk′\n1↑ck1↑:δk+k′\n2,k1+k2δk′+k1,k+k′\n1\n+/summationdisplay\nk′\n1,k1hk(t)/parenleftbig\nG∗\nk′,k,k1,k′\n1(t)−G∗\nk′,k′\n1,k1,k(t)/parenrightbig\n:c†\nk′\n1↑ck1↑:δk′+k1,k+k′\n1\n+/summationdisplay\nk′\n1,k1h∗\nk′(t)/parenleftbig\nGk,k′,k′\n1,k1(t)−Gk,k1,k′\n1,k′(t)/parenrightbig\n:c†\nk′\n1↑ck1↑:δk′+k1,k+k′\n1\n+ linearly independent terms\n+O(U3), (47)\nwhere we only consider terms that have an operator structure proportional to 1or\n:c†\nk′\n1↑ck1↑:. Thus, we have two consistency conditions,\n1!=|hk(t)|2\n+/summationdisplay\nk1,k′\n2,k2|Fk,k1,k′\n2,k2(t)|2/parenleftbig\n(1−nk1)nk′\n2(1−nk2) +nk1(1−nk′\n2)nk2/parenrightbig\nδk+k′\n2,k1+k2\n+O(U3), (48)\n0!=−/summationdisplay\nk′\n2,k2|Fk,k1,k′\n2,k2(t)|2(nk′\n2−nk2)δk+k′\n2,k1+k2\n+hk(t)/parenleftbig\nG∗\nk,k,k1,k1(t)−G∗\nk,k1,k1,k(t)/parenrightbig\n+h∗\nk(t)/parenleftbig\nGk,k,k1,k1(t)−Gk,k1,k1,k(t)/parenrightbig\n+O(U3), (49)\n20SciPost Physics Submission\nwhich are two relations that our three solutions from eqs. (2 5) - (27) should fulfill. We insert\nthe solutions into the first relation and get\n1!= 1\n−U2\nΩ2/summationdisplay\nk1,k′\n2,k21−e−i(∆ǫk,k1,k′\n2,k2)t\n(∆ǫk,k1,k′\n2,k2)2\n×((1−nk1)nk′\n2(1−nk2) +nk1(1−nk′\n2)nk2)δk+k′\n2,k1+k2\n−U2\nΩ2/summationdisplay\nk1,k′\n2,k21−ei(∆ǫk,k1,k′\n2,k2)t\n(∆ǫk,k1,k′\n2,k2)2\n×((1−nk1)nk′\n2(1−nk2) +nk1(1−nk′\n2)nk2)δk+k′\n2,k1+k2\n+U2\nΩ2/summationdisplay\nk1,k′\n2,k2/vextendsingle/vextendsingle/vextendsingle1−ei(∆ǫk,k1,k′\n2,k2)t/vextendsingle/vextendsingle/vextendsingle2\n(∆ǫk,k1,k′\n2,k2)2/parenleftbig\n(1−nk1)nk′\n2(1−nk2) +nk1(1−nk′\n2)nk2/parenrightbig\nδk+k′\n2,k1+k2\n+O(U3). (50)\nWe recognize that the last three terms cancel out. Thus, the fi rst consistency condition is\nfulfilled.\nThe second relation requires\n0!=−U2\nΩ2/summationdisplay\nk′\n2,k2/vextendsingle/vextendsingle/vextendsingle1−ei(∆ǫk,k1,k′\n2,k2)t/vextendsingle/vextendsingle/vextendsingle2\n(∆ǫk,k1,k′\n2,k2)2(nk′\n2−nk2)δk+k′\n2,k1+k2\n−G∗\nk,k1,k1,k(t)\n−Gk,k1,k1,k(t)\n+O(U3), (51)\nwhere we have used that Gk,k,k1,k1(t) = 0 +O(U3). Furthermore, eq. (27) implies\nGk,k1,k1,k(t) =−U2\nΩ2/summationdisplay\nk′\n3,k3∆ǫk′\n3,k3,k1,k\n∆ǫk,k1,k3,k′\n3/parenleftBigg\n1−ei(∆ǫk,k1,k1,k)t\n(∆ǫk′\n3,k3,k1,k)2+ (∆ǫk,k1,k3,k′\n3)2\n−ei(∆ǫk,k1,k3,k′\n3)t−ei(∆ǫk,k1,k1,k)t\n(∆ǫk′\n3,k3,k1,k)2/parenrightBigg\n×(nk′\n3−nk3)δk1+k′\n3,k+k3\n+O(U3)\n=−U2\nΩ2/summationdisplay\nk′\n2,k21−ei(∆ǫk,k1,k′\n2,k2)t\n(∆ǫk,k1,k′\n2,k2)2(nk′\n2−nk2)δk+k′\n2,k1+k2\n+O(U3). (52)\n21SciPost Physics Submission\nWith this result, the second relation reads\n0!=−U2\nΩ2/summationdisplay\nk′\n2,k2/vextendsingle/vextendsingle/vextendsingle1−ei(∆ǫk,k1,k′\n2,k2)t/vextendsingle/vextendsingle/vextendsingle2\n(∆ǫk,k1,k′\n2,k2)2(nk′\n2−nk2)δk+k′\n2,k1+k2\n+U2\nΩ2/summationdisplay\nk′\n2,k21−e−i(∆ǫk,k1,k′\n2,k2)t\n(∆ǫk,k1,k′\n2,k2)2(nk′\n2−nk2)δk+k′\n2,k1+k2\n+U2\nΩ2/summationdisplay\nk′\n2,k21−ei(∆ǫk,k1,k′\n2,k2)t\n(∆ǫk,k1,k′\n2,k2)2(nk′\n2−nk2)δk+k′\n2,k1+k2\n+O(U3), (53)\nwhich is clearly fulfilled due to cancelation of all terms on t he right-hand side.\nA.2 Total spin-up particle number\nThe total spin-up particle number\nN↑def=/summationdisplay\nkc†\nk↑ck↑ (54)\nis a conserved quantity in the Hubbard model, because it comm utes with the Hamiltonian.\nThis provides another consistency check for the solutions f rom eqs. (25) - (27). From eq. (63),\nwe can directly construct the time-evolved total spin-up pa rticle number operator, where we\nonly focus on terms with an operator structure proportional to1or :c†\nk′\n1↑ck1↑:,\nN↑(t) =/summationdisplay\nk|hk(t)|2nk+/summationdisplay\nk,k1,k′\n2,k2|Fk,k1,k′\n2,k2(t)|2nk1(1−nk′\n2)nk2δk+k′\n2,k1+k2\n+/summationdisplay\nk1|hk1(t)|2:c†\nk1↑ck1↑:\n+/summationdisplay\nk′\n1,k1,k′\n2,k2|Fk′\n1,k1,k′\n2,k2(t)|2(1−nk′\n2)nk2:c†\nk1↑ck1↑:δk′\n1+k′\n2,k1+k2\n+/summationdisplay\nk′\n1,k1h∗\nk′\n1(t)/parenleftbig\nGk′\n1,k′\n1,k1,k1(t)−Gk′\n1,k1,k1,k′\n1(t)/parenrightbig\nnk′\n1:c†\nk1↑ck1↑:\n+/summationdisplay\nk′\n1,k1hk′\n1(t)/parenleftbig\nG∗\nk′\n1,k′\n1,k1,k1(t)−G∗\nk′\n1,k1,k1,k′\n1(t)/parenrightbig\nnk′\n1:c†\nk1↑ck1↑:\n+ linearly independent terms\n+O(U3). (55)\n22SciPost Physics Submission\nWe insert the expressions from eqs. (25) - (27) and get\nN↑(t) =/summationdisplay\nknk−2U2\nΩ2/summationdisplay\nk,k1,k′\n2,k21−cos/parenleftbig\n(∆ǫk,k1,k′\n2,k2)t/parenrightbig\n(∆ǫk,k1,k′\n2,k2)2\n×nk/parenleftbig\n(1−nk1)nk′\n2(1−nk2) +nk1(1−nk′\n2)nk2/parenrightbig\nδk+k′\n2,k1+k2\n+2U2\nΩ2/summationdisplay\nk,k1,k′\n2,k21−cos/parenleftbig\n(∆ǫk,k1,k′\n2,k2)t/parenrightbig\n(∆ǫk,k1,k′\n2,k2)2nk1(1−nk′\n2)nk2δk+k′\n2,k1+k2\n+/summationdisplay\nk1:c†\nk1↑ck1↑:\n−2U2\nΩ2/summationdisplay\nk′\n1,k1,k′\n2,k21−cos/parenleftbig\n(∆ǫk′\n1,k1,k′\n2,k2)t/parenrightbig\n(∆ǫk′\n1,k1,k′\n2,k2)2\n×/parenleftbig\n(1−nk′\n1)nk2(1−nk′\n2) +nk′\n1(1−nk2)nk′\n2/parenrightbig\n:c†\nk1↑ck1↑:δk′\n1+k′\n2,k1+k2\n+2U2\nΩ2/summationdisplay\nk′\n1,k1,k′\n2,k21−cos/parenleftbig\n(∆ǫk′\n1,k1,k′\n2,k2)t/parenrightbig\n(∆ǫk′\n1,k1,k′\n2,k2)2(1−nk′\n2)nk2:c†\nk1↑ck1↑:δk′\n1+k′\n2,k1+k2\n+2U2\nΩ2/summationdisplay\nk′\n1,k1,k′\n2,k21−cos/parenleftbig\n(∆ǫk′\n1,k1,k′\n2,k2)t/parenrightbig\n(∆ǫk′\n1,k1,k′\n2,k2)2\n×nk′\n1/parenleftbig\nnk′\n2(1−nk2)−(1−nk′\n2)nk2/parenrightbig\n:c†\nk1↑ck1↑:δk′\n1+k′\n2,k1+k2\n+ linearly independent terms\n+O(U3). (56)\nWe convince ourselves that after interchanging indices mos t of the terms cancel out, yielding\nN↑(t) =/summationdisplay\nknk+/summationdisplay\nk:c†\nk↑ck↑: +linearly independent terms + O(U3). (57)\nAs this is time-independent, this part of the operator is con sistent with the conservation of\nthe total spin-up particle number.\nA.3 Variance of the total spin-up particle number\nAs the total spin-up particle number N↑is a conserved quantity, also its variance\n/angbracketleftbig\nN2\n↑/angbracketrightbig\n−/angbracketleftbig\nN↑/angbracketrightbig2(58)\nshould be time-independent. We can construct the variance f rom the equal-time connected\ndensity-density correlation function for parallel spins, C↑↑\nx′,x(t), by a summation over x′andx,\n/summationdisplay\nx′,xC↑↑\nx′,x(t) =/summationdisplay\nx′,x∝an}b∇acketle{tnx′↑(t)nx↑(t)∝an}b∇acket∇i}ht−/summationdisplay\nx′,x∝an}b∇acketle{tnx′↑(t)∝an}b∇acket∇i}ht∝an}b∇acketle{tnx↑(t)∝an}b∇acket∇i}ht\n=/angbracketleftbig\nN2\n↑/angbracketrightbig\n−/angbracketleftbig\nN↑/angbracketrightbig2. (59)\n23SciPost Physics Submission\nThe solution for C↑↑\nx′,x(t) from eq. (36) should be consistent with this. The summation over\nx′yields aδk′,kand we get\n/summationdisplay\nx′,xC↑↑\nx′,x(t) =/summationdisplay\nknk(1−nk)\n−4U2\nΩ2/summationdisplay\nknk(1−nk)/summationdisplay\nk1,k′\n2,k21−cos/parenleftbig\n(∆ǫk,k1,k′\n2,k2)t/parenrightbig\n(∆ǫk,k1,k′\n2,k2)2\n×/parenleftbig\n(1−nk1)nk′\n2(1−nk2) +nk1(1−nk′\n2)nk2/parenrightbig\nδk+k′\n2,k1+k2\n+4U2\nΩ2/summationdisplay\nknk(1−nk)/summationdisplay\nk1,k′\n2,k21−cos/parenleftbig\n(∆ǫk,k1,k′\n2,k2)t/parenrightbig\n(∆ǫk,k1,k′\n2,k2)2nk′\n2(1−nk2)δk+k′\n2,k1+k2\n−4U2\nΩ2/summationdisplay\nknk(1−nk)/summationdisplay\nk1,k′\n2,k21−cos/parenleftbig\n(∆ǫk,k1,k′\n2,k2)t/parenrightbig\n(∆ǫk,k1,k′\n2,k2)2nk1(nk′\n2−nk2)δk+k′\n2,k1+k2\n−4U2\nΩ2/summationdisplay\nknk/summationdisplay\nk1,k′\n2,k21−cos/parenleftbig\n(∆ǫk,k1,k′\n2,k2)t/parenrightbig\n(∆ǫk,k1,k′\n2,k2)2(1−nk1)nk′\n2(1−nk2)δk+k′\n2,k1+k2\n+4U2\nΩ2/summationdisplay\nknk/summationdisplay\nk1,k′\n2,k21−cos/parenleftbig\n(∆ǫk,k1,k′\n2,k2)t/parenrightbig\n(∆ǫk,k1,k′\n2,k2)2(1−nk1)nk′\n2(1−nk2)δk+k′\n2,k1+k2\n+O(U3). (60)\nThe last two terms cancel out directly and the other terms can be rearranged such that\n/summationdisplay\nx′,xC↑↑\nx′,x(t) =/summationdisplay\nknk(1−nk)\n−4U2\nΩ2/summationdisplay\nknk(1−nk)/summationdisplay\nk1,k′\n2,k21−cos/parenleftbig\n(∆ǫk,k1,k′\n2,k2)t/parenrightbig\n(∆ǫk,k1,k′\n2,k2)2\n×/parenleftbig\n(1−nk1)nk′\n2(1−nk2) +nk1(1−nk′\n2)nk2/parenrightbig\nδk+k′\n2,k1+k2\n+4U2\nΩ2/summationdisplay\nknk(1−nk)/summationdisplay\nk1,k′\n2,k21−cos/parenleftbig\n(∆ǫk,k1,k′\n2,k2)t/parenrightbig\n(∆ǫk,k1,k′\n2,k2)2\n×/parenleftbig\n(1−nk1)nk′\n2(1−nk2) +nk1(1−nk′\n2)nk2/parenrightbig\nδk+k′\n2,k1+k2\n+O(U3)\n=/summationdisplay\nknk(1−nk)\n+O(U3). (61)\nThis is clearly time-independent and hence consistent with the conservation of the variance\nof the total spin-up particle number.\n24SciPost Physics Submission\nB Calculation of correlation functions\nGiven the general structure of the annihilation operator fr om eq. (15), we first calculate\nc†\nk′↑(t)ck↑(t) =h∗\nk′(t)hk(t) :c†\nk′↑::ck↑:\n+/summationdisplay\nk1,k′\n2,k2h∗\nk′(t)Fk,k1,k′\n2,k2(t) :c†\nk′↑::ck1↑c†\nk′\n2↓ck2↓:δk+k′\n2,k1+k2\n+/summationdisplay\nk1,k′\n2,k2h∗\nk′(t)Gk,k1,k′\n2,k2(t) :c†\nk′↑::ck1↑c†\nk′\n2↑ck2↑:δk+k′\n2,k1+k2\n+/summationdisplay\nk1,k′\n2,k2F∗\nk′,k1,k′\n2,k2(t)hk(t) :c†\nk2↓ck′\n2↓c†\nk1↑::ck↑:δk′+k′\n2,k1+k2\n+/summationdisplay\nk′\ni,kiF∗\nk′,k1,k′\n2,k2(t)Fk,k′\n1,k′\n3,k3(t) :c†\nk2↓ck′\n2↓c†\nk1↑::ck′\n1↑c†\nk′\n3↓ck3↓:δk′+k′\n2,k1+k2δk+k′\n3,k′\n1+k3\n+/summationdisplay\nk1,k′\n2,k2G∗\nk′,k1,k′\n2,k2(t)hk(t) :c†\nk2↑ck′\n2↑c†\nk1↑::ck↑:δk′+k′\n2,k1+k2\n+ ”irrelevant terms”\n+O(U3), (62)\nwhere normal-ordered products of at least four annihilatio n and creation operators that are\nof second order in Uare shifted into the irrelevant terms.\n25SciPost Physics Submission\nThe next step is calculating the products of normal-ordered expressions, which yields\nc†\nk′↑(t)ck↑(t) =|hk′(t)|2nk′δk′,k+/summationdisplay\nk1,k′\n2,k2|Fk′,k1,k′\n2,k2(t)|2nk1(1−nk′\n2)nk2δk′+k′\n2,k1+k2δk′,k\n+h∗\nk′(t)hk(t) :c†\nk′↑ck↑:\n+/summationdisplay\nk′\ni,kiF∗\nk′,k′\n1,k′\n2,k2(t)Fk,k1,k′\n2,k2(t)(1−nk′\n2)nk2:c†\nk′\n1↑ck1↑:δk+k′\n2,k1+k2δk′+k1,k+k′\n1\n+/summationdisplay\nk′\n1,k1h∗\nk′(t)/parenleftbig\nGk,k′,k′\n1,k1(t)−Gk,k1,k′\n1,k′(t)/parenrightbig\nnk′:c†\nk′\n1↑ck1↑:δk′+k1,k+k′\n1\n+/summationdisplay\nk′\n1,k1hk(t)/parenleftbig\nG∗\nk′,k,k1,k′\n1(t)−G∗\nk′,k′\n1,k1,k(t)/parenrightbig\nnk:c†\nk′\n1↑ck1↑:δk′+k1,k+k′\n1\n+/summationdisplay\nk′\n1,k1h∗\nk′(t)Fk,k′,k′\n1,k1(t)nk′:c†\nk′\n1↓ck1↓:δk′+k1,k+k′\n1\n+/summationdisplay\nk′\n1,k1hk(t)F∗\nk′,k,k1,k′\n1(t)nk:c†\nk′\n1↓ck1↓:δk′+k1,k+k′\n1\n+/summationdisplay\nk′\ni,kiF∗\nk′,k2,k′\n2,k′\n1(t)Fk,k2,k′\n2,k1(t)(1−nk′\n2)nk2:c†\nk′\n1↓ck1↓:δk+k′\n2,k1+k2δk′+k1,k+k′\n1\n−/summationdisplay\nk′\ni,kiF∗\nk′,k′\n2,k1,k2(t)Fk,k′\n2,k′\n1,k2(t)nk′\n2nk2:c†\nk′\n1↓ck1↓:δk′+k1,k′\n2+k2δk′+k1,k+k′\n1\n+/summationdisplay\nk1,k′\n2,k2h∗\nk′(t)Fk,k1,k′\n2,k2(t) :c†\nk′↑ck1↑c†\nk′\n2↓ck2↓:δk+k′\n2,k1+k2\n+/summationdisplay\nk′\n1,k′\n2,k2hk(t)F∗\nk′,k′\n1,k2,k′\n2(t) :c†\nk′\n1↑ck↑c†\nk′\n2↓ck2↓:δk′\n1+k′\n2,k′+k2\n+ ”irrelevant terms”\n+O(U3). (63)\nB.1 Antiparallel-spin correlations\nFor anti-parallel spins, the equal-time correlation funct ion has a first-order contribution in U.\nTherefore, we neglect the second-order terms in eq. (63). Th e last two terms only completely\ncontract among each other, which would be of second order, he nce they are irrelevant. We\nonly need\nc†\nk′↑(t)ck↑(t) =h∗\nk′(t)hk(t) :c†\nk′↑ck↑:\n+/summationdisplay\nk′\n1,k1h∗\nk′(t)Fk,k′,k′\n1,k1(t)nk′:c†\nk′\n1↓ck1↓:δk′+k1,k+k′\n1\n+/summationdisplay\nk′\n1,k1hk(t)F∗\nk′,k,k1,k′\n1(t)nk:c†\nk′\n1↓ck1↓:δk′+k1,k+k′\n1\n+ ”irrelevant terms”\n+O(U2). (64)\n26SciPost Physics Submission\nWe calcuclate the contractions between the above equation a nd its spin-down counterpart,\nyielding\nC↑↓\nx′,x(t) =1\nΩ2/summationdisplay\nk′,k,q′,qei(k′−k)(x′−x)h∗\nk′(t)hk(t)h∗\nq′(t)Fq,q′,k,k′(t)nk′(1−nk)nq′δk′+q′,k+q\n+1\nΩ2/summationdisplay\nk′,k,q′,qei(k′−k)(x′−x)h∗\nk′(t)hk(t)hq(t)F∗\nq′,q,k′,k(t)nk′(1−nk)nqδk′+q′,k+q\n+1\nΩ2/summationdisplay\nk′,k,q′,qei(k′−k)(x′−x)hk′(t)h∗\nk(t)h∗\nq(t)Fq′,q,k′,k(t)nk′(1−nk)nqδk′+q′,k+q\n+1\nΩ2/summationdisplay\nk′,k,q′,qei(k′−k)(x′−x)hk′(t)h∗\nk(t)hq′(t)F∗\nq,q′,k,k′(t)nk′(1−nk)nq′δk′+q′,k+q\n+O(U2). (65)\nWhen we insert the coefficients from eqs. (25) and (26), we arri ve at the nonequilibrium solu-\ntion in eq. (32), while the coefficients from eqs. (19) and (20) forB=∞give the equilibrium\nsolution in eq. (33).\nB.2 Parallel-spin correlations\nFor parallel spins, the equal-time correlation function is of second order in Uand hence all\nterms in eq. (63) are relevant. Calculating all full contrac tions of normal-ordered expressions,\nwe find that\nC↑↑\nx′,x(t) =1\nΩ2/summationdisplay\nk′,kei(k′−k)(x′−x)nk′(1−nk)|hk′(t)|2|hk(t)|2\n+2\nΩ2/summationdisplay\nk′,kei(k′−k)(x′−x)nk′(1−nk)\n×/summationdisplay\nk′\ni,kiℜ/parenleftbig\nh∗\nk′(t)hk(t)Fk′\n1,k′,k′\n2,k2(t)F∗\nk1,k,k′\n2,k2(t)/parenrightbig\n(1−nk′\n2)nk2δk′\n1+k′\n2,k′+k2δk′+k1,k+k′\n1\n+2\nΩ2/summationdisplay\nk′,kei(k′−k)(x′−x)nk′(1−nk)\n×/summationdisplay\nk′\n1,k1ℜ/parenleftBig\nh∗\nk′(t)hk(t)hk′\n1(t)/parenleftbig\nG∗\nk1,k′\n1,k′,k(t)−G∗\nk1,k,k′,k′\n1(t)/parenrightbig/parenrightBig\nnk′\n1δk′+k1,k+k′\n1\n+2\nΩ2/summationdisplay\nk′,kei(k′−k)(x′−x)nk′(1−nk)\n×/summationdisplay\nk′\n1,k1ℜ/parenleftBig\nh∗\nk′(t)hk(t)h∗\nk1(t)/parenleftbig\nGk′\n1,k1,k,k′(t)−Gk′\n1,k′,k,k1(t)/parenrightbig/parenrightBig\nnk1δk′+k1,k+k′\n1\n27SciPost Physics Submission\n+1\nΩ2/summationdisplay\nk′,kei(k′−k)(x′−x)nk′(1−nk)\n×/summationdisplay\nk′\ni,kih∗\nk′\n1(t)Fk1,k′\n1,k′,k(t)h∗\nk′\n2(t)Fk2,k′\n2,k,k′(t)nk′\n1nk′\n2δk′\n1+k′\n2,k1+k2δk′+k1,k+k′\n1\n+1\nΩ2/summationdisplay\nk′,kei(k′−k)(x′−x)nk′(1−nk)\n×/summationdisplay\nk′\ni,kih∗\nk′\n1(t)Fk1,k′\n1,k′,k(t)hk2(t)F∗\nk′\n2,k2,k′,k(t)nk′\n1nk2δk′\n1+k′\n2,k1+k2δk′+k1,k+k′\n1\n+1\nΩ2/summationdisplay\nk′,kei(k′−k)(x′−x)nk′(1−nk)\n×/summationdisplay\nk′\ni,kihk1(t)F∗\nk′\n1,k1,k,k′(t)h∗\nk′\n2(t)Fk2,k′\n2,k,k′(t)nk1nk′\n2δk′\n1+k′\n2,k1+k2δk′+k1,k+k′\n1\n+1\nΩ2/summationdisplay\nk′,kei(k′−k)(x′−x)nk′(1−nk)\n×/summationdisplay\nk′\ni,kihk1(t)F∗\nk′\n1,k1,k,k′(t)hk2(t)F∗\nk′\n2,k2,k′,k(t)nk1nk2δk′\n1+k′\n2,k1+k2δk′+k1,k+k′\n1\n+2\nΩ2/summationdisplay\nk′,kei(k′−k)(x′−x)nk′\n×/summationdisplay\nk′\ni,kiℜ/parenleftbig\nh∗\nk′(t)h∗\nk1(t)Fk′\n1,k′,k2,k′\n2(t)Fk,k1,k′\n2,k2(t)/parenrightbig\n(1−nk1)nk′\n2(1−nk2)δk+k′\n2,k1+k2δk′+k1,k+k′\n1\n+1\nΩ2/summationdisplay\nk′,kei(k′−k)(x′−x)/summationdisplay\nk1,k′\n2,k2|hk′(t)|2|Fk,k1,k′\n2,k2(t)|2nk′(1−nk1)nk′\n2(1−nk2)δk+k′\n2,k1+k2\n+1\nΩ2/summationdisplay\nk′,ke−i(k′−k)(x′−x)/summationdisplay\nk1,k′\n2,k2|hk′(t)|2|Fk,k1,k′\n2,k2(t)|2(1−nk′)nk1(1−nk′\n2)nk2δk+k′\n2,k1+k2\n+O(U3). (66)\nAgain, inserting the coefficients from eqs. (25) - (27) will gi ve us the nonequilibrium solution,\nwhile the coefficients from eqs. (19) - (21) for B=∞give the equilibrium solution.\n28SciPost Physics Submission\nB.3 Limit of infinite spatial dimensions\nThe Fourier transformation to momentum space, ˆC↑↑\nq(t)def= Ω−1/summationtext\nx′−xe−iq(x′−x)C↑↑\nx′,x(t), ef-\nfectively yields a factor δq,k′−k, and we get\nˆC↑↑\nq(t) =1\nΩ2/summationdisplay\nknk+q(1−nk)\n−4U2\nΩ4/summationdisplay\nknk+q(1−nk)\n×/summationdisplay\nk1,k′\n2,k21−cos/parenleftbig\n(∆ǫk,k1,k′\n2,k2)t/parenrightbig\n(∆ǫk,k1,k′\n2,k2)2\n×/parenleftbig\n(1−nk1)nk′\n2(1−nk2) +nk1(1−nk′\n2)nk2/parenrightbig\nδk+k′\n2,k1+k2\n+2U2\nΩ4/summationdisplay\nknk+q(1−nk)\n×/summationdisplay\nk′\ni,ki1−cos/parenleftbig\n(∆ǫk+q,k′\n1,k′\n2,k2)t/parenrightbig\n−cos/parenleftbig\n(∆ǫk,k1,k′\n2,k2)t/parenrightbig\n+ cos/parenleftbig\n(∆ǫk+q,k,k1,k′\n1)t/parenrightbig\n(∆ǫk+q,k′\n1,k′\n2,k2)(∆ǫk,k1,k′\n2,k2)\n×nk′\n2(1−nk2)δk+k′\n2,k1+k2δk′\n1−k1,q\n−2U2\nΩ4/summationdisplay\nknk+q(1−nk)\n×/summationdisplay\nk′\ni,ki∆ǫk+q,k,k′\n2,k2\n∆ǫk′\n1,k1,k2,k′\n2/parenleftBigg\n1−cos/parenleftbig\n(∆ǫk+q,k,k′\n1,k1)t/parenrightbig\n(∆ǫk+q,k,k′\n2,k2)2+ (∆ǫk′\n1,k1,k2,k′\n2)2−1−cos/parenleftbig\n(∆ǫk+q,k,k′\n2,k2)t/parenrightbig\n(∆ǫk+q,k,k′\n2,k2)2/parenrightBigg\n×(nk′\n1−nk1)(nk′\n2−nk2)δk′\n1−k1,qδk′\n2−k2,q\n+2U2\nΩ4/summationdisplay\nk/parenleftbig\nnk+q(1−nk) + (1−nk+q)nk/parenrightbig\n×/summationdisplay\nk′\ni,ki∆ǫk,k1,k′\n2,k2\n∆ǫk+q,k′\n1,k′\n2,k2/parenleftBigg\n1−cos/parenleftbig\n(∆ǫk+q,k,k1,k′\n1)t/parenrightbig\n(∆ǫk+q,k′\n1,k′\n2,k2)2+ (∆ǫk,k1,k′\n2,k2)2−1−cos/parenleftbig\n(∆ǫk,k1,k′\n2,k2)t/parenrightbig\n(∆ǫk,k1,k′\n2,k2)2/parenrightBigg\n×nk1(nk′\n2−nk2)δk+k′\n2,k1+k2δk′\n1−k1,q\n+2U2\nΩ4/summationdisplay\nknk+q(1−nk)\n×/summationdisplay\nk′\ni,ki(∆ǫk′\n1,k1,k2,k′\n2) + (∆ǫk+q,k,k2,k′\n2)\n(∆ǫk+q,k,k1,k′\n1)/parenleftBigg\n1−cos/parenleftbig\n(∆ǫk+q,k,k1,k′\n1)t/parenrightbig\n(∆ǫk′\n1,k1,k2,k′\n2)2+ (∆ǫk+q,k,k2,k′\n2)2/parenrightBigg\n×(nk′\n1−nk1)(nk′\n2−nk2)δk′\n1−k1,qδk′\n2−k2,q\n29SciPost Physics Submission\n+2U2\nΩ4/summationdisplay\nk/parenleftbig\nnk+q(1−nk) + (1−nk+q)nk/parenrightbig\n×/summationdisplay\nk′\ni,ki(∆ǫk+q,k′\n1,k′\n2,k2) + (∆ǫk,k1,k′\n2,k2)\n(∆ǫk+q,k,k1,k′\n1)/parenleftBigg\n1−cos/parenleftbig\n(∆ǫk+q,k,k1,k′\n1)t/parenrightbig\n(∆ǫk+q,k′\n1,k′\n2,k2)2+ (∆ǫk,k1,k′\n2,k2)2/parenrightBigg\n×nk′\n1(nk′\n2−nk2)δk+k′\n2,k1+k2δk′\n1−k1,q\n+U2\nΩ4/summationdisplay\nknk+q(1−nk)\n×/summationdisplay\nk′\ni,ki1−cos/parenleftbig\n(∆ǫk+q,k,k′\n1,k1)t/parenrightbig\n−cos/parenleftbig\n(∆ǫk+q,k,k′\n2,k2)t/parenrightbig\n+ cos/parenleftbig\n(∆ǫk′\n1,k1,k2,k′\n2)t/parenrightbig\n(∆ǫk+q,k,k′\n1,k1)(∆ǫk+q,k,k′\n2,k2)\n×(nk′\n1−nk1)(nk′\n2−nk2)δk′\n1−k1,qδk′\n2−k2,q\n−2U2\nΩ4/summationdisplay\nknk+q\n×/summationdisplay\nk′\ni,ki1−cos/parenleftbig\n(∆ǫk+q,k′\n1,k′\n2,k2)t/parenrightbig\n−cos/parenleftbig\n(∆ǫk,k1,k′\n2,k2)t/parenrightbig\n+ cos/parenleftbig\n(∆ǫk+q,k,k1,k′\n1)t/parenrightbig\n(∆ǫk+q,k′\n1,k′\n2,k2)(∆ǫk,k1,k′\n2,k2)\n×(1−nk1)nk′\n2(1−nk2)δk+k′\n2,k1+k2δk′\n1−k1,q\n+4U2\nΩ4/summationdisplay\nknk+q\n×/summationdisplay\nk1,k′\n2,k21−cos/parenleftbig\n(∆ǫk,k1,k′\n2,k2)t/parenrightbig\n(∆ǫk,k1,k′\n2,k2)2(1−nk1)nk′\n2(1−nk2)δk+k′\n2,k1+k2\n+O(U3). (67)\nThe calculation for the equilibrium correlation function i s analogous. Now, we take the limit\nof infinite spatial dimensions [41]. This allows us to introd uce energy integrals,\n/summationdisplay\nki...→/integraldisplay\ndǫi/summationdisplay\nkiδ(ǫi−ǫki)..., (68)\nand make use of\n/summationdisplay\nk1,k2,k3δ(ǫ1−ǫk1)δ(ǫ2−ǫk2)δ(ǫ3−ǫk3)δk+k3,k1+k2\nd→∞=1\nΩ/summationdisplay\nk1,k2,k3δ(ǫ1−ǫk1)δ(ǫ2−ǫk2)δ(ǫ3−ǫk3) (69)\nand\n/summationdisplay\nk1,k2δ(ǫ1−ǫk1)δ(ǫ2−ǫk2)δk1−k2,kd→∞=1\nΩ/summationdisplay\nk1,k2δ(ǫ1−ǫk1)δ(ǫ2−ǫk2) (fork∝ne}ationslash=/vector0).(70)\nThis means that from now on we restrict the domain of the Fourier transformed correlation\nfunction to values q∝ne}ationslash=/vector0.\n30SciPost Physics Submission\nFor the nonequilibrium correlation function, we get\nˆC↑↑\nq(t) =1\nΩ2/summationdisplay\nknk+q(1−nk)\n−4U2\nΩ2/summationdisplay\nknk+q(1−nk)/integraldisplay\ndǫ1/integraldisplay\ndǫ2′/integraldisplay\ndǫ2D(ǫ1)D(ǫ2′)D(ǫ2)\n×1−cos/parenleftbig\n(∆ǫk,1,2′,2)t/parenrightbig\n(∆ǫk,1,2′,2)2/parenleftbig\n(1−n1)n2′(1−n2) +n1(1−n2′)n2/parenrightbig\n+2U2\nΩ4/summationdisplay\nknk+q(1−nk)\n×/summationdisplay\nk′\ni,ki1−cos/parenleftbig\n(∆ǫk+q,k′\n1,k′\n2,k2)t/parenrightbig\n−cos/parenleftbig\n(∆ǫk,k1,k′\n2,k2)t/parenrightbig\n+ cos/parenleftbig\n(∆ǫk+q,k,k1,k′\n1)t/parenrightbig\n(∆ǫk+q,k′\n1,k′\n2,k2)(∆ǫk,k1,k′\n2,k2)\n×nk′\n2(1−nk2)δk+k′\n2,k1+k2δk′\n1−k1,q\n−2U2\nΩ2/summationdisplay\nknk+q(1−nk)/integraldisplay\ndǫ1′/integraldisplay\ndǫ1/integraldisplay\ndǫ2′/integraldisplay\ndǫ2D(ǫ1′)D(ǫ1)D(ǫ2′)D(ǫ2)\n×∆ǫk+q,k,2′,2\n∆ǫ1′,1,2,2′/parenleftBigg\n1−cos/parenleftbig\n(∆ǫk+q,k,1′,1)t/parenrightbig\n(∆ǫk+q,k,2′,2)2+ (∆ǫ1′,1,2,2′)2−1−cos/parenleftbig\n(∆ǫk+q,k,2′,2)t/parenrightbig\n(∆ǫk+q,k,2′,2)2/parenrightBigg\n×(n1′−n1)(n2′−n2)\n+2U2\nΩ4/summationdisplay\nk/parenleftbig\nnk+q(1−nk) + (1−nk+q)nk/parenrightbig\n×/summationdisplay\nk′\ni,ki∆ǫk,k1,k′\n2,k2\n∆ǫk+q,k′\n1,k′\n2,k2/parenleftBigg\n1−cos/parenleftbig\n(∆ǫk+q,k,k1,k′\n1)t/parenrightbig\n(∆ǫk+q,k′\n1,k′\n2,k2)2+ (∆ǫk,k1,k′\n2,k2)2−1−cos/parenleftbig\n(∆ǫk,k1,k′\n2,k2)t/parenrightbig\n(∆ǫk,k1,k′\n2,k2)2/parenrightBigg\n×nk1(nk′\n2−nk2)δk+k′\n2,k1+k2δk′\n1−k1,q\n+2U2\nΩ2/summationdisplay\nknk+q(1−nk)/integraldisplay\ndǫ1′/integraldisplay\ndǫ1/integraldisplay\ndǫ2′/integraldisplay\ndǫ2D(ǫ1′)D(ǫ1)D(ǫ2′)D(ǫ2)\n×(∆ǫ1′,1,2,2′) + (∆ǫk+q,k,2,2′)\n(∆ǫk+q,k,1,1′)/parenleftBigg\n1−cos/parenleftbig\n(∆ǫk+q,k,1,1′)t/parenrightbig\n(∆ǫ1′,1,2,2′)2+ (∆ǫk+q,k,2,2′)2/parenrightBigg\n×(nk′\n1−nk1)(nk′\n2−nk2)\n+2U2\nΩ4/summationdisplay\nk/parenleftbig\nnk+q(1−nk) + (1−nk+q)nk/parenrightbig\n×/summationdisplay\nk′\ni,ki(∆ǫk+q,k′\n1,k′\n2,k2) + (∆ǫk,k1,k′\n2,k2)\n(∆ǫk+q,k,k1,k′\n1)/parenleftBigg\n1−cos/parenleftbig\n(∆ǫk+q,k,k1,k′\n1)t/parenrightbig\n(∆ǫk+q,k′\n1,k′\n2,k2)2+ (∆ǫk,k1,k′\n2,k2)2/parenrightBigg\n×nk′\n1(nk′\n2−nk2)δk+k′\n2,k1+k2δk′\n1−k1,q\n31SciPost Physics Submission\n+U2\nΩ2/summationdisplay\nknk+q(1−nk)/integraldisplay\ndǫ1′/integraldisplay\ndǫ1/integraldisplay\ndǫ2′/integraldisplay\ndǫ2D(ǫ1′)D(ǫ1)D(ǫ2′)D(ǫ2)\n×1−cos/parenleftbig\n(∆ǫk+q,k,1′,1)t/parenrightbig\n−cos/parenleftbig\n(∆ǫk+q,k,2′,2)t/parenrightbig\n+ cos/parenleftbig\n(∆ǫ1′,1,2,2′)t/parenrightbig\n(∆ǫk+q,k,1′,1)(∆ǫk+q,k,2′,2)\n×(n1′−n1)(n2′−n2)\n−2U2\nΩ4/summationdisplay\nknk+q\n×/summationdisplay\nk′\ni,ki1−cos/parenleftbig\n(∆ǫk+q,k′\n1,k′\n2,k2)t/parenrightbig\n−cos/parenleftbig\n(∆ǫk,k1,k′\n2,k2)t/parenrightbig\n+ cos/parenleftbig\n(∆ǫk+q,k,k1,k′\n1)t/parenrightbig\n(∆ǫk+q,k′\n1,k′\n2,k2)(∆ǫk,k1,k′\n2,k2)\n×(1−nk1)nk′\n2(1−nk2)δk+k′\n2,k1+k2δk′\n1−k1,q\n+4U2\nΩ2/summationdisplay\nknk+q/integraldisplay\ndǫ1/integraldisplay\ndǫ2′/integraldisplay\ndǫ2D(ǫ1)D(ǫ2′)D(ǫ2)\n×1−cos/parenleftbig\n(∆ǫk,1,2′,2)t/parenrightbig\n(∆ǫk,1,2′,2)2(1−n1)n2′(1−n2)\n+O(U3), (71)\nwith the density of state D(ǫ). 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M¨ uller-Hartmann, Correlated fermions on a lattice in high dimensions , Zeitschrift f¨ ur\nPhysik B Condensed Matter 74, 507 (1989), doi:10.1007/BF01311397.\n35"    },    {        "title": "1909.07740v1.Majorana_representation_for_mixed_states.pdf",        "content": "Majorana representation for mixed states\nE. Serrano-Ens\u0013 astiga\u0003and D. Brauny\nInstitut f ur Theoretische Physik\nUniversit at T ubingen\n72076 T ubingen, Germany\nWe generalize the Majorana stellar representation of spin- spure states to mixed states, and in\ngeneral to any hermitian operator, de\fning a bijective correspondence between three spaces: the\nspin density-matrices, a projective space of homogeneous polynomials of four variables, and a set\nof equivalence classes of points (constellations) on spheres of di\u000berent radii. The representation\nbehaves well under rotations by construction, and also under partial traces where the reduced\ndensity matrices inherit their constellation classes from the original state \u001a. We express several\nconcepts and operations related to density matrices in terms of the corresponding polynomials, such\nas the anticoherence criterion and the tensor representation of spin- sstates described in [1].\nI. INTRODUCTION\nThe Majorana stellar representation [2] enlightens,\namong other properties, an image of any spin- sstate,\nand in consequence provides a glance of the (projective)\nHilbert spaceHsstructure of the pure states. The rep-\nresentation de\fnes a bijection between states j i2Hs\nand 2spoints ( stars ) on the sphere S2, called the con-\nstellation ofj i,C . The spin coherent (SC) states [3, 4],\nwhich are the most `classical' quantum states, have the\nsimplest constellations: all the stars point in the same\ndirection. In the opposite extreme, the most `quantum'\nstates are related to constellations spreading their stars\nover the unit sphere S2, where the `quantum' property\ncan be measured in several ways, e.g., the quantumness\n[5, 6], anticoherence and higher-order multipolar \ructu-\nations [7{10], and states with maximal Wehrl-Lieb en-\ntropy [11]. They have important applications in quan-\ntum metrology, as they contain the most sensitive states\nunder small rotations for a known or unknown rotation\naxis [12{15]. The Hilbert space Hsas a whole can be\nseen as a strati\fed manifold foliated by the SU(2) or-\nbits of all the possible con\fgurations of constellations\n[16]. In addition, the Majorana constellation has been\nuseful in other applications, such as the classi\fcation of\nspinor Bose-Einstein-condensate phases [17{19] or the in-\nvestigation of the thermodynamical limit in the Lipkin-\nMeshkov-Glick model [20, 21]. This representation also\nplays a role in the Atiyah mapping related to his conjec-\nture on `con\fgurations of points' [22]. Other characteri-\nzations of quantum systems via points on a manifold are\nalso commonly used. Examples include the use of zeroes\nof the Husimi function, or zeroes of Haldane's trial wave\nfunction for the fractional quantum Hall e\u000bect [23{25].\nMany representations and parametrizations have been\nfound for pure and/or mixed spin states which also be-\nhave well under rotations [26{31] or Lorentz transforma-\ntions [1]. Moreover, there are complete parametrizations\n\u0003eduardo.serrano-ensastiga@uni-tuebingen.de\nydaniel.braun@uni-tuebingen.deof quantum states for small values of spin- s[32, 33]. How-\never, for the case of mixed states, none of them share all\nthe properties of the standard Majorana representation\nfor pure states as: bijection with a projective space of\npolynomials [34], bijection with a set of points in the\nphysical space, and well{behavior under rotations. The\nMajorana representation for mixed states that we intro-\nduce in this paper has all these properties, with addi-\ntional properties associated with the partial trace and\nthe tensor product. While the bijection of mixed states\nwith polynomials is new and studied here, the bijection\nwith a set of points (constellations) in the physical space\nis described in a little known paper [35], and it uses the\ndecomposition of the density matrix in irreducible rep-\nresentations of the SU(2) group [36]. We call it accord-\ningly the Ramachandran-Ravishankar representation, or\nT-rep for short. The T-rep associates to any density\nmatrix a set of equivalence classes of constellations on\nspheres of di\u000berent radii [35]. The bijective correspon-\ndence between matrices and polynomials implies that the\nirreducible representations in both spaces are equal, and\nhence both of them end up with the same stellar rep-\nresentation as the T-rep. There is another representa-\ntion of mixed states close to the Majorana representation\ndescribed in this paper, given by the tensor product of\nPauli matrices projected in the fully symmetric sector.\nThis tensor representation is described in [1] and it has\nbeen helpful to study the problem of classicality of spin\nstates [37], to establish a relation between entanglement\nand the truncated moment problem [38], and to study\ngenuinely entangled symmetric states [39], among others\nworks [6, 40]. The latter representation will be denoted\nas theS-rep. The connection between the T- andS-\nrepresentations, not known until now, is presented here.\nThe paper is organized as follows: In Sec. II we present\nthe Majorana polynomial for density matrices, the nec-\nessary elements to build it and the translation of the\nphysical operations of interest to this representation. In\nSection III we explain the Ramachandran-Ravishankar\nT-rep, i.e., the bijection between the mixed states and a\nset of equivalence classes of points on the physical space.\nThe way we introduce this bijection is di\u000berent from thearXiv:1909.07740v1  [quant-ph]  17 Sep 20192\npioneering paper [35] but closer to our notation and def-\ninitions. In addition, we deduce the properties of the\nMajorana representation of mixed states with respect to\npartial traces, the polynomial expression of the antico-\nherence criterion, and the connection of the S- andT-\nreps. The relation between the Majorana polynomial for\nmixed states and the Husimi and P- quasiprobability dis-\ntributions is explained in Sec. IV. We end the paper with\nsome \fnal comments in Sec. V.\nII. MAJORANA POLYNOMIAL FOR MIXED\nSTATES\nA. The standard Majorana representation\nThe Majorana stellar representation for pure spin- s\nstates [2, 41] associates one-by-one each point of the\nHilbert spacej i2HswithN= 2spoints on the sphere\nS2that contains the full information of the state since\nthe real dimension of the projective Hilbert space, af-\nter taking out the normalization and global phase factor\nof the state, is dim( Hs) = 2N. E. Majorana [2] de-\n\fned this representation via a polynomial constructed\nwith the expansion of the state j iin theSz-eigenbasis,\nj i=Ps\nm=\u0000s\u0015mjs;mi\np (Z) =sX\nm=\u0000s(\u00001)s\u0000ms\u00122s\ns\u0000m\u0013\n\u0015mZs+m:(1)\nThe complex roots of p (Z) specify uniquely the polyno-\nmial and hence the state j iup to an irrelevant global\ncomplex factor. The polynomial p (Z) has degree at\nmostN= 2s, and by a rule which be clari\fed later, its\nset of rootsf\u0010kgkis always increased to 2 sby adding the\nsu\u000ecient number of roots at the in\fnity. The constella-\ntionC ofj iis the set of points on S2called stars ob-\ntained with the stereographic projection from the South\nPole of the roots f\u0010kgN\nk=1, where the complex plane is\nsituated in the xy-plane and the x- andy- axes are the\nreal and imaginary axes, respectively. The stereographic\nprojection maps the complex number \u0010= tan(\u0012=2)ei\u001eto\nthe pointnon the sphere S2with polar and azimuthal\nangles (\u0012; \u001e).\nIn order to generalize this polynomial to density ma-\ntrices, we work with a similar representation de\fned by\nH. Bacry [42] that associates to each state j ia homoge-\nneous polynomial of two variables, that it can be written\nasp (z1;z2)\u0011h\u0000nBj i, where\nh\u0000nBj\u0011sX\nm=\u0000s(\u00001)s\u0000ms\u00122s\ns\u0000m\u0013\nzs+m\n1zs\u0000m\n2hs; mj:\n(2)\nFollowing the habit in quantum optics, we call jnBithe\nBargmann Spin Coherent (BSC) state, which is propor-\ntional to the Spin Coherent (SC) state pointing in the\ndirectionnassociated with the complex number z1=z2via the stereographic projection. The latter polynomial,\nwhich we call also the Majorana polynomial of j ifor\nsimplicity, has the expression given by\np (z1; z2) =sX\nm=\u0000s(\u00001)s\u0000ms\u00122s\ns\u0000m\u0013\n\u0015mzs+m\n1zs\u0000m\n2:\n(3)\nIn principle, one could work with the zeroes of the new\npolynomial (3) and then one would associate to any state\nj ian algebraic variety in C2. But this is more informa-\ntion than we need to specify a state and it is not easy to\nvisualize. To avoid these complications, we use the fact\nthat the polynomial (3) is homogeneous and hence the\npolynomial is fully factorizable\np (z1; z2) =NY\nk=1(z1\u000bk\u0000z2\fk); (4)\nwhich implies that the polynomial is characterized by N\nrays on C2f(z1; z2)2C2jz1\u000bk\u0000z2\fk= 0gN\nk=1, or equiv-\nalently, by Npointsf\u0010k=\fk=\u000bkgN\nk=1in the projective\ncomplex space P(C2) =CP1, de\fned by the set of (com-\nplex) rays in C2and isomorphic to the extended complex\nplane2C[f1g . The setf\u0010kgN\nk=1obtained here is equal\nto the set of roots de\fned by (1) and hence the same con-\nstellation is obtained using the stereographic projection\nexplained above. On the other hand, any spin- sstate is\na fully symmetric state of Nconstituent spin-1 =2 states\njnki\nj i/X\n\u00192SN\u0019(jn1i\n\u0001\u0001\u0001\njnNi); (5)\nwhere the summation is over all the elements of the\npermutation group of NelementsSNand the spin-1/2\nstates are labeled by its respective Bloch vector nk.\nThe de\fnition of the Majorana polynomial implies that\nthe stars ofC are equal to the directions of the con-\nstituents spin-1 =2 ofj i. In particular, the complex num-\nber\u0010k= tan(\u0012=2)ei\u001ewith stereographic projection nkof\nangles (\u0012k; \u001ek) is associated to the constituent of j i,\njnki=\u000bkj1=2;1=2i+\fkj1=2;\u00001=2iwith\fk=\u000bk=\u0010k\nandj\u000bkj2+j\fkj2= 1. To summarize, the Majorana stel-\nlar representation de\fnes bijective mappings among the\nHilbert spaceHs, the projective space of homogeneous\nbivariate polynomials of degree Nand the set of constel-\nlations onS2withNstars.\nA transformation j 0i=U(R)j iinHswhere the uni-\ntary transformation U(R)\u0011exp(\u0000ie\u0001S\u0011=~) represents\na rotation R2SO(3) with rotation angle \u0011about thee-\naxis of unit norm and angles (\u0002 ;\b), andS= (Sx;Sy;Sz)\nis the vector of angular momentum operators, rigidly ro-\ntates the corresponding constellation C \u001aS2with the\nsame rotation R. The roots of the Majorana polynomial\nofj 0i2Hsare\u00100\nk=M(\u0010k) fork= 1; :::; N [16] and\nM(\u0010) =a\u0010\u0000b\nb\u0003\u0010+a\u0003(6)3\nis the M obius transformation associated to the rota-\ntionRwitha= cos(\u0011=2)\u0000isin(\u0011=2) cos \u0002 and b=\n\u0000isin(\u0011=2) sin \u0002ei\b([43], p.27). The complex numbers\n(a;b) withjaj2+jbj2= 1 are called the Cayley-Klein pa-\nrameters of a rotation R. In polynomials, p 0(z1;z2) =\np (z0\n1;z0\n2) where the new variables are\n\u0012\nz0\n1\nz0\n2\u0013\n=\u0012\na\u0003b\n\u0000b\u0003a\u0013\u0012\nz1\nz2\u0013\n; (7)\nand the matrix\n\u0012\na\u0003b\n\u0000b\u0003a\u0013\n2SU(2) (8)\nis the projective matrix representation of the rotation\nR\u00001and hence the matrix associated to the inverse of\nthe M obius transformation (6). The covariant trans-\nformation of the constellations implies that the point-\ngroup symmetry of C is the point-group symmetry of\nj iunder the respective unitary transformations repre-\nsenting the symmetry operations. A similar statement\nholds true for Lorentz symmetries, where invariants other\nthan the shape of the constellation become relevant, see\n[16, 42, 44]. In this case, a Lorentz transformation is\nassociated with a generic M obius transformation\nM(\u0010) =a\u0010+b\nc\u0010+d; withad\u0000bc= 1: (9)\nThe derivative of the Majorana polynomial\nA not well-known result about the Majorana polyno-\nmial is about the physical meaning of its derivative. Here\nwe explain it brie\ry following Sec. 2.6 of [45]. The state\nj ide\fned in Eq. (5) becomes, after the contraction of\nthe \frst of its constituents with, let us say, the spin-1/2\nstate pointing in the zdirectionjzi=j1=2;1=2i, a state\nj ziof spins0withs0=s\u00001=2 and proportional to\nj zi/NX\nk=1\u000bkX\n\u00192SN\u00001\u0019\u0010\njn1i\n\u0001\u0001\u0001\ndjnki\n\u0001\u0001\u0001\njnNi\u0011\n;\n(10)\nwhere the hat means exclusion in the expression. On the\nother hand, the derivative with respect to z1ofp (z1; z2)\ngiven by (4) is equal to\n@z1p (z1; z2) =NX\nk=1\u000bkY\nj6=k(\u000bjz1\u0000\fjz2)/p z(z1; z2);\n(11)\ni.e., the partial derivative @z1p (z1; z2) of the Majo-\nrana polynomial of j iis, up to an irrelevant global\nfactor, the polynomial p z(z1; z2) of the spin- s0state\nj zi. This result can be generalized in each direction (not\nonly alongz): for a direction mwith angles ( \u0012;\u001e) andits respective spin-1/2 state jmi= cos(\u0012=2)j1=2;1=2i+\nsin(\u0012=2)ei\u001ej1=2;\u00001=2i,\np m(z1; z2) =\u0000\ncos(\u0012=2)@z1\u0000sin(\u0012=2)e\u0000i\u001e@z2\u0001\np (z1; z2):\n(12)\nIn particular p \u0000z(z1; z2) =@z2p (z1; z2) where the\nglobal phase factor is not relevant for the roots of the\nresulting polynomial and hence for the respective \fnal\nstate.\nB. Majorana polynomial for a density matrix and\nits partial traces\nWe reviewed how to associate a bivariate homogeneous\npolynomial of degree Nto a spin-spure statej i, and\nhow the contraction of one of its constituent spin-1 =2 is\nassociated with the derivative of its Majorana polyno-\nmial. Now, we want to generalize this result to spin- s\noperators inB(Hs), in particular to mixed states. To\nachieve this goal, we apply the BSC states (2) to a gen-\neral density matrix \u001afrom the left and right to obtain\np\u001a(za;za) =h\u0000nBj\u001aj\u0000nBi; (13)\nwithza\u0011z\u0003\nathe conjugated complex variables of zafor\na= 1;2. While kets transform covariantly under ro-\ntations via their respective irreps U(R), bras transform\ncontravariantly [46], as well as the BSC states variables,\n\u0012\nz1\nz2\u0013\n!\u0012\na\u0003b\n\u0000b\u0003a\u0013\u0012\nz1\nz2\u0013\n;\n\u0000\nz1z2\u0001\n!\u0000\nz1z2\u0001\u0012\na\u0000b\nb\u0003a\u0003\u0013\n: (14)\nWe consider the set of variables ( z)\u0011(za;za) fora= 1;2\nindependent, i.e.,@azb=@azb= 0 and@azb=@azb=\n\u000eabwhere@a=@zaand@a=@za, and partial derivatives\ntransform as the inverse of their variables. Let us men-\ntion that the Majorana polynomial (13) can be applied to\na general operator C, and in this way we have de\fned a\nmapping between B(Hs) and homogeneous polynomials\npC(z) of degree 2 Nwhere each monomial z\u000b\n1z\f\n2(z1)\r(z2)\u000e\nofpC(z) satis\fes\u000b+\f=\r+\u000e=N. The last property\nimplies that\nza@apC(z) =NpC(z); za@apC(z) =NpC(z);(15)\nfor any operator Cand where from here and forth we\nuse the Einstein sum convention for repeated indices. We\ndenote the vector space of polynomials of four variables\n(z1; z2; z1; z2) asP(N;N)(z) andpC(z) is called the Ma-\njorana polynomial of C. The mapping between B(Hs)\nandP(N;N)(z) is bijective. The space P(N;N)(z) has been\nused before in [47] to calculate the Clebsch-Gordan coef-\n\fcients in terms of the Hahn polynomials.\nThe Majorana polynomial for states \u001apresented here\nis related to the standard Majorana polynomial in the4\ncase of pure states. For instance, the polynomial of \u001a =\nj ih jis equal to\np\u001a (z) =p (za) (p (za))\u0003\u0011p (za)\u0016p (za); (16)\nwhere \u0016p (za) denotes that we only conjugate the coe\u000e-\ncients of the polynomial. Let us give an example of the\nMajorana polynomial for spin-1 =2. The density matrix\n\u001a=jnihnjhas Majorana polynomial\np\u001a(z) = (\u000bz1\u0000\fz2)(\u000b\u0003z1\u0000\f\u0003z2); (17)\nwith\u000b= cos(\u0012=2) and\f= sin(\u0012=2)ei\u001e. In particular,\np\u001a(z) =z1z1andp\u001a(z) =z2z2forn=\u0006z, respectively.\nAs we mentioned before, we can associate a polynomial\nto any operator. For instance, the Pauli matrices \u001b\u0016for\n\u0016=x;y;z and the ladder operators \u001b\u0006=\u001bx\u0006i\u001byhave\npolynomials\np0(z) =zaza; p x(z) =\u0000z1z2\u0000z2z1;\npy(z) =i\u0000\nz1z2\u0000z2z1\u0001\n; pz(z) =z1z1\u0000z2z2;\np+(z) =\u00002z1z2; p\u0000(z) =\u00002z2z1;(18)\nwithp\u0016(z)\u0011p\u001b\u0016(z) for\u0016= 0; x; y; z; +;\u0000and\u001b0=12\nis the 2\u00022 identity matrix. The polynomial of the ad-\njoint of an operator A,pAy(z), is obtained interchanging\nza$zaand conjugating the coe\u000ecients in pA(z). The\npolynomial of an Hermitian operator is invariant under\nthis transformation. We can observe these properties in\nthe Pauli matrices and ladder operators.\nAccording to the discussion of the previous subsection,\nthe reduced density matrix \u001as0withs0=s\u00001=2 ob-\ntained by tracing the spin- sstate\u001aover a constituent\nspin-1=2,\u001as0= Tr 1(\u001a), can be written as the applica-\ntion of the di\u000berential operator ( @a@a) to the Majorana\npolynomial of \u001a. We de\fne the partial trace operator\nL:P(N;N)(z)!P(N\u00001;N\u00001)(z) as\nL(p(z))\u0011N\u00002@a@ap(z); (19)\nwhere, as we will see in Theorem 1, the factor N\u00002guar-\nantees that the trace of the operators is preserved. The\noperatorLis invariant under rotations due to the trans-\nformation laws of the partial derivatives. The application\nof the operator L2(s\u0000k)-times top\u001a(z) yields the asso-\nciated polynomial of the spin- kreduced density matrix\n\u001ak= Tr 2(s\u0000k)(\u001a),\np\u001ak(z) =L2(s\u0000k)(p\u001a(z)): (20)\nC. Operations in B(Hs) in terms of polynomials\nWe are interested in making calculations in terms of\npolynomials, and here we brie\ry deduce the most com-\nmon operations in B(Hs). Let us start with the trace of\nan operator C, given by the action of the partial trace\noperator applied Ntimes\npTr(C)(z) =LN(pC(z)) = (N!)\u00002(@a@a)NpC(z):(21)In particular, the identity matrix polynomial p1(z) =\n(zaza)Nsatis\fespTr(1)(z) =N+ 1.\nAnother basic operation in B(Hs) is the calculation of\nan operator Cgiven by a product of operators C=DE.\nHow to calculate it in terms of polynomials is the result\nof the following\nLemma 1 LetC; D; E2 B(Hs)such thatC=DE\nand with Majorana polynomials pC(z),pD(z),pE(z)2\nP(N;N)(z), respectively. Then\npC(z) = (N!)\u00001pD(za;@a)pE(za;za);\npC(z) = (N!)\u00001pE(@a;za)pD(za;za); (22)\nwhere the order of the variables in each monomial of\npD(za;@a)andpE(@a;za)is such that the partial deriva-\ntives go to the right of the monomial, to a\u000bect only the\npolynomial on the right.\nThe result of lemma 1 can be applied iteratively for a\nproduct of several operators. In particular, an operator\ngiven byC=DEF can be written in terms of di\u000berential\noperators acting on the polynomial pE(za;za),\npC(z) = (N!)\u00002pF(@a;za)pD(za;@a)pE(za;za):(23)\nFor instance, the X=\u001bxchannelX\u001aX has an output\npolynomial equal to\npX\u001aX(z) =\u0000\nz1@2+z2@1\u0001\n(z1@2+z2@1)p\u001a(z):(24)\nThe combination of the trace and the product of opera-\ntors has a simpli\fed expression:\nLemma 2 LetC; D2B(Hs)with Majorana polynomi-\nalspC(z),pD(z)2P(N;N)(z). Then\nTr(CD) = (N!)\u00002pC(@a;@a)pD(za; za): (25)\nIn particular, if the operators are such that Tr (CD) = 0 ,\nhencepC(@a;@a)pD(z) = 0 .\nThe proofs of the previous lemmas can be found in Ap-\npendix A. We end this section writing the expectation\nvalue of an operator Cin a pure statej iwith constella-\ntionC . Using the polynomial of a pure state and Lemma\n2, we obtain that\nh jCj i=(N!)\u00002p (@a)\u0016p (@a)pC(z)\n=(N!)\u00002@nN:::@n1@nN:::@ n1pC(z);(26)\nwherefnkgkis the set of stars of C with angles ( \u0012k; \u001ek)\nand\n@nk= cos(\u0012k=2)@z1\u0000sin(\u0012k=2)ei\u001ek@z2;\n@nk= cos(\u0012k=2)@z1\u0000sin(\u0012k=2)e\u0000i\u001ek@z2: (27)\nThe positive semide\fnite condition of a state \u001acan be\nwritten as the condition that (26) is non-negative for\nanyNpointsfnkg. As an extra result, we obtain\nthat the only polynomials p(z)2P(N;N)(z) such that\np(za;@a)p(za;za) =N!p(za;za) are the polynomials as-\nsociated with a pure state p(z) =p (za)\u0016p (za),i.e.,\npolynomials that are factorizable with respect to the vari-\nables (za) and (za).5\nIII. CONSTELLATIONS FOR MIXED STATES\nThe Majorana representation for pure states allows us\nto visualize any state j ivia the stereographic projec-\ntion of the roots of the polynomial p (z). For the case of\nmixed states \u001a, the equation p\u001a(za;za) = 0 de\fnes an al-\ngebraic variety on C4, orC2taking into account that za=\nz\u0003\na. The algebraic variety is not, in general, the product\nof a set of rays, and hence its projection in the extended\ncomplex plane is not necessarily a set of \fnite points.\nA extreme case is given by the maximally mixed state\n\u001a\u0003= (2s+ 1)\u000011, withp\u001a\u0003(z) = (2s+ 1)\u00001(zaza)2sand\nhence the equation to ful\fll is p\u001a\u0003(z)/\u0000\nj\u0010j2+ 1\u00012s= 0\nwith\u0010=z1=z2. Instead of working with the zeroes of the\nfull Majorana polynomial p\u001a(z), and in order to represent\na state with a \fnite set of points, we work with the irre-\nducible representations (irrep) of SU(2) inP(N;N)(z), or\nequivalently, inB(Hs). TheSU(2)-irreps ofB(Hs) are\nspanned by the well-known (multipolar) tensor operators\nfT(s)\n\u001b\u0016j0\u0014\u001b\u00142s;j\u0016j\u0014\u001bg, and their use to associate to\nany mixed state a set of points in the physical space was\ndiscovered by Ramachandran and Ravishankar [35], lead-\ning to what we called the Ramachandran-Ravishankar\nrepresentation or T-rep for short. In order to make the\npaper self-contained and for a better understanding of\nthe next sections, we explain the T-rep in terms of Ma-\njorana constellations. The T-rep has been used recently\nin Quantum Information [31, 48].\nA.T-representation\nA tensor operator T(s)\n\u001b\u0016:Hs!Hs[43, 46, 49] of rank\n\u001bis an element of a set of linear operators fT(s)\n\u001b\u0016g\u001b\n\u0016=\u0000\u001b\nthat transforms under a unitary transformation U(R)\nrepresenting a rotation R2SO(3) according to an ir-\nrepD(\u001b)(R) ofSO(3) (or equivalent, of SU(2)),\nU(R)T(s)\n\u001b\u0016U\u00001(R) =\u001bX\n\u00160=\u0000\u001bD(\u001b)\n\u00160\u0016(R)T(s)\n\u001b\u00160; (28)\nwhereD(\u001b)\n\u00160\u0016(R)\u0011h\u001b; \u00160je\u0000i\u000bSze\u0000i\fSye\u0000i\rSzj\u001b; \u0016iis the\nWigner D-matrix [43] of a rotation Rwith Euler angles\n(\u000b; \f; \r ), and\u001b= 0;1;2;:::labels the irrep. The explicit\nexpression of T(s)\n\u001b\u0016can be given in terms of the Clebsch-\nGordan coe\u000ecients Cjm\nj1m1j2m2,\nT(s)\n\u001b\u0016=sX\nm;m0=\u0000s(\u00001)s\u0000m0C\u001b\u0016\nsm;s\u0000m0js;mihs;m0j:(29)\nFrom now on, we omit the super index ( s) when there is\nno possible confusion. It is easy to deduce that 0 \u0014\u001b\u0014\n2s,j\u0016j\u0014\u001band the following properties:\nTr(Ty\n\u001b1\u00161T\u001b2\u00162) =\u000e\u001b1\u001b2\u000e\u00161\u00162; Ty\n\u001b\u0016= (\u00001)\u0016T\u001b\u0000\u0016:\n(30)The setfT\u001b\u0016: 0\u0014\u001b\u00142s;\u0000\u001b\u0014\u0016\u0014\u001bgforms hence\nan orthonormal basis over the complex numbers for the\ncomplex square matrices of order N+1 satisfying (28). In\nother words, the set of T\u001b\u0016is the matrix analogue of the\nspherical harmonic functions Ylm(\u0012; \u001e), which span the\nspace of real-valued functions on the sphere f(\u0012;\u001e). A\ndensity matrix \u001a2B(Hs) has then a block decomposition\nin theT\u001b\u0016basis\n\u001a=2sX\n\u001b=0\u001a\u001b\u0001T\u001b; (31)\nwhere\u001a\u001b= (\u001a\u001b\u001b;:::;\u001a\u001b\u0000\u001b)2C2\u001b+1with\u001a\u001b\u0016=\nTr(\u001aTy\n\u001b\u0016),T\u001b= (T\u001b\u001b;:::;T\u001b;\u0000\u001b) is a vector of matrices,\nand the dot product is short forP\u001b\n\u0016=\u0000\u001b\u001a\u001b\u0016T\u001b\u0016. Each\nvector\u001a\u001bcan be associated to a constellation \u0012 a la Ma-\njorana (3) consisting of 2 \u001bpoints onS2obtained with\nthe stereographic projection of the complex roots of the\npolynomial p(\u001b)\n\u001a(z1=\u0010;z2= 1) de\fned as\np(\u001b)\n\u001a(\u0010) =\u001bX\n\u0016=\u0000\u001b(\u00001)\u001b\u0000\u0016s\u00122\u001b\n\u001b\u0000\u0016\u0013\n\u001a\u001b\u0016\u0010\u001b+\u0016:(32)\nThe respective constellation is denoted as C(\u001b)\n\u001aorC(\u001b)\nwhen there is no possible confusion. The vector \u001a0=\n(\u001a00) does not have an associated constellation and its\nvalue is \fxed to \u001a00= (2s+ 1)\u00001=2by Tr\u001a= 1. On the\nother hand, the hermiticity condition implies that\n\u001a\u001b\u0016= (\u00001)\u0016\u001a\u0003\n\u001b\u0000\u0016;for allj\u0016j\u0014\u001b; (33)\nwhich in turn implies that every constellation C(\u001b)has\nantipodal symmetry. For a proof it is enough to show\nthat if\u0010=\u0018is a root of p(\u001b)\n\u001a(\u0010), the corresponding an-\ntipodal complex number \u0018A\u0011\u00001=\u0018\u0003is also a root:\np(\u001b)\n\u001a(\u0018) =X\n\u0016(\u00001)\u001b\u0000\u0016s\u00122\u001b\n\u001b\u0000\u0016\u0013\n\u001a\u001b\u0016\u0018\u001b+\u0016\n=\u00182\u001b X\n\u0016(\u00001)\u001b\u00002\u0016s\u00122\u001b\n\u001b+\u0016\u0013\n\u001a\u001b\u0016\u0018\u0003\u0000\u001b\u0000\u0016!\u0003\n=(\u00001)\u001b\u00182\u001b\u0010\np(\u001b)\n\u001a(\u0018A)\u0011\u0003\n; (34)\nwhere in the second equality we use (33). Hence, the\nproof is done for any root \u00186= 0 but the statement also\nholds in the case of \u0018= 0 and its corresponding antipodal\npoint\u0018=1: Let us suppose that the constant term in\np(\u001b)\n\u001a(\u0010) is zero, and hence there is a root \u0018= 0. The her-\nmiticity property (33) implies that the coe\u000ecient of the\nhighest exponent \u00102\u001bis also zero, implying that p(\u001b)\n\u001a(\u0010)\nhas an extra root at in\fnity. Hence, the roots at zero\nand in\fnity come also in pairs.\nThe standard Majorana representation associates to\neach pure spin- sstate a unique polynomial up to a global6\nfactor, which does not change its roots and is of no con-\ncern as the state can always be assumed normalized and\nthe global phase is irrelevant. But now a pre-factor of a\npolynomial p(\u001b)\n\u001a(\u0010) is a relative factor that carries impor-\ntant information about the relative weights and phases\nof di\u000berent irreps contained in the state. Hence the set of\nconstellations of a state is not su\u000ecient yet to specify the\nstate uniquely: Two states \u001aand\u001a0with the same con-\nstellationC(\u001b)\n\u001a=C(\u001b)\n\u001a0have the same vector \u001a\u001bonly up to\narbitrary complex weights w\u001bei\u001e\u001b(in polar coordinates)\nthat need to be given in addition to the constellations\nin order to fully specify the state. In order to do so, we\nspecify for each constellation C(\u001b)the absolute value of\nthe weight with respect to a vector ~\u001a\u001bwith unit norm,\n\u001a\u001b=w\u001b~\u001a\u001b. The state \u001acan then be written as\n\u001a=1\n2s+ 1+2sX\n\u001b=1w\u001b~\u001a\u001b\u0001T\u001b; (35)\nwith\nw\u001b= \u001bX\n\u0016=\u0000\u001b\u001a\u001b\u0016\u001a\u0003\n\u001b\u0016!1=2\n; (36)\nand in particular w0=\u001a00= (2s+ 1)\u00001=2. For the\nphase factor ei\u001e\u001b, one could de\fne a \\gauge\" for each \u001b-\nblock, i.e., for each constellation C(\u001b)one could specify a\nparticular normalized vector ~\u001ag\n\u001bthat works as a reference\nto the phase factor ~\u001a\u001b=ei\u001e\u001b~\u001ag\n\u001b. A disadvantage of\n\fxing the gauge in this way is that under rotations, the\nphase factor can have non-trivial transformation laws.\nIn fact, we know that a generic spin state may pick up\nan extra global phase after it traces a closed trajectory\nin the quantum states space by a sequence of rotations,\nwhich is the so-called geometric phase [50]. The best\nway to handle the phase factor is the following: First,\nlet us remark that two normalized (2 \u001b+ 1)\u0000vectors ~\u001a\u001b\nand ~\u001a0\n\u001bthat represent the \u001b-block of a physical state\nwith equal constellation C(\u001b), can di\u000ber only by a phase\nfactor ~\u001a\u001b=ei\u001e~\u001a0\n\u001bwithei\u001e=\u00061, otherwise one of the\nvectors does not satisfy the hermiticity condition (33).\nOn the other hand, again by the hermiticity condition,\nthe constellation C(\u001b)de\fned by the (2 \u001b+ 1)\u0000vector ~\u001a\u001b\nhas antipodal symmetry. This implies that there exists\n\u001bstarsc\u0011(n1; :::;n\u001b) inC(\u001b)such that\nfcg[f\u0000 cg=C(\u001b); (37)\nwith\u0000c\u0011(\u0000n1; :::;\u0000n\u001b) and where cis a tuple and\nfcgits respective unordered set, and the same for \u0000c\nandf\u0000cg. In general, the tuple cthat satis\fes (37) is\nnot unique. The other choices can be written with re-\nspect to cinverting the direction of some of its stars\n\rc\u0011(\r1n1; :::; \r\u001bn\u001b) with\rk= 1 or\u00001. For sim-\nplicity, we refer to the unordered set f\rcgwith the same\nsymbol as the tuple \rc, and we call it a subconstellation\nofC(\u001b). Now, we can de\fne a spin- \u001bstate for each sub-\nconstellation \rcgiven by the projected bipartite stateP\u001bj\u001e; \u001eAi\u0011P\u001b(j\u001ei\nj\u001eAi), withP\u001bthe projection op-\nerator in the fully symmetric subspace of spin- \u001bstates,\nj\u001eithe spin-\u001b=2 state with Majorana constellation \rc,\nandj\u001eAi\u0011Aj\u001ei, whereAis the time-reversal operator\nde\fned by\nAj\u001ei\u0011X\nm(\u00001)s+m\u0015\u0003\n\u0000mjs; mi;forj\u001ei=X\nm\u0015mjs; mi:\n(38)\nWe also call Atheantipodal operator because the con-\nstellation ofj\u001eAiis\u0000\rc. The projector operator P\u001bis\na function with domain the states space of 2 \u001bspins-1=2\n(H1=2)2\u001band image the set spanned by the symmetric\nDicke statesjD(m)\n2\u001bi\njD(m)\n2\u001bi=KX\n\u0019\u0019(jzi\n\u0001\u0001\u0001\njzi|{z}\n2\u001b\u0000m\nj\u0000zi\n\u0001\u0001\u0001\nj\u0000zi|{z}\nm);\n(39)\nwithK=\u00002\u001b\n\u001b\u0000m\u0001\u00001=2and where the sum runs over the\npermutations of 2 \u001bobjects of two types, with 2 \u001b\u0000mof\nthe \frst type and mof the second one. The symmetric\nDicke states coincide with the Sz-eigenbasisj\u001b;\u0016i. From\nnow on, we consider the projector operators restricted to\nits imageP\u001b: (H1=2)2\u001b!H\u001b, and the direction of its\naction, left or right, is implicitly given in the equation.\nThe expansion of the state P\u001bj\u001e; \u001eAiin theSz-eigenbasis\nj\u001b;\u0016iconstitutes a (2 \u001b+1)\u0000vector that satis\fes the her-\nmiticity condition (33) and produces the same constella-\ntion of ~\u001a\u001b,C(\u001b),\nP\u001bj\u001e; \u001eAi/j\u0006n1; :::;\u0006n\u001bi; (40)\nwithj\u0006n1; :::;\u0006n\u001bithe spin-\u001bstate with constellation\nC(\u001b). Moreover, if one changes j\u001eiby a phase factor\nei\u000ej\u001ei, the coe\u000ecients of (40) are invariant. On the other\nhand, if one turns the direction \rk!\u0000\rkof a star of\n\rc, the state (40) remains equal times a global factor \u00001\nbecauseP1\u0000\njni\njnAi\u0001\n=\u0000P1\u0000\njnAi\nj(nA)Ai\u0001\nand\nthe statesj\u001eiandj\u001eAiare fully symmetric \u001b=2-states.\nThe last result suggests us to split the subconstellations\nc\u001a C(\u001b)\n\u001asatisfying (37) into two equivalence classes,\nwhere two subconstellations are equivalent if they di\u000ber\nby an even number of stars. Both equivalence classes can\nbe de\fned with respect to a particular subconstellation\nc=fnkgk\n(\n\rc\u001aC(\u001b)\f\f\f\rk= 1 or\u00001 and\u001bY\nk=1\rk= +1)\n;\n(\n\rc\u001aC(\u001b)\f\f\f\rk= 1 or\u00001 and\u001bY\nk=1\rk=\u00001)\n:(41)\nAny element of any class produces the same state (40),\nbut only elements of the same class produce the same\n(2\u001b+ 1)-vector, i.e., the same state and the same phase\nfactor of the (2 \u001b+ 1)-vector. In particular, for a state \u001a\nand for each \u001b= 1;:::; 2s, the vector ~\u001a\u001bbelongs to one7\nof these classes, with the respective constellations of \u001a.\nWe denote the belonging subconstellation class of~\u001a\u001bof\nthe state\u001aby [c(\u001b)\n\u001a], with ca representative element of\nthe class. The components of ~\u001a\u001bcan be written as\n~\u001a\u001b\u0016=N\u001eh\u001b;\u0016jP\u001bj\u001e;\u001eAi; (42)\nwherej\u001ei=jn1;:::;n\u001biis a state with constellation\nlying in the class [ c(\u001b)\n\u001a], andN\u001ea positive factor that\nguarantees ~\u001a\u001bis a normalized vector, namely\nN\u001e=jh\u0006n1; :::;\u0006n\u001bjn1\n\u0000n1\n\u0001\u0001\u0001\nn\u001b\n\u0000n\u001bij\njhn1;::: ;n\u001bjn1\n\u0001\u0001\u0001\nn\u001bij2;\n(43)\nwithjn1\n\u0001\u0001\u0001\nn\u001bi=jn1i\n\u0001\u0001\u0001\njn\u001bi. The scalar\nproduct (42) is given by\nh\u001b;\u0016jP\u001bj\u001e;\u001eAi=h\u001ejTy(\u001b=2)\n\u001b\u0016j\u001ei: (44)\nWe conclude that any density matrix \u001ais uniquely\nspeci\fed through 2 ssubconstellation classes [ c(\u001b)] and\n2snon-negative real numbers w\u001bconsidered as the radii\nof the spheres where each subconstellation class lies,\nrespectively. The (continuous) degrees of freedom that\nparametrize the subconstellation classes [ c(\u001b)] are 2\u001b,\nand therefore the number of free continuous parameters\ninfw\u001b;[c(\u001b)]g2s\n\u001b=1is 4s2+ 4s, the same as the number of\nreal degrees of freedom of the mixed states \u001a2B(Hs).\nThe correspondence also works for any Hermitian op-\neratorH, where in this case the component Tr( HT00) is\nnot \fxed. In addition, the correspondence between phys-\nical states \u001aand the setfw\u001b;[c(\u001b)]gis bijective. The\nparameters domain is restricted by the positive semidef-\ninite conditionh j\u001aj i\u00150 for allj i2Hs, which with\nthe unit trace condition Tr \u001a= 1 implies that all eigen-\nvalues of\u001aare in [0,1]. This condition is considerably\nmore complicated to impose compared to hermiticity and\nunit trace. One necessary condition for positivity is that\nTr\u001a2\u00141, which gives an inequality independent of the\nsubconstellation classes,\n2sX\n\u001b=1w2\n\u001b\u00142s\n2s+ 1: (45)\nHowever, the positivity condition leads in general to a\ndependence of the allowed range of the radii w\u001bon the\nclasses [ c(\u001b)].\nAs an example, let us consider the s= 1=2 case. Any\nvectorr= (rx; ry; rz) with norm r\u00141 is associated\nwith a density matrix \u001a\n\u001a=1\n2(1+r\u0001\u001b); (46)\nwhere\u001b= (\u001bx;\u001by;\u001bz) are the Pauli matrices and ris\ncalled the Bloch vector of \u001a. For a general spin values>0, the necessary tensor operators with \u001b= 1 are [51]\nT(s)\n10=\u00123\ns(s+ 1)(2s+ 1)\u00131=2\nSz; (47)\nT(s)\n1;\u00061=\u0007\u00123\n2s(s+ 1)(2s+ 1)\u00131=2\nS\u0006: (48)\nThe state\u001awritten in the T-rep has a unique vector \u001a(1)\nequal to\n\u001a(1)=1\n2\u0010\n\u0000rx+iry;p\n2rz;rx+iry\u0011\n: (49)\nThe radius w1is obtained after normalization of the\nvector (49), yielding that w1=r=p\n2. The condition\n(45) imposes thatp\n2w1=r\u00141, while the constel-\nlationC(1)is speci\fed with the roots f\u0010gof the Ma-\njorana polynomial associated to the vector \u001a(1). It is\nobtained that \u00101= tan(\u0012=2)ei\u001eand\u00102=\u0010A\n1the an-\ntipodal complex number of \u00101, with (\u0012;\u001e) the spheri-\ncal angles of the vector r. Therefore, the stars of C(1)\npoint in the parallel and anti-parallel directions of the\nBloch vector\u0006r. Lastly, the subconstellation classes\n[c(1)] are [r] and [\u0000r], where each class has a unique\nelement. We deduce the class to which the state (49)\nbelongs by comparing with the coe\u000ecients of the state\nP1\u0000\njri\njrAi\u0001\n. For instance, with the parametrization\njri= cos(\u0012=2)j1=2;1=2i+sin(\u0012=2)ei\u001ej1=2;\u00001=2i, its co-\ne\u000ecients in thejs= 1;mibasis are\n1\n2\u0010\n\u0000sin(\u0012)e\u0000i\u001e;p\n2 cos\u0012;sin(\u0012)ei\u001e\u0011\n; (50)\nwhich are the same coe\u000ecients as in (49) for \u001a1in spher-\nical coordinates and hence its class is [ r]. Conversely,\ngiven a particular set fw1;[c(1)]g, we can obtain the re-\nspective density matrix, i.e., the Bloch vector r. We\nremark that states \u001amay di\u000ber only by some subconstel-\nlation classes [ c(\u001b)\n\u001a], as in our s= 1=2 example where the\nstates with Bloch vector \u0006rhave the same constellation\nC(1)but di\u000berent class [ \u0006r]. We can generalize the re-\nlation between antipodal states \u001aand\u001aA=A\u001aAyusing\nthe fact that Ais anti-unitary and A2j i= (\u00001)2sj i,\n\u001aA=A\u001aAy=2sX\n\u001b=0(\u00001)\u001b\u001a\u001b\u0001T\u001b: (51)\nTherefore, the states \u001aand\u001aAdi\u000ber only by the subcon-\nstellation classes [ c(\u001b)] of\u001bodd.\nAt \frst sight, it seems that how we deal with the rela-\ntive phase factors in (35) via the subconstellation classes\nis rather complicated compared to other gauges that one\ncould use. However, as we already mentioned, the phase\nfactors can not be invariant under rotations, and could\nhave complicated transformations laws in other gauges\nas well. The main advantage to associate the relative\nphase factors with the subconstellation classes is that\ntheir transformation laws under rotations are the same8\nas for all the subconstellations. In addition, when one\nparametrizes the whole set of density matrices, the sub-\nconstellation classes can be also counted. Let us dis-\ncuss this at the hand of the s= 1 case. Here the\nstates are labeled with two radii w\u001b(\u001b= 1;2) and they\nhave two associated constellations: C(1)is the pair of an-\ntipodal points\u0006rwith subconstellation classes [ r] and\n[\u0000r] andC(2)is given by two axes that span a rectan-\ngle (see Fig. 1) with classes [ n1;n2] = [\u0000n1;\u0000n2] and\n[\u0000n1;n2] = [n1;\u0000n2]. Let us orient the coordinate sys-\ntem such that the sides of the rectangle C(2)are parallel\nto thexandy-axes. We denote by \u001ethe angle between\nthex-axis and the star n1in the \frst quadrant and spec-\nify the class [ c(2)] with the vectors n1andn2(see Fig.\n1). As we can observe, to parametrize all the possible\nclasses [ c(2)] we must consider \u001e2[0;\u0019=2]. The associ-\nated vectors of the subconstellations are\n~\u001a(1)=N1\u0010\n\u0000rx+iry;p\n2rz;rx+iry\u0011\n;\n~\u001a(2)=N2\u0012\n1;0;\u00002p\n6cos(2\u001e);0;1\u0013\n; (52)\nwith\nN1=1p\n2r; N 2=\u0012\n2 +2\n3cos2(2\u001e)\u0013\u00001=2\n: (53)\nTherefore, we have parametrized the whole set of spin\ns= 1 states modulo the semide\fnite positive condition.\nThe \frst question regarding the semide\fnite positive\ncondition is whether there is a set of classes f[c(\u001b)]g2s\n\u001b=1\nsuch that for any possible radii w\u001b, the respective density\nmatrix does not represent a physical state. We can prove\nthat in a ball close enough to the maximally mixed state\n\u001a\u0003= (2s+ 1)\u000011, there exist states with any subconstel-\nlation classesf[c(\u001b)]g2s\n\u001b=1. The statement is proved by\nMehta's lemma ([16] p. 466):\nLemma 3 LetMbe a Hermitian matrix of size Dand\nlet\u000e=TrM=p\nTr(M2). If\u000e\u0015p\nD\u00001thenMis\npositive.\nFor a density matrix (35), \u000e=\u0010P2s\n\u001b=0w2\n\u001b\u0011\u00001=2\nand\nhence if\n2sX\n\u001b=1w2\n\u001b\u00141\n2s(2s+ 1); (54)\nthen\u001arepresents a physical state, independent of its sub-\nconstellation classes [ c(\u001b)\n\u001a].\nExamples\nLet us study some spin-state families. Some of these\nfamilies are also described in [31] using the T-rep without\ntaking into account the subconstellation classes.\nFIG. 1. The constellation C(2)\n\u001aof\u001afor\u001b= 2 ands= 1.C(2)\n\u001a\nis oriented such that the constellation lies in the xy-plane.\nThe black points are an element of the class [ c(2)\n\u001a].\nSpin Coherent (SC) states : Let us consider \frst the\nstatej i=js;si, which is the SC state pointing in the\nzdirection. We use the expression of (29) to obtain the\ndecomposition of \u001a=js;sihs;sj\n\u001a\u001b\u0016=\u000e\u0016;0(2s)!\u00142\u001b+ 1\n(2s+\u001b+ 1)!(2s\u0000\u001b)!\u00151=2\n:(55)\nTherefore\n\u000fThe components of \u001a\u001bare zero except \u001a\u001b0.\n\u000fEvery constellation C(\u001b)has\u001bstars in each Pole,\nwhich are the simplest constellations with antipo-\ndal symmetry.\n\u000f[c(\u001b)] = [z; :::;z],i.e., an element of the class\n[c(\u001b)] is the subconstellation formed by \u001bstars\nalong thezdirection.\n\u000fThe radiiw\u001bhave the values\nw\u001b= (2s)!\u00142\u001b+ 1\n(2s+\u001b+ 1)!(2s\u0000\u001b)!\u00151=2\n: (56)\nThe density matrix of the pure SC state in direction\nn(\u0012;\u001e) is obtained rotating the state \u001azby a rotation\nwith Euler angles ( \u001e; \u0012; 0). Using the equations in [43] (\np. 113), we obtain that\n\u001an=\u001a00T00+2sX\n\u001b=1w\u001b\u001bX\n\u0016=\u0000\u001bD(\u001b)\n\u0016;0(\u001e;\u0012;0)T\u001b\u0016; (57)\n=1\n2s+ 1+2sX\n\u001b=1w\u001br\n4\u0019\n2\u001b+ 1\u001bX\n\u0016=\u0000\u001bY\u0003\n\u001b\u0016(\u0012;\u001e)T\u001b\u0016;\nwithY\u001b\u0016(\u0012;\u001e) the spherical harmonics. The respective\nsubconstellation classes are [ n; :::;n] for each\u001b. The9\nSC GHZ W\nw1w2w3w1w2w3w1w2w3\n\u001a3\n2p\n51\n21\n2p\n501\n21p\n21\n2p\n51\n23\n2p\n5\n\u001a11p\n21p\n601p\n61\n3p\n21p\n6\n\u001a1=21p\n201\n3p\n2\nTABLE I. The T-representation of the SC, GHZ and W states for s= 3=2 in a particular orientation. Top: The radiiw\u001b\nof\u001aand their reduced density matrices \u001a1and\u001a1=2after loss of one or two particles (Eq. (66)), respectively. Bottom: A\nrepresentative element of each subconstellation class f[c(\u001b)\n\u001a]g\u001bfor the three states where for each value of \u001b= 1;2;3 we assign\nthe color red, yellow and blue to the respective sphere with radius w\u001b(\u001a). We add the degeneracy number of each star in case\nit is degenerate. The reduced density matrices \u001akinherit the constellations of \u001aup to\u001b= 2kwith di\u000berent radii.\nstatesj\u0006nionly di\u000ber by the classes [ c(\u001b)] of\u001bodd (see\nEq. (51)).\nGeneral pure state : Let us take a spin- sstatej iand\nits density matrix \u001a =j ih j. The state \u001aexpanded in\ntheT-rep is given by\n\u001a =X\n\u001b\u0016h\u001b;\u0016jP\u001bj ; AiT(s)\n\u001b\u0016; (58)\nwhere we use the bipartite notation j ;  Ai=j i\nj Ai\nand the antipodal state j Aiis de\fned as (38). We can\nobserve that the constellations of the T-rep come from\nthe irrep decompositions of the bipartite state j ;  Ai,\nwhere the antipodal state j Aiappears from the fact\nthat it transforms in the same way as the bra h junder\nrotations [46]. In particular, the standard Majorana con-\nstellationC of the pure spin- sstatej iis an element\nof the class [ c(2s)]. However, only with the knowledge of\nthe class [ c(2s)], we cannot specify the state j i. An algo-\nrithm to recover the standard Majorana polynomial from\n[c(2s)\n\u001a ] is the following: calculate the overlap between \u001a \nand the SC states pointing to a star nof an element of\n[c(2s)]. Ifhnj\u001a jni= 0, then\u0000n2C , otherwisen2C .\nDicke state : The Dicke states \u001am=js;mihs;mjwith\nm=\u0000s;:::;s satisfy Tr(\u001amT\u001b\u0016) = 0 for\u00166= 0. For\u001am,\nw\u001b=jhs; mjT\u001b0js; mij=jC\u001b0\nsms\u0000mj. We conclude the\nfollowing results:\n\u000fThe constellations C(\u001b)\n\u001amare the same for all m=\n\u0000s;:::s , with\u001bstars in the each Pole.\u000fThe respective classes [ c(\u001b)] are obtained calculat-\ning the sign of the coe\u000ecients in (58),\nh\u001b;\u0016jP\u001bj ; Ai= (\u00001)s\u0000m\u000e\u00160C\u001b0\nsms\u0000m:(59)\nIn Table I we observe the Dicke states for spin-3 =2\nstates which are equivalent up to a rotation to the\nSC and W states.\n\u000fThe antipodal states \u001amand\u001a\u0000mjust di\u000ber by\nsome classes [ c(\u001b)] of\u001bodd, as we show in (51).\nB. The polynomials of T\u001b\u0016\nThe polynomials of the tensor operators T(s)\n\u001b\u0016are the\nirreps ofSU(2) inP(N;N)(z), where one can compare\nand multiply polynomials of di\u000berent degrees ( i.e., ele-\nments of di\u000berent spaces P(N;N)(z)) more easily than for\ntheir matrix counterpart, which involves tensor product\nand projections in the fully symmetric sector. The \frst\nresult regarding this property is associated with the com-\nparison of the tensor operators of di\u000berent spin- s. Before\nexplaining the general results, let us compare the Majo-\nrana polynomials for T(s)\n10fors= 1=2;1. From equation\n(13) we obtain that\np(1)\n10(z) = (zaza)p(1=2)\n10(z): (60)\nWe observe that the binomial ( zaza) is the factor be-\ntween polynomials representing the same operator but10\nfor di\u000berent spin. In addition, it is easy to observe that\nL(p(1=2)\n10) = 0. We summarize these results in the follow-\ning theorem. Its proof can be found in Appendix B.\nTheorem 1 The polynomials p(s)\n\u001b\u0016(z)associated to the\noperatorsT(s)\n\u001b\u0016have the following properties\n1. The action of the partial trace operator Lunder\np(s)\n\u001b\u0016(z)is equal to\nL\u0010\np(s)\n\u001b\u0016(z)\u0011\n=(p\n(2s+\u001b+1)(2s\u0000\u001b)\n2sp(s\u00001=2)\n\u001b\u0016 (z)ifs>\u001b= 2\n0 otherwise:\n(61)\nIn particular,\n(2s+ 1)\u00001=2L\u0010\np(s)\n00(z)\u0011\n= (2s)\u00001=2p(s\u00001=2)\n00 (z);(62)\nand therefore Lleaves the trace of the respective\noperator invariant.\n2. For any value of \u001b\u00142s,\np(s)\n\u001b\u0016(z) =l(s;\u001b)\u00001(zaza)2s\u0000\u001bp(\u001b=2)\n\u001b\u0016(z); (63)\nwith\nl(s;\u001b)\u0011s\n(2s+\u001b+ 1)!(2s\u0000\u001b)!\n(2\u001b+ 1)!\u001b!\n(2s)!: (64)\nInherited constellations in the T-rep\nBy construction, the action of a SU(2) transformation\non\u001arigidly rotates all its classes [ c(\u001b)\n\u001a] while their radii\nw\u001bare invariant. In addition to the well-behaviour under\nrotations of the T-rep and a visual representation of our\nstates (see Table I), there is an additional property as-\nsociated to their reduced matrices \u001ak. From Theorem 1,\nthe spin-s0(withs0=s\u00001=2) reduced state \u001as0= Tr 1(\u001a)\nis equal to\n\u001as0=2s\u00001X\n\u001b=0p\n(2s+\u001b+ 1)(2s\u0000\u001b)\n2sw\u001b~\u001a\u001b\u0001T(s0)\n\u001b:(65)\nAs we can observe, each component is re-scaled by a fac-\ntor independent of \u0016leaving the subconstellation classes\ninvariant, i.e., the reduced density matrices inherit the\nlowest classes of \u001a,f[c(\u001b)\n\u001as0]gs0\n\u001b=1=f[c(\u001b)\n\u001a]gs0\n\u001b=1. The re-\nscaled factor can be absorbed in the radius w\u001b\nw\u001b(\u001as0) =p\n(2s+\u001b+ 1)(2s\u0000\u001b)\n2sw\u001b(\u001a); (66)\nwhere we write the weights as a function of the density\nmatrix. The radius w\u001bincreases with respect to a par-\nticle loss if \u001b(\u001b+ 1)<2s. If the state loses more than\none particle, the lowest classes are still inherited and theradii are re-scaled with a product of factors of the front\nof (66) with successively reduced spin s. In Table I we\nplot the radii and classes of \u001afw\u001b;[c(\u001b)]gfor\u001aequal to\nthe SC, GHZ and W states with s= 3=2. We only plot a\nrepresentative element of [ c(\u001b)] to simplify the visualiza-\ntion in the \fgures. The table also includes the radii w\u001b\nfor each reduced density matrix \u001akwithk= 1=2;1.\nTo study another example, let us discuss the constel-\nlation di\u000berences between the quantum linear superpo-\nsition\u001aQ=j ih jwithj i= (js;si+js;\u0000si)=p\n2 (a\n\\Schr odinger cat\" state) and a classical mixture of the\nsame states \u001aC= (js;sihs;sj+js;\u0000sihs;\u0000sj)=2 (which\nwe will call \\classical cat state\" for short). \u001aQhas an\nadditional term with respect to \u001aC,\n\u001aQ=\u001aC+1\n2(js;sihs;\u0000sj+js;\u0000sihs;sj)\n=\u001aC+1\n2((\u00001)2sT(s)\n2s;2s+T(s)\n2s;\u00002s); (67)\nand it yields that the constellations set of these two states\nwill be equal except for C(N)\n\u001a, and hence [ c(N)\n\u001a], withN=\n2s. Let us calculate the constellations of \u001aC\frst. Using\nequation (55) and that js;\u0000sihs;\u0000sj=Ajs;sihs;sjAy,\nwe obtain that\n(\u001aC)\u001b\u0016=8\n<\n:\u000e\u0016;0(2s)!h\n2\u001b+1\n(2s+\u001b+1)!(2s\u0000\u001b)!i1=2\n\u001beven\n0 \u001bodd;\n(68)\nand hence\u001aCdoes not have constellations for \u001bodd. In\nparticular,C(N)\n\u001aCforNodd does not exist and for Neven\nit is equal to Npoints in each Pole. On the other hand,\nthe vector\u001a(N)of\u001aQand the respective polynomials are\ngiven by\n\u001a(N)\n\u001aQ=8\n><\n>:\u0012\n1\n2;0;:::; 0;(2s)!p\n(4s)!;0;:::; 0;1\n2\u0013\nforNeven\n\u0010\n(\u00001)2s\n2;0;:::; 0;1\n2\u0011\nforNodd;\np(N)\n\u001aQ(\u0010) =(\n1\n2(z2s+ 1)2forNeven\n(\u00001)2s\n2(z4s+ 1) forNodd: (69)\nThe roots of the polynomials (69) draw on the sphere a\n4s-agon in the odd case and a 2 s-agon with all the stars\ndoubly-degenerate in the even case. The radii wNfor\neach case are equal to\nwC\nN=((2s)!p\n(4s)!forNeven\n0 forNodd;\nwQ\nN=8\n<\n:q\n1\n2+((2s)!)2\n(4s)!forNeven\n1p\n2forNodd: (70)\nOur calculations are in agreement with the results in [31]\nwere the authors also calculated the constellations of the11\ns= 5=2 s= 3\u001aCs= 3\u001aQ\nFIG. 2.T-rep for the density matrices of Schr odinger cat states \u001aQand classical cat states \u001aCfors= 5=2;3.Left:\u001aQfor\ns= 5=2.\u001aCis equal to\u001aQwith the highest class [ c(2s)\n\u001aQ] taken out, which corresponds in the \fgures to the stars lying on the blue\nsphere. Center:\u001aCfors= 3. Right:\u001aQfors= 3. The states \u001aCand\u001aQdi\u000ber only by [ c(2s)]: For integer spin s, the highest\nconstellation for \u001aCshrinks to a small value given by eq.(70), whereas for half-integer sthe radius of the sub-constellation\nvanishes and hence does not contribute. For any spin value, the states \u001aCand\u001aQafter the partial trace of one of its constituent\nspins-1/2 are the same, \u001aQ\ns\u00001=2=\u001aC\ns\u00001=2.\nclassical and quantum cat states for a general spin value\ns. In \fgure 2 we plot the states \u001aQand\u001aCfors=\n5=2;3 with an element of their respective classes [ c(\u001b)].\nIn addition, by the results of the previous subsection, the\nstates after the reduction of one constituent spin-1 =2 have\nthe same subconstellation classes and radii and therefore\nthey are equal, \u001aQ\ns\u00001=2=\u001aS\ns\u00001=2. As a consequence, we\nobtain the old known result that the GHZ state after the\nloss of a particle is separable [52].\nC. Tensor product and the S-rep\nSome operators C2B(Hs) are the projection of the\ntensor product of Nspin-1=2 operators C=PsC1\n\u0001\u0001\u0001\nCNPs, where, again, the projector operator Psis\nconsidered to be restricted to its image. The polynomials\nof these operators are factorizable\npC(z) =NY\nk=1pCk(z); (71)\nwhere the proof consists in the calculation of\nh\u0000nBjPsCPsj\u0000nBiin terms of the symmetric Dicke\nstates (39). In particular, the set of operators given\nby the tensor product of NPauli matrices \u001b\u0016with\n\u0016= 0;x;y;z projected in the fully symmetric space is\na tight frame ofB(Hs) [1] that we called the S-rep . In an\nequivalent way and following the same reasoning as in [1],\nthe set of projected tensor products of the spin-1 =2 opera-\ntorsf\u001b0; \u001b\u0000; \u001bz; \u001b+g,S\u001c1:::\u001cN\u0011Ps\u001b\u001c1\n\u0001\u0001\u0001\n\u001b\u001cNPswith\n\u001ck= 0;\u0000; z;+, is a tight frame. The operator S\u001c1:::\u001cN\nis independent of the order of its indices \u001ck, and the onlyrelevant information can be encoded in a 4-vector of nat-\nural numbers ~ \u0017= (\u00170; \u0017\u0000; \u0017z; \u0017+), whereP\nj\u0017j=Nand\n\u0017jis the number of times that jappears in the indices of\nS\u001c1:::\u001cN. Following the previous result, the polynomial of\nS~ \u0017is factorized in powers of the polynomials of \u001bjwith\nj= 0;\u0006; z,\npS~ \u0017(z) =Y\nj(pj(z))\u0017j: (72)\nD. Connection between T- andS-reps\nIn this subsection we will obtain an explicit formula for\nwriting the T\u001b\u0016operators in terms of the S-rep, using\ntheir respective polynomials. The operators in the T-\nrep andS-rep share the property that their polynomials\ncontain the factor ( zaza)k, wherek= 2s\u0000\u001bforT(s)\n\u001b\u0016and\nk=\u00170forS~ \u0017. Both of them are a basis of B(Hs). In\nparticular, the operators T(s)\n\u001b\u0016can be written in terms of\ntheS-rep\nT(s)\n\u001b\u0016=X\n~ \u0017A~ \u0017\n\u001b\u0016S~ \u0017: (73)\nLemma 2 and Theorem 1 yields that\nTr\u0012\u0010\nT(s)\n\u001b\u0016\u0011y\nS~ \u0017\u0013\n/Y\nj\u0000\npj(@b;@b)\u0001\u0017jp(s)\n\u001b\u0016(za;za) (74)\n/Y\nj6=0\u0000\npj(@b;@b)\u0001\u0017jp(s\u0000\u00170=2)\n\u001b\u0016 (za;za)\n/Tr\u0012\u0010\nT(s\u0000\u00170=2)\n\u001b\u0016\u0011y\nS(0;\u0017\u0000;\u0017z;\u0017+)\u0013\n;12\nand henceA~ \u0017\n\u001b\u0016= 0 for\u00170>2s\u0000\u001b. The resolution of the\nT\u001b\u0016operators in the S-rep is not unique because the S\nmatrices form a tight frame instead of a basis. However,\nit is possible to write a resolution only with one running\nindex and\u00170= 2s\u0000\u001b\fxed,\nT(s)\n\u001b\u0016=\u001bX\nk=\u0016A(s; \u001b; \u0016; k )S(2s\u0000\u001b;k\u0000\u0016;\u001b+\u0016\u00002k;k);(75)\nwith\nA(s; \u001b; \u0016; k ) =s\n(\u001b+\u0016)!(\u001b\u0000\u0016)!\n(2\u001b)!l(s; \u001b)\u00001(\u00001)k2\u0016\u00002k(\u001b!)\nk!(k\u0000\u0016)!(\u001b+\u0016\u00002k)!:\n(76)\nThe proof of this equation is in Appendix C. The S-\nrep has also an additional property under partial traces\n[1]: the coe\u000ecients c0\n(\u00170;\u0017\u0000;\u0017z;\u0017+)= Tr(\u001akS(\u00170;\u0017\u0000;\u0017z;\u0017+))\nwithP\nj\u0017j= 2kof the reduced spin- kstate\u001akare equal\nto a subset of coe\u000ecients c~ \u0017of the original state \u001a\nc0\n(\u00170;\u0017\u0000;\u0017z;\u0017+)=c(\u00170+2(s\u0000k);\u0017\u0000;\u0017z;\u0017+): (77)\nWe can prove that the latter result of the S-rep is related\nto the property of the inherited constellations of the T-\nrep discussed in subsection III B by using the connection\nbetween the representations: A state \u001a=P\u001a\u001b\u0016T(s)\n\u001b\u0016has\nreduced state \u001as\u00001=2equal to eq. (65), and the same eq.\n(77) fork=s\u00001=2 can be obtained using that\nTr\u0010\nT(s)\n\u001b\u0016S(\u00170+1;\u0017\u0000;\u0017z;\u0017+)\u0011\n(78)\n=p\n(2s+\u001b+ 1)(2s\u0000\u001b)\n2sTr\u0010\nT(s\u00001=2)\n\u001b\u0016S(\u00170;\u0017\u0000;\u0017z;\u0017+)\u0011\n:\nThe last equation is proved by Theorem 1.\nE. Anticoherence order in terms of polynomials\nWe end this section writing the criterion for the anti-\ncoherent states in terms of polynomials. Zimba [7] de-\n\fned an anticoherent state of order- t, ort-anticoherent\nfor short, if the expectation value h(n\u0001S)kiis indepen-\ndent of the unit vector nfor anykwith 0\u0014k\u0014t.\nThe criterion of anticoherence in terms of the S- and\nT- representations were obtained in [1]. A state is t-\nanticoherent if and only if its spin- t=2 reduced state \u001at=2\nis the maximally mixed state \u001a\u0003= (2t+ 1)\u000011which\nis equivalent to that hT\u001b\u0016i= 0 for all 1\u0014\u001b\u0014tand\n\u0000\u001b\u0014\u0016\u0014\u001b. In terms of the Majorana polynomial\nof\u001a,p\u001a(z), a state\u001aist\u0000anticoherent if and only if\nL2s\u0000t(p\u001a(z))/p1(z) = (zaza)t.\nIV. THE HUSIMI- AND P-FUNCTIONS OF \u001a\nSeveral quasiprobability distributions are expressed in\nterms of the coe\u000ecients \u001a\u001b\u0016[51], and we are going tostudy two of them: The Husimi- and the P-functions\n[51]. The Husimi function of a state \u001a,H\u001a(n)\u0011hnj\u001ajni,\nis related to the Majorana polynomial of p\u001a(z) as\nH\u001a(\u0000n) =p\u001a(z)\n(zaza)N; (79)\nwithnthe direction associated to the complex number\n\u0010=z1=z2via the stereographic projection. As we can\nobserve, the variables ( z1; z2) (and hence ( za) = (z\u0003\na)) are\nde\fned up to a common factor. In particular, if one takes\nz1= cos(\u0012=2) andz2= sin(\u0012=2)ei\u001e, the denominator of\nthe last equation is one and hence H\u001a(\u0000n) =p\u001a(z). On\nthe other hand, the P-function of a state \u001ais de\fned as\nthe function P\u001a(n) such that\n\u001a=Z\nP\u001a(n)jnihnjd\n; (80)\nwith d\n the volume element of the 2-sphere. The P-\nfunction of a state is not unique and the notion of classical\nstates for spin systems can be expressed in terms of the\nP-function [53]: A state \u001ais classical i\u000b a representation\nof the form (80) with non-negative P-function exists. If\none restricts the P-function to a linear combination of the\n\frst 2sspherical harmonics fY\u001b\u0016(\u0012;\u001e)g2s\n\u001b=1, one obtains\na unique P-function for each state [51]\nP\u001a(\u0012; \u001e)\u00112sX\n\u001b=0X\n\u0016(\u00001)\u001b\u0000\u0016l(s;\u001b)p\n(2\u001b+ 1)!p\n4\u0019(\u001b!)\u001a\u001b\u0016Y\u001b\u0016(\u0012;\u001e):\n(81)\nUsing Theorem 1, we can calculate the P-function of the\nspin-kreduced density matrices \u001akin terms of the coef-\n\fcients of the original state \u001a, yielding that\nP\u001ak(\u0012; \u001e) =2kX\n\u001b=0X\n\u0016(\u00001)\u001b\u0000\u0016l(s;\u001b)p\n(2\u001b+ 1)!p\n4\u0019(\u001b!)\u001a\u001b\u0016Y\u001b\u0016(\u0012;\u001e);\n(82)\ni.e., the P-function of the reduced density matrices is\nequal to the P-function of the original state omitting the\nhigher multipolar terms.\nV. SUMMARY AND CONCLUDING REMARKS\nWe have generalized the Majorana stellar representa-\ntion of pure states to Hermitian operators, in particular\nto density operators and hence mixed states. The map-\nping is a bijective correspondence between states \u001a2\nB(Hs), polynomials p\u001a(z)2P(N;N)(z) and a set of sub-\nconstellation classes on the Euclidean space R3, where\nthe latter is equal to the Ramachandran-Ravishankar\nrepresentation [35], called here the T-rep. The repre-\nsentation behaves well under rotations by construction.\nIn addition, it has also attractive properties such as: de\f-\nnition of polynomials for any operator C2B(Hs); inher-\nited constellations under partial traces; the tensor prod-\nuct of operators in the fully symmetric sector is reduced13\nto the product of their polynomials; and any other op-\neration inB(Hs) can be written as di\u000berential operation\nacting on the corresponding polynomials. Some of these\nresults have been found previously in the T- andS- rep-\nresentations, and now, with the Majorana polynomial,\nthe bridge between them has been explained and their\nresults can be translated from one to another. In addi-\ntion, we discussed the T-representation in terms of sub-\nconstellation classes that allows us to completely follow\nthe state under rotations and, with the results derived\nhere, also under the partial trace. Each subconstella-\ntion class represents the \u001b-block of the state \u001a, and its\nradiusw\u001brepresents its magnitude. The states written\nin theT-rep have been used to study the quantum po-\nlarization of light [10]. The results presented here helps\nto represent each block easily and track its changes un-\nder partial traces. We also wrote the relation between\nthe Majorana representation of a state \u001aand its Husimi\nand P-functions. We hope that this new representation,\nas the standard Majorana representation for pure states,\ncan give the readers more intuition about the space of\nthe mixed states and the action of the SU(2) group on\nit.\nACKNOWLEDGEMENTS\nESE thanks the University T ubingen and its T@T fel-\nlowship. The authors thank John Martin for fruitful cor-\nrespondence.\nAppendix A: Proofs of some Lemmas\nProof of Lemma 1 . Let us consider \frst the polynomi-\nalspD(z0) andpE(z) written in di\u000berent variables, with\nproduct equal to\npD(z0)pE(z) =h\u0000n0\nBjDj\u0000n0\nBih\u0000nBjEj\u0000nBi:(A1)\nTo obtain the polynomial of C=DE,pC(z), we have\nto apply a di\u000berential operator Odependent only on the\nvariablesza0andza, such thatO(j\u0000n0\nBih\u0000nBj) =1.\nNote thatj\u0000n0\nBih\u0000nBjcan be seen as a matrix with\nentries\nhs;m0j\u0000n0\nBih\u0000nBjs;mi= (\u00001)2s\u0000m\u0000m0(A2)\n\u0002s\u00122s\ns\u0000m\u0013\u00122s\ns\u0000m0\u0013\nzs+m\n1zs\u0000m\n2(z10)s+m0(z20)s\u0000m0;\nand the operator Oacts entry by entry. The en-\ntries are equal to the Majorana polynomial of the op-\neratorjs;mihs;m0jwritten in the respective variables,(j\u0000n0\nBih\u0000nBj)m0m=h\u0000nBjs;mihs;m0j\u0000n0\nBi. The\noperatorOhas to produce a Kronecker-delta \u000emm0, which\nis equivalent to saying that it has to act as a trace op-\nerator onjs;mihs;m0j. Hence,Ois similar to the trace\noperator (21),\nO= (N!)\u00002(@10@1+@20@2)N: (A3)\nWe can calculate the action of Oin two steps: we evaluate\n\frst the derivatives of the prime variables, yielding that\nO(hs;m0j\u0000n0\nBih\u0000nBjs;mi) = (\u00001)2s\u0000m\u0000m0(N!)\u00001\n\u0002s\u00122s\ns\u0000m\u0013\u00122s\ns\u0000m0\u0013\n@s+m0\n1@s\u0000m0\n2 (zs+m\n1zs\u0000m\n2);(A4)\nand then we let the remaining derivatives act. The last\nresult showed us that the action of Ois equivalent to\ninterchange the prime variables ( z10;z20) by (@1;@2), and\nthen we apply the remaining derivatives in the second\nfactor of the r.h.s. of Eq. (A1). pC(z) is obtained, after\nOacts on (A1), by making the substitution h\u0000n0\nBj!\nh\u0000nBj,\npC(z) = (N!)\u00001pD(za;@a)pE(za;za); (A5)\nwhere the derivatives only act on pE(z), which can be\nensured by writing the variables in each monomial of\npD(za;@a) such that the partial derivatives go to the\nright of the monomial, to a\u000bect only the polynomial on\nthe right. In a similar way, we can do the same pro-\ncedure evaluating \frst the derivatives over the variables\nzainstead of the prime variables za0, obtaining a similar\nequation as the previous one,\npC(z) = (N!)\u00001pE(@a;za)pD(za;za):\u0003 (A6)\nProof of Lemma 2\n(N!)3Tr (CD) = (@a@a)N[pC(zb;@b)pD(z)]\n= (@c1:::ckpC(zb;@b))\u0000\n@c1:::cN@ck+1:::c2spD(z)\u0001\n= (@c1:::cNpC(zb;@b)) (@c1:::c2spD(z))\n=\u0000\n(@c@c)2spC(zb;@b)\u0001\n(pD(z))\n=(N!)pC(@b;@b) (pD(z)); (A7)\nwhere the repeated indices cjrun from 1 to 2 and @c1:::ck\nis short notation for @c1:::@ck, and where in the second\nline there are no derivatives @kacting inpD(z), otherwise\nthe number of partial derivatives exceeds the degree of\npD(z) in thezavariables. The last equation is equivalent\nto the application of the operator O, and it yields the\n\fnal result. \u000314\nAppendix B: Proof of Theorem 1\n1:We use the equation (29) to calculate explicitly its polynomial using (13)\np\u001b\u0016(z) =X\nm;m0(\u00001)3s\u00002m0\u0000mC\u001b\u0016\nsm;s\u0000m0s\u00122s\ns\u0000m\u0013\u00122s\ns\u0000m0\u0013\nzs+m\n1zs\u0000m\n2(z1)s+m0(z2)s\u0000m0: (B1)\nThe action of Lin the last equation yields\n(2s)2L\u0010\np(s)\n\u001b\u0016\u0011\n=X\nm;m0(\u00001)3s\u00002m0\u0000mC\u001b\u0016\nsm;s\u0000m0s\u00122s\ns\u0000m\u0013\u00122s\ns\u0000m0\u0013\n\u0002\nh\n(s+m)(s+m0)zs+m\u00001\n1zs\u0000m\n2(z1)s+m0\u00001(z2)s\u0000m0+ (s\u0000m)(s\u0000m0)zs+m\n1zs\u0000m\u00001\n2 (z1)s+m0(z2)s\u0000m0\u00001i\n;\n=X\nm;m02s(\u00001)s\u0000m0C\u001b\u0016\nsm;s\u0000m0\u0010p\n(s+m)(s+m0)h\u0000nBjs\u00001=2;m\u00001=2ihs\u00001=2;m0\u00001=2j\u0000nBi\n+p\n(s\u0000m)(s\u0000m0)h\u0000nBjs\u00001=2;m+ 1=2ihs\u00001=2;m0+ 1=2j\u0000nBi\u0011\n;\n= 2sX\nm;m0(\u00001)s\u0000m0h\u0000nBjs\u00001=2;m\u00001=2ihs\u00001=2;m0\u00001=2j\u0000nBi\u0002\n\u0010\nC\u001b\u0016\nsm;s\u0000m0p\n(s+m)(s+m0)\u0000C\u001b\u0016\nsm\u00001;s\u0000m0+1p\n(s\u0000m+ 1)(s\u0000m0+ 1)\u0011\n= 2sp\n(2s\u0000\u001b)(2s+\u001b+ 1)X\nm;m0(\u00001)s\u0000m0C\u001b\u0016\ns\u00001=2m\u00001=2;s\u00001=2\u0000m0+1=2\u0002\nh\u0000nBjs\u00001=2;m\u00001=2ihs\u00001=2;m0\u00001=2j\u0000nBi\n= 2sp\n(2s\u0000\u001b)(2s+\u001b+ 1)p(s\u00001=2)\n\u001b\u0016; (B2)\nwhere\nh\u0000nBjs;mi= (\u00001)s\u0000ms\u00122s\ns\u0000m\u0013\nzs+m\n1zs\u0000m\n2;hs;m0j\u0000nBi= (\u00001)s\u0000m0s\u00122s\ns\u0000m0\u0013\n(z1)s+m0(z2)s\u0000m0;(B3)\nand we use the following properties of the Clebsh-Gordan coe\u000ecients ([43], p.254).\n(2s+ 1)(s+m0)1=2C\u001b\u0016\nsm;s\u0000m0= [(s+m)(2s\u0000\u001b)(2s+\u001b+ 1)]1=2C\u001b\u0016\ns\u00001=2m\u00001=2;s\u00001=2\u0000m0+1=2\n+ [(s\u0000m+ 1)\u001b(\u001b+ 1)]1=2C\u001b\u0016\ns+1=2m\u00001=2;s\u00001=2\u0000m0+1=2; (B4)\n(2s+ 1)(s\u0000m0+ 1)1=2C\u001b\u0016\nsm\u00001;s\u0000m0+1=\u0000[(s\u0000m+ 1)(2s\u0000\u001b)(2s+\u001b+ 1)]1=2C\u001b\u0016\ns\u00001=2m\u00001=2;s\u00001=2\u0000m0+1=2\n+ [(s+m)\u001b(\u001b+ 1)]1=2C\u001b\u0016\ns+1=2m\u00001=2;s\u00001=2\u0000m0+1=2: (B5)\nIn particular, L(p(\u001b=2)\n\u001b\u0016(z)) = 0. Now, for p(s)\n00(z) = (2s+\n1)\u00001=2(zaza)2s,p\n2sL(p(s)\n00(z)) =p2s+ 1p(s\u00001=2)\n00 (z), or\nequivalent, (2 s+ 1)\u00001=2T(s)\n00!L(2s)\u00001=2T(s\u00001=2)\n00 , both\nof them with unit trace. Because T(s)\n00is the only non-\ntraceless operator in the basis fT(s)\n\u001b\u0016g\u001b\u0016for each (s), we\nconclude that the partial trace operator preserves the\ntrace.\u0003\n2:The setfp(s)\n\u001b\u0016(z)g\u001b\u0016ofP(N;N)(z) is an orthonor-\nmal basis due to its bijection with the tensor op-\neratorsfT(s)\n\u001b\u0016g\u001b\u0016, and hence ( zaza)2s\u0000\u001bp(\u001b=2)\n\u001b\u0016(z) =P\n\u001c\u0017c\u001c\u0017p(s)\n\u001c\u0017(z) where the coe\u000ecients c\u001c\u0017can be calcu-\nlated using Lemma 2\nc\u001c\u0017=(\u00001)\u0017(N!)\u00002(@a@a)N\u0000\u001bp(\u001b=2)\n\u001b\u0016(@a;@a)\u0010\np(s)\n\u001c\u0000\u0017(za;za)\u0011\n/(\u00001)\u0017p(\u001b=2)\n\u001b\u0016(@a;@a)\u0010\np(\u001b=2)\n\u001c\u0000\u0017(za;za)\u0011\n/Tr(T(\u001b=2)\n\u001b\u0016Ty(\u001b=2)\n\u001c\u0017 ) =\u000e\u001b\u001c\u000e\u0016\u0017; (B6)\nimplying that p(s)\n\u001b\u0016(z) =K(zaza)2s\u0000\u001bp(\u001b=2)\n\u001b\u0016(z), withKa\nproportional factor. Using Eq. (61) of Theorem 1 (2 s\u0000\u001b)\ntimes, we obtain that15\nL(2s\u0000\u001b)(p(s)\n\u001b\u0016(z)) =K(2s+\u001b+ 1)(2s\u0000\u001b)\n(2s)2L(2s\u0000\u001b\u00001)\u0010\n(zaza)2s\u0000\u001b\u00001p(\u001b=2)\n\u001b\u0016(z)\u0011\n=Kl(s;\u001b)2p(\u001b=2)\n\u001b\u0016(z); (B7)\nwhere we conclude that K=l(s; \u001b)\u00001.\nAppendix C: T(s)\n\u001b\u0016in theS-rep\nIn this appendix, we prove the equations (75)-(76).\nFirst, we calculate eq. (75) with s=\u001c=2 and\u001b=\u001c.\nThe next equation (from [54], p.90) helps us to write\nthe expansion of T(\u001c=2)\n\u001c\u0016 in terms of the Soperators with\n\u00170= 0,\nT(\u001c=2)\n\u001c\u0016 =\u0014(\u001c+\u0016)!\n(2\u001c)!(\u001c\u0000\u0016)!\u00151=2h\nS\u0000;T(\u001c=2)\n\u001c\u001ci\n(\u001c\u0000\u0016);(C1)\nwhereS\u0000is the ladder operator in the ( \u001c=2)-irrep, and\n[A;B]q\u0011[A;[A;:::; [A;B]]:::]|{z}\nq; (C2)\nis the nested commutator. The operators S\u0000andT(\u001c=2)\n\u001c\u001c\nin terms of the S-rep are equal to\nT(\u001c=2)\n\u001c\u001c =(\u00002)\u0000\u001cS(0;0;0;\u001c)\n=(\u00002)\u0000\u001cP\u001c=2(\u001b+\n\u0001\u0001\u0001\n\u001b+|{z}\n\u001c)P\u001c=2; (C3)\nS\u0000=\u001c\n2S(\u001c\u00001;1;0;0);\n=\u001c\n2P\u001c=2(\u001b\u0000\n\u001b0\n\u0001\u0001\u0001\n\u001b0|{z}\n\u001c\u00001)P\u001c=2: (C4)\nThe commutator in eq. (C1) can be calculated with the\nfollowing\nLemma 4 Letp(z)2P(N;N)(z), hence\n1:)zc1:::zck@c1:::ckp(z) =N!\n(N\u0000k)!p(z); (C5)\n2:)zc1:::zckza@c1:::ckbp(z) =(N\u00001)!\n(N\u0000k)!za@bp(z);(C6)\nwhere@c1:::ckis short notation for @c1:::@ckand repeated\nindices run from 1 to 2.Proof. 1.) By induction: k= 2 is easy to prove. Let us\nassume the result is valid for kand prove it for k+ 1,\nzc1:::zck+1@c1:::ck+1p(z)\n=zc1:::zck@c1(zck+1@c2:::ck+1p(z))\n\u0000zck+1zc2:::zck@c2:::ck+1p(z)\n=zc1:::zck@c1:::ck(zck+1@ck+1(p(z))\u0000kp(z))\n=N!\n(N\u0000(k+ 1))!p(z): (C7)\nThe proof of 2.) is analogue. \u0003\nThe commutator G\u0011[S\u0000;S~ \u0017] is calculated with poly-\nnomials using the previous result, Lemma 1 and eq. (71),\npG(z) =(\u001c!)\u00001(pS\u0000(za;@a)\u0000pS\u0000(@a;za))pS~ \u0017(z)\n=1\n2(p\u0000(za;@a)\u0000p\u0000(@a;za))Y\nj(pj(z))\u0017j(C8)\n=(p0)\u00170(p\u0000)\u0017\u0000(pz)\u0017z\u00001(p+)\u0017+\u00001(\u0017zp\u0000p+\u00002\u0017+p2\nz);\nwhere we use the commutators of the Pauli matrices and\nladder operators\n[\u001b\u0000;\u001bz] = 2\u001b\u0000; [\u001b\u0000;\u001b+] =\u00004\u001bz: (C9)\nWe obtain that\n[S\u0000;S~ \u0017] =\u0017zS(\u00170;\u0017\u0000+1;\u0017z\u00001;\u0017+)\u00002\u0017+S(\u00170;\u0017\u0000;\u0017z+1;\u0017+\u00001):\n(C10)\nThe constants \u0017zand\u0017+can be thought as the number\nof possible operators \u001bzand\u001b+where one can apply the\ncommutator of \u001b\u0000. The next step to do is the calculation\nof the equation (C1) applying iteratively the latter result.\nThis implies that T(\u001c=2)\n\u001c\u0016 is a linear combination of S-\noperators satisfying the following: \u00170= 0,\u0017+\u0000\u0017\u0000=\u0016\nand\u0017\u0000+\u0017z+\u0017+=\u001c. Hence\nT(\u001c=2)\n\u001c\u0016 =\u001cX\nk=\u0016A(\u001c=2;\u001c;\u0016;k )S(0;k\u0000\u0016;\u001c+\u0016\u00002k;k);(C11)\nwith the condition that \u001c+\u0016\u00002k\u00150.A(\u001c=2;\u001c;\u0016;k )\naccumulates the constant factors of eqs. (C1), (C3), the\nfactor (\u00002)\u001c\u0000kfrom eq. (C10), where the exponent is the\ndi\u000berence between the initial and \fnal values of \u0017+, and\na combinatorial number given by: the number of ways\nto choose ( \u001c\u0000k)\u001b+operators from a set of \u001c,\u0000\u001c\n\u001c\u0000k\u0001\n(to apply [\u001b\u0000;\u000f] and obtain \u001bz); the number of ways to\nchoose (k\u0000\u0016)\u001bzoperators from a set of ( \u001c\u0000k),\u0000\u001c\u0000k\nk\u0000\u0016\u0001\n(to\napply [\u001b\u0000;\u000f] and obtain \u001b\u0000); and the number of possible\norders to apply the ( \u001c\u0000\u0016)\u001b\u0000operators to obtain the\nrespective operator S(0;k\u0000\u0016;\u001c+\u0016\u00002k;k), (\u001c\u0000\u0016)!=2k\u0000\u0016. 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Louck, Angular momentum\nin quantum physics (Cambridge University Press, 1981)."    },    {        "title": "1909.13478v1.Effect_of_Coulomb_Interaction_and_Disorder_on_Density_of_States_in_Conventional_Superconductors.pdf",        "content": "arXiv:1909.13478v1  [cond-mat.supr-con]  30 Sep 2019Journal of the Physical Society of Japan DRAFT\nEffect of Coulomb Interaction and Disorder on Density of\nStates in Conventional Superconductors\nTakanobu Jujo∗\nDivision of Materials Science, Graduate School of Science a nd Technology, Nara\nInstitute of Science and Technology, Ikoma, Nara 630-0101, Japan\nThedensityofstatesofthedisordereds-wavesuperconductor iscalculatedperturbatively. Theeffect\nof Coulomb interaction on diffusively moving electrons in the normal st ate has been known before,\nbut in the superconducting state both diffuson and the screened C oulomb interaction are modified.\nTherefore, the correction to the density of states in the superc onducting state exhibits an energy\ndependence different from that of the normal state. There is a dip structure in the correction part\nbecause the interaction has a peak at twice the energy of the supe rconducting gap. The Coulomb\ninteraction and the superconducting fluctuation cannot be treat ed separately because the density\nfluctuation is coupled to the phase fluctuation in the superconduct ing state. This coupling results\nin the absence of divergence around the gap edge in the correction part, which suggests the validity\nof this perturbation calculation.\n1. Introduction\nThe conventional s-wave superconductor is not affected by the im purity scattering\nitself because nonmagnetic impurities do not break the symmetry of s-wave supercon-\nductors.1)In general there exist interactions between electrons in superco nductors,\nand the Coulomb interaction changes low-energy properties of elec trons moving diffu-\nsively by disorder.2–4)This is how the scattering by nonmagnetic impurities reduces the\ntransition temperature of s-wave superconductors.5–7)Thus, the correlation between\ninteractions and disorder in superconductors has been an interes ting research subject.\nStudies on correlation between the Coulomb interaction and impurity scattering\nhave been mainly conducted inthe normal state, and physical quan tities such asspecific\nheat and conductivity have been calculated not only in the three-dim ensional case,3,4)\nbut also in the two-dimensional system.8–10)The deviation of physical properties from\n∗E-mail address: jujo@ms.aist-nara.ac.jp\n1/22J. Phys. Soc. Jpn. DRAFT\nthoseofaFermi liquid iscausedby thesuppression oflow-energyele ctronic statesowing\nto the Coulomb interaction enhanced by diffuson. This suppression o f the density of\nstates (DOS) near the Fermi level is known as the Altshuler-Arono v effect. Not only\nthe screened Coulomb interaction but the superconducting fluctu ation is also enhanced\nby the diffusive motion of electrons, and this effect results in the sup pression of the\nDOS above the superconducting transition temperature.11,12)\nThere have been several measurements on the DOS both in the ultr athin\nfilm13,14)(whose thickness is comparable to the coherence length) and the t hree-\ndimensional system.15–17)These studies mainly focuson thephysical properties near the\nsuperconductor-insulator transition, especially the variation of t he size of the supercon-\nducting gap and its spatial distribution when the disorder is increase d. For this reason,\nalthoughtheDOSexhibitsanenergydependencesimilartothatofth eAltshuler-Aronov\neffect both above and below the superconducting transition tempe rature, this energy\ndependence is treated as a uniform background. Therefore, the dependence of the DOS\non energy in the superconducting state has not been precisely inve stigated.\nIn this study, we calculated the correction to the DOS in the superc onducting state\nof a three-dimensional system. We considered the weakly localized r egime in which\nthe expansion parameter of the perturbation is 1 /kFl(kFandlbeing the Fermi wave\nnumber and the mean free path, respectively). We also assume the dirty limit (∆ τ≪1.\n∆ andτbeing the superconducting gap and the relaxation time, respective ly). In the\ncalculation the Coulomb interaction is included consistently with the su perconducting\ncorrelation.\nAlthough the Altshuler-Aronov effect in the superconducting stat e has been studied\nwith use of the Coulomb interaction and diffuson of the normal state ,18,19)the Coulomb\ninteraction and the effect of disorder are modified in the supercond ucting state. The\ndensity fluctuation couples to the fluctuation of the phase of the s uperconducting order\nparameter.20)In addition, because there is an energy gap in the superconducting state,\nthe diffusive motion of quasiparticles is modified and the calculation in th e normal state\ndoes not hold at low energy. Therefore, in the vicinity of the energy gap, the correction\nto the DOS also differs from that of the normal state.\nThis paper is organized as follows. In Sect. 2, the expression for DO S is derived,\nafter discussing the model and the approximations required to calc ulate the correction\nto the DOS. In Sect. 3, after discussing the temperature depend ence and diffuson in\nthe superconducting state, the results of numerical calculations at absolute zero are\n2/22J. Phys. Soc. Jpn. DRAFT\npresented. In Sect. 4, a short summary is provided along with a disc ussion of the effects\nthat are not included in this paper.\n2. Formulation\nThe Hamiltonian is given by\nH=/summationdisplay\nk,σξkc†\nk,σck,σ+/summationdisplay\nqωqb†bq+gph√\nN3/summationdisplay\nk,q,σ(bq+b†\n−q)c†\nk+q,σck,σ+1√\nN3/summationdisplay\nk,k′,σuk−k′c†\nk,σck′,σ\n+1\n2N3/summationdisplay\nk,k′,q,σ,σ′vqc†\nk,σck+q,σc†\nk′,σ′ck′−q,σ′.\n(1)\nξkandωqare the dispersions of electrons and phonons, respectively. The t hird and\nfourth terms represent the interaction between electrons and p honons and the effect of\nimpurity scattering, respectively. We assume that ωqdoes not depend on qand that\nit takes a constant value ωq=ωE. The fifth term represents the Coulomb interaction\nbetween electrons and vq= 4πe2/q2.N3is the number of sites. We consider the three-\ndimensional system, and kandqare wave number vectors in this space. We set /planckover2pi1= 1\nin this paper.\nThe correction to the DOS is given by\nρ′(ǫ) =−1\nπIm1\nN3/summationdisplay\nkTr[ˆGkˆG′\nkˆGk]iǫn→ǫ+i0+. (2)\nHereafter, we use the notation k= (k,ǫn), where kis a wave number vector in the\nthree dimensional space and ǫn=πT(2n−1) is the Matsubara frequency with Tthe\ntemperature. The term Im indicates the imaginary part, and iǫn→ǫ+i0+means the\nanalytic continuation, with 0+an infinitesimal positive quantity ( i=√−1).ˆGkis the\nGreen function of electrons and includes the effects of the impurity scattering and the\nelectron-phonon interaction with Born and mean-field approximatio ns,21)respectively,\nˆGk=1\n(i˜ǫn)2−ξ2\nk−˜∆2\ni˜ǫn+ξk˜∆\n˜∆i˜ǫn−ξk\n. (3)\nHere, ˜ǫnand˜∆ are determined by the following equation:\n(iǫn−i˜ǫn)ˆτ3+(˜∆−∆)ˆτ1=niu2\nN3/summationdisplay\nkˆτ3ˆGkˆτ3 (4)\nwhere ˆτ3= (1 0\n0−1) and ˆτ1= (0 1\n1 0). (niandurepresent the concentration of impurities\nand the magnitude of the impurity potential, respectively.) ∆ is the su perconducting\n3/22J. Phys. Soc. Jpn. DRAFT\n(a)\nI I\n(b)\nI\n(c)\nFig. 1. (a) The diagrammatic representation of the correction to the DOS . The solid line indicates\nthe propagator of electrons ˆGk, and the shaded square includes the effects of interactions. (b) T he\ninteraction effect is obtained by solving this equation. The square wit h “I” included indicates the\nirreducible part. (c) The irreducible part. The dotted line with a cros s represents the scattering by im-\npurities. The dashedline meansthe electron-phononinteraction.T he wavyline representsthe Coulomb\ninteraction.\ngap determined by the gap equation,\n∆ˆτ1=g2\nph\nωE2T\nN3/summationdisplay\nkˆτ3ˆGkˆτ3. (5)\nThe effects of interactions beyond the mean-field approximation ar e included in ˆG′\nk,\nand its diagrammatic representation is shown in Fig. 1. The three inte raction terms\nin Fig. 1(c) combined with the equation represented by Fig. 1(b) give the physical\neffects that are predominant at low energy. The first term (the sc attering by impurities)\ninduces the diffusive motion of electrons, the second term (the inte raction of electrons\nwith phonons) results in the superconducting fluctuation, and the third term gives the\n4/22J. Phys. Soc. Jpn. DRAFT\nscreened Coulomb interaction.\nWe obtain ˆG′\nkas follows. The components of ˆG′\nkare given by\n(ˆG′\nk)jj′=−2T\nN3/summationdisplay\nq˜γij,i′j′\nk,k−q(ˆGk−q)ii′ (6)\nin which i,j,i′,j′are indices specifying rows and columns of 2 ×2 matrices; hereafter\nthe summation is taken over repeated indices. ˜ γij,i′j′\nk,k−qis given by\n˜γij,i′j′\nk,k−q=/bracketleftig\nδi,sδj,t+niu2Mij,lm\nǫn,ǫn−ωl(ˆτ3)sl(ˆτ3)mt/bracketrightig\nγst,s′t′\nq/bracketleftig\nδi′,s′δj′,t′+(ˆτ3)l′s′(ˆτ3)t′m′niu2Ml′m′,i′j′\nǫn,ǫn−ωl/bracketrightig\n,\n(7)\nwhereδi,sis Kronecker’s delta function. γij,i′j′\nqandMij,i′j′\nǫn,ǫn−ωlare given by the following\nequations.\nγij,i′j′\nq=/bracketleftbiggg2\nph\nωE(ˆτ3)i′i(ˆτ3)jj′+vq\n2(ˆτ3)ij(ˆτ3)i′j′/bracketrightbigg\n+/bracketleftbiggg2\nph\nωE(ˆτ3)li(ˆτ3)jm+vq\n2(ˆτ3)ij(ˆτ3)lm/bracketrightbigg\n2T/summationdisplay\nǫnMlm,l′m′\nǫn,ǫn−ωlγl′m′,i′j′\nq(8)\nand\nMij,i′j′\nǫn,ǫn−ωl=1\nN3/summationdisplay\nk(ˆGk)jj′(ˆGk−q)i′i+niu2\nN3/summationdisplay\nk(ˆGk)jm(ˆGk−q)li(ˆτ3)l′l(ˆτ3)mm′Ml′m′,i′j′\nǫn,ǫn−ωl.\n(9)\nThese equations are solved by introducing 4 ×4 matrices such as\nˆM:=\nM11,11M11,22M11,12M11,21\nM22,11M22,22M22,12M22,21\nM12,11M12,22M12,12M12,21\nM21,11M21,22M21,12M21,21\n. (10)\nThen, for example,\n(ˆτ3)i′i(ˆτ3)jj′=\n1 0 0 0\n0 1 0 0\n0 0−1 0\n0 0 0 −1\n(11)\nand\n(ˆτ3)ij(ˆτ3)i′j′=\n1−1 0 0\n−1 1 0 0\n0 0 0 0\n0 0 0 0\n. (12)\n5/22J. Phys. Soc. Jpn. DRAFT\nBy solving Eq. (9), the 4 ×4 matrix corresponding to 2 T/summationtext\nǫnMij,i′j′\nǫn,ǫn−ωlis written\nas follows:\nπρ0\n2\n(χ3+χ0)ˆτ0/2−(χ3−χ0)ˆτ1/2 χ′(ˆτ0−ˆτ1)\nχ′(ˆτ0−ˆτ1) ( χ2+χ1)ˆτ0/2−(χ2−χ1)ˆτ1/2\n.(13)\nHere, ˆτ0= (1 0\n0 1), andρ0=mkF/π2is the noninteracting density of states at the Fermi\nlevel.\nχi= 2T/summationdisplay\nǫnXǫn+ωl,ǫn/α\n1−2Xǫn+ωl,ǫn(hi+gǫn+ωlgǫn+h′\nifǫn+ωlfǫn)−2\nπ(δi,3+δi,0) (14)\n(the second term is necessary when the integration over ξkis performed before the\nsummation over ǫn22)), and\nχ′=T/summationdisplay\nǫnXǫn+ωl,ǫn/α\n1−2Xǫn+ωl,ǫn(gǫn+ωlfǫn−fǫn+ωlgǫn). (15)\ngǫn=−iǫn/ζǫn,fǫn=−∆/ζǫn,ζǫn=/radicalbig\nǫ2\nn+∆2,α:=niu2mkF/2π,\nh3=h0=h′\n0=h′\n1= 1, (16)\nand\nh′\n3=h2=h′\n2=h1=−1. (17)\nαis related to the relaxation time by the impurity scattering: τ= 1/2α= 1/πρ0niu2.\nXǫn,ǫn′:=/integraldisplay\nFS2α+ζǫn+ζǫn′\n(2α+ζǫn+ζǫn′)2+(vk·q)2=2α\nvFqarctan/parenleftbiggvFq\n2α+ζǫn+ζǫn′/parenrightbigg\n.(18)\n(/integraltext\nFSindicates the integration over the Fermi surface.) In the case of a dirty limit\n(vFq/2α≪1, (ζǫn+ζǫn′)/2α≪1)\nXǫn,ǫn′≃2α−(Dαq2+ζǫn+ζǫn′)\n4α(19)\nwith the diffusion constant Dα=v2\nFτ/3 (vFis the Fermi velocity).\nThe indices iofχicorrespond to those of Pauli matrices (ˆ τi). Using Eq. (13),\n/parenleftigπρ0\n2/parenrightig−1\n(2T/summationdisplay\nǫnMij,i′j′\nǫn,ǫn−ωl)(ˆτ0,1)ii′=χ0,1(ˆτ0,1)jj′, (20)\nand\n/parenleftigπρ0\n2/parenrightig−1\n(2T/summationdisplay\nǫnMij,i′j′\nǫn,ǫn−ωl)(ˆτ3+iˆτ2)ii′= (χ3+2χ′)(ˆτ3)jj′+(χ2+2χ′)(iˆτ2)jj′.(21)\nˆτ2= (0−i\ni0). These equations indicate that the density fluctuation (ˆ τ3) couples to the\nphase fluctuation (ˆ τ2) in the presence of a finite value of the superconducting gap (the\nmixing term χ′vanishes when ∆ = 0), and the amplitude fluctuation (ˆ τ1) decouples\n6/22J. Phys. Soc. Jpn. DRAFT\nfrom other modes in the presence of a particle-hole symmetry.\nThen, the solution for Eq. (8) is written in the 4 ×4 matrix form as follows:\nˆγq=/parenleftigπρ0\n2/parenrightig−1\nΓ3(q)(ˆτ0−ˆτ1)+Γ0(q)(ˆτ0+ ˆτ1) Γ′(q)(ˆτ0−ˆτ1)\nΓ′(q)(ˆτ0−ˆτ1) Γ 2(q)(ˆτ0−ˆτ1)+Γ1(q)(ˆτ0+ ˆτ1)\n.\n(22)\nHere,\nΓ3(q) =(p+cq)(1/p+χ2)/2\n(1/p+χ2)[1−(p+cq)χ3]+4(p+cq)(χ′)2, (23)\nΓ2(q) =−[1−(p+cq)χ3]/2\n(1/p+χ2)[1−(p+cq)χ3]+4(p+cq)(χ′)2, (24)\nΓ′(q) =(p+cq)χ′\n(1/p+χ2)[1−(p+cq)χ3]+4(p+cq)(χ′)2, (25)\nΓ0(q) =p/2\n1−pχ0, (26)\nand\nΓ1(q) =−1/2\n1/p+χ1. (27)\nHere,p:=mkFg2\nph/2πωEindicatesthecouplingconstantbetweenelectronsandphonons\nandcq:=mkFvq/2π.\nUsing the above results, the correction to the DOS is written as follo ws.\n−1\nπN3/summationdisplay\nkTr[ˆGkˆG′\nkˆGk]≃ρ03√\n3τ\n2π(kFl)22T/summationdisplay\nωl/integraldisplay\ndx√x\n×Γi(q)(hi+gǫngǫn−ωl+h′\nifǫnfǫn−ωl)+2Γ′(q)(fǫngǫn−ωl−gǫnfǫn−ωl)\n(x+ζǫn+ζǫn−ωl)2gǫn.(28)\n(x=Dαq2.) Here we use the approximate expression Eq. (19), and introduce the upper\nlimits of |ωl|andDαq2(which are on the order of 2 αand will be specified when the\nnumerical calculation is performed in Sect. 3). (The high energy par ts from|ωl|/2α≫1\norvFq/2α≫1 are assumed to be included in the parameters of the electronic sta tes.\nIn fact, 1 /(1−2Xǫn,ǫn−ωl)≃1 in this range, and the correction term is reduced to the\nusual Fock term because the diffuson propagator is absent.)\n2.1 Normal state\nIn this subsection, we show that the expressions previously studie d in the normal\nstate3,4,11,12)are obtained by setting ∆ = 0 in the above expressions. For ∆ = 0 and\n7/22J. Phys. Soc. Jpn. DRAFT\nafter analytic continuation ( iωl→ω+i0+)χi(i= 0,1,2,3) andχ′are written as\nfollows.\nχ3=χ0=2\nπ−Dαq2\nDαq2−iω, (29)\n1\np+χ2=1\np+χ1=2\nπ/integraldisplay\ndǫ/bracketleftbiggtanh(ǫ/2Tc)\n2ǫ+−tanh(ǫ/2T)\n2ǫ+ω+iDαq2/bracketrightbigg\n≃2\nπ/bracketleftbigg\nln/parenleftbiggT\nTc/parenrightbigg\n+π\n8T(Dαq2−iω)/bracketrightbigg\n(30)\n(Tcis the superconducting transition temperature) and χ′= 0. Then, Γ i(q) and Γ′(q)\nare given by\nΓ3(q) =(p+cq)(1/p+χ2)/2\n(1/p+χ2)[1−(p+cq)χ3]≃−1/2\nχ3, (31)\nΓ2(q) =−1/2\n1/p+χ2= Γ1(q), (32)\nΓ0(q) =p/2\n1−pχ3, (33)\nand Γ′(q) = 0.\nThe correction to the DOS in the normal state is given by the following equation:\nρ′(ǫ) =ρ′\nsf(ǫ)+ρ′\ncl(ǫ) (34)\nwith\nρ′\nsf(ǫ)≃ρ012√\n3τ\n(2πkFl)2/integraldisplay\ndω/integraldisplay\ndx√xIm/braceleftbigg2icoth(ω\n2T)Im[Γ2(q)]+tanh(ǫ−ω\n2T)Γ2(q)\n[x−i(2ǫ−ω)]2/bracerightbigg\n(35)\nand\nρ′\ncl(ǫ)≃ρ06√\n3τ\n(2πkFl)2/integraldisplay\ndω/integraldisplay\ndx√xIm/braceleftbiggtanh(ǫ−ω\n2T)[Γ3(q)+Γ0(q)]\n(x−iω)2/bracerightbigg\n.(36)\nρ′\nsf(ǫ) andρ′\ncl(ǫ) include the effects of the superconducting fluctuation above Tc11,12)\nand the screened Coulomb interaction enhanced by diffuson,3,4)respectively.\n3. Results\n3.1 The temperature dependence of the correction to the dens ity of states\nIn this subsection, we show that the temperature dependence of the correction to\nDOS is small at low temperature T≪∆.\nAfter analytic continuation, Eq. (14) is written as follows.\nχi=/integraldisplaydǫ\n2πi/bracketleftbigg\ntanh/parenleftigǫ\n2T/parenrightig\n(κi\n++−κi\n+−)+tanh/parenleftbiggǫ+ω\n2T/parenrightbigg\n(κi\n+−−κi\n−−)/bracketrightbigg\n−2\nπ(δi,3+δi,0)\n(37)\n8/22J. Phys. Soc. Jpn. DRAFT\nwith\nκi\nss′=Xss′\nǫ+ω,ǫ/α\n1−2Xss′\nǫ+ω,ǫ(hi+gs\nǫ+ωgs′\nǫ+h′\nifs\nǫ+ωfs′\nǫ). (38)\nχ′is obtained by replacing κi\nss′in Eq. (37) with i∝ne}ationslash= 3,0 by\nκ′\nss′=Xss′\nǫ+ω,ǫ/α\n1−2Xss′\nǫ+ω,ǫ(gs\nǫ+ωfs′\nǫ−fs\nǫ+ωgs′\nǫ)/2. (39)\nHere,s,s′= + (retarded) or −(advanced), gs\nǫ=−ǫ/ζs\nǫ, andfs\nǫ=−∆/ζs\nǫwithζ±\nǫ=\n√\n∆2−ǫ2θ(∆−|ǫ|)−isgn(±ǫ)√\nǫ2−∆2θ(|ǫ|−∆) [θ(·) is a step function].\nXss′\nǫ,ǫ′=/integraldisplay\nFS2α+ζs\nǫ+ζs′\nǫ′\n(2α+ζs\nǫ+ζs′\nǫ′)2+(vk·q)2≃2α−(Dαq2+ζs\nǫ+ζs′\nǫ′)\n4α.(40)\nFrom Eq. (37),\nImχi=/integraldisplaydǫ\n2π/bracketleftbigg\ntanh/parenleftbiggǫ+ω\n2T/parenrightbigg\n−tanh/parenleftigǫ\n2T/parenrightig/bracketrightbigg\nRe(κi\n++−κi\n+−). (41)\nRe(κi\n++−κi\n+−) takes finite values only for |ǫ+ω|>∆ and|ǫ|>∆. Then, Im χiis\nexponentially small for |ω|<2∆ except for T≃TC, and is negligible in this region.\nWe consider the correction to the DOS for |ǫ|<∆ and|ǫ|>∆ separately in\nthe following. First, we consider the case of |ǫ|<∆. After performing the analytic\ncontinuation of Eq. (28), the imaginary part is written as follows.\nρ′(ǫ)≃ρ0ǫ√\n∆2−ǫ2−6√\n3τ\n(2πkFl)2/integraldisplay\ndω/integraldisplay\ndx√x/bracketleftbigg\ncoth/parenleftigω\n2T/parenrightig\n+tanh/parenleftbiggǫ−ω\n2T/parenrightbigg/bracketrightbigg\n×Im/braceleftigIm[Γi(q)](hi+gǫg+\nǫ−ω+h′\nifǫf+\nǫ−ω)+2Im[Γ′(q)](fǫg+\nǫ−ω−gǫf+\nǫ−ω)\n(x+ζǫ+ζ+\nǫ−ω)2/bracerightig(42)\n(ζǫ=√\n∆2−ǫ2,gǫ=−ǫ/ζǫ, andfǫ=−∆/ζǫ). The imaginary part is finite (Im {·} ∝ne}ationslash=\n0) only for |ǫ−ω|>∆. For|ω|<2∆, ImΓ iand ImΓ′are exponentially small at\nlow temperature, as noted above. The factor coth( ω/2T) + tanh[( ǫ−ω)/2T] is also\nexponentially small for |ǫ|<∆ and|ω|>2∆. Then, the correction to the DOS is\nnegligible for |ǫ|<∆ except for T≃TC.\nOn the other hand, for |ǫ|>∆, the imaginary part of Eq. (28) after the analytic\n9/22J. Phys. Soc. Jpn. DRAFT\ncontinuation is written as follows:\nρ′(ǫ)≃ρ0|ǫ|√\nǫ2−∆2−3√\n3τ\n(2πkFl)2/integraldisplay\ndω/integraldisplay\ndx√x\n×Im/braceleftig\n2coth/parenleftigω\n2T/parenrightigIm[Γi(q)](hi+g+\nǫg+\nǫ−ω+h′\nif+\nǫf+\nǫ−ω)+2Im[Γ′(q)](f+\nǫg+\nǫ−ω−g+\nǫf+\nǫ−ω)\n(x+ζ+ǫ+ζ+\nǫ−ω)2\n+tanh/parenleftbiggǫ−ω\n2T/parenrightbigg/summationdisplay\ns=±sΓi(q)(hi+g+\nǫgs\nǫ−ω+h′\nif+\nǫfs\nǫ−ω)+2Γ′(q)(f+\nǫgs\nǫ−ω−g+\nǫfs\nǫ−ω)\n(x+ζ+\nǫ+ζs\nǫ−ω)2/bracerightig\n.\n(43)\nIn this equation the coefficient of coth( ω/2T) is exponentially small for |ω|<2∆ owing\nto the existence of ImΓ iand ImΓ′, and the coefficient of tanh[( ǫ−ω)/2T] vanishes for\n|ǫ−ω|<∆ (the imaginary part is absent). This indicates that the dependenc e ofρ′(ǫ)\nfor|ǫ|>∆ on temperature is weak for T≪∆. This small dependence of ρ′(ǫ) on\ntemperature is consistent with exponentially small values of ρ′(ǫ) for|ǫ|<∆ at low\ntemperature. Thus, we perform the numerical calculations at T= 0 andǫ >∆ in Sect.\n3.3.\n3.2 Diffuson in the superconducting state\nThe diffuson propagator is usually represented by 1 /(Dαq2−iω). However, in the\nsuperconducting state [Eq. (43), x=Dαq2] it is given by 1 /(x+ζ+\nǫ+ζ±\nǫ−ω) = 1/{x−\ni[sgn(ǫ)√\nǫ2−∆2±sgn(ǫ−ω)/radicalbig\n(ǫ−ω)2−∆2]}for|ǫ|,|ǫ−ω|>∆ (the diffusive motion\nof quasiparticles is effective above the superconducting gap). Ano ther singularity exists\natω= 2ǫin the case of 1 /(x+ζ+\nǫ+ζ+\nǫ−ω) in addition to the pole at ω= 0 in 1/(x+\nζ+\nǫ+ζ−\nǫ−ω). In this subsection, we illustrate that the divergence by this addit ional pole\nis absent when the particle-number conservation is preserved in th e integration of Eq.\n(43).\nBy performing the analytic calculation,\nχ3(q=0) =−8∆2arcsin(ω/2∆)\nπω√\n4∆2−ω2θ(2∆−ω)+/bracketleftbigg8∆2arcosh(ω/2∆)\nπω√\nω2−4∆2+i−4∆2\nω√\nω2−4∆2/bracketrightbigg\nθ(ω−2∆)\n(44)\n(ω >0) and there are following relations between χi(i= 0,1,2,3) andχ′atq=0:\n1/p+χ2= (ω/2∆)2χ3,χ′= (−ω/4∆)χ3, 1/p+χ1= [(ω/2∆)2−1]χ3andχ0=0. Then,\n−(1/p+χ2)χ3+4(χ′)2= 0 atq=0.\nWith use of a relation cq=πω2\npτ/2Dαq2≫p(ωpis the plasma frequency: ω2\np=\n4πnee2/mwithne=k3\nF/3π2electron density and mthe electron mass), Eqs. (23), (24),\n10/22J. Phys. Soc. Jpn. DRAFT\nand (25) are approximately written as follows.\nΓ3(q)≃(1/p+χ2)/2\n−(1/p+χ2)χ3+4(χ′)2, (45)\nΓ2(q)≃χ3/2\n−(1/p+χ2)χ3+4(χ′)2, (46)\nand\nΓ′(q)≃χ′\n−(1/p+χ2)χ3+4(χ′)2. (47)\nThis expressions show that Γ 3,Γ2, and Γ′are proportional to 1 /x= 1/(Dαq2) because\nthe denominator of these quantities vanishes at q=0. The above relations between χi\n(i= 3,2) andχ′indicate that Γ 3/Γ′=−ω/2∆ and Γ 2/Γ′=−2∆/ωatq=0. Then, in\nEq. (43)thetermcontaining 1 /(x+ζ+\nǫ+ζ+\nǫ−ω)2isproportional tothefollowing equation:\n/integraldisplay\ndω/integraldisplay\ndx√x/summationtext\ni=3,2Γi(q)(hi+g+\nǫg+\nǫ−ω+h′\nif+\nǫf+\nǫ−ω)−2Γ′(q)(g+\nǫf+\nǫ−ω−f+\nǫg+\nǫ−ω)\n(x+ζ+ǫ+ζ+\nǫ−ω)2.(48)\nAfter the integration over xwith use of Γ ∝1/x, Eq. (48) is proportional to\n/integraldisplay\ndωω2−4∆2+(ω2+4∆2)(g+\nǫg+\nǫ−ω−f+\nǫf+\nǫ−ω)+4ω∆(g+\nǫf+\nǫ−ω−f+\nǫg+\nǫ−ω)\n(ζ+ǫ+ζ+\nǫ−ω)3/2.(49)\nBoth the numerator and the denominator of this expression vanish atω= 2ǫ, and\nthen the integration over ωresults in a finite correction to the DOS. Therefore, by\npreserving the particle-number conservation, we obtain a finite re sult even when an\nadditional singularity exists in the diffuson propagator in the superc onducting state.\n(As for the case of the pole at ω= 0 in Eq. (43), we obtain a finite result simply\nbecause Γ 3∝ω2/x, Γ′∝ω/x, andh2+g+\nǫg−\nǫ−ω+h′\n2f+\nǫf−\nǫ−ω= 0 atω= 0. The relation\nbetween Γ 3, Γ2and Γ′is irrelevant in this case.)\nIn the case of Γ 0,1, the long-range part 1 /xis absent. As for the terms containing\nΓ0,1, the integration over xis proportional to 1 /(ζ+\nǫ+ζ±\nǫ−ω)1/2, which results in a finite\nvalue after the integration over ωis performed.\n3.3 Numerical calculation\nAs discussed above, the dependence of ρ′(ǫ) on temperature is weak for T≪Tc,\nand so we perform a numerical calculation at T= 0. We consider the superconducting\ngap atT= 0 as the unit of energy (∆ = 1). pis determined by the gap equation.\nThedependences ofΓ i(q)andΓ′(q)[Eqs. (45) −(47),(26)and(27)]on ωareshownin\nFig. 2. (The value of αis implicitly included in Dαq2and the result does not depend on\nαwhen the value of Dαq2/∆ is fixed.) ImΓ iand ImΓ′take finite values above ω >2∆\n11/22J. Phys. Soc. Jpn. DRAFT\n-60-50-40-30-20-10 0 10 20 30 40 50\n 0  2  4  6  8 10(a)ReΓ\nω/∆Γ3\nΓ2\nΓ,\n-2-1.5-1-0.5 0 0.5\nΓ1Γ0\n-160-140-120-100-80-60-40-20 0 20 40 60\n 2 3 4 5 6 7 8 9 10(b)ImΓ\nω/∆Γ3Γ2\nΓ,-3-2-1 0\nΓ1Γ0\nFig. 2. The dependences of Γ iand Γ′onωatT= 0 and Dαq2/∆ = 0.055. (a) The real part of Γ i\nand Γ′. (b) The imaginary part of Γ iand Γ′. The ranges of ω/∆ of the insets are the same as those of\nthe main graphs.\nowing to the finite excitation of quasiparticles across the supercon ducting gap. This\nleads to a peak in ReΓ around ω≃2∆. Forω≫∆, the dependence of Γ on ωshould\nbecome close to that of the normal state. The large value of ImΓ 3forω≫∆ is related\nto Γ3(q)≃(π/4)(1−iω/Dαq2) in the normal state obtained from Eq. (31). The sharp\npeak in Γ 1aroundω= 2∆ indicates the existence of the amplitude mode. The density\nand phase fluctuations (Γ 3, Γ2, and Γ′), however, are quantitatively predominant over\n12/22J. Phys. Soc. Jpn. DRAFT\n-60-50-40-30-20-10 0 10 20 30 40 50\n 0  2  4  6  8 10(a)ReΓ\nDq2/∆Γ3\nΓ2\nΓ,\n-20-10 0 10\nΓ3Γ2\nΓ,\n-30-25-20-15-10-5 0 5 10 15 20 25\n 0  2  4  6  8 10(b)ImΓ\nDq2/∆Γ3\nΓ2\nΓ,\n-0.6-0.4-0.2\nΓ1Γ0\nFig. 3. The dependences of Γ iand Γ′onDαq2atT= 0. (a) The realpart of Γ iand Γ′atω/∆ = 2.5.\nThe inset shows the results at ω/∆ = 1.48. (b) The imaginary part of Γ iand Γ′atω/∆ = 2.5. The\nranges of Dαq2/∆ of the insets are the same as those of the main graphs.\nΓ1,0. These large values come from the long-range part ( ∝1/q2).\nThe dependences of Γ i(q) and Γ′(q) [Eqs. (45) −(47), (26) and (27)] on Dαq2are\nshown in Fig. 3. Γ 3, Γ2and Γ′are proportional to 1 /Dαq2. The results show that these\nthree terms (the density and phase fluctuations) are quantitativ ely comparable to each\nother. This validates the argument about diffuson in the previous su bsection.\nNext, we calculate the correction to the DOS numerically. From Eq. ( 43), we write\n13/22J. Phys. Soc. Jpn. DRAFT\nthe correction to DOS as follows:\nρ′(ǫ) =ρ0|ǫ|√\nǫ2−∆2δρǫ. (50)\nIn the case of the normal state, δρǫ=ρ′(ǫ)/ρ0from Eqs. (34) -(36). The calculation\nin the superconducting state is performed at T= 0 as noted above. In the case of\nthe normal state, the superconducting fluctuation depends on t he temperature. We\nfixT= 1.1TCin Eq. (30) and assume T= 0 in other terms. We take |ω|<1/τ\nandx=Dαq2<4/τas the range of integrations in Eqs. (35), (36) and (43). The\nenergy dependence of δρǫis mainly determined by the low-energy part |ω|,x≪1/τ.\nWhen we change the upper limits of |ω|andx, only the magnitude of |δρǫ|is shifted. We\nconsider theweak-coupling case forthe interactionbetween elect rons andphonons. This\ninteraction istaken tovanish outsidethecutoff frequency ( ωc),andthen Γ i(q),Γ′(q)∝ne}ationslash= 0\n(i= 0,1,2)onlyfor |ǫ|,|ǫ−ω|< ωc[Γ3(q)isfiniteoutsidethisregion.]Wetake ωc= 10∆\nin the numerical calculation. We specify the relation between α= 1/2τandkFlin Eqs.\n(35), (36) and (43) by putting kFl/2τ=EF= 300∆ ( EFis the Fermi energy).\nThe calculated results of the correction to the DOS are shown in Fig. 4. “SC”\nand “N” are the results calculated in the superconducting state an d the normal state,\nrespectively. “N 0” is the calculated result with only the term Γ 3(q) included in Eq.\n(36). The dependence of δρǫonǫchanges slightly with increasing α, and it is written as\nδρǫ∝√ǫfor “N 0”. As for the dependence of the magnitude of δρǫonα,δρǫ∝1/(kFl)2\nholds in both the superconducting and the normal states. This is re lated to the ǫ-\ndependence of δρǫbecause the equation\nρ′\ncl(ǫ)≃ρ0−3/radicalbig\n3τ/2\n(2kFl)2/integraldisplay1/τ\nǫdω1√ω∝√τ\n(kFl)2/parenleftbigg\n−1√τ+√ǫ/parenrightbigg\n(51)\nis derived from Eq. (36).\nTheǫ-dependences of δρǫare not exactly written as δρǫ∝√ǫfor “SC” and “N”.\nThe result for “SC” shows that a dip structure appears around ǫ= 3∆. This structure\nis resulted from the peak in Γ( q) around ω≃2∆. The reason for the overall sup-\npression in “SC” as compared to “N” is the enhancement of Γ 2and Γ′owing to the\ncoupling of the phase fluctuation to the density fluctuation. The re sult for “N” shows\nthat the superconducting fluctuation suppresses the DOS at low e nergy. The δρǫvalues\nof “SC” and “N” approach that of “N 0” at high energy owing to the weakening of the\nsuperconducting correlation for ǫ≫∆.\nThe difference inmagnitudebetween Γ iandΓ′shown inFig.2 isdirectly reflected in\n14/22J. Phys. Soc. Jpn. DRAFT\n-0.32-0.31-0.3-0.29-0.28-0.27-0.26-0.25-0.24-0.23\n 0 1 2 3 4 5 6 7 8(a)δρ\nε/∆SC\nN\nN0\n-0.08-0.075-0.07-0.065-0.06-0.055-0.05\n 0 1 2 3 4 5 6 7 8(b)δρ\nε/∆SC\nN\n-0.02-0.017-0.014\n 1 3 5 7\nFig. 4. The dependences of the correction to the DOS on ǫatT= 0. (a) α/∆ = 120 ( kFl= 2.5).\n(b)α/∆ = 60 ( kFl= 5.0). The inset shows the result for α/∆ = 30 ( kFl= 10.0). The meanings of\n“SC”, “N” and “N 0” are given in the text.\nδρǫ.δρǫin the superconducting state is decomposed into several terms, a nd the results\nare shown in Fig. 5. The decomposition is done according to Γ iand Γ′contained in\nEq. (43). For example, “3 ,2,,” in Fig. 5 represents the contribution from Γ 3, Γ2and\nΓ′toδρǫ. The calculated results show that the phase and density fluctuatio ns majorly\ncontribute to δρǫbecause they contain the long-range part ( ∝1/q2). The contribution\nfrom the amplitude fluctuation is small, as illustrated in Fig. 2.\n15/22J. Phys. Soc. Jpn. DRAFT\n-0.3-0.25-0.2-0.15-0.1-0.05 0\n 1 2 3 4 5 6 7 8δρ\nε/∆sum\n3,2,,\n0\n1\nFig. 5. The decomposition of δρǫinto severalterms according to Γ iand Γ′contained in δρǫ. “3,2,,”,\n”0” and “1” correspond to the suffixes of Γ iand Γ′. “sum” indicates the summation of these three\nquantities. α= 120∆ ( kFl= 2.5) andT= 0.\nEquation (43) seemingly includes a divergence proportional to 1 /√\nǫ2−∆2inδρǫ.\nTo clarify the reason for the absence of this divergence in Fig. 3, we decompose Eq.\n(43) as follows:\nρ′(ǫ) =ρ0|ǫ|√\nǫ2−∆2/parenleftbig\nδρsf\nǫ+δρcl\nǫ/parenrightbig\n(52)\nwith\nδρsf\nǫ=−3√\n3τ\n(2πkFl)2/integraldisplay\ndω/integraldisplay\ndx√x\n×Im/braceleftig\n2coth/parenleftigω\n2T/parenrightigIm[Γi(q)](hi+g+\nǫg+\nǫ−ω+h′\nif+\nǫf+\nǫ−ω)+2Im[Γ′(q)](f+\nǫg+\nǫ−ω−g+\nǫf+\nǫ−ω)\n(x+ζ+ǫ+ζ+\nǫ−ω)2\n+tanh/parenleftbiggǫ−ω\n2T/parenrightbiggΓi(q)(hi+g+\nǫg+\nǫ−ω+h′\nif+\nǫf+\nǫ−ω)+2Γ′(q)(f+\nǫg+\nǫ−ω−g+\nǫf+\nǫ−ω)\n(x+ζ+ǫ+ζ+\nǫ−ω)2/bracerightig\n(53)\nand\nδρcl\nǫ=−3√\n3τ\n(2πkFl)2/integraldisplay\ndω/integraldisplay\ndx√x\n×Im/braceleftig\ntanh/parenleftbiggǫ−ω\n2T/parenrightbiggΓi(q)(hi+g+\nǫg−\nǫ−ω+h′\nif+\nǫf−\nǫ−ω)+2Γ′(q)(f+\nǫg−\nǫ−ω−g+\nǫf−\nǫ−ω)\n(x+ζ+ǫ+ζ−\nǫ−ω)2/bracerightig\n.\n(54)\n16/22J. Phys. Soc. Jpn. DRAFT\n-0.4-0.3-0.2-0.1 0 0.1\n 1 2 3 4 5 6 7 8δρ\nε/∆sum\nsf\ncl\nFig. 6. The decomposition of δρǫ. “sf” and “cl” indicate δρsf\nǫandδρcl\nǫ, respectively. “sum” indicates\nthe summation of these two quantities. α= 120∆ ( kFl= 2.5) andT= 0.\nThe calculated results for these quantities are shown in Fig. 6. When ∆ = 0, Eqs. (53)\nand (54) reduce to Eqs. (35) and (36) (except for the factor ρ0), respectively. Both δρsf\nǫ\nandδρcl\nǫinclude the effects of the superconducting fluctuation and the Cou lomb inter-\naction in the case of ∆ ∝ne}ationslash= 0.δρsf(cl)\nǫincludes only the “retarded (advanced)” quantities\n(ζ+(−)\nǫ−ω,g+(−)\nǫ−ωandf+(−)\nǫ−ω). The calculated results show that the absence of the diver-\ngence proportional to 1 /ζ+\nǫ=i/√\nǫ2−∆2inδρǫis caused by the cancellation between\nthe retarded and the advanced parts [terms proportional to 1 /(x+ζ+\nǫ+ζ+\nǫ−ω)2and\n1/(x+ζ+\nǫ+ζ−\nǫ−ω)2].\nThe DOS with the correction included is written as follows:\nρ(ǫ) =ρ0|ǫ|√\nǫ2−∆2+ρ′(ǫ) =ρ0|ǫ|(1+δρǫ)√\nǫ2−∆2. (55)\nThe calculated result of this expression is shown in Fig. 7. In the norm al state, ρ(ǫ) =\nρ0(1+δρǫ). The result shows that ρ(ǫ) increases with increasing ǫfor large α. For small\nα,ρ(ǫ) decreases as |ǫ|/√\nǫ2−∆2because of the small values of δρǫ. This indicates that,\nalthough the ǫ-dependence of δρǫis almost independent of α, as shown in Fig. 4, the\nincreasing DOS with |ǫ|is observable only for large α.\n17/22J. Phys. Soc. Jpn. DRAFT\n 0.7 0.75 0.8 0.85 0.9\n 1 2 3 4 5 6 7 8ρ(ε)/ρ0\nε/∆SC\nN 0.9 1 1.1 1.2\n 1 3 5 7\nFig. 7. The DOS with the correction included. α= 120∆ ( kFl= 2.5). “SC” and “N” indicate\nthe result for the superconducting and the normal state, respe ctively. The inset shows the result for\nα= 30∆ (kFl= 10.0).\n4. Summary and Discussion\nInthisstudy, we calculatedthecorrectionto theDOSperturbativ ely. Thecorrection\nterm is given by the Coulomb interaction and the electron-phonon int eraction, with\nvertices of these interactions modified by the impurity scattering. The modification\nenhances these interactions at low energy. The energy dependen ce of the correction to\nDOS in the superconducting state is different from that in the norma l state, and a dip\nstructure appears at low energy. This structure is caused by the interaction which has\na peak at about twice the energy of the superconducting gap. (Th e dip structure in the\none-particle spectrum is also observed in cuprates, but its origin is d ifferent.23–25))\nThere are two differences between the superconducting state an d the normal state.\nFirst, the diffuson is modified because the opening of the supercond ucting gap changes\nthe dispersion of quasiparticles. This gives rise to another pole in the diffuson propa-\ngator, and this pole is treated correctly by including the coupling of t he density and\nphase fluctuations. Second, the correction to DOSdoes not affec t the gap-edge singular-\nity in the superconducting state. This is because the cancellation be tween the retarded\nand advanced parts occurs around the gap edge. In the normal s tate, the supercon-\nducting fluctuation and the Coulomb interaction separately contrib ute to the retarded\nand advanced parts, respectively. In the superconducting stat e we cannot treat them\n18/22J. Phys. Soc. Jpn. DRAFT\nseparately and need to include both parts simultaneously in the corr ection to DOS.\nRegarding the validity of perturbation expansion, if we consider the perturbation\nexpansioninthecaseofthescatteringbynonmagneticimpurities,t hecorrectiontoDOS\nis proportional to Im/summationtext\nk,k′Tr[ˆGkˆτ3ˆGk′ˆτ3ˆGk] = 0. The nonmagnetic impurities do not\naffect the DOSin theBornapproximation. In contrast, forparama gnetic impurities, the\ncorrection to DOS is proportional to Im/summationtext\nk,k′Tr[ˆGkˆτ0ˆGk′ˆτ0ˆGk]∝∆2|ǫ|/(ǫ2−∆2)3/2.\nThis means that the perturbation expansion is invalid around |ǫ| ≃∆, and the gap\nedge in the DOS changes qualitatively.26,27)The calculation in this paper shows that\nthe correction to DOS does not diverge around the gap edge. This in dicates that the\nperturbation expansion is valid within our approximations.\nWe calculated the Fock term with its vertices modified by diffuson (for exam-\nple, Fig. 3 (a) in Ref. 6, with the wavy line in this figure replaced by the C oulomb\ninteraction and the superconducting fluctuation in our calculation) . It is possible\nto consider other types of diagrams. For example, these are the F ock terms with\nits vertices modified by Cooperon and the Hartree term (Figs. 3 (b) −(d) in Ref.\n6 ). The correction to DOS by the Fock term with Cooperon is propor tional to\n−Im/summationtext\nk,qTr[ˆGk···/summationtext\nk1,k2Γk1−k2ˆGk1ˆτ3ˆGk2···ˆGq−k···ˆGq−k1ˆτ3ˆGq−k2···ˆGk]. The singu-\nlar part Γ q∝1/q2(which majorly contributes to the correction to DOS in our cal-\nculation) is weakened when the summations are performed. Thus, w e can omit this\ntype of diagram. There is a similar term in the case of the Hartree diag ram mod-\nified by diffuson or Cooperon. (In the case of the Fock term modified by diffuson,\n−Im/summationtext\nk,qTr[ˆGk···/summationtext\nk1,k2ΓqˆGk1ˆτ3ˆGk1−q···ˆGk−q···ˆGk2−qˆτ3ˆGk2···ˆGk].)\nThis study considers the case of low temperatures ( T≪∆). The superconducting\ngap ∆ was taken as the unit of energy, and we did not consider the int eraction effect on\nthe superconducting gap. When the temperature is comparable to the superconducting\ngap, the self-consistency through the gap equation becomes impo rtant.\nFinally, we comment on the possibility of observing a dip structure in ex periments.\nExperimentally, it is known that the superconducting state become s inhomogeneous\nwith decreasing kFl,17)and the one-particle spectrum is averaged over these inhomoge-\nneous states. (There are also theoretical studies on inhomegene ities in superconductors\nwithout Coulomb interaction.28,29)In addition, the perturbative calculation should be\nmodified for small values of kFlnear the insulating state, and the renormalization-\ngroup method30)will be required.) Thus, it is difficult to observe the dip structure in\nthe case of large values of α. Figures 4 and 7 show, however, that the dip structure is\n19/22J. Phys. Soc. Jpn. DRAFT\npossibly observed even for small values of α(kFl≫1, but in the dirty limit ∆ τ≪1)\nwhen the overall factor |ǫ|/√\nǫ2−∆2is removed. The dip structure originates from the\ninteractions in the superconducting state, and therefore the diff erence between our cal-\nculation and the calculations using the Coulomb interaction and diffuso n of the normal\nstate18,19)appears in this quantity.\nAcknowledgment\nThe numerical computation in this work was carried out at the Yukaw a Institute\nComputer Facility.\n20/22J. Phys. Soc. Jpn. 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B 93, 205432 (2016).\n22/22"    },    {        "title": "1910.11573v2.Experimental_study_on_the_bifurcation_of_a_density_oscillator_depending_on_density_difference.pdf",        "content": "Experimental study on the bifurcation of a density oscillator depending on density\ndi\u000berence\nHiroaki Ito,\u0003Taisuke Itasaka, Nana Takeda, and Hiroyuki Kitahatay\nDepartment of Physics, Graduate School of Science, Chiba University, Chiba 263-8522, Japan\n(Dated: September 6, 2021)\nHydrodynamic instabilities often cause spatio-temporal pattern formations and transitions be-\ntween them. We investigate a model experimental system, a density oscillator, where the bifurcation\nfrom a resting state to an oscillatory state is triggered by the increase in the density di\u000berence of\nthe two \ruids. Our results show that the oscillation amplitude increases from zero and the period\ndecreases above a critical density di\u000berence. The detailed data close to the bifurcation point provide\na critical exponent consistent with the supercritical Hopf bifurcation.\nI. INTRODUCTION\nLimit-cycle oscillations have been investigated inten-\nsively since they are related to a wide variety of dynam-\nical phenomena both in natural and arti\fcial systems.\nThere are mainly two scenarios for the realization of os-\ncillations. One is a harmonic-like oscillation realizing in\nan energy conservative system, and the other is a limit-\ncycle oscillation in which energy is alternately supplied\nand dissipated in one cycle. Thus, the observation of os-\ncillation in open systems is the mark of an existence of\na limit-cycle oscillation. Since energy dissipation is un-\navoidable in macroscopic and mesoscopic systems, many\noscillatory phenomena observed in such systems can be\nregarded as limit-cycle oscillations.\nOne of the most famous experimental systems that\nexhibit limit-cycle oscillation is a chemical oscillation\nrepresented by Belousov-Zhabotinsky (BZ) reaction, in\nwhich oxidation and reduction processes occur alter-\nnately and the solution color changes according to such\noxidation/reduction reactions [1, 2]. Other types of os-\ncillators have also been reported; e.g., Briggs-Rauscher\n(BR) reaction, glycolysis, and electrolysis [3]. Another\nclass is hydrodynamical systems such as the B\u0013 enard con-\nvection system and the density oscillator. These hy-\ndrodynamical systems are often discussed in association\nwith the atmospheric circulation and thermohaline circu-\nlation [4, 5]. Martin \frstly reported the density oscillator,\nand he discussed it as a simpli\fed experimental system\nof thermohaline circulation [7].\nThe limit-cycle oscillations in the hydrodynamical sys-\ntems are important not only for the relation to such at-\nmospheric and thermohaline circulations but also as the\nsuitable model systems to investigate the bifurcation phe-\nnomena seen in dynamical systems. The hydrodynamical\nsystems have many degrees of freedom, but they often\nshow successive bifurcation structure from a trivial state\nto oscillatory states, quasi-periodic oscillation states, and\nchaotic states that can be described by a model with a\nsmall number of variables [6].\n\u0003ito@chiba-u.jp\nykitahata@chiba-u.jpOne of the simple and appropriate experimental sys-\ntems that exhibit hydrodynamic limit-cycle oscillations\nis a density oscillator. In this system, the gravitational\ninstability originating from an upset density pro\fle of\nhigher- and lower-density \ruids induces upstream and\ndownstream alternation, which can be regarded as a\nlimit-cycle oscillation. From Martin's \frst report in 1970,\nthere have been experimental and theoretical papers on\nthe density oscillators. Some are on the theoretical anal-\nysis on the mechanism of the oscillation [8{15] and others\nare on the coupling between two-or-more oscillators [16{\n21]. For the bifurcation phenomenon, some theoretical\nstudies have been performed based on the reduced ordi-\nnary di\u000berential equations, which have predicted various\ntypes of bifurcations according to various bifurcation pa-\nrameters [10, 11] However, the experimental analyses on\nthe bifurcation structure with physical parameters be-\ntween the resting and oscillatory states are missing, and\nthe detailed behaviors close to the bifurcation point re-\nmain unclear.\nIn the present paper, we report the experimental re-\nsults on the bifurcation between the resting and oscilla-\ntory states by changing the density of the higher-density\n\ruid as a bifurcation parameter. We measured the os-\ncillation amplitude and period depending on the density\nof the higher-density \ruid, which indicate the bifurcation\nstructure between the resting and oscillatory states.\nII. EXPERIMENTAL SETUP\nAll aqueous solutions and distilled water were prepared\nwith Elix water puri\fcation system Elix UV 3 (Merck,\nDarmstadt, Germany). Sodium chloride (NaCl) aque-\nous solution as a higher-density \ruid was prepared by\ndissolving NaCl (Wako Pure Chemicals, Tokyo, Japan)\ninto the distilled water with various weight/volume con-\ncentrations c. All the aqueous solutions and pure water\n(c= 0 g=L, for control) were degassed for 1 hour in a\nvacuum chamber before use. An observation chamber of\na density oscillator was made of acrylic plates, which is\ncomposed of a smaller container surrounded by a larger\ncontainer for higher- and lower-concentration solutions,\nrespectively. The acrylic plates were 10 mm in thick-arXiv:1910.11573v2  [nlin.PS]  25 Feb 20202\nTop view(a)\n(b)50 mm\nSide view\n90 mm70 mm70 mm 70 mm\n230 mm70 mm\n170 mm0 µm\n2 mmSurface\nheight hLaser displacement meterMeasured\nspot\nø1 mm1 mm\nLaser displacement meter LED light source\n(behind the container)\nAcrylic container\nVibration isolated tableScreenh\nFIG. 1. Experimental setup. (a) Dimensions of the chamber\nmade of acrylic plates. The laser displacement meter and the\nmeasured spot are also shown. (b) Overview of the surface-\nheight measurement and observation.\nness except for the bottom of the smaller container with\n2 mm thickness. The center of the bottom of the smaller\ncontainer had a small cylindrical hole with a diameter of\n1 mm and a length of 2 mm to connect these two con-\ntainers. They were kept isolated by blocking the hole\nwith a needle before the measurements. The dimensions\nof the observation chamber and a photograph of the ex-\nperimental setup are shown in Fig. 1.\nWe put 245 mL of NaCl aqueous solution in the smaller\ncontainer, and 1400 mL of pure water in the larger con-\ntainer to reach the same water levels in these two con-\ntainers. After removing the blocking of the small hole,\nthe \row was induced through the hole if the concentra-\ntion di\u000berence was greater than a critical value. The\ntime series of the surface height of the solution in the\nouter container was measured with a laser displacement\nmeter (32.77 fps, LT9010M, Keyence, Japan) [12]. We\nmeasured the surface height for 4000 s. As the surfaceheight adequately relaxed to a steady resting or oscilla-\ntory state in 2000 s, we analyzed the data from 2000 s\nto 4000 s. It should be noted that the steady oscillatory\nstate with the repetitive macroscopic \row is character-\nistic for limit-cycle oscillations. To visualize the pro\fle\nof the density di\u000berence between higher- and lower-NaCl\nconcentration, we utilized the optical-index di\u000berence in\na similar manner with the shadowgraph method [22] us-\ning a light emitting diode (LED) as a light source and\na plastic sheet (TAMIYA Inc., Shizuoka, Japan) as a\nscreen. The obtained time series of the surface height\nwere processed as follows: The surface height linearly de-\ncreased due to the evaporation, and thus the linear trend\nwas reduced by subtracting the least-squares \ftted linear\nfunction. Then, the data were smoothed by applying a\nband-pass \flter between 0.003 and 0.019 Hz. Using the\nsmoothed data and their time derivatives, we detected\nthe local maximum and minimum points. The time se-\nries of the time derivative were calculated as the slope\nof adjacent\u000610 points (local data points for 0.64 s) ob-\ntained by the least-squares \ftting. We obtained the mean\npeak-to-peak distances and the mean time intervals of the\nlocal minima as the oscillation amplitude and the period,\nrespectively, for each condition of NaCl concentration c.\nIII. EXPERIMENTAL RESULTS\nFigure 2 shows the upstream and downstream series\nthrough the small hole connecting the higher- and lower-\nconcentration solutions. Characteristic oscillatory be-\nhavior of the density oscillator is observed. The con-\ncentration of the NaCl aqueous solution was c= 30 g=L.\nIn this setup, the upstream of lower concentration solu-\ntion and the downstream of higher concentration solution\nwere visualized as dark and light images, respectively,\nsince the cylindrical \row shape acts as concave or convex\noptical lenses. The upstream and downstream series were\nrepeated with a typical period of \u001850 s in this condi-\ntion. In this timescale, the cylindrical \rows did not mix\nwith the surrounding solutions, i.e., di\u000busion of solutes\nis negligible. Figures 2(b) and 2(c) show the details for\nthe switching of these opposite \rows, where the arrow-\nlike shaped \row enters the other solution at an almost\nconstant velocity.\nFor the quantitative measurement of the oscillation,\nwe used a time series of the surface height of the solu-\ntion in the outer container obtained by the laser displace-\nment meter. The characteristic time series of the surface\nheight is shown in Fig. 3. Figure 3(a) shows the result for\nc= 0 g=L as a control condition. The unprocessed time\nseries (blue line) shows a constantly-decreasing trend due\nto the slow evaporation of water, even though a total\nvolume change of the aqueous phase is negligibly small.\nBy subtracting the linear trend (black line), which is ob-\ntained by the least-squares method, time evolution of the\nsurface height h(t) is obtained. h(t) directly re\rects the\n\ruid \row through the small hole between the inner and3\n(b) (a)\n10 sUpstream Downstream Upstream Downstream20 mm \n2 s30 mm (c) (d)Time\nTime Time• • • \nFIG. 2. Snapshots of the oscillatory behavior of the density oscillator. The concentration of the NaCl aqueous solution was\nc= 30 g =L. The oscillatory \row was visualized by the optical-index di\u000berence. (a) Displayed area. (b) Typical oscillatory\n\row. (c) Details for the switching from upstream to downstream. (d) Details for the switching from downstream to upstream.\nSnapshots every (b) 1.67 s and (c,d) 0.33 s are shown. The background-subtracted images are available in Fig. S1 in the SM.\nouter containers. While the time series for c= 0 g=L is al-\nmost constant with small \ructuation (Fig. 3(a)), that for\nc= 30 g=L exhibits steady oscillation (Fig. 3(b)), which\ncorresponds to the oscillatory \row through the hole.\nNote that the linear trends for both resting ( c= 0 g=L)\nand oscillatory ( c= 30 g=L) states were measured in 3\nhours under the condition with similar humidity, and\nshowed the similar decreasing rate with the slopes of\n\u00000:00746\u0016m\u0001s\u00001and\u00000:00772\u0016m\u0001s\u00001, respectively,\nwhich are compatible with the typical rate for water evap-\noration. For further quanti\fcation of the amplitude and\nperiod of the oscillation, we applied a band-pass \flter in\na frequency range 0.003{0.019 Hz (red-dashed line).\nFigure 4 shows the concentration dependence of the\ntypical time series of the surface height for c= 0 g=L\n(control),c= 1:5 g=L,c= 3 g=L,c= 10 g=L, and\nc= 30 g=L. In addition to the original trend-subtracted\ndata (gray points), we plotted smoothed data obtained by\na moving average with adjacent \u000664 points (3.91 s) (black\nline) and the data with a band-pass \flter (red-dashed\nline) (Figs. 4(a-e)). The corresponding whole dynamics\nincluding the initial transient states are shown in Fig. S2\nin the SM. While the surface heights for small c, e.g.,\nc= 0 g=L, are in a resting state, those for c\u00151:5 g=L ex-\nhibit characteristic oscillation. According to the increase\ninc, the amplitude becomes larger and the period be-\ncomes shorter. The detailed wave pro\fles, shown in each\ninset, also change from smooth sinusoidal-like oscillations\nto relaxation oscillations. Figures 4(f{j) show the corre-\nsponding trajectories of the trend-subtracted smoothed\nTime t (s) c = 0 g/L (control) \nc = 30 g/L Unprocessed(a)\n(b)Linear fitting\nTrend-subtracted Band-pass filteredHeight h (µm) Height h (µm) 0 250010 \n-1020 30 \n500 750 1000 1250 1500 1750 2000\nTime t (s) 0 250010 \n-1020 30 \n500 750 1000 1250 1500 1750 2000FIG. 3. Time series of the surface height measured with\na laser displacement meter for (a) c= 0 g =L (control) and\n(b)c= 30 g =L. Unprocessed data (blue lines), linear \ftting\nfor the unprocessed data (black lines), trend-subtracted data\n(gray lines), and smoothed data with a band-pass \flter (red-\ndashed lines) are shown.4\nt (s)\nTime t (s) h (µm)dh/dt (µm/s) (a) (f)\nData\nM. A. \nB.-P.Height h (µm) \nh (µm) \n0 001\n-1 5 10 -5 -10010 20 \n500 1000 1500 20000\n-5 5\n500 250c = 0 g/L (control)\nt (s)\nTime t (s) h (µm)dh/dt (µm/s) (b) (g)\nData\nM. A. \nB.-P.Height h (µm) \nh (µm) \n0 001\n-1 5 10 -5 -10010 20 \n500 1000 1500 20000\n-5 5\n500 250c = 1.5 g/L\nt (s)\nTime t (s) h (µm)dh/dt (µm/s) (c) (h)\nData\nM. A. \nB.-P.Height h (µm) \nh (µm) \n0 001\n-1 5 10 -5 -10010 20 \n500 1000 1500 20000\n-5 5\n500 250c = 3 g/L\nt (s)\nTime t (s) h (µm)dh/dt (µm/s) (d) (i)\nData\nM. A. \nB.-P.Height h (µm) \nh (µm) h (µm) 0 001\n-1 5 10 -5 -10010 20 \n500 1000 1500 20000\n-5 5\n500 250c = 10 g/L\nt (s)\nh (µm)dh/dt (µm/s) (j)\n001\n-1 5 10 -5 -10\nTime t (s)(e)\nData\nM. A. \nB.-P.Height h (µm) \n00\n-5 5\n010 20 \n500500 250\n1000 1500 2000c = 30 g/L\nFIG. 4. Oscillations for di\u000berent NaCl concentrations c. (a)\n0 g=L, (b) 1 :5 g=L, (c) 3 g =L, (d) 10 g =L, and (e) 30 g =L.\nTrend-subtracted data (Data, gray dots), moving-average (M.\nA., black lines), and band-pass \fltered data (B.-P., red-\ndashed lines) are plotted. Each inset shows a magni\fed re-\ngion of 250 s \u0014t\u0014500 s. (f{j) Corresponding trajectories in\na phase space ( h;dh=dt).\ndata in a phase space ( h;dh=dt). According to the in-\ncrease inc, the trajectories corresponding to the limit-\ncycle orbits with larger amplitudes appear. Also in the\ntrajectories for c\u001510 g=L, we can clearly see a char-\nacteristic shape of relaxation oscillation. Though the\nsmoothed plot for small oscillation in c= 1:5 g=L shows\nno clear cyclic trajectory due to the long-term \ructu-\nation of the baseline, the plot after a band-pass \flter\nshows su\u000eciently large oscillatory amplitude, which will\n(a)\n(b)\nc (g/L)T (s) \n0040 \n20 60 100140\n80 120160\n5 10 15 20 25 30 \nT (s) \n040 80 120160\nc (g/L) 0 1 2 3 4 5c (g/L)h (µm) \n00246810 12 \n5 10 15 20 25 30 h (µm) \n012\nc (g/L) 0 1 2 3 4 5FIG. 5. Amplitude and period of the density oscillator for\nvarious concentrations of NaCl aqueous solution as a higher-\ndensity \ruid. (a) Amplitude. The dashed line in the inset\nshows the threshold value for the resting state. (b) Period.\nEach inset shows the magni\fed region around the bifurcation\npoint.\nbe discussed later.\nFrom the band-pass \fltered data, we obtained the am-\nplitudes and the periods of the oscillations for various\nconcentrations c, as shown in Figs. 5(a) and 5(b), re-\nspectively. The results in Fig. 5(a) show that the ampli-\ntude shows a steep increase above a critical concentra-\ntioncc'1 g=L, and it monotonically increases for larger\nconcentrations. Here, we determined one standard de-\nviation of the amplitude in the control condition ( c=\n0 g/L) as the threshold amplitude between the resting\nand oscillatory states. By this standard, c\u00141 g=L is\nin a resting state, and c\u00151:125 g=L shows a limit-cycle\noscillation. For the conditions with limit-cycle oscilla-\ntionsc\u00151:125 g=L, the oscillation period Tis plotted\nagainst various cin Fig. 5(b). The period Tmonotoni-\ncally decreases with the increase in the concentration c,\nand converges to T'50 s for larger c.5\nIV. DISCUSSION\nThe driving force of the density oscillator is the grav-\nitational energy. In fact, the density pro\fle is upset in\nthe initial stage; in other words, the higher-density \ruid\nis located above the lower-density \ruid. This instability\ncan induce the limit-cycle oscillation through the dissipa-\ntion of the gravitational energy. From the experimental\nresults, the limit-cycle oscillation was observed when the\nconcentration of NaCl aqueous solution, c, was above a\ncritical value cc'1 g/L. Considering that the gravita-\ntional energy is proportional to the density, the results\nsuggest that the oscillatory \row is induced by the upset\ndensity pro\fle but it should su\u000ber from the resistance\ndue to the \ruid viscosity. The gravitational energy due\nto the upset pro\fle can overcome the viscosity for c>c c,\nwhile it cannot for c<c c.\nThe existence of the critical value ccfor the instability\nis also interpreted from the transport phenomenon. Since\nthe solute transportation is dominated by di\u000busion in the\nresting state and by the advection in the oscillatory state,\nthe Rayleigh number Ra= (gL3\u0001\u001a=\u001a)=(\u0017D) provides a\ncritical value for the density di\u000berence \u0001 \u001ato drive the os-\ncillation. Here, g(\u001810 m\u0001s\u00002) is the gravitational accel-\neration,L(\u00180:5 mm) is the hole radius as a characteris-\ntic length, \u0001 \u001a=\u001a(\u001810\u00002) is the ratio of the density di\u000ber-\nence,\u0017(\u00181:0\u000210\u00006m2\u0001s\u00001) is the kinematic viscosity\nof water at room temperature, and D(\u001810\u00009m2\u0001s\u00001) is\nthe di\u000busion constant of solutes in water at room temper-\nature. If we take a critical Rayleigh number Ra(\u0018103)\ntypical for Rayleigh-B\u0013 enard instability, the critical den-\nsity di\u000berence \u0001 \u001ac, namely, the critical solute concentra-\ntioncccan be estimated as cc\u00180:8 g=L, which agrees\nwell with the bifurcation point \u00181:0 g=L obtained in\nour experiment. Note that this value is much smaller\ncompared to the parameter ranges in the previous exper-\niments [8, 12]. Actually, the decreases in the density of\nsalt water per a cycle are roughly estimated as \u0018\u00003%\nin Steinbock et al. [8] and \u0018\u00000:2% in Ueno et al. [12],\nwhich is much larger than our experiment, \u0018\u00000:003%.\nThese previous experiments showed continuous shifts of\nthe water surface in mm{cm length scale during the mea-\nsurements due to the decrease in the density di\u000berence\nby mixing of the solutions between the two containers. In\ncontrast, our experiment measured the motion of surface\nheight in\u0016m length scale, and therefore, the bifurcation\nphenomenon at the small density di\u000berence as well as the\nwater evaporation could be detected.\nFrom the viewpoint of the dynamical systems, this\ntransition can be discussed in terms of bifurcation. There\nare several scenarios known for the transition from a rest-\ning state to an oscillatory state; supercritical Hopf bifur-\ncation, saddle-node bifurcation of cycles (subcritical Hopf\nbifurcation), in\fnite-period bifurcation (saddle-node ho-\nmoclinic bifurcation, saddle-node bifurcation on an in-\nvariant circle (SNIC)), homoclinic bifurcation, and so on\n[23{26]. For a supercritical Hopf bifurcation, the am-\nplitude of oscillation raises from a bifurcation point in\n0.0010 2\n10 0\n10 -2 \n10 -4 \n10 -6 c = 0 g/L \n(control)(a)\n(b)0.02 0.04MeanBand-pass filter\n(0.003–0.019 Hz)\nCharacteristic\nmodeS.D.\nc = 1.5 g/L\nMean\nS.D.0.06\nFrequency f (Hz)Power P (µm 2)\n0.08 0.10\n0.0010 2\n10 0\n10 -2 \n10 -4 \n10 -6 \n0.02 0.04 0.06\nFrequency f (Hz)Power P (µm 2)\n0.08 0.10\n(c)\nc = 3 g/L\nMean\nS.D.\n0.0010 2\n10 0\n10 -2 \n10 -4 \n10 -6 \n0.02 0.04 0.06\nFrequency f (Hz)Power P (µm 2)\n0.08 0.10\n(d)\nc = 10 g/L\nMean\nS.D.\n0.0010 2\n10 0\n10 -2 \n10 -4 \n10 -6 \n0.02 0.04 0.06\nFrequency f (Hz)Power P (µm 2)\n0.08 0.10\n(e)\nc = 30 g/L\nMean\nS.D.\n0.0010 2\n10 0\n10 -2 \n10 -4 \n10 -6 \n0.02 0.04 0.06\nFrequency f (Hz)Power P (µm 2)\n0.08 0.10FIG. 6. Power spectra for di\u000berent NaCl concentrations c.\n(a) 0 g =L, (b) 1 :5 g=L, (c) 3 g =L, (d) 10 g =L, and (e) 30 g =L.\nMean (black line) \u0006standard deviation (S.D., gray region) of\nthe samples is plotted. Characteristic frequency modes for the\nlimit-cycle oscillation are indicated by the black solid arrows.\nThe spectra include the harmonics for these modes\nproportion to the square root of the distance from the\nbifurcation point, while the amplitude has a \fnite value\nat the bifurcation point for the other three types.\nIn the experiments, the amplitude of the oscillation\nseems to increase from zero with an increase in cfor\nc > c cas shown in Fig. 5(a). If we assume that the\nsystem undergoes the supercritical Hopf bifurcation, the\ndependence of the amplitude on the concentration cis\nsuggested to be proportional to ( c\u0000cH\nc)1=2when the sys-\ntem is close to the bifurcation point. Here, cH\ncis the6\nc (g/L) c – cc (g/L)~ (c – cc )0.482cc (a)\n(c) (d)(b)h2 (µm 2)T (s)\nT–2 ( ×10 -4 s–2)h (µm)\n0012345\n1 2 3 4 5\nc (g/L)0 1 2 3 4 5Noise level\n10 -1 10 -1 10 0\n10 22×10 2\n2\n1\n6×10 1 010 0H\nH\nc – cc (g/L)10 -1 10 0\nHH\n1\n2= 1.03 (g/L)\nccI= 4.87×10 –2  (g/L)\nFIG. 7. Critical behavior of the bifurcation with respect\nto the concentration. (a) Linear \ftting of squared ampli-\ntude in (1 :25 g=L\u0014c\u00142 g=L), close to the bifurcation\npoint. The intersection estimates the Hopf bifurcation point\ncH\nc= 1:03 g=L. (b) Double logarithmic plot of amplitude\nhversus concentration di\u000berence c\u0000cH\nc. The power ex-\nponent for (1 :25 g=L\u0014c\u00142 g=L) is 0.482. (c) Double\nlogarithmic plot of period Tversus concentration di\u000berence\nc\u0000cH\nc.Tapproaches constant as capproaches the Hopf bi-\nfurcation point cH\nc. Slope is the eye guide for the compar-\nison with the scaling \u0018(c\u0000cH\nc)\u00001=2. (d) Linear \ftting of\nT\u00002in (1:25 g=L\u0014c\u00142 g=L) by assuming the scaling for\nthe in\fnite-period bifurcation \u0018(c\u0000cI\nc)\u00001=2. The intersec-\ntion estimates the possible in\fnite-period bifurcation point\ncI\nc= 4:87\u000210\u00002g=L, which is apparently inconsistent with\nFig. 5(a).\ncritical concentration for the supercritical Hopf bifurca-\ntion. We performed power-law \fttings of the experimen-\ntal data to determine the bifurcation point and the power\nexponent. First, we checked the power spectra for each\nconcentration cto determine the range of the \\close-to-\nbifurcation-point region\", where the critical behavior is\nexpected. Figure 6 shows the power spectra correspond-\ning to the conditions shown in Fig. 4. Above cc, a clear\n\frst peak for the characteristic limit-cycle oscillation ap-\npears within a frequency range of the band-pass \flter,\n0.003{0.019 Hz. For higher concentrations, c\u00153 g=L,\nharmonics originating from the nonlinear waveform of\nrelaxation oscillation appear, i.e., the system is in the\n\\far-from-bifurcation-point region\". Thus, we performed\nlinear \ftting of the squared amplitude in the \\close-to-\nbifurcation-point region\" as shown in Fig. 7(a), and ob-\ntained the critical concentration as cH\nc= 1:03 g=L. Here,\nthe closest oscillatory condition c= 1:125 g=L was elim-\ninated from the \ftting, because the amplitude was closeto the noise level. Then, the double logarithmic plot of\nthe amplitude versus c\u0000cH\ncshown in Fig. 7(b) provides\nthe power exponent close to the bifurcation point, result-\ning in 0:482, which is close to 1 =2. It suggests that the\nbifurcation would be classi\fed into the supercritical Hopf\nbifurcation.\nThe supercritical Hopf bifurcation is also character-\nized by a \fnite angular velocity in the phase space at\nthe bifurcation point. In Fig. 5(b), the period increased\nascapproached the bifurcation point from higher con-\ncentrations c > c c. This behavior recalls in\fnite-period\nbifurcation, in which the amplitude is \fnite while the pe-\nriod diverges at the bifurcation point cI\ncwith the scaling\nof (c\u0000cI\nc)\u00001=2. We further checked the critical behavior\nof the period versus c\u0000cH\ncin the double logarithmic plot\nshown in Fig. 7(c), and con\frmed that the period does\nnot seem to diverge. Instead, it remains constant around\nthe bifurcation point, which is also consistent with the\nsupercritical Hopf bifurcation. Moreover, we also es-\ntimated another critical concentration cI\ncby assuming\nthe in\fnite-period bifurcation through the critical be-\nhavior of the inversed square period. This test resulted\nincI\nc= 4:87\u000210\u00002g=L as shown in Fig. 7(d), which\nis apparently inconsistent with the experimentally indi-\ncatedcc'1:0 g=L as observed in the amplitude shown\nin Fig. 5(a). While our experimental results and anal-\nyses strongly support the supercritical Hopf bifurcation,\nthe precise asymptotic behaviors are still not clear due\nto the experimental limitations. From the experimen-\ntal observation only, it is di\u000ecult to completely exclude\nthe possibilities for other types of bifurcations. From\nthe present results, the saddle-node bifurcation of cycles\nwas dismissed since the bistability between the resting\nand oscillatory states was not observed. Further study\nis needed to identify the bifurcation class of the density\noscillator depending on the density di\u000berence.\nV. SUMMARY\nWe investigated the transition from the resting state\nto the oscillatory state depending on the density di\u000ber-\nence in a density oscillator. The limit-cycle oscillation,\nwhere the upstream and downstream alternations occur,\nwas observed for the higher density di\u000berence, while no\nhydrodynamic \row was observed for the lower density dif-\nference. The detailed data close to the bifurcation point\nprovide a critical exponent close to 1 =2 and a \fnite period\naround the bifurcation point, which is consistent with the\nsupercritical Hopf bifurcation. Further experimental and\ntheoretical studies should be performed to exactly iden-\ntify the bifurcation class since the experimental results\nstill showed the ambiguity.\nThe density oscillator is an excellent experimental sys-\ntem for the limit-cycle oscillation with hydrodynamic in-\nstability since it can be treated from the standpoint of\nphysics; complex processes such as chemical reaction and\nphase transition are not included, and thus only hydro-7\ndynamics is taken into consideration. The present exper-\nimental study will give fundamental knowledge on the\nnonlinear oscillation with hydrodynamic instability from\nthe viewpoint of bifurcation theory in dynamical systems,\nand help further studies using density oscillators.\nACKNOWLEDGMENTS\nThis work was supported by JSPS KAKENHI Grant\nNumbers JP19K14675, JP16H03949. It was also sup-ported by the Japan-Poland Research Cooperative Pro-\ngram \\Spatio-temporal patterns of elements driven by\nself-generated, geometrically constrained \rows\" and the\nCooperative Research Program of \\Network Joint Re-\nsearch Center for Materials and Devices\" (No. 20191030).\n[1] R. Kapral and K. Showalter, Chemical Waves and Pat-\nterns (Kluwer Academic, Dordrecht, 1995).\n[2] A. N. Zaikin and A. M. Zhabotinsky, Nature 225, 535\n(1970).\n[3] A. S. Mikhailov and G. Ertl, Chemical Complexity: Self-\nOrganization Processes in Molecular Systems (Springer,\nBerlin, 2017).\n[4] M. 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Holmes, Nonlinear oscilla-\ntions, dynamical systems, and bifurcations of vector \felds\n(Springer, New York, 1983).\n[25] E. M. Izhikevich, Dynamical systems in neuroscience:\nThe geometry of excitability and bursting (MIT Press,\nCambridge, MA, 2007).\n[26] Y. Murayama, H. Kori, C. Oshima, T. Kondo,\nH. Iwasaki, and H. Ito, Proc. Nat. Acad. Sci. USA 114,\n5641 (2017).8\n(b) (a)\n10 sUpstream Downstream Upstream Downstream20 mm \n2 s30 mm (c) (d)Time\nTime Time• • • \nFIG. S1. Background-subtracted images corresponding to Fig. 2 in the main text. The concentration of the NaCl aqueous\nsolution was c= 30 g =L. (a) Displayed area. (b) Typical oscillatory \row. (c) Details for the switching from upstream to\ndownstream. (d) Details for the switching from downstream to upstream.9\n(a)\n0-20-10010 20 \n500 1000 1500 2500 3000 3500 4000 2000\nTime t (s)Data\nMoving-averagec = 0 g/L (control)Height h (µm) \n(b)\n0-20-10010 20 \n500 1000 1500 2500 3000 3500 4000 2000\nTime t (s)Data\nMoving-averagec = 1.5 g/LHeight h (µm) \n(c)\n0-20-10010 20 \n500 1000 1500 2500 3000 3500 4000 2000\nTime t (s)Data\nMoving-averagec = 3 g/LHeight h (µm) \n(d)\n0-20-10010 20 \n500 1000 1500 2500 3000 3500 4000 2000\nTime t (s)Data\nMoving-averagec = 10 g/LHeight h (µm) \n(e)\n0-20-10010 20 \n500 1000 1500 2500 3000 3500 4000 2000\nTime t (s)Data\nMoving-averagec = 30 g/LHeight h (µm) \nFIG. S2. Typical results of the surface-height measurements for di\u000berent NaCl concentrations c. (a) 0 g =L, (b) 1 :5 g=L, (c)\n3 g=L, (d) 10 g =L, and (e) 30 g =L. Trend-subtracted data (Data, gray dots) and moving-averages (black lines) are plotted. The\nsystem reached the steady states within \u00181000 s. Each trend was obtained by the linear \ftting for t= 2000{4000 s."    },    {        "title": "1910.12380v3.A_Dixmier_trace_formula_for_the_density_of_states.pdf",        "content": "arXiv:1910.12380v3  [math-ph]  4 Mar 2020A DIXMIER TRACE FORMULA FOR THE DENSITY OF\nSTATES\nN.AZAMOV, E.MCDONALD, F.SUKOCHEV, D.ZANIN\nAbstract. A version of Connes trace formula allows to associate a measu re\non the essential spectrum of a Schr¨ odinger operator with bo unded potential.\nIn solid state physics there is another celebrated measure a ssociated with such\noperators — the density of states. In this paper we demonstra te that these\ntwo measures coincide. We show how this equality can be used t o give explicit\nformulae for the density of states in some circumstances.\nContents\n1. Introduction 1\n2. Example computations 5\n2.1. Radially homogeneous potentials 5\n2.2. Stability of the density of states 8\n2.3. Asymptotically homogeneous potentials 10\n3. Preliminaries 11\n3.1. Trace ideals 11\n3.2. Double operator integrals 12\n4. Cwikel type estimates 13\n5. A residue formula 18\n5.1. Abstract result 19\n5.2. The main residue formula 21\n6. Formula for the density of states 24\n7. Acknowledgements 28\nReferences 28\n1.Introduction\nLetd≥2, and let\n(1.1) H=−∆+V\nbe a Schr¨ odinger operator on Rd, where ∆ =/summationtextd\nj=1∂2\n∂x2\njis the Laplace operator\nandVis a bounded real-valued measurable potential V∈L∞(Rd).Thedensity of\nstates(orDOS) is a Borel measure νHonRnaturally associated to H,see e.g.\n[1,31,14,28], defined as follows. Let L >0, and let HLbe the restriction of Hto\nthe cube ( −L,L)dwith Dirichlet boundary conditions (for a definition see e.g. [ 27,\n§VI.4.4], [ 34,§XIII.15]).\n12 N.AZAMOV, E.MCDONALD, F.SUKOCHEV, D.ZANIN\nLetI⊂Rbe a bounded interval, and let NL(I) be the number of eigenvalues\nwith multiplicities of HLinI(which is necessarily finite, since HLhas compact\nresolvent). The density of states measure νHofIis defined as\n(1.2) νH(I) = lim\nL→∞NL(I)\nVol((−L,L)d).\nThe DOS measure does not always exist, see e.g. [ 41, p.513]. However, it is well\nknown to exist for certain classes of Hamiltonians important for solid state physics\nsuch as those corresponding to periodic, almost periodic and ergod ic potentials, see\ne.g. [1,7,31,34,41]. Another point to mention here is that the density of states\nmeasure νHhas several definitions. The difference is in the choice of domain in the\nlimit (1.2): one can replace the cubes {(−L,L)d}L>0with a family of balls or other\ndomains. There is also some variation in the choice of boundary condit ions used to\ndefineHL(such as Dirichlet, Neumann, periodic, etc.). For our purposes it will be\nconvenient to use yet another definition (see e.g. [ 41, (C41)], [ 20, (1.2)]) in terms\nof the spectral projections of H,as follows\n(1.3) νH(I) = lim\nR→∞1\nVol(B(0,R))Tr/parenleftbig\nMχB(0,R)χI(H)MχB(0,R)/parenrightbig\n,\nwhereB(0,R) is the ball of radius Rcentred at zero, Mfis the operator of mul-\ntiplication by a function fonL2(Rd),Tr is the standard operator trace and χA\nis the indicator function of a set A.(We use notation Mfwhen it is necessary to\ndistinguish a function ffrom the operator of multiplication by f,however occa-\nsionally when there is no danger of confusion we write fmeaning Mf,especially\nin the Schr¨ odinger operator −∆+V). It is known that, assuming existence, these\ndifferent definitions of DOS coincide at least for such important class esof potentials\nas periodic or ergodic, the latter including random and almost periodic potentials,\nsee e.g. [ 41, Theorem C.7.4] and [ 20]. In this paper we will assume existence of the\nlimit (1.3).\nThe density of states has attracted substantial attention in the mathematical\nliterature. Items of particular interest have been the existence o fνHin various\ncircumstances, the asymptoticbehaviourofthe integrateddens ityofstatesfunction\nνH((−∞,λ]) asλ→ ∞and its continuity properties [ 39,11]. As an example of\nthese results, we mention in particular that it follows from the work o f Shubin [ 39,\nTheorem 4.5] that if Vis smooth and almost periodic (see [ 39,§1.2] for details)\nthenνHexists and\nνH((−∞,λ]) =ωd\n(2π)dλd\n2+O(λd\n2−1)\nasλ→ ∞, where\nωd:=2πd/2\nΓ/parenleftbigd\n2/parenrightbig\nis the (d−1)-volume of the unit sphere Sd−1. More recently, Bourgain and Klein\n[11, Theorem 1.1] proved among other results that in dimensions d= 1,2,3 the\ndensity of states (defined in terms of cubes, as in ( 1.2)) satisfies the following local\nlog-H¨ older continuity property:\nνH([E,E+ε])≤Cd,V,Elog(ε−1)−2−d, E∈R,ε≤1\n2\nwhenever Vis bounded and νHexists.A DIXMIER TRACE FORMULA FOR THE DENSITY OF STATES 3\nThe second measure which can be associated with Hcomes from a version of\nConnes’ trace formula [ 15], [23, Corollary 7.21], [ 30, Theorem 11.7.5]. One form\nof Connes trace formula asserts that for all continuous and comp actly supported\nfunctions fonRd, we have:\n(1.4) Tr ω/parenleftBig\nMf(1−∆)−d/2/parenrightBig\n=ωd\nd(2π)d/integraldisplay\nRdf(t)dt,\nwhere Tr ωis a Dixmier trace on the ideal L1,∞(L2(Rd)).For our purpose it is\ndesirable to rewrite this formula in the Fourier transform picture, a s follows:\n(1.5) Tr ω/parenleftBig\nf(−i∇)M−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight/parenrightBig\n=ωd\nd(2π)d/integraldisplay\nRdf(t)dt,\nwhere∇= (∂1,...,∂ d) is the gradient operator, f(−i∇) is defined via functional\ncalculus, and\n/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht= (1+|x|2)1/2.\nWe would like to rewrite this formula in terms of the Laplacian −∆ rather than the\ngradient operator ∇. To this end, consider the case where fis a radial function,\nand therefore f(−i∇) can be written as g(−∆) for some continuous compactly\nsupported function gon [0,∞). Then by switching to polar coordinates we have\nTrω/parenleftBig\ng(−∆)M−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight/parenrightBig\n=ωd\nd(2π)d/integraldisplay\nRdg(|x|2)dx\n=ωd\nd(2π)d/integraldisplay\nSd−1/integraldisplay∞\n0g(r2)rd−1drds\n=ω2\nd\nd(2π)d/integraldisplay∞\n0g(r2)rd−1dr\n=ω2\nd\n2d(2π)d/integraldisplay∞\n0g(λ)λd\n2−1dλ.\nNow,let V∈L∞(Rd)beareal-valuedpotential. Ourmainresultisthefollowing:\nTheorem 1.1. LetH=−∆+MVbe a Schr¨ odinger operator on L2(Rd), whereV\nis a bounded real-valued measurable potential. For any g∈Cc(R)the operator\ng(H)M−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight\nbelongs to the weak trace-class ideal L1,∞(L2(Rd)). If we assume that the density\nof states of H(defined according to (1.3)) exists and is a Borel measure νHonR,\nthen for every Dixmier trace TrωonL1,∞there holds the equality\n(1.6) Tr ω/parenleftBig\ng(H)M−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight/parenrightBig\n=ωd\nd/integraldisplay\nRgdνH.\nItisinstructivetoconsiderthesimplestcase, V= 0,whichalsoservestocompute\nthe constant in ( 1.6).\nExample 1.2. ForH0=−∆we have\nTrω(g(H0)M−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight) =ωd\nd/integraldisplay\nRg(λ)dνH0(λ)\nfor allg∈Cc(R).4 N.AZAMOV, E.MCDONALD, F.SUKOCHEV, D.ZANIN\nProof.We shall verify that:\nTrω(e−tH0M−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight) =ωd\nd/integraldisplay∞\n0e−tλdνH0(λ), t >0.\nThis suffices to ensure that the equality holds for all g∈Cc(R) (see the argumentin\nRemark 6.3) provided both sides exist. The existence of the left-hand side follo ws\nfrom classical Cwikel estimates, or may be derived from the Cwikel e stimates given\nbelow in Section 4. That there is indeed an explicitly computable density of states\nmeasure for this case is well known (see e.g. [ 41, Theorem C.7.7]).\nConnes’ trace formula in the form ( 1.5) yields:\nTrω(et∆M−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight) =ωd\nd(2π)d/integraldisplay\nRde−t|x|2dx\n=ωd\nd(2π)d/parenleftBigπ\nt/parenrightBigd/2\n=ωd\nd(4πt)−d/2.\nWe may comparethis to/integraltext∞\n0e−tλdνH0(λ) using (1.3). Accordingto [ 41, Proposition\nC.7.2], it suffices to compute:\nlim\nR→∞1\n|B(0,R)|Tr(MχB(0,R)e−tH0).\nThe semigroup e−tH0has integral kernel (see e.g. [ 33,§IV.7, Example 3]):\nKt(x,y) = (4πt)−d/2e−|x−y|2\n4t.\nHenceKt(x,x) = (4πt)−d/2is constant, and we have:\n1\n|B(0,R)|Tr(MχB(0,R)e−tH0) =1\n|B(0,R)|/integraldisplay\nB(0,R)(4πt)−d/2dx= (4πt)−d/2.\n/square\nTheorem 1.1observes a direct connection between two measures which can nat -\nurally be associated with the operator ( 1.1). Since these two measures a priori\nhave very different definitions, this connection ought to be conside red as some-\nwhat surprising. At the same time, both measures do share some ob vious common\nproperties. Indeed, both measures are invariants of a Schr¨ odin ger operator ( 1.1),\nboth are suppported on the essential spectrum of Hand they both exhibit certain\nrobustness. Namely, the Dixmier trace Tr ω,used in the definition of one of these\nmeasures, is insensitiveto trace classperturbations, while the den sity ofstates mea-\nsureνHis insensitive to localised perturbations V0+Vof the bounded potential V0,\n[41, Theorems C.7.7 and C.7.8] reflecting the fact that DOS is a property of the\nbehaviour of the potential Vat infinity.\nAs mentioned above, throughout this paper we assume existence o f the density\nof states. That is, we assume existence of the limit ( 1.3). It is noteworthy however\nthat the left hand side of ( 1.6) is meaningful for an arbitrary bounded potential\nVand defines a Borel measure on the spectrum of H=−∆ +MV. In the event\nthat the density of states does indeed exist, ( 1.6) implies that the value of the\nDixmier trace Tr ω(f(H)M−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight) is independent of the extended limit ω. In general,\nan operator Tin the weak trace ideal is called Dixmier measurable if Tr ω(T) is\nindependent of the choice of Dixmier trace Tr ω. Hence, existence of the densityA DIXMIER TRACE FORMULA FOR THE DENSITY OF STATES 5\nof states implies that f(H)M−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ightis Dixmier measurable for all f∈Cc(R). We\nconjecture that the converse holds.\n2.Example computations\nBefore giving the full proof of Theorem 1.1, we can demonstrate its utility to\nexplicitly compute the density of states in a number of examples. To t he best of\nour knowledge, these formulae are new. In these examples we have not attempted\nto prove that the density of states exists, nonetheless using ( 1.6) we can compute\nit conditional on the assumption of existence.\nGiven a potential V, we denote the density of states measure of H=−∆+V\nbyνH(assuming that the measure exists). We will also write H=−∆ +MV\nwhenever there is potential for confusion between the pointwise m ultiplierMVand\nthe function V.\n2.1.Radially homogeneous potentials. In this subsection we consider poten-\ntialsVwhich are positively homogeneous in the sense that V(tx) =V(x) for all\nt >0 and all x∈Rd.\nTheorem 2.1. DenoteH0=−∆and letH=H0+MV,whereV∈L∞(Rd)is\nsuch that V(tx) =V(x)for allt >0and allx∈Rd. We assume that the density\nof states of Hexists. Then the density of states of His the average over ξ∈Sd−1\nof the density of states of H0+V(ξ),in other words:\nνH0+V=1\nωd/integraldisplay\nSd−1νH0+V(ξ)dξ.\nThe spectral and scattering theory of Schr¨ odinger operators with positively ho-\nmogeneous potentials have been studied by Herbst and Skibsted [ 24,25,26], al-\nthough we are not aware of any results which imply the existence of t he density of\nstates for these potentials.\nThe computation in this case is based on a version of Connes trace th eorem\nproved by two of the authors in [ 44], which is stated as follows (c.f. [ 44, Theorem\n1.2]). Let Π be the C∗-subalgebra of B(L2(Rd)),generated by the algebras of\npointwise multipliers/braceleftbig\nMf:f∈L∞(Rd)/bracerightbig\nand homogeneous Fourier multipliers\n/braceleftbigg\ng/parenleftbiggD1\n(−∆)1/2,...,Dd\n(−∆)1/2/parenrightbigg\n:g∈L∞/parenleftbig\nSd−1/parenrightbig/bracerightbigg\n,\nwhereDk=1\ni∂\n∂xkandSd−1is thed−1 dimensional unit sphere. Theorem 1.2 of\n[44] asserts that there exists a unique C∗-algebra homomorphism\nsym: Π→L∞/parenleftbig\nRd×Sd−1/parenrightbig\n,\nsuch that sym( Mf) =f⊗1 and sym/parenleftBig\ng/parenleftBig\nD1√−∆,...,Dd√−∆/parenrightBig/parenrightBig\n= 1⊗g.Theorem 1.5\nof [44] asserts that for T∈Π andg∈L∞(Rd) is compactly supported, then the\nopreator TMg/a\\}b∇acketle{t∇/a\\}b∇acket∇i}ht−dbelongs to L1,∞and\n(2.1) Tr ω/parenleftbig\nTMg/a\\}b∇acketle{t∇/a\\}b∇acket∇i}ht−d/parenrightbig\n=/integraldisplay\nRd×Sd−1sym(T)g.\nThis theorem is a version of Connes’ trace theorem [ 15, Theorem 1].6 N.AZAMOV, E.MCDONALD, F.SUKOCHEV, D.ZANIN\nProof of Theorem 2.1.Lets >0.For each N >0, performing N-fold iteration of\nDuhamel’s formula gives the Dyson expansion with remainder term:\ne−sH=e−sH0+N/summationdisplay\nn=1(−s)n\nn!/integraldisplay\n∆ne−θ0sH0MVe−θ1sH0MV···MVe−θnsH0dθ\n+(−s)N+1\n(N+1)!/integraldisplay\n∆N+1e−θ0sH0MV···e−θNsH0MVe−θN+1sHdθ\nwhere ∆ n={(θ0,...,θ n):θ0+···+θn= 1}is then-simplex, and dθis the\nnormalised Lebesgue measure (so/integraltext\n∆ndθ= 1). For each fixed s >0, sinceVis\nbounded it is easy to see that the remainder term tends to zero in th e operator\nnorm as N→ ∞, so we have the operator norm convergent Dyson series:\ne−sH=e−sH0+∞/summationdisplay\nn=1(−s)n\nn!/integraldisplay\n∆ne−θ0sH0MVe−θ1sH0MV···MVe−θnsH0dθ.\nTaking the Fourier transform, we have:\ne−s(M2\n|x|+V(−i∇))=e−sM|x|2\n+∞/summationdisplay\nn=1(−s)n\nn!/integraldisplay\n∆ne−θ0sM2\n|x|V(−i∇)e−θ1sM2\n|x|···V(−i∇)e−θnsM2\n|x|dθ.\nFor each n≥1 the integral:\nIn:=/integraldisplay\n∆ne−sθ0M2\n|x|V(−i∇)e−sθ1M2\n|x|V(−i∇)···e−sθnM2\n|x|dθ\nconverges in the C∗-algebra Π. Indeed, at each θ∈∆nthe integrand belongs to Π\nand is norm continuous as a function of θ. Since the principal symbol function sym\nis norm continuous, we have:\nsym(In)(x,ω) =/integraldisplay\n∆ne−s(θ0+···+θn)|x|2V(ξ)ndθ\n=e−s|x|2V(ξ)n, x∈Rd, ω∈Sd−1.\nThe norm convergence of the Dyson series and the norm continuity of the principal\nsymbol mapping imply that e−s(M2\n|x|+V(−i∇))∈Π, and:\nsym(e−s(M2\n|x|+V(−i∇)))(x,ω) =e−s|x|2+∞/summationdisplay\nn=1(−s)n\nn!sym(In)(x,ω)\n=e−s|x|2∞/summationdisplay\nn=0(−sV(ξ))n\nn!\n=e−s(|x|2+V(ξ)).\nLetgbe a compactly supported smooth function on Rwithg(0) = 1, and let\nε >0. Applying [ 44, Theorem 1.5] (and the unitary invariance of the Dixmier\ntrace), we have\nTrω(e−sHM−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ightg(εH0)) = Tr ω(g(εM2\n|x|)e−s(M2\n|x|+V(−i∇))(1−∆)−d\n2)\n=1\nd(2π)dsd/2/integraldisplay\nSd−1e−sV(ξ)dξ/integraldisplay\nRdg(ε|x|2)e−|x|2dx.A DIXMIER TRACE FORMULA FOR THE DENSITY OF STATES 7\nAsε→0 the integral/integraltext\nRde−|x|2g(ε|x|2)dxconverges to πd/2by the dominated\nconvergence theorem. Therefore, for all s >0:\nlim\nε→0Trω(e−sHM−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ightg(sεH0)) =1\nd(4π)d/2/integraldisplay\nSd−1s−d/2e−sV(ξ)dξ.\nWe need to show that the limit on the left is:\nTrω(e−sHM−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight).\nTo achieve this, write e−sH=e−sH/2e−sH/2and use the cyclic property of Tr ω\nTrω(e−sHM−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ightg(sεH0)) = Tr ω(e−sH/2M−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ightg(sεH0)e−sH/2).\nThe operator e−sH/2M−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ightis in the ideal L1,∞(this is a consequence of Theorem\n4.4, to be proved in Section 4). We claim that:\nlim\nε→0g(sεH0)e−sH/2=e−sH/2\nin the uniform norm. To see this, note that the operator ( H+i)e−sH/2is bounded,\nand the resolvent identity implies that ( H0+i)(H+i)−1is bounded (see the proof\nof Theorem 4.4), so it follows that ( H0+i)e−sH/2is also bounded. By functional\ncalculus for self-adjoint operators, we have:\nlim\nε→0g(sεH0)(H0+i)−1= (H0+i)−1\nin the operator norm. Hence as ε→0 we have\ng(sεH0)e−sH/2=g(sεH0)(H0+i)−1(H0+i)e−sH/2\n→(H0+i)−1(H0+i)e−sH/2\n=e−sH/2\nin the operator norm. Therefore as ε→0,\ne−sH/2M−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ightg(sεH0)e−sH/2→e−sH/2M−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ighte−sH/2\nin theL1,∞topology. Since Tr ωis continuous,\nlim\nε→0Trω(e−sH/2M−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ightg(εsH0)e−sH/2) = Tr ω(e−sH/2M−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ighte−sH/2) = Tr ω(e−sHM−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight).\nTherefore we have proved that:\nTrω(e−sHM−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight) =1\nd(4πs)d/2/integraldisplay\nSd−1e−sV(ξ)dξ.\nHence,assuming that the density of states νHforHexists, it follows from The-\norem1.1that:/integraldisplay\nRe−sλdνH(λ) =d\nωdTrω/parenleftBig\ne−sHM−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight/parenrightBig\n=1\nωd/integraldisplay\nSd−1(4πs)−d/2e−sV(ξ)dξ.\nRecall from Example 1.2that the Laplace transform of the density of states for H0\nis given by:/integraldisplay\nRe−sλdνH0(λ) = (4πs)−d/2.8 N.AZAMOV, E.MCDONALD, F.SUKOCHEV, D.ZANIN\nThus for a constant perturbation H0+c, wherec∈R, we have\n/integraldisplay\nRe−sλdνH0+c(λ) = (4πs)−d/2e−sc.\nTherefore,/integraldisplay\nRe−sλdνH(λ) =1\nωd/integraldisplay\nSd−1/integraldisplay\nRe−sλdνH0+V(ξ)(λ)dξ.\nFinally, Fubini’s theorem and the uniqueness of the Laplace transfor m (c.f. Remark\n6.3) yields\nνH=1\nωd/integraldisplay\nSd−1νH0+V(ξ)dξ\nin the sense that if I⊆Ris Borel, then we have\nνH(I) =1\nωd/integraldisplay\nSd−1νH0+V(ξ)(I)dξ.\n/square\nThe result of Theorem 2.1can be put into a more explicit form by observing\nthat the density of states for a Hamiltonian H0+cwherec∈Ris given by:\nνH0+c((−∞,t]) =ωd\nd(2π)d(t−c)d\n2\n+\nwheret+= max{t,0}. Hence if Vis positively homogeneous and νH0+Vexists,\nthen the result of Theorem 2.1states that\nνH0+V((−∞,t]) =1\nd(2π)d/integraldisplay\nSd−1(t−V(ξ))d\n2\n+dξ.\n2.2.Stability of the density of states. As another example of the utility of\nTheorem 1.1, we also show how the theorem can be used to prove the stability of\nthe density of states under “small” perturbations. Let V∈L∞(Rd) be real valued,\nand let\nH:=−∆+MV\nbe the corresponding Schr¨ odinger operator. We are interested in perturbations\nH+MV0whereV0∈L∞(Rd) is such that\nMV0es∆\niscompact onL2(Rd) for any s >0. We have the following:\nTheorem 2.2. If the density of states measure exists for both HandH+MV0,\nthen the two measures are equal.\nThis theorem reflects the stability of the density of states under “ localised”\nperturbations. Similar statements are already known in the literatu re [41, Theorem\nC.7.7], but under different assumptions on VandV0.\nTo give some feeling for the class of potentials which satisfy the stat ed condition,\nthe following is a sufficient condition.\nLemma 2.3. LetV0∈L∞(Rd)be a potential such that for all t >0,\n|{x∈Rd:|V0(x)| ≥t}|<∞.\nThenMV0es∆is compact for all s >0.A DIXMIER TRACE FORMULA FOR THE DENSITY OF STATES 9\nProof.Letε >0, and let\nAε={x∈Rd:|V0(x)| ≥ε}.\nThen/ba∇dbl(1−χAε)V0/ba∇dbl∞≤ε, and hence:\n/ba∇dblMV0es∆−MχAεV0es∆/ba∇dbl∞=/ba∇dblM(1−χAε)Ves∆/ba∇dbl∞≤ε/ba∇dbles∆/ba∇dbl∞=ε.\nTherefore in the operator norm we have\nlim\nε→0MχAεMV0es∆=MV0es∆.\nOn the other hand, Aεhas finite measure and V0is bounded, so it follows\nthat the function χAεV0is square integrable. Hence the operator MχAεV0es∆is\nHilbert-Schmidt and in particular, compact (see our review of classic al Cwikel-type\nestimates below in Section 4). Thus, MV0es∆is the limit in the operator norm of\na sequence of compact operators, so is itself compact. /square\nRecall that H=−∆+MVis an arbitrary Schr¨ odinger operator with bounded\nreal-valued potential V.\nLemma 2.4. IfV0∈L∞(Rd)is such that MV0es∆is compact for all s >0, then\nMV0e−sHis also compact for all s >0.\nProof.By taking the adjoint, it suffices to check that e−sHMV0is compact. Using\nDuhamel’s formula:\ne−sH=es∆−s/integraldisplay1\n0e−sθHMVes(1−θ)∆dθ\nIt follows that\ne−sHMV0−es∆MV0=−s/integraldisplay1\n0e−sθHMVes(1−θ)∆MV0dθ.\nWe will complete the proof by showing that the integral on the right h and side\nis a convergent K(L2(Rd))-valued Bochner integral, and hence in particular is an\nelement of K(L2(Rd)). Note that the integrand belongs to K(L2(Rd)) for each\nθ∈(0,1), andhasuniformlyboundedoperatornorm. Since K(L2(Rd))isseparable,\nthe Bochner theorem implies that to ensure Bochner integrability it is enough to\nshowthat θ/mapsto→e−sθHMVe−s(1−θ)∆MV0is weaklymeasurable. Since the semigroups\ne−sθHandes(1−θ)∆are strongly continuous and uniformly bounded, it follows that\nthe integrandis weakly continuousand hence weaklymeasurable. Th us the integral\ndefines an element of K(L2(Rd)), and this completes the proof. /square\nLemma 2.5. IfV0∈L∞(Rd)is such that MV0es∆is compact for all s >0, then\nthe difference\ne−s(H+MV0)−e−sH\nis compact for all s >0.\nProof.Using Duhamel’s formula:\ne−s(H+MV0)=e−sH−s/integraldisplay1\n0e−sθ(H+MV0)MV0e−s(1−θ)Hdθ.\nLemma2.4implies that the integrand e−sθ(H+MV0)MV0e−s(1−θ)His compact for\neachθ∈[0,1] and all s >0. Similar reasoning to the proof of Lemma 2.4implies\nthat the integral converges as a K(L2(Rd))-valued Bochner integral, and hence the\ndifference e−s(H+MV0)−e−sHis compact. /square10 N.AZAMOV, E.MCDONALD, F.SUKOCHEV, D.ZANIN\nWe now prove the equality of the left hand sides of ( 1.6) for the potentials H\nandH+MV0.\nProposition 2.6. Suppose that H=−∆ +MVis a Schr¨ odinger operator with\nbounded real-valued potential V, and let V0∈L∞(Rd)be a real-valued potential\nsuch that MV0es∆is compact for all s >0. Then for all Dixmier traces Trωand\nalls >0we have:\nTrω(e−sH(1+M2\n|x|)−d/2) = Tr ω(e−s(H+MV0)(1+M2\n|x|)−d/2).\nProof.ItfollowsfromTheorem 4.4, tobeprovedbelow, thattheoperators e−sH(1+\nM2\n|x|)−d/2ande−s(H+MV0)/2(1+M2\n|x|)−d/2individually belong to L1,∞so each side\nof the equality is meaningful. Using the cyclic property of the Dixmier t race, we\nhave:\nTrω(e−sH(1+M2\n|x|)−d/2) = Tr ω(e−sH/2(1+M2\n|x|)−d/2e−sH/2)\nand similarly\nTrω(e−s(H+MV0(1+M2\n|x|)−d/2) = Tr ω(e−s(H+MV0)/2(1+M2\n|x|)−d/2e−s(H+MV0)/2).\nFor each s, we have the identity:\ne−s(H+MV0)/2(1+M2\n|x|)−d/2e−s(H+MV0)/2−e−sH/2(1+M2\n|x|)−d/2e−sH/2\n= (e−s(H+MV0)/2−e−sH/2)(1+M2\n|x|)−d/2e−s(H+MV0)/2\n+e−sH/2(1+M2\n|x|)−d/2(e−sH0/2−e−sH/2).\nThe operators e−sH/2(1+M2\n|x|)−d/2and (1+ M2\n|x|)−d/2e−s(H+MV0)belong to L1,∞,\nand Lemma 2.5implies that the difference e−s(H+MV0)/2−e−sH/2is compact.\nHence,\ne−s(H+MV0)/2(1+M2\n|x|)−d/2e−s(H+MV0)/2−e−sH/2(1+M2\n|x|)−d/2e−sH/2∈ K·L 1,∞+L1,∞·K.\nThe product of a compact operator and a weak trace-class opera tor belongs to\n(L1,∞)0– the separable part of the ideal L1,∞– and in particular is in the kernel\nof every Dixmier trace. Thus,\nTrω(e−s(H+MV0)/2(1+M2\n|x|)−d/2e−s(H+MV0)/2) = Tr ω(e−sH/2(1+M2\n|x|)−d/2e−sH/2).\nOnceagainusingthecyclicpropertyofthetrace,theresultimmedia telyfollows. /square\nThe proof of Theorem 2.2now follows immediately from Theorem 1.1and the\nuniqueness property of the Laplace transform (see Remark 6.3), if we assume the\nexistence of the density of states for both HandH+MV0.\n2.3.Asymptotically homogeneous potentials. As a straightforward combina-\ntion of the preceding two sections, we can also present a formula fo r potentials\nV∈L∞(Rd) such that there exists a “uniform radial limit at infinity” in the sense\nthat for all x∈Rd\\{0}, the limit:\nVh(x) := lim\nr→∞V(rx)\nexists, and converges uniformly over x∈Sd−1. The function Vhis positively\nhomogeneous in the sense that Vh(tx) =Vh(x) for allx∈Rd\\{0}andt >0. Then\nif the DOS measures ν−∆+Vandν−∆+Vhexist,ν−∆+Vis given by the formula:\nν−∆+V=1\nωd/integraldisplay\nSd−1ν−∆+Vh(ξ)dξ.A DIXMIER TRACE FORMULA FOR THE DENSITY OF STATES 11\nIndeed, the assumption of uniform convergence on x∈Sd−1of the limit Vh(x) =\nlimr→∞V(rx) implies that V−Vhsatisfiesthe assumptionofLemma 2.3, andhence\nby Theorem 2.2it follows that VandVhhave the same DOS measure. Since Vhis\npositively homogeneous, the formula for ν−∆+Vthen follows from Theorem 2.1.\n3.Preliminaries\n3.1.Trace ideals. The following material is standard; for more details we refer\nthe reader to [ 30,40]. LetHbe a complex separable Hilbert space, and let B(H)\ndenote the set of all bounded operators on H, and let K(H) denote the ideal of\ncompact operators on H. GivenT∈ K(H), the sequence of singular values µ(T) =\n{µ(k,T)}∞\nk=0is defined as:\nµ(k,T) = inf{/ba∇dblT−R/ba∇dbl: rank(R)≤k}.\nLetp∈(0,∞).The Schatten class Lpis the set of operators TinK(H) such that\nµ(T) isp-summable, i.e. in the sequence space ℓp. Ifp≥1 then the Lpnorm is\ndefined as:\n/ba∇dblT/ba∇dblp:=/ba∇dblµ(T)/ba∇dblℓp=/parenleftBigg∞/summationdisplay\nk=0µ(k,T)p/parenrightBigg1/p\n.\nWith this norm Lpis a Banach space, and an ideal of B(H).\nThe weak Schatten class Lp,∞is the set of operators Tsuch that µ(T) is in the\nweakLp-spaceℓp,∞, with quasi-norm:\n/ba∇dblT/ba∇dblp,∞= sup\nk≥0(k+1)1/pµ(k,T)<∞.\nAs with the Lpspaces,Lp,∞is an ideal of B(H). We also have the following form\nof H¨ older’s inequality,\n(3.1) /ba∇dblTS/ba∇dblr,∞≤cp,q/ba∇dblT/ba∇dblp,∞/ba∇dblS/ba∇dblq,∞\nwhere1\nr=1\np+1\nq, for some constant cp,q. Indeed, this follows from the definition of\nthe weak Lp-quasinorms and the inequality µ(2n,TS)≤µ(n,T)µ(n,S) forn≥1\n[21, Proposition 1.6], [ 22, Corollary 2.2].\nNote that if r > p, then we have the inequality:\n(3.2) /ba∇dblT/ba∇dblr\nr=∞/summationdisplay\nk=0µ(k,T)r≤∞/summationdisplay\nk=0(k+1)−r\np/ba∇dblT/ba∇dblr\np,∞=ζ/parenleftbiggr\np/parenrightbigg\n/ba∇dblT/ba∇dblr\np,∞\nwhereζis Riemann’s zeta function.\nForq∈[1,∞), we also consider the ideal Lq,1, defined as the set of compact\noperators TonHsatisfying:\n/ba∇dblT/ba∇dblLq,1:=/summationdisplay\nn≥0µ(n,T)\n(n+1)1−1\nq<∞.\nWe have the following H¨ older-type inequality: if1\np+1\nq= 1 then\n(3.3) /ba∇dblTS/ba∇dbl1≤ /ba∇dblT/ba∇dblp,∞/ba∇dblS/ba∇dblq,1\n(see e.g. [ 16, p. 303]).\nFor this paper, the relevant continuous embeddings between thes e ideals are\n(3.4) Lp,∞⊂ Lq,Lp,∞⊂ Lq,1,0< p < q≤ ∞\n(see e.g. [ 16,§IV.2.α, Proposition 1]).12 N.AZAMOV, E.MCDONALD, F.SUKOCHEV, D.ZANIN\nAmong ideals of particular interest is L1,∞, and we are concerned with traces\non this ideal. For more details, see [ 30, Section 5.7] and [ 38]. A linear functional\nϕ:L1,∞→Cis called a trace if it is unitarily invariant. That is, for all unitary\noperators Uand for all T∈ L1,∞we have that ϕ(U∗TU) =ϕ(T). It follows that\nfor all bounded operators Bwe have ϕ(BT) =ϕ(TB).\nA Dixmier trace Tr ωis a trace on L1,∞defined in terms of an extended limit\nω∈ℓ∞(N)∗(i.e., a continuous extension of the limit functional to ℓ∞(N)). Given\na positive operator T∈ L1,∞, Trω(T) is defined as:\nTrω(T) =ω/parenleftBigg/braceleftBigg/summationtextN\nk=0µ(k,T)\nlog(2+N)/bracerightBigg∞\nN=0/parenrightBigg\n.\nIfωisdilation invariant , that is, if for all n≥1 we have ω◦σn=ω, whereσn\nis the dilation semigroup σn({aj}∞\nj=0) ={a⌊j\nn⌋}∞\nj=0, then Tr ωis called a Dixmier\ntrace and extends to a linear functional on L1,∞.\nWe note that it can however be proved that Tr ωextends to a trace on L1,∞with\nno extra invariance conditions on ω(see [37, Theorem 17])1.\nMoregenerally, anextended limit isabounded linearfunctional ωonL∞((0,∞))\nwhich extends the limit functional from the subspace of functions h aving limit at ∞\nto all ofL∞((0,∞)).\n3.2.Double operator integrals. In this paper we will make brief use of the\ntechnique of double operator integrals for unitary operators. Se e e.g. [2,4,6,9,10,\n19].\nGiven two unitary operators UandVonH, a double operator integral with\nsymbolφ∈L∞(T2) is a linear map TU,V\nφ:L2→ L2defined as follows. The\noperators UandValso act as unitary operators of left and right multiplication on\nthe Hilbert-Schmidt space L2:\nLUX=UX, R VX=XV, X ∈ L2.\nAs linear operators on L2,LUandRVare commuting unitary operators and hence\nthereisajointfunctionalcalculus φ/mapsto→φ(LU,LV)∈ B(L2)forφaboundedfunction\non the torus T2. Denote TU,V\nφ:=φ(LU,RV). For a Lipschitz class function f\nonT, denote by f[1]the divided difference function f[1](z,w) =f(z)−f(w)\nz−wset to an\narbitrary value on the diagonal.\nA short computation based on a Fourier decomposition of f(see [2, Theorem\n1.1.3]) shows that\n(3.5) /ba∇dblTU,V\nf[1]|L1/ba∇dblL1→L1≤ /ba∇dbl/hatwidef′/ba∇dblℓ1(Z)≤ /ba∇dblf′/ba∇dblL2(T)+/ba∇dblf′′/ba∇dblL2(T).\nProvided the above right hand side is finite, we also have that TU,V\nf[1]extends by\nduality to B(H), and an interpolation argument (as described in e.g. [ 3, p. 5225])\nimplies that if p >1 we have\n/ba∇dblTU,V\nf[1]/ba∇dblLp,∞→Lp,∞≤ /ba∇dblf′/ba∇dblL2(T)+/ba∇dblf′′/ba∇dblL2(T)\nand similarly if q >1 we have\n(3.6) /ba∇dblTU,V\nf[1]/ba∇dblLq,1→Lq,1≤ /ba∇dblf′/ba∇dblL2(T)+/ba∇dblf′′/ba∇dblL2(T).\n1Moreover it can be proved that Tr ωis a Dixmier trace for every extended limit ω. This will\nappear in the upcoming second edition of [ 30].A DIXMIER TRACE FORMULA FOR THE DENSITY OF STATES 13\nIfX∈ B(H), then we also have the following identity (see [ 10, Theorem 8.5], [ 2,\nTheorem 3.5.4] or [ 9, Theorem 4.1] for the related self-adjoint case):\n(3.7) TU,U\nf[1]([U,X]) = [f(U),X].\nIfgisaboundedfunctiononthespectrumof U, thenitfollowsfromthedefinition\nofTU,U\nf[1]that we also have:\n(3.8)\ng(U)TU,U\nf[1](X) =TU,U\nf[1](g(U)X), TU,U\nf[1](X)g(U) =TU,U\nf[1](Xg(U)), X∈ B(H).\n4.Cwikel type estimates\nWe will extensively use the notation\n/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht= (1+|x|2)1/2\nforx∈Rdand|x|denotes the ℓ2-norm of x, so that x/mapsto→ /a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−1∈Ld,∞(Rd). Recall\nthatVis a bounded measurable real valued function on Rd, andH=−∆+MV\nis the Schr¨ odinger operator associated to the potential V. We exclusively consider\nd≥2.\nThis section is devoted to a proof of the claim that for integers p≥1 and\nz∈C\\R, we have that ( H+z)−pM−p\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ightis in the ideal Ld/p,∞. This is a crucial\ncomponent to proving that the operator inside the Dixmier trace in T heorem1.1\nis indeed in the ideal L1,∞. Somewhat similar estimates on the singular values of\noperators of the form f(H)Mgare also in [ 41, Section B.9].\nOurproofsarebasedon the followingclassicalCwikelestimate (see [40, Theorem\n4.2] or for the p= 2 case see the more recent [ 29, Corollary 1.2, Theorem 5.6]).\nThe function spaces ℓ2,∞(L4)(Rd) andℓ2,log(L∞)(Rd) are defined by the norms:\n/ba∇dblg/ba∇dblℓ2,∞(L4)(Rd)=/ba∇dbl{/ba∇dblg/ba∇dblL4(k+[0,1]d)}k∈Zd/ba∇dblℓ2,∞,\n/ba∇dblf/ba∇dblℓ2,log(L∞)(Rd)=\n/summationdisplay\nk∈Zd(1+log(1+ |k|))/ba∇dblf/ba∇dbl2\nL∞(k+[0,1]d)\n1/2\nwhere|k|denotes the ℓ2-norm of k∈Zd.\nProposition 4.1. For2< p <∞,iff∈Lp(Rd)andg∈Lp,∞(Rd), then the oper-\natorMfg(−i∇)is in the ideal Lp,∞. Ifg∈ℓ2,∞(L4)(Rd)andf∈ℓ2,log(L∞)(Rd),\nthenMfg(−i∇)∈ L2,∞(Rd).\nWe begin with a lemma of elementary operator theory, required for t he proof of\nthe main result of this section (Theorem 4.4).\nLemma 4.2. LetA,B,Cbe bounded operators such that A=B−AC.IfB∈ Lp0,∞\nandC∈ Lp1,∞,for0< p0,p1<∞thenA∈ Lp0,∞.\nProof.By induction, for each n≥1 we have:\nA=n−1/summationdisplay\nk=0(−1)kBCk+(−1)nACn.\nSinceLp0,∞is an ideal, for all k≥0 we have BCk∈ Lp0,∞.Choosensufficiently\nlarge such that p1< np0.From H¨ older’s inequality ( 3.1), it follows that\nACn∈ Lp1\nn,∞⊂ Lp0,∞.14 N.AZAMOV, E.MCDONALD, F.SUKOCHEV, D.ZANIN\nHence,A∈ Lp0,∞. /square\nThe following lemma contains a crucial piece of the proof of Theorem 4.4, the\nmain result in this section. For uniformity of notations, we set H0=−∆.\nLemma 4.3. For all integers p≥0and for any ε >0,we have:\nMp\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight/bracketleftBig\nH0,M−p\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight/bracketrightBig\n(H0+1)−1∈ Ld+ε,∞.\nProof.By definition, on the space S(Rd) of Schwartz-class functions, we have\nH0=−d/summationdisplay\nm=1∂2\nm.\nIff∈C∞(Rd) is bounded, with all derivatives up to all order bounded, then a\nstraightforward calculation shows that\n[∂2\nm,Mf] = 2M∂mf∂m+M∂2mf,1≤m≤d\nwhich gives\n(4.1) [ H0,Mf] =−2d/summationdisplay\nm=1M∂mf∂m−MH0f.\nwhich is also valid on S(Rd).\nDefine the function fpbyfp(x) :=/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−p, x∈Rd.For 1≤m≤d, we have (here\nx= (x1,···,xd)∈Rd):\n∂mfp(x) =−pxm/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−p−2, (4.2)\n∂2\nmfp(x) =p(p+2)x2\nm/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−p−4−p/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−p−2,\nH0fp(x) =−p(p+2−d)/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−p−2.\nUsing (4.1) (once again on the domain S(Rd)), we write\nMp\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight[H0,M−p\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight] =Mf−1\np[H0,Mfp]\n=Mf−1\np(−2d/summationdisplay\nm=1M∂mfp∂m−MH0fp)\n=−2d/summationdisplay\nm=1Mf−1\np∂mfp∂m−Mf−1\npH0fp. (4.3)\nSince for any ε >0\nf−1\np∂mfp(x) =−pxm/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−2∈Ld,∞(Rd)∩L∞(Rd)⊂Ld+ε(Rd)\nand∂m(H0+1)−1=g(−i∇) with\ng(t) =−itm\n1+|t|2∈Ld,∞(Rd)∩L∞(Rd)⊂Ld+ε,∞(Rd)\nit follows from Proposition 4.1that for any ε >0\np/summationdisplay\nm=1Mf−1\np∂mfp∂m(H0+1)−1∈ Ld+ε,∞. (4.4)\nSince\nf−1\npH0fp=−p(p+2−d)/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−2∈Ld/2,∞(Rd)∩L∞(Rd)A DIXMIER TRACE FORMULA FOR THE DENSITY OF STATES 15\nand (H0+ 1)−1=g(−i∇) withg(t) =1\n1+|t|2∈Ld/2,∞(Rd)∩L∞(Rd),it follows\nfrom Proposition 4.1that for any ε >0\nMf−1\npH0fp(H0+1)−1∈ Ld/2+ε,∞. (4.5)\nCombining ( 4.3), (4.4) and (4.5) gives for any ε >0\nMp\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight[H0,M−p\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight](H0+1)−1∈ Ld+ε,∞.\n/square\nWe now present the main Cwikel estimate of this section:\nTheorem 4.4. For any z∈C\\Randp= 1,2,...\n(H+z)−pM−p\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight∈ Ld/p,∞.\nProof.We introduce the notation:\nAp(z) = (H+z)−pM−p\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight,\nBp(z) = (H+z)1−pM−p\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight(H+z)−1,\nCp(z) =Mp\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight[H0,M−p\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight](H+z)−1.\nWe prove the assertion by induction on p, our goal being to prove that Ap(z)∈\nLd/p,∞for allp≥1. The resolvent identity gives\n(H+z)−1(H0+z) = 1−(H+z)−1MV∈ B(H).\nConsider the base case p= 1. Ifd >2, then since x/mapsto→ /a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−1∈Ld,∞(Rd) andy/mapsto→\n(y2+z)−1∈Ld/2,∞(Rd)∩L∞(Rd)⊂Ld(Rd),the Fourier dual of Proposition 4.1\nyields:\n(H0+z)−1M−1\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight∈ Ld,∞.\nOn the other hand, if d= 2, then we shall verify that x/mapsto→ /a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−1∈ℓ2,∞(L4)(R2),\nand that y/mapsto→(|y|2+z)−1∈ℓ2,log(L∞)(R2). There is a constant Cdsuch that for\nk∈Z2we have:\n/ba∇dbl/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−1/ba∇dblL4(k+[0,1]d)≤Cd/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht−1\nand therefore,\n/ba∇dbl/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−1/ba∇dblℓ2,∞(L4)(R2)≤Cd/ba∇dbl{/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht−1}k∈Z2/ba∇dblℓ2,∞<∞.\nSimilarly, for k∈Z2, there is a constant Cd,zsuch that:\n/ba∇dbl(|y|2+z)−1/ba∇dblL∞(k+[0,1]d)≤Cd,z/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht−4\nand therefore,\n/ba∇dbl(|y|2+z)−1/ba∇dblℓ2,log(L∞)(R2)≤Cd,z/parenleftBigg/summationdisplay\nk∈Z2(1+log(1+ |k|))/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht−4/parenrightBigg1/2\n<∞.\nIt then follows from Proposition 4.1that (H0+z)−1M−1\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight∈ L2,∞(L2(R2)) when\nd= 2. So in all cases d≥2, we have ( H0+z)−1M−1\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight∈ Ld,∞.\nHence,\nA1(z) = (H+z)−1(H0+z)·(H0+z)−1M−1\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight∈ B(H)·Ld,∞=Ld,∞.\nThis proves the p= 1 case.16 N.AZAMOV, E.MCDONALD, F.SUKOCHEV, D.ZANIN\nSuppose the estimate holds for p≥1.Let us prove it for p+1.IfXandYare\nlinear operators where YandX−1are bounded and Ypreserves the domain of X,\nthen we have the identity:\n(4.6) [ X−1,Y] =−X−1[X,Y]X−1.\nThis identity applies with Y=M−p−1\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ightandX=H+z, since the domain of Xis\nequal to the domain of H0[36, Theorem 8.8], and clearly Ypreserves the domain\nofH0. Thus we may write\nAp+1(z)−Bp+1(z) = (H+z)−p·[(H+z)−1,M−p−1\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight]\n=−(H+z)−p−1[H0,M−p−1\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight](H+z)−1\n=−Ap+1(z)Cp+1(z).\nWe now verify that the operators Ap+1(z), Bp+1(z) andCp+1(z) satisfy the as-\nsumptions in Lemma 4.2.\nWith the identity\nBp+1(z) =Ap(z)A1(¯z)∗,\nH¨ older’s inequality ( 3.1) and the inductive assumption yield\nBp+1(z)∈ Ld\np,∞·Ld,∞⊂ L d\np+1,∞.\nAlso,\nCp+1(z) =Mp+1\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight[H0,M−p−1\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight](H0+z)−1·(H0+z)(H+z)−1.\nLemma4.3states that:\nMp+1\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight[H0,M−p−1\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight](H0+z)−1∈ L2d,∞.\nSince\n(H0+z)(H+z)−1= 1−MV(H+z)−1∈ B(H)\nit follows that\nCp+1(z)∈ L2d,∞·B(H) =L2d,∞.\nThat is, Bp+1(z)∈ Ld/(p+1),∞andCp+1(z)∈ L2d,∞, so applying Lemma 4.2to\nthe operators Ap+1(z), Bp+1(z) andCp+1(z) yieldsAp+1(z)∈ Ld/(p+1),∞, so the\nassertion follows by induction on p. /square\nAs a useful corollary, we also include the following:\nProposition 4.5. Letε >0. Then\nlimsup\nr↓d/ba∇dbl[M−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight,e−ε(r−1)H]/ba∇dbl1<∞\nand\nlimsup\nr↓d/ba∇dbl[M1−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight,e−ε(r−1)H]/ba∇dbld\nd−1,1<∞.\nProof.LetUbe the unitary operatorH+i\nH−i, and let φεbe a smooth function on the\nunit circle such that for t≥ −/ba∇dblV/ba∇dbl∞we have\nφr/parenleftbiggt+i\nt−i/parenrightbigg\n=e−1\n2ε(r−1)t.A DIXMIER TRACE FORMULA FOR THE DENSITY OF STATES 17\nSinceφris smooth, the transformer TU,U\nφ[1]\nris bounded from L1toL1(see (3.5)),\nand one can compute that the L2(T) norms of φ′\nrandφ′′\nrare bounded above by a\nconstant multiple of r−1 and (r−1)2respectively, so in particular:\n(4.7) limsup\nr↓d/ba∇dblTU,U\nφ[1]\nr/ba∇dblL1→L1<∞.\nSimilarly, ( 3.6) yields:\n(4.8) limsup\nr↓d/ba∇dblTU,U\nφ[1]\nr/ba∇dblLd\nd−1,1→L d\nd−1,1<∞.\nUsing the semigroup property of e−ε(r−1)Hand the Leibniz rule, we have:\n[M−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight,e−ε(r−1)H] =e−1\n2ε(r−1)H[M−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight,e−1\n2ε(r−1)H]+[M−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight,e−1\n2ε(r−1)H]e−1\n2ε(r−1)H.\nSincee−1\n2ε(r−1)H=φr(U), from (3.7), we have:\n[M−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight,e−1\n2ε(r−1)H] =TU,U\nφ[1]\nr([M−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight,U]).\nCombining the preceding two displays, from ( 3.8) it follows that\n[M−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight,e−ε(r−1)H] =TU,U\nφ[1]\nr(e−1\n2ε(r−1)H[M−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight,U]+[M−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight,U]e−1\n2ε(r−1)H).\nNow (4.7) implies that there is a constant Cd,εsuch that:\n/ba∇dbl[M−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight,e−ε(r−1)H]/ba∇dbl1≤Cd,ε(/ba∇dble−1\n2ε(r−1)H[M−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight,U]/ba∇dbl1+/ba∇dbl[M−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight,U]e−1\n2ε(r−1)H/ba∇dbl1).\nAn identical argument yields:\n/ba∇dbl[M1−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight,e−ε(r−1)H]/ba∇dbld\nd−1,1≤Cd,ε(/ba∇dble−1\n2ε(r−1)H[M1−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight,U]/ba∇dbld\nd−1,1+/ba∇dble−1\n2ε(r−1)H[M1−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight,U]/ba∇dbld\nd−1,1)\nfor a possibly larger constant Cd,ε. We now concentrate on determining the L1\nandLd\nd−1,1norms of e−1\n2ε(r−1)H[M−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight,U] ande−1\n2ε(r−1)H[M1−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight,U] respectively.\nIdentical arguments will control the L1andLd\nd−1,1norms of [ M−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight,U]e−1\n2ε(r−1)H\nand [M1−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight,U]e−1\n2ε(r−1)Hrespectively.\nUsing (4.6) (once again justified by the fact that M−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ightpreserves the domain of\nH), we have the following computations for the commutator [ M−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight,U]:\n[M−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight,H+i\nH−i] = [M−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight,1+2i(H−i)−1]\n= 2i[M−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight,(H−i)−1]\n=−2i(H−i)−1[M−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight,H](H−i)−1\n=−2i(H−i)−1[M−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight,H0](H−1)−1.\nIn view of ( 4.1) and the computations leading to ( 4.4), we have:\n(H−i)−1[M−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight,H0](H−i)−1\n= 2rd/summationdisplay\nm=1(H−i)−1Mxm/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight−r−2∂m(H−i)−1+r(r+2−d)(H−i)−1M/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight−r−2(H−i)−1.18 N.AZAMOV, E.MCDONALD, F.SUKOCHEV, D.ZANIN\nIt follows now from the triangle inequality that:\n/ba∇dble−1\n2ε(r−1)H[M−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight,U]/ba∇dbl1≤Cd,ε(rd/summationdisplay\nm=1/ba∇dbl(H−i)−1e−1\n2ε(r−1)HM−r−1\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ightMxm/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight−1∂m(H−i)−1/ba∇dbl1\n+r(r+d+2)/ba∇dbl(H−i)−1e−1\n2ε(r−1)HM/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight−r−2(H−i)−1/ba∇dbl1).\nUsing the fact that xm/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−1,∂m(H−i)−1and (H−i)−1are bounded, we arrive\nat the bound:\n(4.9) /ba∇dble−1\n2ε(r−1)H[M−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight,U]/ba∇dbl1≤Cd,εr2/ba∇dble−1\n2ε(r−1)HM−r−1\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight/ba∇dbl1.\nHere, once again the size of the constant may have increased. An id entical argu-\nment, replacing the L1norm by the Ld\nd−1,1and using ( 4.8) leads to the bound:\n(4.10) /ba∇dble−1\n2ε(r−1)H[M1−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight,U]/ba∇dbld\nd−1,1≤Cd,εr2/ba∇dble−1\n2ε(r−1)HM−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight/ba∇dbld\nd−1,1.\nSincer > d, (4.9) yields\n/ba∇dble−1\n2ε(r−1)H[M−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight,U]/ba∇dbl1≤Cd,εr2/ba∇dble−ε(r−d)\n2H/ba∇dbl∞/ba∇dblMd−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight/ba∇dbl∞/ba∇dble−1\n2ε(d−1)HM−d−1\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight/ba∇dbl1.\nand (4.10) yields\n/ba∇dble−1\n2ε(r−1)H[M1−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight,U]/ba∇dbld\nd−1,1≤Cd,εr2/ba∇dble−ε(r−d)\n2H/ba∇dbl∞/ba∇dblMd−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight/ba∇dbl∞/ba∇dble−1\n2ε(d−1)HM−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight/ba∇dbld\nd−1,1.\nUsing the fact that ( H+i)Ne−1\n2ε(d−1)His bounded for any N≥0, we have:\n/ba∇dble−1\n2ε(d−1)HM−d−1\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight/ba∇dbl1≤Cd,ε/ba∇dbl(H+i)−(d+1)M−(d+1)\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight/ba∇dbl1.\nfor a potentially different constant Cd,ε. Theorem 4.4now provides the desired\nbounds, since Ld\nd+1,∞⊂ L1. Similarly,\n/ba∇dble−1\n2ε(d−1)HM−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight/ba∇dbld\nd−1,1≤Cd,ε/ba∇dbl(H+i)−dM−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight/ba∇dbld\nd−1,1.\nSinceL1,∞⊂ L d\nd−1,1, Theorem 4.4again yields the desired bound.\n/square\n5.A residue formula\nWe now proceed to the proof of the claim that:\nTrω(e−sHM−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight) = lim\nr↓1(r−1)Tr(e−sHM−dr\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight).\nThat the left hand side makes sense is ensured by Theorem 4.4; indeed, it implies\nthat (H+i)−dM−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight∈ L1,∞, and since the operator e−sH(H+i)dhas bounded\nextension it follows that e−sHM−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight∈ L1,∞. That the right hand side makes sense\nwill be a consequence of the arguments in this section. The proofs o f this sec-\ntion are achieved with some recently developed techniques in operat or integration,\ndeveloped originally in [ 18] and later extended in [ 45, Section 5.2].A DIXMIER TRACE FORMULA FOR THE DENSITY OF STATES 19\n5.1.Abstract result. The following is an abstract operator theoretic result. It is\nsimilar to [ 17, Theorem B.1], although the assumptions are slightly different and\nthe result is stated with an explicit norm bound.\nTheorem 5.1. Letd≥2,r >1letp >d\n2≥1and select q∈(1,∞)such that\n1\np+1\nq≤1. LetA,B∈B(H)be positive operators satisfying the following four\nconditions:\n(i) [BA1\n2,A1\n2]∈ Lp,∞;\n(ii)Br−1Ar−1∈ Lq,1;\n(iii)Br−1[B,Ar−1]A∈ L1;\n(iv) (A1/2BA1/2)r−1∈ Lq,1;\nThenBrAr−(A1/2BA1/2)r∈ L1and for some constants cr,p,d>0, we have\n/ba∇dblBrAr−(A1\n2BA1\n2)r/ba∇dbl1≤cr,p,d/parenleftBig\n/ba∇dbl[BA1\n2,A1\n2]/ba∇dblp,∞/ba∇dbl(A1/2BA1/2)r−1/ba∇dblq,1\n+/ba∇dblBr−1Ar−1/ba∇dblq,1/ba∇dbl[BA1\n2,A1\n2]/ba∇dblp,∞+/ba∇dblBr−1[BA1\n2,Ar−1]A1\n2/ba∇dbl1/parenrightBig\n.\nThe proof of Theorem 5.1is based on the formula given in [ 45, Section 5.2],\nstated in terms of a mapping Tr:R→ B(H) defined as follows:\nDefinition 5.2. LetAandBbe positive bounded operators, and let\nY:=A1/2BA1/2.\nWe define the mapping Tr:R→ B(H)by,\nTr(0) :=Br−1[BA1\n2,Ar−1\n2]+[BA1\n2,A1\n2]Yr−1,\nTr(s) :=Br−1+is[BA1\n2,Ar−1\n2+is]Y−is+Bis[BA1\n2,A1\n2+is]Yr−1−is, s/\\e}atio\\slash= 0.\nWe now collect some auxiliary results for the proof of Theorem 5.1.\nLemma 5.3. Letpanddbe as in the statement of Theorem 5.1. Suppose that A\nandBare bounded positive operators such that [BA1\n2,A1\n2]∈ Ld\n2,∞. Then for all\ns∈R,[BA1\n2,A1\n2+is]∈ Lp,∞, and we have:\n/ba∇dbl[BA1\n2,A1\n2+is]/ba∇dblp,∞≤cp(1+|s|)/ba∇dbl[BA1\n2,A1\n2]/ba∇dblp,∞.\nProof.The main result in [ 32] asserts that for every Lipschitz continuous function\nfonRand every 1 < p′<∞, we have a constant Cp′such that:\n/ba∇dbl[X,f(Y)]/ba∇dblp′≤Cp′/ba∇dblf′/ba∇dbl∞/ba∇dbl[X,Y]/ba∇dblp′,\nwhenever Yis a self-adjoint operator and Xis a bounded operator such that\n[X,Y]∈ Lp′. The method of proof in [ 32] was to construct a linear operator\nTY,Y\nf[1]which is bounded from Lp′toLp′for every 1 < p′<∞and such that\nTY,Y\nf[1]([X,Y]) = [X,f(Y)]. Since p >1, the ideal Lp,∞is an interpolation space\nbetween Lp′andLp′′for suitable 1 < p′< p < p′′<∞. An interpolation argument\n(see e.g. [ 3, p. 5225]) further implies that TY,Y\nf[1]is bounded from Lp,∞toLp,∞, and\nwe have the inequality:\n/ba∇dbl[X,f(Y)]/ba∇dblp,∞≤cp/ba∇dblf′/ba∇dbl∞/ba∇dbl[X,Y]/ba∇dblp,∞\nfor some constant cp.\nTakeX=BA1\n2, Y=A1\n2andf(t) =|t|1+2is, t∈R.Since/ba∇dblf′/ba∇dbl∞≤1+2|s|,the\nassertion follows. /square20 N.AZAMOV, E.MCDONALD, F.SUKOCHEV, D.ZANIN\nLemma 5.4. LetA, B, d, r, p andqbe as in Theorem 5.1. Then the map R∋\ns/mapsto→Tr(s)takes values in the trace class, and there is a constant cp>0such that\nfor alls∈R:\n/ba∇dblTr(s)/ba∇dbl1≤cp(1+|s|)/ba∇dbl[BA1\n2,A1\n2]/ba∇dblp,∞/ba∇dblYr−1/ba∇dblq,1\n+cp(1+|s|)/ba∇dblBr−1Ar−1/ba∇dblq,1/ba∇dbl[BA1\n2,A1\n2]/ba∇dblp,∞\n+/ba∇dblBr−1[BA1\n2,Ar−1]A1\n2/ba∇dbl1.\nProof.We prove this for s/\\e}atio\\slash= 0, the proof for s= 0 is identical. By the triangle\ninequality, we have\n/ba∇dblTr(s)/ba∇dbl1≤ /ba∇dblBr−1+is[BA1\n2,Ar−1\n2+is]Y−is/ba∇dbl1+/ba∇dblBis[BA1\n2,A1\n2+is]Yr−1−is/ba∇dbl1\n≤ /ba∇dblBr−1[BA1\n2,Ar−1\n2+is]/ba∇dbl1+/ba∇dbl[BA1\n2,A1\n2+is]Yr−1/ba∇dbl1\n=: (I)+(II).\nUsing the Leibniz rule, we have\n[BA1\n2,Ar−1\n2+is] =Ar−1[BA1\n2,A1\n2+is]+[BA1\n2,Ar−1]A1\n2+is.\nTherefore, using ( 3.3) we get\n(I) =/ba∇dblBr−1[BA1\n2,Ar−1\n2+is]/ba∇dbl1\n≤ /ba∇dblBr−1Ar−1[BA1\n2,A1\n2+is]/ba∇dbl1+/ba∇dblBr−1[BA1\n2,Ar−1]A1\n2+is/ba∇dbl1\n≤ /ba∇dblBr−1Ar−1/ba∇dblq,1/ba∇dbl[BA1\n2,A1\n2+is]/ba∇dblp,∞\n+/ba∇dblBr−1[BA1\n2,Ar−1]A1\n2/ba∇dbl1.\nUsing Lemma 5.3, we have\n(I)≤cp(1+|s|)/ba∇dblBr−1Ar−1/ba∇dblq,1/ba∇dbl[BA1\n2,A1\n2]/ba∇dblp,∞\n+/ba∇dblBr−1[BA1\n2,Ar−1]A1\n2/ba∇dbl1.\nOn the other hand, using ( 3.3), we have:\n(II) =/ba∇dbl[BA1\n2,A1\n2+is]Yr−1/ba∇dbl1≤ /ba∇dbl[BA1\n2,A1\n2+is]/ba∇dblp,∞/ba∇dblYr−1/ba∇dblq,1.\nAgain applying Lemma 5.3, it follows that:\n(II)≤cd(1+|s|)/ba∇dbl[BA1\n2,A1\n2]/ba∇dblp,∞/ba∇dblYr−1/ba∇dblq,1.\nHence,Tr(s)∈ L1with the appropriate norm bound.\n/square\nBy means of Lemma 5.4and an integral formula from [ 45, Theorem 5.2.1], we\nobtain a proof of Theorem 5.1.\nProof of Theorem 5.1.Define the function gr:R→Rby\ngr(0) := 1−r\n2,\ngr(2t) := 1−sinh(rt)\n2sinh(t)cosh((r−1)t), t/\\e}atio\\slash= 0.A DIXMIER TRACE FORMULA FOR THE DENSITY OF STATES 21\nThe function gris Schwartz class on R(see [45, Remark 5.2.2]). According to [ 45,\nTheorem 5.2.1], the mapping Tris continuous in the weak operator topology and\n(5.1) BrAr−Yr=Tr(0)−/integraldisplay\nRTr(s)/hatwidegr(s)ds,\nwhere the integral on the left convergesin the weak operatortop ology, and/hatwidegris the\nFourier transform of gr, scaled so that gr(t) =/integraltext\nR/hatwidegr(s)eitsds.Our result is based\non the estimate:\n(5.2) /ba∇dblBrAr−Yr/ba∇dbl1≤ /ba∇dblTr(0)/ba∇dbl1+/integraldisplay\nR/ba∇dblTr(s)/ba∇dbl1|/hatwidegr(s)|ds.\nand a corresponding implication that if the right hand side of ( 5.2) is finite then\nBrAr−Yr∈ L1. This result does not immediately follow from the integral formula\n(5.1) since the integral only converges in the weak operator topology, however it\ncan be justified by the argument presented in [ 45, Lemma 2.3.2]. Granted ( 5.2), we\napply the bound on /ba∇dblTr(s)/ba∇dbl1from Lemma 5.4to obtain:\n/ba∇dblBrAr−Yr/ba∇dbl1≤RHS(0)+RHS(0)·/integraldisplay\nR(1+|s|)|/hatwidegr(s)|ds,\nwhereRHS(s) is the right hand side of the inequality in Lemma 5.4. Asgris a\nSchwartz class function, then so is the Fourier transform /hatwidegr.Hence, the integral on\nthe right hand side converges and the assertion follows. /square\n5.2.The main residue formula. To apply Theorem 5.1to the problem at hand,\nwe need to verify its assumptions for the relevant operators.\nLemma 5.5. LetA=e−εHandB=M−1\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight,whereε >0. LetY=A1\n2BA1\n2. Let\np >d\n2andq >d\nd−1. We have the following:\n(i) [BA1\n2,A1\n2]∈ Lp,∞andY∈ Ld,∞;\n(ii)/ba∇dblBr−1Ar−1/ba∇dblq,1=O(1)asr↓d;\n(iii)/ba∇dblBr−1[B,Ar−1]A/ba∇dbl1=O(1)asr↓d;\n(iv)Ifr > d, we have (A1/2BA1/2)r−1∈ Lq,1.\nProof.First we verify ( i). We begin by noting that:\n[M−1\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ighte−1\n2εH,H] = [M−1\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight,H]e−1\n2εH= [M−1\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight,H0]e−1\n2εH.\nAs in the proof of Lemma 4.3, specifically ( 4.1), we have:\n[M−1\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight,H0] = 2d/summationdisplay\nm=1M∂m(/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight−1)∂m+MH0(/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight−1).\nWe may evaluate each ∂m(/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−1) andH0(/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−1) using (4.2), to arrive at:\n[M−1\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight,H0] =−2/parenleftBiggd/summationdisplay\nm=1Mxm/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight−1M−2\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight∂m/parenrightBigg\n−(3−d)M−3\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight.\nIt follows that\n[M−1\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ighte−1\n2εH,H] = [M−1\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight,H0]e−1\n2εH\n=−2/parenleftBiggd/summationdisplay\nm=1Mxm/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight−1M/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight−2∂m/parenrightBigg\ne−1\n2εH−(3−d)M−3\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ighte−1\n2εH.22 N.AZAMOV, E.MCDONALD, F.SUKOCHEV, D.ZANIN\nThe latter term is in Ld\n3,∞, since (H+i)3e−1\n2εHis bounded and Theorem 4.4with\np= 3 applies. For the former term, we instead use the fact that ( H0+i)3e−1\n2εHis\nbounded, and then Theorem 4.4withp= 2 implies that:\n[M−1\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ighte−1\n2εH,H]∈ Ld\n2,∞⊂ Lp,∞.\nLetU=H+i\nH−i. It follows that\n[M−1\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ighte−1\n2εH,U] = 2i[M−1\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ighte−1\n2εH,(H−i)−1] =−2i(H−i)−1[M−1\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ighte−1\n2εH,H](H−i)−1∈ Lp,∞.\nWe now use the smooth function φ2introduced in the proof of Proposition 4.5\nand (3.7) to arrive at:\n[M−1\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ighte−1\n2εH,e−1\n2εH] =TU,U\nφ[1]\n2([M−1\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ighte−1\n2εH,U]).\nSincep >1, the ideal Lp,∞is an interpolation space between L1andL∞, and\nhence (3.5) implies the boundedness of TU,U\nφ[1]\n2fromLp,∞toLp,∞.\nThus,\n[M−1\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ighte−1\n2εH,e−1\n2εH] = [BA1\n2,A1\n2]∈ Lp,∞.\nThis yieldsthe first partof ( i). To see the secondpart (that Y∈ Ld,∞), simply note\nthat:AB=e−εH(H+i)·(H+i)−1M−1\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight,so that Theorem 4.4yieldsAB∈ Ld,∞,\nand:\nY=A1\n2BA1\n2=BA−[BA1\n2,A1\n2].\nBy takingd\n2< p < d, we have already proved that [ BA1/2,A1/2]∈ Ld,∞. Hence\nY∈ Ld,∞.\nNow we prove ( ii). Ifr > d, we have:\n/ba∇dblBr−1Ar−1/ba∇dblq,1≤ /ba∇dblBr−d/ba∇dbl∞/ba∇dblAr−d/ba∇dbl∞/ba∇dblBd−1Ad−1/ba∇dblq,1.\nThen\n/ba∇dblBd−1Ad−1/ba∇dblq,1≤ /ba∇dble−ε(d−1)H(H+i)d−1/ba∇dbl∞/ba∇dblM1−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight(H+i)1−d/ba∇dblq,1.\nSinceq >d\nd−1, we have:\n/ba∇dblM1−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight(H+i)1−d/ba∇dblq,1≤ /ba∇dblM1−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight(H+i)1−d/ba∇dbld\nd−1,∞\nThe assertion ( ii) follows now from Theorem 4.4.\nNext we prove ( iii). We write Br−1[B,Ar−1]Aas\nBr−1[B,Ar−1]A= [Br,Ar−1]·A−[Br−1,Ar−1]·BA.\nThus, using ( 3.3) and our previous observation that BA∈ Ld,∞, we have\n/ba∇dblBr−1[B,Ar−1]A/ba∇dbl1≤ /ba∇dbl[Br,Ar−1]/ba∇dbl1/ba∇dblA/ba∇dbl∞+/ba∇dbl[Br−1,Ar−1]/ba∇dbld\nd−1,1/ba∇dblBA/ba∇dbld,∞.\nSincer > d, Proposition 4.5yields the desired uniform control on /ba∇dbl[Br,Ar−1]/ba∇dbl1\nand/ba∇dbl[Br−1,Ar−1]/ba∇dbld\nd−1,1asr↓d.\nFinally, we prove ( iv). We have already proved in ( i) thatY∈ Ld,∞. It follows\nthatYr−1∈ L d\nr−1,∞. Ifr > d, thenLd\nr−1,∞⊂ L d\nd−1,∞, and by our assumption\nonq, (3.4) implies the inclusion Ld\nd−1,∞⊂ Lq,1. This establishes ( iv).\n/squareA DIXMIER TRACE FORMULA FOR THE DENSITY OF STATES 23\nThe following lemma is essential to our results in this section, and its pr oof is\nthe ultimate purpose of Lemmas 5.1and5.5. The importance of this result is that\nit allows us to replace the limit of eisHM−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ightwith the limit of e−(s−εd)HYr, whereY\nis a certain compact operator. This allows the application of zeta-fu nction residue\ntechniques.\nLemma 5.6. Lets >0and0< ε <s\n2d.Denoting Y=e−ε\n2HM−1\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ighte−ε\n2Has in\nLemma5.5, we have\n/vextenddouble/vextenddouble/vextenddoublee−sHM−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight−e−(s−εd)HYr/vextenddouble/vextenddouble/vextenddouble\n1=O(1), r∈[d,s\n2ε]\nProof.LetAandBas in Lemma 5.5. Splitting up e−sHase−(s−εr)HAr, we have\ne−sHM−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight=e−(s−εr)HArBr\n=e−(s−εr)H·(ArBr−Yr)+(e−(s−εr)H−e−(s−εd)H)Yr\n+e−(s−εd)HYr.\nFrom the triangle inequality,\n/vextenddouble/vextenddouble/vextenddoublee−sHM−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight−e−(s−εd)HYr/vextenddouble/vextenddouble/vextenddouble\n1\n≤ /ba∇dble−(s−εr)H/ba∇dbl∞/ba∇dblArBr−Yr/ba∇dbl1+/ba∇dble−(s−εr)H−e−(s−εd)H/ba∇dbl∞/ba∇dblYr/ba∇dbl1.\nConsider the first summand. In view of Lemma 5.5, Theorem 5.1is applicable and,\nmoreover, the right hand side in Theorem 5.1is bounded in r∈[d,s\n2ε].So, the first\nsummand is bounded for r∈[d,s\n2ε].The second summand vanishes at r=d,and\nforr > d,it is estimated via ( 3.2) as:\n/vextenddouble/vextenddouble/vextenddoublee−(s−εr)H−e−(s−εd)H\nr−d/vextenddouble/vextenddouble/vextenddouble\n∞·(r−d)/ba∇dblYr/ba∇dbl1\n≤sup\nt>−/⌊a∇d⌊lV/⌊a∇d⌊l∞/vextendsingle/vextendsingle/vextendsinglee−(s−εr)t−e−(s−εd)t\nr−d/vextendsingle/vextendsingle/vextendsingle·(r−d)ζ/parenleftBigr\nd/parenrightBig\n/ba∇dblY/ba∇dblr\nd,∞.\nSince (r−d)ζ(r\nd) is bounded in the interval ( d,s\n2ε], the result follows. /square\nThe next theorem is the main result in this section. The proof relies on the\nnotion of a zeta-function residue [ 30, Definition 8.6.1], a proximate notion to a\nDixmier trace. If ωis an extended limit on L∞(0,∞), and 0≤A∈ L1,∞andBis\nan arbitrary bounded linear operator, then the functionals ζωandζω,Bare defined\nas\nζω(A) :=ω({t−1Tr(A1+t−1)}t>0), ζω,B(A) :=ω({t−1Tr(A1+t−1B)}t>0).\nIt is not obvious that ζωis additive, but in fact ζωextends to a linear functional\nonL1,∞and is a trace (so that ζω(AB) =ζω(BA)) [30, Theorem 8.6.4 and Lemma\n2.7.4]. Theorem 8.6.5 of [ 30] states that ζω,B(A) =ζω(AB).\nThe key result to which we refer is [ 30, Theorem 9.3.1], which is a zeta-function\nresidue formula for the Dixmier trace. In particular, the result implie s that for a\nlinear operator 0 ≤A∈ L1,∞, the following are equivalent:\n(i) Trω(A) =cfor all dilation invariant extended limits ω,\n(ii) There exists a limit\nc= lim\nt→∞t−1Tr(A1+t−1).24 N.AZAMOV, E.MCDONALD, F.SUKOCHEV, D.ZANIN\nIn particular, the theorem (combined with Theorem 8.6.5 of [ 30] mentioned above)\nentails that if 0 ≤A∈ L1,∞andBis bounded, and the limit\nc= lim\nt→∞t−1Tr(A1+t−1B)\nexists, then Tr ω(AB) =cfor all dilation-invariant extended limits ω. We would\nlike to apply this result to the following theorem, with A=M−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ightandB=e−sH,\nbut the theorem does not directly apply since M−1\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ightis not even compact, let alone\ninL1,∞. To overcome this difficulty, we apply Lemma 5.6. The full details are as\nfollows.\nTheorem 5.7. Assume that there exists the limit\nE=d−1lim\nr↓d(r−d)Tr(e−sHM−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight)\nthen for any Dixmier trace Trωand alls >0we have\nTrω(e−sHM−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight) =E.\nProof.LetY=e−ε\n2HM−1\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ighte−ε\n2H(as in Lemmas 5.5and5.6). Lemma 5.6implies\nE=d−1lim\nr↓d(r−d)Tr(e−(s−εd)HYr).\nHence for every extended limit ω(f) =ω−limt→∞f(t) onL∞(0,∞) , we have\n(takingr\nd−1 =1\nt)\nE=ω/parenleftBig1\ntTr(e−(s−εd)H(Yd)1+1\nt/parenrightBig\n.\nAccording to [ 30, Theorem 8.6.5] (with A=YdandB=e−(s−εd)H), it follows that\nE=ζω(e−(s−εd)HYd).\nwhereζω:L1,∞→Cis the zeta function associated with the extended limit ω(see\n[30, Definition 8.6.1 and Theorem 8.6.4]). Appealing to Lemma 5.6withr=dand\ntaking into account that ζωvanishes on L1,we obtain\nE=ζω(e−sHM−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight).\nSinceζωis a trace (by [ 30, Theorem 8.6.4 and Lemma 2.7.4]), it follows that\nE=ζω(e−1\n2sHM−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ighte−1\n2sH).\nCombining this with [ 30, Theorem 9.3.1] gives\nE= Trω(e−1\n2sHM−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ighte−1\n2sH).\nIt follows that E= Trω(e−sHM−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight). /square\n6.Formula for the density of states\nOur next step is to show that for all s >0 we have\nlim\nR→∞1\n|B(0,R)|Tr(e−sHMχB(0,R)) =d\nωdlim\nr↓1(r−1)Tr(e−sHM−dr\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight).A DIXMIER TRACE FORMULA FOR THE DENSITY OF STATES 25\nFor each s >0, the operator e−sHis an integral operator [ 42, Corollary 25.9],\ndenote its kernel by Ks,V. We shall prove the equivalent statement that\nlim\nR→∞1\n|B(0,R)|/integraldisplay\nB(0,R)Ks,V(x,x)dx=d\nωdlim\nr↓1(r−1)/integraldisplay\nRd/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−drKs,V(x,x)dx.\nAlthough the kernel Ks,Vis onlya prioridefined pointwise-almost everywhere, we\nunderstand the meaning of Ks,V(x,x) in a Lebesgue averaged sense, as justified by\nBrislawn’s theorem [ 13, Theorem 3.1]. The following is a routine abelian theorem.\nLemma 6.1. LetFbe a bounded measurable function on Rdand assume that there\nisc∈Csuch that:/integraldisplay\nB(0,R)F(t)dt=cRd+o(Rd), R→ ∞.\nThen: /integraldisplay\nRd/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht−drF(t)dt=c\nr−1+o/parenleftbigg1\nr−1/parenrightbigg\n, r↓1.\nMore concisely, we have:\nlim\nR→∞1\n|B(0,R)|/integraldisplay\nB(0,R)F(t)dt=1\n|B(0,1)|lim\nr↓1(r−1)/integraldisplay\nRd/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht−drF(t)dt,\nwhenever the left hand side exists.\nProof.Write/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht−dras an integral of an indicator function:\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht−dr=/integraldisplay/ang⌊∇a⌋ketleftt/ang⌊∇a⌋ket∇ight−dr\n0dθ=/integraldisplay1\n0χ[0,/ang⌊∇a⌋ketleftt/ang⌊∇a⌋ket∇ight−dr)(θ)dθ\n=/integraldisplay1\n0χ[0,(θ−2\ndr−1)1/2)(|t|)dθ.\nThus by Fubini’s theorem:\n/integraldisplay\nRd/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht−drF(t)dt=/integraldisplay1\n0/integraldisplay\nRdχ[0,(θ−2\ndr−1)1/2)(|t|)F(t)dtdθ\n=/integraldisplay1\n0/parenleftBigg/integraldisplay\nB(0,(θ−2\ndr−1)1/2)F(t)dt/parenrightBigg\ndθ.\nWith the change of variable θ=/a\\}b∇acketle{tR/a\\}b∇acket∇i}ht−dr,we have:\n/integraldisplay\nRd/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht−drF(t)dt=/integraldisplay∞\n0drR\n/a\\}b∇acketle{tR/a\\}b∇acket∇i}htdr+2/integraldisplay\nB(0,R)F(t)dtdR\n=dr/integraldisplay∞\n0Rd+1\n/a\\}b∇acketle{tR/a\\}b∇acket∇i}htdr+2/parenleftBigg\n1\nRd/integraldisplay\nB(0,R)F(t)dt/parenrightBigg\ndR.\nOur assumption is that:\n1\nRd/integraldisplay\nB(0,R)F(t)dt=c+ρ(R)\nwhereρ(R) =o(1) asR→ ∞. Therefore:\n/integraldisplay\nRd/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht−drF(t)dt=drc/integraldisplay∞\n0Rd+1\n/a\\}b∇acketle{tR/a\\}b∇acket∇i}htdr+2dR+dr/integraldisplay∞\n0Rd+1\n/a\\}b∇acketle{tR/a\\}b∇acket∇i}htdr+2ρ(R)dR.26 N.AZAMOV, E.MCDONALD, F.SUKOCHEV, D.ZANIN\nThe former integral evaluates to1\n2Γ(d(r−1)/2)Γ(1+d/2)\nΓ(1+dr/2). Since the gamma function\nhas a pole with residue 1 at 02, we have:\n/integraldisplay\nRd/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht−drF(t)dt=c\nr−1+O(1)+dr/integraldisplay∞\n0Rd+1\n/a\\}b∇acketle{tR/a\\}b∇acket∇i}htdr+2ρ(R)dR, r↓1.\nIt remains to show that the last summand is o/parenleftBig\n1\nr−1/parenrightBig\n.\nIt is enough to prove that\n(6.1) lim\nr→1+(r−1)/integraldisplay∞\n1Rd+1\n/a\\}b∇acketle{tR/a\\}b∇acket∇i}htdr+2ρ(R)dR= 0.\nBy assumption, ρ(R)→0 asR→ ∞. Ifρhas compact support, then ( 6.1) is\nautomatically true by the dominated convergence theorem. Note t hat if|ρ(R)| ≤a,\nthen:\n(r−1)/integraldisplay∞\n1Rd+1\n/a\\}b∇acketle{tx/a\\}b∇acket∇i}htdr+1|ρ(R)|dR≤a/d.\nChoose any ε >0 and write ρasρ=ρc+ρswhereρchas compact support and\n|ρs|is bounded by εd.Then by the triangle inequality:\n|(r−1)/integraldisplay∞\n1Rd+1\n/a\\}b∇acketle{tR/a\\}b∇acket∇i}htdr+2ρ(R)dR| ≤(r−1)/integraldisplay∞\n1Rd−rd−1|ρc(R)|dR+ε.\nChoosersufficiently close to 1 such that the first term is less than εilon.We can\ndo this, since ( 6.1) holds for ρc.But then:\n|(r−1)/integraldisplay∞\n1R−rρ(R)dR| ≤2ε\nand this gives the result.\n/square\nCorollary 6.2. For alls >0, if the density of states measure νHexists, then:\n/integraldisplay\nRe−sλdνH(λ) =1\n|B(0,1)|lim\nr↓1(r−1)/integraldisplay\nRd/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−drKs,V(x,x)dx.\nThat is,/integraldisplay\nRe−sλdνH(λ) =1\n|B(0,1)|lim\nr↓1(r−1)Tr(e−sHM−dr\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight).\nwhenever νHexists.\nProof.By definition (c.f. ( 1.3)), we have:\n/integraldisplay\nRe−sλdνH(λ) = lim\nR→∞1\n|B(0,R)|Tr(MχB(0,R)e−sH)\n= lim\nR→∞1\n|B(0,R)|/integraldisplay\nB(0,R)Ks,V(x,x)dx.\nWe now conclude the proof by an application of Lemma 6.1. The only condition of\nLemma6.1which needs to be checked is that x/mapsto→Ks,V(x,x) is essentially bounded\nonRd. This is [ 42, Corollary 25.9]. /square\n2This follows from the identity Γ( z) =1\nzΓ(z+1)A DIXMIER TRACE FORMULA FOR THE DENSITY OF STATES 27\nRemark 6.3.Our proof crucially relies on the following well-known property of the\nLaplace transform of measures [ 46, Theorem II.6.3]: if νandµare complex Borel\nmeasures supported on some semiaxis [ −C,∞) such that:\n/integraldisplay\nRe−stdν(t) =/integraldisplay\nRe−stdµ(t)\nfor alls >0 (in particular, both integrals as Lebesgue integrals for all s >0), then\nν=µ.\nAn easy wayto see this is as a consequence ofthe Stone-Weierstra sstheorem [ 35,\n§5.7]. Without loss of generality, C= 0 and µ= 0. Then we have a measure νon\n[0,∞) such that/integraltext∞\n0e−stdν(t) = 0 for all s >0. It follows that/integraltext∞\n0g(t)dν(t) = 0\nfor all functions gwhich are a finite linear span of functions in {e−st}s>0.\nHowever the linear span of {e−st}s>0is a subalgebra of the set C0([0,∞)) which\nseparates points, hence every f∈C0([0,∞)) is a uniform limit of functions in the\nlinear span of {e−st}s>0. Letf∈Cc([0,∞)) be a continuous compactly supported\nfunction, and select a sequence {gn}n≥0of functions in the linear span of {e−st}s>0\nwhich uniformly approximate the continuous compactly supported f unctiont/mapsto→\nf(t)etasn→ ∞. Thus we have:\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay∞\n0f dν−/integraldisplay∞\n0gn(t)e−tdν(t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤sup\nt≥0|etf(t)−gn(t)|/integraldisplay∞\n0e−td|ν|(t).\nSince each/integraltext∞\n0gn(t)e−tdνvanishes and also/integraltext∞\n0e−td|ν|(t)<∞by the assumption\nthat each e−stisν-integrable in the Lebesgue sense, it follows that/integraltext∞\n0f(t)dν= 0\nfor all continuous compactly supported continuous functions f. Hence by the Riesz\ntheorem, ν= 0.\nProof of Theorem 1.1.Corollary 6.2yields:\n/integraldisplay\nRe−sλdνH(λ) =1\n|B(0,1)|lim\nr↓1(r−1)Tr(e−sHM−dr\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight).\nTheorem 5.7identifies the limit above as being exactly:\nlim\nr↓1(r−1)Tr(e−sHM−dr\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight) =1\ndlim\nr↓d(r−d)Tr(e−sHM−r\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight)\n= Trω(e−sHM−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight).\nTherefore,/integraldisplay\nRe−sλdνH(λ) =1\n|B(0,1)|Trω(e−sHM−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight).\nTheorem 4.4implies that if fis a Borel function on Rsuch that t/mapsto→ |f(t)|/a\\}b∇acketle{tt/a\\}b∇acket∇i}htdis\nbounded, then:\n(6.2) |Trω(f(H)M−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight)| ≤Csup\nt∈R|f(t)|/a\\}b∇acketle{tt/a\\}b∇acket∇i}htd\nfor some constant C. From the Riesz theorem, it follows that the functional f/mapsto→\nTrω(f(H)M−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight) is represented by a Borel measure µonR,\nTrω(f(H)M−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight) =/integraldisplay\nRf dµ, f ∈Cc(R).28 N.AZAMOV, E.MCDONALD, F.SUKOCHEV, D.ZANIN\nThis identity is only a priori valid for continuous compactly supported functions,\nbut we mayinclude the function f(t) =e−st, fors >0, asfollows. Select a sequence\n{fn}∞\nn=0⊂Cc(R) such that as n→ ∞we have:\nsup\nt>−/⌊a∇d⌊lV/⌊a∇d⌊l∞|e−st−fn(t)|/a\\}b∇acketle{tt/a\\}b∇acket∇i}htd→0.\nIt follows that/integraltext\nRfndµ→/integraltext\nRe−stdµ(t) and (6.2) implies that Tr ω(fn(H)M−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight)→\nTrω(e−sHM−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight). Hence,\n/integraldisplay\nRe−stdµ(t) = Tr ω(e−sHM−d\n/ang⌊∇a⌋ketleftx/ang⌊∇a⌋ket∇ight) =|B(0,1)|/integraldisplay\nRe−stdνH(t), s >0.\nUniqueness for the Laplace transform (Remark 6.3) gives the equality of measures,\nµ=|B(0,1)|νH=ωd\ndνH, and this is the desired equality. /square\n7.Acknowledgements\nThe authors wish to thank the anonymous referee for helpful com ments and\nsuggestions. F. S. is partially supported by the Australian Researc h Council grant\nFL170100052.\nReferences\n[1] M. Aizenman and S. Warzel. Random Operators: disorder effects on Quantum Spectra and\nDynamics. Graduate Studies in Mathematics, 168. American Mathematic al Society, Provi-\ndence, RI, 2015. xiv+326 pp.\n[2] A. Aleksandrov and V. Peller. Operator Lipschitz functi ons.Uspekhi Mat. Nauk 71 (2016),\nno. 4(430), 3–106; translation in Russian Math. Surveys 71 (2016), no. 4, 605–702\n[3] A. Aleksandrov, V. Peller, D. Potapov and F. Sukochev. Fu nctions of normal operators under\nperturbations. Adv. 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Surveys 34(1979), no. 2, 109–158.\n[40] B. Simon. Trace ideals and their applications. Second edition. Mathematical Surveys and\nMonographs, 120. American Mathematical Society, Providen ce, RI, 2005. viii+150 pp.\n[41] B. Simon. Schr¨ odinger semigroups. Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 3, 447–526.30 N.AZAMOV, E.MCDONALD, F.SUKOCHEV, D.ZANIN\n[42] B. Simon. Functional integration and quantum physics. Second edition. AMS Chelsea Pub-\nlishing, Providence, RI, 2005. xiv+306 pp.\n[43] F. Sukochev and D. Zanin. Which traces are spectral? Adv. Math. 252(2014), 406–428.\n[44] F. Sukochev and D. Zanin. A C∗-algebraic approach to the principal symbol. I. J.Oper.\nTheory80:2 (2018), 101-142.\n[45] F. Sukochev and D. Zanin. The Connes character formula f or locally compact spectral triples.\narXiv:1803.01551\n[46] D. Widder. The Laplace Transform. Princeton Mathematical Series, v. 6. Princeton Univer-\nsity Press, Princeton, N. J., 1941. x+406 pp.\nUniversity of New South Wales, Kensington, NSW, 2052, Australi a\nE-mail address :nurulla.azamov@unsw.edu.au\nE-mail address :edward.mcdonald@unsw.edu.au\nE-mail address :f.sukochev@unsw.edu.au\nE-mail address :d.zanin@unsw.edu.au"    },    {        "title": "1911.05237v1.Estimation_of_Cooper_pair_density_and_its_relation_to_the_critical_current_density_in_hole_doped_high_Tc_cuprate_superconductors.pdf",        "content": "Estimation of Cooper pair density and its relation to the critical \ncurrent density in hole doped high- Tc cuprate superconductors \nNazir Ahmad, S. H. Naqib* \nDepartment of Physics, University of Rajshahi, Rajshahi 6205 \n*Corresponding author; Email: salehnaqib@yahoo.com   \n \nAbstract \nHole concentration in the CuO 2 plane largely controls all the electronic properties in the normal \nand superconducting states of high- Tc cuprates. The critical current density, Jc, is no exception. \nPrevious hole content dependent studies have demonstrated the role of intrinsic depairing \ncurrent density in determining the observed critical current density in copper oxide \nsuperconductors.  It is also widely agreed upon that the temperature and magnetic field \ndependent vortex pinning energy plays a major role in determining the Jc of a system. This \npinning energy depends directly on the superconducting condensation energy. \nSuperconducting condensation energy, on the other hand, is proportional to the Cooper pair \ndensity (superpair density), which is found to be highly dependent on the hole concentration, p, \nwithin the CuO 2 plane. We have calculated the Cooper pair density, ρs, of YBCO (Y123), a typical \nhole doped cuprate, as a function of p, in this study. A triangular pseudogap (PG), pinned at the \nFermi level, in the quasiparticle spectral density has been considered. The low-temperature \ncritical current density of a number of Y(Ca)BCO superconductors over wide range of \ncompositions and hole concentrations have been explored. The normalized values of the \nsuperpair density and the critical current density exhibit a clear correspondence as the in-plane \nhole content is varied. This systematic behavior provides us with strong evidence that the \ncritical current density of hole doped cuprates is primarily dependent on the superpair density, \nwhich in turn depends on the magnitude of the PG energy. The agreement between the \nestimated p-dependent superpair density and the previously experimentally determined \nsuperfluid density of Y(Ca)BCO is quite remarkable.  \nKeywords:  Hole doped cuprates; Superconductivity; Superpair density; Critical current density; \nPseudogap \n \n1. Introduction  \nThe phenomenon of superconductivity is now a considerable focus of attention to the scientific \ncommunity as one of the most promising technologies that can address the global energy crisis by facilitating energy utilization with minimal loss and high degree of efficiency. The possibility \nof every large scale practical application of superconductivity depends primarily on the \nmaximum current density which superconductors can carry (the critical current density, Jc), in \nsome way or other. The value of losses incurred within the superconductors, the maximum \nmagnetic field strength in which superconductors can be used, etc are the other important \nfactors all intimately linked with Jc. All these factors are directly related to the pinning ability of \nthe quantized magnetic flux lines (magnetic vortices) of superconductors in the \nsuperconducting state. Maximizing Jc and the magnetic field under which the superconductor \ncan perform, as well as minimizing losses, have been important goals of scientists working in \nthe field of applied superconductivity.   \n \nHigh-Tc cuprates discovered more than three decades back [1], belongs to the group of the \nmost promising superconductors capable of sustaining high critical currents under high applied \nmagnetic fields [2] even though serious theoretical and technical challenges remain [2, 3]. The \ninitial enthusiasm surrounding the vision for extremely high critical current density, powerful \nmagnets, motors, generators, and loss-less transmission lines working at liquid nitrogen \ntemperatures (77 K) was based on the high superconducting transition temperatures of \ncuprates. The belief that a high- Tc itself ensures a high Jc is oversimplified. In reality applications \nat 77 K have turned out to be much more challenging than at 4.2 K, irrespective of the values of \nTc and the upper critical field, Hc2.  Strong electronic correlations, d-wave order parameter, \nquasi-two dimensional structural features leading to high level of structural and electronic \nanisotropy, and small correlation length in high- Tc cuprates make these systems fascinating \nmaterials with diverse and competing electronic orders [4 – 7]. These complexities, at the same \ntime, hinders any attempt to make such materials useful, which involves compromises among \nconflicting requirements, defining the parameters of merit depending on the operating \nconditions and also on the specific form of the application [2, 3].  Since the early days of \nsuperconductivity in cuprates, it was realized that the effect of strong thermal fluctuations of \nvortices will make pinning a challenging problem at and above the boiling point of liquid \nnitrogen. The supercurrent carrying capability becomes limited by fluctuations of the order \nparameter and thermally activated hopping of the flux lines. The optimized condition is \nexpected to be found in samples with high superfluid density and low anisotropy factor [2, 3, 8].  \nIt also appeared that the low- T Jc might be mainly governed by the intrinsic superconducting \nparameters and the performance cannot be improved significantly via extrinsic modifications \nlike introducing defects as pinning centers, especially in case of optimally doped compounds \nwith maximum possible Tc [2, 9, 10]. In a previous study we have gathered indications that \nindeed the intrinsic depairing current density sets the value of the experimental Jc to a large \ndegree [9].  \n Due to its comparatively high superfluid density and the lowest level of structural and \nelectronic anisotropy, YBa 2Cu3O7- (Y123 or YBCO) remains among one of the most promising \nhigh-Tc cuprates for potential applications. In this study we will focus on the zero-field critical \ncurrent density at zero temperature, as a function of hole content for YBCO and                         \nY1-xCaxBa2Cu3O7- (Y(Ca)BCO) over a wide range of compositions. Zero-field and zero-\ntemperature critical current density is expected to be dominated by the intrinsic effects as \nmaximally developed superconducting (SC) energy gap and order parameter mask extrinsic \neffects to a large extent. Ca substituted compounds have been used because fully oxygenated \n( ~ 0) YBCO is slightly overdoped (OD) and full oxygen loading is difficult; deeply OD regions \ncan be accessed when trivalent Y3+ is replaced by divalent Ca2+ [9, 11 – 13].  \n \nIt is known that the supercurrent circulates due to phase angle twist  (measured through ) of \nthe SC order parameter. This twist, in turn, depends on the phase stiffness  of the SC compound. \nThe phase stiffness, on the other hand, varies linearly with the superfluid density [14]. \nTherefore, from intrinsic consideration, the Jc of a superconductor is expected scale with the \nsuperfluid density or the superpair density for that matter.  \n \nA number of prior studies revealed that a pseudogap (PG) correlation, competing with \nsuperconductivity, depletes superfluid density most effectively [7, 15, 16]. This arises because \nof the removal of the low-energy quasiparticle (QP) spectral weight around the Fermi level due \nto the formation of the PG.  In this paper we intend to investigate this proposal via the \ncalculations of superpair density for Y(Ca)BCO within a triangular PG scenario pinned at the \nFermi level [17, 18]. The superpair density as function of hole content, ρs(p), has been \ncalculated from the previously estimated [10, 18 – 20] p-dependent PG energy scale, Eg(p), \nusing the triangular PG model. The p-dependent zero-field and zero-temperature critical \ncurrent density, J0(p), shows clear correspondence to ρs(p). These are the central results of this \ninvestigation. \n \nRest of the paper is organized as follows. Section 2 comprises of the description of the \nmethodology used to calculate the superpair density within a simple triangular PG model. \nSection 3 deals with the correspondence between ρs(p) and J0(p). Results are discussed and \nfinally, conclusions are drawn in Section 4.  \n \n2. Theoretical methodology for calculation of ρs(p) \nThe PG is a suppression of electronic density of states (EDOS) near the Fermi level. One of the \ndistinct features of this gap in the QP energy spectrum is that it has states non-conserving \ncharacter [21 – 23] and does not show any distinct coherence peak-like feature as observed at the onset of phase coherent superconductivity. Variety of earlier studies have demonstrated \nthe success of a states non-conserving triangular PG model pinned at the Fermi energy to \nexplain diverse class of normal and SC state experimental results including temperature \ndependent resistivity [24, 25], bulk magnetic susceptibility [18, 26], impurity induced magnetic \nbehavior [17, 27, 28], electronic heat capacity [21], and NMR Kinight shift data [29]. The \nschematic diagram of such a simple triangular PG is shown in Fig. 1 below. \n \nFigure 1: The linearly vanishing triangular pseudogap with energy Eg. Δsc marks the amplitude of \nthe zero-temperature superconducting gap. \nFigure 1 also shows the magnitude of the superconducting gap (SCG). It should be noted that \nthe SC coherence peaks at either side of the Fermi level have not been shown in this figure. \nThese coherence peaks are formed in the EDOS at the expense of condensed (coherently \npaired) charge carriers residing within the low-energy electronic density of states. The PG \ndepletes these low-energy EDOS and reduces the superconducting condensate. In other words, \nin the absence of the PG, the QP spectral weight under the coherence peaks would have been \nmuch larger since in this case there would be a lot more QP spectral weight available at low-\nenergies to take part in the SC pairing condensate. \nA clear understanding of the nature of the spectral gaps remain as one of the most challenging \nproblems in hole doped cuprates. Contrary to conventional Fermi-liquid superconductors \nwhere the gap in the QP spectral density around the Fermi level vanishes at the SC critical \ntemperature, as described by the Bardeen-Cooper-Schrieffer (BCS) theory [14, 30], in cuprates \nan energy gap exists much above the critical temperature Tc in the underdoped (UD) and \noptimally doped (OPD) compounds [11, 18, 19, 22, 31 – 34]. Distinguishing this normal state PG \nnear Tc from the superconducting (coherence) energy gap is a challenging issue. There is \ngrowing evidence that the PG is distinct from the SCG and the characteristic PG temperature, T*(p) goes below the Tc(p) dome in the slightly OD side of the phase diagram and terminating \n(T*(p) = 0 K) at a critical hole concentration, pc ~ 0.19 [11, 18, 19, 22, 31 – 35].  \nIt has been observed from a variety of experimental probes [36, 37] that the magnitude of the \nSCG does not vary much over an extended region of the hole content, particularly in the region \nfrom slightly OD to moderately underdoped parts of the T-p electronic phase diagram. In this \nparticular region, the magnitude of the SCG remains larger than the PG and the spectral gaps \nconform to the schematic diagram shown in Fig. 1. For simplicity, we will perform our \ncalculations of superpair density mostly in this region. Within the simple d-wave SCG formalism, \n2sc/kBTc = 4.28 [38]. In the strong coupling regime (prevalent to the deeply UD compounds), \nthis ratio tends to increase. This is because Tc goes down in the underdoped region, but sc \nstays high [36, 37]. In fact, over the range of hole content considered here, sc = 2.14kBTc0, to a \nreasonable approximation [36, 37] for YBCO. Tc0 denotes the maximum SC transition \ntemperature at the optimum doping ( popt = 0.16). For YBCO, Tc0 = 93 K. We discuss the possible \nimplications of this approximation in Section 4.  \nTo be specific, the EDOS profile centered at the Fermi energy as shown in Fig. 1, can be \nmodeled as, \nN(E) = N0(E/Eg)  for E  Eg \n        (1) \n= N0  for E > Eg \nwhere N0 is the EDOS in the flat region outside the spectral gaps. \nBy definition, the superpair density can be expressed as, \n      𝜌s(𝑝)= 〈𝑁(𝐸F)〉Δsc            (2)  \nwhere 〈𝑁(𝐸F)〉 is the normal state average electronic density of states pinned at the Fermi-level. \nIt should be noted that, for the conventional superconductors with weakly energy dependent \nEDOS around the Fermi energy, this quantity is almost identical to the N(𝐸F). Due to the \nsymmetrical nature of the EDOS above and below 𝐸F within a window of energy of width ± Δsc, \none may write, \n<𝑁(𝐸ி)>=∫𝑁(𝐸)𝑑𝐸∆ೞ\n\n∫𝑑𝐸∆ೞ\n \nIn the UD side, where the PG amplitude can be greater in magnitude compared to the SCG, we \nget (using the triangular PG model as described by Eqn. 1), <𝑁(𝐸ி)>=∫ಿబಶ\nಶௗா∆ೞ\nబ\n∫ௗா∆ೞ\nబ=ேబ∆ೄ\nଶா     (3) \nFor the second condition, 𝐸g  Δsc, of the model gap, as shown in Fig. 1, we obtain, \n<𝑁(𝐸ி)>=∫ಿబಶ\nಶௗா∆ೞ\nబ\n∫ௗா∆ೞ\nబ=𝑁(1− ா\nଶ∆ೄ)    (4) \nThe characteristic pseudogap temperature, T* can be expressed as, T* Eg/kB [11, 12, 17 – 19, \n26]. Therefore, one can express the superpair density from Eqns. 2, 3, and 4 as follows, \n𝜌௦(𝑝)= 2.29𝑁𝑘்బమ\n்∗()               for T* > 2.14T c0 \n         (5) \n𝜌௦(𝑝)=𝑁𝑘(2.14𝑇−்∗()\nଶ)                                        for T* 2.14Tc0    \nIt follows from Eqns. 5 that within the proposed scenario, the hole content dependence of the \nsuperpair density arises from the p-dependent PG energy scale.  \nWe have used Eqns. 5 to calculate the doping dependent superpair density of YBCO and Ca \nsubstituted YBCO. It should be noted that reliable independent estimate of N0 does not exist in \nthe literature. Therefore, we have calculated the normalized superpair density. A large body of \nexperimental studies has demonstrated that 𝜌௦(𝑝) becomes maximum at p ~ 0.19 [15, 31, 39] \nwhere the PG vanishes quite abruptly. Hence we have fixed it to unity and calculated 𝜌௦(𝑝) \nwith respect to this value at other hole concentrations. It is worth noting that the PG energy \nscale (and consequently T*) does not depend on the level of Ca substitution in Y(Ca)BCO and \ndepends solely on the number of doped holes in the CuO 2 planes [11, 12, 19]. It is also \nimportant to realize that T*(p) is insensitive to the crystalline state of the material [12, 19]; \nexcept for highly disordered compounds [20, 40], T*(p) is same in bulk single and polycrystals \nand thin films [12, 19] for a given value of p.    \nThe T*(p) values for pure YBCO and Ca doped YBCO compounds are taken from prior published \nsources [11, 12, 19, 20]. Estimated values of normalized 𝜌௦(𝑝) obtained by employing Eqns. 5 \nare presented in Fig. 2.          \nFigure 2: Variation of the normalized superpair density with number of doped hole content in \nthe CuO 2 plane of YBCO. The hole contents are accurate within ± 0.004. The errors in the \nnormalized superpair density come primarily from the uncertainty in the values of T*(p) [9, 11, \n16, 19]. \n3. Superpair density and the critical current density \n𝐽c can be enhanced by increasing the vortex pinning force per unit volume. However, 𝐽c cannot \nbe increased indefinitely even if it were possible to prevent vortex motion completely. There is \nan intrinsic limit to the maximum achievable supercurrent density. This limiting value is termed \nas the depairing current density, Jdp [41]. It sets an upper limit to 𝐽c. This depairing critical \ncurrent density is an intrinsic characteristic of superconductors. It is directly related to the \nphase stiffness of the superconducting wavefunction. Actually the observed critical current \ndensity is primarily determined by this depairing current density in cuprates, particularly in \nYBCO and Ca substituted YBCO [9, 42]. \n \n \n \n 0.00.200.400.600.801.0\n0.080 0.10 0.12 0.14 0.16 0.18 0.20Normalized superpair density\nHole content, pThe depairing current density, Jdp, corresponds to the current density at which the kinetic \nenergy of the Cooper pair and the condensation energy become equal. Jdp can be expressed as \n[43] \n𝐽dp=𝜙\n 3√3𝜋𝜇𝜆ଶ𝜉                                                                                                (6)  \nwhere 𝜙 is the flux quantum, 𝜇 is the permeability of vacuum and  is the SC coherence \nlength. \nFor high-Tc cuprates Jdp is approximately 10ଽA/cmଶ[42]. Currently, the critical current density, \nJc, reaches about 5−10 %  of the Ginzburg–Landau depairing current density at 4.2 K for the \nbest high- Tc specimens [43]. Nevertheless, it has become evident from a variety of hole content \ndependent critical current density studies that it is this depairing contribution which sets the \nlow-T limit of Jc [8 – 10, 42, 44]. \nThe critical current density varies strongly with temperature. As temperature increases the \nextrinsic effects associated with defects of different nature within the compound start to \ndominate the temperature dependent Jc. The extrapolated zero-temperature Jc at zero \nmagnetic field, J0, gives the true reflection of the intrinsic depairing contribution [9, 10]. In this \ninvestigation, we have presented the zero temperature critical current density of high-quality c-\naxis oriented thin films of Y(Ca)BCO as a function of hole content. The thicknesses of these \nepitaxial thin films lied within 2800 ± 300 Å. Hole content was varied for fixed level of Ca \nsubstitution via oxygen annealing under different temperatures and partial pressures. J0 was \nextracted by fitting the hole content dependent zero-field critical current density employing the \nfollowing relation [3, 45]  \n \n n\nc tJtJ )1()(0 0             (7) \n \nwhere, t = (T/Tc), is the reduced temperature and Jc0 is the zero-field critical current density \nobtained from the M-H hysteresis loops via the modified critical state formalism [9, 46]. Value \nof the exponent, n, in Eqn. 7 is dependent on the level of anisotropy, defect distribution, \nmicrostructure, and level of homogeneity in chemical composition [3, 9, 10]. Magnetic field was \napplied along the c-direction. Therefore, the critical current flowed in the ab-plane of the \ncompounds. Details regarding the samples used in this study and magnetization measurements \ncan be found in Refs. [9, 12, 47]. The extrapolated values of J0 together with their normalized \nvalues are presented in Table 1 for a large number of Y(Ca)BCO thin films. The hole contents \nreported in this paper are determined from the room-temperature thermopower measurements [11, 19, 48, 49] and also via the application of the widely employed parabolic Tc-\np relation [50]. The reported values are accurate within ± 0.004. \nTable 1 \nZero-field and zero-temperature critical current density of Y 1-xCaxBa2Cu3O7- thin films. \n \nCompound Hole content ( p) Critical current \ndensity, J0 (106 \nA/cm2) Normalized critical \ncurrent density \nYBa2Cu3O7- 0.162 20.62 0.668 \n 0.146 14.84 0.481 \n 0.102 5.99 0.194 \n    \nY0.95Ca0.05Ba2Cu3O7- 0.184 30.88 1.000 \n 0.170 26.04 0.843 \n 0.156 22.10 0.716 \n 0.123 12.10 0.392 \n    \nY0.90Ca0.10Ba2Cu3O7- 0.198 23.73 0.831 \n 0.188 28.54 1.000 \n 0.162 23.50 0.823 \n 0.160 24.41 0.855 \n 0.126 13.08 0.458 \n    \nY0.80Ca0.20Ba2Cu3O7- 0.201 17.08 0.921 \n 0.186 18.54 1.000 \n 0.166 17.01 0.917 \n 0.150 12.98 0.700 \n 0.144 12.03 0.649 \n 0.136 10.09 0.544 \n   \nFor meaningful comparison, we have plotted normalized superpair density and normalized \nzero-field, zero-temperature critical current density in Fig. 3 as function of number of doped \nholes in the CuO 2 planes. A clear correspondence between the calculated superpair density \nbased on the triangular PG model and J0 obtained from experimental critical current density is \nseen over an extended range of hole content from p ~ 0.10 to 0.19. \n \n \n  \nFigure 3: Variation of the normalized superpair density and normalized J0 with hole content of \nY(Ca)BCO superconductors.  \n3. Discussion and conclusions \nIn the preceding section, we have shown that a simple model based on a triangular PG in the \nQP spectral density pinned at the Fermi level can be used to estimate the superpair density \nquite easily. The normalized values of 𝜌௦(𝑝) closely follows the normalized J0(p) extracted from \nthe experimental critical current density data over a substantial region of SC composition of \nY(Ca)BCO. By definition, the superpair density should have the same physical significance as the \nsuperfluid density, ns. As far as experimental studies are concerned, two of the most direct \nmethods to obtain ns (actually ns/m*) are the magnetic penetration depth measurement and \nmeasurement of the muon spin resonance ( μSR) depolarization rate [51, 52]. Both these \nmeasurements estimate ns/m*, where m* is the effective mass of the super-carriers. It is \ninstructive to note that the both the μSR depolarization rate, σ and λ-2 are directly proportional \nto each other [15, 53], and the magnetic penetration depth is related to the superfluid density \nthrough λ = [m*/(μ0nse2)]1/2. Bernhard et al. [15] have calculated the hole content dependent \nsuperfluid density and SC condensation energy, U0, of Y(Ca)BCO and Tl 0.5-yPb0.5+ySr2Ca1-xYxCu2O7. \nWe have plotted the results of p-dependent normalized ns/m* of Y(Ca)BCO together with our 0.00.200.400.600.801.0\n0.080 0.10 0.12 0.14 0.16 0.18 0.20Normalized superpair density\nNormalized J\n0 (x = 0.00)\nNormalized J\n0 (x = 0.05)\nNormalized J\n0 (x = 0.10)\nNormalized J0 (x = 0.20)Normalized superpair density; Normalized J0\nHole content, presults of normalized ρs in Fig. 4. Considering the simplicity of the triangular PG model, the \nagreement between the theoretically estimated ρs(p) and experimentally determined ns/m*(p) \nis remarkable. The inset of Fig. 4 exhibits the hole content dependent normalized SC \ncondensation energy of Y(Ca)BCO. The p-dependent features of the condensation energy is \nsimilar to that of ρs(p) and ns/m*(p). This has important consequence on the observed hole \ncontent dependence of the critical current density. The SC condensation energy shown below \nwas calculated from the electronic heat capacity results for Y(Ca)BCO [22]. \n                 \n \nFigure 4: Main panel: Hole content dependent normalized superfluid density ( ns/m*\nab) obtained \nfrom the μSR measurements [15] and the calculated normalized superpair density of Y(Ca)BCO. \nInset: Hole content dependent normalized SC condensation energy, U0, extracted from the heat \ncapacity measurement [22] of Y(Ca)BCO superconductors. \nThe observed correspondence between the normalized ρs(p) and the normalized J0(p) as \nillustrated in Fig. 3, can be understood from two different but related point of views. It follows \nfrom Eqn. 6 that the intrinsic depairing critical current density is directly proportional to λ-2, and \ntherefore, to ns/m*\nab, which in turn is a measure of the superpair density as shown in Fig. 4. It is \nworth noticing that Jdp also varies with the SC coherence length . But like the SC energy gap, \nthe variation of  with hole content is weaker than that of ρs(p) or ns/m*\nab(p). Therefore, the \nhole content dependent behavior of Jc is dominated by the p-dependence of ρs or ns/m*\nab. 0.00.200.400.600.801.0\n0.080 0.10 0.12 0.14 0.16 0.18 0.20Normalized superpair density\nNormalized n\ns/m*\nabNormalized superpair density; Normalized ns/m*\nab\nHole content, p0.00.200.400.600.801.0\n0.080 0.12 0.16 0.20Normalized U0\nHole content, pIt is known that the magnetic flux lines are pinned at locations (pinning sites) where the SC \norder parameter is partially or almost completely suppressed. Under this situation the pinning \nenergy of the vortex core reveals itself as the energy barrier to the dissipative movement of the \nflux line and gives a measure of the flux activation energy Ua [54]. It is this activation energy \nwhich sets the values of Jc and the irreversibility magnetic field [45, 54] of a superconductor. \nEmploying a simple heuristic scaling, Yeshurun and Malozemoff [55] and Tinkham [56] have \nfound that Ua ~ Hc2, where Hc is the thermodynamical critical magnetic field. On the other hand, \nthe SC condensation energy can be expressed as U0 ~ Hc2, which in consequence implies that, Ua \n~ U0 ~ Hc2 [45, 54]. Furthermore, the SC condensation energy can also be expressed as U0 = \n<N(EF)>sc2. This expression for U0 establishes a direct link between vortex dynamics with a \ncharacteristic energy scale Ua, and the superpair density  𝜌s(𝑝) (= 〈𝑁(𝐸F)〉Δsc). The arguments \npresented here are quite general in nature and do not depend significantly on the precise \nnature of the mechanism leading to Cooper pairing in a particular type of superconductor. For \nexample, we have shown recently that the variation of the critical current density of heavy \nfermion superconductors (HFSCs) [57] with pressure follows strikingly similar pattern to the p-\ndependent variation of the Jc of hole doped cuprates in Ref. [10]. The common thread is the \npressure dependent variation in the SC condensation energy in the HFSCs and its p-dependent \nvariation in the hole doped cuprates. \nWithin the proposed scenario, both J0(p) and 𝜌s(𝑝) are maximized at a hole content ( p ~ 0.19) \nwhere the PG vanishes, as found experimentally [11, 18, 19, 22, 31 – 35, 58]. For further \noverdoping, both these parameters decrease [8, 9, 15, 39, 42]. The origin of this reduction in \nthe superfluid density in the deeply OD side is still unclear [39]. There is no PG in this particular \nside of the T-p phase diagram. This behavior in the OD side implies that electronic correlations \nof different type (from the correlations giving rise to the PG in the UD to slightly OD region) \ncompetes with superconductivity and weakens the Cooper pairing correlations.     \nAs far as the estimation of 𝜌s(𝑝) based on the triangular PG model is concerned, there are a few \npoints which needs some further elaboration. We have used a single, p-independent value of \nthe SC energy gap given by sc = 2.14kBTc0 (with Tc0 = 93 K) for the calculations over a range of \nhole content from p = 0.10 – 0.19. This approximation is reasonable for the hole contents p > \n0.12 [36, 37]. For Y(Ca)BCO compounds with lower level of hole content, the SC gap exceeds \nthis value. This enhanced SC gap reduces the estimated value of the superpair density to some \nmeasure (at the level of ~ 5 – 10%). To account for this reduction, we have incorporated the \nappropriate error bars to the 𝜌s(𝑝) of the two most underdoped compounds in Figs. 2 – 4. The \ntriangular PG model assumes a flat EDOS outside the QP spectral energy gap region. This \nassumption is supported reasonably well by the p- and T-dependent coefficient of electronic \nheat capacity data [21, 22, 36]. It is instructive to point out that the p-dependent superfluid densities of Y(Ca)BCO,                 \nTl0.5-yPb0.5+ySr2Ca1-xYxCu2O7, Bi2212, and LSCO demonstrate almost identical features [15, 39], \nwhich imply that hole content dependent PG plays the prime role in determining this \nparameter in the hole doped high- Tc cuprates and the model calculations presented in this \nstudy can readily be extended to other families of hole doped cuprate superconductors. \nTo summarize, we have employed a simple PG model to calculate the superpair density of high-\nTc cuprates. 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Lett. \n61, 1658 (1988). \n[57] Soon-Gil Jung et al., A peak in the critical current for quantum critical superconductors. \nNature communications (2018), DOI: 10.1038/s41467-018-02899-5. [58] Islam, R. S., Cooper, J. R., Loram, J. W. & Naqib, S. H., The pseudogap and doping \ndependent magnetic properties of La 2-xSrxCu1-yZnyO4. Phys. Rev. B 81, 054511 (2010). \n \nAuthor Contributions  \nS. H. N. designed the project and wrote the manuscript. N. A. and S. H. N. performed the \ntheoretical analysis. Both the authors reviewed the manuscript. \n \nAdditional Information \n \nCompeting Interests \nThe authors declare no competing interests. "    },    {        "title": "1911.06609v2.Direct_Measurement_Methods_of_Density_Matrix_of_an_Entangled_Quantum_State.pdf",        "content": "arXiv:1911.06609v2  [quant-ph]  5 Jan 2020Direct Measurement Methods of Density Matrix of an Entangle d Quantum State\nYusuf Turek1,∗\n1School of Physics and Electronic Engineering, Xinjiang Nor mal University, Urumqi, Xinjiang 830054, China\nIn general, the state of a quantum system represented by the d ensity operator and its determi-\nnation is a fundamental problem in quantum theory. In this st udy, two theoretical methods such\nas using postselected measurement characterized by modula r value and sequential measurements of\ntriple products of complementary observables to direct mea surement of matrix elements of density\noperator of a two photon entangled quantum state are introdu ced. The similarity and feasibility of\nthose two methods are discussed by considering the previous experimental works.\nPACS numbers: 03.65.Ta, 06.20.Dk, 03.65.−w.\nI. INTRODUCTION\nIn quantum mechanics the state can represent a quan-\ntum system, and the improvement of the state and its\ndetermination has vital importance in obtaining any in-\nformation about that system. Because of the collapse\nof wave function due to the decoherence, the conven-\ntional quantum measurements cannot be directly used\nin some hot topics of quantum information science such\nas quantum state based high precision measurements, re-\nconstruction of unknown quantum state, etc. However,\nthe advance of the research in fundamentals of quantum\nphysics provided an effective method to solve the above\nproblems by using the simple and easily manipulable pre-\nand post-selected quantum weak measurement technique\nwhich is characterized by the weak value [ 1]. In the weak\nmeasurement the induced weak value of the observable\non the measured system is usually a complex number,\nand can be beyond the usual range of eigenvalues of\nthat observable. This property of weak value is referred\nas the amplification effect for weak signal which accom-\npanied by the decrease of the postselection probability.\nSince the weak signal amplification property experimen-\ntally demonstrated in 1991 [ 2], it have been widely used\nand solved plenty of fundamental problems in quantum\nmechanics and related sciences. For details about the\nweak measurement theory and its applications in weak\nsignal amplification processes, we refer the reader to the\nrecent overview of the field [ 3,4].\nAnother main application of postselected weak mea-\nsurement technique is quantum state tomography. The\nsignificant advantageous of postselected weak measure-\nment based state tomography technique than conven-\ntional one [ 5–9] is that in weak measurement technique\nthe tomographic procedures is easy and can get the all\nglobal phase information of unknown state than conven-\ntional schemes. Since J. Lundeen et al. [ 10] firstly in-\nvestigated the reconstruction of transversal spatial wave\nfunction of polarized photon beams by using the post-\nselected weak measurement technique, the direct mea-\nsurement of unknown quantum states have been stud-\n∗yusufu1984@hotmail.comied theoretically and experimentally by using weak and\nstrong measurement techniques [ 11–29]. In particular,\nthe direct measurement of a photon polarization state\nin two dimensional system [ 14] and direct measurement\nof density matrix of a single photon polarization state in\npure and mixed state cases [ 26] showed the power of weak\nmeasurement technique in state determination processes.\nQuantum entanglement is a main feature of quantum\nmechanics, and most of the mysterious phenomena in\nquantum world caused by entangled systems. Thus, the\nstate determination of entangled systems have significant\nimportance in quantum theory. The direct measurement\nof general quantum state by using weak measurement\nhas been studied in Refs. [ 11,15]. Furthermore, in re-\ncent innovative work of Guo-Guang Can et al. [ 30], they\ninvestigated the direct measurement of a two photon en-\ntangled state by using postselected weak measurement\nand used the modular value in reading results in stead of\nweak value. However, in general, the state of a quantum\nsystem is represented by density operator, and the direct\nmeasurement of density matrix of an entangled system\nby using weak measurement technique have not been ex-\nplicitly studied until now.\nIn this paper, as an extension of previous works\n[26,30], we study the two kinds of reconstruction meth-\nods of a two photon entangled state. We take the spatial\n(paths) and polarization degrees of freedom of unknown\nentangled state as pointer and measured system, respec-\ntively, and the joint (or sequential ) projection operators\nof two subsystems considered as measured observables of\nmeasured system. In first method, we follow the theo-\nretical part of Ref. [ 30] and use the postselected weak\nmeasurement technique to measure the matrix elements\nof a two photon entangled state. It is noticed that the\ndensity matrix elements proportional to the weak values\nof appropriate joint projection operators of two subsys-\ntems, and the pre- and post-selected states are the ele-\nments in two mutually unbiased bases. Since the weak\nmeasurement of joint projection operators of two sub-\nsystems can not be measured directly, it is founded in\nterms of the modular values of corresponding operators.\nBased on the theoretical analysis of Ref. [ 30], the real\nand imaginary parts of a matrix element can be readout\nfrom detection probability after taking appropriate pro-2\njection operations before detection on the final state of\nthe pointer.\nIn the second method, the technique introduced by J.\nLundeen et al. [ 11] is used. Three sequential measure-\nments on three projection operators of two subsystems\nwhere each complementary to the last are taken to find\nthe matrix elements of a two photon entangled state. It\nis found that the result of these sequential measurements\nproportional to the value of matrix elements. In order\nto read out the value of matrix elements, it is assumed\nthat the spatial degree of freedom of every pointer of two\nsubsystems have xandydirectional zero mean Gaussian\ndistribution, and initially there have no any correlation\nbetween them. After taking the two sequential weak mea-\nsurements with projection operators where complemen-\ntary each other, and followed by a strong measurement\non another projection operator where complementary to\nthe last, the weak average equal to the expectation values\nof products of annihilation operators (can be defined in\nterms of position and momentum operator) of four Gaus-\nsian pointer states. Thus, the real and imaginary parts\nof corresponding matrix elements can be found by cal-\nculating the joint positions and momentum shifts of the\nfinal pointer state. Here, we have to mention that previ-\nous two sequential weak measurements caused a spatial\nshifts on different directions of the pointer, respectively .\nThe rest of the paper is organized as follows: we briefly\nreview the basic concepts of direct measurement of a\nquantum state by using postselected weak measurement\nbased state tomography technique in Section. II. In Sec-\ntion.III, we give the details of two methods to determine\nthe matrix elements of a two photon entangled system,\nseparately, and take comparison between them and dis-\ncuss their feasibility. We give the conclusion to our study\nin Section. IV.\nII. DIRECT MEASUREMENT OF A STATE\nVIA WEAK MEASUREMENT\nFrom the quantum mechanics we know that the two di-\nmensional photon polarization state |ψ/an}bracketri}htin Hilbert space\ncan be expressed in the A={|H/an}bracketri}ht,|V/an}bracketri}ht}basis as\n|ψ/an}bracketri}ht=/summationdisplay\nici|i/an}bracketri}ht, i∈(H,V) (1)\nwhereci=/an}bracketle{ti|ψ/an}bracketri}htis the probability amplitude. The weak\nvalue of projection operator πi=|i/an}bracketri}ht/an}bracketle{ti|with the pres-\nelected and postselected states, |ψ/an}bracketri}htand|α/an}bracketri}ht, is defined\nas\n/an}bracketle{tπi/an}bracketri}htw\nα=/an}bracketle{tα|πi|ψ/an}bracketri}ht\n/an}bracketle{tα|ψ/an}bracketri}ht=1\nνci. (2)\nThus, it is evident that the probability amplitude ciof\nunknown state |ψ/an}bracketri}htis directly related with the weak value\nof projection operator πi, and the state vector |ψ/an}bracketri}htcan bere-expressed as\n|ψ/an}bracketri}ht=/summationdisplay\niν/an}bracketle{tπi/an}bracketri}htw\nα|i/an}bracketri}ht. α∈(D,A) (3)\nHere,α=D,H is the element in B={|D/an}bracketri}ht=1√\n2(|H/an}bracketri}ht+\n|V/an}bracketri}ht),|A/an}bracketri}ht=1√\n2(|H/an}bracketri}ht − |V/an}bracketri}ht)}diagonal and anti-diagonal\nbasis, andν=/angbracketleftα|ψ/angbracketright\n/angbracketleftα|i/angbracketrightis independent of iand can be de-\ntermined by the normalization condition. Since the real\nand imaginary parts of weak value /an}bracketle{tπi/an}bracketri}htw\nαcan be found\nsimultaneously[ 31], the unknown state vector |ψ/an}bracketri}htcan be\nreconstruct by optical experiments. The most important\npart of this reconstruction technique is the choice of post-\nselection, and from the Eq.( 3) we can know that to deter-\nmine the unknown pure state vector |ψ/an}bracketri}ht, we can scan only\non definite |α/an}bracketri}htinBbasis at postselection process. How-\never, if we want to reconstruct the density matrix of a two\ndimensional unknown state by using weak measurement\ntechnique, we have to take scan through all elements in\nbothAandBbases since its unknown parameters more\nthan the corresponding pure state. The reconstruction of\ndensity matrix of two dimensional system had been stud-\nied experimentally in Refs.[ 14,26]. We have to mention\nthat the two bases AandBare mutually unbiased for all\nbasis|i/an}bracketri}htinAand all basis |α/an}bracketri}htinBin two dimensional\nHilbert space, i.g. |/an}bracketle{ti|α/an}bracketri}ht|2=1\n2.\nIII. THE METHODS OF DIRECT\nMEASUREMENT OF DENSITY OPERATOR OF\nAN ENTANGLED QUANTUM STATE\nLet us consider a system consisting of two subsystems,\nand designate the corresponding state vector as\n|Ψ/an}bracketri}ht=/summationdisplay\nijCij|i/an}bracketri}ht1⊗|j/an}bracketri}ht2=/summationdisplay\ni,jCij|ij/an}bracketri}ht, (4)\nwherei,j∈(H,V), andCij=/an}bracketle{tij|Ψ/an}bracketri}htis complex proba-\nbility amplitude and |/an}bracketri}ht1and|/an}bracketri}ht2represent to subsystem\none and subsystem two, respectively. To reconstruct the\nunknown pure state |Ψ/an}bracketri}ht, we have to find the correspond-\ning amplitudes Cijand this task is not very easy as single\ntwo dimensional pure state case. However, recently the\nGuo-Guang Can et al.[ 30] successfully accomplished this\ntask by using modular value in stead of weak value of\njoint projection operators of two subsystems. In general,\nthe state of a quantum system is characterized by density\nmatrix and up to now the determination of density ma-\ntrix of a two photon entangled state has not been investi-\ngated explicitly yet. The matrix elements of an entangled\nstate described by ρinA′={|HH/an}bracketri}ht,|HV/an}bracketri}ht,|VH/an}bracketri}ht,|VV/an}bracketri}ht}3\nbasis of two subsystems is given by\nρ=|Ψ/an}bracketri}ht/an}bracketle{tΨ|=/summationdisplay\nji,klCijC∗\nkl|ij/an}bracketri}ht/an}bracketle{tkl|=/summationdisplay\nij,klρij,kl|ij/an}bracketri}ht/an}bracketle{tkl|\n=\nρHH,HHρHV,HHρVH,HHρVV,HH\nρHH,HVρHV,HVρVH,HVρVV,HV\nρHH,VHρHV,VHρVH,VHρVV,VH\nρHH,VVρHV,VVρVH,VVρVV,VV\n.(5)\nwhereρij,kl=/an}bracketle{tij|ρ|kl/an}bracketri}htis matrix element of ρand a com-\nplex number, and i,j,k,l∈(H,V). Thus, to find the\ncomplex matrix elements of an entangled state ρ, we have\nto find the real and imaginary parts of each elements,\nρij,kl, respectively. Next we will study this problem with\ntwo different methods.\nA. Method one: Based on modular value scheme\nAs mentioned in Section. II, the weak value of pro-\njection operator πi=|i/an}bracketri}ht/an}bracketle{ti|under the density operator ρ\nwith postselected state |α/an}bracketri}htis defined as\n/an}bracketle{tπi/an}bracketri}htw\nα=/an}bracketle{tα|πiρ|α/an}bracketri}ht\n/an}bracketle{tα|ρ|α/an}bracketri}ht(6)\nFurthermore, If we want to measure the joint projec-\ntion operators of two subsystems, π1\niπ2\nj=|ij/an}bracketri}ht/an}bracketle{tij|, where\nπ1\ni=|i/an}bracketri}ht/an}bracketle{ti|andπ2\nj=|j/an}bracketri}ht/an}bracketle{tj|are represent the projection\noperators of subsystem one and two, then the correspond-\ning weak value of π1\niπ2\njunder the density operator ρwith\npostselected state |αβ/an}bracketri}ht=|α/an}bracketri}ht1|β/an}bracketri}ht2can be written as\n/an}bracketle{tπ1\niπ2\nj/an}bracketri}htw\nαβ=/an}bracketle{tαβ|π1\niπ2\njρ|αβ/an}bracketri}ht\n/an}bracketle{tαβ|ρ|αβ/an}bracketri}ht. (7)\nHere,i,j,k,l∈(V,H)is inA′basis andα,β∈(D,A)is\ninB′={|DD/an}bracketri}ht,|DA/an}bracketri}ht,|AD/an}bracketri}ht,|AA/an}bracketri}ht}basis, respectively.\nBy using the definition of weak value of joint operators,\nevery matrix element of ρwhich is written in Eq.( 5) can\nbe expressed in terms of the weak value of joint project\noperatorπ1\niπ2\njinA′basis as\nρij,kl=/an}bracketle{tij|ρ|kl/an}bracketri}ht=/summationdisplay\nαβpαβ/an}bracketle{tαβ|kl/an}bracketri}ht\n/an}bracketle{tαβ|ij/an}bracketri}ht/an}bracketle{tπ1\niπ2\nj/an}bracketri}htw\nαβ,(8)\nwherepαβ=/an}bracketle{tαβ|ρ|αβ/an}bracketri}htis the probability to find the sys-\ntem in postselected state |αβ/an}bracketri}ht,/angbracketleftαβ|kl/angbracketright\n/angbracketleftαβ|ij/angbracketrightis independent to\nthe above summation and can be determined by using\nnormalization condition. Thus, if we take weak mea-\nsurement on joint projection operators π1\niπ2\njinA′ba-\nsis following take strong measurement on all elements in\nB′basis of both subsystems, respectively, then can get\nthe value of every complex elements of density matrix ρ.\nFurthermore, we can define the density matrix ρof an\nentangled state in B′as well, and the expressions of itsmatrix elements can be written as\nραβ,α′β′=/an}bracketle{tβα|ρ|α′β′/an}bracketri}ht=/summationdisplay\nijpα′β′/an}bracketle{tαβ|ij/an}bracketri}ht\n/an}bracketle{tα′β′|ij/an}bracketri}ht/an}bracketle{tπ1\niπ2\nj/an}bracketri}htw\nα′β′,(9)\nwhere|α′β′/an}bracketri}htalso belong to the B′basis too, i.e., α′,β′∈\n(D,A), andpα′β′=/an}bracketle{tβ′α′|ρ|α′β′/an}bracketri}htis the probability of\nsuccess for postselection of |α′β′/an}bracketri}htbasis. As shown in\nEq. (8) and Eq. ( 9), to get the matrix elements ρij,kl(\nραβ,α′β′) we should find the weak values /an}bracketle{tπ1\niπ2\nj/an}bracketri}htw\nαβwith\nconsider all elements in bases B′(A′), respectively.\nIn postselected weak measurement technique, the weak\nvalue of nonlocal joint operators can not be obtained\nexactly and the efficieny is too low for entangled state\ncase[32]. However, in recent study of Guang-Can- Guo\n[30], they showed that the joint weak values /an}bracketle{tπ1\niπj/an}bracketri}htw\nαβ\ncan be found in terms of modular values. In remaining\npart of this subsection, we will calculate the weak values\n/an}bracketle{tπ1\niπ2\nj/an}bracketri}htw\nαβto find the matrix elements of ρ.\nThe modular value of an observable ˆFwith pre- and\npostselected states, |ψin/an}bracketri}htand|ψfi/an}bracketri}htcan be written as[ 33]\n/an}bracketle{tF/an}bracketri}htm\nψfi=/an}bracketle{tψfi|e−igF|ψin/an}bracketri}ht\n/an}bracketle{tψfi|ψin/an}bracketri}ht. (10)\nwheregis represent the coupling strength between mea-\nsured system and pointer, and the modular value is valid\nfor any weak and strong coupling cases. If we take\nˆF= ˆπ=|i/an}bracketri}ht/an}bracketle{ti|is a projection operator in two dimen-\nsional Hilbert space, then\n/an}bracketle{tπ/an}bracketri}htm\nψfi=/an}bracketle{tψfi|(/summationtext\nie−igλiπi)|ψin/an}bracketri}ht\n/an}bracketle{tψfi|ψin/an}bracketri}ht\n=/an}bracketle{tψfi|((1−πi+e−igπi)|ψin/an}bracketri}ht\n/an}bracketle{tψfi|ψin/an}bracketri}ht\n= 1+(e−ig−1)/an}bracketle{tψfi|πi|ψin/an}bracketri}ht\n/an}bracketle{tψfi|ψin/an}bracketri}ht\n= 1+s/an}bracketle{tπi/an}bracketri}htw\nψfi(11)\nwhereλi= 0,1are eigenvalues of projection operator πi,\nands=e−ig−1.\nFurthermore, if we extend our concern to an entan-\ngled state composed of two subsystems, i.e., consider\n|Ψ/an}bracketri}ht(Eq.(2)) as preselection state of the system, then\nthe modular value of projection operators π1\ni+π2\nj=\n|i/an}bracketri}ht1/an}bracketle{ti|+|j/an}bracketri}ht2/an}bracketle{tj|of total system with postselected state4\n|αβ/an}bracketri}htcan be calculated as\n/an}bracketle{tπ1\ni+π2\nj/an}bracketri}htm\nαβ=/an}bracketle{tβα|e−ig(π1\ni+π2\nj)|Ψ/an}bracketri}ht\n/an}bracketle{tαβ|Ψ/an}bracketri}ht\n=/an}bracketle{tβα|e−igπ1\nie−igπ2\nj|Ψ/an}bracketri}ht\n/an}bracketle{tαβ|Ψ/an}bracketri}ht\n=/an}bracketle{tβα|(1+sπ1\ni)(1+sπ2\nj)|Ψ/an}bracketri}ht\n/an}bracketle{tαβ|Ψ/an}bracketri}ht\n=/an}bracketle{tβα|(1+sπ1\ni)(1+sπ2\nj)|Ψ/an}bracketri}ht\n/an}bracketle{tαβ|Ψ/an}bracketri}ht\n= 1+s/an}bracketle{tπ1\ni/an}bracketri}htw\nαβ+s/an}bracketle{tπ2\nj/an}bracketri}htw\nαβ+s2/an}bracketle{tπ1\niπ2\nj/an}bracketri}htw\nαβ\n=−1+/an}bracketle{tπ1\ni/an}bracketri}htm\nαβ+/an}bracketle{tπ2\nj/an}bracketri}htm\nαβ+s2/an}bracketle{tπ1\niπ2\nj/an}bracketri}htw\nαβ.\n(12)\nIn the last line of above expression we use relations be-\ntween weak value and modular value (see Eq. ( 11)).\nBy taking the modular value of projection operator\nπi=|i/an}bracketri}ht/an}bracketle{ti|which is given in Eq.( 11) into account, from\nthis above equation we can read the weak value /an}bracketle{tπ1\niπ2\nj/an}bracketri}htw\nαβ\nas\n/an}bracketle{tπ1\niπ2\nj/an}bracketri}htw\nαβ=s−2[/an}bracketle{tπ1\ni+π2\nj/an}bracketri}htm\nαβ−/an}bracketle{tπ1\ni/an}bracketri}htm\nαβ−/an}bracketle{tπ2\nj/an}bracketri}htm\nαβ+1].(13)\nFrom this theoretical result we can deduce that if we\ncan measure the modular values of projection operators\nπ1\ni,π2\njandπ1\ni+π2\nj, respectively, the weak value of joint\noperatorsπ1\niπ2\njcan be found easily, after that we can de-\ntermine the matrix elements ρij,klandραβ,α′β′of density\nmatrixρby using Eq.( 8) and Eq.( 9).\nAs studied in Ref.[ 30], after a two photon entangled\nstate generated by nonlinear optical devices, during the\npropagation in space (interferometer for example) the\ntwo photons entangled in their polarization degrees of\nfreedom and paths degrees of freedom, respectively. We\ntake the paths degrees of freedom( | ↑/an}bracketri}htand| ↓/an}bracketri}ht) as\npointer, and polarization degrees of freedom ( |H/an}bracketri}htand\n|V/an}bracketri}ht) take as measured system, respectively. Suppose that\ninitially both paths and polarization degrees of two sub-\nsystems are entangled but there is no any entanglement\nbetween these two degrees of freedoms. Thus, the initial\nstate of the total system can be expressed as\n|Ψms/an}bracketri}ht=|ϕ/an}bracketri}ht⊗|Ψ/an}bracketri}ht, (14)\nwhere,|Ψ/an}bracketri}htis given in Eq. ( 4) and\n|ϕ/an}bracketri}ht=µ| ↑↓/an}bracketri}ht+η| ↓↑/an}bracketri}ht,|µ|2+|η|2= 1 (15)\nis correspond to paths degree of freedom of two compo-\nnent systems. Here, | ↑↓/an}bracketri}ht represent the first photon in\nthe↑path and second photon in ↓path, respectively. To\nget modular value of π1\ni,π2\njandπ1\ni+π2\nj, in Ref.[ 30] they\nintroduced the three interaction Hamiltonians between\ntwo composed pointer state and measured system as\nH1=gδ(t−t0)(π1\n↓π1\ni+π2\n↑π2\nj), (16)\nH2=gδ(t−t0)π1\n↓π1\ni, (17)\nH3=gδ(t−t0)π2\n↑π2\nj. (18)Here,π1\n↓=| ↓/an}bracketri}ht/an}bracketle{t↓ | andπ2\n↑=| ↑/an}bracketri}ht/an}bracketle{t↑ | are represent the\nprojection operators of paths degree of freedom of two\nsubsystems, respectively.\nIf we consider the intrinsic properties of projection op-\nerators of paths degrees of freedom, the evolution op-\nerators corresponding to above interaction Hamiltonians\nbecomes as\nU1= exp[−i\n/planckover2pi1/integraldisplay\nHdτ] =e−ig(π1\n↓π1\ni+π2\n↑π2\nj)\n= [1+(e−igπ1\ni−1)π1\n↓][1+(e−igπ2\nj−1)π2\n↑]\n= (e−igπ1\ni−1)π1\n↓+(e−igπ2\nj−1)π2\n↑+1\n+[e−ig(π1\ni+π2\nj)+1−e−igπ1\ni−e−igπ2\nl]π1\n↓π2\n↑,(19a)\nU2=e−igπ1\n↓π2\nj= 1+(e−igπ1\ni−1)π1\n↓, (19b)\nU3=e−igπ2\n↑π2\nj= 1+(e−igπ2\nj−1)π2\n↑, (19c)\nrespectively. In above calculations we use the for-\nmulaeθˆF=/summationtext\nneθλn|φn/an}bracketri}ht/an}bracketle{tφn|of operator ˆFwith\nˆF|φn/an}bracketri}ht=λn|φn/an}bracketri}ht.\nStart from the initial state of the total system, Eq.( 14),\nand take the above time evolution operators ( see\nEqs.(19a-19c)) and post-selection onto |αβ/an}bracketri}htin basis B′\ninto account, the final states of the pointers can be ob-\ntained as\n|Φ1/an}bracketri}ht=N1[η/an}bracketle{tπ1\ni+π2\nj/an}bracketri}htm\nαβ| ↓↑/an}bracketri}ht+µ| ↑↓/an}bracketri}ht],(20)\n|Φ2/an}bracketri}ht=N2[η/an}bracketle{tπ1\ni/an}bracketri}htm\nαβ| ↓↑/an}bracketri}ht+µ| ↑↓/an}bracketri}ht], (21)\nand\n|Φ2/an}bracketri}ht=N3[η/an}bracketle{tπ2\nj/an}bracketri}htm\nαβ| ↓↑/an}bracketri}ht+µ| ↑↓/an}bracketri}ht], (22)\nwhereN1= [|µ|2+|η/an}bracketle{tπ1\ni+π2\nj/an}bracketri}htm\nαβ|2]−1\n2,N1= [|µ|2+\n|η/an}bracketle{tπ1\ni/an}bracketri}htm\nαβ|2]−1\n2andN3=N1= [|µ|2+|η/an}bracketle{tπ2\nj/an}bracketri}htm\nαβ|2]−1\n2are\nnormalization coefficients, respectively.\nIf we project these above final states of the pointer onto\n|ϕ1/an}bracketri}ht=1√\n2(| ↑/an}bracketri}ht+| ↓/an}bracketri}ht)⊗1√\n2(| ↑/an}bracketri}ht+| ↓/an}bracketri}ht)and|ϕ2/an}bracketri}ht=1√\n2(| ↑\n/an}bracketri}ht+i| ↓/an}bracketri}ht)⊗1√\n2(| ↑/an}bracketri}ht+i| ↓/an}bracketri}ht), respectively, the probabilities\nto find the final states |Φ1/an}bracketri}ht,|Φ2/an}bracketri}htand|Φ3/an}bracketri}hton|ϕ1/an}bracketri}htand5\n|ϕ2/an}bracketri}htare\nP1=|/an}bracketle{tϕ1|Φ1/an}bracketri}ht|2(23a)\n=|N1|2\n2{|µ|2+|η|2|/an}bracketle{tπ1\ni+π2\nj/an}bracketri}htm\nαβ|2+2ℜ[µ∗η/an}bracketle{tπ1\ni+π2\nj/an}bracketri}htm\nαβ]}\n(23b)\nP2=|/an}bracketle{tϕ2|Φ1/an}bracketri}ht|2(23c)\n=|N1|2\n2{|µ|2+|η|2|/an}bracketle{tπ1\ni+π2\nj/an}bracketri}htm\nαβ|2+2ℑ[µ∗η/an}bracketle{tπ1\ni+π2\nj/an}bracketri}htm\nαβ]}\n(23d)\nP3=|/an}bracketle{tϕ1|Φ2/an}bracketri}ht|2(23e)\n=|N2|2\n2{|µ|2+|η|2|/an}bracketle{tπ1\ni/an}bracketri}htm\nαβ|2+2ℜ[µ∗η/an}bracketle{tπ1\ni/an}bracketri}htm\nαβ]}\n(23f)\nP5=|/an}bracketle{tϕ2|Φ2/an}bracketri}ht|2(23g)\n=|N2|2\n2{|µ|2+|η|2|/an}bracketle{tπ1\ni/an}bracketri}htm\nαβ|2+2ℑ[µ∗η/an}bracketle{tπ1\ni/an}bracketri}htm\nαβ]}\n(23h)\nand\nP5=|/an}bracketle{tϕ1|Φ3/an}bracketri}ht|2\n=|N3|2\n2{|µ|2+|η|2|/an}bracketle{tπ1\nj/an}bracketri}htm\nαβ|2+2ℜ[µ∗η/an}bracketle{tπ1\nj/an}bracketri}htm\nαβ]}\n(23i)\nP6=|/an}bracketle{tϕ2|Φ3/an}bracketri}ht|2\n=|N3|2\n2{|µ|2+|η|2|/an}bracketle{tπ1\nj/an}bracketri}htm\nαβ|2+2ℑ[µ∗η/an}bracketle{tπ1\nj/an}bracketri}htm\nαβ]},\n(23j)\nrespectively. If we assume that initially the probability\nof first photon in path ↓and second photon in path ↑is\nsmaller than the probability of first photon in path ↑and\nsecond photon in path ↓, i.e,|η|2≪1, then\nP1≈µℜ[/an}bracketle{tπ1\ni+π2\nj/an}bracketri}htm\nαβ]+1\n2, (24a)\nP2≈µℑ[/an}bracketle{tπ1\ni+π2\nj/an}bracketri}htm\nαβ]+1\n2, (24b)\nP3=µℜ[/an}bracketle{tπ1\ni/an}bracketri}htm\nαβ]+1\n2, (24c)\nP4=µℑ[/an}bracketle{tπ1\ni/an}bracketri}htm\nαβ]+1\n2, (24d)\nP5=µℜ[/an}bracketle{tπ2\nj/an}bracketri}htm\nαβ]+1\n2, (24e)\nP6=µℑ[/an}bracketle{tπ2\nj/an}bracketri}htm\nαβ]+1\n2. (24f)\nThese probabilities can be determine by the detectors\nin the Lab, then we can find the modular values π1\ni,π2\nj\nandπ1\ni+π2\nj. Finally, the real and imaginary parts of\nweak value of π1\niπ2\nj( Eq.( 13)) can be written as\nℜ[/an}bracketle{tπ1\niπ2\nj/an}bracketri}htw\nαβ] =s−2η−1[P1−P3−P5+η+1\n2](25)and\nℑ[/an}bracketle{tπ1\niπ2\nj/an}bracketri}htαβ] =s−2η−1[P2−P4−P6+1\n2], (26)\nrespectively. With these processes finally we can deter-\nmine the matrix elements of density operator by using\nEq.(8) and Eq.( 9), respectively.\nIn Ref.[ 30], they investigated the direct measurement\nmethod of a pure two photon polarization entangled state\ntheoretically and experimentally. In their work to get the\ncomplex amplitude Cijin Eq.( 2) we only need to scan a\ndefinite element of B′basis, i.e.\nCij=χ/an}bracketle{tπ1\niπ2\nj/an}bracketri}htw\nDD (27)\nwhereχ=/angbracketleftDD|Ψ/angbracketright\n/angbracketleftDD|ij/angbracketrightis independent of |ij/an}bracketri}htand can be\nobtained by normalization condition. However, to re-\nconstruct the density matrix of two component entan-\ngled state in A′basis, we need more strong measurement\nsteps inB′basis to determine every matrix elements. For\nexample, if we want to get the matrix element ρHH,HV ,\naccording to Eq.( 8) we should find all weak values of π1\niπ2\nj\nwith postselection states in B′basis. i.e.\nρHH,HV=pDD/an}bracketle{tπ1\nHπ2\nV/an}bracketri}htw\nDD+pDA/an}bracketle{tπ1\nHπ2\nV/an}bracketri}htw\nDA\n+pAD/an}bracketle{tπ1\nHπ2\nV/an}bracketri}htw\nAD+pAA/an}bracketle{tπ1\nHπ2\nV/an}bracketri}htw\nAA.(28)\nOn the other hand, if we want to get the matrix elements\nραβ,DA for example, according to Eq.( 9) we should find\nall weak values of π1\niπ2\njinA′basis with definite postse-\nlection state |DA/an}bracketri}htinB′basis,i.e.\nρDD,DA= 2pDA[/an}bracketle{tπ1\nHπ2\nH/an}bracketri}htw\nDA+/an}bracketle{tπ1\nVπ2\nH/an}bracketri}htw\nDA−1\n2].(29)\nHere, we use the relation of weak value of projection op-\neratorsπ1\niπ2\njin whole Hilbert space of our system, i.e.,\n/summationdisplay\nij/an}bracketle{tπ1\niπ2\nj/an}bracketri}htw\nαβ= 1. (30)\nFrom the above examples we can deduce that the deter-\nmination of matrix elements ρij,klinA′basis need more\nexperimental steps than the matrix elements ραβ,α′β′in\nB′basis.\nThus, if we want to get the matrix elements of density\noperatorρinA′basis, it could be realized in experiment\nbased on the the experimental setup of Guang-Can- Guo\n[30] by extend the chooses of the post-selection state to\nall elements in B′basis rather than only scan on one\ndefinite element in B′.\nB. Method Two: Based on three sequential\nmeasurements [ 11]scheme\nAs Lundeen and his co-workers [ 11] studied, the matrix\nelements of a quantum system can be obtained by consid-\nering the weak measurement of an observable composed\nof three incompatible projection operator:\nΠij=πjπDπi. (31)6\nHere,πi=|i/an}bracketri}ht/an}bracketle{ti|,πj=|j/an}bracketri}ht/an}bracketle{tj|withi,j∈(H,V)inA\nbasis, andπD=|D/an}bracketri}ht/an}bracketle{tD|where|D/an}bracketri}ht=1√\n2(|H/an}bracketri}ht+|V/an}bracketri}ht)is\nelement in Bbasis. The basis vectors in AandBare\nmaximally incompatible, and /an}bracketle{ti|D/an}bracketri}ht=/an}bracketle{tj|D/an}bracketri}ht=1√\n2. The\nmatrix elements ρijof unknown density operator ρcan\nbe found as\nρij= 2/an}bracketle{tΠij/an}bracketri}hts= 2Trs[πjπDπiρ]. (32)\nSinceΠijis non-Hermitian, generally the weak average\n/an}bracketle{tΠij/an}bracketri}htsis a complex number. Thus, according to the\nEq.(32) we can get the complex density matrix elements\nofρif one can find the /an}bracketle{tΠij/an}bracketri}hts. In the recent work of Lun-\ndeen and his co-workers, they investigated their proposal\nwhich introduced in Ref.[ 11], and experimentally recon-\nstruct the density matrix elements of pure and mixed\nstates of 2-dimensional system[ 26]. In this study as ex-\npansion of their work [ 26], we will study how to determine\nthe matrix elements of two photon entangled state.\nFor an entangled state composed of two subsystems,\nthe observable defined in Eq.( 31), can be redefined as\nΠij,kl=πklπαβπij (33)\nwhereπij=π1\niπ2\nj=|i/an}bracketri}ht1/an}bracketle{ti| ⊗ |j/an}bracketri}ht2/an}bracketle{tj|=|ij/an}bracketri}ht/an}bracketle{tij|,πkl=\nπ1\nkπ2\nl=|k/an}bracketri}ht1/an}bracketle{tk|⊗|l/an}bracketri}ht2/an}bracketle{tl|=|kl/an}bracketri}ht/an}bracketle{tkl|withi,j,k,l∈(H,V)in\nA′basis, andπαβ=π1\nαπ2\nβ=|α/an}bracketri}ht1/an}bracketle{tα|⊗|β/an}bracketri}ht2/an}bracketle{tβ|=|αβ/an}bracketri}ht/an}bracketle{tαβ|\nwithα,β∈(D,A)inB′basis. The basis vectors in\nA′andB′are maximally incompatible, and /an}bracketle{tij|αβ/an}bracketri}ht=\n/an}bracketle{tkl|αβ/an}bracketri}ht=1\n2. The matrix elements ρij,klof unknown\ndensity operator ρof an entangled state can be found as\nρij,kl= 4/an}bracketle{tΠij,kl/an}bracketri}hts= 4Trs[πklπo\nαβπijρ]. (34)\nwhereπo\nαβrepresent the two composed project opera-\ntor with definite value of αandβinB′basis. Here,\nwe will only consider the α=β=Dcase with follow-\ning the method of Lundeen[ 11], but other cases such as\nα=β=Aalso can be used to find the matrix elements\nwith similar processes described in this study. To find\nthe matrix elements ρij,kl, we have to find the value of\nTrs[πklπo\nαβπijρ], and remaining part of this subsection\nwe will study this problem.\nWe assume that the initial state of total system is\n|Ω/an}bracketri}ht=|Φ/an}bracketri}ht/an}bracketle{tΦ|⊗ρ,\nwhere the initial state of the pointer Φ(r1,r2)is composed\nby two Gaussian beams of two photons which have xand\nytransverse spatial distributions separately, i.e.\n/an}bracketle{tr|Φ/an}bracketri}ht=ϕ1(x1,y1)ϕ2(x2,y2) (35)where\nϕ1(x1,y1) =/parenleftbigg1\n2πσx1σy1/parenrightbigg1\n2\nexp/parenleftbiggx2\n1\n4σ2x1/parenrightbigg\nexp/parenleftbiggy2\n1\n4σ2y1/parenrightbigg\n(36)\nand\nϕ2(x2,y2) =/parenleftbigg1\n2πσx2σy2/parenrightbigg1\n2\nexp/parenleftbiggx2\n2\n4σ2x2/parenrightbigg\n)exp/parenleftbiggy2\n2\n4σ2y2/parenrightbigg\n.\n(37)\nare represent the spatial distributions of first and second\nphotons, respectively. ρis given in Eq.( 5), and considered\nas measured system.\nWe assume that the interaction Hamiltonian between\neach pointer and measured systems are\nH=H1+H2+H3+H4, (38)\nwith\nH1=g1π1\nix1,H2=g2π2\njx2, i,j∈(V,H)(39)\nand\nH3=g3π1\nDy1,H4=g4π2\nDy2, (40)\nrespectively, and gn(n= 1,2,3,4) represent the coupling\nstrength between the pointer and measuring device, and\nfor simplicity can be taken them as equal quantity, i.e.,\ng1=g2=g3=g4=g. If we assume that the polariz-\ners represented by the projection operator πijandπDD\ncausing displacement along xandydirections, respec-\ntively, we can get the weak average of πDDπijby using\nthe method introduced in Refs.[ 11,32] as\n/an}bracketle{tπDDπij/an}bracketri}hts=1\ng4/an}bracketle{ta2Da1Da2ja1i/an}bracketri}htf, (41)\nwhere\na1i=x1i+ι2σ2\n/planckover2pi1p1xi, a 2j=x2j+ι2σ2\n/planckover2pi1p2xj,(42)\nand\na1D=y1D+ι2σ2\n/planckover2pi1p1yD, a2D=y2D+ι2σ2\n/planckover2pi1p2yD,(43)\nare represent the annihilation operators of every spa-\ntial transversal components of each photons, respectively ,\nand/an}bracketle{t/an}bracketri}htfindicate to find the expectation value of variables\nunder the final state of the pointer state. Here, we have to\nnote thatπijandπDDare non-commute, but as showed\nin Ref.[ 34] the Eq.( 41) still valid for non-commuting ob-\nservables if they are measured sequentially as measuring\ntheπDDfollowed by πij. Since last measurement is will\nbe taken over the projection operators πklare strong,\nthen\nTrs[πklπαβπijρ] =1\ng4Trs[πkla2Da1Da2ja1iρ].(44)\nWith these processes we can obtain the matrix elements\nρij,klof density operator ρas7\nρij,kl= 4Tr[πklπDDπijρ] = 4Tr[πkla1Da2Da2ja1iρ]\n=4\ng4/angbracketleftigg/parenleftigg\ny1D+ι2σ2\ny1\n/planckover2pi1p1yD/parenrightigg/parenleftigg\ny2D+ι2σ2\ny2\n/planckover2pi1p2yD/parenrightigg/parenleftbigg\nx2j+ι2σ2\nx2\n/planckover2pi1p2xj/parenrightbigg/parenleftbigg\nx1i+ι2σ2\nx1\n/planckover2pi1p1xi/parenrightbigg/angbracketrightigg\nf(45)\nThen, the real and imaginary parts of the matrix elements of d ensity operator ρare\nℜ[ρij,kl] =4\ng4[/an}bracketle{ty1Dy2Dx1ix2j/an}bracketri}htf−σ2\nσ2p/an}bracketle{ty1Dy2Dp1xip2xj/an}bracketri}htf−σ2\nσ2p/an}bracketle{tx1ix2jp2yDp1yD/an}bracketri}htf+σ4\nσ4p/an}bracketle{tp1xDp2xjp2yDp1yD/an}bracketri}htf\n−σ2\nσ2p/an}bracketle{ty1Dp2yDx2jp1xi/an}bracketri}htf−σ2\nσ2p/an}bracketle{ty1Dp2yDx1ip2xj/an}bracketri}htf−σ2\nσ2p/an}bracketle{ty2Dp1yDx2jp1xi/an}bracketri}htf−σ2\nσ2p/an}bracketle{ty2Dp1yDx1ip2xj/an}bracketri}htf](46)\nand\nℑ[ρij,kl] =4\ng4σ\nσp[/an}bracketle{ty1Dy2Dx2jp1xi/an}bracketri}htf+/an}bracketle{ty1Dy2Dx1ip2xj/an}bracketri}htf+/an}bracketle{tx1ix2jy1Dp2yD/an}bracketri}htf+/an}bracketle{tx1ix2jy2Dp1yD/an}bracketri}htf\n−σ2\nσ2p/an}bracketle{tp2yDp1yDx2jp1xi/an}bracketri}htf−σ2\nσ2p/an}bracketle{tp2yDp1yDx1ip2xj/an}bracketri}htf−σ2\nσ2p/an}bracketle{tp1xip2xjy1Dp2yD/an}bracketri}htf−σ2\nσ2p/an}bracketle{tp1xip2xjy2Dp1yD/an}bracketri}htf](47)\n, respectively. Here, for simplicity we assume that\nthe width of every Gaussian beam is equal to σ, i.e.,\nσx1=σy1=σx2=σy2=σ, andσpis the momentum\nspace width of the pointer state with σσp=/planckover2pi1\n2. Based\non the experimental results and methods of Ref.[ 26] for\nread out the real and imaginary parts of matrix ele-\nments of single photon polarization state by measuring\nthe probabilities of transmitted photons via optical ap-\nparatuses, in the Lab we may also measure the prob-\nabilities of entangled photons transmitted through the\nfinal polarizers which represented by the projection op-\neratorsπkl=|kl/an}bracketri}ht/an}bracketle{tkl|of two subsystems. In general, these\nprobabilities are functions of positions and momenta of\ntwo photons, i.e, P=P(x1,y1,y2,x2,p1x,p2x,p2y,p2y).\nThen, the elements of density operator of two entan-\ngled photon state can be reconstructed by determining\nthe expectation values /an}bracketle{t/an}bracketri}htfin Eq.( 46) and Eq.( 47) via/integraltext\nABCDPdτ=/an}bracketle{tABCD/an}bracketri}htf, respectively.\nIV. CONCLUSION AND REMARKS\nIn this study we investigated how to reconstruct the\nunknown density operator of two component entangled\nquantum state by using postselected weak measurement\nmethod and three sequential measurements where each\ncomplementary to the last, respectively, and discussed\nits feasibility by taking into account the recent related\nexperimental works. The similarity of these methods is\nthat in each scheme we take the weak and strong sequen-\ntial measurements in two bases during the getting of real\nand imaginary parts of elements of density operator, re-\nspectively, and the postselection is the key to determinewhich matrix element we want to readout from the final\nstate of the pointer state. However, in second method\nit is enough to scan over one definite elements in bases\nB′but in first method we usually need to scan all ele-\nments in B′. Thus, in the same measurement process the\nmethod one may need more resources than method two.\nSince the Hilbert space of two entangled photon state\nis larger than single photon case, during the readout of\nthe matrix elements in the Lab of a two entangled pho-\nton processes we would need more and some complicated\nexperimental setups and need more resources in method\none rather than method two. However, if we consider the\nwide applications of entangled photon states in every field\nof quantum theory, it is worthy to study this vital prob-\nlem. Based on the experimental works of direct measure-\nment of single two dimensional systems and its density\noperators[ 26], and direct measurement of pure two en-\ntangled photon state[ 30], we anticipate that in the near\nfuture the experts can do experiments by taking those\ntwo innovative works into account, and realize the direct\nmeasurement of density operator by considering the the-\noretical results of our current work. In our schemes any\nmatrix elements of an entangled state can be obtained ef-\nficiently via proper weak and strong measurements. Since\nthe density operator representation of a quantum state is\ngeneral than the wave function, if consider the open sys-\ntem cases it suggested that the determination of density\noperator of an unknown two photon entangled state may\nhave more practical applications rather than the direct\nmeasurement of complex probability amplitude of state\nvector of a pure two photon entangled sate.8\nACKNOWLEDGMENTS\nThis work was supported by the National Natural\nScience Foundation of China (Grant No. 11865017,\nNo.11864042).\n[1] Y. Aharonov, D. Z. Albert, and L. Vaidman,\nPhys. Rev. Lett. 60, 1351 (1988) .\n[2] N. Ritchie, J. Story, and R. Hulet,\nPhys. Rev. Lett. 66, 1107 (1991) .\n[3] A. G. Kofman, S. Ashhab, and F. Nori,\nPhys. Rep. 520, 43 (2012) .\n[4] J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and\nR. W. Boyd, Rev. Mod. Phys. 86, 307 (2014) .\n[5] K. Vogel and H. Risken, Phys. Rev. A 40, 2847 (1989) .\n[6] D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani,\nPhys. Rev. 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A 77, 052102 (2008) ."    },    {        "title": "1912.05080v2.Emergence_and_spectral_weight_transfer_of_electronic_states_in_the_Hubbard_ladder.pdf",        "content": "arXiv:1912.05080v2  [cond-mat.str-el]  6 Jan 2020Emergence and spectral-weight transfer of electronic stat es in the Hubbard ladder\nMasanori Kohno∗\nNational Institute for Materials Science, Tsukuba 305-000 3, Japan\n(Dated: January 7, 2020)\nThe number of electronic bands is usually considered invari ant regardless of the electron density in\na band picture. However, in interacting systems, the spectr al-weight distribution generally changes\ndependingon the electron density, and electronic states ca n even emerge or disappear as the electron\ndensity changes. Here, to clarify how electronic states eme rge and become dominant as the electron\ndensity changes, the spectral function of the Hubbard ladde r with strong repulsion and strong\nintrarunghoppingis studied using the non-Abelian dynamic al density-matrix renormalization-group\nmethod. A mode emerging in the low-electron-density limit g ains spectral weight as the electron\ndensity increases and governs the dimer Mott physics at quar ter-filling. In contrast, the antibonding\nband, which is dominant in the low-electron-density regime , loses spectral weight and disappears at\nthe Mott transition at half-filling, exhibiting the momentu m-shifted magnetic dispersion relation in\nthe small-doping limit. This paper identifies the origin of t he electronic states responsible for the\nMott transition and brings a new perspective to electronic b ands by revealing the overall nature of\nelectronic states over a wide energy and electron-density r egime.\nPACS numbers: 71.30.+h, 71.10.Fd, 74.72.Gh, 79.60.-i\nI. INTRODUCTION\nIn band theory, an electron is assumed to hop from\none atomic orbital to another in an effective periodic po-\ntential, forming a band [1]; the number of bands is con-\nsidered essentially determined by the number of atomic\norbitals in a unit cell, which does not change with the\nelectron density. In Fermi-liquid theory, electronic exci-\ntations other than the quasiparticle band are regarded as\nincoherent [2]; the incoherent excitations are usually con-\nsidered almost featureless and unimportant regardless of\nthe electron density.\nHowever, in interacting systems, some electronic exci-\ntations generally become dominant among many excited\nstates, and the number of dominant modes can change\ndepending on the electron density. Electronic excitations\naway from the Fermi level can also become dominant\nand exhibit significant characteristics. Thus, revealing\nthe overall nature of electronic states over a wide energy\nand electron-density regime is important in the deeper\nunderstanding of the effects of strong electronic correla-\ntions. In particular, strong correlations significantly af-\nfect the electronic states near the Mott transition, which\nhave attracted considerable attention in relation to high-\ntemperature superconductivity [3–5].\nIn this paper, to clarify how electronic states emerge,\nchange, and disappear as the electron density changes\nin strongly correlated systems, the spectral function of\nthe Hubbard ladder, which is one of the simplest mod-\nels containing the essence of electronic correlations, is\ninvestigated in the regime of strong Coulomb repulsion\nand strong intrarung hopping. The qualitative features\nof the results would be generally true for coupled dimer\n∗Electronic address: KOHNO.Masanori@nims.go.jpsystems, such as the dimer Mott insulators of molecular\nsolids [6, 7] regardless of the lattice structure or dimen-\nsionalityaslongastheCoulombrepulsionandintradimer\nhopping are much stronger than the interdimer hopping.\nIn particular, the perturbative arguments shown in this\npaper can be straightforwardly extended to bilayer sys-\ntems.\nThe main features we focus on in this paper are the\n(1) emergent electronic states in the low-electron-density\nregime [Sec. V], (2) spectral-weight transfer from the\ndominantmodestotheemergentmodes, whichmakesthe\nemergent modes dominant, whereas the dominant modes\nsignificantly lose spectral weight as the electron density\nincreases to half-filling [Sec. III], (3) dimer Mott gap\nat quarter-filling, whose value is significantly limited by\nthe intrarung hopping in the strong-Coulomb-repulsion\nregime [Sec. VI], and (4) emergent electronic states upon\ndoping a Mott insulator by which the Mott transition is\ncharacterized [Sec. VIII]. The above features are con-\ntrasted with conventional views, such as a band picture.\nII. MODEL AND METHOD\nWe consider the Hubbard ladder defined by the follow-\ning Hamiltonian:\nH=−t/bardbl/summationdisplay\ni,α,σ(cα†\ni,σcα\ni+1,σ+H.c.)−t⊥/summationdisplay\ni,σ(c1†\ni,σc2\ni,σ+H.c.)\n+U/summationdisplay\ni,αnα\ni,↑nα\ni,↓−µ/summationdisplay\ni,α,σnα\ni,σ, (1)\nwherecα\ni,σandnα\ni,σ, respectively, denote the annihilation\nandnumberoperatorsofanelectronwithspin σ(=↑,↓)at\nthe site of leg α(= 1,2) and rung i. Hereafter, the num-\nber of sites in a leg, total number of sites, and number of\nelectrons are denoted by L,Ns(= 2L), andNe, respec-2\ntively. The dopingconcentration δis defined as δ= 1−n,\nwherendenotes the electron density ( n=Ne/Ns).\nThe single-particle spectral function A(k,ω) and the\ndynamical spin structure factor S(k,ω) are defined as\nA(k,ω) =/braceleftBigg\n1\n2/summationtext\nl,σ|/an}bracketle{tl|c†\nk,σ|GS/an}bracketri}ht|2δ(ω−εl) forω>0,\n1\n2/summationtext\nl,σ|/an}bracketle{tl|ck,σ|GS/an}bracketri}ht|2δ(ω+εl) forω<0,\nS(k,ω) =1\n3/summationdisplay\nl,γ|/an}bracketle{tl|Sγ\nk|GS/an}bracketri}ht|2δ(ω−εl),\n(2)\nwhereεlrepresents the excitation energy of the eigen-\nstate|l/an}bracketri}htfrom the ground state |GS/an}bracketri}ht. Here,c†\nk,σand\nSγ\nkdenote the Fourier transform of c†\ni,σand that of the\nγ(=x,y,z) component of the spin operator Si, respec-\ntively. Forladders, k= (k/bardbl,k⊥), wherek/bardblandk⊥denote\nthe momenta in the leg and rung directions, respectively.\nBecause {ck,σ,c†\nk,σ}= 1,A(k,ω) satisfies the following\nsum rule at each k:/integraldisplay∞\n−∞dωA(k,ω) = 1. (3)\nWe consider the case of 0 ≤n≤1 without loss of gen-\nerality because A(k,ω) for 1<n≤2 can be obtained as\nA(k+π,−ω) at the electron density 2 −nby using the\nparticle-hole transformation. The Hubbard ladder has\nbeen studied primarilyon the ground-stateproperties[8–\n12], spin and chargeexcitations[13–15], spectralfunction\naround half-filling [10, 15, 16], charge and photo dynam-\nics [17, 18], and ferromagnetism[19–21]. In this paper, to\nclarify the evolution of electronic states as a function of\nthe electron density, we investigate the spectral function\nin the overall electron-density regime primarily for U≫\nt⊥≫t/bardbl>0 (Ut/bardbl/t2\n⊥is not too largefor the ground state\nto have spin 0 or 1/2 [19, 20]) based on the numerical re-\nsults forU/t/bardbl= 16 andt⊥/t/bardbl= 2 obtained using the\nnon-Abelian dynamical density-matrix renormalization-\ngroup (DDMRG) method [5, 22–27]. The DDMRG cal-\nculationswereperformedona120-siteclusterunderopen\nboundaryconditionswith 240statesretainedfor the den-\nsity matrix. The truncation errors are negligibly small in\nthe scales used for the figures in this paper.\nIII. OVERALL SPECTRAL FEATURES\nThe spectral-weight distributions of electronic states\nfrom the low-electron-density regime ( n≈0.083) to half-\nfilling (n= 1) are shown in Fig. 1 for k⊥= 0 [Figs. 1(a)–\n1(h)]andk⊥=π[Figs. 1(i)–1(p)]. Onemightnaivelyex-\npect thatthedominantmodesinthe low-electron-density\nregime [0 /lessorsimilarω/t/bardbl/lessorsimilar4 in Fig. 1(a); 4 /lessorsimilarω/t/bardbl/lessorsimilar8 in\nFig. 1(i)] continuously deform into those at half-filling\n[−3/lessorsimilarω/t/bardbl/lessorsimilar0 in Fig. 1(h); 11 /lessorsimilarω/t/bardbl/lessorsimilar14 in Fig.\n1(p)] as the electron density increases. However, they\nare different in origin.\nThe dominant mode in the low-electron-densityregime\nfork⊥= 0[Fig. 1(a)]graduallylosesspectralweightwiththe electron density [Figs. 1(b) and 1(c)]. At quarter-\nfilling (n= 1/2), this mode is located below ω= 0, sep-\narated by a small gap from the mode above ω= 0 [Figs.\n1(d) and2(d)]. Thespectralweightfurtherdecreasesand\nalmost disappears at half-filling [Figs. 1(e)–1(h)].\nAs fork⊥=π, the dominant mode in the low-electron-\ndensity regime [Fig. 1(i)] gradually loses spectral weight\nwith the electron density [Figs. 1(j)–1(o)], and com-\npletely disappears at half-filling [Fig. 1(p)].\nInstead, because of the sum rule [Eq. (3)], emergent\nmodes in the low-electron-density regime [Figs. 1(a) and\n1(i)] gradually gain spectral weight [Figs. 1(b)–1(g),\n1(j)–1(o)], and become dominant at half-filling [Figs.\n1(h) and 1(p)]. Thus, a significant amount of the spec-\ntral weight transfers from the dominant modes to the\nemergent modes as the electron density increases.\nItshouldbenotedthatnotonlythehigh-energymodes\nofO(U) but also an intermediate-energy mode emerges\n[ω/t/bardbl≈7 andk/bardbl≈πin Fig. 1(a)], which is separated\nfrom the low-energybonding band by an energy gap, and\nbecomes the most dominant at half-filling for k⊥= 0\n[−3/lessorsimilarω/t/bardbl/lessorsimilar0 in Fig. 1(h)].\nThese features are due to strong electronic correlations\nandcontrastwith arigid-bandpictureinwhichthebond-\ning and antibonding bands remain dominant regardless\nof the electron density.\nIV. ZERO ELECTRON DENSITY\nTo understand the nature of such complicated spec-\ntral features, we consider the properties of characteristic\nmodes from the low-electron-density side. At zero elec-\ntron density ( n= 0), the electron-addition spectra show\nthe noninteracting dispersion relations because the in-\nteraction term does not work with the added electron.\nThus, the Hamiltonian ( U= 0) can be diagonalized in\nthe momentum space as\nH0=/summationdisplay\nk,σǫ0\nkc†\nk,σck,σ. (4)\nThe noninteracting dispersion relation ǫ0\nkis obtained as\nǫ0\nk=−2t/bardblcosk/bardbl−t⊥cosk⊥−µ, (5)\nwherek⊥= 0 for the bonding band and k⊥=πfor the\nantibonding band (solid black curves in Fig. 3). In the\nlow-electron-density limit ( n→0),µ→ −2t/bardbl−t⊥.\nV. LOW ELECTRON DENSITY\nA. High-energy emergent modes\nIn addition to the dominant modes originating from\nthe noninteracting bonding and antibonding bands [Eq.\n(5)], small spectral weights emerge when the electron3\nFIG. 1:A(k,ω)t/bardblfor [(a)–(h)] k⊥= 0 and [(i)–(p)] πat [(a), (i)] n≈0.083, [(b), (j)] 0.25, [(c), (k)] 0.483, [(d), (l)] 0.5, [(e),\n(m)] 0.517, [(f), (n)] 0.75, [(g), (o)] 0.967, and [(h), (p)] 1 forU/t/bardbl= 16 andt⊥/t/bardbl= 2 obtained using the non-Abelian DDMRG\nmethod. The green lines indicate ω= 0. Gaussian broadening with a standard deviation of 0 .1t/bardblhas been used.\ndensity becomes nonzero [Figs. 1(a) and 1(i)]. The emer-\ngent modes at high energies of O(U) can be regarded as\nthe upper Hubbard bands. To capture their character-\nistics, we consider a two-electron system. The energyof a spin-singlet eigenstate ǫkis generally obtained as a4\nFIG. 2: (a), (c), (d) A(k,ω)t/bardblfork⊥= 0 at [(a)] n≈0.483,\n[(c)]n≈0.517, and [(d)] n= 0.5 forU/t/bardbl= 16 andt⊥/t/bardbl= 2\nobtained using the non-Abelian DDMRG method [closeup of\nFigs. 1(c), 1(e), and 1(d) near the Fermi level, respectivel y].\n(b)S(k,ω)t/bardblfork⊥= 0 atn= 0.5 forU/t/bardbl= 16 and\nt⊥/t/bardbl= 2 obtained using the non-Abelian DDMRG method.\nThe green lines indicate ω= 0. Gaussian broadening with a\nstandard deviation of 0 .1t/bardblhas been used.\nFIG. 3: Dispersion relation of electronic excitations in th e\nlow-electron-density limit for U/t/bardbl= 16 andt⊥/t/bardbl= 2 at\nk⊥= 0 [(a)] and π[(b)]. The dotted orange curves indicate\nthe high-energy solutions of Eq. (6). The solid black curves\nindicateǫ0\nk[Eq. (5)]. The dashed-dotted blue curve in (a)\nindicates the high-energy solutions of Eq. (6) for the effect ive\nHubbard chain with U−\neff=E−+ 2t⊥[Eq. (14)]. The green\nlines indicate ω= 0.\nsolution of the following equation [28]:\n1 =U\nNs/summationdisplay\np1\nǫk−ǫ0\nk−p−ǫ0p. (6)\nBy expanding Eq. (6) in powers of U, the dispersion\nrelations of the high-energy modes for U≫t⊥,t/bardblcan beTABLE I: Eigenstates and energies on a rung.\nEigenstate Energy\n|0/angbracketright=|0,0/angbracketright E0= 0\n|Bσ/angbracketright= (|σ,0/angbracketright+|0,σ/angbracketright)/√\n2EB=−t⊥−µ\n|Aσ/angbracketright= (|σ,0/angbracketright−|0,σ/angbracketright)/√\n2EA=t⊥−µ\n|ψ+/angbracketright=−ζ−|S/angbracketright+ζ+|D+/angbracketrightEψ+=E+−2µ\n|ψ−/angbracketright=ζ+|S/angbracketright+ζ−|D+/angbracketrightEψ−=E−−2µ\n|T+/angbracketright=| ↑,↑/angbracketright ET+=−2µ\n|T−/angbracketright=| ↓,↓/angbracketright ET−=−2µ\n|T0/angbracketright= (| ↑,↓/angbracketright+| ↓,↑/angbracketright)/√\n2ET0=−2µ\n|D−/angbracketright ED−=U−2µ\n|Gσ/angbracketright= (|σ,↑↓/angbracketright+| ↑↓,σ/angbracketright)/√\n2EG=U+t⊥−3µ\n|Fσ/angbracketright= (|σ,↑↓/angbracketright−| ↑↓,σ/angbracketright)/√\n2EF=U−t⊥−3µ\n|W/angbracketright=| ↑↓,↑↓/angbracketright EW= 2U−4µ\n|S/angbracketright= (| ↑,↓/angbracketright−| ↓,↑/angbracketright)/√\n2,|D±/angbracketright= (| ↑↓,0/angbracketright±|0,↓↑/angbracketright)/√\n2,\nζ±=/radicalbigg\n1\n2/parenleftBig\n1±U//radicalBig\nU2+16t2\n⊥/parenrightBig\n,E±=U\n2±/radicalBig\nU2+16t2\n⊥\n2.\nobtained as\nǫ(k/bardbl,0)=U+2t⊥+4t/bardbl+J/bardbl/parenleftbig\ncosk/bardbl+1/parenrightbig\n+J⊥,\nǫ(k/bardbl,π)=U+2t⊥+4t/bardbl+J/bardbl/parenleftbig\ncosk/bardbl+1/parenrightbig\n,(7)\nup toO(t2\n/bardbl/U) andO(t2\n⊥/U), whereJ/bardbl= 4t2\n/bardbl/Uand\nJ⊥= 4t2\n⊥/U(dotted orange curves in Fig. 3).\nThe above results can also be explained in terms of\nthe modes of double occupancy. We define |ψ+(k/bardbl)/an}bracketri}htand\n|D−(k/bardbl)/an}bracketri}htfork⊥= 0 andπ, respectively, as\n|X(k/bardbl)/an}bracketri}ht=1√\nLL/summationdisplay\nj=1eik/bardblrj|X/an}bracketri}htjL/productdisplay\nl/negationslash=j|0/an}bracketri}htl,(8)\nwhererjdenotesthe coordinateofrung jin the legdirec-\ntion, andXrepresentsψ+andD−. Here,|ψ+/an}bracketri}htj,|D−/an}bracketri}htj,\nand|0/an}bracketri}htjdenote the eigenstates of the jth rung defined\nin Table I. The effective hoppings of |ψ+/an}bracketri}htand|D−/an}bracketri}htare\nobtained using second-orderperturbation theory with re-\nspect tot/bardblas\ntψ+\n/bardbleff=−4t2\n/bardbl\nU+2t2\n/bardbl/radicalbig\nU2+16t2\n⊥,\ntD−\n/bardbleff=−2t2\n/bardbl\nU,(9)\nrespectively, which reduce to\ntψ+\n/bardbleff≈tD−\n/bardbleff≈ −J/bardbl\n2, (10)\nforU≫t⊥. By taking into account the bond energy\nbetween |ψ+/an}bracketri}htand|0/an}bracketri}ht,ξψ+0(=−tψ+\n/bardbleff), and that between\n|D−/an}bracketri}htand|0/an}bracketri}ht,ξD−0(=−tD−\n/bardbleff), the energies of the high-\nenergy modes are obtained as\nEψ+\n/bardbleff≈J/bardbl(cosk/bardbl+1)+E+−2µ,\nED−\n/bardbleff≈J/bardbl(cosk/bardbl+1)+U−2µ,(11)5\nup toO(t2\n/bardbl/U) forU≫t⊥≫t/bardbl. Here,\nE±=U\n2±/radicalbig\nU2+16t2\n⊥\n2, (12)\nwhich reduces to\nE+≈U+J⊥,\nE−≈ −J⊥,(13)\nforU≫t⊥. By putting µ=−2t/bardbl−t⊥in the low-\nelectron-density limit (the ground-state energy of the\none-electron system is zero), Eq. (11) reduces to Eq.\n(7) forU≫t⊥≫t/bardbl. This result implies that the high-\nenergy modes in the low-electron-density limit can be\ninterpreted as the modes of double occupancy [Eq. (8)]\nforU≫t⊥≫t/bardbl.\nB. Intermediate-energy emergent mode\nAs mentioned in Sec. III, not only the high-energy\nmodes ofO(U) but also an intermediate-energy mode\nemerges[ω/t/bardbl≈7andk/bardbl≈πinFig. 1(a)]. Toclarifythe\nnatureofthis mode, weconsideraneffective modelfor U,\nt⊥≫t/bardbl. In the low-electron-density regime, an electron\nwithk⊥= 0 on a rung, |Bσ/an}bracketri}ht, hops almost freely along\nthe leg. When two electrons with opposite spins sit on\nthe samerung, they areexcitedto oneofthe two-electron\neigenstates with k⊥= 0 on a rung, |ψ±/an}bracketri}ht(Table I). Thus,\nthe effective model for k⊥= 0 can be obtained as the\nHubbard chain with the following effective interaction\nU±\neff[7] and effective chemical potential µeff:\nU±\neff=E±+2t⊥,\nµeff=µ+t⊥,(14)\nby equating EBandEψ±with the effective single-site\nenergies −µeffandU±\neff−2µeff, respectively.\nIn the Hubbard chain, the dispersion relation of the\nhigh-energy mode in a two-electron system can also be\nobtained from Eq. (6) as\nǫk/bardbl=U+4t/bardbl+J/bardbl/parenleftbig\ncosk/bardbl+1/parenrightbig\n, (15)\nup toO(t2\n/bardbl/U) in the large- U/t/bardblregime [t⊥=J⊥= 0 in\nEq. (7)]. By putting U±\neff[Eqs. (13) and (14)] into Eq.\n(15), we obtain\nǫ+\n(k/bardbl,0)=U+ 2t⊥+J⊥+4t/bardbl+J/bardbl/parenleftbig\ncosk/bardbl+1/parenrightbig\n,\nǫ−\n(k/bardbl,0)= 2t⊥−J⊥+4t/bardbl+J/bardbl/parenleftbig\ncosk/bardbl+1/parenrightbig\n,(16)\nup toO(t2\n⊥/U) andO(t2\n/bardbl/U) forU≫t⊥≫t/bardbl. The\nformer [ǫ+\n(k/bardbl,0)] corresponds to the high-energy mode of\nO(U) [Eqs. (7) and (11)]. The latter [ ǫ−\n(k/bardbl,0)] corresponds\nto the emergent mode in the intermediate-energy regime\n[ω/t/bardbl≈7 andk/bardbl≈πin Fig. 1(a); dashed-dotted blue\ncurve in Fig. 3(a)].\nFIG. 4: (a) U−\neff/t/bardblfort⊥/t/bardbl= 2 [Eq. (14)] (dotted red\ncurve). Charge gap ∆ cof the Hubbard ladder at n= 1/2\nfort⊥/t/bardbl= 2 determined as the chemical-potential difference\nobtained using the non-Abelian DDMRG method in a 120-\nsite cluster (blue diamonds), and ∆ cof the effective Hubbard\nchain with U−\neffatneff(= 2n) = 1 (solid cyan curve) obtained\nusing (b). (b) ∆ cof the Hubbard chain at n= 1 obtained\nusing the Bethe ansatz [53].\nFrom the above analysis, the emergent mode in the\nintermediate-energy regime can be interpreted as the up-\nper Hubbard band of the effective Hubbard chain with\nU−\neff=E−+2t⊥≈ −J⊥+2t⊥[dotted red curve in Fig.\n4(a)]. Thus, the energy of this mode is not O(U) but\nO(t⊥) forU≫t⊥≫t/bardbl.\nC. Remarks on the upper Hubbard band\nIn a conventional band picture, the splitting of a band\ninto upper and lower bands is considered a result of sym-\nmetry breaking. For example, antiferromagnetic order-\ning, which causes folding of the Brillouin zone, is con-\nsidered responsible for the formation of the gap in an\nantiferromagnetic insulator in a band picture. Neverthe-\nless, the emergence of the high-energy states is a general\ncharacteristicof stronglyinteractingsystems on a lattice,\nwhich does not require symmetry breaking or long-range\norder.\nThe mechanism of the emergence of the upper Hub-\nbard band can, instead, be interpreted as the formation\nof a pair or a bound state [28–33]. The simplest case\nis the limit of the low-electron-density and strong repul-\nsion where the high-energy mode can be interpreted as\na mode of double occupancy (a bound state of electrons\nwith opposite spins [28]) [Eq. (8)] as shown in Sec. VA.\nMore generally, the interpretation as the formation of a\npair can be justified in terms of string solutions in one-\ndimensional (1D) systems. Among Bethe-ansatz solu-\ntions, there are solutions involving a string which can\nbe regarded as a pair of particles [34, 35]. In the Hub-\nbard chain, the upper Hubbard band has been identi-\nfied as thek-Λ string solutions [29]. Similarly, the high-\nenergy states of the antiferromagnetic Heisenberg chain\nin a magnetic field in S+−(k,ω) (excitation of flipping\na majority spin to a minority spin) have been identified6\nas the two-string solutions [30], which correspond to the\nupper Hubbard band of the interacting hard-core bosons\n[31] and interacting spinless fermions on a chain [32, 33].\nThus, these high-energy states can be identified as states\ninvolving a pair of particles.\nIt should be noted that the quasiparticle responsible\nfor the upper Hubbard band is not exactly the double oc-\ncupancy, in general. In fact, the double occupancy exists\neveninthegroundstateathalf-fillingfor U <∞, andthe\ndouble occupancy is not specified by a quantum number\nof eigenstates. In the Hubbard chain, the quasiparticle\nresponsible for the upper Hubbard band has been iden-\ntified in terms of the quantum number for the k-Λ string\n[29] similarlyto the spinon and holon (defined in terms of\nthe quantum numbers for spin and charge, respectively).\nIn higher dimensions, the quasiparticle responsible for\nthe upper Hubbard band could be interpreted as a pair\nof particles [28] or that of a chain deformed by interchain\nhopping (Sec. VB) [36, 37].\nVI. QUARTER-FILLING\nAt quarter-filling ( n= 1/2), the system becomes a\nMott insulator [Figs. 1(d), 1(l), and 2(d)] because the\nMott transition occurs at neff(=Ne/L= 2n) = 1 in the\neffective Hubbard chain with U−\nefffork⊥= 0 [Sec. VB].\nTo understand the nature of this dimer Mott insulator,\nwe consider the electronic states of the upper Hubbard\nband of the effective Hubbard chain. As mentioned in\nSec. VB, the two values of the effective U,U±\neff=E±+\n2t⊥[Eq. (14)], reflect the two doubly occupied states on\na rung (Table I),\n|ψ+/an}bracketri}ht=−ζ−|S/an}bracketri}ht+ζ+|D+/an}bracketri}ht,\n|ψ−/an}bracketri}ht=ζ+|S/an}bracketri}ht+ζ−|D+/an}bracketri}ht,(17)\nwhere\nζ±=/radicaltp/radicalvertex/radicalvertex/radicalbt1\n2/parenleftBigg\n1±U/radicalbig\nU2+16t2\n⊥/parenrightBigg\n. (18)\nIn the large- U/t⊥limit,|ψ+/an}bracketri}ht → |D+/an}bracketri}htand|ψ−/an}bracketri}ht → |S/an}bracketri}ht\nbecauseζ+→1 andζ−→0. Hence, the high-energy\nmode ofO(U) has a larger component of the doubly oc-\ncupied sites [ |D+/an}bracketri}ht= (| ↑↓,0/an}bracketri}ht+|0,↓↑/an}bracketri}ht)/√\n2], whereas\nthe low-energy mode of O(t⊥) has a larger component\nof the spin-singlet state without doubly occupied sites\n[|S/an}bracketri}ht= (| ↑,↓/an}bracketri}ht − | ↓,↑/an}bracketri}ht)/√\n2] in the large- U/t⊥regime.\nThus, in the dimer Mott insulator whose gap is essen-\ntially determined by U−\neff, the doubly occupied state |ψ−/an}bracketri}ht\ncan primarily be regarded as the spin-singlet state with-\nout doubly occupied sites on a rung; the Mott gap is not\nofO(U) but ofO(t⊥) (U−\neff=E−+ 2t⊥≈ −J⊥+ 2t⊥)\nforU≫t⊥≫t/bardbl[blue diamonds and solid cyan curve in\nFig. 4(a)] [19].\nThe Mott gap and the effective Uare relevant not only\nto the charge excitation but also to spin fluctuationsand electronic excitations. The effective spin coupling\nat quarter-filling is J−\n/bardbleff= 4t2\n/bardbl/U−\neff(≫J/bardbl= 4t2\n/bardbl/U)\nforU≫t⊥≫t/bardbl. Hence, the spin degrees of freedom\nfork⊥= 0 can be described as the effective Heisen-\nberg chain with J−\n/bardbleff, which shows a two-spinon contin-\nuum:ω=πJ−\n/bardbleff\n2(sinp1\n/bardbl+sinp2\n/bardbl), wherek/bardbl=p1\n/bardbl+p2\n/bardblfor\n0≤p1\n/bardbl< p2\n/bardbl≤π, and the spin-wave mode at the lower\nedge:ω=πJ−\n/bardbleff\n2|sink/bardbl|[38] [Fig. 2(b)].\nBy doping the dimer Mott insulator with a hole (an\nelectron), the spin-wave mode emerges in the electronic-\nexcitation spectrum with the dispersion relation shifted\nby the Fermi momenta k/bardblF=±π/2 forω >0 (ω <0)\n[Figs. 1(c), 1(e), 2(a), and 2(c)] as in the Hubbard chain\n[29]. The properties of the doping-induced states and the\nrelationship to the Mott transition are discussed in detail\nin Sec. VIII.\nIt should be noted that the derivation of the effective\nU,U±\neff[Eq. (14)], is valid for U,t⊥≫t/bardbl(not only\nU≫t⊥≫t/bardbl) forn≤1/2. In the case of t⊥≫U≫t/bardbl,\nU+\neff= 4t⊥+U\n2+U2\n16t⊥,\nU−\neff=U\n2−U2\n16t⊥,(19)\nup toO(U2/t⊥). The dimer Mott gap determined by\nU−\neffatneff(= 2n) = 1 is of O(U) fort⊥≫U≫t/bardbl.\nThus, in the case of U,t⊥≫t/bardbl, the dimer Mott gap at\nn= 1/2 as well as U−\neffforn≤1/2 is limited by 2 t⊥\nforU≫t⊥[Fig. 4(a)] and by U/2 fort⊥≫U[Fig.\n4(a) with 2 t⊥↔U/2]. This implies that the effective\nHubbard model with Ueff>2t⊥(Ueff> U/2) is not\nrelevant to the low-energy properties of a dimer Mott\ninsulator for U≫t⊥(t⊥≫U). BecauseU±\neffis obtained\nin adimer, the aboveargumentwouldgenerallyholdtrue\nfor coupled-dimer systems [6, 7] regardless of the lattice\nstructure or dimensionality as long as U,t⊥≫t/bardbl.\nVII. HALF-FILLING\nA. Electronic excitation\nAthalf-filling( n= 1), the groundstateoftheHubbard\nladder forU≫t⊥≫t/bardblcan be effectively approximated\nas\n|GS/an}bracketri}ht ≈L/productdisplay\nj=1|ψ−/an}bracketri}htj. (20)\nThe dominant modes excited from the ground state are\nobtained by replacing one of the |ψ−/an}bracketri}ht’s with|B/an}bracketri}ht,|A/an}bracketri}ht,\n|G/an}bracketri}ht, and|F/an}bracketri}htas\n|X(k/bardbl)/an}bracketri}ht=1√\nLL/summationdisplay\nj=1eik/bardblrj|X/an}bracketri}htjL/productdisplay\nl/negationslash=j|ψ−/an}bracketri}htl,(21)7\nwhereXrepresentsB,A,G, andF. The excitation\nenergies up to the second order in t/bardblare obtained as\nǫX\nk/bardbl=−2tX\n/bardblcosk/bardbl+EX−Eψ−+2ξXψ−−2ξψ−ψ−,(22)\nwhereEXdenotes the rung energy of |X/an}bracketri}ht(Table I), and\ntB\n/bardbl=−tF\n/bardbl=t+\n/bardblandtA\n/bardbl=−tG\n/bardbl=t−\n/bardbl, where\nt±\n/bardbl=−t/bardbl\n2/parenleftBigg\n1±4t⊥/radicalbig\nU2+16t2\n⊥/parenrightBigg\n. (23)\nHere,ξXψ−denotes the bond energy between |X/an}bracketri}htand\n|ψ−/an}bracketri}htobtained in the second-order perturbation theory,\nξψ−ψ−=−2t2\n/bardblU2\n(U2+16t2\n⊥)3/2,\nξTψ−=−4t2\n/bardbl\nU+2t2\n/bardbl/radicalbig\nU2+16t2\n⊥,\nξBψ−=−t2\n/bardbl\n4t⊥−t2\n/bardbl/radicalbig\nU2+16t2\n⊥\n8t⊥U+ξψ−ψ−\n8\n+t2\n/bardbl\nU+t2\n/bardbl/radicalbig\nU2+16t2\n⊥−2t2\n/bardblt⊥\nU/radicalbig\nU2+16t2\n⊥,\nξAψ−=ξBψ−|t⊥↔−t⊥,\nξGψ−=ξAψ−,\nξFψ−=ξBψ−,(24)\nwhereTrepresentsT+,T−, orT0.\nWecanalsoconstructtwo-particlestatesbyusing X(=\nB,A,G, andF) andY(=T+,T−, orT0) as\n|XT(k/bardbl;p/bardbl)/an}bracketri}ht=1/radicalbig\nL(L−1)/summationdisplay\nm/negationslash=nei(k/bardbl−p/bardbl)rmeip/bardblrn\n|X/an}bracketri}htm|Y/an}bracketri}htnL/productdisplay\nl/negationslash=m,n|ψ−/an}bracketri}htl, (25)\nwhose effective excitation energies for L→ ∞are ob-\ntained as\nǫXT\nk/bardbl;p/bardbl=−2tX\n/bardblcos(k/bardbl−p/bardbl)+Jeff\n/bardblcosp/bardbl+EX−2E−\n+2µ+2ξXψ−+2ξTψ−−4ξψ−ψ−,(26)\nwhere\nJeff\n/bardbl=8t2\n/bardbl\nU−4t2\n/bardbl/radicalbig\nU2+16t2\n⊥, (27)\nwhich reduces to Jeff\n/bardbl≈J/bardblforU≫t⊥.\nThe dominant modes and the continua carrying con-\nsiderable spectral weights in Figs. 1(h) and 1(p) can\nbasically be identified with the above modes [Eqs. (21)\nand (22); dotted brown curves and solid black curves in\nFig. 5] and the two-particle states [Eqs. (25) and (26);\nlight orange regions and light gray regions in Fig. 5].\nFIG. 5: Dispersion relation of electronic excitations at n= 1\nforU/t/bardbl= 16 andt⊥/t/bardbl= 2 atk⊥= 0 [(a)] and π[(b)]. In\n(a),ω=ǫG\nk/bardbl(dotted brown curve), −ǫB\nk/bardbl(solid black curve),\nǫFT\nk/bardbl;p/bardbl(lightorangeregion), and −ǫAT\nk/bardbl;p/bardbl(lightgrayregion). In\n(b),ω=ǫF\nk/bardbl(dotted brown curve), −ǫA\nk/bardbl(solid black curve),\nǫGT\nk/bardbl;p/bardbl(light orange region), and −ǫBT\nk/bardbl;p/bardbl(light gray region).\nThe green lines indicate ω= 0. The chemical potential µis\nset so that ǫB\nπ= 0.\nFIG. 6: (a) A(k,ω)t/bardblfork⊥=πatn≈0.967 forU/t/bardbl= 16\nandt⊥/t/bardbl= 2 obtained using the non-Abelian DDMRG\nmethod [closeup of Fig. 1(o) near the Fermi level]. (b)\nS(k,ω)t/bardblfork⊥=πatn= 1 forU/t/bardbl= 16 andt⊥/t/bardbl= 2\nobtained using the non-Abelian DDMRG method. The green\nlines indicate ω= 0. Gaussian broadening with a standard\ndeviation of 0 .1t/bardblhas been used.\nB. Spin excitation\nThe spin excited state is similarly obtained as in Eq.\n(21) withX=T+,T−, orT0, whose excitation energy\nup to the second order in t/bardblcan be obtained as\nǫspin\nk/bardbl=Jeff\n/bardblcosk/bardbl−E−+2ξTψ−−2ξψ−ψ−.(28)\nThis excitation well explains the mode in S(k,ω) for\nk⊥=πat half-filling [Fig. 6(b)].8\nVIII. MOTT TRANSITION\nA. Doping-induced states\nThe most remarkable spectral feature is the emergence\nof electronic states in the Mott gap by doping a Mott\ninsulator [Figs. 1(o) and 1(p)]. From the low-electron-\ndensity side, the dominant mode for k⊥=πin the low-\nelectron-density regime loses spectral weight and disap-\npearsathalf-filling[Figs. 1(i)–1(p)], butitsdispersionre-\nlationremainsdispersinguntil the Mott transitionoccurs\n[Figs. 1(o) and 6(a)]. In the small-doping limit, the dis-\npersion relation reduces to the magnetic dispersion rela-\ntionshiftedbytheFermimomentum kF= (π,0)(Fig. 6).\nThus, the antibonding band in the low-electron-density\nregime gradually loses spectral weight with the electron\ndensity and eventually leads to the magnetic excitation\nat half-filling. This implies that the charge degrees of\nfreedom freeze, whereas the spin degrees of freedom re-\nmain active toward the Mott transition [5, 27, 29, 36].\nFrom the Mott-insulator side, this feature can be de-\nscribed as follows: the spin excited states at half-filling\nappearintheelectron-additionspectrumwiththedisper-\nsion relation shifted by the Fermi momentum upon dop-\ning a Mott insulator because the charge characteristic is\nadded to the spin excited states by doping [5, 27, 29, 36].\nA simple explanation for this feature has been given us-\ning effective eigenstates of the t-Jladder as well as based\non general arguments on quantum numbers in Ref. [27].\nHere, we briefly review the explanation in the case of\nthe Hubbard ladder. The ground state at half-filling for\nU≫t⊥≫t/bardblcan be effectively approximated as in Eq.\n(20). The spin excited states are obtained as |T(k/bardbl)/an}bracketri}ht\n(T=T+,T−, orT0) [Eq. (21)] whose dispersion rela-\ntion is expressed as ω=ǫspin\nk/bardbl[Eq. (28); Fig. 6(b)]. The\none-hole-doped ground state is essentially |B(π)/an}bracketri}ht, which\nhas momentum k= (π,0). In the small-doping limit, the\nchemical potential is set so that the energy of the one-\nhole-doped ground state is zero. When an electron with\nmomentum( p/bardbl,π) isaddedtothe one-hole-dopedground\nstate, theobtainedstatehasoverlapwiththespinexcited\nstate at half-filling |T(p/bardbl+π)/an}bracketri}ht. Thus,A((p/bardbl,π),ω) ex-\nhibits a mode along ω=ǫspin\np/bardbl+π[Eq. (28); Fig. 6(a)].\nThis argument shows that the spin excited states at half-\nfilling appear in the electron-addition spectrum with the\ndispersion relation shifted by the Fermi momentum upon\ndoping a Mott insulator.\nThis feature of the doping-induced states has also been\npointed out in the Hubbard chain [29], two-dimensional\n(2D) Hubbard model [36], t-Jchain [39], 2D t-Jmodel\n[26], andt-Jladder [27], as well as in a system with anti-\nferromagnetic order [40]. In the Hubbard ladder consid-\nered in this paper, the mode of the doping-induced sates\nhas an energy gap because the spin excitation at half-\nfilling has an energy gap (Fig. 6). In the case where the\nspin excitation is gapless in a Mott insulator, the modeof the doping-induced states should be gapless, as shown\nin Fig. 2, and in the Hubbard and t-Jchains, and the\n2D Hubbard and t-Jmodels [5, 26, 27, 29, 36, 39–41].\nThe emergence of electronic states upon doping a\nMott insulator was recognized soon after the discovery\nof cuprate high-temperature superconductors [3, 42, 43].\nHowever, interpretations are controversial. Various in-\nterpretations other than the above interpretation have\nbeen proposed primarily for the 2D Hubbard model [44–\n48], suggesting that the mode of the doping-induced\nstates is essentially separated by a (pseudo-)gap from\nthe low-energy band even though the Mott insulator\nexhibits gapless spin excitation [44–48]. In contrast,\nthe interpretation described above as well as in Refs.\n[5, 26, 27, 29, 36, 39–41] can naturally and collectively\nexplain the behavior of the doping-induced states in the\nHubbardand t-Jchains[29,39]andthe Hubbardand t-J\nladders [27] (Figs. 2 and 6) as well as in the 2D Hubbard\nandt-Jmodels [5, 26, 36], which implies that this inter-\npretation captures the essence of the Mott transition.\nB. What characterizes the Mott transition\nTo discuss how to characterizethe Mott transition, the\ndefinition of the Mott transition must be clarified. In\nparticular, the definition should be what can distinguish\nthe Mott transition from the transition between a metal\nand a band insulator. Hence, a clear distinction between\na Mott insulator and a band insulator is required.\nOne might consider that a Mott insulator could be de-\nfined by the value of the charge gap; if the charge gap\nis primarily determined by the Coulomb repulsion [ O(U)\nin the Hubbard model], the insulating state could be re-\ngarded as a Mott insulator. However, as shown in Sec.\nVI, even though the value of the charge gap is limited\nby the intradimer hopping for U≫t⊥≫t/bardbl, the insulat-\ning state at quarter-filling should be regarded as a Mott\ninsulator because it is essentially the same as the Mott\ninsulator of the Hubbard chain (Fig. 2) [Sec. VI]. Thus,\na Mott insulator is not necessarily well-defined in terms\nof the value of the charge gap.\nAnother definition could be that a Mott insulator is\nan insulator exhibiting gapless spin excitation. Such an\ninsulator is not a band insulator because the spin gap\nis basically equal to the charge gap in a band insulator.\nHowever, this definition appears too narrow; in practice,\na system having low-energy spin excitation with a large\ncharge gap is generally called a Mott insulator regard-\nless of whether a small spin gap opens or not. Hence, a\nMott insulator can be better defined as an insulator with\n∆s≪∆c(or ∆s<∆cif necessary), where ∆ sand ∆ c\ndenote the lowest excitation energies for spin and charge,\nrespectively. This implies that a Mott insulator can be\ndefined in terms of the spin-charge separation (∆ s/ne}ationslash= ∆c)\n[27]. In 1D systems, the spin-charge separation is con-\nsidered to occur even in a metallic phase: The prop-\nerties in the low-energy limit are described in terms of9\nspin and charge excitations independent of each other\nwith different velocities rather than electronlike quasi-\nparticles [49–52]. In a Mott insulator, regardless of the\nlattice structure or dimensionality, the spin-charge sepa-\nration (∆ s≪∆c) occurs more clearly than in a 1Dmetal\n[∆s/ne}ationslash= ∆c=O(1/L)].\nIf a Mott insulator is characterized by ∆ s≪∆c, what\ncharacterizes the Mott transition should also reflect it.\nIf only the ground-state properties are considered, the\nMott transition in a dimerized system, such as the Hub-\nbard ladder for U≫t⊥≫t/bardblat half-filling, is essentially\nthe same as the transition to a band insulator [Figs. 1(g)\nand 1(h)]. However, reflecting ∆ s≪∆cof the Mott\ninsulator [Figs. 1(h), 1(p), and 6(b)], electronic states\nexhibitingthemomentum-shiftedmagneticdispersionre-\nlation emerge in the Mott gap by doping [Figs. 1(o) and\n6(a)]. This characteristic of the Mott transition reflects\nthe characteristic of the Mott insulator (spin-charge sep-\naration: ∆ s≪∆c) and does not appear in the tran-\nsition from a band insulator. Hence, this characteristic\nshouldbegeneralandfundamentaltotheMotttransition\n[5, 26, 27, 29, 36, 39–41]. The above argument implies\nthat the Mott transition is characterized by the doping-\ninduced states that exhibit the momentum-shifted mag-\nnetic dispersion relation rather than critical exponents or\norder parameters.\nIX. SUMMARY\nThe emergence, disappearance, and spectral-weight\ntransferofelectronicstatesareillustratedintheHubbard\nladder in the strong repulsion and strong intrarung hop-\nping regime. The dominant modes in the low-electron-\ndensity regime significantly lose spectral weight as the\nelectron density increases to half-filling, whereas the\nemergent modes in the low-electron-density regime be-\ncome dominant at half-filling; the dominant modes in\nthe low-electron-density regime and those at half-filling\nare different in origin.\nOne of the emergent modes, which has an energy of\nthe order of the intradimer hopping, significantly gains\nspectral weight and governs the dimer Mott physics at\nquarter-filling; the dimer Mott gap is limited by the in-\ntradimer hopping even in the strong repulsion regime.\nIn contrast, one of the dominant modes in the low-\nelectron-density regime, which originates from a nonin-\nteracting band, gradually loses spectral weight as the\nelectron density increases and completely disappears at\nhalf-filling. However, the dispersion relation remains\ndispersing until the Mott transition occurs; the disper-\nsion relation reduces to the magnetic dispersion rela-\ntion shifted by the Fermi momentum in the small-doping\nlimit. Thus, the mode originating from a noninteracting\nband continuously leads to the mode of the spin excita-\ntion at half-filling.\nThese features would be basically true for generalcoupled-dimer systems regardless of the lattice structure\nor dimensionality as long as U≫t⊥≫t/bardbl. As the dimer\nMott gap is limited by the intradimer hopping, a Mott\ninsulator is not necessarily well-defined as an insulator\nhaving a charge gap of the order of the Coulomb repul-\nsion; a Mott insulator is better characterized in terms\nof the existence of spin excitation whose energy is much\nlower than the charge gap, i.e., the spin-charge separa-\ntion. By reflecting this characteristicof a Mott insulator,\nthe Mott transition can be characterized: The spin ex-\ncited states in a Mott insulator emerge in the Mott gap\nas electronic excitation by doping the Mott insulator, ex-\nhibiting the momentum-shifted magnetic dispersion rela-\ntion. This feature does not appear in the transition from\na band insulator and should be general and fundamental\nto the Mott transition.\nInthesmall- t⊥/t/bardblregime, thespectral-weightdistribu-\ntion can be affected by band hybridization. Nevertheless,\nthe overall spectral features such as emergence and dis-\nappearance of spectral weight should generally appear in\nstrongly correlated systems.\nThe emergence, disappearance, and spectral-weight\ntransfer, which have almost been overlookedin band the-\nory and Fermi-liquid theory, play particularly important\nroles in understanding the physics around the Mott tran-\nsition, such as the origin of the Mott gap and the doping-\ninduced states. This would be one of the reasons why\nthe electronic properties near the Mott transition have\nappeared elusive from the conventional viewpoints. This\npaper also brings an unconventional perspective to elec-\ntronic bands. In a conventional band picture, electronic\nbands are usually identified as those primarily originat-\ning from atomic orbitals; the number of bands is con-\nsidered invariant regardless of the electron density as\nlong as symmetry breaking does not occur. However, as\nshown in this paper, a noninteracting band at zero elec-\ntron density, which corresponds to a conventional band,\ncan disappear at the Mott transition, whereas emergent\nelectronic bands in the low-electron-density regime can\nbecome dominant near the Mott transition. The number\nof electronic bands carrying significant spectral weight\ncan vary depending on the electron density even with-\nout symmetry breaking in strongly correlated systems.\nExperimental confirmation of these features over a wide\nenergy and electron-density regime as well as applica-\ntions of the emergence of electronic states to electronic\nor optical devices is desired.\nAcknowledgments\nThe author would like to thank S. Uji and S. Tsuda for\nhelpful discussions. This work was supported by JSPS\nKAKENHI Grant No. JP26400372 and the JST-Mirai\nProgram Grant No. JPMJMI18A3, Japan. The numeri-\ncal calculations were partly performed on the supercom-\nputer at the National Institute for Materials Science.10\n[1] N. W. Ashcroft and N. D. Mermin, Solid State Physics\n(Cengage Learning, Boston, 1976).\n[2] P. Nozi` eres, Theory of Interacting Fermi Systems (W. A.\nBenjamin, New York, 1964).\n[3] E. Dagotto, Correlated electrons in high-temperature su-\nperconductors , Rev. Mod. 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Lett. 20, 1445 (1968)."    },    {        "title": "1912.08096v1.A_new_state_of_dense_matter_in_neutron_stars_with_nucleon_structure.pdf",        "content": "A new state of dense matter in neutron stars with nucleon structure\nVikram Soni\u0003\nCentre for Theoretical Physics, Jamia Millia Islamia, New Delhi, India\nThe existence of stars with a large mass of 2 solar masses means that the equation of state is\nsti\u000b enough to provide high enough pressure at large central densities. Previous work shows that\nsuch a sti\u000b equation of state is possible if the ground state has nucleons as its constituents. We\n\fnd this to be so in a chiral soliton ( skyrmion ) model for a composite nucleon which has bound\nstate quarks. The strong binding of the quarks in this composite nucleon is plausibly the origin\nof the nucleon-nucleon hard core. In this model we \fnd a new state of superdense matter at high\ndensity which is a 'topological'cubic crystal of overlapping composite nucleons that are solitons with\nrelativistic quark bound states. The quarks are frozen in a \flled band of a unique state, which not\nan eigenstate of spin or isospin but an eigenstate of spin plus isospin, ~S+~I= 0.\nIn this alternative model we \fnd that all neutron stars have no regular `free'quark matter. Neutron\nstars whose central density crosses a threshold baryon density of approximately, nb\u00181=fm3, will\nbecome unstable and go through a decompression (sudden) density discontinuity to conventional\nquark matter. Sequentially, this contraction of the core of the star will soften the equation of state\nrelease a large amount of gravitational potential energy which can give rise to a shock wave and\nmatter ejection. Since the merger of two neutron stars gives a compact state whose mass is larger\nthan the allowed maximum mass, this will be followed by a jet and a short gamma ray burst while\ntransiting into a black hole .\nI. INTRODUCTION\nThe object of this work is to reconcile and relate some\nrecent observations on neutron stars and their merger in\na consistent manner.\ni) Till the recent \fndings of the high mass ( \u00182 solar\nmass ) neutron stars [1, 2] , neutron stars were expected\nto have nuclear matter in the outer regions of lower den-\nsity giving into quark matter cores in the interior region\nof high density, close to their centres. However, such\nstars have a sti\u000b, non relativistic nucleon matter exterior\npushing into a softer relativistic quark interior - an un-\nstable situation. In this case a star with a quark core is\nstable only if the nuclear matter to quark matter tran-\nsition takes place in a small window at low pressure [3].\nIt is also for this reason that most neutron stars with\nconventional quark matter cores and in particular with\nmeson condensates have smaller maximum masses ( 1.6\nsolar mass), as has been pointed in the work above. A\nrecent review [4] also points out that pure conventional\n(for example, MIT bag) quark matter stars are very un-\nlikely to ever have an EOS as high as , Mmax\u00182 solar\nmass.\nIt is well known that there are many purely nucleon\nbased neutron star models that have neutron stars with\nmaximum mass slightly above 2 solar masses, for exam-\nple, the APR 98 equation of state (EOS) of Akmal, Pand-\nharipande and Ravenhall [5]. All such equations of state\nhave one common characteristic and that is a hard core\nthat becomes operative at high density. In view of the\nforegoing, we investigate the following question; Is matter\nin neutron stars be entirely composed of nucleon degrees\nof freedom ?\n\u0003vsoni.physics@gmail.comThere is a plurality of equations of state (EOS) for\npurely nuclear matter at high density - APR, Bonn po-\ntential, Paris potentials, Reid potential, Skyrme poten-\ntials, nuclear mean \feld theory, Bruckner Hartree-Fock\netc. Similarly, there are many EOSs for quark matter\n- the MIT bag model, linear sigma model, NJL model,\nPNJL, PQCD models and many variants of these [4{15].\nThese equations of state have not only numerical un-\ncertainties but even more fundamental conceptual ones.\nNone of them are testable at directly at high density.\nBesides, all these come with a large set of parameters,\nthat allows a lot freedom of \ftting, but without much\nconviction.\nWe refer the reader to the review of Baym et al [4]\nand the references therein, where an attempt is made to\nwork out a hybrid model that can interpolate between\nnuclear EOS at low density (nuclear saturation densities\nand above) and quark matter at high density. By appro-\npriately including various repulsive interactions between\nquarks such hybrid equations of state can be pushed to\naccommodate high mass ( \u00182 solar mass) neutron stars.\nHowever, these works do not explicitly use the structure\nof nucleons with quark bound states.\nii) We know that the nucleon is a composite object\nmade of 3 valence quarks. If we could work out a ground\nstate made up of such nucleons that has a nucleon nu-\ncleon hard core, we also know that \fnally, at some thresh-\nold density, it should dissolve into quark matter. A faith-\nful model of the composite nucleon then is obliged to re-\nproduce these features. We shall use our knowledge of\nnucleon structure and a possible new solitonic nucleon\ncrystal ground state to work some insight into the high\ndensity EOS and the threshold density for the transition\nto quark matter.\niii) We have recent data on the remarkable merger of\ntwo neutron stars [16, 17] whose end state is an objectarXiv:1912.08096v1  [nucl-th]  16 Dec 20192\nof rather large mass, \u00182:7 solar mass. Though we do\nnot know with certainty if this object is a neutron star\nor a black hole, we do know that this event does pro-\nduce a kilonova and a weak gamma ray burst. Such an\nevent could be associated with a 'collapse' to a denser ob-\nject and thus the EOS undergoes a drastic change - that\nis, goes soft, engineering an abrupt contraction. This can\nrelease a lot of gravitational energy that could be respon-\nsible for the matter ejection that is seen after the merger.\nIt must be kept in mind that the usually accepted EOS's\nof neutron stars have a hard core repulsion between nu-\ncleons as the density goes up. Such a 'collapse' would\nmean that after a threshold density the hard core barrier\nbetween nucleons will dissolve into quark matter.\nOur attempt here is to work with an alternative model\nfor the EOS of dense matter. We use a chiral theory\nwhich can describe quark matter at high density and\nalso give a very representative model for a nucleon with\nquark bound states[3]. This can account for the nucleon\nnucleon hard core interactions, without introducing any\nadditional interactions or parameters, in the high density\ninterval before nucleons give way to quark matter.\nIn Section II we review the chirally symmetric mean\n\feld theory of quarks and a chiral multiplet of pions and\n\felds which can describe both the composite nucleon and\nquark matter [3] . Section III, addresses the \frst order\ntransition, through a mixed phase, from nuclear matter\nto conventional quark matter via the Maxwell construc-\ntion between the two phases using the popular APR 98\nnuclear equation of state for nuclear matter. This indi-\ncates that the phase transition from nuclear matter to\nquark matter occurs at densities close to the central den-\nsity of\u00182 solar mass stars.\nHowever, this analysis assumes point particle nucle-\nons; it does not take account of the structure and the\nquark binding inside the nucleon. In section IV, we \fnd\nthat the strong binding of quarks in the nucleon can\nchange the nature of the phase transition and move the\nthe nuclear matter-quark matter transition to apprecia-\nbly higher density. In section V we review earlier work\npointing to a possible new 'topological' crystalline ground\nstate of composite nucleons for dense matter. Section,\nVI and VII are a heuristic attempt at writing down a\nsolid crystal EOS for composite nucleons. In section VIII\nwe show how in the passage to increasing density, we\novercome the nucleon-nucleon hard core barrier at some\nthreshold density and make the transition to a soft EOS\nof quark matter, releasing enough gravitational energy to\npower a shock wave that can eject matter. We would like\nto state at the outset that in proposing this alternative\nmodel we shall bring into this context many earlier works\nthat are of relevance.\nII. THE THEORY\nIn this work we look at nucleon structure in an e\u000bective\nchiral symmetric theory for the strong interactions thatis QCD coupled to a chiral sigma model. The theory thus\npreserves the symmetries of QCD. In this e\u000bective theory\nchiral symmetry is spontaneously broken and the degrees\nof freedom are constituent quarks which couple to a color\nsinglet, sigma and pion \felds as well as gluons [3, 18].\nFurthermore, since we do not have exact solutions for a\ntheory of the strong interactions, we work in mean \feld\ntheory in which in the \frst approximation we assume\nthat mean \felds associated the gluon \felds are absent\nand perturbative QCD e\u000bects are ignored.\n L =\u00001\n4Ga\n\u0016vGa\u0016vjcolor\u0000X\n (D+gy(\u001b+i\r5~ \u001c~ \u0019)) \n\u00001\n2(@\u0016\u001b)2\u00001\n2(@\u0016~ \u0019)2\u00001\n2\u00162(\u001b2+~ \u00192)\n\u0000\u00152\n4(\u001b2+~ \u00192)2+ const (1)\nThe masses of the scalar (PS) and fermions follow on\nthe minimization of the potentials above. This minimiza-\ntion yields\n\u00162=\u0000\u00152<\u001b>2(2)\nIt follows that\nm2\n\u001b= 2\u00152<\u001b>2(3)\nFor the vacuum of the theory the constant is adjusted\nto yield,<\u001b> =f\u0019;<~ \u0019> = 0.\nThe nucleon in such a theory is a color singlet quark\nsoliton in the skyrmion background with three valence\nquark bound states [19, 20]. The quark meson couplings\nare set by matching the mass of the nucleon to its ex-\nperimental value and the meson self coupling is set from\npi-pi scattering, which in turn sets the tree level sigma\nparticle mass to be of order 800 - 850 MeV. For details\nwe refer the reader to ref. [3].\nThis is one of the simplest e\u000bective chiral symmetric\ntheories for the strong interactions at intermediate scale\nand we use this consistently to describe, both, the com-\nposite nucleon of quark bound states and quark matter.\nLater, we attempt to look at a ground state that is a crys-\ntal composed of these skyrmion like composite quark soli-\nton nucleons. We \fnd that the strong binding of quarks\nin the nucleon could move the transition from the nuclear\nto the quark phase to appreciably higher density.\nTo reiterate we work at the mean \feld level where the\ngluon interactions are subsumed in the color singlet sigma\nand pion \felds they generate. Since we will be working\nwith quark matter at high density con\fnement is not an\nissue. We could further add perturbative gluon medi-\nated corrections but they are not expected to make an\nappreciable di\u000berence.3\nIII. THE MAXWELL CONSTRUCTION FOR\nTHE NUCLEAR MATTER TO QUARK MATTER\nTRANSITION\nIn this section we examine the Maxwell construction\nfor the \frst order transition from the nuclear phase to\nthe quark matter phase using some typical equations of\nstate for the two phases.\nTo describe the purely nuclear phase we employ the\ntried and tested APR 98 [5] equation of state. For quark\nmatter we use the simple e\u000bective chiral symmetric the-\nory which has been used to describe, both, the composite\nnucleon of quark bound states and quark matter [3, 18].\nVariationally, one of lowest energy ground states at high\nbaryon density that we \fnd in such chiral models is quark\nmatter with a neutral pion condensate [21, 22]. The equa-\ntion of state for neutron stars for such a state has been\nobtained in [3, 18]\nA simple way to look at the transition from nucleons\ninto quark matter is to plot, EB, the energy per baryon,\nin the ground state of both, the quark matter and the\nnuclear phases, versus 1 =nB, wherenBis the baryon den-\nsity. For the quark matter equation of state see Fig.1 [3]\nin which the quark matter EOS is indicated by the solid\ncurves and the APR [5] nucleon EOS by the dashed line.\nThe slope of the common tangent between the two phases\nthen gives the pressure at the phase transition and the\nintercept, the common baryon chemical potential.\nFIG. 1. The Maxwell construction: Energy per baryon plot-\nted against the reciprocal of the baryon number density for\nAPR98 equation of state (dashed line) and the 3-\ravour pion-\ncondensed phase (PC) for three di\u000berent values of m\u001b(solid\nlines). As this \fgure indicates, the transition pressure moves\nup with increasing m\u001b, and at m\u001bbelow\u0018750 MeV a com-\nmon tangent between these two phases cannot be obtained.\n(From Fig. 2 of Soni and Bhattacharya [3] )\nAs can be seen from Fig.1 [3] , it is the tree level value\nof the sigma mass that determines the intersection of the\ntwo phases; the higher the mass the higher the density at\nwhich the transition to quark matter will take place. In\n[3] it was found that above, m\u001b\u0018850 MeV, stars withquark matter cores become unstable as their mass goes\nup beyond the allowed maximum mass.\nFrom Fig. 1, for the tree level value of the sigma\nmass\u0018850 MeV, the common tangent in the two phases\nstarts at 1=nB\u00181:75 fm3(nB\u00180:57=fm3) in the nu-\nclear phase of APR [A18 + dv +UIX] [5] and ends up at\n1=nB\u00181.25 fm3(nB\u00180:8=fm3) in the quark matter\nphase.\nIn the density interval between the two phases, there is\na mixed phase at a pressure given by the slope of the com-\nmon tangent and at a baryon chemical potential given by\nthe intercept of the common tangent on the vertical axis.\nGoing back to the APR phase in in \fg 11 of APR [5] we\n\fnd that for the APR [A18 + dv +UIX] the central den-\nsity of a star of 1.9 solar mass is, nB\u00180:7=fm3, which\nfalls in the middle range of the phase transition. On the\nother hand,for APR [A18 +UIX] the central density of a\nstar of 1.9 solar mass is, nB\u00180:57=fm3.\nIdeally we would want the central density of the star\nto be a little less than the initial density at which the\nabove phase transition begins in the nuclear phase. We\nhave found that even with the simple quark matter EOS\nthese densities are in the same ball park.\nIn the following, we shall present arguments to show\nthat the phase transition to quark matter is likely to\noccur at higher density.\nIV. THE NUCLEON\nThe above analysis assumes point particle nucleons.\nIt does not take account of the structure and the quark\nbinding inside the nucleon This is not captured by the\nMaxwell construction. Our attempt is to take this further\nby looking at the well accepted quark soliton (skyrmion)\nmodel of the nucleon that is amenable to investigating\nquark binding properties. We now go on to show that\nthis could move the transition from the nuclear to the\nquark phase to appreciably higher density.\nFollowing , Kahana, Ripka and Soni[19], we have an\napproximate and simple expression for the energy, EB, of\na color singlet nucleon soliton, with three colored bound\nstate quarks. In accordance with the skyrmion con\fgu-\nration the VEV for the pion and sigma \felds are [3, 18]\n<\u001b> =f\u0019Cos\u0012 (r);<~ \u0019> = ^rf\u0019Sin\u0012 (r) (4)\nwhere,\u0012(r!1 ) = 0, from the \fnite energy condition\nand\u0012(r!0) =\u0000\u0019, for the pion \feld to be well de\fned\nat the origin and, f\u0019= 93 Mev is the pion decay constant.\nThe energy expression for the soliton with quark bound\nstates is given below.The \frst term below is the quark\nbound state(Dirac) energy eigenvalue in the skyrmion\nbackground. In this background there is a single valence\nquark bound eigenstate of spin plus isospin, ~I+~S= 0,\nwith a color degenaracy of 3. The second term is the ki-\nnetic term from the mesonic part. For this calculation we4\nmake the simplifying assumption, \u001b2+~ \u00192=f2\n\u0019. In this\ncase the potential term corresponding to the spontaneous\nsymmetry breaking is identically zero.\nE=(gf\u0019) =N(3:12\nX\u00000:94) + 2\u0019(1 +\u00192=3):X\ng2(5)\nwhere , g is quark meson (Yukawa) coupling and N, the\nnumber of bound state quarks. In this section we work\nwith the dimensionless parameter, X=Rgf\u0019, where R\nis the soliton radius. This follows from a simple param-\neterization for radial dependence of, \u0012(r) =\u0019(r=R\u00001),\nin a soluble model [19](see \fg. 2). The 'mass' of a 'free'\nquark in this model is given by, mq=gf\u0019.\nMinimizing this with respect to , X\nX2=3:12g2N\n27(6)\nOn substitution of this value\nEmin=(gf\u0019) = 2(s\n3:12N:27\ng2)\u00000:94N (7)\nFor the nucleon soliton we must set , N= 3 as all three\nquarks sit in the bound state. Also, the total degeneracy\nof the single, 0+, bound state is 3 - the number of colors.\nThe soluble model above is very useful in understanding\nthe quark bound state structure of the solitonic nucleon.\nHowever, as can be seen from Ref.[19](Section 6 ), com-\npared with the soluble model an exact solution brings\ndown the the soliton energy by close to 25 percent. The\nvalue of the coupling , g, that \fts the nucleon mass also\ngoes down proportionately.\nIn the interests of consistency with the following sec-\ntion we shall choose the coupling to be, g\u00187:55, as given\nin ref[23], which corresponds to the isolated soliton mass,\nMsol\u0018976 Mev.\nThe above formula allows us to also look at the energy\nof the con\fguration in which two quarks sit in the bound\nstate and one is moved up to the continuum. Such a state\nwill give a measure of the energy required to unbind the\nnucleon.\nWe can easily check the possible bound states by eval-\nuating the ratio of the energy of bound states with 2 and\n3 quarks, which is given by, Emin=(Ngf\u0019) . If the answer\nis less than, 1, we have a bound state, otherwise not.\nEmin=(Ngf\u0019) = 2(r27\u00013:12\nNg2)\u00000:94\n\u00180:464 forN= 3\n\u00180:78 forN= 2\n\u00181:49 forN= 1 (8)\nThis indicates that we have bound states only for N =\n2 and 3. Given the value of , g, we can \fnd the energyrequired to unbind a quark from such a nucleon. The\nenergy of a two quark bound state and an unbound quark\nis 2:56gf\u0019\u00181797 MeV in comparison to the energy of\na 3 quark bound state nucleon which is , 1 :39gf\u0019\u0018976\nMeV.\nWe use the results from the parametrization for the\nabove 'soluble' model to make some heuristic estimates\nbelow.\ni) The di\u000berence between the two states above gives\nthe binding energy of the quark in the nucleon, 1 :17gf\u0019\u0018\n821 MeV. The quark binding in this model is very high. It\nshould be noted that this is the origin of a hardcore when\nwe bring two nucleons together. The greater the binding\nof quarks the greater the energy required to liberate them\nwhen we squeeze two nucleons.\nii) In this model the quark bound state eigenvalue (Fig.\n2) [19] is described by the \fgure given below.\nFIG. 2. Dependence of the quark energy on the soliton size\nXin the quark soliton model\n(From Fig. 2 of Kahana, Ripka and Soni [19])\nWe can see that the quarks will become unbound ( go\nto the continuum) when the energy eigenvalue is larger\nthan the unbound mass of the quark which is given by\nmq=gf\u0019. This happens roughly when, in the dimension-\nless units used in Fig. 2, the energy eigenvalue, \u000f\u00181:\n\u000f\u00151;at X = 3.12/1.94 = 1.6 : (9)\nwhich translates into R\u00180:46fmfor, g = 7.55 ( R,\ndepends inversely on, g )\nThis is a rough estimate of the e\u000bective radius of the\nsqueezed nucleon at which the bound state quarks are\nliberated to the continuum. By assuming that the nucle-\nons are stacked in a cubic lattice and inverting the volume\noccupied by a nucleon this translates to a nucleon density\nof,5\nnB=1\n(2R)3\u00181:29fm\u00003(10)\nThis section has shown that the internal structure of\nquark binding in the nucleon not only provides the hard\ncore but changes the nature of the phase transition that\nis captured by the Maxwell construction and indicates\nthat the transition to free quark matter is likely to be\ndelayed till the quark bound states meet the continiuum.\nMore evidence of this comes from the next section.\nFor quark bound states in nucleons we have found\nabove that the coupling is strong and the binding en-\nergy is rather large \u0018821 MeV, whereas the energy scale\ncorresponding to the inverse size of the nucleon ( \u00182\nfermi) is much smaller, \u0018100 MeV. This indicates that\neven when nucleons overlap the the quarks will not dis-\nsociate into a quark plasma and the nucleons get com-\npressed but retain their identity. This is in total contrast\nto atomic physics, for example the hydrogen atom, where\nthe binding energy is much smaller than the energy scale\nof corresponding to the inverse size of the atom.\nThus the quark bound states in the nucleon may per-\nsist until a much higher density nB\u0018(1\u00001:29)=fm3.\nIn other words, nucleons can survive above the density\nrange of the Maxwell phase transition and appreciably\nabove the central density of the APR 2-solar-mass star.\nIt is useful to recall that our parametrization here is rudi-\nmentary.\nV. THE SKYRME SOLITON COMPOSITE\nNUCLEON CRYSTAL STATE I\nNow we move to completely di\u000berent perspective on\nthe EOS. One of the ground states of dense nuclear mat-\nter that has been popular even in nuclear physics, where\nthe nucleons are assumed structureless, is a neutron crys-\ntal [24, 25]. Of course, such a ground state is viable\nmuch beyond saturation density as nucleus matter does\nnot show any such tendency even for very large nuclei.\nSuch a crystalline state can be also treated in a single cell\nWigner Sietz approximation with appropriate boundary\nconditions.\nWe shall \frst review such a calculation that was car-\nried out by Banerjee, Glendenning and Soni [23] with\nsome interesting \fndings. This is a relativistic( Dirac)\nband structure calculation of a cubic lattice of solitonic\ncomposite nucleons, with quarks bound in a skyrme soli-\nton background. The quark bound state in the skyrme (\n'topological') soliton is an eigenstate of spin plus isospin,\n~I+~S= 0, that we encountered in the last section. The\nrelevant quark band is the 0+ relativistic positive parity\nvalence band that emerges and can be tracked as func-\ntion of baryon density, which is plotted in their Fig 1 .\nIn the band the quark wave functions peak at the centres\nof the soliton. It is important to note that this state has\na color degeneracy of 3 and is completely occupied and\nFIG. 3. Eigenvalues of the valance (0+) and sea (0-) orbitals\nof quarks in soliton matter as a function of Wigner-Seitz cell\nradius, R. The band of levels that develops as the spacing\ndecreases is shown by the shaded region. ( From Fig. 1 in\nB. Banerjee, N. Glendenning and V. Soni Physics Letters B,\nVolume 155, Issue 4,(1985))\nthus the 0+ band is full. Below, we highlight issues of\nthis ground state that were not emphasized in Ref.[23].\nThere is a large gap between the top of this band and\nthe next energy states which belong to the positive en-\nergy continuum. Thus quarks are frozen or fermi blocked\nand cannot behave as regular quark matter till this band\nreaches the the positive energy continuum which happens\nat a radius of\u00180:5 fermi (or a cell length of 1 fermi).\nFor a cubic lattice this translates into a baryon density\n[23] of , 1=fm3. (We note that this calculation uses a\ncoupling constant g = 7.55, which yields a soliton mass\nM = 976 MeV, and an equilibrium R 1.22 fm.).\nIt be seen from the \fgure, the band spreads out above\nand below the single bound state we found in the pre-\nceding section. Thus the density at which the top of\nthe band meets the continuum is slightly lower than the\ndensity at which the single bound state merges with the\ncontinuum. As in last section, this work employs the\nsame approximate soluble model parametrization of the\nsigma and pion \felds. An exact solution will yield a lower\nvalue of, g, in turn increasing the value of, R ( R\u00181=g\n), and lowering the density at which band gets to the\ncontinuum.\nThis is an independent validation of the fact that in\nthis model the onset of conventional quark matter occurs\nat much higher density than indicated by the Maxwell\nconstruction of the earlier section. Till this density at6\nwhich the bands intersect the medium behaves as a color\ninsulator.\nThis is a new state of dense matter that is quite dif-\nferent from a regular, 'free' quark matter state and from\nconventional nuclear matter. Such a state is a direct con-\nsequence of our composite soliton nucleon structure.\nThe quarks live in continuum relativistic Bloch states\nbut due to the \flled band and a large band gap they are\nblocked out. It should be noted that this work does not\ninclude the interaction between nucleons and also does\nnot take account of the quantization of the solitons to\nyield well de\fned nucleon / neutron states . However,\nthis is an independent con\frmation of the fact that ac-\ntive ( 'free') quark matter comes into play only well above\nthe density indicated by the the Maxwell construction in\nSec. III. The last two sections have established the exis-\ntence of this new ground state that exists till a threshold\ndensity of approximately, nB\u0018(1)=fm3. The following\ntwo sections are devoted to the listing of the features of\nthe EOS for the new ground state and a heuristic esti-\nmate of the same.\nVI. SOLITON - SOLITON INTERACTION AND\nQUANTIZATION ENERGY IN A PURE\nSKYRMION MODEL\ni) Klebanov[26] considers a cubic crystal of pure\nskyrmions (without quark bound states). This paper\nworks out the most favourable spin/isopin con\fgura-\ntion, the so called attractive 'tensor' interaction, between\nskyrmions. However, ref [26] works in the chiral limit (\nm\u0019= 0 ), whereas realistically the the tensor interac-\ntion is not long range as it is modulated by the factor,\nexp(\u0000m\u0019r)=(r3). This will strongly reduce the attractive\ntensor interaction at larger, r. He also estimates the en-\nergy of canonical quantization (or isorotational energy)\nof the whole crystal to yield states of good isospin and\nthe third component of isospin.\nFigs 1 of [26] calculates the classical energy per baryon\n(skyrmion), E1=Mcl, which includes the 'tensor' interac-\ntion between skyrmions, versus volume per baryon, where\nthefree skyrmion classical mass (864 MeV in their case)\nhas been subtracted. This work goes on to include the\nenergy of canonical quantization in their Fig. 2, which\ncalculates the sum of classical and isorotational energies,\nE2=Mcl+ 1=(8\u0015I) (\u0015Iis the moment of inertia ),\nper baryon versus volume per baryon, where the nucleon\nmass (938 MeV) has been subtracted. The di\u000berence be-\ntween the two, E2\u0000E1, then gives us the contribution\nof the isorotational energy of canonical quantization. We\nshall use these estimates in the following section.\nHowever, there is a caveat. As can be seen from Fig.2\nin [26] the minimum energy per nucleon occurs around\n, 1=nB\u00184fm3ornB\u00180:25=fm3. Below that density\nthe crystal is not a stable state. We should therefore\ntreat this as a variational ground state only above this\ndensity.ii) The Projection approach There is point of con-\ntention here. Many authors have an alternative approach\nand project out good spin, isospin states, ~J=~I=1\n2cor-\nresponding to the nucleon. The soliton is considered to\nbe a coherent wave packet - a linear super position of all,\n~J=~I= (n+1\n2) states (see Ref[18] ). The maximum\nweight comes from the lowest, ~J=~Istates. We may\nthen make the approximation that the soliton is an equal\nlinear superposition of the nucleon, N, and \u0001 states and\nset the soliton energy to be midway between MNand\nM\u0001.\nMsoliton =MN+1\n2(M\u0001\u0000MN) (11)\nor\nEB=MN=Msoliton\u00001\n2(M\u0001\u0000MN) (12)\nUnlike the former case of the isorotational energy of\ncollective quantization, which is additive and raises the\nnucleon mass, in this case, the nucleon energy is well\nbelow that of the soliton. This matter is still not a settled\nissue. In passing, it should also be pointed out that in\nmost works on solitons with quark bound states [18, 23]\nthe attractive 'tensor ' interaction between solitons has\nnot been taken into account.\nVII. THE SKYRME SOLITON COMPOSITE\nNUCLEON CRYSTAL STATE II\nUsing the learning from the last sections, we shall make\na heuristic attempt to write down the EOS for a cubic\ncrystal of composite solitons with quark bound states\nthat we have introduced earlier with some assumptions.\nGiven all the approximations in the previous sections,\nthe following should be viewed as a pedagogical exercise.\nA more complete version is in progress and will be pre-\nsented in a later work.\nFirst we write down the energy, ECS, of an isolated\ncomposite soliton. This follows from chirally symmetric\nlinear sigma model[3] used to construct the soliton with\nquark bound states, where m\u001b= 850 MeV [3, 18]. The\n\frst term below is the quark bound state eigenvalue en-\nergy (of the soluble model), the second term is the kinetic\nterm from the mesonic part. In contrast to Sec IV, here\nwe relax the constraint on \fxed the vacuum expectation\nvalue (VEV) for the meson \felds; <\u001b>2+<\u0019>2=F2\nis the sum of the square the expectation value of the\nsigma and pion \felds which can be density dependent\nand therefore di\u000berent to, F2=f2\n\u0019. We therefore in-\nclude the potential or symmetry energy term which is\nthe last term below.\nECS=(f\u0019) = 3(3:12\nY\u00000:94(g)Z) + 2\u0019(Y)(Z2)(1 +\u00192=3)\n+\u0019=3(\u00152)(Y3)(Z2\u00001)2(13)7\nwhere ,Y=Rf\u0019,Z=F\nf\u0019,f\u0019= 93 Mev is the pion\ndecay constant and and we take \u00152= 42 corresponding\nto a sigma mass of 850 Mev.\nWe can then calculate, ECSat a given, R, which cor-\nresponds to a cell length, a= 2Rand baryon density,\nnB=1\n(2R)3. We then minimise the energy with respect\nto Z(F) . This shows a trend that as the density goes up,\nthe value of F increases, which indicates that the sponta-\nneous breaking of chiral symmetry is enhanced as baryon\ndensity increases.\nECS, is the energy per soliton in the crystal, with-\nout any inter soliton interaction or canonical quantiza-\ntion which we attempt compute below\ni) Realistically, the tensor interaction energy between\nthe solitons in the chiral limit ( m\u0019= 0) needs to be mod-\nulated by a factor exp( \u0000m\u0019)a)=(a3) , where , a, is the\ncell length, a= 2R. In the chiral limit the tensor inter-\naction From Fig. 1 [26] can be normalised at a cell length\na= 2R= 2:15 fm, which is equivalent to a cell volume\nV= (nb)\u00001= 10fm3, and is found to be , \u0018\u000070 Mev.\nWhen we apply the pion mass correction, exp( \u0000m\u0019a), to\nthis it comes down to - 15.4 Mev. We use this as the nor-\nmalisation for the inter soliton tensor interaction which\nis modulated by the factor, exp( \u0000m\u0019a)=(a3) as the cell\nlength decreases. The energy of tensor interaction that\nfollows is\nET= exp(\u0000m\u0019a)=(a\n2:15)3\u000170Mev (14)\nWe must mention here that the above tensor interac-\ntion is an asymptotic form but we have persisted with it\nat separations of , a\u00181 fm, where this assumption may\nnot be valid.\nii) The energy of canonical quantization for our com-\nposite soliton is very similar to the skyrme soliton above.\nAs stated before the di\u000berence between the two, \u0001 EQ=\nE2\u0000E1,[26] gives us the contribution of the energy of\ncanonical quantization.\nIncluding i) and ii) above, the total energy per baryon\nfor our case of a crystal of composite solitons with quark\nbound states is given by\nEB=ECS+ \u0001EQ+ET (15)\nWe note again that the crystal state above is a stable\nstate only well above nuclear density.\nA. Remarks\ni) First, this is indeed a new state of dense matter\nthat has not been explored before. Whereas, conven-\ntional quark matter is a fermi liquid in the presence of\na stationary wave neutral pion condensate [3, 18], in our\ncomposite nucleon soliton crystal state the quarks live in\na 'topological' crystal, in a frozen fully occupied , 0+,\nAPRg=7.55\ng=81 2 3 4nB-1(fm3)900100011001200130014001500EB(MeV)FIG. 4. Energy per baryon, EB, for the APR EOS, and the\nquark soliton crystal EOS for, g = 7.55 and g= 8\nband up to the threshold baryon density. This is also\nvery di\u000berent to conventional nuclear matter.\nii) It is important to note , as can be seen from (Section\n6 ) in Ref.[19], that an exact solution brings down the\nsoliton energy by close to 25 percent.\niii) The inter soliton attractive tensor interaction must\nbe added. This can be a fairly large negative contribu-\ntion has been added as indicated above from the pure\nskyrmion.\niv) In our estimate ( see Fig, 4 ) we use the additive\npositive contribution of the isorotational energy of collec-\ntive quantization. It is a moot question if we should have\nused the large negative contribution that follows from the\nprojection procedure.\nv) Next,there is the question of the zero point energy.\nIn the pure Skyrme soliton in the previous section each\ncell carries a skyrme soliton with unit baryon number\nwhich is then localised and will carry zero point energy.\nFrom the band structure in ref [23] we \fnd that actually\nthe quark wave functions are not localised but are Bloch\nfunctions which occupy a \flled band. The baryon num-\nber is carried by the quarks and not localised in a cell.\nAlso, the pion and sigma \felds are like a stationary wave\ncondensate. Thus, in our model we do not have any zero\npoint energy.\nvi) The APR equation works with point nucleons with\nthe repulsive ( hard core) interaction carried by, for ex-\nample, the !meson interaction potential. Our repulsive\n( hard core interaction) has di\u000berent origin - the deep\nquark bound states. Once we have \fnite size composite\nnucleons the space between nucleons is squeezed. The\ncomposite nucleons have a size which makes them over-\nlap at high density, generating a hard core. Thus we\nexpect them to have a crystalline ground state. Such a\nstate is not accessible for the APR nucleons which are\npoint particles. Point like nucleons would be di\u000ecult to\nlocalise due their large zero point energy.\nvii) Till now, we have set our VEV's ( Sec. (III) to(VI))\nfor our soliton model with quark bound states in accor-\ndance with the single skyrmion con\fguration where the8\nVEV for the pion and sigma \felds are[3, 23]\n<\u001b> =f\u0019Cos\u0012 (r);<~ \u0019> = ^rf\u0019Sin\u0012 (r) (16)\nwhere,\u0012(r) = 0, at the cell boundary, r = R, and\n\u0012(r!0) =\u0000\u0019, for the pion \feld to be well de\fned at\nthe origin.\nBut the constraint at the cell boundary is not required\nfor the crystal which allows the pion \feld to be non zero\nat the cell boundary between the cells. As the density\nis increased the pion \feld will goes up gradually at the\ncell boundary and goes to zero only at the centres of the\nsolitons, thus doubling the 'wavelength' of the pion \feld\nat very high density. This will reduce the energy per\nbaryon, as both the quark bound state eigenvalue and\ngradient energy can come down substantially. This exer-\ncise was not carried out in ref [23] and will be presented\nin a later work.\nInterestingly, though, very di\u000berent from our solitonic\ncrystal lattice, Pandharipande and Smith[24, 25] do \fnd\na nucleon matter EOS that is a crystalline solid of neu-\ntrons with a neutral pion condensate. Neutral pion con-\ndensation is also a feature of our 3 \ravour quark matter\nground state[3]. It thus seems that a pion condensate is\na uniform feature of both the quark soliton state and the\n\fnal quark matter state at high density.\nviii) The Goldberger Trieiman relation which follows\nfrom PCAC introduces a renormalisation factor for nu-\ncleons that ups the axial current coupling, gA, from 1\nto\u00181:36 (see section 6.5 in Baym [25]). The quarks in\nour model are similar to nucleons and acquire their mass\nfrom the spontaneous breaking of chiral symmetry which\ngives them a large constituent mass. We may thus expect\na corresponding increase in the coupling, g.\nix) We have assumed a simple cubic crystal till now\nand a corresponding baryon density, nB\u0018(1=2R)3. A\nhexagonal close packed structure is also possible, which\nwill have higher density for the same , R, compared to the\ncubic structure. This can reduce the threshold density at\nwhich the quark matter transition occurs.\nFurthermore, as in the previous sections, we have used\nthe same approximate soluble model parametrization of\nthe sigma and pion \felds. An exact solution will yield\na lower value of the coupling, g, in turn increasing the\nvalue of, R ( R\u00181=g), and reducing the density at which\nband gets to the continuum.\nx) Though, we have not included, (vii). (viii) and (ix),\nbut used the reduction for the exact solution indicated in\n, (ii), above, we \fnd an EOS that is similar to the APR\n( see Fig. 4) but somewhat above it. With a slight incre-\nment in ,g\u00188, as suggested in (viii) we can recover an\nEOS that is close to the APR ( see Fig. 4). Recall, that\nthe EOS for the relativistic crystalline (quark solitonic)\nnucleon state is good only well above nuclear density.\nOur rough estimates should be viewed as a demon-\nstration that at high baryon density an APR like EOS\nis possible for a crystal of composite solitons with quark\nbound states. As posted earlier we are engaged in anongoing work in which we use a full solution for, ECS,\nincluding all the contributions listed above.\nxi) The maximum mass for the neutron star for our\nmodel can then be taken to be similar to that for the APR\nEOS. From the APR EOS [5] this the maximum mass at\nsuch central densities is \u00182:1\u00002:3 solar masses. We note\nthat for the solid nucleon crystal model of Pandharipande\nand Smith[24] with a pion condensate the maximum mass\nis of the same order.\nOnce the density hits a threshold, where the quarks are\nno longer bound or frozen in the 0+band, we can transit\ninto normal or conventional quark matter. As indicated\nearlier this happens roughly when nB\u00181=fm3. Once\nthe barrier at this threshold is overcome, we expect that\nnuclear matter to make a sudden transition into pion con-\ndensed quark matter. This is the point when the EOS\nbecomes soft through a decompression. The sudden in-\ncrease in density can mimic a collapse generating a shock\nwave which ejects matter.\nVIII. ENERGY RELEASE IN MERGER OF\nNEUTRON STARS\nThe sudden phase transition from the relativistic crys-\ntalline ( quark soliton) state to 'free' quark matter will\nresult in a contraction or the core. To illustrate this\nwe calculate the pressure using the APR nuclear mat-\nter EOS from \fg.1 in Ref. [18] and the quark matter\nEOS from our Fig. 1. We note that the pressure in the\nAPR EOS we use goes up sharply at high density. If the\nsudden phase transition to 'free' quark matter occurs at\naround,nB\u00181=fm3, we \fnd that the pressure in the\nnuclear phase at this density is , P\u0018600Mev=fm3. Fig\n5 illustrates the transition from APR nuclear matter to\nquark matter at this pressure at nB\u00180:95=fm3. Other\nequations of state are softer as is the crystalline state in\nFig. 4. This would move the APS curve to the right\nand reduce the pressure at which this density occurs.The\ncontinuous line tracks the evolution as the system goes\nto higher density. Of course, the pressure, P, and energy\nper baryon, EB, in the 'free' quark matter state at this\ndensity are much lower.\nSince we need to balance the pressure in both phases it\nis pertinent to \fnd the density at which the same pressure\noccurs in the 'free' quark matter' state. From Fig. 5 ,this\nis found to be, nB\u00181:5=fm3, and the corresponding,\nEB\u00181260MeV . Thus, as nuclear matter clears the\nthreshold barrier set by the soliton crystal, there will be\na sudden contraction followed by a consequent increase\nin density from, nB1\u00180:95=fm3tonB2\u00181:5=fm3,\nas the system attains the same pressure. As pointed out\nbefore, if the EOS is softer than the APS the pressure and\nthreshold density at which the quark matter transition\noccurs will come down.\nIn a high mass neutron star or in the merger of 2 neu-\ntron stars such a major change in compressibility, K,\nwould cause a contraction of the core and bring down the9\nthree\nflavor\nPCAPR\n0.5 1.0 1.5nB(1/fm3)20040060080010001200P(MeV/fm3)\nFIG. 5. Pressure, P(Mev=fm3), vsnB(1=fm3), for the APR\nEOS, and the pion condensed 3 \ravour quark matter. Illus-\ntration of the transition that occurs at, nB(0:95=fm3), in the\nnuclear phase\ngravitational potential energy. A rough estimate( New-\ntonian) of the gravitational energy release is provided by\nconsidering a neutron star of mass Mwhose potential\nenergy is, (3 =5)GM2=R. Keeping the mass \fxed we can\nwrite down the energy di\u000berence as we change the den-\nsity indicated above( see Fig. 5) from , \u001a1\u00181:6\u00011015\ngm/cc to\u001a2\u00182:53\u00011015gm/cc , for a uniform density\nstar, where, R= (3M\n4\u0019\u001a)1=3\n\u0001EG= (3=5)(GM2)[1=(R2)\u00001=(R1)] (17)\nOn substituting the the values of a 2 solar mass star,\nM\u00184\u00011033gm and the above density change for the\ncorresponding radii, we can get a sudden release of gravi-\ntational energy of, \u0001 EG\u00180:7\u00011053, ergs which can yield\na matter ejecting shock wave.\nIX. DISCUSSION\nOne signi\fcant di\u000berence with most of the equations of\nstate in the literature and this work is that we have com-\nposite nucleons where the structure of the nucleon plays\nan essential role in the transition from nuclear matter\nto quark matter at high density. Working with nucle-\nons that are chiral solitons with relativistic quark bound\nstates we have presented evidence for plausible new crys-\ntalline ground state for dense matter, at densities where\nnucleons overlap, to show that conventional quark mat-\nter does not occur till such density at which the quark\nbound states get compressed and merge with the contin-\nuum. In this ground state it is topology and chiral quark\ninteractions in the nucleon that determine the thresholdtransition density. This is in contrast to most other equa-\ntions of state. Our motivation in Sec. VII is to make a\nrudimentary estimates for the EOS of our composite soli-\ntons is to show that it can be potentially similar to the\nAPR 98 EOS. As we have stated a more convincing cal-\nculation of the EOS which includes several improvements\nwill be presented separately.\nThese works indicate that strongly bound quarks in the\nquark soliton model of the nucleon translate into a 'hard\ncore' interaction between nucleons, resulting in an equa-\ntion of state that provides even a stronger nucleon nu-\ncleon repulsion than the hard core repulsion encountered\nin the APR EOS. Such a hard core interaction provides a\npotential barrier between the solitonic crystal phase and\nthe normal quark matter phase. These considerations\nmodify the simple minded Maxwell construction above.\nWe would like to emphasize that the composite soli-\ntonic crystal state is a a new state of matter that is nei-\nther conventional quark matter nor conventional nuclear\nmatter. One notable di\u000berence is that whereas the con-\nventional quark matter state is a fermi liquid in the pres-\nence of a stationary wave neutral pion condensate, in our\ncomposite nucleon soliton crystal state, the quarks live\nin a special, frozen, fully occupied, relativistic, 0+, band\nup to the threshold baryon density.\nIf the maximum mass of the neutron star for a partic-\nular EOS occurs below this density then we can say that\nneutron stars exist entirely in the solitonic crystal phase,\nand become unstable even before the transition to quark\nmatter. On the other hand if the maximum mass for a\nparticular EOS occurs above this threshold central den-\nsity we conclude that matter is unstable to transiting to\nquark matter even before the maximum mass of the star\nin the nucleonic phase. In any case, conventional high\nmass quark matter stars (for example, MIT bag) [3, 4]\nare unstable and very unlikely to ever have an EOS that\ncan go up to a, Mmax\u00182, solar mass.\nFor example, a neutron star based on the APR [A18\n+ dv +UIX] [5] EOS has a maximum mass, Mmax\u0018\n2:2Msolar, that occurs at a central density slightly larger\nthannb\u00181=fm3. We expect the the star can be unstable\nand transit into quark matter even below this maximum\nmass. For a softer EOS, the maximum mass will come\ndown, but the transition density to quark matter will\ngo up and thus the star will most likely again become\nunstable even before the maximum mass is reached. .\nFor a sti\u000ber EOS, corresponding to APR [A18 + UIX],\nits maximum mass is slightly higher , Mmax\u00182:3Msolar,\nthan that of APR[A18 + dv +UIX], but at a central\ndensity which is lower than, nB\u00181=fm3. Thus the\nstar becomes unstable before the quark matter transition\ntakes place. For the unrealistic case of an EOS that is\nassumed to be incompressible at a density above 3 times\nnuclear saturation density ( see \fg. 14 ref.[5]) we can\nexpect a higher Mmaxbut not greater than \u00182:5Msolar.\nOur analysis indicates that all stable neutron stars re-\nmain in the nucleonic soliton state and that their Mmax\nwill not exceed\u00182:5Msolar. Secondly, when the mass10\nof a solitonic star (or coalesced stars) exceeds the max-\nimum mass or the central density exceeds the threshold\ndensity, which ever happens earlier, there will be a sud-\nden decompression ( or contraction ) transition to an\nunstable quark matter state with an abrupt change in\ndensity. The hybrid crossover models [4] do not have\nsuch a density 'discontinuity' as in these models nuclear\nand quark matter can co exist with a smooth journey to\nthe maximum mass.\nAfter the merger of two neutron stars the contraction\nof the core due to the sharp change in the compressibil-\nity of EOS at the threshold density of our model would\nresult in a di\u000berent post merger scenario, in contrast to\nhybrid crossover models where there is no such e\u000bect.\nThis may be observable. This abrupt change in density\ncould result in a shock wave and matter ejection. It is\nalso possible that fast rotating binary mergers support a\nmetastable, 'hypermassive', intermediate state. As has\nbeen outlined in some earlier work the passage to highmass stars is likely to produce magnetars [27] beyond a\ncertain mass which can carry very high magnetic \felds\nthat can catalyse the formation of jets in this event.\nIn our model if the mass of the merger of two neutron\nstars exceeds Mmax\u00182:5Msolar then the \fnal state will\nbe a black hole. Given that the observed merger resulted\nin a \fnal state that was, \u00182:7Msolar, either way the\nmerger will \fnally transit to a black hole. To conclude,\nin the alternative model we have presented, all neutron\nstars should have no regular `free'quark matter and that\nthe transition for neutron stars with masses over the max-\nimum mass, Mmax, or for central baryon density larger\nthan,nb\u00181=fm3, will become unstable and transit to\n`free'quark matter and then onto black holes .\nAcknowledgement: We are happy to acknowledge dis-\ncussions and a partial collaboration with Dipankar Bhat-\ntachrya, Pawel Haensel and Mitja Rosina. The author\nthanks the centre for Theoretical Physics, Jamia Millia\nIslamia and ICTP, Trieste for hospitality.\n[1] Paul Demorest, Tim Pennucci, Scott Ransom, Mallory\nRoberts, Jason Hessels Nature 467, (2010)1081-1083\n[2] J. Antoniadis P. C.C. Freire et al, arXiv:1304.6875 [astro-\nph.HE]\n[3] V. Soni and D. Bhattacharya, Phys. Lett. B 643(2006)\n158.\n[4] G. Baym, et al, Reports on Progress in Physics 81(5)\n(2018), arXiv:1707.04966v3 [astro-ph.HE] 2017\n[5] A. Akmal, V. R. Pandharipande and D. G. Ravenhall,\nPhys. Rev. C 58(1998) 1804.\n[6] J. M. Lattimer and M. Prakash, Astrophysical J. 550\n(2001) 426;\n[7] T. Klahn, R. astowiecki,and D. Baschke, Phys. Rev. D\n88,(2013) 085001\n[8] S. Benic et al. Astron.Astrophys. 577 (2015) A40\n[9] R. Lastowiecki, D. Blaschke, T. Fischer and T. Klhn\narXiv:1503.04832v1 (2015)\n[10] K. Masuda, T. Hatsuda, and T. Takatsuka, Prog. Theor.\nExp. Phys. (2012)\n[11] G. Baym, T. Hatsuda, M. Tachibana, N. Yamamoto, J.\nPhys. G35, 104021 (2008).\n[12] K. Fukushima, Phys. Lett. B 591, 277 (2004)\n[13] K. Yamazaki, T. Matsui, and G. Baym, Nucl. Phys. A\n933, (2015), 245[14] T. Hell, N. Kaiser, W. Weise, S. Schulte, B. R ottgers,\n'New Constraints from Neutron Stars'- indico.cern.ch\n[15] S. Fiorilla, N. Kaiser, W. Weise, Nucl.Phys.A880, 65\n(2012)\n[16] B. P. Abbott et al, Phys. Rev. Lett. 119, 161101:1-18\n(2017).\n[17] B. P. Abbott et al, Astrophys. J. Letters 848, (2017)\n[18] V. Soni and D. Bhattacharya, arXiv. hep-ph/0504041 v2.\n[19] S. Kahana, G. Ripka and V. Soni, Nuclear Physics A 415\n(1984) 351.\n[20] M. C. Birse and M. K. Banerjee, Phys. Lett. B 136(1984)\n284.\n[21] F. Dautry, and E. M. Nyman, Nucl. Phys. A319 (1979)\n323\n[22] M. Kutschera, W. Broniowski, and Kotlorz, A. 1990,\nNucl. Phys. A516, 566\n[23] B. Banerjee, N. Glendenning and V. Soni Physics Letters\nB, Volume\n[24] V. R. Pandharipande and R. A. Smith, Nuclear Physics\nA237 (1975) 507-532\n[25] G. Baym, Neutron Stars and the Properties of matter at\nHigh Density, Lecture Notes NBI and NORDITA (1977)\n[26] I. Klebanov, Nuclear Physics B 262 (1985) 133-143\n[27] V. Soni and N. D.Haridass, MNRAS 425 (2):(2012) 1558-\n1566."    },    {        "title": "1912.11131v1.Empirical_constraints_on_the_high_density_equation_of_state_from_multi_messenger_observables.pdf",        "content": "arXiv:1912.11131v1  [nucl-th]  23 Dec 2019Empirical constraints on the high-density equation of stat e\nfrom multi-messenger observables\nM´ arcio Ferreira,1,∗M. Fortin,2Tuhin Malik,3B. K. Agrawal,4,5and Constan¸ ca Providˆ encia1\n1CFisUC, Department of Physics, University of Coimbra, P-30 04 - 516 Coimbra, Portugal\n2N. Copernicus Astronomical Center, Polish Academy of Scien ce, Bartycka,18, 00-716 Warszawa, Poland\n3BITS-Pilani, Department of Physics, K.K. Birla Goa Campus, GOA - 403726, India\n4Saha Institute of Nuclear physics, Kolkata 700064, India\n5Homi Bhabha National Institute, Anushakti Nagar, Mumbai - 4 00094, India\n(Dated: December 25, 2019)\nWe search for possible correlations between neutron star ob servables and thermodynamic quanti-\nties thatcharacterize highdensitynuclearmatter. Wegene rate aset ofmodel-independentequations\nof state describing stellar matter from a Taylor expansion a round saturation density. Each equation\nof state which is a functional of the nuclear matter paramete rs is thermodynamically consistent,\ncausal and compatible with astrophysical observations. We find that the neutron star tidal de-\nformability and radius are strongly correlated with the pre ssure, the energy density and the sound\nvelocity at different densities. Similar correlations are a lso exhibited by a large set of mean-field\nmodels based on non-relativistic and relativistic nuclear energy density functionals. These model\nindependent correlations can be employed to constrain the e quation of state at different densities\nabove saturation from measurements of NS properties with mu lti-messenger observations. In par-\nticular, precise constraints on the radius of PSR J0030+045 1 thanks to NICER observations would\nallow to better infer the properties of matter around two tim es the nuclear saturation density.\nI. INTRODUCTION\nThe properties of the equation of state (EoS) of\nnuclear matter at supra-saturation densities, that is\nabove the nuclear saturation density n0∼0.16 fm−3,\nstill remain an open question in nuclearphysics. Neutron\nstars (NSs) are unique astrophysical objects through\nwhich the properties of super-dense neutron-rich nuclear\nmatter at zero temperature can be studied. Constraining\nthe EoS requires combining astrophysics and nuclear\nphysics. Astrophysical observations are important\nprobes for the dense nuclear matter properties. Several\nNSs with a mass about two-solar masses detected during\nthe last decade set quite stringent constraints on EoS of\nnuclear matter. The pulsar PSR J1614 −2230 is, among\nthe most massive observed pulsars, the one with the\nsmallest uncertainty on the mass M= 1.906±0.016M⊙\n[1–3] (masses are reported with 1 σerror-bars or equiva-\nlently 68.3% credibility intervals throughout this work).\nOther two pulsars with a mass above two solar masses\nare PSR J0348+0432 with M= 2.01±0.04M⊙[4] and\nthe recently detected MSP J0740+6620 with a mass\n2.14+0.10\n−0.09M⊙[5].\nDetecting gravitational waves (GWs) emitted during\nthe coalescence of binary NS systems is also one of the\nmost promising way to probe high density behavior of\nthe EoS for dense stellar matter. The analysis of the\ncompact binary inspiral event GW170817 has placed\nupper bounds on the NS combined dimensionless tidal\ndeformability [6]. Using a low-spin prior (consistent\n∗marcio.ferreira@uc.ptwith the observed NS population), the combined dimen-\nsionless tidal deformability of the two NSs that merged\nduring the event was determined to be ˜Λ≤800 with\n90% confidence. A follow up reanalysis [7] assuming the\nsame EoS for the two NSs and for a spin range consistent\nwith the one observed in Galactic binary NSs obtained\n˜Λ≤900 and the tidal deformability of a 1 .4 solar mass\nNS was estimated to be 70 <Λ1.4M⊙<580 at the\n90% level. The detection of GWs from the GW170817\nevent was followed by the electromagnetic counterpart,\nthe gamma-ray burst (GRB) GRB170817A [8], and the\nelectromagnetic transient AT2017gfo [9], that set extra\nconstraints on the lower limit of the tidal deformability\n[10–14]. This last constraint seems to rule out very\nsoft EoS: the lower limit of the tidal deformability of a\n1.37M⊙star set by the above studies limits the tidal\ndeformability to Λ 1.37M⊙>210 [12], 300 [11], 279 [13],\nand 309 [14].\nIn Ref. [15] Lattimer and Prakash have empirically\nobserved that for densities between 1.5 n0and 2−3n0,\nthe radius of the star scales with p1/4withpthe pressure\nat these densities. This means that knowing the radius\nof a NS with sufficient precision will constrain the EoS of\nstellar matter in this specific range of densities. It also\nraises the question whether other correlations between\nthe thermodynamic properties of nuclear matter in\nβ-equilibrium and NS properties such as the radius\nor tidal deformability could exist. In this case further\nconstraints on the NS EoS could be obtained thanks\nto new measurements of the NS tidal deformability\nwith future LIGO/Virgo detection of gravitational\nwaves emitted from binary NS mergers. The precise\ndetermination of the radius of NSs, in addition to their\nmass, expected from the currently-operating NICER2\nmission [16], and future X-ray observatories like the\nAthena X-ray telescope [17] and eXTP [18] would also\nallow to constrain the EoS in various ranges of density.\nNS properties such as the radius and mass can be\nobtained by solving the Tolman-Oppenheimer-Volkoff\n(TOV) equations [19, 20] for a static and spherical star\nin hydrostatic equilibrium, which requires the EoS as a\ninput. As a consequence a one-to-one correspondence\nis established between the NS mass and radius and the\nEoS ofβ-equilibrated stellar matter. The possibility of\ninverting this mapping, allowing the determination of\nthe EoS from the measurement of the mass and radius\nof a large number of stars was discussed by Lindblom\n[21]. Later, it was proposed that a smaller number of\nastrophysical observations is required if realistic EoS are\nparametrized using piecewise polytropes with transition\ndensities from one polytrope to another chosen at\nwell selected densities [22]. A different approach was\ndiscussed in [23], where it was shown that the determi-\nnation of the pressure at three fiducial densities could\nbe obtainedfrom the measurementofthreedifferent NSs.\nCorrelations between nuclear matter parameters\nand NS properties have been explored using several\nnuclear models [24–32]. These studies, however, show\na considerable model dependence since different models\nwith similar values of the nuclear matter parameters\nmay result in different EoSs. In the present work, we\nstudy the correlationof various astrophysicalobservables\ndirectly with the thermodynamical variables of the EoS\nto avoid the model dependence [33]. The main objective\nof the present work is to look for further correlations\nthat could allow to establish constraints on the EoS of\nnuclear matter from the observation of NS. We use a\nlarge set of so-called meta-models that satisfy a given\nnumber of well defined nuclear matter and NS properties\n[34, 35].\nThe paper is organized as follows. In Sec. II, we intro-\nduce the EoS parametrization and generating process for\nthe meta-models. The correlation analysis on the gen-\nerated set of EoS is developed in Sec. III. Finally, the\nconclusions are drawn in Sec IV.\nII. EOS PARAMETRIZATION\nWe start from the generic functional form for the en-\nergy per particle of homogeneous nuclear matter\nE(n,δ) =e0(n)+esym(n)δ2(1)\nwheren=nn+npis the baryonic density and δ=\n(nn−np)/nis the asymmetry with nnandnpbeing\nthe neutron and proton densities, respectively. This\napproach has been applied recently in several works,\n[34, 36, 37]. We consider a Taylor expansion of this en-\nergy functional around the saturation density nsatuntilfourth order as in [34, 36]:\ne0(n) =Esat+1\n2Ksatx2+1\n6Qsatx3+1\n24Zsatx4(2)\nesym(n) =Esym+Lsymx+1\n2Ksymx2+1\n6Qsymx3(3)\n+1\n24Zsymx4\nwherexis defined as x= (n−nsat)/(3nsat). The em-\npirical parameters can be identified as the coefficients of\nthe expansion. The isoscalar empirical parameters are\ndefined as proportional to successive density derivatives\nofe0(n),\nP(k)\nIS= (3nsat)k∂ke0(n)\n∂nk/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n{δ=0,n=nsat},(4)\nwhereasthe isovectorparametersmeasuredensityderiva-\ntives ofesym(n),\nP(k)\nIV= (3nsat)k∂kesym(n)\n∂nk/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n{δ=0,n=nsat}.(5)\nThe corresponding empirical parameters are then\n{Esat,Ksat,Qsat,Zsat} → {P(0)\nIS,P(2)\nIS,P(3)\nIS,P(4)\nIS}(6)\nand\n{Esym,Lsym,Ksym,Qsym,Zsym}\n→ {P(0)\nIV,P(1)\nIV,P(2)\nIV,P(3)\nIV,P(4)\nIV}.\nThe coefficients of low orders are already quite\nwell constrained experimentally [38–43], however\nQsat, ZsatandKsym, Qsym, Zsymare only poorly\nknown [30, 32, 36, 44–47]. The saturation energy\nEsatand saturation density nsatbeing rather well\nconstrained, we fix their values throughout this work:\nEsat=−15.8 MeV (the current estimated value is\n−15.8±0.3 MeV [34]), and nsat= 0.155 fm−3.\nWith this approach, each meta-model is represented\nby a point in the 8-dimensional space of parameters. In-\nstead of analyzing the models on a fixed grid, we will\nemploy random sampling of models through a multivari-\nate Gaussian with zero covariance:\nEoSi={Esym,Lsym,Ksat,Ksym,Qsat,Qsym,Zsat,Zsym}i\n∼N(µ,Σ)\nwhere the mean value vector and covariance matrix are,\nrespectively,\nµT= (Esym,Lsym,Ksat,Ksym,Qsat,Qsym,Zsat,Zsym)\nand\nΣ=diag(σEsym,...,σZsym).3\nIn the present approach, as discussed in [34], no\na-priori correlations exist between the different param-\neters of the EoS. However, as we will see, imposing\nexperimental and observational constraints will give\nrise to correlations. The physical correlations among\nthe empirical parameters arise from a set of physical\nconstraints [34, 36]. The parameters of the Gaussian\ndistributions for each parameter are in Table I.\nPiInitial dist. Final dist.\nPi√σPiPi√σPi\nKsat 230 20 233.35 18.24\nQsat 300 400 56.04 122.31\nZsat-500 1000 -178.46 141.26\nEsym 32 2 33.33 1.89\nLsym 60 15 51.45 11.83\nKsym -100 100 -44.24 63.24\nQsym 0 400 237.52 299.42\nZsym -500 1000 372.98 698.72\nTable I. The mean Piand standard deviation√σPiof the\nmultivariate Gaussian, where σPiis the variance of the pa-\nrameter Pi. Our EoSs are sampled using the initial distribu-\ntion forPiassuming that there are no correlations among the\nparameters. The final distribution for Piare obtained after\nimposing the filters as listed in the text. All the quantities\nare in units of MeV. The values of Esatandnsatare fixed to\n−15.8 MeV and 0 .155 fm−3, respectively.\nWe impose the following conditions to get a valid EoS:\ni) be monotonically increasing (thermodynamic stabil-\nity); ii) the speed of sound must not exceed the speed of\nlight (causality); iii) supports a maximum mass at least\nas high as 1 .97M⊙[1–4] (observational constraint); iv)\npredicts a tidal deformability of 70 <Λ1.4M⊙<580 [7]\n(observational constraint); and v) the symmetry energy\nesym(n) is positive. All the EoS are in β-equilibrium. We\nuse the SLy4 EoS for the low density region [48]. A valid\nEoS must cross the SLy4 EoS in the P(µ) plane below\nn <0.10 fm−3consistently with the range of core-crust\ntransition densities for a large set of nuclear models [27].\nThe SLy4 EoS is matched with the generated EoSs by\nrequiring PSLy4(µ) =PEoS(µ) withµthe chemical po-\ntential.\nIII. RESULTS\nIn the present section we first discuss the properties\nof the set of EoS we have built after imposing the con-\nstraints listed above. Using these EoSs, we then study\ncorrelations between NS observables and the EoS prop-\nerties at given densities.A. Empirical parameters values and NS properties\nAfter applying all the filters indicated above to 107\nsampled EoS, we obtain 2121 valid EoS. This number\nis quite small and it is mainly due to the constraint of\ncausality and the requirement that the generated EoS\nand the crust SLy4 EoS intersect in the P−µplane.\nUsing a simple interpolation between the crust and the\ncore at some specific density is much less restrictive\nand the number of valid EoS would be much larger.\nHowever, we consider it is important to carry the\ninformation contained on the Taylor expansion not only\nto supra-saturation densities but also to sub-saturation\ndensities.\nIn Table I, the mean values and standard deviations\nof the EoS parameters for the final distribution, after\nthe constraints on the EoS were imposed, are compared\nwiththerespectiveinitialinput. Itisinterestingtonotice\nthatwellconstrainedparameterslike KsatandEsym, and\nevenLsym, do not change much from the initial distribu-\ntion, while the parameters connected to the high orders,\nsuch asQsymandZsym, convergeto quite different mean\nvalues.\nWith this set of EoS, we obtain the relation between\nthe radius Rand the mass Mof the NS, solving the TOV\nequations [19, 20], and calculate the dimensionless tidal\ndeformability Λ\nΛ =2\n3k2/parenleftbiggR\nM/parenrightbigg5\n, (7)\nwherek2its quadrupole tidal Love number, following\nRef. [49].\nIn Fig. 1, we plot the M−Rand the Λ −Mrelations\nfor the set of EoS. In what follows, these results will be\nused to study the correlations between NS observables\nand thermodynamic quantities. As an example, we\npresent in Table II the mean value and standard devi-\nation for the tidal deformability Λ Miand radius RMi\nof stars with masses Mi= 1.0,1.2,1.4,1.6,1.8M⊙.\nThe results obtained for M1.4are well inside the limits\nimposed by GW170817 [7] for the tidal deformability\n70<Λ1.4M⊙<580 (but notice that this is also true\nwithout imposing maximum star mass of 1.97 M⊙) and\nR= 11.9±1.4 km. On the other hand, if the constraints\nset by the electromagnetic counterpart are also consid-\nered then our EoSs satisfies the lower limit determined\nin [12], Λ 1.37M⊙>210, but not the limit calculated in\n[11, 13, 14]: 300, 279 and 309 respectively. However, in\naverage and within a 95% confidence interval these lower\nconstraints are all satisfied. The set of EoSs also satisfies\nthe condition obtained for R1.6M⊙from an existing\nuniversal relation between the critical merger remnant\nmass to a prompt collapse and the compactness of the\nmaximum mass star, i.e., R1.6M⊙/greaterorsimilar10.7 km [50, 51].\nThe obtained results are also in agreement with4\n[52], where the maximum value R1.4M⊙= 13.6 km\nand the minimum value Λ 1.4M⊙= 120 were reported,\nusing a generic family of EoS that interpolate between\nchiral effective field theory results at low densities\nand perturbative QCD at high densities. Further-\nmore, our results are compatible with [53] (an extra\ncondition on the allowed maximum NS mass was\nimposed, Mmax<2.16M⊙, though), in which a mean\nvalue of R1.4M⊙= 12.39 km and a 2 σconfidence of\n12.00< R1.4M⊙/km<13.45 were determined using a\npiecewise polytrope parametrization of the EoS, which\ntook into account nuclear matter calculations of the\nouter crust, near saturation densities, and perturbative\nQCD.\nmean std min max\nΛ1.0M⊙2967.88 283.78 1677.51 3597.70\nΛ1.2M⊙1129.15 118.90 620.60 1377.93\nΛ1.4M⊙467.53 57.89 243.53 579.93\nΛ1.6M⊙201.96 31.36 93.31 267.29\nΛ1.8M⊙87.54 18.48 29.41 126.26\nR1.0M⊙11.96 0.19 10.99 12.34\nR1.2M⊙12.09 0.19 11.07 12.48\nR1.4M⊙12.18 0.21 11.13 12.59\nR1.6M⊙12.20 0.25 11.09 12.68\nR1.8M⊙12.14 0.31 10.85 12.73\nTable II. Sample statistics for Λ MiandRMi(km): mean,\nstandard deviation, maximum, and minimum values.\nInterestingly, the minimum value obtained for Λ 1.4M⊙\nis 243.53 and, furthermore, only a very small percentage\nof the EoS failed to reproduce Λ 1.4M⊙<580 (the max-\nimum value reached for Λ 1.4M⊙was 651.85). In other\nwords, the present set of EoS describes NSs with a nar-\nrow region of Λ 1.4M⊙with a mean value of 467.53, and\nfulfill 70 <Λ1.4M⊙<580 [7].\nB. Correlation between thermodynamic quantities\nand NS observables\nIn the present section, we study the possible existing\ncorrelations between the thermodynamic properties of\ndensestellarmatterin β-equilibrium andNS observables.\nIn the following analysis we use the Pearson correlation\ncoefficient\nCorr[X,Y] =/an}bracketle{t(X−µX)(Y−µY)/an}bracketri}ht\nσXσY,\nwhere/an}bracketle{t.../an}bracketri}htistheexpectationvalueand σXandσYarethe\nstandard deviations of variables XandY, respectively.\nIn particular, we consider the correlation between the\npressure in Fig. 2, the energy density in Fig. 3, and the012\n10 11 12 13 14\nR [km]M [M⊙]\n101001000\n1.0 1.5 2.0 2.5\nM [M⊙]Λ\nFigure 1. Mass vs. radius (left) and the mass vs. tidal de-\nformability (right) diagrams for set of EoS built in the pres ent\nstudy.\nspeed of sound in Fig. 4, at each baryonic density with\nthe radius, tidal deformability, and Love number of NSs\nwithM= 1.0,1.2,1.4,1.6 and 1.8M⊙.\n0.00.51.0\n0.2 0.4 0.6\nn [fm−3]Corr[P(n),R M]\n0.00.51.0\n0.2 0.4 0.6\nn [fm−3]Corr[P(n), ΛM]\n1.0M⊙\n1.2M⊙\n1.4M⊙\n1.6M⊙\n1.8M⊙0.00.51.0\n0.2 0.4 0.6\nn [fm−3]Corr[P(n),k 2]\nFigure 2. The density dependence of correlation coefficient o f\nthe pressure, P(n) with radius R(left), tidal deformability Λ\n(middle), and Love number k2(right) for different NS masses\nas indicated obtained for Meta models.\nWe first discuss the correlation of the NS properties\nwith the pressure as shown in Fig. 2. Interestingly one\ncan identify strong correlations between the pressure\nand the various NS properties that we considered, at\ncertain densities. The left panel, for example, shows how\nthe correlation between P(n) andRMremains quite\nhigh in a relatively small range of densities for all the\nfive masses considered. The maximum of the correlation\nshifts to larger densities, from n≈0.25 to 0.35 fm−3,\nas the mass of the star increases. This is precisely the\nempirical correlation identified by Lattimer and Prakash\nin [15]. A similar and an even stronger correlation, with\na coefficient very close to 1, is observed between the\npressure and the tidal deformability in the same range\nof densities (center panel). The Love number k2shows\na quite strong correlation at n≈0.45 fm−3but only for5\nthe larger masses (right panel).\n0.00.51.0\n0.2 0.4 0.6\nn [fm−3]Corr[E(n),R M]\n0.00.51.0\n0.2 0.4 0.6\nn [fm−3]Corr[E(n), ΛM]\n1.0M⊙\n1.2M⊙\n1.4M⊙\n1.6M⊙\n1.8M⊙0.00.51.0\n0.2 0.4 0.6\nn [fm−3]Corr[E(n),k 2]\nFigure 3. The density dependence of correlation coefficient o f\nthe energy density, E(n), with radius R(left), tidal deforma-\nbility Λ (middle), and Love number k2(right) for different NS\nmasses as indicated obtained for Meta models.\nThe correlations between NS observables and the en-\nergy density are shown in Fig. 3. Again, strong corre-\nlations are identified with all the NS observables but at\nlarger densities, n≈0.32 to 0.5 fm−3with the smaller\n(larger) masses closer to the smaller (larger) value. Sim-\nilar to the case of pressure, the correlation between the\nenergy density and Λ remains strong for all NS masses.\nThe maximum correlation is obtained at higher densities\nwhen massive NS are considered.\nFinally, for the speed of sound shown in Fig. 4 strong\ncorrelations are again present but occurring at smaller\ndensities than the ones obtained for the pressure, i.e. at\ndensities ≈0.2−0.3 fm−3.\nA strong correlation, i.e., Corr ≈1, means that the\nsample variance of the NS observable is almost entirely\nlinearly explained by the variance of the thermodynami-\ncal quantity. Therefore, one can use the NS observable\nas a way to constrain the thermodynamical quantities\nat different baryon densities.\nIt is interesting to compare our results, obtained from\na Taylor expansion parametrized around saturation den-\nsity, with nuclearmodels EoS.We useadatasetofunified\nEOS based on 24 Skyrme interactions and 26 relativis-\ntic mean-field nuclear parametrizations (some of them\nincluding a transition to hyperonic matter at high den-\nsity) [54]. The constraints 70 <Λ1.4M⊙<580 and\nMmax>1.97M⊙are fulfilled by the following models:\nBSk20, BSk21 [55], BSk25, BSk26 [56], SKa, SKb [57],\nSkI4 [58], SkI6 [59], SkMP [60], SKOp [61], SLy2, SLy9\n[62], SLy230a [63], SLy4 [64]. In Fig. 5 we show the\nvariation with the density of the correlation coefficient\nbetween P,E, andvsand the NS properties R, Λ and\nk2for different masses. When we compare the pres-\nsure correlations in Fig. 2 and in top panel of Fig. 5,\nwe notice that the main difference occurs at high den-\nsitiesn >0.6 fm−3, where the nuclear models show a0.00.51.0\n0.2 0.4 0.6\nn [fm−3]Corr[v s(n),R M]\n0.00.51.0\n0.2 0.4 0.6\nn [fm−3]Corr[v s(n),ΛM]\n1.0M⊙\n1.2M⊙\n1.4M⊙\n1.6M⊙\n1.8M⊙0.00.51.0\n0.2 0.4 0.6\nn [fm−3]Corr[v s(n),k 2]\nFigure 4. The density dependence of correlation coefficient o f\nthe sound velocity, vs(n), with radius R(left), tidal deforma-\nbility Λ (middle), and Love number k2(right) for different NS\nmasses as indicated obtained for Meta models.\nstronger correlation with all NS observables. We indi-\ncate in Table III the maximum correlation obtained for\nM= 1.4M⊙and the densities at which they occur for\nthe nuclear models. For reference, the correlation coef-\nficient at the same densities obtained for our EoSs from\nconstrained meta-models are also given. We see that the\nmaximum correlations happen at nearly the same densi-\nties. The correlations of NS properties with the energy\ndensity obtained for our set of EoSs from meta-models\n(Fig 3) and those for nuclear models (middle panel of\nFig. 5) show overall similar trends. The fact that for our\nEoSs, the correlations are more suppressed at high den-\nsities may be due to the differences in the high density\nbehavior.\nModels Nuclear Meta\nn corr corr\nP(n) Λ 1.4M⊙0.320 0.992 0.891\nP(n)R1.4M⊙0.281 0.980 0.937\ne(n) Λ 1.4M⊙0.544 0.978 0.914\ne(n)R1.4M⊙0.442 0.995 0.974\nvs(n) Λ 1.4M⊙0.219 0.987 0.870\nvs(n)R1.4M⊙0.195 0.964 0.844\nTable III. Densities of maximum correlations for the nuclea r\nmodels and the correlation value at those densities for our s et\nobtained from meta-models (last column).\nC. Constraining the thermodynamical properties\nIn the previous section, we saw that all thermody-\nnamical quantities show strong correlations with NS\nobservables, specially the radius and the tidal deforma-\nbility, in some specific range of density. We now study\nhow to use these linear dependences to constrain the\nthermodynamical quantities from future measurements6\n0.00.51.0\n0.2 0.4 0.6\nn [fm−3]Corr[P(n),R M]\n0.00.51.0\n0.2 0.4 0.6\nn [fm−3]Corr[P(n), ΛM]\n1.0M⊙\n1.2M⊙\n1.4M⊙\n1.6M⊙\n1.8M⊙ 0.00.51.0\n0.2 0.4 0.6\nn [fm−3]Corr[P(n),k 2]\n0.00.51.0\n0.2 0.4 0.6\nn [fm−3]Corr[E(n),R M]\n0.00.51.0\n0.2 0.4 0.6\nn [fm−3]Corr[E(n), ΛM]\n0.00.51.0\n0.2 0.4 0.6\nn [fm−3]Corr[E(n),k 2]\n0.00.51.0\n0.2 0.4 0.6\nn [fm−3]Corr[v s(n),R M]\n0.00.51.0\n0.2 0.4 0.6\nn [fm−3]Corr[v s(n),ΛM]\n0.00.51.0\n0.2 0.4 0.6\nn [fm−3]Corr[v s(n),k 2]\nFigure 5. The density dependence of correlation coefficient\nof the pressure P(n) (top row), energy density E(n) (middle\nrow), and sound velocity vs(n) (bottom row) with radius R\n(left), tidal deformability Λ (middle), and Love number k2\n(right) for different NS masses as indicated obtained for a\ndiverse set of nuclear models.\nof the radius and the tidal deformability for a canonical\nNS,M= 1.4M⊙.\nFig. 6 shows the density at which the correlation is\nmaximumbetween both the tidaldeformability(top pan-\nels) and radius (bottom panels) with the energy density\n(left), pressure (middle), and sound velocity (right) of a\n1.4M⊙star. The regression analysis is summarized in\nTable IV for three NS masses: 1 .0M⊙, 1.4M⊙(shown\nin Fig. 6), and 1 .8M⊙. Recently, the first simultaneous\ndetermination of the radius and mass of a NS with the\nNICER mission was obtained after the modeling the pul-\nsating X-ray emission from the isolated millisecond pul-\nsar PSR J0030+0451: R∼11.5−14 km for M∼1.4M⊙\n[65–67]. If a more precise radius measurement becomes\navailable, one wouldthen be ableto immediately get con-\nstraints on the EoS at three different densities: the en-\nergy density at 0.413 fm−3, the pressure at 0.311 fm−3\nandthe soundvelocityat0.250fm−3. Similarconstraints\ncould also be derived from the observation of another\nNS, one of the primary targets PSR J0437 −4715 with\na mass 1.44 ±0.07M⊙[16, 68]. If besides also informa-\ntion on the radiusofthe massivepulsarPSR J1614 −2230\n(M= 1.908±0.018M⊙) would be measured we would be\nable to get further constraints on the EoS for other dif-ferent densities.\nFrom Table IV, using the maximum and the mini-\nmum Λ 1.4M⊙calculated from our EoS set (see Table II),\nwe constrain the thermodynamical quantities at different\ndensities\nP(n= 0.340 fm−3) = [17.82,34.56] MeV/fm3(8)\nE(n= 0.456 fm−3) = [452.22,468.22] MeV/fm3(9)\nvs(n= 0.266 fm−3) = [0.325,0.446] units of c .(10)\nDoing the same analysis for R1.4M⊙, using our sample\nmax/min values, we get\nP(n= 0.311 fm−3) = [11.66,25.27] MeV/fm3(11)\nE(n= 0.413 fm−3) = [403.98,416.78] MeV/fm3(12)\nvs(n= 0.242 fm−3) = [0.278,0.395] units of c .(13)\nThese results are summarized in Table V. We also in-\nclude, in the same Table, the constraints on P,E, and\nvsobtained from the LIGO/Virgo GW170817 analysis\n[7]. Confidence intervals (on the marginalized posterior)\nfor the pressure were determined in [7]. The 90% credi-\nble level is P(n= 0.311 fm−3) = [8.79,31.09] MeV/fm3\nandP(n= 0.340 fm−3) = [12 .01,41.39] MeV/fm3,\nwhile the 50% credible level is P(n= 0.311 fm−3) =\n[11.95,23.66] MeV/fm3andP(n= 0.340 fm−3) =\n[16.37,32.00] MeV/fm3. Our results, Eqs. (8) and (11),\nare in good agreement even at the 50% credible level.\nWe should note, however, that the EoSs used in [7]\nhave lower Λ 1.4M⊙values, which are are not represented\nin our dataset. Assuming that the correlations that\nwe have obtained are model-independent, as the anal-\nysis of the nuclear models seems to suggest, one may\nuse the regression analysis in Table IV to constrain the\npressure at any given value of Λ 1.4M⊙(the same ap-\nplies to R1.4M⊙). We get for 70 <Λ1.4M⊙<580,\nP(0.340 fm−3) = [9.19,34.56] MeV/fm3, which is close\ntotheintervaldeterminedin[7]. Furthermore,additional\nconstraints on the energy density and sound velocity can\nbe determined from 70 <Λ1.4M⊙<580. They are listed\nin Table V.\nIn Fig. 7 we show the constraints on the pressure at\ndifferent densities which one can obtain from measure-\nments of the radius and tidal deformability for NSs of\nvarious masses. For the radius, we adopt the masses of\nM= 1.3,1.4and1.5M⊙, whichcoversthe rangeofvalues\nfor most observed NSs in particular the NICER targets\nPSR J0030+0451 and PSR J0437 −4715 and M= 1.8\nand 1.9M⊙to explore the consequence of the radius de-\ntermination of a massive NS such as PSR J1614 −2230.\nFor the tidal deformability we restrict ourselves to M=\n1.3 and 1.4M⊙as this corresponds to the mass range of\nNSs currently observed in a binary with another NS [69].\nThe vertical error bars (hardly visible) for constraints\nfrom the tidal deformabilties account for the errors in\nthe determination ofthe slope and the interception in the\nlinear regression with the pressure (parameters mandb\nin Table IV). For medium-range masses M∼1.4M⊙the\nfuture determination of the tidal deformability will allow7\ncorr=0.97\n455460465470\n300 400 500\nΛ1.4M⊙Energy [MeV/fm3]\n3]corr=0.98\n20253035\n300 400 500\nΛ1.4M⊙Pressure [MeV/fm3]\nn=0.268 [fm ]corr=0.95\n0.300.350.400.45\n300 400 500\nΛ1.4M⊙Sound velocity [units of c]\n3]\n404408412416\n11.5 12.0 12.5\nR1.4M⊙ [km]Energy [MeV/fm3]\ncorr=0.96\n152025\n11.5 12.0 12.5\nR1.4M⊙ [km]Pressure [MeV/fm3]\n0.280.320.360.40\n11.5 12.0 12.5\nR1.4M⊙ [km]Sound velocity [units of c]\nFigure 6. Λ 1.4M⊙(top) and R1.4M⊙(bottom) as a function of the energy density (left), the pres sure (middle) and the speed of\nsound (right) for all meta models at the densities correspon ding to the maximum correlations. These densities and corre lation\ncoefficient are indicated in each of the panels.\nn Q Z Corr[Q,Z] m b\n0.275\nP(n)Λ1.0M⊙ 0.984 (4 .453±0.017)×10−30.827±0.052\n0.340 Λ 1.4M⊙ 0.982 (49 .389±0.209)×10−35.898±0.098\n0.430 Λ 1.8M⊙ 0.970 (43 .10±0.23)×10−224.78±0.21\n0.254 R1.0M⊙ 0.943 (503 .89±3.86)×10−2−49.04±0.46\n0.311 R1.4M⊙ 0.962 (942 .68±5.85)×10−2−91.11±0.70\n0.403 R1.8M⊙ 0.939 (1885 .9±15.0)×10−2−171.6±1.8\n0.377\nE(n)Λ1.0M⊙ 0.955 (4 .4±0.0)×10−3359.8±0.1\n0.456 Λ 1.4M⊙ 0.973 (47 .4±0.2)×10−3440.9±0.1\n0.564 Λ 1.8M⊙ 0.973 (374 .7±1.9)×10−3564.8±0.2\n0.333 R1.0M⊙ 0.992 (4583 .9±12.3)×10−3271.3±0.1\n0.413 R1.4M⊙ 0.984 (8823 .4±34.4)×10−3306.8±0.4\n0.529 R1.8M⊙ 0.960 (1628 .0±10.3)×10−2354.5±1.3\n0.209\nvs(n)Λ1.0M⊙ 0.960 (4 .908±0.031)×10−50.160±0.001\n0.268 Λ 1.4M⊙ 0.953 (0 .361±0.002)×10−30.239±0.001\n0.346 Λ 1.8M⊙ 0.934 (1 .918±0.016)×10−30.360±0.001\n0.192 R1.0M⊙ 0.921 (66 .371±0.610)×10−3−0.507±0.007\n0.242 R1.4M⊙ 0.930 (80 .609±0.694)×10−3−0.604±0.008\n0.323 R1.8M⊙ 0.898 (93 .086±0.992)×10−3−0.620±0.012\nTable IV. Maximum correlations, Corr[ Q,Z], between the thermodynamic properties, Q={P(n),E(n),vs(n)}, and the NS\nproperties, Z={ΛMi,RMi}. We show the linear regression analysis for Q=m×Z+bat a fixed density n. In the table bhas\nthe units of Q(MeV/fm3forPandEandcforvs) andmhas the units of Z/Q, i.eQ−1forZ= Λ and km/Q−1forZ=R.\nto constrain the pressure at densities ∼10% larger than\nthat for the radius. In addition, measuring the radius ofa massive NS would allow us to put limits at the pres-\nsure for larger densities. Thus observational constraints8\nBoundsP(0.340)\n[MeV fm−3]E(0.456)\n[MeV fm−3]vs(0.268)\n[c]\nmin max min max min max\nΛ1.4M⊙243.53 – 579.93 17.82 34.56 452.22 468.22 0.325 0.446\n70 – 580 [7] 9.19 34.56 443.96 468.23 0.262 0.446\nBoundsP(0.311)\n[MeV fm−3]E(0.413)\n[MeV fm−3]vs(0.242)\n[c]\nmin max min max min max\nR1.4M⊙11.13 – 12.59 11.66 25.27 403.98 416.78 0.278 0.395\n10.5 – 13.3 [7] 5.74 31.93 398.40 423.05 0.227 0.453\nTable V. Constraints on pressure P(n), energy density E(n) and sound velocity vs(n) forβ- equilibrium matter at density\nn(fm−3) obtained from bounds on tidal deformability and radius for neutron star with canonical mass 1 .4M⊙using our EoSs.\nConstraints obtained from the LIGO/Virgo analysis are also shown (see text).\non NS properties from multi-messenger astrophysics will\nenableustoinferthepropertiesofNSmatterintherange\n1−3n0which we cannot probein terrestriallaboratories.\nFigure 7. Constraints on the pressure obtained at different\ndensities (expressed in units of the nuclear saturation den -\nsity) from measurements of the radius (full symbols) and tid al\ndeformabilities (empty symbols) of NSs of different masses\n(shown by different types of symbols). The outer (inner) gray\nregion is the 90% credible level (50% credible level) from [7 ].\nSee text for details.\nIV. CONCLUSIONS\nWe found strong correlations between the thermody-\nnamical properties, i.e., the pressure, energy density and\nsound velocity with the radius and the tidal deformabil-\nity of NS over a wide range of masses. These correlations\nwere obtained from a set of EoS parametrized by a Tay-\nlor expansion around the saturation density. Similar cor-\nrelations are confirmed using EoS obtained from nuclearmodels, indicatingthattheyaremodel-independent. Itis\nshown that for a given NS mass, there is alwaysa density\nwhere the tidal deformability and the radius are highly\ncorrelated with the pressure, energy density, and speed\nof sound. A single determination of the tidal deforma-\nbility (Λ Mi) or radius ( RMi) of a NS of mass Miallows\nto constrain the thermodynamic properties at three dis-\ntinct but close densities. For a 1 .4M⊙NS, the pressure\ncould be constrained at n≈2n0, the energy density at\nn≈2.7n0, and the sound velocity at n≈1.7n0.\nWe showthatthe radiusand tidaldeformabilityofNSs\nfor different masses are strongly correlated with thermo-\ndynamic variables of EoS at supra-saturation densities\nin the range of n≈1−3n0. The precise measurement\nof the radius of the pulsars PSR J0030+0451 and PSR\nJ0437−4715withamass ∼1.4M⊙bytheNICERmission\nwould allow us to get immediately information on the\nEoS at three different densities, the pressure at ∼2n0,\nthe energy density at ∼2.5n0, and the sound velocity\nat∼1.5n0. Complementing this information with the\nfurther radius measurement, for example, of PSR J1614-\n2230 with M≃1.9M⊙and constraints on the tidal de-\nformability of various NSs from the observations of GW\nfrommergingbinary NS systems wouldset verystringent\nconstraints on the EoS at supra-saturation densities. 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Gerasimenko∗\nInstitute of mathematics of the NAS of Ukraine\nKyiv, Ukraine\nJanuary 7, 2020\nAbstract\nIn this survey the possible approaches to the description of the evolution of states of\nquantum many-particle systems by means of the possible modi fications of the density\noperator which kernel known as density matrix are considere d. In addition, an ap-\nproach to the description of the evolution of states by means of the state of a typical\nparticle of a quantum system of many particles is discussed, or in other words, the\nfoundations of describing the evolution of states by kineti c equations are considered.\nPACS 03.65.-w, 05.30.d, 05.20.Dd.\nKeywords: density operator (matrix), correlation operator, reduced density operator,\nvon Neumann equation, von Neumann hierarchy, BBGKY hierarc hy, kinetic equation.\nContents\n1. Introduction 2\n2. The density operator 2\n3. Cluster expansions of the density operator 4\n4. Reduced density operators 5\n5. Reduced correlation operators 7\n6. On the description of the evolution of states by the one-pa rticle correlation\noperator 10\n7. On the scaling limits of reduced density operators 11\n8. Conclusion 13\nReferences 13\n∗E-mail address: gerasym@imath.kiev.ua\n12 V . I. Gerasimenko\n1. Introduction\nThe paper deals with the mathematical problems of describin g the evolution of states of\nquantum many-particle systems by means of operators genera ted by the density operator\nwhich kernel is known as a density matrix.\nAs known, a quantum system is described in terms of such notio ns as an observable\nand a state. The functional for the mean value of observables determines a duality between\nobservables and states. In a consequence of this there exist two approaches to the descrip-\ntion of the evolution of a quantum system of finitely many part icles, namely, in terms of\nobservables that are governed by the Heisenberg equation, o r in terms of states governed by\nthe von Neumann equation for the density operator, respecti vely [1],[2],[3].\nAn alternative approach to the description of states of a qua ntum system of finitely many\nparticles is given by means of operators determined by the cl uster expansions of the density\noperator. They are interpreted as correlation operators. T he evolution of such operators is\ngoverned by the von Neumann hierarchy [3],[4],[5].\nOne more approach to describing a state of many-particle sys tems is to describe a state\nby means of a sequence of so-called reduced density operator s (marginal density operators)\ngoverned by the BBGKY (Bogolyubov–Born–Green–Kirkwood–Y von) hierarchy [6]. An\nalternative approach to such a description of a state is base d on operators determined by the\ncluster expansions of reduced density operators. These ope rators are interpreted as reduced\ncorrelation operators that are governed by the hierarchy of nonlinear evolution equations\n[3]. On a microscopic scale, the macroscopic characteristi cs of fluctuations of observables\nare directly determined by the reduced correlation operato rs. The mention approaches are\nallowed to describe the evolution of states of quantum syste ms both with a finite and infinite\nnumber of particles, in particular, systems in condensed st ates [7].\nIn addition, an approach to the description of the evolution of states by means of the\nstate of a typical particle of a quantum system of many partic les is discussed, or in other\nwords, the foundations of describing the evolution of state s by kinetic equations are consid-\nered [8].\nHereinafter we denote the n-particle Hilbert space which is a tensor product of nHilbert\nspacesHby theHn=H⊗nand we use the usual convention that H⊗0=C. The Fock\nspace over the Hilbert space Hwe denote by the FH=/circleplustext∞\nn=0Hn. The self-adjoint op-\neratorfndefined in the space Hn=H⊗nwill also be denoted by the following symbol\nfn(1,...,n).\nLetL(Hn)be the space of bounded operators fn≡fn(1,...,n)∈L(Hn)equipped\nwith the operator norm /bardbl./bardblL(Hn). Accordingly, let L1(Hn)be the space of trace class\noperatorsfn≡fn(1,...,n)∈L1(Hn)equipped with the norm: /bardblfn/bardblL1(Hn)=\nTr1,...,n|fn(1,...,n)|,whereTr1,...,n are partial traces over 1,...,n particles. Below we\ndenote by L1\n0(Hn)the everywhere dense set of finite sequences of degenerate op erators\nwith infinitely differentiable kernels with compact suppor ts.\n2. The density operator\nFor generality, we consider a quantum system of non-fixed, i. e. arbitrary, but finite\nnumber of identical (spinless) particles, obeying the Maxw ell–Boltzmann statistics, in\nthe space R3. For this system observables can be described by means of the sequences\nA= (A0,A1(1),...,An(1,...,n),...)of self-adjoint operators An∈L(Hn).\nIn this case the mean value (expectation value) of an observa ble is determined by the\npositive continuous linear functional, which is represent ed by the following series expan-Operators generated by density matrix 3\nsion:\n/a\\}bracketle{tA/a\\}bracketri}ht= (I,D)−1∞/summationdisplay\nn=01\nn!Tr1,...,nAnDn, (1)\nwhere the sequence D= (I,D1,...,Dn,...)of self-adjoint positive operators Dn∈\nL1(Hn)is a sequence of density operators, describing all possible states of a quantum\nsystem of non-fixed number of particles, and (I,D) =/summationtext∞\nn=01\nn!Tr1,...,nDnis a normaliza-\ntion factor. The functional (1) that defines a duality of obse rvables and states, exists if if\nDn∈L1(Hn)andAn∈L(Hn).\nWe note that in case of a system of fixed number N <∞of particles the ob-\nservables and states are one-component sequences A(N)= (0,...,0,AN,0,...)and\nD(N)= (0,...,0,DN,0,...), respectively, and hence, mean value functional (1) takes\nthe conventional representation\n/a\\}bracketle{tA(N)/a\\}bracketri}ht= (Tr1,...,NDN)−1Tr1,...,NANDN,\nand it is usually assumed that the normalization condition Tr1,...,NDN= 1, holds.\nIf initial state is specified by the sequence of density opera torsD(0) =\n(I,D0\n1(1),...,D0\nn(1,...,n),...), then the evolution of all possible states, i.e. the sequenc e\nD(t) = (I,D1(t,1),...,Dn(t,1,...,n),...)of the density operators Dn(t), n≥1, is\ndetermined by the following groups of operators:\nDn(t) =G∗\nn(t)D0\nn.=e−itHnD0\nneitHn, n≥1, (2)\nwhere the self-adjoint operator Hnis then-particle Hamiltonian, and we used units where\nh= 2π/planckover2pi1= 1is a Planck constant. In the sense of the mean value functiona l (1), the group\nG∗\nn(t)is conjugated to the group of operators Gn(t), which describes the evolution of the\nobservables.\nThe one-parameter mapping G∗\nn(t)is defined on the space of trace class operators\nL1(Hn), and it is an isometric strongly continuous group of operato rs that preserves pos-\nitivity and self-adjointness of operators [2]. In the seque l the inverse group to the group\nG∗\nn(t)we will denote by symbol (G∗\nn)−1(t) =G∗\nn(−t). On its domain of the definition the\ninfinitesimal generator N∗\nnof the group of operators G∗\nn(t)is determined in the sense of the\nstrong convergence of the space L1(Hn)by the generator of the von Neumann equation\n(quantum Liouville equation), namely\nlim\nt→01\nt/parenleftbig\nG∗\nn(t)fn−fn/parenrightbig\n=−i(Hnfn−fnHn).=N∗\nnfn. (3)\nThe operator N∗\nnhas the following structure: N∗\nn=/summationtextn\nj=1N∗(j)+/summationtextn\nj1<j2=1N∗\nint(j1,j2),\nwhere the operator N∗(j)is a free motion generator of the von Neumann equation, the\noperatorN∗\nintis defined by means of the operator of a two-body interaction p otentialΦby\nthe formula: N∗\nint(j1,j2)fn.=−i(Φ(j1,j2)fn−fnΦ(j1,j2)).\nIfD0\nn∈L1(Hn), n≥1, then fort∈Rthe sequence of density operators (2) is a\nunique solution of the Cauchy problem of the von Neumann equa tions [2]:\n∂\n∂tDn(t) =N∗\nnDn(t), (4)\nD(t)n/vextendsingle/vextendsingle\nt=0=D0\nn, n≥1, (5)\nwhere the generators N∗\nn, n≥1,of equations (4) are adjoint operators to generators of\nthe Heisenberg equations for observables in the sense of mea n value functional (1), and are\ndefined by formula (3).4 V . I. Gerasimenko\nWe remind that the density operator is represented as a conve x linear combination of a\nrank one projectors, though this representation is not uniq ue. A density operator that is a\nrank one projection Dn(t) =Pψn(t), ψn∈ Hn,is known as a pure quantum state, and all\nquantum states that are not pure are interpreted as mixed sta tes.\nIn consequence of the validity for the projector of the equal ityPψn(t) =Pψn(t), where\nψn(t) =e−itHnψn, we conclude that the evolution of pure states can be also des cribed by\nmeans of the Cauchy problem for the Schr¨ odinger equation:\ni∂\n∂tψn(t) =Hnψn(t),\nψ(t)n/vextendsingle/vextendsingle\nt=0=ψn, n≥1,\nwhere the operator Hnis then-particle Hamiltonian.\n3. Cluster expansions of the density operator\nAn alternative approach to the description of states of a qua ntum system of finitely many\nparticles is given by means of operators determined by the cl uster expansions of the density\noperator. They are interpreted as correlation operators.\nWe introduce the sequence of correlation operators g(t) = (I,g1(t,1),...,gs(t,\n1,...,s),...)using cluster expansions of a sequence of density operators D(t) =\n(I,D1(t,1),...,Dn(t,1,...,n),...):\nDn(t,1,...,n) =gn(t,1,...,n)+/summationdisplay\nP : (1,...,n) =/uniontext\niXi,\n|P|>1/productdisplay\nXi⊂Pg|Xi|(t,Xi), n≥1, (6)\nwhere/summationtext\nP:(1,...,n)=/uniontext\niXi,|P|>1is the sum over all possible partitions Pof the set (1,...,n)\ninto|P|>1nonempty mutually disjoint subsets Xi⊂(1,...,n).\nSolutions of recursion relations (6) are given by the follow ing expansions:\ngs(t,1,...,s) =Ds(t,1,...,s)+/summationdisplay\nP : (1,...,s) =/uniontext\niXi,\n|P|>1(−1)|P|−1(|P|−1)!/productdisplay\nXi⊂PD|Xi|(t,Xi), s≥1. (7)\nThe structure of expansions (7) is such that the correlation operators can be treated as cu-\nmulants (semi-invariants) of the density operators (2).\nThus, correlation operators (7) are to enable to describe of the evolution of states of\nfinitely many particles by the equivalent method in comparis on with the density operators,\nnamely within the framework of dynamics of correlations [4] ,[5].\nIf initial state described by the sequence of correlation op eratorsg(0) = (I,g0\n1(1),...,\ng0\nn(1,...,n),...)∈ ⊕∞\nn=0L1(Hn), then the evolution of all possible states, i.e. the se-\nquenceg(t) = (I,g1(t,1),...,gs(t,1,...,s),...)of the correlation operators gs(t), s≥\n1, is determined by the following group of nonlinear operator s [5]:\ng(t,1,...,s) =G(t;1,...,s|g(0)).= (8)/summationdisplay\nP:(1,...,s)=/uniontext\njXjA|P|(t,{X1},...,{X|P|})/productdisplay\nXj⊂Pg0\n|Xj|(Xj), s≥1,Operators generated by density matrix 5\nwhere/summationtext\nP:(1,...,s)=/uniontext\njXjis the sum over all possible partitions Pof the set (1,...,s)into\n|P|nonempty mutually disjoint subsets Xj, the set({X1},...,{X|P|})consists from el-\nements of which are subsets Xj⊂(1,...,s), i.e.|({X1},...,{X|P|})|=|P|. The\ngenerating operator A|P|(t)in expansion (8) is the |P|th-order cumulant of the groups of\noperators (2) which is defined by the expansion\nA|P|(t,{X1},...,{X|P|}).= (9)\n/summationdisplay\nP′:({X1},...,{X|P|})=/uniontext\nkZk(−1)|P′|−1(|P′|−1)!/productdisplay\nZk⊂P′G∗\n|θ(Zk)|(t,θ(Zk)),\nwhereθis the declusterization mapping: θ({X1},...,{X|P|}).= (1,...,s).\nIn particular case of the absence of correlations between pa rticles at the initial time\n(known as initial states satisfying a chaos condition [11], [12],[13]) the sequence of initial\ncorrelation operators has the form gc(0) = (0,g0\n1(1),0,...,0,...)(in case of the Maxwell–\nBoltzmann statistics in terms of a sequence of density opera tors it means that Dc(0) =\n(I,D0\n1(1),D0\n1(1)D0\n1(2),...,/producttextn\ni=1D0\n1(i),...)). In this case expansions (8) are represented\nas follows:\ngs(t,1,...,s) =As(t,1,...,s)s/productdisplay\ni=1g0\n1(i), s≥1,\nwhereAs(t)is thesth-order cumulant of groups of operators (2) defined by the expa nsion\nAs(t,1,...,s) =/summationdisplay\nP:(1,...,s)=/uniontext\niXi(−1)|P|−1(|P|−1)!/productdisplay\nXi⊂PG∗\n|Xi|(t,Xi),(10)\nand it was used notations accepted in formula (2).\nIfg0\ns∈L1(Hs), s≥1, then fort∈Rthe sequence of correlation operators (8) is a\nunique solution of the Cauchy problem of the quantum von Neum ann hierarchy [4],[5]:\n∂\n∂tgs(t,1,...,s) =N∗\nsgs(t,1,...,s)+ (11)\n/summationdisplay\nP:(1,...,s)=X1/uniontextX2/summationdisplay\ni1∈X1/summationdisplay\ni2∈X2N∗\nint(i1,i2)g|X1|(t,X1)g|X2|(t,X2),\ngs(t,1,...,s)/vextendsingle/vextendsingle\nt=0=g0\ns(1,...,s), s≥1, (12)\nwhere/summationtext\nP:(1,...,s)=X1/uniontextX2is the sum over all possible partitions Pof the set (1,...,s)into\ntwo nonempty mutually disjoint subsets X1andX2, and the operator N∗\nsis defined on the\nsubspace L1\n0(Hs)by formula (3). It should be noted that the von Neumann hierar chy (11)\nis the evolution recurrence equations set.\n4. Reduced density operators\nFor the description of quantum systems of both finite and infin ite number of particles an-\nother approach to describe of states and observables is used , which is equivalent to the\napproach formulated above in case of systems of finitely many particles [6],[7].\nIndeed, for a system of finitely many particles mean value fun ctional (1) can be repre-\nsented in one more form\n/a\\}bracketle{tA/a\\}bracketri}ht= (I,D)−1∞/summationdisplay\nn=01\nn!Tr1,...,nAnDn= (13)\n∞/summationdisplay\ns=01\ns!Tr1,...,sBs(1,...,s)Fs(1,...,s),6 V . I. Gerasimenko\nwhere, for the description of observables and states, the se quence of so-called re-\nduced observables B= (B0,B1(1),...,Bs(1,...,s),...)(other used terms: marginal\nors-particle observable) was introduced and reduced density o peratorsF=\n(I,F1(1),...,Fs(1,...,s),...)(other used terms: marginal or s-particle density operators\n[6],[11]), respectively. Thus, the reduced observables ar e defined by means of observables\nby the following expansions [9],[10]:\nBs(1,...,s).=s/summationdisplay\nn=0(−1)n\nn!s/summationdisplay\nj1/ne}ationslash=.../ne}ationslash=jn=1As−n((1,...,s)\\(j1,...,jn)), s≥1,(14)\nand the reduced density operators are defined by means of dens ity operators as follows [7]\nFs(1,...,s).= (I,D)−1∞/summationdisplay\nn=01\nn!Trs+1,...,s+nDs+n(1,...,s+n), s≥1.(15)\nWe emphasize that the possibility of describing states with in the framework of reduced\ndensity operators naturally arises as a result of dividing t he series in expression (1) by the\nseries of the normalization factor, i.e. in consequence of r edefining of mean value functional\n(13).\nIf initial state specified by the sequence of reduced density operatorsF(0) =\n(I,F0\n1(1),...,F0\nn(1,...,n),...), then the evolution of all possible states, i.e. a sequence\nF(t) = (I,F1(t,1),...,Fs(t,1,...,s),...)of the reduced density operators Fs(t), s≥1,\nis determined by the following series expansion [14],[15]:\nFs(t,1,...,s) = (16)\n∞/summationdisplay\nn=01\nn!Trs+1,...,s+nA1+n(t,{1,...,s},s+1,...,s+n)F0\ns+n(1,...,s+n),\ns≥1,\nwhere the generating operator\nA1+n(t,{1,...,s},s+1,...,s+n) = (17)/summationdisplay\nP:({1,...,s},s+1,...,s+n)=/uniontext\niXi(−1)|P|−1(|P|−1)!/productdisplay\nXi⊂PG∗\n|θ(Xi)|(t,θ(Xi))\nis the(1+n)th-order cumulant of groups of operators (2) [15]. In expansio n (17) the sym-\nbol/summationtext\nPmeans the sum over all possible partitions Pof the set ({1,...,s},s+1,...,s+n)\ninto|P|nonempty mutually disjoint subsets Xi⊂({1,...,s},s+ 1,...,s+n)and we\nuse notations accepted in formula (8).\nIfF(0)∈ ⊕∞\nn=0αnL1(Hn)andα > e , then fort∈Rthe sequence of reduced den-\nsity operators (16) is a unique solution of the Cauchy proble m of the quantum BBGKY\nhierarchy [6]:\n∂\n∂tFs(t,1,...,s) =N∗\nsFs(t,1,...,s)+ (18)\ns/summationdisplay\nj=1Trs+1N∗\nint(j,s+1)Fs+1(t,1,...,s,s +1),\nFs(t,1,...,s)|t=0=F0\ns(1,...,s), s≥1, (19)\nwhere we used notations accepted in formula (3).Operators generated by density matrix 7\nWe note that traditionally [6],[11],[7],[16] the reduced d ensity operators are represented\nby means of the perturbation theory series of the BBGKY hiera rchy (18)\nFs(t,1,...,s) =\n∞/summationdisplay\nn=0t/integraldisplay\n0dt1...tn−1/integraldisplay\n0dtnTrs+1,...,s+nG∗\ns(t−t1)s/summationdisplay\nj1=1N∗\nint(j1,s+1))G∗\ns+1(t1−t2)...\nG∗\ns+n−1(tn−1−tn)s+n−1/summationdisplay\njn=1N∗\nint(jn,s+n))G∗\ns+n(tn)F0\ns+n(1,...,s+n), s≥1,\nwhere we used notations accepted in formula (3). The nonpert urbative series expansion for\nreduced density operators (16) is represented in the form of the perturbation theory series\nfor suitable interaction potentials and initial data as a re sult of the employment of analogs\nof the Duhamel equation to cumulants (17) of the groups of ope rators (2).\nAn equivalent definition of reduced density operators can be formulated based on cor-\nrelation operators (8) of systems of finitely many particles [5], namely\nFs(t,1,...,s).=∞/summationdisplay\nn=01\nn!Trs+1,...,s+ng1+n(t,{1,...,s},s+1,...,s+n), s≥1,(20)\nwhere the correlation operators of clusters of particles g1+n(t),n≥0,are defined by the\nexpansions\ng1+n(t,{1,...,s},s+1,...,s+n) = (21)/summationdisplay\nP:({1,...,s},s+1,...,s+n)=/uniontext\niXiA|P|/parenleftbig\n−t,{θ(X1)},...,{θ(X|P|)}/parenrightbig/productdisplay\nXi⊂Pg0\n|Xi|(Xi),\nn≥0,\nandA|P|(t)is the|P|th-order cumulant (9) of the groups of operators (2). Owing tha t\ncorrelation operators g1+n(t), n≥0,are governed by the corresponding von Neumann\nhierarchy, for reduced density operators (20) we can derive the quantum BBGKY hierarchy.\nThus, as follows from the above, the cumulant structure of co rrelation operator expan-\nsion (21) induces the cumulant structure of series expansio ns for reduced density operators\n(16), i.e. in fact, dynamics of correlations is generated dy namics of infinitely many parti-\ncles.\n5. Reduced correlation operators\nAnother approach to the description of states of quantum sys tems of both finite and in-\nfinite number of particles is can be formulated as in above by m eans of operators de-\ntermined by the cluster expansions of the reduced density op erators. Such operators are\ninterpreted as reduced correlation operators of states (ma rginal ors-particle correlation op-\nerators) [6],[17],[18].\nTraditionally reduced correlation operators are introduc ed by means of the cluster ex-\npansions of the reduced density operators (20) as follows:\nFs(t,1,...,s) =/summationdisplay\nP : (1,...,s) =/uniontext\niXi/productdisplay\nXi⊂PG|Xi|(t,Xi), s≥1, (22)8 V . I. Gerasimenko\nwhere/summationtext\nP:(1,...,s)=/uniontext\niXiis the sum over all possible partitions Pof the set (1,...,s)into|P|\nnonempty mutually disjoint subsets Xi⊂(1,...,s). As a consequence of this, the solution\nof recurrence relations (22) represented through reduced d ensity operators as follows\nGs(t,1,...,s) =/summationdisplay\nP : (1,...,s) =/uniontext\niXi(−1)|P|−1(|P|−1)!/productdisplay\nXi⊂PF|Xi|(t,Xi),(23)\ns≥1,\nare interpreted as the operators that describe correlation s of states in many-particle systems.\nThe structure of expansions (23) is such that the reduced cor relation operators can be treated\nas cumulants (semi-invariants) of the reduced density oper ators (16).\nAssuming as a basis an alternative approach to the descripti on of the evolution of states\nof quantum many-particle systems within the framework of co rrelation operators (8), we\ncan define the reduced correlation operators by means of a sol ution of the Cauchy problem\nof the von Neumann hierarchy (11),(12) as follows [17],[18] :\nGs(t,1,...,s).=∞/summationdisplay\nn=01\nn!Trs+1,...,s+ngs+n(t,1,...,s+n), s≥1,(24)\nwhere the operator gs+n(t,1,...,s+n)is defined by expansion (7). We emphasize that ev-\nery term of the expansion (24) of reduced correlation operat or is determined by the (s+n)-\nparticle correlation operator (8) as contrasted to the expa nsion of reduced density operator\n(20) which is determined by the (1+n)-particle correlation operator of clusters of particles\n(21).\nIfG(0) = (I,G0\n1(1),...,G0\ns(1,...,s),...)is a sequence of reduced correlation op-\nerators at initial instant, then the evolution of all possib le states, i.e. a sequence G(t) =\n(I,G1(t,1),...,Gs(t,1,...,s),...)of the reduced correlation operators Gs(t), s≥1, is\ndetermined by the following series expansion [18]:\nGs(t,1,...,s) =∞/summationdisplay\nn=01\nn!Trs+1,...,s+nA1+n(t;{1,...,s},s+1,...,s+n|G(0)),(25)\ns≥1,\nwhere the generating operator A1+n(t;{1,...,s},s+1,...,s+n|G(0)) of this series is\nthe(1+n)th-order cumulant of groups of nonlinear operators (2):\nA1+n(t;{1,...,s},s+1,...,s+n|G(0)).= (26)/summationdisplay\nP:({1,...,s},s+1,...,s+n)=/uniontext\nkXk(−1)|P|−1(|P|−1)!G(t;θ(X1)|...\nG(t;θ(X|P|)|G(0))...), n≥0,\nand where the composition of mappings (2) of the correspondi ng noninteracting groups of\nparticles was denoted by G(t;θ(X1)|...G(t;θ(X|P|)|G(0))...), for example,\nG/parenleftbig\nt;1| G(t;2|G(0))/parenrightbig\n=A1(t,1)A1(t,2)G0\n2(1,2),\nG/parenleftbig\nt;1,2| G(t;3|G(0))/parenrightbig\n=A1(t,{1,2})A1(t,3)G0\n3(1,2,3) +\nA2(t,1,2)A1(t,3)/parenleftbig\nG0\n1(1)G0\n2(2,3)+G0\n1(2)G0\n2(1,3)/parenrightbig\n.\nWe will adduce examples of expansions (26). The first order cu mulant of the groups of\nnonlinear operators (2) is the group of these nonlinear oper ators\nA1(t;{1,...,s} |G(0)) =G(t;1,...,s|G(0)).Operators generated by density matrix 9\nIn case ofs= 2the second order cumulant of nonlinear operators (2) has the structure\nA1+1(t;{1,2},3|G(0)) =G(t;1,2,3|G(0))−G/parenleftbig\nt;1,2| G(t;3|G(0))/parenrightbig\n=\nA1+1(t,{1,2},3)G0\n3(1,2,3) +/parenleftbig\nA1+1(t,{1,2},3)−A1+1(t,2,3)A1(t,1)/parenrightbig\nG0\n1(1)G0\n2(2,3) +/parenleftbig\nA1+1(t,{1,2},3)−A1+1(t,1,3)A1(t,2)/parenrightbig\nG0\n1(2)G0\n2(1,3) +\nA1+1(t,{1,2},3)G0\n1(3)G0\n2(1,2)+A3(t,1,2,3)G0\n1(1)G0\n1(2)G0\n1(3),\nwhere the operator\nA3(t,1,2,3) =A1+1(t,{1,2},3)−A1+1(t,2,3)A1(t,1)−A1+1(t,1,3)A1(t,2)\nis cumulant (10) of groups of operators (2) of the third order .\nIn the case of the initial state specified by the sequence of re duced correlation operators\nG(c)= (0,G0\n1,0,...,0,...), that is, in the absence of correlations between particles a t\nthe initial moment of time [11],[12],[13], according to defi nition (26), reduced correlation\noperators (25) are represented by the following series expa nsions:\nGs(t,1,...,s) =∞/summationdisplay\nn=01\nn!Trs+1,...,s+nAs+n(t;1,...,s+n)s+n/productdisplay\ni=1G0\n1(i), s≥1,(27)\nwhere the generating operator As+n(t)is the(s+n)th-order cumulant (10) of groups of\noperators (2).\nIfG(0)∈ ⊕∞\nn=0L1(Hn), then fort∈Rthe sequence of reduced correlation operators\n(25) is a unique solution of the Cauchy problem of the hierarc hy of nonlinear evolution\nequations (known as the nonlinear quantum BBGKY hierarchy) [18]:\n∂\n∂tGs(t,1,...,s) =N∗\nsGs(t,1,...,s)+ (28)\n/summationdisplay\nP:(1,...,s)=X1/uniontextX2/summationdisplay\ni1∈X1/summationdisplay\ni2∈X2N∗\nint(i1,i2)G|X1|(t,X1)G|X2|(t,X2))+\nTrs+1/summationdisplay\ni∈YN∗\nint(i,s+1)/parenleftbig\nGs+1(t,1,...,s+1)+\n/summationdisplay\nP : (1,...,s+1) =X1/uniontextX2,\ni∈X1;s+1∈X2G|X1|(t,X1)G|X2|(t,X2)/parenrightbig\n,\nGs(t,1,...,s)/vextendsingle/vextendsingle\nt=0=G0\ns(,1,...,s), s≥1, (29)\nwhere we use accepted in hierarchy (11) notations.\nWe note that the reduced correlation operators give an equiv alent approach to the de-\nscription of the evolution of states of quantum many-partic le systems as compared with the\nreduced density operators. Indeed, the macroscopic charac teristics of fluctuations of ob-\nservables are directly determined by the reduced correlati on operators on the microscopic\nscale [6],[17], for example, the functional of the dispersi on of an additive-type observable,\ni.e. the sequence A(1)= (0,a1(1),...,/summationtextn\ni1=1a1(i1),...), is represented by the formula\n/a\\}bracketle{t(A(1)−/a\\}bracketle{tA(1)/a\\}bracketri}ht)2/a\\}bracketri}ht(t) = Tr 1(a2\n1(1)−/a\\}bracketle{tA(1)/a\\}bracketri}ht2(t))G1(t,1)+Tr 1,2a1(1)a1(2)G2(t,1,2),\nwhere/a\\}bracketle{tA(1)/a\\}bracketri}ht(t) = Tr 1a1(1)G1(t,1)is the mean value functional of an additive-type ob-\nservable.10 V . I. Gerasimenko\n6. On the description of the evolution of states by the one-\nparticle correlation operator\nFurther, we shall consider systems which the initial state s pecified by a one-particle reduced\ncorrelation (density) operator, namely, the initial state specified by a sequence of reduced\ncorrelation operators satisfying a chaos property stated a bove, i.e. by the sequence G(c)=\n(0,G0\n1,0,...,0,...). We remark that such an assumption about initial states is in trinsic in\nkinetic theory of many-particle systems.\nThe following statement is true. In the case of the initial st ate specified by a one-particle\ncorrelation (density) operator G(c)the evolution that described within the framework of\nthe sequence G(t) = (I,G1(t),...,Gs(t),...)of reduced correlation operators (25), is\nalso be described by the sequence G(t|G1(t)) = (I,G1(t),G2(t|G1(t)),...,Gs(t|\nG1(t)),...)of reduced (marginal) correlation functionals: Gs(t,1,...,s|G1(t)), s≥2,\nwith respect to the one-particle correlation operator G1(t)governed by the generalized\nquantum kinetic equation [8],[19].\nIn the case under consideration the reduced correlation fun ctionalsGs(t|G1(t)), s≥\n2, are represented with respect to the one-particle correlat ion operator\nG1(t,1) =∞/summationdisplay\nn=01\nn!Tr2,...,1+nA1+n(t,1,...,n+1)n+1/productdisplay\ni=1G0\n1(i), (30)\nwhere the generating operator A1+n(t)is cumulant (10) of the groups of operators (2) of\nthe(1+n)th-order, by the following series:\nGs/parenleftbig\nt,1,...,s|G1(t)/parenrightbig\n= (31)\n∞/summationdisplay\nn=01\nn!Trs+1,...,s+nVs+n/parenleftbig\nt,θ({1,...,s}),s+1,...,s+n/parenrightbigs+n/productdisplay\ni=1G1(t,i), s≥2.\nThe generating operator Vs+n(t), n≥0, of the(s+n)th-order of this series is determined\nby the following expansion [19]\nVs+n/parenleftbig\nt,θ({1,...,s}),s+1,...,s+n/parenrightbig\n= (32)\nn!n/summationdisplay\nk=0(−1)kn/summationdisplay\nn1=1...n−n1−...−nk−1/summationdisplay\nnk=11\n(n−n1−...−nk)!×\nˆAs+n−n1−...−nk(t,θ({1,...,s}),s+1,...,s+n−n1−...−nk)×\nk/productdisplay\nj=1/summationdisplay\nDj:Zj=/uniontext\nljXlj,\n|Dj| ≤s+n−n1−···−nj1\n|Dj|!s+n−n1−...−nj/summationdisplay\ni1/ne}ationslash=.../ne}ationslash=i|Dj|=1/productdisplay\nXlj⊂Dj1\n|Xlj|!ˆA1+|Xlj|(t,ilj,Xlj).\nwhere/summationtext\nDj:Zj=/uniontext\nljXljis the sum over all possible dissections [19] of the linearly ordered\nsetZj≡(s+n−n1−...−nj+ 1,...,s+n−n1−...−nj−1)on no more than\ns+n−n1−...−njlinearly ordered subsets, the (s+n)th-order scattering cumulant is\ndefined by the formula\nˆAs+n(t,θ({1,...,s}),s+1,...,s+n).=As+n(t,1,...,s+n)s+n/productdisplay\ni=1A−1\n1(t,i),\nand notations accepted above were used. A method of the const ruction of reduced correla-\ntion functionals (31) is based on the application of the so-c alled kinetic cluster expansions\n[19] to the generating operators (10) of series (27).Operators generated by density matrix 11\nWe adduce simplest examples of generating operators (32):\nVs(t,θ({1,...,s})) =As(t,1,...,s)s/productdisplay\ni=1A−1\n1(t,i),\nVs+1(t,θ({1,...,s}),s+1) =As+1(t,1,...,s+1)s+1/productdisplay\ni=1A−1\n1(t,i)−\nAs(t,1,...,s)s/productdisplay\ni=1A−1\n1(t,i)s/summationdisplay\nj=1A2(t,j,s+1)A−1\n1(t,j)A−1\n1(t,s+1).\nWe note that reduced correlation functionals (31) describe all possible correlations gen-\nerated by the dynamics of quantum many-particle systems in t erms of a one-particle corre-\nlation operator.\nIfG0\n1∈L1(H), then for arbitrary t∈Rone-particle correlation operator (30) is a weak\nsolution of the Cauchy problem of the generalized quantum ki netic equation [19]\n∂\n∂tG1(t,1) =N∗(1)G1(t,1)+Tr 2N∗\nint(1,2)G1(t,1)G1(t,2)+ (33)\nTr2N∗\nint(1,2)G2/parenleftbig\nt,1,2|G1(t)/parenrightbig\n,\nG1(t,1)/vextendsingle/vextendsingle\nt=0=G0\n1(1), (34)\nwhere the second part of the collision integral in (33) is det ermined in terms of the two-\nparticle correlation functional represented by series exp ansion (31).\n7. On the scaling limits of reduced density operators\nThe conventional philosophy of the description of the kinet ic evolution consists of the fol-\nlowing. If the initial state specified by a one-particle corr elation operator, then the evolution\nof states can be effectively described by means of a one-part icle correlation operator gov-\nerned by the nonlinear kinetic equation in a suitable scalin g limit.\nFurther, we consider a scaling asymptotic behavior of the co nstructed reduced correla-\ntion operators in particular case of a mean field limit for ini tial states specified by a one-\nparticle correlation operator mentioned above [11],[12], [16].\nWe will assume the existence of a mean field limit of the initia l reduced correlation\noperatorG0,ǫ\n1scaled by the parameter ǫ≥0in the following sense\nlim\nǫ→0/vextenddouble/vextenddoubleǫG0,ǫ\n1−g0\n1/vextenddouble/vextenddouble\nL1(H)= 0, (35)\nand the operator N∗\nintin hierarchy (28) scaled in such a way that ǫN∗\nint.\nSince thenthterm of series (27) for the s-particle correlation operator is determined by\nthe(s+n)th-order cumulant of asymptotically perturbed groups of oper ators (2), then the\nproperty of the propagation of initial chaos holds\nlim\nǫ→0/vextenddouble/vextenddoubleǫsGs(t)/vextenddouble/vextenddouble\nL1(Hs)= 0, s≥2. (36)\nThe equality (36) is derived by the following assertions. If fs∈L1(Hs), then for\narbitrary finite time interval for asymptotically perturbe d first-order cumulant (10) of the12 V . I. Gerasimenko\ngroups of operators (2), i.e. for the strongly continuous gr oup (2) the following equality\ntakes place\nlim\nǫ→0/vextenddouble/vextenddouble/vextenddoubleG∗\ns(t,1,...,s)fs−s/productdisplay\nj=1G∗\n1(t,j)fs/vextenddouble/vextenddouble/vextenddouble\nL1(Hs)= 0.\nHence for the (s+n)th-order cumulants of asymptotically perturbed groups of ope rators\n(2) the following equalities true:\nlim\nǫ→0/vextenddouble/vextenddouble/vextenddouble1\nǫnAs+n(t,1,...,s+n)fs+n/vextenddouble/vextenddouble/vextenddouble\nL1(Hs+n)= 0, s≥2. (37)\nIf for the initial one-particle correlation operator equal ity (35) holds, then in case of\ns= 1for series expansion (27) the following equality is true\nlim\nǫ→0/vextenddouble/vextenddoubleǫG1(t)−g1(t)/vextenddouble/vextenddouble\nL1(H)= 0,\nwhere for arbitrary finite time interval the limit one-parti cle correlation operator g1(t,1)is\nrepresented by the series\ng1(t,1) = (38)\n∞/summationdisplay\nn=0t/integraldisplay\n0dt1...tn−1/integraldisplay\n0dtnTr2,...,n+1G∗\n1(t−t1,1)N∗\nint(1,2)2/productdisplay\nj1=1G∗\n1(t1−t2,j1)...\nn/productdisplay\nin=1G∗\n1(tn−tn,in)n/summationdisplay\nkn=1N∗\nint(kn,n+1)n+1/productdisplay\njn=1G∗\n1(tn,jn)n+1/productdisplay\ni=1g0\n1(i).\nThen we conclude that limit one-particle correlation opera tor (38) is a weak solution of\nthe Cauchy problem of the quantum Vlasov kinetic equation\n∂\n∂tg1(t,1) =N∗(1)g1(t,1)+Tr 2N∗\nint(1,2)g1(t,1)g1(t,2), (39)\ng1(t,1)|t=0=g0\n1(1). (40)\nFor pure states limit one-particle correlation operator (3 8) is governed by the Hartree\nequation. Indeed, in terms of the kernel g1(t,q;q′) =ψ(t,q)ψ(t,q′)of operator (38),\ndescribing a pure state, in the configuration space represen tation, kinetic equation (39) is\nconverted into the Hartree equation\ni∂\n∂tψ(t,q) =−1\n2∆qψ(t,q)+/integraldisplay\ndq′Φ(q−q′)|ψ(t,q′)|2ψ(t,q),\nwhere the function Φis the two-body potential of interaction.\nWe remark that in case of pure states kinetic equation (39) ca n be also transformed into\nthe nonlinear Schr¨ odinger equation [20] or into the Gross– Pitaevskii kinetic equation [21].\nWe remark that some other approaches to the derivation of qua ntum kinetic equations\n[22], in particular, quantum systems with initial correlat ions were developed in papers\n[10],[23],[24].\nIn the last decade, other scaling limits (weak coupling, low -density, semiclassical) of\nthe reduced density operators constructed by means theory o f perturbations were rigorously\nestablished in numerous papers, for example, in articles [1 1],[16], [20],[21],[25],[26] and\npapers cited therein.Operators generated by density matrix 13\n8. Conclusion\nThis article deals with a quantum system of non-fixed, i.e. ar bitrary but finite average\nnumber of identical (spinless) particles obeying Maxwell– Boltzmann statistics. The above\nresults are extended to quantum systems of many bosons or fer mions, as in paper [5].\nIt was considered some approaches to the description of the e volution of states of quan-\ntum many-particle systems employing the possible modificat ions of the density operator\nwhich kernel is known as a density matrix. One of these approa ches is allowed to describe\nthe evolution of quantum systems of both finite and infinite av erage number of particles\nthrough the reduced density operator (16) or reduced correl ation operators (25) which are\ngoverned by the dynamics of correlations (8).\nAbove it was established that the notion of cumulants (9) of g roups of operators (2)\nunderlies non perturbative expansions of solutions for the fundamental evolution equations,\nnamely for the von Neumann hierarchy (11) of correlation ope rators, for the BBGKY hi-\nerarchy (18) of reduced density operators and for the nonlin ear BBGKY hierarchy (28) of\nreduced correlation operators, as well as it underlies the k inetic description of the evolution\nof states (31).\nWe emphasize that the structure of expansions for correlati on operators (21), in which\nthe generating operators are corresponding order cumulant (9) of the groups of operators\n(2), induces the cumulant structure of series expansions fo r reduced density operators (16),\nreduced correlation operators (25) and marginal correlati on functionals (31). Thus, in fact,\nthe dynamics of systems of infinitely many particles is gener ated by the dynamics of corre-\nlations.\nThe origin of the microscopic description of the collective behavior of quantum many-\nparticle systems by a one-particle correlation operator th at is governed by the generalized\nquantum kinetic equation (33) was also considered. One of th e advantages of such an\napproach to the derivation of kinetic equations from underl ying dynamics consists of an\nopportunity to construct the kinetic equations with initia l correlations, which makes it pos-\nsible to describe the propagation of initial correlations i n the scaling limits [27],[28]. In\naddition, it was established that in particular case of a mea n field approximation for ini-\ntial states specified by a one-particle correlation operato r the asymptotic behavior of the\nconstructed reduced correlation operators (27) is governe d by the quantum Vlasov kinetic\nequation (39).\nReferences\n[1] von Neumann, J. Mathematical Foundations of Quantum Mechanics. Princeton\nUniversity Press, 2018.\n[2] Dautray, R. and Lions, J. L. Mathematical Analysis and Numerical Methods for\nScience and Technology .1, Springer-Verlag: Berlin, Heidelberg, 2000.\n[3] Gerasimenko, V .I. (2012). Hierarchies of quantum evolu tion equations and dynamics\nof many-particle correlations. Statistical Mechanics and Random Walks: Principles,\nProcesses and Applications. N.Y.: Nova Science Publ., Inc. , 233-288.\n[4] Gerasimenko, V . I. and Shtyk, V . O. (2008). Evolution of c orrelations of quantum\nmany-particle systems. J. Stat. Mech. Theory Exp. , 3, P03007.\n[5] Gerasimenko, V . I. and Polishchuk, D. O. (2011). Dynamic s of correlations of Bose\nand Fermi particles. Math. Meth. Appl. Sci. , 34 (1): 76-93.14 V . I. Gerasimenko\n[6] Bogolyubov, M.M. Lectures on Quantum Statistics. Problems of Statistical Me chanics\nof Quantum Systems . Rad. Shkola, Kiev, 1949 (in Ukrainian).\n[7] Cercignani, C., Gerasimenko, V . I. and Petrina, D. Ya. Many-Particle Dynamics and\nKinetic Equations . Springer: The Netherlands, 2012.\n[8] Gerasimenko, V . I. (2017). On the description of quantum correlations by means of a\none-particle density operator. Transactions Inst. Math. NASU , 14 (1): 116-127.\n[9] Borgioli, G. and Gerasimenko, V . I. (2010). Initial-val ue problem of the quantum dual\nBBGKY hierarchy. Nuovo Cimento , 33 C (1): 71-78.\n[10] Gerasimenko, V . I. (2011). Heisenberg picture of quant um kinetic evolution in mean-\nfield limit. Kinet. Relat. Models , 4 (1): 385-399.\n[11] Benedikter, N., Porta, M. and Schlein, B. Effective Evolution Equations from Quan-\ntum Dynamics . SpringerBriefs in Mathematical Physics, 2016.\n[12] Spohn, H. (1980). Kinetic equations from Hamiltonian d ynamics: Markovian limits.\nRev. Modern Phys. , 52 (3): 569-615.\n[13] Benedetto, D., Castella, F., Esposito, R. and Pulviren ti, M. (2007). A short review on\nthe derivation of the nonlinear quantum Boltzmann equation s.Commun. Math. Sci. ,\n5: 55-71.\n[14] Gerasimenko, V . I. and Shtyk, V . O. (2006). Initial-val ue problem of the Bogolyubov\nhierarchy for quantum systems of particles. Ukrain. Math. J. , 58 (9): 1175-1191.\n[15] Gerasimenko, V . I., Ryabukha, T. V . and Stashenko, M. O. (2004). On the structure\nof expansions for the BBGKY hierarchy solutions. J. Phys. A: Math. Gen. , 37: 9861-\n9872.\n[16] Golse, F. (2016). On the dynamics of large particle syst ems in the mean field limit.\nIn: Macroscopic and large scale phenomena: coarse graining , mean field limits and\nergodicity. Lect. Notes Appl. Math. Mech. , Springer, 3: 1-144.\n[17] Gerasimenko, V . I. and Polishchuk, D. O. (2013). A nonpe rturbative solution of the\nnonlinear BBGKY hierarchy for marginal correlation operat ors.Math. Methods Appl.\nSci., 36 (17): 2311-2328.\n[18] Gerasimenko, V . I. (2017). Evolution of correlation op erators of large quantum parti-\ncle systems. Methods Funct. Anal. Topology. , 23 (2): 123-134.\n[19] Gerasimenko, V . I. and Tsvir, Zh. A. (2010). A descripti on of the evolution of quantum\nstates by means of the kinetic equation. J. Phys. A: Math. Theor ., 43 (48): 485203.\n[20] Erd¨ os, L., Schlein, B. and Yau, H.-T. (2007). Derivati on of the cubic nonlinear\nSchr¨ odinger equation from quantum dynamics of many-body s ystems. Invent. Math. ,\n167 (3): 515-614.\n[21] Erd¨ os, L., Schlein, B. and Yau, H.-T. (2010). Derivati on of the Gross–Pitaevskii\nEquation for the Dynamics of Bose–Einstein Condensate. Ann. of Math. , 172: 291-\n370.\n[22] Gerasimenko, V . I. (2009). Approaches to derivation of quantum kinetic equations.\nUkr. Phys. J. , 54 (8-9): 834-846.Operators generated by density matrix 15\n[23] Gerasimenko, V . I. (2015). New approach to derivation o f quantum kinetic equations\nwith initial correlations. Carpathian Math. Publ. , 7 (1): 38-48.\n[24] Gerasimenko, V . I. (2016). Processes of creation and pr opagation of correlations in\nquantum many-particle systems. Reports NAS of Ukraine , (5): 58-66.\n[25] Pezzotti, F. and Pulvirenti, M. (2009). Mean-field limi t and semiclassical expansion\nof quantum particle system. Ann. Henri Poincar ´e, 10: 145-187.\n[26] Golse, F., Mouhot, C. and Paul, T. (2016). On the mean-fie ld and classical limits of\nquantum mechanics. Commun. Math. Phys. , 343: 165-205.\n[27] Gerasimenko, V . I. and Tsvir, Zh. A. (2012). On quantum k inetic equations of many-\nparticle systems in condensed states. Physica A: Stat. Mech. Appl. , 391 (24): 6362-\n6366.\n[28] Gerasimenko, V . I. (2014). Mean field asymptotic behavi or of quantum particles with\ninitial correlations. Transactions Inst. Math. NASU , 11 (1): 46-66."    },    {        "title": "2001.03202v2.___bf_2k_F___Density_Wave_Instability_of_Composite_Fermi_Liquid.pdf",        "content": "2kFDensity Wave Instability of Composite Fermi Liquid\nShao-Kai Jian1and Zheng Zhu2, 3,\u0003\n1Condensed Matter Theory Center, Department of Physics,\nUniversity of Maryland, College Park, Maryland 20742, USA\n2Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China\n3Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA\n(Dated: September 23, 2020)\nWe investigate the 2 kFdensity-wave instability of non-Fermi liquid states by combining exact\ndiagonalization with renormalization group analysis. At the half-\flled zeroth Landau level, we\nstudy the fate of the composite Fermi liquid in the presence of the mass anisotropy and mixed\nLandau level form factors. These two experimentally accessible knobs trigger a phase transition\ntowards a unidirectional charge-density-wave state with a wavevector equal to 2 kFof the composite\nFermi liquid. Based on exact diagonalization, we identify such a transition by examining both the\nenergy spectra and the static structure factor of charge density-density correlations. Moreover,\nthe renormalization group analysis reveals that gauge \ructuations render the non-Fermi liquid state\nunstable against density-wave orders, consistent with numerical observations. Possible experimental\nprobes of the density-wave instability are also discussed.\nI. INTRODUCTION\nNon-Fermi liquids (NFLs) are among the most exotic\nquantum states in condensed matters. One class of NFL\nstates is realized at quantum critical points (QCPs)1{4\nwith gapless collective mode. The most well-known ex-\nample is the strange metal, which has been intensively\ninvestigated after the discoveries of high-temperature\nsuperconductors5and heavy-fermion materials6. More\nrecently, Moir\u0013 e materials such as the twisted bilayer\ngraphene7have created new excitement. Instead of ap-\npearing at QCPs, the NFL state can also arise as a stable\nphase at zero temperature. A prominent example is the\ntwo-dimensional (2D) electrons under a strong magnetic\n\feld: when the zeroth Landau level (LL) is half-\flled,\nit becomes a fractionalized gapless state8,9with a large\nFermi surface formed by composite fermions (CFs)10,11.\nFathoming the instabilities of NFL is the very essence\nof understanding various phenomena in strongly corre-\nlated systems. For example, the high transition temper-\nature and the complex orders of the high-temperature\nsuperconductors are all believed to result from a NFL\nmother state12{15. Theoretically a stable and controllable\nplatform is crucial and urgently needed for investigating\nthe intriguing properties of the NFL states. In particular,\nthe compressible NFL state at the half-\flled LL is well es-\ntablished both experimentally16{18and numerically19{21,\nwhich provides a promising platform. More importantly,\nthe physical setup also comes with various tuning knobs\nsuch as the magnetic \feld, the geometry, and the number\nof components including layers, subbands, spins and/or\nvalleys. With these knobs, plenty of states adjacent to\nthe composite Fermi liquid (CFL) are discovered, con-\nsequently revealing various instabilities of CFL. For in-\nstance, the Cooper instability22,23leads to the p+ip\npaired Moore-Read (MR) state24{26(we brie\ry review it\nin Appendix A); the Pomeranchuk instability27,28results\nin nematic quantum Hall states29,30; the Stoner instabil-\nity of CFL gives rise to spin or valley polarizations31{33;and the instability towards the Halperin 331-state34,35in\nquantum Hall bilayers.\nIn this paper, we propose one mechanism to reap yet\nanother instability of CFL: the 2 kFdensity-wave insta-\nbility,which is of equal importance to the previously dis-\ncovered CFL instabilities and is likely to exhibit distinct\nphysics from ordinary Fermi liquids36{38. Based on an\nexact diagonalization (ED) and renormalization group\n(RG) analysis, we propose one possible mechanism to\ntrigger the density-wave instability of CFL on half \flled\nLLs: tuning the interactions via the mixed LL form fac-\ntors from an anisotropic CFL state. We demonstrate\nsuch an instability numerically and reveal the underlying\nmechanism by RG analysis. We \fnd the density-wave\ninstability would be dominant over the pairing instabil-\nity via increasing the gauge \ructuations, which can be\nachieved by breaking the rotational symmetry. Impor-\ntantly, the mixed form factor is experimentally accessible\nin Dirac materials, e.g., in bilayer graphene, by tuning\nthe interlayer electric bias and the magnetic \feld39{42,\nrendering it possible to examine our \fndings.\nII. NUMERICAL SETUP AND RESULTS\nWe consider 2D electrons on a torus with a strongly\nperpendicular magnetic \feld piercing through its surface.\nThe Hamiltonian is given by\nH=1\n2AX\nqV(q)F(q)F(\u0000q) :\u001ay(q)\u001a(q) :; (1)\nwhereV(q) is the Fourier transform of the un-projected\nCoulomb interaction, F(q) denotes the density form fac-\ntor introduced by projection, \u001a(q) is the guiding cen-\nter density operators, and Arepresents the area of the\n2D plane. Below we consider the mixed form factors\nF(q) = cos2\u0002F0(qm) + sin2\u0002F1(qm) to tune the inter-\nactions39{42, whereF0;1(qm) = exp(\u0000q2\nm=4)L0;1[q2\nm=2]arXiv:2001.03202v2  [cond-mat.str-el]  22 Sep 20202\n2468100.00.20.40.60.81.0sin24my/mxMRCharge Density Waves (CDW)Composite Fermi Liquid (CFL)0.00.20.40.60.81.00.000.040.08CDWCFL(II) K=(8,8) K=(0,3) K=(1,2) other K En-E0sin24Ne=16  my/mx=8\nCFL(I)(a)(b)\n0.00.20.40.60.81.0-10123\n5101520CDWCFL(II)CFL(I)dE0/d(sin24)\nsin24\n-d2E0/d(sin24)2 (c)(d)\n024680.000.040.080.12sin24=0.96E(Kx,Ky)-E0  [e2/elB]\nKxDq \n2468100.00.20.40.60.81.0sin24my/mxMRCharge Density Waves (CDW)Composite Fermi Liquid (CFL)0.00.20.40.60.81.00.000.040.08CDWCFL(II) K=(8,8) K=(0,3) K=(1,2) other K En-E0sin24Ne=16  my/mx=8\nCFL(I)(a)(b)\n0.00.20.40.60.81.0-10123\n5101520CDWCFL(II)CFL(I)dE0/d(sin24)\nsin24\n-d2E0/d(sin24)2 (c)(d)\n024680.000.040.080.12sin24=0.96E(Kx,Ky)-E0  [e2/elB]\nKxDq \n2468100.00.20.40.60.81.0sin24my/mxMRCharge Density Waves (CDW)Composite Fermi Liquid (CFL)0.00.20.40.60.81.00.000.040.08CDWCFL(II) K=(8,8) K=(0,3) K=(1,2) other K En-E0sin24Ne=16  my/mx=8\nCFL(I)(a)(b)\n0.00.20.40.60.81.0-10123\n5101520CDWCFL(II)CFL(I)dE0/d(sin24)\nsin24\n-d2E0/d(sin24)2 (c)(d)\n024680.000.040.080.12sin24=0.96E(Kx,Ky)-E0  [e2/elB]\nKxDq \n2468100.00.20.40.60.81.0sin24my/mxMRCharge Density Waves (CDW)Composite Fermi Liquid (CFL)0.00.20.40.60.81.00.000.040.08CDWCFL(II) K=(8,8) K=(0,3) K=(1,2) other K En-E0sin24Ne=16  my/mx=8\nCFL(I)(a)(b)\n0.00.20.40.60.81.0-10123\n5101520CDWCFL(II)CFL(I)dE0/d(sin24)\nsin24\n-d2E0/d(sin24)2 (c)(d)\n024680.000.040.080.12sin24=0.96E(Kx,Ky)-E0  [e2/elB]\nKxDq \nFig. 1. (Color online) The phase diagram and the energy spectra. Depending on the mass anisotropy my=mx, we identify the\npairing instability and density-wave instability of CFL when tuning the interaction via sin2\u0002, and the corresponding phase\ndiagram is shown in panel (a). For a \fxed mass ratio, e.g., my=mx= 8 in panel (b-d), the phase boundary is consistently\nidenti\fed from the evolution of energy spectra with sin2\u0002 (b) and the derivatives of the ground-state energy (c). In the charge\ndensity wave phase, the energy spectra along momentum Kxexhibits the quasidegenerate states that di\u000ber by a momentum\n\u0001q(d). Here, we consider a half-\flled Landau level with Ne= 16 electrons.\n(a)\n (b)\n(c)\n (d)\nFig. 2. (Color online) The static structure factors N(q). The nature of the di\u000berent phases in Fig. 1(b-c) can be identi\fed from\nthe static structure factor N(q) of the density-density correlation. Panels (a-d) show N(q) in the CFL phase with sin2\u0002 = 0:16\n(a) and sin2\u0002 = 0:48 (b) , as well as N(q) in the charge density wave phase with sin2\u0002 = 0:68 (c) and sin2\u0002 = 0:96 (d).The\ndashed line in (a), (b) and (c) indicates the Fermi surface of CFs. Note a factor of two between the scattering momentum and\nthe momentum of CFs. Here, we consider a half-\flled Landau level with Ne= 16 electrons and mass ratio my=mx= 8.3\nare the form factors for n= 0 andn= 1 Galilean\nLLs, respectively. Ln(x) is the Laguerre polynomial.\nThe anisotropic CFL can be achieved by introducing\nthe mass anisotropy, where q2\nm=gab\nmqaqbincludes the\nmetricgm= diag[p\nmy=mx;p\nmx=my] derived from the\nband mass tensor. In the isotropic limit (i.e., my=mx),\nthe CFL and MR states are stabilized at sin2\u0002 = 08,10\nand sin2\u0002 = 124{26, respectively. The corresponding\npairing instability in this limit, such as tuning sin2\u0002,\nhas been theoretically con\frmed43{45, though the nature\nof this transition is still controversial22{26. The mass\nanisotropy explicitly breaks the spatially rotational sym-\nmetry46{55, concealing another factor to trigger the insta-\nbility of CFL. Previous studies have demonstrated that\nCFL is remarkably robust against mass anisotropy when\nsin2\u0002 = 054, while the MR state is fragile against mass\nanisotropy and \fnally translates to a stripe state56when\nsin2\u0002 = 155. Then it is natural to investigate the pos-\nsible density-wave instability of CFL by tuning the in-\nteractions via sin2\u0002 from an anisotropic CFL state at\nsin2\u0002 = 0. Below we will detect such a possibility by\nsolving the Hamiltonian by ED57.\nOur numerical results are depicted in the phase dia-\ngram shown in Fig. 1(a). In the isotropic limit, we have\ncon\frmed the pairing instability of CFL when tuning the\ninteraction via sin2\u0002, consistent with previous studies.\nIn the presence of mass anisotropy, we \fnd the pairing\ninstability only survives in a small regime in the phase\nspace, and instead, the density-wave instability becomes\nthe dominant instability of CFL after rotational sym-\nmetry breaking, which can be triggered more easily by\nincreasing the mass anisotropy [see Fig. 1(a)].\nThe phase boundaries in Fig. 1(a) are identi\fed from\nboth the energy spectra and the derivatives of the\nground-state energy. Figure 1(b) shows an example of\nthe energy spectra as a function of sin2\u0002 for anNe= 16\nsystem with my=mx= 8. Further results of Ne= 12;14\nare given in Appendix C. The CFL state is robust up\nto sin2\u0002\u00190:64 upon tuning the interaction, which can\nbe further con\frmed from the derivatives of the ground-\nstate energy in Fig. 1(c). The energy gap in the spectra\nof CFL is induced by the shell-\flling e\u000bect on a \fnite\nsized system, which can be identi\fed by comparing the\nquantum number of the ground state obtained by ED and\nthe CFL wavefunctions on a torus33,43,57{59. The energy\nlevel crossing near sin2\u0002\u00190:32 represents the change\nof the CFL ground-state momentum sectors, in contrast\nto the phase transitions around sin2\u0002\u00190:64. We fur-\nther con\frm the nature of these phases by studying the\nstatic structure factor N(q) of the density-density cor-\nrelation,N(q) =1\nNh\u001aq\u001a\u0000qi=1\nNP\ni;jheiq\u0001Rie\u0000iq\u0001Rji,\nwhere\u001aq=PN\ni=1eiq\u0001Riis the Fourier transform of the\nguiding center density. As shown in Fig. 2(a-b) for\nsin2\u0002.0:64,N(q) exhibits a strong 2 kFscattering fea-\nture induced by the scattering among CFs close to the\nFermi surface. At sin2\u0002>0:64, there are two sharp\npeaks inN(q) in the same direction, which can be re-\ngarded as the hallmark of charge ordering with the wavevector determined by the position of the peaks. Here,\nN(q) displays a stripe feature.\nFurther increasing sin2\u0002&0:88, the peaks rotate from\n(qx;qy) = (0;\u0006q\u0003) to (qx;qy) = (\u0006q\u0003\u0003;0) as shown in\nFig. 2(c-d). Here, the wave vector \u0006q\u0003\u0003also can be iden-\nti\fed from the low energy spectra of such resulting phase\n[see Fig. 1(d)], where there is no recognizable gap sepa-\nrating the ground-state manifold from the excited states,\nand instead, the energy spectra displays a conspicuous\nset of quasi-degenerate states which di\u000ber by momentum\n\u0001qand satisfy \u0001 q=\u0006q\u0003\u0003. The line connecting the low-\nest energy states in each momentum sector has a zigzag\nstructure as shown in Fig. 1 (d), which only appears in\nthe energy spectra in one momentum direction, implying\na unidirectional charge density wave state.\nIII. RG ANALYSIS FROM CFL\nAs the Fermi surface and its instability are indicated\nin Fig. 2, it is natural to understand it within the con-\ntext of the Halperin-Lee-Read (HLR) theory8. Because\nthe instability is peaked at two antipodal Fermi points\n[see Fig 2 (c)], we use the patch theory (cf. Chapter\n18 of3for a review) to analyze the competing \ructu-\nations. The composite-Fermi surface is approximated\nby two patches60,61near the antipodal Fermi points:\nHf=\u0000isvF@x\u00001\n2K@2\ny, wheres=\u0006denotes the two\npatches, srefers to CF, and x;yare the normal and\ntangent directions of the Fermi surface, respectively. vF\nandKcapture the CF Fermi velocity and the curvature\nof the patch. Including the gauge \feld \ructuation, the\ntotal e\u000bective action is given by S=Sf+Sa+Sint,\nSf=X\nsZ\nd3x y\ns(@\u001c+Hf) s; (2)\nSa=Z\nkjkyj1+\u000fja(k)j2; (3)\nSint=X\nsseZ\nd3xa(x) y\ns(x) s(x); (4)\nwhereR\nk\u0011Rd3k\n(2\u0019)3,a(x) is the emergent gauge \feld,\nandeis the Yukawa coupling between the fermion and\ngauge boson. \u000fis the expansion parameter, and \u000f= 0\ncorresponds to the long-range Coulomb interaction60,61.\nThe patch theory is an e\u000bective description in the\nrangejkxj;k2\ny<\u0003 (note that kxandkyscale di\u000berently).\nWe address the IR properties of the theory by integrating\nout the high energy mode, \u0003 e\u0000l< k2\ny<\u0003 to generate\nRG equations, where l > 0 is the running parameter.\nThere is no renormalization to the boson propagator be-\ncause it is nonlocal. The rationale for using a nonlocal\nbare kinetic term for the gauge boson lies in the fact that\nthe boson kinetic potential does not receive corrections\nup to three-loop62. Taking into account of the fermion\nself-energy \u0006 s(p) =\u0000e2R\nkD(k)Gs(k+p)\u0019\u0000ie2\n4\u00192vFp0,4\nthe RG equation reads (see Appendix B)\ndg\ndl=\u000f\n2g\u0000g2\n4; (5)\nwhereg\u0011e2\n\u00192vF\u0003\u000f=2captures the e\u000bective Yukawa cou-\npling. The presence of a nontrivial stable \fxed point\ng\u0003= 2\u000fcorresponds to the NFL interacting strongly with\nthe gauge \feld.\nNext, we analyze the density-wave instability in the\nCFL. Because we are interested in the 2 kFinstability\nconnecting antipodal Fermi points, we can consider the\nscattering processes within the patch theory, namely,\nS=Sf+Sa+Sint+S4,S4=UR\nd3x y\n+ + y\n\u0000 \u0000.\nIn the patch theory, the four-body interaction is irrel-\nevant, which is consistent with the fact that the forward-\nscattering process does not a\u000bect the existence of the\nFermi surface63, and the perturbative calculation should\nbe valid. As indicated in Fig. 3(a), the renormalization\nto the four-body interaction reads\n\u0000(a)\n4=\u00002U2Z\nkG+(k)G\u0000(k)\u0019\u000b0p\n2\u00192p\n\u0003KU2\nvFl;\nwhere\u000b0\u0011\u0000(0;1)\u00190:219, and \u0000( n;x)\u0011R1\nxdttn\u00001e\u0000t\nis the incomplete Gamma function. Without gauge \ruc-\ntuation, the RG equation of dimensionless coupling con-\nstantu\u0011p\nK\u0003\n\u00192vFUis\ndu\ndl=\u0000u\n2+p\n2\u000b0u2; (6)\nwhich shows that an instability only occurs at \fnite inter-\naction strength. When uis large enough, i.e., u>1\n2p\n2\u000b0,\nit develops a wave-density instability with the 2 kForder\nparameter\u001e= y\n+ \u0000.\nNow we consider the e\u000bect of gauge \ructuations. As\nshown in Figs. 3(b), 3(c), the corrections from gauge \ruc-\ntuation read\n\u0000(b)\n4=\u00002e2\n3Z\nkGR(k)GL(k)D(k)\u0019\u000b0\n3\u00192e2u\nvFl;\n\u0000(c)\n4=\u0000e4\nN2Z\nkGR(k)GL(k)D2(k)\u0019\u000b0\n2p\n2\u00192e4\np\nK\u0003vFl:\nThere is no backreaction from the short-ranged interac-\ntion to the gauge \ructuation at one-loop order. Thus, in\nthe presence of \ructuating gauge bosons, the RG equa-\ntion becomes\ndu\ndl=\u00001\n2u+p\n2\u000b0u2+\u0010\u000b0\n3+3\n8\u0011\ngu+\u000b0\n2p\n2g2:(7)\nIn the RG equations, there are four \fxed points in the\n(u;g)-plane, including the Gaussian \fxed point (0 ;0),\nthe density-wave transition point (1\n2p\n2\u000b0;0) in the ab-\nsence of gauge bosons, and two new \fxed points emerging\nfrom the interplay between gauge \ructuations and short-\nranged interactions: FPCFL=\u00106\u0000(9+8\u000b0)\u000f\u0000p\nC(\u000f)\n24p\n2\u000b0;2\u000f\u0011\n\u0019(2p\n2\u000b0\u000f2;2\u000f) andFPT=\u00106\u0000(9+8\u000b0)\u000f+p\nC(\u000f)\n24p\n2\u000b0;2\u000f\u0011\n, where\nC(\u000f) = (81 + 144 \u000b0\u00001088\u000b2\n0)\u000f2\u000012(9 + 8\u000b0)\u000f+ 36 is a\nquadratic function in \u000f. When 0<\u000f<\u000fc,C(\u000f)>0, all of\nthe four \fxed points are physically accessible, and FPCFL\n(FPT) corresponds to the CFL \fxed point (density-wave\ntransition point). Here \u000fc\u00116(9\u00008(3p\n2\u00001)\u000b0)\n81+144\u000b0\u00001088\u000b2\n0is a posi-\ntive number. When \u000f < \u000fcthe blue points in Fig. 4(a)\ncorrespond to the Gaussian and the density-wave transi-\ntion point without gauge \ructuation, while the red points\ncorrespond to FPCFLandFPT.\nWe also note that, in the presence of gauge \ructua-\ntions, the critical coupling strength of the 2 kFdensity-\nwave transition is signi\fcantly reduced. More exotically,\nwhen\u000f=\u000fc,C(\u000fc) = 0, the CFL \fxed point and the tran-\nsition point collide with each other, as shown in Fig. 4(b).\nThe CFL transition \fxed point is unstable against 2 kF\ndensity-wave instability. We would like to point out that\nsuch a \fxed point collision is also found in previous liter-\natures64{66. When\u000f>\u000fc, the CFL is totally preempted\nby density-wave orders as shown in Fig. 4(c). These re-\nsults indicate that the NFL \fxed point is unstable if the\ngauge \ructuation is strongly enough. We also note that\ntheFPCFLis well controlled by \u000fexpansion as can be seen\nfromFPCFL\u0019(2p\n2\u000b0\u000f2;2\u000f) at small\u000f. Moreover, at the\nmerging point \u000fc, the values of two \fxed points FPCFL\nandFPTare coincident, and so are controlled provided\n\u000fcto be small. In the one-loop calculation, \u000fc\u00190:32<1,\nindicates the scenario of the \fxed-point collision is under\ncontrol.\nIV. DISCUSSIONS\nThe large portion of CFL in the phase diagram in\nFig. 1 suggests \u000f < \u000fc. Although it is unclear how\nthe bare interaction strengths, namely, the gauge cou-\npling and the short-ranged interaction, change with the\nmixed form factors, the RG analysis is able to predict\nthe wavevector of the density wave in the presence of\nthe mass anisotropy. This is because the bare gauge\ncoupling is enhanced by the mass anisotropy through\nthe Fermi velocity. Assuming ( u;g) = (u0;g0) for the\nisotropic CFL, we have ( u;g) = (u0;~\u000b1=4g0) at the\npatches k= (\u0006p2 ~mx\u0016;0) of the anisotropic Fermi sur-\nface, where ~ \u000b\u0011~mx=~mydenotes the mass anisotropy\nof CFs (we will consider ~ \u000b\u00151, since the opposite case\nis equivalent). ~ \u000bis related to the mass anisotropy of\nelectrons\u000bthrough ~\u000b=p\u000b67. It is easy to see that\n~\u000b1=4g0is the largest bare value in the elliptic Fermi sur-\nface, therefore, the above RG analysis predicts that the\n2kFinstability occurs at 2 kF= 2p2 ~mx\u0016, which connects\nthe Fermi points with the smallest Fermi velocity. This\nobservation is consistent with N(q) in Fig. 2(c) near the\ntransition point. Note that this is a gauge \ructuation\ninduced stripe transition.\nDeep inside the charge density wave phase, we also \fnd\nthe switch of stripe orientations, as shown in Figs. 2(c-5\n(a)\n (b)\n (c)\nFIG. 3. (Color online) The corrections to short-ranged four-fermion interactions within the patch theory. Panel (a) denotes\nthe correction from the four-fermion interaction, and panels (b-c) denote corrections from the gauge \ructuations.\n0 0.5 100.511.5\nu/u0*g/g*\n● ●● ●\n(a)\n0 0.5 100.511.5\nu/u0*g/g*\n● ●● ● (b)\n0 0.5 100.511.5\nu/u0*g/g*\n● ● (c)\nFIG. 4. (Color online) The RG \row diagrams of ( u;g) at di\u000berent \u000f. The blue points show the Gaussian \fxed point and stripe\ntransition point without gauge \ructuations. Red points show the NFL \fxed point and stripe transition point in the presence\nof gauge \ructuations. Fig. 4(a) shows four \fxed points when \u000f < \u000f c. Figs. 4(b), 4(c) show the RG \row for \u000f=\u000fc,\u000f > \u000f c,\nrespectively. The dashed line indicates the trajectory of two nontrivial \fxed points in the presence of gauge \ructuations. After\ntheir collision, the \fxed points become imaginary values, and disappeare from the \row diagram.\nd). This phenomenon might be beyond the CFL physics\nsince it is further away from the critical point, however,\nit can be attributed to the reduction of Hartree energy\ncost when the stripe orientation coincides with the di-\nrection of the smaller mass55. Moreover, from our ED\nresults, the energy level crossing [see Fig. 1(b)] and the\nsudden jump in the \frst order derivatives [see Fig. 1(c)]\nsuggest the transition from CFL to charge-density wave\nmight be \frst order. We should note that it is still under\ndebate whether the 2 kFdensity-wave transition is con-\ntinuous. While Altshuler et al.36argues that a \frst-order\ntransition occurs due to the strong 2 kF\ructuation at low\nenergies, a more recent article by Sykora et al.37shows\na second-order transition is also possible. It will also be\nan excellent task to investigate the critical phenomena in\n2kFtransitions of NFL, which we leave for future works.\nHere, the RG analysis does not rely on the particle-hole\nsymmetry. Therefore, the density-wave instability should\nbe possible to extend to CFL at other \flling factors, such\nas\u0017= 1=4. A recent numerical study78suggests that, for\n\u0017= 1=4 with mass anisotropy, the CFL state in n= 0LL\nis robust while a stripe-like phase emerges in n= 1LL,\nthen one can expect that tuning the mixed form factor\nwould trigger the density-wave instability of CFL at \u0017=\n1=4 by the same mechanism proposed here.\nThe experimental probe of the various instabilities of\nCFL still presents many challenges and under intensive\ninvestigations. Previous studies mainly focus on detect-ing the pairing instability of CFL, which has been pro-\nposed by tuning the subband level crossings68,69or ap-\nplying hydrostatic pressure70{72in GaAs quantum wells,\nor by tuning either the perpendicular magnetic \feld\nor the interlayer electric bias in bilayer graphene40{42.\nIn particular, the hydrostatic pressure experiments70{72\nhave found that tuning the pressure through Pc1would\ntrigger the transition from MR to an anisotropic com-\npressible phase, which is consistent with either a stripe\nphase43,55,56or nematic phase28. Interestingly, further\nincreasing the pressure to Pc2leads to a transition to an\nisotropic compressible phase, which might be relevant to\nthe density-wave instability, particularly considering that\nthe pressure is believed to change the LL mixing parame-\nters72. However, we should also note the pressure-driven\nplatform is hard to be captured by an ideal Hamiltonian\nmicroscopically, which would be an interesting direction\nfor a future study. Moreover, the mixed form factor could\nbe realized and tunable in bilayer graphene by an inter-\nlayer electric bias and magnetic \feld39{42, then break-\ning the rotational symmetry may potentially probe the\ndensity-wave instability of CFL. The mass anisotropy ex-\nists in AlAs quantum wells73,74in nature or could be\nintroduced by applying an in-plane \feld75or uniaxial\nstrain76,77, so then realizing a density-wave instability on\ntop of an anisotropic CFL is also a promising direction\nto pursue experimentally.6\nACKNOWLEDGEMENTS\nWe thank Lukas Janssen, Sung-Sik Lee, D. N. Sheng,\nInti Sodemann and Brian Swingle for helpful discussions.\nS.-K.J. is supported by the Simons Foundation via the\nIt From Qubit Collaboration. We are grateful to D. N.\nSheng for kindly providing some computational resources\nto \fnish some numerical calculations in this work. The\npart of this work carried out at Harvard was supportedby the funding via Ashvin Vishwanath. The part of\nthis work carried out at KITS was supported by the\nFundamental Research Funds for the Central Universi-\nties, the start-up funding of KITS at UCAS, and the\nStrategic Priority Research Program of CAS (Grant No.\nXDB33000000).\nShao-Kai Jian and Zheng Zhu contributed equally to\nthis work.\nAppendix A: Composite fermi liquid\nThe action of two patches is given by\nS=Sf+Sa+Sint (A1)\nSf=X\nsZ\nd3x y\ns(@\u001c\u0000isvF@x\u00001\n2K@2\ny) s (A2)\nSa=Zd3k\n(2\u0019)3jkyj1+\u000fja(k)j2(A3)\nSint=X\nsZ\nd3xsep\nNa y\ns s (A4)\nwheres=\u0006denotes the two patches,  sandarefer to composite fermion and emergent gauge \feld, respectively.\nvFandKcapture the fermi velocity and curvature of the patch, and eis the Yukawa coupling between fermion and\ngauge boson. The above action is believed to describe various interesting systems, such as U(1) quantum spin liquids\nwith a large spinor fermi surface and composite fermi liquids in the half-\flled Landau level. Here, we mainly focus on\nthe latter case, and the above action is a patch description of the Halperin-Lee-Read (HLR) theory8. The summation\noverN\ravors of the patch fermion is implicit, and \u000fis the expansion parameter. \u000f= 0 corresponds to the long-range\nCoulomb interaction60,61. In the noninteracting limit, the action is invariant under scaling transformation dictated\nby the scaling dimensions,\n[kx] = 1;[ky] =1\n2;[!] = 1;[ ] =3\n4;[a] = 1\u0000\u000f\n4;[e] =\u000f\n4: (A5)\nThe RG calculation is controllable in the large- Nand small\u000f\u00181\nNexpansion60. The patch theory is an e\u000bective\ndescription in the range jkxj;k2\ny<\u0003. In the following, we integrate out the high energy mode,p\n\u0003e\u0000l<jkyj<p\n\u0003\nto generate RG equations, where l >0 is the running parameter. There is no renormalization to boson propagators\nbecause it is nonlocal. The rationale for using a nonlocal bare kinetic term for the gauge boson lies in the fact that\nthe boson kinetic potential does not receive corrections up to three-loop62. The fermion self-energy is (Fig. 5(a))\n\u0006s(p) =\u0000e2\nNZd3k\n(2\u0019)3D(k)Gs(k+p) =\u0000e2\nNZd3k\n(2\u0019)31\njkyj1+\u000f1\n\u0000i(k0+p0) +svF(kx+px) +1\n2K(ky+py)2(A6)\n=\u0000e2\nNZd3k\n(2\u0019)3i\u0019sgn(k0+p0)\u000e(svF(kx+px) +1\n2K(ky+py)2)\njkyj1+\u000f(A7)\n=\u0000ie2\n2(2\u0019)2NvFZ\ndkydk0sgn(k0+p0)\njkyj1+\u000f=\u0000ie2\n2\u00192NvFp0Zp\n\u0003\np\n\u0003e\u0000ldky1\njkyj1+\u000f\u0019\u0000ie2\n4\u00192NvFp0l; (A8)\nThe vertex correction is (Fig. 5(b))\n\u00003=X\nsZd3k\n(2\u0019)3G2\ns(k)D(k) =X\nsZd3k\n(2\u0019)31\njkyj1+\u000f1\n[\u0000i(k0+p0) +svF(kx+px) +1\n2K(ky+py)2]2; (A9)\nwhich vanishes because the poles of k0lie in the same plane. In terms of the dimensionless coupling constant\ng\u0011e2\n\u00192vF\u0003\u000f=2that captures the e\u000bective Yukawa coupling, we have following RG equations,\ndg\ndl=\u000f\n2g\u0000g2\n4: (A10)7\n(a)\n (b)\nFIG. 5. The Feynman diagrams: (a) fermion self-energy, and (b) fermion-boson vertex.\n(a)\n (b)\nFIG. 6. The Feynman diagrams in the particle-particle channel. (a) is the one-loop corrections of from the four-fermion BCS\ninteraction. (b) is the interpatch interaction resulting from integrating out high energy gauge \rucuation.\nThe presence of a nontrivial stable \fxed point g\u0003= 2\u000fcorresponds to the non-fermi liquid (NFL) interacting strongly\nwith the gauge \feld.\nAppendix B: Cooper instability and 2kFdensity-wave instability\nDespite the long-range interactions between the composite fermion mediated by the gauge \feld, there are local\ninteractions between the composite fermion that might generate pairing or stripe instability. Thanks to the Pauli\nexclusion principle of fermions, among in\fnite channels of four-fermion interactions only BCS and the forward-\nscattering channel survive in the low energy63. For simplicity, we will send N= 1 in the following and consider\nfour-fermion interactions. We \frst consider the BCS Hamiltonian for the nondegenerate fermi surface,\nHBCS=\u0000Zd2k\n(2\u0019)2d2k0\n(2\u0019)2V(k;k0) y(k) y(\u0000k) (k0) (\u0000k0); (B1)\nwhere denotes the fermi surface, and Vis the strength of the BCS interaction. It is well known that the pairing\ninstability is marginally relevant for fermi liquids63. The RG equation is given by (Fig. 6(a), and we consider spherical\nfermi surface for simplicity),\ndvj\ndl=\u0000v2\nj; (B2)\nwherevj=kF\n2\u0019vfVjandVj=Rd\u0012\n2\u0019V(\u0012)ei\u0012j. Di\u000berent from fermi liquids, the presence of an emergent gauge boson\nin a composite fermi liquid suppress the pairing instability. Indeed, as shown in Fig. 6(b), integrating out the high-\nenergy mode of gauge \ructuation will generate an interpatch interaction23. Here we review the calculations23. For a\nsmall-angle BCS interaction, \u0012\u00180, we have the correction from the gauge \ructuation,\n\u000eV(k1;k2) =e2\n2ND(k1\u0000k2); (B3)\nwhile it also contributes to V(\u0012\u0018\u0019). Taking both of these into consideration, the corrections to the BCS interaction\nare\n\u000evj=kF\n2\u0019vfZd\u0012\n2\u0019\u000eV(\u0012)ei\u0012j\u00191\n\u0019vfe2\nNZp\n\u0003\np\n\u0003e\u0000ldk\n2\u00191\nk1+\u000f\u0019e2\n4\u00192vfNl: (B4)\nTherefore, including the gauge \ructuation, the RG equation reads\ndv\ndl=\u0000v2+g\n4: (B5)8\nBecause of the suppression from gauge \ructuation, the BCS instability is no longer marginally relevant. Instead, it\nrequires a \fnite bare BCS interaction to drive the composite fermi liquid into the paired state. Note that in the\ncontext of the half-\flled Landau level, for example, in the \u0017= 5=2 \flling fraction, the system favors p+ippairing,\nwhich is the famous Moore-Read Pfa\u000ean state24,26.\nOn the other hand, we consider the four-fermion interaction within the patch theory in the following,\nS=Sf+Sa+Sint+S4; (B6)\nS4=UZ\nd3x y\n+ + y\n\u0000 \u0000: (B7)\nIn the patch theory, the four-body interaction is irrelevant, which is consistent with the fact that forward-scattering\ndoes not a\u000bect the existence of a fermi surface, and the perturbative calculation should be valid. The correction reads\n\u0000(a)\n4=\u00002U2Zd3k\n(2\u0019)3G+(k)G\u0000(k) =2p\n2KU2\nvFZd3q\n(2\u0019)31\nq0+i(qx+q2y)1\nq0\u0000i(qx\u0000q2y)(B8)\n=4p\n2KU2\n(2\u0019)2vFZp\n\u0003\np\n\u0003e\u0000ldqyZ1\nq2ydqxe\u0000q2\nx=\u00032\nqx\u0019p\n2\u0000(0;1)\n\u00192p\n\u0003KU2\nvFl; (B9)\nwhere \u0000(n;x)\u0011R1\nxdttn\u00001e\u0000tis the incomplete Gamma function, and \u0000(0 ;1)\u00190:219. In the calculation, we have\nintroduced a regularization function e\u0000q2\nx=\u00032to regularize the UV divergence. Without gauge \ructuation, the RG\nequation of the dimensionless coupling constant u\u0011p\nK\u0003\n\u00192vFUis\ndu\ndl=\u00001\n2u+p\n2\u0000(0;1)u2; (B10)\nwhich shows that an instability only occurs at a \fnite interaction strength, and the fermi liquid is perturbatively\nstable.\nThe corrections from gauge \ructuation read\n\u0000(b)\n4=\u00002e2\n3Zd3k\n(2\u0019)3GR(k)GL(k)D(k) =2p\n2Ke2u\n3vFZd3q\n(2\u0019)31\nq0+i(qx+q2y)1\nq0\u0000i(qx\u0000q2y)1\njp\n2Kqyj1+\u000f(B11)\n=4e2u\n3(2\u0019)2vFZp\n\u0003\np\n\u0003e\u0000ldqy\njqyj1+\u000fZ1\nq2ydqxe\u0000q2\nx=\u00032\nqx\u0019\u0000(0;1)\n3\u00192e2u\nvFl; (B12)\nand\n\u0000(c)\n4=\u0000e4Zd3k\n(2\u0019)3GR(k)GL(k)D2(k) =p\n2Ke4\nvFZd3q\n(2\u0019)31\nq0+i(qx+q2y)1\nq0\u0000i(qx\u0000q2y)1\njp\n2Kqyj2(1+\u000f)(B13)\n=2e4\n(2\u0019)2N2p\n2KvFZp\n\u0003\np\n\u0003e\u0000ldqy\njqyj2(1+\u000f)Z1\nq2ydqxe\u0000q2\nx=\u00032\nqx\u0019\u0000(0;1)\n2p\n2\u00192N2e4\np\nK\u0003vFl: (B14)\nThese corrections lead to the RG equations in the main text.\nAppendix C: Finite Size E\u000bect\nIn the main text, we mainly show the results for Ne= 16 systems, but we also have checked the other system and\nfound similar results, as shown in Fig. 7 and Fig. 8 for Ne= 12 andNe= 14 systems. 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(Color online) (a)The evolution of energy spectra as a function of sin2\u0002 for a half-\flled Landau level with Ne= 12\nelectrons and mass ratio my=mx= 8. The nature of the di\u000berent phases can be identi\fed from the static structure factor N(q)\nof the density-density correlation. Panels (b-c) show N(q) in the CFL phase with sin2\u0002 = 0:1 (a) and in the charge density\nwave phase with sin2\u0002 = 0:6 (c).\n0.0 0.2 0.4 0.6 0.8 1.00.000.040.080.12\nCDW K=(7,0)\n K=(0,7)\n other K En-E0\nsin2Ne=14  mx/my=8\nCFL(a) (b) (c)\n-8\n-4\n0\n4\n8-8-4048\nqy qxsin2=0.2 \n023\n-8\n-4\n0\n4\n8-8-4048\nqy qxsin2=0.8 \n023\nFig. 8. (Color online) (a)The evolution of energy spectra as a function of sin2\u0002 for a half-\flled Landau level with Ne= 14\nelectrons and mass ratio my=mx= 8. The nature of the di\u000berent phases can be identi\fed from the static structure factor N(q)\nof the density-density correlation. 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Lett. 119, 016402\n(2017).\n78Zheng Zhu, Liang Fu, to appear."    },    {        "title": "2003.09657v2.Magnetic_breakdown_and_charge_density_wave_formation__a_quantum_oscillation_study_of_the_rare_earth_tritellurides.pdf",        "content": "Magnetic breakdown and charge density wave formation: a quantum oscillation study\nof the rare-earth tritellurides\nP. Walmsley,1, 2S. Aeschlimann,1, 2, 3, 4J. A. W. Straquadine,1, 2P. Giraldo-Gallo,5\nS. C. Riggs,6M. K. Chan,7R. D. McDonald,7and I. R. Fisher1, 2\n1Department of Applied Physics and Geballe Laboratory for Advanced Materials,\nStanford University, Stanford, California 94305, USA\n2Stanford Institute of Energy and Materials Science, SLAC National Accelerator Laboratory,\n2575 Sand Hill Road, Menlo Park 94025, CA 94305, USA.\n3Institute of Physical Chemistry, Johannes Gutenberg-University Mainz, Duesbergweg 10-14, 55099 Mainz, Germany\n4Graduate School Materials Science in Mainz, Staudingerweg 9, 55128, Mainz, Germany\n5Department of Physics, Universidad de Los Andes, Bogot\u0013 a, Colombia\n6National High Magnetic Field Laboratory, Tallahassee, Florida 32310, USA\n7Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA\n(Dated: May 12, 2020)\nThe rare-earth tritellurides ( RTe3, whereR= La, Ce, Pr, Nd, Sm, Gd, Tb, Dy, Ho, Er, Tm,\nY) form a charge density wave state consisting of a single unidirectional charge density wave for\nlighterR, with a second unidirectional charge density wave, perpendicular and in addition to the\n\frst, also present at low temperatures for heavier R. We present a quantum oscillation study in\nmagnetic \felds up to 65 T that compares the single charge density wave state with the double charge\ndensity wave state both above and below the magnetic breakdown \feld of the second charge density\nwave. In the double charge density wave state it is observed that there remain several small, light\npockets with the largest occupying around 0.5% of the Brillouin zone. By applying magnetic \felds\nabove the independently determined magnetic breakown \feld, the quantum oscillation frequencies\nof the single charge density wave state are recovered, as expected in a magnetic breakdown scenario.\nMeasurements of the electronic e\u000bective mass do not show any divergence or signi\fcant increase on\nthe pockets of Fermi surface observed here as the putative quantum phase transition between the\nsingle and double charge density wave states is approached.\nI. INTRODUCTION\nThe Fermiology of compounds that harbour charge-\ndensity wave (CDW) order has attracted renewed inter-\nest due to the discovery of CDW order in several cuprate\nhigh-temperature superconductors6{13. The results of\nquantum oscillation studies in the cuprates appear to\nbe consistent with a divergence of the electronic e\u000bec-\ntive mass,m\u0003, on approach to optimal doping and possi-\nbly also on the very underdoped region approaching the\nMott transition14{16. In both cases this e\u000bect is coin-\ncident with a dome of CDW order17. Close to optimal\ndoping it remains unclear as to whether this divergence\noccurs as a result of a CDW quantum critical point, a\nquantum critical point associated with the pseudogap,\nor indeed some other mechanism, whereas on the un-\nderdoped side of the cuprate phase diagram, there are\nquantum phase transitions between CDW, spin density\nwave, and a Mott insulating phase that could lead to a\ndivergingm\u0003. In addition, as a point of principle, the\ne\u000bect of disorder (implicit due to chemical substitution)\non a unidirectional incommensurate CDW in tetragonal\nmaterials leaves only a nematic phase transition, raising\nthe possibility that the putative CDW quantum critical\npoint mentioned above would have a nematic character18.\nAs there is no clear precedent for an enhancement of m\u0003\naround a CDW quantum critical point, it is clear that\nthere is a need for a model system in which to study\nsuch a scenario. In this work we examine a promisingcandidate material.\nThe rare-earth tritellurides ( RTe3whereRcan be La,\nCe, Pr, Nd, Sm, Gd, Tb, Dy, Ho, Er, Tm or Y) form\na family of materials in which the CDW transition tem-\nperature can be smoothly tuned by lanthanide contrac-\ntion without doping the system or introducing disorder1.\nAs shown in the phase diagram in Fig. 1, the transition\ntemperature of the primary, unidirectional CDW order is\ntuned from greater than 450 K in LaTe 3down to 244 K in\nTmTe 3, but of particular interest here is the emergence\nof a second, perpendicular, unidirectional CDW that \frst\nappears in a stoichiometric compound in TbTe 3at 41 K\nand strengthens with lanthanide contraction up to 186 K\nin TmTe 31,2. As the local rare-earth moments don't seem\nto a\u000bect the formation of the CDW states, the choice of\nRacts principally as chemical pressure, and as such the\nphase diagram can be reframed in terms of the lattice\nparameter as in Fig. 1. This framing implies that contin-\nuous compression (expansion) of the lattice from GdTe 3\n(TbTe 3) could drive the system through a quantum phase\ntransition, potentially yielding a quantum critical point\nacross which to search for an enhanced m\u0003.\nThe beauty of this system is that, as a stoichiometric\nseries, disorder does not need to be introduced in or-\nder to tune the CDW phases and thus quantum oscil-\nlations can be observed down to low magnetic \felds for\nallR. The Fermi surface of RTe3has been studied pre-\nviously by ARPES in both the single and double-CDW\nstates in CeTe 3and ErTe 3respectively3{5, by positronarXiv:2003.09657v2  [cond-mat.str-el]  10 May 20202\n4.284 .304 .324 .340100200300400q\n1 ≈ 2c*/7q\n2 ≈ a*/3 T (K)a\n (Å)q1 ≈ 2c*/7 a)TmErH oDyTbG dS m(Nd, Ce, La)R  =0\n.00.5Double-CDWi )i )i\ni)c)b )N\no CDW0\n.00.5X X Δ\n1Δ2Δ2q\n2Δ1-\n0.50 .00 .5-0.5S\ningle-CDWSingle-C\nDWii)kx / a*k\nz / c*-0.50 .00 .5-0.5X Δ1X\nq1k\nx / a*k\nz / c*Δ1\nFIG. 1. a) Phase diagram of RTe3shown without the low-\ntemperature magnetic phases (see App. A)1,2as a function of\nin-plane lattice parameter a. The top axis marks the speci\fc\nrare-earth ( R) ions that yield these lattice parameter values.\nThere are two unidirectional, incommensurate CDWs with\nq1\u00192/7c\u0003andq2\u00191/3a\u0003. Both are present at low tem-\nperatures in the heaviest R(shortesta, Tm - Tb), but just\nq1for lighterR(longera, Gd, Sm, Nd, Ce, La). The com-\npounds where R=La, Ce and Nd are known to have a unidi-\nrectional CDW ordering with q1that onsets at temperatures\nabove those measured. b) & c) A 2-D tight-binding model\nthat captures the essential structure of the Fermi surface of\nRTe3is shown in gray, with the \frst Brillouin zone shown\nas a light green box. Note that this model ignores a subtle\nb-axis warping and bilayer splitting. The full Fermi surface\nwould be re\rected across the kx= 0 line. In b.ii) a portion\nof the Fermi surface observed by ARPES measurements3in\nthe single-CDW state is sketched in green (shown in the un-\nfolded zone). c) Illustration of the remaining portions of the\nunfolded Fermi surface resolved by ARPES once gaps \u0001 1and\n\u00012have opened3{5. c.i) shows this for the double-CDW state\n(orange) and c.ii) for the single-CDW state (green) with the\nCDW vectors q1andq2illustrated by the green and orange\narrows respectively. Figures b) & c) are adapted from Brouet\net al.3and Moore et al.5.\nannihilation in GdTe 319, and by quantum oscillations in\nLaTe 320and GdTe 321. A 2D tight-binding model has\nbeen found to provide a good description of the Fermi\nsurface and is shown in gray in Figs. 1b) & c) (bilayer\nsplitting has been omitted). The Fermi surface derives\nfrom thepxandpzorbitals of the nearly-square Te net\nbilayer (remembering that the baxis is out-of-plane in\nRTe3), forming almost perpendicular, quasi 1-D sheets\nwith weak hybridisation at their crossing points and bi-\nlayer splitting3. The primary CDW, with q1\u00192=7c,leaves the material metallic due to an imperfect nesting\ncondition. ARPES and quantum oscillation studies have\nshown that the diamond-shaped pocket at the X point\nis una\u000bected by the folding, that there is likely an elon-\ngated pocket along an imperfectly nested sheet, and then\nanother small pocket elsewhere in the zone. Figure 1b.ii)\nillustrates a portion of Fermi surface resolved by ARPES\nin the single-CDW state that con\frmed the survival of\nthe X pocket, as well as a larger irregular pocket as a\nproduct of the zone folding. Fig. 1c.ii) shows where\nthe primary CDW gap \u0001 1opens on the unfolded Fermi\nsurface3. Further ARPES data shows that as the sec-\nond CDW, q2\u00191=3c, folds the Brillouin zone again, a\nsecondary gap \u0001 2also opens on the largest remaining\npockets as illustrated in orange Fig.1c.i)5.\nIn this study we use quantum oscillation measure-\nments to study the folding and gapping of the Fermi\nsurface across the implied single to double-CDW quan-\ntum phase transition. Key to understanding the data\nare magnetic breakdown phenomena. We observe that\nabove the independently calculated breakdown \feld for\nthe second CDW gap the quantum oscillation frequen-\ncies match those of the singly folded zone, whereas be-\nlow the breakdown \feld only some very low frequencies\nremain, consistent with Fermi surface composed of just\nvery small pockets. The quantum oscillation spectrum\nchanges very little upon Lanthanide contraction in the\nsingle-CDW state, indicating that the choice of Rhas a\nnegligible e\u000bect on the area of the Fermi surface despite\nthe size of the gap changing signi\fcantly. The temper-\nature dependence of the quantum oscillation amplitudes\nshow that there is no observed enhancement of m\u0003on\napproach to the putative quantum phase transition.\nII. METHODS\nSingle crystals of RTe3were grown via a self-\rux tech-\nnique described elsewhere23. Quantum oscillation mea-\nsurements up to 14 T and 16 T were performed in com-\nmercially available magnets from Quantum Design and\nCryogenic Ltd respectively. Measurements to 35 T DC\n\felds were performed at the National High Magnetic\nField Laboratory in Tallahassee and measurements to\n65 T at the pulsed-\feld facility at Los Alamos National\nLaboratory. In DC \felds, quantum oscillations were mea-\nsured in the electrical resistivity (Shubnikov-de Haas Os-\ncillations) along the crystallographic b-axis measured by\na quasi-montgomery technique. The resistivity was mea-\nsured via a Stanford Research SR830 lock-in ampli\fer\nand a Princeton Applied Research Model 1900 Low Noise\nTransformer that added gains of 100 or 1000 owing to the\nvery low resistance of the samples. The excitation was\ntypically 1 mA at 10 - 200 Hz. Measurements in pulsed\nmagnetic \felds utilised a mutual inductance technique24,\nwhereby the sample is mounted on top of a \rat-wound\ninductive coil that forms a tank circuit in combination\nwith the coaxial line capacitance. In this con\fguration,3\n05 1 01 5202 53 03 50481216d\n)c )b) \nTbTe3 (1.7K) \nGdTe3 (1.8K) \nHoTe3 (1.5K)R (B) / R 0B\n (T)a)0\n.040 .080 .120 .160 .20-4-2024 TbTe3 (1.7K) \nGdTe3 (offset by -2) (1.8K) \nHoTe3 (offset by -4) (1.5K)[ R (B) - Rbgrd ] / R 0B\n -1 (T -1)0\n2 04 06 01.841.861.881.90 TmTe3 (1.5K)f (MHz)B\n (T)0.050 .100 .150 .20-400-2000200400 TmTe3 (1.5K)f - fbgrd (Hz)B\n -1 (T -1)\nFIG. 2. a)b-axis resistance as a function of magnetic \feld, R(B), normalised by the zero-\feld value R0for three representative\nmeasurements (TbTe 3, GdTe 3and HoTe 3as red, black and dark blue lines respectively), one from each DC magnet (see\nsection II). Smooth, non-oscillating background estimates, Rbgrd=R0for each measurement are shown as dashed lines. b) The\noscillating component of the data extracted from a) by subtraction of the smooth background from the data plotted as a\nfunction of inverse \feld to reveal periodic oscillations indicative of quantum oscillations. c) Representative mutual inductance\ndata shown as the mixed-down frequency of the tank circuit for TmTe 3, with the smooth background estimate shown as a\ndashed line. d) The oscillating component of the signal shown by subtraction of the non-oscillating background, fbgrdas a\nfunction of inverse magnetic \feld, again revealing periodic oscillations that are consistent with quantum oscillations.\nchanges in the resonant frequency fof the tank circuit\nre\rect changes in the average in-plane conductivity of\nthe sample. The magnetic \feld was oriented parallel\nto the crystallographic baxis (out-of-plane) in all mea-\nsurements The presence of rare-earth magnetism in most\nmembers of the RTe3series meant that the samples had\nto be encased in epoxy (Devcon 5-minute epoxy) to pre-\nvent delamination and to secure the crystals against the\nlarge magnetic torques that can arise due to the large\ncrystal \feld anisotropy. Encasing the samples in epoxy\ndoes not appear to signi\fcantly change their resistivity or\nCDW transition temperatures (constant to within 0.5 K,\naround 0.1% ), indicating that any pressure applied by\nthe epoxy must be small.\nIII. RESULTS\nA. Magnetic Breakdown\nRepresentative magnetoresistance data is shown in Fig.\n2(a) for each of the DC magnets used in this study, with\nrepresentative mutual inductance data taken in pulsed\n\feld also shown in Fig. 2(c). In order to isolate the pe-\nriodic oscillations associated with quantum oscillations,\na smooth background (dashed lines in Figs. 2(a)&(c) ) isremoved and the data plotted versus inverse \feld. The\nlow temperature magnetic phases do not appear to sig-\nni\fcantly a\u000bect the data, as shown in Appendix A, and\nare therefore not considered in the following analysis and\ndiscussion.\nIt can be seen even in the raw data in Fig. 2 that\nthe dominant quantum oscillation frequencies in HoTe 3\nand TmTe 3are much lower than those in GdTe 3, as ex-\npected from the additional folding of the Fermi surface\nby the second CDW for heavier R(Fig. 1). TbTe 3on the\nother hand, with two CDWs, seems to yield quantum os-\ncillation frequencies that are more similar to GdTe 3, de-\nspite having a second CDW. This can be understood by\nconsidering the approximate \feld-scale associated with\nmagnetic breakdown of the second CDW gap. By in-\nvoking the Blount criterion for magnetic breakdown25,\n\u0016h!c> E2\ng=EF, where \u0016h!cis the cyclotron frequency,\nEgthe gap energy, and EFthe Fermi energy, the mag-\nnetic breakdown \feld of the second CDW gap, B0, can\nbe estimated independently of the present data26.Egis\nobtained from the single particle excitation gap of the\nsecond CDW as measured by optical spectroscopy by Hu\net al2728. Note that there has been no direct measure-\nment of the single particle excitation gap for the second\nCDW of TbTe 327and so the gap magnitude has been as-\nsumed to scale proportionately with the transition tem-4\n05 001 0001 5000 5 001 0001 5000 5 001 0001 5004.274 .284 .294 .304 .314 .324 .3454.40020406080T\nbTe3: 2T - 7TFFT amplitude (arb. u.)TmTe3: 6T - 65TE\nrTe3: 2.8T - 14TH\noTe3 #1: 3T - 16TH\noTe3 #2: 6T - 21.6TF\n (T)DyTe3: 6T - 12T/s98/s32+/s100/s50/s100/s98\n/s100/s50\n/s97/s51/s97∗ 1/s982/s103\n/s97/s98\n1Two CDWs, B < B0G\ndTe3 #1: 4.5T - 14TF\n (T)GdTe3 #2: 15T - 34.5T7\nT - 34.5TS\nmTe3: 9T - 14T5\n.3T - 14TN\ndTe3: 11T - 14T4\nT - 14TL\naTe3: 4.5T - 14TOne CDW only/s50\n/s982/s97/s50\n/s98/s50/s98/s98/s32/s43/s32/s100/s100/s32/s45/s32/s98/s100\n/s32/s45/s32/s98/s100\n/s32/s45/s32/s98/s51/s98/s103/s50/s98/s45/s100/s50\n/s97/s50/s98/s32/s43/s32/s100/s97\n/s32+/s98/s50/s98/s51/s98/s50/s98/s98\n/s32+/s32/s100/s98\n/s32+/s100/s103 /s50/s100/s50\n/s100/s103/s50/s100/s98\n +/s97/s50/s98/s50\n/s98/s98 +/s100/s100\n/s32/s45/s32/s98/s50/s98/s103\n/s51/s98/s98\n +/s100/s103/s50\n/s98/s98/s100 c)d )b )T wo CDWs, B > B0T\nbTe3 #1: 9.6T - 14TT\nbTe3 #2: 14T - 34.5THoTe3 #2: 40T - 65TF\n (T)DyTe3: 13.6T - 16Ta)/s51\n/s98/s50\n/s100  B0 (measured 2Δ) \n B0 (estimated 2Δ) \n One CDW \n Two CDWs,  B < B0 \n Two CDWs,  B > B0B (T)a\n (Angstroms)#1TmErH oD yT bG dS mNdL a#\n2#2#\n1#2#\n1\nFIG. 3. (a) A guide showing the \feld ranges across which the FFTs shown in (b), (c) and (d) were obtained for each member\nof theRTe3series studied here. Each R(top axis) is placed according to its alattice parameter at 300 K (bottom axis)22,\nnoting that overlapping ranges have been symmetrically o\u000bset in xfor clarity. For compounds with two CDWs, the calculated\ncharacteristic breakdown \feld for the lower temperature (smaller gap) CDW is shown as a star. The FFTs are thus grouped into\nthree categories; measurements in the double-CDW state with FFTs obtained from a magnetic \feld range below B0(orange\npanel and lines, panel (b) ), measurements in the double-CDW state with FFTs derived from a magnetic \feld range above B0\n(blue panel and lines, panel (c) ), and measurements in the single-CDW state (green panel and lines, panel (d) ),. Note that\npanels (b), (c) and (d) are all plotted to the same xscale for comparison. Peaks in the FFTs shown in panel (c) and (d) are\nlabelled to identify primary frequencies from mixing frequencies and harmonics as discussed further in the main text. The\nprimary\u000b,\fand\u000efrequencies are highlighted by yellow ribbons to more clearly show the consistency in their values as R\nchanges. In panel (d), NdTe 3, SmTe 3and GdTe 3#2 have an additional FFT of the same data restricted to a higher range\nof \felds to highlight high frequency components (frequencies below 700 T are omitted from these curves for clarity). Data in\nthese plots were taken at \fxed temperatures between 1.5 K and 2 K.5\n04 08 01 201 602 00TbTe3: 2T - 7TTwo CDWs, B < B0T\nmTe3: 6T - 65TE\nrTe3: 2.8T - 14TH\noTe3 #1: 3T-16TH\noTe3 #2: 6T - 21TFFT amplitude (arb. u.)F\n (T)DyTe3: 6T - 12T\nFIG. 4. An expanded view of the quantum oscillation fre-\nquency spectrum for B<B 0in compounds with two CDWs.\nperature relative to its neighbour, DyTe 3. The Fermi\nmomentum and e\u000bective mass are obtained from previ-\nous quantum oscillation measurements by Ru et al20to\ncalculateEF, and we use the \ffrequency for this analy-\nsis because it is easily resolvable and reliably ascribed to\nthe X pocket, which is known to gapped by the second\nCDW from ARPES measurements (as illustrated in Fig.\n1c).i))5,20.\nFigure 3(a) shows the resulting values of B0as stars.\nWhileB0is not a sharp transition line, but rather the\npoint at which the probability of tunneling via magnetic\nbreakdown has reached 1 =e, the data can nonetheless be\nplaced into three groups; data at \felds below B0in the\ndouble-CDW state (orange in Fig. 3), data at \felds above\nB0in the double-CDW state (blue), and data taken in\nthe single-CDW state (green). The quantum oscillation\nfrequency spectrum is then determined in each regime\nby taking the Fast Fourier Transform (FFT) of the os-\ncillating component of the data as a function of inverse\n\feld con\fned to the relevant magnetic \feld range. Figure\n3(a) shows the ranges used to calculate the FFTs shown\nin Figures 3(b),(c)&(d).\nBefore discussing the frequency spectra in greater de-\ntail, it's worth highlighting the key trends. Compared to\nthe data taken in the single-CDW state, only a few low\nfrequency quantum oscillations are observed ( F\u0014100 T)\nin the double-CDW state when the applied \feld is below\nB0, which is consistent with an additional folding of theBrillouin zone and gapping of the Fermi surface. Upon\napplying a \feld greater than B0the frequency spectrum\nin the double-CDW state appears very similar to that in\nthe single-CDW state. This is consistent with the mag-\nnetic breakdown scenario whereby quantum oscillation\nfrequencies associated with an unfolded brillouin zone\nare recovered when the \feld scale exceeds the breakdown\n\feld of the hybridisation gap25,29. The quantitative suc-\ncess of the independently determined B0in delineating\nthese \feld scales is quite striking.\nA detailed quantum oscillation study of LaTe 3has\nbeen performed previously by Ru et al. using torque\nand magnetic susceptibility measurements20. It is to be\nexpected that di\u000berent techniques might be more or less\nsensitive to di\u000berent regions of the Fermi surface, as well\nas observing di\u000berent rules for the mixing of frequencies\nand appearance of harmonics. However, the primary fre-\nquencies that correspond to the real area of the Fermi\nsurface should be reproduceable and so a comparison be-\ntween techniques is an instructive place to start. In Fig.\n3(d) the frequencies observed by Ru et al. in LaTe 3,\n\u000b,\f1,\f2, 2\fand\rare indicated along with an addi-\ntional frequency observed but not labelled by Ru et al.\nat around 1.45 kT marked in 3(d) as \u00031. In addition, two\nmore peaks are observed at low frequency that coincide\napproximately with where the second and third harmon-\nics of\u000bshould be, labelled 2 \u000b, 3\u000b. Highlighted in yellow\nbut not marked is a very subtle peak at around 864 T.\nBased on the LaTe 3data along this small peak might be\nignored, and it was not observed by Ru et al. However,\nthis subtle feature does coincide with a frequency that\nbecomes quite clear in the other datasets in this group.\nAs this peak is not obviously related to any harmonics,\nor addition and subtraction frequencies, we shall refer to\nit as\u000e.\nLooking at all of the traces in Figure 3(d) the \ffre-\nquencies can be clearly seen throughout the group and\nare highlighted in yellow. Owing to the slight warping of\nthe Fermi surface and bilayer splitting there are expected\nto be several frequencies very close to one another (Ru\net al. observe up to six for the \ffrequencies) and so\nfor comparative purposes we don't distinguish between\ndi\u000berent\ffrequencies but can assume that they all orig-\ninate from the same pocket of the Fermi surface. Any\nsystematic change in these frequencies with Ris within\nthe width of the group (around 440 T to 550 T). There is\nalso consistently a group of low frequencies in the 40 T to\n150 T range that are highighted in yellow and labelled \u000b.\nIn GdTe 3the\u000bfrequencies look very similar to those in\nLaTe 3, but NdTe 3and SmTe 3appear to have some addi-\ntional frequencies in this range. It isn't obvious whether\nthe Fermi surface has additional small pockets in these\ncompounds or whether there are simply some addition,\nsubtraction or harmonic frequencies that happen to be\nprominent in these materials. It is worth remembering\nthat low frequencies are particularly sensitive to e\u000bects\nof background subtraction and windowing e\u000bects and so\nthe relative amplitudes and precise frequencies should be6\ninterpreted with a degree of caution.\nThere are two signi\fcant di\u000berences between the LaTe 3\nspectrum and the others in this group ( R= Nd, Sm, Gd).\nThe \frst of which is the signi\fcant reduction, or even\ndisappearance, of the \rfrequency which is only clearly\nobservable in LaTe 3. The second is the appearance in the\nother members of the group of another strong frequency,\n\u000e, that is not clearly observable in LaTe 3and doesn't ap-\npear to be a product of other fundamental frequencies.\nTherefore\u000eseems to be another fundamental frequency.\nIt may not be a coincidence that the \rfrequency is very\nclose to where 2 \u000emay appear, and so while the possible\n(weak)\rpeaks are marked on the plot, the expected lo-\ncation of any 2 \u000epeaks is also indicated by a grey bracket\nwhere there is ambiguity (some 3 \ffrequencies may also\nappear in this frequency range). Only LaTe 3unambigu-\nously shows a \rpeak.\nThe other frequencies in the spectra can be described\nby the addition and subtraction of the primary frequen-\ncies and their harmonics as marked on the plot. It is no-\ntable that LaTe 3doesn't show addition and subtraction\nfrequencies, with the obvious di\u000berence being that LaTe 3\nlacks a large local moment. This is consistent with fre-\nquency mixing due to oscillations of the magnetisation.\nAlthough the other available rare-earths that could be\nplaced in this group, Ce and Pr, are not studied here,\nthere is no reason to expect any variation from the broad\ntrends described here.\nThe next group of FFTs, shown in Fig. 3(b), are taken\nbelowB0in the double-CDW state. The data clearly\nshows that the \f,\rand\u000epeaks are all absent, even in\ndata up to 65 T in TmTe 3. A cluster of low frequencies\nremain and these are shown in more detail in Fig. 4. It\nis di\u000ecult to identify trends between the spectra, other\nthan that the highest fundamental frequency is probably\nno higher than around 100 T (assuming that the higher\npeaks unique to the TmTe 3data are likely to be har-\nmonics given that this dataset extends to considerably\nhigher magnetic \felds), which corresponds to approxi-\nmately 0.5% of the unfolded Brilluoin zone. It isn't ob-\nvious based on this data whether the \u000bfrequency is still\npresent because of the density of peaks in the data being\ngreater than our resolution. But the qualitative state-\nment is clear; this data is consistent with a Fermi surface\ncomposed of multiple small pockets following a second\nfolding of the Brillouin zone by the second CDW.\nThe \fnal group of FFTs, representing data obtained at\nmagnetic \felds greater than B0in the double-CDW state,\nis shown in Fig. 3(c). Despite having a second CDW,\nthese datasets look much more similar to those for the\nsingle-CDW case (Fig. 3(d)) than the double-CDW case\n(Fig. 3(b)), with \u000b,\fand\u000efrequencies clearly observed.\nThis is precisely what is expected to occur as the mag-\nnetic \feld scale exceeds the breakdown \feld of the sec-\nond CDW gap. The addition and subtraction frequencies\nare also clearly observed in the data taken at the high-\nest \felds in TbTe 3and HoTe 3and marked on the plot.\nTaken at face value, HoTe 3appears to show a \rpeakthat is quite well separated from the expected position\nof its 2\u000epeak, which would suggest that this frequency is\nalso recovered upon magnetic breakdown. However, the\nbroad single \u000epeak is almost certainly representative of\nsplit peaks that are unresolved due to limited bandwidth\nin 1=B. The potential \rpeak is within the range of 2 \u000e\npeaks that may be anticipated given the breadth of the\n\u000epeak, and so its origin remains ambiguous. Identifying\nthe\u000bpeaks clearly is problematic in these data because\nlower frequencies tend to be more prominent at lower\n\felds, which by necessity are not analysed in this dataset.\nAt intermediate \felds, the frequencies of both the high\nand low \feld regimes can be expected to be present in the\nFFT, and so in these data we can't di\u000berentiate between\nthe recovery of the \u000bfrequency by magnetic breakdown\nand low frequencies observed in the low \feld regime. In-\ndeed, the highest \feld data may be beyond the quantum\nlimit for the lowest frequencies present.\nB. E\u000bective Masses\nFigure 5 shows the temperature dependence of the\nFFT amplitudes for (a) ErTe 3, (b) DyTe 3, (c) TbTe 3,\n(d) GdTe 3and (e) NdTe 3, with \fts to the temperature\ndependent term of the Lifshitz-Kosevitch formula (that\nyieldsm\u0003) shown as lines. Dashed lines and open symbols\ncorrespond to breakdown frequencies. The data is typi-\ncally more spread for the lower frequencies in part due to\na greater sensitivity to the background subtraction but\nalso due to the crowded spectrum at low frequencies. It\nshould also be noted that many split frequencies are not\nindividually resolvable at all temperatures, and so it's\nlikely that many of these values are in fact an averaged\nvalue across bilayer-split Fermi surfaces or between neck\nand bellies of slightly warped pockets. In principle there\ncould be some deviation from the Lifshitz-Kosevitch for-\nmula for the breakdown frequencies owing to tempera-\nture dependence of the second CDW gap, particularly in\nTbTe 3and DyTe 3, but this is not resolved in the data.\nThe error values shown in the legend are the standard\nerrors derived from the \ftting routine.\nThe variation of m\u0003withRis shown in Fig. 6(a) with\nguides to the eye shown as dashed lines. There is no evi-\ndence of mass enhancement associated with the \u000b,\fand\n\u000eon approach to the second CDW phase, with the most\nclear trends actually an apparent reduction of m\u0003on the\n\u000band\u000efrequencies. Note that no trendline is shown in\nthe double-CDW state because it isn't clear whether the\nsame pocket is being tracked for each Rbecause of the\ncrowded frequency spectrum and low resolution at low\nfrequencies. Taking all of the data together, Fig. 6(b)\nshows that there is an overall trend that m\u0003tracks with\nthe size of the orbit such that smaller pockets of the Fermi\nsurface are lighter than larger ones.7\n01020300 1020300.00.20.40.60.81.01.21.41.6 97T (3-13.6T): m*/me=0.081(6) \n9T (3-13.6T):  m*/me=0.088(3) \n15T (3-13.6T): m*/me=0.097(7) \n69T (3-13.6T): m*/me=0.073(7) \n57T (3-13.6T): m*/me=0.090(5) \n49T (3-13.6T): m*/me=0.14(1)E\nrTe3A/A0T\n (K)0102030e)d )c )a ) \n26T (4-16T): m*/me=0.12(2) \n36T (4-16T): m*/me=0.15(1) \n800T (12-16T): m*/me=0.16(1) \n480T (12-16T): m*/me=0.14(1) \n86T (12-16T): m*/me=0.076(4)D\nyTe3T\n (K)b)0\n102030 837T (9-14T): m*/me=0.24(1) \n520T (9-14T): m*/me=0.22(1) \n24T (3-7T): m*/me=0.16(4)T\nbTe3T\n (K) 522T (4-14T): m*/me=0.194(3) \n488T (4-14T): m*/me=0.200(6) \n840T (10-14T): m*/me=0.20(2) \n60T (4-14T): m*/me=0.11(2) \n108T (4-14T): m*/me=0.08(1)T\n (K)GdTe30\n102030 480T (6-14T): m*/me=0.19(1) \n515T (6-14T): m*/me=0.22(1) \n840T (6-14T): m*/me=0.31(2) \n869T (6-14T): m*/me=0.24(2) \n54T (4-14T): m*/me=0.09(2) \n124T (4-14T): m*/me=0.096(7) \n140T (4-14T): m*/me=0.085(7)N\ndTe3T\n (K)\nFIG. 5. Temperature dependence of the quantum oscillation amplitudes for primary orbits in (a) ErTe 3, (b) DyTe 3, (c) TbTe 3,\n(d) GdTe 3and (e) NdTe 3. Fits to the Lifshitz-Kosevitch formula are shown by solid lines and yield the e\u000bective mass m\u0003.\nOpen symbols and dashed lines correspond to breakdown frequencies. The \feld range used for the FFT and the frequency of\nthe orbit are shown in the legend.\n4.284 .324 .364 .400.00.10.20.30.40\n5 001 0001 5000.00.10.20.30.4double-CDW /s97 \n/s98 \n/s103 \n/s100 \ndouble-CDW m* / mea\n (Å)single-CDWb\n) \n/s97 \n/s98 \n/s103/s100\n \ndouble-CDW m* / meF\n (T)a)E rDyTbGdN dL a\nFIG. 6. Summary of e\u000bective mass data. (a) shows the ef-\nfective masses established for various R, with dashed lines a\nguide to the eye to highlight trends upon approach to the\ntransition to a double-CDW state. No trendline is included\nin the double-CDW state because it's less clear whether like\norbits are being compared for di\u000berent R. (b) shows the e\u000bec-\ntive masses plotted as a function of the quantum oscillation\nfrequency, establishing a trend that the e\u000bective mass scales\nwith the size of the pocket.IV. DISCUSSION\nThe data presented here shows strong evidence that\nmagnetic breakdown of the second CDW gap occurs with\na characteristic breakdown \feld consistent with that cal-\nculated from independent measurements. If one dataset\nwere viewed in isolation, it would be reasonable to ques-\ntion whether there were simply di\u000berent Dingle terms on\nthe higher frequency pockets to those at low frequency.\nThis could give a similar staggered onset of frequencies\nwith increasing magnetic \feld, but the systematic trends\ne\u000bectively eliminate this theory. Besides, it would be\nhighly unexpected for the Fermi surface to be unchanged\nfollowing the additional folding by the second CDW, as\nthe data at \felds B >B 0would imply if magnetic break-\ndown were not invoked.\nHaving accounted for magnetic breakdown, it is clear\nfromB <B 0data that the Fermi surface is signi\fcantly\naltered by the presence of the second CDW with the\nlargest pockets all disapearing. There remains just a\nseries of small pockets with the largest at most 100 T\nin area, or 0.5% of the unfolded BZ. This is consistent\nwith previous ARPES results on CeTe 3and ErTe 33,5that\nshow a moderately large pocket at the X point (thought\nto be the origin of the \ffrequencies20) that is ungapped\nby the \frst CDW becoming gapped by the second CDW\n(as illustrated in Fig.1c)). While the exact origin of the\n\rand\u000epockets isn't known, they also seem to be gapped\nby the second CDW owing to their absence from the\nB <B 0data even up to 65 T in TmTe 3. The fate of the\n\u000bpocket is unclear as other similar frequencies appear\nand it isn't clear whether any of them are the original \u000b\nfrequency.\nThe quantum oscillation frequencies in the single-8\nCDW state and the breakdown frequencies in the double-\nCDW state do not resolvably change with R, implying\nthat there is no doping e\u000bect and that any changes in q1\nmust be small so as not to signi\fcantly a\u000bect the folding\nof the BZ. However, the magnitude of the primary CDW\ngap, \u0001 1, does change signifcantly with Rand warrants\ndiscussion. Firstly, any portions of the Fermi surface that\nare not gapped by the primary CDW will be una\u000bected,\nwhich includes the \ffrequencies that are thought orig-\ninate from the ungapped diamond at the X point. The\norigin of the \u000efrequencies is not known, but the fact\nthat they are not strongly a\u000bected by \u0001 1suggests that\nthey are either ungapped by the primary CDW, or poorly\nnested byq1such that their area is only weakly depen-\ndent on \u0001 1. This may be instructive when considering\nthe exception, which is that the \rfrequency that is quite\nprominent in LaTe 3is far less obvious for other R, while\nan additional primary frequency \u000ethat isn't obviously\npresent in LaTe 3becomes strong. The \rfrequency is\nvery close to where 2 \u000eis expected to occur, as indicated\nin Fig. 3(d), which makes it possible that \rmay only oc-\ncur in LaTe 3. Ru et al. argue that the \rfrequency may\noriginate from a thin, elongated portion of the folded\nFermi surface, consistent with the poor-nesting scenario\nput forward for the \u000epockets above whereby changes\nin \u0001 1would only tweak the tips of a long, thin pocket.\nGiven that only one of \ror\u000eseem to be prominent for a\ngivenRin the single-CDW state, it would be consistent\nwith the data to suggest that these two frequencies may\noriginate from the same elongated piece of Fermi surface,\nwith a change in q1or \u0001 1as a function of R`pinching'\nthe long pocket in the middle and approximately halving\nits size asRchanges. Further work is required to test\nthis hypothesis.\nAs discussed in the introduction, a large part of the\nmotivation for this work was to explore RTe3as a model\nsystem to look for mass enhancement close to a CDW\nquantum critical point. In that respect this data presents\na null result. The cyclotron mass associated with the \f\nfrequency may rise moderately on approach to the pu-\ntative QCP, but the m\u0003values derived from the \u000band\n\u000efrequencies actually seem to get lighter. It is how-\never interesting to consider why this might be the case\nand what the implications may be in a wider context.\nThe \frst point to make is that while there is an implied\nquantum phase transition to the double-CDW state at\na value of the lattice parameter between those of R=Tb\nandR=Gd, it is not known whether this phase transi-\ntion would remain continuous or whether it may become\n\frst order. In the latter case, there is no reason to expect\nquantum critical \ructuations that would lead to mass en-\nhancement, although some remain in the case of a weakly\n\frst order transition. Also, without having located the\nquantum phase transition's location in phase space ex-\nactly, it is possible that these measurements simply were\nnot performed in close enough proximity to the quantum\nphase transition to be strongly in\ruenced by it; a factor\nthat would depend on the critical exponents.A second point of note is that quantum oscillations\nyield a cyclotron e\u000bective mass i.e. the e\u000bective mass\naveraged around the cyclotron orbit. As only the parts\nof the Fermi surface nested by q2may be expected to\nbe renormalised then if the Fermi surface is only poorly\nnested the majority of the orbit may not be renor-\nmalised, thus leaving the cyclotron e\u000bective mass mostly\nunchanged. In support of this idea, the peaks in the Lind-\nhard function are quite localised in k-space30, implying\nthat the renormalised portion of the Fermi surface would\nform a small portion of an orbit-averaged value, however\nARPES shows that a signi\fcant portion of the X pocket\n(\ffrequency) is gapped, implying that the same portion\nof the Fermi surface is well nested. In quantum oscilla-\ntion measurements, it is also always possible that there\nis another peak that is unobserved on which the mass\nenhancement is to be found, and it would be consistent\nwith the present data if, for example, the \rpeak becomes\nless prominent due to an increase in its e\u000bective mass on\napproach to the quantum phase transition damping its\namplitude. It would still remain somewhat surprising\nhowever that no mass enhancement at all would be ob-\nserved on the \fpocket as it is clearly gapped and there-\nfore must be somewhat nested by the critical boson.\nAn interesting trend that emerges from the data is that\nthe e\u000bective mass scales with the frequency of the peak\nand hence the size of the Fermi surface pocket. In gen-\neral, small Fermi surface pockets tend to be close to band\nedges, and in e\u000bect folding of the BZ by a CDW intro-\nduces new band edges where the qvector connects states\nthat are slightly above or below the Fermi level. There-\nfore one explanation is that the states closest to the hy-\nbridisation points are lighter than those that are farther\nfrom them due to the dispersion relation in the vicinity\nof the hybridisation point. Alternatively, the areas of\nFermi surface that are well nested by qmay be expected\nto have the largest electron-phonon coupling, and there-\nfore also the largest m*, but these are the same states\nthat are gapped out when the system orders, hence leav-\ning lighter carriers at the Fermi level as the folding of the\nBZ succesively reduces the size of the pockets.\nV. CONCLUSION\nTo conclude, this study presents a characterisation of\nquantum oscillations in the RTe3system that identi\fes\nmagnetic breakdown of the second CDW as the origin of\nthe observed magnetic \feld dependences of the quantum\noscillation frequencies. The Fermi surface in the double-\nCDW state is observed to consist of just small pockets\nwith a maximum area of around 0.5% of the Brillouin\nzone, consistent with previous ARPES measurements.\nThe e\u000bective mass is not observed to be renormalised\nclose to the quantum phase transition between the sin-\ngle and double-CDW states for the observed orbits, and\nwhile the reason for this is not clear, several possible av-\nenues through which to explain the absence of this e\u000bect9\nare discussed. Although further work is required to fully\nunderstand the observed lack of mass enhancement, this\nwork nonetheless narrows the parameters in which the\nsearch should take place in the RTe3family and provides\na greater understanding of their Fermi surfaces.\nVI. ACKNOWLEDGEMENTS\nThis work was supported by the Department of En-\nergy, O\u000ece of Basic Energy Sciences under contract DE-\nAC02-76SF00515. The National High-Magnetic Field\nLaboratory and related technical support are funded by\nthe National Science Foundation Cooperative Agreement\nNumber DMR-1157490 and DMR-1644779, the State of\nFlorida and the U.S. Department of Energy. Pulse \feld\nmeasurements were supported by the US Department\nof Energy Science of 100 tesla BES program. S.A. is\na recipient of a DFG-fellowship through the Excellence\nInitiative by the Graduate School Materials Science in\nMainz (GSC 266). JAWS acknowledges support as an\nABB Stanford Graduate Fellow.\nVII. APPENDICES\nA. In\ruence of magnetic order on quantum\noscillation frequencies.\nThe rare-earth moments in RTe3order at low tem-\nperatures, but do not appear to signi\fcantly in\ruence\nour results. To explicitly check this, we have mapped\nout the magnetic phase diagram as a function of mag-\nnetic \feld for the relevant \feld orientation Bkbfor\nGdTe 3, which can be considered a `worst case' scenario\nas it has the highest ordering temperature and the largest\nde Gennes factor. Figs. 7(a)&(b) respectively show re-\nsistivity curves at \fxed magnetic \felds and their second\nderivatives in temperature as a means to identify fea-\ntures that may be associated with phase boundaries. Fig-ure 7(c) shows the results of this analysis, with peaks in\nthe second derivative of the resistivity plotted as solid\nsymbols, and thermal hysteresis loops, which may be in-\ndicative of \frst-order transitions, marked as open sym-\nbols. The symbols are coloured based on tracking similar\nlooking features. The magnetic phase appears to be fully\nsuppressed just slightly above 14 T.\nIn order to test whether the magnetic phase is a\u000becting\nthe quantum oscillation data, FFTs (presented in Fig.\n7(d) ) were taken in the magnetic phase (2 K, 4.5 T -\n12 T) and outside of the magnetic phase in both higher\n\felds (2 K,15 T - 24 T) and higher temperatures (17 K,\n10 T - 14 T). The resultant frequency spectra are quali-\ntatively very similar, with the variation principally due to\nreduced bandwidth and increased noise in the high \feld\ntrace, and temperature damping of the quantum oscilla-\ntion at 17 K. There may be a small shift in the frequency\nof the\u000epeak, but not to the extent that it a\u000bects any\nof the discussion of our results. Panel (e) in Fig.7 shows\ndata from Ru et al.31showing the scaling of the mag-\nnetic phase at zero \feld as a function of the de Gennes\nfactor, highlighting that Gd has the strongest magnetic\nordering. It is in general di\u000ecult to determine the mag-\nnetic polarisation \felds in RTe3from magnetoresistance\ndata because the signal is often dominated by quantum\noscillations, but estimates are shown for GdTe 3, TbTe 3\nand HoTe 3. The polarisation \feld in GdTe 3is estimated\nfrom Figure 7c) with the two points being two di\u000ber-\nent extrapolations to 2 K (vertical and linear), the two\npoints shown for TbTe 3are the two kinks observable in\nthe raw data (Figure 2(a)), and in HoTe 3the peaks in the\n\frst and second derivative of the contactless conductivity\nmeasurement are both shown. While this is a crude anal-\nysis, it shows that the polarisation \feld scales with the\nzero-\feld transition temperature thus justifying the as-\nsertion that GdTe 3represents a worst-case scenario and\nthat the majority of the data analysed here is outside of\nthe magnetically ordered phase. 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M.\nSchoop, Science Advances 6, 1 (2020).\n22C. D. Malliakas and M. G. Kanatzidis, Journal of the\nAmerican Chemical Society 128, 12612 (2006).\n23N. Ru and I. R. Fisher, Physical Review B 73, 033101\n(2006).\n24C. T. Van Degrift, Review of Scienti\fc Instruments 46,\n599 (1975).\n25D. Shoenberg, \\Magnetic breakdown,\" in Magnetic Os-\ncillations in Metals , Cambridge Monographs on Physics\n(Cambridge University Press, 1984) pp. 331{368.\n26Note that the magnetic breakdown reduction factor has\nthe formRMB = (ip\nP)l\u0017(p\n1\u0000P)l\u0011wherel\u0017andl\u0011\ncount the number of magnetic breakdown tunnelling and\nBragg re\rections that occur around the orbit with trans-\nmitted amplitudes ip\nPandp\n1\u0000Prespectively. P=\nexp(\u0000B0=Bcos\u0012) describes the magnetic breakdown prob-\nability with B0the characteristic breakdown \feld discussed\nin the main text. Without further knowledge of the Fermi\nsurface,\u0017and\u0011cannot be known and are not discussed fur-\nther here. However as we are estimating B0independently\n(as oppose to extracting it from \fts of RMBto the quan-\ntum oscillation data) it remains a meaningful parameter\nas an estimate of the \feld at which appreciable magnetic\nbreakdown should occur. Note also that as this system is\nquasi-2D with the magnetic \feld out of plane in all mea-\nsurements, cos \u0012= 1 for all of the present data25?.\n27B. F. Hu, B. Cheng, R. H. Yuan, T. Dong, and N. L.\nWang, Physical Review B 90, 085105 (2014).\n28Gap magnitudes obtained by Hu et al.27are consistent\nwith those obtained elsewhere3,5? ?, but this reference is\nfavoured here because it provides a consistent measure of\nthe gap magnitudes for all Rand for both CDWs.\n29N. Harrison, J. Caul\fed, J. Singleton, P. H. P. Reinders,\nF. Herlach, W. Hayes, M. Kurmoo, and P. Day, Journal\nof Physics Condensed Matter 8, 5415 (1996).\n30M. D. Johannes and I. I. Mazin, Physical Review B 77,\n165135 (2008).\n31N. Ru, J. H. Chu, and I. R. Fisher, Physical Review B 78,\n012410 (2008).11\n04 8 12164 8 1 21 60\n5 001000150020000 2468101214024681012140\n2 4 6 8 101214160246810120TT (K)R (arb. units)14TG dTe314T0\nTT (K)d\n 2R / dT 2 (arb. units) 14T \n13T \n12T \n11T \n10T \n9T \n8T \n7T \n6T \n5T \n4T \n3T \n2T \n1T \n0Td\n)F\nFT amplitude (arb. units)F\n (T) 2K, 4.5T-12T \n2K, 15T-24T \n17K 10-14TT (K)B\n (T) \nMagnetic Transition Temperature at zero-field \nMagnetic Transition Field at 2Ke)c)b)C\neNdS\nmHoDyTbT (K)d\ne Gennes FactorGda)0\n481216B\n (T)\nFIG. 7. a)baxis resistivity data in \fxed magnetic \feld ( Bkb)\nin GdTe 3taken on both warming and cooling every 1 T be-\ntween 0 T and 14 T (data o\u000bset for clarity). b) Second deriva-\ntive in temrperature of the data in a) (o\u000bset for clarity). c)\ntheB\u0000Tphase diagram derived from the data in b); closed\nsymbols represent peaks in the second derivative, open sym-\nbols are taken from the centre of the thermal hysteresis loops\nin (b), implying a \frst-order phase transition. The phase\nboundaries seems to be approaching 0 K just slightly above\n14 T. d) Three FFTs taken in di\u000berent regions of B\u0000Tphase\nspace in GdTe 3; one from within the magnetic phase (2 K,\n4.5 T - 12 T, red line), one above the magnetic phase bound-\nary in \feld (2 K, 15 T - 24 T, black line), and one from above\nthe magnetic phase in temperature (17 K, 10-14 T, blue line).\nThis illustrates that any alteration of the quantum oscillation\nspectrum by magnetic ordering must be subtle enough not\nto a\u000bect our conclusions. e) the magnetic phase transitions\ndetermined previously without applied magnetic \feld by Ru\net al. (black circles, left axis)31compared to the magnetic\npolarisation \felds observed in applied \felds at 2 K for R=Gd,\nTb and Ho (red squaress, right axis) as a function of the de\nGennes factor. Determination of the polarisation \felds is dis-\ncussed in the text. The polarisation \felds seem to scale with\nthe zero-\feld transition temperatures."    },    {        "title": "2004.14205v2.Nuclear_Symmetry_Energy_and_Neutron_Skin_Thickness_of____208_Pb___using_a_finite_range_effective_interaction.pdf",        "content": "Nuclear Symmetry Energy and Neutron Skin Thickness of208Pb using a finite range\neffective interaction\nD. Behera\u0003, S. K. Tripathyy, T. R. Routrayzand B. Beherax\nWe use a finite range simple effective interaction to construct nuclear equations of state for the\nstudyofthedensitydependenceofthenuclearsymmetryenergy. TheEoSsprovidegooddescriptions\nof the nuclear symmetry energy at a subsaturation density \u001ac= 0:11fm\u00003and at a density around\ntwo times the saturation density \u001a0. We obtain a correlation between the neutron skin thickness in\n208Pb and the density slope parameter at the subsaturation density. A linear relation is obtained\nbetween the neutron skin thickness and the parameter \f0=L(\u001ac)\n3Es(\u001a0), whereEs(\u001a0)andL(\u001ac)are\nrespectively the nuclear symmetry energy at saturation density and the density slope parameter at\nthe subsaturation density.\nPACS number : 21.65.Ef,24.30.Cz\nI. INTRODUCTION\nThe nuclear symmetry energy (NSE) Es(\u001a), defined as the energy cost per particle for changing all the protons\ninto neutrons in symmetric nuclear matter (SNM), is a fundamental quantity in nuclear physics and astrophysics.\nIt provides a link to the properties of atomic nuclei with the structure and dynamics of neutron stars. An accurate\nknowledge of NSE is essential to understand the equation of state of (EoS) of isospin asymmetric nuclear matter\n(ANM). In fact, the understanding and constraining the EoS of dense neutron-rich matter has remained a major\ngoal in nuclear science research [1]. In neutron stars, NSE controls the proton fraction of the beta stable matter\nand decides the cooling mechanism and thickness of the neutron star crusts [2–4]. Also, Es(\u001a)has a great role\nin obtaining the mass-radius relation and tidal deformability in neutron stars [5]. In astrophysics, its behaviour is\nrequired to understand the supernova explosion mechanism and stellar nucleosynthesis [6–8]. In nuclear physics,\nNSE has important consequences for understanding the dynamics in heavy-ion reaction processes involving isospin\nasymmetric nuclei, prediction of properties of exotic nuclei with large neutron excess, etc. Experimentally, Es(\u001a)\nis not a directly measurable quantity and has to be extracted from isospin sensitive observables. The experimental\ndetermination of NSE therefore depends on the reliability of the model in describing the experimental observables.\nAlthough, it remains a quite challenging task, in the last one and half decade, a lot of theoretical and experimental\nefforts have been made to constrain the density dependence of NSE. In recent years, significant progress has been\nmade to constrain the NSE at low densities from dynamical behaviour [9, 10], resonances and excitations [11–14],\nstatic properties of finite nuclei [15–18], neutron skin thickness [19–25] and electric dipole polarisability [14, 26–28].\nThe density dependence of Es(\u001a)is largely unknown, except the value of Es(\u001a0)and its slope parameter L(\u001a0)at\nsaturation density \u001a0. While data from nuclear experiments and astrophysical observations prior to 2013 constrain\nthese two parameters as Es(\u001a0) = 31:6\u00062:7MeV andL(\u001a0) = 58:9\u000616MeV [29], some recent work extracted the\nvaluesEs(\u001a0) = 31:7\u00063:2MeV andL(\u001a0) = 58:7\u000628:1MeV [30]. However, our present knowledge of the curvature\nparameterKsym(\u001a0)is rather poor. Usually, one needs to assume some models and then the data of stable nuclei to\nthe parameters of the assumed models are fitted. This process may not constrain the isovector part of the nuclear\nequations of state in a precise way [20].\nNeutron skin thickness (NST) has proved to be a better tool to improve our knowledge in the isovector channels of\nnuclear effective interaction [31–35]. The Lead Radius Experiment (PREX) is able to determine the neutron radius\nin208Pb up to 1%accuracy through the measurement of the parity violating asymmetry at low momentum transfer\nin the polarised elastic electron scattering performed at an angle 50[36]. This experiment provides the first purely\nelectroweak, model-independent measurement of the weak charge form factor which is closely related to the neutron\nskin thickness of208Pb. A measurement of the form factor of208Pb at a momentum transfer of q'0:475fm\u00001,\nyielded the PREX results \u0001rnp= 0:33+0:16\n\u00000:18fm [36]. Another run of the same has been proposed (PREX-II) to reduce\nthe uncertainty to 0:06fm [37]. The nuclear droplet model (DM) [38] suggested that NST in heavy nucleus is related\nto different symmetry energy parameters. On the basis of DM, it has been predicted from different mean field models\n\u00031. Department of Physics, Indira Gandhi Institute of Technology, Sarang, Dhenkanal, Odisha-759146, India\n2. School of Physics, Sambalpur University, Jyotivihar, Sambalpur, Odisha-768019, India,\nE-mail:dipadolly@rediffmail.com\nyDepartment of Physics, Indira Gandhi Institute of Technology, Sarang, Dhenkanal, Odisha-759146, India, E-mail:tripathy_\nsunil@rediffmail.com\nzRetired Professor, School of Physics, Sambalpur University, Jyotivihar, Sambalpur, Odisha-768019, India, E-mail:trr1@rediffmail.com\nxRetired Professor, School of Physics, Sambalpur University, Jyotivihar, Sambalpur, Odisha-768019, IndiaarXiv:2004.14205v2  [nucl-th]  30 Aug 20202\nin208Pb that, NST is linearly correlated with the density slope parameter [40]. In recent years, different correlation\nsystematics of isovector properties of finite nuclei with NST have yielded constraints on the density slope parameter\nand the curvature symmetry parameter [19–21, 39, 40]. However so far a consensus has not yet been reached in this\ndirection.\nThe purpose of the present work is two fold. First, we construct nuclear equations of state using a finite range\neffective interaction so that the EoSs reproduce the basic features of SNM at saturation density and the nuclear\nsymmetry energy at the subsaturation cross density \u001ac= 0:11fm\u00003as constrained in a recent work from analysing\nthe binding energy difference of heavy isotope pairs[23]. Similar method has been adopted in a recent work [41] to\nconstrain the density dependence of NSE at high density. Second, through a comprehensive and quantitative analysis,\nwe correlate the neutron skin thickness and the energy constant of isovector giant dipole resonance (IVGDR) of208Pb\nto the isovector indicators of the finite range effective interaction. The paper is organised as follows: in Section II,\nthe basic formalism of the finite range effective interaction and the method of parameter fixation are discussed. In\nSection III, we discuss the mass dependence of symmetry energy coefficient of finite nuclei. In Section IV, we carry\nout a comprehensive analysis for the correlation of the neutron skin thickness with the density slope parameter at\nsaturation density and at a reference density \u001ac= 0:11fm\u00003. The correlation of IVGDR energy constant of208Pb\nwith the isovector indicators of the effective interaction is presented in Section-V. At the end, the conclusion and\nsummary of the present work are presented in Section-VI.\nII. MODEL AND METHOD\nA. Nuclear Symmetry Energy and Nuclear Equation of state\nFor an isospin asymmetric nuclear matter (ANM) with neutron-proton asymmetry \u000e, neutron density \u001an, proton\ndensity\u001ap, the energy per nucleon e(\u001a;\u000e)can be written as [42]\ne(\u001a;\u000e) =1\n\u00192\u001a[f(kn) +f(kp)] +V(\u001a;\u000e); (1)\nwhere\u001a=\u001an+\u001apis the nucleon density, kn;pare the neutron(proton) Fermi momenta. V(\u001a;\u000e)is the interaction part\nof the equation of state, e(\u001a;\u000e). The kinetic part of the EoS is treated in the relativistic Fermi gas model and the\nfunctionalf(ki)withi=n;pis expressed as [42]\nf(ki) =kfZ\n0\u0000\nc2~2k2+M2c4\u00011\n2k2dk: (2)\nHere,Mc2is the rest mass of the nucleon. The interaction part of e(\u001a;\u000e)has a complicated dependence on \u001aand\n\u000ebecause of the presence of finite range exchange interaction between nucleons. However, the isospin exchange\nsymmetry allows us to expand the EoS in even powers of the neutron-proton asymmetry \u000eas\ne(\u001a;\u000e) =e0(\u001a) +Es(\u001a)\u000e2+O(\u000e4); (3)\nwheree0(\u001a) =e(\u001a;\u000e= 0)is the energy per particle in symmetric nuclear matter (SNM). Es(\u001a)is nuclear symmetry\nenergy defined as\nEs(\u001a) =1\n2!@2e(\u001a;\u000e)\n@\u000e2j\u000e=0: (4)\nAssuming that the contribution from higher order terms in \u000eis small, the NSE can be expressed as the difference in\nthe energy per particle in pure neutron matter en(\u001a) =e(\u001a;\u000e= 1)and that in SNM,\nEs(\u001a) =en(\u001a)\u0000e0(\u001a): (5)\nAn expansion of Es(\u001a)around the saturation density \u001a0reads as\nEs(\u001a)\u0019Es(\u001a0) +L\n3\u0012\u001a\u0000\u001a0\n\u001a0\u0013\n+Ksym\n18\u0012\u001a\u0000\u001a0\n\u001a0\u00132\n+\u0001\u0001\u0001; (6)\nwhereL(\u001a0) = 3\u001a0@Es(\u001a)\n@\u001aj\u001a=\u001a0andKsym(\u001a0) = 9\u001a2\n0@2Es(\u001a)\n@\u001a2j\u001a=\u001a0are the slope and curvature parameters of Es(\u001a)at\u001a0.\nUp to 2nd order in the deviation from saturation density, the density dependence of NSE depends on the parameters\nL(\u001a0)andKsym(\u001a0).3\nB. Finite range effective interaction\nIn the present work, we consider a finite range simple effective interaction (SEI)\nveff(r) =t0(1 +x0P\u001b)\u000e(r) +1\n6t3(1 +x3P\u001b)\u0014\u001a(R)\n1 +b\u001a(R)\u0015\r\n\u000e(r) + (W+BP\u001b\u0000HP\u001c\u0000MP\u001bP\u001c)f(r);(7)\nwheref(r)represents the form factor of a finite range Yukawa interactione\u0000r=\u000b\nr=\u000b. Here\u000bis the range of the interaction.\nrandRare respectively the relative and centre of mass coordinates of the two interacting nucleons. W;B;H and\nMare the strength of the Wigner, Bartlett, Heisenberg and Majorana components. P\u001bandP\u001care the spin and\nisospin exchange operators respectively. The simple effective interaction is similar to the Skyrme type interaction\nexcept the fact that the t1andt2terms of the Skyrme interaction are replaced by a finite range term. Also, the t3\nterm has been modified. The replacement of the t1andt2terms by a finite range term is essential to account for the\ncorrect momentum dependence of the nuclear mean field as extracted from the optical model fits in heavy-ion collision\nstudies at intermediate energies [43–50]. In the denominator of the t3term we have considered a factor (1 +b\u001a(R))\r\nto avoid the supraluminous behaviour of the nuclear matter equation of state at high density region. The simple\neffective interaction contains altogether 11 adjustable parameters namely t0;x0;t3;x3;b;\r;W;B;H;M and\u000b. This\nSEI has already been used to study the momentum and density dependence of the isoscalar part of the nuclear mean\nfield at zero and finite temperature [51–53], isovector part of the nuclear mean field at zero temperature [54] and the\ntemperature dependence of nuclear symmetry energy [55, 56]. The SEI has also been used to calculate the half-lives\nof spherical proton emitters [57].\nThe energy density H(\u001a;yp;T)in ANM at a density \u001a, proton fraction ypand temperature Tcan be obtained from\nSEI as\nH(\u001a;yp;T) =Z\n[fn\nT(k) +fp\nT(k)]\u0000\nc2~2k2+M2c4\u0001\nd3k\n+1\n2\"\n\"l\n0\n\u001a0+\"l\n\r\n\u001a\r+1\n0\u0012\u001a\n1 +b\u001a\u0013\r#\n\u0000\n\u001a2\nn+\u001a2\np\u0001\n+\"\n\"ul\n0\n\u001a0+\"ul\n\r\n\u001a\r+1\n0\u0012\u001a\n1 +b\u001a\u0013\r#\n\u001an\u001ap\n+\"l\nex\n2\u001a0Z Z\u0002\nfn\nT(k)fn\nT(k0) +fp\nT(k)fp\nT(k0)gex(jk\u0000k0j)\u0003\nd3k d3k0\n+\"ul\nex\n2\u001a0Z Z\u0002\nfn\nT(k)fp\nT(k0) +fp\nT(k)fn\nT(k0)gex(jk\u0000k0j)\u0003\nd3k d3k0; (8)\nwheref\u001c\nT(k);\u001c=n;pare the respective Fermi-Dirac distribution functions and gex(jk\u0000k0j) =1\n1+jk\u0000k0j2\n\u00032.\u0003is inversely\nrelated to the range parameter i.e. \u0003 =1\n\u000b. The study of ANM involves nine parameters: \r;b;\"l\n0;\"ul\n0;\"l\n\r;\"ul\n\r;\"l\nex;\"ul\nex\nand the range parameter \u000b. The parameters \"l\n0;\"ul\n0;\"l\n\r;\"ul\n\r;\"l\nex;\"ul\nexare related to the parameters of SEI [58].\nThe energy per particle in symmetric nuclear matter at zero temperature ( T= 0) for the Yukawa type finite range\neffective interaction becomes\ne0(\u001a) =3Mc2\n8x3\nf\u0002\n2xfu3\nf\u0000xfuf\u0000ln(xf+uf)\u0003\n+\"0\n2\u001a\n\u001a0+\"\r\n2\u001a\n\u001a\r+1\n0\u0012\u001a\n1 +b\u001a\u0013\r\n+\"ex\n2\u001a0\u001aJ0(\u001a); (9)\nwherexf=~kf\nMc,uf= (1 +xf)1\n2andkf=\u0000\n1:5\u00192\u001a\u00011\n3is the Fermi momentum in SNM. The functional J0(\u001a)is given\nby\nJ0(\u001a) =R\u0010\n3j1(kfr)\nkfr\u00112e\u0000r=\u000b\nr=\u000bd3r\nRe\u0000r=\u000b\nr=\u000bd3r: (10)\nHerej1(kfr)is the first order spherical Bessel function and \"0=1\n2\u0000\n\"l\n0+\"ul\n0\u0001\n;\"\r=1\n2\u0000\n\"l\n\r+\"ul\n\r\u0001\n,\"ex=1\n2\u0000\n\"l\nex+\"ul\nex\u0001\n.\nThe energy per particle in pure neutron matter (PNM) at zero temperature is obtained from the SEI as\nen(\u001a) =3Mc2\n8x3n\u0002\n2xnu3\nn\u0000xnuf\u0000ln(xn+un)\u0003\n+\"l\n0\n2\u001a\n\u001a0+\"l\n\r\n2\u001a\n\u001a\r+1\n0\u0012\u001a\n1 +b\u001a\u0013\r\n+\"l\nex\n2\u001a0\u001aJn(\u001a); (11)\nwherexn=~kn\nMc,un= (1 +xn)1\n2andkn=\u0000\n3\u00192\u001a\u00011\n3is the Fermi momentum in PNM. The functional Jn(\u001a)is given\nby\nJn(\u001a) =R\u0010\n3j1(knr)\nknr\u00112e\u0000r=\u000b\nr=\u000bd3r\nRe\u0000r=\u000b\nr=\u000bd3r: (12)4\nTABLE I: Values of the interaction parameters in SNM.\nParameters \rb\u000b\"0\"\r\"ex\nin SNM (fm3)(fm)(MeV)(MeV)(MeV)\n0:50.56680.4044-57.8676.91-121.84\nTABLE II: Values of the interaction parameters in PNM corresponding to three different values of Es(\u001a0)andEs(\u001ac) = 26:65MeV. The table\nincludes the predicted values of E0\ns(\u001a0)andEs(2\u001a0).\nParameter sets Es(\u001a0)E0\ns(\u001a0)Es(2\u001a0)\"l\n0\"l\n\r\"l\nex\nin PNM (MeV)(MeV)(MeV) (MeV) (MeV) (MeV)\nSet I 3314.7238.65- 1.7401 21.0099 -81.2299\nSet II 3418.3544.76-11.2489 33.0473 -81.2299\nSet III 352250.90-20.8451 45.1762 -81.2299\nC. Constraining the interaction parameters\nThe complete description of SNM requires only the knowledge of six parameters \r;b;\u000b;\" 0;\"\rand\"ex. However,\nthe equation of state in PNM requires the splitting of the strength parameters \"0;\"\rand\"exinto like (l) and unlike\n(ul) channels. The parameters \u000band\"exare obtained from a simultaneous optimization procedure so as to provide\na correct momentum and density dependence of the nuclear mean field in SNM as demanded by the optical model\nfits to the heavy-ion collision data at intermediate energies. During the optimization procedure, it is kept in view\nthat, the nuclear mean field in SNM at saturation density vanishes for a kinetic energy of 300MeV=n.\"0and\"\rare\ndetermined from the saturation condition in normal nuclear matter. Here we have used the fact that, Mc2= 939\nMeV, energy per nucleon in SNM e0(\u001a0) = 923 MeV,\u0010\nc2~2k2\nf0+M2c4\u00111\n2= 976MeV and the saturation density\n\u001a0= 0:1658fm\u00003. The stiffness of the equation of state in SNM determines the exponent \r. In the present work, we\nhave used\r=1\n2corresponding to the incompressibility in normal nuclear matter K= 240MeV. The parameters as\nconstrained from different physical basis to determine the EoS of SNM are given in Table I.\nThere are no experimental or empirical constraints on the splitting of \"0;\"\rand\"exinto the corresponding like\n(l) and unlike ( ul) components. However, from an analysis of the entropy density in PNM and SNM, \"l\nexhas been\nconstrained in a recent work as \"l\nex=2\n3\"ex[55]. For this constrained value of \"l\nex, the neutron effective mass in\nneutron-rich matter is predicted to pass over the proton effective mass. Even though controversy prevails among the\nresults from different theoretical models about the neutron-proton effective mass splitting, there is almost a consensus\nreached on the fact that, neutron effective mass in neutron rich matter will go over the proton effective mass. For\na given splitting of \"exinto\"l\nexand\"ul\nex, the splitting of other two strength parameters \"0and\"\rinto like and\nunlike components requires the knowledge of the zero temperature nuclear symmetry energy Es(\u001a0)and its slope\nE0\ns(\u001a0) =\u001adEs(\u001a)\nd\u001aj\u001a=\u001a0=1\n3L(\u001a0)at saturation density. During the last one or two decades, there have been a lot\nof experimental and theoretical efforts to constrain these parameters. However, amidst all these efforts, still there\nexist uncertainties in the values of Es(\u001a0)andL(\u001a0)[3, 10, 30, 59–64]. While the Es(\u001a0)is constrained to a narrow\nrange ofEs(\u001a0) = 32\u00064MeV, there is a large uncertainty in the values of L(\u001a0). Different theoretical models predict\nsimilarEs(\u001a)at saturation density but widely differ in the values of L(\u001a0)and therefore predict quite different density\ndependence of NSE. The slope parameter has been constrained recently in the range L(\u001a0) = 58:7\u000628:1MeV from an\naveragingoveragoodnumberofexperimentalandobservationaldata[30]. ConstraintsontheNSEhavebeenobtained\nin recent times from the properties of finite nuclei such as the nuclear binding energy [65–67], neutron skin thickness\n[12, 20, 21, 33, 68–70], electric response ( dipole polarisability)[14]. Properties of finite nuclei provide stringent\nconstraints on Es(\u001a)andL(\u001a)at subsaturation densities. Using different microscopic and phenomenological models,\nFuchs and Wolter have obtained the NSE at a subsaturation density around \u001ac'0:6\u001a0to beEs(\u001ac)'24MeV [71].\nBrown used the properties of doubly magic nuclei to constrain the nuclear equation of state at a density of \u001ac= 0:1\nfm\u00003[72]. From an analysis of the binding energy difference of heavy isotope pairs, Zhang and Chen have obtained a\ntighterconstraintonthesymmetryenergyatsubsaturationdensity \u001ac= 0:11fm\u00003i.e.Es(\u001ac) = 26:65\u00060:20MeV[23].\nIt is worth to mention here that, the average density of heavy nuclei such as208Pb is around 0:11fm\u00003. On the other\nhand, density dependence of NSE at high density is quite uncertain. However there have been a significant progress in\nunderstanding the high density behaviour of NSE from terrestrial nuclear laboratories and astrophysical observations.\nThe upper limit of tidal deformability for canonical neutron stars \u00031;4= 580extracted from GW170817 by LIGO and\nVIRGO collaborators [73] has allowed to constrain the high density nuclear equation of state to a narrow range. From\nthe analysis of these observations for a constant maximum mass of Mmax= 2:01M\fand radius R1;4= 12:83, the\nhigh density behaviour of NSE has been constrained at a density \u001a= 2\u001a0to beEs(2\u001a0) = 46:9\u000610:1MeV [74, 75].\nUsing the extracted upper limit on the tidal deformability of the gravitational waves from GW170817, Tong et al.\nhave also obtained a constraint on the NSE at twice saturation density as Es(2\u001a0)\u001460:7\u000610:9MeV [76]. Bayesian\nanalysis of the radii of canonical Neutron Stars predicts Es(2\u001a0) = 39:2+12:1\n\u00008:2MeV [77]. Zhang et al. constrained the\nNSE from an analysis of heavy ion collision data, neutron skin of208Pb, tidal deformability and maximum mass of5\n0.00 .51 .01 .52 .02 .53 .0-4004080120e0 (ρ),  en (ρ)  [MeV]ρ\n/ρ0  SNM \n Set III \n Set II \n Set I\nFIG. 1: Energy per particle in SNM and PNM for three different parameter sets of the finite range simple effective interaction. The dotted curve\nis for SNM and the solid curves represent the energy per particle in PNM.\nneutron stars and obtained Es(2\u001a0)in the range 35\u000055MeV [78]. More accurate determination of the millisecond\npulsars from pulse-profile modelling of accretion hot spots by NASA’s Neutron Star Interior Composition Explorer\n(NICER) mission is expected to advance our understanding of the EoS of cold dense matter [79–83]. It appears that\nthe NICER results and the newly discovered heavy neutron star PSR J0740+6620 [84] support stiffer rather than\nsofter choices of the asymmetric nuclear matter EoS. In order to constrain our interaction parameters so as to provide\na good description of the nuclear symmetry density both at a subsaturation cross density as well as at a reasonably\nhigh density (as applicable to neutron stars), in the present work, we have varied the slope parameter for a given\nEs(\u001a0)within its acceptable range to obtain the Es(\u001ac) = 26:65\u00060:2fm\u00003. The slope parameter that fixes the\nconstrained Es(\u001ac)is then checked so as to predict the NSE at 2\u001a0close to the limit Es(2\u001a0) = 46:9\u000610:1MeV. In\nthis procedure, we obtained the splitting of the strength parameters \"0and\"\rinto like and unlike components. We\nhave considered three different values of Es(\u001a0)namelyEs(\u001a0) = 33;34and35MeV and obtained the corresponding\nsplitted parameter sets. The parameters of the given finite range SEI as constrained above for the EoS of PNM are\ngiven in Table II. The slope parameter for all the sets lies in the acceptable range 44\u0014L(\u001a0)\u001466MeV. For all the\nsets of nuclear equation of state, the values of Ksym(\u001a0)lie in the range\u0000201\u0014Ksym(\u001a0)\u0014\u0000126MeV. The NSE at\n2\u001a0lies in the range 38:65\u0014Es(2\u001a0)\u001452:15MeV. These values are well within those predicted from terrestrial and\nastrophysical observations [74, 75, 78]. For completeness, we have shown the EoSs for SNM and PNM for the three\ndifferent parameter sets of the finite range effective interaction in Figure 1.\nD. Density dependence of Nuclear Symmetry Energy\nIn Figure 2, we have shown the density dependence of NSE for three different sets of parameters. Also, we have\nplotted the NSE as obtained from a comparison of ASY-EOS data concerning the elliptic-flow ratio of neutrons with\nrespect to charged particles with ultra relativistic quantum molecular dynamics (UrQMD) transport model [85]. In\nthat study, Russotto et al. have used parametrized form of NSE\nEs(\u001a)[MeV] = 12\u0012\u001a\n\u001a0\u00132=3\n+ 22\u0012\u001a\n\u001a0\u0013\u001b\n; (13)\nand constrained the exponent as \u001b= 0:72\u00060:19[85]. This is an improvement over a similar study from FOPI-LAND\nexperiment that suggested a moderately soft to linear NSE characterised by an exponent \u001b= 0:9\u00060:4[86]. This\nstudy ruled out the supersoft scenarios for NSE. It is worth to mention here that, the supersoft scenarios for NSE\nwere ruled out after the parameter dependence of the ASY-EOS result was investigated by Cozma et al. [87]. In Fig.\n2, we have also included the behaviour of NSE at subnormal densities as obtained from different studies [9, 62]. Our\nconstructed sets of NSE are quite compatible with these results at low density region. However, in the high density6\n0.00 .51 .01 .52 .00204060  Set III \n Set II \n Set IEs(ρ) [Me V]ρ\n/ρ0\nFIG. 2: Nuclear symmetry energy Es(\u001a)as a function of the reduced density\u001a\n\u001a0. The results at subnormal densities of Ref.[9, 62] and the\nUrQMD results (shaded grey region) of ASY-EOS data of Ref. [85] are also shown in the figure. The shaded region in orange is for the HIC\nresults of Sn+Sn reaction and the pink region is for the results of IAS.\nregion beyond the normal nuclear matter density, our results for the sets with Es(\u001a0) = 33and34MeV pass below the\npredicted results from the UrQMD study of ASY-EOS data [85]. Only the set with Es(\u001a0) = 35MeV passes through\nthe shaded region (refer to Fig. 2) and therefore may comprise a more suitable equation of state for neutron-rich\nasymmetric nuclear matter. At normal nuclear matter density, our NSE for set III goes above the ASY-EOS data\nbecause of the fact that the parametrized form used in Ref.[85] has a sharp value of Es(\u001a0) = 34MeV. Regarding\nthe density dependence behaviour, the NSE for the other two sets can not simply be ruled out on the basis of the\nASY-EOS data. Recently, Zhou and Chen [41], on the basis of the discovery of millisecond PSR J0740+6620 with\nMass 2:14+0:10\n\u00000:09M\f[84] together with the data of finite nuclei ruled out super soft NSE that becomes negative at\nsuprasaturation densities in neutron stars. On that basis, the set II with Es(\u001a0) = 34MeV will pass the test. The set\nI withEs(\u001a0) = 33MeV can be viable according to the constraints of Refs. [74, 75].\nIn order to understand the density dependence of NSE as calculated from the finite range SEI, we can split it into\nits kinetic contribution and the contribution from the interaction part\nEs(\u001a) =Ekin\ns(\u001a) +Epot\ns(\u001a); (14)\nwhere\nEkin\ns(\u001a) =3Mc2\n8\"\n2xnu3\nn\u0000xnuf\u0000ln(xn+un)\nx3n\u00002xfu3\nf\u0000xfuf\u0000ln(xf+uf)\nx3\nf#\n; (15)\nand\nEpot\ns(\u001a) =(\"l\n0\u0000\"0)\n2\u001a\n\u001a0+(\"l\n\r\u0000\"\r)\n2\u001a\n\u001a\r+1\n0\u0012\u001a\n1 +b\u001a\u0013\r\n+[\"l\nexJn(\u001a)\u0000\"exJ0(\u001a)]\n2\u001a\n\u001a0: (16)\nIn Figure 3, the kinetic and potential contributions to the NSE are shown as function of the reduced nuclear density\n\u001a\n\u001a0. It is obvious that, the kinetic part has a positive contribution behaving as\u0010\n\u001a\n\u001a0\u00112=3\n. But the potential contribution\ndepends on the value of Es(\u001a0)andL(\u001a0). For a given pair of Es(\u001a0)andL(\u001a0), the potential part Epot\ns(\u001a)increases to\nattainapeakandthendecreaseswithanincreaseinthereducednucleardensity. Thepeakvalueandthecorresponding\nreduced density\u0010\n\u001a\n\u001a0\u0011\nmaxincrease with the increase in the values of Es(\u001a0)andL(\u001a0). Consequently, the Epot\ns(\u001a)\nwith low values of Es(\u001a0)andL(\u001a0)quickly becomes negative and thereby makes the NSE soft or supersoft. Since\nthe exchange strength parameters \"exand\"l\nexare fixed for all the sets, the contribution coming from the exchange\nterm remains the same. In fact, this exchange contribution to NSE increases with an increase in the nuclear matter7\n0123051015202530Eskin(ρ) [MeV]ρ\n/ρ0(a)0\n123-100102030 \n Set III \n Set II \n Set IEspot(ρ) [MeV]ρ\n/ρ0(b)0\n1230102030405060 \n Set III \n Set II \n Set IEs(ρ) [MeV](ρ\n/ρ0(c)\nFIG. 3: Components of Nuclear symmetry energy Es(\u001a)as a function of the reduced density\u001a\n\u001a0. In the extreme left panel we have shown the\nkinetic energy contribution to the NSE. In the middle panel, the contribution coming from the interaction part is shown and in the right panel,\nthe total NSE is plotted.\ndensity. One can note that, the contribution from the term involving the parameters \"0and\"l\n0is positive and the\ncontribution from the term involving the parameters \"\rand\"l\n\ris negative. This negative contribution decreases as\nwe increase the Es(\u001a0)value. Consequently, the potential contribution to NSE becomes more and more negative after\ncertain nuclear matter density leading to a softer NSE.\nThe slope parameter L(\u001a0) = 3E0\ns(\u001a0)as obtained from the fitting procedure can be correlated with the Es(\u001a0). In\nfact these two parameters decide the density dependence of NSE and are highly correlated. From different mean field\nmodels, it is found that, these two parameters are linearly correlated. Roca-Maza et al. have obtained a correlation\nbetweenEs(\u001a0)andL(\u001a0)from the experimentally measured values 19:6\u00060:6fm3of the dipole polarisability of208Pb\nasEs(\u001a0) = (24:5\u00060:8) + (0:168\u00060:007)L(\u001a0)[68]. Lattimer and Steiner [88] have obtained a similar but slightly\ndifferent correlation relation from the experimental value of the dipole polarisability of208Pb as reported by Tamii et\nal. [12]. In Figure 4, we have shown these experimentally extracted correlation of Es(\u001a0)andL(\u001a0). In the figure,\nwe have also shown the correlation from IAS results as extracted from Ref. [88]. The red dots are the results of\nthe present work as constrained from the fitting procedure of the interaction parameters of the finite range effective\ninteraction. At a given Es(\u001a0), we have obtained three different values of L(\u001a0)corresponding to the fitting of the\nNSE to the value of Es(\u001ac) = 26:65\u00060:2MeV. The correlation between Es(\u001a0)andL(\u001a0)from our calculations are\nmore or less similar and compatible to that of Roca-Maza et al. It may be noted from the figure that all the points\nof our calculation lie in the black hatched region even though the results corresponding to Es(\u001a0) = 35MeV overlap\nwith the region extracted by Lattimer and Steiner.\nIn order to have an idea of the behaviour of the symmetry energy at a reference density \u001ac<\u001a 0, we may expand\nthe NSE around Es(\u001ac)as\nEs(\u001a) =Es(\u001ac) +L(\u001ac)\"+Ksym(\u001ac)\n2!\"2+O(\"3); (17)\nwhere\"=\u001a\u0000\u001ac\n3\u001ac.L(\u001ac) = 3\u001acdEs(\u001a)\nd\u001aj\u001a=\u001acis the density slope parameter and Ksym(\u001ac) = 9\u001a2\ncd2Es(\u001a)\nd\u001a2j\u001a=\u001acis the curva-\nture parameter at the reference density \u001ac. Knowledge of L(\u001ac)besides being an important quantity in determining\nthe density dependence of the NSE at low density region, plays an important role in determining the neutron skin\nthickness of heavy nuclei. In view of this, more precise constraints on L(\u001ac)will help a lot in the understanding of\nthe nuclear equation of state. We have plotted the NSE as a function of the parameter \"in Figure 5 for the three sets\nassuming the reference density as the central density for208Pb. In the figure, the NSE of Russotto et al. [85] with a\nparametrized form as in Eq.(13) and the extracted value of the exponent as \u001b= 0:72is also shown for comparison.\nThe values of the parameters L(\u001ac)andKsym(\u001ac)have also been calculated from the expression of the NSE. The\ncalculated values of L(\u001ac)andKsym(\u001ac)are given in the Table III. Recently Zhang and Chen [14] have extracted the\nvalue ofL(\u001ac)from the analysis of the electric dipole polarisability in208Pb asL(\u001ac) = 47:3\u00067:8MeV. From the8\n204 06 08 0202428323640Es(ρ0) [MeV]L\n(ρ0) [MeV]IAS resultsR\noca-Maza et al.\nFIG. 4: Nuclear symmetry energy Es(\u001a)as a function of L(\u001a0)as extracted from different works. The black hatched region represents the results\nof Roca-Maza et al. [68]. The region between the two thick blue lines represent the results of Lattimer and Steiner [88] . The red ruled area is the\ncorrelation from IAS results as extracted from Ref.[88]. The red dots are the calculated values from the present work.\n-0.4- 0.20 .00 .20 .40 .60 .81 .01 .2020406080Es(ρ) [MeV]ε\n  Set III \n Set II \n Set I \n Russotto et al.ρc=0.11 fm-3\nFIG. 5: Nuclear symmetry energy Es(\u001a)as a function of \"=\u001a\u0000\u001ac\n3\u001ac. The results of Ref. [85] are also shown in the figure for comparison.\npresent calculations, the L(\u001ac)are predicted to be 46:69\u0014L(\u001ac)\u001455:63MeV which are well within the extracted\nvalues of Zhang and Chen [14]. Using the finite range effective interaction, we have obtained Ksym(\u001ac)in the range\n\u0000120:69\u0014Ksym(\u001ac)\u0014\u000082:8MeV. It is worth to mention here that, while empirical constraints on L(\u001ac)are available\nin literature there are no such constraints on the parameter Ksym(\u001ac).9\nTABLE III: Prediction of different isovector indicators from the finite range effective interaction.\nParameter sets L(\u001a0)L(\u001ac)Ksym(\u001a0)Ksym(\u001ac)assacsQ\n(MeV)(MeV) (MeV) (MeV) (MeV)(MeV)(MeV)\nSet I 44.1646.69-201.32 -120.69 43.8836.4652.59\nSet II 55.0551.15-168.86 -101.82 52.4451.1746.81\nSet III 66.055.63-136.16 -82.80 61.0565.9742.42\nIII. SYMMETRY ENERGY COEFFICIENT\nTounderstandthebehaviourofnuclearsymmetryenergyinnuclearmasses, itisessentialtohaveacorrelationofthe\nsymmetry energy coefficient in finite nuclei asym(A)and the NSE Es(\u001a). Within the droplet model (DM), Centelles\net al. [19, 89] have found that the symmetry energy coefficient asym(A)of finite nuclei with mass number Ais equal\nto the nuclear symmetry energy, Es(\u001aA)calculated at the central density \u001aAof the nucleus i.e asym(A)'Es(\u001aA).\nThey have shown that, this relation not only works for heavy nuclei such as208Pb but also holds for medium mass\nnuclei [89]. Such a relation is helpful in providing a direct correlation of the isospin observables of finite nuclei with\nEs(\u001a)at subnormal densities.\nThe symmetry energy coefficient asym(A)in finite nuclei can also be expressed as [89]\nasym(A) =Es(\u001a0)\n1 +xA; (18)\nwhere\nxA=9\n4Es(\u001a0)\nQA\u00001=3: (19)\nQis the surface stiffness parameter that measures the resistance of the nucleus against separation of neutrons from\nprotons to form a skin. This quantity is directly related to the nuclear surface symmetry energy [38, 67]. Usually Q\nis obtained from semi-infinite nuclear matter calculations [90, 91]. The surface stiffness parameter can be calculated\nfrom the expression [92, 93]\nQ=9\n4\u0012\nL(\u001a0)\u0000Ksym(\u001a0)\n12\u0013\u00001\nE2\ns(\u001a0): (20)\nHowever, in the present work, we have used the expression\nQ=9\n4\u0012Es(\u001a0)\nasym(A)\u00001\u0013\u00001\nEs(\u001a0)A\u00001=3: (21)\nThe calculated values of the surface stiffness coefficient are given in Table III. One can note that, with an increase in\nthe values of L(\u001a0)andEs(\u001a0), the value of Qdecreases.\nAssuming the relation asym(A)'Es(\u001aA)to be valid in a wide range of nuclear mass, we have calculated the sym-\nmetry energy coefficient for the three sets of nuclear EoSs. For this purpose we have used the density parametrization\n\u001aA=\u001a0\n1 +\u0014A\u00001=3; (22)\nwhere the constant \u0014is fixed to reproduce the density of208Pb. In Figure 6, the reference density \u001aAas calculated\nfrom Eq. (22) is shown as a function of mass number A. The reference density of heavy nuclei are usually considered in\nliterature as close to 0:1fm\u00003. The same has been reflected in the figure. The reference density increases with Aand\nalmost saturates for large nuclear masses. In Figure 7, the symmetry energy coefficient of finite nuclei (as obtained\nin an approximate manner) is plotted as a function of the mass number for the three different sets of interaction\nparameters. In the plot, we have extrapolated the results to large values of nuclear mass A= 1000to get an idea\nof the asymptotic value of asym(A). As the mass number increases, asym(A)increases slowly to reach its asymptotic\nvalue. In the figure, we have also shown the asym(A)as calculated by using SLy4 interaction [94, 95]. Even though\nthe behaviour of the curve for SLy4 is the same as that of the curves for finite range effective interaction SEI, it lies\nbelow the SEI curves. The reason may lie in the fact that, the finite range interaction parameters are fitted in such\na manner to reproduce the empirically constrained value of Es(\u001ac) = 26:65\u00060:2MeV for208Pb [23] whereas for the\nSLy4 interaction asym(208)is around 22:90MeV. It can be observed that, the three sets namely Set I, Set II and\nSet III predict almost similar values for asym(A)with a little spread both at the high mass and low mass region.\nIn a leptodermous expansion within liquid drop model, the symmetry energy coefficient can be expanded in powers\nofA\u00001=3as\nasym(A) =Es(\u001a0)\u0000assA\u00001=3+acsA\u00002=3; (23)10\n04 08 01 201 602 002 402 800.070.080.090.100.110.12ρΑ [fm-3]A\nFIG. 6: Reference density \u001aAas a function of mass number A. The\nreference density for208Pb is taken as 0:11fm\u00003.\n02 004 006 008 001 00015202530asym(A) [MeV]A\n Set III \n Set II \n Set I \nSLy4asym(208)=26.65 MeVFIG. 7: Symmetry energy coefficient asym (A)as a function of mass\nnumberAfor the three EoSs. The results of SLy4 interaction [94] are\nalso shown in the graph for comparison.\n404 55 05 56 03035404550556065 \n extracted from fit of asym(A) \n Linear fitass [MeV]L\n(ρc) [MeV]\nFIG. 8: Surface symmetry coefficient assas extracted from the leptodermous expansion of asym (A)as a function of the density slope parameter\nL(\u001ac)at reference density. Here the L(\u001ac)values are considered for208Pb as given in Table III. The plot also shows a linear fit between assand\nL(\u001ac).\nwhereassis the coefficient of the surface symmetry term and acsis the coefficient of the curvature symmetry term.\nWe have fitted the asym(A)as calculated from Set I, Set II and Set III to Eq.(23) and extracted the coefficients ass\nandacs. The values of these coefficients are given in Table III. One can note that, the values of assandacsincrease\nwith an increase in the value of L(\u001ac).\nKeeping up to first order in A\u00001=3in Eq. (23), we can express the surface symmetry term as\nass= [Es(\u001a0)\u0000asym(A)]A1=3: (24)\nSubstituting for asym(A)from Eq.(17) in Eq. (23), we obtain\nass= [Es(\u001a0)\u0000Es(\u001ac)\u0000L(\u001ac)\"]A1=3: (25)\nHere the relation asym(A)'Es(\u001aA)is assumed and the expansion in Eq.(17) is truncated after first order term in\n\". This equation (25) clearly indicates a linear relation between the surface symmetry term and the density slope11\nparameter at a reference density. In Figure 8, we have shown the linear correlation between the extracted assand\nL(\u001ac). A linear fit of the data points provides the relationship\nass=\u000045:872 + 1:922L(\u001ac): (26)\nIV. NEUTRON SKIN THICKNESS OF208Pb\nThe neutron skin thickness (NST) is defined as the difference between the rms radii for the density distributions of\nthe neutrons and protons in the nucleus,\n\u0001rnp=\nr2\u000b1=2\nn\u0000\nr2\u000b1=2\np; (27)\nwhich can be expressed as [21]\n\u0001rnp=\u0001rbulk\nnp+\u0001rCoul\nnp+4rsurf\nnp: (28)\nThe bulk part \u0001rbulk\nnp =q\n3\n5(Rn\u0000Rp)is proportional to the distance between the neutron and proton radii of\nuniform sharp distributions. \u0001rCoul\nnp =\u0000q\n3\n5e2Z\n70Es(\u001a0)is a correction due to the Coulomb repulsion and \u0001rsurf\nnp =\n2:5q\n3\n5\u0010\nb2\nn\nRn\u0000b2\np\nRp\u0011\n'q\n3\n55\n2R\u0000\nb2\nn\u0000b2\np\u0001\nis a correction due to the difference in the surface widths bnandbpof the\nneutron and proton profiles.\nIt is well known that, the liquid drop model provides a useful tool to correlate the NST with different NSE\nparameters. In the present section, we wish to find out the correlations among the neutron skin thickness of208Pb\nand different isovector parameters of the nuclear symmetry energy. Within the purview of DM [38] and neglecting\nthe shell correction, the bulk part of NST can be expressed as\n\u0001rbulk\nnp=r\n3\n5t; (29)\nwhere\nt=3\n2r0Es(\u001a0)\nQ\u0012I\u0000Ic\n1 +xA\u0013\n: (30)\nHereR=r0A1=3,r0=\u00004\n3\u0019\u001a0\u0001\u00001=3andIc=e2Z\n20Es(\u001a0)Ris the Coulomb correction to the symmetry energy coefficient.\nSubstituting the expressions of xAfrom Eq. (18) and Qfrom Eq. (21) into the above relation (30), we obtain,\nt=2\n3r0\u0014\n1\u0000asym(A)\nEs(\u001a0)\u0015\nA1=3(I\u0000Ic): (31)\nUsing Eq.(6) and asym(A) =Es(\u001aA), Eq.(31) can be reduced to\nt'\u00002r0\u000fA\f\u0012\n1 +Ksym(\u001a0)\n2L(\u001a0)\u000fA\u0013\nA1=3(I\u0000Ic); (32)\nwhere\f=L(\u001a0)=3\nEs(\u001a0)=E0\ns(\u001a0)\nEs(\u001a0)and\u000fA=\u001ac\u0000\u001a0\n\u001a0. Obviously, Eqs.(31) and (32) suggest a correlation between the bulk part\nof the NST in finite nuclei and some isovector indicators such as 1\u0000asym (A)\nEs(\u001a0);\fandKsym (\u001a0)\nEs(\u001a0). The close correlations\namong different isovector observables in finite nuclei with symmetry energy parameters have been studied and are\nreported in literature [13, 19–21, 23, 39, 40, 68, 89, 96, 97]. In most of the studies, the correlation between \u0001rnpor its\nbulk part of a given nucleus with Es(\u001a0)=QorL(\u001a0)have been established. In some cases, \u0001rnphas been correlated\nwithEs(\u001a0)\u0000asym(A). The bulk part of NST, or more specifically the quantity t, has a dominant contribution to\nthe neutron skin thickness and depends on the nuclear symmetry energy parameters. Basing upon the expressions\noftin Eqs. (31) and (32), it is more reasonable to correlate twith the isovector indicators such as 1\u0000asym (A)\nEs(\u001a0);\f\nandKsym (\u001a0)\nEs(\u001a0). In the present work, we have considered three different values of Es(\u001a0)namely 33;34and35MeV\nand constrained the EoSs so as to reproduce the symmetry energy for208Pb in the range 26:65\u00060:2MeV. In view of\nthis, in the correlation plots, we have considered the combined parameter 1\u0000asym (A)\nEs(\u001a0)taking into account the role of\nEs(\u001a0). In Figure 9, we plot \u0001rnpfor208Pb as a function of 1\u0000asym (A)\nEs(\u001a0). In Figure 10, we show the correlation of\nthe quantity twith 1\u0000asym (A)\nEs(\u001a0). In these figures, the results of all the nine EoSs are shown. The shaded bands in the\nfigures depict the predictions from the regression procedure. The respective Pearson correlation coefficients CPare12\n0.180 .190 .200 .210 .220 .230 .240 .250.160.170.180.190.200.210.22Δrnp [fm][\n1-asym(A) / Es(ρ0)]CP= 0.9995\nFIG. 9: The neutron skin thickness in208Pb is plotted as a function of\nthe quantity 1\u0000asym(A)\nEs(\u001a0). The shaded region shows the correlation\nband with a Pearson correlation coefficient CP= 0:9995.\n0.180 .190 .200 .210 .220 .230 .240 .250.140.150.160.170.180.190.200.210.22t [fm][\n1 - asym(A) / Es(ρ0)]CP = 0.99996FIG. 10: The quantity tfor208Pb is plotted as a function of the\nquantity 1\u0000asym(A)\nEs(\u001a0). The shaded region shows the correlation band\nwith a Pearson correlation coefficient CP= 0:99996.\n0.400 .450 .500 .550 .600 .650 .700.160.170.180.190.200.210.22Δrnp [fm]β\nCP = 0.99945\nFIG. 11: The neutron skin thickness in208Pb is plotted as a function\nof the quantity \f. The shaded region shows the correlation band with\na Pearson correlation coefficient CP= 0:99945.\n0.400 .450 .500 .550 .600 .650 .700.140.150.160.170.180.190.200.210.22t [fm]β\nCP = 0.99995FIG. 12: The quantity tfor208Pb is plotted as a function of the\nquantity\f. The shaded region shows the correlation band with a\nPearson correlation coefficient CP= 0:99995.\nmentioned in the figures. One may note that, the Pearson correlation coefficient for Fig. 9 is 0.9995 and that for Fig.\n10 is 0.99996. In other words, the correlation for the bulk part of the NST is much better as compared to that of the\nwhole of the neutron skin thickness.\nOne should note that the quantity tis directly proportional to the parameter \f=E0\ns(\u001a0)\nEs(\u001a0). Another fact is that, EoSs\nwith same value of NSE at saturation density may have different slopes. In view of this, we emphasize the correlation\nbetween\u0001rnpand\finstead of finding a compact relation between \u0001rnpandL(\u001a0). For the EoSs employed in this\nwork,\u0001rnpis calculated to be in the range 0:17\u00000:21fm for the range of 0:425\u0014\f\u00140:675with a spreading of\naround 0:04fm. In Figures 11 and 12, we have shown respectively the correlation of \u0001rnpandtwith the parameter\n\f. It is interesting to note that, these quantities are highly correlated in a linear relationship. From a linear fit to the\nplots, we obtain the relationships\n\u0001rnp= (0:0934\u00060:0012) + (0:1796\u00060:0023)\f[fm]; (33)\nt= (0:6196\u00060:0005) + (0:2167\u00060:0008)\f[fm]: (34)\nThe values of \u0001rnpas estimated from the above relationship are in conformity with the predictions from a large\nsets of relativistic and non relativistic nuclear mean field models [13]. Usually L(\u001a0)shows a linear relationship with\nthe NST of finite nuclei. Roca-Maza et al. have extracted a linear relationship among \u0001rnpandL(\u001a0)for208Pb as\n\u0001rnp= 0:101 + 0:00147L(\u001a0)[40]. Substituting the values of L(\u001a0)in this extracted relation from our work we obtain\nthe NST in the range 0:16\u0014\u0001rnp\u00140:2fm. For completeness, we have plotted the \u0001rnpas a function of L(\u001a0)in\nFigure 13 and compared our results with that of Ref. [40]. Results of some mean field calculations are also shown in\nthe figure for comparison. The correlation coefficient in this case is CP= 0:99943a bit less than that of the \u0001rnp\u0018\f\nplot. A linear fit of the results returns us the relation \u0001rnp= (0:1066\u00060:00107) + (0 :0015\u00060:000019)L(\u001a0). This13\n304 05 06 07 08 00.100.150.200.25N\nL3,1D\nDME2N\nL3,2 Present work \n Roca-Maza et al., 2011Δrnp [fm]L\n (ρ0) [MeV]SLy4FSU GoldD\nDME1\nFIG. 13: The neutron skin thickness in208Pb is plotted as a function of the density slope parameter at saturation density. The solid triangles are\nthe extracted results of some of the mean field calculations of Ref.[40]. The results are compared with the linear fit of the neutron skin thickness\nin208Pb with the density slope parameter as obtained in Ref. [40]. The grey shaded region corresponds to the predicted region from the\nregression procedure of Ref.[40].\nexpression predicts a bit higher value of the NST as compared to the prediction of the linear fit of Roca-Maza et\nal.[40]. However, our results are well within the predicted regions of Ref.[40].\nWe next explore the correlation between the NST and the density slope parameter at a reference density \u001ac<\u001a 0.\nReplacing\u001aby\u001a0in Eq. (17) and keeping up to second order in \", we get\nEs(\u001a0)\u0000asym(A)\n3Es(\u001a0)=\f0\"\u0014\n1 +1\n2Ksym(\u001ac)\nL(\u001ac)\"\u0015\n; (35)\nso that the distance between the neutron and proton mean surface locations becomes\nt= 2r0\"\f0\u0014\n1 +1\n2Ksym(\u001ac)\nL(\u001ac)\"\u0015\nA1=3(I\u0000Ic); (36)\nwhere\f0=L(\u001ac)\n3Es(\u001a0)and the present value of \"is\u001a0\u0000\u001ac\n3\u001ac. Here we have used the fact that \u001aA'\u001ac< \u001a 0andA\ncorresponds to the mass number of the nucleus with central density \u001aA. Eq. (36) suggests a correlation between\nthe NST with the parameters \f0andKsym(\u001ac). The correlation of the NST with \f0andKsym(\u001ac)for the results of\nfinite range effective interaction are shown in Figures 14 and 15. As expected these quantities are highly correlated\nas depicted by the respective correlation coefficients. Linear fits to the correlations provide the relationships\n\u0001rnp= (\u00000:09263\u00060:00319) + (0 :56413\u00060:00637)\f0fm; (37)\nand\n\u0001rnp= (0:27682\u00060:0025) + (0:00085\u00060:00002)Ksym(\u001ac)fm: (38)\nTheneutronskinthickness \u0001rnpshowsastrongcorrelationamongcertainisovectorobservablesofheavyfinitenuclei\nsuch as208Pb. Therefore accurate determination of the NST of208Pb from experiments would provide constraints\non the density dependence of NSE and on the slope parameter. The Lead Radius Experiment (PREX) [36] is able to\ndetermine the neutron radius in208Pb upto 1%accuracy through the measurement of the parity violating asymmetry\nat low momentum transfer. The PREX results for the neutron skin thickness in208Pb are\u0001rnp= 0:33+0:16\n\u00000:18fm\n[36]. Experiments with hadronic probes constrained the NST in208Pbas\u0001rnp= 0:16\u0006(0:02) (stat)\u0006(0:04) (systfm\n[33] and\u0001rnp= 0:211+0:054\n\u00000:063(Osaka-RCNP)[34]. Measurements from coherent pion photoproduction yield a value\n\u0001rnp(208Pb) = 0:15\u00060:03fm (Mainz experiment)[98]. Our results from the finite range effective interactions are\nin conformity with these experimental results. In Figures 16 (a) and (b), we have shown the fitted results from our\ncalculation and compared with those of the Mainz experiment [98] and Osaka-RCNP results [34]. The Mainz results\nconstrain the parameters \fand\f0respectively in the range 0:145\u0014\f\u00140:48and0:38\u0014\f0\u00140:485. Similarly, the14\n0.440 .460 .480 .500 .520 .540.160.170.180.190.200.210.22Δrnp [fm]L\n(ρc) / 3Es(ρ0)CP = 0.99955\nFIG. 14: Neutron skin thickness in208Pb is correlated with the\nquantity\f0=L(\u001ac)\n3Es(\u001a0). The shaded region shows the correlation band\nwith a Pearson correlation coefficient CP= 0:99955.\n-130- 120- 110- 100- 90- 80- 700.160.170.180.190.200.210.22Δrnp [fm]K\nsym(ρc) [MeV]CP =0.99719FIG. 15: Neutron skin thickness in208Pb is correlated with the\nquantityKsym (\u001ac). The shaded region shows the correlation band\nwith a Pearson correlation coefficient CP= 0:99719.\n0.00.20.40.60.81.00.100.150.200.250.30 Linear FitP\nresent work Δrnp [fm]β\nMainzOsaka-RCNP(a)0\n.40 .50 .60 .70 .8Osaka-RCNPM\nainz Linear FitP\nresent work β\n'(b)\nFIG. 16: (a)The neutron skin thickness of208Pb is shown as a function of the parameter \f=L(\u001a0)\n3Es(\u001a0). (b) The same as a function of the\nparameter\f0=L(\u001ac)\n3Es(\u001a0), (\u001ac= 0:11fm\u00003). The solid dots are the calculations from the finite range effective interaction. The blue line is a linear\nfit to the solid dots. Our results are compared with the extracted results for NST of208Pb from Mainz experiment [98] and Osaka-RCNP\nresults[34].\nconstraints as obtained from a comparison of the Osaka-RCNP results with our results are 0:315\u0014\f\u00140:93and\n0:43\u0014\f0\u00140:625. The quoted errors of 0:06fm for the NST as expected from PREX II are too large to make a\ncontribution. Even the quoted errors of 0:03fm of the Mainz experiment define a 1\u001binterval of 0:06fm which spans\nthe full range of cases plotted in Fig. 13. It is worth to mention here that, from an analysis of Skyrme type forces,\nBrown [72] has obtained a linear relationship between the derivative of the NSE at \u001acand the neutron skin thickness\nof208Pb asdEs(\u001a)\nd\u001aj\u001ac=pa\u0001rnp, wherepa= 882\u000632MeV fm2. This relation can be transformed as \f0=\u001acpa\nEs(\u001a0)\u0001rnp.\nFor a given value of Es(\u001a0) = 35MeV, we get 0:41\u0014\f0\u00140:735from the results of Osaka-RCNP and 0:33\u0014\f0\u00140:5\nfrom the results of Mainz experiment. These values are compatible with the constraints obtained from Fig.16.\nV. CORRELATION OF NST WITH ISOVECTOR GIANT DIPOLE RESONANCE\nIn medium-to-heavy neutron rich nuclei, the dipole response of a nucleus to an externally applied electric field is\nmostly dominated by the giant dipole resonance (GDR) of width 2\u00004MeV [99, 100]. The isovector giant dipole15\nresonance is perceived as an out of phase collective oscillation of neutrons against protons. The excess neutrons in\nthe neutron-rich nuclei may form a skin and oscillate against the isospin-saturated core giving rise to a low-energy\nE1mode [101]. The nuclear symmetry energy at subsaturation densities acts as a restoring force in this collective\nexcitation phenomena. It is a well known fact that, the properties of IVGDR depend on the nuclear symmetry energy\n[12,102–105]. CorrelationswithdifferentcollectiveexcitationmodessuchastheIVGDRmayalsoputsomeconstraints\non the density dependence of the nuclear symmetry energy. Correlations of nuclear symmetry energy parameters with\nother isovector modes of collective excitations such as electric dipole polarisability [13, 14, 40, 100, 106], pygmy dipole\nresonance (PDR) [107–109] and isovector giant quadrupole resonance (IVGQR) [110] also exist in literature.\nThe IVGDR energy constant can be expressed in a semiclassical framework as [111]\nD=vuut8~2\nmr2\n0\"\nEs(\u001a0)\n1 + 3Es(\u001a0)\nQA\u00001=3#\n: (39)\nThe IVGDR energy constant depends on the factor Es(\u001a0)=Qand consequently depends on the neutron skin\nthickness\u0001rnp(see Eqs. (29) and (30)). Using the Qvalues calculated in the present work, we obtained the IVGDR\nenergy constant for208Pb in the range 79:77\u000081:1MeV in close agreement with the experimental value Dexp\u001980\nMeV for heavy nuclei. In the left panel of Figure 17, we show Das a function of the neutron skin thickness. The\nIVGDR energy constant appears to decrease with \u0001rnp. Since the NST depends on the density slope parameter at\na reference density, it is expected that, Dwill have certain relation with L(\u001ac). In the right panel of Figure 17, the\nIVGDR energy constant is plotted as a function of L(\u001ac)obtained from the finite range effective interactions. It\nappears that Ddecreases with an increase in the value of L(\u001ac). The experimental value of Dis exactly reproduced\nforL(\u001ac) = 51:9MeV corresponding to a neutron skin thickness of \u0001rnp= 0:194fm in208Pb.\nWe can write Eq. (39) as\nD=s\n8~2\nmr2\n0\u00123\nA1=3Q\u0013\u00001=2\u0014\n1 +A1=3Q\n3Es(\u001a0)\u0015\u00001=2\n: (40)\nExpanding the square-bracketed term in powers ofA1=3Q\n3Es(\u001a0)and retaining up to the 1st order we get\nD's\n8~2\nmr2\n0\u00123\nA1=3Q\u0013\u00001=2\u0014\n1\u0000A1=3Q\n6Es(\u001a0)\u0015\n; (41)\nwhich may be expressed as\nD=BA\u0012asym(A)\nt\u00131=2\u0014\n1\u0000A1=3Q\n6Es(\u001a0)\u0015\n: (42)\nHere we have substituted the expression for Qfrom the relationship of tin Eq. (30) and defined a constant BA=q\n4~2\nmr0A1=3(I\u0000Ic)for a given nucleus. In Eq.(42), the leading term is proportional to\u0010\nasym (A)\nt\u00111=2\n. This expression\nin Eq.(42) allows us to draw a linear correlation between the IVGDR energy and the quantity\u0010\nasym (A)\nt\u00111=2\n. In Figure\n18, we show Das a function of\u0010\nasym (A)\nt\u00111=2\n. The Pearson correlation coefficient for the present case is CP= 0:82. In\nthe figure we show a linear fit to the correlated data points that reads as D= 73:833 + 0:538\u0010\nasym (A)\nt\u00111=2\n. Assuming\nasym(A) = 26:65MeV, with an IVGDR energy constant of 80MeV, we obtain t= 0:202fm. The linear fit formula\nprovides a reasonable estimate of the NST for208Pb.\nVI. CONCLUSION\nInthepresentwork, wehavestudiedthedensitydependenceofnuclearsymmetryenergyusingafiniterangeeffective\ninteraction. The effective interaction has a finite range part so as to describe the correct momentum dependence of\nthe nuclear mean field as extracted from the optical model fits in the heavy-ion collision studies at intermediate\nenergies. By exploring the recently constrained nuclear symmetry energy at a subsaturation cross density ( \u001ac= 0:11\nfm\u00003), we have constructed three different sets of nuclear EoSs from the SEI. The EoSs constructed in the present\nwork satisfactorily reproduce the properties of SNM at saturation density. We have considered the nuclear symmetry\nenergy at saturation in the accepted range of 32\u00064MeV and obtained the density slope parameter in the range\n44\u0014L(\u001a0)\u001466MeV and the curvature parameter in the range \u0000201\u0014Ksym(\u001a0)\u0014\u0000126MeV. These values are\nwell within the constrained values from different analysis. The EoSs obtained in the present work not only satisfy the16\n0.160 .180 .200 .227879808182D [MeV]Δ\nrnp [fm]455 05 56 07879808182D [MeV]L\n(ρc) [MeV]\nFIG. 17: (Left panel): The IVGDR energy constant Din208Pb is plotted as a function of neutron skin thickness. (Right panel): The IVGDR\nenergy constant Dis shown as a function of the density slope parameter at a reference density \u001ac= 0:11fm\u00003. The horizontal line shows the\nexperimental value of D.\n101 11 21 31 479808182 \n inear fit \nPresent workD [MeV][\nasym(A) / t]1/2 [MeV/fm]1/2\nFIG. 18: The IVGDR energy is correlated with the quantity\u0010asym(A)\nt\u00111=2. The horizontal line shows the experimental value of D.\nconstraints for NSE in the subnormal densities but also pass within the experimentally extracted region of Russotto\net al.[85]. The constraints coming from astrophysical and terrestrial laboratories limit the NSE at a density twice the\nnormal nuclear matter density as Es(2\u001a0) = 46:9\u000610:1MeV [74, 75]. The predictions for NSE from our EoSs lie well\nwithin these limits at 2\u001a0and therefore viable for different nuclear studies for a wide range of density domain. For the\ngiven range of Es(\u001a0)andEs(\u001ac), we have obtained the density slope parameter in the range 46:69\u0014L(\u001ac)\u001455:63\nMeV and the curvature symmetry energy at a subsaturation density in the range \u0000120:69\u0014Ksym(\u001ac)\u0014\u000082:80MeV.17\nIt is worth to mention here that while constraints on L(\u001ac)are available in literature there are no reliable constraints\nonKsym(\u001ac). The predicted values of L(\u001ac)are in conformity with the available constraints [14].\nWe have analysed the correlation of the neutron skin thickness in208Pb with different isovector parameters. In order\nto calculate the NST, we required the values of the symmetry coefficient asym(A)as a function of mass number. We\nassumed the relationship asym(A)'Es(\u001aA). It is worth to mention here that, this relation has already been verified\nby many workers for different mean field calculations. From the calculated values of asym(A)we have extracted the\nvalues of surface symmetry energy coefficient in a leptodermous expansion. We were able to find a linear relationship\nbetween the surface symmetry energy coefficient with the density slope parameter at a subsaturation density L(\u001ac).\nThe NST is calculated within the framework of droplet model using the finite range effective interaction. Instead of\ncorrelating the NST with the usual density slope parameter at saturation or the quantity Es(\u001a0)\u0000asym(A), we find it\nreasonable to have a correlation between \u0001rnpwith the isovector indicators such as 1\u0000asym (A)\nEs(\u001a0)and\f=E0\ns(\u001a0)\nEs(\u001a0). The\nNSTaspredictedfromourEoSsliesintherange 0:17\u00000:21fmwithaspreadof 0:04fm. Thesevaluesareinconformity\nwith the experimentally extracted values of NST from different works. We have explored the correlation between the\nNST with the density slope parameter at a subsaturation density. A linear fit of \u0001rnpto the quantity \f0=L(\u001ac)\n3Es(\u001a0)\nis obtained from the correlation procedure. A similar correlation between the NST and the curvature symmetry\nparameter at subsaturation density is obtained from the calculation. From a comparison with the Mainz results, we\nconstrained the parameters \fand\f0respectively in the range 0:145\u0014\f\u00140:48and0:38\u0014\f0\u00140:485. Similarly,\nthe constraints as obtained from a comparison of the Osaka-RCNP results with our results are 0:315\u0014\f\u00140:93\nand0:43\u0014\f0\u00140:625. We calculated the IVGDR energy constant in208Pb and correlated the results with the NST\nand the density slope parameter at a subsaturation density. As a final remark, we say that, the parameters \fand\f0\nshould be accurately fixed up for an understanding of the NSE. Future experiments with accurate determination of\nthe neutron skin thickness of nuclei may pin down these parameters.\nAcknowledgement\nDB and SKT thank IUCAA, Pune (India) for providing hospitality and support during an academic visit where a\npart of this work is accomplished.\nReferences\n[1] B. A. Li, P. G. Krastev, D. H. Wen and N. B. Zhang, Eur. Phys. J. A ,55, 217 (2019).\n[2] J. Lattimer and M. Prakash, Phys. Rep. ,333-334 , 121 (2000).\n[3] A. W. Steiner, M. Prakash, J. Lattimer, P. Ellis, Phys. Rep. ,41, 325 (2005).\n[4] A. W. Steiner, S. Gandolfi, Phys. 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Fischer,1and Titus Neupert1\n1Department of Physics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland\n(Dated: September 23, 2020)\nDensity functional theory underlies the most successful and widely used numerical methods for\nelectronic structure prediction of solids. However, it has the fundamental shortcoming that the\nuniversal density functional is unknown. In addition, the computational result|energy and charge\ndensity distribution of the ground-state|is useful for electronic properties of solids mostly when re-\nduced to a band structure interpretation based on the Kohn-Sham approach. Here, we demonstrate\nhow machine learning algorithms can help to free density functional theory from these limitations.\nWe study a theory of spinless fermions on a one-dimensional lattice. The density functional is\nimplicitly represented by a neural network, which predicts, besides the ground-state energy and\ndensity distribution, density-density correlation functions. At no point do we require a band struc-\nture interpretation. The training data, obtained via exact diagonalization, feeds into a learning\nscheme inspired by active learning, which minimizes the computational costs for data generation.\nWe show that the network results are of high quantitative accuracy and, despite learning on random\npotentials, capture both symmetry-breaking and topological phase transitions correctly.\nI. INTRODUCTION\nMaterials with strong electronic correlations host a\nvariety of intriguing phenomena and quantum phases.\nModeling and understanding these systems are among\nthe greatest challenges in theoretical condensed matter\nphysics. For quantitative and predictive results, numeri-\ncal calculations are indispensable. The most widely and\nsuccessfully used numerical approach to the electronic\nstructure problem is based on density functional the-\nory (DFT). In condensed matter physics, DFT is often\nlinked to band structure calculations, while it is in prin-\nciple much more powerful than that. The Hohenberg-\nKohn theorems guarantee that a (potentially correlated)\nmany-body ground state is uniquely determined by its\nenergy and charge density distribution1. However, for\npractical implementations and a physical interpretation\nof calculated results, the Kohn-Sham ansatz is commonly\nused, producing the band structure of a di\u000berent, non-\ninteracting system with the same energy and density2.\nThe implicit assumption is that this band structure cap-\ntures the essential physics of the original system, at least\nif correlations are weak enough.\nA critical shortcoming of DFT is that its eponymous\nfunctional is not known; instead, approximations on var-\nious levels of complexity are commonly employed3. It\nis important to emphasize that most of the functional\nisuniversal , representing the many-particle Schr odinger\nequation. The only nonuniversal input in a DFT calcu-\nlation for a crystal is the potential landscape within the\nunit cell induced from the ions and core electrons as well\nas the particle number, both of which do not a\u000bect the\nuniversal part of the functional.\nThe recent rise of machine-learning methods used to\nmodel physical systems has sparked hopes to use these\nmethods for improving DFT calculations4. The ap-\nproaches interject the DFT work\row at various stages,\nranging from improving the Kohn-Sham scheme by rep-\nresenting the exchange-correlation functionals5{10or ap-proximating the unknown energy functional and its\nderivatives11{19. More recent works bypass the Kohn-\nSham-solution scheme by directly learning the map-\nping between material parameters and ground-state\nproperties20{31, or constructing the ground-state wave-\nfunction from a corresponding density distribution32.\nDespite these advancements, previous approaches are ei-\nther limited by complicated, non-scalable networks, suf-\nfer from ine\u000ecient training data generation or struggle in\napplications to the di\u000berent physical phases of the used\nmodels.\nIn this work, we take an approach that follows three\nguiding principles: (i) implicit knowledge representation\nis the key strength of neural networks. Therefore, we\nuse a neural network to implicitely represent the (mini-\nmized) DFT functional . (ii) We aim at solving for phases\nof quantum matter beyond the band structure paradigm.\nTo that end, we train the neural network to directly out-\nput correlation functions32, which can be used to char-\nacterize phases and phase transitions. (iii) A balanced\ndataset is the key challenge as data acquisition { theo-\nretical or experimental { is costly. In such settings, ac-\ntive learning schemes33{37o\u000ber better results with fewer\ntraining instances. In general, active learning describes\nan algorithm which can actively choose the data it wants\nto learn from during training. Here we employ a proce-\ndure inspired by active learning, to incorporate data from\ndi\u000berent system sizes in the a priori data generation. In\nparticular, our approach uses costly data of larger sys-\ntems only in situations where large \fnite-size e\u000bects are\ndetected.\nFigures 1 (a) and (b) summarize our model, neural\nnetwork, and work\row. We choose to work with a one-\ndimensional lattice model of spinless fermions. The hop-\nping and interaction terms of the model de\fne our `uni-\nversal Schr odinger equation' and are therefore left unal-\ntered throughout the study. Input to the neural network\nis the problem-speci\fc potential and particle number. Its\noutput is the ground-state energy EGSas well as thearXiv:2005.03014v2  [cond-mat.dis-nn]  22 Sep 20202\nFIG. 1. (a) Schematic of the dense neural network used to learn the map from unit-cell potentials and \flling to ground-state\nenergy and density-density correlators. (b) Data generation inspired by active learning allows to check the energy deviations\nof di\u000berent system sizes and continuing to larger systems only if necessary. (c) Exact versus predicted ground-state energies\non the test data set of the actively learning network (ALN). The inset shows the absolute error on the test-energy values as a\ndensity histogram n. (d) Mean absolute error per correlator entry on the test data set for the ALN. (e) Exact versus predicted\nground-state energies for the network trained on a single system size (PLN), with the inset showing the density nof absolute\nerrors on the test data set. (f) Mean absolute error per correlator entry on the test data set for the PLN.\ndensity-density correlation function. We start by intro-\nducing the employed learning scheme and demonstrate\nthe quantitative accuracy of network predictions, after\ntraining it on random potentials. The active training\nshows superior performance compared to conventional\ntraining. We obtain mean squared errors of the energy of\n3:08\u000110\u00004in units of the hopping integral. Finally, we ap-\nply the trained network to a topological and a symmetry-\nbreaking phase transition. Our results demonstrate a\nscalable architecture, able to capture interacting lattice\nmodels, with successful applications to structured phases.\nII. MODEL\nWhile density functional theory was originally formu-\nlated as a continuum theory, it has also been successfully\napplied to lattice models38. We consider a Hamiltonian\nfor spinless fermions on a one-dimensional lattice with\nsites labelled by i= 1;\u0001\u0001\u0001;Lunder periodic boundaryconditions,\n^H=\u0000tX\ni\u0010\n^cy\ni^ci+1+ h.c.\u0011\n+UX\ni^ni^ni+1\n+U0X\ni^ni^ni+2+X\niVi^ni; (1)\nwhere ^cy\niand ^ciare the fermion creation and annihilation\noperators on site iand ^ni= ^cy\ni^ciis the corresponding den-\nsity operator. Nearest-neighbor hopping is parametrized\nbyt, which will serve as the energy unit throughout. The\nparticles are subject to a repulsive interaction on nearest-\nand next-nearest-neighbor sites which we \fx to U= 1\nandU0= 0:5 so as to model a lattice analogue of the\nCoulomb interaction. This parameter choice places the\nsystem in a metallic, but strongly correlated phase in\nabsence of a potential Vi, even at half \flling39.\nMotivated by the Hohenberg-Kohn theorems, we con-\nsider the kinetic term and the electron-electron interac-\ntions as universal, such that the external or ionic poten-\ntial^Vext=P\niVi^nitogether with the \flling uniquely de-\ntermine the ground state and all of its properties. We\nonly consider potentials with periodicity of four sites.\nThat is, the four values Vi,i= 1;:::;4, completely spec-\nify the Hamiltonian for any lattice size L= 4Nuc, with3\nNucthe number of unit cells and Vi=Vi+4for alli. This\nfour-site unit cell can be thought of as the discretized unit\ncell of a periodic crystal, while Viis the ionic potential\nin this analogy. We restrict it to the range Vi2[\u00004;4].\nWe further denote by the real number 0 < n e<4 the\nparticle \flling per unit cell. We emphasize that despite\nimposing this periodicity to the potential, our approach\nis able to capture phases which spontaneously break the\nfour-site translation symmetry.\nIII. LEARNING\nThe supervised-machine-learning algorithm we use by-\npasses the Kohn-Sham scheme by directly learning the\nmap from the external parameters neandVito the cor-\nresponding ground-state energy and density-density cor-\nrelatorsh^ni^njiGS. The density-density correlators are\ncalculated for two adjacent unit cells40. The chosen fully\nconnected neural network41consists of four hidden layers\nwhich increase in size towards the output as depicted in\nFig. 1 (a) (see also Appendix A).\nA central challenge in machine learning is unbiased and\ne\u000ecient data generation; one usually deals with limited\ncomputational or experimental resources. Here, we gen-\nerate data by \fnite-size exact diagonalization (ED) of\nsystems with randomly chosen ne; Vi. In order to reduce\n\fnite size e\u000bects, these examples should naively be gen-\nerated with as large systems as possible. However, the\ncomputational cost for data generation with ED grows\nexponentially with system size. For this reason, we em-\nploy a procedure inspired by active learning , performing\ncostly large system ED, as depicted in Fig. 1 (b), only if\nnecessary. Using random values for neandViand start-\ning with a comparison between Nuc= 3;4, the scheme\niteratively computes larger systems until the \fnite size\ndeviation between ground-state energies lies below a pri-\norly chosen threshold \u0012. Correspondingly, the fast com-\nputation of smaller systems is used as often as possi-\nble, while providing more accurate data in critical cases\n(see Appendix B). The samples are further augmented by\napplying translations within the unit cell and inversion,\nallowing the network to capture the symmetries of the\nuniversal part of the Hamiltonian.\nWe contrast the active learning approach outlined\nabove with a passive learning scheme using training data\ngenerated for systems of \fxed size Nuc= 5, with \flling ne\nand on-site potentials Vichosen randomly. This system\nsize is still solvable e\u000eciently by ED, while su\u000eciently re-\nducing \fnite size e\u000bects. The data are again symmetry-\naugmented. Both learning procedures were run with an\nequal time budget to ensure comparability.\nA mean absolute error loss function is then optimized\nto obtain the weights and biases of the actively (ALN)\nand passively (PLN) learning neural network. The result-\ning performance is evaluated on unseen data, consisting\nof 20 % of the full data set. Over\ftting was avoided\nfor both systems by suitable hyperparameter choices (see\nFIG. 2. Neural network results for a transition between dif-\nferent (obstructed) atomic limit insulators. (a) Compressibil-\nity\u0014for various potential strengths at quarter \flling, as calcu-\nlated from the actively and passively learned neural network,\nexact diagonalization (ED) and density matrix renormaliza-\ntion group (DMRG) of several system sizes. (b) Schematic\ndepiction of the potential in the four-site unit cell: depending\non the strength and sign, two obstructed atomic limits and a\nmetallic phase can be realized. (c) Corresponding observable\nCas calculated from the 8x8 density-density correlator for the\nsame numerical methods as used for the compressibility. The\ninsets show the correlator as obtained from the ALN in the\n\frst (lower left) and second atomic limit (upper right) with\nthe unit cell depicted in red.\nAppendix C). The absence of signi\fcant deviations in the\nenergy correlation plot in Fig. 1 (c) shows that the ALN\nperforms well on random data, with an absolute error\npeaked at 1 :2\u000110\u00002. Similarly, Fig. 1 (d) shows only small\nerrors in the correlator prediction, with an overall mean\nabsolute error of 1 :8\u000110\u00003. The PLN performs worse in\npredicting energy values and correlators, as shown in the\ninset of Fig. 1 (e) and in Fig. 1 (f), with errors at least\ntwice as large as in the case of the ALN. This compari-\nson shows the advantage of intelligent data generation at\n\fxed computational time budget.\nRandom potentials rarely represent a relevant physical\nscenario. We therefore present in the following how the\nrandomly trained ALN outperforms the PLN for struc-\ntured systems, where Viobey further symmetries. Inter-\nestingly, including the smaller samples, which are su\u000ber-4\ning from \fnite-size e\u000bects, with a reduced sample weight\nin the ALN training leads to a more stable model predic-\ntion, in particular for physical systems (see Appendix B).\nWe attribute this feature to a regularization of the model.\nIV. LEARNABILITY OF OBSTRUCTED\nATOMIC LIMITS\nWe consider the model introduced in Eq. (1) for a po-\ntential choice ( V1;V2;V3;V4) = (0;V;V; 0) at quarter \fll-\ning (ne= 1). The system is metallic for V= 0 which\nseparates two distinct insulating phases for V > 0 and\nV < 0. ForV > 0 (V < 0), Wannier functions are lo-\ncalized between the unit cells (in the middle of the unit\ncell). This corresponds to two topologically distinct (ob-\nstructed) atomic-limit insulators, with the intra-unit-cell\nhopping being e\u000bectively reduced (enhanced) compared\nto the inter-unit-cell hopping. The insulating nature of\nthese phases can be shown by calculating the compress-\nibility\n\u0014=1\nn2e\u0012@2EGS(ne)\n@n2e\u0013\u00001\n; (2)\nwhereneis the electron \flling and EGS(ne) the cor-\nresponding ground-state energy. Figure 2 (a) shows\nthat, as one approaches the critical metallic state around\nV= 0,\u0014increases rapidly. We emphasize that since\n\u0014is the second derivative of the energy, it is extremely\nsusceptible to errors. Note, further, that the ED data\nshow a strong even-odd e\u000bect in Nuc. We also calculated\n\u0014(V) with the density matrix renormalization group algo-\nrithm (DMRG)4243forNuc= 28. Compared with these\nexact results, the ALN produces a meaningful \u0014(V) with\na peak value \u0014(V= 0) interpolating between the results\nof even and odd Nuc. On the contrary, the PLN is worse\nwith a less pronounced and non-symmetric peak. Even\nthough only trained with at most Nuc= 6 data, we see\nthat the ALN also resembles the Nuc= 28 DMRG result\nreasonably well.\nThe location of Wannier centers can be used to dif-\nferentiate between the two phases. De\fning C=\n(h^n2^n2i\u0000h^n2^n3i)\u0000(h^n4^n4i\u0000h^n4^n5i)44, the trivial\natomic limit with localization in the unit cell is obtained\nforC > 0, the phase transition happens at C= 0 and\nthe non-trivial atomic limit has C < 0. Figure 2 (c) high-\nlights that both networks are able to capture Cacross the\ntransition well, but the ALN results are markedly more\naccurate than the PLN results when compared with the\nED and DMRG data. This supports the statement that\nour active learning scheme delivers quantitatively better\nresults.\nFIG. 3. Neural network results for a spontaneous symmetry-\nbreaking phase. (a) Ground-state energy for V= 0 for sev-\neral electron \fllings neas calculated from the actively and\npassively learned neural network and ED. The inset displays\nthe noninteracting band structure. (b) Ground-state energy\nas a function of the electron \flling nein the symmetry broken\nphase (V= 4). The kink at ne= 1 signals an interaction-\ninduced incompressible phase. The inset reveals the band\n\rattening of the non-interacting system. (c) Phase diagram,\nschematic of the potential, and density-density correlation\nfunctions. The latter are obtained from the ALN for V= 0\n(left) andV= 4 (right). O\u000b-diagonal terms, whose inequiv-\nalence signals the symmetry-breaking phase (right) are high-\nlighted in red and green.\nV. LEARNABILITY OF SPONTANEOUSLY\nSYMMETRY-BROKEN PHASES\nSpontaneous breaking of translation symmetry can\nbe triggered by introducing a potential of the form\n(V1;V2;V3;V4) = (\u0000V;V;\u0000V;V) at quarter \flling ( ne=\n1). The symmetry-broken phase arises from the compe-\ntition between the next-nearest-neighbor interaction U0\nand the increasing potential V. The four-site transla-\ntional symmetry of Hamiltonian (1) is broken sponta-\nneously at Vc\u00191:8 (see Appendix D) by the two-site\nrelations of the emerging degenerate ground-states. The\nmetallic system shows a smooth dependence of EGSon\nnearound quarter \flling [Fig. 3 (a)]. With increasing\nV,EGS(ne) develops a kink at ne= 1, signalling the\nemergence of the symmetry-broken charge-density wave5\n(CDW) insulator [Fig. 3 (b)]. Both neural networks rep-\nresent the di\u000berent phases very well, the deviations at\nsmall \fllings are attributed to the limited amount of\ntraining samples in this limit.\nFigure 3 (c) shows that the correlation in the metal-\nlic phase is short-ranged and fast decaying, whereas\nthe symmetry broken phase possesses a distinct or-\nder. The corresponding order parameter hCSSBi=\n1\nNuchP\ni(\u00001)i^n2i+1iis, however, zero, since the two de-\ngenerate ground states have opposite imbalance in elec-\ntron density between the \frst and third site of each unit\ncell. Instead, the order can be diagnosed by comput-\ning the square of the order parameter from the density-\ndensity correlation functions. This amounts to hC2\nSSBi=\n2\nN2ucP\ni6=jh^n4i+1^n4j+1i\u00002\nN2ucP\ni;jh^n4i+1^n4j+3i+C0, with\nan overall shift C0=1\nN2uchP\ni^n2i+1i. Its nonzero value\nin the symmetry broken phase is implied by the inequiv-\nalence between the \frst (+) and second (-) term in the\nabove expression, highlighted in the density-density cor-\nrelator in Fig. 3 (c) by the red (+) and green (-) squares.\nThis behavior is well captured by the ALN, producing\nquantitatively accurate correlations in both phases.\nVI. CONCLUSION\nWe presented a supervised learning approach for lat-\ntice DFT, bypassing the Kohn-Sham solution scheme.\nEmploying a procedure inspired by active learning al-\nlowed us to improve our results at \fxed computational\ncost regarding data generation. Focussing on correlation\nfunctions on a subsystem and taking only the potential\nlandscape in the unit cell and particle number as input\nresults in a scalable architecture. Besides veri\fcation of\nour algorithm on unseen random potentials, we demon-\nstrated that the trained networks reliably solve for dif-\nferent structured phases.\nLooking ahead, it is highly desirable to construct\nsimilar implicit (neural network) representations of DFT\nfor systems in continuous space and higher dimensions,\nin particular to attack the electronic structure problem\nin strongly correlated regimes. The main challenge\nis the generation of valid and balanced data sets,\nand the incorporation of data from various sources,\nincluding conventional DFT, Monte Carlo calculations,\nexperiments, and future quantum simulation devices.\nTwo concepts on which our study relies, (1) focus on\ncorrelation functions instead of quantum states and (2)\nthe use of e\u000ecient learning, should prove useful in this\nfuture venture.\nACKNOWLEDGMENTS\nWe thank Giuseppe Carleo and Xi Dai for insightful\ndiscussions. This project has received funding from theEuropean Research Council (ERC) under the European\nUnions Horizon 2020 research and innovation program\n(ERC-StG-Neupert-757867-PARATOP).\nAppendix A: Network parameters\nThe supervised-machine-learning algorithm we pro-\npose in this paper uses dense neural networks of iden-\ntical architecture, for both the active and passive learn-\ning scheme. The layers are connected by Softplus( x) =\nln (1 +ex) activation functions, except for the output\nlayer. The last layer takes into account that correlator\nvalues and ground-state energies have di\u000berent ranges, by\nemploying a linear activation function. Table I indicates\nthe relevant parameters used in this paper.\nTABLE I. Relevant parameters used to create the neural\nnetworks in this paper.\nParameter Value\nNeurons Layer 1 50\nActivation 1 Softplus\nWeight init. 1 lecun uniform\nNeurons Layer 2 125\nActivation 2 Softplus\nWeight init. 2 lecun uniform\nNeurons Layer 3 150\nActivation 3 Softplus\nWeight init. 3 lecun uniform\nNeurons Layer 4 200\nActivation 4 Softplus\nWeight init. 4 lecun uniform\nNeurons Layer 5 65\nActivation 5 Linear\nWeight init. 5 lecun uniform\nOptimizer Adam\nBatch size 100\nLearning rate 0.001\nEpochs 1500\nAppendix B: Training Data\nThe choice of training examples is crucial in a machine\nlearning setting, as data is precious. An intelligent data\ngeneration procedure is advantageous if computational\ntime is \fnite, removing the necessity to always calculate\nas large systems as possible. Naively, the latter is the way\nto go in order to avoid \fnite size biases in the training\nexamples. We contrast these two approaches as active\nand passive learning schemes.\nThe naive approach generates data with exact diago-\nnalization of a \fve unit cell system, with electron \fllings\nneand potentials Viin the unit cell chosen randomly.6\nThe potentials are in the range jVij\u00144, and the \flling\ncan assume values between 0 :6\u0014ne\u00143:4 electrons per\nunit cell. Data augmentation is then used to enlarge the\ndataset and allow the neural network to capture the un-\nderlying symmetries of the physical problem. This means\napplying translations Vi!Vi+1and inversion within the\nunit cell, where the former leads to a shift of the cor-\nrelator data. The resulting dataset consists of 12.020\npairs of \fllings and external potentials, mapped to the\ncorresponding ground-state energies and density-density\ncorrelators. The split in training, validation and test sets\nis illustrated in Tab. II.\nTABLE II. Data for the passively trained neural network.\nParameter Value\nTotal number of samples 12020\nNumber of training samples 7212\nNumber of validation samples 2404\nNumber of test samples 2404\nNumber of samples from 20 site ED 12020\nSample weight 1\nHowever, large system diagonalization is not always\nnecessary to obtain accurate data. This is the basis for\nan active learning scheme, which performs the diagonal-\nization of large systems only if strong \fnite size e\u000bects\nare detected. The input to this procedure is the random\nchoice of electron \fllings neand potentials Viin the unit\ncell. The former assuming values between 0 :5\u0014ne\u00143:5\nelectrons per unit cell and the latter being chosen in the\nrangejVij\u00144. The training data for both networks was\nchosen asjVij\u00144 in order to ensure that the transition\nto the symmetry broken phase lies within the trained po-\ntential range.\nIf the ground-state energy for a Nuc= 3 andNuc= 4\nunit cell system, calculated with exact diagonalization,\ndeviates more than a priorly chosen threshold \u0012, a sys-\ntem with one additional unit cell is being calculated. This\nprocedure is repeated up to Nuc= 6 unit cells if neces-\nsary. This means that the neural network can query a\nlarger system whenever the deviation of ground-state en-\nergies exceeds the chosen threshold \u0012, thereby reducing\n\fnite size e\u000bects. Naively one would remove the small\ninaccurate samples from the dataset, however we found\nthat including them with a reduced sample weight yields\nsimilar results on random test data. However, we obtain\nbetter and more stable results on physical phases, mean-\ning that the presence of the less accurate data points acts\nas a regularization of the model. As we strive for mean-\ningful results on physical phases and not on random po-\ntentials, we consider this approach to be more promising.\nData augmentation is again used not only to enlarge the\ndataset to 74.500 input-output data pairs (see Tab. III),\nbut also to allow the network to capture the underlying\nsymmetries.\nThe choice of the threshold \u0012is a crucial parameter of\nthe active learning scheme. We therefore investigate theTABLE III. Data for the actively trained neural network.\nParameter Value\nTotal number of samples 74500\nNumber of training samples 44700\nNumber of validation samples 14900\nNumber of test samples 14900\nNumber of samples from 16 site ED 68850\nNumber of samples from 20 site ED 5500\nNumber of samples from 24 site ED 150\nSample weight 16 site data 1a\nSample weight 20 site data 2b\nSample weight 24 site data 3\naThis weight is increased to 3 if no 20 or 24 site system had to\nbe calculated for this sample.\nbThis weight is increased to 3 if no 24 site system had to be\ncalculated for this sample.\nimplications of di\u000berent thresholds (see Tab. IV), and\ntest the performance of the trained models on unseen ex-\namples. These examples are not part of the training set\nand were generated with random \fllings neand poten-\ntialsViin the rangejVij\u00144 for systems of Nuc= 5;6.\nTABLE IV. Number of samples for di\u000berent choices of the\nthreshold\u0012, calculated with exact diagonalization and an\nequal time budget for each parameter choice.\nNuc\u0012= 0:0\u0012= 0:001\u0012= 0:0025\u0012= 0:005\u0012= 0:01\u0012= 1:5\n4 1350 4200 13500 18850 81075 83250\n5 1335 575 1030 790 850 0\n6 245 10 30 15 20 0\nThe resulting mean absolute error is presented in Fig.\n4, indicating that the error decreases with increasing \u0012.\nWhen considering random potentials, it is therefore ad-\nvantageous to use training data generated with as large\n\u0012as possible, resulting in the largest possible dataset.\nConsequently, this means larger systems with Nuc= 5;6\nare never calculated.\nThe random data generation with potentials in the\nrangejVij\u00144 causes however mostly situations where\nthe electrons are localized in the potential landscape, due\nto large potentials Vi. Finite size deviations are however\nnegligible in such a case. This means that a good per-\nformance on this data does not necessarily correspond\nto a good elimination of \fnite size e\u000bects in a realistic\nphysical potential. Figure 4 highlights this with the eval-\nuation of a dataset with potentials jVij\u00141, with the\nminimum of the obtained mean absolute error shifting\ntowards smaller values of \u0012.\nConsequently, a trade-o\u000b is needed between the total\nnumber of training samples and the number of samples\nwithNuc= 5;6. The best compromise is reached for\n\u0012= 0:0025, with a large enough size of the dataset, while\ncontaining signi\fcant information about larger systems.\nThis parameter was therefore used to generate the \fnal7\ndataset used to train the actively learning neural network\nin this paper.\nMean absolute error\nFIG. 4. Mean absolute error of neural network predictions for\nvarious datasets, trained on examples generated with di\u000berent\nthreshold\u0012as illustrated in Tab IV. Random potentials were\nused to generate the evaluation data, with a Nuc= 5;jVij\u00144\ndataset consisting of 5600 samples, Nuc= 6;jVij\u00144 contain-\ning 220 datapoints and 1205 Nuc= 5;jVij\u00141 input - output\npairs. The evaluation sets were not used in the training pro-\ncess of the neural networks.\nAppendix C: Training Performance\nThe actively and passively constructed datasets are\nsplit into training, validation and test sets. Achieving\nconsistent performance on the optimized and unseen data\nindicates that the network has not been over\ftted. The\ncorresponding results for both training schemes are pre-\nsented in Tab. V and VI, highlighting that the intended\nmapping from electron \fllings and potentials to ground-\nstate energy and density-density correlator is well cap-\ntured.\nTABLE V. Performance of the passively trained neural net-\nwork. The last table section shows the performance on the\nALN test dataset.\nParameter Value\nMAE onEtest\nGS 2.8e-2\nMSE onEtest\nGS 1.5e-3\nMAE onh^ni^njitest\nGS 3.6e-3\nMSE onh^ni^njitest\nGS 5.9e-5\nMAE onEvalidation\nGS 2.6e-2\nMSE onEvalidation\nGS 1.3e-3\nMAE onh^ni^njivalidation\nGS 3.5e-3\nMSE onh^ni^njivalidation\nGS 4.9e-5\nMAE onEtraining\nGS2.5e-3\nMSE onEtraining\nGS1.2e-3\nMAE onh^ni^njitraining\nGS3.2e-3\nMSE onh^ni^njitraining\nGS4.2e-5\nMAE onEALNtest\nGS 3.7e-2\nMSE onEALNtest\nGS 2.7e-3\nMAE onh^ni^njiALNtest\nGS 4.5e-3\nMSE onh^ni^njiALNtest\nGS 7.0e-5TABLE VI. Performance of the actively trained neural net-\nwork. The last table section shows the performance on the\nPLN test dataset.\nParameter Value\nMAE onEtest\nGS 1.2e-2\nMSE onEtest\nGS 2.4e-4\nMAE onh^ni^njitest\nGS 1.8e-3\nMSE onh^ni^njitest\nGS 1.8e-5\nMAE onEvalidation\nGS 1.2e-2\nMSE onEvalidation\nGS 2.3e-4\nMAE onh^ni^njivalidation\nGS 1.8e-3\nMSE onh^ni^njivalidation\nGS 1.5e-5\nMAE onEtraining\nGS1.2e-2\nMSE onEtraining\nGS2.3e-4\nMAE onh^ni^njitraining\nGS1.8e-3\nMSE onh^ni^njitraining\nGS1.2e-5\nMAE onEPLNtest\nGS 1.8e-2\nMSE onEPLNtest\nGS 5.2e-4\nMAE onh^ni^njiPLNtest\nGS 2.9e-3\nMSE onh^ni^njiPLNtest\nGS 3.6e-5\nAppendix D: Determination of the point of\nspontaneous symmetry-breaking\nThe four-site translational symmetry of the consid-\nered extended Hubbard model can be spontaneously bro-\nken by the introduction of an external potential ^Vext=P4\ni=1Vi^ni. Choosing a potential of the form V1=\u0000V2=\nV3=\u0000V4=\u0000Vcauses a competition between next-\nnearest neighbor repulsion U0and the potential Vat\nquarter \flling. As a result, a symmetry-breaking phase\nwith two degenerate ground-states emerges, correspond-\ning to ordering the electrons either to site one or to site\nthree in each unit cell.\nWe consider two approaches in order to identify the\npotential strength at which the spontaneous symmetry-\nbreaking occurs. For each of these we investigate a \fnite\nsize scaling plot, to derive the extrapolated point of the\nphase transition in the thermodynamic limit.\nAs stated above, the symmetry broken phase possesses\na distinct order in the density-density correlator, occu-\npying only sites with negative potential. By adding a\nsmall potential on one site, e.g. V1=\u0000V\u0000\u000eV; \u000eV\u001cV,\none of the degenerate ground-states is favoured. Conse-\nquently, the occupation h^niiGSwill concentrate on the\n\frst lattice site of each unit cell. We therefore inves-\ntigate the di\u000berence in the occupation of site one and\nthree,C=h^n1iGS\u0000h^n3iGS. In the metallic phase for\na vanishing potential V= 0, all sites are equally occu-\npied, despite the small o\u000bset \u000eV. This changes around\nthe point of the phase transition, where explicitly one of\nthe symmetry-breaking ground states is selected. The\nposition of this jump in Ccan then be studied with\nvarious system sizes and extrapolated to the thermody-\nnamic limit. Since exact diagonalization already reaches\ncomputational boundaries for quite small systems, we\nadditionally employ the density-matrix renormalization8\nFIG. 5. (a) Finite size scaling plot of the critical potential Vc, obtained with a small potential \u000eVto break the ground-state\ndegeneracy. Density matrix renormalization group (DMRG) calculations of open boundary conditions are combined with the\nexact diagonalization (ED) of a 16 site system. (b) Schematic of the potential landscape in the unit cell. (c) Finite size\nscaling plot of the critical potential Vc, obtained with an additional half unit cell and electron to break the ground-state\ndegeneracy. Density matrix renormalization group (DMRG) calculations of open boundary conditions are combined with the\nexact diagonalization (ED) of an 18 site system. (d) Schematic of the potential landscape with additional two lattice sites.\ngroup algorithm (DMRG)4543. In order to ensure con-\nvergence also for large systems, open boundary condi-\ntions are considered. The in\ruence of the boundary can\nhowever be neglected when considering density-density\ncorrelations in the middle of the system. Figure 5(a) in-\ndicates the transition point as extrapolated from several\nsystem sizes, leading to a potential of Vc= 1:744.\nAdditionally, one can study the emergence of the spon-\ntaneously symmetry-breaking phase in the energy spec-\ntrum. 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V\u0013 azquez-Mayagoitia, arXiv:1910.10254 [cond-mat.mtrl-\nsci] (2019), arXiv:1910.10254.\n36J. Yao, Y. Wu, J. Koo, B. Yan, and H. Zhai, Phys. Rev.\nResearch 2, 013287 (2020).\n37G. Teichert, A. Natarajan, A. Van der Ven, and\nK. Garikipati, arXiv:2002.02305 [cs.LG] (2020),\narXiv:2002.02305.\n38K. Sch onhammer, O. Gunnarsson, and R. M. Noack, Phys.\nRev. B 52, 2504 (1995).\n39L. Markhof, B. Sbierski, V. Meden, and C. Karrasch,\nPhysical Review B 97, 235126 (2018).\n40In order to also use small systems ( Nuc= 4) with periodic\nboundary conditions as part of the training data, at most\ntwo adjacent unit cells from the interior of the density-\ndensity correlator are a valid representative.\n41F. Chollet et al. , \\Keras,\" https://keras.io (2015).\n42Calculations were performed using the TeNPy Library\n(version 0.5.0), for a \fnite but periodic lattice with maxi-\nmum bond dimension \u001fmax= 2000, max. truncation error\n\u000f= 10e\u00008 and energy convergence criterion \u0001 E= 10e\u00006.\n43J. Hauschild and F. Pollmann, SciPost Phys. Lect.\nNotes 5 (2018), 10.21468/SciPostPhysLectNotes.5,\ncode available from https://github.com/tenpy/tenpy ,\narXiv:1805.00055.\n44Note that we choose Cto highlight that two neighboring\nsites are always occupied by just one electron. That dis-\ntinguishes it from the band insulator case of \flling ne= 2\nand a spontaneously translation symmetry broken phase\natne= 1 (bothC= 0).\n45Calculations were performed using the TeNPy Library\n(version 0.5.0), for a \fnite lattice with maximum bond di-\nmension\u001fmax= 1000, max. truncation error \u000f= 10e\u000010\nand energy convergence criterion \u0001 E= 10e\u00006."    },    {        "title": "2006.03200v1.Maximizing_optical_production_of_metastable_xenon.pdf",        "content": "Maximizing optical production of metastable\nxenon\nH.P. L AMSAL , J.D. F RANSON ,AND T.B. P ITTMAN\nPhysics Department, University of Maryland Baltimore County, Baltimore, MD 21250\nAbstract: Thewiderangeofapplicationsusingmetastablenoblegasatomshasledtoanumber\nof different approaches for producing large metastable state densities. Here we investigate a\nrecently proposed hybrid approach that combines RF discharge techniques with optical pumping\nfrom an auxiliary state in xenon. We study the effect of xenon pressure on establishing initial\npopulationinboththeauxiliarystateandmetastablestateviatheRFdischarge,andtheroleofthe\nopticalpumpingbeampowerintransferringpopulationbetweenthestates. Wefindexperimental\nconditions that maximize the effects, and provide a robust platform for producing relatively large\nlong-term metastable state densities.\n1. Introduction\nMetastable noble gas atoms are used in diverse areas ranging from fundamental physics\nexperiments [1 –4], to applications in plasma display panels [5 –7], ultralow-power nonlinear\nopticaldevices[8 –10],andraregaslasers[11 –16]. Inmanyoftheseapplications,theproduction\nof high densities of metastable states is desirable, and techniques for efficiently promoting a\nsubstantial fraction of the ground state population to the metastable state are of paramount\nimportance. Althoughrelativelyhighmetastablestatedensities( \u00181013cm\u00003)canbeachievedfor\nshorttimescalesusingpulsedelectricaldischargetechniques[5 –7],therealizationofcomparable\nlong-termsteady-statedensitiesremainsasignificantchallenge. Standardapproachesinclude\nsteady state RF excitation (see, for example, [17 –19]), optical pumping techniques [20 –22], and\narecentlyproposedhybridmethodwhichcombinesboth[23,24]. Inthispaper,weinvestigate\nthe experimental conditions needed to optimize this hybrid approach to maximize the production\nof metastable state densities. Our study specifically investigates the production of metastable\nxenon (Xe*), but the general results are expected to apply to other noble gas species as well.\nTheenergyleveldiagramsinFigure1provideasimplifiedcomparisonofthevariousmethods\nused to promote the ground state, jgi, of a noble gas atom to the metastable state of interest,\njmi. Here,jmilies roughly\u001810eV abovejgi, and optical transitions between jgiandjmiare\nforbiddenbyelectricdipole selectionrules. Noble gasatoms alsohave anearbyauxiliary state,\njai,whichdoeshaveadirectopticaltransitionfrom jgiataresonancewavelengthinthedeep\nUV (\u0018100nm). Finally, we consider an upper-level state, jui, which has NIR optical transitions\n(\u0018800 nm ) between both jaiandjmi.\nPanel (a)ofFigure1 representsa traditionalelectricaldischargetechnique [17 –19]. Herean\nRF(orDC)fieldisappliedtoagasofatoms,resultinginvariouscollisionalmechanismsthat\ntransferpopulation(bothdirectlyandindirectly)from jgitojmi. Incontrast, panel(b)shows\nan “all optical” approach in which resonant VUV light (typically from an external lamp) excites\njgi!j ai[20–22]. Populationisthenopticallypumpedfrom jai!j uiusinganarrowbandlaser,\nandspontaneousemissionthencausesatransitionfrom jui!j mi. Panel(c)showsanalternative\n“all-optical” approach that relies on a direct two-photon transition from jgi!j ui[25,26].\nPanel (d) of Figure 1 shows the hybrid approach of interest here [23]. The key idea is that the\nsameelectricaldischargethatcreatespopulationin jmi(cf. panel(a))alsocreatespopulation\nintheauxiliarystate jai. Thiselectrically-producedauxiliarypopulationcanthenbeoptically\npumped fromjai!j ui!j mito supplement the existing electrically-produced jmipopulation.\nAtthesametime,spontaneousemissionfromthedipole-allowed jai!j gitransitionprovides\na source of resonant VUV photons that can be subsequently absorbed by nearby ground statearXiv:2006.03200v1  [physics.optics]  5 Jun 2020⟩|u\n⟩|g⟩|a⟩|mRF⟩|u⟩|g⟩|a⟩|m⟩|u\n⟩|g⟩|a⟩|m⟩|u\n⟩|g⟩|a⟩|mRF(a)(b)(c)(d)VUVNIRNIRUVUVNIRNIRNIRVUVFig. 1. Comparative overview of 4 different methods used to promote ground state jgi\npopulationtothemetastablestate jmiinnoblegasatoms: (a)standardRFdischarge\ntechnique, (b) “all-optical” technique involving transitions to an auxiliary state jai\nand upper statejui[20–22], (c) alternative “all-optical” technique [25,26], and (d) the\nhybrid technique of interest here [23]. The hybrid technique in panel (d) combines the\nelectrical discharge of panel (a) with the optical pumping of panel (b).\natoms. Because the overall number of atoms that are excited by the RF discharge in a typical\nexperimental setup is much larger than the number of atoms in the cross-section of the NIR\nopticalpumpingbeam,thiseffectgivesasignificantboosttothepopulationthatcanbeoptical\npumped fromjai!j mi. The combination of all of these effects results in the strong optical\nenhancement to the electrically-produced population of jmithat was first observed in [23].\nIn this paper we investigate the experimental conditions that maximize this optically-enhanced\nelectrical production of metastable states. We use a strong RF discharge and spectroscopic\ntechniques to study and compare the initial populations of both the metastable state jmiand the\nauxiliarystatejai,andfindthattheyareeachmaximizedunderthesameexperimentalconditions.\nIn addition, we experimentally observe that the metastable state population can be enhanced by\nan amount greater than the steady-state population of the auxiliary state due to the drastically\ndifferent effective lifetimes of these states. The end result is a convenient and robust method\nto experimentally optimize the overall steady-state metastable state population using the hybrid\ntechnique illustrated in Figure 1(d).\n2. Metastable xenon\nOurstudyusesalowpressure( \u0018100mTorr)gasofxenonatoms. Anoverviewoftherelevant\nenergy levels is shown in Figure 2. Here, the metastable state jmi(6s»32¼2state) has a natural\nlifetime of\u001843 s [27] which is reduced to the order of ms due to collisions under our typical\nexperimentaloperatingconditions[28 –30]. Theauxiliarystate jai(6s»32¼1state)hasalifetime\nof\u00184 ns, which is significantly shorter than the lifetime of jmi. In xenon, the UV optical\ntransitionbetweenjaiandthegroundstate jgi(5P6state)isat147nm,andtheNIRtransition\nbetweenjaiand the upper state jui(6p»32¼1state) is at 916 nm. This 916 nm transition is\ndriven by a narrowband pump laser in our experiments.\nOncepopulated,juicandecaytothedesiredmetastablestate jmiviaspontaneousemission\nat 841 nm with a probability of 80%, or back to the auxiliary state jaiwith a probability of\n20%. The lifetime of the upper state juiis\u001838 ns [31]. In addition to the four key atomic states\nof interest, we also utilize an extra upper state ju0i(6p»32¼2state) to perform spectroscopic\nmeasurements that characterize the metastable state (Xe*) density as a function of various\nexperimental parameters.\nAswillbedescribedbelow,thebaselinesteady-stateXe*densityunderoptimizedRFdischarge\nconditions in our experiment is \u00181011cm\u00003. Under the same conditions, the baseline density of\natoms in the auxiliary state jaiis\u00181010cm\u00003. The goal of optically enhanced production of823 nm(τ~43s)147 nm5P(916 nm841 nm\nRFρ*~10-.cm12ρ3~10--cm12pumpprobe⟩|6683/2;(τ~38ns)⟩|>6?3/2-⟩|>′6?3/2;\n(τ~4ns)⟩|A683/2-⟩|B80 %20 %Fig.2. Relevantenergyleveldiagramandassociatedparametersforxenon. Here,the\nmetastable statejmihas an intrinsic lifetime of \u001c\u001843 sec, while the auxiliary state\njaihas a lifetime of \u001c\u00184 ns. In our experiments, the baseline densities (due to the\nRF discharge alone) of the metastable state and auxiliary state are \u001am\u00181011cm\u00003and\n\u001aa\u00181010cm\u00003,respectively. Thegoaloftheexperimentsistoopticallypumpasmuch\nofthejaipopulationtojmiaspossible. Themetastablestatedensity \u001amismeasuredby\nperformingspectroscopyonan823nmtransitionbetween jmiandasecondupper-level\nstateju0i.\nXe* is simply to transfer as much of this jaipopulation as possible into jmi[23]. Intuitively,\nthe requirements are (1) starting with a large population in jai, and (2) driving the process\nwith a strong optical pumping beam at 916 nm. In the following sections, we summarize a\nseries of experimental measurements that characterize the optimization of this process. The key\nparametersthatinfluencetheprocessaretheoverallpressureoftheneutralxenongas,andthe\nstrength of the 916 nm optical pumping beam.\n3. Experimental setup\nFigure 3 provides an overview of the pump-probe experimental setup. The pump and probe\nlasers are independent narrowband tunable diode lasers (Toptica DL Pro; linewidths \u0018300 KHz)\ncenteredat916nmand823nm,respectively. Forthespectroscopicmeasurementsofinterest,\nthe lasers are scanned over ranges of \u001810 GHz, and the laser frequencies are measured using\na calibrated wavelength meter with 100 MHz resolution (High Finesse WSU-30). The pump\nand probe lasers are combined into a single spatial mode using a single-mode fiber, and then\nlaunchedintothexenonvacuumchamberasacommonfree-spacebeamwithadiameterof \u00182\nmm. A power meter that can be inserted into the free-space beam before the vacuum chamber is\nusedtomeasurethelaserpowers. Toavoidatomicsaturationeffects,theexperimentsaredone\nwith 823 nm probe beam powers of only \u00187.5 uW [32]. As will be described below, the 916 nm\npump beam power is a key experimental parameter that is varied in the range of 0 - 10 mW.\nThe vacuum system is comprised of a standard 4.5\" stainless steel ConFlat cube with glass\nviewportsthatletthepumpandprobebeamspassthroughthechamber. Thechamberistypically\npumped down to a pressure of \u001810\u00006Torr and then backfilled with xenon gas. A capacitancewavelengthmeter823 nm(probe)single-modefiberPDfree-spacebeamPower meterRF amplifierRF coil130 MHzfilter916 nm(pump)vac. and XesupplyVUVpumping beamRF fieldXeatomsZoom-inExp. cross-sectionFig. 3. Overview of the pump-probe setup used to characterize the experimental\nconditionsthatmaximizetheopticalproductionofXe*. Thepumpandprobelasers(at\n916 nmand 823nm, respectively) arecombined intoa commonspatial modeusing a\nsingle-mode fiber, and then launched as a free-space beam through a vacuum chamber\ncontainingagasofxenonatoms. Axenonglowdischargeplasmaisproducedusing\nan RF coil mounted on the outside of a glass window on the vacuum chamber. The\ndashed-box inset represents a “zoom-in” on a cross-section of the experiment. The RF\nfieldexcitesatomsthroughouttheentirechamber,whilethe2mmdiameterpumping\nbeam (and probe beam) only interact with a small subset of the atoms. The 2 key\nexperimental parameters are the neutral xenon pressure and the 916 nm pumping beam\npower.\nmanometer gauge fitted on the chamber is used to monitor the neutral xenon pressure. The\nneutralxenonpressureisakeyexperimentalparameterthatitisvariedtherangeof \u001810-600\nmTorr in the experiments.\nA large 4.5\" glass viewport is also used on one side of the cube to allow RF excitation of\nthexenongas. AnRFcoilismountedontheviewport,anddrivenwithastandardtankcircuit\n(consisting of the coil and capacitors). This system has an RF resonance frequency of 130 MHz,\nand is driven by a low power signal generator that is amplified to produce a glow discharge\nplasma in the chamber. For our full range of neutral xenon pressures, an amplified RF power of\n\u001820 W was found to maximize the initial Xe* density [10] (ie. the baseline jmipopulation; via\nthe process shown illustrated in Figure 1(a)).\nThe inset to Figure 3 illustrates a cross-section of the gas of atoms under the influence of both\ntheRFexcitationfield,andtheopticalpumpingbeamat916nm. Thishighlightsthefactthat\nwhile a large\u001810 cm diameter cross section undergoes RF excitation, only a small subset of the\natoms interact with the optical pumping beam ( \u00182 mm diameter). These surrounding atoms\nprovide a source of spontaneously emitted 147 nm photons that boost the baseline population of\njaiin Figure 2.\nA photodiode (PD) preceded by various filters is used to measure the transmitted laser powers\nduring the spectroscopic measurements. For the baseline spectroscopic measurements of the jai\ntojuitransitionat916nm,orthe jmitoju0itransitionat823nm,onlyasinglelaserisused. For\nthe full pump-probe experiments using both lasers, a narrow bandpass filter (centered at 823\nnm) is used to block the transmitted 916 nm pump beam, while passing the 823 nm probe beamduring the measurements.\n4. Experimental results\nWith the absence of the RF field, nearly all of the atoms in the gas are in the ground state\nand the initial populations of jaiandjmiare roughly zero. The RF discharge contribution\nto the population of jmican then be determined by applying the RF field, and performing\ntransmission-spectroscopymeasurementsonthe jmi!j u0itransitionat823nm. Anexample\nresult at a neutral xenon pressure of 15 mTorr is shown in Figure 4(a). Here, the 6 “dips” in\ntransmissionareduetoabsorptionfromthe variousisotopesofXe[19]. Weobtainanestimate\nof the Xe* density to be \u001am\u00182 x1011cm\u00003by fitting these transmission lineshapes using the\nmodels of reference [33], while taking into account the natural abundances of the various xenon\nisotopes and branching ratios of the hyperfine transitions [34]. This value of \u001am\u00182 x1011cm\u00003\nrepresents the baseline steady-state Xe* density (ie. jmipopulation) that can subsequently be\nenhanced by optical pumping population from the auxiliary state jai.\nWeestimatethebaselinepopulationof jaibyperforminganalogoustransmission-spectroscopy\nmeasurementsonthe jai!j uitransitionat916nm. Figure4(b)showsatypicalresultusingthe\nsame RF discharge conditions and xenon pressure of 15 mTorr. Here, the large central “dip” and\nsmaller side dips are once again due to the various isotopes of Xe and the hyperfine splittings of\nthe relevant states [35]. Fitting the lineshapes in Figure 4(b)providesan estimated baseline jai\ndensityof\u001aa\u00184x1010cm\u00003. Notethatthispopulationof jairesultsfromboththeRFdischarge,\naswellastheabsorptionof147nmphotonsspontaneouslyemittedfromsurroundingatomsin\nthe large-scale gas.\nNext,werepeattheseindependentmeasurementsforarangeofneutralxenonpressuresranging\nfrom 10 to 600 mTorr. The results are shown in Figure 5, where each data point represents a\nbaseline density value extracted from fitting a transmission spectrum similar to those in Figure 4.\nThe key result here is the nearly identical behavior of the jmiandjaipopulations as function\nofthexenonpressureinoursystem. Asthepressureisincreasedtherearesimplymoreatoms\nto promote from the ground state and the populations increase, while at higher pressures the\nrole ofcollisions playsa moredominant roleand limitsthe populations. Note thatthe trade-off\nbetweentheseeffectsresultsinthesameoptimalpressureof15mTorrforboththe jmiandjai\npopulationsinoursystem. Atpressureshigherthan \u0018600mTorr(orlowerthan10mTorr),we\nwere unable to produce a stable discharge in our system.\n1.00Transmission0.5-4-2Detuning (GHz)(a)402ω\"#$%&\n-41.0Transmission0.50-2(b)02Detuning (GHz)ω\"'(\"\n4\nFig.4. Singlelaserspectroscopyof(a)the jmi!j u0itransitionat823nm,and(b)the\njai!j uitransition at 916 nm. In panel (a) a detuning of zero corresponds to !probe\n= 364,095 GHz. In panel (b), zero detuning corresponds to !pump= 327,099 GHz.\nThroughout the work, transmission spectra lineshapes like these are fitted to determine\nthe metastable state jmiand auxiliary statejaidensities\u001amand\u001aa.Xepressure (mTorr)5050020010020104\n0213!\"×10&&'()*Density!+×10&,'()*!+!\"Fig.5. Baselinedensities(duetotheRFdischargealone)ofthemetastablestate jmi\n(bluetrace)andauxiliarystate jai(redtrace)asafunctionofneutralxenonpressure.\nThekeyresulthereisthesimilardependenceonpressure;optimizingthesystemfor\nthe largest baseline \u001amvalue also provides the largest baseline \u001aavalue.\nFrom a practical point of view, the significance of the nearly identical behavior in Figure 5 is\nthat the same experimental conditions that give the largest baseline Xe* density also provide the\nlargestauxiliarystatedensity. BecausetheopticalenhancementofXe*productionintuitively\nrequiresalargebaselineauxiliarystatepopulation,thisresultsuggeststhattheconditionsneeded\nformaximizingtheoptically-enhancedproductionofXe*canbeachievedbysimplymaximizing\nthe starting baseline density of Xe*.\nIn Figure 6 we investigate the role of the 916 nm optical pumping beam power, Ppump, for\ntransferringpopulationfrom jaitojmi. Weperformaseriesofpump-probeexperimentsfora\nvariety of neutral xenon pressures. Here, 823 nm (probe) transmission spectra similar to those in\nFigure 4(a) are measured, but now with the 916 nm optical pumping beam turned on (and with\n!pumpfixedat zerodetuning). Figure6(a) showsanexample ofresultsobtainedat arelatively\nhighneutralxenonpressureof300mTorr. As Ppumpisincreased,morepopulationistransferred\nfromjaitojmi, resulting in stronger absorption of the 823 nm probe beam (ie. increasingly\n3\n0211006842Pump power (mW)(b)100 mTorr300 mTorr500 mTorr15 mTorr\n600 mTorrρ\"×10&&cm)*\n1.0\n0Transmission0.5(a)-4Detuning (GHz)-2049.5 mW23 mW5 mWPump off1 mW\nFig. 6. (a) example of full pump-probe experimental results at a neutral xenon pressure\nof 300 mTorr. The data shows 823 nmprobe spectroscopy for several differentvalues\nof916nmpumpbeampower Ppumprangingfrom0mW(pumpoff)to9.5mW.As\nPpumpis increased, the increasing absorption “dips” are indicative of more population\nbeing pumped from jaitojmi. (b) quantification of the desired increase in \u001amas\na function of Ppumpfor several different neutral xenon pressures in the range of 15\nmTorr to 600 mTorr.deeper “dips” in the transmission spectra).\nIn order to quantify this population transfer, we fit each of the spectra in Figure 6(a) to obtain\nestimates of the Xe* density \u001amas a function of Ppump. We then repeat these pump-probe\nmeasurements for a several different neutral xenon pressures ranging from 15 to 600 mTorr,\nwith the results summarized in Figure 6(b). In all cases, a dramatic increase of \u001amis seen as\na function of increasing Ppump. In many ways, the results shown in Figure 6(b) represent the\nmainresultofthestudy. Weseethatthegoalofmaximizingthemetastablestatedensity \u001amis\nachieved by using the neutral xenon pressure that maximizes the initial baseline values of \u001am\n(andthus\u001aa),andutilizingasmuchpumpbeampoweraspossible. Asaturation-likeeffectis\nalso seen for all cases of neutral xenon pressure [23,24].\nTheoverallutilityofthishybridapproachisseenmostclearlyfornon-optimizedneutralxenon\npressures which result in very small initial baseline densities \u001am(for example, the 600 mTorr\ncase in Figure 6(b)). Here, the application of pump beam powers of only a few mW can provide\na relative increase in \u001amby roughly an order of magnitude [23]. However, one significant result\nwhich emerges from the data in Figure 6(b) is that these order-of-magnitude increases do not\nhold for more optimized cases of larger initial baseline densities (for example, the 15 mTorr case\ninFigure6(b)). Rather,thedatainFigure6(b)showthatthe netincreasein \u001amasfunctionof\nPpumpis roughly constant over a large range of neutral xenon pressures. Figure 7 examines this\neffectinmore detailbyplottingthenet increaseinmetastablestatedensity, \u0001\u001am,as afunction\nof neutral xenon pressure, for several different values of Ppump. The data shows that over the\nrange of xenon pressures from 15 mTorr to \u0018300 mTorr, \u0001\u001amis essentially flat, with a value\ndetermined solely by Ppump.\nItisinterestingtonotethatthevalueof \u0001\u001am(forallcasesinFigure7)actually exceedsthe\ninitialsteady-statebaselineauxiliarystatedensity \u001aa. Forexample,usingourmaximumavailable\npump power of Ppump= 9.5 mW (upper trace in Figure 7), the change in Xe* density is roughly\n\u0001\u001am\u00188.5x 1010cm\u00003overtherangeofxenonpressuresfrom15mTorrto \u0018300mTorr. Overthat\nsamepressurerange,thebaselineauxiliarystatedensity \u001aavariesfromamaximumofonly3.9x\n1010cm\u00003downto0.1x 1010cm\u00003(cf.Figure5). Thisapparentexcessin \u0001\u001amisduetothefact\nthatthemetastablestateeffectivelifetime( \u0018ms)isseveralordersofmagnitudelargerthanthe\nauxiliarystatejaiandupperstatejuilifetimes(\u0018ns),resultinginadesirablecontinuous-wave\nlaser-pumped steady-state “build up” of population in the metastable state [36]. In Figure 7\n5 mW7 mW9.5 mW8.5 mW4286\nXepressure (mTorr)5050020010020100∆ρ#×10'(cm+,10\nFig.7. Netchangeinmetastablestatedensityasafunctionofneutralxenonpressure\nfor 4 different values of pump beam power between a moderate value of 5 mW and our\nmaximum available power of 9.5 mW. The data shows that \u0001\u001amis fairly constant over\na large pressure range, with a value largely determined by the pump power.the small drop-off in \u0001\u001amvalues for pressures higher than \u0018300 mTorr is most-likely due to\ninsufficient initial \u001aavalues in our set-up at higher pressures ( cf.Figure 5).\n5. Summary and conclusions\nWe have investigated the experimental conditions needed to maximize the optical production\nof metastable xenon using the hybrid method of Figure 1(d) [23,24]. Here, conventional\nRF-excitationiscombinedwithopticalpumpingfromanauxiliarystatetoprovideadramatic\nenhancement of the metastable state density. We find that the same RF discharge conditions\nthat optimize the initial metastable state density also optimize the auxiliary state density. While\nthese initial baseline densities (due to RF excitation alone) vary significantly with the neutral\nxenon pressure, we find that the subsequent net increase in metastable state density (due to\noptical pumping from the auxiliary state) is relatively constant over a large pressure range, and is\ndetermined solely by the pumping beam power. In addition, the net increase in thesteady-state\nmetastable state density can exceed the baseline auxiliary state density due to the vastly different\nlifetimes of these states.\nThese results suggest that the hybrid technique of Figure 1(d) can provide a robust method for\nproducing relatively large long-term metastablestate densities. Itis important to note, however,\nthat the long-term densities achieved here ( \u00181011cm\u00003) are still lower than those obtained using\nshort pulsed electrical discharge techniques ( \u00181013cm\u00003) [5–7]. Nonetheless, the techniques\ndescribedhereimpactapplicationsrequiringhighdensitiesforlongercontinuoustimescales[16],\nand studying the transient dynamics of this hybrid approach in the pulsed regime may also be of\nfundamental interest.\nFunding\nNational Science Foundation (Grant No. 1402708); The Office of Naval Research (Grant\nN00014-15-1-2229)\nReferences\n1.David M. Giltner, Roger W. 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Happer, “Optical Pumping\", Rev. Mod. Phys. 44, 169 (1972)."    },    {        "title": "2006.11461v1.PDE_based_Dynamic_Density_Estimation_for_Large_scale_Agent_Systems.pdf",        "content": "arXiv:2006.11461v1  [eess.SY]  20 Jun 2020PDE-based Dynamic Density Estimation for\nLarge-scale Agent Systems\nTongjia Zheng, Student Member, IEEE , Qing Han, and Hai Lin, Senior Member, IEEE\n©2012 IEEE. Personal use of this material is permitted. Perm ission from IEEE must be obtained for all other uses, in any cu rrent or future media, including\nreprinting/republishing this material for advertising or promotional purposes, creating new collective works, for r esale or redistribution to servers or lists, or\nreuse of any copyrighted component of this work in other work s.Abstract —Large-scale agent systems have foreseeable applica-\ntions in the near future. Estimating their macroscopic dens ity is\ncritical for many density-based optimization and control t asks,\nsuch as sensor deployment and city traffic scheduling. In thi s\npaper, we study the problem of estimating their dynamically\nvarying probability density, given the agents’ individual dynamics\n(which can be nonlinear and time-varying) and their states\nobserved in real-time. The density evolution is shown to sat isfy\na linear partial differential equation uniquely determine d by\nthe agents’ dynamics. We present a density filter which takes\nadvantage of the system dynamics to gradually improve its\nestimation and is scalable to the agents’ population. Speci fically,\nwe use kernel density estimators (KDE) to construct a noisy\nmeasurement and show that, when the agents’ population is\nlarge, the measurement noise is approximately “Gaussian”. With\nthis important property, infinite-dimensional Kalman filte rs are\nused to design density filters. It turns out that the covarian ce\nof measurement noise depends on the true density. This state -\ndependence makes it necessary to approximate the covarianc e in\nthe associated operator Riccati equation, rendering the de nsity\nfilter suboptimal. The notion of input-to-state stability i s used\nto prove that the performance of the suboptimal density filte r\nremains close to the optimal one. Simulation results sugges t\nthat the proposed density filter is able to quickly recognize the\nunderlying modes of the unknown density and automatically\nignore outliers, and is robust to different choices of kerne l\nbandwidth of KDE.\nIndex Terms —Large-scale systems, estimation, Kalman filter-\ning, stochastic systems\nI. I NTRODUCTION\nEFFICIENTLY estimating the continuously varying prob-\nability density of the states of large-scale agent systems\nis an important step for many density-based optimization\nand control tasks, such as sensor deployment [1] and city\ntraffic scheduling. In such scenarios, the agents (such as\nUA Vs) are built by task designers and follow the specified\ncontrol commands, which means their dynamics are known.\nWhile density estimation is a fundamental problem extensiv ely\nstudied in statistics, we are particularly interested in th e case\nwhere the samples (i.e. the agent states) are governed by\nknown dynamics (which can be nonlinear and time-varying).\nWe study how to take advantage of their dynamics to obtain\nefficient and convergent density estimates in real time.\nThis work was supported in part by the National Science Found ation\nunder Grant IIS-1724070, and Grant CNS-1830335, and in part by the Army\nResearch Laboratory under Grant W911NF-17-1-0072.\nTongia Zheng and Hai Lin are with the Department of Electrica l Engi-\nneering, University of Notre Dame, Notre Dame, IN 46556, USA (e-mail:\ntzheng1@nd.edu, hlin1@nd.edu.).\nQing Han is with the Department of Mathematics, University o f Notre\nDame, Notre Dame, IN 46556, USA (e-mail: Qing.Han.7@nd.edu .).In general, density estimation algorithms are classified as\nparametric and non-parametric. Parametric algorithms ass ume\nthat the samples are drawn from a known parametric family\nof distributions with a fixed set of parameters, such as the\nGaussian mixture models [2]. Performance of such estimator s\nrely on the validity of the assumed models, and therefore\nthey are unsuitable for estimating an evolving density. In\nnon-parametric approaches, the data are allowed to speak\nfor themselves in determining the density estimate. As a\nrepresentative, kernel density estimation (KDE) [2] has been\nwidely used for estimating the global density in the study\nof swarm robotic systems [3], [4], [5]. It is known that\nthe performance of KDE largely depends on a smoothing\nparameter called the bandwidth [2]. When the samples are\nstationary, many bandwidth selection techniques have been\nproposed, such as cross-validation [2], adaptive bandwidt h [6]\nand the plug-in technique [7]. Such techniques are heuristi c in\ngeneral, since any predefined optimality of bandwidth selec -\ntion requires certain information of the unknown density. T hey\nare also not suitable for dynamic estimation which requires\nthe algorithm to be computationally efficient and adapt to\nthe density evolution. For dynamic density estimation, man y\nadaptations of KDE have been proposed in the domain of\ndata stream mining [8]. In such problems, the dynamics of\nthe density are usually unknown, and therefore little can be\nclaimed about the convergence of the algorithm.\nTo estimate the global distribution of large-scale agents\nwith known dynamics, an alternative way is to formulate it\nas a filtering problem, for which there exists a large body\nof literature, such as the celebrated Kalman filters and thei r\nvariants [9], [10], the more general Bayesian filters and the ir\nMonte Carlo approach, also known as particle filters [11].\nUnfortunately, all these methods are known to suffer from th e\ncurse of dimensionality , especially for large-scale nonlinear\nsystems. A potential solution is to use the so-called consensus\nfilters [12], [13], [14], which obtain local distribution estimate s\nof the agent system using local Kalman or Bayesian filters,\nand then compute the global distribution by averaging local\nestimates through some consensus protocols. However, few\nconclusions are made concerning the relationship between t he\ntrue distribution and the estimated global distribution. M ore-\nover, stability analysis is difficult when the agents’ dynam ics\nare nonlinear and time-varying.\nIn summary, considering efficiency and convergence re-\nquirements, existing methods are unsuitable for estimatin g\nthe time-varying density of large-scale agent systems. Thi s\nmotivates us to propose a dynamic and scalable density esti-\nmation algorithm that can perform online and take advantageof the dynamics to guarantee its convergence. Specifically,\nwe show that the agents’ density is governed by a linear\npartial differential equation (PDE), called the Fokker-Pl anck\nequation, which is uniquely determined by the agents’ dy-\nnamics. KDE is used to construct a noisy measurement of\nthe density (the state of this PDE). The measurement noise\nis approximately “Gaussian” (more precisely, asymptotica lly\nGaussian when the agents’ population tends to infinity), so\nthat infinite-dimensional Kalman filters can be used to desig n\ndensity filters. It turns out that the covariance of measurem ent\nnoise depends on the true density, for which approximating\nthe covariance is required and the density filter becomes\nsuboptimal. We then use the notion of input-to-state stabil ity\nto prove the stability of the suboptimal density filter and th e\nassociated operator Riccati equation.\nOur contributions contain the following aspects: (i) By usi ng\na density filter, we can (largely) circumvent the problem of\nbandwidth selection; (ii) The density filter takes advantag e of\nthe agents’ dynamics to improve its density estimation, in t he\nsense of minimizing the covariance of estimation error; (ii i)\nThe density filter is proved to be convergent and is scalable\nto the population of agents; (iv) All the results hold even if\nthe agents’ dynamics are nonlinear and time-varying.\nThe rest of the paper is organized as follows. Section II\nintroduces some preliminaries. Problem formulation is giv en\nin Section III. Section IV is our main results, in which we\npresent a density filter and then study its stability/optima lity.\nSection V performs an agent-based simulation to verify the\neffectiveness of the density filter. Section VI summarizes t he\ncontribution and points out future research.\nII. P RELIMINARIES\nA. Infinite-Dimensional Kalman Filters\nThe Kalman filter is an algorithm that uses the system’s\nmodel, known control inputs and sequential measurements to\nform a better state estimate [15]. It was later extended to\ninfinite-dimensional systems, represented using linear op era-\ntors [16], [17]. Suppose the signal x(t)and its measurement\ny(t), both in a Hilbert space, are generated by the stochastic\nlinear differential equations\ndx=A(t)xdt+B(t)q(t)dv, x(t0) =x0, E[x0] = 0\ndy=C(t)xdt+r(t)dw, y(t0) =y0\nwheredvanddware infinite-dimensional Wiener processes\nwith incremental covariance operators VandW, respectively\n[18]. Assume Cov[v(t),w(τ)] = 0 andE[/an}b∇acketle{tv(t),w(τ)/an}b∇acket∇i}ht] = 0\nfor allt/ne}ationslash=τ. Denote Q(t) =q(t)Vq∗(t)andR(t) =\nr(t)Wr∗(t). The Kalman filter is an algorithm that minimizes\nthe covariance of the estimation error, given by [16]\ndˆx=A(t)ˆxdt+L(t)[y−C(t)ˆx]dt\nwhereL(t) =P(t)C∗(t)R−1(t)is the called the optimal\nKalman gain, and P(t)is the solution of the Riccati equation\n˙P=AP+PA∗−PC∗R−1CP+BQB∗\nwithP(t0) = Cov[x0,x0].B. Input-to-state stability\nInput-to-state stability (ISS) is a stability notion widel y used\nto study stability of nonlinear control systems with extern al\ninputs [19]. Roughly speaking, a control system is ISS if it i s\nasymptotically stable in the absence of external inputs and if\nits trajectories are bounded by a function of the magnitude o f\nthe input. To define the ISS concept for infinite-dimensional\nsystems we need to introduce the following classes of com-\nparison functions [19].\nK:={γ:R+→R+|γis continuous and strictly\nincreasing ,γ(0) = 0}\nL:={γ:R+→R+|γis continuous and strictly\ndecreasing with lim\nt→∞γ(t) = 0}\nKL:={β:R+×R+→R+|βis continuous ,β(·,t)∈ K,\nβ(r,·)∈ L,∀t≥0,∀r >0}.\nDefinition 1: (ISS [20]). Consider a control system Σ =\n(X,U,φ)consisting of normed linear spaces (X,/ba∇dbl·/ba∇dblX)and\n(U,/ba∇dbl·/ba∇dblU), called the state space and the input space, endowed\nwith the norms /ba∇dbl·/ba∇dblXand/ba∇dbl·/ba∇dblUrespectively, and a transition\nmapφ:R+×X×U→X. The system is said to be ISS if\nthere exist β∈ KL andγ∈ K, such that\n/ba∇dblφ(t,x0,u)/ba∇dblX≤β(/ba∇dblx0/ba∇dblX,t−t0)+γ( sup\nt0≤τ≤t/ba∇dblu(τ)/ba∇dblU)\nholds∀x0∈X,∀t≥t0and∀u∈U. It is called locally input-\nto-state stable (LISS), if there also exists constants ρx,ρu>0\nsuch that the above inequality holds ∀x0:/ba∇dblx0/ba∇dblX≤ρx,∀t≥\nt0and∀u∈U:/ba∇dblu/ba∇dblU≤ρu.\nIII. P ROBLEM FORMULATION\nThis paper studies the problem of estimating the dynam-\nically varying probability density of large-scale stochas tic\nagents. The dynamics of the agents are assumed to be known\nand satisfy the following stochastic differential equatio ns\ndXi=v(Xi,t)dt+σ(Xi,t)dBt, i= 1,...,n, (1)\nwhereXi∈RNrepresents the state of the i-th agent (e.g. po-\nsitions),v= (v1,...,v N)∈RNis the deterministic dynamics\n(e.g. velocity fields), Btis anM-dimensional standard Wiener\nprocess, and σ= [σjk]∈RN×Mrepresents the stochastic\ndynamics. We also assume the collection of states {Xi(t)}n\ni=1\nare observable.\nThe probability density p(x,t)of agent states {Xi(t)}n\ni=1\nis known to satisfy the Fokker-Planck equation [21]:\n∂p(x,t)\n∂t=−N/summationdisplay\ni=1∂\n∂xi[vi(x,t)p(x,t)]\n+N/summationdisplay\ni=1N/summationdisplay\nj=1∂2\n∂xi∂xj[Dij(x,t)p(x,t)],\np(·,0) =p0,(2)\nwherep0is the initial density and\nDij(x,t) =1\n2M/summationdisplay\nk=1σik(x,t)σjk(x,t).If the agent states are confined within a bounded domain Ω⊆\nRN, one can impose a reflecting boundary condition\nn·(g−vp) = 0,on∂Ω, (3)\nwhereg= (/summationtextN\nj=1∂\n∂xj(D1jp),...,/summationtextN\nj=1∂\n∂xj(DNjp)),∂Ωis\nthe boundary of Ωandnis the outward normal to ∂Ω.\nRemark 1: Note that the dynamics of agents’ density (2)\nare uniquely determined by their individual dynamics (1) an d\nare therefore known. This relationship holds even if (1) is\nnonlinear and time-varying. Therefore, the density filter t o be\npresented applies to a very wide class of systems. It is worth\npointing out that PDE (2) is always linear even if (1) is not.\n(It is time-varying if (1) is time-varying.)\nNow, we can formally state the problem to be solved as\nfollows:\nProblem 1: Given the density dynamics (2) and agent states\n{Xi(t)}n\ni=1, we want to estimate their density p(x,t).\nIV. M AINRESULTS\nIn this section, we design a density filter to estimate the\nstate (i.e. the density) of (2) by combining KDE and infinite-\ndimensional Kalman filters. Specifically, we use KDE to\nconstruct a noisy measurement of the unknown density and\nshow that the measurement noise is approximately “Gaussian ”,\nwhich enables us to use infinite-dimensional Kalman filters t o\ndesign a density filter. Stability analysis is also presente d.\nA. Density Filter Design\nIn this section, we use KDE to construct a measurement\nwith Gaussian noise and design a density filter.\nKDE is a non-parametric way to estimate an unknown\nprobability density function (pdf) [2]. The agents’ states\n{Xi(t)}n\ni=1⊆Rdat timetcan be seen as a set of n\nindependent samples drawn from the pdf p(x,t). Fixtand\ndenotef(x) =p(x,t). The density estimator is given by\nfn(x) =1\nnhdn/summationdisplay\ni=1K/parenleftBig1\nh(x−Xi)/parenrightBig\n, (4)\nwhereK(x)is a kernel function [2] and his the bandwidth,\nusually chosen as a function of nsuch that limn→∞h= 0\nandlimn→∞nh=∞. The Gaussian kernel is frequently used\ndue to its infinite order of smoothness, given by\nK(x) =1\n(2π)d/2exp/parenleftBig\n−1\n2x⊺x/parenrightBig\n.\nIt is known that the fn(x)is asymptotically normal and that\nfn(xi)andfn(xj)are asymptotically uncorrelated for any\nxi/ne}ationslash=xj. These two properties are summarized as follows.\nLemma 1: (Asymptotic normality [22]) Under conditions\nlimn→∞h= 0 andlimn→∞nh=∞, if the bandwidth h\ntends to zero faster than the optimal rate, i.e.,\nh∗=o/parenleftbigg1\nn/parenrightbigg1/(d+4)\n.\nthen asn→ ∞ , we have\n√\nnhd(ˆf(x)−f(x))→ N/parenleftBig\n0,f(x)/integraldisplay\n[K(u)]2du/parenrightBig\n. (5)Lemma 2: (Asymptotic uncorrelatedness [22]) Let xi\nandxjbe two distinct continuity points of f. Then under\nlimn→∞h= 0, asn→ ∞ , the asymptotic covariance of\nfn(xi)andfn(xj)satisfies\nnhpCov[fn(xi),fn(xj)]→0.\nIn practice, nis finite. Hence, the kernel density estimator\nis biased (i.e. E[ˆfn(x)]/ne}ationslash=f(x)). Also,fn(xi)andfn(xj)\nare correlated. However, one can always choose a smaller\nbandwidth hto reduce the bias and the covariance [22], [2],\nwhich means that fn(x)−f(x)can be made approximately\nGaussian with independent components when nis large. It is\nknown that the performance of KDE largely depends on its\nbandwidth [2]. Any predefined optimality of the bandwidth\nselection requires certain information of the unknown dens ity\nf. For example, the one that balances the estimation bias and\nvariance depends on the second-order derivatives of f. An\nexcessively large (small) bandwidth may cause the problem o f\noversmoothing (undersmoothing). For the density filter to b e\npresented, we tend to choose a smaller bandwidth. We recall\nthat “filters” essentially combine past outputs to produce b etter\nestimates, which can ease the problem of undersmoothing. In\nthis regard, by using a density filter, we can largely circum-\nvent the problem of optimal bandwidth selection. Simulatio n\nstudies will show that the performance of the proposed densi ty\nfilter is robust to different choices of bandwidth.\nWe are now ready to present a density filter using the\ninfinite-dimensional Kalman filters. We rewrite the PDE of\ndensity evolution (2) in the form of an evolution equation an d\nuse KDE to construct a noisy measurement y(t):\n˙p(t) =A(t)p(t)\ny(t) =pKDE(t) =p(t)+w(t)(6)\nwhereA(t) =−/summationtextN\ni=1∂\n∂xi(vi·) +/summationtextN\ni=1/summationtextN\nj=1∂2\n∂xi∂xj(Dij·)\nis a linear operator, pKDE(t)represents a kernel density es-\ntimator using the states {Xi(t)}n\ni=1at timet, andw(t)is\nthe measurement noise which is approximately Gaussian with\ncovariance operator R(t) =kdiag(p(t))wherek >0is a\nconstant depending on nandh.\nAccording to Section II-A, the optimal density filter can be\ndesigned as\n˙ˆp=A(t)ˆp+L(t)(y−ˆp),ˆp(t0) =pKDE(t0), (7)\nwhereL(t) =P(t)R−1(t)is the optimal Kalman gain and\nP(t)is a solution of the following operator Riccati equation\n˙P=AP+PA∗−PR−1P. (8)\nThe Riccati equation (8) actually depends on the unknown\nstate/density p(t)becauseR(t) =kdiag(p(t)), which means\nwe have to approximate R(t)at the same time. Intuitively,\n¯R(t) =kdiag(ˆp(t))would be a reasonable approximation.\nHowever, this would make (7) and (8) strongly coupled, for\nwhich it is very difficult to analyze the stability. Therefor e,\nwe use¯R(t) =¯kdiag(pKDE(t)), where¯kis computed as ¯k=\n(/integraltext[K(u)]2du)/(nhd)according to (5). In this way, we can\ntreat the approximation error as an external disturbance an d\nuse the concept of ISS to study its stability. We note that(5) should be understood to be accurate in the limit ( n→ ∞ ).\nWhennis finite, the estimate may be inaccurate. However, we\nshall point out that the estimation error of ¯kis also included\nas part of the approximation error of ¯R(t).\nBy approximating R(t)with¯R(t) =¯kdiag(pKDE(t)), the\n“suboptimal” density filter is correspondingly given by\n˙ˆp=A(t)ˆp+¯L(t)(y−ˆp),ˆp(t0) =pKDE(t0) (9)\nwhere¯L(t) =¯P(t)¯R−1(t)is the suboptimal Kalman gain and\n¯P(t)is a solution of the approximated Riccati equation\n˙¯P=A¯P+¯PA∗−¯P¯R−1¯P. (10)\nThe optimal density filter (7) essentially combines linearl y\nall past density estimations {ˆp(τ)}t0≤τ<tand the most recent\nKDE measurement pKDE(t)to produce a density estimate ˆp(t)\nwith minimum estimation error covariance. The approximate d\ndensity filter (9) is called “suboptimal” in the sense that th e\nsuboptimal gain ¯Lremains “close” to the optimal gain L\n(which will be proved in Corollary 1).\nB. Stability Analysis of the Suboptimal Density Filter\nIn this section, we study the stability of the suboptimal\ndensity filter (9) and the associated Riccati equation (10).\nLetP∗>0be the minimum (but unknown) covariance\natt=t0. We denote by Π(t)the solution of (8) with the\ninitial condition P∗, which represents the flow of minimum\ncovariance and also generates the optimal gain. Let P0>0\nbe a “guessed” initial condition. Denote by ¯Π(t)the solution\nof (10) with the initial condition P0. We will show that ¯Π(t)\nconverges to and remains close to Π(t)in the presence of\napproximation error on R.\nDefineΓ =¯Π−Π. Using (8) and (10) we have\n˙Γ =AΓ+ΓA∗−¯Π¯R−1¯Π+ΠR−1Π,\nΓ(t0) =P0−P∗.(11)\nDefine˜p= ˆp−p. Then along ¯Π(t)we have\n˙˜p= (A−¯Π¯R−1)˜p+¯Π¯R−1w. (12)\nOur idea is to show that (11) is ISS with respect to the\napproximation error on R(more precisely /ba∇dbl¯R−1−R−1/ba∇dbl,\nwhich equals 0if and only if R=¯R). In this way, the\nsuboptimal gain ¯Lremains close to Lwhen there is any ap-\nproximation error, and converges to 0when the approximation\nerror vanishes. We also show that the suboptimal density filt er\n(9) is stable even though we use an approximation for R.\nRemark 2: To avoid confusions, we point out that if we\nuse (8) to compute the gain for (7), then the estimation error\ncovariance Cov[˜p,˜p]alongΠ(t)also satisfies (8), which is a\nproperty of Kalman filters [15]. However, if we use (10) to\ncompute the gain for (9), then the covariance Cov[˜p,˜p]along\n¯Π(t), denoted by Q(t)in this case, actually satisfies\n˙Q=AQ+QA∗−¯Π(t)¯R−1R¯R−1¯Π(t). (13)\nIt would be desirable to prove that the solutions of (8) and\n(13) remain close, which is much harder because it involves\nthree Riccati equations and will be left as our future work.The following assumption is required for proving stability .\nAssumption 1: Assume that /ba∇dblΠ(t)/ba∇dbland/ba∇dbl¯Π(t)/ba∇dblare uni-\nformly bounded, and that there exist positive constants c1and\nc2such that for all t≥t0,\n0< c1I≤R−1(t),¯R−1(t),Π−1(t),¯Π−1(t)≤c2I. (14)\nRemark 3: The assumption for R−1(t)is an imposed\nrequirement for p(t)considering R(t) =kdiag(p(t)), which\nroughly speaking, requires that p(t)has positive upper bounds\nand lower bounds. The assumption for ¯R−1(t)can be easily\nsatisfied because ¯R(t) =¯kdiag(pKDE(t))and we construct\npKDE(t). For finite-dimensional systems, the assumptions for\n/ba∇dblΠ(t)/ba∇dbl,/ba∇dbl¯Π(t)/ba∇dbl,Π−1(t)and¯Π−1(t)follow the assumption\nforR−1(t)and¯R−1(t). This is because when Π(t)and¯Π(t)\nare large, the solutions of (8) and (10) will be dominated by\nthe negative second-order term and decay. Although intuiti vely\ncorrect, we find it much harder to prove for the infinite-\ndimensional case and thus leave it as our future work.\nStability results for (11) and (12) are given as follows.\nTheorem 1: Under Assumption 1, the unforced part of (12),\ngiven by\n˙˜p= (A−¯Π¯R−1)˜p, (15)\nis uniformly exponentially stable, and (11) is locally inpu t-to-\nstate stable (LISS) with respect to the approximation error in\nthe form of /ba∇dbl¯R−1−R−1/ba∇dbl.\nProof: To prove the first statement, consider a Lyapunov\nfunctional V1=/an}b∇acketle{t¯Π−1˜p,˜p/an}b∇acket∇i}ht. We have\n˙V1=/angbracketleftbig¯Π−1˙˜p,˜p/angbracketrightbig\n+/angbracketleftbig¯Π−1˜p,˙˜p/angbracketrightbig\n−/angbracketleftbig¯Π−1˙¯Π¯Π−1˜p,˜p/angbracketrightbig\n=/angbracketleftbig¯Π−1(A−¯Π¯R−1)˜p,˜p/angbracketrightbig\n+/angbracketleftbig\n(A∗−¯R−1¯Π)¯Π−1˜p,˜p/angbracketrightbig\n−/angbracketleftbig¯Π−1(A¯Π+¯ΠA∗−¯Π¯R−1¯Π)¯Π−1˜p,˜p/angbracketrightbig\n=−/an}b∇acketle{t¯R−1˜p,˜p/an}b∇acket∇i}ht.\nIn view of (14), we conclude that (15) is uniformly exponen-\ntially stable. Similarly, by considering a Lyapunov functi onal\ndefined by V2=/an}b∇acketle{tΠ−1˜p,˜p/an}b∇acket∇i}ht, one can show that the following\nsystem along Π(t)is also uniformly exponentially stable:\n˙˜p= (A−ΠR−1)˜p. (16)\nWe also note that the inner products in V1andV2are\nequivalent because of the assumption (14). To prove the seco nd\nstatement, we rewrite (11) as\n˙Γ =AΓ+ΓA∗−¯Π¯R−1Γ−¯Π¯R−1Π−ΓR−1Π+¯ΠR−1Π\n= (A−¯Π¯R−1)Γ+Γ(A∗−R−1Π)−¯Π(¯R−1−R−1)Π\n(17)\nWe note that (17) is essentially a linear equation. Now fix q\nwith/ba∇dblq/ba∇dbl= 1. Since (15) and (16) are uniformly exponen-\ntially stable, and /ba∇dbl¯Π/ba∇dbland/ba∇dblΠ/ba∇dblare assumed to be uniformly\nbounded, there exist constants λ,c >0such that\n/ba∇dblΓ(t)q/ba∇dbl ≤e−λ(t−t0)/ba∇dblΓ(t0)q/ba∇dbl\n+/integraldisplayt\nt0e−λ(t−τ)c/ba∇dbl¯R−1(τ)−R−1(τ)/ba∇dbl/ba∇dblq/ba∇dbldτ≤e−λ(t−t0)/ba∇dblΓ(t0)q/ba∇dbl\n+csup\nt0≤τ≤t/ba∇dbl¯R−1(τ)−R−1(τ)/ba∇dbl/integraldisplayt\nt0e−λ(t−s)ds\n≤e−λ(t−t0)/ba∇dblΓ(t0)q/ba∇dbl+c\nλsup\nt0≤τ≤t/ba∇dbl¯R−1(τ)−R−1(τ)/ba∇dbl\n=:β(/ba∇dblΓ(t0)q/ba∇dbl,t−t0)+γ( sup\nt0≤τ≤t/ba∇dbl¯R−1(τ)−R−1(τ)/ba∇dbl),\nwhereβ∈ KL andγ∈ K. Under the uniform boundedness\nassumption of Πand¯Π, we conclude that (11) is LISS. /square\nWe can conclude from Theorem 1 that the suboptimal gain\n¯Lalso remains close to the optimal gain L, given as follows.\nCorollary 1: Under Assumption 1, there exist functions\nβ1∈ KL andγ1∈ K such that\n/ba∇dbl¯L−L/ba∇dbl ≤β1(/ba∇dblΓ(t0)/ba∇dbl,t−t0)+γ1(/ba∇dbl¯R−1−R−1/ba∇dbl).\nProof: Observe that\n/ba∇dbl¯L−L/ba∇dbl=/ba∇dbl¯Π¯R−1−ΠR−1/ba∇dbl\n≤ /ba∇dbl¯Π¯R−1−Π¯R−1+Π¯R−1−ΠR−1/ba∇dbl\n≤ /ba∇dbl¯R−1/ba∇dbl/ba∇dbl¯Π−Π/ba∇dbl+/ba∇dblΠ/ba∇dbl/ba∇dbl¯R−1−R−1/ba∇dbl\nIn view of Definition 1, Theorem 1 and the uniform bound-\nedness of /ba∇dblΠ/ba∇dbland/ba∇dbl¯R−1/ba∇dbl, we obtain the desired result. /square\nV. S IMULATION STUDIES\nIn this section, we study the performance of the proposed\ndensity filter. Consider a collection of 300agents given by\ndXi=∇·D∇f(x)\nf(x)dt+DdBt, i= 1,...,300,(18)\nwhere the states {Xi}300\ni=1are restricted within Ω = [0,1]2,\nD= 0.05andf(x)is a continuous pdf over Ωto be specified.\nThe initial conditions {Xi(t0)}300\ni=1are drawn from the uniform\ndistribution over Ω. Therefore, the ground truth density of the\nagents satisfies\n∂tp(x,t) =−∇·Dp(x,t)∇f(x)\nf(x)+1\n2D2∆p(x,t),\np(·,t0) = 1,\nwith a reflecting boundary condition (3). For illustra-\ntion purpose, we let f(x)be a Gaussian mixture of\ntwo components with the common covariance matrix\ndiag(0.015,0.015) but different time-varying means [0.5 +\n0.35cos(0.2t),0.5+0.35sin(0.2t)]⊺and[0.5+0.35cos(0.2t+\nπ),0.5 + 0.35sin(0.2t+π)]⊺. Under this design, the agents\nare nonlinear and time-varying and their states will conver ge\nto the two “spinning” Gaussian components.\nWe use the finite difference method [23] to numerically\nsolve the density filter (9) and the operator Riccati equatio n\n(10). Specifically, Ωis equally divided into a 30×30grid.\nThe pdfs ˆpandpKDE are represented as 900×1vectors.\nThe operators A,¯Pand¯Rare represented as 900×900\nmatrices. We set the initial conditions to be ¯P(t0) =Iand\nˆp(t0) =pKDE(t0). The time difference is dt= 0.1sand the\nbandwidth is h= 0.05. Note that Ais highly sparse and ¯Ris\ndiagonal, so the computation is very fast in general.Simulation results are given in Fig. 2. We see that the\nestimate by KDE is usually rugged, especially in the early\nstage when the true density is nearly uniform. (If we had\nknown that it is nearly uniform, we would have chosen a\nmuch larger bandwidth hfor better smoothing.) In the late\nstage, the estimate by KDE has two major components since\nthe samples are more concentrated. But it still has undesire d\nmodes due to outliers. The density filter, by taking advantag e\nof the dynamics, quickly ”recognizes” the two underlying\nGaussian components and gradually catches up with the evo-\nlution of the ground truth density. In Fig. 1, we compare the\nL2norms of estimation errors of the KDE and the density\nfilter under different choices of bandwidth. It shows that th e\nestimation error of the density filter quickly converges and the\nperformance is robust to different choices of bandwidth.\n0 5 10 15 20 25 30 35 40\nTime (s)00.10.20.30.40.50.60.70.8Density estimation errorh=0.04\nh=0.04\nh=0.05\nh=0.05\nh=0.06\nh=0.06\nh=0.07\nh=0.07\nFig. 1: Estimation errors of the KDE (solid line) and the\ndensity filter (dashed line) with different choices of bandw idth.\nVI. C ONCLUSION\nWe have presented a density filter for estimating the dy-\nnamic density of large-scale agent systems with known dy-\nnamics by combining KDE with infinite-dimensional Kalman\nfilters. The density filter improved its estimation using the\ndynamics and the real-time states of the agents. It was\nscalable to the population of agents and was proved to be\nconvergent. All results held even if the agents’ dynamics ar e\nnonlinear and time-varying. This algorithm can be used for\nmany density-based optimization and control tasks of large -\nscale agent systems when density feedback information is\nrequired, and can be potentially extended to other dynamic\ndensity estimation problems when certain prior knowledge o f\nits spatiotemporal evolution is available, such as populat ion\nmigration. Our future work includes addressing the remaini ng\nproblems noted in Remark 2 and 3, decentralizing the density\nfilter and integrating it into density feedback control for l arge-\nscale agent systems.\nREFERENCES\n[1] G. Foderaro, P. Zhu, H. Wei, T. A. Wettergren, and S. Ferra ri, “Dis-\ntributed optimal control of sensor networks for dynamic tar get tracking,”\nIEEE Transactions on Control of Network Systems , vol. 5, no. 1, pp.\n142–153, 2016.\n[2] B. W. Silverman, Density Estimation for Statistics and Data Analysis .\nCRC Press, 1986, vol. 26.0 0.2 0.4 0.6 0.8 1\nx100.20.40.60.81x2\n0 0.2 0.4 0.6 0.8 1\nx100.20.40.60.81x2\n0 0.2 0.4 0.6 0.8 1\nx100.20.40.60.81x2\n0 0.2 0.4 0.6 0.8 1\nx100.20.40.60.81x2\n0\n12\n14\nx20.5\nx10.56\n000\n12\n14\nx20.5\nx10.56\n000\n12\n14\nx20.5\nx10.56\n000\n12\n14\nx20.5\nx10.56\n00\n0\n12\n14\nx20.5\nx10.56\n000\n12\n14\nx20.5\nx10.56\n000\n12\n14\nx20.5\nx10.56\n000\n12\n14\nx20.5\nx10.56\n00\n00.51\nx100.51\nx20246\n0\n12\n14\nx20.5\nx10.56\n000\n12\n14\nx20.5\nx10.56\n000\n12\n14\nx20.5\nx10.56\n00\nFig. 2: States of the large-scale agent systems (1st row), de nsity estimates using KDE pKDE(x,t)(2nd row), outputs of the\ndensity filter ˆp(x,t)(3rd row) and the ground truth density p(x,t)(4th row).\n[3] G. Foderaro, S. Ferrari, and M. Zavlanos, “A decentraliz ed kernel density\nestimation approach to distributed robot path planning,” i nProceedings\nof the Neural Information Processing Systems Conference, L ake Tahoe,\nNV, 2012.\n[4] V . Krishnan and S. Mart´ ınez, “Distributed control for s patial self-\norganization of multi-agent swarms,” SIAM Journal on Control and\nOptimization , vol. 56, no. 5, pp. 3642–3667, 2018.\n[5] M. H. de Badyn, U. Eren, B. Ac ¸ikmes ¸e, and M. Mesbahi, “Op timal mass\ntransport and kernel density estimation for state-depende nt networked\ndynamic systems,” in 2018 IEEE Conference on Decision and Control\n(CDC) . IEEE, 2018, pp. 1225–1230.\n[6] S. R. Sain, “Multivariate locally adaptive density esti mation,” Compu-\ntational Statistics & Data Analysis , vol. 39, no. 2, pp. 165–186, 2002.\n[7] S. J. Sheather and M. C. Jones, “A reliable data-based ban dwidth\nselection method for kernel density estimation,” Journal of the Royal\nStatistical Society: Series B (Methodological) , vol. 53, no. 3, pp. 683–\n690, 1991.\n[8] A. Qahtan, S. Wang, and X. Zhang, “Kde-track: An efficient dynamic\ndensity estimator for data streams,” IEEE Transactions on Knowledge\nand Data Engineering , vol. 29, no. 3, pp. 642–655, 2016.\n[9] S. J. Julier and J. K. Uhlmann, “Unscented filtering and no nlinear\nestimation,” Proceedings of the IEEE , vol. 92, no. 3, pp. 401–422, 2004.\n[10] I. Arasaratnam and S. Haykin, “Cubature kalman filters, ”IEEE Trans-\nactions on automatic control , vol. 54, no. 6, pp. 1254–1269, 2009.\n[11] Z. Chen et al. , “Bayesian filtering: From kalman filters to particle filters ,\nand beyond,” Statistics , vol. 182, no. 1, pp. 1–69, 2003.\n[12] R. Olfati-Saber, “Distributed kalman filtering for sen sor networks,” in\n2007 46th IEEE Conference on Decision and Control . IEEE, 2007, pp.\n5492–5498.\n[13] S. Bandyopadhyay and S.-J. Chung, “Distributed estima tion using\nbayesian consensus filtering,” in 2014 American control conference .\nIEEE, 2014, pp. 634–641.\n[14] G. Battistelli and L. Chisci, “Kullback–leibler avera ge, consensus onprobability densities, and distributed state estimation w ith guaranteed\nstability,” Automatica , vol. 50, no. 3, pp. 707–718, 2014.\n[15] R. E. Kalman and R. S. Bucy, “New results in linear filteri ng and\nprediction theory,” Journal of basic engineering , vol. 83, no. 1, pp. 95–\n108, 1961.\n[16] P. L. Falb, “Infinite-dimensional filtering: The kalman -bucy filter in\nhilbert space,” Information and Control , vol. 11, no. 1-2, pp. 102–137,\n1967.\n[17] A. Bensoussan, Filtrage optimal des syst` emes lin´ eaires . Dunod, 1971.\n[18] G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions .\nCambridge university press, 2014.\n[19] E. D. Sontag and Y . Wang, “On characterizations of the in put-to-state\nstability property,” Systems & Control Letters , vol. 24, no. 5, pp. 351–\n359, 1995.\n[20] S. Dashkovskiy and A. Mironchenko, “Input-to-state st ability of infinite-\ndimensional control systems,” Mathematics of Control, Signals, and\nSystems , vol. 25, no. 1, pp. 1–35, 2013.\n[21] G. A. Pavliotis, Stochastic processes and applications: diffusion pro-\ncesses, the Fokker-Planck and Langevin equations . Springer, 2014,\nvol. 60.\n[22] T. Cacoullos, “Estimation of a multivariate density,” Annals of the\nInstitute of Statistical Mathematics , vol. 18, no. 1, pp. 179–189, 1966.\n[23] J. Chang and G. Cooper, “A practical difference scheme f or fokker-\nplanck equations,” Journal of Computational Physics , vol. 6, no. 1, pp.\n1–16, 1970."    },    {        "title": "2006.14399v2.Modifications_on_parameters_of__Z_4430___in_a_dense_medium.pdf",        "content": "arXiv:2006.14399v2  [hep-ph]  23 Nov 2020Modifications on parameters of Z(4430)in a dense medium\nK. Azizi∗\nDepartment of Physics, University of Tehran, North Karegar Ave. Tehran 14395-547, Iran and\nDepartment of Physics, Doˇ gu¸ s University, Acibadem-Kadi k¨ oy, 34722 Istanbul, Turkey\nN. Er†\nDepartment of Physics, Bolu Abant ˙Izzet Baysal University, G¨ olk¨ oy Kamp¨ us¨ u, 14980 Bolu, T urkey\n(Dated: November 24, 2020)\nThe charmonium-like resonance Zc(3900) and its excited state Z(4430) are among the particles\nthat are serious candidates for double heavy tetraquarks. C alculations of different parameters\nassociated with these states both in the vacuum and the mediu m with finite density are of great\nimportance. Such investigations help us clarify their natu re, internal quark-gluon organization\nand quantum numbers. In this accordance, we extend our previ ous analyses on the ground state\nZc(3900) to investigate the medium modifications on different p arameters of the excited Z(4430)\nstate. In particular, we calculate the mass, vector self-en ergy and current coupling of Z(4430) in\nterms of density, up to a density comparable to the density of the cores of massive neutron stars.\nThe obtained results may help experimental groups aiming to study the behavior of exotic states at\nhigher densities.\nI. INTRODUCTION\nOver the past two decades, the results of many ex-\nperimental observations on some resonances have shown\nthat these states can not be put in the class of the stan-\ndard mesons and baryons since their spectrum and decay\npatterns differ from the standard hadrons, considerably.\nThese results have led us to collect these states under\nthe new title: the exotic states made of multiquarks, an-\ntiquarks and valence gluons. Among these states, the\ntetraquarks have been relatively more in the focus of the\nexperimental and theoretical studies. The charmonium-\nlike resonances made of a c¯cand a light q1¯q2pair receive\nmuch attentions as they demonstrate different proper-\nties. Investigation of these states can help us not only\nclarify their nature and internal structure but also get\nuseful knowledge on the nature of strong interaction in-\nside these particles.\nThe observation of the ground states Z±\nc(3900) were\nreported simultaneously by the BESIII [ 1] and Belle [ 2]\nCollaborations in 2013. Many aspects of these states\nhave been previously investigated. For details see for\ninstance Refs. [ 3–7] and references therein. Hence, we\nwon’t spend time on this state in the present work and\nwill focus on the state Z(4430). For a first time in 2008,\nthe Belle Collaboration reported a distinct peak in the\nπ±ψ′invariant mass distribution in B→Kπ±ψ′decay\nwith the statistical significance of 6 .5σ. The measured\nmass and width were M= 4433±4(stat)±2(syst) MeV\nand Γ = 45+18\n−13(stat)30\n−13(syst) MeV [ 8]. In an ampli-\ntude analysis of B0→ψ′K+π−decays [9], the quan-\ntum numbers of the Z(4430) were set to JP= 1+and\nthe results for the mass and width were obtained as\n∗kazem.azizi@ut.ac.ir\n†nuray@ibu.edu.tr4485+22+28\n−22−11MeV/c2and 200+41+26\n−46−35MeV, respectively.\nThe Belle Collaboration then [ 10], during the observa-\ntion of a new charmonium like state Z+\nc(4200), in addi-\ntion, found evidence for Z+(4430)→J/ψπ+. In the first\nindependent confirmation by LHCb Collaboration [ 11],\nits spin-parity was assigned as 1+. The mass of the reso-\nnance, 4475 ±7+15\n−25MeV, and its width, 172 ±13+37\n−34MeV,\nwere measured. In the same mass region, an alternative\nand model-independent confirmation of the existence of\naψ(2S)πresonance is later presented by the LHCb, as\nwell [12].\nIn theoretical side, various assignments have been\nmade for the nature and quark-gluon structure of\nZ(4430) resonance. Using different phenomenological\nand computational models, the mass and some other\nquantities of this state have been studied. Considering\na molecular structure, the Z(4430) state was studied in\n[13–21]. The compact tetraquark or diquark-antidiquark\nstructure was assigned for this state in Refs. [ 5,22–\n29] to calculate various observables associated with the\nZ(4430)resonance. Other theoreticalinterpretations can\nbecollectedascuspeffects[ 30–32]andhadrocharmonium\n[33,34]. For more information, one can see Refs. [ 35–43].\nIn the present work, we shall adopt the com-\npact tetraquark/diquark-antidiquark interpretation of\nthe stateZ(4430) and consider it as the radial excita-\ntion of the ground state Zc(3900) with the same spin-\nparity. In Ref. [ 25], the authors have made the hypothe-\nsisthat the spacing(massdifference) in radialexcitations\ninthese channelscouldcloselyresemblethoseobservedin\nthe standard P-wave charmonia. Therefore, the authors\nhave considered the state Z(4430) as the first radial exci-\ntation of the Zc(3900) state. The same scenario has been\napplied in different studies [ 5,26,44,45], as well. We\ncalculate the mass and current coupling of Z(4430) state\nin the medium with high density, comparable with the\ndensity of the neutrons star cores. We discuss the modi-\nfications on the considered physical quantities due to the2\nnuclear dense medium. The medium effects are repre-\nsented by various operators that enter the nonperturba-\ntive parts of the calculations (for details see, for instance,\nRef. [46] and references therein). We calculate the values\nof the mass and current coupling at saturation density.\nThese quantities are also obtained in vacuum setting the\ndensity of the medium to zero. We compare the obtained\nresults at zero density limit with the existing theoretical\npredictions and the experimental data.\nThehighenergyheavyioncollisionexperimentsareex-\ncellent platforms to produce the heavy standard hadrons\nas well as the exotic particles. With upgraded detec-\ntors at the Relativistic Heavy Ion Collider (RHIC) at the\nBookhaven National Laboratory (BNL) and the Large\nHadron Collider (LHC), now it is possible to investigate\nnot only the ground state particles but also their excited\nstates. Thus, these experiments provide opportunities to\nstudy exotic hadrons like Zc(3900) and its excited state\nZ(4430). The study on other exotic states that are ei-\nther molecular states made of various standard hadrons\norcompactobjectsofquarksandantiquarksarenowpos-\nsible, as well. Due to a large number of heavy quarks and\nantiquarks produced in these experiments, many exotic\nstates could be formed. Hence, it will be a good op-\nportunity to study the exotic particles and their higher\nstates in order to determine their nature and internal\nquark-gluon structure as well as fix their quantum num-\nbers (for more information see for instance [ 47,48]). At\nRHIC and heavy ion collision experiments at LHC it will\nbealsopossibletoinvestigatedifferentphasesofhadronic\nmatter and search for quark-gluon-plasma (QGP) as a\nnew possible phase of matter . Some in-medium exper-\niments like ¯PANDA at Facility for Antiproton and Ion\nResearch (FAIR) at GSI and the Nuclotron-based Ion\nCollider Facility at JINR (NICA) at Dubna aim to study\nthe parametersofexotic states in terms of density. These\nexperiments can investigate the the Z+(4430) even fur-\nther by switching to studies ofthe Z+(4430)in formation\nmode. Due to the charge of the Z, this is only possible by\nannihilating the antiprotons on a neutron in a deuterium\nor hydrogen gas target. They can study the Z+(4430)\nstate in both production and formation experiments [ 49].\nThe paper is organized as follows: In next section the\nparametersofthe Z(4430)iscalculatedin densemedium.\nIn section 3 the numerical analyses of the physical quan-\ntities under study is performed. The last section includes\nthe summary and our concluding notes.\nII. PARAMETERS OF Z(4430)IN COLD\nNUCLEAR MEDIUM\nIn this section, we aim to calculate the mass and cur-\nrent coupling aswell as the scalarand vectorself-energies\nof the exotic Z(4430) (in what follows we denote it by Z\n) state as an exited state of the Zc(3900) (Zc) in cold\nnuclear medium. To this end, the positively charged\nZ+(4430) state with quark content ¯ ccu¯dis considered.But, the parameters of the negatively charged state with\n¯cc¯udcontent do not change as a result of Chiral limit\nused in the calculations. In this context, our starting\npoint is to consider the following density dependent two-\npoint correlation function:\nΠµν(p) =i/integraldisplay\nd4xeip·x/angbracketleftψ0|T[JZ(c)\nµ(x)JZ(c)†\nν(0)]|ψ0/angbracketright,(1)\nwhere|ψ0/angbracketrightisthegroundstateofnuclearmatterand JZ(c)\nµ\nisthe interpolatingcurrentof Z/Zcstatesthatcouplesto\nthese states, simultaneously. We should note that in the\npresent study we choose to work at zero momentum limit\nofthe particle’sthreemomentum, |p|→0, forwhich the\nlongitudinal and transverse polarizations become degen-\nerate, and there is no way to distinguish these two po-\nlarizations. For non-zero and finite three momentum in\nnuclear matter, however, the longitudinal and transverse\npolarizationstatesaredistinguishableandtheyshouldbe\ncalculated, separately (for more details see for instance\n[50]). At zero three momentum limit, p2=p2\n0and the\ncalculation goes forward in the standard way with the\nabove single function (see also [ 51]). ForJP= 1+, the\ninterpolating current is given by\nJZ(c)\nµ(x) =iǫabcǫdec\n√\n2/braceleftBigg/bracketleftBig\nuT\na(x)Cγ5cb(x)/bracketrightBig/bracketleftBig\n¯dd(x)γµC¯cT\ne(x)/bracketrightBig\n−/bracketleftBig\nuT\na(x)Cγµcb(x)/bracketrightBig/bracketleftBig\n¯dd(x)γ5C¯cT\ne(x)/bracketrightBig/bracerightBigg\n,(2)\nwhere,a,b,c,d,e are color indices and Cis the charge\nconjugation operator.\nWe need to derive the in-medium sum rules for the\nmassmZand current coupling fZof the excited state\nZ. For this purpose, the previously obtained expressions\nfor the mass and current coupling of Zc(see Ref. [ 6])\nare considered as input parameters. The procedure is\ndone in two steps: First, we derive the sum rules for\nthe in-medium mass mZcand in-medium current cou-\nplingfZcof the ground state exotic Zc. For this, we\nuse the ”ground state + continuum” approximation by\nincluding the Z(4430) state into the list of ”higher reso-\nnances” and extract sum rules and correspondingnumer-\nical values for Zcas discussed in Ref. [ 6]. At the next\nstep, we increase the threshold and adopt the ”ground\nstate+radially excited state+continuum” scheme, and\nperform the required standard calculations. In the sec-\nond case, the parameters of both the ZcandZappear\nin the expression of the hadronic side. To extract the\nparameters of Z, we use those of the Zcas inputs.\nThus, the phenomenological side of the correlation\nfunction is obtained by saturating it with the complete\nsets of hadronic states with the same quantum num-\nbers as the interpolating current. Isolating the ground\nstateZcand radially excited Zresonances, considering\nthe ”ground state+radially excited state+continuum”\nscheme, and integrating over x, we obtain the phe-3\nnomenological side of the correlation function in momen-\ntum space as\nΠPhe\nµν(p) =−/angbracketleftψ0|Jµ|Zc(p)/angbracketright/angbracketleftZc(p)|J†\nν|ψ0/angbracketright\np∗2−m∗2\nZc\n−/angbracketleftψ0|Jµ|Z(p)/angbracketright/angbracketleftZ(p)|J†\nν|ψ0/angbracketright\np∗2−m∗2\nZ+...,(3)\nwherep∗is the in-medium momentum; and mZcand\nmZare the masses of ZcandZstates, respectively. In\nEq. (3), contributionsarisingfromhigherresonancesand\ncontinuum are represented by the dots. The current me-\nson couplings of the states under consideration are ex-\npressed in terms of the polarization vectors εµand ˜εµ,\nrespectively, as\n/angbracketleftψ0|Jµ|Zc(p)/angbracketright=f∗\nZcm∗\nZcεµ,\n/angbracketleftψ0|Jµ|Z(p)/angbracketright=f∗\nZm∗\nZ˜εµ. (4)\nUsing Eq. ( 4) in Eq. ( 3) and summing over the polar-\nization vectors, we can construct the new form of the\nfunction ΠPhe\nµν(p) as\nΠPhe\nµν(p) =−m∗2\nZcf∗2\nZc\np∗2−m∗2\nZc/bracketleftBig\n−gµν+p∗\nµp∗\nν\nm∗2\nZc/bracketrightBig\n−m∗2\nZf∗2\nZ\np∗2−m∗2\nZ/bracketleftBig\n−gµν+p∗\nµp∗\nν\nm∗2\nZ/bracketrightBig\n+....(5)\nAt this point, we should note that a particle in nuclear\nmedium gain two kinds of self-energies: The scalar self-\nenergy, which is defined as the difference between the\nin-medium and vacuum masses, Σ s=m∗\nZ−mZ, and\nthe vector self-energy that enters the expression of the\nin-medium momentum, p∗\nν=pν−Συuν, where Σ υis\nthe vector self-energy and uνis the four-velocity of thenuclear medium. Our calculations take place in the rest\nframe of the medium, uν= (1,0), and so pν·uν=p0.\nConsequently, the correlation function is written as\nΠPhe\nµν(p) =−f∗2\nZc\np2−µ2\nZc/bracketleftBig\n−gµνm∗2\nZc+pµpν\n−ΣZc\nυpµuν−ΣZc\nυpνuµ+ΣZc2\nυuµuν/bracketrightBig\n−f∗2\nZ\np2−µ2\nZ/bracketleftBig\n−gµνm∗2\nZ+pµpν−ΣZ\nυpµuν\n−ΣZ\nυpνuµ+ΣZ2\nυuµuν/bracketrightBig\n+...,\n(6)\nwhereµ2\nZ(c)=m∗2\nZ(c)−ΣZ(c)2\nυ+2p0ΣZ(c)\nυ. After applying\nthe Borel transformation with respect to the parameter\np2to both sides of Eq. ( 6), the phenomenological side of\nthe correlation function yields,\nΠPhe\nµν(p) =f∗2\nZce−µ2\nZc/M2/bracketleftBig\n−gµνm∗2\nZc+pµpν\n−ΣZc\nυpµuν−ΣZc\nυpνuµ+ΣZc2\nυuµuν/bracketrightBig\n+f∗2\nZe−µ2\nZ/M2/bracketleftBig\n−gµνm∗2\nZ+pµpν−ΣZ\nυpµuν\n−ΣZ\nυpνuµ+ΣZ2\nυuµuν/bracketrightBig\n+..., (7)\nwhereM2is the Borel mass parameter.\nThe next stepis to calculatethe QCDsideofthe corre-\nlation function. This is done via the usage of the explicit\nform of the interpolating current in the correlation func-\ntion. The quark fields are contracted in the presence of\nthe dense medium via the Wick’s theorem. This results\nin the expression of the correlation function in terms of\nthe in-medium heavy and light propagators:\nΠQCD\nµν(p) =−i\n2εabcεa′b′c′εdecεd′e′c′/integraldisplay\nd4xeipx/braceleftBig\nTr/bracketleftBig\nγ5˜Saa′\nu(x)γ5Sbb′\nc(x)/bracketrightBig\nTr/bracketleftBig\nγµ˜Se′e\nc(−x)γνSd′d\nd(−x)/bracketrightBig\n−Tr/bracketleftBig\n[γµ˜Se′e\nc(−x)γ5Sd′d\nd(−x)/bracketrightBig\nTr/bracketleftBig\nγν˜Saa′\nu(x)γ5Sbb′\nc(x)/bracketrightBig\n−\nTr/bracketleftBig\nγ5˜Saa′\nu(x)γµSbb′\nc(x)/bracketrightBig\nTr/bracketleftBig\nγ5˜Se′e\nc(−x)γνSd′d\nd(−x)/bracketrightBig\n+\nTr/bracketleftBig\nγν˜Saa′\nu(x)γµSbb′\nc(x)/bracketrightBig\nTr/bracketleftBig\nγ5˜Se′e\nc(−x)γ5Sd′d\nd(−x)/bracketrightBig/bracerightBig\n|ψ0/angbracketright,\n(8)\nwhere˜Sq(c)=CST\nq(c)C. The in-medium heavy and light\nquarks’ propagatorsare used in coordinate space and the\ncalculations are transferred to the momentum space by\nperforming the Fourier integrals. To suppress the con-\ntributions of the higher states and continuum the Borel\ntransformations are applied based on the standard pre-\nscriptions of the method. To further suppress the con-tributions of the unwanted states, continuum subtrac-\ntion supplied by the quark-hadron duality assumption is\napplied. All of these procedures are described in Ref.\n[6]. We should stress that we take into account the non-\nperturbative operators up to five dimensions in the QCD\nside. The contributions of the six and upper dimen-\nsional operators are expected to be small and they are4\nneglected. Thus, the QCD side of the correlation func-\ntion in terms of the selected Lorentz structures is written\nas\nΠQCD\nµν(M2,s∗\n0) =ΥQCD\n1(M2,s∗\n0)(−gµν)\n+ΥQCD\n2(M2,s∗\n0)pµpν+ΥQCD\n3(M2,s∗\n0)(−pµuν)\n+ΥQCD\n4(M2,s∗\n0)(−pνuµ)+ΥQCD\n5(M2,s∗\n0)uµuν.\n(9)\nThe Borel transformed invariant functions\nΥQCD\ni(M2,s∗\n0) in Eq. ( 9) are represented in terms\nof the two-point spectral densities, ρQCD\ni(s), related to\nthe imaginary parts of the selected coefficients:\nΥQCD\ni(M2,s∗\n0) =/integraldisplays∗\n0\n4m2cdsρQCD\ni(s)e−s\nM2,(10)\nwheres∗\n0is the in-medium continuum threshold param-\neter separating the contributions of the ground state Zc\nand the first excited state Zfrom the higher resonances\nand continuum. iruns from 1 to 5. The spectral density\ncorresponding to each structure is expressed in terms of\nperturbative and non-perturbative contributions as fol-\nlows:\nρQCD\ni(s) =ρpert\ni(s)+ρqq\ni(s)+ρgg\ni(s)+ρqgq\ni(s),(11)\nwhereqq,ggandqgqdenotethetwoquark,twogluonand\nmixed quark-gluon condensates as the non-perturbative\neffects, respectively. As an example, the spectral density\ncorresponding to the structure gµνis given in Ref. [ 6].\nAfter all these lengthy calculations, when the phe-\nnomenological and QCD results of each structure are\ncompensate, the sum rules for the mass, vector self-\nenergy and current coupling constant of Zstate are ob-\ntained as follows:\nm∗2\nZcf∗2\nZce−µ2\nZc\nM2+m∗2\nZf∗2\nZe−µ2\nZ\nM2=ΥQCD\n1(M2,s∗\n0),(12)\nf∗2\nZce−µ2\nZc\nM2+f∗2\nZe−µ2\nZ\nM2=ΥQCD\n2(M2,s∗\n0),(13)\nΣZc\nυf∗2\nZce−µ2\nZc\nM2+ΣZ\nυf∗2\nZe−µ2\nZ\nM2=ΥQCD\n3(M2,s∗\n0),(14)\nΣZc\nυf∗2\nZce−µ2\nZc\nM2+ΣZ\nυf∗2\nZe−µ2\nZ\nM2=ΥQCD\n4(M2,s∗\n0),(15)\nΣZc2\nυf∗2\nZce−µ2\nZc\nM2+ΣZ2\nυf∗2\nZe−µ2\nZ\nM2=ΥQCD\n5(M2,s∗\n0).(16)\nThemasssumrulesfor Zstatecanbederivedbydiffer-\nent ways from the above equations. We apply derivative\nwith respect to −1\nM2to both sides of Eq. ( 12) and elimi-\nnatef∗2\nZby dividing both sides of the resultant equation\nto both sides of the original one. As a result we get\nµ2\nZ=∂ΥQCD\n1(M2,s∗\n0)\n∂/parenleftBig\n−1\nM2/parenrightBig−m∗2\nZcf∗2\nZcµ2\nZce−µ2\nZc\nM2\nΥQCD\n1(M2,s∗\n0)−m∗2\nZcf∗2\nZce−µ2\nZc\nM2,(17)where, the in-medium mass of Zstate is found as\nm∗2\nZ=µ2\nZ+ΣZ2\nυ−2p0ΣZ\nυ. (18)\nBy rearranging the equations ( 16) and (13) and dividing\nboth sides of these equations to each other we get ΣZ2\nυ\nas\nΣZ2\nυ=ΥQCD\n5(M2,s∗\n0)−ΣZc2\nυf∗2\nZce−µ2\nZc\nM2\nΥQCD\n2(M2,s∗\n0)−f∗2\nZce−µ2\nZc\nM2.(19)\nTo obtain the in-medium current coupling constant, f∗\nZ,\nthere aredifferent possibilities. One wayis to find it from\nEq. (12), which leads to\nf∗2\nZ=ΥQCD\n1(M2,s∗\n0)−m∗2\nZcf∗2\nZce−µ2\nZc\nM2\nm∗2\nZe−µ2\nZ\nM2.(20)\nAs it is seen, the parameters of Zccalculated in Ref. [ 6]\nare entered as the inputs to the sum rules for Zstate in\nthe dense medium.\nIII. NUMERICAL RESULTS\nThe expressions obtained in Eqs. ( 17-20) for the phys-\nical observables in cold nuclear matter contain quark and\nbaryon masses, nuclear matter density and different op-\nerators both in vacuum and dense medium as input pa-\nrameters. The operators appearing in the calculations in\na dense medium are presented in Ref. [ 46]. The numer-\nical results for expectation values of different operators\ntogether with the values of other input parameters are\ncollected in table ( I).\nThe obtained in-medium sum rules for the mass, cur-\nrent coupling and vector self-energy of Zstate contain\ntwo more auxiliary parameters, as well. These parame-\nters are fixed based on the standard requirements of the\nmethod: mild variations of the physical quantities with\nrespect to the changes in the values of these parameters,\ndominance of first two resonances over the higher states\nand continuum and convergence of the series of the oper-\natorproduct expansion(OPE).To this end, we introduce\nthe quantities\nFTRC =ΥQCD\ni(M2,s∗\n0)\nΥQCD\ni(M2,∞), (21)\nwhere FTRC stands for first two resonance contribution,\nand\nR=ΥDim5\ni(M2,s∗\n0)\nΥQCD\ni(M2,s∗\n0). (22)\nInEq.(22)ΥDim5\ni(M2,s0)isthesumofalloperatorswith\nmassdimension5, thehighestdimensionthatweconsider\ninthecalculations. TheFTRCisusedtofixupperbound5\nParameter Value UnitRef. #\nρsat0.113GeV3[52]\np0 4.478+0.015\n−0.018 GeV [53]\nmu 2.16+0.49\n−0.26 MeV [53]\nmd 4.67+0.48\n−0.17 MeV [53]\nmc 1.27±0.02 GeV [53]\n/an}bracketle{tq†q/an}bracketri}htρ3\n2ρ GeV3[51]\n/an}bracketle{t¯qq/an}bracketri}ht0 −[272(5)]3MeV3[54–59]\n/an}bracketle{t¯qq/an}bracketri}htρ /an}bracketle{t¯qq/an}bracketri}ht0+σπN\n2mqρMeV3[60]\nmq 0.00345 GeV [53]\n/an}bracketle{tαs\nπG2/an}bracketri}ht0 0.012±0.004 GeV4[54,61,62]\n/an}bracketle{tαs\nπG2/an}bracketri}htρ/an}bracketle{tαs\nπG2/an}bracketri}ht0+ρ/an}bracketle{tαs\nπG2/an}bracketri}htNGeV4[54,63,64]\n/an}bracketle{tαs\nπG2/an}bracketri}htN−8\n9(MN−σπN−σsN)GeV[52,54]\nMN 939.49±0.05 MeV [53]\nσπN 45(6) MeV [65]\nσsN 21(6) MeV [65]\n/an}bracketle{tq†iD0q/an}bracketri}htρ3\n2mqρ GeV4[54,60]\n/an}bracketle{tq†gsσGq/an}bracketri}htρm2\n0/an}bracketle{tq†q/an}bracketri}htρ GeV5[54]\n/an}bracketle{t¯qgsσGq/an}bracketri}ht0 m2\n0/an}bracketle{t¯qq/an}bracketri}ht0 GeV5[54]\nm2\n0 0.8±0.2 GeV2[54,66]\n/an}bracketle{t¯qgsσGq/an}bracketri}htρ/an}bracketle{t¯qgsσGq/an}bracketri}ht0+ρm2\n0σπN\n2mqGeV5[54,60]\n/an}bracketle{t¯qiD0iD0q/an}bracketri}htρ0.3GeV2ρ−1\n8/an}bracketle{t¯qgsσGq/an}bracketri}htρGeV5[54,60]\nTABLE I. Input parameters and their numerical values used\nin calculations.forM2, whereasR(M2) is necessaryto fix the lowerlimit\nfor the Borel parameter. To fix the working regions, we\nimpose FTRC ≥0.5 andR >0.1 in order to guarantee\nthe first two resonances dominance over the higher states\nand continuum and the OPE convergence. Our analyses\nshowthatalloftheaboverequirementsaresatisfiedwhen\n/braceleftBigg\nM2in [3−5] GeV2,\ns∗\n0in [21.9−23.8] GeV2.(23)\nFIG. 1. The in-medium mass of Zstate as functions of Borel\nmass parameter M2at the nuclear matter saturation density\nand at fixed values of the continuum threshold.\nmZ m∗\nZ fZ.102f∗\nZ.102\nPS 4486+112\n−113MeV3311+203\n−185MeV 1.27+0.53\n−0.36GeV41.34+0.78\n−0.45GeV4\nExp. [53]4478+15\n−18MeV - - -\n[5] 4452+182\n−228MeV - 1.48+0.31\n−0.42GeV4-\n[26]4.51+0.17\n−0.09GeV - (5.75+98\n−78)/mZGeV4-\n[15]4.40±0.10 GeV - - -\n[27]4.52±0.09 GeV - (3.75±0.48)/mZGeV4-\n[44]4.53+0.16\n−0.10GeV - - -\nTABLE II. The vacuum and in-medium masses and current\ncouplings of Zstate. PS stands for present study.\nIn Fig. ( 1), we demonstrate the in-medium mass m∗\nZ\nas a function of Borel mass parameter M2at the nu-\nclear matter saturation density, ρsat= 0.113GeV3, and\nat three fixed values of the continuum threshold. As is\nseen,thein-mediummassofthe Z(4430)resonanceshows\npretty good stability against variations of both of Borel\nmass parameter M2and in-medium continuum thresh-\nolds∗\n0. In Fig. ( 2), the vacuum mass ( mZ) which is\nobtained at ρ→0 limit, the in-medium mass ( m∗\nZ) and\nvectorself-energy(ΣZ\nυ) ofZstate aredisplayedas afunc-\ntion ofM2at average value of continuum threshold andat nuclear matter saturation density. From this figure we\nsee a considerable negative shift in the mass of the state\nunderstudydue tothe medium effects. This shitis called\nthe scalar self-energy. Strictly speaking, the in-medium\nmass of the Zresonance reduces to approximately 74%\nof its vacuum value at the nuclear matter saturation den-\nsity and amounts as 3311+203\n−185MeV. Thus, the negative\nshift in the mass due to the cold nuclear matter is ap-\nproximately 26%. This state gains a vector self-energy\nwith the value 1310+101\n−86MeV at nuclear matter satura-\ntion density. At saturation density the current coupling6\nFIG. 2. The vacuum mass, in-medium mass and vector self-\nenergy ofZstate as functions of M2at average value of con-\ntinuum threshold and at nuclear matter saturation density.\nFIG. 3. In-medium mass and vector self-energy as functions\nofρ/ρsatat mean value of the continuum threshold and Borel\nparameter.\n¶¶¶¶¶¶¶¶¶¶¶¶¶¶¶¶¶\n£££££££££££££££££¶mz*+ΣνZSRresult\nmz*+ΣνZfitting\n£Σs+ΣνZSRresult\nΣs+ΣνZfitting\n0123450246810012345\n0246810\nρ/ρsat\nFIG. 4. The Σ s+ΣZ\nυandm∗\nZ+ΣZ\nυas functions of ρ/ρsatat\nmean value of the continuum threshold and Borel parameter.\nof the state Zis found as 1 .34+0.78\n−0.45GeV4. The numeri-\ncal results for the masses and current couplings obtainedFIG. 5. The current coupling of Zstate as a function of\nρ/ρsatat mean value of the continuum threshold and Borel\nparameter.\nFIG. 6. The mass m∗\nZas a function of density when the\nuncertainties of the auxiliary and other input parameters a re\nconsidered.\nin vacuum and medium are collected in table ( II). For\ncomparison, we demonstrate the existing values from the\nexperiment and other theoretical studies in the same ta-\nble, as well. From this table, we see that the vacuum\nmass obtained in the present study is nicely consistent\nwith the world experimental average [ 53]. This value is\nalso in accord with other theoretical predictions within\nthe errors. The vacuum current coupling is consistent\nwith other theoretical predictions within the presented\nuncertainties. Our predictions for the in-medium mass\nand current coupling as well as the vector self-energy can\nbe tested in future in-medium experiments and by other\ntheoretical studies.\nThe main aim in the present study, is to obtain the\nbehavior of the physical quantities under study with re-\nspect to the density, specially at higher densities. In the\npresent study, we consider the range from zero to 5 ρsat,\ncorresponding to the density of the cores of massive neu-\ntron stars. Neutron stars, as natural QCD matter labo-\nratories, are very dense and compact that may produce\nstrange and heavy baryons as well as the exotic states\nregarding the processes that may occur in their inside.7\n¶\n¶¶\n¶\n¶\n¶¶\n¶¶\n¶¶\n¶¶¶¶¶SRresult\nfitting\n0123450246810012345\n0246810\nρ/ρsatTerms~ \n\u0001q\n-q〉ρ×10\n\u00006\n¶¶¶¶¶¶¶¶¶¶¶¶¶¶¶¶SRresult\nfitting\n0123450102030405060012345\n0102030405060\nρ/ρsatTerms~〈q\nq〉ρ×10-9\n¶¶¶¶¶¶¶¶¶¶¶¶¶¶¶¶SRresult\nfitting\n012345012345012345\n012345\nρ/ρsatTerms~〈αs\nπG2〉ρ×10-8\n¶¶¶¶¶¶¶¶¶¶¶¶¶¶¶¶SRresult\nfitting\n0123450102030405060012345\n0102030405060\nρ/ρsatTerms~〈q†iD0q\n\u0003\n\u0002\n\u0004×10-12\n¶¶¶¶¶¶¶¶¶¶¶¶¶¶¶¶SRresult\nfitting\n012345\n0 \u0005 \u0006\n\u0007 \b \t\n1 \n \u000b\n\f \r \u000e\n2 \u000f \u0010012345\u0011 \u0012 \u0013\n\u0014 \u0015 \u0016\n\u0017 \u0018 \u0019\n\u001a \u001b \u001c\n\u001d \u001e \u001f\nρ/ρsatTerms~〈q-iD0iD0q〉ρ×10-6\n¶¶\n¶\n¶\n¶\n¶\n¶¶¶¶¶¶¶¶¶¶SRresult\nfitting\n012345\n ! \"\n# $ %\n& ' (\n) * +\n, . /0123434 6 7\n8 9 :\n; < =\n> ? @\nA B C\nρ/ρsatTerms~〈q-gsσ\nGD 〉ρ×10-6\nFIG. 7. The dependence of terms proportional to various\ncondensates, entering the QCD side of the calculations, on\nρ/ρsatat average values of the auxiliary parameters.\nFor the neutron stars with mass ∼1.5M/circledottext, the corre-\nsponding core density is approximately (2-3) ρsatand for\nthe mass ≥2M/circledottextthe density is about 5 ρsatof the nu-\nclearmatter [ 67]. To understandthe processoccurringat\nhigher densities with the aim of getting knowledgeon the\ninternal structures of the neutron stars as well as analyz-\ning of the future data in the heavy ion collision exper-\niments, we need to investigate the properties of various\nhadrons under extreme conditions. In the present study,\nwe deal with cold nuclear matter ( T≃0), hence there\nis no phase transition expected up to the density that\nwe consider in the calculations. The phase transition to\nQGPisexpectedtooccuratpoints ≥5.8ρsatatzerotem-\nperature [ 68]. The QGP state of matter is studied in cur-\nrent heavy ion collision experiments at the LHC/CERN\nand RHIC at BNL. Presently, the QCD phase diagram\nof quark matter is not well known, either experimentally\nor theoretically, but there are some information from the\nlattice QCD and other phenomenological models. Many\nexperiments are designed with the aim of investigation\nof the hadronic properties and possible phase transitions\nat finite temperatures and densities. As previously men-\ntioned, the FAIR at GSI and the NICA at Dubna aim to\nprovide useful information in this respect.\nTo discuss the dependence of the physical quantities\non the density, in Fig. ( 3) we display m∗\nZand ΣZ\nυas\nfunctions of ρ/ρsatat average values of the continuum\nthreshold and Borel parameter. As it is clear, the sum\nrules for both the in-medium mass and vector self-energy\ngive reliable results up to ρ= 1.6ρsatGeV3. To ex-\ntend our calculations for higher densities, we use some\nfit functions. The best fit for the mass is the followingexponential function:\nm∗\nZ(ρ) = 4.384e−0.233xGeV, (24)\nwherex=ρ/ρsat. For vector self-energy we obtain\nΣZ\nυ(ρ) = 0.099x2+0.917x+0.272 GeV.(25)\nIn orderto better understand how Z(4430)feels the total\npotential, and how much energy would be necessary to\nproduce this state in experiments using nuclear targets,\nin Fig. (4), we show the dependence of Σ s+ΣZ\nυ, orm∗\nZ+\nΣZ\nυonρ/ρsat. We observe that both of these quantities\ngrow with increasing in the density of the medium.\nIn Fig. (5), we plot the density-dependent current cou-\npling. The following function best describes the current\ncoupling:\nf∗\nZ(ρ) =−0.002x2+0.002x+0.015GeV4.(26)\nThe current coupling becomes zero at x= 3.2. The vec-\ntor self-energy increases with the increasing in the den-\nsity, while the mass decreases with increasing in the den-\nsity and reaches to roughly 1 .4GeVatx= 5 in average.\nFig. (6) shows the m∗\nZ−xgraphic when all the uncer-\ntainties in the auxiliary parameters as well as in other\ninputs are considered. As it is clear from the presented\nfigures, the parameters of the Z(4430) state depend on\nthe density, considerably. In [ 46,51,69,70], which study\nthe behavior of the nucleon, hyperon and light Θ+pen-\ntaquark with respect to density up to ρ≃1.5ρsat, the\ndependence of the parameters on density are relatively\nsmall. In the present study we considerto investigate the\nbehaviorofmanyparametersofthe doubly-heavyexcited\nZ(4430) tetraquark state up to 5 ρsat, corresponding to\nthe density of the cores of massive neutron stars, and\nobserve that the considered parameters depend on the\ndensity, remarkably. These behavior can be attributed\nto the considerable dependence of the various terms pro-\nportional to different condensates in the QCD side of the\npresent calculations on the density as depicted in Fig.\n(7).\nIV. SUMMARY AND CONCLUSIONS\nThe vacuum properties of different tetraquark states\nhave been discussed in many theoretical studies as well\nas different experiments. With the developments in the\nexperimental side, we hope we will be able to study\nthe behavior of such states in in-medium experiments\nin near future. Thus, theoretical studies on the spec-\ntroscopic properties of exotics can play crucial roles in\nconducting the related experiments. Such studies can\nalso help us fix the quantum numbers and determine the\nnature and internal structures of the famous candidates\nof charmonium-like tetraquarks. In this accordance, we\ntook into consideration the Z(4430) state in the present\nstudy and calculated some spectroscopic parameters of8\nthis state both in vacuum and a medium with finite\ndensity. Based on some experimental information, we\ntreatedthisstateasthe firstexcitedstatein the Zc(3900)\nchannel and constructed the related sum rules based on\nthe standard prescriptions of the method. As we saw,\nthe parameters of the ground state Zc(3900) were en-\ntered as input parameters to the sum rules for Z(4430)\nstate. We fixed the entering auxiliary parameters in ac-\ncordance with the requirements of the method and used\nthe expectation values of the different in-medium oper-\nators available in the literature to extract the values of\nthe parameters at saturation density and determine the\nbehavior of these parameters with respect to the density.\nThe numericalcalculations showedthat the massgains\na negative shift (scalar self-energy) due to the medium\neffects: this shifts amounts −26% at saturation den-\nsity. This state gains a repulsive vector self-energy which\namounts 1310+101\n−86MeV at saturation density. We also\nfound the values of the current coupling as 1 .34+0.78\n−0.45\nGeV4at nuclear matter saturation density. This value\ncan be used in determinations of different parameters re-\nlated to the electromagnetic, weak and strong interac-\ntions/decays of this state in medium.\nWe obtained the behavior of the physical quantities\nin the interval ρ∈[0,5]ρsat, which its upper limit\ncorresponds to the core density of the neutron stars.We observed that the current coupling becomes zero at\nρ= 3.2ρsat, while the vector self-energy increases with\nthe density. The mass exponentially reduces with the in-\ncreasinginthevalueofthedensityandreachestoroughly\n1.4GeVat the end point. With the progress in the ex-\nperimental side we hope that the heavy ion collision ex-\nperiments at CERN and BNL as well as the FAIR at\nGSI and NICA at Dubna will be able to provide use-\nful information on the behavior of the hadronic matter\nunder extreme conditions as well as possible phase tra-\nditions to QGP. 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D 92(2015) no.5, 054026"    },    {        "title": "2006.15840v1.Analyticity_of_density_of_states_for_the_Cauchy_distribution.pdf",        "content": "arXiv:2006.15840v1  [math-ph]  29 Jun 2020Analyticity of density of states for the Cauchy\ndistribution\nWerner Kirsch\nFakult¨ at Mathematik\nFern Universit¨ at Hagen\nHagen , Germany\nemail:werner.kirsch@fernuni-hagen.de\nM Krishna\nAshoka University\nPlot No 2, Rajiv Gandhi Education City\nRai, Haryana 131029, India\nemail: krishna.maddaly@ashoka.edu.in\nJune 30, 2020\nAbstract\nWe compute the density of states for the Cauchy distribution for\na large class of random operators and show it is analytic in a s trip\nabout the real axis.\n1 The Models and Results\nIn this short note we show that many random operators with the Ca uchy\ndistribution have analytic density of states, independent of disord er. This\ntheorem applied to the Anderson model shows analyticity through t he mo-\nbility edge (or the mobility band) on the Bethe lattice for small disorde r and\nsome models of Random Schr¨ odinger operators.\n1The density of states of random operators is of interest for many random\nmodels. We are concerned in particular with random models coming fro m\ni.i.d random variables with the Cauchy distribution. For these models th ere\nis an exact formula for the density of states.\nOne of the earliest models with the Cauchy distribution was the Lloyd\nmodel studied by Lloyd [8] on ℓ2(Z3), in which he obtained an exact formula\nforthe density of states. Inthe bookof Carmona-Lacroix[4], pag e329andin\nproblemVI.5.5onefindscalculationsgiving theexact valuefortheinte grated\ndensity of states (IDS) for the Anderson model on the lattice ℓ2(Zd) with the\nCauchy distribution. On the Bethe lattice Accosta-Klein [1], proved t hat the\ndensity of states is analytic in a strip about the real axis for any dist ribution\nclose to and including the Cauchy distribution, in a function space. We note\nthat Accosta-Klein [1] result is the first known result where smooth ness of\nDOS is shown in the region where absolutely continuous spectrum is pr esent\nfor random operators.\nIn this note we show that the exact expressions for the IDS are va lid in\na much larger context both for the discrete and some continuous m odels,\nwhenever i.i.d random variables with the Cauchy distribution are involve d.\nWe use the Trotter product formula for proving our result.\nWe consider a separable Hilbert space Hand a self-adjoint operator H0\nwith domain D(H0)⊂ H, a sequence {Pn}of mutually orthogonal projec-\ntionssuchthat/summationtext\nn∈NPn=Idandconsidertheoperator Nα=/summationdisplay\nn∈NnαPn, α>1,\nwith domain D(Nα)⊂ H. We consider i.i.d random variables {ωn, n∈N}\nwith the Cauchy distribution whose density is given by ψλ(x) =1\nπλ\nλ2+x2, λ>\n0. If we consider an α >1, then under the assumption on the random\nvariables {ωn}, an application of the Borel-Cantelli Lemma implies that the\nset\nΩ0={(ωn) :|ωn| ≤ |n|α,for all but finitely many n} (1)\nhas probability 1. Therefore the domain of the self-adjoint operat orVω=/summationtext\nn∈NωnPncontainsD(Nα) for every ω∈Ω0. We assume that D(H0)∩\nD(Nα) is dense in Hand that the random operators\nHω=H0+Vω, Vω=/summationdisplay\nn∈NωnPn (2)\nare essentially self-adjoint on D(H0)∩D(Nα) for every ω∈Ω0. In the\n2following we denoteby EA(), thespectral projectionof aself-adjoint operator\nA.\nThen we have, with Edenotes integration with respect to ωand∗denotes\nconvolution :\nTheorem 1.1. LetHbe a separable Hilbert space and H0,Vωbe self-adjoint\noperators on H, as in equation (2), with D(Vω)⊃ Dfor some dense set D\nfor almost every ω. Assume that D(H0)∩Dis dense in HwithHωessentially\nself-adjoint on D(H0)∩ Dfor allmost all ω. Letφ,ψ∈ Hbe unit vectors\nand setµφ,ψ=/an}bracketle{tφ,EH0()ψ/an}bracketri}ht. Then,\nN(·) =E/parenleftbigg\n/an}bracketle{tφ,EHω(·)ψ/an}bracketri}ht/parenrightbigg\n=ψλ∗µφ,ψ, (3)\nas measures.\nWe have a few corollaries of this theorem applied to the Anderson mod el\nonZdand the Bethe lattice, where we denote by δ0the unit vector supported\nat 0 inZdor a chosen and fixed root in the Bethe lattice. In both these\ncases, the operator H0, being the adjascency matrix, is bounded. Therefore\nthe conditions of the Theoremm 1.1 are satisfied and the following cor ollary\nresults by setting φ=ψ=δ0in the Theorem 1.1.\nCorollary 1.2. Consider the Anderson model on ℓ2(Zd)orℓ2(B),Bthe\nBethe lattice, with the single site distribution ψλ(x)dx, λ > 0. Then the\ndensity of states, which is the density of the measure\nN(·) =E/parenleftbig\n/an}bracketle{tδ0,EHω(·)δ0/an}bracketri}ht/parenrightbig\n,\nis analytic in a strip {z:ℑ(z)<λ}about the real axis.\nRemark 1.3. The above Corollary was proved for the three dimensional cas e\nby Lloyd [8] and a proof is indicated in the problem VI.5.5 of C armona [4]\nin the case of the Anderson model on ℓ2(Zd), for any disorder λ >0. Our\nresult as that of Accosta-Klein [1], shows smoothness of DOS through the\nmobility edge and everywhere in the spectrum , when applied t o the Bethe\nlattice where existence of absolutely continuous spectrum is proved see Klein\n[7], Froese et.al. [5] and Aizenman-Warzel [2]. Our results may also apply\nto the models considered by Anantharaman et.al. [3].\n3The second corollary is for Random Schr¨ odinger Operators on L2(R).\nConsider a collection of smooth bump functions {un,n∈Z}supported in a\nneighbourhood of unit cubes centered at n∈Z. Let,\nHω=−∆+Vω, Vω(x) =/summationdisplay\nn∈Zωnun(x),/summationdisplay\nnun(x) = 1,∀x∈R,(4)\nwhere−∆ is the Laplacian defined on its maximal domain D(−∆),{ωn}are\ni.i.d random variables with Cauchy distribution ψλ(x)dx.\nThen we have the corollary, where as before ∗denotes convolution. The\nonlystatement to verify to prove the corrolaryisthe essential se lf-adjointness\nof the random Schr¨ odinger operators involved, which follows from a theorem\nof Kirsch-Martinelli [6]. We note that if the essential self-adjointne ss holds\nfor the operators on L2(Rd), then the same result extends to that case also.\nCorollary 1.4. Consider the random Schr¨ odinger operators given in equa-\ntion (4). Then there exists a set Ω0of probability 1 such that for each ω∈Ω0,\nthe operators Hωare essentialy self-adjoint on D(−∆)∩D(|x|α), for some\nα>1and the integrated density of states Nis given by\nN(E) =ψλ∗tr/parenleftbigg\nu0EH0((−∞,E]))/parenrightbigg\n/integraltext\nu0(x)dx, E∈R. (5)\n2 The Proofs\nProof of Theorem 1.1: We first note that for any unit vector φ∈ H,µφ\nis a probability measure and its convolution with ψλis also a probability\nmeasure.\nThe proof of this is by the use of Trotter product formula.\nWe start by choosing a sequence QMof finite rank orthogonal projections\nsuch that\nlim\nM→∞QM=Id,[QM,Pn] = 0, QM,n=QMPn∀n∈N (6)\n∀M∃N(M)s.t. Rank (QM,n)/ne}ationslash= 0,∀n≤N(M), N(M)→ ∞asM→ ∞,\n(7)\n4where the limits are in the stong sense. The existence of such QMis not hard\nto see, since Pns are mutually orthogonal and add to the identity.\nWe will prove the theorem by showing that the Fourier transform of Nis\nthe product of Fourier transforms of µφ,ψandψλ. By the spectral theorem\nfor self-adjoint operators, the Fourier transform of Nis given by\n/integraldisplay\neitxdN(x) =E/parenleftbig\n/an}bracketle{tφ,eitHωψ/an}bracketri}ht/parenrightbig\n. (8)\nWe setVω=/summationtextωnPnand use the Trotter product formula [9, Theorem\nVIII.31], to write the right hand side of equation (8) as\nE/parenleftbig\n/an}bracketle{tφ,eitHωψ/an}bracketri}ht/parenrightbig\n= lim\nk→∞E/parenleftbig\n/an}bracketle{tφ,/parenleftbig\neit\nkH0eit\nkVω/parenrightbigkψ/parenrightbig\n= lim\nk→∞lim\nM→∞E/parenleftbig\n/an}bracketle{tφ,/parenleftbig\neit\nkH0eit\nkVωQM/parenrightbigkψ/an}bracketri}ht/parenrightbig\n= lim\nk→∞lim\nM→∞N(M)/summationdisplay\nn1,...,nk=1E/parenleftbigg\n/an}bracketle{tφ,/parenleftbigk\n→/productdisplay\nj=1eit\nkH0eit\nkVωQM,nj/parenrightbig\nψ/an}bracketri}ht/parenrightbigg\n,(9)\nwhere in the second equality we used the Lebesgue dominated conve rgence\ntheoremto interchange thelimit andexpectation andinthenext equ ality the\narrowonthe product denotes anordered product withincreasing indexjand\nfinally the sum and the expectation are interchanged using Fubini’s th eorem.\nSince we have from definitions of Vω,QM,nthateit\nkVωQM,nj=eit\nkωnj, the\nabove equality becomes\nE/parenleftbig\n/an}bracketle{tφ,eitHωψ/an}bracketri}ht/parenrightbig\n= lim\nk→∞lim\nM→∞N(M)/summationdisplay\nn1,...,nk=1E/parenleftbigg\n/an}bracketle{tφ,/parenleftbigk\n→/productdisplay\nj=1eit\nkH0eit\nkωnjQM,nj/parenrightbig\nψ/an}bracketri}ht/parenrightbigg\n= lim\nk→∞lim\nM→∞N(M)/summationdisplay\nn1,...,nk=1e−λ|t|/an}bracketle{tφ,/parenleftbigk\n→/productdisplay\nj=1eit\nkH0QM,nj/parenrightbig\nψ/an}bracketri}ht/parenrightbigg\n= lim\nk→∞lim\nM→∞e−λ|t|/an}bracketle{tφ,/parenleftbigk\n→/productdisplay\nj=1eit\nkH0QM/parenrightbig\nψ/an}bracketri}ht/parenrightbigg\n=e−λ|t|/an}bracketle{tφ,eitHωψ/an}bracketri}ht, (10)\n5where in the second equality we used the fact that when the produc t is\nexpanded for a fixed configuration of ( n1,...,nk), we will get an expression\nof the form\neit\nk(α1ωn1+α2ωn2+···+αkωnk),\nwith theαj’s denoting the number of times the ωnjoccurs in the product and\nfor any configuration of ( n1,...,nk) we have/summationtextαj=k. This fact together\nwith the fact that ωnjs are independent random variables for distint njand\nthe expectation with respect to the Cauchy distribution satisfies\n/integraldisplay\neisxψλ(x)dx=e−λ|s|,Eeisωneiwωm=e−λ|s+w|, n/ne}ationslash=m\nshows that the expectation always gives the value e−λ|t|for any configuration\nof (n1,...,nk) in the sum, so we could take that factor out and resum the\nexpression.\nThe equation (10) shows that E/an}bracketle{tφ,eitHωψ/an}bracketri}htis integrable as a function of\ntand so by Fourier inversion, the Theorem follows.\nAcknowledgement: We thank K B Sinha and Peter Hislop for conversa-\ntions on the Trotter product formula.\nReferences\n[1] Victor Acosta and Abel Klein. Analyticity of the density of states in the\nAnderson model on the Bethe lattice. J. Statist. Phys. , 69(1-2):277–305,\n1992.\n[2] Michael Aizenman and Simone Warzel. Resonant delocalization for r an-\ndom Schr¨ odinger operators on tree graphs. J. Eur. Math. Soc. (JEMS) ,\n15(4):1167–1222, 2013.\n[3] Nalini Anantharaman, Maxime Ingremeau, Mostafa Sabri, and Br ian\nWinn. Absolutely continuous spectrum forquantum trees. arXiv preprint\narXiv:2003.12765 , 2020.\n[4] Ren´ e Carmona and Jean Lacroix. Spectral theory of random Schr¨ odinger\noperators . Springer Science & Business Media, 2012.\n6[5] Richard Froese, David Hasler, and Wolfgang Spitzer. Absolutely c ontin-\nuous spectrum for the anderson model on a tree: a geometric pro of of\nklein’s theorem. Communications in mathematical physics , 269(1):239–\n257, 2007.\n[6] Werner Kirsch and Fabio Martinelli. On the essential selfadjointne ss of\nstochastic Schr¨ odinger operators. Duke Math. J. , 50(4):1255–1260, 1983.\n[7] Abel Klein. Extended states in the anderson model on the bethe lattice.\nAdvances in Mathematics , 133(1):163–184, 1998.\n[8] P. Lloyd. Exactly solvable model of electronic states in a three di-\nmensional disordered hamiltonian: non existence of localized states .J.\nPhysics (C) , 2:1717–1725, 1969.\n[9] Michael Reed and Barry Simon. Methods of modern mathematical\nphysics: Functional analysis , volume 1. Academic Press Inc., 1980.\n7"    },    {        "title": "2007.10771v2.Density_profile_of_a_semi_infinite_one_dimensional_Bose_gas_and_bound_states_of_the_impurity.pdf",        "content": "Density profile of a semi-infinite one-dimensional Bose gas and bound states of the impurity\nAleksandra Petkovi ´c, Benjamin Reichert, and Zoran Ristivojevic\nLaboratoire de Physique Th ´eorique, Universit ´e de Toulouse, CNRS, UPS, 31062 Toulouse, France\n(Dated: October 23, 2020)\nWe study the effect of the boundary on a system of weakly interacting bosons in one dimension. It strongly\ninfluences the boson density which is completely suppressed at the boundary position. Away from it, the density\nis depleted over the distances on the order of the healing length at the mean-field level. Quantum fluctuations\nmodify the density profile considerably. The local density approaches the average one as an inverse square of the\ndistance from the boundary. We calculate an analytic expression for the density profile at arbitrary separations\nfrom the boundary. We then consider the problem of localization of a foreign quantum particle (impurity) in\nthe potential created by the inhomogeneous boson density. At the mean-field level, we find exact results for\nthe energy spectrum of the bound states, the corresponding wave functions, and the condition for interaction-\ninduced localization. The quantum contribution to the boson density gives rise to small corrections of the\nbound state energy levels. However, it is fundamentally important for the existence of a long-range Casimir-like\ninteraction between the impurity and the boundary.\nI. INTRODUCTION\nBoundaries play an important role in one-dimensional\nquantum liquids affecting, e.g., correlation and response func-\ntions, as well as the ground-state energy [1–8]. The boundary\nhas also an impact on the density profile of particles n(x),\nwhich is suppressed at the boundary position x= 0. Away\nfrom it, the particle density in fermionic systems shows so-\ncalled Friedel oscillations [9]. They describe an oscillatory\ndecay ofn(x)\u0000n0, wheren0is the average density. The\nenvelop of Friedel oscillations follows 1=xKlaw for spinless\nfermions [3, 10]. Here Kis the Luttinger liquid parameter,\nwhich is determined by the interaction between fermions. The\nperiodicity of oscillations is controlled by the average fermion\ndensity.\nFriedel oscillations in fermionic systems are studied within\nthe harmonic Tomonaga-Luttinger liquid description [3, 10].\nThis is the low-energy theory of both, interacting fermions\nand bosons in one dimension [11]. It was applied for bosons\nin Ref. [12] where the same pattern of density oscillations\nwas found for separation longer than the inverse mean den-\nsity of bosons, 1=n0. Applied to the special case of weakly-\nrepulsive bosons where K\u001d1, the result of Ref. [12] implies\nvery rapid saturation of n(x)\u0000n0. Such fast recovery of the\ndensity is not expected to occur in weakly interacting superflu-\nids where the density should change on the scale comparable\nto the healing length \u0018\u0018K=n 0, which is much longer than\n1=n0. Thus, in order to describe the density profile of bosons\none needs to go beyond the harmonic Tomonaga-Luttinger liq-\nuid theory.\nIn this paper we study weakly-interacting bosons in a semi-\ninfinite system. We consider the equation of motion for the\nfield operator. At the mean-field (classical) level it reduces\nto the Gross-Pitaevskii equation [13]. Its solution reveals that\nthe spatial extent of the depletion of the boson density im-\nposed by the boundary is controlled by the healing length \u0018.\nAt distances longer than \u0018, the mean-field density exponen-\ntially rapidly reaches the constant value n0. Accounting for\nthe effect of quantum fluctuations around the mean-field so-\nlution within the Bogoliubov-de Gennes formalism, we find\nthat the density reaches n0much slower, following 1=x2law.Our approach enables us to obtain an analytic expression for\nthe density profile at all distances.\nThe nonuniform density profile of the Bose gas determines\nthe potential for a weakly-coupled quantum impurity intro-\nduced in the system. For repulsive interaction between the\nimpurity and the bosons, the impurity can be localized. We\nsolve the Schr ¨odinger equation and characterize the impurity\nby the energy spectrum of the bound states, their wave func-\ntions, and the mean position. We find the condition for the ap-\npearance of the bound states, which is a threshold for a single\ndimensionless parameter that involves the masses of the im-\npurity and of the particles of the Bose gas as well as the inter-\naction strengths. We note that a related phenomenon of self-\ntrapping, i.e., the localization of a single impurity in homo-\ngeneous Bose-Einstein condensates was studied in Refs. [14–\n19]. Contrary to the case of the self-trapping where a bound\nstate is created due to the significant distortion of the density\nof the host system due to the coupling with the impurity, in\nour case it is the boundary that critically modifies the density.\nOther related phenomena include the localization of bosonic\natoms by fermionic ones in attractive Bose-Fermi mixtures ex-\namined in Ref. [20]. The formation of bounds states of two\nimpurities immersed in one-dimensional liquid has been stud-\nied recently in Refs. [21–23], while the higher-dimensional\ncases are studied, e.g., in Ref. [24–26]. We eventually men-\ntion the study of a ionic impurity in a condensate [27].\nThe paper is organized as follows. In Sec. II we introduce\nthe model of interacting bosons in a semi-infinite geometry. In\nSec. III we solve the equation of motion for the single-particle\nfield operator. We find the mean-field solution and the first\ntwo quantum corrections. This enables us to evaluate the den-\nsity of bosons, including its quantum contribution, in Sec. IV.\nIn Sec. V we consider the problem of a mobile quantum im-\npurity interacting with the system. We solve the Schr ¨odinger\nequation for the impurity and study its properties. Section VI\nis devoted to discussions, while in Sec. VII we summarize our\nresults. In Appendix A we present a simplified procedure that\nleads to the density in the regime of large separations from the\nboundary.arXiv:2007.10771v2  [cond-mat.quant-gas]  22 Oct 20202\nII. THE SEMI-INFINITE BOSE GAS\nWe are interested in the influence of the boundary at x=\n0on the physical quantities in a one-dimensional system of\ninteracting bosons. We thus study a long system described by\nthe Hamiltonian\nH=ZL\n0dx\u0012\n\u0000^\ty~2@2\nx\n2m^\t +g\n2^\ty^\ty^\t^\t\u0013\n: (1)\nHeremis the mass of bosons, while the coupling constant g>\n0describes the repulsive contact interaction between them.\nThe system size is L; however we will eventually consider the\nthermodynamic limit. The bosonic single particle operators\n^\tand^\tysatisfy the usual equal time commutation relations\n[^\t(x;t);^\ty(x0;t)] =\u000e(x\u0000x0), while the other commutators\nvanish. The model (1) is characterized by the dimensionless\nparameter [28] \r=mg=~2n0, wheren0is the mean particle\ndensity. The boundary of the system imposes the nullification\nof the single-particle operator at its position,\n^\t(x= 0;t) = 0: (2)\nThe condition (2) implies that the boson density also vanishes\natx= 0.\nOur first goal in this work is to calculate the density profile\nof the bosons\nn(x) =h^\ty(x)^\t(x)i; (3)\nwhere the average is with respect to the ground state of the\nHamiltonian (1). For simplicity, we introduce the dimension-\nless coordinates for the position and the time, respectively,\ndefined as\nX=x\n\u0018\u0016; T =t\n~=\u0016: (4)\nHere\u0018\u0016=~=pm\u0016denotes the healing length, while \u0016is the\nchemical potential. Assuming the single particle operator in\nthe form\n^\t(x;t) =r\u0016\ng^ (X;T)e\u0000iT; (5)\nits equation of motion i~@t^\t = [ ^\t;H]in the dimensionless\nunits becomes\ni@T^ (X;T) =\u0014\n\u0000@2\nX\n2\u00001 +^ y(X;T)^ (X;T)\u0015\n^ (X;T):\n(6)\nWe will solve Eq. (6) at \r\u001c1, corresponding to the regime\nof weak interaction. In this case, one can expand the field\noperator as [13, 29]\n^ (X;T) = 0(X) +\u000b^ 1(X;T) +\u000b2^ 2(X;T) +:::; (7)\nsuch that [^ (X;T);^ y(X0;T)] =\u000b2\u000e(X\u0000X0). Here the\nsmall parameter is \u000b= (\rgn 0=\u0016)1=4\u0019\r1=4\u001c1. In thelatter estimate we used the expression \u0016=gn0, which is valid\nat weak interaction, \r\u001c1.\nThe function  0(X)describes the time-independent wave\nfunction of the system in the absence of fluctuations. The field\noperators ^ 1and^ 2account for its first and the second quan-\ntum correction. Note that the Bose-Einstein condensate does\nnot exist in one dimension in the thermodynamic limit due\nto strong effect of long-wavelength fluctuations. However,\nin finite-size systems, the inverse system size provides an in-\nfrared cutoff. The perturbative expansion (7) is justified as\nlong as [29, 30] ln(L=\u0018\u0016)\u001c1=p\rwhereLis the length of\nthe system. The latter inequality shows that for weakly inter-\naction bosons [i.e., at \r\u001c1] the system size can actually be\nhuge. We point out that the density n(x)[see Eq. (3)] which\nis to be calculated, is well defined (i.e., cutoff–independent)\nin the thermodynamic limit [31].\nIII. SOLUTION OF THE EQUATION OF MOTION\nA. Wave function in the absence of fluctuations\nThe form (7) substituted into Eq. (6) leads to a hierarchy\nof equations controlled by the small parameter \u000b\u001c1. The\nequation of motion for  0(X)is obtained at order \u000b0. It reads\nbL1(X) 0(X) = 0; (8)\nwhere we have introduced the operator\nbLj(X) =\u0000@2\nX\n2+jj 0(X)j2\u00001: (9)\nThe expression (8) is known as the Gross-Pitaevskii equation\n[13]. This second order differential equation should be sup-\nplemented by two boundary conditions. One of them follows\nfrom Eq. (2) and becomes  0(0) = 0 . The other condition\narises from the physical requirement that the density, which\nis proportional toj 0(X)j2, is unaffected by the boundary at\nlong separations from it and thus becomes a constant. The real\nsolution [32] of Eq. (8) satisfying such boundary conditions is\n 0(X) = tanhX: (10)\nAt weak interaction the chemical potential of the Bose gas is\n\u0016=gn0. This gives the density (3) at the mean-field level to\nbe\nn(x) =n0tanh2(x=\u0018); \u0018 = 1=n0p\r: (11)\nThis expression shows that the density quickly saturates at dis-\ntances beyond the healing length \u0018(which is\u0018\u0016in the limit\n\r\u001c1), provided one takes only the leading order term from\nthe expansion (7).3\nB. First quantum correction\nWe now consider the effects of quantum fluctuations and\ndetermine ^ 1. Its equation of motion is\ni@T^ 1(X;T) =bL2(X)^ 1(X;T) + 0(X)2^ y\n1(X;T):\n(12)\nWe seek a solution for ^ 1(X;T)as a superposition of excita-\ntions of energy \u000fkusing the ansatz based on Bogoliubov trans-\nformation [13]\n^ 1(X;T) =X\nk>0Nkh\nuk(X)^bke\u0000i\u000fkT\u0000v\u0003\nk(X)^by\nkei\u000fkTi\n:\n(13)\nSince Eq. (12) is linear and homogeneous, we must account\nfor the normalization factor Nkin Eq. (13), which should be\ndetermined in such a way to satisfy the proper commutation\nrelations between ^ 1fields. This is discussed further below.\nThe bosonic operators ^bkand^by\nksatisfy the standard commu-\ntation relations [^bk;^by\nk0] =\u000ek;k0and[^bk;^bk0] = 0 . The bound-\nary condition (2) at order \u000bis given in terms of ukandvk\nby\nuk(0) =vk(0) = 0: (14)\nSubstitution of Eq. (13) into Eq. (12) leads to two coupled\nequations for ukandvk. They are known as the Bogoliubov-\nde Gennes equations and are given by\n\u000fkuk(X) =bL2(X)uk(X)\u0000 2\n0(X)vk(X); (15a)\n\u0000\u000fkvk(X) =bL2(X)vk(X)\u0000 2\n0(X)uk(X): (15b)\nIn order to simplify them, we introduce the functions\nS(k;X) =uk(X)+vk(X)andD(k;X) =uk(X)\u0000vk(X).\nThis enables us to decouple the equations and lead to the\nfourth order differential equation\n\u000f2\nkS(k;X) =bL3(X)bL1(X)S(k;X): (16a)\nNote thatDis given in terms of Sas\nD(k;X) =1\n\u000fkbL1(X)S(k;X): (16b)\nFour independent solutions of the forth-order equations (16)\nare [33]\nSn(X) = (\u0000ikn+ 2 tanhX)eiknX; (17a)\nDn(X) =\u0000ikn\n\u000fkn1\ncosh2XeiknX+k2\nn\n2\u000fknSn(X);(17b)\nwheren2f1;2;3;4g, while\u000fk=p\nk2+k4=4is the energy\ndispersion. The four roots entering Snarek1;2=\u0006kand\nk3;4=\u0006ip\n4 +k2in terms ofk=p\n2qp\n\u000f2\nk+ 1\u00001. The\ngeneral solution of Eqs. (16) is a linear combination\nS(k;X) =AS1(k;X) +S2(k;X) +BS3(k;X);(18a)\nD(k;X) =AD1(k;X) +D2(k;X) +BD3(k;X):(18b)Note thatS4andD4do not appear since they diverge at large\nX. The two unknown coefficient AandBare determined\nusing the boundary condition (14) that is terms of SandD\nbecomesS(k;0) =D(k;0) = 0 . We findA= 1 andB=\n0. Therefore the general solution of Eqs. (15) that satisfy the\nboundary condition (14) are the real functions\nuk(X) =k\u0012\n1 +k2+ 2 cosh\u00002X\n2\u000fk\u0013\nsin(kX)\n+ 2\u0012\n1 +k2\n2\u000fk\u0013\ncos(kX) tanhX; (19a)\nvk(X) =k\u0012\n1\u0000k2+ 2 cosh\u00002X\n2\u000fk\u0013\nsin(kX)\n+ 2\u0012\n1\u0000k2\n2\u000fk\u0013\ncos(kX) tanhX: (19b)\nThe normalization in Eq. (13) is obtained by requiring [13]\nNkNqZL\n\u0018\u0016\n0dX[uk(X)uq(X)\u0000vk(X)vq(X)] =\u000ek;q:\n(20)\nThis leads to Nk= (\u0018\u0016=4L\u000fk)1=2atL\u001d\u0018\u0016. One can then\nverify the equal time commutation relation [^ 1(X);^ 1(Y)] =\n0. The evaluation of the other commutation relation is more\ninvolved:\n[^ 1(X);^ y\n1(Y)] =X\nk>0N2\nk[uk(X)uk(Y)\u0000vk(X)vk(Y)]\n=\u0018\u0016\nLX\nk>0[cos (k(X\u0000Y))\u0000cos(k(X+Y))]\n=\u000e(X\u0000Y); (21)\nsince\u000e(X+Y)always equals zero for X;Y > 0. The second\nequality in Eq. (21) is obtained after performing the integral\noverkof the function [uk(X)uk(Y)\u0000vk(X)vk(Y)]=4\u000fk\u0000\n2 sin(kX) sin(kY), which turns out to be zero. The remaining\npart after integration over kthen gives the delta function in\nEq. (21). There we useP\nk>0(\u0001\u0001\u0001) = (L=\u0019\u0018\u0016)R1\n0dk(\u0001\u0001\u0001).\nC. Second quantum correction\nThe first quantum contribution to the density (3) is propor-\ntional to\u000b2and thus is determined by the first two correc-\ntions of the field operator in Eq. (7). We thus now consider\nthe second quantum correction to the field operator ^ denoted\nby^ 2. Its equation of motion is obtained from Eq. (6) at or-\nder\u000b2. Since 0(X)is real [see Eq. (10)], it is sufficient\nto consider the real part of the expectation value h^ 2iwhich\nenters inton(x). For this purpose we introduce the notation\n 2=Reh^ 2i. The equation of motion for  2is\nbL3(X) 2(X) =f(X); (22)4\nwhere\nf(X) =\u00002 0h y\n1 1i\u0000 0h 2\n1i: (23)\nAt zero temperature one has\nh y\n1 1i=X\nk>\u0015N2\nkv2\nk;h 2\n1i=\u0000X\nk>\u0015N2\nkukvk: (24)\nNote that the source term f(X)in Eq. (22) is time-\nindependent and thus  2is only a function of X. We note\nthat evaluation of f(X)requires a small- kcutoff (\u0015\u0018\u0018\u0016=L)\nsince it is divergent at k!0.\nThe solution of the linear equation (22) can be expressed as\n 2(X) =Z1\n0dYG(X;Y )f(Y); (25)\nwhereGis the Green’s function of the operator bL3(X). It\nsatisfies\nbL3(X)G(X;Y ) =\u000e(X\u0000Y): (26)\nMoreover, the Green’s function is symmetric G(X;Y ) =\nG(Y;X)and satisfies\nG(Y+;Y) =G(Y\u0000;Y); (27a)\n@XG(X=Y+;Y)\u0000@XG(X=Y\u0000;Y) =\u00002;(27b)\nto account for the \u000e(X\u0000Y)on the right hand side of Eq. (26).\nHereY\u0006= lim\u000e!0+Y\u0006\u000e. Also, the Green’s function obeys\nG(0;Y) = 0 due to the boundary condition imposed by the\nend of the system, i.e.,  2(0) = 0 . For the solution of Eq. (26)\nwe find\nG(X;Y ) =g(Y)\u0012(X\u0000Y) +g(X)\u0012(Y\u0000X)\ncosh2Xcosh2Y;(28a)\nwhere\ng(Y) =12Y+ 8 sinh(2Y) + sinh(4Y)\n16: (28b)Therefore, the solution of Eq. (22) for  2(X)is obtained from\nEq. (25), where one should substitute the Green’s function and\nthe source function (23). Since the latter depends on the in-\nfrared cutoff,  2has also this feature. In Appendix A we de-\nrive 2(X)in the regime X\u001d1.\nIV . LOCAL DENSITY\nHaving solved the equation of motion for the terms  0,^ 1,\nand^ 2of the single-particle operator (7), we are now in posi-\ntion to evaluate the spatial density profile of the semi-infinite\nBose gas. It is given by the expansion\nn(x) =\u0016\ngh\nn(0)(X) +\u000b2n(1)(X) +O(\u000b3)i\f\f\f\f\nX=x\n\u0018\u0016:(29)\nHere the mean-field contribution is\nn(0)(X) =j 0(X)j2= tanh2X; (30)\nwhile the quantum contribution to the density has the form\nn(1)(X) =h^ y\n1(X;T)^ 1(X;T)i+ 2 0(X) 2(X);(31)\nor more explicitly\nn(1)(X) =Z1\n0dk\n4\u0019\u000fk\u001a\nvk(X)2\u00002 tanhX\n\u0002Z1\n0dYG(X;Y )vk(Y) [2vk(Y)\u0000uk(Y)] tanhY\u001b\n:\n(32)\nIn Eq. (32) one should use Eqs. (19) and (28), while we re-\ncall\u000fk=p\nk2+k4=4. Unlike 2that requires an infrared\ncutoff, the density contribution (32) does not. Each of the two\nterms in Eq. (31) is divergent, however their sum is finite. This\nshould be the case since the density fluctuations are expected\nto be finite unlike the fluctuations of the phase of the single\nparticle operator [34].\nThe evaluation of Eq. (32) is rather tedious. We first per-\nform the integration over Yusing various trigonometric iden-\ntities and the integration by parts. As a result, we obtain\na cumbersome expression involving several hypergeometric\nfunctions. Such expression is an integrable function of k.\nTo perform the integration we split the integrand into several\nsummands. We should stress that, unlike the whole integrand,\nthe summands can be nonintegrable due to singularities and\none should regularize them by adding and subtracting some\nfunctions. As a result of such procedure, we obtain the local\ndensity that can be expressed in the form\nn(x) =\u0016\ng\u001a\ntanh2(x=\u0018\u0016) +\u000b2\u0014tanh2(x=\u0018\u0016)\n\u0019+(x=\u0018\u0016) tanh(x=\u0018\u0016)\n\u0019cosh2(x=\u0018\u0016)+h(x=\u0018\u0016)\u0015\u001b\n; (33)5\nwhere\u0018\u0016=~=pm\u0016and\nh(z) =2\n\u0019\u00003\u0000ztanhz\n\u0019cosh2z+\u0019z(2\u0000cosh 2z)\n8 cosh4z\u00148z\n\u001922F3\u0000\n1;1; 1=2;3=2;2; 4z2\u0001\n\u0000I1(4z)L0(4z) +I0(4z)L1(4z)\u0015\n+ tanhz(1\u00004ztanhz) [I0(4z)\u0000L0(4z)]\u00001\n2[1\u0000ztanhz(3 + 5 tanh2z)] [I1(4z)\u0000L\u00001(4z)]: (34)\nHere 2F3is the hypergeometric function while I\u0017(z)and\nL\u0017(z)are the modified Bessel function of the first kind and\nthe modified Struve function, respectively.\nThe density (33) is obtained for a fixed value of the chem-\nical potential. We eliminate \u0016by using the mean density\nn0that is given by n0=RL\n0dxn(x)=L. In the thermo-\ndynamic limit, the mean density of the Bose gas is n0=\n(\u0016=g)(1 +\u000b2=\u0019+:::). Inverting this relation, one obtains\nthe chemical potential as a function of the mean density [28],\n\u0016=gn0(1\u0000p\r=\u0019+:::), where\r=mg=~2n0\u001c1. Sub-\nstituting\u0016as a function of n0in Eq. (29) yields the density\nprofile of the weakly-interacting semi-infinite Bose gas\nn(x) =n0\u0002\ntanh2(x=\u0018) +p\rh(x=\u0018)\u0003\n; (35)\nwhere\u0018= 1=n0p\r, whilehis given by Eq. (34).\nWe now analyze the behavior of the local density (35). At\nshort separations, x\u001c\u0018, the quantum fluctuation contribu-\ntions to the density can be neglected and the leading contribu-\ntion is given by\nn(x) =n0tanh2(x=\u0018): (36)\nThis classical result originates from the mean-field contribu-\ntion (30) to the density. At longer separations, x\u001d\u0018, the\nclassical result (36) quickly saturates to a constant n0. How-\never, the approach of the local density toward n0is actually\nmuch slower than one obtains from the mean-field result (36).\nAtx\u001d\u0018the quantum contribution (31) is the dominant term\nin the density deviation n(x)\u0000n0since it decays algebraically\nwith the distance. This follows from the asymptotic expansion\nh(z) =\u00001\n16\u0019z2\u0014\n1 +1\nz+15\n16z2+O(z\u00003)\u0015\n: (37)\nTherefore, at x\u001d\u0018the leading correction to the local density\nis given by\nn(x) =n0\u0012\n1\u0000p\r\n16\u0019\u00182\nx2\u0013\n: (38)\nThe crossover distance xcbetween the two regimes can be\nestimated by equating the two expressions (36) and (38). As a\nresult we get\nxc\u0019\u0018ln\u00128p\u0019\n\r1=4\u0013\n: (39)\nNote that the crossover distance depends only logarithmically\non\rand practically is on the order of few \u0018. This is illustrated\nin Fig. 1 where the density profile of the Bose gas is shown as\nFigure 1. Density deviation from the mean density [n0\u0000n(x)]=n0\nof the Bose gas for three values of the dimensionless interaction pa-\nrameter\r. The dashed curve denotes the mean-field result.\na function of the distance from the end of the system at several\nvalues of\r. The density in the crossover regime between the\nclassical and the quantum one is given by Eq. (35). A simpli-\nfied derivation of n(1)(X)atX\u001d1is given in Appendix A.\nWe finally notice that the ground-state energy calculation of\nRef. [35] leads to the limiting forms (36) and (38).\nV . INTERACTION-INDUCED LOCALIZATION OF A\nSINGLE IMPURITY\nIn this section we study the problem of a quantum impurity\nin the semi-infinite Bose gas. We consider the impurity that\nis locally coupled to the boson density (35). In the case of re-\npulsion, the density profile of the Bose gas forms an attractive\npotential for the impurity, which can lead to the bound states\nin the spectrum. In that case the particle becomes localized by\nthe surrounding medium.\nIn the case of a weakly coupled impurity, one can find\nits wave function in the approximation of unperturbed Bose\ngas density (as further discussed in Sec. VI). The impurity\nwave function is a function of xanttand is governed by the\nSchr ¨odinger equation. After separation of variables it reduces\nto the eigenvalue problem on the positive semi-axis x>0\n\u0014\n\u0000~2\n2Md2\ndx2+Gn(x)\u0000E\u0015\n imp(x) = 0; (40a)\n imp(0) = 0: (40b)6\nHereMis the impurity mass, G > 0denotes the coupling\nconstant, while the density of the Bose gas n(x)is given by\nEq. (35). An alternative formulation of the eigenvalue prob-\nlem (40) is to consider unrestricted xwith the symmetric po-\ntentialGn(jxj), and to account only for odd eigenfunctions.\nThey have a node at x= 0and thus automatically satisfy the\nboundary condition (40b).\nIn order to simplify the notations, we rewrite Eq. (40) in the\nform\n\u0014\n\u0000d2\ndz2\u0000\u0015(\u0015\u00001)\ncosh2z+p\r\u0015(\u0015\u00001)h(z) +{2\u0015\nf(z) = 0;\n(41a)\nf(0) = 0; (41b)\nwhere we have introduced\nf(z) = imp(z\u0018); (42)\n\u0015=1\n2+1\n2s\n1 +8GM\ngm; (43)\n{2=2GM\ngm\u0012\n1\u0000E\nGn0\u0013\n: (44)\nIn the following we will find the bound state spectrum for the\neigenvalue problem (41).\nA. Bound states of P ¨oschl-Teller potential\nIn the limit where the quantum correction to the density\nh(z)is neglected, the potential of Eq. (41a) describes a hole\nof modified P ¨oschl-Teller type which admits an exact solu-\ntion [36, 37]. We now derive the bound states for this special\neigenproblem defined by\n\u0014\n\u0000d2\ndz2\u0000\u0015(\u0015\u00001)\ncosh2z+{2\u0015\nf(z) = 0; (45a)\nf(0) = 0: (45b)\nFrom the definition (43) follows \u0015>1and thus the potential\n\u0000\u0015(\u0015\u00001)=cosh2zin Eq. (45a) is negative. We also notice\nthat for unrestricted z, the eigenstates in such symmetric po-\ntential can be even or odd functions of z. For the problem (45)\nonly odd solutions have a node at z= 0. They are of the form\n[37]\nf(z) = cosh\u0015zsinhz\n\u00022F1\u0012\u0015\u0000{+ 1\n2;\u0015+{+ 1\n2;3\n2;\u0000sinh2z\u0013\n;\n(46)\nwhere 2F1is the Gauss’s hypergeometric function.\nIn order to discuss the appearance of bound states, we con-\nsider the asymptotic expansion of Eq. (46) in the limit z!1 ,\nwhich is given by\nf(z)'2\u0000{\u0000({)\u0000(3=2)e{z\n\u0000\u0000\u0015+{+1\n2\u0001\n\u0000\u00002\u0000\u0015+{\n2\u0001+2{\u0000(\u0000{)\u0000(3=2)e\u0000{z\n\u0000\u0000\u0015\u0000{+1\n2\u0001\n\u0000\u00002\u0000\u0015\u0000{\n2\u0001:\n(47)We notice that there are two exponential terms /e\u0006{zat\nz! 1 . Taking into account the definition of {given by\nEq. (44), we conclude that a bound state can be realized only\nif{2>0, i.e., ifGn0>E . In the opposite case, Gn0<E ,\nthe solution at long separations from the boundary is an oscil-\nlating trigonometric function since {is purely imaginary. In\nthe following we focus on the localized (bound) states and de-\nrive their energy spectrum. We assume without loss of gener-\nality that {>0. Thus, the coefficient in front of the term e{z\nmust vanish in order to obtain a nondivergent (normalizable)\nwave function at large z. This occurs when the arguments of\nthe Gamma functions in the denominators are 0 or negative\nintegers. Since \u0015 > 1and by assumption {>0, only the\nargument of \u0000\u00002\u0000\u0015+{\n2\u0001\nmatters. This yields\n{=\u0015\u00002\u00002\u0011; (48)\nwhere\u0011= 0;1;2;:::. From the condition {>0we obtain\nthat integer\u0011satisfies\n0\u0014\u0011<\u0015\n2\u00001: (49)\nEquation (49) determines the condition \u0015 > 2necessary for\nthe appearance of the first bound state [38]. The number \u0011de-\nnotes the number of nodes of the odd bound-state eigenfunc-\ntion in the region of interest z >0. Thus, the localized solu-\ntions of the eigenvalue problem Eq. (45) are given by Eq. (46)\nwhere {satisfies Eq. (48), while \u0011is a nonnegative integer\nsatisfying the condition (49). The latter enables us to reex-\npress the eigenfunction (46) in the form\nf\u0011(z) =\u0011! \u0000(\u0015\u0000\u0011\u00001)sinhz\ncosh\u0015\u00001z\n\u0002\u0011X\nj=0(\u00004)jsinh2jz\n(2j+ 1)! (\u0011\u0000j)! \u0000(\u0015\u0000\u0011\u00001\u0000j)(50)\nusing 2F1(c\u0000a;c\u0000b;c;z) = (1\u0000z)a+b\u0000c2F1(a;b;c;z).\nThe preceding expressions enable us to find the spectrum\nof bound states of the impurity defined by Eq. (40) in the case\nwhen the quantum fluctuations of the density are neglected.\nThe eigenfunctions are  imp;\u0011(x) =f\u0011(x=\u0018)[see Eq. (50)],\nwhere one should substitute Eq. (43) for \u0015. The corresponding\nenergies [cf. Eq. (48)] are\nE\u0011=gn0m\nM\"\u00123\n4+\u0011\u0013s\n1 +8GM\ngm\u00005\n4\u00003\u0011\u00002\u00112#\n;\n(51)\nprovided the system has bound states. The condition for that\nis sufficiently strong coupling between the impurity and the\nBose gas, which should satisfy\nGM\ngm>(\u0011+ 1)(2\u0011+ 1): (52)\nThe latter follows from Eq. (49). The set of allowed values for\n\u0011(non-negative integers) in the spectrum (51) are determined7\nby the inequality (52). In particular, the first bound state (cor-\nresponding to \u0011= 0) occurs for GM=gm > 1. The second\none (corresponding to \u0011= 1) exists forGM=gm > 6, etc.\nWe introduce an important quantity\nEb\n\u0011=Gn0\u0000E\u0011; (53)\nwhich denotes the binding energy, Eb\n\u0011>0.\nIn Fig. 2 we show the bound state energies for the first five\nlevels. We notice that the specific bound state energy ex-\npressed in units of Gn0, which is the value of the potential\nat infinity, is a function of the single parameter GM=gm . In\nFig. 3 we show the normalized wave functions for the bound\nstate levels for the specific value GM=gm = 30 . We notice\nthat the wave functions for energies near the top of the po-\ntential,Gn0, are weakly localized as their spatial extension\nincreases. The level (quantum) number \u0011corresponds to the\nnumber of nodes of the wave function. For a given wave func-\ntion, its spatial extension decreases with increasing GM=gm ,\nsince in that case the potential becomes deeper. This is re-\nflected in the mean distance of the particle from the origin,\nhxi\u0011=\u0018R1\n0dxxjf\u0011(x)j2\nR1\n0dxjf\u0011(x)j2: (54)\nWe plot this dependence in Fig. 4. Close to the threshold for\nthe appearance of the bound state, the mean distance diverges\naccording to the law\nhxi\u0011\n\u0018=1\n2{=\u0011+3\n4\nGM\ngm\u0000(\u0011+ 1)(2\u0011+ 1)+O(1): (55)\nThe denominator of Eq. (55) denotes the distance from the\nthreshold for the appearance of the bound state f\u0011measured\nfrom the “localized” side where the particle is in the bound\nstate. Equation (55) can be also expressed as function of the\nbinding energyhxi\u0011=\u0018=q\nm\u0016=8MEb\u0011using Eqs. (44) and\n(53). Indeed,hxi\u0011determines the localization length. This\neasily follows from the asymptotic expansion of the wave\nfunctionf\u0011(z)\u0018exp (\u0000{z)[cf. Eq. (47)], where the local-\nization length is 1={close to the “transition”. The localiza-\ntion length diverges with the first power of the distance from\nthe “transition”, which physically denotes the disappearance\nof the bound state as GM=gm is decreased (and the bind-\ning energy approaches zero). More generally, the universal\nform of the wave function enables us to find any moment as\nhxji\u0011=\u0018j=j!=(2{)jclose to the threshold.\nIn the preceding part we considered a localized impurity,\ni.e., its bound states which exist when the eigenenergies are\nsmaller than Gn0(corresponding to the value of the po-\ntential at infinity). We should stress that the impurity has\na continuum of scattering states with energies higher than\nGn0. They are characterized by the oscillating wave functions\n[cf. Eq. (47)].\nB. Quantum correction to the potential\nIn the previous subsection we have solved the eigenproblem\nfor the impurity using the mean-field contribution to the po-\nFigure 2. The first five bound state energies E\u0011=Gn 0as a function\nof the dimensionless parameter GM=gm . They occur, respectively,\nat ratiosGM=gm = 1;6;15;28;45. Notice that the energies start\nfromGn0, which is the value of the potential at infinity.\nFigure 3. Normalized wave functions for the first four states at\nGM=gm = 30 . Notice that the wave function corresponding to\n\u0011= 3 is spatially less localized since the value of GM=gm is close\nto the threshold for its appearance.\nFigure 4. Mean distance hxi\u0011=\u0018of the impurity from the origin for\nthe first four bound states as a function of GM=gm . At the threshold\nvalue for a given \u0011, the mean distance diverges according to Eq. (55),\nsignaling the disappearance of the bound state.8\nFigure 5. Quantum correction to the bound state energies\n\u000eE\u0011=p\rGn 0for different \u0011as a function of GM=gm for the first\nfive levels.\ntential. We now account for the quantum correction h(z), see\nEq. (41). We were not able to solve analytically the whole\neigenproblem, since the form of h(z)is very complicated.\nHowever we can take advantage of the small parameterp\r\nthat controls it and find the corrections of the bound state en-\nergy (51) using perturbation theory. The energy correction is\ngiven by\n\u000eE\u0011=p\rGn 0R1\n0dxjf\u0011(x)j2h(x)R1\n0dxjf\u0011(x)j2; (56)\nwherehis defined in Eq. (34), while f\u0011by Eq. (50). The\nquantum correctionp\rGn 0h(x=\u0018)to the effective potential\n[see, e.g., Eq. (40a)] becomes practically important at dis-\ntancesx\u001d\u0018(see Fig. 1) with respect to the classical one\n\u0000Gn0=cosh2(x=\u0018). Contrary to the function h(x=\u0018)that has\na long-range tail/\u00001=x2, the wave function is localized and\nexponentially small at x\u001d\u0018. As a result the numerator in\nEq. (56) is small thus the quantum correction to the bound\nstate energy is negligible. Notice that very near the threshold\nfor the bound state appearance (see, e.g., the curve for \u0011= 3\nin Fig. 3) the wave function may have some overlap with the\ntail ofh(x=\u0018), and the correction \u000eE\u0011can be somewhat im-\nportant with respect to E\u0011\u0000Gn0. Nevertheless, \u000eE\u0011is nega-\ntive since the quantum correction to the potential broadens the\nwell. The quantum corrections to the different energy levels\nof the bound states are numerically evaluated using Eq. (56)\nand shown in Fig. 5 [39].\nVI. DISCUSSION\nWe have calculated the bound states of the impurity in\na semi-infinite Bose gas. We assumed that the density of\nthe Bose gas it not affected by the presence of the impurity\n[cf. Eq. (40)]. This approach is valid in cases of weak cou-\nplingGbetween the impurity and the particles of the Bose\ngas. A more general model would be the Hamiltonian (1) sup-plemented by the impurity part\nZL\n0dx\u0012\n\u0000^\timp~2@2\nx\n2M^\timp+G^\ty\nimp^\timp^\ty^\t\u0013\n:(57)\nFrom the solution of the whole problem one could obtain the\nspectrum of the bound states at any G. However this is in prac-\ntice difficult to do analytically. The parameter region where\nour approach applies can be obtained by calculating the cor-\nrection to the Bose gas density (35) due to the coupling with\nthe impurity. For a heavy impurity such correction is small at\nG\u001cg=p\r[35]. In this regime is therefore justified to study\nthe simplified problem (45). Notice that for weakly interact-\ning bosons we have \r\u001c1and thus our work covers a big\nrange of values for G.\nThe quantum contribution to the boson density that univer-\nsally behaves as 1=x2atx\u001d\u0018[see, e.g., Eq. (38)] leads\nto a small energy correction to the bound state levels of the\nlocalized particle. However, the knowledge of such spatial\ndependence of the density of particles far from the boundary\nis fundamentally important for quantities where the density\ngradient matters. An example is the Casimir-like interaction\n[31, 40–43] between the impurity and the boundary of the sys-\ntem mediated by density fluctuations of the medium. In the\nregimex\u001d\u0018its form was found in Ref. [35], while our\nresult (35) determines the interaction law at all distances. It\ntakes the form\nU(x) =G[n(x)\u0000n0]\n=\u0000Gn0\u00141\ncosh2(x=\u0018)\u0000p\rh(x=\u0018)\u0015\n: (58)\nIt is expected that under the influence of the long-range poten-\ntial (58), a heavy impurity immersed in the system far from\nthe boundary slowly drifts towards it and eventually get lo-\ncalized. During this process, the impurity energy that equals\nGn0at long distances is dissipated by exciting the Bose gas.\nThe boundary in our system can exist naturally as the sys-\ntem’s end. Alternatively, it can be created by a heavy impu-\nrity strongly coupled to the Bose gas that creates impenetra-\nble potential and leads to the complete depletion of the boson\ndensity at its position. Thus the effective interaction between\nthe boundary and the impurity can also be interpreted as the\nCasimir-like interaction between two very different impuri-\nties. We notice two related very recent experimenal works\n[44, 45], which have demonstrated the existence of the in-\nduced interaction between quantum particles mediated by the\nsurrounding quantum gas.\nThe Hamiltonian (1) can be studied within the dual model\nof attractive fermions [46, 47]. The density of bosons in the\noriginal model is equal to the density of fermions in the dual\none. Such mapping is particularly useful in the regime of\nstrong interaction between bosons, corresponding to \r\u001d1\nand the Luttinger liquid parameter K'1 + 4=\r. In this case\nthe fermions are weakly-attractive, which enables one to study\ntheir density, obtain the characteristic form with Friedel oscil-\nlations and go beyond the Tomonaga-Luttinger liquid descrip-\ntion. It is interesting to study the density and the fate of Fridel9\noscillations as the value of Kincreases and find a connec-\ntion with our result (35). A numerical study of finite number\nof bosons [48] also suggests the above picture. Note that the\nmodel (1) is integrable in the box geometry [5] and thus it is\nin principle possible to use the exact Bethe ansatz solution to\nfind the exact density profile in the thermodynamic limit and\nanswer the above question. We are not aware of such study in\nthe literature.\nRecent experimental realizations of cold bosons trapped in\na box potential [49–52] provide a possibility to observe our\nresults. In order to see the characteristic long-range tail of the\ndensity [cf. Eq. (38)], the system length has to be much longer\nthan the crossover distance (39). In addition, the constraint on\nthe temperature comes from the fact that the thermal length\n`T=~v=2\u0019Thas to be longer than the the crossover dis-\ntance (39). Here v=p\ngn0=mdenotes the sound velocity of\nweakly interacting bosons. Thus we obtain T=\u0016\u001clog (\r\u00001).\nConcerning the observation of the bound states, the condition\n(52) could be realized for a given mass ratio M=m by tuning\nthe interaction strengths using the Feshbach resonance. Tem-\nperature should be sufficiently small to avoid smearing of the\nenergy levels.\nVII. SUMMARY\nIn this paper we first calculated the density profile of the\nweakly-interacting semi-infinite one-dimensional Bose gas.\nWe solved the equation of motion for the field operator. At\nthe mean-field level, it leads to the density profile (11). Tak-\ning into account the effect of quantum fluctuations around the\nmean-field solution we calculated the quantum contribution to\nthe density, see Eqs. (34) and (35). It shows the universal 1=x2\nbehavior at long distances.\nWe then studied the spectrum of the bound states of a quan-\ntum impurity which experiences the potential created by the\nsurrounding Bose gas. We were able to exactly solve this\nproblem accounting for the mean-field boson density. The\ndiscrete spectrum is given by Eq. (51), while the correspond-\ning wave function are written in Eq. (50). Bound states exist\nfor sufficiently strong repulsion between the impurity and par-\nticles of the Bose gas, see Eq. (52). We also found how the\nmean particle distance from the boundary behaves, and in par-\nticular its divergence (55) as one approaches the threshold for\nthe appearance of the bound state levels. We showed that the\nquantum contribution to the density gives rise to the negligible\ncorrections of the bound state levels. However, the quantum\ncontribution to the density is important since it leads to the\nlong-range Casimir-like interaction between the impurity and\nthe boundary.\nACKNOWLEDGMENTS\nThis study has been partially supported through the EUR\ngrant NanoX ANR-17-EURE-0009 in the framework of the\n“Programme des Investissements d’Avenir”.Appendix A: The density contribution at long distances\nHere we present derivation of the quantum contribution to\nthe density at long separation from the boundary. We begin\nwith the expression for  2(X)that simplifies at X\u001d1. In\norder to obtain it we first notice that Eq. (25) leads to the first-\norder differential equation\nd 2(X)\ndX=\u00002 tanhX 2(X)\n+ 2 cosh2XZ1\nXdYf(Y)\ncosh2Y: (A1)\nIn the regime X\u001d1one can set tanhXto 1 and cosh\u00002X\nto 0 in Eqs. (19), such that the source term becomes\nf(Y) =\u0000X\nk>\u0015N2\nk\u0012\n1\u00002k2\n\u000fk+3k4\n4\u000f2\nk\u0013\n\u0002[ksin(kY) + 2 cos(kY)]2: (A2)\nOne can then perform the integration in Eq. (A1) for which\nit suffices to approximate cosh2Ybye2Y=4and solve an el-\nementary integral. Solving the differential equation we then\nfind\n 2(X) =\u00001\n4X\nk>\u0015N2\nk\u0012\n1\u00002k2\n\u000fk+3k4\n4\u000f2\nk\u0013\u0014\n4 +k2\n+4\u0000k2\n1 +k2cos(2kX) +4k\n1 +k2sin(2kX)\u0015\n(A3)\natX\u001d1(corresponding to x\u001d\u0018).\nThe quantum contribution to the density (31) can now be\nfound more easily using Eq. (A3). It leads to\nn(1)(X) =Z1\n0dk\n2\u0019\u001a\n1\u0000kp\n4 +k2+\u00145\n3(4 +k2)\n\u00002\n3(1 +k2)\u0000k\n(4 +k2)3=2\u0015\n\u0002\u0014\u0012\n1\u0000k2\n4\u0013\ncos(2kX) +ksin(2kX)\u0015\u001b\n:\n(A4)\nThe fist two terms in braces give a constant after the inte-\ngration. 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We\n\fnd that topologically nontrivial density wave states can induce stable Dzyaloshinskii-Moriya (DM)\ninteractions among the localized spins of the lattice in the presence of an external magnetic \feld,\nand we study the resulting spin models for both antiferromagnetic and ferromagnetic backgrounds.\nWhile the density wave state itself can contribute to the the thermal Hall e\u000bect, as shown by Li\n& Lee1(arXiv:1905.04248v3), symmetry considerations preclude the resulting spin excitations from\ninducing a further thermal Hall e\u000bect. We utilize a Holstein-Primako\u000b (HP) substitution about the\nclassical mean-\feld ground state to calculate the magnon dispersion for LSCO and \fnd that the\ndensity wave induces a weak dx2\u0000y2anisotropy; upon calculating the non-Abelian Berry curvature\nfor this magnon branch we show explicitly that the magnon contribution to \u0014xyis zero. Finally,\nwe calculate corrections to the magnetic ground state energy, spin canting angles, and the spin-\nwave dispersion due to the topological density wave for ferromagnetic backgrounds. We \fnd that\nterms linear in the HP bosons can a\u000bect the critical behavior, a point previously overlooked in the\nliterature.\nI. INTRODUCTION\nDespite concerted e\u000borts to illuminate the precise\nnature of the pseudogap phase of the cuprate high-\ntemperature superconductors2{5, it remains unclear\nwhich of a host of competing order parameters is re-\nsponsible for the interesting behavior of this phase. One\npromising candidate6is the`= 2 spin-singlet order, the\nd-density wave (DDW), which gives rise to a dx2\u0000y2gap\nand currents that alternate between adjacent plaquettes\non a square lattice. The relevance of this state is certainly\nbelievable given the proximity of the pseudogap phase\nto the antiferromagnetic Mott insulator at low doping,\nwhich doubles the Brillouin zone in the same way and is\nsusceptible to singlet pairing.\nThis density wave state of nonzero angular momen-\ntum belongs to a larger class of such states7, and it\nis worth exploring other, more exotic members of this\nclass related to the singlet DDW which maintain the\nkey characteristics necessary for relevance to the pseu-\ndogap phase. Such states are also of some interest due\nto their topological properties.8We focus on a mixed\ntriplet-singlet DDW order, which has generated inter-\nest recently due to promising transport calculations con-\nsistent with surprising physics found in the pseudogap\nphase of the cuprate superconductor La 2\u0000xSrxCuO 4and\nrelated compounds.1,9Namely, for nonzero hole doping,\nthe mixed triplet-singlet DDW state generates a nonva-\nnishing thermal Hall conductivity \u0014xy, and hosts hole\npockets on the reduced Brillouin zone boundaries consis-\ntent with Hall coe\u000ecient measurements.10,11\nAt the mean-\feld level a general density wave state\nmay be described by the Hamiltonian\nHDDW =X\nk;Qcy\nk+Q[\b\u0016\nQ(k)\u001c\u0016]ck+ h.c.; (1)\nwhere Qis the wave vector at which the density wave\ncondensation occurs; \u001c1;\u001c2;and\u001c3are the Pauli ma-trices; and \u001c0=1. This Hamiltonian can be thought\nof as arising from a mean-\feld decomposition of nearest\nneighbor electron-electron interaction terms in the most\ngeneral problem12{15in which the order parameter\nhcy\nk+Q;\u000bck;\fi= [\b\u0016\nQ(k)\u001c\u0016]\u000b\f (2)\nacquires a nonzero value for some nonzero Q. In our\nwork we assume that all terms which transform nontriv-\nially under rotations and translations are captured by\nthis mean-\feld decomposition.\nHere we consider a speci\fc example of Eq. (1), namely\nthe the triplet-singlet DDW wave8(denotedi\u001bdx2\u0000y2+\ndxy)\n\bi\nQ(k)/iW0Ni(coskx\u0000cosky)\n\b0\nQ(k)/\u00010sinkxsinky;(3)\nwhereNiis a unit vector pointing along the spin quanti-\nzation direction, i= 1;2;3, and Q= (\u0019=a;\u0019=a ). In real\nspace the Hamiltonian is written as\nHDDW =Ht+Hs (4)\nwith\nHt=iW0\n4X\ni;\u000b;\f(\u00001)m+n(N\u0001\u001b)\u000b\f\n\u0002[cy\ni+a^x;\u000bci;\f\u0000cy\ni+a^y;\u000bci;\f] + h.c.(5)\nand\nHs=\u00010\n4X\ni;\u000b;\f\u000e\u000b;\f(\u00001)m+n\n\u0002h\ncy\ni+a^x+a^y;\u000bci;\f\u0000cy\ni+a^x\u0000a^y;\u000bci;\fi\n+ h.c.(6)\nThe Hamiltonian H0+HDDW, describes a topological\nMott insulator7,8with a quantized spin Hall conduc-arXiv:2007.11719v1  [cond-mat.str-el]  23 Jul 20202\ntance; it is a variant of the singlet d-density wave model\nhypothesized6to explain the pseudogap phase of the\ncuprates. Unlike the singlet d-density state, however, the\nmixed triplet-singlet i\u001bdx2\u0000y2+dxy-density wave state\ndoes not inherently break time reversal symmetry, yet it\nretains most of the signatures of the singlet d-density\nwave state. For example, the i\u001bdx2\u0000y2+dxy-density\nwave wave state possesses similar hole pockets centered\non the Brillouin zone diagonals which are consistent with\nboth the measured Hall coe\u000ecient9and some aspects of\nquantum oscillation experiments16{18. Recently, second-\nharmonic generation experiments have suggested that\nan inversion symmetry breaking is responsible for large\nsecond harmonic generation signatures in YBa 2Cu3Oy19\nbut we note that this could be due to, in principle,\nthe quadrupole moment induced via a triplet d-density\nwave7.\nThis model was shown by Li & Lee to produce a\nnonzero thermal Hall e\u000bect{however, despite consider-\nable e\u000bort, we have not managed to exactly reproduce\ntheir plots using their parameter values and instead \fnd a\nthermal Hall e\u000bect which is an order of magnitude smaller\nfor nonzero temperatures shown here in Fig. 1. Details\nof the calculation are highlighted in the Appendix.\n0.1 0.2 0.3 0.4 0.5T\n-0.006-0.005-0.004-0.003-0.002-0.001κxy/T\nFIG. 1: Thermal Hall conductivity \u0014xy=Tas a function\nof temperature Tproduced by the triplet-singlet DDW state\nde\fned by Equation (3) with \u0001 0= 0:3t, magnetic \feld B=\n0:0075t=\u0016B,t0=\u00000:1t,W0= 2tat a doping of p= 0:06.\u0014xy\nis listed here in units of k2\nB=~, andTin units oft=kB.\nWe now ask ourselves, what e\u000bect does this density\nwave state have on the localized spins of the lattice? The\nmagnitude of the experimentally-measured thermal Hall\ne\u000bect exceeds the maximum possible contribution from\nthe density wave state alone by an order of magnitude; it\nis possible that magnetic excitations induced by the den-\nsity wave state could contribute further. In our work we\nassume that at nonzero doping the density wave will sur-\nvive in the presence of long range magnetic order1. The\ntriplet part of the density wave order parameter induces\na staggered spin current13on the lattice, and hence, for\nneighboring lattice points A and B, this intrinsic spin\ncurrent implies that there exists no center of inversion at\nany point C on the bond connecting A and B, therebyallowing an antisymmetric exchange among the localized\nspins.20,21These types of (staggered) antisymmetric ex-\nchanges have been considered in the literature22, but to\nour knowledge have never been considered in the context\nof being generated via the spin currents associated with\ndensity wave states.\nWe \fnd that the spin currents intrinsic to the triplet\n\ravored density wave states induce a Dzyaloshinskii-\nMoriya (DM) interaction between the underlying neigh-\nboring spins20, and we investigate the e\u000bect that this DM\ninteraction has on antiferromagnetic and ferromagnetic\nspin textures, using a Holstein-Primako\u000b transformation\nexpanded about the ground state. It has been previously\ndemonstrated22{24that certain DM interactions can lead\nto a thermal Hall e\u000bect. We \fnd that the particular DM\ninteraction induced by triplet-singlet DDW states can not\ncontribute to \u0014xy, which is consistent with speculations\non the nature of the neutral excitation responsible for the\nsizable thermal Hall conductivity seen in the cuprates.9\nThere are strong constaints and unique properties asso-\nciated with the DM vectors that are generated by triplet\ndensity waves. Because triplet density wave states break\nspin-rotational invariance the associated Goldstone bo-\nson excitations will destroy the two dimensional triplet\ndensity wave order at \fnite temperatures unless there is\nsome external mechanism which stabilizes the triplet den-\nsity wave order parameter like interlayer coupling. How-\never, we \fnd that when the underlying band structure\nis su\u000eciently topologically nontrivial insofar as it hosts\na nonzero spin Hall conductance, and an external mag-\nnetic \feld is turned on, the triplet density wave induced\nDM vectors are energetically stable in the absence of in-\nterlayer coupling. Furthermore, these DM vectors are\npinned to be collinear with the magnetic \feld, regard-\nless of its direction, and the DM interaction will have the\nsame symmetry as the form factor of the triplet density\nwave.\nIn the following we derive the DM coe\u000ecients induced\nby triplet density waves and investigate the e\u000bects they\nhave on the physics of the underlying spin textures of\nthe lattice. We \fnd that for a ferromagnetic background,\nthe ground state remains perfectly collinear below some\ncritical strength of the density wave; above the critical\nstrength, the ground state acquires a nonzero canting\nangle, and we show that linear boson terms shift the\nclassically-predicted threshold for nonzero canting angle.\nAssuming a small DMI the dispersion of the magnons\nin this case develop a characteristic dx2\u0000y2gap. For\nan antiferromagnetic background we \fnd that the den-\nsity wave-induced DMI has a small e\u000bect unless there\nis some weak ferromagnetism present. We compute the\nspin-wave spectrum approptiate to LSCO and \fnd that\nfor modest density wave strengths there exists a small\ndx2\u0000y2anisotropy.3\nII. THE EFFECTIVE MAGNETIC\nHAMILTONIAN\nFor any type of mixed triplet-singlet density wave con-\ndensation the mean-\feld Hamiltonian in the absence of\non-site repulsion or ferromagnetic coupling can be writ-\nten in the suggestive manner\nH=X\nijcy\ni\u000b(tij\u000e\u000b;\f+i\u0015ij\u0001\u001b)cj\f (7)\nwhere all singlet density wave terms are absorbed into\nthe de\fnition of tij, and \u0015ijare the triplet density wave\nterms which couple to \u001b. It can be shown20,25that\nthis\u0015ijinduces a DM interaction in the underlying\nspin structure (be it antiferromagnetic or ferromagnetic)\nwhose coe\u000ecients are given by\nDij=\u0015ijTr\u001bNji; (8)\nwhereNji\u0011hcy\nicji=\u00001\n\u0019REF\n\u00001ImGji(E)dE,Gji(E) is\nthe Green function de\fned by H, andEFis the Fermi\nenergy. Tracing the spin index over an expansion of\nGji(E) =f(E)tji=t+g(E)\u0015ji\u0001\u001b=t+O(\u00152=t) reveals\nthat to leading order Gji(E) should have the same sym-\nmetry astjiunder translations and rotations. In this\nwork we consider a speci\fc example of Eq. (1), written\nin real space as\nHDDW =H0+Ht+Hs (9)\nwhereH0is the tight binding Hamiltonian of the un-\nderlying crystal lattice which is some union of all square\nplanar lattices which host the triplet-singlet-DDW. For\nthis particular triplet-DDW case, because \u0015ijonly con-\nnects nearest neighbors, tjiis simply the tight-binding\nkinetic energy coe\u000ecient which we will assume to trans-\nform trivially under rotation{thus we write\nDij=\u000b\u0015ij (10)\nfor some constant \u000b. Because we will allow the density\nwave strength to be a tunable parameter we will hence-\nforth absorb \u000b, and all other constant numerical prefac-\ntors into the de\fnition of W0. The DM coe\u000ecients for\nthe tripletdx2\u0000y2-density wave therefore become\nDN\ni;i\u0006a^x= (\u00001)ix+iyW0\nDN\ni;i\u0006a^y=\u0000(\u00001)ix+iyW0;(11)\nwherei=ix+iy, and the superscript Ndenotes that\nthe DM vector points along the Ndirection. We stress\nthat the method implemented here can be applied, in\ngeneral, to triplet density waves in any angular momen-\ntum channel. The direction of the DM vector is along\nthe triplet quantization axis, and the form factor associ-\nated with the triplet density wave dictates the symmetry\nof the DM vector on the lattice. To understand the or-der of magnitude of the DMI induced by density waves\none can directly use Moriya's perturbation theory result\n(including the on-site repulsion U)26\nD\u00192tW0\nU; (12)\nwhich implies that a density wave-mediated DMI is\nroughly on the scale of 10-100 meV for LSCO at low\ndoping for density wave stengths W0\u0018t.\nFor a density wave-induced DM interaction to not be\ndisordered by Goldstone modes at \fnite temperatures\nthere must be some mechanism which externally stabi-\nlizes the triplet density wave's quantization axis, i.e. the\ndirection of N. Previously it was suggested that inter-\nlayer coupling was needed to stabilize the direction of\nN8, however it was recently shown1that the direction of\nNfor the triplet-singlet DDW in two dimensions can be\nstabilized by the bulk orbital magnetization's coupling to\nthe magnetic \feld. Explicitly, a magnetic \feld induces a\nbulk orbital magnetization, M, which is given by27\nM=\u0000X\n\u000b=N\u0001\u001b=\u00061e\nhcC\u000b\u0001EZ;\u000b; (13)\nwhereC\u000bis the Chern number of the band of spin \u000b,eis\nthe electron charge his Planck's constant, cis the speed\nof light, and \u0001 EZ;\u000bis the Zeeman splitting\n\u0001EZ;\u000b=\u0000\u000b\u0016Bsgn(W0)N\u0001B; (14)\nwhere\u0016Bis the Bohr magneton. For the case of the\ntriplet-singlet DDW the resulting energy density due to\nthe orbital magnetization-magnetic \feld is1\n\u0001EZeeman =\u0000\u0016BB2\n\u0019csgn(W0\u00010)(N\u0001^B); (15)\nwhich implies that it is energetically most favorable for\nW0\u00010NkB. Thus, for \u0001 0>0,B6= 0, Eq. 11 necessar-\nily becomes\nDB\ni;i\u0006a^x= (\u00001)ix+iyjW0j\nDB\ni;i\u0006a^y=\u0000(\u00001)ix+iyjW0j:(16)\nFrom this argument alone we see that stable density\nwave-induced DM interactions in two dimensions can\nonly arise from topological density waves with nonvan-\nishing spin Hall conductance{that is, given \u0001 EZ;+1=\n\u0000\u0001EZ;\u00001, stability is only guarenteed if C+1=\u0000C\u00001.\nFurthermore, because density wave-induced DM vectors\nmust be collinear with the magnetic \feld, they will trans-\nform like the magnetic \feld under rotations and time-\nreversal. This immediately implies that the correspond-\ning magnons in the problem will have no contribution to\nany thermal Hall e\u000bect because of the spin rotation and\ntime-reversal symmetry considerations if we utilize the4\nsymmetry arguments of23\n\u0014xy[J;Dij;B] =\u0014xy[J;R\u001e^BDij;R\u001eB]\n\u0014xy[J;Dij;B] =\u0000\u0014xy[J;\u0000^BDij;\u0000B];(17)\nwhereR\u001eis the vector representation of spin rotation\nby some angle \u001eabout the axis de\fned by ^\u001e. Rotating\nthe system about an angle \u0019about an axis perpendicu-\nlar to ^BmapsR\u001e^Bto\u0000^Band hence\u0014xy=\u0000\u0014xy= 0:\nThe bulk magnetization (Eq. 13) would, in principle,\nproduce a small ferromagnetic-like signal detectable in\npolar Kerr measurements so long as the external mag-\nnetic \feld is not exactly zero for weak disorder at small\nenough temperatures. More detailed calculations involv-\ning interlayer coupling, the inclusion of magnetic impu-\nrities, and nonzero temperatures should be considered in\nfuture work to quantitatively compare this triplet-singlet\nDDW bulk magnetization signal to the polar Kerr rota-\ntion data previously gathered28. Interesting questions to\nask are how the Goldstone modes would disorder the DM\nvectors in the absence of an external magnetic \feld, and\nhow the DM vectors behave for density wave states with\nvanishing spin Hall conductances.\nWe now study the e\u000bect of this dynamically generated\nDM interaction on the isotropic two dimensional Heisen-\nberg ferromagnet and antiferromagnet. Namely, we con-\nsider\nH=JX\ni;jSi\u0001Sj+X\ni;jDij\u0001(Si\u0002Sj)\u0000B\u0001X\niSi;(18)\nwhereJis the spin exchange, and the DM interaction\nincludes the contribution from the density wave.\nIII. THE ANTIFERROMAGNETIC\nBACKGROUND\nThe triplet-singlet density wave induced DMI will typ-\nically have little e\u000bect on perfectly antiferromagnetic\nbackgrounds in the linear spin wave approximation. This\nis because, upon turning on a magnetic \feld to stabilize\nthe DMI, the localized spins will \rop perpendicular to\nthe magnetic \feld direction. Terms which couple to W0\nin this case are proportional to higher order terms in the\nHP bosons. When this happens only very large density\nwave strengths will cause distortions in the magnetic or-\ndering. On the other hand, if there exists some small fer-\nromagnetic component associated with the otherwise an-\ntiferromagnetic ordering, the density wave-induced DMI\nappears in terms quadratic in the HP bosons and thus\nwill a\u000bect the magnon dispersion. This is the case for\nLSCO which we will consider in the following.\nTaking B=B^zthe DM matrix for LSCO can be writ-ten as Di= (\u00001)ix+iyD, where\nD=0\n@p\n2Dcos\u0012dp\n2Dsin\u0012dW0\n\u0000p\n2Dsin\u0012d\u0000p\n2Dcos\u0012d\u0000W0\n0 0 01\nA: (19)\nThexandyspin direction entries are due to the buckling\nof the oxygen atoms out of the copper oxide plane and\ninduce a weak net ferromagnetic moment out of the cop-\nper oxide plane.29{31We \fnd the mean-\feld ground state\nby summing the classical energy over the four sublattices\nand then numerically minimizing this energy with respect\nto the four sets of spherical angles which characterize the\nclassical spin directions. Here we parameterize the spins\nas\nhnii= (cos\u001ei;sin\u001ei;nz) (20)\nwherenzis the weak ferromagnetic canting due to D.\nFor density wave strengths on the order of tthe ground\nstate is unchanged{characterized by antiferromagnetic\nspin \ropping in the plane perpendicular to the magnetic\n\feld with a weakly ferromagnetic component pointing\nalongBinduced by D. Closely following the work of\nHan, Park, and Lee24, we choose the form of the classical\nground state as\nh\u0016nii=\u0000(\u00001)ix+iy\np\n2cos\u0012c(^x+ ^y) + sin\u0012c^z: (21)\nWe expand our spin operators about this mean-\feld\nground state32\nSi=aihnii+ti (22)\nso that we can perform the appropriate Holstein-\nPrimako\u000b (HP) substitution. The amplitudinal reduc-\ntion along the mean-\feld state given as\nai=S\u0000by\nibi; (23)\nand, to leading order in boson density operators, the\ntransverse \ructuation operator tiis given by\nti=tx0\ni^x0\ni+ty0\ni^y0\ni; (24)\nwith\ntx0\ni=r\nS\n2(by\ni+bi)\nty0\ni=ir\nS\n2(by\ni\u0000bi)(25)\nwhere the primed coordinates are de\fned such that\n^x0\ni\u0002^y0\ni=hnii. Upon substitution of these operators\ninto Eq. (16) the terms quadratic in boson creation and\nannihilation operators in the Hamiltonian can be written5\nin real space as (rede\fning the couplings to absorb S)\nH= (4J0+Bsin(\u0012c))X\niby\nibi\u0000X\nhi;jiJtx0\nitx0\nj+J0ty0\nity0\nj\n+X\niD0[tx0\nity0\ni+x+ty0\nitx0\ni+x\u0000tx0\nity0\ni+y+ty0\nitx0\ni+y]\n+iX\niW0(\u00001)ix+iy[biby\nj\u0000by\nibj];(26)\nwhereJ0=J=cos(2\u0012c) andD0= cos(\u0012c)D(cos\u0012d+\nsin\u0012d). We take the Fourier transformation and write\nour Hamiltonian in Nambu form as\nH=X\nk1\n2 y\nkHk k (27)\nwhere y\nk= (by\nk;by\nk+Q;b\u0000k;b\u0000k+Q) and\nHk=0\nB@Ak2iWksin\u0012cBk 0\n0\u0000Ak 0\u0000Bk\n0 0 Ak\u00002iWksin\u0012c\n0 0 0 \u0000Ak1\nCA+ h.c.;(28)\nwhere\nAk= 2J0+B\n2sin\u0012c+1\n2(J0\u0000J)(coskx+ cosky)\nBk=\u00002iD0(coskx\u0000cosky);\n\u0000(J0+J)(coskx+ cosky);\nWk=W0(coskx\u0000cosky):(29)\nThe spectrum is given by the absolute value of the eigen-\nvalues of the dynamic matrix K= (\u001b3\nI2)Hk33{ these\neigenvalues correspond to what are called particle and\nhole bands for the positive and negative eigenvalues re-\nspectively. We plot the dispersion in Fig. 2 for a set of\nparameter values. Taking into account both the particle\nand hole bands the dispersion consists of one four-fold\ndegenerate magnon branch very similar to that which\nis obtained when W0= 0; because W0couples to the\nweak ferromagnetic moment its e\u000bects on the spectrum\nare small if the density wave strength is not very large.\nNamely, tuning W0from zero induces a weak anisotropy\nin the the magnon dispersion along the kxandkydirec-\ntions. The dispersion along kd=kx=kyis unchanged\nby increasing W0whereas the energy is increased along\nthek= (kx;0),k= (0;ky) directions. This anisotropy is\ndemonstrated in Fig. 3, where the largest deviation from\nthe unperturbed magnon dispersion is roughly 0 :4 meV\nwhenk= (0;\u0019) andk= (\u0019;0).\n-3 -2 -1 1 2 3kx1234ΔωkFIG. 2: Magnon dispersion !kforky= 0 in units of Jfor\nW0= 0.3J,D= 0.1J,\u0012d= 0.05,B= 0:05J. The spectrum\nremains antiferromagnetic with small corrections increasing\nwith the value of kx.\n-3 -2 -1 1 2 3kx0.10.20.30.4Δωk\nFIG. 3: The di\u000berence in the magnon dispersions listed\nin meV with ky= 0,D= 12 meV, \u0012d= 0.05, and B= 6\nmeV. The orange, and blue curves correspond to the di\u000berence\nbetween the W0= 100 meV and W0= 0 dispersions and the\ndi\u000berence between the W0= 50 meV and W0= 0 dispersions\nrespectively.\nFIG. 4: Non-Abelian Berry curvature calculated on a\n200\u0002200 lattice with W0= 0.3J,D= 0.1J,\u0012d= 0.05,\nB= 0:05J.6\nThe thermal Hall conductivity is given by34\n\u0014xy\nT=\u0000X\nnk2\nB\n~Zdk\n(2\u0019)2\n\u0014\nc2(nB(!n;k))\u0000\u00192\n3\u0015\n\nz\nn(k)(30)\nwhere\nc2(x) =Zx\n0ds\u0012\nln\u00141 +s\ns\u0015\u00132\n; (31)\nnBis the Bose distribution function, \nz\nnis the Berry\ncurvature of the n'th band, and the sum is taken over\nthe particle bands. Because the particle band in this\nmodel is twofold degenerate we calculate its non-Abelian\nBerry curvature using the discretized link method.35We\nde\fne the Berry curvature as\n\nz(k) =\u0000iln[~U1(k)~U2(k+e1)~U1(k+e2)\u00001~U2(k)\u00001]\n(32)\nwhere the vectors e1= 2\u0019(1;0)=N,e2= 2\u0019(0;1)=N, and\nN2is the total number of lattice sites. The link variables\nare de\fned as\n~U\r(k) =detU\r(k)\njdetU\r(k)j(33)\nwhere the matrix entries of U\r(k) are the eigenstate over-\nlap elements in the degenerate subspace, which for the\nmagnon case take the form36\nU\r(k) =\u0012\nh 1(k)j~ 1(k+e\r)i h 1(k)j~ 2(k+e\r)i\nh 2(k)j~ 1(k+e\r)i h 2(k)j~ 2(k+e\r)i\u0013\n:\n(34)\nHere the magnon eigenstates j i(k)iare the normal-\nized eigenvectors of Kthat correspond to the positive\nenergy eigenvalues{i.e. the particle bands, and j~ ii=\n(\u001b3\nI2)j ii. We plot the non-Abelian Berry curva-\nture calculated on an 200 \u0002200 lattice in Fig. 4 for a\ncharacteristic set of parameter values. Assuming that\nEq. (30) can be generalized trivially for the case of\nnon-Abelian Berry curvature it can be seen immediately\nthat\u0014xy= 0 because our numerically calculated \nz(k)\nobeys \nz(kx;ky) = -\nz(\u0000kx;ky) = -\nz(kx;\u0000ky) whereas\n!(kx;ky) =!(\u0000kx;ky) =!(kx;\u0000ky){thereby causing the\nintegral to vanish.\nIV. THE FERROMAGNETIC BACKGROUND\nIt has previously been shown15via a one-loop renor-\nmalization group analysis of the extended U-V-Jmodel\nthat triplet dx2\u0000y2-density wave condensation is energet-\nically favorable for a range of interaction strengths given\nJ=U < 0. Furthermore, it was theoretically predicted37\nand recently experimentally con\frmed38that the highly\noverdoped cuprates show ferromagnetic ordering in theCuO 2planes. Due to these reasons we investigate the\nmixed triplet-singlet density wave DMI e\u000bects on a two-\ndimensional Heisenberg ferromagnet. Taking B=B^z\nthe symmetric exchange term favors mean-\feld states of\nthe form\nh\u0016Sii=nz^z; (35)\nand the antisymmetric exchange favors mean-\feld states\nof the form\nh~Sii=\u0018x(rix;iy)^x+\u0018y(rix;iy)^y;\n\u0018x(ri) =\u00180[(\u00001)ix+ (\u00001)iy]\n2\n\u0018y(ri) =\u00180[(\u00001)ix\u0000(\u00001)iy]\n2:(36)\nThus, the mean-\feld state which occurs in the presence\nof both types of exchange is\nhnii=h\u0016Six;iyi+h~Six;iyi: (37)\nWe have checked that this is the true mean-\feld ground\nstate by summing the classical energy over the four sub-\nlattices and then numerically minimizing the energy with\nrespect to the four sets of spherical angles which charac-\nterize the classical spin directions. The mean-\feld energy\nper site in this case is (restoring S)\nE0\nN=\u0000jJjzS2cos2(\u0012)=2\u00002S2jW0jsin2(\u0012)\n\u0000BScos(\u0012);(38)\nwhereNis the number of lattice sites, z= 4 (6) in\ntwo (three) dimensions, and \u0012is de\fned as the angle\nbetweenh\u0016Six;iyiandhnii. For the square lattice case the\nground state is minimized about \u0012= 0 for all 2 W0<\nzjJj=2 +B=S, whereas the ground state is minimized at\n\u0012= cos\u00001[B=(4S(W0\u0000J))] for 2W0>zjJj=2 +B=S.\nFollowing the procedure highlighted in the previous\nsection we expand the operators about the mean-\feld\nground state and substitute them into Eq. (18) to yield\nthe real space Hamiltonian\nH=E0+H0+H0; (39)\nwhere the classical mean \feld energy E0is de\fned in Eq.\n(20),H0is\nH0=X\ni\u0016by\nibi+X\nhi;ji[\u0016Z\u0012g(j)by\nibj+~Z\u0012g(j)by\niby\nj\n+iJScos(\u0012)(\u00001)ix+iybiby\nj+ h.c.];(40)\nwithg(j) = +1 forj=i+ ^x, andg(j) =\u00001 forj=i+ ^y,7\nand the coe\u000ecients are de\fned as\n\u0016Z\u0012\u0011JS\n2sin2(\u0012) +W0S\n2(cos2(\u0012) + 1)\n~Z\u0012\u0011JS\n2sin2(\u0012) +W0S\n2(cos2(\u0012)\u00001)\n\u0016\u00114JScos2(\u0012) + 4W0Ssin2(\u0012) +Bcos(\u0012);(41)\nandH0is\nH0=X\ni(\u00001)ixA\u0012(by\ni+bi); (42)\nwhere\nA\u0012= sin(\u0012)\"r\nS\n2B+\u0010\nJ(2S)3=2\u0000W0(2S)3=2\u0011\ncos(\u0012)#\n:\n(43)\nTerms linear in boson creation and annihilation opera-\ntors imply spin-wave creation and annihilation from the\nground state. Thus, assuming that the system is in its\nground state, it is typically argued in the literature that\nthis coe\u000ecient A\u0012must vanish at each point ion the lat-\ntice; in the following we show that allowing small A\u0012has\na nontrivial e\u000bect on the critical behavior of the system.\nThere exist two unique solutions for vanishing A\u0012: the\nperfectly ferromagnetic case of \u0012= 0, and\nj\u0012j= cos\u00001\u0014B\n4S(W0\u0000J)\u0015\n(44)\nwhich is the aforementioned magnetic ground state cant-\ning angle that occurs at the classical mean-\feld level\nwhen 2W0>2J+B=S. Anticipating quantum correc-\ntions to the ground state canting angle we opt to in-clude the e\u000bects of A\u00126= 0{however, to maintain consis-\ntency with the Holstein-Primako\u000b substitution about the\nmean-\feld ground state it is understood that A\u0012is nec-\nessarily either small or exactly zero, i.e. we are expand-\ning su\u000eciently close to the classical mean-\feld theory's\npredicted relationship between the parameters. Thus,\ninstead of taking A\u0012= 0, we eliminate terms linear in\nbosonic creation and annihilation operators by perform-\ning the canonical transformation\nbi=~bi\u0000(\u00001)ixx\nby\ni=~by\ni\u0000(\u00001)ixx;(45)\nwherexiis the C-number\nx=\u0000A\u0012\n4\u0016Z\u0012+ 4~Z\u0012\u0000\u0016: (46)\nNote that this transformation is well de\fned when the\ndenominator 4 \u0016Z\u0012+ 4~Z\u0012\u0000\u00166= 0{this is indeed the case\nwhen we investigate the physics in close proximity to the\nmean-\feld behavior. The Hamiltonian then becomes\nH=E0\u0000NxA\u0012+H0\n0=E0\n0+H0\n0; (47)\nwhereH0\n0is identical to the Hamiltonian written in Eq.\n(27) but in terms of the transformed bosonic operators ~b.\nUpon Fourier transformation the total Hamiltonian can\nbe written in terms of the Nambu basis as\nH=E00\n0\u0000NxA\u0012+X\nk1\n2 y\nkHk k (48)\nwhere y\nk= (~by\nk;~by\nk+Q;~b\u0000k;~b\u0000k+Q),\nHk=0\nBB@\u0016Z\u0012;k+\u0016=2iJkcos(\u0012) 2 ~Z\u0012;k 0\n0\u0000\u0016Z\u0012;k+\u0016=2 0 \u00002~Z\u0012;k\n0 0 \u0016Z\u0012;k+\u0016=2\u0000iJkcos(\u0012)\n0 0 0 \u0000\u0016Z\u0012;k+\u0016=21\nCCA+ h.c.; (49)\nE00\n0\nN=\u00002S(S+ 1)[Jcos2(\u0012) +jW0jsin2(\u0012)]\n\u0000B(S+ 1=2)cos(\u0012);(50)\nand\u0016Z\u0012;k=\u0016Z\u0012[coskx\u0000cosky]=2,~Z\u0012;k=~Z\u0012[coskx\u0000cosky],\nJk=JS[coskx+ cosky].\nThe spectrum is given by the absolute value of the\neigenvalues of the dynamic matrix K= (\u001b3\nI2)Hk.33.\nThe Hamiltonian, written in terms of the appropriate\nBogoliubov operators \rk, is then\nH=Eg+X\nk;n!k;n\ry\nk;n\rk;n; (51)wherenis the band index and the ground state energy\nEgis\nEg=E00\n0\u0000NxA\u0012+X\nk;n!k;n\n2: (52)\nThe canting angle is now determined by minimizing Eg\nwith respect to \u0012. The e\u000bects of the linear boson terms\ncan be seen by comparing the critical value of W\u0003\n0ob-\ntained byE00\n0and byE00\n0\u0000NxA\u0012in the absence of quan-\ntum corrections. Upon expanding E00\n0to leading order in8\n\u0012we yield (setting S= 1/2)\nE00\n0(\u0012)=N\u0019\u00003J\n2\u0000B+\u0014B\n2+3J\n2\u00003W0\n2\u0015\n\u00122;(53)\nwhereas the expansion of E00\n0=N\u0000xA\u0012is\nE00\n0(\u0012)=N\u0000xA\u0012\u0019\u00003J\n2\u0000B+\u0014B\n4+J\u0000W0\u0015\n\u00122:(54)\nThe critical value of W\u0003\n0can be obtained by \fnding its\nvalue when the \u00122coe\u000ecient changes sign. Hence, with-\nout linear boson e\u000bects W\u0003\n0=J+B=3, whereas includ-\ning the linear boson e\u000bects reduces the critical value to\nW\u0003\n0=J+B=4. This technique can be applied to gen-\neral noncollinear spin systems to identify shifts in critical\nvalues of the parameters in the theory.\nThe dispersion for modest values of W0consists of\ntwo two-fold degenerate branches with a characteris-\nticdx2\u0000y2gap{this is due to the translation symme-\ntry breaking nature of the DMI. For higher period in-\ncommensurate triplet density wave states the number of\nmagnon branches will be equal to the period of incom-\nmensurability because it is precicely that period which\ndetermines the number of sites contained per unit cell.\nFor\u0012= 0 andS= 1/2 the magnon dispersion is\n!k;n=1\n2B+J\u0006\n1\n2q\nJ2(coskx+ cosky)2+W2\n0(coskx\u0000cosky)2(55)\nwhere the\u0006corresponds to the upper and lower band\nrespectively. The dispersion is plotted along the high\nsymmetry directions for some representative values of B,\nW0,\u0012in Figs. 5-8.\n0.5 1.0 1.5 2.0 2.5 3.0kx0.51.01.52.0ωk\nFIG. 5: Magnon dispersion !kwithky= 0 in units of Jfor\nvarious values of density wave strength with B= 0:05J. The\nblue, orange, green, and red curves correspond to W0!0,\nW0= 0:25J, andW0= 0:5J,W0=Jrespectively, all below\nW\u0003\n0. AsW0is increased the gap at k= (\u0019;0) increases as\n2W0.\n0.5 1.0 1.5kd0.51.01.52.0ωkFIG. 6: Magnon dispersion !kalong the line kx=ky=kd\nin units ofJfor various values of density wave strength with\nB= 0:05J. TuningW0does not alter the dispersion in this\ndirection.\n0.5 1.0 1.5 2.0 2.5 3.0kx0.51.01.52.02.5ωk\nFIG. 7: Magnon dispersion !kwithky= 0 in units of Jfor\nvarious values of density wave strength with B= 0:1Jwith\nthe appropriate canting angles determined by minimization\nof Eq. 39. The blue, orange, and green curves correspond to\nW0!1:25J,W0= 1:35J, andW0= 1:45Jrespectively, all\naboveW\u0003\n0. AsW0is increased the gap at k= (\u0019;0) increases\nas 2W0. The low energy excitations are now governed solely\nby the points k= (0;\u0019) and (\u0019;0).\n0.5 1.0 1.5kd1.11.21.31.41.5ωk\nFIG. 8: Magnon dispersion !kalong the line kx=ky=kd\nin units of JwithB= 0:1Jwith the appropriate canting\nangles determined by minimization of Eq. 39. The blue,\norange, and green curves correspond to W0!1:25J,W0=\n1:35J, andW0= 1:45Jrespectively, all above W\u0003\n0. Nonzero\ncanting shifts the spectrum upwards in energy.9\nAsW0is tuned from zero the low energy physics is\ngovened by the k= (0;0) point and gaps develop at\nk= (\u0019;0),k= (0;\u0019) with an energy di\u000berence of\n2W0. AsW0approaches its critical value the lowest en-\nergy excitations are goverend by k= (0;0),k= (\u0019;0),\nk= (0;\u0019). TuningW0beyond the critical value of den-\nsity wave strength the low energy excitations are de-\nscribed entirely by the points k= (\u0019;0),k= (0;\u0019),\nand the spectrum is shifted upwards in energy due to the\ncanting of the localized moments.\nV. DISCUSSION\nWe have shown that a triplet density wave state in-\nduces a DM interaction in the host spin system. The\ndensity wave-mediated DM vector is stabilized in topo-\nlogical systems by the direction of the magnetic \feld, and\nthe symmetry such a DMI is governed entirely by the an-\ngular momentum channel of the triplet density wave. Al-\nthough it has been shown that the triplet-singlet density\nwave state produces a nonzero thermal Hall e\u000bect1, the\nmagnitude of the experimentally-measured thermal Hall\ne\u000bect exceeds the maximum possible contribution from\nthe density wave state alone by an order of magnitude.\nThe excitations of a spin system including DM interac-\ntions can, in principle, contribute to the thermal Hall\nconductivity22{24; however, we have shown the particular\nform of DM interaction generated by the triplet density\nwave does not seem to produce a nonzero \u0014xy, and thus\nno additional contribution can be found through the in-\n\ruence of the density wave state on the underlying spin\nsystem.\nTriplet-singlet density wave order is notoriously di\u000e-\ncult to detect directly8, and so it is important to explore\npossible in\ruences that the state might have on its host\nsystem. Experimental detection of such features could,\nfor example, help to assess the importance of the triplet-\nsinglet DDW state in the description of the pseudogap\nphase of the cuprates. The magnetic structure of LSCO\nat low doping is Neel order with a small ferromagnetic\nmoment. We have shown that in such a system, the pres-\nence of the DDW induces anisotropy in the spin-wave\ndispersion, re\recting the anisotropy of the DDW. Fur-\nthermore, the magnon branch for such a system has a\nnon-Abelian berry curvature that vanishes upon integra-\ntion in such a way that \u0014xy= 0.\nAdditionally, a two patch RG analysis of the U-V-\nJmodel indicates that triplet dx2\u0000y2-density wave or-\nder is can be energetically favorable in a \fnite region of\ncoupling space given J=U < 0.15Ferromagnetic order-\ning was also predicted37to emerge in the highly over-\ndoped cuprates and experimentally con\frmed38to exist\nin the CuO 2planes of the cuprates. We \fnd that the\ni\u001bdx2\u0000y2+dxy-density wave-induced DM interaction in a\n2D ferromagnetic system generically produces a magnon\nspectrum with two branches with a characteristic dx2\u0000y2\ngap. For higher period incommensurate triplet densitywave states in such a spin system the number of magnon\nbranches is equal to the density wave's period of incom-\nmensurability.\nWe have also found that the inclusion of terms in the\nHamiltonian linear in Holstein-Primako\u000b boson opera-\ntors has a nontrivial e\u000bect on the critical behavior of the\nHamiltonian. These terms are typically ignored in the lit-\nerature, which is justi\fed when considering models that\nare far from the critical regime; however, we have shown\nthat they induce shifts in the critical parameter values\nwhich control collinear to noncollinear phase transitions.\nACKNOWLEDGMENT\nThis work was supported in part by funds from David\nS. Saxon Presidential Term Chair at UCLA.\nAppendix: Calculating the thermal Hall coe\u000ecient\nIn our calculation we compute \u0014xyusing the Mott-like\nformula34,39{42\n\u0014xy\nT=1\nT2Z(\u000f\u0000\u0016)2\ncosh2\u0000\u000f\u0000\u0016\n2T\u0001\u001bxy(\u000f)d\u000f (A.1)\nwhere\u0016is the chemical potential and \u001bxy(\u000f) is the Hall\ncoe\u000ecient for the system at zero temperature with chem-\nical potential \u000f. We implement the linear-in-\feld approx-\nimation where calculation of \u0014xyis greatly simpli\fed at\nlow magnetic \feld:43\n\u001bxy(\u000f)\u0019B@B\u001bxy(\u000f)jB=0=\u0000B~B(\u000f) (A.2)\nwhereBis magnitude of the magnetic \feld, and ~B(\u000f) is\nthe e\u000bective Berry curvature density given by\n~B(\u000f) =X\nnksBnsk\u000e(\u000f\u0000\u000fnsk): (A.3)\nHeren=\u00061 for the lower and upper bands, and the spin\nindexs=\u00061. 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Zhang, Physical Re-\nview Letters 124, 186602 (2020)."    },    {        "title": "2007.12052v1.Interpretations_of_ground_state_symmetry_breaking_and_strong_correlation_in_wavefunction_and_density_functional_theories.pdf",        "content": "\t1\tInterpretations of ground-state symmetry breaking and strong correlation in wavefunction and density functional theories   John P. Perdew1,2,*, Adrienn Ruzsinszky1, Jianwei Sun3, Niraj K. Nepal1, and Aaron D. Kaplan1,# 1 Department of Physics, Temple University. Philadelphia, PA 19122. 2 Department of Chemistry, Temple University, Philadelphia, PA 19122. 3 Department of Physics, Tulane University, New Orleans, LA 70118. * perdew@temple.edu # kaplan@temple.edu  Strong correlations within a symmetry-unbroken ground-state wavefunction can show up in approximate density functional theory as symmetry-broken spin-densities or total densities, which are sometimes observable. They can arise from soft modes of fluctuations (sometimes collective excitations) such as spin-density or charge-density waves at non-zero wavevector. In this sense, an approximate density functional for exchange and correlation that breaks symmetry can be more revealing (albeit less accurate) than an exact functional that does not.  The examples discussed here include the stretched H2 molecule, antiferromagnetic solids, and the static charge-density wave/Wigner crystal phase of a low-density jellium. It is shown that (and in what sense) the static charge density wave is a soft plasmon.  This article presents examples and interpretations of broken symmetry and strong correlation, complexities that sometimes show up in quantum mechanical systems of many particles, including molecules and solids. Figure 1 suggests why several different perspectives can be helpful to understand complex phenomena. The density functional theory (DFT) of Kohn and Sham [1] is an exact-in-principle mean-field-like theory for the ground-state (lowest) energy and electron density (or spin densities) of any N-electron system with a Hamiltonian or energy operator of the form    𝐻\t=[−∇&'/2*&+,+𝑣(𝒓&)]+𝒓&−𝒓34,35&&   (1)                (taking fundamental constants to be 1).\t𝑣(𝒓) is an external scalar potential, such as the Coulomb attraction to the nuclei, and the electron-electron Coulomb repulsion is explicitly included. A ground-state N-electron wavefunction is an eigenstate of the Hamiltonian for the lowest energy eigenvalue. Since electrons are spin-1/2 fermions, the physical wavefunctions must be antisymmetric, changing sign under the exchange of any two electrons. Squaring a wavefunction yields a positive probability distribution. Integrating the square of the wavefunction over N-1 electron position vectors and N-1 spin coordinates yields a one-electron spin density 𝑛7(𝒓) (where 𝜎=\t↑  or ↓  along the z-axis of spin quantization) and total electron density 𝑛𝒓=𝑛↑𝒓+𝑛↓𝒓 that define the starting point for DFT. Integrating over all but two electron positions 𝒓 and spins 𝜎 yields an electron pair density 𝜌77=(𝒓,𝒓?) that shows explicitly how the electrons avoid one another due to wavefunction antisymmetry (which makes 𝜌77(𝒓,𝒓)=0) and Coulomb repulsion. Although the Hamiltonian of Eq. (1) does not depend explicitly on spin, the spin-state importantly affects the ground-state energy through the antisymmetry of the wavefunction.         There is an exact ground-state density functional for the energy, including non-interacting kinetic, \t2\texternal, Hartree, and exchange-correlation terms, whose minimization leads to exact one-electron Schrödinger equations for the orbitals or one-electron wavefunctions used to construct the non-interacting kinetic energy and the electron density. A practically useful method is attained by approximating only the negative exchange-correlation energy functional 𝐸BC[𝑛↑,𝑛↓], sometimes based on exact constraints and on appropriate norms such as the uniform electron gas for which all functionals discussed here are exact.         Coulomb correlation means all effects beyond the symmetry-unbroken self-consistent mean-field Hartree-Fock approximation [2]. The mean-field appearance of Kohn-Sham theory too often creates the impression that this theory lacks correlation or at least strong correlation. But Coulomb correlation is about as strong as exchange even in ordinary sp-bonded molecules, and this correlation is well described by non-empirical approximate functionals [1,4-7], as Table 1 shows. Kohn-Sham DFT achieves computational efficiency by not calculating a correlated many-electron wavefunction, but it still includes the Coulomb-repulsion-driven correlation among electrons in its functional or rule for finding the exchange-correlation term of the total energy from the electron up- and down-spin densities, and in its related exchange-correlation potential in the one-electron Schrödinger equation that shapes the orbitals and the spin densities in a self-consistent calculation.                  “Strong correlation” is sometimes used to mean “everything that density functional theory gets wrong”. Yet hybrid functionals (including part of the Hartree-Fock exchange) like HSE06 [9,10] (for non-metallic states) and meta-GGA’s like SCAN [6,11-17] are yielding quantitatively correct ground-state (and we emphasize “ground-state”) results for some systems that have long been regarded as strongly correlated. In the cuprate high-temperature superconducting materials, for example, the SCAN meta-GGA [6] is able to do what simpler density functionals (LSDA, GGA) cannot [13], by creating the correct spin moments on the copper atoms, their antiferromagnetic order, and a correctly non-zero band gap in the undoped material that correctly disappears under the doping that also leads to superconductivity [12-14]. Full but more reliable self-interaction corrections [7] might eventually further improve approximate DFT for other strongly-correlated systems. And, as will be argued here, even the simplest DFT approximations yield a qualitative insight into some such systems.         The first interpretation to be expressed here is that certain “strong correlations” that are present as fluctuations in the exact symmetry-unbroken ground-state wavefunction are “frozen” in symmetry-broken electron densities or spin-densities of approximate DFT. For finite systems, this would not happen with the exact functional. So, while the exact functional would always be exact for the ground-state energy and density, it would not always be as qualitatively revealing as the approximate one.             This first interpretation (but without the term “strong correlations”) is eloquently expressed in P.W. Anderson’s essay “More is Different” [18], which explains that, while a ground-state wavefunction is necessarily static, it must describe fluctuations in the expectation values of operators that are not diagonalized along with the Hamiltonian, and that these fluctuations can freeze as the number of particles in the system grows large, leading to an observable symmetry breaking that would not be possible in a few-particle system. Symmetry-breaking in approximate DFT can thus be more correct for solids than it is for small molecules, but in either case it can reveal a strong correlation in a symmetry-unbroken wavefunction. The possibility of observable symmetry breaking is compatible [19] with the continued existence of a symmetry-unbroken exact ground-state wavefunction. Symmetry breaking is surprising only in quantum physics; in classical physics, it is familiar and intuitive.               The second interpretation involves the wave-like fluctuations and collective excitations of a solid. A wave in the total electron density 𝑛𝒓=𝑛↑𝒓+𝑛↓(𝒓) is a plasma wave (quantized as plasmons), and a wave in the net electron spin density 𝑚𝒓=𝑛↑𝒓−𝑛↓𝒓\tis a spin density wave. Like other waves, these have amplitude, wavelength 𝜆 or wavevector q = 2𝜋/𝜆, and frequency 𝜔, and at small amplitudes can be simply superposed or added together. However, their frequencies are not sharply defined. By the uncertainty principle, a smaller \t3\tfrequency width leads to a longer lifetime. Collective excitation modes are distinguished from other fluctuation modes by much smaller frequency widths. The second interpretation explains how some symmetry breakings and strong correlations can arise: Under a variation of the external potential, a collective excitation or fluctuation of the electrons of non-zero wavevector, such as a charge-density wave or a spin-density wave, can soften, with an excitation energy or frequency tending to zero, until it appears as a static wave in the symmetry-broken density or spin-density of an approximate functional. Since the frequency is positive, the frequency width must go to zero when the frequency does. This mechanism is analogous to the softening of a phonon mode of non-zero wavevector that can lead to a distortion or structural phase transition of an ionic lattice. The ground-state energy of a quantum lattice nearly equals that of the classical lattice plus the zero-point energy of its lattice vibrational modes. Via the fluctuation-dissipation theorem [20,21], the ground-state total energy of a system of interacting electrons, including the Coulomb correlation energy, has contributions from fluctuations of various wavevectors, including the non-negative zero-point energies ℏω/2 of its collective excitations. Collective excitations, like observable or physical symmetry breakings, are emergent phenomena, arising only in systems with large numbers of particles [22].           The Hamiltonian of Eq. (1) can break spatial symmetries through its external potential, but that is not the subject of this discussion. We also note that this Hamiltonian, while a good starting point, is not exact. It omits relativistic effects including the spin-orbit interaction, and it neglects the fact that the external potential comes from nuclei that can move and, in a solid, can break lattice symmetries. These physical effects will be ignored here for the sake of clarity and simplicity.            We know that it is possible to find the eigenstates of a complete set of commuting observables including the Hamiltonian. The states with the lowest eigenvalue of the Hamiltonian are the ground states. If the ground state is non-degenerate, the ground-state density will have all the symmetries of the external potential. For example, the ground-state of the Ne atom is non-degenerate, and its electron density has the spherical symmetry of the 𝑣𝑟=−10/𝑟\t\tCoulomb attraction to the nucleus. If the ground state is degenerate, an equally-weighted average (equi-ensemble) of all ground-state densities or spin densities will have the symmetries of the external potential. “The symmetry of the Hamiltonian is the symmetry of the ground state” [19].           The Hamiltonian of Eq. (1) is independent of electron spin.  Then the complete set of commuting observables can include the square and z-component of the total electron spin.  When the ground-state is non-degenerate, as in a closed-shell atom or molecule, or presumably in a finite non-ferromagnetic and non-ferrimagnetic crystal, the square of the total spin must be zero (𝑆=0 or spin singlet state) and thus the local net spin density must be zero, and the exact ground-state density is expected to have all the symmetries of the external potential.  When there is a net spin moment or a net current density, of course, the ground-state must be degenerate, and a single ground state from a degenerate set need not have the symmetry of the external potential, although the state of thermal equilibrium in the zero-temperature limit still should have it.             A closed-shell molecule like H2 has a non-degenerate ground-state whose exact electron density must remain spin-unpolarized at all bond lengths. In 1976, Gunnarsson and Lundqvist [23] made an LSDA calculation for the ground-state H2 molecule at various bond lengths. For bond lengths less than a critical value (1.7Å), they found a spin-unpolarized density and a realistic binding energy curve. Above the critical bond length, they found an unobservable spin symmetry breaking, with the net spin up near one nuclear center and down near the other (a kind of molecular precursor of the antiferromagnetism to be discussed in the next section). The whole binding-energy curve was realistic only when the symmetry of the spin density was allowed to break under bond stretching. A constraint that prevented spin symmetry breaking led in the limit of infinite bond length to an LSDA total energy much higher than the LSDA energy of two individual hydrogen atoms.            The exact wavefunction tells us that, at a stretched but finite bond length, when an electron with a given spin direction is near one nucleus, an electron \t4\twith the opposite spin direction is highly likely to be near the other. This is a strong correlation within a symmetry-unbroken wavefunction of zero total spin, which is revealed to us by the breaking of the symmetry of the spin density in the LSDA calculation.             Similarly, symmetry suggests that the exact ground-state density of any finite system with zero net spin moment should be spin-unpolarized. But approximate spin-density functional calculations, e.g., [11-15,17], find that many materials and especially transition-metal oxides are antiferromagnetic, with alternating net spin moments on alternating atomic sites. This is presumably a strong correlation within the exact symmetry-unbroken ground-state wavefunction that freezes into the symmetry-broken ground-state spin densities of the approximate density functional. Strong correlations within an exact symmetry-unbroken wavefunction might tell us that there are local spin moments on the transition-metal atoms that are correlated, so that for example a spin-up moment on a given atom has spin-down neighbors on the nearest-neighbor atoms, although the direction of the spin moment on a given atom is not fixed.         A connection between spin-density waves and antiferromagnetism was proposed by Overhauser [24]. Our interpretation is that a spin-density wave drops down (as a function of lattice constant) in energy and frequency to create the broken-symmetry static spin density. Under extreme compression of a given lattice, any material will be a non-magnetic metal with zero energy gap and no local magnetic moment. As this compression is relaxed, we imagine that a spin density wave will soften. To create an ordered antiferromagnetic state, the non-zero wavevector of the soft spin density wave must extend from the center of the first Brillouin zone of the underlying ionic lattice to the center of one of the Brillouin-zone faces, typically doubling the size of the unit cell in real space and opening an energy gap that makes the material an insulator. In approximate Kohn-Sham density functional theory, the metallic, non-magnetic state of unbroken symmetry remains as a solution of the Kohn-Sham equations, but not as the solution of lowest energy. Many correlations become stronger and symmetries tend to break as the electron density and electron kinetic energy decrease.               Anderson [25] found an accurate estimate of the ground-state energy for an antiferromagnetic Heisenberg model Hamiltonian describing the effective interaction between localized spin moments on a simple cubic lattice. He started from the energy of a classical antiferromagnetic ground state, then included the zero-point energy of the transverse spin-wave excitations. Here the transverse directions are perpendicular to the z direction, i.e., to the spin direction on the first of the two spin sub-lattices. The frequency is of course zero for any long-wavelength mode (Goldstone mode) that simply rotates the spin direction, since there is no restoring force. The expectation value of the spin moment vector on each site of a sublattice can point in any direction, as long as the expectation value on each site of the other sublattice points in the opposite direction. Anderson estimated the time (proportional to the number of atoms present) for the spin direction to rotate by 90 degrees as three years, much longer than the time scale of a neutron diffraction experiment. Thus the singlet spin symmetry is well broken by the soft spin density wave of non-zero wavevector, and the spin-isotropy symmetry is well broken by the soft Goldstone mode of zero wavevector.               Interestingly, the local spin moment on a transition-metal atom can persist even above the magnetic disordering temperature, both in ferromagnetic iron [26] and in transition metal monoxides and other strongly-correlated materials (where it can still give rise to an insulating gap in the Kohn-Sham density of one-electron states) [11-15].             Now consider a standard model for a simple metal, an infinite jellium in its ground state. The positive background charge is uniform and rigid, and there should always be a ground-state wavefunction with a uniform electron density.  But, at low density (𝑟M≈69,\twhere the density is 𝑛=3/[4𝜋𝑟MS]=𝑘US\t/3𝜋' and thus 𝑟M  is the radius of a sphere containing on average one electron), where the jellium is greatly expanded from its equilibrium density (𝑟M≈4), an exchange-correlation kernel-corrected linear-response density functional calculation [27] finds a static charge-density wave. Approximate density functionals like LSDA also predict [28] a charge-density wave instability of the uniform electron \t5\tdensity, but often at the wrong density. The charge-density wave is an incipient Wigner lattice, in the sense that it has a wavevector 𝑞≈2.28𝑘U, close to the first reciprocal lattice vector of a bcc lattice, so that a bcc Wigner lattice can be constructed by superposing charge density waves with wavevectors along the twelve (110) directions. At the density where the Wigner lattice first appears (𝑟M≈85±20\tfrom a quantum Monte Carlo calculation [29]), each electron is spread out over its whole Wigner-Seitz cell, but in the 𝑟M\t\t→\t∞ limit, each electron further localizes to the center of its cell. The Wigner lattice is a strong correlation within the exact symmetry-unbroken ground-state wavefunction at low density, and is justified by the theory of strictly-correlated electrons [30].           The charge density wave in jellium is a case in which we can computationally identify the collective mode or fluctuation and see how its excitation energy or frequency approaches zero as the external potential is varied. The charge-density wave appears [27] to arise from a soft plasmon: a collective density-wave excitation of the system that drops down in excitation energy or frequency and eventually freezes in the broken-symmetry density of an approximate density functional calculation. The argument for this in Ref. [27] was based on extrapolation of the plasmon frequency 𝜔](𝑞) into the range of wavevectors 𝑞 where the plasmon frequency is mixed with the continuum of single particle-hole excitations, and the plasmon is no longer a true collective excitation. Figures 2 and 3, however, show that this extrapolation is well justified. Figure 2 shows the average frequency of a density fluctuation of wavevector q, and Fig. 3 shows the root mean square deviation from this average, both computed from the structure factor [20,21] or spectral function [22] 𝑆𝑞,𝜔\tof time-dependent density functional theory [22].  The refinement of approximate density functionals toward the exact functional offers the possibility of suppressing symmetry breaking [31], but the result would not be an unmixed blessing since qualitative insights into strong correlation could be lost.              We can now propose a definition of “strong correlation”: Strong correlation in a symmetry-constrained wavefunction is any correlation between electrons that results in an exceptionally structured electron pair density, or is otherwise qualitatively different from the “normal” Coulomb correlation found in simple sp-bonded materials in their ground states near equilibrium nuclear geometries and having Hamiltonians of the form of Eq. (1). This definition has little in common with “everything that DFT gets wrong”. Indeed, DFT with standard approximate functionals perfectly captures the energetic consequences of strict correlation [30] in a uniform electron gas of very low density. Strong correlation by this definition is sometimes displayed by symmetry breaking in a symmetry-unconstrained wavefunction or density. DFT increasingly describes the ground-state energies and densities of systems with d or f electrons that are widely regarded as strongly-correlated.             There is another valid interpretation [32] of spin symmetry breaking in spin-density functional theory: Approximate spin density functionals reliably predict the total electron density, but predict the on-top electron pair density 𝜌7,7=7,7=𝒓,𝒓\t(yielding the probability to find two electrons together at the same position in space) more reliably than they predict the net spin density; such functionals predict both reliably when the correlation is not strong. This interpretation is now being used to merge the wavefunction and density-functional approaches [33].  References:  1. W. Kohn and L. J. Sham, Self-consistent equations including exchange and correlation effects. Phys. Rev. 140A, 1133-1138 (1965). 2. A. Szabo and N. S. Ostlund, Modern Quantum Chemistry (Macmillan, New York, 1982). 3. B. J. Lynch and D. G. Truhlar, Small representative benchmarks for thermochemical calculations. J. Phys. Chem. A 107, 8996-8999 (2003). 4. J. P. Perdew and Y. Wang, Accurate and simple analytic representation of the electron-gas correlation energy. Phys. Rev. B 45, 13244-13249 (1992). \t6\t5. J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865-3868 (1996). 6. J. Sun, A. Ruzsinszky, and J. P. Perdew, Strongly constrained and appropriately normed semilocal density functional. Phys. Rev. Lett. 115, 036402 (2015). 7. P. Bhattarai, K. Wagle, C. Shahi, Y. Yamamoto, S. Romero, B. Santra, R. R. Zope, J. E. Peralta, and J. P. Perdew, A step in the direction of resolving the paradox of Perdew-Zunger self-interaction correction. II. Gauge consistency at three levels of approximation. J. Chem. Phys. 152, 214109 (2020). 8. J. Heyd, G. E. Scuseria, and M. Ernzerhof, Erratum: “Hybrid functionals based on a screened Coulomb potential”. J. Chem. Phys. 124, 219906 (2006). 9. P. J. Hay, R. L. Martin, J. Uddin, and G. E. Scuseria, Theoretical study of CeO2 and Ce2O3 using a screened hybrid density functional. J. Chem. Phys. 125, 034712 (2006). 10. X.-D. Wen, R. L. Martin, T. M. Henderson, and G. E. Scuseria, Density functional theory studies of the electronic structure of solid state actinide oxides. Chem. Rev. 113, 1063-1096 (2013). 11. H. Peng and J. P. Perdew, Synergy of van der Waals and self-interaction corrections in transition metal monoxides. Phys. Rev. B 96, 100101(R) (2017). 12. J. W. Furness, Y. Zhang, C. Lane, I. G. Buda, B. Barbiellini, R. S. Markiewicz, A. Bansil, and J. Sun, An accurate first-principles treatment of doping-dependent electronic structure of high-temperature cuprate superconductors. Communications Physics 1, 11 (2018). 13. K. Pokharel, C. Lane, J. W. Furness, R. Zhang, J. Ning, B. Barbiellini, R. S. Markiewicz, Y. Zhang, A. Bansil, and J. Sun, Ab initio description of the electronic structure of high-temperature cuprate superconductors: A comparative density functional study. arXiv:2004.08047v1 (2020). 14. Y. Zhang, C. Lane, J. W. Furness, B. Barbiellini, J. P. Perdew, R. S. Markiewicz, A. Bansil, and J. Sun, Competing stripe and magnetic phases in the cuprates from first principles. Proc. Natl. Acad. Sci. U.S.A. 117, 68-72 (2020). 15. Y. Zhang, J. Furness, R. Zhang, Z. Wang, A. Zunger, and J. Sun, Symmetry-breaking polymorphous descriptions for correlated materials without interelectronic U. Phys. Rev. 102, 045112 (2020). 16. A. Pulkkinen, B. Barbiellini, J. Nokelainen, V. Sokolskiy, D. Baigutlin, O. Miroshkina, M. Zagrebin, V. Buchelnikov, C. Lane, R., S. Markiewicz, A. Bansil, J. Sun, K. Pussi, and E. Lahderanta, Coulomb correlation in noncollinear antiferromagnetic 𝛼−Mn. Phys. Rev. B 101, 075115 (2020). 17. C. Lane, Y. Zhang, J. W. Furness, R. S. Markiewicz, B. Barbiellini, J. Sun, and A. Bansil, First-principles calculation of spin and orbital contributions to magnetically ordered moments in Sr2IrO4. Phys. Rev. B 101, 155110 (2020). 18. P. W. Anderson, More is different. Science 177, 393-396 (1972). 19. R. Shankar, Principles of Quantum Mechanics (Kluwer, New York, ed. 2, 1994). 20. P. Nozières and D. Pines, A dielectric formulation of the many-body problem – Application to the free electron gas. Nuovo Cimento 9, 470-490 (1958). 21. D. C. Langreth and J. P. Perdew, Exchange-correlation energy of a metallic surface. Solid State Commun. 17, 1425-1429 (1975). 22. G. F. Giuliani and G. Vignale, Quantum Theory of the Electron Liquid (Cambridge University Press, Cambridge, 2005). 23. O. Gunnarsson and B. I. Lundqvist, Exchange and correlation in atoms, molecules, and solids. Phys. Rev. B 13, 4274-4298 (1976). 24. A.W. Overhauser, Spin-density-wave mechanisms of antiferromagnetism, J. Appl. Phys. 34, 1019-1024 (1963). 25. P. W. Anderson, An approximate quantum theory of the antiferromagnetic ground state. Phys. Rev. 86, 694-701 (1952). 26. A. J. Pindor. J. Staunton, G. M. Stocks, and H. Winter, Disordered local moment state of magnetic transition metals – a self-consistent KKR CPA calculation. J. Phys. F 13, 979-989 (1983). \t7\t27. A. Ruzsinszky, N. K. Nepal, J. M. Pitarke, and J. P. Perdew, Constraint-based wavevector and frequency dependent exchange-correlation kernel for the uniform electron gas. Phys. Rev. B 101, 245135 (2020). 28. J. P.  Perdew and T. Datta, Charge and spin density waves in jellium. Phys. Stat. Sol. B 102, 283-293 (1988). 29. D. M. Ceperley and B. J. Alder, Ground-state of the electron gas by a stochastic method. Phys. Rev. Lett. 45, 566-569 (1988). 30. M. Seidl, P. Gori-Giorgi, and A. Savin, Strictly correlated electrons in density functional theory: A general formulation with applications to spherical densities. Phys. Rev. A 75, 042511 (2007). 31. N. Q. Su, C. Li, and W. Yang, Describing strong correlation with fractional spin in density functional theory. Proc. Natl. Acad. Sci. U.S.A. 115, 9678-9683 (2018). 32. J. P. Perdew, A. Savin, and K. Burke, Escaping the symmetry dilemma through a pair-density interpretation of spin density functional theory. Phys. Rev. A 51, 4531-4541 (1995). 33. S. Ghosh, P. Verma, C. J. Cramer, L. Gagliardi, and D. G. Truhlar, Combining wave function methods with density functional theory for excited states. Chem. Rev. 118, 7249-7292 (2018). 34. N. K. Nepal, S. Adhikari, B. Neupane, S. Ruan, S. Neupane, and A. Ruzsinszky, Understanding plasmon dispersion in nearly-free electron metals: Relevance of exact constraints for exchange-correlation kernels within time-dependent density functional theory. Phys. Rev. B 101, 195137 (2020). 35. P. Gori-Giorgi and J. P. Perdew, Phys. Rev. B 66, 165118 (2002).  Acknowledgments:  JPP and JS acknowledge stimulating discussions of strong correlation with Alex Zunger.  Funding:  The work of JPP was supported by the U.S. National Science Foundation under grant number DMR-1939528. The work of AR and NKN was supported by the U.S. National Science Foundation under grant number DMR-1553022. The work of JS was supported by the Department of Energy, Office of Science, Basic Energy Sciences grant number DE-SC0019350. The work of ADK was supported by the Department of Energy, Basic Energy Sciences, through the Energy Frontier Research Center for Complex Materials from First Principles, grant number DE-SC0012575.  Author contributions:  J.P.P., A.R., and J.S. designed the research, performed the research, and wrote the manuscript. N.K.N. and A.D.K. designed some of the research and performed the calculations and analysis.     \t8\t \n Fig. 1. The parable of the blind men and the elephant suggests that scientists who study the same problem from different perspectives should pool their insights. This article touches four perspectives on broken symmetry and strong correlation in many-electron systems: ground-state and time-dependent density functional, wavefunction, and model Hamiltonian. For brevity, the important Green’s function and dynamical mean field perspectives are not discussed here. Artwork by Liyu Ye, with her permission.   \n\t9\t Fig. 2. Charge density wave in jellium as a soft plasmon. The frequency <𝜔]𝑞>\tof a density fluctuation as a function of its wavevector 𝑞=2𝜋/𝑤𝑎𝑣𝑒𝑙𝑒𝑛𝑔𝑡ℎ in a uniform electron gas at its equilibrium density (𝑛=3/[4𝜋𝑟MS,]=𝑘US/(3𝜋');\t\t𝑟M=4) and at a much lower density (𝑟M=69) where a static charge density wave appears and breaks translational symmetry. At 𝑟M=4, the frequency increases with 𝑞, but at 𝑟M=69 the frequency softens, dropping to zero around 𝑞=2.28𝑘U, the critical wavevector of the static charge-density wave. The solid part of the 𝑟M=69 curve was computed as in Ref. [27], at wavevectors where the plasmon has not yet penetrated into the continuum of single particle-hole excitations. The dashed parts of the curves were computed here at all wavevectors using the frequency distribution (structure factor [20,21] or spectral function [22,34]) 𝑆𝑞,𝜔=[−1/(𝜋𝑛)]𝐼𝑚𝜒,𝑞,𝜔>0; 𝜒, is the density response function at full coupling strength defined in Ref. [27). The dashes show the average frequency <𝜔]𝑞>=𝜔𝑆(𝑞,𝜔)/𝑆𝑞,𝜔.klkl The bulk plasma frequency is 𝜔]0=[4𝜋𝑛],/'. For a plot of 𝑆𝑞=𝑆𝑞,𝜔kl, refer to Fig. S6.   \t10\t Fig. 3. An analogous plot to Fig. 2 presenting the standard deviation in the frequency of a density fluctuation, with the variance defined as <∆\t𝜔]𝑞>'=𝜔'𝑆(𝑞,𝜔)𝑑𝜔/𝑆𝑞,𝜔𝑑𝜔−\tkl𝜔]𝑞'kl, as a function of its wavevector 𝑞. Some noise is present at small 𝑞, where the spectral function is small for all values of 𝜔.  At 𝑟M=4, the variance is monotonic, but at 𝑟M=69, the variance begins decreasing as 𝑞 approaches its critical value (2.28𝑘U) for the static charge-density wave. There is a rather sudden increase in the width of the spectral function as the plasmon enters the continuum of single particle-hole excitations (at q/𝑘U>0.9 and 1.4 for 𝑟M = 4 and 69 respectively) that is not reflected in the variance. The variance is controlled by a much higher and narrower central peak. See the contour plots of 𝑆𝑞,𝜔 (and the dielectric function) in the Supplementary Materials.   \t11\tTable 1. Mean absolute errors (MAE) in electron volts for the atomization energies of the six representative AE6 [3] sp-bonded molecules, for Hartree-Fock exchange (x) and for exchange-correlation (xc) functionals on the first three rungs of a ladder of approximations (none of them fitted to bonded systems). Hartree-Fock results from Lynch and Truhlar [3]. Local spin density approximation (LSDA) [1,4], Perdew-Burke-Ernzerhof (PBE) [5] generalized gradient approximation (GGA), and strongly constrained and appropriately normed (SCAN) [6] meta-GGA results from Bhattarai et al. [7]. Note that the semilocal approximations LSDA, PBE, and SCAN are much better here for xc together than for x or c separately, due to an understood cancellation between the full nonlocalities of exact x and exact c. The LSDA exchange-correlation energy density depends only on the local spin densities, the GGA further includes the density gradients, and the meta-GGA still further includes the orbital kinetic energy densities.  Approximation AE6 MAE (eV) Hartree-Fock x 6.3 LSDA xc 3.3 PBE GGA xc 0.6 SCAN meta-GGA xc 0.1    \t12\tSupplementary Materials for: Interpretations of ground-state symmetry breaking and strong correlation in wavefunction and density functional theories  John P. Perdew, Adrienn Ruzsinszky, Jianwei Sun, Niraj K. Nepal, and Aaron D. Kaplan  \n Fig. S1. Spectral function or structure factor 𝑆𝑞,𝜔 as defined in the main text for a jellium of bulk density parameter 𝑟M=4.   \t13\t Fig. S2. Spectral function or structure factor 𝑆𝑞,𝜔 as defined in the main text for a jellium of bulk density parameter 𝑟M=69.   \t14\t Fig. S3. Same as Fig. S2 but plotting the logarithm of the spectral function.   \t15\t Fig. S4. Real (left) and imaginary (right) parts of the dielectric function 𝜖𝑞,𝜔=\t1−\tpqrs+𝑓BC𝑞,𝜔𝜒uv(𝑞,𝜔) in a jellium of 𝑟M=4. 𝑓BC𝑞,𝜔 is the exchange-correlation kernel of the MCP07 functional, and 𝜒uv(𝑞,𝜔) is the Lindhard susceptibility [27]. Note that the interacting response function satisfies 𝜒𝑞,𝜔=𝜒uv𝑞,𝜔/𝜖𝑞,𝜔. In the upper panels, ℑ𝜔=\t𝜔](0)×104p, and in the bottom panels, \tℜ𝜔=\t𝜔](0)×104p. The real or imaginary parts of 𝜖𝑞,𝜔 vanish on the black dashed curves. The upper black dashed curve in the top left panel resembles the real part of the complex plasmon frequency 𝜔(𝑞) defined by 𝜖𝑞,𝜔=0 for complex 𝜔 in Ref. [27].   \t16\t Fig. S5. Same as Fig. S4, but with 𝑟M=69.   \t17\t Fig. S6. The static structure factor, 𝑆𝑞=\t𝑆𝑞,𝜔𝑑𝜔kl for jellium, which should vary from 0 at 𝑞→0 to 1 at 𝑞→∞ (the gray dashed line is 𝑆𝑞=\t1). At 𝑟M=4 (roughly the valence electron density of solid sodium), 𝑆𝑞 exhibits smooth, slowly-varying behavior in 𝑞. At 𝑟M=69 (a much lower density), 𝑆𝑞 peaks up sharply around the wavevector of the incipient static charge-density wave. For the relationship between 𝑆𝑞 and the pair density 𝑛'𝑔(|𝒓?−𝒓|), see Eq. (6) of Ref. [35].  \t18\t Fig. S7. From the sum rule of Eq. 3.141 of Ref. [22], the exact <𝜔](𝑞)> as defined in Fig. 2 of the main text equals 𝑞'/[2𝑆𝑞]. This plot of 𝑞'/2𝑆𝑞 (dashed) and <𝜔](𝑞)> (solid) demonstrates that the approximate susceptibility used throughout this work [27] quite accurately recovers the sum rule on the exact susceptibility for most values of 𝑞.  "    },    {        "title": "2008.01103v1.Effects_of_density_dependent_scenarios_of_in_medium_nucleon_nucleon_interactions_in_heavy_ion_collisions.pdf",        "content": "arXiv:2008.01103v1  [nucl-th]  3 Aug 2020Effects of density-dependent scenarios of in-medium nucleo n-nucleon interactions in\nheavy-ion collisions\nGao-Feng Wei,1,2,∗Chang Xu,3,†Wei Xie,1,2Qi-Jun Zhi,1,2Shi-Guo Chen,1and Zheng-Wen Long4\n1School of Physics and Electronic Science, Guizhou Normal Un iversity, Guiyang 550025, China\n2Guizhou Provincial Key Laboratory of Radio Astronomy and Da ta Processing,\nGuizhou Normal University, Guiyang 550025, China\n3School of Physics, Nanjing University, Nanjing 210008, Chi na\n4College of Physics, Guizhou University, Guiyang 550025, Ch ina\nUsing a more reasonable separate density-dependent scenar io instead of the total density-\ndependent scenario for in-medium nn,ppandnpinteractions, we examine effects of differences\nof in-medium nucleon-nucleon interactions in two density- dependent scenarios on isospin-sensitive\nobservables in central197Au+197Au collisions at 400 MeV/nucleon. It is shown that the symme-\ntry potentials and resulting symmetry energies in two densi ty-dependent scenarios indeed become\nto deviate at nonsaturation densities, especially at supra saturation densities. Naturally, several\ntypical isospin-sensitive observables such as the free neu tron-proton ratios and the π−/π+ratios\nin heavy-ion collisions are affected significantly. Moreove r, to more physically detect the differ-\nences between the nucleon-nucleon interactions in two dens ity-dependent scenarios, we also map\nthe nucleon-nucleon interaction in the separate density-d ependent scenario into that in the total\ndensity-dependent scenario through fitting the identical c onstraints for symmetric nuclear matter\nas well as the identical slope parameter of nuclear symmetry energy at the saturation density. It is\nshown that two density-dependent scenarios also lead to ess entially different symmetry potentials\nespecially at high densities although they can lead to the id entical equation of state for the sym-\nmetry nuclear matter as well as the identical symmetry energ y for the isospin asymmetric nuclear\nmatter. Consequently, these isospin-sensitive observabl es are also appreciably affected by the differ-\nent density-dependent scenarios of in-medium nucleon-nuc leon interactions. Therefore, according to\nthese findings, it is suggested that effects of the separate de nsity-dependent scenario of in-medium\nnucleon-nucleon interactions should be taken into account when probing the high-density symmetry\nenergy using these isospin-sensitive observables in heavy -ion collisions.\nI. INTRODUCTION\nSimulations of heavy-ion collisions (HICs) as well as\ncomparisonswith the correspondingexperiments provide\nan important tool to explore the properties of strong in-\nteracting nucleonic matter at extreme conditions. As the\nimportant inputs in simulations of HICs, the density-\ndependent nucleon-nucleon interactions as well as the re-\nsulting nuclear mean field have been paid much atten-\ntion in the past few decades [ 1–12]. However, the nuclear\nmean field especially its isovector part, i.e., the symme-\ntry potential, is still incompletely understood at present.\nEssentially, the symmetry potential is determined by\nthe competition between the isospin singlet and isospin\ntriplet channels of nucleon-nucleon interactions [ 13–15].\nThe symmetry potential is also found to be sensitive\nto the in-medium effects such as the in-medium nuclear\neffective many-body force and the tensor force due to\nthe in-medium ρ-meson exchange [ 16]. In nonrelativistic\nmodels, the in-medium many-body forces effects are usu-\nally taken into account by a density-dependent term in\nthe two-body effective interactions [ 17–19], and the rel-\nativistic models generate the similar density-dependent\nterm in the two-body effective interactions as in the non-\n∗E-mail: wei.gaofeng@gznu.edu.cn\n†E-mail: cxu@nju.edu.cnrelativistic models through dressing of the in-medium\nspinors [20]. Nevertheless, how exactly does the resulting\ntwo-body effective interaction depend on the in-medium\nnucleon densities remains an open question. For exam-\nple, a total density-dependent scenario without distin-\nguishing the density dependence for in-medium nn,pp\nandnpinteractions is usually assumed in the Skyrme,\nM3Y and Gogny forces and then adopted in some the-\noretical simulations of HICs [ 21–27]. However, within\nthe Brueckner theory [ 28–30], Brueckner and Dabrowski\npointed out that the G-matrix of nucleon-nucleon inter-\nactions depends strongly on the respective Fermi mo-\nmenta of neutrons and protons in isospin asymmetric nu-\nclear matter. Actually, the separate density-dependent\nscenario for in-medium nucleon-nucleon interactions has\nbeen used in studying the structure of finite nuclei as\nwell as properties of infinite nuclear matter by some au-\nthors such as Negele [ 31], Sprung and Banerjee [ 32] as\nwell as Brueckner and Dabrowski [ 28–30]. Of partic-\nular interest, authors in Refs. [ 33] and [34] employing\nthe Gogny effective interactions, respectively, studied ef-\nfects of the separate density dependence of in-medium\nnucleon-nucleon interactions on the symmetry potential\nand energy, they found consistently that the resulting\nsymmetry potential and energy at nonsaturation densi-\nties in the separate density-dependent scenario indeed\nbecome to deviate significantly from those in the total\ndensity-dependent scenario. Stimulated by these studies,2\nwe examine effects of differences of in-medium nucleon-\nnucleon interactions in these two density-dependent sce-\nnarios on isospin-sensitive observables in HICs at in-\ntermediate energies. The main purpose of this article\nis to answer whether one needs to consider the sepa-\nrate density-dependent scenario for in-medium nucleon-\nnucleon interactions in HICs, especially for probing the\ndensity-dependent nuclear symmetry energy using these\nisospin-sensitive observables in HICs, which is seldom\nconsidered in simulations of HICs to our best knowledge.\nII. THE MODEL\nFor completeness, we first recall the total density-\ndependent scenario for in-medium nucleon-nucleon inter-\nactions according to the original Gogny effective interac-\ntion [35],\nv(r) =/summationdisplay\ni=1,2(W+BPσ−HPτ−MPσPτ)ie−r2/µ2\ni\n+t0(1+x0Pσ)/bracketleftbig\nρ/parenleftbigri+rj\n2/parenrightbig/bracketrightbigαδ(rij), (1)\nwhereW,B,H,M, andµare five parameters, and\nPτandPσare the isospin and spin exchange operators,\nrespectively; while αis the density-dependent parame-\nter used to mimic in-medium effects of the many-body\ninteractions, particularly, the case with α= 1 corre-\nsponds to an effective density-dependent two-body in-\nteraction deduced from a three-body contact interaction\nin spin-saturated nuclear matter [ 17,34]. Based on the\nHatree-Fock approximation using the original Gogny ef-\nfective interaction, i.e., Eq. ( 1), Daset al.derived a\nmomentum-dependent interaction (MDI) single-nucleon\npotential for the Boltzmann-Uehling-Uhlenbeck (BUU)\ntransport model expressed as [ 6,36]\nU(ρ,δ,/vector p,τ) =Au(x)ρ−τ\nρ0+Al(x)ρτ\nρ0\n+B(ρ\nρ0)σ(1−xδ2)−8τxB\nσ+1ρσ−1\nρσ\n0δρ−τ\n+2Cl\nρ0/integraldisplay\nd3p′fτ(/vector p′)\n1+(/vector p−/vector p′)2/Λ2\n+2Cu\nρ0/integraldisplay\nd3p′f−τ(/vector p′)\n1+(/vector p−/vector p′)2/Λ2, (2)\nwhereτ= 1/2 for neutrons and −1/2 for protons; while\nparameters Au(x) andAl(x) are determined as\nAu(x) =−95.98−x2B\nσ+1, Al(x) =−120.57+x2B\nσ+1.\n(3)\nHere, the parameter xis related to the spin(isospin)-\ndependent parameter x0viax= (1 + 2 x0)/3 in the\ndensity-dependent term of original Gogny effective in-\nteractions, which controls the relative contributions of\nthe density-dependent term to the total energy in theisospin singlet channel [ ∝(1+x0)ρα+1] and triplet chan-\nnel [∝(1−x0)ρα+1] [35]. Therefore, varying the xpa-\nrameter can cover uncertainties of the spin(isospin) de-\npendence of in-medium many-body forces which are re-\nsponsible for the divergent density dependence of nu-\nclear symmetry energy in the Gogny Hatree-Fock calcu-\nlations [7,15,34]. However, it should be emphasized\nthat the xparameter does not affect the equation of\nstate of symmetric nuclear matter as well as the sym-\nmetry energy at the saturation density due to the con-\ntributions of different channels are cancelled out exactly,\ni.e.,∝(1 +x0)ρα+1+ (1−x0)ρα+1= 2ρα+1. The pa-\nrameters B= 106.35 MeV and σ= 4/3 in the MDI\nsingle-nucleon potential are related to t0andαin the\noriginal Gogny effective interactions via t0=8\n3B\nσ+11\nρσ\n0\nandσ=α+ 1 [15,33]. While Cu=−103.4 MeV and\nCl=−11.7 MeV are the interaction strength parame-\nters for a nucleon with isospin τinteracting, respectively,\nwith unlike and like nucleons in the nuclear matter, and\nthusaccountforthemomentumdependenceofthesingle-\nnucleon potential. These parameters are all obtained by\nfitting the reached consensuses on properties of nuclear\nmatter at the saturation density ρ0= 0.16 fm−3includ-\ning the binding energy E0(ρ0) =−16 MeV, the incom-\npressibility K0= 212MeVfor symmetricnuclearmatter,\nas well as the symmetry energy Esym(ρ0) = 30.5 MeV,\nfor more details about the MDI interaction, see, e.g.,\nRefs. [6,36].\nWhile for the separate density-dependent scenario for\nin-medium nucleon-nucleon interactions, we follow the\nRef. [33] to replace the density-dependent term in the\noriginal Gongy effective interaction by the following\ndensity-dependent term\nVD=t0(1+x0Pσ)[ρτi(ri)+ρτj(rj)]αδ(rij).(4)\nHere, the interaction explicitly depends on densities of\ntwo nucleons at positions riandrjinstead of the to-\ntal density of two-nucleon central position ( ri+rj)/2\nas in the original Gogny effective interaction. With\nthis density-dependent scenario for in-medium nucleon-\nnucleon interactions, the resulting single-nucleon poten-\ntial labelled as the improved MDI single-nucleon poten-\ntial (IMDI) is changed as [ 33]\nU′(ρ,δ,/vector p,τ) =A′\nu(x)ρ−τ\nρ0+A′\nl(x)ρτ\nρ0+B\n2/parenleftbig2ρτ\nρ0/parenrightbigσ(1−x)\n+2B\nσ+1/parenleftbigρ\nρ0/parenrightbigσ(1+x)ρ−τ\nρ/bracketleftbig\n1+(σ−1)ρτ\nρ/bracketrightbig\n+2Cl\nρ0/integraldisplay\nd3p′fτ(/vector p′)\n1+(/vector p−/vector p′)2/Λ2\n+2Cu\nρ0/integraldisplay\nd3p′f−τ(/vector p′)\n1+(/vector p−/vector p′)2/Λ2, (5)\nand the corresponding parameters Au(x) andAl(x) are3\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s51/s48/s54/s48/s57/s48/s69\n/s115/s121/s109/s32/s40/s77/s101/s86/s41/s32/s77 /s68/s73/s40/s45/s49/s41/s32/s38/s32/s73/s77 /s68/s73/s40/s45/s48/s46/s53/s56/s55/s52/s41\n/s32/s77 /s68/s73/s40/s48/s41/s32/s32/s38/s32/s73/s77 /s68/s73/s40/s48/s46/s50/s48/s54/s51/s41\n/s32/s77 /s68/s73/s40/s49/s41/s32/s32/s38/s32/s73/s77 /s68/s73/s40/s49/s41\n/s73/s77/s68/s73/s40/s48/s41/s73/s77/s68/s73/s40/s45/s49/s41\nFIG. 1. (Color online) The density dependencies of nuclear\nsymmetry energies calculated from the MDI and IMDI single-\nnucleon potentials.\nchanged as\nA′\nu(x) =−95.98−2B\nσ+1/bracketleftbig\n1−2σ−1(1−x)/bracketrightbig\n,(6)\nA′\nl(x) =−120.57+2B\nσ+1/bracketleftbig\n1−2σ−1(1−x)/bracketrightbig\n.(7)\nIt should be mentioned that the properties of symmetric\nnuclearmatter arenot changedfrom the MDI interaction\nto the IMDI interaction due to the isospin scalar poten-\ntialsU0(ρ,0,/vector p,τ) =U′\n0(ρ,0,/vector p,τ) by setting δ= 0 and\nρn=ρp=1\n2ρ. While for the isospin asymmetric nuclear\nmatter, the properties are expected to change from the\nMDI interaction to the IMDI interaction.\nShowninFig. 1arethe densitydependenciesofnuclear\nsymmetry energy calculated from the MDI and IMDI\nsingle-nucleon potentials. For parameter x= 1, it is\nseen that the symmetry energy calculated from the IMDI\nsingle-nucleon potential is the same as that calculated\nfrom the MDI single-nucleon potential. This is because\nthe fourth term in the IMDI single-nucleon potential is\nzerowith x= 1whileothertermsareunchangedasinthe\nMDI single-nucleon potential. However, for parameters\nofx=−1 and 0, the symmetry energies calculated from\nthe IMDI single-nucleon potential become stiffer (softer)\nat suprasaturation(subsaturation)densities comparedto\nthose calculated from the MDI single-nucleon potential.\nUndoubtedly, these differences of the symmetry energy\nare resulting from different symmetry potentials gener-\nated in the MDI and IMDI single-nucleon potentials.\nIt should be emphasized that even with the same sym-\nmetry energies, the corresponding symmetry potentials\ncould be very different due to the fact that the symme-\ntry potentials depend not only on the nucleon density\nbut also on the nucleon momentum or energy. There-\nfore, to physically distinguish the symmetry potentials\nderived from the MDI and IMDI single-nucleon poten-\ntials, it is useful to map the nucleon-nucleon interaction\nin the separate density-dependent scenario (i.e., IMDI\nsingle-nucleon potential) into that in the total density-dependent scenario (i.e., MDI single-nucleon potential).\nThis is carried out by fitting the identical constraints for\nsymmetric nuclear matter as well as the identical slope\nparameter of symmetry energy at ρ0, the corresponding\nresultsarealsoshowninFig. 1. Itisseenthatthesymme-\ntry energy calculated from the IMDI single-nucleon po-\ntentialwithmappedparameters x=−0.5874and0 .2063,\nrespectively, is completely identical with that calculated\nfrom the MDI single-nucleon potential with parameters\nx=−1 and 0.\nShown in Fig. 2are the momentum-dependent sym-\nmetry potentials at ρ= 0.5ρ0,ρ0, 1.5ρ0and 2ρ0calcu-\nlated from the MDI and IMDI single-nucleon potentials\nwith parameters x=−1, 0 and 1 as well as the mapped\nparameters x=−0.5874 and 0 .2063 used in the IMDI\nsingle-nucleon potential. Again, with parameter x= 1,\nthe symmetry potentials are completely identical to each\nother in calculations using the MDI and IMDI single-\nnucleon potentials at either low densities or high densi-\nties. While for parameter x=−1, the symmetry po-\ntentials are significantly stronger especially at high den-\nsities in calculations using the IMDI single-nucleon po-\ntential compared to those in calculations using the MDI\nsingle-nucleon potential. However, for the case of pa-\nrameterx= 0, the symmetry potentials calculated from\nthe IMDI single-nucleon potential change from strong to\nweak(weaktostrong)atsuprasaturation(subsaturation)\ndensities with the increase of nucleons momenta, com-\npared to those calculated from the MDI single-nucleon\npotential. As to the mapped symmetry potentials cal-\nculated from the IMDI single-nucleon potential with pa-\nrameters x=−0.5874and0 .2063,itcanbeobservedthat\nthey are also appreciably different from those in calcu-\nlations using the MDI single-nucleon potetnial with pa-\nrameters x=−1 and 0, respectively. Actually, according\nto the relation between the symmetry energy and the\nsingle-nucleon potential [ 15,28–30], i.e.,\nEsym(ρ)≈1\n3t(kF)+1\n6∂U0\n∂k|kFkF+1\n2Usym(kF),(8)\nwitht(k) denotes the nucleon kinetic energy and kF\nrepresents the Fermi momentum of nucleons in sym-\nmetry nuclear matter, one can know that these are\nmainly due to the differences between the symmetry\npotential calculated from the MDI single-nucleon po-\ntential and the mapped symmetry potential calculated\nfrom the IMDI single-nucleon potential. Therefore, ef-\nfects of two density-dependent scenarios for in-medium\nnucleon-nucleon interactions on isospin-sensitive observ-\nables in HICs can be reflected through examining ef-\nfects of the symmetry potential calculated from the MDI\nsingle-nucleon potential and the mapped symmetry po-\ntential calculated from the IMDI single-nucleonpotential\non these observables in HICs. Nevertheless, it should be\nemphasized that the differences between the MDI sym-\nmetry potential and the IMDI mapped symmetry po-\ntential are essentially resulting from different density-\ndependent scenarios because the momentum-dependent4\n/s45/s50/s48/s45/s49/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48\n/s48 /s49 /s50 /s51 /s52/s45/s52/s48/s45/s50/s48/s48/s50/s48/s52/s48/s54/s48\n/s48 /s49 /s50 /s51 /s52/s73/s77 /s68/s73/s40/s48/s41\n/s73/s77 /s68/s73/s40/s45/s49/s41/s32/s77/s68/s73/s40/s45/s49/s41/s32/s32 /s32/s73/s77/s68/s73/s40/s45/s48/s46/s53/s56/s55/s52/s41\n/s32/s77/s68/s73/s40/s48/s41/s32/s32/s32 /s32/s73/s77/s68/s73/s40/s48/s46/s50/s48/s54/s51/s41\n/s32/s77/s68/s73/s40/s49/s41/s32/s32/s32 /s32/s73/s77/s68/s73/s40/s49/s41/s40/s97/s41/s32/s32/s85\n/s115/s121/s109/s32/s40/s77/s101/s86/s41/s73/s77 /s68/s73/s40/s48/s41/s73/s77 /s68/s73/s40/s45/s49/s41/s40/s98/s41/s32/s32\n/s45/s49/s48/s48/s49/s48/s50/s48/s51/s48\n/s73/s77 /s68/s73/s40/s48/s41/s73/s77 /s68/s73/s40/s45/s49/s41/s40/s99/s41/s32/s32\n/s112/s32/s40/s71/s101/s86/s47/s99/s41/s73/s77 /s68/s73/s40/s48/s41/s73/s77 /s68/s73/s40/s45/s49/s41/s40/s100/s41/s32/s32\n/s112/s32/s40/s71/s101/s86/s47/s99/s41/s45/s56/s48/s45/s52/s48/s48/s52/s48/s56/s48\nFIG. 2. (Color online) The momentum dependencies of symmetr y potentials calculated from the MDI and IMDI single-nucleo n\npotentials.\nbut thexparameter independent Cterms in Eq. ( 5) are\ncompletely identical to that in Eq. ( 2). As a result, ac-\ncording to the formula of nucleon effective mass, i.e.,\nm∗\nτ/m=/bracketleftbig\n1+m\nkτdUτ\ndk/bracketrightbig−1, (9)\nwhich is only related to the Cterms in Eq. ( 2) and\nEq. (5), one can know that the nucleon effective mass\nas well as its isospin splitting are not changed from the\nMDI interaction to the IMDI interaction.\nIII. RESULTS AND DISCUSSIONS\nNow, wecompareeffectsofthesymmetrypotentialcal-\nculated from the MDI single-nucleon potential and the\nmapped symmetry potential calculated from the IMDI\nsingle-nucleon potential on isospin-sensitive observables\ninHICs. Ascomparisons,wealsoincludethecorrespond-\ning results calculated from the IMDI single-nucleon po-\ntential with parameters x=−1 and 0 in the following\ndiscussions./s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s49/s50/s51/s52/s53/s54/s40/s110/s47/s112/s41\n/s102/s114/s101/s101\n/s116/s32/s40/s102/s109/s47/s99/s41/s32/s77 /s68/s73/s40/s45/s49/s41\n/s32/s77 /s68/s73/s40/s48/s41\n/s32/s77 /s68/s73/s40/s49/s41\n/s32/s73/s77 /s68/s73/s40/s45/s48/s46/s53/s56/s55/s52/s41\n/s32/s73/s77 /s68/s73/s40/s48/s46/s50/s48/s54/s51/s41\n/s32/s73/s77 /s68/s73/s40/s49/s41/s73/s77 /s68/s73/s40/s45/s49/s41\n/s73/s77 /s68/s73/s40/s48/s41\nFIG. 3. (Color online) Evolutions of free neutron-proton ra -\ntios in calculations with the MDI and IMDI single-nucleon\npotentials.\nShow in Fig. 3are the free neutron-proton ratios\ngenerated in central197Au +197Au collisions at 400\nMeV/nucleonwherethefreeneutronsandprotonsarede-\nfined as those with local densities less than ρ0/8. First,5\nas expected, with parameter x= 1, the free neutron-\nproton ratio generated in simulations using the IMDI\nsingle-nucleon potential is completely identical with that\nin simulations using the MDI single-nucleon potential.\nSecond, with parameter x=−1 and 0, the free neutron-\nproton ratios generated in simulations using the IMDI\nsingle-nucleon potential are larger especially at the com-\npressionstagecomparedtothose in simulationsusing the\nMDI single-nucleon potential, reflecting the free neutron-\nproton ratios are indeed sensitive to high-density behav-\niorsofnuclearsymmetrypotentialand/orenergybecause\nthe stronger positive symmetry potentials at high densi-\nties get more neutrons to be spread but more protons to\nbe gathered. Certainly, for the case of parameter x= 0,\nthe competition of symmetry potentials at high densities\nbetween low nucleon momentum and high nucleon mo-\nmentum as aforementioned causes the observed effects\nto be not so obvious as those in the case of parameter\nx=−1. However, it should be emphasized that, besides\nthe different density-dependent scenarios of in-medium\nnucleon-nucleoninteractions, these effects are alsoresult-\ning from the different symmetry energy settings because\nthe slope values Lof nuclear symmetry energy at ρ0are\ncompletely different although the identical parameters x\nare used in the MDI and IMDI single-nucleon poten-\ntials. Therefore, to more physically detect the effects\nof differences of in-medium nucleon-nucleon interactions\nin two density-dependent scenarios on the free neutron-\nproton ratios, we compare the free neutron-proton ratios\ngenerated in calculations using the MDI single-nucleon\npotential with parameters x=−1 and 0 and those in\ncalculation using the IMDI single-nucleon potential with\nmapped parameters x=−0.5874 and 0 .2063 as well as\nthose in calculations using the IMDI single-nucleon po-\ntential with parameters x=−1 and 0. As shown also\nin Fig.3, the free neutron-proton ratios in calculations\nusing the IMDI single-nucleon potentials with mapped\nparameters x=−0.5874 and 0 .2063 are obviously lower\nthan those in calculations using the IMDI single-nucleon\npotentials with parameters x=−1 and 0, respectively.\nCertainly, the values of free neutron-proton ratios in cal-\nculations using the IMDI single-nucleon potentials with\nmapped parameters x=−0.5874 and 0 .2063 are also\nlarger than those in calculations using the MDI single-\nnucleon potentials with parameters x=−1 and 0, re-\nspectively. Actually, the symmetry energy is not directly\nentering into the reaction process, and thus does not di-\nrectly affect the free neutron-proton ratios. On the con-\ntrary, the different symmetry energy is originated from\ndifferent symmetry potential, i.e., the isovector part of\nsingle-nucleon potential under different xparameter set-\ntings. Naturally, these different single-nucleon potentials\ndominate the different reaction dynamics as well as dif-\nferent free neutron-proton ratios which thus indirectly\nreflect different symmetry energy settings. This can be\nconfirmed by comparing the reduced maximum densities\nρmax/ρ0reached at the maximum compression stages in\ncollisions with the IMDI single-nucleon potential under/s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48/s50/s46/s50/s48/s50/s46/s50/s50/s50/s46/s50/s52/s50/s46/s50/s54/s109/s97/s120/s102/s109/s45/s51\n/s116/s32/s40/s102/s109/s47/s99/s41/s32/s77 /s68/s73/s40/s45/s49/s41\n/s32/s73/s77 /s68/s73/s40/s45/s48/s46/s53/s56/s55/s52/s41\n/s32/s73/s77 /s68/s73/s40/s45/s49/s41\nFIG. 4. (Color online) Evolutions of reduced maximum den-\nsitiesρmax/ρ0reached at the maximum compression stages in\ncollisions with the MDI and IMDI single-nucleon potentials .\nsetting two different parameters x=−1 (IMDI(-1)) and\nx=−0.5874 (IMDI(-0.5874)) as shown in Fig. 4. Never-\ntheless, for the case with the MDI single-nucleon poten-\ntial under setting x=−1 (MDI(-1)) and that with the\nIMDI single-nucleonpotential under setting x=−0.5874\n(IMDI(-0.5874)), the resulting ρmax/ρ0are also differ-\nent although their corresponding symmetry energies are\ncompletely identical as shown in Fig. 1. Actually, ac-\ncording to the formula ( 8) as well as the Fig. 2, this is\nexactly the difference of in-medium nucleon-nucleon in-\nteractions in two density-dependent scenarios that leads\nto the different ρmax/ρ0in collisions, and thus gener-\nates different free neutron-proton ratios. Correspond-\ningly, the kinetic energy distributions of free neutron-\nproton ratios at the end of reactions are also affected by\nthe density-dependent scenarios of in-medium nucleon-\nnucleoninteractionsasshowninFig. 5. Therefore,effects\nof the separate density-dependent scenario of in-medium\nnucleon-nucleon interactions should be carefully consid-\nered in studies of using the free neutron-proton ratio as\na probe of nuclear symmetry energy especially at high\ndensities.\nOntheotherhand, accordingtotheproductionmecha-\nnism of pions, i.e., pions areproduced mainly at the com-\npression stages during collisions and π−is mainly from\nnninelasticcollisionsbut π+mainlyfrom ppinelasticcol-\nlisions, one naturally expects that the effects of density-\ndependent scenarios of in-medium nucleon-nucleon inter-\nactions also hold for the π−/π+ratio, which has been\nindicated to be very sensitive to the symmetry energy\nand potential at high densities [ 37–41] but still affected\nby some incompletely known uncertainties [ 42–50]. In\nHICs at intermediateenergies, pions areproduced during\ncollisions mostly from the decay of ∆(1232), therefore,\nit is useful to examine the effects of density-dependent\nscenarios of in-medium nucleon-nucleon interactions on6\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48/s48/s46/s56/s49/s46/s48/s49/s46/s50/s49/s46/s52/s49/s46/s54\n/s32/s77 /s68/s73/s40/s45/s49/s41\n/s32/s77 /s68/s73/s40/s48/s41\n/s32/s77 /s68/s73/s40/s49/s41/s32/s73/s77 /s68/s73/s40/s45/s48/s46/s53/s56/s55/s52/s41\n/s32/s73/s77 /s68/s73/s40/s48/s46/s50/s48/s54/s51/s41\n/s32/s73/s77 /s68/s73/s40/s49/s41/s40/s110/s47/s112/s41\n/s102/s114/s101/s101\n/s69\n/s107/s105/s110/s32/s40/s77/s101/s86/s41/s73/s77 /s68/s73/s40/s45/s49/s41\n/s73/s77 /s68/s73/s40/s48/s41\nFIG. 5. (Color online) Kinetic energy distributions of free\nneutron-proton ratios at the end of reactions with the MDI\nand IMDI single-nucleon potentials.\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s50/s46/s50/s50/s46/s52/s50/s46/s54/s50/s46/s56/s51/s46/s48\n/s73/s77 /s68/s73/s40/s48/s41\n/s73/s77 /s68/s73/s40/s45/s49/s41/s40 /s41\n/s108/s105/s107/s101\n/s116/s32/s40/s102/s109/s47/s99/s41/s32/s77/s68/s73/s40/s45/s49/s41/s32 /s32/s73/s77/s68/s73/s40/s45/s48/s46/s53/s56/s55/s52/s41\n/s32/s77/s68/s73/s40/s48/s41/s32/s32 /s32/s73/s77/s68/s73/s40/s48/s46/s50/s48/s54/s51/s41\n/s32/s77/s68/s73/s40/s49/s41/s32/s32 /s32/s73/s77/s68/s73/s40/s49/s41\nFIG. 6. (Color online) Evolutions of π−/π+ratios in calcula-\ntions with the MDI and IMDI single-nucleon potentials.\ndynamic pion ratio ( π−/π+)like, i.e.,\n(π−/π+)like=π−+∆−+1\n3∆0\nπ++∆+++1\n3∆+.(10)\nCertainly, because all the ∆ resonances will eventually\ndecay into nucleons and pions, the ratio ( π−/π+)likewill\nnaturallybecometothefree π−/π+ratioattheendofre-\nactions. ShowninFig. 6aretheevolutionsof( π−/π+)like\nratios generated in central197Au +197Au collisions at\n400 MeV/nucleon. First, it is obvious to see that the\nπ−/π+ratio is indeed sensitive to the density depen-\ndence of nuclear symmetry energy regardlessof using the\nIMDI single-nucleon potential or the MDI single-nucleon\npotential, and a softer symmetry energy usually leads to\na higher π−/π+ratio, reflecting a more neutron-rich par-\nticipant region formed in the reaction [ 37–41]. Second,\nwith parameter x= 1, the π−/π+ratio calculated from\nthe IMDI single-nucleon potential is completely identical\nwith that calculated from the MDI single-nucleon poten-\ntial. Third, for the case of identical parameter x=−1,\nit is seen that the stronger positive symmetry potentialat high densities gets the π−/π+ratio in calculations us-\ning the IMDI single-nucleon potential to be significantly\nsmaller than that in calculations using the MDI single-\nnucleon potential. Actually, due to the stronger posi-\ntive symmetry potential with parameter x=−1 causes\nmore neutrons to be spread but more protons to be gath-\nered, naturally, according to the production mechanism\nof pions, i.e., π−is produced mainly from the channel\nn+n→π−+pbutπ+from the channel p+p→π++p,\nwe can observea smaller π−/π+ratios in calculations us-\ning the IMDI single-nucleon potential. Certainly, due to\nthe symmetry potential calculated from the IMDI single-\nnucleon potential with parameter x= 0 changes from\nstrong to weak at suprasaturation densities with the in-\ncrease of nucleons momenta compared to that calculated\nfrom the MDI single-nucleon potential with parameter\nx= 0,wecanalsoseethatthedifferencesof π−/π+ratios\ninthiscasearenotaslargerasthoseinthecaseofparam-\neterx=−1. Again, these effects are resulting from both\ndifferent density-dependent scenarios and different sym-\nmetry energy settings. While comparing the π−/π+ra-\ntios in calculations using the IMDI single-nucleon poten-\ntial with mapped parameters x=−0.5874 and 0 .2063\nwith those in calculations using the MDI single-nucleon\npotential with parameters x=−1 and 0, respectively,\nwe find again the similar observations as those in free\nneutron-proton ratios due to the IMDI mapped symme-\ntry potential is after all appreciably different from the\nMDI symmetry potential. Especially, due to the differ-\nences of the IMDI mapped symmetry potential with pa-\nrameter x=−0.5874 and the MDI symmetry potential\nwith parameter x=−1 are relative large at both low nu-\ncleon momentum and high nucleon momentum, the cor-\nrespondingeffectsonthe π−/π+ratiosarerelativeappre-\nciable. Certainly, it should be emphasized that this effect\nis independent of nuclear symmetry energy but is exactly\nresultingfromdifferentdensity-dependentscenariosofin-\nmedium nucleon-nucleon interactions. Therefore, effects\nof the separate density-dependent scenario of in-medium\nnucleon-nucleoninteractionsshouldalsobecarefullycon-\nsidered in studies of using the π−/π+ratio as a probe of\nnuclear symmetry energy especially at high densities.\nBefore ending this part, we give two useful remarks.\nFirst, although there are currently no physical stud-\nies based on first principles to illustrate more accuracy\nof the separate density-dependent scenario, some results\nrelevant to nuclear structure studies have been shown to\nyield very satisfactory agreement with the corresponding\nexperiments such as the binding energies, single-particle\nenergies, and electron scattering cross sections for16O,\n40Ca,48Ca,90Zr, and208Pr [31,51]. Moreover, as in-\ndicated in Ref. [ 51], the separate density dependence\nof effective two-body interactions is originated from the\nrenormalization of multibody force effects, and the latter\nmay extend the density dependence of effective interac-\ntions for calculations beyond the mean-field approxima-\ntion and open a new freedom in the effective interactions.\nSecond, it is well known that the double neutron-7\nproton and/or π−/π+ratios from two reaction systems\nhave the advantage of reducing both systematic errors\nand the influences of isoscalarpotentials in HICs [ 52,53].\nThis could enlarge the contribution of the isovector po-\ntentials and better discriminate between the two sce-\nnarios. Therefore, the double ratios of these observ-\nables from two reactions as well as the cross examina-\ntionsoftheseobservablesusingvariousexperimentaldata\nsuch as the FOPI data [ 40] and that from the symme-\ntry energy measurement experiment at RIBF-RIKEN in\nJapan [54] could be good candidates in probing the ef-\nfects of density-dependent scenarios in HICs in future.\nIV. SUMMARY\nIn conclusion, we have studied effects of differences\nof in-medium nucleon-nucleon interactions in the sepa-\nrate and total density-dependent scenarios on isospin-\nsensitive observables in HICs within a transport model.\nConsistent with the previous studies, the nuclear sym-\nmetry energy and potential at nonsaturation densities in\nthe separate density-dependent scenario indeed become\nto deviate significantly from those in the total density-\ndependent scenario for the identical xparameters except\nfor the parameter x= 1. Two typical isospin-sensitive\nobservables including the free neutron-proton ratios and\ntheπ−/π+ratios in HICs are affected significantly. Nev-\nertheless, it should be emphasized that these effects\nare resulting from both the different symmetry energysettings and the different density-dependent scenarios\nof in-medium nucleon-nucleon interactions. Therefore,\nto more physically detect the differences of in-medium\nnucleon-nucleoninteractionsas well as the resulting sym-\nmetry potential in two density-dependent scenarios, we\nhave also mapped the nucleon-nucleon interaction in the\nseparatedensity-dependentscenariointothatin thetotal\ndensity-dependent scenario through fitting the identical\nconstraints for symmetric nuclear matter as well as the\nidentical slope parameter of symmetry energy at the sat-\nuration density. It is shown that the mapped symmetry\npotentials calculated from the IMDI single-nucleon po-\ntentialindeeddeviatefromthoseincalculationsusingthe\nMDI single-nucleon potential especially at high densities.\nConsequently, these isospin-sensitiveobservables in HICs\ncould also be appreciably affected. Therefore, according\nto these findings as well as the Brueckner theory and\nprevious findings in nuclear structure studies, we con-\nclude that effects of the separate density-dependent sce-\nnario for in-medium nucleon-nucleon interactions might\nbe very important and thus should be taken into account\nwhen probing the high-density symmetry energy using\nthese isospin-sensitive observables in HICs.\nACKNOWLEDGMENTS\nThis work is supported by the National Natural Sci-\nence Foundation of China under grant Nos.11965008,\n11822503, U1731218, 11847102, and Guizhou Provin-\ncial Science and Technology Foundation under Grant\nNo.[2020]1Y034,and the PhD-funded project of Guizhou\nNormal university (Grant No.GZNUD[2018]11).\n[1] V. R. Pandharipande, V. K. Garde, Phys. Lett. B 39,\n608 (1972).\n[2] J. A. McNeil, J. R. Shepard, and S. J. Wallace, Phys.\nRev. Lett. 50, 1439 (1983).\n[3] R. B. Wiringa, V. Fiks, and A. Fabrocini, Phys. Rev. C\n38, 1010 (1988).\n[4] M. Kutschera, Phys. Lett. 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Nuclear Instru-\nments and Methods in Physics Research A 784, 513\n(2015)."    },    {        "title": "2008.01324v2.Electrically_induced_charge_density_waves_in_a_two_dimensional_electron_channel__Beyond_the_Local_Density_Approximation.pdf",        "content": "arXiv:2008.01324v2  [cond-mat.mes-hall]  22 Nov 2020Electrically induced charge-density waves in a two-dimens ional electron channel:\nBeyond the Local Density Approximation\nErica Hroblak, Mohammad Zarenia, and Giovanni Vignale\nDepartment of Physics and Astronomy, University of Missour i, Columbia, Missouri 65211, USA\nIn a previous paper we suggested that a macroscopic force fiel d applied across a two-dimensional\nelectron gas channel could induce a microscopic charge dens ity wave as soon as the proper compress-\nibility becomes negative, which happens at densities much h igher than the critical density for the\nWigner crystal transition. The suggestion was based on a cal culation of the ground state energy in\nthe local density approximation. In this paper we refine our c alculation of the energy by including\na self-consistent gradient correction to the kinetic energ y. Due to the increased energy cost of rapid\ndensity variations, we find a much lower critical density for the onset of the charge density wave.\nThis critical density coincides with the result of a linear s tability analysis of the uniform ground\nstatein the absence of the electric field .\nI. INTRODUCTION\nSpontaneous breaking of translational symmetry in\nelectron liquids is a classic problem in condensed matter\nphysics. The localization of electrons at sufficiently low\ndensity was first proposed by Wigner in 1934. [ 1,2] He\nnoticed that in this regime the kinetic energy would play\na secondary role, allowing the electrons to crystalize in a\nform that he termed “an inverted alkali metal”. A few\ndecades later Overhauser showed that even at the high\ndensities of (real) alkali metals, exchange and correlation\neffects would encourage the formation of a charge den-\nsity wave state (CDW) with wave vector q∼2kF, where\nkFis the Fermi wave vector.[ 3] Small ionic displacement\nwould cancel out a large part of the Coulombic energy\narising from the non-uniform charge distribution, thus\nstabilizing the CDW ground state. More recently, bro-\nken translational symmetry has been predicted in stripe\nand bubble phases in electronic systems at high magnetic\nfield, where the magnetic field plays a crucial role in sup-\npressing the kinetic energy of the electrons.[ 4,5] Stripes\nhave also been predicted to occur in high- Tcsupercon-\nductors, of which the copper-oxide superconductors are\nthe most prominent representatives [ 6].\nA different route to stabilizing a CDW state at rela-\ntivelyhighdensitywasproposedinRef. [ 7]. Theideawas\nto “polarize”a two-dimensionalelectron channel(2DEC)\nby applying a uniform in-plane electric field perpendicu-\nlar to the edges of the channel: current flow is prevented\nby insulating barriers running along the channel. Un-\nder “normal” conditions, the electron gas responds to\nthe field by a small rearrangement of its density, re-\nsulting in a small accumulation of electrons along one\nedge and depletion on the other: translational symme-\ntry is not broken on the microscopic scale. The situa-\ntion may change dramatically when the bulk density n\nof the 2DEC falls below the value ( rs∼2.1) for which\nthe thermodynamic compressibility κ= (1\nn2)(∂n\n∂µ) be-\ncomes negative. [ 8–13] A negative compressibility sim-\nply means that, due to the dominance of exchange and\ncorrelation effect, the chemical potential of the charge-\nneutral electron gas decreases with increasing density.By itself, this is not a signal of instability, since the\ndensity of the electronic system is stabilized by the re-\nquirement of macroscopic charge neutrality. And indeed,\nin the last few decades, negative electronic compressibil-\nity has been experimentally observed in two-dimensional\nmaterials through spectroscopy and measurements of\nquantum capacitance in transition metal dichalcogenide\nmonolayers,[ 14] carbon nanotubes [ 15,16], oxide inter-\nfaces, and quantum heterostructures.[ 8,16–18] This ef-\nfect has also been observed in graphene when placed un-\nder a strong magnetic field. [ 13]\nOur proposal in Ref. [ 7] was to combine the negative\ncompressibility with a uniform polarizing electric field.\nWe were surprised to discover that, as soon as the com-\npressibility becomes negative, the equilibrium density in\nthe presence of the field (obtained from a force balance\nequation which is equivalent to minimizing the ground-\nstate energy in a local density approximation) develops\nfast oscillations, which break translational symmetry on\nthe microscopic scale . This may happen at densities\nfar higher (by orders of magnitude) than the ones usu-\nally considered in the context of Wigner crystallization,\nCDWs etc..\nHowever, there is a problem with this prediction. It is\nwell known that the local density approximation for the\nenergy is justifiable only in the limit in which the den-\nsity is slowly varying on the microscopic scale, which is\nthe scale of the local Fermi wavelength k−1\nF. This condi-\ntion was not satisfied in the calculations of Ref. [ 7]: on\nthe contrary - the oscillations of the equilibrium density\nwere found to occur on the scale of the screening length\nλ=4πǫ\ne2/vextendsingle/vextendsingle/vextendsingle∂µ\n∂n/vextendsingle/vextendsingle/vextendsingle, which vanishes at the onset of the instabil-\nity (meaning that the onset of the instability cannot be\ndetermined in a physically reliable manner) and becomes\nat most as large as ∼2.8k−1\nFat the lowest densities.\nThe local density approximation is particularly dan-\ngerous when applied to the kinetic energy (in which case\nit is better known as the Thomas-Fermi approximation).\nIn this paper we continue to use the local density approx-\nimation for exchange and correlation, which is generally\nfound to work well in ab-initio calculations for reasons\nthat are quite well understood, but we improvethe treat-2\nment of the kinetic energy by introducing a “gradient\ncorrection”, which severely discourages rapid variations\nof the density. The gradient correction for the kinetic\nenergy density ǫkin[n(r)] (regarded as a functional of the\ndensityn(r), wherr= (x,y) is the position in space) has\nthe form\nδǫkin[n(r)] =C/planckover2pi12\n2m|∇n(r)|2\nn(r)(1)\nwhereCis a dimensionless constant that can be derived\nfrom linear response theory, as detailed in the Appendix.\nFor small density variations the value C= 1/4 provides\nan upper bound to the energy, meaning that by adopting\nthis value of Cwe almost certainly overestimate the ki-\nnetic energy penalty for forming a charge density wave.\nIndeed, a significantly smaller value of Cwill be argued\nbelow to be physically more accurate.\nAfter including the gradient correction term we repeat\nthe minimization of the total energy of the electron gas\nin the presence of the force field and find that micro-\nscopic translational symmetry is broken at a value of\nthe Wigner-Seitz radius rs≃21.7 that is much larger\nthan the onset of negative compressibility (yet consid-\nerably lower than the typical values associated with the\nonset of Wigner crystallization in the absence of external\nfields). The rapid variation of the density in our solution\nis no longer an issue, since its energy cost is taken into\naccount by the gradient correction term. Unfortunately,we also find that the critical density calculated in this\nmanner coincides with the result of the standard linear\nstability analysis of the uniform 2DEG in the absence of\nthe electric field, i.e., with the critical density of a spon-\ntaneous CDW instability driven by exchange-correlation\neffectsinthelocaldensityapproximation. Thustheques-\ntion whether the uniform electric field actually helps the\nCDWinstabilitytooccurathigherdensityremainsunan-\nswered.\nII. DESCRIPTION OF THE PHYSICAL\nSYSTEM AND FORCE BALANCE EQUATION\nAsinourpreviouspaper, weconsideranelectronchan-\nnel of width Lin thexdirection, with −L/2< x < L/ 2\nand infinite in the ydirection. The system is assumed\nto be translationally invariant in the ydirection, so that\nthe two-dimensional electron density n(x) is a function\nofxonly. This density is expressed as n(x) =n0+δn(x)\nwhere the uniform component is exactly neutralized by\na positively charged background of density n0and\n/integraldisplay+L/2\n−L/2dx δn(x) = 0. (2)\nFrom now on we express xin units of L/2, i.e. we set\nx= ¯xL/2. The total energy per unit area of the system\nis expressed as a functional of the electronic density n(x)\nas follows\nE[¯n(¯x)]\nRy/a2=1\n2/integraldisplay1\n−1d¯x/braceleftbigg\n¯ǫ(¯n(¯x))+4C\n¯L2|¯n′(¯x)|2\n¯n(¯x)+¯V\n2¯x¯n(¯x)−¯L/integraldisplay1\n−1d¯x′δ¯n(¯x)δ¯n(¯x′)ln|¯x−¯x′|/bracerightbigg\n, (3)\nwhereRy=e2\n2ais the Rydberg unit of energy ( ∼13.6\neV),a=/planckover2pi12\nme2is the Bohr radius, eis the absolute value\nof the electron charge. We have introduced the following\ndimensionless variables (all marked by an overbar): (i)\nthe dimensionless 2D density ¯ n=na2=1\nπr2s, and simi-\nlarlyδ¯n= (δn)a2; (ii) thedimensionlessenergy densityof\na 2DEGofuniform electron density ¯ n: ¯ǫ(¯n) =ǫ(¯n)\nRy/a2; (iii)\nthe dimensionless width of the channel ¯L=L/a; (iv) the\ndimensionless potential energy difference ¯V=−FL/Ry\ngenerated by a uniform force Facross the width of the\nchannel. ¯ n′(¯x)denotesthederivativeof ¯ n(¯x)withrespect\nto ¯x. The first term in Eq. ( 3) is the local densityapprox-\nimation, where ¯ ǫincludes the kinetic energy density and\nthe exchange-correlation energy densities [ 9]. The latter\nis accuratelyknownfromthe QMC workofAttaccaliteet\nal. [19]. The second term is the gradient correctionto the\nkineticenergy[ 20–22], discussedintheintroduction. The\nthird term is the potential energy created by the force\napplied across the channel and the fourth term is theHartree energy arising from the inter-particle Coulomb\nrepulsion.\nThe energy functional must be minimized with re-\nspect to the density deviation δ¯n(¯x), subject to the con-\nstraint (2). This gives us the equation\nδE[¯n(¯x)]\nδ¯n(¯x)=µ, (4)\nwhereδE[¯n(¯x)]\nδ¯n(¯x)is the functional derivative of the energy\nfunctional with respect to ¯ n(¯x), andµis a constant La-\ngrange multiplier required to satisfy the constraint ( 2).\nAlternatively, we can write\nd\nd¯xδE[¯n(¯x)]\nδ¯n(¯x)= 0. (5)\nCarrying out the prescribed operations and keeping only\nterms linear in δ¯n(under the assumption that δ¯n≪¯n0)3\n0 0.05 0.1 0.15 0.2 0.25510152025\n1.522.533.5\nFigure 1. Critical Wigner-Seitz radius rsc(blue curve) and\nthe wave vector of the CDW instability at the critical Wigner -\nSeitzradius qc/(2kFc)(redcurve)vscoefficient ofthegradient\ncorrection term C.\nwe arrive at the integrodifferential equation\n¯ǫ′′(¯n0)δ¯n′(¯x)\n2¯L−4C\n¯n0δ¯n′′′(¯x)\n¯L3−−/integraldisplay1\n−1d¯x′δ¯n(¯x′)\n¯x−¯x′=−¯V\n4¯L,(6)\nwhere ¯ǫ′′(¯n0) is the second derivative of the energy den-\nsity evaluated at the background density. This second\nderivative is related to the compressibility K(n) by the\nwell-known formula ǫ′′(n) = 1/(n2K(n)) and becomes\nnegative (passing through zero) for rsgreaterthan about\n2.1.[9] In the above equation ¯ xis restricted to the inter-\nval|¯x|<1 and the integral is evaluated according to the\nCauchy principal value prescription (denoted by a strike\nacross the integral sign). For C= 0 Eq. ( 6) reduces to\nthe force balance equation used in Ref. [ 7]. Notice that\nthe equation is linear in δ¯n, so the induced density is\ndirectly proportional to the force applied, F=−V/L.III. APPROXIMATE SOLUTION\nIn the limit of large channel width (i.e., ¯L→ ∞) it\nmay appear that only the Hartree term on the left side\nof Eq. (6) is relevant. Neglecting the remaining terms\nwould give the elegant classical solution\nδ¯n(¯x) =−¯V\n4¯L¯x\nπ√\n1−¯x2. (7)\nThis conclusion is correct only as long as the density\nremains slowly varying on the macroscopic scale ( L). In\nour previous paper we showed that a negative compress-\nibility can induce rapid variations of the density on a\nmicroscopic scale ( a), so that the derivatives of the den-\nsity on the left hand side of Eq. ( 6) must be taken into\naccount. Unfortunately, rapid variation of the density\nalso means that the validity of the local density approxi-\nmation for the energy (i.e., the first term on the the left\nhand side of Eq. ( 6)) is dubious. Hence the importance\nof including the gradient correction term, which strongly\ndiscourages such rapid variations by imposing a large en-\nergy penalty.\nBased on these considerations, we seek a solution of\nEq. (6) in the form\nδ¯n(¯x) =−¯V\n4¯L/braceleftbigg¯x\nπ√\n1−¯x2+fsin(¯q¯x)/bracerightbigg\n,(8)\nthat is to say the classical solution plus an oscillat-\ning term with dimensionless wave vector ¯ q=qL/2 =\n(qa)¯L/2. We assume q >0, while the real amplitude f\ncan have either sign as needed. This Ansatz has been\nshown in Ref. [ 7] to yield excellent results vis-a-vis the\nexact numerical solution of the integro-differential equa-\ntion.\nWe substitute Eq. ( 8) into Eq. ( 6) and focus on the\nregion|¯x| ≪1 (which, we emphasize, can be macroscop-\nically large if ¯Lis large). In this region we have\n−/integraldisplay+1\n−1d¯x′sin(¯q¯x′)\n¯x−¯x′≃ −πcos(¯q¯x)+π−2Si(¯q),|¯x| ≪1,\n(9)\nwhere Si(¯ q)≡/integraltext¯q\n0sin(t)\ntdtis the Sine Integral function\nand the strike across the integral sign mandates that we\ntake the principal value of the integral. Thus, we arrive\nat the following equation for ¯ qandf:\n/braceleftbigg/bracketleftbigg¯ǫ′′(¯n0)\n2π/parenleftBig¯q\n¯L/parenrightBig\n+4C\nπ¯n0/parenleftBig¯q\n¯L/parenrightBig3\n+1/bracketrightbigg\ncos(¯q¯x)−[1−2Si(¯q)/π]/bracerightbigg\nf+¯ǫ′′(¯n0)\n2π2¯L= 0 (10)\n(we have omitted a term proportional to C/¯L3, which is\nnegligible in the limit ¯L→ ∞).The consistency of our Ansatz requires the coefficient\nofcos(¯q¯x) to vanish, thus determining ¯ q/¯Lasthe positive4\nsolution of the equation\n¯ǫ′′(¯n0)\n2π/parenleftBig¯q\n¯L/parenrightBig\n+4C\nπ¯n0/parenleftBig¯q\n¯L/parenrightBig3\n+1 = 0.(11)\nNoticethat ¯ q/¯L=qa/2soweexpect asolutionwith qa/2\non the order 1, independent of Lin the limit L→ ∞.\nThis corresponds to a microscopic charge density wave.\nGiven that C >0, it is immediately evident that a\nsolution can exist only if the compressibility is negative,\ni.e. for ¯ǫ′′(¯n0)<0. The plain local density approxima-\ntionC= 0 yields a solution for any density such that\n¯ǫ′′(¯n0)<0, but then the wave vector of the solution di-\nverges when the density is close to the the critical value\nfor which ¯ ǫ′′vanishes. On the other hand, a sufficiently\nlarge value of Cmay completely prevent the existence\nof a solution, by making the left hand side of Eq. ( 11)\nalways positive for ¯ q >0.\nFor a given value of Ca solution of Eq. ( 11) first ap-\npears at a critical Wigner-Seitz ratio rscand wave vector\n¯qcsuch that\n¯ǫ′′(rsc)\n2π+12Cr2\nsc/parenleftBig¯qc\n¯L/parenrightBig2\n= 0, (12)\ni.e., when the zero of the left hand side of Eq. ( 11), re-\ngarded as a function of ¯ q, yields the minimum of that\nfunction. Combining Eqs. ( 11) and (12) we find\n¯qc\n¯L=−3π\n¯ǫ′′(rsc), C=/parenleftbigg\n−¯ǫ′′(rsc)\n6πr2/3\nsc/parenrightbigg3\n.(13)\nIt is convenient to express the wave vector qcin terms of\nthe Fermi wave vector kFcat the critical density. Noting\nthatkFca=√\n2/rscwe get\nqc\nkFc=−6πrsc√\n2¯ǫ′′(rsc). (14)\nIn Fig.1we plotqc/kFcand the critical Wigner-Seitz\nradiusrscas functions of C. Clearly the plot is mean-\ningful only for ¯ ǫ′′(rsc)<0, i.e., for rsc>2. Numerical\nvalues of ¯ ǫ′′(rs) are taken from Ref. [ 19]. We note in\npassing that an excellent approximation (for the present\npurpose) to ¯ ǫ′′(rs) forrs>2 is\n¯ǫ′′(rs)≃ −1.35π(rs−2), r s>2.(15)\nFor a given value of Cin the range0 < C <1/4, where\nC= 0 neglects the gradient correction and C= 1/4 is\nan upper bound (see next section) the blue curve in Fig.\n1allows us to read the value of the critical rs, which is\nthen used in Eq. ( 14) to determine qc/(2kFc) (red curve).\nWe see that even in the most unfavorable case, C= 1/4,\nour model predicts a charge density wave instability with\nqc≃3.38kFcand critical rs≃27, which is smaller than\nthe critical rsfor Wigner crystallization in 2D. However,\nthere are good reasons to believe that the appropriate\nvalue of Cis significantly smaller than 1 /4. Indeed, a\nself-consistent gradient correction, described in the nextsection, produces a lower value of rsc. For completeness,\nwe also calculate the amplitude fof the charge density\nwave (normalized by ¯V/(4¯L)). This is given by\nf=¯ǫ′′(¯n0)\n2π2¯L[1−2Si(¯q)/π](16)\nIn the limit of large ¯Lwe make use of the limiting form\n[1−2Si(¯q)/π]→2\nπ¯qcos¯q,¯q≫1,(17)\nto arrive at\nf=¯ǫ′′(¯n0)(qa)\n8πcos(qL/2)(18)\nwhich is finite, but exhibits a wild non-analytic depen-\ndenceon L. Wewillreturntothispointintheconcluding\nsection.\nIV. SELF-CONSISTENT GRADIENT\nCORRECTION\nIn this section we analyze more closely the nature of\nthe gradient correction to the kinetic energy density of a\nweakly inhomogeneous 2DEG ( δn(r)≪n0). To second\norder in δn, the kinetic energy density relative to the\nhomogeneous state is given by\nǫkin[δn] =−1\n2A/integraldisplayd2q\n(2π)2χ−1\n0(q)|δ˜n(q)|2,(19)\nwhereχ0(q) is the static Lindhard function (i.e., the\nstatic density-density response function of the noninter-\nacting 2DEG), δ˜n(q) is the Fourier transform of δn(r)\nat wave vector q, andAis the area of the system. The\nstatic Lindhard function is known analytically:\nχ0(q) =−m\nπ/planckover2pi12\n1−ℜe/radicalBigg\n1−4k2\nF\nq2\n.(20)\nIt is well-known that the local density approxima-\ntion for the kinetic energy is obtained (to second in δn)\nby ignoring the q-dependence of χ0(q), i.e., by setting\nχ−1\n0(q) =χ−1\n0(0) in Eq. ( 19). The exact correction to\nthe local density approximation for the kinetic energy is\ntherefore given by\nδǫkin[δn] =−1\n2A/integraldisplayd2q\n(2π)2/bracketleftbig\nχ−1\n0(q)−χ−1\n0(0)/bracketrightbig\n|δ˜n(q)|2,\n(21)\nThe quantity −/bracketleftbig\nχ−1\n0(q)−χ−1\n0(0)/bracketrightbig\nis plotted in Fig. 2\n(in units of m/(π/planckover2pi12). It is strictly zero for q <2kFand\nremainsalwayslowerthan the parabola q2/2k2\nF, to which\nit tends for q→ ∞. It is easy to verify that the gradient\ncorrection of Eq. ( 1), withC= 1/4 is obtained precisely\nbyapproximating −/bracketleftbig\nχ−1\n0(q)−χ−1\n0(0)/bracketrightbig\nbyitshigh- qlimit5\n0 1 2 3 4 5 605101520\n0 2 400.10.2\nFigure 2. Plot of −m\nπ/planckover2pi12/bracketleftbig\nχ−1\n0(q)−χ−1\n0(0)/bracketrightbig\n(blue-solid line)\nand its large- qasymptote q2/2k2\nF(red-dashed line) vs q/kF.\nInset shows C(q) from Eq. ( 22) vsq/kF.\nπ/planckover2pi12q2\n2mk2\nF. ThereforeEq.( 1), withC= 1/4,definitelyoveres-\ntimates the kinetic energy cost of the inhomogeneity. We\nemphasize that Eq. ( 1) with the choice C= 1/4 provides\na rigorous upper bound to the noninteracting kinetic en-\nergy functional in the linear response regime. This fact\ndoes not seem to have been recognized in the literature\nso far.\nFrom Fig. 2we also see that density waves with wave\nvectors smaller than 2 kFare adequately described by the\nlocal density approximation. Therefore the choice C=\n1/4 is inconsistent, since it predicts a density wave with\nq≃3.38kFc(atrs≃27), which in turn would yield a\nvalue of Csmaller than 0.2 as can be seen in the inset\nof Fig. 2. However, the choice C= 0 would also be\ninconsistent, since it would predict wave vectors much\nlarger than 2 kF, which would imply C= 1/4. These\nconsiderations lead us to propose that the value of the\ncoefficient Cin the kinetic gradient correction can be\nchosen consistently with the value of qcthat it predicts.\nIn other words, we set\nC(q) =−m\nπ/planckover2pi12χ−1\n0(q)−χ−1\n0(0)\n2(q/kF)2, (22)\nsuch that C→1/4 forq→ ∞andC= 0 forq <2kF.\nSubstituting Eq. ( 22) into ourEq. ( 11) and noting that\nχ−1\n0(0) =−π/planckover2pi12\nm¯ǫ′′\n0where ¯ǫ′′\n0is the second derivative of the\ndimensionless kinetic energy density, we see that Eq. ( 11)\ncan be rewritten as\nm\nπ/planckover2pi12χ−1\n0(q/kF)−¯ǫ′′\nxc\n2π−√\n2rs\nq/kF= 0 (23)\nor, after restoring physical units, as\nχ−1(q)≡χ−1\n0(q)−ǫ′′\nxc−2πe2\nq= 0.(24)2 2.5 3 3.5 400.511.522.5\nFigure 3. Plot of theinverse density-densityresponse func tion\n(left hand side of Eq. ( 23), plotted on the vertical axis) vs\nq/kF(horizontal axis) for three different values of rs, from\ntop to bottom, rs= 21.4 (blue), rs= 21.7 (red), and rs=\n22.0 (green). The inverse response function first vanishes at\nrs≃21.7 andq/kF≃3.1.\nThe quantity on the left hand side is recognizable as\nthe inverse of the density-density response function when\nexchange-correlation effects are treated in the local ap-\nproximation[ 9](i.e., whentheexactexchange-correlation\nfield created by a density fluctuation δn(r) is approx-\nimated as ǫ′′\nxcδn(r)). Thus our instability criterion,\nEq. (11), reduces to the requirement that the approx-\nimate density-density response function tends to infin-\nity at some value of q, signifying an instability with re-\nspect to the formation of a charge density wave at that\nwave vector. This result is not surprising in the frame-\nworkofthelinearresponseapproximation,whichwehave\nadopted here as well as in Ref. 7. Plotting the left hand\nside of Eq. ( 23) vsqfor different values of rs(see Fig.\n3) we find that a zero first appears at rs≃21.7, with\nq≃3.1kFandC= 0.17. Notice that qandCare no\nlonger exactly given by Eqs. ( 13) and (14), since the self-\nconsistent q-dependence of Cinvalidates the derivation\nof those equations.\nV. DISCUSSION AND OUTLOOK\nThe calculations presented in this paper show that the\nsuggestion of Ref. [ 7], namely that a macroscopic electric\nfield can induce a microscopic charge density wave in a\n2DEG at relatively high density, remains unproven. The\nself-consistent gradient approximation for the kinetic en-\nergy density in the linear response regime turns out to be\nequivalent to the standardlinear stabilityanalysisfor the\nuniform jellium in the absence of an electric field. The\ncritical values of rsandqare obtained from the “ear-\nliest” zero of the inverse density-density response func-6\ntionχ−1(q,rs)in the absence of an electric field. This is\na spontaneous instability driven by exchange-correlation\neffects and not attributable to the presence of the macro-\nscopic electric field. In addition, the amplitude of the\ncharge density wave could not be pinpointed within our\nlinear response approach, which is not surprising given\nthat the appearance of a microscopic CDW in the ab-\nsence of an external field of the same wave vector is an\nintrinsically nonlinear phenomenon.\nThe way out of these difficulties is to work out the full\nnumerical solution of the Kohn-Sham equation, which is\nnonlinear in the electric field, and map the threshold ofthe CDW instability as a function of both density and\nelectric field. This remains a project for the future.\nVI. ACKNOWLEDGEMENT\nWe gratefully acknowledge support for this work from\nNSF Grant No. DMR-1406568.\nVII. REFERENCES\n[1]E. Wigner, Trans. Faraday Soc. 34, 678 (1938).\n[2]E. Wigner, Phys. Rev. 46, 1002 (1934).\n[3]A. W. Overhauser, Phys. Rev. 167, 691 (1968).\n[4]A.A. Koulakov, M. M. Fogler, andB. I.Shklovskii, Phys.\nRev. Lett. 76, 499 (1996).\n[5]M. M. Fogler, A. A. Koulakov, andB. I.Shklovskii, Phys.\nRev. B54, 1853 (1996).\n[6]V. J. Emery, S. A. Kivelson, and J. M. Tranquada, Pro-\nceedings of the National Academy of Sciences 96, 8814\n(1999).\n[7]E. E. Hroblak, A. Principi, H. Zhao, and G. Vignale,\nPhys. Rev. B 96, 075422 (2017).\n[8]J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Phys.\nRev. Lett. 68, 674 (1992).\n[9]G. Giuliani and G. Vignale, Quantum Theory of the Elec-\ntron Liquid (Cambridge University Press, 2005).\n[10]M. S. Bello, E. I. Levin, B. I. Shklovskii, and A. L. Efros,\nJETP53, 822 (1981).\n[11]K. Steffen, R. Fr´ esard, and T. Kopp, Phys. Rev. B 95,\n035143 (2017).\n[12]S. V. Kusminskiy, J. Nilsson, D. K. Campbell, and A. H.\nCastro Neto, Phys. Rev. Lett. 100, 106805 (2008).\n[13]B. Skinner, G. L. Yu, A. V. Kretinin, A. K. Geim, K. S.\nNovoselov, and B. I. Shklovskii, Phys. Rev. B 88, 155417(2013).\n[14]J. M. Riley, W. Meevasana, L. Bawden, M. Asakawa,\nT. Takayama, T. Eknapakul, T. K. Kim, M. Hoesch, S.-\nK. Mo, H. Takagi, T. Sasagawa, M. S. Bahramy, and\nP. D. C. King, Nature Nanotechnology 10, 1043 (2015).\n[15]S. Ilani, L. A. K. Donev, M. Kindermann, and P. L.\nMcEuen, Nature Physics 2, 687 (2006).\n[16]Y. Wu, X. Chen, Z. Wu, S. Xu, T. Han, J. Lin, B. Skin-\nner, Y. Cai, Y. He, C. Cheng, and N. Wang, Phys. Rev.\nB93, 035455 (2016).\n[17]L. Li, C. Richter, S. Paetel, T. Kopp, J. Mannhart, and\nR. C. Ashoori, Science 332, 825 (2011).\n[18]S. Larentis, J. R. Tolsma, B. Fallahazad, D. C. Dillen,\nK. Kim, A. H. MacDonald, and E. Tutuc, Nano Letters\n14, 2039 (2014), pMID: 24611616.\n[19]C. Attaccalite, S. Moroni, P. Gori-Giorgi, and G. B.\nBachelet, Phys. Rev. Lett. 88, 256601 (2002).\n[20]N. Choudhury and S. K. Ghosh, Phys. Rev. B 51, 2588\n(1995).\n[21]M. Zarenia, D. Neilson, B. Partoens, and F. M. Peeters,\nPhys. Rev. B 95, 115438 (2017).\n[22]M. Zarenia, D. Neilson, and F. M. Peeters, Scientific Re-\nports7, 11510 (2017)."    },    {        "title": "2008.12573v2.Extending_solid_state_calculations_to_ultra_long_range_length_scales.pdf",        "content": "Extending solid-state calculations to ultra long-range length scales\nT. M uller\nMax-Planck-Institut f ur Mikrostrukturphysik,\nWeinberg 2, 06120 Halle, Germany\nS. Sharma\nMax-Born-Institut f ur Nichtlineare Optik und Kurzzeitspektroskopie,\nMax-Born-Strasse 2A, 12489 Berlin, Germany\nE. K. U. Gross\nFritz Haber Center for Molecular Dynamics, Institute of Chemistry,\nThe Hebrew University of Jerusalem, 91904 Jerusalem, Israel\nJ. K. Dewhurst\nMax-Planck-Institut f ur Mikrostrukturphysik,\nWeinberg 2, 06120 Halle, Germany\u0003\n(Dated: October 27, 2020)\nAbstract\nWe present a method which enables solid-state density functional theory calculations to be\napplied to systems of almost unlimited size. Computations of physical e\u000bects up to the micron\nlength scale but which nevertheless depend on the microscopic details of the electronic structure,\nare made possible. Our approach is based on a generalization of the Bloch state which involves\nan additional sum over a \fner grid in reciprocal space around each k-point. We show that this\nallows for modulations in the density and magnetization of arbitrary length on top of a lattice-\nperiodic solution. Based on this, we derive a set of ultra long-range Kohn-Sham equations. We\ndemonstrate our method with a sample calculation of bulk LiF subjected to an arbitrary external\npotential containing nearly 3500 atoms. We also con\frm the accuracy of the method by comparing\nthe spin density wave state of bcc Cr against a direct super-cell calculation starting from a random\nmagnetization density. Furthermore, the spin spiral state of \r-Fe is correctly reproduced and the\nscreening by the density of a saw-tooth potential over 20 unit cells of silicon is veri\fed.\n1arXiv:2008.12573v2  [cond-mat.mtrl-sci]  25 Oct 2020I. INTRODUCTION\nDensity functional theory (DFT)1has had a tremendous impact on solid-state physics\nand is, due to its computational e\u000eciency, at the heart of modern computer based material\nresearch. Since its original proposal, further developing DFT has been an ongoing process.\nExtensions to DFT typically include extra densities in addition to the charge density, such\nas the magnetization2, current density3or the superconducting order-parameter4. Another\nfundamental extension of DFT was the generalization to time-dependent systems5enabling\naccurate calculations of dynamical properties of molecules and solids. While these extensions\nallowed for a more in-depth understanding of microscopic properties, not much progress has\nbeen made in applying DFT to e\u000bects in solids occurring on larger, mesoscopic length scales.\nSuch e\u000bects include long-ranged quasiparticles, magnetic domains or spatially dependent\nelectric \felds. As DFT is a formally exact theory, the underlying physics for such phenomena\nare readily at hand, yet actual calculations remain very di\u000ecult. In a typical calculation,\na single unit cell is solved with periodic boundary conditions, thus e\u000bects extending far\nbeyond the size of a single unit cell are lost. While it is, in principle, possible to use ever\nlarger super-cells, in practice one quickly reaches the limit of computational viability. This\nis mostly due to the poor scaling with the number of atoms, \u0018O(N3\natom), which plagues all\ncomputer programs with a systematic basis set and limits calculations to systems containing\na maximum of\u00181000 atoms. Recent progress based on linear scaling approaches6was able\nto increase the computable system size considerably. Linear scaling approaches, however,\nrequire a \\nearsightedness\" of the system. While this might be ful\flled for e\u000bects strictly\nrelated to the charge density, this is certainly not ful\flled for large magnetic systems, such\nas magnetic domains.\nIn this work we propose a fundamentally di\u000berent approach to drastically extend the\nlength scale of DFT calculations without signi\fcantly increasing the computational cost.\nOur approach relies on altered Bloch states and can be understood as a generalization of\nthe spin-spiral ansatz7, which emerges as a special case of our ansatz. In the spin-spiral\nansatz, a momentum-dependent phase is added to the normal Bloch state. It then becomes\npossible to compute a large, extended spiraling magnetic moment with a single unit cell.\nWhile this is computationally very e\u000ecient, it is, at the same time, the biggest limitation\nof the spin-spiral ansatz: It allows only for a change in the direction of the magnetization\n2while the magnitude of the magnetization and the charge density remain unaltered. We\novercome this limitation by introducing an additional sum in the Bloch states over a \fner\ngrid in reciprocal space around each k-point. The resulting densities then become a Fourier\nseries with a controllable periodicity, which may extend far beyond the length scale of a\nsingle unit cell.\nII. ULTRA LONG-RANGE ANSATZ\nThe systems we will focus on in this article are described by the Kohn-Sham (KS) Hamil-\ntonian of spin-density functional theory (atomic units are used throughout):\n^H0=\u0000r2\n2+vs(r) +Bs(r)\u0001\u001b: (1)\nThe KS potential vs(r) =vext(r) +vH(r) +vxc(r) consists of an external potential vext,\na Hartree potential vHand an exchange-correlation (xc) potential vxc. Similarly, the KS\nmagnetic \feld Bs(r) =1\n2cBext(r)+Bxc(r) can be decomposed into an external \feld Bextand\nan xc-\feld Bxc.\nWe will start o\u000b by extending the KS wave functions. From that we will derive altered\ncharge and magnetization densities. Finally we will derive a long-range Hamiltonian and\nthe matrix elements associated with it.\nA. Wave function and densities\nBloch states of the form 'ik(r) =uik(r)eik\u0001r, whereuikis a lattice-periodic spinor func-\ntion, are used in standard solid-state calculations. The central idea of our approach is a\ngeneralization of this Bloch state to include long-range \ructuations. A similar idea was put\nforward with the spin-spiral ansatz7, where a momentum-dependent phase is applied to the\nnormal Bloch spinor state. Our ultra long-range ansatz employs, in addition, momentum-\ndependent expansion coe\u000ecients which allow for changes in magnitude of the densities from\ncell to cell. For a \fxed k-vector our new Bloch-like state reads:\n\bk\n\u000b(r) =1pNuX\ni\u0014c\u000b\nik+\u00140\n@u\"\nik(r)\nu#\nik(r)1\nAei(k+\u0014)\u0001r(2)\n3(a)\n (b)\nFIG. 1: (a) Schematic of the \u0014-point grid. For each k-point (black dashed line) all bands\n(blue) are augmented with a \fne grid of \u0014-points (green). Three di\u000berent types of\ncouplings between \u0014-points corresponding to di\u000berent length scales are possible. (i) A\ncoupling between two identical \u0014-points but with di\u000berent band indices (ii) a coupling\nbetween di\u000berent \u0014-points sharing the same band index, and (iii) a coupling between\ndi\u000berent \u0014-points with di\u000berent band indices. The maximum length scale of the\ncalculation may be chosen by adjusting the \u0014-point grid. (b) A schematic of the long range\napproach. The red lines indicate unit cells. The lattice periodic density \u001aQ(blue) is altered\nby aQ-dependent modulation (orange) with a di\u000berent periodicity. The result (lower\ngraph) depends on both, the long-range modulation and the lattice periodic solution. ais\nthe lattice constant of a unit cell and Ais the lattice constant of the ultracell, which is the\nsmallest cell that contains the full long-range solution.\nwhereu\"#\nikare the normalized orbitals of a lattice-periodic system, iis a band index and\nka reciprocal space vector, c\u000b\nik+\u0014are complex coe\u000ecients to be determined variationally\n(or by propagating in time) and \u000blabels a particular long-range state. The vectors \u0014live\non a \fner grid around each k-point in reciprocal space (Fig. 1a), which we use to sample\nlong-range e\u000bects. Finally Nuis a normalization factor which is equal to the number of\nunit cells on which \bk\n\u000bis periodic. Note that we have used the lattice periodic parts of the\norbitals at kand not k+\u0014. In principle, both are complete basis sets capable of expanding\nany lattice-periodic function. In practice, the choice of using u\"#\nikoveru\"#\nik+\u0014is more e\u000ecient\nfor determining the density, magnetization and Hamiltonian matrix elements.\n4From this wave function, we can construct a charge and magnetization density:\n\u001a(r) =1\nNkX\nk;\u000bfk\n\u000b\bky\n\u000b(r)\bk\n\u000b(r) (3)\nm(r) =1\nNkX\nk;\u000bfk\n\u000b\bky\n\u000b(r)\u001b\bk\n\u000b(r) (4)\nwith the number of k-pointsNkand the ultra long-range occupation numbers fk\n\u000bassociated\nwith the orbitals \bk\n\u000b. The charge and magnetization density obtained from this wave function\ntake the form\n\u001a(r) =X\nQ\u001aQ(r)eiQ\u0001r;\nm(r) =X\nQmQ(r)eiQ\u0001r(5)\nwithQ=\u0014\u0000\u00140. The partial densities \u001aQandmQin Eq. (5) are complex in general and\nact as lattice-periodic Fourier coe\u000ecients. The resulting real-space densities \u001a(r) and m(r)\nare real functions, which, depending on the values of Q, will have a periodicity larger than\nthe length scale of a unit cell (Fig. 1b). By adjusting the underlying \u0014-lattice, it is therefore\npossible to change the Q-vectors and hence allow for variations of arbitrary length in the\nsystem. The Q= 0 term deserves special mention, as it corresponds to the full lattice-\nperiodic solution. We emphasize that there is no restriction on the magnitude of \u001aQ;mQ\nand we are thus able to expand arbitrary modulations in the charge and magnetization\ndensities. This is a key di\u000berence compared to the spin-spiral ansatz7.\nThe Fourier coe\u000ecients \u001aQandmQcan be calculated e\u000eciently by \frst calculating the\nwave function in Eq. (2) for a subset of unit cells given by a set of real-space lattice vectors\nfRig. We choose the Ri-vectors to be the conjugate real-space vectors of the Q-vectors.\nThe wave function in a single unit cell is then given by a sum over nand a fast Fourier\ntransform in \u0014of the coe\u000ecients c\u000b\nnk+\u0014:\n\bk\n\u000b(r+Ri)\u0019eik\u0001(r+Ri)X\nn0\n@u\"\nnk(r)\nu#\nnk(r)1\nAX\n\u0014c\u000b\nnk+\u0014ei\u0014\u0001Ri(6)\nwhere ris restricted to a single unit cell and we have assumed that j\u0014\u0001rj\u001c1. Note also\nthat the normalization constant 1 =pNuhas been removed. This ensures that observables\nsuch as charge and energy are calculated per unit cell rather than per ultracell. From this,\n5we compute the charge and magnetization densities on the same grid, i.e. \u001ai=\u001a(r+Ri)\nandmi=m(r+Ri). This set can then be partially (fast) Fourier transformed to reciprocal\nspace to obtain \u001aQ(r) and mQ(r):\n\u001aQ(r) =1\nNRX\ni\u001a(r+Ri)e\u0000iQ\u0001Ri\nmQ(r) =1\nNRX\nim(r+Ri)e\u0000iQ\u0001Ri(7)\nHereNRdenotes the number of R-vectors chosen. With the densities at hand, we will\nnow focus on generalizing the Hamiltonian such that meaningful, non-trivial values for the\nexpansion coe\u000ecients c\u000b\nnk+\u0014in Eq. (2) are obtained.\nB. Long-range Hamiltonian\nThe ultra long-range Hamiltonian retains the full lattice periodic KS Hamiltonian ^H0\ngiven in Eq. (1), but also has an additional \\modulation\" term\n^H=^H0+X\nQ^HQ(r)eiQ\u0001r: (8)\nThe total Hamiltonian ^His thus decomposed in the same way as the charge and magneti-\nzation densities in Eq. (5). For a KS system like Eq. (1), our \\modulation\" Hamiltonian\nreads\n^HQ(r) =VQ(r) +BQ(r)\u0001\u001b; (9)\nwhereVQ(r) and BQ(r) are again complex, lattice periodic Fourier coe\u000ecients and contribute\nto long-ranged versions of the scalar potential and the magnetic \feld, respectively. In the\nfollowing we will discuss these coe\u000ecients and how to compute them in more detail. We will\nstart with the scalar potential, which can again be decomposed into an external potential\nVext\nQ(r), a Hartree potential VH\nQ(r) and an xc-potential Vxc\nQ(r). The coe\u000ecients Vext\nQ(r) of\nan external, long-ranged potential can be freely chosen. The coe\u000ecients for the long-ranged\nHartree potential VH\nQ(r) are obtained from the long-range density in Eq. (5):\nVH\nQ(r) =Z\nd3r0\u001aQ(r0)\njr\u0000r0je\u0000iQ\u0001(r\u0000r0): (10)\nThis may be performed e\u000eciently by further Fourier transforming \u001aQ(r) to\u001aQ(G) where\nGis a reciprocal lattice vector. The Hartree potential is then determined directly via\n6VH\nQ(G) = 4\u0019\u001aQ(G)=jG+Qj2and can be subsequently Fourier transformed back to real-\nspace. This is easily extended to the case of the augmented plane wave basis by using the\nmethod of Weinert8.\nNext we will determine the coe\u000ecients associated with the xc-interaction. An important\ndi\u000berence compared to the Hartree potential is that the xc-functional is inherently non-\nlinear, therefore the naive approach Vxc\nQ=Vxc[\u001aQ] may introduce a mixing of the real and\nimaginary part of \u001aQ. Instead we \frst Fourier transform the density to real-space, \u001aRi(r), and\nthen evaluate the xc-potential separately for each R-vector. The inverse Fourier transform\nis then applied to obtain\nVxc\nQ(r) =1\nNRX\niVxc[\u001aRi] (r)e\u0000iQ\u0001Ri: (11)\nIt is worth noting that de\fning the long-range xc-functional this way does not change how\nlocal an xc-functional inherently is, it is merely a Fourier interpolation.\nThe magnetic \feld BQin Eq. (9) consists of an external \feld, an xc-\feld and a dipole-\ndipole \feld:\nBQ(r) =1\n2cBext\nQ(r) +Bxc\nQ(r) +1\n2cBD\nQ(r): (12)\nAgain, the external magnetic \feld may be chosen arbitrarily and the xc-\feld can be com-\nputed analogously to the xc-potential\nBxc\nQ(r) =1\nNRX\niBxc[\u001aRi;mRi] (r)e\u0000iQ\u0001Ri: (13)\nThe last term in Eq. (12) corresponds to the magnetic \feld associated with the magnetostatic\ndipole-dipole interaction\nBD\nQ(r) =1\n2cZ\nd3r03er\u0000r0(mQ(r0)\u0001er\u0000r0)\u0000mQ(r0)\njr\u0000r0j3e\u0000iQ\u0001(r\u0000r0); (14)\nwhere er\u0000r0is the unit vector along the direction r\u0000r0. The contribution of the dipole-dipole\ninteraction is typically neglected in DFT calculations as it is usually small in comparison with\nBxc, which originates from the Coulomb exchange interaction. As the coulomb exchange\ninteraction is inherently short ranged, the magnetic dipole-dipole interaction is expected to\nhave a signi\fcant contribution at larger length scales. We therefore include this term in the\n\\modulation\" Hamiltonian. The derivation of a truly non-local, Q-dependent xc-potential\nis beyond the scope of this article but has been addressed by Pellegrini, et. al9for the dipole\ninteraction.\n7We conclude this section with a remark on the kinetic energy. The kinetic energy operator\n^ p2=2 does not explicitly depend on the periodicity of the problem at hand. As the kinetic\nenergy operator is already included in ^H0(Eqs. (1), (8)), it should not be added to ^HQ. It\nis important to note, however, that the kinetic energy is sensitive to the shifts in reciprocal\nspace of the wave function (Eq. (2)) k!k+\u0014which should be taken into account.\nC. Hamiltonian Matrix Elements\nWe will now focus on diagonalizing the long-range Hamiltonian in Eq. (8). For that we\ncompute the matrix elements for a \fxed k-point in the orbital basis of Eq. (2) to evaluate\nh'ik+\u0014j^H0+^HQj'jk+\u00140i=\u000e\u0014;\u00140\u0010\nOy\nk+\u0014;k\u000f0\nk+\u0014Ok+\u0014;k\u0011\nij+h'ik+\u0014j^HQj'jk+\u00140i: (15)\nHere Q=\u0014\u0000\u00140,\u000f0\nk+\u0014is the diagonal matrix of eigenvalues of ^H0atk+\u0014andOk+\u0014;kis\nthe unitary overlap matrix between the orbitals at kandk+\u0014, i.e.\n(Ok+\u0014;k)ij=X\nsZ\nd3r'\u0003\nisk+\u0014(r) exp(i\u0014\u0001r)'jsk(r): (16)\nThis overlap matrix is required because our chosen basis is the set of orbitals at kand not\nthose at k+\u0014. The overlap matrix Ok+\u0014;kmay however not be strictly unitary in practice.\nThis may be because of numerical inaccuracies but also because the basis is \fnite and there\ncould be bands of a particular character at some ( k+\u0014)-points but not at others. Unitarity is\nnecessary for preserving the eigenvalues \u000f0\nk+\u0014and we ensure this by \frst performing a singular\nvalue decomposition Ok+\u0014;k=U\u0006Tyand then making the substitution Ok+\u0014;k!UTy. One\ncan show that this new matrix is the closest (in the sense of the Frobenius norm) unitary\nmatrix to the original.\nWhat remains to be done is the calculation of the matrix elements of ^HQin Eq. (9). We\nstart with the the scalar potential and \fnd:\nh'\u0014\nnkj^VQ\f\f\f'\u00140\nn0kE\n=X\nsZ\nunitd3ru\u0003\nnsk(r)un0sk(r)VQ(r); (17)\nwheres=\";#is a spin index. Here we have de\fned '\u0014\nnk(r)\u0011'nk(r)ei\u0014\u0001r. In the \frst step we\nconverted the integral over the ultracell into an integral over a unit cell and a sum over all\nunit cells in the ultracellR\nultrad3r!P\nRuR\nunitd3rand made use of the lattice periodicity\nofunk(r). In the second step we then carried out the sum over Rufollowed by the sum\n8overQ. The matrix elements for the ultracell can thus be expressed by a simple unit cell\nintegration. Similarly we \fnd for the magnetic \feld contribution:\nh'\u0014\nnkj^BQ\u0001\u001b\f\f\f'\u00140\nn0kE\n=Z\nunitd3ru\u0003\n\"nk(r)u#n0k(r)\u0000\nBx\nQ(r)\u0000iBy\nQ(r)\u0001\n+u\u0003\n#nk(r)u\"n0k(r)\u0000\nBx\nQ(r) +iBy\nQ(r)\u0001\n+\u0000\nu\u0003\n\"nk(r)u\"n0k(r)\u0000u\u0003\n#nk(r)u#n0k(r)\u0001\nBz\nQ(r):(18)\nIII. NUMERICAL IMPLEMENTATION\nIn this section we will address how to implement the ultra long-range ansatz in practice.\nThe discussions in this section are based on our implementation in the Elk electronic struc-\nture code10, which is an all-electron code using the full potential linearized augmented plane\nwave (FP-LAPW) method.\nA. Self-consistent solution\n^Hin Eq. (8) is a KS system in which the potentials are functionals of the partial\ndensities\u001aQ(r) and mQ(r) in Eq. (7), which in turn depend on the orbitals \bk\n\u000b(r) from Eq.\n(2). Equation (8) thus needs to be solved self-consistently. We employ an iteration scheme\nas it is usually done when solving KS systems:\n91. Solve the lattice periodic ground state, Eq. (1) and obtain the spinor orbitals0\n@u\"\nnk(r)\nu#\nnk(r)1\nAas well as all Eigen energies \u000f0\nnk+\u0014associated with the k+\u0014-points.\n2. Initialize the external long-range potentials via VQandBQand the occupation num-\nbersfk\n\u000b.\n3. (a) Compute the matrix elements of ^HQin Eq. (9). Diagonalize ^Hin Eq. (8) to\nobtain the expansion coe\u000ecients c\u000b\nnk+\u0014as well as the long-range Eigen-energies\n\u000fk\n\u000bfor each k-point.\n(b) Concurrently with the step above, accumulate the long-range densities \u001aQ(r)\nandmQ(r) from Eqs. (3) and (4). This is performed most e\u000eciently by \frst\ncalculating the long-range orbitals explicitly in real-space: \bk\n\u000b(r+Ri).\n4. Calculate the new occupation numbers fk\n\u000b.\n5. Calculate new long-range potentials V0\nQandB0\nQ. Mix the new potentials with the\npotentials from the previous iteration. Monitor the relative change in the potentials.\n6. Repeat steps 3 to 5 until the change in the potentials is su\u000eciently small.\nWe will discuss two steps in this self-consistent cycle in more detail.\nFirst we will explain the order of calculating the energies \u000fk\n\u000b\frst, the densities \u001aQ(r)\nandmQ(r) second and the occupation numbers fk\n\u000bthird. This seems counter-intuitive,\nas the densities depend on the occupation numbers (Eqs. (3) and (4)). However, as we\nare performing a self-consistent cycle, the occupation numbers will converge to the correct\nvalue as self-consistency is achieved. Computing the occupation numbers last enables us\nto parallelize step 3 over the k-point set in a single loop: For each k-point, we diagonalize\n^Hk=^Hk\n0+^Hk\nQand simultaneously compute \u001ak\nQ(r) and mk\nQ(r). These are added to the total\ndensity and magnetization. The central computational gain is that this ordering is much\nless demanding when it comes to memory: the coe\u000ecients c\u000b\nnk+\u0014do not have to be stored\nbut can be calculated and used on-the-\ry instead.\nSecond, we note that some care has to be taken during the mixing. We choose to mix\nthe complex Fourier coe\u000ecients VQ(r) and BQ(r) rather than their real-space counterparts.\n10We also want to emphasize that in a typical calculation a rather slow mixing should be\napplied. The Coulomb interaction in a large system will react very strongly to any external\nperturbation because of the divergence of 1 =Q2. This can lead to substantial charge sloshing\nduring convergence necessitating the use of a small mixing parameter. This is an aspect of\nthe method which would bene\ft from further investigation and improvement. One possibility\nis to use a screened Coulomb interaction to remove the divergence. This screening could be\nslowly reduced to zero during the self-consistent loop to improve the rate of convergence.\nB. k-point grids\nThe underlying grids have to be chosen carefully in order to avoid computational artifacts\nand to achieve a most e\u000ecient calculation. Ideally, the smallest distance between k-points\nshould be greater than the largest distance between \u0014-points, i.e.j\u0014\u0000\u00140j<jk\u0000k0j. This\nwill ensure that the set k+\u0014does not overlap for any two k-points, which may lead to\ndouble counting and an over-complete basis set. Physically speaking, the length scales in\nthe system should be well separated, i.e. the modulation should be far larger than the size\nof a unit cell. Ifj\u0014\u0000\u00140j\u0019jk\u0000k0j, however, the system tends to have a size which can and\nshould be solved with a super-cell instead.\nMany Fourier transformations need to be carried out during each self-consistent step: with\ne\u0000i\u0014\u0001Riwhen calculating the wave function in Eq. (6), and with e\u0000iQ\u0001Rwhen calculating\nthe densities in Eqs. (3) and (4); and the xc-potential and -\feld, Eqs. (11) and (13). This\nconstitutes a major part of the computational e\u000bort and it is therefore highly bene\fcial to\ncarry out all Fourier transformations via a Fast Fourier Transform (FFT). This requires the\nunderlying grid to be FFT compatible (having, in our case, radices 2, 3, 5 and 7). Owing to\nQ=\u0014\u0000\u00140theQ-point and the \u0014-point grids are dependent on one another. The number\nofQ-pointsnQalong a given direction iisni\nQ= 2ni\n\u0014\u00001. In our implementation, we ensure\nthat the input Q-grid snaps to the next FFT compatible grid. We then choose the \u0014-point\ngrid such that 2 ni\n\u0014\u00001\u0014ni\nQ. This grid choice can sometimes result in unmatched Q-points,\ne.g. ifnQ= 20 andn\u0014= 10 then the Q-vectors are not symmetric around zero. While\nthe unmatched Q-point is \\dead-weight\" and remains zero throughout the calculation, the\nspeed up obtained by using a FFT outweighs having additional Q-points.\n11C. Computation of the Hartree and dipole interaction\nWe will brie\ry address how to calculate the complex integrals appearing in the scalar\npotential, Eq. (10), and the magnetic \feld, Eq. (14). When computing the Hartree-\npotential in Eq. (10), we solve Poisson's equation using the method by Weinert8, which can\nbe generalized to complex densities relatively easily.\nThe dipole interaction, Eq. (14), can be solved for in a similar way by evaluating Poisson's\nequation component-wise for the vector potential. From classical electrodynamics, the vector\npotential associated with a magnetization is given by:\nAdip(r) =1\ncZ\nd3r0r\u0002m(r0)\njr\u0000r0j(19)\nWe partially Fourier transform both sides and obtain:\nX\nQAdip\nQ(r)eiQ\u0001r=1\ncZ\nd3r0r\u0002P\nQmQ(r0)eiQ\u0001r0\njr\u0000r0j: (20)\nWe thus obtain for the coe\u000ecients of the vector potential:\nAdip\nQ;j(r) =1\ncX\nkl\u000fjklZ\nd3r0e\u0000iQ\u0001(r\u0000r0)@kmQ;l(r0) +iQkmQ;l(r0)\njr\u0000r0j(21)\nHerej;k;l indicate vector components and \u000fjklis the Levi-Civita symbol. The coe\u000ecients\nAdip\nQ(r) now have the same form as the Hartree potential, Eq. (10), and can also be computed\nby a complex version of Weinert's method8. From this it is easy to obtain the magnetic \feld\nof the dipole interaction via Bdip(r) =r\u0002Adip(r). We \fnd for the coe\u000ecients:\nBdip\nQ;j=X\nkl\u000fjklh\n@kAdip\nQ;l(r) +iQkAdip\nQ;l(r)i\n: (22)\nWe point out that if we were to consider an exact theory for the current density, the dipole\nvector potential, Eq. (19), should be included in the Hamiltonian, corresponding to a Lorentz\nforce generated by the dipole-dipole interaction.\nIV. RESULTS\nThree calculations for which the ultracell is small enough to be amenable to super-cell\ncalculations so that a detailed comparison is possible. We also performed a calculation which\nwould be considered too large to be treated as a super-cell.\n12A. Spin-spirals in \r-Fe\n(a)\n75 76 77 78 79 80 81 82\nUnit cell volume (a.u.3)0.811.21.41.61.822.22.42.62.8Moment per unit cell ( µB)Ultracell calculation\nSpin-spiral calculation(b)\nFIG. 2: (a) Ultra long-range magnetization density of \r-Fe plotted in the plane\nperpendicular to [001]. The color indicates the magnitude of the magnetization and the\narrows indicate direction. The modulation encompasses 32 unit cells in the [100] direction.\n(b) Plot of moment against unit cell volume for both the long-range and spin-spiral ansatz.\nThe \frst numerical test deals with the so called \rphase of Fe. Previous calculations11have\nshown that the spin-spiral state has the lowest energy compared to several commensurate\nferromagnetic and anti-ferromagnetic structures. The ultra long-range method allows us to\naddress the question whether the much larger variation freedom associated with ultra-cell\nstill yields the spin-spiral as ground state. We performed a traditional spin-spiral calculation\nand an ultra-cell calculation for this materials. The parameters used are as follows: Ultracell\nk-point grid: 1\u000212\u000212,Q-point grid: 32\u00021\u00021, ultracell: 32\u00021\u00021 unit cells. A single\nunit cell was used for the spin-spiral calculation with a 12 \u000212\u000212k-point grid and a\nQ-vector of 1 =32.\nAn initial magnetic \feld is required to break the spin symmetry. To ensure an unbiased\ncalculation, we applied a random \feld to the ultracell calculation and subsequently reduced\nit to zero. Throughout the calculation, we enforced the constraintR\nunitd3rmQ=0(r) = 0.\nThis ensures that the system is not drawn to a lattice-periodic ferromagnetic solution. The\nmagnetization converged to an ordered state where the magnitude was constant over the\nultracell and only the direction varied (Fig. 2(a)). This corresponds precisely to the spin-\n13spiral state i.e. the ultra-cell calculations shows that the spin-spiral state is still the lowest\nenergy solution. The overall magnitude of the magnetization is sensitive to the lattice\nparameter and undergoes a transition from \u00181\u0016Bto\u00182:5\u0016Bfor this relatively small Q-\nvector. As may be seen in Fig. 2(b), this behavior is observed for both the ultracell and\nspin-spiral calculations.\nB. Spin density wave in bcc Cr\n0 20 40 60 80 100\nr (a.u.)-0.12-0.08-0.0400.040.080.12mz(r)Ultracell magnetisation\nSupercell magnetisationMagnetisation density of bcc Cr\n(a)\n0 20 40 60 80 100\nr (a.u.)-0.009-0.006-0.00300.0030.0060.009∆ρ(r)Ultracell density\nSupercell densityDensity difference of bcc Cr (b)\nFIG. 3: (a) Magnetization density for bcc Cr over 21 unit cells. (b) Change in density over\nthe same range. For the ultracell, this was generated by setting \u001aQ=0(r) in Eq. (7) to zero.\nFor the super-cell, the lattice-periodic density was subtracted leaving just the modulated\ndensity.\nIn a second test, we aim at calculating the spin density wave (SDW) state in Cr. The\nexistence of a SDW in Cr is well known and the \frst research dates back to around 196012{14.\nDespite this, computing the SDW state within DFT remains di\u000ecult and has been the topic\nof many studies15{27, with partially con\ricting results28. It is likely that a SDW is not\nthe true ground state of Cr within DFT29, however we will not focus here on the inherent\ncomplexities of the system. This state is not achievable by the spin-spiral ansatz because\nthe magnitude of the moment changes but not its direction, thus a super-cell calculation\nis required. Cr is an excellent test scenario, as the periodicity of the SDW is \u001820:83 unit\ncells, which is still well within computational reach of the super-cell approach.\n14For our comparison, we use the LSDA and a lattice parameter of 2 :905/RingA as suggested by\nCottenier et al.28. We consider 21\u00021\u00021 unit cells of bulk Cr for both super-cell and ultracell\ncalculations. For the super-cell we used a 1 \u000212\u000212k-point grid. A randomized symmetry\nbreaking magnetic \feld was used to start the calculation and subsequently reduced to zero.\nSpin-orbit coupling is also included. Our super-cell calculation reproduces the result by\nCottenier et al.28. For the ultracell we also used a 1 \u000212\u000212k-point grid with a 21 \u00021\u00021\nQ-points grid corresponding to a grid of 11 \u00021\u00021\u0014-points to obtain the best possible\nsampling of the xc-potential and -\feld. Around 60 empty states in the lattice-periodic basis\nare used to provide enough degrees of freedom during the convergence. We started with\na randomized initial \feld that was reduced after each step. Throughout the calculation,\nwe enforced the constraintR\nMTd3rmQ=0(r) = 0 for each mu\u000en-tin. This ensures that the\nsystem is not drawn to a lattice-periodic anti-ferromagnetic solution.\nOur results are shown in Fig. 3. Speci\fcally, Fig. 3(a) shows the comparison of the\nmagnetization in the SDW state, as obtained from the super-cell and ultracell calculations.\nThe maximum moment of the ultracell calculations is larger than that of the super-cell,\n1.174\u0016Band 0.712\u0016B, respectively. This we attribute to the fact that the ultra long-range\ncalculation is performed in the basis of Kohn-Sham states and not in the original LAPW\nbasis for which the linearization energies are optimally adjusted. It is also known that LSDA\ncalculations this of system are particularly sensitive to the basis and the moment depends\nstrongly on the lattice parameter28.\nIn Fig. 3(b) we present the charge density wave (CDW) which is known to stabilize\nalongside the SDW with twice the period. While obtaining the CDW in the ultracell is\nstraight-forward (as all \u001aQ(r) are known), it is numerically more challenging to extract it for\nthe super-cell. We did this by subtracting the density from the calculation of a single unit\ncell. We obtain the same periodicity in both calculations as well as a comparable magnitude.\nC. Saw-tooth potential in Si\nThe previous two examples involved modulations in long-range magnetic order, but we\nalso need to test the method with long-range, external electrostatic \felds. This is in antici-\npation of a future development where ultra long-range TDDFT calculations are performed\nin conjunction with Maxwell's equations. In such a scenario, an electromagnetic wave prop-\n150 20 40 60 80 100 120 140\nr (a.u.)-1.5-1-0.500.511.5V(r) (Ha)Ultracell potential\nSupercell potentialSawtooth potential applied to Si(a)\n0 20 40 60 80 100 120 140\nr (a.u.)00.050.10.150.20.25ρ(r)Supercell density\nUltracell densityElectronic density of Si (b)\nFIG. 4: (a) Saw-tooth potential applied to 20 unit cells of silicon in the ultracell and\nsuper-cell. The ultracell potential is smoother than that of the super-cell because of the\nrelatively small number of Qvectors used to expand it. (b) The resultant ground state\ndensity in the ultracell and super-cell. This was plotted along a line which was slightly\no\u000b-set from the atomic centers in order to avoid the very high densities near the nuclei.\nagating through the solid could have a periodicity of many hundreds of unit cells. The\nlong-range ansatz should be ideal for performing such a simulation. With that goal in mind,\nwe apply a simple saw-tooth potential to silicon over a range of 20 unit cells to check if the\nultracell calculation agrees with its super-cell equivalent. This corresponds to a constant\nelectric \feld, at least near the center of the saw-tooth. Both calculations were performed\nwith a 4\u00024\u00024k-point grid. The ultracell calculation used a Q-point grid of 20\u00021\u00021\nwith a basis of 60 empty states per k-point. The applied electric \feld was 0 :01 in atomic\nunits and the corresponding saw-tooth potential is plotted in Fig. 4(a). As can been seen,\nthere is a di\u000berence between the ultracell and super-cell potentials. The ultracell potential\nis expanded in a \fnite set of Qvectors and thus contains oscillations on the length scale of\nthe longest vector. The super-cell potential is a sharp saw-tooth. Despite this di\u000berence,\nthe two densities plotted in Fig. 4(b) are broadly the same with strong screening near the\ncenter and charge accumulation and depletion near the edges. We note that for the intended\npurpose of describing propagating light through solids, the resulting electric and magnetic\n\felds will be well expanded with a \fnite number of Qvectors.\n16D. Long-range electrostatic potential in LiF\nFIG. 5: Self-consistent density without the \u001aQ=0(r) term for a 3456 atom ultracell of LiF\nwith an arti\fcial external potential. The plotting plane is perpendicular to [001] and\ncontains 48\u000236 unit cells.\nLastly we perform a calculation which is too large for a super-cell. Rather than attempt-\ning to model a physical phenomenon at this stage, we simply apply an arbitrarily chosen\nelectrostatic potential to an insulator, in this case LiF. An ultracell of 48 \u000236\u00021 unit cells\nwas constructed with an equivalent Q-point grid. The number of empty states per k-point\nwas taken to be 4 to keep the memory requirements to within those of our computer. The\nresultant change in density away from unit cell periodicity is plotted in Fig. 5. As the poten-\ntial is arti\fcial, the important metric here is the computational e\u000bort expended in reaching\nthe self-consistent solution. The rate of convergence is fairly slow because of the e\u000bect of the\nlong-range Coulomb interaction, and thus we performed 170 iterations of the self-consistent\nloop. The calculation was performed on 480 CPU cores and each iteration took about 40\nminutes. This level of performance for an all-electron calculation indicates that physical\nphenomena involving modulations of the electronic state over hundreds or thousands of unit\ncells are within reach of this approach.\n17V. CONCLUSION AND OUTLOOK\nWe have developed a method which makes possible the ab-initio treatment of hitherto\nuncomputable length-scales in solids. This consists of a modi\fed Bloch ansatz and a set\nof Kohn-Sham equations which have to be solved self-consistently. The underlying lattice\nof nuclear charges is still periodic on the unit cell length scale but the electronic state can\naccommodate arbitrary modulations on any length scale. Based on our experience with the\nall-electron Elk code, we are con\fdent that this method can be e\u000eciently implemented in\nmost existing solid-state electronic structure codes. We demonstrated the capabilities of\nthe novel method by solving an arbitrary external potential applied to nearly 3500 atoms\nof LiF. Additionally, we showed that our method can reproduce the results obtained by\nsuper-cell calculations on smaller length scales for both insulators and magnetic solids. The\nmethod presented in this paper opens up exciting possibilities of future research: on the\ntechnical level, a derivation of long-range and explicitly Q-dependent xc-potentials (see,\nfor example, Pellegrini et al.9). On the applied level, our method could pave the way to\ncalculations of mesoscopic systems, such as magnetic domain walls or skyrmions, which have\nso far been out of reach for ab-initio methods like DFT. Furthermore, the novel technique is\nstraightforwardly incorporated in real-time TDDFT calculations which, when combined with\nthe solution of Maxwell's equations, will give access to the propagation of electromagnetic\nradiation through extended solids within a genuine ab-initio description.\nAcknowledgments: SS and TM would like to thank DFG for funding though QUTIF\nproject. EKUG acknowledges \fnancial support by European Research Council Advanced\nGrant Fact (ERC-2017-AdG-788890).\n\u0003dewhurst@mpi-halle.mpg.de\n1P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).\n2U. von Barth and L. Hedin, Journal of Physics C: Solid State Physics 5, 1629 (1972).\n3G. Vignale and M. Rasolt, Phys. Rev. Lett. 59, 2360 (1987).\n4L. N. Oliveira, E. K. U. Gross, and W. Kohn, Phys. Rev. Lett. 60, 2430 (1988).\n5E. Runge and E. K. U. Gross, Phys. Rev. Lett. 52, 997 (1984).\n6S. Goedecker, Rev. Mod. Phys. 71, 1085 (1999).\n187L. M. Sandratskii, physica status solidi (b) 136, 167 (1986).\n8M. Weinert, Journal of Mathematical Physics 22, 2433 (1981).\n9C. Pellegrini, T. M uller, J. K. Dewhurst, S. Sharma, A. Sanna, and E. K. U. Gross, Phys. Rev.\nB101, 144401 (2020).\n10\\The Elk Code,\" http://elk.sourceforge.net/ .\n11E. Sj ostedt and L. Nordstr om, Phys. Rev. B 66, 014447 (2002).\n12L. M. Corliss, J. M. Hastings, and R. J. Weiss, Phys. Rev. Lett. 3, 211 (1959).\n13V. N. Bykov, V. S. Golovkin, N. V. Ageev, V. A. Levdik, and S. I. Vinogradov, Soviet Physics\nDoklady 4, 1070 (1960).\n14A. W. Overhauser, Phys. Rev. 128, 1437 (1962).\n15V. Moruzzi, J. Janak, and A. Williams, Calculated Electronic Properties of Metals , edited by\nV. Moruzzi, J. Janak, and A. Williams (Pergamon, 1978) pp. 30 { 159.\n16J. K ubler, Journal of Magnetism and Magnetic Materials 20, 277 (1980).\n17H. L Skriver, Journal of Physics F: Metal Physics 11, 97 (1981).\n18N. I Kulikov and E. Kulatov, Journal of Physics F: Metal Physics 12, 2291 (1082).\n19N. I. Kulikov, M. Alouani, M. A. Khan, and M. V. Magnitskaya, Phys. Rev. B 36, 929 (1987).\n20J. Chen, D. Singh, and H. Krakauer, Phys. Rev. B 38, 12834 (1988).\n21V. L. Moruzzi and P. M. Marcus, Phys. Rev. B 46, 3171 (1992).\n22D. J. Singh and J. Ashkenazi, Phys. Rev. B 46, 11570 (1992).\n23K. Hirai, Journal of the Physical Society of Japan 66, 560 (1997).\n24G. Y. Guo and H. H. Wang, Phys. Rev. B 62, 5136 (2000).\n25G. Bihlmayer, T. Asada, and S. Bl ugel, Phys. Rev. B 62, R11937 (2000).\n26J. Sch afer, E. Rotenberg, S. Kevan, and P. Blaha, Surface Science 454-456 , 885 (2000).\n27R. Hafner, D. Spis\u0013 ak, R. Lorenz, and J. Hafner, Journal of Physics: Condensed Matter 13,\nL239 (2001).\n28S. Cottenier, B. De Vries, J. Meersschaut, and M. Rots, J. Phys.: Condens. Matter 14, 3275\n(2002).\n29R. Hafner, D. Spi\u0014 s\u0013 ak, R. Lorenz, and J. Hafner, Phys. Rev. B 65, 184432 (2002).\n19"    },    {        "title": "2008.13722v1.State_densities_of_heavy_nuclei_in_the_static_path_plus_random_phase_approximation.pdf",        "content": "State densities of heavy nuclei in the static-path plus random-phase approximation\nP. Fanto and Y. Alhassid\nCenter for Theoretical Physics, Sloane Physics Laboratory,\nYale University, New Haven, Connecticut 06520, USA\n(Dated: September 1, 2020)\nNuclear state densities are important inputs to statistical models of compound-nucleus reac-\ntions. State densities are often calculated with self-consistent mean-\feld approximations that do\nnot include important correlations and have to be augmented with empirical collective enhance-\nment factors. Here, we benchmark the static-path plus random-phase approximation (SPA+RPA)\nto the state density in a chain of samarium isotopes148\u0000155Sm against exact results (up to statis-\ntical errors) obtained with the shell model Monte Carlo (SMMC) method. The SPA+RPA method\nincorporates all static \ructuations beyond the mean \feld together with small-amplitude quantal\n\ructuations around each static \ructuation. Using a pairing plus quadrupole interaction, we show\nthat the SPA+RPA state densities agree well with the exact SMMC densities for both the even-\nand odd-mass isotopes. For the even-mass isotopes, we also compare our results with mean-\feld\nstate densities calculated with the \fnite-temperature Hartree-Fock-Bogoliubov (HFB) approxima-\ntion. We \fnd that the SPA+RPA repairs the de\fciencies of the mean-\feld approximation associated\nwith broken rotational symmetry in deformed nuclei and the violation of particle-number conser-\nvation in the pairing condensate. In particular, in deformed nuclei the SPA+RPA reproduces the\nrotational enhancement of the state density relative to the mean-\feld state density.\nI. INTRODUCTION\nNuclear level densities, which measure the average\nnumber of nuclear levels per unit energy, are impor-\ntant inputs to the statistical Hauser-Feshbach theory\nof compound-nucleus reactions [1, 2]. Neutron capture\nrates, which a\u000bect the predicted isotopic abundances in\nr-process nucleosynthesis [3, 4] and the precision of i-\nprocess simulations [5], are particularly sensitive to level\ndensities. Uncertainties in nuclear reaction rates also\nhave important implications for nuclear technology and\nstockpile stewardship applications [6].\nLevel densities are extracted from various experimen-\ntal data, including level counting at low excitation ener-\ngies, neutron resonance data at the neutron separation\nenergy, the Oslo method [7, 8], and particle evaporation\nspectra [9]. Rare-isotope beam facilities and novel ex-\nperimental techniques such as the \f-Oslo method [10]\npromise to extend level density measurements to unstable\nnuclei. However, available data to determine nuclear level\ndensities remain limited, and theoretical calculations of\nlevel densities are required to describe many compound-\nnucleus reactions.\nThe calculation of level densities in the presence of\ncorrelations is a challenging many-body problem. Most\ntheoretical approaches are based on phenomenological\nmodels \ftted to experimental data [2, 11]. Such mod-\nels cannot be reliably extrapolated to regions in which\ndata are scarce. Moreover, these models are limited by\nuncertainties in the experimental data [9].\nConsequently, it is important to develop microscopic\nmodels of the level density that are based on the un-\nderlying nuclear interaction. Widely used methods that\nrely on mean-\feld approximations [12{14] must be aug-\nmented with phenomenological models to describe rota-\ntional and vibrational enhancements [13, 14]. In contrast,the con\fguration-interaction (CI) shell model describes\nboth single-particle and collective excitations within the\nsame framework. However, conventional CI shell model\nmethods are limited by the combinatorial growth of the\nmany-particle model space with the number of nucleons\nand/or the number of single-particle states. The shell-\nmodel Monte Carlo (SMMC) method is capable of cal-\nculating level densities exactly (up to controllable sta-\ntistical errors) in model spaces that are far beyond the\nreach of conventional CI methods, and has been applied\nto nuclei as heavy as the lanthanides [15{18]; for a re-\nview, see Ref. [19]. Most applications of the SMMC to\nthe calculation of level densities have used e\u000bective nu-\nclear interactions with a good Monte Carlo sign [19{21],\nalthough smaller bad-sign components of more general\ne\u000bective nuclear interactions can be treated using the\nmethod of Ref. [21].\nAnother method to calculate level densities based on\nthe CI shell mode is the moment method [22{24]. This\napproach has been applied to light and mid-mass nuclei\nbut becomes costly in heavy nuclei and is limited by the\nneed to calculate independently an accurate ground-state\nenergy. Similarly, the stochastic estimation method of\nRef. [25] and the methods of Ref. [26] based on the Lanc-\nzos algorithm have been used to calculate CI shell model\nlevel densities in mid-mass nuclei, but these methods can-\nnot be applied to heavy nuclei due to their prohibitive\ncomputational cost.\nThe static-path plus random-phase approximation\n(SPA+RPA) is a promising method for including cor-\nrelation e\u000bects beyond the mean \feld within the CI\nshell model framework. The approach includes large-\namplitude static \ructuations beyond the mean \feld\nand small-amplitude time-dependent quantal \ructua-\ntions around each static \ructuation [27{34]. In solv-\nable models, the SPA+RPA was found to give nearlyarXiv:2008.13722v1  [nucl-th]  31 Aug 20202\nexact results for thermodynamic quantities above a low\ntemperature below which the method breaks down [29{\n31, 33, 34]. The SPA+RPA can also be applied to cer-\ntain interactions for which the SMMC has a sign problem\nand thus can be complementary to the SMMC. However,\nthere have been few applications of the SPA+RPA to\nmany-particle systems with realistic forces. In nuclear\nphysics, the SPA+RPA was used in Ref. [34] to calcu-\nlate thermal quantities in erbium isotopes with a pure\npairing force. The SPA+RPA was used to study the\npairing properties of molybdenum isotopes with a pair-\ning force [35], as well as the level density of56Fe with a\npairing plus quadrupole interaction [36]. The SPA+RPA\nhas also been applied to calculate thermodynamic prop-\nerties, such as the heat capacity and spin susceptibility,\nin nanoscale metallic grains in the presence of pairing\nand exchange correlations [37]. However, the SPA+RPA\nhas not been benchmarked systematically against exact\nresults in heavy nuclei.\nHere, we benchmark SPA+RPA nuclear state densi-\nties1against SMMC state densities for a chain of samar-\nium isotopes148\u0000155Sm, which describe the crossover\nfrom vibrational to rotational collectivity [16, 38, 39]. For\nthe even-mass samarium isotopes, we also compare our\nresults with mean-\feld state densities calculated with the\n\fnite-temperature Hartree-Fock-Bogoliubov (HFB) ap-\nproximation. We implement a Monte Carlo method to\ncalculate SPA+RPA thermodynamic observables. The\nSPA+RPA method breaks down below a certain low tem-\nperature, and we use the partition function extrapolation\nmethod of Ref. [40] to extract the ground-state energy\nfrom the SPA+RPA excitation partition function above\nthe breakdown temperature.\nUsing a pairing plus quadrupole interaction, we \fnd\nthat the SPA+RPA canonical entropy and state density\nare in good agreement with the corresponding SMMC\nresults for each of the even- and odd-mass samarium\nisotopes. In the even-mass deformed isotopes, the\nSPA+RPA method reproduces the enhancement of the\nstate density relative to the HFB density due to rota-\ntional collectivity. In addition, the SPA+RPA entropy\nremains nonnegative at low temperatures, whereas the\npairing phase of the HFB approximation is character-\nized by an unphysical negative entropy. We study the\nevolution with neutron number of the enhancement of\nthe SPA+RPA and SMMC state densities relative to the\nHFB density, as was done in Ref. [16]. We \fnd that the\nSPA+RPA enhancement factors are in good agreement\nwith those extracted from the SMMC and are consistent\nwith a crossover from pairing-dominated collectivity to\nrotational collectivity.\nThe outline of this paper as follows. In Sec. II, we de-\nrive the SPA+RPA expressions for the grand-canonical\n1In the state density, all 2 J+ 1 degenerate states associated with\na nuclear level of spin Jare counted, whereas in the level density\neach level with spin Jis counted only once.and approximate canonical partition functions, and we\ndiscuss the calculation of the state density. In Sec. III,\nwe present the Monte Carlo method used to evaluate\nthermodynamic quantities in the SPA+RPA. We also\nsummarize the partition function extrapolation method\nused to extract the SPA+RPA ground-state energy. In\nSec. IV, we apply the SPA+RPA to calculate the state\ndensities in a chain of samarium isotopes148\u0000155Sm and\ncompare our results with the SMMC state densities. For\nthe even-mass samarium isotopes, we also compare the\nSPA+RPA densities with the HFB densities and extract\nthe collective enhancement factors. Finally, in Sec. V,\nwe summarize our conclusions, discuss the advantages\nand limitations of the SPA+RPA method, and provide\nan outlook for future developments of this method.\nII. STATIC-PATH PLUS RANDOM-PHASE\nAPPROXIMATION TO STATE DENSITIES\nA. General formulation\nHere, we brie\ry review the SPA+RPA formalism for\nthe partition function in the grand-canonical ensemble.\nFor similar derivations, see Refs. [33, 34]. We consider a\nHamiltonian in which the two-body residual interaction\nis written as a sum of separable terms,\n^H=^H1\u00001\n2X\n\u000bv\u000b^O2\n\u000b; (1)\nwhere ^H1is a one-body operator and ^O\u000bare Hermitian\nand bilinear in fermion creation and annihilation opera-\ntors. Any Hermitian two-body interaction can be decom-\nposed in this way [20]. We assume that the interaction is\npurely attractive when written in the form of Eq. (1), i.e.,\nall thev\u000bare positive. An approximate SPA+RPA treat-\nment of repulsive interactions was proposed in Ref. [41]\nbut is not investigated here.\nThe Hubbard-Stratonovich transformation [42, 43] ex-\npresses the Gibbs density operator at inverse temperature\n\fas a functional integral\ne\u0000\f^H=Z\nD[\u001b(\u001c)]e\u0000R\f\n0d\u001cP\n\u000bv\u000b\u001b2\n\u000b(\u001c)=2Te\u0000R\f\n0d\u001c^h\u001b(\u001c);\n(2)\nwhere\u001b\u000b(\u001c) (with 0\u0014\u001c\u0014\f) are real-valued auxiliary\n\felds,Tdenotes time ordering, and ^h\u001b(\u001c) is a Hermitian\none-body Hamiltonian given by\n^h\u001b(\u001c) =^H1\u0000X\n\u000bv\u000b\u001b\u000b(\u001c)^O\u000b: (3)\nNext, each auxiliary \feld is separated into a static and\n\u001c-dependent part\n\u001b\u000b(\u001c) =\u001b\u000b+X\nr6=0\u0011\u000brei!r\u001c; (4)3\nwhere!r= 2\u0019r=\f (r=\u00061;\u00062;:::) are the bosonic Mat-\nsubara frequencies. The grand-canonical partition func-\ntion at inverse temperature \fand chemical potentials\n\u0016p;\u0016n(for protons and neutrons) can then be written as\nZ(\f;\u0016p;\u0016n) = Tre\u0000\f(^H\u0000P\n\u0015=p;n\u0016\u0015^N\u0015)\n=Z\"Y\n\u000b\u0012\fv\u000b\n2\u0019\u00131=2\nd\u001b\u000b#\ne\u0000\fP\n\u000bv\u000b\u001b2\n\u000b=2Z(\u001b)\n\u0002Z\nD\u0011e\u0000\fP\n\u000b;r6=0v\u000bj\u0011\u000brj2=2D\nTe\u0000R\f\n0d\u001c^V\u0011(\u001c)E\n\u001b;\n(5)\nwhereZ(\u001b) is the partition function for static auxiliary\n\felds\u001b\nZ(\u001b) = Tre\u0000\f(^h\u001b\u0000P\n\u0015=p;n\u0016\u0015^N\u0015): (6)\nIn Eq. (6), the one-body Hamiltonian ^h\u001bis given by\nEq. (3), with the time-dependent auxiliary \felds \u001b\u000b(\u001c)\nreplaced by the static \felds \u001b\u000b. The operator ^V\u0011(\u001c) =\n\u0000P\n\u000b;r6=0v\u000b\u0011\u000brei!r\u001c^O\u000b(\u001c) in Eq. (5) ( ^O\u000b(\u001c) is the inter-\naction picture representation of ^O\u000bwith respect to the\nstatic Hamiltonian ^h\u001b) accounts for the contribution of\nthe time-dependent \ructuations of the auxiliary \felds to\nthe one-body Hamiltonian. The angular brackets h:::i\u001b\ndenote the expectation value with respect to the static\ndensity operator e\u0000\f(^h\u001b\u0000P\n\u0015=p;n\u0016\u0015^N\u0015).\nIn the SPA+RPA, the logarithm of the integrand in\nEq. (5) is expanded to second order in the amplitudes \u0011\u000br\nof the time-dependent auxiliary-\feld \ructuations, and\nthe resulting Gaussian integral over \u0011\u000bris evaluated an-\nalytically [29{34]. The \fnal result is\nZ(\f;\u0016p;\u0016n) =Z\"Y\n\u000b\u0012\fv\u000b\n2\u0019\u00131=2\nd\u001b\u000b#\ne\u0000\fv\u0001\u001b2=2Z(\u001b)C(\u001b);\n(7)\nwherev\u0001\u001b2=P\n\u000bv\u000b\u001b2\n\u000bandC(\u001b) is the RPA correction\nfactor for static \felds \u001bgiven by [27, 33, 34]\nC(\u001b) =Q\nk>l1\n~Ek\u0000~Elsinh\u0010\n\f(~Ek\u0000~El)=2\u0011\nQ\n\u0017>01\n\n\u0017sinh (\f\n\u0017=2):(8)\nIn Eq. (8), ~Ekare the generalized quasiparticle energies\nobtained by diagonalizing ^h\u001b, and \n\u0017are the eigenvalues\nof the\u001b-dependent matrix\nSkl;k0l0= (~Ek\u0000~El)\u000ekk0\u000ell0\u00001\n2(~fl\u0000~fk)X\n\u000bO\u000b;klO\u000b;l0k0:\n(9)\nHereO\u000bis the matrix representation of the one-body\noperator ^O\u000bin the quasiparticle basis diagonalizing ^h\u001b,\nand ~fkis the generalized thermal quasiparticle occupa-\ntion number associated with generalized quasiparticle en-ergy ~Ek.2The matrixSin Eq. (9) has the same form as\nthe thermal RPA matrix derived from considering small\noscillations around a self-consistent mean-\feld solution\nat \fnite temperature [44]. Consequently, we refer to the\neigenvalues \n \u0017as the RPA frequencies. These frequen-\ncies come in pairs of opposite sign, andQ\n\u0017>0in Eq. (8)\ndenotes the product over half the frequencies with a \fxed\nsign.\nSome of the RPA frequencies can become imaginary\nfor certain static auxiliary \feld con\fgurations. The\nSPA+RPA is well de\fned at inverse temperature \fpro-\nvided that there exists no imaginary frequency ~\n\u0017satis-\nfyingj~\n\u0017j\u00152\u0019=\f[33, 37]. Below a certain temperature,\nthis condition no longer holds, and the SPA+RPA breaks\ndown. In our calculations, we \fnd that this breakdown\ntemperature is very low and does not limit our ability to\ncalculate state densities. However, the breakdown of the\nSPA+RPA makes it challenging to estimate the ground-\nstate energy, which is needed to determine the excitation\nenergy. We have overcome this challenge using the par-\ntition function extrapolation method [40]; see Sec. III B.\nB. Pairing plus quadrupole Hamiltonian\nHere we apply the SPA+RPA to a CI shell\nmodel Hamiltonian containing pairing and quadrupole-\nquadrupole interaction terms. The CI shell model single-\nparticle space consists of a set of spherical orbitals a\u0015=\n(na\u0015la\u0015ja\u0015), where\u0015=p;ndenotes protons and neu-\ntrons, respectively, and the orbitals can be di\u000berent for\nprotons and neutrons. The orbital single-particle ener-\ngies\u000fa\u0015correspond to a central Woods-Saxon potential\nplus spin-orbit interaction [45]. The Hamiltonian is given\nby\n^H=X\na\u0015\u000fa\u0015^na\u0015\u0000\u001f\n22X\n\u0016=\u00002: (\u0000)\u0016^O2\u0000\u0016^O2\u0016:\u0000X\n\u0015=p;ng\u0015^Py\n\u0015^P\u0015:\n(10)\nIn Eq. (10), : : denotes normal ordering and the\nquadrupole operator ^O2\u0016=^Op\n2\u0016+^On\n2\u0016is de\fned by\n^O\u0015\n2\u0016= (dVWS=dr)\u0015Y2\u0016;\u0015, whereVWSis the Woods-Saxon\ncentral potential. Also, ^P\u0015=P\n(a\u0015m)>0ca\u0015mca\u0015mis the\npair annihilation operator, where ja\u0015miis a shell-model\nsingle-particle state with magnetic quantum number m\nfor particle species \u0015, (a\u0015m)>0 runs over half the states,\nandja\u0015mi= (\u0000)ja\u0015+la\u0015+mja\u0015\u0000miis the time-reversed\npartner ofja\u0015mi. The interaction in Eq. (10) has a sim-\nilar form to the interaction used in Refs. [15{18], except\n2There are 2 Nsgeneralized quasiparticle energies and thermal\noccupation numbers, where Nsis the number of single-particle\nstates. Let k= 1;:::;NsandEkbe the positive quasiparticle\nenergy. Then, ~Ek=Ekand ~Ek+Ns=\u0000Ek. Similarly, ~fk=\n(1 +e\fEk)\u00001, and ~fk+Ns= 1\u0000~fk= (1 +e\u0000\fEk)\u00001.4\nthat the latter also included octupole and hexadecupole\nterms in the multipole-multipole part of the Hamiltonian.\nThe static auxiliary \felds consist of \fve complex \felds\n\u000b2\u0016(\u0016=\u00002;:::;2) satisfying \u000b\u0003\n2\u0016= (\u0000)\u0016\u000b2\u0000\u0016that cou-\nple to the corresponding quadrupole operators ^O2\u0016, to-\ngether with two complex \felds \u0001 p;\u0001nthat couple to the\ncorresponding pair operators ^Pp;^Pn. We transform \u000b2\u0016\nto their intrinsic frame components ~ \u000b2\u0016de\fned by\n~\u000b20=\f2cos\r;~\u000b21= ~\u000b2\u00001= 0;~\u000b22= ~\u000b2\u00002=\f2p\n2sin\r;\n(11)\nand the three Euler angles characterizing the orientation\nof the intrinsic frame. The integrand in Eq. (7) is inde-\npendent of the Euler angles and the phases of the pair-\ning \felds. After integrating over the Euler angles and\npairing \feld phases, the remaining integration variables\nare the intrinsic deformation parameters \f2>0 and\n0\u0014\r\u0014\u0019=3, and the pairing \felds \u0001 p;n>0. The\nSPA+RPA grand-canonical partition function is then\ngiven by\nZ(\f;\u0016p;\u0016n) =Z1\n0d\f2Z\u0019=3\n0d\r\n\u0002Z1\n0Z1\n0d\u0001pd\u0001nM(\u001b)Z(\u001b)C(\u001b);(12)\nwhere\u001b= (\f2;\r;\u0001p;\u0001n) and the measure M(\u001b) is\nM(\u001b) =(\f\u001f)5=2\n(2\u0019)1=2\u00122\f\ng\u00132\n\f4\n2sin(3\r)\u0001p\u0001n\n\u0002e\u0000\f\u001f\f2\n2=2\u0000\fP\n\u0015=p;n\u00012\n\u0015=g\u0015:(13)\nZ(\u001b) is given by Eq. (6), where the static one-body\nHamiltonian ^h\u001bis\n^h\u001b=^H1\u0000\u001f\f2\u0014\ncos\r^O20+1p\n2sin\r(^O22+^O2\u00002)\u0015\n\u0000X\n\u0015=p;n\u0014\n\u0001\u0015\u0010\n^Py\n\u0015+^P\u0015\u0011\n+g\u0015\n2\u0012Ns;\u0015\n2\u0000^N\u0015\u0013\u0015\n;\n(14)\nwhere ^H1is the one-body Hamiltonian in Eq. (10) that\nincludes a shift due to uncoupling the normal-ordered\nquadrupole-quadrupole term in Eq. (10), and Ns;\u0015is the\nnumber of single-particle states for particle species \u0015. In-\ncluding the contribution of the chemical potentials and\nusing a Bogoliubov transformation to diagonalize (14)\nyields [37]\n^h\u001b\u0000X\n\u0015=p;n\u0016\u0015^N\u0015=X\n\u0015=p;nX\nk>0\u0014\nEk;\u0015(ay\nk;\u0015ak;\u0015+ay\n\u0016k;\u0015a\u0016k;\u0015)\n+ (\"k;\u0015\u0000\u0016\u0015\u0000Ek;\u0015)\u0015\n;\n(15)wherea;ayare annihilation and creation quasiparti-\ncle operators, \"k;\u0015are the eigenvalues of the particle-\nnumber-conserving term in Eq. (14), and the quasiparti-\ncle energies Ek;\u0015are given by\nEk;\u0015=q\n(\"k;\u0015\u0000\u0016\u0015\u0000g\u0015=2)2+ \u00012\n\u0015: (16)\nThe static partition function Z(\u001b) is then given by\nZ(\u001b) =Y\n\u0015=p;nY\nk>0e\u0000\f(\"k;\u0015\u0000\u0016\u0015)4 cosh2\u0012\fEk;\u0015\n2\u0013\n:(17)\nThe RPA correction factor C(\u001b) in Eq. (12) can be cal-\nculated using Eq. (8). For the single-particle Hamiltonian\n(14), the RPA matrix (9) connects only quasiparticle-\nstate pairs with the same total parity and total z-\nsignature. These symmetries render the RPA matrices\nblock-diagonal and consequently make their diagonal-\nization computationally more e\u000ecient. We note that\nthe computational cost of diagonalizing the RPA ma-\ntrix presents a signi\fcant challenge for extending the\nSPA+RPA method to larger model spaces and to more\ngeneral interactions; see Sec. V.\nC. Approximate canonical partition function\nEq. (12) gives the SPA+RPA partition function in the\ngrand-canonical ensemble. However, to calculate state\ndensities of nuclei with given numbers of protons and\nneutrons, it is necessary to determine the canonical par-\ntition function [46]. We do so approximately in the\nSPA+RPA by combining exact number-parity projection\nwith a saddle-point approximation for particle-number\nprojection [36, 37].\nThe number-parity projection operator is given by\n^P\u0011=Y\n\u0015=p;n1 +\u0011\u0015ei\u0019^N\u0015\n2; (18)\nwhere\u0011p;\u0011n= +1(\u00001) for even(odd) numbers of pro-\ntons or neutrons, respectively. In the SPA+RPA, the\nnumber-parity projected grand-canonical partition func-\ntion is given by\nZ\u0011(\f;\u0016p;\u0016n) =Z\nd\u001bM (\u001b)Z\u0011(\u001b)C\u0011(\u001b);(19)\nwhere\nZ\u0011(\u001b) =Y\n\u0015=p;nTr\u0010\ne\u0000\f(^h\u001b;\u0015\u0000\u0016\u0015^N\u0015)\u00111\n2\u0010\n1 +\u0011\u0015hei\u0019^N\u0015i\u001b\u0011\n(20)\nis the number-parity projected partition function for a\nstatic auxiliary-\feld con\fguration \u001b, and ^h\u001b;\u0015is the one-\nbody Hamiltonian (15) for particle species \u0015.3The op-\n3We note that ^h\u001bin Eqs. (14) and (15) is the sum of proton and\nneutron terms.5\neratorei\u0019^N\u0015commutes with the Hamiltonian (15), so\nEq. (20) can be evaluated explicitly [47] to give\nhei\u0019^N\u0015i\u001b=Y\nk>0tanh2\u0012\fEk;\u0015\n2\u0013\n: (21)\nC\u0011(\u001b) in Eq. (19) is the number-parity projected RPA\ncorrection factor, which is obtained by replacing the\nquasiparticle occupation numbers in the RPA matrix (9)\nwith the number-parity-projected occupation numbers\nf\u0011\nk;\u0015=fk;\u0015+\u0011\u0015hei\u0019^N\u0015i\u001bf\u0019\nk;\u0015\n1 +\u0011\u0015hei\u0019^N\u0015i\u001b; (22)\nwherefk;\u0015= (1 +e\fEk;\u0015)\u00001andf\u0019\nk;\u0015= (1\u0000e\fEk;\u0015)\u00001.\nThe canonical partition function Zc(\f;Np;Nn),\nwhereNpandNnare, respectively, the number\nof valence protons and neutrons, is related to the\nnumber-parity-projected grand-canonical partition func-\ntionZ\u0011(\f;\u0016p;\u0016n) by an inverse Laplace transform\nZc(\f;Np;Nn) =(2\f)2\n(2\u0019i)2Zi\u0019=2\f\n\u0000i\u0019=2\fd\u0016pd\u0016ne\u0000\fP\n\u0015=p;n\u0016\u0015N\u0015\n\u0002Z\u0011(\f;\u0016p;\u0016n):\n(23)\nThe leading factors of 2 in the integrals over \u0016pand\u0016n\nin Eq. (23) arise because the number-parity projection\nexcludes half the possible particle numbers [34, 48]. Fol-\nlowing Ref. [37], we insert Eq. (19) into Eq. (23), change\nthe order of the integrations over \u001band over\u0016p;\u0016n, and\nevaluate the integrals over \u0016p;\u0016nby applying the saddle-\npoint approximation to the unprojected single-particle\npartition function (6). We \fnd\nZc(\f;Np;Nn)\u0019Z\nd\u001bM (\u001b)Z\u0011(\u001b)\n\u0002eP\n\u0015=p;n(ln\u0010\u0015\u0000\f\u0016\u0015N\u0015)C\u0011(\u001b);\n(24)\nwhere\n\u0010\u0015= 2\u00122\u0019\n\f2@2lnZ(\u001b)\n@\u00162\n\u0015\u0013\u00001=2\n; (25)\nand [37, 47]\n@2lnZ(\u001b)\n\f2@\u00162\n\u0015=X\nk>0\fEk;\u0015(\"k;\u0015\u0000\u0016\u0015\u0000g\u0015\n2)2+ \u00012\n\u0015sinh(\fEk;\u0015)\n2\fE3\nk;\u0015cosh2\u0010\n\fEk;\u0015\n2\u0011 :\n(26)\nThe chemical potential \u0016\u0015in (26) for each particle species\n\u0015is determined by the saddle-point condition [37]\nN\u0015=X\nk>0\u0014\n1\u0000\u0012\"k;\u0015\u0000\u0016\u0015\u0000g\u0015\n2\nEk;\u0015\u0013\ntanh\u0012\fEk;\u0015\n2\u0013\u0015\n:\n(27)D. State density\nThe state density is the inverse Laplace transform of\nthe canonical partition function\n\u001a(E;Np;Nn) =1\n2\u0019iZi1\n\u0000i1d\fe\fEZc(\f;Np;Nn):(28)\nWe determine the average state density by evaluating\nthe integral in Eq. (28) in the saddle-point approxima-\ntion [46]\n\u001a(E;Np;Nn)\u0019\u00122\u0019\n\f2C(\f)\u0013\u00001=2\neSc(\f); (29)\nwhere\nSc(\f) =\fEc(\f) + lnZc(\f) (30)\nis the canonical entropy and C=dEc=dT is the canon-\nical heat capacity.4In Eq. (29), \fis a function of E\ndetermined by the saddle-point condition\nE=Ec(\f) =\u0000@lnZc(\f)\n@\f: (31)\nIII. PRACTICAL METHODS FOR\nCALCULATING SPA+RPA STATE DENSITIES\nA. Monte Carlo method\nIn previous applications of the SPA+RPA, the integra-\ntion over the static \felds \u001bwas evaluated by quadrature\nmethods [34{37]. Although quadrature methods could\nin principle be used for the pairing plus quadrupole in-\nteraction, the computational cost of such methods scales\nas the exponent of the number of static \felds and thus\nbecomes prohibitive for more general e\u000bective nuclear in-\nteractions, such as the interaction used in Refs. [15{18].\nSince our purpose is to show that the SPA+RPA can be\na practical alternative to existing methods for calculat-\ning CI shell model state densities in heavy nuclei, it is\nimportant to use a computational method that can be\napplied to more general interactions. Here we introduce\na Monte Carlo method to calculate the SPA+RPA canon-\nical energy and heat capacity, from which we obtain the\npartition function, canonical entropy, and state density.\nWe evaluate the canonical energy Ecde\fned by Eq. (31)\nusing the approximate canonical partition function Zcin\nEq. (24). Using dimensionless integration variables\nx=\u0012p\n\f\u001f\f 2;\r;q\n\f=gp\u0001p;p\n\f=gn\u0001n\u0013\n; (32)\n4For simplicity of notation, we have omitted the explicit depen-\ndence onNp;NninZc,Ec,Sc, andC.6\nwe rewrite Eq. (24) in the form\nZc(\f)\u0019Z\ndxM (x)Z\u0011(x) (33)\nwhere\nZ\u0011(x) =Z\u0011(x)eP\n\u0015(ln\u0010\u0015\u0000\f\u0016\u0015N\u0015)C\u0011(x); (34)\nandM(x) is the measure in Eq. (13). M(x) becomes in-\ndependent of \fwhen expressed as a function of x. Using\nEq. (33) in Eq. (31), we \fnd5\nEc(\f) =\u0000R\ndxW (x)h\nC\u0011(x)@lnZ\u0011(x)\n@\fi\nR\ndxW (x)C\u0011(x); (35)\nwhereW(x) is the positive-de\fnite weight function\nW(x) =M(x)Z\u0011(x)\nC\u0011(x)=M(x)Z\u0011(x)eP\n\u0015(ln\u0010\u0015\u0000\f\u0016\u0015N\u0015):\n(36)\nEq. (35) can be rewritten as\nEc(\f) =\u0000D\nC\u0011(x)@lnZ\u0011(x)\n@\fE\nW\nhC\u0011(x)iW; (37)\nwhere the expectation value hf(x)iWof a function f(x)\nwith respect to a weight function W(x) is de\fned by\nhf(x)iW=R\ndxW (x)f(x)R\ndxW (x): (38)\nTo evaluate the expectation values in Eq. (37), we use\na standard Monte Carlo method in which the values of\nxare sampled according to the weight function W(x).\nSpeci\fcally, we perform a random walk in the space of\nintegration variables x, updating con\fgurations of xac-\ncording to the Metropolis-Hastings algorithm [49]. We\ncalculate the observables at sample con\fgurations sep-\narated by a su\u000ecient number of steps to ensure that\nthe samples are decorrelated. We then use the jackknife\nmethod [50] to calculate expectation values and statisti-\ncal errors of the thermodynamic observables, such as Ec\nin Eq. (37).6Further details are provided in the Sup-\nplemental Material repository that accompanies this ar-\nticle [51]. This Monte Carlo method is similar to the one\nimplemented in the SMMC method [19].\nHaving calculated Ec(\f) using Eq. (37) for a su\u000ecient\nnumber of\fvalues, we obtain the partition function by\nintegrating Eq. (31)\nlnZc(\f) =Sc(0)\u0000Z\f\n0d\f0Ec(\f0): (39)\n5We note that the canonical energy Ec(\f) =\u0000@lnZc=@\fcalcu-\nlated in Eq. (35) is not the same as the thermal expectation value\nof the Hamiltonian in the SPA+RPA. In our approach, the basic\nquantity is the partition function, and the canonical energy is\nde\fned by the saddle-point condition (31).\n6In practice, @lnZ\u0011(x)=@\f in Eq. (37) for a given sample xis\nevaluated using a \fnite-di\u000berence formula.Sc(0) in Eq. (39) is the canonical entropy at \f= 0 given\nby\nSc(0) =X\n\u0015=p;nln\u0012\nNs;\u0015\nN\u0015\u0013\n; (40)\nwhereNs;\u0015is the dimension of the single-particle model\nspace andN\u0015is the number of valence particles for parti-\ncle species\u0015. The heat capacity C(\f) =\u0000\f2@2lnZc=@\f2\ncan also be expressed in terms of expectation values of\nthe form (38) via\n@2lnZc\n@\f2\u0019\u001c\nC\u0011(x)\u0014\n@2lnZ\u0011(x)\n@\f2+\u0010\n@lnZ\u0011(x)\n@\f\u00112\u0015\u001d\nW\nhC\u0011(x)iW\n\u00002\n4D\nC\u0011(x)@lnZ\u0011(x)\n@\fE\nW\nhC\u0011(x)iW3\n52\n:\n(41)\nUsing Eqs. (37), (39-41) together with Eqs. (30) and (29),\nwe calculate the canonical entropy and state density, re-\nspectively.\nFor importance sampling, it would have been prefer-\nable to include the RPA correction factor C\u0011(x) in the\nweight function. However, diagonalizing the RPA ma-\ntrix is the most computationally intensive part of the\ncalculation. We have therefore chosen W(x) in Eq. (36)\nthat does not include C\u0011(x). We \fnd that the Monte\nCarlo method with W(x) as the weight function samples\nthe con\fguration space e\u000eciently enough for our calcu-\nlations.\nB. Ground-state energy\nEq. (29) expresses the state density as a function of the\nabsolute energy E. However, in statistical reaction the-\nory, state densities have to be known as functions of the\nexcitation energy Ex=E\u0000E0, whereE0is the ground-\nstate energy. However, the SPA+RPA breaks down at\nlow but nonzero temperature, and therefore the ground-\nstate energy cannot be calculated directly. A similar\nproblem occurs in SMMC calculations in nuclei with an\nodd number of valence protons or neutrons with good-\nsign interactions, where the projection on an odd num-\nber of particles introduces a Monte Carlo sign problem\nat low temperatures [19, 52] and the ground-state energy\ncannot be accessed directly. In Ref. [40], the partition\nfunction extrapolation method was introduced to deter-\nmine the ground-state energy E0of such nuclei from the\ncalculated SMMC partition function at higher tempera-\ntures. We apply this method, summarized below, to the\nSPA+RPA partition function in order to estimate the\nground-state energy.\nWe de\fne the excitation partition function Z0\nc(\f;Eref)\nwith respect to some reference energy Erefby\nZ0\nc(\f;Eref) =Zc(\f)e\fEref: (42)7\nIn particular, if Eref=E0, the excitation partition func-\ntion is the Laplace transform of the state density \u001a(Ex)\ngiven as a function of the excitation energy\nZ0\nc(\f;E0) =Zc(\f)e\fE0=Z1\n0dEx\u001a(Ex)e\u0000\fEx:(43)\nThe excitation partition function for an arbitrary refer-\nence energy is related to the excitation partition function\nin Eq. (43) by\nlnZ0\nc(\f;Eref) = lnZ0\nc(\f;E0)\u0000\f(E0\u0000Eref):(44)\nThe main idea behind the partition function extrapo-\nlation method is to use a reliable model for the state den-\nsity in Eq. (44). For even-even nuclei, it was shown [15]\nthat the state density is well described by the composite\nformula [53]\n\u001acomp(Ex) =(\ne(Ex\u0000E1)=T1Ex<EM\n\u001aBBF(Ex)Ex>EM; (45)\nwhereEMis a matching energy, and \u001aBBFis the back-\nshifted Bethe formula [54]\n\u001aBBF(Ex) =p\u0019\n12a1=4e2p\na(Ex\u0000\u0001)\n(Ex\u0000\u0001)5=4: (46)\nBelowEM, the state density is described by the constant-\ntemperature formula with parameters E1;T1, which are\ndetermined from the parameters aand \u0001 of the BBF\nformula by the conditions that the state density and its\nderivative are continuous at EM. Inserting the compos-\nite state density formula (45) into Eq. (43), we can \ft\nEq. (44) to the SPA+RPA excitation partition function\nabove the method's breakdown temperature.\nWe carry out the \ft in two steps [40]. In the \frst step,\nwe evaluate Eq. (43) in the saddle-point approximation\nusing the BBF formula for \u001a(Ex) to \fnd\nlnZ0\nc(\f;Eref)\u0019a\n\f+ ln\u0012\u0019\f\n6a\u0013\n\u0000\fs; (47)\nwheres=E0\u0000Eref+\u0001. Choosing Erefsu\u000eciently close\nto the yet-to-be-determined ground-state energy E0, we\ncalculate ln Z0\nc(\f;Eref) from the SPA+RPA data and \ft\nEq. (47) to this data at moderate temperatures (for which\nthe BBF holds). This yields the \ftted parameters (^ a;^s).\nIn the second step, we use a \ft function obtained by\nsubstituting the complete composite formula \u001acomp(Ex)\nfor\u001a(Ex) in Eqs. (43) and (44)\nlnZ0\nc(\f;Eref)\u0019lnZ1\n0dEx\u001acomp(Ex)e\u0000\fEx\u0000\f(E0\u0000Eref);\n(48)\nwhere ^ais \fxed, \u0001 = ^ s\u0000(E0\u0000Eref), and the integral\nis carried out numerically. Eq. (48) depends on only two\nparameters E0andEM, which we determine with a \u001f2\n\ft.If the backshift parameter \u0001 is negative, we \fnd that\nthe \ftted value of EMis small, and the composite formula\nis essentially equivalent to the BBF (46). In practice,\nwe then use \u001aBBFinstead of\u001acomp in the second step of\nEq. (48). We note that for negative \u0001, the BBF is well\nde\fned down to Ex= 0.\nIV. APPLICATION TO SAMARIUM ISOTOPES\nHere, we apply the SPA+RPA method discussed\nin Secs. II and III to a chain of samarium isotopes\n148\u0000155Sm, which describes the crossover from spherical\nto well-deformed nuclei. We use a model space consist-\ning of the following orbitals: 0 g7=2, 1d5=2, 1d3=2, 2s1=2,\n0h11=2, and 1f7=2for protons; 0 h11=2, 0h9=2, 1f7=2, 1f5=2,\n2p3=2, 2p1=2, 0i13=2, and 1g9=2for neutrons. The se-\nlection of these orbitals is discussed in Ref. [15]. The\nsingle-particle energies and wave functions correspond to\na Woods-Saxon central potential plus spin-orbit inter-\naction with the parameters described in Ref. [15]. The\nquadrupole interaction parameter in Eq. (10) is given by\n\u001f=k2\u001f0, where\u001f0is determined self-consistently [55]\nandk2is a renormalization factor accounting for core po-\nlarization. The pairing strengths gp(n)=\rgp(n), where\ngp(n)= 10:9=Z(N) MeV (ZandNare the total num-\nber of protons and neutrons, respectively), and \ris a\nrenormalization factor. k2and\rare parametrized by\n\r= 1:225\u0012\n0:72\u00000:5\n(N\u000090)2+ 5:3\u0013\nk2= 2:15 + 0:0025(N\u000087)2;(49)\nThis parameterization is similar to the one used in the\nSMMC calculations of Refs. [16, 18], except that \ris\nincreased by 22.5% to account for the absence of higher-\norder multipoles in the interaction.\nIn the SPA+RPA, we calculate the canonical energy\nEcand heat capacity Cwith the Monte Carlo method\ndescribed in Sec. III A, using Eqs. (37) and (41), respec-\ntively. We de\fne a sweep as an update of each of the\nfour integration variables x. For each\fvalue, we initially\ncarry out 50 sweeps to make sure that the Monte Carlo\nwalk is thermalized, i.e., the walk has reached a repre-\nsentative region of the con\fguration space according to\nthe weight function W(x). We then calculate the ob-\nservables every 70 sweeps to ensure that their values are\nsu\u000eciently decorrelated. In the Supplemental Material\nrepository, we show that these numbers of sweeps yield\nacceptable thermalization and decorrelation [51]. We use\n\u00181000\u00002000 samples per \fvalue in the calculation of\nthe observables.\nFrom the energy Ec(\f), we calculate ln Zc(\f) using\nEq. (39) and the canonical entropy Sc(\f) using Eq. (30).\nWe then calculate the state density from Eq. (29). The\nMonte Carlo results and computer codes used to analyze\nthem are provided in the Supplemental Material reposi-\ntory [51].8\n0 1 2 3 4\n (MeV1)\n0102030405060Entropy\n246810\n (MeV1)\n04812Entropy148Sm\n0 1 2 3 4\n (MeV1)\n246810\n (MeV1)\n04812Entropy150Sm\n0 1 2 3 4\n (MeV1)\n246810\n (MeV1)\n04812Entropy152Sm\n0 1 2 3 4 5\n (MeV1)\n246810\n (MeV1)\n04812Entropy154Sm\nFIG. 1. The canonical entropy as a function of inverse temperature \ffor the even-mass samarium isotopes148;150;152;154Sm.\nFor each isotope, the SPA+RPA entropy (orange circles) is compared with the SMMC entropy (blue squares) and the HFB\nentropy (green dashed-dotted line). The error bars on the SMMC and SPA+RPA results are statistical errors due to the Monte\nCarlo sampling. The insets show an expanded scale at large values of \f.\nBelow, we compare the SPA+RPA results with exact\n(up to statistical errors) SMMC results and with mean-\n\feld \fnite-temperature HFB results. In the SMMC, we\ncalculate the thermal canonical energy as the expecta-\ntion value of the Hamiltonian h^Hifor \fxed proton and\nneutron numbers; for details, see Ref. [19]. We also cal-\nculate the heat capacity using the method of Ref. [56],\nwhich reduces signi\fcantly the statistical errors. Using\nthe canonical energy and heat capacity, we obtain the\nentropy and state density using Eqs. (30) and (29), re-\nspectively.\nFor the \fnite-temperature HFB, we calculate the\nself-consistent HFB solution at each temperature us-\ning the code of Ref. [57], together with the particle-\nnumber-projected partition function using the method of\nRef. [58]. We then apply Eqs. (31), (30), (29) to obtain,\nrespectively, the energy, entropy, and state density.\nA. Even-mass samarium isotopes\nIn Fig. 1, we show the canonical entropy Scas a func-\ntion of inverse temperature \ffor the SPA+RPA (or-\nange circles), SMMC (blue squares), and HFB (green\ndashed-dotted line) for the even-mass samarium isotopes\n148;150;152;154Sm. The SPA+RPA entropies are shown up\nto values of \fclose to the value above which the approx-\nimation breaks down. For each of the isotopes, we \fnd\nthe SPA+RPA entropy to be in excellent agreement with\nthe SMMC entropy. The two kinks in the HFB entropy\nfor148Sm indicate the proton and neutron pairing phase\ntransitions, and the additional kink at lower \ffor the\nother isotopes is due to the shape phase transition from\na spherical to a deformed mean-\feld solution.\nAt\fvalues above the shape transition, the HFB en-\ntropy signi\fcantly underestimates the SPA+RPA and\nSMMC entropies because the HFB does not describe the\ncontribution of rotational bands that are built on intrin-\nsic mean-\feld band heads [46]. The SPA+RPA restoresthe rotational symmetry that is broken in the HFB and\nthus reproduces this rotational enhancement of the en-\ntropy. Furthermore, in the pairing phase, the HFB en-\ntropy becomes unphysically negative because of the in-\nherent breaking of particle-number conservation in the\nHFB approximation [58]. In contrast, the SPA+RPA\nentropy remains nonnegative (within statistical errors)\nbecause the SPA+RPA repairs the intrinsic violation of\nparticle-number conservation. Finally, as the neutron\nnumber increases, the SPA+RPA and SMMC entropies\nremain nonzero to increasingly large values of \f, indi-\ncating the presence of a rotational enhancement down to\nlower temperatures in nuclei with larger deformation.\nWe used the partition function extrapolation method\nsummarized in Sec. III B to estimate the ground-state\nenergy from the SPA+RPA partition function above\nthe breakdown temperature of the approximation. For\n148;150Sm, we used the composite formula (45) in the sec-\nond step of the \ft, whereas for152;154Sm the back-shift\nparameter \u0001 is negative, and it was simpler to use the\nBBF formula (46). In Table I, we compare the SPA+RPA\nground-state energy estimates to the SMMC and HFB\nground-state energies. We calculated the SMMC ground-\nstate energies by taking a weighted average of the ther-\nmal energy at large \fvalues (\f\u00198\u000020 MeV\u00001). Ta-\nble I shows that the SPA+RPA misses at most \u0018600\nkeV of ground-state correlation energy, whereas the HFB\nmisses a few MeV of correlation energy in each isotope.\nThe agreement between the SPA+RPA estimate and the\nSMMC ground-state energy improves with decreasing de-\nformation, and the two agree with each other for the\nspherical isotope148Sm. In Table II, we show the pa-\nrametersa;\u0001;EMof the state density formulas obtained\nfrom the ground-state energy \fts.\nUsing the ground-state energies in Table I, we calcu-\nlated the SMMC, SPA+RPA, and HFB state densities\nfor148;150;152;154Sm. Fig. 2 shows these densities, using\na similar convention as in Fig. 1. In each isotope, the\nSPA+RPA state density is in good agreement with the9\n024681012\nEx (MeV)1001021041061081010 (MeV1)\n148Sm\n024681012\nEx (MeV)150Sm\n024681012\nEx (MeV)\n152Sm\n02468101214\nEx (MeV)154Sm\nFIG. 2. The state density \u001aas a function of excitation energy Exfor the even-mass samarium isotopes. The SPA+RPA state\ndensity (orange circles) is compared with the SMMC density (blue squares) and the HFB density (green dashed-dotted lines)\nfor each isotope. The grey dashed lines show the composite formula (45) for148;150Sm and the BBF (46) for152;154Sm, with\nthe parameters from Table II (see text). Experimental neutron resonance data (red triangles) and low-energy level counting\ndata (black histograms) are also shown. The error bars show statistical errors in the SPA+RPA and SMMC results.\nTABLE I. The ground-state energies (in MeV) for the\nSMMC, SPA+RPA, and HFB for148;150;152;154Sm. The\nSPA+RPA estimates are obtained with the partition func-\ntion extrapolation method described in Sec. III B, with errors\narising from the \u001f2\ft to the SPA+RPA partition function\ndata. The SMMC values are obtained by taking a weighted\naverage of the thermal energies at large \fvalues.\nSMMC SPA+RPA HFB\n148Sm -234.180 \u00060.016 -234.131 \u00060.021 -230.979\n150Sm -254.019 \u00060.014 -253.859 \u00060.015 -251.127\n152Sm -273.756 \u00060.010 -273.242 \u00060.017 -271.153\n154Sm -293.292 \u00060.010 -292.680 \u00060.017 -290.449\nTABLE II. The parameters obtained from the partition func-\ntion extrapolation method discussed in Sec. III B applied to\nthe SPA+RPA. For148;150Sm, we used the composite formula\n(45) in the second step of the \ft, while for152;154Sm, we used\nthe BBF (46).\na(MeV\u00001) \u0001 (MeV) EM(MeV)\n148Sm 17.09 \u00060.11 1.07 \u00060.03 1.452\n150Sm 18.28 \u00060.06 0.62 \u00060.02 0.95\n152Sm 19.14 \u00060.10 -0.17 \u00060.03 {\n154Sm 18.89 \u00060.13 -0.38 \u00060.03 {\nSMMC state density. In contrast, the HFB state density\nsigni\fcantly underestimates the SMMC and SPA+RPA\ndensities. As the neutron number increases, the enhance-\nment of the SMMC and SPA+RPA state densities over\nthe HFB state density persists to higher excitation en-\nergy. This enhancement originates in the contribution\nof rotational bands that are included in the SMMC and\nSPA+RPA densities but are not described by the HFB\napproximation. We observe an additional enhancement\nof the SPA+RPA and SMMC state densities over theHFB density at very low excitation energies, which is due\nto the unphysical negative entropy in the pairing phase\nof the HFB. This latter enhancement is particularly ap-\nparent in the spherical nucleus148Sm.\nIn Fig. 2 we also compare the calculated state den-\nsities with experimental state densities obtained from\nlevel counting at low excitation energies [59] (black his-\ntograms) and the average s-wave neutron resonance spac-\ningsD0at the neutron threshold [60] (red triangles). We\nused a spin cuto\u000b model [61, 62] with the rigid-body mo-\nment of inertia to convert D0values to state densities.\nThe agreement between the data and the SPA+RPA and\nSMMC state densities is good overall, in particular for\n148;150Sm. In152;154Sm, the calculated densities overesti-\nmate the experimental data. In contrast, the mean-\feld\nHFB densities do not agree well with the experimental\ndata.\nIn Fig. 2 we also show the phenomenological compos-\nite or BBF state densities calculated with the parame-\nters reported in Table II. We \fnd good agreement be-\ntween these \ftted parameterizations and the SPA+RPA\nstate densities. This agreement demonstrates that the\npartition function extrapolation method to extract the\nground-state energy described in Sec. III B is reliable.\nTo demonstrate even more clearly how well the\nSPA+RPA describes correlations that are missing in the\nmean-\feld approximation, we show in Fig. 3 the state\ndensity enhancement factor K=\u001a=\u001aHFBfor the SMMC\n(blue squares) and SPA+RPA (orange circles). The\nSPA+RPA enhancement factors are in good agreement\nwith the SMMC enhancement factors. In the spheri-\ncal nucleus148Sm, the enhancement factor di\u000bers sig-\nni\fcantly from one only at the lowest excitation energies\nand is due entirely to the unphysical negative entropy\nin the pairing phase of the HFB. In the deformed iso-\ntopes150;152;154Sm, a signi\fcant rotational enhancement\nof\u001810 appears and persists to increasing excitation\nenergy as the neutron number increases. This change10\n0 5 10 15\nEx (MeV)100101102Enhancement K148Sm\n0 5 10 15\nEx (MeV)\n150Sm\n0 5 10 15\nEx (MeV)\n152Sm\n0 5 10 15 20\nEx (MeV)\n154Sm\nFIG. 3. The enhancement factor K=\u001a=\u001aHFBfor the SPA+RPA (orange circles), the SMMC (blue squares), and the neutron\nresonance data (red triangles) for the even-mass samarium isotopes. The error bars indicate statistical errors from the Monte\nCarlo sampling.\n0 1 2 3 4\n (MeV1)\n0102030405060Entropy\n246810\n (MeV1)\n04812Entropy149Sm\n0 1 2 3 4\n (MeV1)\n246810\n (MeV1)\n04812Entropy151Sm\n0 1 2 3 4\n (MeV1)\n246810\n (MeV1)\n04812Entropy153Sm\n0 1 2 3 4 5\n (MeV1)\n246810\n (MeV1)\n04812Entropy155Sm\nFIG. 4. The canonical entropy as a function of inverse temperature \ffor the odd-mass samarium isotopes149;151;153;155Sm.\nThe SPA+RPA entropy (orange circles) is compared with the SMMC entropy (blue squares) for each isotope. The error bars\nshow statistical errors arising from the Monte Carlo sampling. The insets show an expanded scale at large values of \f.\nin the enhancement factor indicates the crossover from\npairing-dominated to rotational collectivity in the chain\nof samarium isotopes [16, 38, 39]. Fig. 3 also shows that\nthe SPA+RPA and SMMC results are in good agree-\nment with the neutron resonance data (red triangles)\nin148;150Sm. The calculated enhancement factors some-\nwhat overestimate the neutron resonance data in152Sm.\nB. Odd-mass samarium isotopes\nHaving established the accuracy of the SPA+RPA\nstate densities for the even-mass samarium isotopes,\nwe next benchmark the SPA+RPA state densities for\nthe odd-mass samarium isotopes149;151;153;155Sm. In\nFig. 4, we compare the SPA+RPA canonical entropy\n(orange circles) with the SMMC canonical entropy (blue\nsquares) for the odd-mass isotopes. In each isotope, the\nSPA+RPA entropy is in excellent agreement with the\nSMMC entropy. The odd-mass sign problem leads to\nlarge \ructuations of the SMMC entropy at high values of\n\f, as is shown in the insets of Fig. 4. The SPA+RPA en-tropy remains reliable to slightly lower temperatures than\nthe SMMC entropy. Both the SPA+RPA and SMMC en-\ntropies appear to converge to a nonzero limit, indicating\nthe magnetic degeneracy of the nonzero-spin ground state\nof the odd-mass system.\nThe projection on the odd number of neutrons intro-\nduces a Monte Carlo sign problem in the SMMC at low\ntemperatures [19, 52] that prevents the ground-state en-\nergy from being calculated directly. To obtain the SMMC\nand SPA+RPA state densities as functions of excita-\ntion energy, we used the partition function extrapolation\nmethod, which is summarized in Sec. III B, to determine\nthe ground-state energies E0in both approaches. We\nused the BBF (46) in the second step of the \fts. Table III\nshows the extracted values of E0and the BBF state den-\nsity parameters a;\u0001 for the SMMC and SPA+RPA. The\nagreement between the SMMC and SPA+RPA ground-\nstate energy estimates is even better than for the even-\nmass isotopes, with the largest discrepancy of \u0018300 keV\nin149Sm.\nIn Fig. 5, we compare the state densities calculated\nwith the SMMC (blue squares) and SPA+RPA (orange11\n024681012\nEx (MeV)10010210410610810101012 (MeV1)\n149Sm\n024681012\nEx (MeV)151Sm\n024681012\nEx (MeV)\n153Sm\n02468101214\nEx (MeV)155Sm\nFIG. 5. The state density as a function of excitation energy Excalculated with the SPA+RPA (orange circles) and SMMC\n(blue squares) for the odd-mass samarium isotopes149;151;153;155Sm. The BBF calculated with parameters obtained from \ftting\nthe SPA+RPA ground-state energies (grey dashed lines) and \ftting the SMMC ground-state energies (black dotted lines) are\nalso shown. Experimental neutron resonance data (red triangles) and low-energy level counting data (black histograms) are\nshown as well.\nTABLE III. Ground-state energies E0anda;\u0001 values from\n\ftting the BBF (46) to the SPA+RPA and SMMC excitation\npartition functions for the odd-mass samarium isotopes.\nE0(MeV) a(MeV\u00001) \u0001 (MeV)\n149Sm SPA+RPA -242.957 \u00060.008 18.36 \u00060.04 -0.13 \u00060.01\nSMMC -243.327 \u00060.019 17.97 \u00060.04 -0.04 \u00060.02\n151Sm SPA+RPA -262.913 \u00060.006 19.24 \u00060.07 -0.39 \u00060.02\nSMMC -262.909 \u00060.047 18.63 \u00060.06 -0.77 \u00060.05\n153Sm SPA+RPA -282.384 \u00060.005 19.57 \u00060.12 -0.84 \u00060.02\nSMMC -282.449 \u00060.031 18.78 \u0006.09 -1.25 \u00060.05\n155Sm SPA+RPA -301.949 \u00060.003 19.07 \u00060.12 -1.00 \u00060.03\nSMMC -302.077 \u00060.021 18.27 \u00060.10 -1.39 \u00060.04\ncircles). The results are in good agreement with each\nother and with available experimental data from level\ncounting [59] (black histograms) and the average s-wave\nneutron resonance spacings [60] (red triangles). The\nagreement between the calculated and experimental state\ndensities degrades somewhat as the neutron number in-\ncreases and is of similar quality to the agreement found in\nRef. [18] using an interaction that included contributions\nfrom higher-order multipoles. We also show in Fig. 5 the\nBBF state densities calculated with the parameters tab-\nulated in Table III. These \ftted BBF densities agree well\nwith the calculated state densities.\nV. CONCLUSION AND OUTLOOK\nHere we benchmarked state densities calculated with\nthe SPA+RPA in the CI shell model framework against\nexact (up to controllable statistical errors) SMMC state\ndensities for a chain of samarium isotopes148\u0000155Sm.\nWe implemented a Monte Carlo method to calculate the\ncanonical energy and heat capacity in the SPA+RPA,from which we determined the canonical entropy and\nstate density. The SPA+RPA ground-state energy was\nestimated from the excitation partition function above\nthe SPA+RPA breakdown temperature using the parti-\ntion function extrapolation method [40].\nWe found good agreement between the SPA+RPA\nstate densities and SMMC state densities for all the iso-\ntopes considered. For the even-mass samarium isotopes,\nwe also calculated mean-\feld state densities using the\n\fnite-temperature HFB approximation. The main de\f-\nciencies of the mean-\feld approximation arise from the\nbroken rotational symmetry in deformed nuclei and the\ninherent violation of particle-number conservation in the\npairing condensate. Consequently, the mean-\feld ap-\nproximation cannot reproduce the contribution of rota-\ntional bands that are characteristic of deformed nuclei\nand yields an unphysical negative entropy in the pairing\nphase of the HFB. The SPA+RPA resolves these de\f-\nciencies of the mean-\feld approximation. In particular,\nthe SPA+RPA reproduces well the rotational collective\nenhancement of the state density relative to the mean-\n\feld density in deformed nuclei. This enhancement per-\nsists to higher excitation energies as the neutron number\nincreases, showing that the importance of rotational col-\nlectivity increases with deformation. Overall, our results\nshow that the SPA+RPA provides state densities in the\nCI shell model framework that are in agreement with ex-\nact SMMC densities.\nA signi\fcant limitation of the SPA+RPA method is\nthe computational cost of diagonalizing the RPA matrix\nat each sampled con\fguration of the static \felds. The di-\nmension of the RPA matrix scales as \u0018N2\ns;p+N2\ns;n(where\nNs;p(n)is the number of proton(neutron) single-particle\nstates), and the cost of diagonalizing this matrix scales\nas the cubic power of this dimension.7Calculating the\n7For the case considered here, calculating the RPA correction fac-12\ncanonical energy and heat capacity in the SMMC scales\nas a lower power of the number of single-particle states,\nspeci\fcally as\u0018N4\nsfor each particle species. It would\ntherefore be useful to investigate methods for speeding\nup the calculation of the RPA correction factor. One\nsuch method was proposed in Ref. [63].\nIn comparing the SPA+RPA to the SMMC, it is also\nuseful to consider the limits of the applicability of each\nmethod. The SPA+RPA method requires that the single-\nparticle Hamiltonian ^h\u001bin Eq. (3) be a Hermitian oper-\nator for any con\fguration \u001bof the static auxiliary \felds.\nThis condition is guaranteed if all terms in the Hamilto-\nnian are attractive when written in the separable form ^O2\n\u000b\nof Eq. (1), where each operator ^O\u000bis Hermitian. More-\nover, this condition guarantees that, at temperatures\nabove the breakdown temperature of the SPA+RPA, the\nweight function W(\u001b) of the Monte Carlo method dis-\ncussed in Sec. III A and the RPA correction factor C\u0011(\u001b)\nare both positive de\fnite for any static \feld con\fgura-\ntion\u001b. Consequently, W(\u001b) can be used as a weight\nfunction to sample the static \felds, and the Monte Carlo\nmethod described in Sec. III A will not have a sign prob-\nlem. In contrast, for the SMMC method to have a good\nMonte Carlo sign, the Hamiltonian must be invariant un-\nder time reversal, and all of its interaction terms must be\nattractive when written as a sum over terms of the form\nf^O\u000b;\u0016O\u000bgwhere \u0016O\u000bis the time reverse of ^O\u000b[19{21]. It\ncan be shown that the time-reversal and Hermitian con-\njugate of a tensor one-body density operator are related\nby a sign. Thus, in some cases, either the SPA+RPA\nor SMMC would have good sign while the other method\nwould have a sign problem, and the two methods wouldcomplement each other. Furthermore, the SPA+RPA\ncan be applied if time-reversal symmetry is broken, e.g.,\nin the presence of a cranking term \u0000!^Ji(i=x;y;z ),\nwhich would cause a sign problem in the SMMC.\nA method for approximately including repulsive inter-\nactions in the SPA+RPA framework was proposed in\nRef. [41]. It would be interesting to benchmark this\nmethod for realistic nuclear interactions that include re-\npulsive components.\nFinally, statistical reaction codes require as input spin-\nand parity-dependent level densities, rather than just\nstate densities. To calculate these level densities, it is\nnecessary to extend the SPA+RPA formalism to include\nspin and parity projections.\nACKNOWLEDGMENTS\nWe gratefully acknowedge W. Ryssens for the use of the\nHF-SHELL code [57] in the calculation of the HFB state\ndensities, and for providing the parameters of the CI shell\nmodel interaction used in this work. We also thank G.\nF. Bertsch for useful discussions and for providing com-\nments on the manuscript. 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We study the manifestations of unconventional\ntriplet pairing in the density of states of a disordered TMD based monolayer. The conventional\nsinglet pairing attraction is assumed to be dominant paring interaction. We map the phase diagrams\nof disordered Ising superconductors in the plane of temperature and the in-plane magnetic \feld. The\nlatter suppresses singlet and promote triplet correlations. The triplet order parameters of a trivial\n(non-trivial) symmetry compete (cooperate) with singlet order parameter which gives rise to a rich\nphase diagram. We locate the model dependent phase boundaries and compute the order parameters\nin each of the distinct phases. With this information, we obtain the density of states by solving the\nGorkov equation. The triplet components of the order parameters may change an apparent width\nof the density of states by signi\fcantly increasing the critical \feld. The triplet components of the\norder parameters lead to the density of states broadening signi\fcantly exceeding the broadening\ninduced by magnetic \feld and disorder in singlet superconductor.\nI. INTRODUCTION\nThe progress in growth and fabrication techniques\nmade it possible to create high quality ultra-thin multi-\nlayer systems with individual layers held together by\nweak Van der Waals forces [1]. In particular, few- and\nmonolayers of transition metal dichalcogenides (TMD)s\nwith highly tunable properties have been fabricated [2].\nMany such systems turned out to be superconducting\ndown to the monolayer limit [3{10] greatly stimulating\nthe research of two-dimensional superconductivity. Very\nrecently, the TMD based systems have been proposed as\na platform for controlled studies of the intrinsic or exter-\nnally induced unconventional superconductivity [11, 12].\nMany TMD monolayers such as 1H-NbSe 2lack the in-\nversion center even though the bulk (2H-NbSe 2) has such\na center. This gives rise to a spin splitting of electron\nbands in the presence of atomic spin-orbit interaction.\nDue to the basal mirror plane symmetry \u001bh, the electron\nspins are polarized out-of-plane. The superconducting\nproperties of such systems referred to as Ising supercon-\nductors [3, 5, 6, 13] are determined to a large extent by\nthe spin splitting, \u0001 SOtypically exceeding the supercon-\nducting gap by few orders of magnitude.\nOne of the experimentally con\frmed signatures of Ising\nsuperconductivity is its remarkable stability to the in-\nplane magnetic \feld, B?^z. The in-plain critical \feld,\nBcis demonstrated to greatly exceed the Pauli limit\n[3, 5, 6, 8{10, 14]. In fact, Bcis in\fnite at zero tem-\nperature,T= 0 unless the disorder is present in the\nsystem [15{18] or a random Rashba spin-orbit coupling\nis produced by ripples breaking \u001bhsymmetry [19]. Here-\ninafter, we absorb a half product of a g-factor and the\nBohr magneton in the de\fnition of Bsuch that the Zee-\nman spin splitting is 2 B. In few-layer systems we neglect\nthe coupling of the in-plane \feld to the orbital motion.\nSuch coupling is dominant for out-of plane \feld [20].\nA largeBccan be explained by an in-plane spin sus-\nceptibility being close to that of a normal state [6]. The\nsuperconductivity is then stabilized as the magnetic po-larization energy is excluded from the energy di\u000berence\nbetween the normal and the superconducting states. In\ncontrast to the case of conventional superconductors, the\nelectrons with momenta in between the spin split Fermi\nsurfaces reorient their spins in response to an applied in-\nplane \feld [21]. The net spin polarization along the \feld\nis independent of \u0001 SOand is determined by the Pauli\nsusceptibility in the normal state.\nA smooth adjustment of paired electrons to the ap-\nplied \feld is secured by a \u001bhTsymmetry combining \u001bh\nwith the time reversal symmetry T[22]. At \fnite Bthe\nspins of paired electrons related to each other by \u001bhT\noperation are no longer anti-parallel. It follows that as\nelectrons are polarized by the \feld, the wave function of\nthe Cooper pairs they form inevitably acquires a triplet\ncomponent [23]. Such \feld induced triplets, here referred\nto as non-trivial, have a symmetry lower than the sym-\nmetry of the crystal. Therefore, it is meaningful to assign\nthe two distinct transition temperatures, TcsandTctto\nthe leading singlet and subleading non-trivial triplet in-\nteraction channels, respectively.\nCrucially, the non-trivial triplets are distinct by sym-\nmetry from the triplets coexisting with singlets in the\nabsence of inversion center at B= 0 [24{26]. We refer\nto the latter triplets as trivial as they transform trivially\nunder all symmetry operations. In the limit studied here\n\u0001SO\u001cEF, whereEFis the Fermi energy, trivial triplets\ndecouple from singlets [27]. For this reason, we charac-\nterize a possible attraction in a Cooper pairs forming a\ntrivial triplet by a separate transition temperature, Tctz.\nThe trivial triplets are insensitive to moderate \felds and\nare suppressed by a minute disorder [18]. In contrast,\nnon-trivial triplets are induced by the \feld and are sta-\nbilized against the disorder due to the strong coupling to\nthe leading singlet order parameter (OP) [18, 28].\nFor the trivial triplets to be observed the critical tem-\nperature should be large enough, Tctz.Tcsas other-\nwise the singlet correlations dominate at experimentally\naccessible \felds. In contrast, the non-trivial triplets no-\nticeably a\u000bect the Bcand OPs already for Tct\u001cTcs.arXiv:2010.08224v1  [cond-mat.supr-con]  16 Oct 20202\nIn both cases, for triplet correlations to come into play,\nelectrons forming a triplet Cooper pair should attract.\nThe indirect evidence for attraction in triplet channel\ncomes from the very recent Density Functional Theory\ncalculations performed on the NbSe 2monolayers either\nfree standing or on a substrate. These studies \fnd a large\nStoner enhancement of the magnetic susceptibility in-\ndicative of strong ferromagnetic interactions [21] and/or\nferromagnetic ground state [29].\nThe ferromagnetic \ructuations revealed by Density\nFunctional Theory enhance the pairing interaction in the\ntriplet channel and suppress the interactions in the sin-\nglet channel [30{33]. As argued in Ref. [21] signi\fcant at-\ntraction in the triplet channel results from ferromagnetic\n\ructuations with correlation length exceeding vF=\u0001SO,\nwherevFis the Fermi velocity and we set ~=kB= 1.\nIn the alternative scenario considered in Ref. [34] appro-\npriate to the gated TMDs with small Fermi pockets both\nsinglet and triplet instabilities arise from repulsion. In\nthis approach TcsandTct=Tctzare promoted by distinct\ninter-pocket pair hopping processes, and can be both \f-\nnite.\nHere we assume \fnite Tct(z)<Tcsand study the e\u000bect\nof triplet correlations on the density of states (DOS),\nN(!), where!is the quasi-particle energy. Typically, the\nDOS curve, N(!) is inferred from the low-temperature\ndi\u000berential conductance in devices contacted via a tunnel\nbarrier [8]. A recent tunneling data in gated MoS 2is in\nfact, indicative of an unconventional symmetry of the OP\nin gated TMDs [35].\nThe essential conclusion of this work is that an admix-\nture of a triplet component to the superconducting OP\nmay result in apparent broadening of the N(!) curves\nas the applied \feld increases. Indeed, as detailed in\nSec. IV A for \u0001 SO\u001cEF, the scalar disorder charac-\nterized by the scattering rate \u0000 acts as a spin \ripping\ndisorder with the e\u000bective rate, \u0000 e\u000b/\u0000B2=\u00012\nSOin the\nexperimentally relevant regime, B.\u0001SO. This holds\nalso in the opposite limit, \u0001 SO&EF, Ref. [16]. The\ntriplet components describing the Cooper pairs with par-\nallel spins do not add to broadening per se. In other\nwords, when the triplet component of OP is added and\nall the other parameters are kept \fxed the DOS broad-\nening inferred from the Gorkov equation does not change\nappreciably. Nevertheless, triplet components push the\ncritical \feld Bc(T) up. This, in turn, leads to a stronger\nbroadening/B2\nceven in systems that are nominally in\nthe clean limit, \u0000 \u001cTcs, see Fig. 1.\nWe may express this point yet di\u000berently (see Sec. VI).\nAt \fxedBc(T), addition of the triplet components of OP\nenforces the adjustment of the other model parameters\nvia self-consistency condition. And this leads to extra\nbroadening compared with the pure singlet OP.\nThese results are illustrated in Fig. 1, where the com-\nputed DOS is compared for systems with purely singlet\nOP (panel (a)) achieved for Tct(z)= 0 and the OP which\ncontains a \feld induced triplet correlations present for\nTct>0. Qualitatively, as the Fig. 1 demonstrates, abroadening growing rapidly towards Bcin very clean sys-\ntems with \u0000 = 0 :1Tcsmight be an indication of a \fnite\ntriplet component of the OP.\nThe \feld dependence of the DOS broadening is a\nsalient feature of Ising superconductors, along with the\nenhancedBc. As such it applies equally to pure singlet\nand to the mixed parity superconductors. Its concrete\nmanifestation, however, di\u000bers in these two cases. To\nhighlight these di\u000berences which are of a potential ex-\nperimental importance we investigate in detail the ( T;B)\nmean \feld phase diagrams of a monolayer TMD for a sub-\nset of a relevant symmetry constrained OPs. The phase\ndiagrams are constructed for the complimentary models\nof the nodal and nodeless \u0001 SO, and for systems with dif-\nferent degree of purity. In each case the representative\nDOS curve is presented.\nThe paper is structured as follows. In Sec. II we spec-\nify the model Hamiltonian including the kinetic energy,\ninteraction constrained by symmetries and the disorder\npotential. Section III contains a summary of main results\nintended for the reader not interested in the details of the\nderivation. In this section, we show phase diagrams for\nselected sets of parameters as well as few representative\nDOS curves computed along di\u000berent lines on these phase\ndiagrams. The Gorkov equation employed for \fnding the\nDOS is detailed in Sec. IV. Qualitative picture of inter-\nrelation between the triplet correlations and the DOS is\ngiven in Sec. IV A. The Landau expansion of the ther-\nmodynamic potential we rely upon in order to map the\nphase diagram is obtained in Sec. V. Finally in Sec. VI\nwe discuss the results in light of the recent tunneling ex-\nperiments.\nII. THE MODEL HAMILTONIAN\nConsider a disordered monolayer superconductor with-\nout an inversion center. The appropriate Hamiltonian\nH=H0+Hdis+Hint (1)\nincludes the kinetic energy, random disorder potential\nand pairing interaction term, respectively. Kinetic en-\nergy,\nH0=X\nk;s\u0018kcy\nkscks+X\nk;ss0[\r(k)\u0000B]\u0001\u001bss0cy\nkscks0(2)\ncontains the dispersion measured from the chemical po-\ntential,\u0018(k), the anti-symmetric spin-orbit coupling\n(SOC), \r(\u0000k) =\u0000\r(k) due to the lack of the inversion\ncenter, and Zeeman \feld B=B^x, Fig. 2. We denote,\ncy\nks=V\u00001=2R\ndreik\u0001r y\nrs, where y\nrscreates a particle\nwith spin projection son thez-axis at position rin a\nvolumeV. The vector of Pauli matrices is denoted by\n\u001b= (\u001b1;\u001b2;\u001b3), and\u001b0stands for a unit matrix in spin\nspace.\nIn this work we consider the two models of the band\nstructure and SOC both having D3has the point sym-\nmetry group, Fig. 2a. NbSe 2monolayer the Nb derived3\nN(ω)/2N 0(a)\n0.0 0.2 0.4 0.6 0.8 1.0024681012\nB- 0.80 0.85 0.90 0.95 1.000246810\nT\nω\nN(ω)/2N 0(b)\n0.0 0.2 0.4 0.6 0.8 1.0024681012\nB\n+ 0.80 0.85 0.90 0.95 1.000510152025\nT\nω\nFIG. 1. The DOS, N(!) normalized by its normal state value, 2 N0as a function of !given in units of Tcs. In this example\nthe SOC is given by K-model (Sec. II). \u0001 SO= 15Tcs, \u0000 = 0:1Tcs. Panel (a): singlet OP, Tct= 0. Panel (b): singlet-triplet\nOP,Tct= 0:8Tcs. Insets: (T;B) phase diagrams with TandBin units ofTcs. The solid (grey) line shows, Bc(T) separating\nthe normal and superconducting phase shown as shaded (blue) area. The coordinates of a vertically aligned and evenly spaced\ndots, (T= 0:9;B) are the (T;B) pairs for which the DOS curves are shown in the main \fgure using the same color scheme.\nThe dashed black line in the inset of panel (b) is the Bc(T) shown as a solid line in panel (a) for singlet only case. In panels\n(a) and (b) DOS curves broaden with increasing \feld.\nband crossing the Fermi level gives rise to the two distinct\nhole Fermi pockets. In the hexagonal Brillouin Zone one\nof the pockets is centered at \u0000 and the other two discon-\nnected pockets enclose the \u0006Kpoints, Fig. 2b. Although\nboth pockets are present in NbSe 2, here for simplicity we\nconsider two separate models referred to as \u0000- and K-\nmodel with only one type of pockets.\nThe\u001bhsymmetric SOC has a form \r(k) = \u0001 SO^\r(k) ^z.\nAs the axial vector ^ zbelongs toA0\n2irrep ofD3h, the same\nshould be true for the scalar function ^ \r(k). For simplic-\nity we write ^ \r(k) = ^\r('k), where'kis an angle the\nvector kforms with the kxaxis in the Brillouin Zone.\nThe acceptable functions ^ \r('k) are linear combinations\nof Fourier harmonics cos(3 n'k) with integer n6= 0. All\nsuch functions vanish along \u0000 Mas prescribed by the ver-\ntical mirror symmetry. Hence, without loss of generality\nwe take for the \u0000-model ^ \r('k) =p\n2 cos(3'k) [26, 36].\nThe simplest model of SOC for K-model is ^\r('k) =\nsgn[cos(3'k)]. Such a function is constant at each of\nthe\u0006Kpockets. We considered normalized ^ \rfunctions,\n^\r2('k)\u000b\nF= 1, where the angular averaging over the\nFermi surface is denoted as h\u0001\u0001\u0001i F\u0011(2\u0019)\u00001R\nd'k(\u0001\u0001\u0001).\nThe disorder is modeled as a collection of impurities\nof a density nimplocated at random positions, Rj. The\nresulting scattering potential reads,\nHdis=X\njX\ns=1;2X\nk;k0uk\u0000k0eiRj\u0001(k\u0000k0)cy\nksck0s;(3)\nwhereuqis a Fourier transformation of the spin indepen-\ndent, scalar potential produced by a single impurity. For\nsimplicity we consider a short range impurity potential\nsuch thatuq=u0. The resulting elastic scattering time,\n\u001cis given by the Golden Rule, \u001c\u00001= 2\u0019nimpN0u2\n0, where\nN0is the normal state DOS per spin species.Finally, we treat the pairing interaction\nHint=1\n2X\nk;k0X\nfsigVs1s2\ns0\n1s0\n2(k;k0)cy\nks1cy\n\u0000ks2c\u0000k0s0\n2ck0s0\n1(4)\nwithin the mean \feld approximation. To this end we\nintroduce the OP\n\u0001s1s2(k) =1\nVX\nk0;s0\n1;s0\n2Vs1s2\ns0\n1s0\n2(k;k0)\nc\u0000k0;s0\n2ck0;s0\n1\u000b\n(5)\nand make an approximation, Hint\u0019HMF\u0000\u0016Hi, where\nthe mean \feld interaction Hamiltonian is\nHMF=V\n2X\nk;si\u0001\u0003\ns1s2(k)c\u0000ks2cks1+h:c:; (6)\nwhereh:c:stands for Hermitian conjugated term. Equa-\ntion (5) is a self-consistency equation with the right hand\nside computed with the quadratic Hamiltonian (6). The\nexpectation value of the mean \feld Hamiltonian,\n\u0016Hi=1\n2X\nsi;s0\niX\nk;k0Vs1s2\ns0\n1s0\n2(k;k0)D\ncy\nk;s1cy\n\u0000k;s2E\nc\u0000k0;s0\n2ck0;s0\n1\u000b\n:\n(7)\nNext, we introduce a simpli\fed model of interaction,\n(4) which contains a minimum amount of necessary in-\nformation to capture the thermodynamic properties and\n\feld induced phase transitions in the superconducting\nTMD monolayer. To this end we invoke the arguments\nbased on symmetry considerations.4\nA. Symmetry and the choice of the OPs\nWe now consider the momentum and spin dependence\nof the superconducting OP written in the standard form\nas\n\u0001 (k) = [ (k)\u001b0+d(k)\u0001\u001b]i\u001b2: (8)\nHere (k) and d(k) parametrize the singlet and triplet\ncomponents of the OP.\nIn our description the leading OP is singlet, Tcs>\nTct(z). Here we neglect it's anisotropy setting  (k) =\n\u00110^ 0with the basis function ^ 0= 1. At B= 0 the triplet\nOP coexisting with the singlet one is determined by the\naxial vector dA0\n1(k) transforming as A0\n1. Since d(k) and\nthe SOC transform in the same way, the reasoning \fxing\n\r(k) applies, and we write dA0\n1(k) =\u0011A^\r(k) ^z. In addi-\ntion to dA0\n1we include the triplet OP induced by the \feld,\npreviously introduced in Ref. [28]. Since the in-plane \feld\nB= (Bx;By) belongs to E00it couples to the triplet OPs\nof the same symmetry. For the \u0000-model within the con-\nsidered space of the Fourier harmonics the pairs of E00-\npartners ( d1\nE00;d2\nE00) are (cos 3'k^x;cos 3'k^y), (cos'k^x\u0000\nsin'k^y;sin'k^x+ cos'k^y), [11] and (sin 3 'k^x;sin 3'k^y).\nThe \frst OP in the full list above written in the form\n(^\r('k)^x;^\r('k)^y) applies to both \u0000- and K-models. In\ncontrast to other E00combinations, it couples to the \felds\nvia the SOC induced polarization of the bands even when\nthe interaction has a full rotational symmetry such that\ndi\u000berent Fourier harmonics decouple. Although in the\ngeneric situation all E00triplets condense together, here\nwe consider the model with rotational invariant interac-\ntion. This narrows the list of E00triplet OPs down to\none entry, ( d1\nE00;d2\nE00) = [\u0011E1^\r('k)^x;\u0011E2^\r('k)^y]. We\ndo not include the triplet OPs given by the d-vectors\ncos'k^x+ sin'k^y, sin 3'k^z, sin'k^x\u0000cos'k^y, and be-\nlonging to A00\n1,A0\n2andA00\n2symmetry, respectively. We\nalso omit the E0triplet assuming no strain.\nIn summary, in our model the d-vector characterizing\nthe triplet component of the OP, Eq. (8) reads,\nd(k) = ^\r(k) (\u0011E1^x+\u0011E2^y+\u0011A^z); (9)\nand our list of OPs includes A0\n1singlet\u00110,A0\n1triplet\u0011A,\nandE00triplet (\u0011E1;\u0011E2), see Fig 2c.\nThe above symmetry arguments lead us to the e\u000bective\ninteraction\nVs1s2\ns0\n1s0\n2(k;k0) =vs[i\u001b2]s1s2[i\u001b2]\u0003\ns0\n1s0\n2(10)\n+X\nj=1;2vt[^\r(k)\u001bji\u001b2]s1s2[^\r(k0)\u001bji\u001b2]\u0003\ns0\n1s0\n2\n+vtz[^\r(k)\u001b3i\u001b2]s1s2[^\r(k0)\u001b3i\u001b2]\u0003\ns0\n1s0\n2:\nEquation (10) is the minimal Hamiltonian capturing the\ninteraction in the A0\n1singlet channel of a strength, vs,\nthe interaction in the A0\n1triplet channel of a strength\nvtz, and \fnally the E00channel with interaction vt.\nFIG. 2. a) The crystal structure of 1H-NbSe 2, where the\nNb are blue and Se yellow. The gray region shows the unit\ncell in which there is no inversion center. This \fgure was\nmade with Vesta [37]. b) Schematic Fermi surface of metallic\nmonolayer TMDs at zero magnetic \feld. SOC vanishes along\nthe dashed \u0000 Mlines. The sign of SOC alternates between\nadjacent dashed lines. c) The directions of the Zeeman \feld\n(red), SOC (blue) and the OPs (black).\nWith the model Hamiltonian, (10) the OP, Eq. (8) is\nfully determined by the four OPs of a de\fnite symmetry,\n\u00110=vs\nVX\nk0;s0\n1s0\n2[i\u001b2]\u0003\ns0\n1s0\n2\nc\u0000k0;s0\n2ck0;s0\n1\u000b\n(11)\n\u0011E1(2)=vt\nVX\nk0;s0\n1s0\n2\u0002\n^\r(k0)\u001b1(2)i\u001b2\u0003\u0003\ns0\n1s0\n2\nc\u0000k0;s0\n2ck0;s0\n1\u000b\n\u0011A=vtz\nVX\nk0;s0\n1s0\n2[^\r(k0)\u001b3i\u001b2]\u0003\ns0\n1s0\n2\nc\u0000k0;s0\n2ck0;s0\n1\u000b\n:\nIt is convenient to use the observable transition tem-\nperatures than the interaction amplitude. We hence in-\ntroduce the three transition temperatures Tcs,Tct(z)cor-\nresponding to the three terms in Eq. (10). These temper-\natures are de\fned under conditions that only one interac-\ntion amplitude is non-zero, the system is clean, SOC and\nmagnetic \feld are turned o\u000b. In this case, we have the\nstandard relations, Tcs= 2\u0003e\rE\u0019\u00001exp(\u00001=2N0jvsj),\nandTct(z)= 2\u0003e\rE\u0019\u00001exp(\u00001=2N0jvt(z)j) where \u0003 is a\nhigh energy cuto\u000b for the attraction and \rEis Euler's\nconstant [38].\nIII. SUMMARY OF RESULTS\nIn this section we present the main \fndings of this\nwork. The end result is the calculated DOS of a disor-\ndered Ising superconductor throughout the ( T;B) phase\ndiagram. More speci\fcally, we focused on the role played\nby the triplet correlations. Although the \u0000- and K-model\nof SOC give overall similar results, there are qualitative5\n(a)\n+ \n0.2 0.4 0.6 0.8 1.0020406080100Tctz Tct\nTB(b)\n+ + \n0.2 0.4 0.6 0.8 1.0010203040Tct Tctz\nTB\nFIG. 3. The ( T;B) phase diagrams with both axes given in units of Tcsfor theK-model with \u0001 SO= 15Tcs;\u0000 = 0:2Tcs. The\nlight shaded (blue) regions mark the superconducting phase of coexisting singlet and \u0011E2triplet OP ( s+ifphase). The second\norder normal to superconducting transition lines are denoted by thin (blue) lines. The TctandTctzare marked by vertical\ndashed lines. (a) Tct= 0:7Tcs,Tctz= 0:5Tcs. (b)Tct= 0:5Tcs,Tctz= 0:7Tcs. The shaded (green) region in panel (b) marks\nsuperconducting phase with \u0011Atriplet OP ( s0phase). The \frst order transition between s+ifands0phases is denoted by\na thick (blue) line. The coordinates of vertically aligned and evenly spaced colored dots, at T= 0:4TcsandT= 0:8Tcsin\npanel (a)[(b)] de\fne the ( T;B) pairs for which the DOS is shown in Fig. 4(a,b) [Fig. 4(c,d)], respectively, using the same colors\nscheme.\ndi\u000berences in some regions of the phase diagram. We,\ntherefore, consider these two models separately.\nFurthermore, for each of the two models the two phys-\nically distinct situations arise depending on the symme-\ntry of the dominant triplet channel. The \frst scenario\nfor which the \feld induced E00triplet channel domi-\nnates over the symmetric A0\n1triplet channel is realized\nforTct> Tctz. And the opposite scenario is realized for\nTctz>Tct. Below we consider these two scenarios sepa-\nrately.\nA. TheK-model of SOC\n1.Tct>Tctz: dominant E00triplet channel\nThe representative phase diagram in the situation\nwhenE00triplet channel dominate the A0\n1triplets, is\nshown in Fig. 3a. In this case there is a normal and super-\nconducting phases with the superconducting OP having\nbothA0\n1singlet and E00\feld-induced triplet components.\nIn the limit, \u0001 SO\u001cEF, considered here, the A0\n1triplet\ncomponent remains zero. Previously, we have referred\nto this phase as having the s+ifsymmetry [28]. The\ncritical \feld diverges at a \fnite temperature T1. This\ntemperature can be easily computed by noticing that in\nthe high \feld limit, the singlet component of the OP is\nsuppressed. The triplet component of OP, on the other\nhand, has a E00symmetry with dk^ywhich is perpen-\ndicular to B, see Fig. 2c. Therefore, magnetic \feld does\nnot limit this triplet OP. Although din this case alsoorthogonal to \r(k) in the high \feld limit, B\u001d\u0001SOthe\nlimiting of the E00triplet by SOC is eliminated. As a\nresult, in the considered limit T1is determined by the\nstandard relation,\nlnT1\nTct= \t\u00121\n2\u0013\n\u0000\t\u00121\n2+\u0000\n2\u0019T1\u0013\n; (12)\nwhere \t is the digamma function. Clearly, T1=Tctin\nthe clean system and is suppressed by the disorder. For\ninstance, for the set of parameters used in Fig. 3a Eq. (12)\ngivesT1\u00190:536. At \u0000>\u0019T cte\u0000\r=2,Bcis \fnite for all\ntemperatures, and the high \feld part of the supercon-\nducting phase is eliminated. Much stronger disorder is\nneeded to bring Bcto the Pauli limit, see Ref. [18] for\ndetails.\nBoth OPs and DOS are plotted along two constant\ntemperature cuts in the phase diagram Fig. 4a and\nFig. 4b, respectively. Low Tcut, Fig. 4a shows the sat-\nuration of the triplet OP and vanishing of singlet OP at\nhigh \feld con\frming the statements made above. The\nDOS similarly is broadened with increasing the \feld and\nsaturates in the same limit. At higher Tthe DOS shows\na strong smearing e\u000bect of the \feld, while the gap is still\nvisible, Fig. 4b.\n2.Tctz>Tct: dominant A0\n1triplet channel\nFig. 3b features a phase diagram of the system with\nA0\n1triplet pairing dominating E00triplet correlations. In6\nN(ω)/2N 0(a)\n0 1 2 3 40246810T=0.4Tcs\n|η0| |ηE2|\n0 20 40 60 80 1000123\nB\nω\nN(ω)/2N 0(c)\n0 1 2 3 40246810T=0.4Tcs\n|ηA|\n0 510 15 20 25 300123\nB\nωN(ω)/2N 0(b)\n0.0 0.5 1.0 1.5 2.00246810T=0.8T cs\n0246810121400.40.81.2\nB\nω\nN(ω)/2N 0(d)\n0.0 0.5 1.0 1.5 2.00246810T=0.8T cs\n0 2 4 6 8 1000.40.81.2\nB\nω\nFIG. 4. Normalized DOS, N(!)=2N0. Panels (a),(b) pertain to T= 0:4TcsandT= 0:8Tcscolumns of dots in Fig.3(a),\nrespectively. Panels (c),(d) pertain to T= 0:4TcsandT= 0:8Tcscolumns of dots in Fig.3(b), respectively. The DOS curves\nhave the same color as the corresponding dots in Fig.3, and broaden with increasing \feld. Insets: the evolution of OPs with\nB. The magnitude of singlet \u00110, triplet\u0011E2and triplet \u0011AOPs are shown by solid, dashed and dotted lines, respectively. The\nvalues ofBfor which the DOS is plotted in the main \fgure is marked by a vertical bar of the same color as the DOS curve\nattached to a horizontal axis. OPs, !andBare in units of Tcs.\nthis case the low \feld second order transition line bifur-\ncates at the tricritical point as the \feld increases. Among\nthe two high \feld transition lines emanating from the tri-\ncritical point one marks the \frst-order phase transition\nbetween the s+ifstate and the s0state where the OP is\na pureA0\n1symmetric triplet. The other is a vertical line,\nT=Tvof the second order transitions between the nor-\nmal state and s0superconducting state. The transition\ntemperature Tvis determined by the instability towards\ntheA0\n1triplet. For this triplet the dk^z, see Fig. 2c\nwhich is parallel to \r(k) and orthogonal to B. This im-\nplies that the A0\n1transition temperature, Tvis insensitive\nneither to SOC nor to the magnetic \feld, and is given by\na standard equation,\nlnTv\nTctz= \t\u00121\n2\u0013\n\u0000\t\u00121\n2+\u0000\n2\u0019Tv\u0013\n; (13)\nwhere \t is the digamma function. For the parameters\nused in Fig. 3a, Eq. (13) gives Tv\u00190:536. Note that\nTvaccidentally coincides with T1found in Sec. III A 1.More generally, the temperature at which the critical\n\feld diverges is determined by max fTct;Tctzg. For both\nFig. 3a and Fig. 3b max fTct;Tctzg= 0:7Tcs. For this rea-\nson for the current set of critical temperatures, Tv=T1.\nAnother manifestation of the similarity of the two triplet\ninstabilities at high \felds can be seen by comparison\nof the OP \feld dependence in Fig. 4a and Fig. 4c at\nT= 0:4Tcs. TheE00triplet in the inset of Fig. 4a and\ntheA0\n1triplet in the inset of Fig. 4c saturate to the same\nvalue at high \feld. To avoid confusion we stress, however\nthat in general Tv6=T1. The most general statement is\nthat both the temperature at which the critical \feld di-\nverges and the magnitude of the dominant triplet OP at\nhigh \feld are determined by max fTct;Tctzg. In this form\nthe above statement applies equally to both K- and the\n\u0000-model considered in Sec. III B.\nThe DOS and the variation of the OPs for the two con-\nstantTcuts of the phase diagram in Fig. 3b is shown in\nFig. 4c and Fig. 4d. The OPs along the low temperature\ncut crossing the \frst order transition line are discontinu-7\n■ ■■ ■■ ■■ ■■ ■(a)\n+ \n0.2 0.4 0.6 0.8 1.001020304050Tctz Tct\nTB+ + (b)\n0.2 0.4 0.6 0.8 1.002468101214Tct Tctz\nTB\nFIG. 5. The ( T;B) phase diagrams with both axes given in units of Tcsfor the \u0000-model with \u0001 SO= 15Tcs;\u0000 = 0:2Tcs. The\nlight shaded (blue) regions mark the superconducting phase of coexisting singlet and \u0011E2triplet OP ( s+ifphase). The second\norder normal to superconducting transition lines are denoted by thin (blue) lines. The TctandTctzare marked by vertical\ndashed lines. (a) Tct= 0:7Tcs,Tctz= 0:5Tcs. (b)Tct= 0:5Tcs,Tctz= 0:7Tcs. The shaded (green) region in panel (b) marks\nsuperconducting phase with \u0011Atriplet OP ( s0phase). The thin elongated (red) region in between the s+ifands0phases\ndenotes an intermediate s+if+is0phase. The coordinates of vertically aligned and evenly spaced colored dots, at T= 0:4Tcs\nandT= 0:8Tcsin panel (a)[(b)] de\fne the ( T;B) pairs for which the DOS is shown in Fig. 7(a,b) [Fig. 7(c,d)], respectively,\nusing the same colors scheme.\nous as shown in the inset of Fig. 4c. Similarly, the DOS\ndoes not evolve smoothly across the \frst order transi-\ntion. Nevertheless, it is not easily seen in Fig. 4c. This\nis explained by the smearing of the DOS as well as the\nsmallness of the jump in the quasi-particle spectral gap\nacross the transition. At higher \felds the OPs as well as\nthe shape of the DOS deep in the s0phase reach satu-\nration as can be seen from the last two DOS curves in\nFig. 4c corresponding to the largest \felds shown.\nNaturally, the DOS for the high temperature cut\nFig. 4d is qualitatively similar to Fig. 4b.\nB. The \u0000-model of SOC\n1.Tct>Tctz: dominant E00triplet channel\nA representative phase diagram of the Ising supercon-\nductor with nodal SOC is shown in the Fig. 5a. Here as in\ntheK-model in the same regime, Tct>Tctzthe OP has\nans+ifsymmetry all over the superconducting phase\nprovided the disorder scattering rate is not substantially\nsmaller than Tcs. In clean system there is a small island\nofs0phase separated from the s+ifphase by the inter-\nmediates+if+is0phase and at higher Tbordering the\ndomain of the normal state, see Fig. 6. Even when the\ndisorder eliminates the s0phase the critical temperature\nremains non-monotonic function of the \feld, Fig. 5a.\nThe DOS shown for T < T ctzandT > T ctin the\nFig. 7a and Fig. 7b is characterized by the peak follow-\ning the \feld dependent OPs and strongly broadened atelevated magnetic \felds. In addition, for \felds B >B\u0003,\nwhereB\u0003=\u00110(T;B\u0003) the spectral gap is closed along the\n\u0000Mline, where 12 nodes are formed causing topological\nedge states in stripe shaped samples [39]. So that above\nthe characteristic \feld, B\u0003slightly exceeding the Pauli\ncritical \feld obtained for \u0001 SO= 0,N(!) is \fnite for all\npositive!>0. In the clean limit it results in a V-shaped\nDOS. At \fnite disorder concentration the spectral gap\ncloses, and the \u0000-model of Ising superconductor provides\nus with yet another example of gapless superconductivity\natB > B\u0003. At weak disorder, N(0)\u0018exp(\u0000\u0019\u00110=\u0000) is\n\fnite, and yet suppressed exponentially [40].\nIn general, the self-consistent Born approximation fails\nat low energies [41], and the detailed analysis of the DOS\nin this regime as well as the zero-energy Majorana sur-\nface states is beyond the scope of the present work. Still,\nwe note that the overall low-energy behavior of the bulk\nDOS stays nearly una\u000bected by moderately weak disor-\nder, consistent with Fig. 7.\n2.Tctz>Tct: dominant A0\n1triplet channel\nIn distinction with the K-model the high \feld \frst or-\nder transition between the s+ifands0phases proceeds\nvia an intermediate s+if+is0phase, see Fig. 5b. In the\nintermediate phase the OP contains two types of triplets.\nTheE00(if) triplets areTodd as they are induced by the\nexternally applied \feld breaking the Tsymmetry explic-\nitly. In contrast, A0\n1(is0) triplets are present due to the\nspontaneousTsymmetry breaking. Despite this di\u000ber-8\n+ + \n0.2 0.4 0.6 0.8 1.00102030Tctz Tct\nTB\nFIG. 6. The ( T;B) phase diagrams with both axes given in\nunits ofTcsfor the \u0000 model with \u0001 SO= 15Tcsand no disor-\nder. TheTctz= 0:5Tcs,Tct= 0:7Tcs, are indicated by vertical\ndashed lines. The light shaded (blue) regions mark the su-\nperconducting phase of coexisting singlet and \u0011E2triplet OP\n(s+ifphase). The shaded (green) region marks supercon-\nducting phase with \u0011Atriplet OP ( s0phase). The thin cres-\ncent shaped (red) region in between the s+ifands0phases\ndenotes an intermediate s+if+is0phase. The normal state\noccupies the high- Tpart of the phase diagram. All the phase\nboundaries shown as solid (blue) lines mark the second order\nphase transitions.\nence the triplet components have the same phase and the\nOP is unitary all over the phase diagram. Although the\nsuperconducting state breaks Tsymmetry, both extrin-\nsically and spontaneously, because the OP stays unitary\nthere is no net Cooper pair spin polarization.\nFor the \u0000-model with Tctz< T ctthe DOS curves for\n\fxedT= 0:4Tcs(T= 0:8Tcs) and ascending sequence\nofBis shown in in Fig. 7a (Fig. 7b). Similarly, the\nresults for the case Tct<Tctzfor \fxedT= 0:4Tcs(T=\n0:8Tcs) and ascending sequence of Bis shown in in Fig. 7c\n(Fig. 7d).\nIV. GORKOV EQUATION AND QUALITATIVE\nCONSIDERATIONS\nOur goal is to compute the DOS at any given part\nof the phase diagram in the ( T;B) plane. This goal is\nachieved in two steps. First the OPs are found by the\nprocess of minimization of the mean \feld thermodynamic\npotential. Then with these OPs as an input we calculate\nthe DOS by solving the Gorkov equation. The \frst step\nis detailed in Sec. V and covered partially in our pre-\nvious work [18]. This section is devoted to the actual\ncalculation of the DOS.\nTo this end we introduce the normal and anomalous\nMatsubara Green functions as 2 \u00022 matrices in the spinspace,\nGss0(r;r;\u001c;\u001c0) =\u0000D\nT\u001c rs(\u001c) y\nr0s0(\u001c0)E\n;\nFss0(r;r;\u001c;\u001c0) =\u0000hT\u001c rs(\u001c) r0s0(\u001c0)i: (14)\nHere, rs(\u001c) =eH\u001c rse\u0000H\u001care the \feld operators in the\nHeisenberg representation, where \u001cis the imaginary time.\nThe symbol T\u001cstands for the time-ordering operator, and\nh\u0001\u0001\u0001i indicates thermal averaging.\nThe disorder averaged Green function is a 4 \u00024 matrix\n^G(k;!n) =\u0014\nG(k;!n)F(k;!n)\n\u0000F\u0003(\u0000k;!n)\u0000G\u0003(\u0000k;!n)\u0015\n;(15)\nwhere we have introduced the Fourier transformed Green\nfunction\nG(k;!n) =Z\nVdrZ\f\n0d\u001ce\u0000ik\u0001r+i!n\u001cG(r;\u001c) (16)\nwith the Matsubara frequencies !n= (2n+ 1)\u0019Tand\nsimilarly for F(k;!n). The matrix ^Gis de\fned in the\ndirect product of particle-particle or Nambu and spin\nspaces. This implies that each of the Green functions\nin Eq. (15) is a matrix in spin space in accordance with\nEq. (14).\nThe spin unresolved DOS is then expressed in terms of\nthe normal Green function,\nN(!) =\u0000N0\n\u0019X\nsImZ\nd\u0018k\nGss\u0000\nk;!+i0+\u0001\u000b\nF;(17)\nwhere 0+is a positive in\fnitesimal.\nThe Green function, ^Gsatis\fes the Gorkov equation\nwhich can be written ash\ni!n^\u001b0\u0000^HBdG(k)\u0000^\u0006 (!n)i\n^G(k;!n) = ^\u001b0;(18)\nwhere ^\u001b0= diag (\u001b0;\u001b0) is a unit matrix of rank 4. The\nBogoliubov-de Gennes Hamiltonian,\n^HBdG(k)=\u0014\u0018k+[\r(k)\u0000B]\u0001\u001b \u0001 (k)\n\u0001y(k)\u0000\u0018k+[\r(k)+B]\u0001\u001b\u0015\n;(19)\nwhere the superconducting OP, \u0001( k) is given by Eq. (8).\nThe e\u000bect of the disorder scattering is described within\nthe self-consistent Born approximation by the self-energy\nappearing in Gorkov equation (18),\n^\u0006 (!n) = \u0000Zd'k\n2\u0019Zd\u0018k\n\u0019^\u001bz^G(k;!n) ^\u001bz; (20)\nwhere \u0000 = (2 \u001c)\u00001, and ^\u001bz= diag (\u001b0;\u0000\u001b0).\nTo obtain the DOS we solved Eqs. (18) and (20) nu-\nmerically by the method of iterations. With the initial\nguess of the self-energy the matrix inversion in Eq. (18)\ngives the Green function which in turn is used in order to\n\fnd an updated self-energy from Eq. (20). These steps\nare repeated until convergence is reached.\nBefore we proceed to the discussion of the calculated\nDOS for various representative parts of the phase dia-\ngram we present qualitative picture of the e\u000bect of the\ncombined action of SOC, magnetic \feld and triplet cor-\nrelation on the DOS in the next section.9\nN(ω)/2N 0(a)\n0 1 2 3 40246810T=0.4Tcs\n|η0|\n|ηE2|\n0 20 400123\nB\nN(ω)/2N 00 1 200.511.5\nω\nω\nN(ω)/2N 0(c)\n0 1 2 3 40246810T=0.4Tcs\n|ηA|\n0 2 4 6 8 100123\nB\nωN(ω)/2N 0(b)\n0.0 0.5 1.0 1.5 2.00246810T=0.8T cs\n0 1 2 300.40.81.2\nB\nω\nN(ω)/2N 0(d)\n0.0 0.5 1.0 1.5 2.00246810T=0.8T cs\n0 1 2 300.40.81.2\nB\nω\nFIG. 7. Normalized DOS, N(!)=2N0for the \u0000-model. Panels (a),(b) pertain to T= 0:4TcsandT= 0:8Tcscolumns of dots\nin Fig.5(a), respectively. Panels (c),(d) pertain to T= 0:4TcsandT= 0:8Tcscolumns of dots in Fig.5(b), respectively. The\nDOS curves have the same color as the corresponding dots in Fig.5, and broaden with increasing \feld. The horizontal (black)\ndashed line in panel (a) is the DOS for normal state points in the phase diagram in Fig.5(a) indicated by black squares. Panel\n(a): The right inset shows the high-\feld DOS. All panels: the left inset shows the evolution of OPs with \feld. The magnitude\nof singlet\u00110, triplet\u0011E2and triplet \u0011AOPs are shown by solid, dashed and dotted lines, respectively. OPs, !andBare in\nunits ofTcs. Panel (b): The critical \feld in the Pauli limit is 0 :86Tcsand the gape closes at B\u0003\u00191:4Tcs.\nA. Qualitative picture of the DOS broadening\nTo understand how triplet correlations a\u000bect N(!), it\nis useful to consider the singlet Ising superconductor \frst\nwith the OP \u0001( k) =\u00110i\u001b2. In this case, when B= 0,\nSOC does not show up due to Anderson theorem [42].\nOn the other hand, for \u0001 SO= 0 an in-plane Bsplits the\nBCS peak, without causing its broadening [43]. Never-\ntheless, when both magnetic \feld and SOC are present\nthe disorder causes a \fnite broadening of the DOS.\nTo clarify this we consider the unitary transformation\nofHBdG, Eq. (19) diagonalizing the normal components\nof^G. We consider the K-model of SOC for simplic-\nity. Let us refer to the momenta such that ^ \r(k)>0,\n(^\r(k)<0) as belonging to + Kand\u0000Kpockets, respec-\ntively. The diagonalization is carried out separately formomenta in\u0006Kpockets by the unitary transformation\n^U=\u0014\nU\u00060\n0U\u0007\u0015\n; U\u0006= cos\u0012\n2\u0007i\u001b2sin\u0012\n2; (21)\nwhere the angle \u0012satis\fes sin \u0012=jBj=p\nB2+ \u00012\nSO.\nThe BdG Hamiltonian, Eq. (19) is diagonalized, by the\ntransformation \u0016HBdG=^UHBdG^U\u00001, fork2\u0006K,\n\u0016HBdG=\u0014\n\u0018k\u0006\u001b3p\nB2+\u00012\nSO \u00010(k)\n\u0001y0(k)\u0000\u0018k\u0006\u001b3p\nB2+\u00012\nSO\u0015\n;(22)\nwhere similar to Eq. (8) we have \u00010(k) = [ 0(k)\u001b0+\nd0(k)\u0001\u001b]i\u001b2:The singlet part of the transformed OP is\n\u00110\n0=\u00110cos\u0012, and the triplet part d0(k) =\u0007i\u00110sin\u0012^yfor\nk2\u0006K.\nThe disorder potential, Eq. (3) is similarly transformed10\n\u0016Hdis=^UHdis^U\u00001,\n\u0016Hdis=X\njX\nsX\nk;k00\nuk\u0000k0eiRj(k\u0000k0)cy\nksck0s\n+X\njX\ns;s0X\nk;k000\n(cos\u0012\u0007i\u001b2;ss0sin\u0012)\n\u0002uk\u0000k0eiRj(k\u0000k0)cy\nksck0s0; (23)\nwhereP0denotes the summation over the momenta k\nandk0belonging to the same pocket. This terms ac-\ncounts therefore for the intra-pocket scattering. The\nsummation in the second term,P00accounts for the inter-\npocket scattering. The upper (lower) sign describes the\nscattering from\u0006Kto\u0007Kpockets, respectively.\nThe shape of the N(!) close to the peak, !\u0019\u00110is\ndetermined by the states in the energy interval of the or-\nder\u00110aroundEF. For such low energies the bands split\nby the SOC as appears in Eq. (22) can be considered\nas decoupled. Furthermore, the inter-pocket scattering\nterm in the transformed disorder potential, (23) acquires\na spin \ripping component, /B=\u0001SO. It appears that\nto the leading order in the parameter, B=\u0001SO\u001c1 the\nenergy dependence of the DOS can be captures by a sim-\nple model of the anisotropic magnetic impurity scattering\n[44]. The Gorkov equation for such a problem reads\nh\ni!n^\u001b0\u0000^He\u000b\nBdG(k)\u0000^\u0006e\u000b(!n)i\n^G(k;!n) = ^\u001b0;(24)\nwhere the e\u000bective BdG Hamiltonian takes a simple form,\n^He\u000b\nBdG(k) =\u0014\n\u0018k\u001b0\u00110i\u001b2\n\u0000\u0011\u0003\n0i\u001b2\u0000\u0018k\u001b0\u0015\n; (25)\nand the e\u000bective magnetic disorder gives rise to the self-\nenergy,\n^\u0006e\u000b(!n) = \u0000e\u000b\nmZd'k\n2\u0019Zd\u0018k\n\u0019^\u001b2^G(k;!n) ^\u001b2;(26)\nwhere ^\u001b2= diag (\u001b2;\u001b2) and the scattering o\u000b the mag-\nnetic impurities is characterized by the e\u000bective rate,\n\u0000e\u000b\nm=1\n2\u0000B2\nB2+ \u00012\nSO: (27)\nEven though we have considered the limit B=\u0001SO\u001c\n1 we retained the magnetic \feld in the denominator of\n(27) to stress that the e\u000bective rate cannot exceed the\nscalar impurity scattering rate. The prefactor of a 1 =2 is\nneeded as only half of all the scattering events acquire a\nspin \ripping component. We have checked the validity\nof the e\u000bective model captured by Eqs. (25) and (27)\nnumerically. We have shown that the DOS given by the\ne\u000bective and the original models agree well, see Fig. 8a.\nAlthough the e\u000bective model of magnetic impurities\ncaptures well the shape of the DOS it does not reproduce\nBc. The physical reason for this is the limitation of the\nabove e\u000bective model to energies, !n\u001c\u0001SO. Indeed,when only one of the spin split bands is populated, the Bc\nis determined by the pair breaking equation known from\nthe theory of magnetic scattering with renormalized Tcs\n[16]. In contrast, when both spin split bands are occupied\nthe pair breaking equation di\u000bers from that of the to\nmagnetic impurities model, because the frequencies !n>\n\u0001SOcontribute to Bc. For such frequencies the spin-\nindependent Hamiltonian, (25) is inadequate since the\ntwo spin-split bands cannot be considered separately.\nWe now turn to the discussion of the in\ruence of the\ntriplet correlations on the shape of the DOS. For de\f-\nniteness we focus on the OP, (8) with \fnite singlet com-\nponent\u00110and a \fnite triplet component speci\fed by the\nvector d=d2(k)^y, whered2(k) is an odd function of\nmomentum. The unitary transformation, Eq. (21) trans-\nforms the OP such that the singlet components acquires\na small correction, \u00110\n0=\u00110cos\u0012\u0000id2sin\u0012. This correc-\ntion does not contribute to the broadening to the leading\norder.\nThe triplet components encoded by d0\u0019ddescribe the\npairs of electrons with the parallel spins. Such states dif-\nfer in energy by an amount \u0001 SO. Therefore, these terms\nrepresent a small perturbation in the Gorkov equation in\nthe limit,jdj\u001c\u0001SO. It makes it clear that except for\nthe overall shift of the spectral peak the equal spin triplet\ncomponents make a negligible contribution to its shape.\nWe tested this statement numerically. As is evident from\nFig. 8b the triplet component has little direct e\u000bect on\nthe shape of the DOS. Rather it changes the phase di-\nagram as determined by the self-consistency condition,\nand in this way has a strong indirect e\u000bect on DOS. More\nspeci\fcally, the above arguments show that as the triplet\ncorrelations make the critical \feld higher the broadening\ntowards the critical \feld make the DOS curves progres-\nsively more broadened at elevated magnetic \felds, see\nFig. 1.\nV. THERMODYNAMIC POTENTIAL AND THE\nPHASE DIAGRAM\nWe write the di\u000berence of the thermodynamic poten-\ntial in superconducting and normal state in the form of\na Landau expansion,\n(V2N0)\u00001\n(\u00110;\u0011A;\u0011E1;\u0011E2) = \n(2)+ \n(4): (28)\nWe \frst present the second order terms [18],\n\n(2)=CsAj\u00110j2+CtAj\u0011Aj2+X\nj=1;2CtEjj\u0011Ejj2\n\u00002CsA;tEImf\u0011\u0003\n0(Bx\u0011E2\u0000By\u0011E1)g; (29)\nwhere the last term describes the coupling between the\nA0\n1singlet and the components of the E00triplet OP [28].\nThe structure of this term is \fxed by symmetries. In our\nnotations the pair of components ( \u0011E1;\u0011E2) transforms\nexactly as (\u0000By;Bx) underD3h, and furthermore under\nthe time reversal operation the \ripping of the magnetic11\nN(ω)/2N 0(a)\n0.7 0.8 0.901.23\nω\nN(ω)/2N 0(b)\n0.9 1.1.1 1.2 1.3 1.401234567\nω\nFIG. 8. Normalized DOS, N(!)=2N0as a function of the\nenergy!in units of Tcsfor theK-model. (a): solid line:\nDOS for original model with B= 2:5Tcs, \u0001SO= 15Tcs, \u0000 =\n0:5Tcsandj\u00110j= 0:8Tcs. Dashed line: DOS for the e\u000bective\nmagnetic impurity model with the e\u000bective scattering rate\n(27), and with the rest of the parameters as in the original\nmodel. Panel (b): Dashed line: DOS for singlet OP Solid\nline: DOS for mixed singlet-triplet OP. The parameters are\nB= 2Tcs;\u0001SO= 15Tcs;\u0000 = 0:1Tcsand for illustration we\ntakej\u00110j=Tcsfor the dashed curve and j\u00110j=Tcs;j\u0011E2j=Tcs\nfor the solid.\n\feld is compensated by the OPs conjugation thanks to\nthe imaginary part in Eq. (29).\nIn what follows for de\fniteness we set By= 0. The full\nset of mean \feld equations yields \u0011E1= 0 in this case.\nFor this reason we list only the coe\u000ecients which are not\nmultiplying the \u0011E1to simplify the presentation.\nFor the sake of clarity we present the Landau expan-\nsion of the thermodynamic potential for the K-model,\ni.e. for the nodeless SOC. Employing the notations,\n!n=\u0019T(2n+ 1) for Matsubara frequency labeled by\nan integern, and ~!n=!n+ sgn (!n) \u0000 we have [18],\nCsA=X\n!n>02\u0019TB2~!n\n!n[~!n(B2+!2n)+!n\u00012\nSO]+lnT\nTcs;(30a)\nCtE2=X\n!n>02\u0019T\u0002\n\u0000\u0000\nB2+!2\nn\u0001\n+!n\u00012\nSO\u0003\n!n[~!n(B2+!2n)+!n\u00012\nSO]+lnT\nTct;(30b)\nCtA=X\n!n>02\u0019T\u0000\n!n~!n+ lnT\nTctz; (30c)\nCsA;tE=X\n!n>02\u0019T\u0001SO\n~!n(B2+!2n) +!n\u00012\nSO: (30d)\nThe terms of the fourth order in the OPs can be rep-\nresented in the form,\n\n(4)= \n(4)\nA+ \n(4)\nE+ \n(4)\nAE; (31)where the \frst term describes the coupling of the singlet\nand triplet OPs of A0\n1symmetry. The second term de-\nscribes the contribution of E00triplets. And \fnally the\nlast term describes the coupling of OPs of di\u000berent sym-\nmetry allowed at the fourth order. In the explicit form\neach term of the equation (31) reads\n\n(4)\nA=D1j\u00110j4+D2j\u0011Aj4+D3j\u00110j2j\u0011Aj2\n+ (D4\u0011\u00032\n0\u00112\nA+c:c:); (32a)\n\n(4)\nE=Kj\u0011E2j4; (32b)\n\n(4)\nAE=L0Ej\u00110j2j\u0011E2j2+LAEj\u0011E2j2j\u0011Aj2+n\nL0\n0E\u0011\u00032\n0\u00112\nE2\n+L0\nAE\u0011\u00032\nE2\u00112\nA+M0Ej\u00110j2\u0011\u0003\n0\u0011E2+M0\n0E\u0011\u0003\n0\u00112\nE2\u0011\u0003\nE2\n+N1\u0011\u0003\n0\u0011\u0003\nE2\u00112\nA+N2\u0011\u0003\n0\u0011E2j\u0011Aj2+c:c:o\n; (32c)\nwherec:c:refers to each term in the curly brackets. All\nof the coe\u000ecients appearing in Eq. (32) are derived in\nthe next three sections, Secs. V A, V B and V C, and pre-\nsented in details in the Appendix A for the K-model of\nSOC.\nBelow we describe the derivation of the expansion\ncoe\u000ecients of the thermodynamic potential. The self-\nconsistency equations, (5) are equivalent to the minimiza-\ntion of the thermodynamic potential,\n\n =\u0000\f\u00001ln\u0010\nTre\u0000\f(H0+Hdis+HMF)\u0011\n\u0000\u0016Hi\u0000\n0(33)\nconsidered as a function of the OPs [45]. In Eq. (33) we\nhave subtracted the thermodynamic potential in the nor-\nmal state for convenience. The second term in Eq. (33)\nis readily obtained from the mean \feld Eq. (6) and the\ninteraction, Eq. (4),\n\u0016Hi=\u0000V2N0\u0014\nL(Tcs)j\u00110j2(34)\n+L(Tct)X\nj=1;2j\u0011Ejj2+L(Tctz)j\u0011Aj2\u0015\n;\nwhere we have introduced the notation, L(T) =\nln(2\u0003e\rE=T\u0019).\nTo deal with the disorder we employ a quasi-classical\napproach [38]. To implement this approach, in the next\nsection we express the disorder averaged thermodynamic\npotential, Eq. (33) via the quasi-classical Green function.\nA. Thermodynamic potential and quasi-classical\nGreen function\nIn the presence of the disorder it is convenient to use\nan alternative representation of the thermodynamic po-\ntential, Eq. (33), \n = \n0\u0000\u0016Hi, where\n\n0=Z1\n0d\u0015hHMFi\u0015; (35)12\nandh\u0001\u0001\u0001i\u0015stands for the thermal averaging with respect\nto the quadratic Hamiltonian, H0+Hdis+\u0015HMFwith\nHMFspeci\fed by Eq. (6). Equivalently, at a given \u0015\nthe averaging in Eq. (35) is performed with respect to\nthe original mean \feld Hamiltonian with the OP \u0001( k)\nreplaced by \u0015\u0001(k). Our derivation leading to Eq. (46)\nis a variant of the original one in Ref. [46] adopted to\nthe case of multiple OPs. The present approach in its\ncurrent form has been used recently to study the e\u000bect\nof disorder on the phase diagram of non-superconducting\nsystems with multiple magnetic OPs [47].\nWe de\fne the \u0015-dependent anomalous Green function,\nF\u0015;ss0(r;r0;\u001c;\u001c0) by Eq. (14) with the thermal average,\nh\u0001\u0001\u0001i replaced byh\u0001\u0001\u0001i\u0015. With these de\fnitions Eq. (35)\ntakes the form,\n\n0=V\n\fX\nk;siX\n!nZ1\n0d\u0015Re\u0002\n\u0001\u0003\ns1s2(k)F\u0015;s1s2(k;!n)\u0003\n;(36)\nwhere we have used the explicit form of the mean \feld\nHamiltonian, Eq. (6) and the relation,\nhc\u0000ks2cks1i\u0015=\f\u00001X\n!nF\u0015;s1s2(k;!n): (37)\nTo deal with the disorder we employ a quasi-classical\nformalism. The quasi-classical Green's functions are de-\n\fned by\n^g(kF;!n) =Z1\n\u00001d\u0018k\n\u0019i^\u001b3^G(k;!n) (38)\n=\u0014\ng(kF;!n)\u0000if(kF;!n)\n\u0000if\u0003(\u0000kF;!n)\u0000g\u0003(\u0000kF;!n)\u0015\n;\nwhere ^\u001b3= diag(\u001b0;\u0000\u001b0). We parametrize the quasi-\nclassical Green's functions in terms of Pauli matrices as\ng(kF;!n) =g0(kF;!n)\u001b0+g(kF;!n)\u0001\u001b; (39)\nf(kF;!n) = [f0(kF;!n)\u001b0+f(kF;!n)\u0001\u001b]i\u001b2:(40)\nThe\u0015-dependent quasi-classical Green functions can be\nde\fned similarly to Eqs. (38), (39) and (40) by adding\nthe subscript \u0015to all the quantities. For instance, we\nhave the de\fnition,\nf\u0015(kF;!n) =\u0000Z1\n\u00001d\u0018k\n\u0019F\u0015(k;!n): (41)\nSimilar to Eq. (40) we parametrize\nf\u0015(kF;!n)=[f\u0015;0(kF;!n)\u001b0+f\u0015(kF;!n)\u0001\u001b]i\u001b2:(42)\nThe de\fnition, Eq. (41) allows to write Eq. (36) in the\nform,\n\n0\nV2N0=\u0000\u0019\n\fX\n!nZd'k\n2\u0019\u0002 (43)\nZ1\n0d\u0015Re\"X\ns1;s2\u0001\u0003\ns1s2(kF)f\u0015;s1s2(kF;!n)#\n:For the choice of the OP speci\fed by Eqs. (8) and (9) we\nhave\n1\n2X\ns1;s2\u0001\u0003\ns1s2(kF)f\u0015;s1s2(kF;!n) =\u0011\u0003\n0f\u0015;0(kF;!n) (44)\n+^\r(kF)2\n4X\nj=1;2\u0011\u0003\nEjf\u0015;j(kF;!n) +\u0011\u0003\nAf\u0015;3(kF;!n)3\n5:\nFor brevity we henceforth write f0(kF;!n) =f0,\nf(kF;!n) =f,f\u0003\n0(\u0000kF;!n) =f\u0003\n0,f\u0003(\u0000kF;!n) =f\u0003\nand use the same notations for g0;g;g\u0003\n0;g\u0003. Further-\nmore, naturally we extend the same notation to all the\n\u0015-dependent quasi-classical Green functions to the re-\nmainder of the paper. With these notations, and using\nEq. (44) the expression Eq. (43) takes the form,\n\n0\nV2N0=\u00002\u0019\n\fX\n!nZ1\n0d\u0015Re\u0014\n\u0011\u0003\n0hf\u0015;0iF+\n+\u0011\u0003\nAh^\r(kF)f\u0015;3iF+X\nj=1;2\u0011\u0003\nEjh^\r(kF)f\u0015;jiF\u0015\n:\n(45)\nThe standard regularization,P\n!nj!nj\u00001= (\f=\u0019)L(T)\nalong with the representation \n = \n0\u0000\u0016Hi, allows us to\ncombine the expression (45) with Eq. (34) to obtain for\nthe thermodynamic potential, (33),\n\nV2N0=2\u0019\n\fX\n!nZ1\n0d\u0015Re\u0014\n\u0011\u0003\n0\u0012\u00110\n2j!nj\u0000hf\u0015;0iF\u0013\n(46)\n+\u0011\u0003\nA\u0012\u0011A\n2j!nj\u0000h^\r(kF)f\u0015;3iF\u0013\n+X\nj=1;2\u0011\u0003\nEj\u0012\u0011Ej\n2j!nj\u0000h^\r(kF)f\u0015;jiF\u0013#\n+j\u00110j2lnT\nTcs+X\nj=1;2j\u0011Ejj2lnT\nTct+j\u0011Aj2lnT\nTctz:\nThe Eq. (46) gives an expression of the thermodynamic\npotential in terms of the quasi-classical Green function.\nIt is greatly simpli\fed when expanded to the fourth order\nin OPs. This is done in the next section.\nB. Landau expansion of the thermodynamic\npotential\nIn this section, we perform the expansion of the ther-\nmodynamic potential in the OPs, \u00110,\u0011A,\u0011E1and\u0011E2.\nFor our purposes the expansion up to the fourth order\nsu\u000eces. Clearly, in order to expand the Eq. (46) to the\nfourth order we need to perform the expansion of the \u0015-\ndependent quasi-classical Green functions, f\u0015;0and for f\u0015\nup to the third order. Using the quasi-classical methods,13\nin the next section we will obtain the expansion of the\noriginal,\u0015-independent, functions in the form\nf0=1X\n\u0017=0f(\u0017)\n0;f=1X\n\u0017=0f(\u0017);\ng0=1X\n\u0017=0g(\u0017)\n0;g=1X\n\u0017=0g(\u0017); (47)\nwhere the superscript \u0017denotes the order of expan-\nsion. In Eq. (47) and in what follows we extend our\nconvention of omitting the arguments ( k;!n) in each\nof the functions f0;f;g0;gto the expansion coe\u000ecients\nf(\u0017)\n0;f(\u0017);g(\u0017)\n0;g(\u0017)appearing in Eq. (47).\nSince the dependence of f\u0015;0;f\u0015on\u0015originates ex-\nclusively from changing \u0001 ( k) by\u0015\u0001 (k) the expansion\nof the\u0015-dependent Green functions is straightforwardly\nrelated to the expansion of the original, \u0015-independent\nGreen functions Eq. (47),\nf\u0015;0=1X\n\u0017=0\u0015\u0017f(\u0017);f\u0015=1X\n\u0017=0\u0015\u0017f(\u0017): (48)\nSubstituting (48) to (46) and performing the simple in-\ntegration over \u0015we obtain to the second order in OPs as\nintroduced in Eq. (28),\n\n(2)=\u0019TX\n!nRe\u001a\n\u0011\u0003\n0\u0014\u00110\nj!nj\u0000D\nf(1)\n0E\nF\u0015\n(49)\n+\u0011\u0003\nA\u0014\u0011A\nj!nj\u0000D\n^\r(kF)f(1)\n3E\nF\u0015\n+X\nj=1;2\u0011\u0003\nEj\u0014\u0011Ej\nj!nj\u0000D\n^\r(kF)f(1)\njE\nF\u0015)\n+j\u00110j2lnT\nTcs+j\u0011Aj2lnT\nTctz+X\nj=1;2j\u0011Ejj2lnT\nTct:\nThe \frst and third order terms are forbidden by the\ngauge invariance, \n(1)= \n(3)= 0. Technically, this fol-\nlows from the vanishing ofn\nf(0)\n0;f(0)o\nandn\nf(2)\n0;f(2)o\n,\nrespectively, as shown in Sec. V C.)\nTo the fourth order we obtain,\n\n(4)=\u0000\u0019Tcs\n2X\n!0nRe\u0014\n\u0011\u0003\n0D\nf(3)\n0E\nF+\u0011\u0003\nAD\n^\r(kF)f(3)\n3E\nF\n+X\nj=1;2\u0011\u0003\nEjD\n^\r(kF)f(3)\njE\nF\u0015\n: (50)\nNote that unlike in Eq. (49), in Eq. (50) the temperature\nis set to the critical temperature, T=Tcs. This is nec-\nessary for the consistency of the Landau expansion close\ntoTcs. In result, the Matsubara frequencies in Eq. (50),\n!0\nn=\u0019Tcs(2n+ 1) are also evaluated at T=Tcs.\nIn the next section we compute ff(1)\n0;f(1)\n0gand\nff(3)\n0;f(3)\n0gby solving the quasi-classical Eilenbergerequation. Together with Eqs. (49) and (50) this will allow\nus to complete the Landau expansion of the thermody-\nnamic potential.\nC. Solution of Eilenberger equation\nIn this section, we introduce Eilenberger equation,\n[48, 49] and solve it for the quasi-classical Green func-\ntion, Eq. (38) up the third order in the superconducting\nOPs. This, according to Eqs. (49) and (50) allow us to\ncompute the thermodynamic potential to the fourth or-\nder. For brevity, in this section we denote the momentum\nargument, kFof the quasi-classical Green functions by k,\nand denote the angular average h:::iFbyh:::inot to be\nconfused with the thermodynamic average. The Eilen-\nberger equation in the presence of SOC and Zeeman \feld\nreads [18]\nh\u0010\ni!n\u0000^\u0006 (!n)\u0000^S(k)\u0011\n^\u001b3;^g(k;!n)i\n= 0; (51)\nwhere\n^S(k) =\u0014(\r(k)\u0000B)\u0001\u001b \u0001(k)\n\u0001y(k) ( \r(k) +B)\u0001\u001bT\u0015\n:(52)\nEquation (51) together with the normalization condition\n^g2(k;!n) = ^\u001b0 (53)\ndetermines ^ g(k;!n). The disorder self-energy in Eq. (51)\nis de\fned in Eq. (20) and is expressed in terms of the\nquasi-classical Green function, Eq. (38) as\n^\u0006 (!n) =\u0000i\u0000h^g(k;!n)i^\u001b3: (54)\nThe Eilenberger equation (51) is written in the product\nof the Nambu and spin spaces the same way as the Green\nfunction, Eq. (15). In what follows we denote the four\nNambu blocks by a pair of indices ( i;j), wherei;j=\n1;2. For instance, the (1 ;2)-block of the Green function,\nEq. (15) is an anomalous Green function, F(k;!n). The\n(1;2)-block of Eq. (51) gives four equations which can be\npresented as coupled scalar and vector equations,\n2!nf0= 2if\u0001B (55a)\n+ (k) [g0+g\u0003\n0] +d(k)\u0001[g\u0000g\u0003]\n+ \u0000[\u0000hg0+g\u0003\n0if0+hf0i(g0+g\u0003\n0)\n\u0000hg\u0000g\u0003i\u0001f+hfi\u0001(g\u0000g\u0003)];\n2!nf= 2if0B+ 2\r(k)\u0002f+ (k) [g\u0000g\u0003] (55b)\n+i[g+g\u0003]\u0002d(k) + [g0+g\u0003\n0]d(k)\n+ \u0000[\u0000hg\u0000g\u0003if0+hf0i(g\u0000g\u0003)\n\u0000hg0+g\u0003\n0if+hfi(g0+g\u0003\n0)\n+if\u0002hg+g\u0003i\u0000ihfi\u0002(g+g\u0003)]:14\nThe (1;1)-block of the normalization condition (53) is\nsimilarly presented as\ng2\n0+g2= 1\u0000f0f\u0003\n0+f\u0001f\u0003; (56a)\n2g0g=if\u0002f\u0003+f0f\u0003\u0000f\u0003\n0f: (56b)\nIn Eq. (47) we have introduced the Landau expansion\nforf0,f,g0andgfunctions. It is convenient to separately\nintroduce the expansion of the functions f\u0003\n0,f\u0003,g\u0003\n0and\ng\u0003,\nf\u0003\n0=1X\n\u0017=0(f\u0003\n0)(\u0017);f\u0003=1X\n\u0017=0(f\u0003)(\u0017);\ng\u0003\n0=1X\n\u0017=0(g\u0003\n0)(\u0017);g\u0003=1X\n\u0017=0(g\u0003)(\u0017); (57)\nwhere for clarity we henceforth omit the arguments\n(\u0000k;!n) of the expansion coe\u000ecients, ( f\u0003\n0)(\u0017), (f\u0003)(\u0017),\n(g\u0003\n0)(\u0017)and (g\u0003)(\u0017)in the same way as we did for f\u0003\n0,f\u0003,\ng\u0003\n0andg\u0003before.\nTo solve the Eilenberger equation up to third order it\nsu\u000ece to consider the Eqs. (55) and (56). We show in Ap-\npendix C that the solutions of Eqs. (55) and (56) found\nhere automatically satisfy the equations contained in the\n(1;1)-block of Eq. (51) as well as the equations contained\nin the (1;2)-block of Eq. (53). This statement is a nec-\nessary condition for the consistency of the quasi-classical\nmethod. Furthermore, the (2 ;1)- and (2;2)-blocks of the\nEilenberger equation, (51) are equivalent to the (1 ;2)-\nand (1;1)- blocks, respectively in the sense that the for-\nmer follow from the latter by the complex conjugation\nand replacement of kby\u0000k. The same block-wise equiv-\nalence holds for the normalization condition, Eq. (53).\n1. zero order solutions\nThe zero order solutions that are exact in the normal\nstate are not \fxed by the Eilenberger equation, and can\nbe determined most easily by solving the Gorkov equa-\ntion (18) directly and using the de\fnition of the quasi-\nclassical Green function (38),\nf(0)\n0= 0;f(0)= 0; g(0)\n0= sgn (!n);g(0)= 0:(58)\n2. solutions to the \frst order\nWith the equation (58) as an input, Eq. (55) expanded\nto the \frst order reads\n!nf(1)\n0=if(1)\u0001B+ sgn (!n) (k) (59a)\n+ \u0000sgn (!n)hD\nf(1)\n0E\n\u0000f(1)\n0i\n;!nf(1)=if(1)\n0B+\r(k)\u0002f(1)+ sgn (!n)d(k) (59b)\n+ \u0000sgn (!n)hD\nf(1)E\n\u0000f(1)i\n:\nTo the \frst order the normalization condition, Eq. (56)\nyields\ng(1)\n0= 0;g(1)= 0: (60)\nEquation (59) has been solved in detail in Ref. [18]. Here\nwe outline the procedure used to obtain the solution as\nthe same approach is used here to obtain solutions at\nhigher orders.\nFirst, one solves Eq. (59) for f(1)\n0;f(1)withD\nf(1)\n0E\n;\nf(1)\u000b\nconsidered as given along with the OPs\nappearing in Eq. (59). The solutions then are aver-\naged over angles. This results in a linear equations forD\nf(1)\n0E\n;\nf(1)\u000b\nwhich is easily solved. Finally, the ob-\ntained values of the latter angular averages are used to\nexpressf(1)\n0;f(1)via the OPs.\nThe \frst order solutions obtained by following this pro-\ncedure read,\nf(1)\n0=\u00012\nSO+ ~!2\nn\nj~!njK1\u0010\n\u00110+ \u0000D\nf(1)\n0E\u0011\n+iBsgn (!n)\nK1\u0000D\nf(1)\n1E\n\u0000iB\u0001SO\nj~!njK1\u0011E2; (61a)\nf(1)\n1=isgn (!n)B\nK1\u0010\n\u00110+ \u0000D\nf(1)\n0E\u0011\n+j~!nj\nK1\u0000D\nf(1)\n1E\n\u0000sgn (!n) \u0001SO\nK1\u0011E2; (61b)\nf(1)\n2=iB\r\nj~!njK1\u0010\n\u00110+ \u0000D\nf(1)\n0E\u0011\n(61c)\n+sgn (!n) \u0001SO^\r(k)\nK1\u0000D\nf(1)\n1E\n+\u0000\n~!2\nn+B2\u0001\n^\r(k)\nj~!njK1\u0011E2;\nf(1)\n3=^\r(k)\nj~!nj\u0011A; (61d)\nwhere we have de\fned the denominator K1=B2+\u00012\nSO+\n~!2\nnand the angular averages are\nD\nf(1)\n0E\n=\u0000\n\u00012\nSO+j~!njj!nj\u0001\n\u00110\u0000iB\u0001SO\u0011E2\nj~!nj(B2+!2n) +j!nj\u00012\nSO;(62a)\nD\nf(1)\n1E\n=iB~!n\u00110\u0000\u0001SO!n\u0011E2\nj~!nj(B2+!2) +j!nj\u00012; (62b)\nandD\nf(1)\n2E\n=D\nf(1)\n3E\n= 0.15\nUsing the solutions (61) we obtain for the quantities\nthat, together with (62a), enter the quadratic part of the\nthermodynamic potential, Eq. (49),\nD\n^\r(k)f(1)\n2E\n=iB\u0001SO\u00110+\u0000\nB2+!2\nn\u0001\n\u0011E2\nj~!nj(B2+!2n) +j!nj\u00012\nSO; (63a)\nD\n^\r(k)f(1)\n3E\n=\u0011A\nj~!nj: (63b)\nSubstitution of Eq. (63) in Eq. (49) reproduces Eq. (29)\nwith coe\u000ecients Eq. (30).\n3. solution to the second order\nWith the solutions to the zero and \frst order, Eqs. (58)\nand (60) the Eilenberger equations (55) to the second\norder take the form,\n!nf(2)\n0=if(2)\u0001B+ \u0000sgn (!n)hD\nf(2)\n0E\n\u0000f(2)\n0i\n;(64a)\n!nf(2)=if(2)\n0B+\r(k)\u0002f(2)\n+ \u0000sgn (!n)hD\nf(2)E\n\u0000f(2)i\n: (64b)\nUnlike Eq. (59), Eq. (64) is homogeneous, and as a\nresult, has only trivial solution,\nf(2)\n0= 0;f(2)= 0: (65)\nTo the second order, the normalization condition,\nEq. (56) gives the second order corrections to g0andg\ndirectly in terms of the \frst order solutions\n2sgn (!n)g(2)\n0=\u0000f(1)\n0(f\u0003\n0)(1)+f(1)\u0001(f\u0003)(1);(66a)\n2sgn (!n)g(2)=if(1)\u0002(f\u0003)(1)+f(1)\n0(f\u0003)(1)\u0000(f\u0003\n0)(1)f(1):\n(66b)\nHere we do not present Eq. (66) in the explicit form. The\nangular averages of Eq. (66) are given below for illustra-\ntion,\nD\ng(2)\n0E\n=E(0)\n1j\u00110j2+E(0)\n2j\u0011E2j2+E(0)\n3j\u0011Aj2(67a)\n+\u0010\nE(0)\n4\u00110\u0011\u0003\nE2+c:c:\u0011\n;\nD\ng(2)\n1E\n=E(1)\n1j\u00110j2+E(1)\n2j\u0011E2j2+E(1)\n3\u00110\u0011\u0003\nE2(67b)\n\u0000E(1)\u0003\n3\u0011\u0003\n0\u0011E2+n\nE(1)\n4\u00110\u0011\u0003\nA+E(1)\n5\u0011E2\u0011\u0003\nA+c:c:o\nandD\ng(2)\n2E\n=D\ng(2)\n3E\n= 0. The coe\u000ecients in Eq. (67)\nare listed in Appendix B. A useful property,\ng\u0003(2)\n0=g(2)\n0 (68)\nfollows from Eq. (66a).4. solution to the third order\nTo expand the Eq. (55) to the third order we make use\nof Eqs. (58), (60), (65) and the property (68),\n2!nf(3)\n0= 2if(3)\u0001B+ 2 (k)g(2)\n0 (69a)\n+d(k)\u0001\u0010\ng(2)\u0000(g\u0003)(2)\u0011\n+ \u0000\u0014\n2sgn (!n)\u0010D\nf(3)\n0E\n\u0000f(3)\n0\u0011\n+ 2D\nf(1)\n0E\ng(2)\n0\n\u0000D\ng(2)\u0000(g\u0003)(2)E\n\u0001f(1)\u00002D\ng(2)\n0E\nf(1)\n0\n+D\nf(1)E\n\u0001\u0010\ng(2)\u0000(g\u0003)(2)\u0011\u0015\n;\n2!nf(3)= 2if(3)\n0B+ 2\r(k)\u0002f(3)(69b)\n+ (k)\u0010\ng(2)\u0000(g\u0003)(2)\u0011\n+i\u0010\ng(2)+ (g\u0003)(2)\u0011\n\u0002d(k) + 2g(2)\n0d(k)\n+ \u0000\u0014D\nf(1)\n0E\u0010\ng(2)\u0000(g\u0003)(2)\u0011\n\u0000D\ng(2)\u0000(g\u0003)(2)E\nf(1)\n0\u00002f(1)D\ng(2)\n0E\n+ 2sgn (!n)\u0010D\nf(3)E\n\u0000f(3)\u0011\n+ 2D\nf(1)E\ng(2)\n0\n+if(1)\u0002D\ng(2)+ (g\u0003)(2)E\n\u0000iD\nf(1)E\n\u0002\u0010\ng(2)+ (g\u0003)(2)\u0011\u0015\n:\nIn order to solve Eq. (69), for f(3)\n0;f(3)we follow the same\nprocedure as in Sec. V C 2 for \fnding the \frst order solu-\ntions. With the expressions for f(3)\n0;f(3)we calculate the\nangular averages entering the quartic part of the thermo-\ndynamic potential, Eq. (50),\n\u0011\u0003\n0D\nf(3)\n0E\n=D1(!n)j\u00110j4+L0\n0E(!n)\u0011\u00032\n0\u00112\nE2 (70a)\n+1\n2M0\n0E(!n)\u0011\u0003\n0\u00112\nE2\u0011\u0003\nE2+1\n2N2(!n)\u0011\u0003\n0\u0011E2j\u0011Aj2\n+\u001a\nM0E(!n)\u00110\u0011\u00032\n0\u0011E2+1\n2c:c:\u001b\n+1\n2N1(!n)\u0011\u0003\n0\u0011\u0003\nE2\u00112\nA\n+1\n2L0E(!n)j\u00110j2j\u0011E2j2+1\n2D3(!n)j\u00110j2j\u0011Aj2\n+D4(!n)\u0011\u00032\n0\u00112\nA;16\n\u0011\u0003\nE2D\n^\r(k)f(3)\n2E\n=1\n2L0E(!n)j\u00110j2j\u0011E2j2(70b)\n+L0\n0E(!n)\u0003\u00112\n0\u0011\u00032\nE2+1\n2N1(!n)\u0011\u0003\n0\u0011\u0003\nE2\u00112\nA\n+1\n2M0E(!n)\u0003\u00112\n0\u0011\u0003\n0\u0011\u0003\nE2+1\n2N2(!n)\u0003\u00110\u0011\u0003\nE2j\u0011Aj2\n+\u001a\nM0\n0E(!n)\u0003\u00110\u0011E2\u0011\u00032\nE2+1\n2c:c:\u001b\n+K(!n)j\u0011E2j4\n+1\n2LAE(!n)j\u0011E2j2j\u0011Aj2+L0\nAE(!n)\u0011\u00032\nE2\u00112\nA;\n\u0011\u0003\nAD\n^\r(k)f(3)\n3E\n=D2(!n)j\u0011Aj4+1\n2D3(!n)j\u00110j2j\u0011Aj2\n+D4(!n)\u0003\u00112\n0\u0011\u00032\nA+N1(!n)\u0003\u00110\u0011E2\u0011\u00032\nA (70c)\n+\u001a1\n2N2(!n)\u0011\u0003\n0\u0011E2j\u0011Aj2+c:c\u001b\n+1\n2LAE(!n)j\u0011E2j2j\u0011Aj2+L0\nAE(!n)\u0003\u00112\nE2\u0011\u00032\nA:\nThe coe\u000ecients in Eq. (70) are even functions of the fre-\nquency, and are given explicitly in Appendix A. We mul-\ntiplied the averages with suitable OP, as in the thermo-\ndynamic potential, Eq. (50). Substitution of Eq. (70)\nin Eq. (50) reproduces Eq. (32) with the coe\u000ecients\npresented in Appendix A.\nVI. DISCUSSION AND CONCLUSIONS\nTriplet superconductivity remains elusive in most ma-\nterials. Uranium-based superconductors are perhaps the\nonly uncontroversial exception [33]. Some success in mea-\nsuring the singlet-triplet ratio has been achieved in non-\ncentrosymmetric superconductors by Little-Parks e\u000bect\nexperiments [50]. Identifying triplets in DOS in noncen-\ntrosymmetric superconductors is harder and indirect, be-\ncause the potentially dominant singlet component gaps\nout the low energy states. Here, we showed that exces-\nsive broadening of the coherence peaks, which cannot be\nexplained by SOC, Zeeman e\u000bect and disorder, may be\nindicative of a triplet component induced by a Zeeman\n\feld. This behavior can be most clearly identi\fed by\nobserving the DOS broadening in the vicinity of the co-\nherence peaks.\nIn view of the potential importance of the unconven-\ntional paring in NbSe 2based on the recent theoretical\nand experimental works [11, 12, 21, 29] it is practically\nimportant to understand the implications of the triplet\ncomponents of and OP on the tunneling data in TMDs.\nIn this work we have addressed the question of how\nthe triplet correlations a\u000bect the DOS in TMD monolay-\ners. In practical terms, we argue that the strength of the\ntriplet correlations is an important knob controlling the\nDOS evolution with an in-plane magnetic \feld.\nWe illustrate this point by a concrete example in Fig. 9.\nThe DOS is shown for the two systems with the same\ncritical \feld Bc(T) and the same level of weak disorder,\nN(ω)/2N 0(a)\n0.0 0.2 0.4 0.6 0.8 1.0024681012\nB\n- \n0.70 0.75 0.80 0.85 0.90 0.95 1.0002468\nT\nωN(ω)/2N 0(b)\n0.0 0.2 0.4 0.6 0.8 1.0024681012\nB\n+ 0.70 0.75 0.80 0.85 0.90 0.95 1.0002468\nT\nωFIG. 9. Normalized DOS, N(!)=2N0as a function of the\nenergy!in units of Tcsfor theK-model of SOC, and \u0000 =\n0:1Tcs. The DOS curves broaden as the \feld increases. Panel\n(a): purely singlet OP, Tct= 0, \u0001 SO= 15Tcs. Panel (b):\nmixed singlet-triplet OP, Tct= 0:5Tcs, \u0001SO= 6:28Tcs. Insets\nshow the (T;B) phase diagrams with both axes given in units\nofTcs. The solid (grey) line in the insets is Bc(T) separating\nthe superconducting state shown as shaded (blue) region from\nthe normal state. The dashed black line in the inset of panel\n(b) is the is the Bc(T) shown as the solid line in the inset to\npanel (a). The coordinates of vertically aligned and evenly\nspaced colored dots, at T= 0:9Tcsin insets to panel (a)[(b)]\nde\fne the ( T;B) pairs for which the DOS is shown in the\ncorresponding main \fgure using the same color scheme.\n\u0000\u001cTcs. In one system (Fig. 9a) the order parameter is\na pure singlet, while in the other (Fig. 9b) system there\nis an admixture of the triplet component with the triplet\ntransition temperature Tct= 0:5Tcs. The critical \feld is\nthe same because the SOC in the singlet superconductor,\n\u0001SOC= 15Tcsis larger than the SOC in the superconduc-\ntor of the mixed parity, \u0001 SOC= 6:2Tcs. The comparison\nbetween the two \fgures shows that the \feld evolution of\nthe DOS in the two systems is strikingly di\u000berent. The\nbroadening is much more pronounced in the system with\na small fraction of the triplet component of the OP. The\nabove example highlights the potential practical impor-\ntance of DOS measurements in identi\fcation of triplet\nsuperconducting correlations.\nIn conclusion, the monolayer TMDs are promising17\nplatform for studies and controlled manipulation of\ntriplet components of the superconducting OP. In this\nwork, we demonstrate that the \feld induced triplet cor-\nrelations can be inferred from the \feld evolution of the\ntunnelling data combined with the knowledge of other\nparameters such as SOC and degree of system purity\navailable from transport measurements.\nACKNOWLEDGMENTS\nWe thank M. Aprili, L. Attias, T. Dvir, M. Dzero, A.\nLevchenko, I. Mazin, C. H. L. Quay and H. Steinberg for\nuseful discussions. The authors acknowledge the \fnan-\ncial support from the Israel Science Foundation, Grant\nNo. 2665/20. D.M. acknowledges the \fnancial support\nfrom the Swiss National Science Foundation, Project No.\n184050.\nAppendix A: Fourth order coe\u000ecients of Landau\nexpansion of the thermodynamic potential\nWe present here the forth order coe\u000ecients of the Lan-\ndau expansion of the thermodynamic potential, in the\nform of the summation over the Matsubara frequencies.\nHere we consider the K-model of SOC. As the Landau\nexpansion is performed close to Tcsthe fourth order coef-\n\fcients are evaluated at T=Tcs. Hence we present any\ngiven coe\u000ecient, Cappearing in Eq. (32) in the form,\nC=\u0000\u0019TcsX\n!0n>0C(!0\nn); (A1)\nwhere the summation is over the frequencies !0\nn=\n\u0019Tcs(2n+ 1) labeled by an integer n. We \frst list thecoe\u000ecients in the expansion Eq. (32a),\nD1(!) =D\u00004\n1\n2\u0014\n\u0000!\u0000\n!~!+ \u00012\nSO\u00014\n+2B2!(~!2\u0000\u00012\nSO)(\u00012\nSO+!~!)2\n+B4(3~!4!+ 2\u00012\nSO~!2(\u0000 + ~!)\u0000\u00014\nSO!)\u0015\n;(A2)\nwhere as in the main text ~ !=!+sgn (!) \u0000, and we have\nde\fned the denominator,\nD1(!) =B2~!+!\u0000\n!~!+ \u00012\nSO\u0001\n: (A3)\nThe other coe\u000ecients in Eq. (32a) are\nD2(!) =\u0000!\n2~!4; (A4)\nD3(!) =1\n~!2D2\n1\u0014\nB2\u0000\n\u0000~!2\u00002\u00012\nSO!\u0001\n\u0000(2!+ \u0000)\u0000\n\u00012\nSO+ ~!!\u00012\u0015\n; (A5)\nD4(!)=\u00001\n2~!2D2\n1h\nB2\u0000\n\u00012\nSO!+~!3\u0001\n+~!\u0000\n\u00012\nSO+~!!\u00012i\n:(A6)\nThe coe\u000ecient in the expansion Eq. (32b) is\nK(!) =\u0000!\u0000\nB2+!2\u0001\n2D4\n1\u0002\nB6+B4\u0000\n2\u00012\nSO+ 3!2\u0001\n+B2\u0000\n\u00014\nSO+ 3!4\u00004\u0000\u00012\nSO!\u0001\n\u0000!2\u0000\n3\u00014\nSO+ 2\u00012\nSO\u0000\n~!2\u0000\u00002\u0001\n\u0000!4\u0001\u0003\n:(A7)\nFinally, the coe\u000ecients in the expansion Eq. (32c) are\nL0E(!) =1\nD4\n1\u001a\nB6\u0000\n\u0000~!2\u00002\u00012\nSO!\u0001\n\u0000B4!\u0002\n2\u00012\nSO\u0000\n3\u00002+ 7\u0000!+ 6!2\u0001\n+4\u00014\nSO+ ~!2!(2!\u0000\u0000)\u0003\n(A8)\n\u0000B2!\u0002\n2\u00012\nSO!2\u0000\n4\u00002+10\u0000!+7!2\u0001\n+\u00014!(7\u0000 + 12!)+2\u00016\nSO+ ~!2!3(\u0000 + 4!)\u0003\n\u0000!4(\u0000 + 2!)\u0000\n\u00012\nSO+ ~!!\u00012\u001b\n;\nLAE(!) =\u00001\n~!2D2\n1\u0002\n2B4!+B2\u0000\n\u00012\nSO(\u0000 + 2!) + 4!3\u0001\n\u0000\u0000\u00012\nSO!2+ 2!5\u0003\n; (A9)\nL0\n0E(!) =1\n2D4\n1\u001a\nB6\u0000\n\u00012\nSO!\u0000~!3\u0001\n\u0000B4!\u0002\n3~!3!\u0000\u00012\nSO\u0000\n\u00002+ 2\u0000!+ 2\u00012\nSO+ 3!2\u0001\u0003\n(A10)\n+B2!\u0002\n\u00014\nSO!(\u0000 + 3!)\u0000\u00012\nSO!2\u0000\n2\u00002+ 4\u0000!+!2\u0001\n+ \u00016\nSO\u00003~!3!3\u0003\n\u0000!3\u0000\n\u00012\nSO+ ~!!\u00013\u001b\n;\nL0\nAE(!) =\u0000\nB2+!2\u0001\n2~!2D2\n1\u0002\n!\u0000\nB2+ \u00012\nSO+!2\u0001\n+ \u0000\u00012\nSO\u0003\n; (A11)18\nM0E(!) =i\u0001SOB\nD4\n1\u001a\nB4\u0002\n\u00012\nSO!\u0000~!2(\u0000 + ~!)\u0003\n+B2!\u0002\n2\u00014\nSO+ 2~!2!2+ \u00012\nSO(~!+!)2\u0003\n(A12)\n+!\u0002\n!(2\u0000 + 3!) + \u00012\nSO\u0003\u0002\n\u00012\nSO+ ~!!\u00032\u001b\n;\nM0\n0E(!) =iB\u0001SO!\nD4\n1\u001a\nB6+B4\u0002\n2\u0000\n\u00002+ \u00012\nSO\u0001\n+ 4\u0000!+ 5!2\u0003\n+B2\u0002\u0000\n\u00012\nSO+!2\u0001\u0000\n\u00012\nSO+ 3!2\u0001\n+ 4~!2!2\u0003\n(A13)\n+ \u00012\nSO!2\u0000\n2!2\u0000\u00012\nSO\u0001\n+!4\u0000\n\u00002+ 3~!2\u00002\u0000~!\u0001\u001b\n;\nN1(!) =\u0000iB\u0001SO\n~!2D2\n1\u0002\n!\u0000\nB2+!2\u0001\n+ \u00012\nSO~!\u0003\n;(A14)\nN2(!) =iB\u0001SO\n~!2D2\n1\u0002\n2!\u0000\nB2+ ~!2+!2\u0001\n+ \u00012\nSO(\u0000 + 2!)\u0003\n: (A15)\nAppendix B: Second order Landau expansion\ncoe\u000ecients\nWe present here the the coe\u000ecients to the averages\nover the second order Landau expansions from Eq. (67).\nAs in the main text we denote ~ !n=!n+ sgn (!n) \u0000 and\nde\fne the recurring denominator\nJ1=j!nj\u00012\nSO+j~!nj\u0000\nB2+!2\nn\u0001\n: (B1)\nThe coe\u000ecients are\nE(0)\n1=\u0000sgn(!n)\n2J2\n1\u0014\n2\u0000j!nj\u0000\n\u00012\nSO\u0000B2+!2\nn\u0001\n(B2)\n\u0000B2\u0000\n\u00002\u0000\u00012\nSO+!2\nn\u0001\n+ \u00002!2\nn+\u0000\n\u00012\nSO+!2\nn\u00012\u0015\n;\nE(0)\n2=sgn(!n)\n2J2\n1\u0002\n!2\nn\u00012\nSO\u0000B4\u0000B2\u0000\n\u00012\nSO+ 2!2\nn\u0001\n\u0000!4\nn\u0003\n;\n(B3)\nE(0)\n3=\u0000sgn(!n)\n2j~!nj2; (B4)\nE(0)\n4=\u0000isgn (!n)B\u0001SO\n2J2\n1\u0000\n2\u0000j!nj+B2+ \u00012\nSO+ 3!2\nn\u0001\n;\n(B5)\nE(1)\n1=\u0000iBj~!nj\nJ2\n1\u0000\n\u0000j!nj+ \u00012\nSO+!2\nn\u0001\n; (B6)\nE(1)\n2=iBj!nj\u00012\nSO\nJ12; (B7)E(1)\n3=\u0001SO\n2J2\n1\u0000\n\u0000\u0000\nB2\u0000!2\nn\u0001\n\u0000j!nj\u0000\n\u00012\nSO\u0000B2+!2\nn\u0001\u0001\n;\n(B8)\nE(1)\n4=Bsgn(!n)\u0001SO\n2j~!njJ1; (B9)\nE(1)\n5=\u0000i\u0000\nB2+!2\nn\u0001\nsgn (!n)\n2j~!njJ1: (B10)\nAppendix C: Eilenberder equation and\nnormalization condition: consistency of\nquasi-classical method\nHere we complete the discussion of Eilenberder equa-\ntion, (51), and the normalization condition (53) in\nSec. V C. In this appendix, as in Sec. V C, we denote\nthe momentum argument, kFof the quasi-classical Green\nfunctions by k, and denote the angular average h:::iFby\nh:::i. In Eq. (55) we wrote the (1 ;2)-block of Eilen-\nberder equation, the (1 ;1)-block is\n0 = \u0003(k)f0\u0000 (k)f\u0003\n0+d(k)\u0001f\u0003+f\u0001d\u0003(k) (C1a)\n+ \u0000 [hf\u0003\n0if0\u0000hf0if\u0003\n0+hfi\u0001f\u0003\u0000hf\u0003i\u0001f];\n0 = 2g\u0002(\r(k)\u0000B)\u0000 \u0003(k)f\u0000f\u0003 (k) (C1b)\n+d(k)f\u0003\n0\u0000f0d\u0003(k) +if\u0003\u0002d(k) +id\u0003(k)\u0002f\n+ \u0000[hf\u0003if0\u0000hf0if\u0003+hfif\u0003\n0\u0000hf\u0003\n0if\n\u0000i(hf\u0003i\u0002f+ 2g\u0002hgi+hfi\u0002f\u0003)]:\nIn Eq. (56) we wrote the (1 ;1)-block of the normalization\ncondition, the (1 ;2)-block is\nf0(g0\u0000g\u0003\n0) +f\u0001(g+g\u0003) = 0; (C2a)\nf\u0002(g\u0000g\u0003) +if(g0\u0000g\u0003\n0) +i(g+g\u0003)f0= 0:(C2b)\nWe expect the equations derived by taking the Landau\nexpansion of Eqs. (C1),(C2) up to third order to be con-\nsistent with the expressions for f(\u0017)\n0;f(\u0017);(f\u0003\n0)(\u0017);(f\u0003)(\u0017)\nandg(\u0017)\n0;g(\u0017);(g\u0003\n0)(\u0017);(g\u0003)(\u0017),\u0017= 0;1;2, that are found19\nin the way illustrated in section V C. In the remainder of\nthis appendix we perform this consistency check. Consid-\nering the zero order terms (58) and the \frst order terms\n(60) the \frst order expansion of Eqs. (C1),(C2) and the\nsecond order expansion of Eq. (C2) hold. The second\norder expansion of Eq. (C1) yields\n0 = \u0003(k)f(1)\n0\u0000 (k) (f\u0003\n0)(1)+d(k)\u0001(f\u0003)(1)(C3a)\n+f(1)\u0001d\u0003(k) + \u0000\u0014D\n(f\u0003\n0)(1)E\nf(1)\n0\u0000D\nf(1)\n0E\n(f\u0003\n0)(1)\n+D\nf(1)E\n\u0001(f\u0003)(1)\u0000D\n(f\u0003)(1)E\n\u0001f(1)\u0015\n;0 = 2g(2)\u0002(\r(k)\u0000B)\u0000 \u0003(k)f(1)(C3b)\n\u0000(f\u0003)(1) (k) +d(k) (f\u0003\n0)(1)\u0000f(1)\n0d\u0003(k)\n+i(f\u0003)(1)\u0002d(k) +id\u0003(k)\u0002f(1)+ \u0000\u0014D\n(f\u0003)(1)E\nf(1)\n0\n\u0000D\nf(1)\n0E\n(f\u0003)(1)+D\nf(1)E\n(f\u0003\n0)(1)\u0000D\n(f\u0003\n0)(1)E\nf(1)\n\u0000i\u0010D\n(f\u0003)(1)E\n\u0002f(1)+D\nf(1)E\n\u0002(f\u0003)(1)\u0011\u0015\n:\nThe expressions for f(1)\n0;(f\u0003\n0)(1);f(1);(f\u0003)(1);g(2)derived\nfrom the procedure illustrated in section V C are consis-\ntent with Eq. (C3). This can be seen by direct substi-\ntution. Considering Eqs. 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Box 1033 Blindern, N-0315 Oslo, Nor way\nMarkus Penz\nDepartment of Mathematics, University of Innsbruck,\nTechnikerstraße 13/7, A-6020 Innsbruck, Austria\nErik I. Tellgren†\nHylleraas Centre for Quantum Molecular Sciences, Departme nt of Chemistry,\nUniversity of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo, Nor way\n(Dated: April 13, 2021)\nDensity-functional theory requires an extra variable besi des the electron density in order to prop-\nerly incorporate magnetic-field effects. In a time-dependen t setting, the gauge-invariant, total cur-\nrent density takes that role. A peculiar feature of the stati c ground-state setting is, however, that\nthe gauge-dependent paramagnetic current density appears as the additional variable instead. An\nalternative, exact reformulation in terms of the total curr ent density has long been sought but to\ndate a work by Diener is the only available candidate. In that work, an unorthodox variational\nprinciple was used to establish a ground-state density-fun ctional theory of the total current density\nas well as an accompanying Hohenberg–Kohn-like result. We h ere reinterpret and clarify Diener’s\nformulation based on a maximin variational principle. Usin g simple facts about convexityimplied by\ntheresulting variational expressions, we prove that Diene r’s formulation is unfortunately not capable\nof reproducing the correct ground-state energy and, furthe rmore, that the suggested construction\nof a Hohenberg–Kohn map contains an irreparable mistake.\nI. INTRODUCTION\nThe Hohenberg–Kohn theorem is commonly regarded\nas the theoretical foundation of density-functional theory\n(DFT). Omitting technical points [1–3], it asserts that\ntheelectrondensitydeterminestheexternalpotential(up\nto a constant) and therefore the Hamiltonian and all sys-\ntem properties [4]. To include arbitrary magnetic fields\ninto the formalism, DFT needs to be supplemented by an\nadditional basic variable. In current-density-functional\ntheory (CDFT) the paramagnetic current density takes\nthat role [5]. It is also possible to forego any attempt to\nfind a universal functional independent of the external\npotentials and instead have a formalism that is paramet-\nrically dependent on the magnetic field [6]. A peculiar\nfeature of CDFT is that it is the paramagnetic current\ndensity, andnotthegauge-invarianttotalcurrentdensity,\nthat enters as a basic variable. This leaves a disconnect\nbetween ground-state CDFT and the time-dependent\nversion of the theory, which is naturally formulated us-\ning the total current density [7, 8]. Additionally, the\ntotal current density avoids practical issues arising from\nhaving to extract the gauge invariant part of the para-\nmagnetic current density in approximate density func-\ntionals [9–11]. As far as a CDFT for ground states for-\nmulated with the total current density is concerned, the\nquestion if a Hohenberg–Kohntheorem holds is still open\n∗corresponding author: andre.laestadius@kjemi.uio.no\n†corresponding author: erik.tellgren@kjemi.uio.noand has attracted some recent attention [12–17]. Several\nauthors have realized that the total current density at\nbest fits awkwardly into standard density-functional ap-\nproaches and that, in fact, it is incompatible with the\nstandard energy minimization principle [11, 12, 18, 19].\nHowever, it has been remarked that energy maximiza-\ntion with respect to the current density is not excluded\nby any known result [12] and recent work has shown that\nthe Maxwell–Schr¨ odinger energy minimization principle\nnaturally leads to a density-functional theory that fea-\ntures the total current density [11].\nTo date, a work by Diener [20] is the only candidate\nfor a density-functional theory of the total current that\ndoes not modify the underlying Schr¨ odinger equation.\nLogical gaps in his formulation have been identified be-\nfore [12, 13], although one specific criticism was mistaken\n(Proposition 8 in Ref. 13, which we correct below at the\nend of Sec. IV). Nonetheless, despite the gaps, Diener’s\nunique approach is interesting as it comes tantalizingly\nclosetosucceeding andit has sofarbeen unclearwhether\nthe approach can be rigorously completed.\nIn this work, we first clarify the underlying assump-\ntions in Diener’s approach by reinterpreting it as based\nonamaximinvariationalprinciple. Basedonsimplefacts\nabout convexity of the resulting energy functional, it can\nbe concluded that Diener’s approach is neither capable\nof reproducing the ground-state energy nor the correct\ntotal current density. We also establish that Diener’s\nconstruction of a Hohenberg–Kohn map suffers from an\nirreparable error: the selection of a vector potential via\na stationary search over current densities is not correct.2\nOuranalysisisverygeneralandappliesevenifpreviously\nidentified issues [12, 13] could somehow be resolved.\nII. PRELIMINARIES\nOur point of departure is the time-independent mag-\nnetic Schr¨ odinger equation for electrons with the Hamil-\ntonian(inSI-basedatomicunits, comparedtoDiener[20]\nwe use the convention eA→Aand−eΦ→vfor the po-\ntentials)\nH(v,A) =1\n2/summationdisplay\nj(−i∇j+A(rj))2+/summationdisplay\njv(rj)+W.(1)\nHere (v,A) are the external electromagnetic potentials\nandW=/summationtext\ni<jr−1\nijis the electron–electron repulsion\noperator. We use the short-hand notation H0=H(0,0)\nfor the universal part of the Hamiltonian. Spin has no\nbearing on the present work and we therefore leave out\nall spin degrees-of-freedom from the notation.\nFor pure states ψ(r1,...,rN), where Γ = |ψ/an}b∇acket∇i}ht/an}b∇acketle{tψ|is the\ndensity matrix, the particle density and paramagnetic\ncurrent density are given by, respectively,\nρψ(r1) =N/integraldisplay\n|ψ|2dr2···drN,\njp\nψ(r1) =NIm/integraldisplay\n¯ψ∇1ψdr2···drN,(2)\nand with well-known extensions to mixed states. Under\na gauge transformation A/ma√sto→A+∇f, the paramagnetic\ncurrent density transforms as jp/ma√sto→jp−ρ∇f. The gauge-\ninvariant, total current density is thus given by j=jp+\nρA.\nFrom a direct calculation, using the densities defined\nin Eq. (2),\ntr(H(v,A)Γ) = tr(H0Γ)+/integraldisplay\njp\nΓ·Adr\n+/integraldisplay\nρΓ(v+1\n2|A|2)dr.(3)\nUsing Eq. ( 3) the ground-state energy can be obtained\nfrom the expression\nE(v,A) = inf\nΓtr(H(v,A)Γ)\n= inf\nρ,jp/braceleftbigg\nFVR(ρ,jp)+/integraldisplay\n(ρ(v+1\n2|A|2)+jp·A)dr/bracerightbigg\n,\n(4)\nwhere we have introduced the Vignale–Rasolt universal\nfunctional [5],\nFVR(ρ,jp) = inf\nΓ/mapsto→(ρ,jp)tr(H0Γ). (5)\nArecentresultestablishesthattheinfimuminEq.( 5)can\nbe replaced by a minimum for any physically reasonabledensities [21]. It is known that the paramagnetic current\ndensity(together with ρ) doesnot determine the external\npotentials [22], althoughthe originalproofidea [5] can be\nusedtoestablishamappingfrom( ρ,jp)tonondegenerate\nground states [12]. This was termed a weakHohenberg–\nKohn result in Ref. 23, where the degenerate case was\nfurther analysed.\nAnother formulation is obtained by introducing the\nGrayce–Harris semiuniversal density functional [6],\nG(ρ,A) = inf\nΓ/mapsto→ρtr(H(0,A)Γ)\n=/integraldisplay\n1\n2ρ|A|2dr+inf\njp/braceleftbigg\nFVR(ρ,jp)+/integraldisplay\njp·Adr/bracerightbigg\n,\n(6)\nwhich enables the ground-state energy to be written as\nthe (magnetic field-) B-DFT variational principle,\nE(v,A) = inf\nρ/braceleftbigg\nG(ρ,A)+/integraldisplay\nρvdr/bracerightbigg\n.(7)\nThe semiuniversal nature of G(ρ,A) directly leads to a\ntype of Hohenberg–Kohn result: For every fixed A, a\npositive ground state state ρ(r)>0 determines vup to\na constant [6].\nThe relationship between the above two frameworks\nhas recently been highlighted and analyzed [11, 24], with\nparticular focus on convexity properties and variational\nprinciples connecting the formalisms. [See Appendix A\nfor basic definitions of convexity and related notions.] At\nleast for small vector potentials, the physical interpreta-\ntionisthatconvexityoftheenergyin Aisassociatedwith\ndiamagnetism, while concavity in Ais associated with\nparamagnetism. Here, we note that the mixed-state ver-\nsion ofFVRdefined in Eq. ( 5) is jointly convex in ( ρ,jp)\n(but the pure-state version is not, see Proposition 8 in\nRef. 25). The mixed state version of Gis likewise con-\nvex inρ; however, it is neither convex nor concave in A.\nAs discussed in Ref. 11, the Grayce–Harris functional is\nparaconcave (“concave up to a square”) in A, i.e., the\ndifference ¯G(ρ,A) =G(ρ,A)−/integraltext1\n2ρ|A|2dris concave.\nLoosely interpreted in physical terms this means that\nall systems appear paramagnetic when the diamagnetic\nterm is removed. The corresponding transformation of\nthe ground-state energy E(v,A) is a change of variables\n¯E(u,A) =E(u−1\n2|A|2,A), which makes ¯E(u,A) jointly\nconcave in ( u,A), unlike the original E(v,A).\nThatG(ρ,A) cannot be convex in Ais fairly obvi-\nous from the physical interpretation. However, since this\nproperty will be important in the further results below,\nwe give a full proof.\nProposition 1. For someρ, the Grayce–Harris func-\ntionalG(ρ,A)is not convex in A.\nProof.Consider a ρsuch that for A= 0 one has a\nground-state degeneracy that allows for a current ±jp\ngs/ne}ationslash=\n0. Both signs are possible for jp\ngsdue to time-reversal\nsymmetry. Now, take A/ne}ationslash= 0 such that for one of the3\nground states one has/integraltext\njp\ngs·Adr=−/vextendsingle/vextendsingle/integraltext\njp\ngs·Adr/vextendsingle/vextendsingle<0.\nThen for sufficiently small but nonzero A,\nG(ρ,A) =/integraldisplay\n1\n2ρ|A|2dr+ inf\nΓ/mapsto→ρ/braceleftbigg\ntr(H0Γ)+/integraldisplay\njp\nΓ·Adr/bracerightbigg\n≤G(ρ,0)+/integraldisplay\n1\n2ρ|A|2dr−/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\njp\ngs·Adr/vextendsingle/vextendsingle/vextendsingle/vextendsingle<G(ρ,0).\n(8)\nOn the other hand, invoking time-reversal symmetry,\nnamelyG(ρ,+A) =G(ρ,−A), the assumption of con-\nvexity ofG(ρ,A) inAwould have entailed G(ρ,A) =\n1\n2(G(ρ,A) +G(ρ,−A))≥G(ρ,0), in contradiction with\nEq. (8).\nNote that the above result substantially understates\nthe extent of the non-convexity—it is not restricted at\nall to very special densities ρ. For example, some ρ\ncorrespond to paramagnetic systems that have concave\nG(ρ,A) inA. Moreover, most ρare such that increasing\nthe magnetic-field strength will reorder the energy spec-trum so that states with permanent paramagnetic cur-\nrents eventually become the ground state. These level\ncrossings introduce non-convexity as well.\nIII. DIENER’S FORMULATION AS A\nMAXIMIN VARIATIONAL PRINCIPLE\nNext, we turn to Diener’s unconventional attempt to\nformulate a total current-density-functional theory. Di-\nener’s formalism is greatly simplified and clarified by\nstarting from the ground-state energy and algebraically\nmanipulating the formula until we obtain a variational\nexpression that can be related to his working equations.\nTaking the B-DFT variational principle as the point of\ndeparture, it is indeed sufficient to rewrite the Grayce–\nHarris functional. Letting kdenote an arbitrary current\ndensity, we begin by adding an energy term that clearly\ngives a vanishing net contribution:\nG(ρ,A) =/integraldisplay\n1\n2ρ|A|2dr+inf\njp/braceleftbigg\nFVR(ρ,jp)+/integraldisplay\njp·Adr−inf\nk/integraldisplay|jp+ρA−k|2\n2ρdr/bracerightbigg\n= inf\njpsup\nk/braceleftbigg\nFVR(ρ,jp)+/integraldisplay\nk·Adr−/integraldisplay|jp−k|2\n2ρdr/bracerightbigg\n.(9)\nWhilekisadummyvariablethatisbeingoptimizedover,\nits value at the solution to the above minimax problem\nwill satisfy k=jp+ρAand hence exactly reproduce the\ntotal current density. This way, the issue that the correct\nenergy cannot be obtained from a standard minimization\nprinciple for the total current density is avoided. Using\nthe general fact that inf xsupyf(x,y)≥supyinfxf(x,y),\nwe next obtain\nG(ρ,A)\n≥sup\nkinf\njp/braceleftbigg\nFVR(ρ,jp)+/integraldisplay\nk·Adr−/integraldisplay|jp−k|2\n2ρdr/bracerightbigg\n= sup\nk/braceleftbigg\nFD(ρ,k)+/integraldisplay\nk·Adr/bracerightbigg\n=:GD(ρ,A).\n(10)\nWe have above introduced GD(ρ,A) and identified Di-\nener’s proposed total current-density functional\nFD(ρ,k) = inf\njp/braceleftbigg\nFVR(ρ,jp)−/integraldisplay|jp−k|2\n2ρdr/bracerightbigg\n= inf\nΓ/mapsto→ρ/braceleftbigg\ntr(H0Γ)−/integraldisplay|jp\nΓ−k|2\n2ρdr/bracerightbigg\n.(11)\nThe issue now arises as to whether the above maximin\nprinciplealwaysachievesequalityinEq.( 10). Ifthiswere\ntrue, we would have succeeded in expressing the ground-\nstate energy in terms a universal functional FDof thetotal current density. Unfortunately, this can immedi-\nately be disproven on the basis of convexity properties:\nThe right-hand side of Eq. ( 10), i.e.,GD, is manifestly\nconvex in A, and hence can only describe diamagnetic\nsystems, whereas the Grayce–Harris functional G(ρ,A)\nis nonconvex in A. This establishes the following result:\nProposition 2. For some (ρ,A), we have a strict in-\nequalityG(ρ,A)>GD(ρ,A).\nA remaining issue is whether Diener’s functional\nFD(ρ,k) or the variational principle for GD(ρ,A) are\nuseful for other purposes, such as reconstructing the cor-\nrect external vector potential from an input pair ( ρ,j=\njp+ρA) or delivering the correct total current den-\nsity from a pair ( ρ,A). The former would establish a\nHohenberg–Kohn-type mapping, since then ( ρ,j) deter-\nmines (ρ,A) up to a gauge. In a next step one could use\nthe B-DFT extension of the Hohenberg–Kohn theorem\nto determine v[6, 16, 26]. In Diener’s work, this is in\nfact the primary intended use of the minimization prin-\nciple that defines FD. Moreover, he relies heavily on the\nfact that a state Γ and an arbitrary vector field kcan be\n“related” through the effective vector potential\na(Γ,k) :=k−jp\nΓ\nρΓ. (12)\nBy definition we have k=jp\nΓ+ρΓa(Γ,k), mimicking4\nthe relationship between the total current density, the\nparamagnetic current, and the actual external vector\npotential. If supplying the true total current density\nj=jp+ρAtoFD(ρ,j) always yields a minimizer Γ m\nin Eq. (11) such that a(Γm,k) =A, a Hohenberg–Kohn-\ntype mappingwouldbe established. Moreprecisely, since\ntheinputto FDisgaugeinvariant,theexternalvectorpo-\ntential can at best be determined up to a gauge. Hence,\nwe have to allow for a(Γm,k) =A+∇fand multi-\nple gauge dependent minimizers jp\nmin Eq. (11), one of\nwhich corresponds to a gauge in which a(Γm,k) =A.\nThis weaker statement would be sufficient to establish\nthe Hohenberg–Kohn-type mapping. Unfortunately, the\nnext proposition shows that such an FD-based mapping\ndoes not exist.\nProposition 3. For some (ρ,A), Diener’s current den-\nsity functional FDfails to reconstruct the external poten-\ntial. That is, for any minimizer jp\nmin Eq.(11)we have\nj−jp\nm\nρ/ne}ationslash=A. (13)\nProof.Fix an arbitrary pair ( ρ,A) for which there exist\ncurrentdensities( jp\n0,j0=jp\n0+ρA)thatsolvetheminimax\nproblem Eq. ( 9). Inserting j0into Diener’s functional\nyields\nFD(ρ,j0) = inf\njp/braceleftbigg\nFVR(ρ,jp)−/integraldisplay|jp−j0|2\n2ρdr/bracerightbigg\n=FVR(ρ,jp\nm)−/integraldisplay|jp\nm−j0|2\n2ρdr,(14)\nwherejp\nmis a minimizer. Now assume, arguendo, thatthis minimizer can always be chosen to satisfy\nj0−jp\nm\nρ=A. (15)\nBut this is equivalent to jp\nm=j0−ρA=jp\n0. As a direct\nconsequence, we have the lower bound\nGD(ρ,A) = sup\nk/braceleftbigg\nFD(ρ,k)+/integraldisplay\nk·Adr/bracerightbigg\n≥FD(ρ,j0)+/integraldisplay\nj0·Adr\n=FVR(ρ,jp\n0)+/integraldisplay\nj0·Adr−/integraldisplay|jp\n0−j0|2\n2ρdr\n=FVR(ρ,jp\n0)+/integraldisplay\n(jp\n0+ρA)·Adr−/integraldisplay|ρA|2\n2ρdr\n=G(ρ,A).(16)\nCombining the above bound with the fact that\nGD(ρ,A)≤G(ρ,A) from Eq. ( 10), we have established\nthatGD(ρ,A) =G(ρ,A) forarbitrary( ρ,A). This, how-\never, is impossible in light of Proposition 2. Hence, we\nconclude that the assumption that the minimizer jp\nmcan\nalways be chosen to satisfy Eq. ( 15) is false, which com-\npletes the proof.\nIt shouldbe noted thatwe donotneed toexplicitly im-\npose that the total current density arising from an eigen-\nstate is divergence-free. This condition, ∇·k= 0, is not\nneeded in the minimax principle for G. However, it pos-\nsibly makesa difference in the maximin principle GD, yet\nadding it does not circumvent the problems noted above.\nFinally, it must be remarked that in his original work,\nDiener actually relies on a stationarity principle for a\nquantityGstat, rather than on the above maximin prin-\nciple forGD. However, this difference is inessential and,\nin fact, only adds to the problems identified above. The\nfollowing bounds are immediate:\nG(ρ,A) = inf\njpsup\nk/braceleftbigg\nFVR(ρ,jp)+/integraldisplay\nk·Adr−/integraldisplay|jp−k|2\n2ρdr/bracerightbigg\n≥GD(ρ,A) = sup\nkinf\njp/braceleftbigg\nFVR(ρ,jp)+/integraldisplay\nk·Adr−/integraldisplay|jp−k|2\n2ρdr/bracerightbigg\n≥Gstat(ρ,A) = stat\nkinf\njp/braceleftbigg\nFVR(ρ,jp)+/integraldisplay\nk·Adr−/integraldisplay|jp−k|2\n2ρdr/bracerightbigg\n.(17)\nHence, by Proposition 2it follows that G(ρ,A)>\nGD(ρ,A)≥Gstat(ρ,A), for some ( ρ,A). The problems\nwith the maximin principle for GDthus directly carry\nover to the stat-min principle for Gstat. Naturally, a\npure minimization principle, obtained by replacing the\nmaximization over kby a minimization, can only make\nproblems worse.IV. DIENER’S ORIGINAL FORMULATION\nThe previous section reinterpreted Diener’s formula-\ntion in terms of a maximin principle. In the present sec-\ntion, we provide a direct disproof in terms of Diener’s\noriginal concepts. As was already clear from Proposi-\ntion3, Diener’s proof is unfortunately in error and FD5\ncannot be used for a Hohenberg–Kohn result in CDFT.\nRecall Eq. ( 12), where for given Γ and kwe have the\nvector potential a(Γ,k) = (k−jp\nΓ)/ρΓ. The following\nproposition is a direct consequence of Eq. ( 3).\nProposition 4 (Eq. (6) in Diener [20]) .LetH(v,A)be\nfixed. Then for any Γand any current density k\ntr(H(v,A)Γ) =ED(Γ,k)+/integraldisplay\n(k·A+ρΓv)dr\n+/integraldisplay\n1\n2ρΓ|A−a(Γ,k)|2dr,(18)\nwith\nED(Γ,k) = tr(H0Γ)−/integraldisplay|k−jp\nΓ|2\n2ρΓdr.(19)\nNote that ED(Γ,k) = tr(Ha(Γ,k)Γ) withHa(Γ,k)=\nH0−1\n2/summationtext\nj|a(Γ,k;rj)|2, i.e.,EDcan be viewed as an\nexpectation value over a state-dependent Hamiltonian\nHa(Γ,k). Equation ( 18) can also be stated as\nED(Γ,k)+/integraldisplay\n(k·A+ρΓv)dr= tr(H(v,A)Γ)\n−/integraldisplay\n1\n2ρΓ|A−a(Γ,k)|2dr.(20)\nOn the left-hand side of Eq. ( 20) we have ED(albeit not\na functional of the densities) and a linearcoupling be-\ntween (v,A) and the variables ( ρ,k). This mimics the\nsituation that one has in density-only DFT, however,\nwith one important difference: the expression on the\nleft-hand side of Eq. ( 20) does not equal the expecta-\ntion value tr( H(v,A)Γ). To obtain a density-functional\nsetting, Diener minimized the left-hand side of Eq. ( 20)\nover all Γ /ma√sto→ρand transitioned from ED(Γ,k) to a den-\nsity functional by defining\nFD(ρ,k) := inf\nΓ/mapsto→ρED(Γ,k), (21)\nwhich is equivalent to Eq. ( 11). The existence and\nuniqueness of minimizers of FDwas neverinvestigated by\nDiener—apossibleminimumwassimplytacitlyassumed.\nAs far as the attempt to obtain a Hohenberg–Kohn theo-\nrem is concerned, Diener’s proof cannot be completed, as\nwill be demonstrated here based on Proposition 3. How-\never, we will first make an attempt at providing the best\npossible presentation of Diener’s argument.\nA word on notation: If a minimizer of FD(ρ,k) exists\nwe denote it by Γ mand call it a “Diener minimizer”. For\nsuch a Γ mwe have Γ m/ma√sto→ρand\nFD(ρ,k) =ED(Γm,k). (22)\nThat such a minimizer indeed can be guaranteed to exist\nunder certain assumptions is proven in Appendix B.\nDiener has formulated an unorthodox variational prin-\nciple, Eqs. (10) and (15) in Ref. 20, which we restate in\nthe following proposition.Proposition 5 (Diener’s generalized variational princi-\nple).Letv,Abe fixed. Diener’s functional FDverifies\nfor anyρ, for any Γ/ma√sto→ρ, and any current density k, the\ninequality\nFD(ρ,k)+/integraldisplay\n(k·A+ρv)dr≤tr(H(v,A)Γ)\n−/integraldisplay\n1\n2ρ|A−a(Γ,k)|2dr.(23)\nMoreover, suppose a minimizer ΓmofFD(ρ,k)exists,\nthen\ntr(H(v,A)Γm)−/integraldisplay\n1\n2ρ|A−a(Γm,k)|2dr\n≤tr(H(v,A)Γ)−/integraldisplay\n1\n2ρ|A−a(Γ,k)|2dr.(24)\nProof.The first inequality, Eq. ( 23), follows from mini-\nmizing the left-hand side of Eq. ( 20) over Γ /ma√sto→ρ,\nFD(ρ,k)+/integraldisplay\n(k·A+ρv)dr\n= inf\nΓ/mapsto→ρ/braceleftbigg\nED(Γ,k)+/integraldisplay\n(k·A+ρv)dr/bracerightbigg\n≤tr(H(v,A)Γ)−/integraldisplay\n1\n2ρΓ|A−a(Γ,k)|2dr.(25)\nTo obtain Eq. ( 24), we note that the left-hand side of\nEq. (25) can be rearranged into\nED(Γm,k)+/integraldisplay\n(k·A+ρv)dr\n= tr(H0Γm)+/integraldisplay\n(k·A+ρv)dr−/integraldisplay\n1\n2ρ|a(Γm,k)|2dr\n= tr(H(v,A)Γm)−/integraldisplay\n1\n2ρ|A−a(Γm,k)|2dr.\n(26)\nHere we used FD(ρ,k) =ED(Γm,k) and Eq. ( 3) with\nk=jp\nΓm+ρa(Γm,k).\nIn light of Proposition 5, a natural question to ask is\nthe relation between Diener minimizers Γ mand ground\nstates Γ 0. We can offer the following answer: For a\nHamiltonian H(v,A) where A=a(Γm,k), the Diener\nminimizer Γ mis a ground state. However, a ground state\nΓ0/ma√sto→(ρ0,j0) generally does not need to be a minimizer\nofFD(ρ0,j0) (the proof is given below in Proposition 7).\nCorollary 6. Suppose Γm/ma√sto→ρ0to be a Diener min-\nimizer for FD(ρ0,k)and thatH(v,A), withA=\na(Γm,k), hasρ0as a ground-state density. Then vis\nunique up to a constant. Moreover, for any Γ/ma√sto→ρ0it\nholds (Eq. (15) in Diener)\ntr(H(v,A)Γm)≤tr(H(v,A)Γ)−/integraldisplay\n1\n2ρ|A−a(Γ,k)|2dr,\n(27)6\nand as a consequence Γmis a ground state for H(v,A)\nandk=jp\nΓm+ρ0A=:j0is the ground-state current den-\nsity. Further, a(Γ0,j0) =a(Γm,j0)for any other ground\nstate ofH(v,A)withΓ0/ma√sto→ρ0.\nNote that Corollary 6implies that all ground states\nwith Γ 0/ma√sto→ρ0have the same paramagnetic current den-\nsities, since the diamagneticpart alwaysis ρ0A. [See also\nthe joint-degeneracy theorem in Ref. 27.]\nProof of Corollary 6. Firstly, since the vector potential\nA=a(Γm,k) inH(v,A) is fixed and ρ0is by assumption\na ground-state density, the Hohenberg–Kohn result of B-\nDFT [6] gives that vis determined up to a constant.\nThe inequality in Eq. ( 27) is a direct consequence of\nEq. (24) witha(Γm,k) =A. Since Eq. ( 27) implies\nthe weaker bound tr( H(v,A)Γm)≤tr(H(v,A)Γ) for\nany Γ/ma√sto→ρ0, it follows that Γ mis a ground state of\nH(v,A). In particular, Eq. ( 27) gives for any ground\nstate ofH(v,A) with Γ 0/ma√sto→ρ0,\n/integraldisplay\nρ0|A−a(Γ0,k)|2dr= 0 (28)\nand thusρ0|A−a(Γ0,k)|2= 0 almost everywhere (a.e.).\nBytheunique-continuationpropertyfromsetsofpositive\nmeasure [16, 26], we have |{ρ0= 0}|= 0, soa(Γ0,k) =\nA=a(Γm,k) (a.e.).\nThe main question at this point is, how to guarantee\nthe required A=a(Γm,k). To meet that end, Diener\nsuggested in Ref. 20 to choose Aand, for arbitrary ρ,\nfind the stationary point,\njstat(ρ,A) = arg stat\nk/braceleftbigg\nFD(ρ,k)+/integraldisplay\nk·Adr/bracerightbigg\n,(29)\nwhere it was implicitly assumed that (i) there is a Di-\nener minimizer Γ mand (ii)FD(ρ,k) is differentiable with\nrespect to k. Under these assumptions, Diener claimed\nthat this jstathas the desired property\na(Γm,jstat) =A(up to a gauge) .(30)\nFor a ground state Γ 0/ma√sto→ρ0ofH(v,A) one then has by\nEq. (20)\nFD(ρ0,jstat(ρ0,A))+/integraldisplay\n(jstat(ρ0,A)·A+ρ0v)dr\n≤tr(H(v,A)Γ0),(31)\nwheretheleft-handsideequalstr( H(v,A)Γm). Tosumit\nup, a stationary variation over kis thought to select the\ncorrecta(Γm,jstat) =Awhile in the next step minimiz-\ning over densities gives the ground-state energy because\nof Eq. (31),\nE(v,A) = inf\nρstat\nk/braceleftbigg\nFD(ρ,k)+/integraldisplay\n(k·A+ρv)dr/bracerightbigg\n.\n(32)The attempted proof of Diener for a Hohenberg–Kohn\nresult then relies on the augmented variational principle\nin Eq. (27) that moreover has to be a strict inequality\nfor Γ not being a Diener minimizer Γ m. But since Corol-\nlary6shows that under certain assumptions such Diener\nminimizers are ground states, an additional condition of\nuniqueness of ground states gives a strict inequality. The\nusual Hohenberg–Kohn argument by contradiction could\nthen be completed by means of Eq. ( 27).\nFurthermore, underthe assumptionthat a(Γm,jstat) =\nAgets selected, there is also a more direct argument\navailable. Suppose that ( ρ0,j0) is the ground-state den-\nsity pair of two different Hamiltonians with vector po-\ntentialsAandA′, respectively. Then if for FD(ρ0,k) +/integraltext\nk·AdrandFD(ρ0,k)+/integraltext\nk·A′drDiener’s stationary\nsearch selects j0=jstat(ρ0,A) =jstat(ρ0,A′) that has\na(Γm,j0) equal to both AandA′up to a gauge, then\nthe magnetic field is the same for both systems. The\nHohenberg–Kohn result, i.e., that the scalar potentials\nalso are equal (up to an additive constant), then would\nfollow by the B-DFT result of Grayce and Harris [6].\nAlas, asacorollaryto ourmain Proposition 3, the next\npropositionshowsthatDiener’sstationarysearch(assug-\ngested and erroneously proved in Ref. 20) does notselect\na(Γm,jstat) =Aup to a gauge. Furthermore, we also\nhave, as a corollary to Proposition 3, that ground states\narenotin general minimizers of the Diener functional\nFD.\nProposition 7. (i) LetρandAbe fixed. The Diener\noptimization\nGstat(ρ,A) = stat\nk/braceleftbigg\nFD(ρ,k)+/integraldisplay\nk·Adr/bracerightbigg\n(33)\ndoes not in general select jstatsuch that a(Γm,jstat) =A\n(up to a gauge).\n(ii) A ground state with the density pair (ρ,j)is not in\ngeneral a Diener minimizer of FD(ρ,j).\nProof.For (i), we shall establish GD(ρ,A)≥G(ρ,A) for\narbitrary (ρ,A), which by Proposition 2is a contradic-\ntion. If no stationary point exists there is nothing to\nprove. Therefore assume that the value of Gstat(ρ,A) is\nrealized by a jstatand its contribution via FD(ρ,jstat),\nwhich, in turn, is realized by some state Γ mwith para-\nmagnetic current density jp\nm. Because the input to FDis\ngauge invariant, any gauge transformed state Γ′\nm, with\njp\nm′=jp\nm+ρ∇χand gauge function χ, is an equally\nvalid minimizer. By stipulation, we have ρa(Γm,jstat) =\njstat−jp\nm=ρ(A+∇f), wherefis agaugefunction for A.\nIt follows that choosing χ=freproduces the external\nvector potential exactly, i.e., a(Γ′\nm,jstat) =A, and also7\njstat=jp\nm′+ρA. Hence,\nGD(ρ,A)≥Gstat(ρ,A) = stat\nk/braceleftbigg\nFD(ρ,k)+/integraldisplay\nk·Adr/bracerightbigg\n= tr(H0Γ′\nm)+/integraldisplay\njstat·Adr−/integraldisplay\n1\n2ρ|A|2dr\n= tr(H0Γ′\nm)+/integraldisplay/parenleftbig\njp\nm′·A+1\n2ρ|A|2/parenrightbig\ndr≥G(ρ,A).\n(34)\nFor part (ii), we demonstrate that the assumption that\na ground state also is a Diener minimizer leads to a\ncontradiction. Let A(r) =1\n2Bez×rbe a vector po-\ntential representing a uniform magnetic field along the\nz-axis. Let vB(r) =−Z/|r|+1\n2(ω2\n0−1\n4B2)(x2+y2),\nwithω0/ne}ationslash= 0 and note that the effective scalar potential\nu=vB+1\n2|A|2=−Z/|r|+1\n2ω2\n0(x2+y2) is independent\nofB. The Hamiltonian H(vB,A) =H(v0,0) +1\n2BLz\nhas cylindrical symmetry and the eigenstates therefore\nhave quantized angular momentum component Lz=\n−i/summationtext\nj[rj× ∇j]z. Due to the quantization, the param-\nagnetic term becomes a trivial shift and the ground state\nis piecewise constant as a function of B, with jumps cor-\nresponding to level crossings. Consequently, the ground-\nstate density ρis also piecewise constant in B. For values\nofZthat correspond to an open-shell atom (e.g., a car-\nbonatomwith Z= 6andsixelectrons), thereisaground\nstateΓ −MforB≥0withtr(LzΓ−M) =−M. ForB≤0,\nthe ground state is Γ +M= Γ∗\n−Mwith the same density,\nΓ±M/ma√sto→ρ, but tr(LzΓ+M) = +M. For sufficiently small\n|B|, the Grayce–Harris functional is given by\nG(ρ,A) =G(ρ,0)−1\n2|MB|+/integraldisplay\n1\n2ρ|A|2dr,(35)\nwhich is nonconvex in Bbecause of the term −1\n2|MB|\nand is independent of the sign of B.\nForB >0, the total current density is given by j+=\njp\nΓ−M+ρAand forB <0 it isj−=jp\nΓ+M+ρA=−j+.\nBy stipulation, the ground states Γ ±Mfor all sufficiently\nsmall|B|are also a minimizers of FD(ρ,j±). Then\nGD(ρ,A)≥FD(ρ,j±)+/integraldisplay\nj±·Adr\n= tr(H0Γ∓M)−/integraldisplay\n(1\n2ρ|A|2+j±·A)dr\n=G(ρ,A).(36)\nCombined with the generic fact G(ρ,A)≥GD(ρ,A), we\nnow haveG(ρ,A) =GD(ρ,A) for a whole interval of\nsmall|B|. However, GD(ρ,A) is convex in A(and there-\nfore also in B) and therefore cannot equal the nonconvex\nG(ρ,A) on an interval of small |B|. This contradiction\ncompletes the proof.\nIn the previous section, Propositions 2and3estab-\nlished via a reinterpretation as a minimax principle that\nDiener’s approach cannot work, since it attempts to de-\nrive claims that are false. Proposition 7above furthershows, in the terminology of Ref. 20, that the central\nsteps in Diener’s reasoning towards a Hohenberg–Kohn-\nlike result fail. Our results here are thus definitive and\ngo further than previous critiques, which identified an\nunfounded strict inequality, a self-consistency condition\nthat would requirefurther analysis, and a variationalcol-\nlapse for specific types of ground state densities [12, 25].\nIt mayalsobe instructive to notea casewhereDiener’s\napproach does go through—albeit under extreme restric-\ntions. When only Γ with vanishing jp\nΓare allowed (or\nwhen only real valued states are allowed), we can choose\naH(v,A) with ground state Γ 0and densities ρ0and\nj0=0+ρ0A. In this case, the densities trivially deter-\nmine the external vector potential A=j0/ρ0. However,\nthe correct k=j0=ρ0Ais also recovered from the vari-\national principle\nGstat(ρ0,A) = stat\nk/braceleftbigg\nFD(ρ0,k)+/integraldisplay\nk·Adr/bracerightbigg\n= stat\nk/braceleftbigg\ninf\nΓ/mapsto→ρ0,jp\nΓ=0tr(H0Γ)−/integraldisplay|k|2\n2ρ0dr+/integraldisplay\nk·Adr/bracerightbigg\n.\n(37)\nThis restrictive case works because the coupling term k·\njp\nΓ/ρ0is absent and inf Γ/mapsto→ρ0,jp\nΓ=0tr(H0Γ) is independent\nofk, such that Eq. ( 37) leads to (at k=jstat)\n−jstat\nρ0+A=0⇐⇒a(Γm,jstat) =jstat\nρ0=A.(38)\nBeforeconcluding,wealsotakethisopportunitytocor-\nrect an incorrect claim in a previous publication of one\nof the authors, namely Proposition 8 in Ref. 25, which\nessentially misconstrued the contradiction reached in a\nreductio ad absurdum proof as a problem with the proof\nitself. Specifically, Proposition 8 in Ref. 25 considers two\nground state energies E=E(v,A) andE′=E(v′,A′).\nLeavingaside for the sake ofthe argumentall other prob-\nlems with Diener’s proof idea, one then reaches the con-\ntradictionE+E′<E+E′, but contraryto Proposition8\nin Ref. 25 this is not an additional flaw of the attempted\nproof.\nV. CONCLUSIONS\nWe haverevisited Diener’s attempted construction ofa\ndensity-functional theory featuring the gauge-invariant,\ntotal current density. The underlying crucial assump-\ntions have been clarified by a reformulation in terms of a\nmaximin principle. As Diener’s construction employs a\nnonstandard variational principle, it avoids some of the\nusual difficulties with the total current density asa varia-\ntional parameter. Nonetheless, we have shown here that\nhis attempted construction fails to establish a current-\ndensity-functional theory. Since the correct ground-state\nenergy cannot be obtained within this framework. More-\nover, the attempt to establish a Hohenberg–Kohn map-\nping for total current densities suffers from irreparable8\ngaps in the reasoning. We have shown that there must\nbe counterexamples for which the procedure does not re-\ntrieve the correct external vector potential from a given\ncurrent density.\nOn the other hand, in broad outline, Diener’s formu-\nlation shares notable features with the recently proposed\nMaxwell–Schr¨ odinger DFT (MDFT) [11], though details\ndiffer on crucial points. Diener introduces an effective\nvector potential, which is equivalent to a total current\ndensity, while MDFT takes the induced magnetic field\ninto account that is equivalent to a vector potential or a\ncurrent density and the total current density then arises\nnaturally as a basic variable. In both cases, the total cur-\nrent density is a variational parameter that is varied in-\ndependently from the wave function and the external po-\ntentials. Moreover, in Diener’s approach this variational\nparameter originates from a nonstandard, and unfortu-\nnately mistaken, re-expression of the Schr¨ odinger varia-\ntional principle. In MDFT, it comes from a modified en-\nergy minimization principle that simply adds the energy\nof the induced magnetic field. One can thus view MDFT\nas a proof of concept for deriving density-functional the-\nories of the total current from modified variational prin-\nciples. The very same considerations incorporating a\nfully quantized electromagnetic field lead to quantum-\nelectrodynamical DFT (QEDFT) [14]. Such extended\ndensity-functional theories form a physically better mo-\ntivated and theoretically more sound way for a density-\nfunctional framework including the total current density.\nACKNOWLEDGEMENTS\nA. L. acknowledges support from the Research Coun-\ncil of Norway (RCN) under CoE Grant Nos. 287906 and\n262695 (Hylleraas Centre for Quantum Molecular Sci-\nences). E. I. T. acknowledges support from RCN under\nGrantNo.287950andERC-STG-2014GrantNo.639508.\nM. P. acknowledges support by the Erwin Schr¨ odinger\nFellowship J 4107-N27 of the FWF (Austrian Science\nFund). The authors are thankful to L. Garrigue and\nM. A. Csirik for comments and suggestions that greatly\nimproved the manuscript and moreover acknowledge the\nsupportofthe CentreforAdvancedStudy (CAS) inOslo,\nNorway, which funded and hosted the workshop “Do\nElectron Current Densities Determine All There Is to\nKnow?” during 2018.\nAppendix A: Convex functions\nWe here review basic definitions from convex analy-\nsis. A subset Sof a vector space is said to be convexif\nx1,x2∈Simpliesλx1+(1−λ)x2∈S, for all 0 ≤λ≤1.\nA function f(x) is said to be convexif it is defined on a\nconvex domain and linear interpolation always yields an\noverestimate,\nf(λx1+(1−λ)x2)≤λf(x1)+(1−λ)f(x2),(A1)for all 0≤λ≤1. A function is said to be concaveif the\nreverse inequalities hold.\nA function of two variables f(x,y) can be convex in\neach of the arguments separately,\nf(λx1+(1−λ)x2,y)≤λf(x1,y)+(1−λ)f(x2,y),\nf(x,λy1+(1−λ)y2)≤λf(x,y1)+(1−λ)f(x,y2).\n(A2)\nAstrongerpropertyis joint convexity in both arguments,\nf(λx1+(1−λ)x2,λy1+(1−λ)y2)\n≤λf(x1,y1)+(1−λ)f(x2,y2).(A3)\nThe function\ng(z) = inf\nx(/an}b∇acketle{tx,z/an}b∇acket∇i}ht+f(x)), (A4)\nwhere/an}b∇acketle{tx,z/an}b∇acket∇i}htdenotes ascalarproduct (or bilinearpairing)\nis concave by construction. Changing the infimum to a\nsupremum yields a convex function. This fact is useful in\ndensity-functional theory, since the ground-state energy\nof standard DFT,\nE(v) = inf\nρ/parenleftbigg/integraldisplay\nρvdr+F(ρ)/parenrightbigg\n, (A5)\nis of this form. Convexity properties in B-DFT and\nCDFT are reviewed in Ref. 11.\nAppendix B: Proof of existence of Diener minimizers\nA Previous work pointed out the issue of variational\ncollapse, i.e., for some ρwe haveFD(ρ,k) =−∞[12].\nThis is a serious, but not totally decisive, problem for\nDiener’s approach. Here we show how it can be circum-\nvented at the cost of breaking gauge invariance by intro-\nducing a restriction on the kinetic energy density. Under\nsuch conditions, we prove the existence of a minimizer of\nFD(ρ,k).\nFirst some preparations follow. We limit ourselves to\npure states Γ = |ψ/an}b∇acket∇i}ht/an}b∇acketle{tψ|. Define the trial set of physical\nwave functions by\nWN:=/braceleftBig\nψ∈H1(R3N,C) :/integraldisplay\nR3N|ψ|2dr1···drN= 1/bracerightBig\n.\n(B1)\nThe setWNis an intuitive choice for the wave functions\nbeing minimized overin the pure-state Diener functional.\nWe may thus take\nFD(ρ,k) = inf\nψ∈WN,ψ/mapsto→ρED(ψ,k) (B2)\nas the Diener functional in a more detailed setting. (Also\nnote the slight abuse of notation, as we write ED(ψ,k)\ninstead of the consistent choice ED(|ψ/an}b∇acket∇i}ht/an}b∇acketle{tψ|,k).)9\nThe functional FDcan be defined on the space (see\nRef. 28)\nRN:=/braceleftBig\n(ρ,k) :√ρ∈H1(R3,R),k∈L1∩L3/2(R3,R3),\n/integraldisplay\nR3ρdr=N,/integraldisplay\nR3|k|2\nρdr<+∞/bracerightBig\n.\n(B3)\nThenψ∈ WNimplies (ρψ,jp\nψ)∈ RN. By afeature called\ncompatibility [28], the jpandjbelong to the same space\nRNandkis an arbitrary current in that space.\nFor technical reasons, we introduce a further restricted\nwave-function space. Let the kinetic energy density of a\nstateψ,τψ:R3→[0,+∞], be given by\nτψ(r1) =/integraldisplay\nR3(N−1)1\n2|∇ψ|2dr2···drN.(B4)\nForg≥0 a fixed integrable function with/integraltext\ngdr=C,\nC >0, set\nWC\nN:=/braceleftbig\nψ∈ WN:τψ≤g,τψ/ρψ∈L∞/bracerightbig\n.(B5)\nNote that the kinetic energy/integraltext\nτψdris bounded above by\nCfor allψ∈ WC\nN, so we get a kinetic-energy cutoff for\nall such wave functions. The constraint τψ/ρψ∈L∞is\nto guarantee jp\nψ/ρψ∈L∞, a property used later. In the\nmathematical literature of CDFT ψis typically assumed\nto have finite kinetic energy (here we go further and have\neven imposed a pointwise bound on the kinetic-energy\ndensityτψ).\nThe following lemma is an adaptation of Proposition 5\nin Ref. 25.\nLemma8. Fixρandk. Suppose weak convergence of ψn\ntoψminH1and thatρψn=ρandk=jp\nψn+a(ψn,k)ρ\nhold for all n. Thenρψm=ρand, for a subsequence\n{nk},jp\nψmis the weak L1limit of the {ψnk}’s param-\nagnetic current densities as well as their pointwise limit\n(a.e.),\njp\nψnk⇀jp\nψm(inL1),jp\nψnk→jp\nψm(a.e.) (B6)\nProof.By Theorem 3.3 in Lieb [1] we know that for a\nsubsequence, indexed by nk,ψnk→ψminL2and that\nρψm=ρ. Next we demonstrate that jp\nψnk⇀jp\nψmweakly\ninL1andpointwise(a.e.). Set J(ψ) tobe anycomponent\nof\n/integraldisplay\nR3(N−1)ψ∇ψdr2···drN (B7)\nand note, for φbeing the characteristic function of anymeasurableset M⊂R3, that(jindexingthecomponent)\n/integraldisplay\nR3φ(J(ψn)−J(ψm))dr1\n=/integraldisplay\nR3φ/integraldisplay\nR3(N−1)[(ψn∇ψn−ψm∇ψm)]jdr2···drN\n=/integraldisplay\nR3Nφ/bracketleftBig\n(ψn−ψm)∇ψn−ψm∇(ψm−ψn)/bracketrightBig\njdr1···drN\n=/integraldisplay\nR3Nφ(ψn−ψm)[∇ψn]jdr1···drN\n−/integraldisplay\nR3Nφψm[∇(ψm−ψn)]jdr1···drN\n=:Aj(n)−Bj(n).\n(B8)\nSinceφ∇ψn∈L2we haveAj(nk)→0 ask→ ∞by\nnorm convergence of {ψn}, and from φψ∈L2we obtain\nBj(nk)→0 by weak convergence in L2of{∇ψn}. We\ncan also use this argument (take φ∈L∞) to obtain weak\nconvergence in L1of (a subsequence of) jp\nψntojp\nψm.\nNow, define the kinetic-cutoff Diener functional\nFC\nD(ρ,k) by\nFC\nD(ρ,k) = inf\nψ∈WC\nN,ρψ=ρED(ψ,k).(B9)\nFor this version of the Diener functional we can finally\nestablish existence of Diener minimizers.\nProposition 9. Let(ρ,k)∈ RN∩{(ρ,k) :k/ρ∈L∞}.\nThere exists ψm∈ WC\nNsuch thatFC\nD(ρ,k) =ED(ψm,k)\nandψm/ma√sto→ρ, i.e., the infimum in Eq. (B9)is a minimum.\nProof.Letψnbe a minimizing sequence in WC\nN, i.e.,\nE(ψn,k)→FC\nD(ρ,k), ρψn=ρ,k=jp\nψn+a(ψn,k)ρ.\n(B10)\nThis implies that /ba∇dblψn/ba∇dbl2\nH1= 1 + 2/integraltext\nτψndr≤1 + 2C\nand thus by the Banach–Alaoglu theorem there exists a\nweakly convergentsubsequence and an element ψm∈H1\nsuch thatψnk⇀ψm(weakly in H1). By Lemma 8, the\nlimit function ψmhas the particle density ρ, andjp\nψmis\nthe weakL1limit of the {ψnk}’s paramagnetic current\ndensities as well as the pointwise limit (a.e.) like given\nin Eq. (B6). Then with\na(ψm,k) =k−jp\nψm\nρ, (B11)\nψmcan be taken as a candidate for a minimizer, where\nby definition E(ψm,k)≥FC\nD(ρ,k). What remains to be\nverified is the reverse inequality. To meet that end, set\nan:=a(ψn,k) =k−jp\nψn\nρ, (B12)\nwhich is an element of L∞sincek/ρ∈L∞and since for\nψ∈ WC\nN\njp\nψ\nρψ≤/parenleftbiggτψ\nρψ/parenrightbigg1/2\n∈L∞. (B13)10\nThen, pointwise a.e. and weakly in L1we have for a sub-\nsequence\nk= lim\nk(jp\nψnk+ankρ) =jp\nψm+(lim\nkank)ρ.(B14)\nThis gives that ankρ ⇀a(ψm,k)ρweakly inL1as well\nas pointwise a.e. (for a subsequence). Moreover, using\nagain the fact that k/ρ∈L∞and Lemma 8, we obtain\nby dominated convergence since1\n2|jp\nψnk|2ρ−1≤τψnk≤g∈L1that\nlim\nk/integraldisplay\nρ|ank|2dr\n=/integraldisplay|k|2\nρdr−2lim\nk/integraldisplayk·jp\nψnk\nρdr+lim\nk/integraldisplay|jp\nψnk|2\nρdr\n=/integraldisplay|k−jp\nψm|2\nρdr=/integraldisplay\nρ|a(ψm,k)|2dr.\n(B15)\nConsequently,\nED(ψm,k) =/an}b∇acketle{tψm,H0ψm/an}b∇acket∇i}ht−/integraldisplay\n1\n2ρ|a(ψm,k)|2dr\n≤lim\nk/parenleftbigg\n/an}b∇acketle{tψnk,H0ψnk/an}b∇acket∇i}ht−/integraldisplay\n1\n2ρ|ank|2dr/parenrightbigg\n=FC\nD(ρ,k)\n(B16)\nfollows by Eq. ( B15) above and lower semi-continuity of\nthe quadratic form ψ/ma√sto→ /an}b∇acketle{tψ,H0ψ/an}b∇acket∇i}ht.\n[1]E. H. Lieb, Int. J. Quantum Chem. 24, 243 (1983) .\n[2]P. E. Lammert, J. Math. Phys. 59, 042110 (2018) .\n[3]L. Garrigue, Math. Phys. Anal. Geom. 21, 27 (2018) .\n[4]P. Hohenberg and W. Kohn, Phys. Rev. 136, B864\n(1964).\n[5]G. Vignale and M. Rasolt, Phys. Rev. Lett. 59, 2360\n(1987).\n[6]C. J. Grayce and R. A. Harris, Phys. Rev. A 50, 3089\n(1994).\n[7]G. Vignale, Phys. Rev. B 70, 201102 (2004) .\n[8]G. Vignale and W. Kohn, Phys. Rev. Lett. 77, 2037\n(1996).\n[9]W. Zhu and S. B. Trickey, J. Chem. Phys. 125, 094317\n(2006).\n[10]E. I. Tellgren, A. M. Teale, J. W. Furness, K. K. Lange,\nU. Ekstr¨ om, and T. Helgaker, J. Chem. Phys. 140,\n034101 (2014) .\n[11]E. I. Tellgren, Phys. Rev. A 97, 012504 (2018) .\n[12]E. I. Tellgren, S. Kvaal, E. Sagvolden, U. Ekstr¨ om, A. M.\nTeale, and T. Helgaker, Phys. Rev. A 86, 062506 (2012) .\n[13]A. Laestadius and M. Benedicks, Int. J. Quantum Chem.\n114, 782 (2014) .\n[14]M. Ruggenthaler, (2017), arXiv:1509.01417 [quant-ph] .\n[15]E. I. Tellgren, A. Laestadius, T. Helgaker, S. Kvaal, andA. M. Teale, J. Chem. Phys. 148, 024101 (2018) .\n[16]L. Garrigue, Documenta Math. 25, 869 (2020).\n[17]L. Garrigue, J. Stat. Physik 177, 415 (2019) .\n[18]G. Vignale, C. A. Ullrich, and K. Capelle, Int. J. Quan-\ntum Chem. 113, 1422 (2013) .\n[19]A. Laestadius and M. Benedicks, Phys. Rev. A 91,\n032508 (2015) .\n[20]G. Diener, J. Phys.: Condens. Matter 3, 9417 (1991) .\n[21]S. Kvaal, A. Laestadius, E. I. Tellgren, and T. Helgaker,\n(2020),arXiv:2011.05129 [physics.chem-ph] .\n[22]K. Capelle and G. Vignale, Phys. Rev. B 65, 113106\n(2002).\n[23]A.Laestadius andE. I.Tellgren, Phys.Rev.A 97, 022514\n(2018).\n[24]S. Reimann, A. Borgoo, E. I. Tellgren, A. M. Teale, and\nT. Helgaker, J. Chem. Theory Comput. 13, 4089 (2017) .\n[25]A. Laestadius, Int. J. Quantum Chem. 114, 1445 (2014) .\n[26]A. Laestadius, M. Benedicks, and M. Penz, Int. J. Quan-\ntum Chem. 120, e26149 (2020) .\n[27]K. Capelle, C. A. Ullrich, and G. Vignale, Phys. Rev. A\n76, 012508 (2007) .\n[28]A. Laestadius, E. Tellgren, M. Penz, M. Ruggenthaler,\nS. Kvaal, and T. Helgaker, J. Chem. Theor. Comp. 15,\n4003 (2019) ."    },    {        "title": "2012.13414v2.Enhancement_in_tunneling_density_of_states_in_Luttinger_liquid____role_of_non_local_interaction.pdf",        "content": "Enhancement in tunneling density of states in Luttinger liquid :\nrole of non-local interaction\nAmulya Ratnakar and Sourin Das\nDepartment of Physical Sciences,\nIndian Institute of Science Education and Research (IISER) Kolkata\nMohanpur - 741 246, West Bengal, India\n(Dated: January 9, 2023)\nPower-law suppression of local electronic tunneling density of states (TDOS) in the zero-energy\nlimit is a hallmark of the Luttinger liquid (LL) phase of the interacting one-dimensional electron\nsystem. We present a theoretical model which hosts the LL state with the surprising feature of en-\nhancement rather than suppression in local TDOS originating from non local and repulsive density-\ndensity interactions. Importantly, we \fnd enhancement of TDOS in the manifold of parameter space\nwhere the system is stable in the renormalization group (RG) sense. We argue that enhancement of\nTDOS along with RG stability is possible only when the system has broken parity symmetry about\nthe position of local TDOS enhancement. Such a model could be realized on the edge states of a bi\nlayer quantum Hall system where both intra layer and inter layer density-density interactions are\npresent mimicking the role of local and non local interactions, respectively.\nPACS numbers:\nI. INTRODUCTION\nIt is well-known that the ampliude of electron tunnel-\ning into a Luttinger liquid (LL) state exhibits power-\nlaw suppression in the zero-energy limit owing to its non\nFermi-liquid behavior1{9. The suppression can be at-\ntributed to many-body orthogonality, which is akin to\nan orthogonality catastrophe discussed in the context of\na LL10{18, and it can be understood as follows. When\nan electron-like quasiparticle (with vanishingly small en-\nergy) tunnels locally into the LL prepared in its ground\nstate, interelectron interactions lead to a signi\fcant re-\narrangement of all the other electrons constituting the\nLL state, which results in an excited state which is or-\nthogonal to the corresponding ground state hence leading\nto the suppression of the tunneling process itself. This\nis a direct consequence of the fact that the low-energy\nexcitation spectrum of a LL is devoid of electron like\nquasiparticles19{25.\nIt is worthwhile to explore possibilities of departure\nfrom the observed suppression, which is treated as a hall-\nmark of the LL phase, and this is the main idea behind\nthe present study. Here, we obtain an enhancement of lo-\ncal tunneling density of states (TDOS) in the LL phase.\nWe present a minimal set up that allows for such an en-\nhancement in TDOS of a LL. It should be noted that in\nthe bulk of a LL, any deviation from suppression is un-\nlikely to take place due to the above-stated argument of\nthe orthogonality. However, in the local neighborhood of\nthe boundary between two LLs, we may be able to real-\nize a situation where such deviations could occur. Hence,\nwith the goal of \fnding a deviation from suppression, in\nthis paper, we consider a geometry involving the junction\nof two chiral LLs26{28. The junction of multiple LLs29{56,\nwhether chiral or non chiral, has been an area of theo-\nretical interest owing to the rich physics associated with\nvarious \fxed points that they host and belongs to the\nrealm of the general topic of quantum impurity problemsin low-dimensional electronic systems.\nIn an earlier study45involving one of the present au-\nthors along with others, it was shown that the exotic\n\fxed points of a three-wire junction can lead to an en-\nhancement of electron TDOS in the vicinity of the junc-\ntion when the LL parameter Kis tuned to the repulsive\ninterelectron interaction limit, i.e., K < 1. The origin\nof this TDOS enhancement was attributed to the re\rec-\ntion of a hole current35from the three-wire LL junction\ndue to interaction e\u000bects. A concern that remained is\nthat the \fxed points which allowed for an enhancement\nwere unstable to relevant perturbation in the renormal-\nization group (RG) sense and may not be direct of in-\nterest for experimental exploration. In the follow-up\nwork, the spin degree of freedom was incorporated for\nthe three-wire LL junction52; however, the issue of sta-\nbility remained. Recently, a density-matrix renormaliza-\ntion group (DMRG) study was carried out by one of the\npresent authors along with others which demonstrated a\nTDOS enhancement for the three-wire junction but again\nfor an unstable \fxed point56.\nActually, it is quite logical that all the \fxed points\ndepicting TDOS enhancement in the vicinity of the\nLL junction are unstable \fxed points. To understand\nthis point, let us consider a hypothetical situation con-\nsisting of a two-LL wire junction tuned to a disconnected\n\fxed point. The RG \row for the TDOS for each individ-\nual wire in the vicinity of the junction decides the rate\nat which the tunneling amplitude of an electron into the\nindividual wires diverges or gets suppressed in the zero-\nenergy limit ( E!0). In the presence of a weak tunneling\namplitude between the two wires, the net current \row-\ning from one wire to another, in the linear response limit,\nis proportional to the product of the TDOSs of the two\nwires at the junction. In order for the disconnected \fxed\npoint to be stable against weak interwire electron tun-\nneling perturbation, the tunneling current between the\ntwo wires must vanish in the zero-bias limit. This, inarXiv:2012.13414v2  [cond-mat.mes-hall]  6 Jan 20232\nFigure 1: Stacking of bilayer QH states with \flling fraction\n\u00171and\u00172exposed to a uniform magnetic \feld with a local\ntunnel coupling at the apex which is denoted by S. The top\nleft panel shows the zoomed-in view of the unfolded version\nof the bilayer QH edge states. The bottom panel shows the\n\u000b;\f; and\rinteractions between chiral QH edge states. Here,\nthe subscript \\ I\" and \\O\" stand for the \felds \rowing into\nthe junction and \felds the \rowing out of the junction, re-\nspectively. \\ I1=2;I=O\" stand for the currents on the incoming\nand outgoing edges.\nthe RG sense, implies that the interwire tunneling op-\nerator is an irrelevant perturbation. Hence the stability\nof the disconnected \fxed point along with simultaneous\nTDOS enhancement is achieved only when the rate at\nwhich TDOS gets suppressed in one of the wires is more\nthan the rate at which TDOS is enhanced in the other\nwire asE!0, i.e., simultaneous TDOS enhancement\nand stability are always accompanied by breaking of par-\nity symmetry between the two wires about the junction.\nThis can be achieved by having di\u000berent intrawire inter-\nactions in the two wires.\nIn this paper, we consider a \fxed point of junction be-\ntween two chiral LLs for exploring TDOS enhancement\nwhere the two chiral edge modes could belong to two\ndistinct quantum Hall (QH) states. This is theoreti-\ncally equivalent to having a single quantum point con-\ntact (QPC) in a Hall bar geometry forming a tunnel\njunction between the left and the right region, each of\nwhich in principle could host a distinct quantum Hall\nstate surrounded by its own chiral edge states. Such a\nsetup is simple from a theoretical perspective owing to\nthe fact that a junction between two chiral LLs can hostonly two \fxed points (a connected \fxed point and the\ndisconnected \fxed point29,57), unlike the three-wire case,\nwhich hosts a family of \fxed points35,39,45.\nFurthermore, we are interested in exploring how non-\nlocal density-density interaction between edge states be-\nlonging to the two sides of a QPC in\ruences the TDOS .\nOne way to simulate such interactions is to consider a sit-\nuation where we fold the two-dimensional system about\nthe QPC to form a bilayer system58{61and then consider\nlocal (in the folded 1 D coordinate system) interlayer and\nintralayer edge interaction as shown in Fig. 1. As dis-\ncussed above, to host a \fxed point that is stable and at\nthe same time also shows TDOS enhancement, we need\nto break the junction symmetry. We explore two di\u000berent\nways of breaking the symmetry (i) by having a junction\nbetween the chiral LLs belonging to two layers such that\neach layer has a di\u000berent \flling fraction or, (ii) if the\n\flling fraction is same, then by introducing asymmetric\nintralayer edge interaction.\nII. INTERACTING QH EDGE HAMILTONIAN\nConsider the situation of a bilayer interacting\nQH system with \flling fractions \u00171and\u00172on the two\nlayers as depicted in Fig. 1. To begin with, we consider\nrepulsive density-density interactions between the edges\nsuch that it poses a symmetric situation about the junc-\ntion and is parametrized by \u000b,\f;and\r, where\u000bis the\ninteraction between the counterpropagating edge states\nin the same QH states (intralayer interaction), \fis the\ninteraction between the copropagating edge states of the\ndi\u000berent QH states (interlayer interaction), and \ris the\ninteraction between the counterpropagating edge states\nof the di\u000berent QH states (interlayer interaction).\nThe Hamiltonian for QH edge states can be described\nin terms of bosonic \felds. The fermionic \feld  I=Ofor\nthe electron on the edge can be expressed in terms of\nthe bosonic \felds \u001eI=Ousing the standard bosonization\nformula19,21,23{25,29,62as I=O\u0018FI=Oexp(i\u001eI=O=\u0017),\nwhere subscript I(O) describes in (out) \felds. Here,\n\\in\"(\\out\") is used to index the chiral \felds which \row\ninto the junction (out of the junction). FI=Oare the\ncorresponding Klein factors for in/out \felds. Then the\nbosonized interacting edge Hamiltonian describing our\nsetup is given by\nH=~\u0019vFZ1\n0dx\u0014\u0012\u001a2\n1I+\u001a2\n1O\n\u00171\u0013\n+\u0012\u001a2\n2I+\u001a2\n2O\n\u00172\u0013\n+ 2\u000b\u0012\u001a1I\u001a1O\n\u00171+\u001a2I\u001a2O\n\u00172\u0013\n+2\fp\u00171\u00172(\u001a1I\u001a2I+\u001a1O\u001a2O) +2\rp\u00171\u00172(\u001a1I\u001a2O+\u001a1O\u001a2I)\u0015\n; (1)\nwhere\u001ai;I=O =\u0006(1=2\u0019)@x\u001ei;I=O and they represent the electronic density operator for the in/out bosonic \felds3\ncorresponding to \flling fraction \u0017iof theith QH layer\n(i2 f1;2g).vFis the Fermi velocity, which has been\ntaken to be the same on all the edges. Note that the inter-\naction parameters \u000b,\f, and\rare scaled appropriately in\nthe above Hamiltonian so that the transformation which\ndiagonalizes the above Hamiltonian stays algebraically\nsimple. We use the folded basis to describe the junc-\ntion such that all the QH edge states lie between x= 0\nandx=1with the junction positioned at x= 0. We\napplied the appropriate \fxed-point boundary condition\non the \\in\" and the \\out\" \felds at the junction. The\ninteracting Hamiltonian given in Eq. (1) along with the\nboundary condition describes the total system. In what\nfollows we will closely follow the diagonalization proce-\ndure for the above Hamiltonian as was done in Ref. [63].\nTo begin with, we can rewrite Eq. (1) in a compact form\nas\nH=~vF\n4\u0019Z1\n0dxr\u0016\u001eT(x)Kr\u0016\u001e(x); (2)\nwhere the matrix Kis given by\nK=0\nB@1\f\u0000\u000b\u0000\r\n\f1\u0000\r\u0000\u000b\n\u0000\u000b\u0000\r1\f\n\u0000\r\u0000\u000b \f 11\nCA (3)\nand is written in the basis \u0016\b(x;t), which is given by\n(\u0016\u001e1;\u0016\u001e2;\u0016\u001e3;\u0016\u001e4)(x;t)=\u0012\u001e1Op\u00171;\u001e2Op\u00172;\u001e1Ip\u00171;\u001e2Ip\u00172\u0013\n(x;t):\n(4)\nThen att= 0, the mode decomposition for the \feld\n\u0016\u001eais given by\n\u0016\u001ea(x) =Z1\n0dk\nkh\n\u0016ca;kei\u000fakx+ \u0016cy\na;ke\u0000i\u000fakxi\n;(5)wherea2f1;2;3;4g, with\u000fa= +1 fora=f1;2g(for\nthe outgoing \feld) and \u000fa=\u00001 fora=f3;4g(incoming\n\feld). The commutation relation for the bosonic annihi-\nlation and creation operator is given byh\n\u0016ca;k;\u0016cy\nb;k0i\n=\n\u000eabk\u000e(k\u0000k0), which is consistent with the commuta-\ntion relation of the bosonic \feld in the real-space basis\n\u0016\u001e(x) given by\u0002\u0016\u001ea(x);\u0016\u001eb(y)\u0003\n=i\u0019\u000fa\u000eabsgn(x\u0000y). Since\nthe relation between original interacting \felds \u001ei;I=O and\nthe transformed \feld \u0016\u001eais given by Eq. (4), the anni-\nhilation operators ciI=Ok of the\u001eiI=O \feld and \u0016cakof\nthe\u0016\u001ea\feld are also related as ( c1Ok;c2Ok;c1Ik;c2Ik) =\n(p\u00171\u0016c1k;p\u00172\u0016c2k;p\u00171\u0016c3k;p\u00172\u0016c4k), where\u0017iis the \fll-\ning fraction of the ith QH layer.\nLet the interacting \u0016\u001e(x;t) \feld be related to the Bo-\ngoliubov \feld ~\u001e(x;t) through a real matrix X, such that\n\u0016\u001e(x;t) =X~\u001e(x;t); (6)\nwhere\n~\u001e\u000b(x;t) =Z1\n0dk\nk\u0010\n~c\u000b;kei\u000f\u000bk(x\u0000~v\u000bt)+ ~cy\n\u000b;ke\u0000i\u000f\u000bk(x\u0000~v\u000bt)\u0011\n;\n(7)\nwhere\u000b2 f1;2;3;4gand\u000f\u000b= sgn(~v\u000b). ~c\u000bk(~cy\n\u000bk)\nis the bosonic annihilation (creation) operator for the\n\u000bth Bogoliubov mode, with commutation relations ash\n~c\u000bk;~cy\n\fk0i\n=\u000e\u000b\fk\u000e(k\u0000k0) and [~c\u000bk;~c\fk0] = 0, and this\nis consistent withh\n~\u001e\u000b(x);~\u001e\f(y)i\n=\u0019i\u000f\u000b\u000e\u000b\fsgn(x\u0000y).\nFrom Eq. (2), the Heisenberg equation of motion for\nthe bosonic \felds is given by\nd\ndt\u0016\u001ea(x;t) =\u0000vf\u000fa4X\n\u000b=1Ka\u000bd\ndx\u0016\u001e\u000b(x;t): (8)\nUsing Eqs. (6) and (7) in Eq. (8), we have\n4X\n\u000b=1\"Z1\n0dk\nk(ik\u000f\u000b) \nXa\u000b~v\u000b\u0000vf\u000fa4X\nb=1KabXb\u000b!\u0010\n~c\u000bkeik(x\u0000~v\u000bt)\u0000~cy\n\u000bke\u0000ik(x\u0000~v\u000bt)\u0011#\n= 0: (9)\nEquation (9) implies that\nvf4X\nb=1\u000faKabXb\u000b=Xa\u000b~v\u000b: (10)\nNow we can solve for the Xa\u000band the ~v\u000bby\nsolving the above equation. The ~ v\u000b's are given\nby\u0006p\n(1\u0000\f)2\u0000(\u000b\u0000\r)2and\u0006p\n(1 +\f)2\u0000(\u000b+\r)2,\nwith a + (\u0000) sign for out (in) free \feld. The bosonic\nexcitations are stable if the ~ v\u000b's are real. In order for\nnew \felds to satisfy the bosonic commutation relationswe must impose the following normalized condition:\n4X\n\u000b=1\u000fa\u000f\u000bXa\u000bXb\u000b=\u000eab;\n4X\na=1\u000fa\u000f\u000bXa\u000bXa\f=\u000e\u000b\f: (11)\nOnce we have obtained the Xmatrix, then for all ( x;t)4\nwe have\n\u0016\u001ea(x;t) =4X\n\u000b=1Xa\u000b~\u001e\u000b(x;t);\n~\u001e\u000b(x;t) =4X\na=1\u000fa\u000f\u000bXa\u000b\u0016\u001ea(x;t): (12)The interacting bosonic \feld operator \u0016 cakand the Bo-\ngoliubov \feld operator ~ c\u000bkare related as\n\u0016cak=4X\n\u000b=1Xa\u000b\u0010\nPa\u000b;+~c\u000bkei(\u000f\u000b\u0000\u000fa)kx+Pa\u000b;\u0000~cy\n\u000bke\u0000i(\u000f\u000b+\u000fa)kx\u0011\n;\n~c\u000bk=4X\na=1Xa\u000b\u0010\nPa\u000b;+\u0016cakei(\u000fa\u0000\u000f\u000b)kx\u0000Pa\u000b;\u0000\u0016cy\nake\u0000i(\u000fa+\u000f\u000b)kx\u0011\n; (13)\nwhere the projection operator is given by Pa\u000b;\u0006=\n(1\u0006\u000fa\u000f\u000b)=2. Let \u0016\u001eO=I,~\u001eO=Ibe doublets such that \u0016\u001eO=\n(\u0016\u001e1;\u0016\u001e2),\u0016\u001eI= (\u0016\u001e3;\u0016\u001e4) and ~\u001eO= (~\u001e1;~\u001e2),~\u001eI= (~\u001e3;~\u001e4).\nAlso, ~v1=\u0000~v3>0 and ~v2=\u0000~v4>0. Then, we can\nexpress Eq. (6) as\n\u0012\u0016\u001eO\u0016\u001eI\u0013\n(x;t)=\u0012\nX1X2\nX3X4\u0013\u0012~\u001eO\n~\u001eI\u0013\n(x;t); (14)\nwhere theXi's are 2\u00022 matrices. Now, the original\nincoming \felds \u001eiIand outgoing \feld \u001eiOare related to\neach other through a boundary condition at the junc-\ntion (x= 0). The boundary condition is expressed as\nthe current splitting matrix S, which corresponds to the\ndi\u000berent \fxed points of the junction, such that\n\u0012\n\u001e1O\n\u001e2O\u0013\n(x=0;t)=S\u0012\n\u001e1I\n\u001e2I\u0013\n(x=0;t);\n\u0012\u0016\u001e1\u0016\u001e2\u0013\n(x=0;t)=M\u00001SM\u0012\u0016\u001e3\u0016\u001e4\u0013\n(x=0;t)=\u0016S\u0012\u0016\u001e3\u0016\u001e4\u0013\n(x=0;t);\n(15)\nwhereMis a 2\u00022 matrix, with Mij=p\u0017i\u000eijand\n\u0016S=M\u00001SM. From Eqs. (14) and (15), we have\n\u0010\nX1~\u001eO+X2~\u001eI\u0011\n(x=0;t)=\u0016S\u0010\nX3~\u001eO+X4~\u001eI\u0011\n(x=0;t);\n\u0000\nX1\u0000\u0016SX3\u0001~\u001eO(x=0;t)=\u0000\u0016SX4\u0000X2\u0001~\u001eI(x=0;t);\n(16)\n~\u001eO(x=0;t)=\u0000\nX1\u0000\u0016SX3\u0001\u00001\u0000\u0016SX4\u0000X2\u0001~\u001eI(x=0;t);(17)\nwhich can be translated to \fnite values of xusing the\nfollowing relation,\n~\u001eO(x;t) =\u0000\nX1\u0000\u0016SX3\u0001\u00001\u0000\u0016SX4\u0000X2\u0001~\u001eI(\u0000x;t):(18)\nHere, we have used the fact that in our setup the incom-\ning \felds are left-moving \felds (see Fig. 1). Now, usingEq. (14) and the relation between the \u0016\u001eaand\u001ea\felds,\nwe have\n\u001eO(x;t) =Mh\nT1~\u001eI(\u0000x;t) +T2~\u001eI(x;t)i\n;\n\u001eI(x;t) =Mh\nT3~\u001eI(\u0000x;t) +T4~\u001eI(x;t)i\n;(19)\nwhere,\nT1=X1\u0000\nX1\u0000\u0016SX3\u0001\u00001\u0000\u0016SX4\u0000X2\u0001\n;\nT2=X2;\nT3=X3\u0000\nX1\u0000\u0016SX3\u0001\u00001\u0000\u0016SX4\u0000X2\u0001\n;\nT4=X4: (20)\nHence we have expressed all the interacting bosonic \felds\nin terms of the tilde \felds [Eq. (19)], which are free,\nand this will be used to calculate TDOS and scaling di-\nmensions of tunneling and backscattering operators that\ncould be switched on at the junction for RG analysis.\nBefore we conclude this section, it should be noted that\nfor the setup considered here, there are only two allowed\n\fxed points28,29and, hence, two possible Smatrices,\nwhich are given by\nS1=\u0012\n1 0\n0 1\u0013\n; (21)\nS2=1\n\u00171+\u00172\u0012\n\u00171\u0000\u001722\u00171\n2\u00172\u00172\u0000\u00171\u0013\n; (22)\nwhereS1is the fully re\recting disconnected \fxed point\nandS2is the strongly coupled \fxed point. For \u001716=\u00172,\ntheS2\fxed point may allow for incident current to be\npartially re\rected as a hole current.\nIII. POWER LAW DEPENDENCE OF TDOS\nThe electronic TDOS45,57at energyEat the position\nxis given by5\n\u001a(E) = 2\u0019X\nnjN+1hnj y(x)j0iNj2\u000e(EN+1\nn\u0000EN\n0\u0000E);\nwhereEN+1\nn;jniN+1andEN\n0;j0iNare the energy eigen-\nvalues and eigenstates corresponding to the nth excited\nstate of (N+ 1)-electron system and the ground state of\ntheN-electron system, respectively, for the interacting\nHamiltonian given in Eq. (1) subjected to appropriate\nboundary conditions ( S1orS2)and y(x) is the electron\ncreation operator at position x. In particular, we will\nbe calculating TDOSs only for the outgoing edge as they\nonly carry interesting information about the \fxed point\nto which the junction is tuned. Hence, to obtain the\nTDOS in terms of the bosonic \feld, we rewrite it as\n\u001ai(E) =Z1\n\u00001h0j iO(x;t) y\niO(x;0)j0ie\u0000iEtdt;\nwhich in terms of the bosonic \felds \u001eiOreads as\n\u001ai(E)\u0018Z1\n\u00001dth0jei\u001eiO(x;t)\n\u0017ie\u0000i\u001eiO(x;0)\n\u0017ij0ie\u0000iEt:\nHere, we have suppressed the subscript representing\nthe number of electrons in the ground state given by j0i\nfor notational convenience. Using Eq. (19), we evaluate\nthe above expression in two limits, (i) at the junction\n(x=0), and (ii) far from the junction ( x!1 ). In both\nthese limits, TDOS has a pure power-law dependence of\nthe form of E(\u0001i\u00001). The TDOS power law at the junc-\ntion (x= 0) is denoted by \u00010\ni, and far from the junction,\nx!1 is denoted by \u00011\ni(for details, see Appendix A).\nAfter a straightforward algebra, the TDOS exponent at\nthe junction is found to be given by\n\u00010\ni=1\n\u0017i2X\nj=1\u0010\n[T1]ij+ [T2]ij\u00112\n; (23)\nwhile far from the junction it is given by\n\u00011\ni=1\n\u0017i2X\nj=1\u0010\n[T1]2\nij+ [T2]2\nij\u0011\n: (24)\nTDOS in the zero-energy limit is enhanced when \u0001 i\u00001<\n0, is marginal when \u0001 i= 1, and is suppressed when\n\u0001i\u00001>0. Our primary focus is to study \u00010\ni, but be-\nfore we go ahead, we brie\ry discuss \u00011\ni. \u00011does not\ndepend on the \fxed point that we impose at the junc-\ntion but gets modi\fed only by the bulk interaction be-\ntween the edges, and it always corresponds to suppressed\nTDOS irrespective of the interaction strength which is\nexpected from standard LL physics45. The explicit form\nfor the TDOS exponent \u00011\nicorresponding to our model\nconsidered in Eq. (1) is given by\n\u00011\ni=1\n2\u0017i \n1\u0000\fp\n(1\u0000\f)2\u0000(\u000b\u0000\r)2\n+1 +\fp\n(1 +\f)2\u0000(\u000b+\r)2!\n: (25)Note that in the \u000b;\f;\r!0 limit we recover the expected\n1=\u0017power-law suppression of TDOS for an edge of the\nfractional quantum Hall state33. The power law of 1 =\u0017\nis also recovered when only \u000b;\r!0 while\f6= 0, due to\nthe fact that the nonzero \fcorresponds to a pure forward\nscattering interaction and hence can result only in the\nrenormalization of Fermi velocity but cannot in\ruence\nthe power law of the TDOS. One should also note that\neven in absence of the tunneling between the edges at\nx= 0, the very presence of interaction parameters \u000b;\r\nbreaks translational invariance along the edge while \f\nalone does not a\u000bect translational invariance as expected.\nIV. STABILITY OF THE FIXED POINT\nIn this section, we obtain a general expression for the\nscaling dimension of various tunneling and backscatter-\ning operators which can be switched on at the junction,\nwhere the scaling dimension being greater (less) than\nunity corresponds to an irrelevant (relevant) operator and\nbeing equal to 1 corresponds to being marginal. There\nare two possible \fxed points for the junction described in\nFig. 1: (i) The \frst one is the disconnected \fxed point,\nwhere the tunneling between the two layers at x= 0 is\nfully suppressed. Hence the most important perturbation\nto be analyzed as far as the RG stability of the junction\nis concerned is the electron tunneling operator between\nthe two layers at x= 0. (ii) The second one is the strong\ntunneling \fxed point, where the two layers are strongly\ncoupled atx= 0 and, hence, the most important pertur-\nbation to be analyzed as far as the RG stability of the\njunction is concerned is the quasiparticle backscattering\noperator in each layer at x= 0.\nFurthermore, it should be noted that the relation\nbetween the scaling dimensions of tunneling operators,\nwhich can be switched on at the junction as a pertur-\nbation, and the TDOS in the immediate vicinity of the\njunction (x!0) is not a simple relation which one\nmight naively expect. To understand this point, let us\nconsider the disconnected \fxed point to be speci\fc. In\nthis case, the scaling dimension of interlayer tunneling\noperators is dictated by the correlation function given\nbyG=h0j y\ne;1O(0) e;2I(0) y\ne;2I(t) e;1O(t)j0i, while the\nTDOS in each of the individual edge states is governed by\nthe correlation functions g1=h0j y\ne;1O(0;0) e;1O(0;t)j0i\nandg2=h0j y\ne;2I(0;0) e;2I(0;t)j0i. Here, the subscript\n\\e\" corresponds to the electron operator, while we will\nuse \\qp\" for the quasiparticle operator and the subscript\n1;2 stand for the layer index. Hence one might expect\nthatG=g1g2for the disconnected \fxed point leading to\na simple relation between TDOS and the stability of the\njunction. However, G6=g1g2owing to the fact that one\nhas interlayer interactions such that the ground state of\nthe full edge Hamiltonian, j0i, does not decompose onto\nthe direct product of the ground states of the edge Hamil-\ntonian of individual layers, i.e., j0i6=j0i1j0i2even for\nthe disconnected \fxed point. This fact plays an impor-\ntant role in the interplay of stability of a \fxed point and6\nTDOS enhancement via the various nonlocal interaction\nterms.\nThe expressions for scaling dimension of backscattering\nand tunneling operators are straightforward to calculate\nusing Eq. (19) and are given below:\n(1) The intralayer quasiparticle backscattering opera-\ntor y\nqp;iO(0) qp;iI(0) has a scaling dimension given\nbydO;I\nii=dI;O\nii=1\n2P2\nk=1(\u0003k\nB;i)2, where\n\u0003k\nB;i=p\u0017i(T3+T4\u0000T1\u0000T2)ik; (26)\n(2) The interlayer electron tunneling operator\n y\ne;iO(0) e;jI(0) has a scaling dimension given\nbydO;I\nij=dI;O\nji=1\n2P2\nk=1(\u0003k\nT;ji)2, where\n\u0003k\nT;ij=1p\u0017i(T3+T4)ik\u00001p\u0017j(T1+T2)jk; (27)\n(3) The interlayer electron tunneling operator\n y\ne;iO(0) e;jO(0) has a scaling dimension given by\ndO;O\nij=dO;O\nji=1\n2P2\nk=1(\u0003k\nT;ij)2, where\n\u0003k\nT;ij=1p\u0017i(T1+T2)ik\u00001p\u0017i(T1+T2)jk; (28)\n(4) The interlayer electron tunneling operator\n y\ne;iI(0) e;jI(0) has a scaling dimension given\nbydI;I\nij=dI;I\nji=1\n2P2\nk=1(\u0003k\nT;ij)2, where\n\u0003k\nT;ij=1p\u0017i(T3+T4)ik\u00001p\u0017i(T3+T4)jk: (29)\nV. SIMULTANEOUS TDOS ENHANCEMENT\nAND STABILITY OF S1FIXED POINT\nThe explicit form of the TDOS exponent correspond-\ning to the disconnected \fxed point S1, denoted by \u00010\ni;S1,\nis evaluated on one of the two outgoing QH edge states\nof the bi-layer system and is given by\n\u00010\ni;S1=1\n2\u0017i s\n1 +\u000b\u0000\f\u0000\r\n1\u0000\u000b\u0000\f+\r\n+s\n1 +\u000b+\f+\r\n1\u0000\u000b+\f\u0000\r!\n: (30)\nWe note that though interlayer interactions do exist,\nthe TDOS in each layer only depends on the \flling frac-\ntion (\u0017i) of the respective layer and not on that of the\nother layer. We will see later that this is not the case\nfor theS2\fxed point. It is also clear from the above ex-\npression that increasing \u000bmonotonically increases \u00010\ni;S1,\nwhich leads to the suppression of TDOS. When \f\nand\rare zero in the above expression, \u00010\ni;S1reducesto (1=\u0017i)p\n(1 +\u000b)=(1\u0000\u000b), wherep\n(1 +\u000b)=(1\u0000\u000b) is\nnothing but the inverse of the standard LL parameter64\nwhich is known to suppress the TDOS and the factor 1 =\u0017i\nleads to additional suppression owing to the presence of a\nfractional QH edge state33. Also, it was discussed earlier\nthat the e\u000bect of \falone is trivial as it represents the\nforward scattering interaction. Hence the enhancement\nof TDOS is expected to be induced by the presence of a\n\fnite\r.\nTo have a closer look at the interplay of various inter-\naction parameters leading to the enhancement of TDOS ,\nwe carry out a small \u000b;\r expansion of \u00010\ni;S1(\u000b;\f;\r )\naround (\u000b= 0;\f;\r = 0) to leading order and obtain\n\u00010\ni;S1'1\n\u0017i\u0012\n1 +\u000b\u0000\f\r\n1\u0000\f2\u0013\n: (31)\nFrom now onwards, we will only consider the case of\nrepulsive electron-electron interactions, i.e., \u000b;\f;\r >\n0. Furthermore, we focus on a speci\fc case for ex-\nploring the possibility of observing enhancement of\nTDOS (i.e., \u00010\ni;S1<1) for a junction of a \u00171= 1\nand\u00172= 1=3 QH system. This case could be of rel-\nevance as in this case we break the layer symmetry\n(which is necessary for the observation of simultane-\nous TDOS enhancement and stability of the junction)\nby choosing distinct \u0017for each layer and both \u0017= 1\nand\u0017= 1=3 represent a quantum Hall state which de-\npicts prominent plateaus in experiments65{67. It is ex-\npected that TDOS enhancement for the \u00172= 1=3 edge\nwill be practically impossible due to strong suppression\narising from the 1 =\u0017term in \u00010\ni;S1, and hence we focus\non the\u00171= 1 edge only. Equation (31) implies that\nTDOS enhancement for the \u00171= 1 edge will be possible\nonly if\f\r >\u000b in the small \u000b;\rlimit, which implies that\nthe magnitude of \u000b;\f, and\rhas to follow a speci\fc hi-\nerarchy for TDOS enhancement. However, most impor-\ntantly, this inequality points to the fact that interaction\nparameters \rand\fare essential for TDOS enhancement\nwhile\u000bis not (i.e., \u000bcan be zero). This point is\ndemonstrated numerically in Fig. 2, where the \frst plot\nin Fig. 2(a) shows enhancement of the TDOS in the\n\f-\rplane around the origin whereas the \frst plot in\nFig. 2(b) shows that the region of TDOS enhancement\nstarts shrinking as we turn on small but \fnite \u000b.\nAs far as the stability of the S1\fxed point (FP) is\nconcerned, the most relevant operators are the inter-\nlayer single electron tunneling operators, which are to\nbe considered for the analysis because the scaling dimen-\nsion of back-scattering operators is dI;O\n11=dI;O\n22= 0 for\nall\u000b;\f;\r as expected. Also note that for the S1FP,\ndI;O\n12=dO;I\n12=dI;I\n12=dO;O\n12=dS1. We obtain the expres-\nsion fordS1, which is given by\ndS1=1\n4 s\n1 +\u000b\u0000\f\u0000\r\n1\u0000\u000b\u0000\f+\r\u00121p\u0017i+1p\u0017j\u00132\n+s\n1 +\u000b+\f+\r\n1\u0000\u000b+\f\u0000\r\u00121p\u0017i\u00001p\u0017j\u00132!\n:(32)7\nFigure 2: The schematic pictures on the left show the unfolded version of the S1junction \fxed point of bilayer QH states\nexposed to a uniform magnetic \feld. For a junction of \u00171= 1 and\u00172= 1=3 tuned to the S1\fxed point, (a) and (b) each\nshow three density plots corresponding to \u00010\nS1anddS1and the region of simultaneous TDOS enhancement and stability as we\nmove from left to right in each row for \u000b= 0 and\u000b= 0:02 respectively. (c) shows \u00010\nS1anddS1and the region of simultaneous\nTDOS enhancement and stability as we move from left to right in the row for \u000b= 0:02. The third plot in this row indicates\nthat the region of simultaneous TDOS enhancement and stability is mutually exclusive in this case.\nNote that dS1is a function of both symmetric and\nantisymmetric combination ofp\u00171andp\u00172. The pres-\nence of antisymmetric combination indicates that the\nbroken layer symmetry ( \u001716=\u00172) results in an addi-\ntional contribution to the scaling dimension which is con-\nnected to the essential requirement for tuning simultane-\nous TDOS enhancement and stability. It is also clear\nfrom the above expression that the junction gets more\nand more stable as we increase \u000b; that is, increasing \u000b\nleads to monotonically increasing dS1. Hence \fnite \u000bhas\nan adverse e\u000bect on simultaneous TDOS enhancement\nand stability as its presence, on one hand, leads to greater\nstability but, on the other hand, suppresses the enhance-\nment of TDOS .\nSimilar to the expansion of \u00010\ni;S1above, we now per-\nturbatively expand dS1about (\u000b= 0;\f;\r = 0) to obtain\nthe following expression:\ndS1'\u00171+\u00172\n2\u00171\u00172\u0012\n1 +\u000b\u0000\f\r\n1\u0000\f2+\u00122p\u00171\u00172\n\u00171+\u00172\u0013\u000b\f\u0000\r\n1\u0000\f2\u0013\n(33)\nConsider the speci\fc case of \u00171= 1 and\u00172= 1=3\nwhich was previously discussed in the context of \u00010\ni;S1,where a TDOS enhancement was observed on the \u00171= 1\nedge when \u000b= 0. For this case, with \u000b= 0, we ob-\ntaindS1'2\u00002\r(\f+p\n3)=(1\u0000\f2) using the above\nequation, which implies that even for small \fand\r,\ndS1>1, implying a simultaneous TDOS enhancement\nand stability. This fact is demonstrated clearly in the\nsecond and the third plot in Fig. 2(a). Furthermore, the\nthird plot in Fig. 2(b) shows how the region of simul-\ntaneous TDOS enhancement and stability shrinks as we\nturn on a small but \fnite \u000b. This study established the\nfact that breaking of layer symmetry by taking \u001716=\u00172\nmay lead to TDOS enhancement in one of the two layers\nwhile ensuring stability of the \fxed point as was argued\nin the Introduction. For a \fnite \u000balso, we do \fnd si-\nmultaneous TDOS enhancement and stability provided\nwe proportionately increase the strength of the other in-\nteractions, but this is harder to see from the analytic\nexpressions. Hence we have performed a numerical anal-\nysis to demonstrate that it is indeed possible, which is\ndepicted in Fig. 3(b).\nNow, consider the case when \u00171=\u00172= 1, so that\n\u00010\n1= \u00010\n2= \u00010\nS1. When (\f=\r= 0),dS1= \u00010, which\nis due to the fact that the ground state of the system\ncan be written as the direct product of the ground states\nof individual QH layers. In the presence of interlayer in-8\nFigure 3: The schematic pictures on the left show the unfolded version of the S1junction \fxed point of bilayer QH states\nexposed to a uniform magnetic \feld. For a junction of \u00171= 1 and\u00172= 1 states tuned to the S1\fxed point with asymmetric \u000b\nin two layers, (a) shows three density plots corresponding to \u00010\nS1anddS1and the region of simultaneous TDOS enhancement\nand stability as we move from left to right in the row for \f= 0:6 and\r= 0:4. Here, region A shows the interaction parameters\nfor which TDOS is enhanced on both the edges. Region B (C) shows interaction parameters for which TDOS for the \u00171(\u00172)\nQH edge is enhanced and the junction is stable simultaneously. For a junction of \u00171= 1 and\u00172= 1=3 tuned to the S1\fxed\npoint, in the presence of symmetric interactions, (b) shows three density plots corresponding to \u00010\nS1anddS1and the region of\nsimultaneous TDOS enhancement and stability as we move from left to right in the row for \r= 0:4. Here, regions A, B, and\nC in the rightmost plots correspond to (\u00010\nS1>1;dS1>1), (\u00010\nS1<1;dS1>1), and (\u00010\nS1<1;dS1<1), respectively. Region\nB shows the interaction parameters for which the TDOS is enhanced and the junction is stable simultaneously.\nteraction (\f;\r), \u00010\nS1anddS1both modify themselves,\nanddS1acquires an additional contribution such that\ndS1= \u00010\nS1+ (\u000b\f\u0000\r)=(1\u0000\f2), which is due to the fact\nthat the ground states of the two QH layers are now en-\ntangled in the presence of nontrivial ( \f;\r). Also note\nthat in the presence of only copropagating edge interac-\ntion\f(with\u000b=\r= 0), the ground state of the two\nQH layers is still entangled, but the power laws are not\nmodi\fed, and we have dS1= \u00010\nS1. We do not have simul-\ntaneous TDOS enhancement and stability as expected\nowing to perfect layer symmetry even in the presence\nof\f;\rinteraction as shown in Fig. 2(c). We can break\nthe layer symmetry by taking the interaction parameter\n\u000bin the two QH layers to be asymmetric. The analytic\nexpressions of \u00010\nS1anddS1for the case of asymmetric \u000b\nin the two layers are too cumbersome to be included inthis paper; hence we have performed a numerical analysis\ncorresponding to this case and shown that the asymme-\ntry in\u000bcan indeed result in simultaneous enhancement\nof TDOS and stability though it requires the presence of\nstrong interaction. The result of our numerical analysis\nis presented in Fig. 3(a).\nVI. SIMULTANEOUS TDOS ENHANCEMENT\nAND STABILITY OF S2FIXED POINT\nFor the strongly coupled S2\fxed point, the\nTDOS exponent for the outgoing edge of the ith\nQH layer is denoted by \u00010\ni;S2and has a lengthy analytic\nexpression; hence we \frst focus on performing an expan-\nsion of \u00010\ni;S2to leading orders in \u000b;\r, which is given by\n\u00010\n1;S2'1\n\u00171\u0012\n1 +(\u00171\u0000\u00172)(\u000b\u0000\f\r)\n(1\u0000\f2)(\u00171+\u00172)+2p\u00171\u00172(\r\u0000\f\u000b)\n(1\u0000\f2)(\u00171+\u00172)\u0013\n(34)\n\u00010\n2;S2'1\n\u00172\u0012\n1\u0000(\u00171\u0000\u00172)(\u000b\u0000\f\r)\n(1\u0000\f2)(\u00171+\u00172)+2p\u00171\u00172(\r\u0000\f\u000b)\n(1\u0000\f2)(\u00171+\u00172)\u0013\n(35)\nNote that in the weak ( \u000b;\r) limit, both \u00010\n1;S2and\n\u00010\n2;S2have a term which is proportional to \u00171\u0000\u00172butwith opposite sign. This implies that if \u00171> \u0017 2, then\nthe contribution from this term will tend to suppress9\nTDOS on the \u00171edge while it will enhance it on the\n\u00172edge. Now, we consider the speci\fc case of \u00171= 1\nand\u00172= 1=3 which was discussed earlier in the context\nof theS1\fxed point. Naively, one would expect that\nit is more likely to obtain an TDOS enhancement in the\n\u00171= 1 edge as compared with \u00172= 1=3, because the\n\u00172= 1=3 edge su\u000bers from a strong suppression arising\nfrom the overall factor of 1 =\u0017in the expression for \u00010\n2;S2.\nHence we focus on TDOS enhancement in the \u00171= 1\nQH layer as it will have a higher likelihood of having si-\nmultaneous stability. Substituting \u00171= 1 and\u00172= 1=3\nin the expression of \u00010\n1;S2given above, we get \u00010\n1;S2'\n1 + (1=2)(\u000b\u0000\f\r)=(1\u0000\f2) + (p\n3=2)(\r\u0000\f\u000b)(1\u0000\f2),\nwhich implies that if the second and the third terms in\nthis expression turn out to be negative, then enhance-\nment of TDOS will be possible. This would require that\n\u000b < \f\r and\r < \f\u000b simultaneously, which is impossi-\nble because the interaction parameters are bounded be-\ntween 0 and 1. Hence we need to look for a possibility\nwhere the sum of the two terms is negative, which im-\nplies (\u000b\u0000\f\r) +p\n3(\r\u0000\f\u000b)<0. Now if we take an\nextreme limit of \f, i.e.,\f= 1\u0000\u000f, where\u000fis a small\nnumber which is of the order of \u000b;\r, or smaller and\n\u000b=\r+\u000ewhere\u000e << \u000b;\r then the inequality reduces\nto\u000e(1\u0000p\n3)<0 to leading order in all the small param-\neters hence resulting in TDOS enhancement. However,\none should note that the \f!1 or equivalently the \u000f!0\nlimit of Eq. (35) is problematic as it is itself a pertur-\nbative result and hence we must conform it using exact\nnumerical values. For example, for \u000e= 0:006,\u000f= 0:01,\nand\r= 0:1, we see TDOS enhancement on the \u00171= 1\nQH edge, while for \u000e= 0:00996,\u000f= 0:01, and\r= 0:1,\nwe see TDOS enhancement on the \u00171= 1=3 QH edge. A\nnumerical analysis of possible TDOS enhancement is ex-\nplored in Fig. 4(b), where we \fnd that both for large val-\nues of\f(\r) and small values of \r(\f), TDOS enhancement\nexists for\u000b= 0:2.\nThis observation of enhancement for the case of \u00171= 1\nand\u00172= 1=3 is indeed very interesting when we see it in\nthe light of Ref. [45], which reported TDOS enhancement\nfor a junction of three LL wires in the weak repulsive in-\nterelectron interaction limit. Ref. [45] shows a correla-\ntion between the Andreev-re\rection-like process leading\nto hole current35bouncing o\u000b the LL junction and the\nTDOS enhancement at the junction. Note that even in\nour setup (which is analogous to a junction of two nonchi-\nral LL wires), hole current is generated on the \u0017= 1=3\nedge4,26for the junction of \u00171= 1 and\u00172= 1=3. This can\nbe seen from the expression of the \feld splitting matrix\ngiven in Eq. (22), where one of its diagonal elements turns\nnegative for the choice of \u00171= 1 and\u00172= 1=3. Hence\none would have naively expected that we should observe\nan enhancement only on the \u00172= 1=3 edge, but on the\ncontrary we observe that the enhancement is happening\non both the \u00171= 1 and the \u00171= 1=3 edges. We conclude\nthat Andreev-re\rection-like processes do not necessarily\nlead to TDOS enhancement in general.\nFor theS2\fxed point, the intralayer single-\nquasiparticle backscattering operator represents the mostrelevant perturbation, and the junction is stable when\ndO=I\n11;dO=I\n22>1. The scaling dimension is studied mostly\nnumerically as its exact expression is too lengthy. We\nstart by analyzing the weak ( \u000b;\r) limit by carrying out\na leading order expansion of dO=I\niiin these parameters\nwhich is given by\ndO=I\n11'2\u00171\u00172\n\u00171+\u00172\u0014\n1 +\f\r\u0000\u000b\n1\u0000\f2+\u00122p\u00171\u00172\n\u00171+\u00172\u0013\r\u0000\f\u000b\n1\u0000\f2\u0015\n(36)\nNote thatdO=I\n11is symmetric under \u00171$\u00172, and thus\ndO=I\n11=dO=I\n22as expected. Let dO=I\n11=dO=I\n22=dS2. In the\nlimit\f\r >\u000b , the second term in Eq. (36) dominates over\nthe third term, and dS2tends towards the region where\nthe strongly coupled S2\fxed point is stable. Now, con-\nsider the speci\fc case of \u00171= 1 and\u00172= 1=3.dS2can be\nwritten asdS2= 1=2 +\u0010, where\u0010is a function of \u000b;\f;\r\nand is of the same order as them in the \u000b;\f;\r << 1\nlimit, which implies that the S2\fxed point is an unsta-\nble \fxed point in this limit. Above we have noted that\nTDOS enhancement is possible for large values of some\ninteraction parameters [see Fig. 4(b)] for the \u00171= 1 edge,\nand hence we would like to check whether S2can be si-\nmultaneously stable in this parameter regime; however,\nthis analysis is too complicated to be pursued analyti-\ncally owing to lengthy expressions, and hence we perform\na numerical analysis. The result of our analysis is pre-\nsented in last two plots in Fig. 4(b), where we have shown\nthe existence of a small but \fnite overlap region between\nstability and TDOS enhancement in the strong \rlimit.\nIt is not surprising that unlike the S1\fxed point, the\nregion of simultaneous TDOS enhancement and stability\nof the junction for the S2\fxed point always lies in the\nstrong\rlimit. This arises from the fact that the discon-\nnected \fxed point ( S1) is a stable \fxed point while the\nconnected \fxed point ( S2) is an unstable \fxed point in\nthe presence of \u000balone (i.e.,\f= 0 and\r= 0). Hence it\nrequires large values of \for\ror both to stabilize the S2\n\fxed point. Also, note that for \u00171= 1 and\u00172= 1=3, we\nhavedO;I\n11=dO;I\n22=dO;I\n12=dO=I\n21. This is due to symme-\ntry in the current splitting matrix for the S2\fxed point,\ni.e., transmission = re\rectance = 1/2, when current is\nexcited from the \u00171= 1 side.\nFor\u00171=\u00172=\u0017, the scaling dimensions of the tunnel-\ning operators are zero, i.e., dS2\n12=dS2\n21=dS2\nII=dS2\nOO= 0\nas expected. Also, \u00010\n1;S2= \u00010\n2;S2. The exact expression\nfor the scaling dimension of the backscattering operator\nand the TDOS exponent \u00010\ni;S2is given by\ndS2=\u0017s\n1\u0000\u000b\u0000\f+\r\n1 +\u000b\u0000\f\u0000\r(37)\n\u00010\ni;S2=1\n2\u0017i s\n1 +\r\u0000\f\u0000\u000b\n1\u0000\r\u0000\f+\u000b+s\n1 +\u000b+\f+\r\n1\u0000\u000b+\f\u0000\r!\n(38)10\nFigure 4: The schematic pictures on the left show the unfolded version of the S2junction \fxed point of bilayer QH states\nexposed to a uniform magnetic \feld. (a) shows a junction of \u00171= 1 and\u00172= 1 tuned to the S2\fxed point with asymmetric\n\u000bin the two layers. The three density plots in (a) correspond to the region of \u00010\ni;S2<1 anddS2>1 and the region of\nsimultaneous TDOS enhancement and stability as we move from left to right in the row for \f= 0:6 and\r= 0:9. Region A (B)\nin the third plot shows interaction parameters for which the TDOS for the \u00171(\u00172) QH edge is enhanced and the junction is\nsimultaneously stable. (b) shows a junction of \u00171= 1 and\u00172= 1=3 tuned to the S2\fxed point, in the presence of symmetric\nintralayer interactions. The \frst two density plots in (b) correspond to \u00010\nS2anddS2for\u000b= 0:2 as we move from left to\nright. The last plot in the row shows the region of simultaneous TDOS enhancement, which is marked as region A. (c) shows a\njunction of \u00171= 1 and\u00172= 1=3 tuned to the S2\fxed point, in the presence of asymmetric \u000binteractions. The \frst two density\nplots in (c) correspond to \u00010\nS2anddS2for\f= 0:6 and\r= 0:7. The last plot in the row shows the region of simultaneous\nTDOS enhancement and stability, which is again marked as A.\nExpanding Eqs. (37) and (38) in the weak \u000b;\rlimit,\nwe get\ndS2'\u0017\u0012\n1 +\r\u0000\u000b\n1\u0000\f2\u0013\n\u00010\ni;S2'1\n\u0017i\u0012\n1 +\r\u0000\f\u000b\n1\u0000\f2\u0013\n(39)\nNote that for \u00171=\u00172= 1 there exists a symmetry\nbetween the scaling dimension of the electron tunnel-\ning operator, dS1, of theS1\fxed point and the scal-\ning dimension of the electron backscattering operator,\ndS2, of theS2\fxed point in the \u000b$\rexchange,\nsuch thatdS1(\u000b;\f;\r ) =dS2(\r;\f;\u000b ). Also, we have\n\u00010\ni;S1(\u000b;\f;\r ) = \u00010\ni;S2(\r;\f;\u000b ). From Eq. (39), we note\nthat in the weak \u000b;\rlimit, the junction becomes stable in\nthe\r >\u000b region and the TDOS shows that enhancement\nappears in the \f\u000b>\r region, which is impossible to sat-isfy simultaneously. Similar to the S1\fxed point, we do\nnot expect to see simultaneous TDOS enhancement and\nstability of the junction in this case, as the role of \u000band\n\rgets exchanged but the region of dS2>1 and \u00010\ni;S2<1\nstill remains mutually exclusive.\nNow, we break the parity symmetry or layer symmetry\nof the junction by introducing asymmetric \u000bin the two\nQH layers as we did for the S1\fxed point to investigate\nthe possibility of having dS2>1 and \u00010\nS2<1 simulta-\nneously. We run a numerical search to check the possi-\nbility of simultaneous stability and TDOS enhancement\nin the presence of asymmetric \u000bin the case of both\n\u00171= 1;\u00172= 1 and\u00171= 1;\u00172= 1=3, and the results\nof our \fndings are given in Figs. 4(a) and 4(c). In both\ncases we again \fnd the region of simultaneous stability\nand TDOS enhancement, but it exists only in the strong\ninteraction limit.11\nVII. DISCUSSION AND CONCLUSIONS\nBoth the bulk and boundary of an isolated LL wire\nshow suppression of TDOS for LL parameter K < 1,\ni.e., the repulsive interelectron interaction limit?. The\nminimal modi\fcations which could be added to the\nLL model such that it leads to a deviation for the stan-\ndard paradigm of TDOS suppression are (i) formation\nof a junction of multiple LLs and (ii) switching on non-\nlocal density-density interaction in addition to the lo-\ncal ones. Introducing exotic quantum impurity into the\nLL68could also lead to TDOS enhancement, but such\na scenario is not the focus of this paper. The junction\nof LL wires is a well-studied subject both theoretically\nand experimentally, but the physical setting for motivat-\ning a nonlocal density-density interaction is not obvious.\nThis leads us to consider the bilayer quantum Hall sys-\ntem, which can naturally host such a model. In par-\nticular, a bilayer quantum Hall line junction69could be\na possibility which allows all the four edge states par-\nticipating at the junction (two from the top layer and\ntwo from the bottom layer) to come in the close vicin-\nity of each other hence leading to mutual interactions\nbetween them. Such a system has been in discussion\nrecently owing to the possibility of being a host to lo-\ncalized parafermion zero modes69{72. Also, there exists\na long history in the experimental realization of bilayer\nquantum Hall systems58{61,73,74. Additionally, there has\nbeen signi\fcant experimental progress also in realizing\ngraphene bilayer quantum Hall systems75{80. This exper-\nimental progress indicates that the technology required\nfor designing the proposed setup may not be a far-fetched\none.\nIn general, it is di\u000ecult to \fnd a \fxed point for the\nLL system which leads to TDOS enhancement at the\njunction of LL s and is also stable (in the RG sense)\nagainst perturbations that could be switched on at\nthe junction. This is obvious as an enhancement of\nTDOS naturally implies the presence of relevant pertur-\nbations involving tunneling of electrons at the junction\nwhich could destabilize the junction \fxed point. An im-\nportant realization in this paper was the fact that simul-\ntaneous enhancement of TDOS and RG stability of the\njunction \fxed point is a possibility provided we break the\nlayer symmetry either by having di\u000berent \flling fractions\non the two layers or by choosing a di\u000berent strength forthe two intralayer interaction parameters ( \u000b).\nFurthermore, we would like to point out that the oc-\ncurrence of processes analogous to Andreev re\rection at\nthe junction of LL s does not seem to provide a litmus\ntest for the presence of TDOS enhancement in general\nthough such a connection was observed in Ref. [45] in\nthe context of a junction of three LL s for a repulsive\ninterelectron interaction parameter regime. An invalida-\ntion of such an identi\fcation was demonstrated explicitly\nwhen we considered the S2\fxed point between the \u0017= 1\nand\u0017= 1=3 edge, which supports a process analogous to\nAndreev re\rection at the junction owing to the fact that\nthe quasiparticles on the \u0017= 1=3 edge have fractional\ncharge as opposed to the electron-like quasiparticles on\nthe\u0017= 1 edge. Furthermore, we note that the junc-\ntion of two chiral LL s (not three) is enough to show\nTDOS enhancement provided we switch on interaction\n(like\fand\r) in addition to the routinely considered\ninteraction parameter \u000b.\nLastly, we would like to point out that here we have\ntaken the interaction parameters to be independent of\neach other, which in a general QH setup need not be\ntrue. Codependencies of interaction parameters can be\naccounted for through a distributed circuit model intro-\nduced in Ref. 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Fujisawa Nat. Nanotechnol. 9, 177{181 (2014); M.\nHashisaka, H. Kamata, N. Kumada, K. Washio, R. Mu-\nrata, K. Muraki, and T. Fujisawa, Phys. Rev. B 88, 235409\n(2013); M. Hashisaka and T. Fujisawa, Rev. Phys. 3, 32\n(2018).14\nAppendix A: Tunneling Density of states (TDOS)\nThe local electron TDOS for a chiral outgoing QH edge\nof a 2\u00022 QH edge junction with \flling fraction \u0017iat a\npointxfrom the junction is given by\n\u001ai(E) = 2\u0019X\nnjhnj y\niO(x)j0ij2\u000e(En\u0000E0\u0000E)\n=Z1\n\u00001h0j iO(x;t) y\niO(x;0)j0ie\u0000iEtdt(A1)\nThe fermionic \feld  iI=O denotes the incom-\ning/outgoing chiral edge with \flling fraction \u0017iand can\nbe expressed in terms of the bosonic \feld \u001eiI=O as\n iI=O\u0018Fie\u0013\u001eiI=O=\u0017i, whereFiis the Klein factor. Then\nthe TDOS is given by\n\u001ai(E)\u0018Z1\n\u00001dth0jei\u001eiO(x;t)\n\u0017ie\u0000i\u001eiO(x;0)\n\u0017ij0ie\u0000iEt(A2)\nLet\u0016\u001eO=\u0000\u0016\u001e1O;\u0016\u001e2O\u0001\nand \u0016\u001eI=\u0000\u0016\u001e1I;\u0016\u001e2I\u0001\n. The free\nBogoliubov \felds ~\u001eare related to the interacting \u0016\u001eby\ntheXmatrix, which can now be decomposed as follows:\n\u0012\u0016\u001eO\u0016\u001eI\u0013\n(x;t)=\u0012\nX1X2\nX3X4\u0013\u0012~\u001eO\n~\u001eI\u0013\n(x;t)(A3)\nThe QPC of the 2 \u00022 QH edge system can be ac-\ncounted for by a current splitting matrix at the junction\nwhich relates the incoming interacting bosonic \felds to\nthe outgoing interacting bosonic \felds, such that\n\u0012\n\u001e1O\n\u001e2O\u0013\n(x=0)=S\u0012\n\u001e1I\n\u001e2I\u0013\n(x=0)(A4)\nwhereSdenotes the current splitting matrix given by the\ntwo possible \fxed points, namely, the S1andS2\fxed\npoints.\n\u0012\u0016\u001e1O\u0016\u001e2O\u0013\n(x=0)=M\u00001SM\u0012\u0016\u001e1I\u0016\u001e2I\u0013\n(x=0)=\u0016S\u0012\u0016\u001e1L\u0016\u001e2L\u0013\n(x=0)\n(A5)\nwhereMij=p\u0017i\u000eij. Then the real interacting bosonic\n\felds\u001eOand\u001eIcan be expressed only in terms of the\nleft-moving Bogoliubov \feld ~\u001eI( which are independent\nof each other) as follows:\n\u001eO(x;t) =Mh\nT1~\u001eI(\u0000x;t) +T2~\u001eI(x;t)i\n; (A6)\n\u001eI(x;t) =Mh\nT3~\u001eI(\u0000x;t) +T4~\u001eI(x;t)i\n; (A7)\nwhere\nT1=X1\u0000\nX1\u0000\u0016SX3\u0001\u00001\u0000\u0016SX4\u0000X2\u0001\n; (A8)\nT2=X2; (A9)\nT3=X3\u0000\nX1\u0000\u0016SX3\u0001\u00001\u0000\u0016SX4\u0000X2\u0001\n;(A10)\nT4=X4: (A11)Then\nh0j iO(x;t) y\niO(x;0)j0i \u00182Y\nj=1\u0012i\u000b\n\u0000~vjt+i\u000b\u0013\rij\n\u0002\u0012(i\u000b)2\u00004x2\n(i\u000b\u0000~vjt)2\u00004x2\u0013\u0010ij\n;\n(A12)\nwhere\u000bis the short-distance cuto\u000b, \rij=[T1]2\nij+[T2]2\nij\n\u0017i,\nand\u0010ij=[T1]ij[T2]ij\n\u0017i. Now we calculate the TDOS in\ntwo limits, namely, \frst at the junction with x\u0000!0, in\nwhich case Eq. (A12) becomes\nh0j iO(x;t) y\niO(x;0)j0i\u00182Y\nj=1\u0012i\u000b\n\u0000~vjt+i\u000b\u0013\rij+2\u0010ij\n;\n(A13)\nand the other far from the junction with x\u0000!1 , in\nwhich case Eq. (A12) becomes\nh0j iO(x;t) y\niO(x;0)j0i\u00182Y\nj=1\u0012i\u000b\n\u0000~vjt+i\u000b\u0013\rij\n:(A14)\nNow from Eqs. (A1), (A13), and (A14), we have the\nTDOS integral in the two limits of the form\nZ1\n\u000012Y\nj=1\u0012i\u000b\n\u0000~vjt+i\u000b\u0013\u0001ij\ne\u0000iEtdt/E(\u0001i\u00001);(A15)\nwhere the TDOS exponent \u0001 iis given by\n\u00010\ni=1\n\u0017i2X\nj=1\u0010\n[T1]ij+ [T2]ij\u00112\n(A16)\nand, far from the junction,\n\u00011\ni=1\n\u0017i2X\nj=1\u0010\n[T1]2\nij+ [T2]2\nij\u0011\n: (A17)\nAppendix B: Inter-dependency of Interaction\nParameters\nLet us consider the simplest possible case of two quan-\ntum Hall (QH) systems with \flling fraction \u00171=\u00172= 1\nin bilayer stacking, with two incoming and two outgoing\nedges. Here, we use a simple approach to account for\nthe e\u000bect of Coulomb interactions between the edges in\nterms of a distributed circuit model81. The dynamics of\nedge plasmons traveling along a single edge channel is\nmodeled through the distributed electrochemical capaci-\ntance per unit length between the channel and the ground\n(denoted by Cch-channel capacitance). The interaction\nbetween the two di\u000berent channels is modeled with dis-\ntributed elements, which is expressed by the interedge\ncapacitance.15\nInteredge capacitance per unit length between \u001eiIand\n\u001eiOof the same QH layer is given by C\u000b, between\u001eiI\nand\u001ejIof di\u000berent QH layers ( i6=j) is given by C\f,\nand between \u001eiIand\u001ejOof di\u000berent QH layers is given\nbyC\r. Channel capacitance per unit length C(i)\nch=Cch\nfor all the edges ( i=f1;4g) is taken to be the same (as\nthe Fermi velocity for each edge plasmon is taken to be\nthe same81). Let\u001aiO=I(x;t),ViO=I(x;t), andIiO=I(x;t)\nbe the excess charge density, potential, and current \row-\ning through the out/in edge channel of the ith QH\nlayer, respectively, at position xand timet. The rela-\ntion between the current Ii(x;t) = (I1O;I2O;I1I;I2I)T\nand potential Vi(x;t) = (V1O;V2O;V1I;V2I)Tis given by,\nIiO(x;t) =\u001b(i)\nxyViO(x;t) andIiI(x;t) =\u0000\u001b(i)\nxyViI(x;t).\nThe excess charge density \u001aiI=O is related to the po-\ntential through the matrix CTgiven by\n0\nB@\u001a1O\n\u001a2O\n\u001a1I\n\u001a2I1\nCA\n(x;t)=0\nB@Cp\u0000C\f\u0000C\u000b\u0000C\r\n\u0000C\fCp\u0000C\r\u0000C\u000b\n\u0000C\u000b\u0000C\rCp\u0000C\f\n\u0000C\r\u0000C\u000b\u0000C\fCp1\nCA0\nB@V1O\nV2O\nV1I\nV2I1\nCA\n(x;t)\n(B1)\nwhich in compacted form can be written as \u001a(x;t) =\nCTV(x;t).Cpis given byCp=Cch+C\u000b+C\f+C\r. The\nHeisenberg equation of motion for the coupled system\n[Eq. 8 of the main text] is given by\nd\ndt\u001ea(x;t) =\u0000vf\u000fa4X\n\u000b=1Ka\u000bd\ndx\u001e\u000b(x;t): (B2)\nSince\u001aiI=O=\u0006d\ndx\u001eiI=O andIiI=O=\u0007d\ndt\u001eiI=O, we have\nd\ndtIa(x;t) =\u0000vf4X\n\u000b=1Ka\u000bd\ndxI\u000b(x;t);\nd\ndtI(x;t) =\u0000Ud\ndxI(x;t); (B3)whereUis a 4\u00024 matrix given by\nU=vf0\nB@1\f \u000b \r\n\f1\r \u000b\n\u0000\u000b \r\u00001\u0000\f\n\u0000\r\u0000\u000b\u0000\f\u000011\nCA: (B4)\nNow, using the continuity equation @t\u001ai;I=O +\n@xIi;I=O = 0, Eq. (B1), and the relation IiO=I(x;t) =\n\u0006\u001b(i)\nxyViO=I(x;t), we get\nU=\u001bxyC\u00001\nT; (B5)\nwhere\u001bxy is a diagonal 4 \u00024 matrix with\n(\u001b(1)\nxy;\u001b(2)\nxy;\u0000\u001b(1)\nxy;\u0000\u001b(2)\nxy) being the diagonal elements.\nWe can now express interaction parameters in terms of\ncapacitance as follows:\nvf=1\nCch+1\nCch+ 2 (C\u000b+C\f)+1\nCch+ 2 (C\u000b+C\r)\n+1\nCch+ 2 (C\f+C\r)\n\u000b=1\nCch\u00001\nCch+2(C\u000b+C\f)\u00001\nCch+2(C\u000b+C\r)+1\nCch+2(C\f+C\r)\n1\nCch+1\nCch+2(C\u000b+C\f)+1\nCch+2(C\u000b+C\r)+1\nCch+2(C\f+C\r)\n\f=1\nCch\u00001\nCch+2(C\u000b+C\f)+1\nCch+2(C\u000b+C\r)\u00001\nCch+2(C\f+C\r)\n1\nCch+1\nCch+2(C\u000b+C\f)+1\nCch+2(C\u000b+C\r)+1\nCch+2(C\f+C\r)\n\r=1\nCch+1\nCch+2(C\u000b+C\f)\u00001\nCch+2(C\u000b+C\r)\u00001\nCch+2(C\f+C\r)\n1\nCch+1\nCch+2(C\u000b+C\f)+1\nCch+2(C\u000b+C\r)+1\nCch+2(C\f+C\r)\n(B6)\nAs can be seen from the above equation (B6), interac-\ntions between the edges, in general, cannot be treated as\nindependent parameters in a realistic situation."    },    {        "title": "2102.03841v1.Positive_energy_density_leads_to_no_squeezing.pdf",        "content": "arXiv:2102.03841v1  [quant-ph]  7 Feb 2021Positive energy density leads to no squeezing\nS. Kannan1,∗and C. Sudheesh1,†\n1Department of Physics, Indian Institute of Space Science an d Technology, Thiruvananthapuram, 695 547, India.\n(Dated: February 9, 2021)\nWe consider two kinds of superpositions of squeezed states o f light. In the case of superpositions\nof first kind, the squeezing and all higher order squeezing va nishes. However, in the case of the\nsecond kind, it is possible to achieve a maximum amount of squ eezing by adjusting the parameters\nin the superposition. The emergence and vanishing of squeez ing for the superposition states are\nexplained on the basis of expectation values of the energy de nsity. We show that expectation values\nof energy density of quantum states which show no squeezing w ill be always positive and that of\nsqueezed states will be negative for some values of spacetim e-dependent phase.\nI. INTRODUCTION\nSqueezedstatesareaclassofnonclassicalstatesoflight\nwhichhavelessfluctuationsinonequadraturephasethan\na coherent state at the expense of increased fluctuations\nin the other [1–4]. These states offer intriguing appli-\ncations such as in optical communication systems [5, 6],\nquantum sensing [7], noise-free amplification [8], gravita-\ntional wave detection [9–11] etc. Recently the increase\nin sensitivity using squeezed light in LIGO and VIRGO\ndetectors has been reported [12, 13]. A rapid increase\nin the use of squeezed states in the continuous variable\nquantum information processing is also seen [14]. Engi-\nneering interactions between electric dipoles utilizing the\nantisqueezed quadrature of a squeezed vacuum state has\nbeen predicted recently. This does not require any pho-\ntonic structures [15]. As many such theoretical and prac-\ntical implications exist, investigations of squeezing prop-\nerties of various quantum states is still a central topic in\nquantum optics.\nAs pointed out by Dirac, the linear superposition prin-\nciple is one of the most fundamental features of quantum\nmechanics. It has been shown that quantum superpo-\nsition can give rise to various nonclassical effects such\nas squeezing, higher-order squeezing [16], sub-Poissonian\nstatistics, etc. Superposition of two coherent states [17],\nsuperposition leading to even and odd squeezed coherent\nstates [18, 19], superposition of squeezed coherent states\n[20], binomial state [21, 22], intermediate number-phase\nstate [23] and generalized superposition of two squeezed\nstates [24] are some examples for the nonclassical states.\nDespite the fact that there are a lot of studies in the\nliterature which talk about squeezing due to superposi-\ntion of states but there is no work which clearly states\nwhat type of superpositions leads to squeezing and its\nconnection with any other physical parameter of the sys-\ntem. However, there are some attempts has been made\nin the literature [17, 25] to address the aforementioned\nquestions. In this paper, we analyze different types of\nsuperposition states to find what type of superpositions\n∗Present address: ISRO Inertial Systems Unit, Thiruvananth apu-\nram, 695 013, India.; kns.phys@gmail.com\n†sudheesh@iist.ac.ingive rise to squeezing, and to find a relation between\nsqueezing and expectation value of the energy density.\nAnothermotivationtodothisstudyistodeviseamethod\nto get higher or no squeezing when we add two squeezed\nstates. Thesetwoanalyseswill definitely help researchers\nworking in quantum communication and quantum sens-\ning using squeezed light to choose proper squeezed states\nfor their study. For this purpose, we consider the super-\nposition of various squeezed states such as the squeezed\nvacuum states [3], photon-added coherent states [26] etc.\nWe investigate both the nonclassical and statistical fea-\nturesofthestatesandidentifytheroleofsuperpositionin\nmodifying the quantum fluctuations in the field quadra-\ntures.\nThe paper is organized as follows: in Section II we\nstudy the squeezing and higher-order squeezing of su-\nperposition of various squeezed states, vanishing and en-\nhancing of squeezing for different superposition is seen.\nIn Section III expectation value of energy density is used\nto explain the squeezing properties shown by the super-\nposition states and Section IV is devoted to conclusion.\nII. SUPERPOSITION OF SQUEEZED STATES\nAND ITS SQUEEZING\nSqueezing can be quantified by analyzing the uncer-\ntainties in the quadrature operators. For a single-mode\nscenario, the quadrature operators are defined as\nˆX= (ˆa+ˆa†)/2,ˆP= (ˆa−ˆa†)/2i. (1)\nThey satisfy the commutation relation\n[ˆX,ˆP] =i/2 (2)\nand hence the uncertainty relation\n/angb∇acketleft(∆ˆX)2/angb∇acket∇ight/angb∇acketleft(∆ˆP)2/angb∇acket∇ight ≥1/16. (3)\n(We have set /planckover2pi1= 1.) A given state is said to be quadra-\nture squeezed if the variance in one of the quadrature\nfalls below the vacuum value:\n/angb∇acketleft(∆ˆX)2/angb∇acket∇ight<1/4 or/angb∇acketleft(∆ˆP)2/angb∇acket∇ight<1/4. (4)\nHere we only consider the superposition of the same class\nof quadraturesqueezed states. We can classify the super-\npositionofthesetypesofstatesintotwo: (1)allthestates2\nin the superposition have the same squeezing value, (2) a\ngeneralized superposition where the states have different\nsqueezing values. The first kind of superposition can be\nexpressed as\n|Ψ/angb∇acket∇ightl=Nll−1/summationdisplay\nj=0/vextendsingle/vextendsingle/vextendsingleΦei2πj/l/angbracketrightBig\n, (5)\nwhereNlis the normalization constant and |Φ/angb∇acket∇ightcan be\nany squeezed state of the same class. Here, it is always\npossible to find an observable which can turn one of the\nstates in the superposition to another one in the super-\nposition. The second kind, the generalized superposition\nis of the form:\n|Ψ/angb∇acket∇ightgen=Ngenl−1/summationdisplay\nj=0aj|Φj/angb∇acket∇ight. (6)\nHereNgendenotes the normalization factor, |Φj/angb∇acket∇ightrepre-\nsents squeezed states of the same class, and ajthe corre-\nsponding weight factors.\nLet us now analyze the quadrature variance of super-\npositions of some important squeezed states.\nA. Superpositions of the first kind\n1. Squeezed vacuum state\nThe squeezed vacuum state |ξ/angb∇acket∇ightis generated by the ac-\ntionofsqueezingoperator ˆS(ξ)[2–4]onthevacuumstate:\n|ξ/angb∇acket∇ight=ˆS(ξ)|0/angb∇acket∇ight, (7)\nwhereξ=reiθwith realr(squeezing parameter) and\n0≤θ≤2π. Experimental realization of this state was\nachieved way back in 1985-1986 [27, 28]. In the recent\nyears, great progress in the generation of this state have\nbeen achieved [29–32] that could significantly improve\nquantum information applications.\nIn the Fock basis, |ξ/angb∇acket∇ightcan be expanded as\n|ξ/angb∇acket∇ight=1/radicalbig\ncosh(r)∞/summationdisplay\nm=0/parenleftbig\n−eiθtanh(r)/parenrightbigm/radicalbig\n(2m)!\n2mm!|2m/angb∇acket∇ight.\n(8)\nFor this state, the variance in the both quadratures come\nout to be [33]\n/angb∇acketleft(∆ˆX)2/angb∇acket∇ight=1\n4e−2rand/angb∇acketleft(∆ˆP)2/angb∇acket∇ight=1\n4e2r,(9)\nand, hence, there is a squeezing in the ˆX-quadrature.\nNow, considera superpositionof lsqueezedvacuum state\n|ζ/angb∇acket∇ightlof the first kind:\n|ζ/angb∇acket∇ightl=Nsl−1/summationdisplay\nj=0/vextendsingle/vextendsingle/vextendsingleξei2πj/l/angbracketrightBig\n, (10)whereNlis the normalization constant and is equal to\nNl=/parenleftBigg∞/summationdisplay\nm=0(2lm)!\n22lm(lm)!2(tanh(r))2lm/parenrightBigg−1/2\n.(11)\nIn the Fock basis,\n|ζ/angb∇acket∇ightl=Nl∞/summationdisplay\nm=0/parenleftbig\n−eiθtanh(r)/parenrightbiglm/radicalbig\n(2lm)!\n2lm(lm)!|2lm/angb∇acket∇ight.(12)\nTheabovesuperposedstatescanbeproducedbyutilizing\nthetechniquesin[34]. Wefindthatforthestates |ζ/angb∇acket∇ightl≥2,\n/angb∇acketleftˆa/angb∇acket∇ight=/angb∇acketleftˆa†/angb∇acket∇ight= 0 (13)\nand\n/angb∇acketleftˆa2/angb∇acket∇ight=/angb∇acketleftˆa†2/angb∇acket∇ight= 0. (14)\nSince the aboveaboveexpectation values arezero, we get\n/angb∇acketleft(∆ˆX)2/angb∇acket∇ight=/angb∇acketleft(∆ˆP)2/angb∇acket∇ight=1\n4+N2\nl\n2∞/summationdisplay\nm=0(tanh(r))2lm2lm(2lm)!\n22lm(lm)!2.\n(15)\nSince the RHS of the above equation is always greater\nthan 1/4, there is no squeezing for |ζ/angb∇acket∇ightl≥2in both the\nquadratures. It is one of the important results of this pa-\nper. To check whether this is true for any superposition\nof squeezed states of first kind, we examine superposi-\ntions of other types of squeezed states in the following\nsubsections.\n2. Photon-added coherent state\nm-photon-addedcoherentstates(m-PACS) wereintro-\nduced as the states which are intermediate between the\nFock states and the coherent states. The Fock state rep-\nresentation of m-PACS is\n|α,m/angb∇acket∇ight=e−|α|2/2\n/radicalBig\nLm(−|α|2)m!∞/summationdisplay\nn=0αn\nn!/radicalbig\n(n+m)!|n+m/angb∇acket∇ight,\n(16)\nwhereLmis the Laguerre polynomial of order m,α(co-\nherent parameter) is a complex number and mis an in-\nteger. These states show rich nonclassical properties and\nexhibit quadraturesqueezing for a wide range of αvalues\n[26]. Experimentalgenerationofthe singlephoton-added\ncoherent state has been achieved [35], and there also ex-\nists schemes for the generation of higher-order photon-\nadded coherent states [36, 37]. Consider the lsuperposi-\ntion of the first kind of m-PACS:\n|Γ,m/angb∇acket∇ightl=Nml−1/summationdisplay\nj=0/vextendsingle/vextendsingle/vextendsingleαei2πj/l,m/angbracketrightBig\n, (17)\nwhereNmis the normalization constant. Since we are\nexamining only superposition of squeezed states, we con-\nsider range of αfor which the state |α,m/angb∇acket∇ightshows quadra-\nture squeezing. We then calculate the quadrature vari-\nance for the state |Γ,m/angb∇acket∇ightlfor different mandlvalues.3\nOur calculations show that the quadrature squeezing is\nalso absent for all mvalues for the state |Γ,m/angb∇acket∇ightl≥2. We\nconfirm our result also for superposition of two-mode\nsqueezed states.\n3. Two-mode squeezed vacuum state\nQuadrature squeezing is observed in multi-mode sce-\nnarios also. The most common one is the two-mode\nsqueezed vacuum state |ξ/angb∇acket∇ight2. It is generated by the ac-\ntion of a two-mode squeeze operator ˆS2(ξ) [38] on the\nvacuum state:\n|ξ/angb∇acket∇ight2=ˆS2(ξ)|0,0/angb∇acket∇ight\n=1\ncosh(r)∞/summationdisplay\nn=0/parenleftbig\n−eiθtanh(r)/parenrightbign|n,n/angb∇acket∇ight,(18)\nwhereξ=reiθ. Since there is a correlation between\nthe modes, squeezing exist in the effective quadrature\noperators:\nˆX1= (ˆa+ˆa†+ˆb+ˆb†)/23/2,ˆX2= (ˆa−ˆa†+ˆb−ˆb†)/i23/2.\n(19)\nHere also we consider the first kind of superposition and\nvarianceintheabovequadraturesoperatorsareanalyzed.\nFor the two superposition state\nN(|ξ/angb∇acket∇ight2+|−ξ/angb∇acket∇ight2) =2N\ncosh(r)∞/summationdisplay\nn=0einθ(tanh(r))2n|2n,2n/angb∇acket∇ight,\n(20)\nwith normalization constant N, the variance in the\nquadratures ˆX1andˆX2comes out to be the same and is\nequal to\n∆ˆX2\n1= ∆ˆX2\n2=1\n4/parenleftBigg\n1+/parenleftbigg2N\ncosh(r)/parenrightbigg2∞/summationdisplay\nn=04n(tanh(r))4n/parenrightBigg\n.\n(21)\nSo there is no squeezing in the superposition state and\nwe confirm also the absence of quadrature squeezing for\nall higher numbers (i.e., l >2) of superpositions. In\nthe above three cases, we have shown that no first-order\nsqueezing exists when we take superposition of the first\nkind with squeezed states. In the next section, we extend\nour analysis for higher-order squeezing of superposition\nof squeezed states.\nWe have also calculated both the Hong and Mandel\n[16] and Hillery type [39] higher order squeezing for the\nsuperpositionofsqueezingstates. Ourcalculationsshows\nthat there is no higher-order squeezing exists for the su-\nperposition of the states which we have considered ear-\nlier. In this section, we have shown that superposition of\nsqueezed states can lead to complete vanishing of squeez-\ning properties. This is quite unexpected, since usually\nthe superposition of wavepackets give rise to squeezing\nproperties (e.g. even-cat state [40]).B. Generalized superposition\nWe have defined the generalized superposition state\n|Ψ/angb∇acket∇ightgenin Eq. 6 as the superposition of squeezed states\nwith different squeezing value and weight factors. Here,\nwe analyze the variance and higher-ordermoments in the\nquadrature of |Ψ/angb∇acket∇ightgenfor squeezed vacuum states:\n|Ψ/angb∇acket∇ightgen=Ngenl−1/summationdisplay\nj=0aj|ξj/angb∇acket∇ight, (22)\nwhereξj=rjeiθj,aj’s are the weight factors and the\nnormalization constant\nNgen=\nl/summationdisplay\ni=1l/summationdisplay\nj=1aia∗\nj1/radicalbig\ncosh(ri−rj)\n−1/2\n.(23)\nThe squeezing and statistical properties of the two su-\nperposition cases have been discussed by Barbosa et al.\nin [24]. Here, we also find squeezing properties of more\nthan two superpositions and in turn find suitable weight\nfactors to achieve maximum quadrature squeezing. The\nvariance in the ˆXquadrature for the state given in Eq.\n22 comes out to be\n/angb∇acketleft(∆ˆX)2/angb∇acket∇ight=1\n4+N2\ngen\n4/bracketleftBiggl/summationdisplay\ni=1l/summationdisplay\nj=1∞/summationdisplay\nn=0aiaj(tanh(ri)tanh(rj))n\n/radicalbig\ncosh(ri)cosh(rj)\n×(2n)!\n22n(n!)2/parenleftBigg\n4n−(2n+1)/parenleftBig\ntanh(ri)+tanh(rj)/parenrightBig/parenrightBigg/bracketrightBigg\n.\n(24)\nWithout lossofgenerality,we takerealvaluesfor ξj’sand\naj’s. Usingaminimizationalgorithm[41], theweightfac-\ntors (ai’s) are calculated to achieve the highest squeez-\ning. For fixed squeezing parameters, the weight factors\nare computed such that the variance attains the min-\nimum possible value. We found that the state |Ψ/angb∇acket∇ightgen\ncan have quadrature variance less than1\n4e−2R, where\nR= max(rj). The minimum value of variance obtained\nfor different numbers of superpositions along with the\ncorresponding weight factor is tabulated below.\nIt is clear from the table that it is possible to increase\nthesqueezingbyincreasingthesuperpositioninthestate.\nWe havealsostudied the higher-ordersqueezingthe state\nand found that the state exhibits higher-order squeezing.\nFor example, the state |Ψ/angb∇acket∇ightgenwithl= 3, the fourth and\nsixth-order squeezing comes out to be /angb∇acketleft(∆X)4/angb∇acket∇ight= 0.0068\nand/angb∇acketleft(∆X)6/angb∇acket∇ight= 0.1857, respectively.\nIn this section, we have shown that the two different\ntypesofsuperpositionsgivetwocontrastingresults: com-\nplete vanishing of squeezing to all order and a greater\namount of squeezing by adjusting the weight factors in\nthe superposition. This enables us to control the squeez-\ning by adjusting the parameters in the superpositions\nwhich will have application in quantum information pro-\ncessing. More important than this, the above result4\nTABLE I.\nS/NSqueezing parameters Variance Weight factors\n(rj) /angbracketleft(∆ˆX)2/angbracketright(aj)\n1l= 1,\nr1= 10.0338 a1= 1\n2l= 2,\nr1= 0.5,r2= 10.0268a1=−0.32678,\na2= 1\n3l= 3,\nr1= 0.5,r2= 0.8,r3= 10.0188a1=−0.2317,\na2=−1.0103,\na3= 1\n4l= 4,\nr1= 0.5,r2= 0.7,\nr3= 0.8,r4= 10.0151a1=−0.3464,\na2= 2.4050,\na3=−2.9717,\na4= 1\nleads us to find a connection between energy density and\nsqueezing which we describe in detail in the next section.\nIII. SQUEEZING AND ENERGY DENSITY\nAn obvious question that arises now is, why is the\nsqueezing in some superposition state vanishing? Or why\nis it increasing in certain cases? It is well known that\nsome superposition of coherent state, such as even cat\nstate [40] and Yurke-Stoler states [42] exhibit quadra-\nture squeezing while coherent state doesn’t have such a\nnonclassical property. In this section, we explain these\nproperties based on the expectation value of the energy\ndensity of these quantum states.\nIt is known that all forms of classical matter have non-\nnegative energy density but this is not the case for a gen-\neral quantum state. A superposition of number eigen-\nstates can give a negative expectation value of energy\ndensity in certain spacetime regions due to coherence ef-\nfect. To find the expectation value of the energy density,\nwe calculate the expectation value of the stress-energy\ntensor or energy-momentum tensor [43]. Stress-energy\ntensor describes the density and flux of energy and mo-\nmentum in spacetime. In order to calculate this expec-\ntation value for the stress-energy tensor, the method de-\nscribed in reference [44] is followed. Consider a massless\nand minimally coupled scalar field. Its Lagrangian den-\nsity is given by\nL=1\n2ηµν(∂µφ)(∂νφ), (25)\nwhere the spacelike metric ηµν= diag[-1,1,1,1] and φis\nthe quantum field operator which satisfies the dynamical\nequation\n/parenleftbigg\n−∂2\n∂t2+∇2/parenrightbigg\nφ(x)≡∂µ∂µφ(x) = 0.(26)\nWe have taken c=ℏ= 1.The quantum field operator φ\ncan also be expanded in mode functions as\nφ=/summationdisplay\nk(akfk+a†\nkf∗\nk), (27)whereaksatisfy the usual bosonic commutation relation\nand the mode function fkis given by\nfk= (2L3ω)−1/2eik.x−iωt. (28)\nHere we have assumed periodic boundary condition in a\nthree-dimensional box of side L and ωis the frequency\nof the mode and kthe wave number. For our purpose,\nwe consider the stress tensor corresponding to the single\nmode excitation:\nTµν= (∂µφ)(∂νφ)−1\n2ηµν(∂σφ)(∂σφ).(29)\nWe calculate the normal ordered (same as subtracting\nthe vacuum expectation value) expectation value of this\nstress tensor for the quantum states of our interest. We\nwill be only looking at the expectation value of the tem-\nporal component ( /angb∇acketleft:T00:/angb∇acket∇ight). For coherent state |α/angb∇acket∇ight, this\nnormal ordered expectation value comes out to be\n/angb∇acketleft:T00:/angb∇acket∇ight= 2K00α2(1−cos(2θ)), (30)\nwhereK00=(k0)2\n2ωL3and the spacetime-dependent phase\nθ=kρxρ. Without loss of generality, the above expres-\nsion is derived for real values of α. It is easy to see that\nfor coherent states, /angb∇acketleft:T00:/angb∇acket∇ight ≥0 and increases with in-\ncrease inαvalue for all values of θ. For even cat state\n|ψ/angb∇acket∇ight=2/radicalBig\n2(1+e−2|α|2)∞/summationdisplay\nn=0α2n\n/radicalbig\n(2n)!|2n/angb∇acket∇ight,(31)\nwith realα, the energy density comes out to be\n/angb∇acketleft:T00:/angb∇acket∇ight= 2K00α2tanh(α2)/parenleftBig\n1−coth(α2)cos(2θ)/parenrightBig\n.(32)\nFor fixed values of α, the energy density becomes neg-\native for some values of θin [0,2π]. Thus for the even\ncat state, which showsquadraturesqueezing, energyden-\nsity becomes negative for certain values of spacetime-\ndependent phase θ. In contrast, we have found earlier\nthat a coherent state, which shows no squeezing, never\ngives negative energy density for any values of θ. To\ncheck that whether the above result is a general result\napplicable to all quantum states, we carry out the above\nanalysis to other important class of states and superpo-\nsition states considered in the Section II.\nThe energy density for odd cat state, which shows no\nsqueezing, is always positive. For the squeezed vacuum\nstate defined in Eq. 8the average energy density\n/angb∇acketleft:T00:/angb∇acket∇ight= 2K00sinh(r)/parenleftBig\ncosh(r)cos(2 θ)+sinh(r)/parenrightBig\n.(33)\nAgain, we have considered real values for ξ(=r) without\nloss of generality. Here, the energy density can go below\nzero for certain values of θfor a given rand for larger\nvalues ofrthe energy density tends to zero. This can be\nassociated with larger oscillations in the photon number\ndistribution [45]. For the superposition state of first kind5\ngiven in Eq. 10 with l= 2, which shows no squeezing,\nthe average energy density\n/angb∇acketleft:T00:/angb∇acket∇ight=16K00sech(r)/parenleftBig\n1+/radicalbig\nsech(2r)/parenrightBig/parenleftBig∞/summationdisplay\nn=0n(4n)!\n(2n)!2(tanh(r)/2)2n/parenrightBig\n(34)\nis always greater than zero. Higher order superposition\nwill also give rise to a positive average energy density.\nOur results are also verified for PACSs (Eq. 16) and\ntheir superpositions. Finally for the generalized states\ngiven in TABLE. I, the expectation value for the stress\nenergy tensor is found to have negative values.\nOne of the important results from this section can be\nstated as follows: The expectation value of energy den-\nsity of quantum states which show no squeezing will be\nalwayspositive and that of squeezed states will have neg-\native for some values of spacetime-dependent phase. Our\nanalysis also gives the reason for absence of squeezing in\nsuperposition states of first kind and presence of squeez-\ning in superposition of states of second kind in terms of\nexpectation value of the energy density.IV. CONCLUSION\nWe have studied the squeezing properties of arbitrary\nnumbersofsuperpositionsofvarioussqueezedstatessuch\nas squeezed vacuum state, photon-added coherent states\nand two mode squeezed vacuum state. We have con-\nsidered two kinds of superpositions: first kind and sec-\nond kind. In the case of first kind, the superpositions of\nsqueezed states doesn’t show both squeezing and higher-\norder squeezing of all orders. This is found to be true\nfor any state which has quadrature squeezing and multi-\nmode squeezed states. However, in the case of the super-\npositions of second kind (also called as generalized su-\nperpositions), it has been shown that the superposition\nstatesshowlargeamountsofquadratureandhigher-order\nsqueezing. Thisisachievedbychoosingtheproperweight\nfactors in the superpositions; this method also enables us\nto have a control overthe amount of squeezing produced.\nThe vanishing and appearance of squeezing in super-\nposition state is explained on the basis of expectation\nvalues of energy density or stress-energy tensor. States\nwith squeezing are shown to have a negative expecta-\ntion value for the stress-energy tensor for some values of\nspacetime-dependent phase. 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Ganose2, Max Dylla1, Shashwat Anand1, Junsoo Park2,\nMadison K. Brod1, Jason Munro3, Kristin A. Persson4;5, Anubhav Jain2, and G. Jeffrey Snyder1\n1Department of Materials Science and Engineering, Northwestern University, Evanston, IL 60208, USA.\n2Energy Technologies Area, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA.\n3The Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA.\n4Department of Materials Science and Engineering, University of California, Berkeley, Berkeley, CA 94720, USA. and\n5The Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA.\n(Dated: March 8, 2021)\nThe electronic density of states (DOS) highlights fundamental properties of materials that oftentimes dictate\ntheir properties, such as the band gap and Van Hove singularities. In this short note, we discuss how sharp\nfeatures of the density of states can be obscured by smearing methods (such as the Gaussian and Fermi smearing\nmethods) when calculating the DOS. While the common approach to reach a “converged” density of states\nof a material is to increase the discrete k-point mesh density, we show that the DOS calculated by smearing\nmethods can appear to converge but not to the correct DOS. Employing the tetrahedron method for Brillouin\nzone integration resolves key features of the density of states far better than smearing methods.\nPerhaps the simplest descriptor of the electronic structure of\na material is the density of states, which condenses many fun-\ndamental electrical and optical properties into a single infor-\nmative diagram. Features such as the band gap and the effec-\ntive mass of carriers are directly related to transport properties\nof the material, whereas sharp peaks (i.e. Van Hove singulari-\nties) provide critical information about optical properties such\nas the dielectric constant.\nOftentimes, computational methods based on first-\nprinciples are used to calculate the density of states of mate-\nrials. However, computational approaches require a trade-off\nbetween computational cost and accuracy, contextualizing the\nnotion of “convergence” in which the property of interest is\ncalculated with increasing computational cost until the prop-\nerty no longer appears to change within the desired accuracy.\nFor the density of states, convergence is especially impor-\ntant for generating salient features of the electronic structure.\nSuch numerical artefacts may arise from the technique used to\ncalculate the density of states, which can broadly be divided\ninto two categories: “smearing methods” and the “tetrahedron\nmethod”.\nSmearing methods (as described mathematically in the Ap-\npendix) fix a continuous function at each band and k-point to\napproximate the density of states. The tetrahedron method on\nthe other hand divides the Brillouin zone into tetrahedra, cal-\nculates the eigenenergies at the corners of each tetrahedron,\nand linearly interpolates the eigenenergies inside of each tetra-\nhedron to perform the integration. The tetrahedron method\nis reminiscent of the trapezoidal rule for approximating the\nintegral of single-variable functions. As is the case with\nsingle-variable functions where a linear interpolation overes-\ntimates regions of positive curvature and underestimates re-\ngions of negative curvature, the linear-tetrahedron method is\nprone to similar errors. In response, a correction to the in-\ntegration weights was introduced by Blöchl,[1] and the cor-\nrected Brillouin zone integration method is called the tetrahe-\ndron method with Blöchl corrections.\nIn this short note, we compare the density of states that\nare calculated using smearing methods and the tetrahedron\nmethod. We use first-principles Density Functional Theory(DFT) to calculate the density of states of the half-Heusler\ncompound TiNiSn (Materials Project ID: mp-924130).[2] We\ncompare the density of states calculated using the tetrahe-\ndron method with Blöchl corrections[1] against that calculated\nusing smearing methods. DFT calculations were performed\nusing the Vienna ab-initio simulation package (V ASP).[3–5]\nExchange-correlation effects were treated using the Perdew-\nBurke-Ernzerhof functional[6] for all calculations. We use\nthe recommended Ti_pv, Ni_pv, and Sn_d pseudopotentials\ndistributed by V ASP. Each structure was relaxed using a 520\neV energy cutoff and a G-centered grid to sample k-points in\nthe Brillouin zone.[7] Spin-orbit coupling effects were not in-\ncluded in the calculations. The density of states is calculated\nusing 5001 energy bins. We employ the Gaussian and Fermi\nsmearing methods, where we use a smearing width of 0.05 eV .\nAs shown in Figure 1, key analytical features representa-\ntive of the electronic structure of a material are clearly evi-\ndent when the tetrahedron method with Blöchl corrections is\nemployed. Sharp Van Hove peaks such as those at 0.8 eV\nand 2 eV below the valence band maximum are visible from\nthe density of states calculated using the tetrahedron method,\nwhereas the peaks are obscured by Gaussian and Fermi smear-\nings. Gaps in the density of states, such as the one at 1.6 eV\nabove the valence band maximum observed by the tetrahedron\nmethod, close when smearing methods are employed.\nSmearing methods should in principle reach the true density\nof states by increasing the computational cost by e.g. increas-\ning the k-point mesh density. However, as shown in Figure\n1, stark features such as Van Hove singularities and the band\nedge shape are obscure even at higher k-point densities using\nsmearing methods without carefully adjusting the smearing\nparameters, whereas they appear clearly with the tetrahedron\nmethod. Although the density of states of the Gaussian and\nFermi smearing methods separately converge to a stable dis-\ntribution as the k-point density increases, they do not approach\nthe density of states calculated using the tetrahedron method.\nMoreover, the width parameter of the smearing methods must\nbe carefully chosen to reach the density of states calculated\nusing the tetrahedron method. As shown in Figure 2, using a\ncomparatively large smearing width of 50 meV obscures thearXiv:2103.03469v1  [cond-mat.mtrl-sci]  5 Mar 20212\n−5 −4 −3 −2 −1 0 1 2\nE - EVBM (eV)Density of States(a) Tetrahedron7×7×7\n11×11×11\n15×15×15\n17×17×17\n21×21×21\n−5 −4 −3 −2 −1 0 1 2\nE - EVBM (eV)Density of States(b) GaussianTetrahedon\n7×7×7\n11×11×11\n15×15×15\n17×17×17\n21×21×21\n−5 −4 −3 −2 −1 0 1 2\nE - EVBM (eV)Density of States(c) FermiTetrahedron\n7×7×7\n11×11×11\n15×15×15\n17×17×17\n21×21×21\nFIG. 1: Density of states of TiNiSn using different k-point sampling\nmethods. The width parameter is fixed to 50 meV for the smearing\nmethods. The tetrahedron method (a) shows a clear band gap, van\nHove singularities and flat DOS regions even with a 7×7×7 k-point\nmesh. The Gaussian (b) and Fermi (c) smearing methods appear to\nconverge the density of states above a mesh size of 15, but even the\nconverged density of states blurs the features seen in the tetrahedron\nmethod and can produce artificial features even at a much denser k-\npoint mesh.\npeaks at\u00182 eV below the valence band maximum, whereas\nusing a small width of 10 meV introduces spurious noise. It\nis therefore clear from this comparison that the tetrahedron\nmethod is the preferred computational method for evaluating\nthe electronic structure of a semiconductor, since many fea-\n5\n 4\n 3\n 2\n 1\n 0 1 2\nE - EVBM (eV)Density of StatesTetrahedron\nWidth = 10 meV\nWidth = 20 meV\nWidth = 50 meVFIG. 2: The density of states calculated using the Gaussian smearing\nmethod with different smearing parameters, using a dense k-point\nmesh of 21\u000221\u000221.\ntures of the electronic structure are markedly expressed.\nWe stress the importance of the Brillouin zone integration\nmethod over the k-point mesh density in this note. Specif-\nically, the tetrahedron method for Brillouin zone integration\nis preferred over smearing methods to generate sharp quali-\nties of the density of states. Although many in the commu-\nnity already follow this recommendation, they are important\nto re-emphasize as the user community of electronic structure\nmethods grows.\nAppendix\nThe density of states g(E)is calculated as\ng(E) =1\nVå\nnZ\nVd(E\u0000en;k)dk (1)\nwhere en;kis the energy of band nat k-point k, and Vis the\nvolume of the reciprocal primitive cell. The Gaussian and\nFermi smearing methods approximate the d(E\u0000en;k)func-\ntion in the following ways:\nGaussian smearing:\nd(E\u0000en;k)\u00191\nsppe\u0000\u0012\nE\u0000en;k\ns\u00132\n(2)\nFermi smearing:\nd(E\u0000en;k)\u0019e\u0000E\u0000en;k\ns\ns\u0012\n1+e\u0000E\u0000en;k\ns\u00132(3)\nAcknowledgements\nWe acknowledge support from NSF DMREF award\n#1729487. MYT acknowledges support from the United3\nStates Department of Energy through the Computational Sci-\nence Graduate Fellowship (DOE CSGF) under Grant Number\nDE-SC0020347. This research was supported in part through\nthe computational resources and staff contributions provided\nfor the Quest high performance computing facility at North-\nwestern University which is jointly supported by the Office of\nthe Provost, the Office for Research, and Northwestern Uni-\nversity Information Technology. Work by AG, JP, and AJ\nwas supported by the U.S Department of Energy, Office ofBasic Energy Sciences, and the Early Career Research Pro-\ngram. JM and KAP acknowledge support from the U.S. De-\npartment of Energy, Office of Science, Office of Basic Energy\nSciences, Materials Sciences and Engineering Division under\ncontract no. DE-AC02-05-CH11231 (Materials Project pro-\ngram KC23MP). This research used resources of the National\nEnergy Research Scientific Computing Center (NERSC), a\nU.S. Department of Energy Office of Science User Facility\noperated under Contract No. DE-AC02-05CH11231.\n[1] P. E. Blöchl, O. Jepsen, and O. K. Andersen, Physical Review B\n49, 16223 (1994).\n[2] A. Jain, S. P. Ong, G. Hautier, W. Chen, W. D. Richards,\nS. Dacek, S. Cholia, D. Gunter, D. Skinner, G. Ceder, et al., APL\nMaterials 1, 011002 (2013).\n[3] G. Kresse, Journal of Non-Crystalline Solids 192, 222 (1995).\n[4] G. Kresse and J. Furthmüller, Computational materials science\n6, 15 (1996).[5] G. Kresse and J. Furthmüller, Physical review B 54, 11169\n(1996).\n[6] J. P. Perdew, K. Burke, and M. Ernzerhof, Physical review letters\n77, 3865 (1996).\n[7] H. J. Monkhorst and J. D. Pack, Physical review B 13, 5188\n(1976)."    },    {        "title": "2103.10096v1.Nuclear_energy_density_functionals_from_empirical_ground_state_densities.pdf",        "content": "Nuclear energy density functionals from empirical ground-state\ndensities\nGiacomo Accorto,1Tomoya Naito,2, 3Haozhao\nLiang,2, 3Tamara Nik\u0014 si\u0013 c,1and Dario Vretenar1, 4\n1Department of Physics, Faculty of Science,\nUniversity of Zagreb, HR-10000 Zagreb, Croatia\n2Department of Physics, Graduate School of Science,\nThe University of Tokyo, Tokyo 113-0033, Japan\n3RIKEN Nishina Center, Wako 351-0198, Wako 351-0198, Japan\n4State Key Laboratory of Nuclear Physics and Technology,\nSchool of Physics, Peking University, Beijing 100871, China\n(Dated: March 19, 2021)\nAbstract\nA model is developed, based on the density functional perturbation theory and the inverse Kohn-\nSham method, that can be used to improve relativistic nuclear energy density functionals towards\nan exact but unknown Kohn-Sham exchange-correlation functional. The improved functional is\ndetermined by empirical exact ground-state densities of \fnite systems. A test of the model and\nan illustrative calculation are performed, starting from two di\u000berent approximate functionals, to\nreproduce the parameters and density dependence of a target functional, using exact ground-state\ndensities of symmetric N=Zsystems.\n1arXiv:2103.10096v1  [nucl-th]  18 Mar 2021I. INTRODUCTION\nNuclear energy density functionals (EDFs) have been developed, more or less systemat-\nically, over the last two decades into a universal theoretical framework for the analysis of\nground-state properties, low-energy collective excitations, and reaction dynamics of medium-\nheavy and heavy nuclei [1, 2]. At present no other theoretical approach can be used to\nconsistently describe diverse emergent nuclear phenomena with the same level of simplicity\nand accuracy, and at a comparable computational cost. Based on universal EDFs, a num-\nber of microscopic models, such as the random phase approximation [3{5], the interacting\nboson model [6, 7], and the generator coordinate method [8{10], have also been designed to\nexplore low-energy spectroscopy and large-amplitude dynamics on a quantitative level, and\nto calculate various details for astrophysical applications.\nThe exact but unknown nuclear EDF must be approximated by functionals of powers and\ngradients of ground-state nucleon densities and currents. Even though a great number of\nstudies have been devoted to the microscopic formulation of a universal EDF framework (for\na recent review, see Ref. [11]), the most successful nuclear EDFs are either semi-empirical\nor completely phenomenological. In a semi-empirical approach, one starts from a micro-\nscopically motivated ansatz for the nucleonic density dependence of the energy of a system\nof protons and neutrons. Part of the parameters of such a functional can be determined,\nat least qualitatively, by microscopic calculations of the energy of isospin symmetric and\nasymmetric in\fnite nuclear matter as a function of the nucleonic density (or Fermi momen-\ntum). The remaining parameters are usually adjusted to selected ground-state data, e.g.,\nmasses and charge radii, of an arbitrarily large set of nuclei. Fully phenomenological EDFs,\nfor instance Skyrme, Gogny, and relativistic ones, usually take into account some empirical\nproperties of nuclear matter at saturation, but all parameters are adjusted to ground-state\ndata of \fnite nuclei.\nThe question we address in this work is how to improve a given functional towards the\nexact but unknown nuclear EDF. One could start, for instance, from a general expansion in\npowers of densities and currents and retain all terms allowed by symmetries up to a given\norder. Such a functional could be derived, in principle, from a microscopic theory (low-\nenergy QCD) that describes the underlying many-body dynamics. The problem, however, is\nthat available low-energy nuclear data can only constrain a relatively small subset of terms\n2and determine the corresponding parameters. Moreover, nuclear EDFs are \\sloppy\", that is,\nthey generally exhibit an exponential range of sensitivity to parameter variations, and one\n\fnds many softlinear combinations of bare model parameters that are poorly constrained\nby data. This often indicates the presence of low-dimensional e\u000bective functionals associated\nwith the relevant ( sti\u000b) parameter combinations. In Refs. [12, 13], we considered, in the\ncontext of nuclear structure, the interesting problem of a systematic construction of reduced\nlow-dimensional functionals from a more complete but sloppy framework. Using methods\nof information geometry, it has been shown how to systematically construct e\u000bective EDFs\nof successively lower dimension in parameter space until sloppiness is eventually eliminated\nand the resulting functional contains only sti\u000b combinations of parameters.\nInstead of using low-energy data to reduce the complexity of a very general functional,\none could also start from a relatively simple functional form and improve it towards the\nexact but unknown EDF. Such an expansion must, of course, be constrained by available\ndata. In the spirit of density functional theory (DFT) [14{16], the empirical (exact) ground-\nstate densities should determine the improved EDF. In fact, the inverse problem of DFT is\nformulated as a density-to-potential inversion that, starting from a given exact ground-state\ndensity, determines the true Kohn-Sham (KS) exchange-correlation potential [17{25]. The\n\frst application of the inverse KS method to nuclear EDFs has recently been reported in\nRef. [26]. The inverse Kohn-Sham (IKS) method is often used to benchmark and test the\naccuracy of various approximate exchange-correlation functionals but, since its implementa-\ntion depends on the exact density of a speci\fc system, it does not provide direct information\non the universal EDF.\nThe problem of improving the functional starting from exact ground-state densities has\nrecently been addressed in Ref. [27], where a model was introduced based on a combination\nof the IKS method and the density functional perturbation theory (DFPT). The idea is to\nconsider the di\u000berence, supposedly small, between the known functional and the exact but\nunknown EDF as a \frst-order perturbation. In Ref. [27], the method was successfully tested\nin benchmark calculations of systems of noble-gas atoms.\nIn this work, we apply the novel approach to atomic nuclei and, in particular, to rel-\nativistic nuclear EDFs. The principal reason for considering relativistic functionals is the\nfact that they automatically take into account the spin-orbit potential. The strong coupling\nbetween the orbital angular momentum and nucleon spin accounts for the empirical magic\n3numbers and shell gaps. While in atomic systems the e\u000bect of the spin-orbit coupling is\nperturbative, in nuclei the energy spacings between spin-orbit partner single-nucleon states\ncan be as large as the gaps between major shells. However, because the spin-orbit potential\nis a completely phenomenological addition to the non-relativistic KS potential, it cannot be\ndetermined from the ground-state density using the IKS method [26]. For relativistic EDFs,\nthe spin-orbit potential emerges automatically with the proper strength as the constructive\ncombination of the scalar and vector nucleon self-energies.\nIn Sec. II, we introduce the formalism of DFPT and IKS, and develop the model that will\nbe used in this work. In Sec. III, a test case is discussed, and Sec. IV presents an illustrative\ncalculation. Section V contains a brief summary and outlook for future studies.\nII. NUCLEAR ENERGY DENSITY FUNCTIONALS IMPROVED BY EXACT\nGROUND-STATE DENSITIES\nA. Density functional perturbation theory\nHere, we apply the method of Ref. [27] to improve a relativistic nuclear EDF, starting\nfrom a given empirical ground-state density. It is assumed that the unknown exact Hartree-\nexchange-correlation (interaction) functional can be written in the following form:\nEint[\u001a] =E(0)\nint[\u001a] +\u0015E(1)\nint[\u001a] +O\u0000\n\u00152\u0001\n; (1)\nwhereE(0)\nint[\u001a] denotes the known functional that we wish to make better and \u0015is a small\nparameter. The main premise of this method, therefore, is that the di\u000berence between the\nexact functional Eint[\u001a] and the starting functional E(0)\nint[\u001a] is small enough to be treated\nperturbatively. The correction will be determined by the exact ground-state density, using\nthe IKS approach.\nThe Dirac KS single-nucleon equation, derived from the relativistic EDF, reads\n[\u000b\u0001p+\f(m+S(r)) +V(r)] j(r) =\u000fj j(r); (2)\nwith the scalar and vector KS potentials, respectively,\nS(r) =\u000eE[\u001aV;\u001aS]\n\u000e\u001aS\f\f\f\f\ngsandV(r) =\u000eE[\u001aV;\u001aS]\n\u000e\u001aV\f\f\f\f\ngs: (3)\n4The corresponding scalar and vector densities\n\u001aS,gs(r) =X\nj2occ y\nj(r)\f j(r) and\u001aV,gs(r) =X\nj2occ y\nj(r) j(r) (4)\nare obtained from the self-consistent solutions of the single-nucleon Dirac KS equation in\ntheno-sea approximation, which omits explicit contributions of negative-energy states to\ndensities and currents [31, 33, 34] and, thus, the sums run only over the occupied positive-\nenergy single-nucleon orbitals.\nThe exact Dirac spinors, that is, the solutions of Eq. (2) for the exact EDF, can also be\nexpanded to the \frst order in \u0015:\n j(r) = (0)\nj(r) +\u0015 (1)\nj(r) +O\u0000\n\u00152\u0001\n; (5)\nwhere the \frst-order correction is orthogonal to the zeroth-order spinor\nZ\n (0)y\nj(r) (1)\nj(r)dr= 0: (6)\nThe exact ground-state scalar and vector densities\n\u001aS,gs(r) =X\nj2occ (0)y\nj(r)\f (0)\nj(r)\n+\u0015 X\nj2occ (1)y\nj(r)\f (0)\nj(r) +X\nj2occ (0)y\nj(r)\f (1)\nj(r)!\n+O\u0000\n\u00152\u0001\n; (7a)\n\u001aV,gs(r) =X\nj2occ (0)y\nj(r) (0)\nj(r)\n+\u0015 X\nj2occ (1)y\nj(r) (0)\nj(r) +X\nj2occ (0)y\nj(r) (1)\nj(r)!\n+O\u0000\n\u00152\u0001\n; (7b)\ntake the form:\n\u001aV(S),gs (r) =\u001a(0)\nV(S)(r) +\u0015\u001a(1)\nV(S)(r) +O\u0000\n\u00152\u0001\n; (8)\nwhere the \frst term on the right-hand side denotes the densities derived by the known\nfunctional. We will assume that, given the exact densities \u001aV(S),gs (r), one can use the IKS\nmethod to calculate the exact single-nucleon Dirac spinors  j(r) and energies \u000fj.\nThe ground-state energy can also be decomposed as follows:\nEgs=Ekin+E(0)\nint[\u001aV,gs;\u001aS,gs] +\u0015E(1)\nint[\u001aV,gs;\u001aS,gs]; (9)\n5where the KS kinetic energy reads\nEkin=X\nj2occZ\n y\nj(r)^t j(r)dr; (10)\nand^t=\u000b\u0001p+\fm. By expanding the Dirac spinors as in Eq. (5) and retaining only terms\nlinear in\u0015, we obtain the following expression for the kinetic energy\nEkin=X\nj2occZ\n (0)y\nj(r)^t (0)\nj(r)dr\n+\u0015X\nj2occZ\n (1)y\nj(r)^t (0)\nj(r)dr+\u0015X\nj2occZ\n (0)y\nj(r)^t (1)\nj(r)dr+O\u0000\n\u00152\u0001\n:(11)\nThe second and third terms on the right-hand side of Eq. (9) denote the interaction (Hartree-\nexchange-correlation) contribution to the total energy. Using the expansion [Eq. (8)] for the\nground-state densities up to terms linear in \u0015, one derives:\nE(0)\nint[\u001aV,gs;\u001aS,gs] =E(0)\ninth\n\u001a(0)\nV,gs;\u001a(0)\nS,gsi\n+\u0015Z\u000eE(0)\nint[\u001aV;\u001aS]\n\u000e\u001aV\f\f\f\f\f\ngs(0)\u001a(1)\nV,gs(r)dr\n+\u0015Z\u000eE(0)\nint[\u001aV;\u001aS]\n\u000e\u001aS\f\f\f\f\f\ngs(0)\u001a(1)\nS,gs(r)dr+O\u0000\n\u00152\u0001\n; (12a)\n\u0015E(1)\nint[\u001aV,gs;\u001aS,gs] =\u0015E(1)\ninth\n\u001a(0)\nV,gs;\u001a(0)\nS,gsi\n+O\u0000\n\u00152\u0001\n: (12b)\nInserting the expressions Eqs. (7b) and (7a) for the \frst-order density corrections \u001a(1)\nV,gs\nand\u001a(1)\nS,gs, respectively, into the expansion for the interaction energy Eqs. (12a) and (12b),\ntogether with the expression for the kinetic energy Eq. (11), the ground-state energy reads\nEgs=E(0)\nkin+E(0)\ninth\n\u001a(0)\nV,gs;\u001a(0)\nS,gsi\n+\u0015X\nj2occZ\n (1)y\nj(r)2\n4^t+\u000eE(0)\nint[\u001aV;\u001aS]\n\u000e\u001aV\f\f\f\f\f\ngs(0)+\f\u000eE(0)\nint[\u001aV;\u001aS]\n\u000e\u001aS\f\f\f\f\f\ngs(0)3\n5 (0)\nj(r)dr\n+\u0015X\nj2occZ\n (0)y\nj(r)2\n4^t+\u000eE(0)\nint[\u001aV;\u001aS]\n\u000e\u001aV\f\f\f\f\f\ngs(0)+\f\u000eE(0)\nint[\u001aV;\u001aS]\n\u000e\u001aS\f\f\f\f\f\ngs(0)3\n5 (1)\nj(r)dr\n+\u0015E(1)\ninth\n\u001a(0)\nV,gs;\u001a(0)\nS,gsi\n+O\u0000\n\u00152\u0001\n: (13)\nOne notices that the expression in square brackets represent the zeroth-order (unperturbed)\nDirac Hamiltonian, that is\n2\n4^t+\u000eE(0)\nint[\u001aV;\u001aS]\n\u000e\u001aV\f\f\f\f\f\ngs(0)+\f\u000eE(0)\nint[\u001aV;\u001aS]\n\u000e\u001aS\f\f\f\f\f\ngs(0)3\n5 (0)\nj=\u000f(0)\nj (0)\nj: (14)\n6The corresponding terms in Eq. (13) vanish because of the orthogonality relation [Eq. (6)]\nand, thus, a much simpler relation for the ground-state energy reads\nEgs=E(0)\nkin+E(0)\ninth\n\u001a(0)\nV,gs;\u001a(0)\nS,gsi\n+\u0015E(1)\ninth\n\u001a(0)\nV,gs;\u001a(0)\nS,gsi\n+O\u0000\n\u00152\u0001\n: (15)\nOn the other hand, the ground state energy Egs=Ekin+Eint[\u001ags], can be written in the\nfollowing form\nEgs=X\nj2occ\u000fj+Eint[\u001aV,gs;\u001aS,gs]\u0000Z\u000eEint[\u001aV;\u001as]\n\u000e\u001aV\f\f\f\f\ngs\u001aV,gs(r)dr\u0000Z\u000eEint[\u001aV;\u001as]\n\u000e\u001aS\f\f\f\f\ngs\u001aS,gs(r)dr;\n(16)\nwhere the Dirac KS equation has been used to eliminate the explicit contribution of the\nkinetic energy term, and \u000fjare the exact single-particle energies with the summation over\noccupied states. If we separate the zeroth-order and \frst-order terms of the exact interaction\nfunctionalEint[\u001ags], then\nEgs=X\nj2occ\u000fj+E(0)\nint[\u001aV,gs;\u001aS,gs]\u0000Z\u000eE(0)\nint[\u001aV;\u001as]\n\u000e\u001aV\f\f\f\f\f\ngs\u001aV,gs(r)dr\u0000Z\u000eE(0)\nint[\u001aV;\u001as]\n\u000e\u001aS\f\f\f\f\f\ngs\u001aS,gs(r)dr\n+\u0015E(1)\nint[\u001aV,gs;\u001aS,gs]\u0000\u0015Z\u000eE(1)\nint[\u001aV;\u001as]\n\u000e\u001aV\f\f\f\f\f\ngs\u001aV,gs(r)dr\u0000\u0015Z\u000eE(1)\nint[\u001aV;\u001as]\n\u000e\u001aS\f\f\f\f\f\ngs\u001aS,gs(r)dr:\n(17)\nFrom Eq. (15), we can express the \frst-order correction to the interaction energy as a\nfunction of the zeroth-order ground-state densities\n\u0015E(1)\ninth\n\u001a(0)\nV,gs;\u001a(0)\nS,gsi\n=Egs\u0000E(0)\nkin\u0000E(0)\ninth\n\u001a(0)\nV,gs;\u001a(0)\nS,gsi\n=Egs\u0000E(0)\ngs; (18)\nwhere obviously E(0)\ngsdenotes the ground-state energy calculated with the known functional.\nNext, Eq. (17) for Egsis inserted in this expression and the following relation is obtained,\n\u0015E(1)\ninth\n\u001a(0)\nV,gs;\u001a(0)\nS,gsi\n=X\nj2occ\u000fj+E(0)\nint[\u001aV,gs;\u001aS,gs]\u0000Z\u000eE(0)\nint[\u001aV;\u001as]\n\u000e\u001aV\f\f\f\f\f\ngs\u001aV,gs(r)dr\u0000Z\u000eE(0)\nint[\u001aV;\u001as]\n\u000e\u001aS\f\f\f\f\f\ngs\u001aS,gs(r)dr\n+\u0015E(1)\nint[\u001aV,gs;\u001aS,gs]\u0000\u0015Z\u000eE(1)\nint[\u001aV;\u001as]\n\u000e\u001aV\f\f\f\f\f\ngs\u001aV,gs(r)dr\u0000\u0015Z\u000eE(1)\nint[\u001aV;\u001as]\n\u000e\u001aS\f\f\f\f\f\ngs\u001aS,gs(r)dr\u0000E(0)\ngs:\n(19)\n7Equation (19) is now rearranged so that all terms linear in \u0015(\frst-order corrections to the\ninteraction functional) are collected on the left-hand side\n\u0015E(1)\ninth\n\u001a(0)\nV,gs;\u001a(0)\nS,gsi\n\u0000\u0015E(1)\nint[\u001aV,gs;\u001aS,gs] +\u0015Z\u000eE(1)\nint[\u001aV;\u001aS]\n\u000e\u001aV\f\f\f\f\f\ngs\u001aV,gs(r)dr+\u0015Z\u000eE(1)\nint[\u001aV;\u001aS]\n\u000e\u001aS\f\f\f\f\f\ngs\u001aS,gs(r)dr\n=X\nj2occ\u000fj+E(0)\nint[\u001aV,gs;\u001aS,gs]\u0000Z\u000eE(0)\nint[\u001aV;\u001aS]\n\u000e\u001aV\f\f\f\f\f\ngs\u001aV,gs(r)dr\u0000Z\u000eE(0)\nint[\u001aV;\u001aS]\n\u000e\u001aS\f\f\f\f\f\ngs\u001aS,gs(r)dr\u0000E(0)\ngs:\n(20)\nThe right-hand side of this equation depends only on the exact ground-state densities and\nthe known functional E(0)\nint. For given ground-state empirical densities, therefore, we can\ncalculate all terms on the right-hand side, except the \frst term which is a sum of the\nexact single-particle energies. These energies are, of course, implicit functionals of the exact\nground-state densities. One can, therefore, use the IKS method to calculate the single-\nparticle energies starting from given ground-state densities, a procedure that we describe in\nthe following section.\nIn practical terms, one must assume a certain ansatz for the functional E(1)\nint[\u001a], that will\nalso include parameters to be determined from Eq. (20) for a choice of empirical ground-state\ndensities. There is no guarantee, especially in the case of several undetermined parameters\nfor the \frst-order correction, that the improved functional will reproduce the exact densities\nto a desired level of accuracy. The solution is an iterative procedure [27], in which the\nfunctional improved in the \frst iteration step is considered as the known functional for the\nnext iteration, i.e.,\nEn-th\nint[\u001aV;\u001aS] =E(0)\nint[\u001aV;\u001aS] +nX\nk=1\u0015E(1),k-th\nint [\u001aV;\u001aS]; (21)\nand the operation is repeated until the exact densities are reproduced by the solutions of\nthe resulting n-thiteration Dirac KS equation to a desired accuracy.\nB. Inverse Kohn-Sham method\nDetermining the KS potential for a given density presents an inverse problem. According\nto the Hohenberg-Kohn theorem, this inverse problem has a solution and the KS exchange-\ncorrelation potential for a given system of interacting particles can be calculated starting\n8from its ground-state density. Here, we perform the density-to-potential inversion in order to\ndetermine the single-particle energies that appear on the right-hand side Eq. (20). Starting\nfrom the single-nucleon Dirac KS equation (2), we rewrite the KS potentials: V+(r) =\nV(r) +S(r) andV\u0000(r) =V(r)\u0000S(r), so that Eq. (2) takes the form\n\u0014\n\u000b\u0001p+1\n2(\f\u000011) (m\u0000V\u0000(r)) +1\n2(\f+11) (m+V+(r))\u0015\n j(r) =\u000fj j(r): (22)\nBy multiplying Eq. (22) with  y\nj(r) from the left and summing over the occupied positive-\nenergy states, one obtains\nX\nj2occ y\nj(r) (\u000b\u0001p\u0000\u000fj) j(r)\n+1\n2(m\u0000V\u0000(r))X\nj2occ y\nj(r) (\f\u000011) j(r) +1\n2(m+V+(r))X\nj2occ y\nj(r) (\f+11) j(r) = 0:\n(23)\nThe scalar and vector densities that appear in this expression\n\u001aS,gs(r) =X\nj2occ y\nj(r)\f j(r) and\u001aV,gs(r) =X\nj2occ y\nj(r) j(r) (24)\ncan also be combined in the following form: \u001a+(r) =\u001aV,gs(r) +\u001aS,gs(r) and\u001a\u0000(r) =\n\u001aV,gs(r)\u0000\u001aS,gs(r), so that\nX\nj2occ y\nj(r) (\u000b\u0001p\u0000\u000fj) j(r)\u00001\n2(m\u0000V\u0000(r))\u001a\u0000(r) +1\n2(m+V+(r))\u001a+(r) = 0:(25)\nIf Eq. (22) is multiplied with \u0016 j= y\nj\f, the following expression is obtained:\nX\nj2occ\u0016 j(r) (\u000b\u0001p\u0000\u000fj) j(r) +1\n2(m\u0000V\u0000(r))\u001a\u0000(r) +1\n2(m+V+(r))\u001a+(r) = 0:(26)\nFinally, by adding and subtracting Eqs. (25) and (26), we derive\nX\nj2occ\u0010\n y\nj(r) +\u0016 j(r)\u0011\n(\u000b\u0001p\u0000\u000fj) j(r) + (m+V+(r))\u001a+(r) = 0; (27a)\nX\nj2occ\u0010\n y\nj(r)\u0000\u0016 j(r)\u0011\n(\u000b\u0001p\u0000\u000fj) j(r)\u0000(m\u0000V\u0000(r))\u001a\u0000(r) = 0; (27b)\nfrom which the KS potentials V+andV\u0000are expressed\nV+(r) =\u0000m\u00001\n\u001a+(r)X\nj2occ\u0010\n y\nj(r) +\u0016 j(r)\u0011\n(\u000b\u0001p\u0000\u000fj) j(r); (28a)\nV\u0000(r) = +m\u00001\n\u001a\u0000(r)X\nj2occ\u0010\n y\nj(r)\u0000\u0016 j(r)\u0011\n(\u000b\u0001p\u0000\u000fj) j(r): (28b)\n9The set of IKS equations (28a) and (28b) can be solved iteratively. If we assume that the\ndensities in the denominator are the exact (target) densities, and use Eqs. (27a) and (27b) in\nthe numerator to de\fne the densities and potentials for the n-th step, the resulting potentials\nin the (n+ 1)-th step read\nV(n+1)\n+ (r) =\u001a(n)\n+(r)\n\u001a+(r)V(n)\n+(r) +m\u001a(n)\n+(r)\u0000\u001a+(r)\n\u001a+(r); (29)\nV(n+1)\n\u0000 (r) =\u001a(n)\n\u0000(r)\n\u001a\u0000(r)V(n)\n\u0000(r)\u0000m\u001a(n)\n\u0000(r)\u0000\u001a\u0000(r)\n\u001a\u0000(r): (30)\nIn actual IKS calculations, we have modi\fed an algorithm proposed in Ref. [21], and\nreplaced Eqs. (29) and (30) with\nV(n)\n+(r) =V(n\u00001)\n+ (r) +\r+\u001a(n)\n+(r)\u0000\u001a+(r)\n\u001a+(r); (31)\nV(n)\n\u0000(r) =V(n\u00001)\n\u0000 (r) +\r\u0000\u001a(n)\n\u0000(r)\u0000\u001a\u0000(r)\n\u001a\u0000(r): (32)\nThis algorithm was also used in the \frst nuclear IKS calculation with non-relativistic EDFs\n[26], and justi\fed by the following argument. Equation (29) for the potential V+(r) =\nV(r) +S(r), which is the equivalent of the non-relativistic KS potential, has a simple\ninterpretation: the potential is enhanced in absolute value in those regions where the density\nis larger than the target density, and reduced in regions where the density is smaller than\nthe target density. However, this is what one expects for repulsive potentials (e.g., the\nCoulomb potential for electrons), whereas in the case of attractive potentials (e.g., the\nnuclear potential for nucleons) the opposite should happen. In Ref. [26], it has been shown\nthat this issue can be avoided by adopting the modi\fed algorithm of Eqs. (31) and (32).\nFollowing Ref. [26], here we use the value of 1 MeV for both parameters \r+and\r\u0000. In\nactual calculations we have encountered some stability issues for large values of the radial\ncoordinate, due to small values of denominators in Eqs. (31) and (32) beyond the nuclear\nradius. This problem can be simply solved by introducing a cut-o\u000b radius rcut, and setting\nthe potentials to zero for r > r cut. For the initial KS potential, a realistic Woods-Saxon\npotential [36] is used, and the Broyden mixing procedure [37] is employed to solve Eqs. (31)\nand (32). The convergence criterion used to halt the iterative IKS algorithm is de\fned in\nterms of the absolute variation of the potential, i.e.\n\u0001V(n)\n\u0006\u0011max\nrh\nV(n+1)\n\u0006 (r)\u0000V(n)\n\u0006(r)i\n<\u000b\u0006: (33)\n10III. A TEST CASE\nThe atomic nucleus is a complex quantum mechanical system with two types of con-\nstituent particles of spin one-half and, therefore, a general EDF will be a functional of\nisoscalar, isovector, and spin densities, as well as corresponding currents. Even in the sim-\nplest case of spin-saturated even-even nuclei, the EDF will depend on isoscalar and isovector\ndensities. The problem, of course, is that accurate data exists only on charge (proton) den-\nsities and, thus, adjusting a general functional to empirical densities is not a straightforward\nprocedure. In the particular case of relativistic EDFs that we consider here, the functional\ndepends also on the Lorentz scalar single-nucleon density, which is not an observable. In the\n\fnal section, we will discuss a possible approach that can be used to determine the scalar and\nisovector densities in an indirect way, but, for the purpose of testing the proposed method,\nhere onlyN=Zsystems without Coulomb interaction are considered. For such arti\fcial\nnuclei, to demonstrate the relativistic IKS method with DFPT, we will use an existing rela-\ntivistic EDF as the exact target functional, and apply the method developed in the previous\nsection to improve di\u000berent approximate functionals towards the target functional. In real\nnuclei, the exact functional is, of course, unknown and we will need more than ground-state\ndata to determine the functional dependence on various nuclear densities.\nFor the exact target EDF, we will use the relativistic EDF DD-PC1 [35], for which the\nsingle-nucleon Hamiltonian reads\n^h=\u000b\u0001p+\f(m+S(r)) +V0(r) + \u0006 R(r); (34)\nwhere the scalar potential, vector potential, and rearrangement terms are respectively de-\n\fned by\nS=\u000bS(\u001a)\u001aS+\u000eS4\u001aS;\nV=\u000bV(\u001a)\u001aV+\u000bTV(\u001a)~ \u001aTV\u0001~ \u001c+e1\u0000\u001c3\n2A0;\n\u0006R=1\n2@\u000bS\n@\u001a\u001a2\nS+1\n2@\u000bV\n@\u001a\u001a2\nV+1\n2@\u000bTV\n@\u001a\u001a2\nTV: (35)\nIn these expressions, mis the nucleon mass, \u000bS(\u001a),\u000bV(\u001a), and\u000bTV(\u001a) are density-dependent\ncouplings for di\u000berent space-isospace channels, \u000eSis the coupling constant of the derivative\nterm,eis the electric charge,1\u0000\u001c3\n2A0corresponds to the Coulomb interaction, and the\n11single-nucleon densities \u001aS(scalar-isoscalar density), \u001aV(time-like component of the isoscalar\ncurrent), and \u001aTV(time-like component of the isovector current).\nIn addition to contributions from the isoscalar-vector four-fermion interaction and the\nelectromagnetic interaction, the isoscalar-vector self-energy includes the rearrangement\nterms in \u0006 Rthat arise from the variation of the vertex functionals \u000bS,\u000bV, and\u000bTVwith\nrespect to the nucleon \felds in the vector density operator ^ \u001aV.\nIn a phenomenological construction of a relativistic EDF, one starts from an assumed\nansatz for the medium dependence of the mean-\feld nucleon self-energies, and adjusts the\nfree parameters directly to ground-state data of \fnite nuclei. Guided by the microscopic\ndensity dependence of the vector and scalar self-energies, the following practical ansatz for\nthe functional form of the couplings was adopted in Ref. [35]:\n\u000bS(\u001aV) =as+ (bs+csx)e\u0000dsx;\n\u000bV(\u001aV) =av+bve\u0000dvx; (36)\n\u000bTV(\u001aV) =btve\u0000dtvx;\nwithx=\u001aV=\u001asat, where\u001asatdenotes the nucleon density at saturation in symmetric nuclear\nmatter. The set of 10 parameters was adjusted in a \u001f2\ft to the experimental masses\nof 64 axially deformed nuclei in the mass regions A\u0019150{180 and A\u0019230{250. The\nresulting functional DD-PC1 [35] has been further tested in calculations of binding energies,\ncharge radii, deformation parameters, neutron-skin thickness, and excitation energies of\ngiant monopole and dipole resonances. During the last decade the functional DD-PC1\nhas successfully been applied in a number of studies of various nuclear phenomena, from\nground-state properties to the description of collective spectra, giant resonances, shape-phase\ntransitions, and the dynamics of nuclear \fssion.\nIn the simpli\fed case of N=Zdoubly closed-shell nuclei without Coulomb interaction,\nthere is no contribution of the isovector channel either. We will use four N=Zsystems:\n8\n816,20\n2040,28\n2856, and50\n50100 to improve, starting from the exact ground-state densities, the\napproximate zeroth-order functionals towards DD-PC1. First, we illustrate the accuracy of\nthe IKS scheme, described in the previous section, in determining the KS potentials for given\nscalar and vector densities of the N=Z= 8 system. The densities-to-potentials inversion\nenables the calculation of the single-particle energies that appear on the right-hand side\nEq. (20).\n12Figure 1 compares the densities obtained in the inverse KS scheme to the target DD-\nPC1 densities. We plot four di\u000berent neutron densities: the sum of the scalar and vector\ndensity\u001a+(r) =\u001aV(r) +\u001aS(r) (panel (a)), the di\u000berence between the vector and scalar\ndensity\u001a\u0000(r) =\u001aV(r)\u0000\u001aS(r) (panel (b)), and separately the vector \u001aV(r) (panel (c))\nand scalar \u001aS(r) densities (panel (d)). Without Coulomb interaction the proton densities\nare, of course, identical to the neutron ones. In all four panels the dash-dotted green curves\ndenote the target densities calculated with the DD-PC1 functional, the dashed red curves are\nthe initial densities that correspond to Woods-Saxon potentials and, \fnally, the solid black\ncurves represent the \fnal densities obtained by the inversion method. The corresponding\nresults for the potentials are shown in Fig. 2: the sum of the vector and scalar potentials\nV+(r) =V(r) +S(r) (panel (a)), the di\u000berence between the vector and scalar potential\nV\u0000(r) =V(r)\u0000S(r) (panel (b)), and separately the vector V(r) (panel (c)) and scalar S(r)\n(panel (d)) potentials. Again, the green dash-dotted curves denote the target potentials,\nthe dashed red curves are the initial Woods-Saxon potentials, and the solid black curves\nare the \fnal potentials obtained by the inversion method. Obviously, the result is that one\ncannot distinguish between the target and \fnal IKS densities and potentials. The latter can,\ntherefore, be used to calculate the single-particle energies that are needed in the application\nof the DFPT method.\nIn the test case, we will assume for the known functional E(0)\nint[\u001aV;\u001aS] a simple form that\nis actually a part of the DD-PC1 functional\nE(0)\nint[\u001aV;\u001aS] =1\n2\u000b(0)\ns\u001a2\nS+1\n2\u000b(0)\nv\u001a2\nV+\u000es\u001as4\u001as; (37)\nThe values for the a(0)\nsanda(0)\nvconstants are those used in the DD-PC1 functional: a(0)\ns=\nas=\u000010:4602 fm\u00002anda(0)\nv=av= 5:9195 fm\u00002, and the same choice is made for the\nderivative term: \u000es=\u00000:8149. Note that the \frst two terms of this functional coincide with\nthe simple Walecka mean-\feld model which, with only two parameters, produces a realistic\nequation of state of symmetric nuclear matter. Such a model, in fact, correspond to a local\ndensity approximation (LDA) for the EDF. The derivative term is used in modeling \fnite\nsystems and takes into account the rapid variations of the density in the surface region. The\nstrength parameter of this term can be determined, at least qualitatively, from microscopic\ncalculations of inhomogeneous nuclear matter.\n130 1 2 3 4 5 60.000.050.100.150.20ρv(r)+ρs(r) (fm-3)\nFinal\nInitial\nTarget\n0 1 2 3 4 5 602468ρV(r)-ρS(r) (10-3 fm-3)\n0 1 2 3 4 5 6\nr (fm)0.000.020.040.060.08ρV(r) (fm-3)\n0 1 2 3 4 5 6\nr (fm)00.020.040.060.08ρs(r) (fm-3)N=Z=8(a) (b)\n(c) (d)FIG. 1: (Color online) (a) The sum of neutron vector and scalar densities in the N=Z= 8\nsystem as a function of the radial coordinate. The target density obtained using the DD-PC1\nfunctional (dashed green curve) is compared to the the density calculated in the initial step of the\ninversion method (dot-dashed red) with a Woods-Saxon potential, and to the \fnal IKS density\n(solid black). (b) Same as in panel (a) but for the di\u000berence between the neutron vector and scalar\ndensities. (c) Same as in panel (a) but for the neutron vector density. (d) Same as in panel (a)\nbut for the neutron scalar density.\nFor the remaining unknown part of the functional E(1)\nint[\u001aV;\u001aS] we choose\nE(1)\nint[\u001aV;\u001aS] =1\n2\u000b(1)\ns(\u001aV)\u001a2\nS+1\n2\u000b(1)\nv(\u001aV)\u001a2\nV: (38)\nwhere\u000b(1)\ns(\u001aV) and\u000b(1)\nv(\u001aV) have the functional form of the density-dependent parts of the\nDD-PC1 couplings:\n\u000b(1)\ns(\u001aV) =\u0000\nb(1)\ns+c(1)\nsx\u0001\ne\u0000dsxand\u000b(1)\nv(\u001aV) =b(1)\nve\u0000dvx; (39)\nwithx=\u001aV=\u001asat, and\u001asat= 0:152 fm\u00003. As explained above, the parameters of the func-\ntional DD-PC1 were adjusted to reproduce the nuclear matter equation of state and the\n140 1 2 3 4 5 6-80-60-40-200V(r)+S(r) (MeV)Final\nInitial\nTarget\n0 1 2 3 4 5 60200400600800V(r)-S(r) (MeV)\n0 1 2 3 4 5 6\nr (fm)0100200300400V(r) (MeV)\n0 1 2 3 4 5 6\nr (fm)-500-400-300-200-1000S(r) (MeV)N=Z=8 (a) (b)\n(c) (d)FIG. 2: (Color online) (a) The sum of neutron vector and scalar potentials in the N=Z= 8\nsystem as a function of the radial coordinate. The target DD-PC1 Kohn-Sham potential (dashed\ngreen curve) is compared to the initial Woods-Saxon potential (dot-dashed red), and to the \fnal\nIKS potential (solid black). (b) Same as in panel (a) but for the di\u000berence between the neutron\nvector and scalar potentials. (c) Same as in panel (a) but for the neutron vector potential. (d)\nSame as in panel (a) but for the neutron scalar potential.\nexperimental masses of 64 deformed nuclei. The test of the method proposed in this work\nconsist in trying to determine the parameters of E(1)\nint[\u001aV;\u001aS] shown in Eq. (38) (i.e., b(1)\ns,c(1)\ns,\nandb(1)\nv) by using density functional perturbation theory and the IKS scheme, that is, using\nEq. (20). Because the right-hand side of this equation is just a number that can be evalu-\nated provided the exact single-particle energies and vector and scalar densities are known, a\ndi\u000berent \fnite system is needed for each parameter of the unknown functional. Since this is\nan illustrative test, we will employ three N=Zsystems:8\n816,28\n2856 and50\n50100 to determine\nthe constants b(1)\ns,c(1)\nsandb(1)\nv, whiledsanddvare again \fxed to the DD-PC1 values. Note\nthat, even though the problem has been simpli\fed to a certain extent, nevertheless the test\n15is far from being trivial. Namely, only three arti\fcial systems are used to reproduce the\nvalues of parameters that were originally adjusted to the experimental masses of a large\nnumber of nuclei. Moreover, since our choice for the unperturbed functional is obviously\nnot close to the exact target functional, it is far from obvious that a \frst-order perturbation\nmethod will determine the unknown parameters with su\u000ecient accuracy. Hence, we repeat\nthe calculation in several iterative steps, as described in the previous section [Eq. (21)], and\nat each step improve the values of b(1)\ns,c(1)\ns, andb(1)\nv. Note that, because of the functional\nform ofE(1)\nintexpressed in Eq. (38),\nE(1),n-th\nint [\u001aV;\u001aS] =nX\nk=1\u00141\n2\u0000\nb(1),k-th\ns +c(1),k-th\nsx\u0001\ne\u0000dsx\u001a2\nS+1\n2b(1),k-th\nv\u001a2\nV\u0015\n=1\n2 (nX\nk=1b(1),k-th\ns)\n+(nX\nk=1c(1),k-th\ns)\nx!\ne\u0000dsx\u001a2\nS+1\n2(nX\nk=1b(1),k-th\nv)\n\u001a2\nV\n(40)\nholds. Hence, hereafter,\u0010\nb(1)\ns\u0011\ni,\u0010\nc(1)\ns\u0011\ni, and\u0010\nb(1)\nv\u0011\nisimply denotenPi\nk=1b(1),k-th\nso\n,\nnPi\nk=1c(1),k-th\nso\n, andnPi\nk=1b(1),k-th\nvo\n, respectively.\nFigure 3 displays the values of\u0010\nb(1)\ns\u0011\ni,\u0010\nc(1)\ns\u0011\ni, and\u0010\nb(1)\nv\u0011\niat each iteration step i. As-\nsuming that nothing is known about these parameters, we start with zero values. After\nsome initial oscillations in the \frst few steps, especially between c(1)\nsandb(1)\nv, the parame-\nters converge to the values that correspond to the DD-PC1 target functional, denoted by\nthe horizontal lines in Fig. 3. The results of this test demonstrate not only the feasibility\nof the IKS + DFPT method for nuclear densities, but also the convergence and accuracy of\nthe iteration scheme.\nIV. AN ILLUSTRATIVE CALCULATION\nIn the second example, we again use DD-PC1 as the unknown target functional, and the\ncorresponding exact single-particle energies that appear on the right-hand side of Eq. (20)\nare obtained by the IKS method as described in Sec. II B. Also for the known functional\nE(0)\nint[\u001aV;\u001aS] the simple form of Eq. (37) is adopted, that is\nE(0)\nint[\u001aV;\u001aS] =1\n2as\u001a2\nS+1\n2av\u001a2\nV+\u000es\u001as4\u001as; (41)\n160 1 2 3 4 5 6 7 8 9\nIteration number-40-2002040EDF constants (fm-2)(bs(1))i\n(cs(1))i\n(bv(1))iFIG. 3: (Color online). Values of the constants\u0010\nb(1)\ns\u0011\ni,\u0010\nc(1)\ns\u0011\niand\u0010\nb(1)\nv\u0011\niat di\u000berent iteration\nsteps. The dashed lines denote the target values that correspond to the functional DD-PC1.\nwith the DD-PC1 values of the three parameters. For the remaining unknown part of the\nfunctional\nE(1)\nint[\u001aV;\u001aS] =1\n2\u000b(1)\ns(\u001aV)\u001a2\nS+1\n2\u000b(1)\nv(\u001aV)\u001a2\nV;\nwe choose a polynomial form of the couplings \u000b(1)\ns(\u001aV) and\u000b(1)\nv(\u001aV):\n\u000b(1)\ns(\u001aV) =b(1)\ns(x\u00001) +c(1)\ns(x\u00001)2and\u000b(1)\nv(\u001aV) =b(1)\nv(x\u00001) +c(1)\nv(x\u00001)2;(42)\nwithx=\u001aV=\u001asat, and\u001asat= 0:152 fm\u00003. Therefore, we will examine whether the known\nfunctionalE(0)\nint[\u001a] can be improved towards the exact target functional DD-PC1, by assuming\na polynomial density dependence of the coupling parameters of E(1)\nint[\u001a].\nSince the values of four parameters have to be determined, Eq. (20) requires the input\nfrom four \fnite systems. Here we choose: N=Z= 8,N=Z= 20,N=Z= 28, and\nN=Z= 50. In Fig. 4 the parameters\u0010\nb(1)\ns\u0011\ni,\u0010\nc(1)\ns\u0011\ni,\u0010\nb(1)\nv\u0011\ni, and\u0010\nc(1)\nv\u0011\niare shown at\neach iteration step of the IKS + DFPT procedure. They are compared with the parameters\nof the linear and quadratic term in the Taylor expansion of the DD-PC1 couplings:\n\u000bDD-PC1\ns,v (\u001aV)\u0019\u000bs,v(\u001asat) +d\u000bDD-PC1\ns,v\nd\u001aV\f\f\f\f\f\n\u001aV=\u001asat(\u001aV\u0000\u001asat) +1\n2d2\u000bDD-PC1\ns,v\nd\u001a2\nV\f\f\f\f\f\n\u001aV=\u001asat(\u001aV\u0000\u001asat)2;\n(43)\n17or expressed in terms of x=\u001aV=\u001asat:\n\u000bDD-PC1\ns,v (\u001aV)\u0019\u000bs,v(\u001asat) +\u001asatd\u000bDD-PC1\ns,v\nd\u001aV\f\f\f\f\f\n\u001aV=\u001asat(x\u00001) +1\n2\u001a2\nsatd2\u000bDD-PC1\ns,v\nd\u001a2\nV\f\f\f\f\f\n\u001aV=\u001asat(x\u00001)2:\n(44)\nEven though we start with zero initial values, after only a few iterations the parameters\nof the linear and quadratic couplings of the unknown functional reach values that are very\nclose to the corresponding parameters of the Taylor expansion of the target DD-PC1 cou-\nplings. Using the \fnal values of the parameters, we calculate the total scalar and vector\n0 1 2 3 4 5-2-101234EDF constants (fm-2)\n(bs(1))i\n(cs(1))i\nDD-PC1, lin.\nDD-PC1, quad.\n0 1 2 3 4 5\nIteration number-4-3-2-1012EDF constants (fm-2)\n(bv(1))i\n(cv(1))i\nDD-PC1, lin.\nDD-PC1, quad.(a)\n(b)\nFIG. 4: (Color online). Values of the constants\u0010\nb(1)\ns\u0011\ni,\u0010\nc(1)\ns\u0011\ni(panel (a)),\u0010\nb(1)\nv\u0011\niand\u0010\nc(1)\nv\u0011\ni\n(panel (b)) at di\u000berent iteration steps of the IKS+DFPT calculation. The dashed lines denote the\ncorresponding parameters of the linear and quadratic term in the Taylor expansion of the DD-PC1\ncouplings.\ncouplings\u000bS(\u001aV) and\u000bV(\u001aV) as functions of the vector density, and compare them with\nthe corresponding couplings of the functional DD-PC1 in Fig. 5. While the couplings of\nthe unknown functional E(1)\ninthave been approximated by simple quadratic functions of the\nvector density, nevertheless the \fnal IKS + DFPT scalar and vector couplings accurately\nreproduce the DD-PC1 target couplings over a broad range of densities.\nFinally, in Fig. 6 we compare the vector densities for the four symmetric systems: N=\nZ= 8,N=Z= 20,N=Z= 28, andN=Z= 50, calculated with the IKS+DFPT method\n180.04 0.08 0.12 0.16 0.20 0.24 0.28-20-18-16-14-12-10αs(ρ)\nDD-PC1\nQuadratic\n0.04 0.08 0.12 0.16 0.2 0.24 0.28\nρV (fm-3)810121416αv(ρ)(a)\n(b)FIG. 5: (Color online) Scalar (panel (a)) and vector (panel (b)) IKS+DFPT couplings as functions\nof vector density, compared to the corresponding DD-PC1 target coupling functions.\nand the target functional DD-PC1. The red curves denote the densities that correspond\nto the unperturbed initial functional E(0)\nintand they are, of course, very di\u000berent from those\nobtained with the target functional. However, even when the unknown part of the functional\nis approximated by the simple expressions of Eq. (42), the IKS + DFPT method produces\nground-state densities that are virtually identical to the exact target densities.\nV. SUMMARY\nIn this study we have considered an interesting problem in the framework of nuclear\nenergy density functionals, namely, how to improve a given functional towards an exact\nbut unknown Kohn-Sham exchange-correlation functional. Based on the density functional\nperturbation theory and inverse Kohn-Sham method, a model has been developed that can\nbe used to improve an approximate relativistic EDF.\nUsing the method introduced in Ref. [27] for non-relativistic functionals, and based on the\ndensity functional perturbation theory, we have derived Eq. (20) that is used to compute the\n\frst-order correction to an approximate zeroth-order functional. The input to this equation\nare the exact ground-state densities and the known functional, as well as the exact single-\n19012 3 45 6 7 8 9 100.040.080.120.160.20ρV (fm-3)\nFinal\nInitial\nDD-PC1\n012 3 45 6 7 8 9 100.040.080.120.160.20\n012 3 45 6 7 8 9 10\nr (fm)0.040.080.120.160.20ρV (fm-3)\n012 3 45 6 7 8 9 10\nr (fm)0.040.080.120.160.20N=Z=8 N=Z=20\nN=Z=28 N=Z=50(a) (b)\n(c) (d)FIG. 6: (Color online). The vector densities of the four symmetric systems: N=Z= 8,\nN=Z= 20, N=Z= 28, and N=Z= 50. The dashed red curves are the densities that\ncorrespond to the unperturbed initial functional E(0)\nintshown in Eq. (41). The dot-dashed green\nand solid black curves denote the densities obtained with the target functional DD-PC1 and the\n\fnal results of the IKS + DFPT calculation, respectively.\nparticle energies that are also implicit functionals of the densities. We then use the inverse\nKohn-Sham (IKS) method to calculate these single-particle energies starting from given\nground-state densities.\nIn practice, one must assume a certain ansatz for the \frst-order correction to the Kohn-\nSham exchange-correlation functional, and use empirical exact ground-state densities of\n\fnite systems to determine the corresponding phenomenological parameters. In case the\n\frst-order functional does not reproduce the exact densities to a desired level of accuracy,\nthe functional can be further improved in an iterative procedure, in which the \frst-order\nfunctional obtained in each iteration is considered as the known functional for the next\niteration.\nThe model has been tested using the relativistic functional DD-PC1 as the exact target\nfunctional. A simpli\fed form of DD-PC1 has been employed for the known functional.\nAssuming for the \frst-order correction the same functional form as in the remaining part of\nDD-PC1, the method described above has been applied to determine three parameters. By\n20employing only three \fnite N=Zsystems, and with less than ten iterations, the resulting\nparameters of the \frst-order correction are found in excellent agreement with the original\nparameters of the functional. In a further illustrative calculation the target functional has\nbeen approximated by a di\u000berent functional form, namely, a quadratic polynomial in the\ndensities, determined by four parameters of the scalar and vector KS potentials. Even\nthough the assumed density dependence di\u000bers from that of the target functional DD-PC1,\nnevertheless the model accurately reproduces the density-dependent coupling functions, as\nwell as the target densities of four N=Zsystems.\nAs noted in the introduction, the reason for considering relativistic functionals is that\nthey automatically take into account the nuclear spin-orbit potential. The inclusion of this\nterm in the nuclear KS potential is crucial to reproduce the empirical magic numbers and\nshell gaps, and yet in the non-relativistic case the spin-orbit term cannot be determined\nby the IKS method because there is no information on the corresponding density. The\nrelativistic formulation does not provide a direct solution though. The reason is that the\nspin-orbital potential emerges as a constructive combination of the scalar and vector nucleon\npotentials, but the corresponding scalar density does not represent an observable. This\nbrings us to the fact that accurate data exists only for charge (proton) densities, while in\nthe IKS construction of the potential we need not only the isoscalar vector and scalar, but\nalso the isovector densities. A possible solution would be to combine the model developed in\nthis work, which utilizes empirical exact densities of \fnite nuclei, with the equations of state\nof isospin symmetric and isospin asymmetric nuclear matter. Namely, data on the proton\nvector densities in \fnite nuclei can be used together with the (microscopic) equations of state\nof nuclear matter to determine the isoscalar-scalar and isovector channels of the Kohn-Sham\npotential. Work along these lines is in progress.\nAcknowledgments\nThis work has been supported in part by the Croatian Science Foundation under the\nproject Uncertainty quanti\fcation within the nuclear energy density framework (IP-2018-\n01-5987). It has also been supported by the QuantiXLie Centre of Excellence, a project\nco-\fnanced by the Croatian Government and European Union through the European Re-\ngional Development Fund - the Competitiveness and Cohesion Operational Programme\n21(KK.01.1.1.01.0004), the RIKEN iTHEMS program, the RIKEN Pioneering Project: Evo-\nlution of Matter in the Universe, and the JSPS Grant-in-Aid for Scienti\fc Research (Grant\nNos. 18K13549, 19J20543, and 20H05648).\n[1] T. Nakatsukasa, K. Matsuyanagi, M. Matsuo, and K. Yabana, Rev. Mod. Phys. 88, 045004\n(2016).\n[2] N. Schunck, ed., Energy Density Functional Methods for Atomic Nuclei , IOP Expanding\nPhysics (Bristol, UK: IOP Publishing)\n[3] T. Nakatsukasa, T. Inakura, and K. Yabana, Phys. Rev. C 76, 024318 (2007).\n[4] H. Liang, T. Nakatsukasa, Z. Niu, and J. Meng, Phys. Rev. C 87, 054310 (2013).\n[5] T. Nik\u0014 si\u0013 c, N. Kralj, T. Tuti\u0014 s, D. Vretenar, and P. Ring, Phys. Rev. C 88, 044327 (2013).\n[6] A. Arima and F. Iachello, Phys. Rev. Lett. 35, 1069 (1975).\n[7] K. Nomura, D. Vretenar, Z. P. Li, and J. Xiang, Phys. Rev. C 102, 054313 (2020).\n[8] D. L. Hill and J. A. Wheeler, Phys. Rev. 89, 1102 (1953).\n[9] J. J. Gri\u000en and J. A. Wheeler, Phys. Rev. 108, 311 (1957).\n[10] J. Zhao, T. Nik\u0014 si\u0013 c, D. Vretenar, and S.-G. Zhou, Phys. Rev. C 101, 064605 (2020).\n[11] R. J. Furnstahl, Eur. Phys. J. A 56, 85 (2020).\n[12] T. Nik\u0014 si\u0013 c and D. Vretenar, Phys. Rev. C 94, 024333 (2016).\n[13] T. Nik\u0014 si\u0013 c, M. Imbri\u0014 sak, and D. Vretenar, Phys. Rev. C 95, 054304 (2017).\n[14] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).\n[15] W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).\n[16] W. Kohn, Rev. Mod. Phys. 71, 1253 (1999).\n[17] Y. Wang and R. G. Parr, Phys. Rev. A 47, R1591 (1993).\n[18] Q. Zhao and R. G. Parr, J. Chem. Phys. 98, 543 (1993).\n[19] R. van Leeuwen and E. J. Baerends, Phys. Rev. A 49, 2421 (1994).\n[20] Q. Zhao, R. C. Morrison, and R. G. Parr, Phys. Rev. A 50, 2138 (1994).\n[21] D. S. Jensen and A. Wasserman, Int. J. Quantum Chem. 118, e25425 (2018).\n[22] A. Kumar, R. Singh, and M. K. Harbola, J. Phys. B: At. Mol. Opt. Phys. 52, 075007 (2019).\n[23] B. Kanungo, P. M. Zimmerman, and V. Gavini, Nat. Commun. 10, 4497 (2019).\n[24] S. Nam, R. J. McCarty, H. Park, and E. Sim, \\Kohn-Sham Inversion Toolkit,\" (2020),\n22arXiv:2008.08783 [physics.comp-ph].\n[25] L. Garrigue, \\Building Kohn-Sham potentials for ground and excited states,\" (2021),\narXiv:2101.01127 [math-ph].\n[26] G. Accorto, P. Brandolini, F. Marino, A. Porro, A. Scalesi, G. Col\u0012 o, X. Roca-Maza, and E.\nVigezzi, Phys. Rev. C 101, 024315 (2020).\n[27] T. Naito, D. Ohashi, and H. Liang, J. Phys. B, At. Mol. Opt. Phys. 52, 245003 (2019).\n[28] J. W. Negele and D. Vautherin, Phys. Rev. C 5, 1472 (1972).\n[29] P.-G. Reinhard and C. Toep\u000ber, Int. J. Mod. Phys. E 3, 435 (1994).\n[30] R. M. Dreizler and E.K.U. Gross, Density functional theory: an approach to the quantum\nmany-body problem, Springer-Verlag, Berlin, 1990.\n[31] P.-G. Reinhard, Rep. Prog. Phys. 52, 439 (1989).\n[32] H.-P. Duerr, Phys. Rev. 103, 469 (1956).\n[33] B.D. Serot and J.D. Walecka, Adv. Nucl. Phys. 16, 1 (1986).\n[34] P. Ring, Prog. Part. Nucl. Phys. 37, 193 (1996).\n[35] T. Nik\u0014 si\u0013 c, D. Vretenar, P. Ring, Phys. Rev. C 78, 034318 (2008).\n[36] W. Koepf, P. Ring, Z. Phys. A 339, 81 (1991).\n[37] A. Baran, A. Bulgac, M. McNeil Forbes, G. Hagen, W. Nazarewicz, N. Schunck, M.V. Stoitsov,\nPhys. Rev. C 78, 014318 (2008).\n23"    },    {        "title": "2104.13616v1.Auxiliary_many_body_wavefunctions_for_TDDFRT_electronic_excited_states__Consequences_for_the_representation_of_molecular_electronic_transitions.pdf",        "content": "arXiv:2104.13616v1  [physics.chem-ph]  28 Apr 2021Auxiliary many-body wavefunctions for TDDFRT electronic e xcited states\nConsequences for the representation of molecular electron ic transitions\nThibaud Etienne∗\nICGM, University of Montpellier, CNRS, ENSCM, Montpellier , France\nThis contribution reports the study of a set of molecular ele ctronic-structure reorganization\nrepresentations related to light-induced electronic tran sitions, modeled in the framework of\ntime-dependent density-functional response theory. More precisely, the work related in this paper\ndeals with the consequences, for the electronic transition s natural-orbital characterization, that are\ninherent to the use of auxiliary many-body wavefunctions co nstructed a posteriori and assigned to\nexcited states — since time-dependent density-functional response theory does not provide excited\nstate ansätze in its native formulation. Three types of such auxiliary many-body wavefunctions\nare studied, and the structure and spectral properties of th e relevant matrices (the one-electron\nreduced difference and transition density matrices) is disc ussed and compared with the native\nequation-of-motion time-dependent density functional re sponse theory picture of an electronic\ntransition — we see for instance that within this framework t he detachment and attachment density\nmatrices can be derived without diagonalizing the one-body reduced difference density matrix.\nThe common “departure/arrival” wavefunction-based repre sentation of the electronic transitions\ncomputed with this method is discussed, and two such common “ departure/arrival” density-based\npictures are also compared.\nKeywords: Molecular electronic excited states; electroni c-structure reorganization; one-body\nreduced density matrices.\nI.Introduction\nTime-dependent density-functional response theory\n(TDDFRT [ 1–5]) is probably the most used electronic\nexcited-state calculation method. Such a method\ngives access to difference and transition properties —\nthe first allows the comparison between the properties\nin the ground and in the excited state, while the sec-\nond informs about the coupling between the states,\nand the response of the system to the light-induced\nperturbation of the system — but it does not come\nwith an ansatz for the excited states that are com-\nputed, at least in its equation-of-motion (EOM) for-\nmulation. Knowledge of such an ansatz would how-\never be very useful for analyzing the nature of excited\nstates [ 6–8], but also in the context of non-adiabatic\nexcited-state dynamics [ 5,9,10], for it allows the com-\nputation of quantities such as the non-adiabatic cou-\npling between states [ 11,12]. Therefore, multiple re-\nsearch groups have worked out expressions for approx-\nimate, auxiliary many-body wavefunctions (AMBW)\nthat are constructed a posteriori , i.e., based on the\ndata issued from the excited-state TDDFRT calcula-\ntions. The “existence” of certain of these wavefunc-\ntions is sometimes conditioned by some constraints\nthat will also be recalled in this paper.\nThe first of these AMBW’s [ 13] has been pro-\nposed by Casida in his famous paper where he ex-\nposes his transposition of the time-dependent Hartree-\nFock (TDHF [ 4,14]) excited-state calculation proce-\ndure to the realm of electron densities, introducing the\nTDDFRT method [ 1]. The construction of his AMBW\n— which has the same structure as a configuration in-\nteraction singles (CIS) wavefunction [ 4,15–18] — is\nphysically motivated, and related to the expression\n∗thibaud.etienne@umontpellier.frof the TDDFRT transition moments, but as we will\nsee in this report, the excited states constructed with\nthis procedure are not mutually orthogonal, and an\northogonalization procedure makes the nature of the\nconsidered states dependent on the number and na-\nture of all the states that are initially computed. This\nansatz has been extensively used in the context of\nexcited-state dynamics simulations [ 19–25].\nThe second AMBW was brought by Luzanov and\nco-workers by solving an alternative TDDFRT re-\nsponse equation, yielding states that are normalized\nand mutually orthogonal [ 7]. The drawback of this\nprocedure is that it produces coupled pairsof ansätze\nrather than a single electronic excited-state wavefunc-\ntion.\nThe third AMBW analyzed in this paper comes\nfrom the work of Subotnik and co-authors [ 26]. It is\nbased on Hellmann-Feynman theorem, includes Pulay\nterms associated with an atom-centered basis, as well\nas a correction ensuring translational invariance, and\nensures a proper behavior near the crossing between\ntwo states and in the complete basis limit. However,\nas for the previous AMBW, these ansätze come in\npairs.\nThe analysis and comparison of the EOM-TDDFRT\npicture of an electronic transition with the AMBW\nproperties will be performed in terms of some rele-\nvant one-body reduced density matrices (1–RDM’s):\nthe one-electron reduced unrelaxed difference density\nmatrix (1–DDM) and the one-body reduced transi-\ntion density matrix (1–TDM) [ 3,27–30], and their\nspectral decomposition yielding the so-called natural\ndifference (NDO’s) and transition (NTO’s) orbitals\n[3,15,18,27,28,31].\nSince the principal AMBW dealt with in this report\ntakes the form of a linear combination of Fock states\nthat are singly-excited relatively to the Fermi vacuum,\none section of this report will introduce a quick review\nof the basic properties of such ansätze: the CIS wave-2\nfunction itself will be introduced, and the properties\nof CIS electronic transitions will be given in terms of\nnatural orbitals, before commenting the expression of\nthe CIS transition dipole moments — those consid-\nerations are useful for deriving Casida’s AMBW in\nappendix VI B— and extending those considerations\nbeyond CIS.\nCasida’s AMBW will then be introduced and\nits properties studied. The structure and natural-\norbital properties of this model will be compared\nwith the EOM–TDDFRT picture, and the uniqueness\nof Casida’s AMBW will be discussed, among other\nthings such as the consequences of this representation\nfor the one-body reduced transition density kernel.\nFinally, the AMBW introduced by Luzanov and co-\nworkers, and the one introduced by Subotnik and co-\nworkers, will be exposed and analyzed. The detach-\nment/attachment density matrices [ 3,32] will be de-\nrived for the AMBW of Luzanov and co-workers, while\nthe AMBW of Subotnik and co-workers will be showed\nshowed to elude the native TDDFRT de-excitation\ncontributions when used for deriving the 1–TDM.\nII.Hypotheses and notations\nRefs. [ 28,29] are two strong prerequisites papers for\nthis contribution.\nBold Greek letters and bold, upper case letters from\nLatin alphabet will denote matrices with more than\none column. For instance, while (x)iawill denote the\niacomponent of vector xwhereiais assimilated to an\nindividual integer, (˜X)i,awill denote the i×aelement\nof matrix ˜Xand(A)ia,jbwill denote the (ia)×(jb)\nelement of matrix A, where again iaandjbare as-\nsimilated to individual integers.\nIn this contribution we will consider N–electron sys-\ntems. For any electronic quantum state considered\nbelow, theNelectrons are distributed in Lorthonor-\nmal real-valued spinorbitals ( L≥N), each of them\nbeing denoted by an indexed ϕsymbol and defined\nonS4:=R3×{↑,↓}.\nAny molecular electronic ground-state wavefunc-\ntion value considered in this manuscript is a single\nN–dimensional Slater determinant: When Lspinor-\nbitals are being produced when solving the Kohn-\nSham equations for optimizing the ground state wave-\nfunction, only Nare “occupied” with occupation num-\nber equal to unity, while (L−N)are “unoccupied” (or\n“virtual”) with their occupation number equal to zero\nin the ground state.\nWe define three integer intervals:\nI:=/llbracket1,N/rrbracket,\nC:=/llbracket1,L/rrbracket,\nA:=C\\I.\nThe nomenclature is inspired from the traditional as-\nsignment of iandaindices for “occupied” ( ϕiwith\ni∈I) and “virtual” ( ϕawitha∈A) spinorbitals re-\nspectively. The Cinteger interval is then seen as the\nset of indices covering the total canonical space (oc-cupied plus virtual):\nC:= (ϕr)r∈C.\nWe will also assume that from any of the excited-state\nquantum chemical calculation method considered in\nthis contribution we have produced a finite set of M\nmolecular electronic excited states. We then set\nS:=/llbracket1,M/rrbracket.\nFor any electronic transition from the ground elec-\ntronic state |ψ0/a\\}bracketri}htto an excited electronic state |ψm/a\\}bracketri}ht\n(withm∈S), when “ 0m” is used as a subscript or a\nsuperscript, it is intended to be representative of the\n|ψ0/a\\}bracketri}ht −→ |ψm/a\\}bracketri}httransition — see the related comment\nin ref. [ 28] where it is explained that such an im-\nplicit convention is not universal at all. When there is\nno such “ 0m” subscript or superscript, the transition\nis implicitely supposed to be taking place from the\nground electronic state to a higher-energy electronic\nstate.\nThrough all the text, the “ ⊤” superscript will de-\nnote the transpose of a matrix, and should not be con-\nfused with the “ T” superscript, standing for “transi-\ntion” — often used to denote the one-particle reduced\ntransition density matrix.\nIII.A quick review of CIS(–like) many-body\nwavefunctions features\nIn this section we will start by introducing the CIS(–\nlike) wavefunctions, i.e., wavefunctions being defined\nas or having the same properties as CIS wavefunctions.\nWe will then review some basic properties related to\nthose particular wavefunctions.\nA.CIS(–like) wavefunctions\nWe now examine the case of any single-reference elec-\ntronic excited state wavefunction value written as a\nlinear combination of singly-excited Slater determi-\nnants\n/a\\}bracketle{ts1,...,sN|˜ψζ\nm/a\\}bracketri}ht=N/summationdisplay\ni=1L/summationdisplay\na=N+1(tm)ia/a\\}bracketle{ts1,...,sN|ψa\ni/a\\}bracketri}ht\n(1)\nwith(m∈S),\n∀i∈I,si∈S4,\ntm∈RN(L−N)×1,\nand\n∀(i,a)∈I×A,|ψa\ni/a\\}bracketri}ht:=ˆa†ˆi|ψ0/a\\}bracketri}ht. (2)\nThetmvector in ( 1) is released during the excited-\nstate calculation. In ( 2),ˆiis the second quantiza-\ntion operator annihilating an electron in spinorbital\nϕiandˆa†is the second quantization operator creat-\ning an electron in spinorbital ϕa.\nSince every |ψa\ni/a\\}bracketri}htconfiguration in ( 2) is orthogonal\nto|ψ0/a\\}bracketri}htwe find, using the definition of the general3\n1–TDM in [ 28,29] for an orthonormal basis of real-\nvalued spinorbitals, i.e.,\n∀m∈S,∀(r,s)∈C2,/parenleftbig\nγT\n0m/parenrightbig\ns,r=/a\\}bracketle{tψ0|ˆr†ˆs|ψm/a\\}bracketri}ht,\nthat, for our trial wavefunction ˜ψζ\nm,\n∀(i,a)∈I×A,(˜γT\nζ,0m)a,i= (tm)ia, (3)\n∀(r,i)∈C×I,(˜γT\nζ,0m)r,i= 0, (4)\n∀(a,b)∈A2,(˜γT\nζ,0m)a,b= 0. (5)\nIf|˜ψζ\nm/a\\}bracketri}htis not normalized, we can write the correspond-\ning normalized ansatz as\n|ψζ\nm/a\\}bracketri}ht=z−1/2\nζ,m|˜ψζ\nm/a\\}bracketri}ht, (6)\nwith\nzζ,m:=t⊤\nmt†\nm.\nIn what follows, any normalized ansatz that can be\nwritten as a linear combination of N-electron Fock\nstates singly-excited relatively to the Fermi vacuum\nwill be called a ζ–ansatz (or ζ–type ansatz).\nB.Properties of ζ–type ansätze\nLetWζdenote the set of all N–electronζ–type an-\nsätze defined as in the previous subsection for any\nL–dimensional orthonormal spinorbital space. Any\nelement ofWζhas the properties depicted in this sub-\nsection. For the sake of readability, we will consider\n“any” electronic transition — i.e., a generic transition\namong the Mavailable — so we will drop the “ 0m”\nsubscripts through all this subsection.\nWe define the 1–TDM corresponding to such a ζ–\nansatz as\nγT\nζ:=z−1/2\nζ˜γT\nζ∈RL×L.\nLet0obe theN×Nzero matrix, 0ovtheN×(L−N)\nzero matrix, and 0vthe(L−N)×(L−N)zero matrix.\nWe see from ( 3), (4), and ( 5), thatγT\nζis partitioned\nas\nγT\nζ=/parenleftbigg0ov0ov\nT⊤v0v/parenrightbigg\nwith\nT∈RN×(L−N),\nand\n∀(i,a)∈I×A, z−1/2\nζ(t)ia= (γT\nζ)a,i= (T)i,a−N.\nWe are now going to review few properties mostly re-\nlated with the spectral decomposition of the CIS one-\nbody reduced difference and transition density matri-\nces.\nProposition III.1 Anyζ–type 1–TDM is nilpotent.The proof to this known result is trivial: multiplying\nany of the γT\nζabove by itself yields the L-dimensional\nzero square matrix. If we write γT\nζas theL–tuple of\nits columns, i.e.,\nγT\nζ= (gr)r∈C,\nwe find that\n{gr:r∈C} ⊂ker/parenleftbig\nγT\nζ/parenrightbig\n. (7)\nwhere the kernel of γT\nζis defined as\nker/parenleftbig\nγT\nζ/parenrightbig\n=/braceleftbig\nw∈RL×1:γT\nζw=0L/bracerightbig\n.\nRelation ( 7) means that the columns of γT\nζare column\nvectors that all belong to the kernel of γT\nζ.\nGiven that the second power of any ζ–type 1–TDM\nis theL×Lzero matrix, any nthpower (with na\npositive integer superior to two) of γT\nζwill also be the\nL×Lzero matrix.\nDue to the nilpotency of ζ–type 1–TDM’s, one\nrather performs a singular value decomposition (SVD)\nof its southwest block, the rectangular T⊤. The SVD\nofT⊤reads\n∃(O,V,λ)∈RN×N×R(L−N)×(L−N)×RN×(L−N)\n+,\nT⊤=VλO⊤\n(8)\nwithO⊤=O−1,V⊤=V−1. Theλmatrix satisfies\n∀(p,q)∈I×/llbracket1,(L−N)/rrbracket,(p/\\e}atio\\slash=q) =⇒(λ)p,q= 0.\nWe then set\n∀i∈/llbracket1,q/rrbracket, λi:= (λ)i,i\nwhere\nq= min(N,(L−N)). (9)\nColumns of Vcontain the components of the so-called\nleft (v) NTOs in the (ϕr)r∈Abasis, and the columns\nofOcontain the components of the right ( o) NTOs\nin the(ϕr)r∈Ibasis: for every pin/llbracket1,q/rrbracket, we build\nϕo\np=N/summationdisplay\nl=1(O)l,pϕl\nand\nϕv\np=L−N/summationdisplay\nl=1(V)l,pϕ†\nN+l.\nAccordingly, we see that there exists an exact and\nunique pairing between left and right NTOs through\ntheir singular values, so that the canonical picture of a\nlinear combination of singly-excited configurations in\nthe(ϕr)r∈Cbasis can be transposed in the NTOs ba-\nsis without approximation. Given that the problem is\noriginally expressed in a finite-dimensional local basis\nofLone-body wavefunctions, the NTO picture is the4\nmost compact representation of the electronic transi-\ntion — the expansion is reduced from N×(L−N)\nterms to min(N,(L−N))— with the exact conserva-\ntion of the ζ–ansatz nature.\nAccording to ( 8), we have that\nO⊤TT⊤O=λ⊤λ∈RN×N\n+, (10)\nV⊤T⊤TV=λλ⊤∈R(L−N)×(L−N)\n+ . (11)\nCombining lemma VI.3with time-independent Wick’s\ntheorem for fermions we find\nProposition III.2 ζ–type 1–DDM is the direct sum\nof a negative semidefinite and a positive semidefinite\nmatrix.\nThis was originally stated in ref. [ 32]. Complete proof\nis given in ref. [ 29] where the following structure for\nγ∆\nζwas derived:\nγ∆\nζ=/parenleftbig\n−TT⊤/parenrightbig\n⊕T⊤T. (12)\nUsing again lemma VI.3we find\nCorollary III.1 ζ–type detachment and attachment\ndensity matrices can be derived from the transition\ndensity matrix without matrix diagonalisation.\nDue to the definition of the detachment and attach-\nment density matrices in terms of the sign of the oc-\ncupation numbers, i.e., eigenvalues of the one-body\nreduced difference density matrix, we find that\nTT⊤⊕0v=γd\nζ,\n0o⊕T⊤T=γa\nζ,\nwhich was also originally stated in ref. [ 32].\nProposition III.3 The non-zero natural difference\nand transition orbitals components are identical.\nDue to the structure of γ∆\nζin (12), we have the fol-\nlowing possible matrix of eigenvectors\nU=O⊕V (13)\nand, given ( 10) and ( 11), we find the following possible\neigendecomposition:\nU⊤γ∆\nζU=/parenleftbig\n−λ⊤λ/parenrightbig\n⊕λλ⊤\nwhich means that the CIS natural difference orbitals\nare paired just as the natural transition orbitals are.\nWe finally establish that the trace of ζ–type detach-\nment and attachment 1–RDM’s, i.e., ϑζ, is equal to\nϑζ= tr/parenleftbig\nλλ⊤/parenrightbig\n= tr/parenleftbig\nλ⊤λ/parenrightbig\n=q/summationdisplay\ni=1λ2\ni.\nC.Electronic transition dipole moment of\nCIS(–like) states and beyond\nForζ–type excited-state wavefunctions, the wcompo-\nnent (w∈ {x,y,z}) of the electronic transition dipolemoment between ground state |ψ0/a\\}bracketri}htand excited state\n|ψζ\nm/a\\}bracketri}htreads\n/a\\}bracketle{tψ0|ˆw|ψζ\nm/a\\}bracketri}ht=L/summationdisplay\nr,s=1/a\\}bracketle{tϕr|ˆw|ϕs/a\\}bracketri}ht/a\\}bracketle{tψ0|ˆr†ˆs|ψζ\nm/a\\}bracketri}ht\n=L/summationdisplay\nr,s=1N/summationdisplay\ni=1L/summationdisplay\na=N+1(tm)ia/a\\}bracketle{tϕr|ˆw|ϕs/a\\}bracketri}ht/a\\}bracketle{tψ0|ˆr†ˆsˆa†ˆi|ψ0/a\\}bracketri}ht\nwhere (m∈S), and\nˆw=N/summationdisplay\ni=1wi.\nIf we write npthe occupation number of spinorbital\nϕp(withp∈C), we deduce from\n/a\\}bracketle{tψ0|ˆr†ˆsˆa†ˆi|ψ0/a\\}bracketri}ht=δr,sδi,anrnsnani\n+δr,iδs,anrni(1−ns)(1−na),\nand\n(I∩A=∅)⇐⇒[∀(i,a)∈I×A, δi,a= 0],\nthat thewcomponent of the electronic transition\ndipole moment becomes\n/a\\}bracketle{tψ0|ˆw|ψζ\nm/a\\}bracketri}ht=N/summationdisplay\ni=1L/summationdisplay\na=N+1/a\\}bracketle{tϕi|ˆw|ϕa/a\\}bracketri}ht(tm)ia\nSetting\n∀(i,a)∈I×A,(pw)ia:=/a\\}bracketle{tϕi|ˆw|ϕa/a\\}bracketri}ht,(14)\nwe find\n/a\\}bracketle{tψ0|ˆw|ψζ\nm/a\\}bracketri}ht=p⊤\nwt†\nm.\nNote finally that if the excited-state ansatz did include\nadditional terms with higher level of excitation, as in\n|ψm/a\\}bracketri}ht=N/summationdisplay\ni=1ˆT(i)\nm|ψ0/a\\}bracketri}ht (15)\nwith, for the first two excitation operators,\nˆT(1)\nm=N/summationdisplay\ni=1L/summationdisplay\na=N+1(t(1)\nm)iaˆa†ˆi,\nˆT(2)\nm=N/summationdisplay\ni,j=1L/summationdisplay\na,b=N+1(t(2)\nm)iajbˆb†ˆjˆa†ˆi,\nthe structure of the 1–TDM would have remained un-\nchanged and would solely include components from\nt(1)\nm, and the expression of /a\\}bracketle{tψ0|ˆw|ψm/a\\}bracketri}htwould solely\ninclude the single-excitation contributions and read\np⊤\nwt†(1)\nm.\nIV.Comparison of the EOM and auxiliary\nwavefunction pictures\nThis section introduces and compares the two pictures\n(EOM [ 13,33] and AMBW — with three variants) one\nmeets when dealing with TDDFRT.5\nA.EOM representation of an electronic\ntransition\nLetˆTmbe a nilpotent, number- and norm-conserving\ntransition operator corresponding to the transition be-\ntween two quantum states ( m∈S):\nˆTm=|ψm/a\\}bracketri}ht/a\\}bracketle{tψ0|,ˆT†\nm=|ψ0/a\\}bracketri}ht/a\\}bracketle{tψm|\nwith the constraint that the two states are normalized\nand orthogonal to each other:\n/a\\}bracketle{tψ0|ψm/a\\}bracketri}ht=δ0,m.\nIn these conditions, we can assess the equation-of-\nmotion 1–DDM elements [ (r,s)∈C2]\n/angbracketleftBig\nψ0/vextendsingle/vextendsingle/vextendsingle/bracketleftBig\nˆT†\nm,/bracketleftBig\nˆr†ˆs,ˆTm/bracketrightBig/bracketrightBig/vextendsingle/vextendsingle/vextendsingleψ0/angbracketrightBig\n=/angbracketleftbig\nψm/vextendsingle/vextendsingleˆr†ˆs/vextendsingle/vextendsingleψm/angbracketrightbig\n−/angbracketleftbig\nψ0/vextendsingle/vextendsingleˆr†ˆs/vextendsingle/vextendsingleψ0/angbracketrightbig\n= (γ∆\n0m)r,s\nand 1–TDM elements\n(γT\n0m)s,r=/angbracketleftBig\nψ0/vextendsingle/vextendsingle/vextendsingle/bracketleftBig\nˆr†ˆs,ˆTm/bracketrightBig/vextendsingle/vextendsingle/vextendsingleψ0/angbracketrightBig\n.\nB.The case of EOM-TDDFRT\nIn an EOM formulation (denoted by a “ η” symbol and\nsometimes called native in the following) of TDHF,\nthe transition operator ˆTin the derivation of the ma-\ntrix elements is substituted by\nˆTη:=N/summationdisplay\ni=1L/summationdisplay\na=N+1/parenleftBig\nxiaˆa†ˆi−yiaˆi†ˆa/parenrightBig\n,\ni.e., a truncated CI excitation operator ( x–hole-\nparticle ,hp), corrected by deexcitation contributions\n(y–particle-hole ,ph). Here again, for the sake of\nreadability of the equations, we will consider a general\nelectronic transition between the ground state and any\nexcited state of a molecule. The components of the x\nandyvectors, solutions to the response equation\n/parenleftbigg\nA∗B\nB∗A/parenrightbigg\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nΛ/parenleftbigg\nx\ny/parenrightbigg\n=ω/parenleftbigg\nI−0\n0−I/parenrightbigg\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nJ/parenleftbigg\nx\ny/parenrightbigg\n,(16)\nwith, for every (i,j)inI2and every (a,b)inA2,\n(A)ia,jb=/angbracketleftBig\nψ0/vextendsingle/vextendsingle/vextendsingle/bracketleftBig\nˆi†ˆa,/bracketleftBig\nˆH,ˆb†ˆj/bracketrightBig/bracketrightBig/vextendsingle/vextendsingle/vextendsingleψ0/angbracketrightBig\n, (17)\n(B)ia,jb=/angbracketleftBig\nψ0/vextendsingle/vextendsingle/vextendsingle/bracketleftBig\nˆi†ˆa,/bracketleftBig\nˆH,ˆj†ˆb/bracketrightBig/bracketrightBig/vextendsingle/vextendsingle/vextendsingleψ0/angbracketrightBig\n, (18)\ncan be recast into square matrices:\nX′=/parenleftbigg\n0ovX\n0vo0v/parenrightbigg\n, (19)\nY′=/parenleftbigg\n0ovY\n0vo0v/parenrightbigg\n, (20)\nwith\nX∈RN×(L−N),\nY∈RN×(L−N),and\n∀(i,a)∈I×A,xia= (X′)i,a= (X)i,a−N,(21)\nyia= (Y′)i,a= (Y)i,a−N.(22)\nNote that xandysatisfy\nx⊤x−y⊤y= 1. (23)\nIn (16),ωis the transition energy between the ground\nstate and the excited state of interest and ˆHis the\nmolecular Hamiltonian. Iand0are the[N×(L−\nN)]×[N×(L−N)]identity and zero matrices, re-\nspectively. In equations ( 17) and ( 18),iaandjbare\nassimilated to individual integers (as it is done for the\ncomponents of xandy), for pointing the row (on the\nleft of the comma) and column (on the right of the\ncomma) indices of matrices AandB— see the intro-\nductory comments of this paper.\nThe native TDDFRT eigenproblem has the exact\nsame structure as depicted above. So do the follow-\ning matrices. This also stands for the Bethe-Salpeter\nexcited-state claculation method [ 34]. Hence, the\nstructure and properties of the objects discussed be-\nlow are identical in the three methods.\nIn the conditions detailed above, the transition den-\nsity matrix reads\n/parenleftbigg0oY\nX⊤0v/parenrightbigg\n=γT\nη∈RL×L. (24)\nIts transpose is often seen in the litterature to de-\nnote the same transition — here, “ T” denotes an elec-\ntronic transition with electronic energy promotion, as\nstated at the beginning of this paper. However we\nhave showed and discussed in refs. [ 28,29] that the\ntranspose of the matrix given above corresponds to\nthe transition between the two same states but with\nthe departure/arrival states being permuted.\nProposition IV.1 TheEOM 1–DDM is the direct\nsum of a negative semidefinite matrix and a positive\nsemidefinite matrix.\nIn the conditions detailed above, the difference density\nmatrix is [ 29]\n/parenleftbig\n−XX⊤−YY⊤/parenrightbig\n⊕/parenleftbig\nX⊤X+Y⊤Y/parenrightbig\n=γ∆\nη∈RL×L\n(25)\nHence, we see that the 1–DDM and 1–TDM in\nTDDFRT and in the Bethe-Salpeter method have\nidentical structures to those in equations ( 25) and ( 24)\n[29]. According to lemma VI.4given in the appendix,\n−/parenleftbig\nXX⊤+YY⊤/parenrightbig\n/√recedesequal0,\n−−/parenleftbig\nX⊤X+Y⊤Y/parenrightbig\n/{ollowsequal0,\nwhich confirms the proposition.\nCorollary IV.1 The native TDDFRT detachment\nand attachment density matrices can be derived with-\nout matrix diagonalization.\nAccording to what precedes, and according to the def-\ninition of the detachment and attachment, we can now6\ngive the expression of the EOM detachment and at-\ntachment one-body reduced density matrices:\n/parenleftbig\nXX⊤+YY⊤/parenrightbig\n⊕0v=γd\nη,\n0o⊕/parenleftbig\nX⊤X+Y⊤Y/parenrightbig\n=γa\nη\nWe deduce from the expression of the detachment and\nattachment density matrices that\nProposition IV.2 The native TDDFRT natural dif-\nference orbitals are not paired in general.\nIndeed, unlike the CIS method for which the nega-\ntive semidefinite and positive semidefinite blocks share\nmin(N,(L−N))eigenvalues according to lemma VI.2,\nhere we do not have the same configuration since we\nhave two sums of products of the type that was en-\ncountered in the negative semidefinite and positive\nsemidefinite blocks in CIS, which means here that the\nη–type detachment/attachment 1–RDM’s might not\nshare eigenvalues — so the natural difference orbitals\nare not necessarily paired. We also find that the inte-\ngral of the detachment/attachment density satisfies\nϑη≥1.\nThe square, non-hermitian 1–TDM has non-zero en-\ntries covering the whole canonical space. Hence, it\nis not meaningful — unless the ycomponents are all\nclose to zero — to speak in terms of “occupied” and\n“virtual” NTOs when performing its singular value\ndecomposition. Its left and right singular vectors\nhave components in both the occupied and virtual\nspaces, so in the general case picturing a TDDFRT\nelectronic transition as a composition of right-to-left\nNTOs single-electron promotions as in CIS — the\nright-singular vectors correspond to the “hole” of the\nhole/electron model while the left-singular vectors\ncorrespond to the “electron” of the hole/electron sim-\nplified representation [ 28]. Such an approximate pic-\nture is often met in the litterature: The elements of\nthe one-body reduced transition density matrix are\nseen as the singles coefficients in the single-electron-\nreplacement truncation of a complete expansion of\nthe type encountered in ( 15), and the NTOs used to\npresent this expansion in second quantization as\n|ψη′\nm/a\\}bracketri}ht ≃L/summationdisplay\ni=1λη\niˆl†\niˆr†\ni|ψ0/a\\}bracketri}ht+···\nwhere theλη\nicoefficients are the singular values corre-\nsponding to the ithcouple of left (“ l”) and right (“ r”)\nsingular vectors of γT\nη. In the expression above, we\nhave used the ˆl†\niandˆrioperators to signify that they\nare second quantization operators in the NTOs basis:\nˆl†\nicreates an electron in the ithleft-NTO and ˆrianni-\nhilates an electron in the ithright-NTO. The picture\ndepicted here consists in a series of\ntransition-hole NTO−→transition-electron NTO\nrepresentations. However, such a picture implies that,\nin EOM-TDDFRT, and consistently with what has\njust been written, the pseudo-ansatz given above, oncethe NTOs are backtransformed in the canonical basis\n— we come back to the second-quantization operators\nwe met before —, reads\n|ψη′\nm/a\\}bracketri}ht ≃L/summationdisplay\nr=1L/summationdisplay\ns=1/parenleftbig\nγT\nη/parenrightbig\nr,sˆr†ˆs|ψ0/a\\}bracketri}ht+···\n=N/summationdisplay\ni=1L/summationdisplay\na=N+1/parenleftBig\nX⊤/parenrightBig\na−N,iˆa†ˆi|ψ0/a\\}bracketri}ht\n+N/summationdisplay\nj=1L/summationdisplay\nb=N+1(Y)j,bˆj†ˆb|ψ0/a\\}bracketri}ht+···\nAny 1–TDM element using this pseudo-ansatz will\nread\n/parenleftbig\nγT\nη′/parenrightbig\ns,r=/a\\}bracketle{tψ0|ˆr†ˆs|ψη′\nm/a\\}bracketri}ht,\nwhich will miss all the Ycoefficients. Therefore, the\n1–TDM obtained with this pseudo-ansatz will differ\nfrom the 1–TDM that was used for producing it, which\nis in contradiction with what was stated about any\nsuch ansatz [see the discussion after ( 15)]. This means\nthat, in the general case, picturing an electronic tran-\nsition as a linear combination single-NTO population\ntransfers — or using the 1–TDM elements as CIS-\nlike coefficients — can be problematic. Since such an\nansatz in TDDFRT leads to losing the de-excitation\ncontributions, there has been a quest for proposing\nmeaningful ansätze — this is the so-called TDDFRT\n“assignment problem”, that will be discussed in the\nnext section.\nWhen the approximate picture of the transition is\ngiven in terms of one-body reduced density functions\nrather than one-body wavefunctions, the\ntransition-hole density→transition-electron density\nrepresentation can appear. Since the native TDDFRT\ntransition-hole ( h) and the transition-electron ( e) one-\nbody reduced density matrices have the following\nstructure\nγh\nη:=XX⊤⊕Y⊤Y,\nγe\nη:=YY⊤⊕X⊤X,\nwe see that, though the trace of γd\nη,γa\nη,γh\nη, andγe\nη\nare equal — and superior or equal to unity —, the\ndetachment (respectively, attachment) density matrix\nsolely involves the orbitals that are occupied (respec-\ntively, unoccupied) in the ground state, while both\nthe transition-hole and the transition-electron den-\nsity matrices involve orbitals from the full canonical\nspace. There would be a true interpretative ambi-\nguity when comparing the hypothetical hole/electron\nand the detachment/attachment pictures if some dis-\nambiguation had not been proposed relatively to that\nrepresentation matter [ 28], which suggested that the\nhole/electron density matrices and the NTOs are sim-\nply not meant to bear the same physical information\nas the detachment/attachment density matrices and\nthe NDOs, concluding that, since NDOs are, in gen-\neral, unpaired, there is no exact “departure/arrival”\npicture that can be drawn from natural transition7\nor difference orbitals in the general case, and that\namong the objects discussed here only the detach-\nment/attachment densities should be seen as being\npart of a “departure/arrival” simplified picture.\nIn the absence of de-excitations, Yvanishes, the\nFrobenius norm of Xbecomes equal to unity, and the\nresulting representation of the electronic transition is\nexactly consistent with the one of a ζ–ansatz. We call\nthis the Tamm-Dancoff Approximation (TDA).\nNote finally that ˆTηis not used as a direct ap-\nproximation to an EOM transition operator |ψη/a\\}bracketri}ht/a\\}bracketle{tψ0|\nto write an excited state ansatz. Indeed, ˆTηis\nnot nilpotent, not number-conserving, and not norm-\nconserving — this comment is in line with our remark\nconcerning the use of the 1–TDM elements directly\nfor producing a CIS-like ansatz.\nC.Auxiliary many-body wavefunctions\nIn this section we will discuss three alternative pic-\ntures to the original equation-of-motion TDDFRT, us-\ning auxiliary many-body wavefunctions for the elec-\ntronic excited states.\n1.Casida’s assignment problem\nWe will introduce here the TDDFRT AMBW origi-\nnally proposed by Casida [ 1] and extensively used in\nthe context of non-adiabatic dynamics. After a nec-\nessary normalization, we will inspect the properties of\nthe 1–TDM and 1–DDM one can derive from inspect-\ning such an ansatz and the structure of the related\nobjects. The AMBW proposed by Casida has the fol-\nlowing expression:\n|˜ψζ/a\\}bracketri}ht:=N/summationdisplay\ni=1L/summationdisplay\na=N+1(xia+yia)|ψa\ni/a\\}bracketri}ht (26)\nobtained by recasting ( 16) into a Hermitian eigenprob-\nlem\nR+f+=ω2f+\nwith\nR+:= (A−B)1/2(A+B)(A−B)−1/2,\nf+:= (A−B)−1/2(x+y).\nAll the details for reaching the Hermitian eigenprob-\nlem and for deriving ( 26) from it are given in Ap-\npendix VI B.\nWhen inspecting ( 26), we see that multiplication of\n|˜ψζ/a\\}bracketri}htby the inverse of\nz1/2\nζ:=/radicalBig\n(x+y)†(x+y) (27)\nis required to yield a normalized ansatz |ψζ/a\\}bracketri}htbelong-\ning toWζ, with the corresponding γ∆\nζ1–DDM hav-\ning a zero trace, depicting a number-conserving elec-\ntronic transition. Indeed, without the normalization,\nthe norm of the excited-state wavefunction would bethe square root of zζfrom ( 27), and the trace of the\n1–DDM would be\ntr/parenleftbig˜γ∆\nζ/parenrightbig\n=N(zζ−1)\nwhich might differ from zero.\nThe construction of |˜ψζ/a\\}bracketri}htas in ( 26), hence of |ψζ/a\\}bracketri}ht,\nrelies on the existence of the R+matrix and the f+\nvector, so the existence criterion for |˜ψζ/a\\}bracketri}htas in ( 26)\nrelies on the positive definiteness of (A−B). Note\nthat such a condition is ensured for pure–DFT xc–\nfunctionals for which (A−B)is always diagonal and\npositive definite. The abovementioned criterion is of-\nten forgotten when using hybrid functionals, and the\nAMBW is constructed directly by taking the xand\nyvectors from ( 16) to write an ansatz, as if (A−B)\nwasa priori positive definite. Using the same manip-\nulations as in equations ( 19) to ( 22), the TDDFRT\nrenormalized ansatz |ψζ/a\\}bracketri}htproduces a traceless 1–DDM\nWd⊕Wa=zζγ∆\nζ∈RL×L\nwith\n−Wd:= (X+Y)(X+Y)⊤,\nWa:= (X+Y)⊤(X+Y).\nThe transition density matrix is the nilpotent\n/parenleftbigg\n0ovX+Y\n0vo0v/parenrightbigg\n=z1/2\nζ/parenleftbig\nγT\nζ/parenrightbig⊤∈RL×L.\nAccording to lemma VI.3(see Appendix),\n−(X+Y)(X+Y)⊤/√recedesequal0,\n−−(X+Y)⊤(X+Y)/{ollowsequal0.\nTherefore,\n(X+Y)(X+Y)⊤⊕0v=zζγd\nζ,\n0o⊕(X+Y)⊤(X+Y)=zζγa\nζ\nand theζ–type detachment and attachment 1–RDM’s\nshare eigenvalues [see lemma VI.2and equations ( 10)\nand (11)] and can be derived without requiring the\ndiagonalization of the ζ–type 1–DDM. We also have\nthatϑζ= 1.\nThe transition density matrix is nilpotent, and its\nnon-zero elements can be collected into a rectangular\nT⊤\nζmatrix:\nz−1/2\nζ/parenleftBig\nX⊤+Y⊤/parenrightBig\n=T⊤\nζ∈R(L−N)×N.\nThis matrix can be diagonalized by an SVD to pro-\nduce its leftandrightsingular vectors, i.e., the left\nandright aNTOs ( auxiliary Natural Transition Or-\nbitals), eigenvectors of γa\nζandγd\nζpaired by their\neigenvalue.8\n2.Casida’s AMBW: consequences for the one-electron\ntransition density (kernel)\nThe one-body reduced transition density kernel [ 28]\nfor the TDDFRT ζ–ansatz reads\nγT\nζ(s1;s′\n1) =L/summationdisplay\nr=1L/summationdisplay\ns=1(γT\nζ)r,sϕr(s1)ϕs(s′\n1)\n=N/summationdisplay\ni=1L/summationdisplay\na=N+1(γT\nζ)a,iϕa(s1)ϕi(s′\n1)\n=z−1/2\nmN/summationdisplay\ni=1L/summationdisplay\na=N+1(x+y)iaϕa(s1)ϕi(s′\n1).\nFor the EOM approach, it reads\nγT\nη(s1;s′\n1) =N/summationdisplay\ni=1L/summationdisplay\na=N+1xiaϕa(s1)ϕi(s′\n1)\n+N/summationdisplay\nj=1L/summationdisplay\nb=N+1yjbϕj(s1)ϕb(s′\n1).\nThough the two kernels are different, the correspond-\ning one-body transition densities are identical, up to\na scaling factor:\nz1/2\nmnT\nζ(s1) =nT\nη(s1)\n=N/summationdisplay\ni=1L/summationdisplay\na=N+1(x+y)iaϕi(s1)ϕa(s1).\n3.Orthogonalization of Casida’s AMBW’s\nIn practice there are usually more than one excited\nstate computed at the same time during a calcula-\ntion. Hence, a set of data and matrices is produced as\nthe outcome of such a calculation, and the AMBW’s\none can construct based on the proposed scheme are\nnot orthogonal to each other. After orthogonalizing\nthe set of TDDFT AMBW’s, we verify that the con-\nclusions from the previous paragraph are transferrable\nwhen orthogonalization of the excited states is done a\nposteriori .\nWe consider the Mnormalized but non-orthogonal\nansätze constructed using the procedure depicted in\nthe previous paragraph. Any of these ansätze reads\n|ψζ\nm/a\\}bracketri}ht=N/summationdisplay\ni=1L/summationdisplay\na=N+1(cm)ia|ψa\ni/a\\}bracketri}ht,\nwith(m∈S), and\nz1/2\nm(cm)ia= (xm)ia+(ym)ia\nwhere we have explicitly identified the number of the\nexcited state, m, using subscripts.\nThe overlap between two of these states is\n∀(m,n)∈S2,(˜S)m,n:=/a\\}bracketle{tψζ\nm|ψζ\nn/a\\}bracketri}ht=c†\nmcn.One can orthonormalize the set of primitive AMBW’s\nby considering a linear combination of them ( k∈S)\n|ψ⊥\nk/a\\}bracketri}ht=M/summationdisplay\np=1/parenleftBig\n˜S−1/2/parenrightBig\np,k|ψζ\np/a\\}bracketri}ht\n=M/summationdisplay\np=1N/summationdisplay\ni=1L/summationdisplay\na=N+1/parenleftBig\n˜S−1/2/parenrightBig\np,k(cp)ia|ψa\ni/a\\}bracketri}ht\nwhich is equivalent to writing\n|ψ⊥\nk/a\\}bracketri}ht=N/summationdisplay\ni=1L/summationdisplay\na=N+1(c⊥\nk)ia|ψa\ni/a\\}bracketri}ht. (28)\nLetW⊥be the set of these Morthonormal states:\nW⊥=/braceleftbig\n|ψ⊥\nm/a\\}bracketri}ht:/a\\}bracketle{tψ⊥\nm|ψ⊥\nm/a\\}bracketri}ht= 1,\n(n/\\e}atio\\slash=m) =⇒ /a\\}bracketle{tψ⊥\nm|ψ⊥\nn/a\\}bracketri}ht= 0,/bracketleftbig\n(m,n)∈S2/bracketrightbig/bracerightbig\n.\nObviously,\nW⊥⊂Wζ.\nAccording to ( 28) showing a normalized linear com-\nbination of singly-excited configurations, we see that\nthe same rules for constructing the γT\nζandγ∆\nζcor-\nresponding to elements of Wζalso apply to ele-\nments ofW⊥, so aγT\n⊥matrix can be constructed\nand its non-zero elements can be encoded into an\n(L−N)×Nmatrix leading to the derivation of\nprojected-auxiliary NTOs. One can also obtain γ∆\n⊥\nand detachment/attachment 1–RDM’s from this pic-\nture. Note however that the representation of the elec-\ntronic transitions will rely on the size and nature of\nthe set of excited states ansätze used for construct-\ning the abovementioned objects. In that case, we also\nhave thatϑ⊥= 1.\n4.Uniqueness of Casida’s AMBW’s\nIn many references the TDDFRT AMBW in ( 26)\nreads\n|˜ψζ/a\\}bracketri}ht:=N/summationdisplay\ni=1L/summationdisplay\na=N+1/radicalbigg\nǫa−ǫi\nω(z+)ia|ψa\ni/a\\}bracketri}ht (29)\nwithz+being the normalized f+vector\nz+=√ω(A−B)−1/2(x+y) =√ωf+,\nandǫpthe energy of the pthcanonical spinorbital\n(withp∈C). For pure-DFT xc-functionals, the ex-\npression of |˜ψζ/a\\}bracketri}htin (29) is identical to the one of ( 26)\nsince in that case,\n∀(i,j)∈I2,∀(a,b)∈A2,(A−B)ia,jb= (ǫa−ǫi)δi,jδa,b.\nFrom equations ( 41) and ( 42) we have that\n(A−B)(x−y) =ω2(A+B)−1(x−y).\nMultiplying to the left by (A+B)1/2and inserting\nthe(A+B)1/2(A+B)−1/2matrix between (A−B)\nand(x−y)in the left-hand side gives\nR−f−=ω2f−9\nwith\nR−= (A+B)1/2(A−B)(A+B)1/2,\nf−= (A+B)−1/2(x−y),\nwithf−normalizing into\nz−=√ωf−.\nIn ref. [ 7] by Luzanov and Zhikol, two possible\nAMBW’s are proposed:\n/a\\}bracketle{ts1,...,sN|ψ±\nζ/a\\}bracketri}ht=N/summationdisplay\ni=1L/summationdisplay\na=N+1(z±)ia/a\\}bracketle{ts1,...,sN|ψa\ni/a\\}bracketri}ht.\n(30)\nAccording to the expression of these wavefunctions, we\ndeduce that such ansätze are of ζ–type, implying that\ntheir properties are identical to those of the orthonor-\nmalized TDDFRT AMBW studied above, and that\nthe algebraic properties of the matrices one can build\nfor constructing NTOs and detachment/attachment\ndensity matrices will also be identical to those related\nin paragraph III B(in particular, ϑ±\nζ= 1since the z±\nvectors are normalized).\nThe possibility of writing two ansätze from the\ncentral ( 16) lead the authors to state that the CIS–\nlike representation of the electronic transition derived\nwithin such models is not unique. We would like to no-\ntice here that the proposed ansätze in ( 30) do not have\nthe same definition as those, elements of Wζ, being de-\nrived from Casida’s assignment problem by inserting\nthe expression of the real part of the TDDFRT linear\nresponse of the density matrix into the sum-over-state\nformulation of any component of the dynamic polar-\nizability. They are elements of Wζexpressed in terms\nof the corresponding f+vectors, while |ψ±\nζ/a\\}bracketri}htarecon-\nstructed fromz+andz−. For these reasons the two\npictures should be considered separately and indepen-\ndently.\nAnsätze reported in ( 30) have both the exact same\nproperties in the (ϕr)r∈Cbasis asζ-ansätze [see ( 6)].\n5.The AMBW’s of Luzanov and co-workers\nDue to the non-uniqueness of the electronic-transition\nrepresentation involving z+orz−as nilpotent transi-\ntion operator, Luzanov and Zhikol have proposed an\nalternative scheme for reducing ( 16) into an Hermitian\neigenvalue problem: ( 16), that we can rewrite as\nΛ|u/a\\}bracketri}ht=ωJ|u/a\\}bracketri}ht, (31)\nwith\n|u/a\\}bracketri}ht:=/parenleftbigg\nx\ny/parenrightbigg\n,\nis, upon certain conditions that are detailed below and\nin appendix VI C, isospectral to\nΩ|u⋆/a\\}bracketri}ht=ω|u⋆/a\\}bracketri}ht, (32)\nwith\nΩ:=Λ1/2JΛ1/2,\n|u⋆/a\\}bracketri}ht:=/parenleftbig\nω−1Λ/parenrightbig1/2|u/a\\}bracketri}ht=/parenleftbigg\nx⋆\ny⋆/parenrightbigg\n.The detailed derivation of ( 32) from ( 31) is given in\nAppendix VI C. We see that it involves the Λ1/2ma-\ntrix, which reads\n2Λ1/2=/parenleftbigg\nQ+Q−\nQ−Q+/parenrightbigg\n,\nwith\nQ+= (A+B)1/2+(A−B)1/2,\nQ−= (A+B)1/2−(A−B)1/2,\nwhich implies that (A+B)and(A−B)must be pos-\nitive semidefinite in order to reduce ( 31) to (32).\nWe note here few interesting properties of any |u⋆/a\\}bracketri}ht\nvector:\n/a\\}bracketle{tu⋆|u⋆/a\\}bracketri}ht=ω−1/a\\}bracketle{tu|Λ|u/a\\}bracketri}ht=/a\\}bracketle{tu|J|u/a\\}bracketri}ht=x⊤x−y⊤y= 1,\ni.e.,\nx⊤\n⋆x⋆+y⊤\n⋆y⋆= 1, (33)\nand\n2x⋆=z−+z+\n=ω−1/2/bracketleftBig\n(A+B)1/2(x+y)+(A−B)1/2(x−y)/bracketrightBig\n,\n2y⋆=z−−z+\n=ω−1/2/bracketleftBig\n(A+B)1/2(x+y)+(A−B)1/2(y−x)/bracketrightBig\n.\nFrom these results, Luzanov and Zhikol have sug-\ngested to build two normalized ansätze, namely\n|ψx\n⋆/a\\}bracketri}ht=/parenleftbig\nx⊤\n⋆x†\n⋆/parenrightbig−1/2N/summationdisplay\ni=1L/summationdisplay\na=N+1(x⋆)ia|ψa\ni/a\\}bracketri}ht,\n|ψy\n⋆/a\\}bracketri}ht=/parenleftbig\ny⊤\n⋆y†\n⋆/parenrightbig−1/2N/summationdisplay\ni=1L/summationdisplay\na=N+1(y⋆)ia|ψa\ni/a\\}bracketri}ht.\nThese two ansätze are of ζtype, with two state 1–\nRDM which read, using the same manipulations as in\nequations ( 19) to (22),\nγx\n⋆=γ0+/parenleftbig\nx⊤\n⋆x†\n⋆/parenrightbig−1/bracketleftBig/parenleftBig\n−X†\n⋆X⊤\n⋆/parenrightBig\n⊕/parenleftBig\nX⊤\n⋆X†\n⋆/parenrightBig/bracketrightBig\n,\nγy\n⋆=γ0+/parenleftbig\ny⊤\n⋆y†\n⋆/parenrightbig−1/bracketleftBig/parenleftBig\n−Y†\n⋆Y⊤\n⋆/parenrightBig\n⊕/parenleftBig\nY⊤\n⋆Y†\n⋆/parenrightBig/bracketrightBig\n,\nwhereγ0is the ground-state 1–RDM. They then use\na statistical mixture of the corresponding 1–RDM to\nbuild the total excited-state 1–RDM:\nγ⋆=/parenleftbig\nx⊤\n⋆x†\n⋆/parenrightbig\nγx\n⋆+/parenleftbig\ny⊤\n⋆y†\n⋆/parenrightbig\nγy\n⋆\nwhich we can show leads to the 1–DDM\nγ∆\n⋆=/parenleftBig\n−X†\n⋆X⊤\n⋆−Y†\n⋆Y⊤\n⋆/parenrightBig\n⊕/parenleftBig\nX⊤\n⋆X†\n⋆+Y⊤\n⋆Y†\n⋆/parenrightBig\nand to the detachment/attachment density matrices\nγd\n⋆=/parenleftBig\nX†\n⋆X⊤\n⋆+Y†\n⋆Y⊤\n⋆/parenrightBig\n⊕0v,\nγa\n⋆=0o⊕/parenleftBig\nX⊤\n⋆X†\n⋆+Y⊤\n⋆Y†\n⋆/parenrightBig\n,\nwhich share the same algebraic properties as the ma-\ntrices in the EOM picture, though they are derived\nfrom AMBW’s. The difference with the EOM picture\nis that there are restrictions (positive semidefiniteness\nof the ( A+B) and ( A−B) matrices) to the exis-\ntence of this picture, and, according to ( 33), the fact\nthatϑ⋆= 1.10\n6.The AMBW’s of Subotnik and co-workers\nRecently [ 26], an alternative ansatz was introduced by\nSubotnik and co-workers for computing TDHF and\nTDDFT derivative couplings. It comes with pairs of\nansätze (denoted hereafter by the Greek letter λ):\n∀(m,n)∈S2,|ψλ\nm;n/a\\}bracketri}ht:=/parenleftBig\nˆXm+ˆXmˆXnˆYn/parenrightBig\n|ψ0/a\\}bracketri}ht,\n|ψλ\nn;m/a\\}bracketri}ht:=/parenleftBig\nˆXn+ˆXnˆXmˆYm/parenrightBig\n|ψ0/a\\}bracketri}ht,\nwhere “m;n” reads “mgivenn”, and\n∀m∈S,ˆXm=N/summationdisplay\ni=1L/summationdisplay\na=N+1(xm)iaˆa†ˆi,\nˆYm=N/summationdisplay\ni=1L/summationdisplay\na=N+1(ym)iaˆa†ˆi,\ni.e.,ˆXandˆYare both excitation operators, but with\ncoefficients taken from the components of xandy.\nOne element of the 1–TDM for the |ψ0/a\\}bracketri}ht → |ψλ\nm;n/a\\}bracketri}ht\ntransition reads\n(γ0m;n\nλ)s,r=/a\\}bracketle{tψ0|ˆr†ˆs|ψλ\nm;n/a\\}bracketri}ht,[(r,s)∈C2,(m,n)∈S2],\nwhich develops into\n/a\\}bracketle{tψ0|ˆr†ˆs|ψλ\nm;n/a\\}bracketri}ht=/a\\}bracketle{tψ0|ˆr†ˆsˆXm|ψ0/a\\}bracketri}ht\n+/a\\}bracketle{tψ0|ˆr†ˆsˆXmˆXnˆYn|ψ0/a\\}bracketri}ht. (34)\nThe second term in ( 34) vanishes, and the 1–TDM\nelement reduces to\n/a\\}bracketle{tψ0|ˆr†ˆs|ψλ\nm;n/a\\}bracketri}ht=N/summationdisplay\ni=1L/summationdisplay\na=N+1xia/angbracketleftBig\nψ0/vextendsingle/vextendsingle/vextendsingleˆr†ˆsˆa†ˆi/vextendsingle/vextendsingle/vextendsingleψ0/angbracketrightBig\n=N/summationdisplay\ni=1L/summationdisplay\na=N+1xiaδr,iδs,anr(1−ns)\nleading to\n/parenleftbigg0ov0ov\nX⊤\nm0v/parenrightbigg\n=γ0m;n\nλ∈RL×L,\nwith\n∀(i,a)∈I×A,(xm)ia= (Xm)i,a−N\nwhich has the exact same structure as the ζ–type 1–\nTDM. This eludes the de-excitation coefficients.\nOn the other hand, one element of the 1–DDM for\nthe|ψ0/a\\}bracketri}ht → |ψλ\nm;n/a\\}bracketri}httransition reads\n(γ∆,λ\nm;n)r,s=/a\\}bracketle{tψλ\nm;n|ˆr†ˆs|ψλ\nm;n/a\\}bracketri}ht−/a\\}bracketle{tψ0|ˆr†ˆs|ψ0/a\\}bracketri}ht.\nThe second term reduces to\n/a\\}bracketle{tψ0|ˆr†ˆs|ψ0/a\\}bracketri}ht=δr,snrns,\nbut the first term\n/a\\}bracketle{tψλ\nm;n|ˆr†ˆs|ψλ\nm;n/a\\}bracketri}ht=/angbracketleftBig\nψ0/vextendsingle/vextendsingle/vextendsingle/parenleftBig\nˆY†\nnˆX†\nnˆX†\nm+ˆX†\nm/parenrightBig\nˆr†ˆs\n×/parenleftBig\nˆXm+ˆXmˆXnˆYn/parenrightBig/vextendsingle/vextendsingle/vextendsingleψ0/angbracketrightBighas to be developed into four parts:\n/a\\}bracketle{tψλ\nm;n|ˆr†ˆs|ψλ\nm;n/a\\}bracketri}ht= (Lm;n\n1)r,s+(Lm;n\n2)r,s\n+(Lm;n\n3)r,s+(Lm;n\n4)r,s,\nwith\n(Lm;n\n1)r,s=/a\\}bracketle{tψ0|ˆX†\nmˆr†ˆsˆXm|ψ0/a\\}bracketri}ht,\n(Lm;n\n2)r,s=/a\\}bracketle{tψ0|ˆY†\nnˆX†\nnˆX†\nmˆr†ˆsˆXm|ψ0/a\\}bracketri}ht,\n(Lm;n\n3)r,s=/a\\}bracketle{tψ0|ˆX†\nmˆr†ˆsˆXmˆXnˆYn|ψ0/a\\}bracketri}ht,\n(Lm;n\n4)r,s=/a\\}bracketle{tψ0|ˆY†\nnˆX†\nnˆX†\nmˆr†ˆsˆXmˆXnˆYn|ψ0/a\\}bracketri}ht.\nAfter developing the ˆXandˆYoperators, we see that\nLm;n\n2andLm;n\n3are twoL×Lzero matrices, and that\nLm;n\n1= (−X†\nmX⊤\nm)⊕(X⊤\nmX†\nm),\nwhich is aζ–like 1–DDM. However, there is still the\nLm;n\n4matrix to add to Lm;n\n1for getting the full γ∆,λ\nm;n.\nOne matrix element of Lm;n\n4is a sum of expectation\nvalues of chains of fourteen second quantization op-\nerators with respect to the one-determinant reference\nstate, each expectation value being decomposed into\na minimum of 252 products of Kronecker’s deltas.\nV.Conclusion\nWe have unveiled and discussed properties of molec-\nular electronic transitions representations issued by\nTDDFRT calculations and their post-processing into\nauxiliary many-body wavefunctions for the electronic\nexcited states. The native EOM formulation of\nTDDFRT was extensively discussed in terms of\nnatural-orbital representation of the transitions —\nwith the important result that the related detach-\nment/attachment density matrices can be derived\nwithout requiring the diagonalization of the 1–DDM\n—, and such a representation was compared with the\nCIS-like picture derived by Casida’s ansatz. The NTO\npopulation-transfer approximate picture for a transi-\ntion has also been discussed in the EOM–TDDFRT\nframework, as well as the detachment/attachment and\ntransition-hole/transition-electron pictures, for whic h\na disambiguation has been previously proposed in an-\nother publication.\nTwo alternative types of ansätze were reported: the\none introduced by Luzanov and co-workers, and the\none introduced by Subotnik and co-workers. The\nstructure and properties of the relevant matrices (the\n1–DDM and the 1–TDM) were resolved for these two\ntypes of ansätze. The detachment/attachment density\nmatrices corresponding to the AMBW by Luzanov\nand co-workers, though being issued from a couple\nof CIS-like expansions, have their structure identical\nto the one from the EOM picture. On the other hand,\nthe 1–TDM in the frame proposed by Subotnik and\nco-authors eludes the TDDFRT “de-excitations” con-\ntributions, and the 1–DDM in that model contains\nmulti-excitation contributions.\nWe would like to conclude this contribution\nby mentioning that without the disambiguation11\nin [28] relatively to the interpretation of the de-\ntachment/attachment and transition-hole/transition-\nelectron models, the TDDFRT picture of an elec-\ntronic transition would have remained either equivo-\ncal (EOM–TDDFRT), either incomplete or arbitrary\n(AMBW’s).\nAcknowledgements\nDrs Pierre-François Loos, Anthony Scemama, Em-\nmanuel Fromager, Benjamin Lasorne, Matthieu\nSaubanère, Nicolas Saby, Mario Barbatti, Miquel\nHuix-Rottlant, Christophe Raynaud and Evgeny\nPosenitskiy are gratefully acknowledged for very fruit-\nful discussions on the topic.\nVI.Appendix\nA.Basic results from linear algebra\nFrom the existence of a singular value decomposition\nfor any matrix, we get\nLemma VI.1 LetAbe anm×ncomplex matrix and\nA†its adjoint. Let pbe any strictly positive inte-\nger lower or equal to min(m,n). Then, the pthleft-\nsingular vector wp∈Cm×1ofAcorresponds to the\nsame singular value σpas thepthright-singular vector\nvp∈Cn×1ofA.\nLemma VI.1also reads\n∀p∈/llbracket1,min(m,n)/rrbracket,Avp=σpwp, (35)\nA†wp=σpvp. (36)\nLemma VI.2 LetAbe anm×ncomplex matrix and\nA†its adjoint. Let qbe the strictly positive integer\nequal to min(m,n). Then, AA†andA†Ashareq\neigenvalues. Those eigenvalues are the squared singu-\nlar values of A.\nProof: Let pbe any strictly positive integer lower or\nequal tog= min(m,n). Then, multiplying ( 35) to\nthe left by A†leads, according to ( 36), to\nA†Avp=σ2\npvp, (37)\nwhile multiplying ( 36) to the left by Aleads, accord-\ning to ( 35), to\nAA†wp=σ2\npwp, (38)\n/square\nFrom what precedes we also conclude to the following\nlemmas:\nLemma VI.3 LetAbe anm×ncomplex matrix and\nA†its adjoint. Then,\nAA†/{ollowsequal0,\nA†A/{ollowsequal0,\ni.e., the product of a matrix by its adjoint is a positive\nsemi-definite matrix.Lemma VI.4 LetAandBbe twon×ncomplex\nmatrices. Then,\n(A/{ollowsequal0 and B/{ollowsequal0) =⇒(A+B/{ollowsequal0),\ni.e., the sum of two positive semi-definite matrices is\na positive semi-definite matrix.\nB.Derivation of Casida’s TDDFRT AMBW\nOur starting point here will be ( 16). In a first step\nwe will derive the working equations for a general,\nelectronic transition between the ground state and one\nexcited state, so the target excited state will not have\na particular label (we omit the mindices for a while).\nWe get, for real AandBmatrices,\nAx+By=ωx, (39)\nBx+Ay=−ωy. (40)\nSubtracting ( 39) to (40) gives\n(A−B)(x−y) =ω(x+y)⇐⇒(x−y) = (A−B)−1ω(x+y)\n(41)\nwhile adding them and using ( 41) gives\n(A+B)(x+y) =ω(x−y) =ω2(A−B)−1(x+y).\n(42)\nMultiplication to the left by (A−B)1/2\n(A−B)1/2(A+B)(x+y) =ω2(A−B)−1/2(x+y)\nand insertion of (A−B)1/2(A−B)−1/2between (A+\nB)and(x+y)finally leads to\nR+f+=ω2f+\nwith\nR+:= (A−B)1/2(A+B)(A−B)1/2\nand\nf+:=(A−B)−1/2(x+y) (43)\nFor pure–DFT exchange–correlation (xc) functionals,\nthef+vectors are then used to write the compo-\nnents of the electronic transition dipole moment. In\nTDDFRT, these components read, for a given excited\nstate|ψm/a\\}bracketri}ht,\n/a\\}bracketle{tψ0|ˆw|ψm/a\\}bracketri}ht=p†\nw(αmam),(m∈S)\nwithpwfrom ( 14),\nαm=/parenleftBig\nωmf†\n+,mf+,m/parenrightBig−1/2\n,\nand\nam= (A−B)1/2f+,m.\nDropping again the msubscripts for readability, we\nhave that, if the excited state could be written as a12\nlinear combination of Slater determinants, its single-\nexcitation part would be, according to what precedes,\n/a\\}bracketle{ts1,...,sN|˜ψζ/a\\}bracketri}ht:=αN/summationdisplay\ni=1L/summationdisplay\na=N+1aia/a\\}bracketle{ts1,...,sN|ψa\ni/a\\}bracketri}ht.\n(44)\nThe linear combination of singly-excited Slater deter-\nminants in ( 44) has then been used in the context\nof non-adiabatic dynamics as an auxiliary many-body\nwavefunction assigned to the excited state reached by\nthe linear response treatment of the Kohn-Sham DFT\nsystem: From equations ( 41) and ( 43), we deduce\n(x+y) = (A−B)1/2f+,\n(x−y) =ω(A−B)−1/2f+.\nUsing ( 23), and considering that the AMBW was orig-\ninally derived for pure–DFT xc-functionals, for which\n(A−B)is diagonal, hence (A−B)1/2is self-adjoint,\n1 = (x+y)†(x−y) =f†\n+(A−B)1/2(A−B)−1/2f+ω,\ni.e.,\nf†\n+f+=ω−1=⇒α= 1, (45)\nso that R+andf+satisfy\nf†\n+R+f+=ω.From ( 45), we have that ( 44) transforms into\n|˜ψζ/a\\}bracketri}ht:=N/summationdisplay\ni=1L/summationdisplay\na=N+1(xia+yia)|ψa\ni/a\\}bracketri}ht.\nC.Derivation of the alternative TDDFT\nresponse equation of Luzanov and co-workers\nWe start by multiplying ( 31) byω−1/2, and by rewrit-\ningΛasΛ1/2Λ1/2— the existence criterion for this\nmatrix product is that (A+B)and(A−B)are pos-\nitive semi-definite —, which gives\nΛ1/2/parenleftbig\nω−1Λ/parenrightbig1/2|u/a\\}bracketri}ht/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n|u⋆/a\\}bracketri}ht=√ωJ|u/a\\}bracketri}ht.\nGiven that Jis involutory/parenleftbig\nJ=J−1/parenrightbig\n, we multiply the\nprevious to the left by Λ1/2J\nΛ1/2JΛ1/2\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nΩ|u⋆/a\\}bracketri}ht=ω/parenleftbig\nω−1Λ/parenrightbig1/2|u/a\\}bracketri}ht/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n|u⋆/a\\}bracketri}ht,\ni.e.,\nΩ|u⋆/a\\}bracketri}ht=ω|u⋆/a\\}bracketri}ht.\n[1]M. E. Casida, “Time-Dependent Density Functional\nResponse Theory for Molecules,” in Recent Advances\nin Density Functional Methods , vol. Volume 1 of Re-\ncent Advances in Computational Chemistry , pp. 155–\n192, WORLD SCIENTIFIC, Nov. 1995.\n[2]M. E. Casida, “Time-dependent density-functional\ntheory for molecules and molecular solids,” Journal of\nMolecular Structure: THEOCHEM , vol. 914, pp. 3–\n18, Nov. 2009.\n[3]A. Dreuw and M. Head-Gordon, “Single-Reference ab\nInitio Methods for the Calculation of Excited States\nof Large Molecules,” Chemical Reviews , vol. 105,\npp. 4009–4037, Nov. 2005. Publisher: American\nChemical Society.\n[4]S. Hirata, M. Head-Gordon, and R. J. Bartlett,\n“Configuration interaction singles, time-dependent\nHartree–Fock, and time-dependent density functional\ntheory for the electronic excited states of extended\nsystems,” The Journal of Chemical Physics , vol. 111,\npp. 10774–10786, Dec. 1999. Publisher: American In-\nstitute of Physics.\n[5]N. Ferré, M. 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Jacquemin, “The\nBethe–Salpeter equation in chemistry: relations with\nTD-DFT, applications and challenges,” Chemical So-\nciety Reviews , vol. 47, pp. 1022–1043, Feb. 2018."    },    {        "title": "2105.08849v3.Spatial_order_in_a_two_dimensional_spin_orbit_coupled_spin_1_2_condensate__superlattice__multi_ring_and_stripe_formation.pdf",        "content": "Spatial order in a two-dimensional spin-orbit-coupled spin-1/2 condensate:\nsuperlattice, multi-ring and stripe formation\nS. K. Adhikari\u0003\nInstituto de Física Teórica, Universidade Estadual Paulista - UNESP, 01.140-070 São Paulo, São Paulo, Brazil\n(Dated: August 2, 2021)\nWe demonstrate the formation of stable spatially-ordered states in a uniform and also trapped\nquasi-two-dimensional (quasi-2D) Rashba or Dresselhaus spin-orbit (SO) coupled pseudo spin-1/2\nBose-Einstein condensate using the mean-field Gross-Pitaevskii equation. For weak SO coupling,\none can have a circularly-symmetric (0;+1)- or (0;\u00001)-type multi-ring state with intrinsic vorticity,\nfor Rashba or Dresselhaus SO coupling, respectively, where the numbers in the parentheses denote\nthe net angular momentum projection in the two components, in addition to a circularly-asymmetric\ndegenerate state with zero net angular momentum projection. For intermediate SO couplings, in\naddition to the above two types, one can also have states with stripe pattern in component densities\nwith no periodic modulation in total density. The stripe state continues to exist for large SO\ncoupling. In addition, a new spatially-periodic state appears in the uniform system: a superlattice\nstate, possessingsomepropertiesofa supersolid , withasquare-latticepatternincomponentdensities\nand also in total density. In a trapped system the superlattice state is slightly different with multi-\nring pattern in component density and a square-lattice pattern in total density. For an equal mixture\nofRashbaandDresselhausSOcouplings, inbothuniformandtrappedsystems, onlystripestatesare\nfound for all strengths of SO couplings. In a uniform system all these states are quasi-2D solitonic\nstates.\nI. INTRODUCTION\nThe pursuit of a supersolid [1] has lately gained im-\npetus among research workers in different areas of low-\ntemperature physics. A supersolid is a special form of\nmatter possessing the properties of both a crystalline\nsolid and a superfluid. It is characterized by a spatially-\nordered periodic matter resulting from a breakdown of\ncontinuous translational invariance as in a crystalline\nsolid. It also enjoys a frictionless flow as in a superfluid,\nbreaking continuous gauge invariance. Hence a super-\nsolid is a special type of superfluid and its search has\nbeen confined to a superfluid, e.g., Bose–Einstein con-\ndensate (BEC) [2], and Fermi superfluid [3]. The search\nof a supersolid helium [4] was inconclusive [5]. Follow-\ning theoretical suggestions to create a supersolid with\nfinite-range [6] and dipolar [7] interactions, different ex-\nperimentalgroupsconfirmedsupersolidityinaquasi-one-\ndimensional (quasi-1D) [8] and a quasi-two-dimensional\n(quasi-2D) [9] dipolar trapped BEC.\nThere have been theoretical suggestions for creat-\ning supersolid-like states in a spin-orbit-coupled (SO-\ncoupled) BEC [10]. Although there cannot be a natu-\nral spin-orbit (SO) coupling in a neutral atom, an artifi-\ncial synthetic SO coupling is possible by tuned Raman\nlasers that couple the different spin component states\n[11] of a spinor BEC [12]. Two such possible SO cou-\nplings are due to Rashba [13] and Dresselhaus [14]. An\nequal mixture of these SO couplings has been experi-\nmentally realized in a pseudo spin- 1=2(F= 1=2)23Na\n\u0003sk.adhikari@unesp.br\nhttps://professores.ift.unesp.br/sk.adhikari/[15] and87Rb [16] BEC of Fz= 0;\u00001spin component\nstates. A pseudo spin-1/2 state contains only the com-\nponentsFz= 0;\u00001of a hyperfine spin-1 ( F= 1) state.\nThe mean-field equationof this systemis governedbythe\nPauli spin matrices and hence the name pseudo spin-1/2.\nLater, an SO-coupled spin-187Rb BEC of Fz=\u00061;0\nspin component states was also realized [17]. Recently,\na spatially-periodic supersolid-like state with stripe pat-\ntern in density, called a superstripe state, was observed\nin an SO-coupled quasi-1D pseudo spin-1/2 spinor BEC\nof23Na atoms [18] employing an equal mixture of Rashba\nand Dresselhaus couplings. There have been theoretical\nstudies on the stability of different phases of BECs with\nRashba-Dresselhaus spin-orbit coupling against quantum\nand thermal fluctuations [19].\nIn view of the observed superstripe state in an SO-\ncoupled quasi-1D pseudo spin-1/2 spinor BEC [18], in\nthis paper we look for a supersolid-like state [1] in a\nquasi-2D pseudo spin-1/2 SO-coupled spinor BEC us-\ning the mean-field Gross-Pitaevskii (GP) equation [20].\nPrevious considerations were limited to a stripe state\nin a quasi-1D pseudo spin-1/2 SO-coupled spinor BEC\n[21, 22]. In a quasi-2D trap, different types of spatially-\nordered 2D lattice states seem likely in an SO-coupled\npseudo spin-1/2 spinor BEC, specially after the confir-\nmation of such states in an SO-coupled quasi-2D spin-1\nspinorBEC[23,24]. Wewillconsiderbothuniform(trap-\nless) and trapped states in this study under the action\nof Rashba or Dresselhaus SO coupling, in addition to an\nequal mixture of Rashba and Dresselhaus couplings.\nFirst we will consider a uniform system, where the lo-\ncalized state appears in the form of a quasi-2D soliton.\nIn a scalar BEC, solitons cannot be stabilized in two [25]\nand three dimensions [26] due to a collapse instability.\nHowever, a pseudo spin-1/2 SO-coupled BEC can sustainarXiv:2105.08849v3  [cond-mat.quant-gas]  30 Jul 20212\na quasi-1D [27] or a quasi-2D [28] or a 3D [29] soliton.\nWe identify a variety of solitons in an SO-coupled quasi-\n2D pseudo spin-1/2 spinor BEC. For a weak SO cou-\nplingtherecanbetwotypesofdegeneratesolitonicstates\nwith very large spatial extension: circularly-symmetric\n(0;\u00061)-type multi-ring and circularly-asymmetric soli-\ntons, where the numbers in parenthesis indicate the an-\ngular momentum projection in the two components with\nthe upper (lower) sign corresponding to Rashba (Dres-\nselhaus) SO coupling. The (0;\u00061)-type multi-ring state\nhas 1/2 unit of total angular momentum projection and\noften is classified as a half-vortex state [30, 31]. Each\ncomponent of the circularly-asymmetric soliton hosts an\nantivortex-vortex pair resulting in zero net angular mo-\nmentum projection in each. With the increase of SO\ncoupling, a stripe state with stripe pattern in component\ndensities appears. For strong SO coupling two types of\ndegenerate solitonic states appear: a state with stripe\npattern in density without any modulation in total den-\nsity and a superlattice state with square-lattice pattern\nin both component and total densities. The present soli-\ntons with a 2D square-lattice structure in total density\n[21], viz. figures 6(d)-(f), sharing properties with, and\nmore closely related to, a conventional supersolid, will\nbe termed superlattice solitons in the following as sug-\ngested in Refs. [21–24]. The multi-ring, viz. figures 5(a)-\n(c), and stripe solitons, viz. figures 6(a)-(c), only exhibit\na spatially-periodic pattern in the component densities\nwithout any periodic pattern in the total density. Never-\ntheless, in the literature such a state with stripe pattern\nin component density only, also bearing some similar-\nity to a supersolid, has often been termed a superstripe\nstate [18, 21, 22]. In this paper we call such a state by the\nname stripe state to differentiate it from the superlattice\nstate with square-lattice pattern in total density. For an\nequal mixture of Rashba and Dresselhaus SO couplings\nwe find only stripe solitons for all values of SO couplings;\nthese solitons have a stripe pattern in component den-\nsities with no spatially-periodic pattern in total density.\nTo generate an equal mixture of Rashba and Dresselhaus\ncouplings in a laboratory a pair of Raman lasers coupling\nthe spin-component states are needed [16], whereas to\ncreate a Rashba or a Dresselhaus coupling multiple Ra-\nman laser beams are needed [32], which leads to a more\ncomplex electromagnetic “trapping” potential. Hence it\nseems reasonable that a Rashba or a Dresselhaus SO-\ncoupled BEC may host different types of superlattice\nstates not possible in the case of an equal mixture of\nRashba and Dresselhaus couplings.\nThe scenario of spatially-ordered states in a\nharmonically-trapped SO-coupled quasi-2D pseudo\nspin-1/2 BEC remains almost the same when compared\nwith a uniform system. For a weak SO coupling,\nthe trapped system can host three types of states:\ncircularly-symmetric (0;\u00061)-type multi-ring state,\ncircularly-asymmetric state and stripe state. For large\nSO coupling, one encounters two types of spatially-\nperiodic states: stripe state and a special type ofmulti-ring state. This multi-ring state possesses a multi-\nring pattern in component density with a square-lattice\npattern in total density near the center thus exhibiting\na supersolid-like behavior.\nIn section IIA we present the mean-field GP equa-\ntion that we use in this investigation. In section IIB\nwe demonstrate the plausibility of the appearance of the\n(0;\u00061)-type state, stripe state, and superlattice state\nfrom a consideration of linear version of this model. The\nnumerical result is presented in section III. In section\nIIIA the results for a uniform quasi-2D pseudo spin-\n1/2 Rashba and Dresselhaus SO-coupled self-attractive\nBEC soliton are presented. Those for an equal mix-\nture of Rashba and Dresselhaus couplings are presented\nin section IIIB. The results for a uniform quasi-2D\npseudo spin-1/2 Rashba and Dresselhaus SO-coupled\nself-repulsive BEC soliton are presented in section IIIC.\nTheresultsforaharmonicallytrappedSO-coupledquasi-\n2D BEC are presented in section IIID. Finally, a sum-\nmary of our findings is given in section IV.\nII. MEAN-FIELD MODEL\nA. Gross-Pitaevskii equation\nWe consider a BEC of Natoms, each of mass m, un-\nder a harmonic trap V(r) =m!2(x2+y2)=2 +m!2\nzz2=2\n(!z\u001d!)of frequency !zand!in thezdirection and in\nthex\u0000yplane, respectively. The single particle Hamil-\ntonian of the SO-coupled BEC is [16]\nH0=\u0000~2\n2mr2\nr+V(r) +\r[\u0011py\u001bx\u0000px\u001by];(1)\nwhere r\u0011fx;y;zg,r2\nr= (@2=@x2+@2=@y2+@2=@z2)\u0011\n(@2\nx+@2\ny+@2\nz),\ris the strength of the SO-coupling term\nin square bracket, \u0011= +1for Rashba coupling, \u0011=\u00001\nfor Dresselhaus coupling, and \u0011= 0for an equal mix-\nture of Rashba and Dresselhaus couplings, momentum\ncomponent px=\u0000i~@x;py=\u0000i~@yand the Pauli spin\nmatrices\u001bxand\u001byare\n\u001bx=\u0012\n0 1\n1 0\u0013\n; \u001by=\u0012\n0\u0000i\ni0\u0013\n: (2)\nUnder a strong trap in the zdirection, the system is\nassumed to be frozen in the Gaussian ground state in\nthezdirection and the relevant dynamics in the x\u0000y\nplane, obtained by integrating out the zcoordinate in a\nstraight-forward fashion [33], leads to the following set of\nGP equations at zero temperature for spin components\nFz= 0;\u00001[34]\ni@t 1(\u001a;t) = [H\u001a+c0n1(\u001a;t) +c2n2(\u001a;t)] 1(\u001a;t)\n+\r(\u0000i\u0011@y+@x) 2(\u001a;t); (3)\ni@t 2(\u001a;t) = [H\u001a+c0n2(\u001a;t) +c2n1(\u001a;t)] 2(\u001a;t)\n\u0000\r(i\u0011@y+@x) 1(\u001a;t); (4)3\nH\u001a=\u00001\n2r2\n\u001a+1\n2\u001a2; (5)\nc0=Np\n2\u0019\u0014a 0; c 2=Np\n2\u0019\u0014a 12;(6)\nwhere \u001a=fx;yg,\u0014=!z=!; @t\u0011@=@t,r2\n\u001a=@2\nx+@2\ny,\nnj=j jj2;j= 1;2are the densities of spin components\nFz= 0;\u00001, andn(\u001a) =P\njnj(\u001a)the total density, a0\n(a2) is the intraspecies (interspecies) scattering length.\nAll quantities in (3)-(4) and in the following are dimen-\nsionless; this is achieved by expressing length in units of\noscillator length l0\u0011p\n~=m!, density in units of l\u00002\n0,\nenergy in units of ~!, and time in units of !\u00001. The\nnormalization condition isR\nn(x;y)dxdy = 1:\nThe time-independent version of (3)-(4), appropriate\nfor stationary solutions, can be derived from the energy\nfunctional\nE[ ] =R\ndxdy\bP\nj1\n2jr\u001a jj2+1\n2c0(n2\n1+n2\n2) +c2n1n2\n+1\n2\u001a2n+\r\u0002\n \u0003\n1(@x\u0000i\u0011@y) 2\u0000 \u0003\n2(@x+i\u0011@y) 1\u0003\t\n:\n(7)\nEquations (3), (4), and (7) are valid for a harmonically\ntrapped quasi-2D SO-coupled pseudo spin-1/2 spinor\nBEC. The same for a uniform system can be obtained\nby taking the limit !!0and removing the harmonic\ntrap from these equations. For a uniform system (3) and\n(4) remain valid with\nc0=Np\n2\u0019a0; c 2=Np\n2\u0019a12: (8)\nNowlengthisexpressedinunitsof lz=p\n~=m!z,density\nin units of l\u00002\nz, and energy in units of ~!z, and time in\nunits of!\u00001\nz.\nB. Analytic Consideration\nMany properties of the density distribution can be\nunderstood from a consideration of the single-particle\nHamiltonian in the absence of nonlinear interaction.\nFirst we consider a Rashba or a Dresselhaus SO cou-\npling. The quasi-2D wave function of the single-particle\nHamiltonian\nH2D\n0=\u00001\n2r2\n\u001a\u0000i\r[\u0011@y\u001bx\u0000@x\u001by] (9)\nsatisfies the following eigenvalue equation\n\u0014\u00001\n2r2\n\u001a\r(@x\u0000i\u0011@y)\n\u0000\r(@x+i\u0011@y)\u00001\n2r2\n\u001a\u0015\u0012\n 1\n 2\u0013\n=E\u0012\n 1\n 2\u0013\n;(10)\nwith energyE. In circular coordinates \u001a=f\u001a;\u0012g;x=\n\u001acos\u0012;y=\u001asin\u0012, we have (@x\u0006i@y) = exp(\u0006i\u0012)(@\u001a\u0006\ni@\u0012=\u001a);with@\u001a\u0011@=@\u001a;@\u0012\u0011@=@\u0012. A circularly-\nsymmetric solution of angular momentum projection m\nof (10) will have the form [31]\n m(\u001a) =\u0012\n 1(\u001a)\n 2(\u001a) exp(i\u0011\u0012)\u0013exp(im\u0012)p\n2\u0019:(11)\nFIG. 1: Contour plot of density njof the state (15) for\n\r= 8and\u0011= 1of components (a) j= 1(n1), (b)j= 2\n(n2) and (c) total density after multiplying the state (15)\nby an appropriate Gaussian distribution. The densities are\nnormalized asR\ndxdyn (x;y) = 1. Results in all figures are\nplotted in dimensionless units.\nAs\u0011=\u00061for Rashba and Dresselhaus SO couplings,\nrespectively, we find that the second component carries a\nrelative angular momentum projection \u0011with respect to\nthe first component in addition to a possible overall an-\ngular momentum projection m. All states with jmj>1\nare unstable in general, hence a stable state of type (11)\nis a state of type (0;\u0011)or(0;\u00061)for Rashba or Dres-\nselhaus SO coupling. Considering m=\u00071, we can have\nan equivalent state of type (\u00071;0). These two types are\nall states with angular momentum projection less than\nunity. Of these two types of equivalent states \u0000(0;\u00061)\nand(\u00071;0)\u0000, in this study we consider only the state\nof type (0;\u00061). This state has angular momentum pro-\njectionm= 0and spins= 1=2, hence the total angular\nmomentum projection jz=m+sz= 1=2and this state\nis often called a quantum half-vortex state.\nThe appearance of the stripe state can be understood\nfrom the consideration that (10) has the following degen-\nerate solutions\n\u0012\n 1\n 2\u0013\n=\u0012\ncos(\rx)\n\u0000sin(\rx)\u0013\n; (12)\n\u0012\n 1\n 2\u0013\n=\u0012\ncos(\ry)\n\u0000i\u0011sin(\ry)\u0013\n; (13)\nwith energy\nE=\u0000\r2\n2: (14)\nThe solutions (12) and (13) represent stripes along\nyandxdirections, respectively. There is another\nset of equivalent solutions which we will not con-\nsider: ( 1; 2)T= (sin(\rx);cos(\rx))Tand( 1; 2)T=\n(sin(\ry);i\u0011cos(\ry))T; these states represent stripes in\ndensityalong xandydirectionsinthecomponents. Both\nsets of solutions have uniform density in the sum of the\ntwo components: j 1j2+j 2j2= 1.\nA linear combination of the degenerate states (12) and\n(13), e.g.\n\u0012\n 1\n 2\u0013\n=pn\u0012\ncos(\rx)\u0006cos(\ry)\n\u0000sin(\rx)\u0007i\u0011sin(\ry)\u0013\n;(15)4\n 0 0.5 1\n 0  4  8  12(i)                    (ii)\n(  1,0,±1)−typesuperlattice and stripe\n(0,±1)−type multiring asymmetric|c0| = |c2|\nγ\nFIG. 2: The c0=c2versus\rphase plot showing soliton for-\nmation in a Rashba or a Dresselhaus SO-coupled pseudo spin-\n1/2BECindifferentregionsofparameterspace: (i)formation\nofcircularly-symmetric( 0;\u00061)-typemulti-ringandcircularly-\nasymmetric solitons and (ii) superlattice and stripe solitons.\nis also a valid eigenfunction and represents a square-\nlattice pattern in density of the components as well as\nin the total density. The spatial modulation in density is\nproduced by the sinandcosterms in (15).\nIt is tempting to multiply the functions (12), (13), and\n(15) by a localized Gaussian distribution and see if such a\nstate can simulate the density of a localized stripe or su-\nperlattice state. The answer is affirmative. The density\npattern of the state (12) or (13) after this multiplication\nis quite similar to the density of a localized stripe state\n(result not shown here) obtained by a numerical solu-\ntion of the GP equation. The total density of the stripe\nstate does not have any spatially-periodic modulation.\nIn figure 1, we display a contour plot of the density of\ncomponents (a) j= 1, (b)j= 2, and (c) total density\nof the state (15) for \r= 8;\u0011= +1after multiplying by\nan appropriate Gaussian. The density of this state is\nquite similar to a superlattice soliton with square-lattice\npattern in density for \r= 8;c0=c2=\u00000:5, viz. fig-\nures 6(d)-(f). Even the total density in figure 1(c) has\na square-lattice pattern as in a superlattice BEC. Hence\nan analytic consideration of the single-particle Hamilto-\nnian (9) reveals that it can naturally lead to eigenstates\nwith densities in agreement with those in an actual phys-\nical(0;\u00061)-type state, stripe state and superlattice state\nin a SO-coupled spin-1/2 BEC. In a weakly-attractive\nuniform system, that we consider in this paper, the ener-\ngies of these states are very close to the analytic energy\nE=\u0000\r2=2in most cases, indicating a negligible contri-\nbution from the nonlinear terms c0andc2.\nFor an equal mixture of Rashba and Dresselhaus SO\ncouplings (i\r@x\u001by), (10) becomes\n\u0014\u00001\n2r2\n\u001a\r@x\n\u0000\r@x\u00001\n2r2\n\u001a\u0015\u0012\n 1\n 2\u0013\n=E\u0012\n 1\n 2\u0013\n: (16)\nEquation (16) does not allow (0;\u00061)-type state implied\nby (11). Equation (16) has the spatially-periodic stripe\nsolution (12). This solution (cos(\rx);\u0000sin(\rx))TwithenergyE=\u0000\r2=2represents stripe in density along y\ndirection. There is no degenerate solution, similar to\n(13), with the same energy. Thus there can be no so-\nlution of type (15) in this case, which could represent a\nsuperlattice state with square-lattice pattern in density.\nIII. NUMERICAL RESULT\nTo solve (3) and (4) numerically, we propagate these in\ntime by the split-time-step Crank-Nicolson discretization\nscheme [35] using a space step of dx=dy= 0:05and a\ntime step of dt=dx2\u00020:1for imaginary-time propa-\ngation and a time step of dt=dx2\u00020:05for real-time\npropagation. In all calculations of stationary states,\nwe employ imaginary-time approach with the conserva-\ntion of normalization ( =R\ndxdy [n1(x;y) +n2(x:y)]) dur-\ning time propagation, which finds the lowest-energy so-\nlution of each type. Real-time propagation is used to\ntest the stability of the solitons. The quantity ( M=R\ndxdy [n1(x;y)\u0000n2(x;y)]) is not a good quantum num-\nber and is left to freely evolve during time propagation\nto attain a final converged value consistent with the pa-\nrameters of the problem. The converged value of the\nquantityMwas found to be essentially zero in all un-\ntrapped cases, viz. figures 3-8, indicating an equal num-\nber of atoms in the two components. The same could\nhave a small nonzero value ( /0:1) for the trapped case\nfor small values of \r, viz. figure 9(a)-(f).\nA. Uniform SO-coupled quasi-2D spin-1/2 BEC:\nRashba/Dresselhaus coupling\nWe study the formation of a spatially-ordered periodic\npattern in density of a self-attractive (c0<0)quasi-\n2D SO-coupled uniform pseudo spin-1/2 BEC for dif-\nferent sets of parameters: the nonlinearities c0;c2and\nSO-coupling strength \r. Without losing generality we\nwill consider only the case c2<0withc0=c2. The\nscenario of soliton formation for Rashba or Dresselhaus\nSO coupling is illustrated in the phase plot of c0=c2\nversus\rforc2<0in figure 2 for small values of\nc0(\u00001< c 0<0). Circularly-symmetric (0;\u00061)-type\nmulti-ring solitons and circularly-asymmetric solitons are\nformed in region (i), superlattice and stripe solitons are\nformed in region (ii).\nFirst, we consider a (0;\u00061)-type multi-ring soliton for\na small\r(\r= 0:5) for Rashba (upper sign) or Dressel-\nhaus (lower sign) SO coupling. In figure 3 we display\nthe contour plot of density of the central region of com-\nponents (a) j= 1, (b)j= 2, and (c) total density of\na(0;\u00061)-type multi-ring soliton. In figures 4(a)-(b) we\ndisplay the contour plot of the phase of wave function\ncomponents j= 1;2of the circularly-symmetric (0;+1)-\ntype Rashba SO-coupled soliton of figures 3(a)-(c). In\nfigure 4(b) there is a phase drop of +2\u0019under a complete\nrotation, indicating an angular momentum projection of5\nFIG. 3: Contour plot of density njof a circularly-symmetric\n(0;\u00061)-type spin-1/2 Rashba or Dresselhaus SO-coupled\nBEC soliton for components (a) j= 1(n1), (b)j= 2(n2)\nand (c) total density ( n); the same of a circularly-asymmetric\nspin-1/2 Rashba or Dresselhaus SO-coupled BEC soliton\nwith an antivortex-vortex structure in each component for\n(d)j= 1, (e)j= 2and (f) total density. The parameters are\nc0=c2=\u00000:5;\r= 0:5.\nFIG. 4: Contour plot of phase of components (a) j= 1,\nand (b)j= 2of the quasi-2D soliton of figures 2(a)-(b) for\nRashba SO coupling, and (c)-(d) the same for Dresselhaus\nSO coupling; contour plot of phase of components (e) j= 1,\nand (f)j= 2of the quasi-2D soliton of figures 2(d)-(e) for\nRashba SO coupling, and (g)-(h) the same for Dresselhaus SO\ncoupling.\n+1in component j= 2. There is no such phase drop in\ncomponent j= 1. Although the densities of Rashba and\nDresselhaus SO-coupled solitons are the same, the corre-\nsponding phases for Dresselhaus SO-coupling are differ-\nent, viz. figures 4(c)-(d) showing the phases of compo-\nnentsj= 1;2of the Dresselhaus SO-coupled soliton. In\nthis case in figure 4(d) we find a phase drop of \u00002\u0019un-\nder a complete rotation, indicating an angular momen-\ntum projection of \u00001in this component. The contour\nplot of density of components (a) j= 1, (b)j= 2,\nand (c) total density of a circularly-asymmetric soliton\nis shown in figures 3(d)-(f) for Rashba or Dresselhaus\nSO coupling. Again the phases of the two SO couplings\nare different. In figures 4(e)-(f) we plot the phase of\nFIG. 5: Contour plot of density njof a circularly-symmetric\n(0;\u00061)-type spin-1/2 multi-ring Rashba or Dresselhaus SO-\ncoupled BEC soliton for components (a) j= 1(n1), (b)j=\n2(n2) and (c) total density ( n); the same of a circularly-\nasymmetricspin-1/2 RashbaorDresselhausSO-coupledBEC\nsolitonwithanantivortex-vortexstructureineachcomponent\nfor (d)j= 1, (e)j= 2and (f) total density; the same of a\nspin-1/2 Rashba or Dresselhaus SO-coupled stripe soliton for\n(g)j= 1, (h)j= 2and (i) total density. The parameters\nused arec0=c2=\u00000:5;\r= 2.\ncomponents j= 1;2of the wave function of the soli-\nton of figures 3(d)-(e) for Rashba coupling, the same for\nDresselhaus coupling are shown in figures 4(g)-(h). In all\ncases we find an antivortex-vortex pair of angular mo-\nmentum projection \u00001and +1at two different places\nsuch that the net angular momentum projection in each\ncomponent is zero. Of the vortex-antivortex pair, one is\ncoreless [36]. In imaginary-time propagation, to obtain\na(0;\u00061)-type multi-ring soliton, the appropriate vor-\ntex (antivortex) for Rashba (Dresselhaus) SO coupling\nin component j= 2was imprinted in an initial Gaussian\nfunction as  initial\n2 (x;y) = (x\u0006iy)\u0002\u001eGauss (x;y). To\nobtain the circularly-asymmetric soliton, the initial wave\nfunction was taken to be a localized Gaussian function in\nthe two components. The numerical energy of both types\nof degenerate solitons in figure 3 is E=\u00000:125, in agree-\nment with the analytic result E=\u0000\r2=2 =\u00000:125, viz.\n(14), independent of the type of SO coupling: Rashba or\nDresselhaus.\nTo study multi-ring solitons for medium values of SO\ncoupling (\r= 2), in figure 5 we show the contour plot of\ndensity of components (a) j= 1, (b)j= 2and (c) to-\ntal density of a circularly-symmetric (0;\u00061)-type multi-6\nFIG. 6: Contour plot of density njof a pseudo spin-1/2\nRashba or Dresselhaus SO-coupled stripe soliton of compo-\nnents (a)j= 1and (b)j= 2(c) and total density. The\nsame of a superlattice soliton of components (d) j= 1, (e)\nj= 2, and (f) the total density. The parameters used are\nc0=c2=\u00000:5;\r= 8.\nring soliton for Rashba or Dresselhaus SO coupling for\nc0=c2=\u00000:5. Although, there is a multi-ring struc-\nture in density of the two components, the total density\nshows no spatially-periodic modulation. Multi-ring soli-\ntons were also investigated in a quasi-2D pseudo spin-1/2\nSO-coupled BEC trapped in a radially periodic poten-\ntial [37] which creates a multi-ring modulation in density.\nHowever, the present radial modulation in density with-\noutanyexternaltrapisaconsequenceoftheSOcoupling.\nThe increase of \rfrom figures 3(a)-(c) to figures 5(a)-\n(c) has increased the binding, and hence aids in forming\nthe compact solitons. In figures 5(d)-(f) we display the\ncircularly-asymmetric solitons for c0=c2=\u00000:5;\r= 2.\nThese solitons are more compact and have developed\nasymmetric ringswith theincreaseof \rfromfigures3(d)-\n(f) to figures 5(d)-(f). In addition to these two types of\nsolitons we find a new type of soliton not possible for a\nsmallSOcoupling( \r= 0:5). Thesearethestripesolitons\ndisplayed in figures 5(g)-(i). To obtain these solitons the\nstripe pattern is imprinted on the initial wave functionas: initial\n1 (x;y)\u0018sin(\rx)\u0002\u001eGauss (x;y); initial\n2 (x;y)\u0018\ncos(\rx)\u0002\u001eGauss (x;y). In this case the total density has\nno spatially-periodic modulation. The numerical ener-\ngies of these three types of states for both Rashba and\nDresselhausSOcouplingsareidentical( E=\u00002:000)and\nequal to the analytic energy of (14) ( E=\u0000\r2=2 =\u00002);\nhence these states can be considered degenerate.\nAs\ris increased, the (0;\u00061)-type multi-ring soliton\nceases to exist and gives rise to a new type of soliton: su-\nperlattice soliton with square-lattice modulation in den-\nsity. The stripe soliton and the circularly-asymmetric\nsoliton continue to exist. However, as we are inter-\nested in spatially-periodic states, we will not consider the\ncircularly-asymmetric soliton here. The spatial period of\nthe lattice or stripe increases as \ris reduced. For a small\n\r, the size of the soliton is smaller than this period and\nthe periodic pattern in density is not possible, viz figure 3\nfor\r= 0:5. In figure 6 we show a quasi-2D stripe soliton\nforc0=c2=\u00000:5;and\r= 8through a contour plot of\ndensity of components (a) j= 1, (b)j= 2, and (c) total\ndensity, obtained by imaginary-time propagation using\nan initial localized wave function modulated by appro-\npriate stripes in j= 1andj= 2components. Although,\nthere is a stripe pattern in density in this case, the po-\nsitions of maxima in component j= 1coincide with the\nminima in component j= 2, thus resulting in a total\ndensity without modulation.\nIn figure 6 the superlattice soliton for the same set of\nparameters ( c0=c2=\u00000:5;\r= 8) is displayed through\na contour plot of density of components (d) j= 1, (e)\nj= 2, and (f) total density, obtained by imaginary-time\npropagation using the converged wave function of figures\n5(a)-(c) as the initial state. The distribution of matter\non a 2D square lattice is prominent in this case, not only\nin the component densities but also in the total density,\nviz. figure 6(f) [21]. The present superlattice soliton is\na consequence of the SO coupling and breaks continuous\ntranslational symmetry as required in a supersolid [1].\nThenumericaldensitypatternofthecomponentsandthe\ntotal density in this case are quite similar to the analytic\ncomponent density and total density of figure 1. Again,\nforthesamesetofparameters, c0=c2=\u00000:5;\r= 8, the\nnumerical energies of the stripe and superlattice solitons\nare the same ( E=\u000032:00) and in agreement with the\nanalytic estimate (14) ( E=\u0000\r2=2 =\u000032). Hence the\nspatially-periodic stripe and superlattice solitons can be\nconsidered to be degenerate.\nB. Uniform SO-coupled quasi-2D spin-1/2 BEC:\nEqual-mixture coupling\nAlthough the Rashba or Dresselhaus SO couplings are\nfundamental in nature, all experiments so far employed\nan equal mixture of Rashba and Dresselhaus SO cou-\nplings [15–17] (\u0000\rpx\u001by). This SO coupling is simpler\nin nature, involving partial derivative in one direction\nonly, and is easily realizable in an experiment. Because7\nFIG. 7: Contour plot of density of a pseudo spin-1/2 stripe\nsoliton for an equal mixture of Rashba and Dresselhaus cou-\nplings of components (a) j= 1, (b)j= 2and (c) the total\ndensity for parameters c0=c2=\u00005;\r= 0:5. (d)-(f) The\nsame densities for parameters c0=c2=\u00005;\r= 8:\nof this phenomenological interest, we consider here the\npossibility of the formation of spatially-periodic pattern\nin a uniform quasi-2D pseudo spin-1/2 BEC under the\naction of an equal mixture of Rashba and Dresselhaus\ncouplings. In this case, for all strengths of SO coupling\n\r, one can only have a soliton with a stripe pattern in\ncomponent densities with a total density without such\nmodulation. For a small \r(=0.5), the stripe soliton for\nc0=c2=\u00005is displayed in figures 7(a)-(c), where we\nplot the density of components j= 1,j= 2, and the\ntotal density. In this case as the derivative SO coupling\nis of lower dimension acting only along the xdirection,\nit has a weak localization capacity, hence we had to em-\nploy stronger attractive nonlinearity c0=c2=\u00005, in\nplace ofc0=c2=\u00000:5used for Rashba or Dresselhaus\ncoupling, to obtain a compact soliton. For large \r(=8),\nthe stripe state for c0=c2=\u00005, and for an equal mix-\nture of Rashba and Dresselhaus couplings, has a circular\nboundary as exhibited in figures 7(d)-(f) through a con-\ntour plot of densities of components j= 1,j= 2and\nthe total density. This boundary is elliptic for Rashba or\nDresselhaus SO coupling in figures 6(a)-(c).\nC. Uniform SO-coupled quasi-2D spin-1/2\nself-repulsive BEC\nSo far we considered a uniform SO-coupled quasi-2D\nself-attractive BEC ( c0<0). It is also possible to have\na self-repulsive ( c0>0) BEC. Such a self-repulsive BEC\ncan be formed in a quasi-1D system [38], provided that\nthe interspecies interaction is attractive. In the case of\na uniform SO-coupled pseudo spin-1/2 BEC, the sce-\nnario of soliton formation of a self-repulsive BEC with\nattractive interspecies interaction is the same as a self-\nFIG. 8: Contour plot of density of a self-repulsive Rashba\nor Dresselhaus SO-coupled pseudo spin-1/2 stripe soliton of\ncomponents (a) j= 1, (b)j= 2and (c) the total density.\nThe same of a superlattice soliton of components (d) j= 1,\n(e)j= 2and (f) the total density. The parameters employed\narec0=c2= 0:5;\r= 16.\nattractive BEC. Nevertheless, due to SO coupling, one\ncan also have a quasi-2D self-repulsive BEC soliton for\nrepulsive interspecies interaction; such a soliton is not\npossible in the absence of SO coupling ( \r= 0). Here\nwe demonstrate the formation of a self-repulsive soliton\nin a Rashba or Dresselhaus SO-coupled quasi-2D pseudo\nspin-1/2 BEC for repulsive interspecies interaction. To\nthis end we consider c0=c2= 0:5; these values of non-\nlinearity lead to compact solitons of appropriate size for\na not too small \r. In this case it is possible to have the\nsame types of solitons as in the case of a self-attractive\nBECfordifferentstrengthsofSOcouplingandweexhibit\nonly the solitons for \r= 16with pronounced spatially-\nperiodic pattern in density. In figure 8, we display a\ncontour plot of densities of an SO-coupled quasi-2D self-\nrepulsive stripe BEC of components (a) j= 1, (b)j= 2\nand (c) total density. The same of a superlattice BEC\nis shown in figures 8(d)-(f), where the square-lattice pat-\ntern in the total density is clearly visible. The numerical\nenergies of the stripe and the superlattice solitons are\nE=\u0000127:9and\u0000128:0, respectively, in agreement with\nthe analytic estimate (14) ( E=\u0000\r2=2 =\u0000128).\nD. Trapped SO-coupled quasi-2D BEC:\nRashba/Dresselhaus coupling\nWenowinvestigatetheformationofaspatially-ordered\nstate in a trapped SO-coupled quasi-2D pseudo spin-1/2\nBEC for Rashba or Dresselhaus coupling. The densities\nfor Rashba or Dresselhaus SO couplings are the same\nfor all sets of parameters. The scenario of the states for\ndifferent strengths of SO coupling remain qualitatively\nthe same as in the case of a uniform system. How-8\never, there are differences for large SO-coupling \r. In\nthis case we consider a moderate value of nonlinearity\nc0=c2= 100. For a small \r, one can have a circularly-\nsymmetric (0;\u00061)-type state; however, due to the har-\nmonictrapitisnotpossibletohaveamulti-ringstructure\nin the outer region as in the case of a uniform BEC for\na small\r\u00181. In figures 9(a)-(c) we display the contour\nplot of densities of the circularly-symmetric (0;\u00061)-type\nstate for\r= 1. The same for the circularly-asymmetric\nstate is quite similar to the densities in figures 3(d)-(f)\n(result not shown here). The densities of the stripe state\nfor the same set of parameters are shown in figures 9(d)-\n(f). In this case, the circularly-symmetric (0;\u00061)-type\nstate and the circularly-asymmetric state have the same\nenergyE= 3:375, whereas the stripe state has a smaller\nenergyE= 3:363. Hence the stripe state is the ground\nstate. In fact, it is the ground state for all strengths of\nSO coupling. In the uniform case, considered so far, all\nthese states, for a fixed \r, had approximately the same\nnumerical energy and hence could be considered degen-\nerate. The contour plot of densities of the stripe state for\nlargestrengthofSOcoupling \r= 8andforc0=c2= 100\nis shown in figures 9(g)-(i). For small values of nonlin-\nearityc0andc2(c0;c2/60), the stripe state is the only\npossible stable state for large \r. For medium to large\nvalues of nonlinearity c0=c2= 100, one has a new type\nof excited multi-ring state with concentric ring pattern\nin the density of components j= 1andj= 2as dis-\nplayed in figures 9(j)-(l); however, the total density has\na square-lattice pattern in the central region and is with-\nout modulation in density in the outer region. However,\nthe multi-ring state ( E= 27:99) is an excited state com-\nparedtothestripestate( E= 28:15), whichistheground\nstate. Nevertheless, the multi-ring state has supersolid-\nlike properties. We also studied spatially-periodic states\nin a trapped SO-coupled quasi-2D spin-1/2 BEC for an\nequal mixture of Rashba and Dresselhaus couplings (re-\nsult not presented in this paper). In that case only a\nstripe pattern can be obtained for all strengths of SO-\ncoupling\r.\nIV. SUMMARY AND DISCUSSION\nTo search for a supersolid-like spatially-ordered state\nin a uniform or harmonically-trapped pseudo spin-\n1/2 quasi-2D SO-coupled BEC, we find spatially peri-\nodic multi-ring, stripe, and superlattice states for large\nstrengths of SO coupling \r. We consider Rashba and\nDresselhaus SO couplings and an equal mixture of these\ncouplings in this paper. For a small \rand for Rashba\nand Dresselhaus SO couplings, we could have two degen-\nerate states: a circularly symmetric (0;\u00061)-type multi-\nring state, and a circularly asymmetric state. For a large\n\r, thestripestateisobtainedforalltypesofSOcouplings\nin both uniform and trapped BECs. The stripe state has\nstripe pattern in component densities and no modulation\nin total density. The superlattice state is found only in\nFIG. 9: Contour plot of density of a Rashba or Dresselhaus\nSO-coupled pseudo spin-1/2 harmonically-trapped circularly-\nsymmetric (0;\u00061)-type BEC of components (a) j= 1, (b)\nj= 2and (c) the total density for \r= 1. (d)-(f) The same\ndensities of a stripe state. (g)-(i) The same densities of a\nstripe state for \r= 8. (j)-(l) The same densities of a multi-\nring supersolid-like state for \r= 8. The nonlinearities are\nc0=c2= 100:\na uniform BEC for Rashba or Dresselhaus SO couplings\nand has a square-lattice pattern in component densities\nand also in total density, viz. figures 6(d)-(f) and 8(d)-\n(f). The multi-ring state is found in a uniform system for\nmedium\r(= 2), viz. figures 5(a)-(c), and in a trapped\nsystem for large \r(= 8), viz. figures 9(m)-(o). In the case\nofamulti-ringstateinauniformsystem, thereisnomod-\nulation in total density, whereas in the case of a trapped\nmulti-ring state, there is a square-lattice modulation in\nthetotaldensitynearthecenterindicatingsupersolid-like\nbehavior. Nevertheless, the square-lattice pattern in the\nsuperlatticestatesinauniformsystemismoreprominent\nthan the same in the trapped multi-ring state. In a uni-\nform system, the stripe and the superlattice states have\nthe same numerical energy and hence they are degener-\nate, whereas in a trapped system the stripe state is the\nground state with lowest energy for all \r. For an equal9\nmixture of Rashba and Dresselhaus SO couplings, only\na stripe state is obtained in all cases; for a small \r, no\nsupersolid-like state is obtained. For a small \r, the soli-\ntonicstatesinauniformSO-coupledBEChaveverylarge\nspatialextension. Allthesestatesaredynamicallystable,\nas we verified by real-time simulation over a long period\nof time as in the case of an SO-coupled spin-1 quasi-2D\nBEC [23, 24] (result not presented in this paper). Su-\nperlattice states were also found in an SO-coupled spin-1\nspinor BEC [23]. However, it is more difficult to per-\nform an experiment in an SO-coupled spin-1 spinor BEC\nwith three spin components compared to an SO-coupled\npseudo spin-1/2 spinor BEC with two spin components.Hence we believe that the present study could motivate\nexperiments in the search of superlattice states in an SO-\ncoupled pseudo spin-1/2 spinor BEC. The superlattice\nsoliton is dynamically robust and deserve further theo-\nretical and experimental investigation.\nAcknowledgments\nThe author acknowledges support by the CNPq\n(Brazil) grant 301324/2019-0, and by the ICTP-SAIFR-\nFAPESP (Brazil) grant 2016/01343-7\n[1] A. F. Andreev and I. M. Lifshitz, Zurn. Eksp. Teor. 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In the first part, we formulate the density\nestimation problem as a filtering problem of the associated\nmean-field partial differential equation (PDE), for which we\nemploy kernel density estimation (KDE) to construct noisy\nobservations and use filtering theory of PDE systems to\ndesign an optimal (centralized) density filter. It turns out\nthat the covariance operator of observation noise depends\non the unknown density. Hence, we use approximations\nfor the covariance operator to obtain a suboptimal density\nfilter, and prove that both the density estimates and their\ngradient are convergent and remain close to the optimal\none using the notion of input-to-state stability (ISS). In\nthe second part, we continue to study how to decentralize\nthe density filter such that each agent can estimate the\nmean-field density based on only its own position and local\ninformation exchange with neighbors. We prove that the\nlocal density filter is also convergent and remains close to\nthe centralized one in the sense of ISS. Simulation results\nsuggest that the centralized suboptimal density filter is able\nto generate convergent density estimates, and the local\ndensity filter is able to converge and remain close to the\ncentralized filter.\nIndex Terms — Distributed density estimation, PDE sys-\ntems, large-scale systems, Kalman filtering\nI. INTRODUCTION\nRecent years have seen increased research activities in the\nstudy of large-scale agent systems, which treat the swarm as\na continuum and use its mean-field density to represent the\ncollective effect of all the agents [1]–[4]. More recently, the\ntechnique of mean-field feedback control has been proved to\nbe promising for designing control laws for these mean-field\nmodels. In this approach, the global density and its gradient\nof the swarm is fed back into the algorithms to form a closed-\nloop framework, which provides extra stability and robustness\nproperties in the macroscopic level [5]–[9]. Considering the\nscalability issue of the control algorithms and the privacy issue\n*This work was supported by the National Science Foundation under\nGrant No. IIS-1724070, CNS-1830335, IIS-2007949.\n1Tongia Zheng and Hai Lin are with the Department of Electrical\nEngineering, University of Notre Dame, Notre Dame, IN 46556, USA.\ntzheng1@nd.edu, hlin1@nd.edu.\n2Qing Han is with the Department of Mathematics, University of Notre\nDame, Notre Dame, IN 46556, USA. Qing.Han.7@nd.edu.of data sharing, it is desirable to implement the control strategy\nin a fully distributed manner using only local observations and\ninformation exchange. This motivates the distributed mean-\nfield density (and gradient) estimation problem.\nDensity estimation is a fundamental problem in statis-\ntics and has been studied using various methods, including\nparametric and nonparametric methods. Parametric algorithms\nassume that the samples are drawn from a known parametric\nfamily of distributions and try to determine the parameters\nto maximize the likelihood using the samples [10]. Several\ndistributed parametric techniques exist in the literature for\nestimating a distribution from a data set. For example, in [11],\n[12], the unknown density is represented by a mixture of Gaus-\nsians, and the parameters are determined by a combination\nof expectation maximization (EM) algorithms and consensus\nprotocols. Performance of such estimators rely on the validity\nof the assumed models, and therefore they are unsuitable for\nestimating an evolving density. In non-parametric approaches,\nthe data are allowed to speak for themselves in determining\nthe density estimate. Kernel density estimation (KDE) [10]\nis probably the most popular algorithm in this category. A\ndistributed KDE algorithm is presented in [13], which uses\nan information sharing protocol to incrementally exchange\nkernel information between sensors until a complete and\naccurate approximation of the global KDE is achieved by\neach sensor. All these distributed algorithms are proposed\nfor estimating a static/stationary density. To the best of our\nknowledge, distributed estimation for dynamic/non-stationary\ndensities remains largely unexplored.\nIn the density estimation problem of large-scale systems,\nthe agents’ dynamics are usually available because they are\nbuilt by task designers and follow the designed control laws.\nThis motivates the problem of how to take advantage of\nthe available dynamics to improve the density estimation. A\ncandidate solution is to consider a filtering problem to estimate\nthe distribution of all the agents’ states. There is a large\nbody of literature for the filtering problem, ranging from the\ncelebrated Kalman filters and their variants [14] to the more\ngeneral Bayesian filters and their Monte Carlo approaches, also\nknown as particle filters [15]. Motivated by the development\nof sensor networks, distributed implementation for these filters\nhas also been extensively studied [16]–[18]. The general\nprocedure of these distributed filtering algorithms is that each\nagent implements a local filter using its local observation, and\nexchanges its observation or/and estimates with nearby agents\nto gradually improve its estimate of the global distribution.arXiv:2106.05318v2  [eess.SY]  11 Oct 2021However, stability analysis and numerical implementation are\ndifficult especially when the agents’ dynamics are nonlinear\nand time-varying. Also, it is unclear whether the obtained\ndistribution is indeed the mean-field density of the agents.\nIn summary, considering the requirements of decentraliza-\ntion, convergence, and efficiency, existing methods are un-\nsuitable for estimating the time-varying/non-stationary density\nof large-scale agent systems in a distributed manner. This\nmotivates us to propose a distributed, dynamic, and scalable\nalgorithm that can perform online and use only local obser-\nvation and information exchange to obtain convergent density\nestimates. We tackle this problem by formulating the density\nestimation problem as a state estimation problem of the PDE\nsystem that governs the spatial and temporal evolution of the\nmean-field density and using KDE to construct noisy obser-\nvations for this PDE system. We show that the observation\nnoise is approximately Gaussian but its covariance operator\ndepends on the unknown density. Hence, we use infinite-\ndimensional Kalman filters for PDE systems to design a\n(centralized) density filter and prove its stability and optimality\nwhen approximations are used for the unknown covariance\noperator. Then, we decentralize the density filter such that each\nagent estimates the mean-field density based on only its own\nposition and local information exchange with neighbors, and\nprove that the local density filter has comparable performance\nwith centralized one in the sense of ISS. Our contribution is\nsummarized as follows: (i) we present a centralized density\nfilter and prove that the density estimates and their gradient\nare both convergent; (ii) we present a distributed density filter\nsuch that each agent estimates the global density using only its\nown position and local communication; (iii) all the results hold\neven if the agents’ dynamics are nonlinear and time-varying.\nSome of the results have been reported in our previous work\n[19], [20]. The difference is clarified as follows. In [19], we\npresented a density filter and proved its stability assuming that\nthe filter is well-posed in some Hilbert space. In this work, we\nwill rigorously study the solution properties (such as well-\nposedness and mass conservation) of the density filter and\nuse the obtained properties to improve the stability results.\nIn particular, with the new results we will be able to prove\nthat the gradient of the density estimates is also convergent.\nThis is a significant property considering that the gradient is\nalmost always required in mean-field feedback control. In [20],\nwe presented a distributed density filter, which can be seen as\na special case of the distributed density filter in this work.\nThe rest of the paper is organized as follows. Section II\nintroduces some preliminaries. Problem formulation is given\nin Section III. Section IV and V are our main results, in which\nwe design centralized and distributed density filters and then\nstudy their stability and optimality. Section VI discusses their\nnumerical implementation. Section VII performs an agent-\nbased simulation to verify the effectiveness of the density\nfilters. Section VIII summarizes the contribution and points\nout future research.\nII. P RELIMINARIESA. Notations\nConsiderf:E!R, whereE\u001aRnis a mea-\nsurable set. Denote L2(E) =ffjkfkL2(E)<1g, en-\ndowed with the norm kfkL2(E)= (R\nEjf(x)j2dx)1=2and\ninner producthf;giL2(E)=R\nEf(x)g(x)dx. Let@if=\n@f\n@xibe the weak derivatives of f. DenoteH1(E) =\nffjkfkH1(E)<1g, endowed with the norm kfkH1(E)=\nkfkL2(E)+Pn\ni=1k@ifkL2(E)and inner producthf;giH1(E)=\nhf;giL2(E)+Pn\ni=1h@if;@igiL2(E). We will omit Ein the\nnotations of norms and inner products when it is clear.\nLet\nbe a bounded and connected C1-domain in Rnand\nT > 0be a constant. Denote by @\nthe boundary of \n. Set\n\nT= \n\u0002(0;T).\nB. Infinite-Dimensional Kalman Filters\nThe Kalman filter is an algorithm that uses the system’s\nmodel and sequential measurements to gradually improve its\nstate estimate [21]. We present its extension to PDE systems\ndeveloped in [22].\nLetV,Hbe Hilbert spaces with norms k\u0001kV;k\u0001kHand\ninner productsh\u0001;\u0001iV;h\u0001;\u0001iH, respectively. Vis dense inHand\nkvkH\u0014ckvkVfor allv2V.V\u0003is the dual of Vand hence\nV\u001aH\u001aV\u0003. LetE,Falso be Hilbert spaces. Consider\n8\n<\n:du\ndt+A(t)u(t) =f(t) +B(t)\u0018(t); t2(0;T)\nu(0) =u0+\u0010;(1)\ny(t) =C(t)u(t) +\u0011(t); (2)\nwhere\nA(\u0001)2L1(0;T;L(V;V\u0003))and9\u000b>0;\u0015\u00150such that\nhA(t)z;ziH+\u0015kzk2\nH\u0015\u000bkzk2\nV;8z2V;t2(0;T);\nB(\u0001)2L1(0;T;L(E;H ));C(\u0001)2L1(0;T;L(H;F));\nf(\u0001)2L2(0;T;H); u 0;\u00102H\n\u0018(\u0001)2L2(0;T;E); \u0011(\u0001)2L2(0;T;F):\nThese standard conditions ensure existence and uniqueness of\nsolutions of (1) in the Sobolev space\nW(0;T) =ffjf2L2(0;T;V);df\ndt2L2(0;T;V\u0003)g:\nLet\b :=H\u0002L2(0;T;E)\u0002L2(0;T;F)be the space of\ntriplesf\u0010;\u0018(\u0001);\u0011(\u0001)g. Equip \bwith a cylindrical probability\n\u0016such that\nE[\u0016] =0\n@0\n0\n01\nA;Cov[\u0016] =0\n@P0 0 0\n0Q(t) 0\n0 0R(t)1\nA\nwhereP02L(H;H ),Q(\u0001)2L1(0;T;L(E;E)), and\nR(\u0001)2L1(0;T;L(F;F))are all self-adjoint and positive\nsemi-definite covariance operators, and R(\u0001)is invertible.\nTheorem 1: [22] The minimum mean squared error esti-\nmator ofuis given by the unique solution of\nd^u\ndt+A(t)^u(t) =f(t) +P(t)C\u0003(t)R\u00001(t)\u0000\ny(t)\u0000C(t)^u(t)\u0001\n;\n^u(0) =u0;which satisfies ^u2W(0;T), whereP(t)2L(H;H)is the\nfamily of operators uniquely defined by\n8\n<\n:If'2W(0;T)and\u0000d'\ndt+A\u0003(t)'2L2(0;T;H);\nthenP'2W(0;T);\n(3)\nand8\n<\n:dP\ndt'+AP'+PA\u0003'+PC\u0003R\u00001CP'=BQB\u0003';\nP(0) =P0;for'as in (3);\nwheredP\ndt':=d\ndt(P')\u0000Pd'\ndt.\nC. Input-to-state stability\nInput-to-state stability (ISS) is a stability notion to study\nnonlinear control systems with external inputs [23]. Its exten-\nsion to infinite-dimensional systems is developed in [24]. Let\n(X;k\u0001kX)and(U;k\u0001kU)be the state space and the input\nspace, endowed with norms k\u0001kXandk\u0001kU, respectively.\nDenoteUc=PC(R+;U), the space of piecewise right-\ncontinuous functions from R+toU, equipped with the sup-\nnorm. Define the following classes of comparison functions:\nP:=f\r:R+!R+j\ris continuous ;\r(0) = 0;\nand\r(r)>0;8r>0g\nK:=f\r2Pj\ris strictly increasing g\nK1:=f\r2Kj\ris unboundedg\nL:=f\r:R+!R+j\ris continuous and strictly\ndecreasing with lim\nt!1\r(t) = 0g\nK L :=f\f:R+\u0002R+!R+j\f(\u0001;t)2K;8t\u00150;\n\f(r;\u0001)2L;8r>0g:\nConsider a control system \u0006 = (X;Uc;\u001e)where\u001e:R+\u0002\nR+\u0002X\u0002Uc!Xis a transition map. Here, \u001e(t;s;x;u )\ndenotes the system state at time t2R+, if its state at time\ns2R+isx2Xand the input u2Ucis applied.\nDefinition 1: \u0006is called locally input-to-state stable (LISS) ,\nif9\u001ax;\u001au>0,\f2K L , and\r2K, such that\nk\u001e(t;t0;\u001e0;u)kX\u0014\f(k\u001e0kX;t\u0000t0)+\r\u0010\nsup\nt0\u0014s\u0014tku(s)kU\u0011\nholds8\u001e0:k\u001e0kX\u0014\u001ax;8u2Uc:kukUc\u0014\u001auand8t\u0015t0.\nIt is called input-to-state stable (ISS) , if\u001ax=1and\u001au=1.\nTo emphasize the state space and input space, we will\nsometimes sayk\u001e(t)kXis (L)ISS toku(t)kU. To verify the\nISS property, Lyapunov functions can be exploited.\nDefinition 2: A continuous function V:R+\u0002D!\nR+;D\u001aXis called an LISS-Lyapunov function for\u0006, if\n9\u001ax;\u001au>0, 1; 22K1;\u001f2K, andW2P, such that:\n(i) 1(kxkX)\u0014V(t;x)\u0014 2(kxkX);8t2R+;8x2D\n(ii)8x2X:kxkX\u0014\u001ax;8\u00182U:k\u0018kU\u0014\u001auit holds:\nkxkX\u0015\u001f(k\u0018kU))_Vu(t;x)\u0014\u0000W(kxkX);8t2R+\nfor allu2Uc:kukUc\u0014\u001auwithu(0) =\u0018, where the\nderivative of Vcorresponding to the input uis given by\n_Vu(t;x) = lim\n\u000e!+0V(t+\u000e;\u001e(t+\u000e;t;x;u ))\u0000V(t;x)\n\u000e:IfD=X;\u001ax=1and\u001au=1;then the function Vis called\nanISS-Lyapunov function .\nAn (L)ISS-Lyapunov function theorem for time-invariant\nsystems is given in [24]. The following theorem is an extension\nto time-varying systems, whose proof can be found in [9].\nTheorem 2: [9] Let \u0006 = (X;Uc;\u001e)be a control system,\nandx\u00110be its equilibrium point. If \u0006possesses an (L)ISS-\nLyapunov function, then it is (L)ISS.\nISS is a convenient tool for studying the stability of cascade\nsystems. Consider two systems \u0006i= (Xi;Uci;\u001ei);i= 1;2,\nwhereUci=PC(R+;Ui)andX1\u001aU2. We say they form a\ncascade connection if u2(t) =\u001e1(t;t0;\u001e01;u1).\nTheorem 3: [25] The cascade connection of two ISS\nsystems is ISS. If one of them is LISS, then the cascade\nconnection is LISS.\nIII. P ROBLEM FORMULATION\nThis paper studies the problem of estimating the dynamic\nmean-field density of large-scale stochastic agents. Consider\nNagents within a spatial domain \n\u001aRn. Their dynamics\nare assumed to be known and satisfy a family of stochastic\ndifferential equations:\ndXi\nt=v(Xi\nt;t)dt+\u001b(Xi\nt;t)dWi\nt; i= 1;:::;N; (4)\nwhereXi\nt2\nis the state of the i-th agent,v= (v1;:::;vn)2\nRnis the deterministic dynamics, fWi\ntgN\ni=1is a family of m-\ndimensional standard Wiener processes independent across the\nagents, and \u001b= [\u001bkl]2Rn\u0002mis the stochastic dynamics. In\nthis work, the superscription iis reserved to represent the i-th\nagent. The statesfXi\ntgN\ni=1are assumed to be observable. The\nprobability density p(x;t)of the states is known to satisfy a\nmean-field PDE, called the Fokker-Planck equation:\n@tp=\u0000nX\nj=1@j(vjp) +nX\nj;k=1@j@k(\u0006jkp)in\nT;\np=p0on\n\u0002f0g;\nn\u0001(g\u0000vp) = 0 on@\n\u0002(0;T);(5)\nwherep0is the initial density, \u0006 = [\u0006jk] =1\n2\u001b\u001bT2\nRn\u0002nis the diffusion tensor, g= (g1;:::;gn)withgj=Pn\nk=1@k(\u0006jkp), and nis the outward normal to @\n. By\ndefinition, \u0006is symmetric and positive semidefinite. We al-\nways assume that it is uniformly positive definite (or uniformly\nelliptic), i.e., there exists a constant \u0015>0such that\n\u0018T\u0006(x;t)\u0018\u0015\u0015k\u0018k2;8(x;t)2\nT;8\u00182Rn:\nRemark 1: Note that (5) is uniquely determined by (4)\nwhich is known. This relationship holds even if (4) is nonlinear\nand time-varying. Hence, the centralized/distributed density\nfilters we design are applicable to a large family of systems.\nAlso note that (5) is always a linear PDE.\nWe recall the following result proved in [9] for (5), which\nwill be useful when we study the solution property of the den-\nsity filters. Note that a density function fsatisfieshf;1iL2=R\n\nfdx= 1 where 1is the 1-constant function on \n.\nTheorem 4: [9] Assume that\nvj;\u001b;@j\u001b2L1(\nT)andp02L1(\n): (6)Then\n\u000f(Well-posedness ) Problem (5) has a unique weak solution\np2H1(\nT).\n\u000f(Mass conservation ) The solution satisfies hp(t);1iL2=\n1;8t.\n\u000f(Positivity ) If we further assume that\n@jvj;@2\nj\u001b2L1(\nT); (7)\nthenp0\u0015(>)0impliesp\u0015(>)0for allt2[0;T].\nWe assume the agents can exchange information with\nneighbors and form a time-varying communication topology\nG(t) = (V;E(t)), whereVis the set of Nagents and\nE(t)\u001aV\u0002Vis the set of communication links. We assume\nG(t)is undirected. Now, we can formally state the problems\nto be solved as follows:\nProblem 1 (Centralized density estimation): Given the sys-\ntem (5) and agent states fXi\ntgN\ni=1, we want to estimate their\ndensityp(x;t).\nProblem 2 (Distributed density estimation): Given the sys-\ntem (5), the state Xi\ntof an agent i, and the topology G, we\nwant to design communication and estimation protocols for\neach agent to estimate the global density p(x;t).\nIV. C ENTRALIZED DENSITY ESTIMATION\nIn this section, we study the centralized density estimation\nproblem, assuming all agent states are observable. This central-\nized density filter was reported in our previous work [19]. Our\nnew results include studying its solution properties and using\nthem to improve the stability results (with better regularity).\nHence, we include its development for completeness.\nTo design a filter, we rewrite (5) as an evolution equation\nand use KDE to construct a noisy observation y(t):\n_p(t) =A(t)p(t);\ny(t) =pKDE(t) =p(t) +w(t);(8)\nwhereA(t)p=\u0000Pn\nj=1@j(vjp) +Pn\nj;k=1@j@k(\u0006jkp)is a\nlinear operator, pKDE(t)represents a kernel density estimator\n(see (28)) usingfXi\ntgN\ni=1, andw(t)is the observation noise\nwhich is asymptotically Gaussian with asymptotically diagonal\ncovariance operator R(t) =kdiag(p(t))where diag(p(t))2\nL(H1(\n);H1(\n)) is a diagonal operator and k > 0is a\nconstant depending on N, the kernel K, and its bandwidth h.\n(See Appendices VIII-A for these properties of w(t).)Ris\nalways invertible because of the positivity of pin Theorem 4.\nThe independence of w(t)between different toriginates from\nthe property of the Wiener process in (4).\nA. Optimal density filter\nBy Theorem 1, the optimal density filter is given by\n_^p=A(t)^p+K(t)(y\u0000^p);^p(0) =pKDE(0); (9)\nK(t) =P(t)R\u00001(t); (10)\n_P=A(t)P+PA\u0003(t)\u0000PR\u00001(t)P;P(0) =P0;(11)\nwhereA\u0003(t)is the adjoint operator of A(t), given by\nA\u0003(t) =nX\nj=1vj(x;t)@j+nX\nj;k=1\u0006jk(x;t)@j@k; (12)and (11) should be interpreted as in Theorem 1.\nFirst, we study the well-posedness and mass conservation\nproperties of the density filter. These results are new and will\njustify the stability proof of the density filter.\nFrom Section II-B, it is seen that the well-posedness of (9)\nand (11) relies on the well-posedness of (5), which is already\ngiven in Theorem 4. Then, we have the following results for\nthe solutions of (9) and (11).\nTheorem 5: Assume that (6) and (7) hold. Then we have\n(i) (Well-posedness ) The operator Riccati equation (11) has\na unique weak solution P(t)2L(H1(\n);H1(\n)) such\nthatP'2H1(\nT)for'2H1(\nT)withd'\ndt+A\u0003(t)'2\nL2(0;T;H1(\n)) . The density filter (9) has a unique weak\nsolution ^p2H1(\nT).\n(ii) ( Mass conservation ) The solution satisfies h^p(t);1iL2=\n1;8t.\nProof: (i) The well-posedness follows from a direct\napplication of Theorems 1 and 4. Note that p2H1(\nT)\nimpliesp(t)2H1(\n)and@tp(t)2H1(\n)\u0003for almost all t.\nThis means we can choose V=H=H1(\n)andW(0;T) =\nffjf2L2(0;T;H1(\n));df\ndt2L2\u0000\n0;T;H1(\n)\u0003\u0001\ngfor\n(9). According to Theorem 1, we obtain ^p2H1(\nT)and\nP2L(H1(\n);H1(\n)) .\n(ii) To prove the mass conservation property, first note that\nP01= E[(^p(0)\u0000p(0))\u000e(^p(0)\u0000p(0))]1=0;\nwhere 0is the zero constant function on \n. Using (12), we\nhaveA\u0003(t)1=0;8t, and\n_P1=AP1+PA\u00031\u0000PR\u00001P1=AP1\u0000PR\u00001P1;\ni.e.,P(t)1satisfies a homogeneous equation with zero initial\nconditionP01=0. Hence,P(t)1=0for allt\u00150. Now,\nh_^p;1iL2=hA^p+PR\u00001(y\u0000^p);1iL2\n=h^p;A\u00031iL2+hR\u00001(y\u0000^p);P1iL2=0:\nHence, we have\nh^p(t);1iL2=h^p(0);1iL2= 1 for allt\u00150;\nwhich means (9) conserves mass.\nRemark 2: Again, ^p2H1(\nT)implies ^p(t)2H1(\n)\nfor almost all t. Determining that the state space is H1(\n)\nis crucial because it will enable us to prove in subsequent\nsections that the (distributed) density filters are convergent in\nH1norm, which means that not only the density estimate\nitself but also its gradient is convergent. Finally, the mass\nconservation property of (11) actually implies that 0is a simple\neigenvalue ofP(t)with1being an associated eigenfunction.\nB. Suboptimal density filter\nWe note thatR(t)depends on the unknown density p(t).\nHence, we approximate R(t)using \u0016R(t) =\u0016kdiag(pKDE(t)),\nwhere \u0016kis given by\n\u0016k=h2R\nK(u)2du+Pn\nj=1R\n(@jK(u))2du\nNhn+2\n(according to (30) and (31) in Appendices). We always assume\nthat\u0016Ris invertible, which is easy to satisfy since we constructpKDE. Correspondingly, we obtain a “suboptimal” density\nfilter:\n_^p=A(t)^p+\u0016K(t)(y(t)\u0000^p);^p(0) =pKDE(0); (13)\n\u0016K(t) =\u0016P(t)\u0016R\u00001(t); (14)\n_\u0016P=A(t)\u0016P+\u0016PA\u0003(t)\u0000\u0016P\u0016R\u00001(t)\u0016P;\u0016P(0) = \u0016P0:(15)\nWe have the following results for (13) and (15), whose proof\nis similar with Theorem 5 and thus omitted.\nTheorem 6: Under assumptions (6) and (7), we have\n(i) (Well-posedness ) The operator Riccati equation (15) has\na unique weak solution \u0016P(t)2L(H1(\n);H1(\n)) such\nthat\u0016P'2H1(\nT)for'2H1(\nT)withd'\ndt+A\u0003(t)'2\nL2(0;T;H1(\n)) . The density filter (13) has a unique\nweak solution ^p2H1(\nT).\n(ii) ( Mass conservation ) The solution satisfies h^p(t);1iL2=\n1;8t.\nTo study the stability of the suboptimal filter, define ~p=\n^p\u0000p. Then along \u0016P(t)we have\n_~p= (A(t)\u0000\u0016P(t)\u0016R\u00001(t))~p+\u0016P(t)\u0016R\u00001(t)w(t): (16)\nWe definek\u0016R\u00001(t)\u0000R\u00001(t)kH1as the approximation error\nof\u0016R(t), since it is zero if and only if \u0016R(t) =R(t). We also\nneed the notion of generalized inverse of self-adjoint operators\nbecause of the zero eigenvalues of Pand\u0016P, which is defined\nin Appendices VIII-B.\nWe will show that: (i) (16) is ISS to w(t); (ii) the solution\nof (15) remains close to the solution of (11); and (iii) the\nsuboptimal gain \u0016Kremains close to the optimal gain K, which\njustifies the “suboptimality” of (13). In [19], we proved weaker\nresults assuming that all density filters and operator Riccati\nequations have unique solutions in certain Hilbert spaces. In\nthis work, we use the newly obtained solution properties and\nthe notion of generalized inverse operators to prove stability\nresults in a better functional space (with improved regularity).\nThe stability and optimality results are stated in the following.\nTheorem 7: Assume that (6) and (7) hold, and that\nkP(t)kH1andk\u0016P(t)kH1are uniformly bounded. Also assume\nthat there exist constants c1;c2>0such that for all t\u00150,\n0<c1I\u0014R\u00001(t);\u0016R\u00001(t);Py(t);\u0016Py(t)\u0014c2I; (17)\nwherePy(t)(\u0016Py(t)) is the generalized inverse of P(t)(\u0016P(t))\nandIis the identity operator on H1(\n). Then we have\n(i) (Stability )k^p(t)\u0000p(t)kH1is ISS tokw(t)kH1.\n(ii)k\u0016P(t)\u0000P(t)kH1is LISS tok\u0016R\u00001(t)\u0000R\u00001(t)kH1.\n(iii) ( Optimality )k\u0016K(t)\u0000K(t)kH1is LISS tok\u0016R\u00001(t)\u0000\nR\u00001(t)kH1.\nProof: (i) We study the unforced part of (16), i.e.,\nassumew= 0 . Consider a Lyapunov function V1=\nh\u0016Py(t)~p(t);~p(t)iH1. By the definition of \u0016Py(t), we see that\n0is a simple eigenvalue of \u0016Py(t)for allt, with 1being\nan associated eigenfunction. Then V1= 0 if and only\nif~p=constant, which would necessarily be 0 because\nh~p;1iL2=h^p\u0000p;1iL2= 0(the mass conservation property).By appropriate approximation arguments, we have\n_V1=\n\u0016Py_~p;~p\u000b\nH1+\n\u0016Py~p;_~p\u000b\nH1\u0000\n\u0016Py_\u0016P\u0016Py~p;~p\u000b\nH1\n=\n\u0016Py(A\u0000\u0016P\u0016R\u00001)~p;~p\u000b\nH1+\n(A\u0003\u0000\u0016R\u00001\u0016P)\u0016Py~p;~p\u000b\nH1\n\u0000\n\u0016Py(A\u0016P+\u0016PA\u0003\u0000\u0016P\u0016R\u00001\u0016P)\u0016Py~p;~p\u000b\nH1\n=\u0000h\u0016R\u00001~p;~piH1:\nHence, the unforced part of (16) is uniformly exponentially\nstable. Since (16) is a linear control system, with the uniform\nboundedness of \u0016Pand\u0016R\u00001, we obtain thatk^p(t)\u0000p(t)kH1\nis ISS tokw(t)kH1.\n(ii) Define \u0000 = \u0016P\u0000P . Using (11) and (15) we have\n_\u0000 =A(t)\u0000 + \u0000A\u0003(t)\u0000\u0016P\u0016R\u00001(t)\u0016P+PR\u00001(t)P: (18)\nSimilarly, by considering a Lyapunov function defined by\nV2=hPy(t)~p(t);~p(t)iH1, one can show that the following\nsystem alongP(t)is also uniformly exponentially stable:\n_~p= (A\u0000PR\u00001)~p: (19)\nNote that the inner products in V1andV2are equivalent\nbecause of assumption (17). We rewrite (18) as\n_\u0000 =A\u0000 + \u0000A\u0003\u0000\u0016P\u0016R\u00001\u0000\u0000\u0016P\u0016R\u00001P\u0000\u0000R\u00001P+\u0016PR\u00001P\n= (A\u0000 \u0016P\u0016R\u00001)\u0000 + \u0000(A\u0003\u0000R\u00001P)\u0000\u0016P(\u0016R\u00001\u0000R\u00001)P\nwhich is essentially a linear equation. Now fix q2H1(\n)\nwithkqkH1= 1. Since the unforced part of (16) and (19) are\nuniformly exponentially stable, and k\u0016P(t)kH1andkP(t)kH1\nare uniformly bounded, 9constants\u0015;c> 0such that\nk\u0000(t)qkH1\u0014e\u0000\u0015tk\u0000(0)qkH1\n+c\n\u0015sup\n0\u0014\u001c\u0014tk\u0016R\u00001(\u001c)\u0000R\u00001(\u001c)kH1;\ni.e.,k\u0016P(t)\u0000P(t)kH1is LISS tok\u0016R\u00001(t)\u0000R\u00001(t)kH1.\n(ii) To prove the third statement, observe that\nk\u0016K(t)\u0000K(t)kH1\n=k(\u0016P\u0016R\u00001\u0000PR\u00001)(t)kH1\n\u0014k(\u0016P\u0016R\u00001\u0000P\u0016R\u00001+P\u0016R\u00001\u0000PR\u00001)(t)kH1\n\u0014k\u0016R\u00001(t)kH1k\u0016P(t)\u0000P(t)kH1\n+kP(t)kH1k\u0016R\u00001(t)\u0000R\u00001(t)kH1:\nBy Theorem 3,k\u0016K(t)\u0000K(t)kH1is LISS tok\u0016R\u00001(t)\u0000\nR\u00001(t)kH1.\nRemark 3: A few comments are in order. By taking advan-\ntage of the dynamics, the density filter essentially combines\nall past outputs (constructed using KDE) to produce better and\nconvergent estimates, which largely circumvent the problem\nof bandwidth selection of KDE [10]. The fact that ^p(t)is\nconvergent in H1(\n)implies that the L2norm of its gradient is\nalso convergent. This implication is significant as it is observed\nthat the existing mean-field feedback laws are more or less\nbased on gradient flows [26], which use the gradient of density\nestimates as feedback [5], [7]–[9]. Since the gradient operator\nis an unbounded operator, any algorithm that produces accurate\ndensity estimates may have arbitrarily large estimation error\nfor the gradient. However, by formulating a filtering problem\nand studying its solution property, we are able to obtain density\nestimates whose gradient is also convergent.V. D ISTRIBUTED DENSITY ESTIMATION\nIn this section, we study the distributed density estimation\nproblem. We present a distributed density filter by integrating\ncommunication protocols into (13) and (15), and study its con-\nvergence and optimality. We reformulate (8) in the following\ndistributed form:\n_p(t) =A(t)p(t);\nzi(t) =Ki(t); i= 1;:::;N;(20)\nwhereKi(t) =1\nhnK\u00001\nh(x\u0000Xi\nt)\u0001\nis a kernel density function\ncentered at position Xi\nt. We viewzi(t)as the local observation\nmade by the i-th agent. The challenge of distributed density\nestimation lies in that each agent alone does not have any\nmeaningful observation of the unknown density p(t), because\nits local observation zi(t)is simply a kernel centered at\npositionXi\nt, which conveys no information about p(t).\nA. Local density filter\nInspired by the distributed Kalman filters in [16], we design\nthe local density filter for the i-th agent (i= 1:::;N ) as\n_^pi=A(t)^pi+Li(t)(yi(t)\u0000^pi) +\u0012Pi(t)X\nj2Ni(^pj\u0000^pi);\n(21)\nKi(t) =Pi(t)Ri(t); (22)\n_Pi=A(t)Pi+PiA\u0003(t)\u0000PiR\u00001\ni(t)Pi; (23)\n^pi(0) =yi(0);Pi(0) =Pi0; (24)\nwhere\u0012\u00150,yi(t)is to be constructed later, Ri(t) =\n\u0016kdiag(yi(t)), andNirepresents the neighbors of agent i. The\nalgorithm we reported in [20] can be seen as a special case\n(\u0012= 0) of the above algorithm.\nAn important observation is that the quantities in (13) satisfy\ny(t) =1\nNNX\ni=1zi(t);\u0016R(t) = diag(1\nNNX\ni=1zi(t)):\nThese relations suggest that if we can design algorithms such\nthatyi(t)!1\nNPN\nizi(t)(which then implies Ri(t)!\u0016R(t)),\nthen the local filter (21) and (23) should converge to the\ncentralized filter (13) and (15). The associated problem of\ntracking the average of Ntime-varying signals is called dy-\nnamic average consensus. We use a variant of the proportional-\nintegral (PI) consensus estimator given in [27], which is a low-\npass filter and therefore suitable for our problem. Some useful\nproperties are summarized in Appendices VIII-C. According\nto (32), we construct yiin the following way:\n@t i=\u0000\u000b( i\u0000zi)\n\u0000X\nj2Niaij( i\u0000 j) +X\nj2Nibji(\u001ei\u0000\u001ej);\n@t\u001ei=\u0000X\nj2Nibij( i\u0000 j);\n i(0) =zi(0); \u001ei(0) =\u001ei0;\nyi= i+di\nk i+dikL1;(25)where\u000b;aij;bij>0are constants, \u001ei0needs to be a density\nfunction, and diis defined by\ndi=(\njc\u0000infx i(x)jifinfx i(x)<c;\n0 if i(x)\u0015c;\nwherec >0is a pre-specified small constant. The first two\nequations of (25) are based on (32), i.e., the consensus algo-\nrithm is performed in a pointwise manner. The last equation is\nadded to redistribute the negative mass of  ito other spatial\nareas such that the output becomes positive and conserves\nmass. Note thathyi;1iL2= 1, andyi= iif i\u0015c;8x.\nHence, as long as there exists tc>0such that i(t)\u0015cfor\nt\u0015tc(which should be expected because  iis meant to track\nthe strictly positive density function pKDE(t)), we can assume\nyi= iin the convergence analysis.\nFirst, we show that the internal states of (25) conserve mass.\nTheorem 8: The solution of (25) satisfies h i(t);1iL2=\nh\u001ei(t);1iL2= 1;8i;8t\u00150.\nProof: First, sincezi(t) =Ki(t), we havehzi(t);1iL2=\n1;8t. Now consider change of variables: \u0016 i= i\u00001and\n\u0016\u001ei=\u001ei\u00001. Thenh\u0016 i(0);1iL2=h\u0016\u001ei(0);1iL2= 0, and\nh@t\u0016 i;1iL2=\u0000\u000bh\u0016 i;1iL2\u0000NX\nj=1aijh\u0016 i\u0000\u0016 j;1iL2\n+NX\nj=1bjih\u0016\u001ei\u0000\u0016\u001ej;1iL2;\nh@t\u0016\u001ei;1iL2=\u0000NX\nj=1bijh\u0016 i\u0000\u0016 j;1iL2;\ni.e., (h\u0016 i(t);1iL2;h\u0016\u001ei(t);1iL2)satisfies a family of ho-\nmogeneous equations with zero initial conditions. Hence,\nh\u0016 i(t);1iL2=h\u0016\u001ei(t);1iL2= 0 andh i(t);1iL2=\nh\u001ei(t);1iL2= 1 for allt\u00150.\nNow we show that the local quantities indeed track the\ncentral quantities. Define z(x;t) = (z1(x;t);:::;zN(x;t))\nand\u0005=IN\u00001\nN1N1>\nNwhere IN2RN\u0002Nis the identity\nmatrix and 1N= (1;:::; 1)2RN. We have the following\ntheorem.\nTheorem 9: Assume the communication topology G(t)is\nalways connected. If 9tc>0such that i(t)\u0015cfort\u0015\ntc, thenkyi(t)\u0000y(t)kH1andkRi(t)\u0000\u0016R(t)kH1are ISS to\nk\u0005_z(t)kH1. Moreover, ifkR\u00001\ni(t)kH1andk\u0016R\u00001(t)kH1are\nuniformly bounded, then kR\u00001\ni(t)\u0000\u0016R\u00001(t)kH1is also ISS\ntok\u0005_z(t)kH1.\nProof: According to Lemma 3 in Appendices VIII-C\n(with proper generalizations to infinite-dimensional ODEs),\nkyi(t)\u0000y(t)kL2is ISS tok\u0005_z(t)kL2. By taking the partial\nderivatives in xon both sides of (25), we see that their spatial\nderivatives satisfy the same equations and hence k@jyi(t)\u0000\n@jy(t)kL2is ISS tok\u0005@j_z(t)kL2;8j= 1;:::;n . As a result,\nkyi(t)\u0000y(t)kH1is ISS tok\u0005_z(t)kH1, which also implies that\nkRi(t)\u0000\u0016R(t)kH1is ISS tok\u0005_z(t)kH1. Finally, notice that\nkR\u00001\ni(t)\u0000\u0016R\u00001(t)kH1\n\u0014kR\u00001\ni(t)kH1k\u0016R\u00001(t)kH1kRi(t)\u0000\u0016R(t)kH1\nwhich implieskR\u00001\ni(t)\u0000\u0016R\u00001(t)kH1is also ISS.Remark 4: The above ISS property holds even if the net-\nwork is switching, as long as it remains connected. In practice,\nthe network may not be always connected since the agents are\nmobile. Nevertheless, the transient error caused by permanent\ndropout of agents will be slowly forgotten according to (33).\nThe mass conservation property is also useful in numerical\nimplementation because it helps to avoid error accumulation\ncaused by numerical errors or agent dropout.\nB. Stability and optimality\nSimilar to the centralized filters, we have the following\nresults for the local density filter (21) and the local operator\nRiccati equation (23).\nTheorem 10: Assume that (6) and (7) hold, and that 9tc>0\nsuch that i(t)\u0015cfort\u0015tc. We have\n(i) (Well-posedness ) The local operator Riccati equa-\ntion (23) has a unique weak solution Pi(t)2\nL(H1(\n);H1(\n)) such thatPi'2H1(\nT)for'2\nH1(\nT)withd'\ndt+A\u0003(t)'2L2(0;T;H1(\n)) . The\nlocal density filter (21) has a unique weak solution ^pi2\nH1(\nT).\n(ii) ( Mass conservation ) The solution satisfies\nh^pi(t);1iL2= 1;8t.\nProof: (i) The well-posedness of (23) for each ifollows\nfrom Theorems 1 and 4. The local density filter (21) is a family\nofNPDEs where the coupling only occurs on the low-order\nterms. Hence, it lies in the framework of (1) and Theorem 1,\nand has a unique weak solution ^pi2H1(\nT).\n(ii) By similar arguments as in Theorem 5, we have\nPi(t)1=0;8t\u00150. Then,\nh_^pi;1iL2\n=hA^pi+PiR\u00001\ni(yi\u0000^pi) +\u0012PiX\nj2Ni(^pj\u0000^pi);1iL2\n=h^pi;A\u00031iL2+hR\u00001\ni(yi\u0000^pi);Pi1iL2\n+\u0012hX\nj2Ni(^pj\u0000^pi);Pi1iL2=0:\nHence,h^pi(t);1iL2=h^pi(0);1iL2= 1 for allt\u00150.\nThe remaining task is to show that (21) and (23) indeed\nremain close to (13) and (15), respectively. Towards this end,\ndefine ~pi= ^pi\u0000p. Note that ^pj\u0000^pi= ~pj\u0000~pi. Then along\nPi(t)we have\n_~pi= (A\u0000PiR\u00001\ni)~pi+PiR\u00001\ni(yi\u0000y+w)+\u0012PiX\nj2Ni(~pj\u0000~pi):\nTheorem 11: Let the communication topology G(t)be con-\nnected. Assume that (6) and (7) hold, and that 9tc>0such\nthat i(t)\u0015cfort\u0015tc. Also assume that kPi(t)kH1and\nk\u0016P(t)kH1are uniformly bounded and that there exist constants\nc1;c2>0such that for all t\u00150,\n0<c1I\u0014R\u00001\ni(t);\u0016R\u00001(t);Py\ni(t);\u0016Py(t)\u0014c2I;\nwherePy\ni(t)(\u0016Py(t)) is the generalized inverse of Pi(t)(\u0016P(t)).\nThen we have:\n(i) (Stability )PN\ni=1k^pi(t)\u0000p(t)kH1is ISS toPN\ni=1kyi(t)\u0000y(t)kH1andkw(t)kH1.(ii)kPi(t)\u0000\u0016P(t)kH1is LISS tokR\u00001\ni(t)\u0000\u0016R\u00001(t)kH1.\n(iii) ( Optimality )kKi(t)\u0000\u0016K(t)kH1is LISS tokR\u00001\ni(t)\u0000\n\u0016R\u00001(t)kH1.\nProof: (i) Consider a Lyapunov function V=PN\ni=1hPy\ni(t)~pi(t);~pi(t)iH1. ThenV= 0 if and only if ~pi=\n0;8i. We have\n_V=NX\ni=1\u0010\nhPy\ni_~pi;~piiH1+hPy\ni~pi;_~piiH1\u0000hPy\ni_PiPy\ni~pi;~piiH1\u0011\n=NX\ni=1\u0010\nhPy\ni(A\u0000PiR\u00001\ni)~pi;~piiH1\n+hR\u00001\ni(yi\u0000y+w);~piiH1\n+hPy\ni~pi;(A\u0000PiR\u00001\ni)~piiH1\n+hPy\ni~pi;PiR\u00001\ni(yi\u0000y+w)iH1\n\u0000hPy\ni(APi+PiA\u0003\u0000PiR\u00001\niPi)Py\ni~pi;~piiH1\n+h\u0012X\nj2Ni(~pj\u0000~pi);~piiH1\n+hPy\ni~pi;\u0012PiX\nj2Ni(~pj\u0000~pi)iH1\u0011\n=NX\ni=1\u0010\n\u0000hR\u00001\ni~pi;~piiH1+ 2hR\u00001\ni(yi\u0000y+w);~piiH1\u0011\n+\u0012NX\ni=1X\nj2Nih(~pj\u0000~pi);~piiH1;\nwhere the last term is negative according to consensus theory\n[16]. Let\r2(0;1). Then,\n_V\u0014NX\ni=1\u0010\n\u0000(1\u0000\r)hR\u00001\ni~pi;~piiH1\u0000\rhR\u00001\ni~pi;~piiH1\n+ 2hR\u00001\ni(yi\u0000y+w);~piiH1\u0011\n\u0014\u0000(1\u0000\r)NX\ni=1hR\u00001\ni~pi;~piiH1;\nif\nNX\ni=1k~pikH1\u00152c2\n\rc1NX\ni=1kyi\u0000y+wkH1:\nSincekyi\u0000y+wkH1\u0014kyi\u0000ykH1+kwkH1, we obtain thatPN\ni=1k^pi(t)\u0000p(t)kH1is ISS toPN\ni=1kyi(t)\u0000y(t)kH1and\nkw(t)kH1.\n(ii) Define \u0000i=\u0016P\u0000Pi. We have\n_\u0000i=A\u0000i+ \u0000iA\u0003\u0000P\u0016R\u00001\u0016P+PiR\u00001\niPi;\nwhich is formally equivalent to (18). Hence we omit the proof.\n(iii) The proof is also similar to Theorem 7 (iii).\nFinally, by Theorem 3, we have the following corollary.\nCorollary 1: Under the assumptions in Theorems 9 and 11,\nwe have:\n(i) (Stability )PN\ni=1k^pi(t)\u0000p(t)kH1is ISS tok\u0005_z(t)kH1\nandkw(t)kH1.\n(ii)kPi(t)\u0000\u0016P(t)kH1is LISS tok\u0005_z(t)kH1.\n(iii) ( Optimality )kKi(t)\u0000\u0016K(t)kH1is LISS tok\u0005_z(t)kH1.Remark 5: The distributed density filter allows each agent\nto estimate the global density of all agents using only its own\nposition and local information exchange. Like the centralized\ndensity filter, the density estimates and their gradient are both\nconvergent. Notice that we allow \u0012= 0 in (21), in which\ncase each agent only communicates zi=Kiwith neighbors.\nThus, it only needs to exchange its real-time position Xi\nt\nwith neighbors since Kiis uniquely determined by Xi\nt,\nwhich is very efficient from a communication perspective but\nmay result in slow convergence. When \u0012 > 0, each agent\nadditionally exchange its density estimate ^piwhich results in\nfaster convergence by using extra synchronization but also\nincreases communication cost. The accelerated convergence\ncan be observed from the proof of Theorem 11 where the \u0012\nterm provides additional negative mass to _V.\nVI. N UMERICAL IMPLEMENTATION\nIn this section, we discuss the numerical implementation\nof the proposed algorithms. The simplest implementation is\nprobably the one based on finite difference approximations of\nderivatives [28], although more delicate numerical methods are\nalso possible. Assume the spatial domain \nis discretized as\na grid with Mcells. Then the density functions like ^piare\napproximated by M\u00021vectors. The operators like A,Piand\nRiare approximated by M\u0002Mmatrices. The time evolution\nis based on Euler discretization. Hence, (21) and (23) can be\napproximated by:\n^pi(k+ 1) = (I+\u000eTA(k))^p(k)\n+\u000eTPi(k)Ri(k)(yi(k)\u0000pi(k))\n+\u000eT\u0012Pi(k)X\nj2Ni(^pj(k)\u0000^pi(k));\nPi(k+ 1) = (I+\u000eTA)Pi(k) +\u000eTPi(k)A>(k)\n\u0000\u000eTPi(k)R\u00001\ni(k)Pi(k);\nwhere ^pi;yi2RM,A;Pi;Ri2RM\u0002M, and\u000eTis the step\nsize. The consensus algorithm (25) can be approximated in a\nsimilar way, with a step size \u000et. The selection of \u000ethas been\nstudied in many papers [29]. We may let \u000eT=l\u000et, i.e., perform\nmultiple steps of information exchange between two succes-\nsive updates of the filters. With this type of asynchronous\nupdates, the local filters can potentially achieve better tracking\nof the centralized filter. This setup of asynchronous updates\nis left as future work. In terms of computational cost, the\ncomputational bottleneck of Kalman filters is on the inverse\nof the covariance operators/matrices. We also notice that M\nis usually large in the discretization scheme. However, Riis\ndiagonal and Ais highly sparse. Hence, the numerical iteration\nof the filter is very efficient and can be carried out online.\nIn terms of communication cost, when \u0012= 0, each agent\nonly sends its own position to neighbors during each iteration,\nwhich is very efficient. If \u0012>0, each agent additionally sends\nanM\u0002Mmatrix. In this case, each agent may decrease the\nresolution/dimension of ^piby downsampling before sending it\nto neighbors, and increase the resolution of a received density\nestimate by interpolation.VII. S IMULATION STUDIES\nTo verify the performance of the proposed central-\nized/distributed filters, we perform a simulation on Matlab\nusing 100agents. The agents’ positions are generated by the\nfollowing family of equations:\ndXi\nt=r\u0001Drf(x;t)\nf(x;t)dt+DdWi\nt; i= 1;:::; 100;(26)\nwhereD= 0:03andf(x;t)is a time-varying density function\nto be specified. We set \n = (0;1)2with a reflecting boundary.\nThe initial positions fXi\n0g100\ni=1are drawn from the uniform\ndistribution. Therefore, the ground truth density satisfies\n@tp=\u0000r\u0001Dprf\nf+1\n2D2\u0001p;\np(\u0001;0) =1;(27)\nwhich will be used for comparison. We design f(x;t)to be\na Gaussian mixture of two components with the common\ncovariance matrix diag (0:02;0:02)but different time-varying\nmeans [0:5 + 0:3 cos(0:04t);0:5 + 0:3 sin(0:04t)]Tand[0:5 +\n0:3 cos(0:04t+\u0019);0:5 + 0:3 sin(0:04t+\u0019)]T. Under this\ndesign, the agents are nonlinear and time-varying and their\nstates will converge to two “spinning” Gaussian components.\nThe numerical implementation is based on the procedures\noutlined in Section VI. Specifically, \nis discretized as a\n30\u000230grid. All density functions are approximated by 900\u00021\nvectors and all operators are approximated by 900\u0002900\nmatrices. The density updates and information exchange are\nperformed in a synchronous manner with a step size \u000eT=\n0:1s. The bandwidth of KDE is h= 0:08. For the PI consensus\nalgorithm (25), we set \u000b= 0:2,aij= 0:4, andbij= 0:04.\nA graphical illustration of the simulation results is presented\nin Fig. 1, where each column represents the same time step and\neach row represents the evolution of an investigated quantity.\nAs already observed in [19], the centralized filter, by taking\nadvantage of the dynamics, quickly ”recognizes” the two\nunderlying Gaussian components and gradually catches up\nwith the evolution of the ground truth density, outperforming\nthe outputs of KDE. Then, we randomly select an agent and\ninvestigate the performance of its local density filter when\n\u0012= 0 and\u0012= 0:4. We observe that the two local filters\nalso gradually catch up with the ground truth density, however\nin a slower rate compared with the centralized filter due\nto the delay effect of information exchange. In Fig. 2, we\ncompare the L2norms of density and gradient estimation\nerrors of KDE, the centralized filter, and 5 randomly selected\nlocal filters. We observe that all the filters outperform KDE\nas the filtering process proceeds. Both the density estimates\nand the gradient estimates are convergent. The local filters\nhave larger initial density/gradient estimation errors because\nof limited observation on the initial step. Through information\nexchange, they gradually produce comparable performance\nwith the centralized filter. In particular, the one with \u0012 > 0\nconverges faster than the one with \u0012= 0 because of the\nadditional exchange of density estimates with neighbors.Fig. 1 :1st row: the collection of states fXi\ntg100\ni=1of the systems (26), where the red circle is the randomly selected representative\nagent and the red dots are its neighbors; 2nd row: the evolution of the ground truth density p(x;t)computed using (27); 3rd\nrow: density estimates pKDE(x;t)using KDE; 4th row: outputs ^p(x;t)of the centralized filter (13); 5th row: outputs ^pi(x;t)\nof the local filter (21) when \u0012= 0;6th row: outputs ^pi(x;t)of the local filter (21) when \u0012>0.\nVIII. C ONCLUSION\nWe presented centralized/distributed density filters for es-\ntimating the mean-field density of large-scale agent systems\nwith known dynamics by a novel integration of mean-fieldmodels, KDE, infinite-dimensional Kalman filters, and com-\nmunication protocols. With the distributed filter, each agent\nwas able to estimate the global density using only the mean-\nfield dynamics, its own state/position, and local information(a) Density estimation errors.\n(b) Gradient estimation errors.\nFig. 2 :Black solid line: density/gradient estimation errors of\nKDE; black dashed line: density/gradient estimation errors\nof the centralized density filter; blue line: density/gradient\nestimation errors of local filters when \u0012= 0 ;red line:\ndensity/gradient estimation errors of local filters when \u0012>0.\nexchange with its neighbors. It was scalable to the number\nof agents, convergent in not only the density estimates but\nalso their gradient, and efficient in terms of numerical imple-\nmentation and communication. These algorithms can be used\nfor many density-based distributed optimization and control\nproblems of large-scale agent systems when density feedback\ninformation is required. Our future work is to integrate the\ndensity filters into mean-field feedback control for large-scale\nagent systems and study the closed-loop stability.\nAPPENDICES\nA. Kernel density estimation\nKernel density estimation is a non-parametric way to\nestimate an unknown probability density function. Let\nX1;:::;XN2Rnbe independent identically distributed\nrandom variables having a common probability density f. The\nkernel density estimator for fis given by [10], [30]\nfN(x) =1\nNhnNX\ni=1K\u00101\nh(x\u0000Xi)\u0011\n; (28)\nwhereK(x)is a kernel function [10] and his the bandwidth,\nusually chosen as a function of Nsuch that limN!1h= 0andlimN!1Nhn=1. The Gaussian kernel is frequently\nused, given by\nK(x) =1\n(2\u0019)n=2exp\u0010\n\u00001\n2x>x\u0011\n:\nIt is known that the fN(x)is asymptotically normal and that\nfN(xi)andfN(xj)are asymptotically uncorrelated for xi6=\nxj, asN!1 , which is summarized as follows.\nLemma 1 (Asymptotic normality [31]): IflimN!1h= 0,\nthen at any continuity point x,\nlim\nN!1E[fN(x)]\u0000f(x) = 0: (29)\nIf in addition limN!1Nhn=1, then asN!1 ,\np\nNhn(fN(x)\u0000E[fN(x)])!N\u0010\n0;f(x)Z\nK(u)2du\u0011\n(30)\nin distribution, where Nrepresents the Gaussian distribution.\nLemma 2 (Asymptotic uncorrelatedness [31]): Letxi;xj\nbe two distinct continuity points. If limN!1h= 0, then the\ncovariance of fN(xi)andfN(xj)satisfies\nNhnCov[fN(xi);fN(xj)]!0;asN!1:\nThese two lemmas together implies that as N!1 , the\nestimation error fN\u0000fis asymptotically Gaussian with zero\nmean and diagonal covariance. Similar asymptotic normality\nand uncorrelatedness properties hold for the derivatives of\nfN\u0000f[32]. In particular, as N!1 ,\np\nNhn+2(@jfN(x)\u0000E[@jfN(x)])\n!N\u0010\n0;f(x)Z\u0000\n@jK(u)\u00012du\u0011 (31)\nin distribution.\nB. Generalized inverse operators\nWe introduce the notion of generalized inverse operators\nof self-adjoint operators defined in [33], which is based on\nan orthogonal decomposition. Let Hbe a Hilbert space and\n\u0002 :D(\u0002)\u0012H!H be a self-adjoint operator, where D(\u0002)\nrepresents the domain of \u0002. LetR(\u0002) andN(\u0002) to be the\nrange and kernel of \u0002, respectively. Since \u0002is self-adjoint,\nN(\u0002)?=R(\u0002) (and we always have \u0002(D(\u0002)\\R(\u0002))\u0012\nR(\u0002)). Thus, under the decomposition H=N(\u0002)\bR(\u0002),\nwe have the following representation for \u0002:\n\u0002 =\u00120 0\n0b\u0002\u0013\nwhereb\u0002 :D(\u0002)\\R(\u0002)\u0012R(\u0002)!R(\u0002) is self-adjoint.\nNow, the generalized inverse \u0002yis defined by:\n\u0002y=\u00120 0\n0b\u0002\u00001\u0013\nwith domain\nD(\u0002y) =N(\u0002) +R(\u0002)\n\u0011fu0+u1ju02N(\u0002);u12R(\u0002)g\u0013R (\u0002):\nWe have the following facts:\n(i)\u0002yis self-adjoint.(ii) One has that\n\u0002\u0002y\u0002 = \u0002;\u0002y\u0002\u0002y= \u0002y;(\u0002y)y= \u0002\n(iii) The operator \u0002\u0002y:D(\u0002y)!H is an orthogonal pro-\njection ontoR(\u0002). We may naturally extend it, still denoted\nit by itself, toD(\u0002y) =H. Hence, \u0002\u0002y:H!R(\u0002)\u0012H\nis the orthogonal projection onto R(\u0002). Similarly, we can\nextend \u0002y\u0002to be an orthogonal projection from Honto\nR(\u0002y) =N(\u0002y)?=N(\u0002)?=R(\u0002). Therefore, we have\n\u0002\u0002y= \u0002y\u0002\u0011PR(\u0002)\u0011orthogonal projection onto R(\u0002):\nC. Dynamic average consensus\nWe introduce the PI consensus estimator presented in [27].\nConsider a group of Nagents where each agent has a local\nreference signal ui(t) : [0;1)!R. The dynamic average\nconsensus problem consists of designing an algorithm such\nthat each agent tracks the time-varying average uavg(t) :=\n1\nNPN\ni=1ui(t). The PI consensus algorithm is given by [27]:\n_\u0017i=\u0000\u000b(\u0017i\u0000ui)\u0000NX\nj=1aij(\u0017i\u0000\u0017j) +NX\nj=1bji(\u0011i\u0000\u0011j)\n_\u0011i=\u0000NX\nj=1bij(\u0017i\u0000\u0017j) (32)\nwhereuiis agenti’s reference input, \u0011iis an internal state,\n\u0017iis agenti’s estimate of uavg,[aij]N\u0002Nand[bij]N\u0002Nare\nadjacency matrices of the communication graph, and \u000b>0is\na parameter determining how much new information enters the\ndynamic averaging process. The Laplacian matrices associated\nwith [aij]and[bij]are represented by LP(proportional)\nandLI(integral) respectively. The PI estimator solves the\nconsensus problem under constant (or slowly-varying) inputs,\nand remains stable for varying inputs in the sense of ISS.\nDefine the tracking error of agent iby\u000fi(t) =\u0017i(t)\u0000\nuavg(t);i= 1;:::;N . Decompose the error into the consensus\ndirection 1Nand the disagreement directions orthogonal to\n1N. Define the transformation matrix T= [(1=p\nN)1NR]\nwhere R2RN\u0002(N\u00001)is such that T>T=TT>=IN.\nConsider the change of variables\n\u0016\u000f=\u0014\u0016\u000f1\n\u0016\u000f2:N\u0015\n=T>\u000f;w=\u0014w1\nw2:N\u0015\n=T>\u0011;\ny=w2:N\u0000\u000b(R>L>\nIR)\u00001R>_u\nto write (32) in the equivalent form\n_w1=0;2\n4_y\n_\u0016\u000f1\n_\u0016\u000f2:N3\n5=2\n40 0\u0000R>LIR\n0\u0000\u000b 0\nR>L>\nIR 0\u0000\u000bI\u0000R>LPR3\n5\n| {z }\nA2\n4y\n\u0016\u000f1\n\u0016\u000f2:N3\n5\n+2\n4\u0000\u000b(R>L>\nIR)\u00001\n0\n03\n5\n|{z}\nBR>_u:The stability result is given as follows.\nLemma 3: [29] Let LIandLPbe Laplacian matrices of\nstrongly connected and weight-balanced digraphs. 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S´ a de Melo2†\n1Department of Physics, Renmin University of China, Beijing 100872, China and\n2School of Physics, Georgia Institute of Technology, Atlant a, Georgia 30332, USA\nWe discuss the evolution from BCS to Bose superconductivity versus carrier density in gated\ntwo-dimensional s-wave superconductors. We investigate the carrier density dependence of the\ncritical temperature, superfluid density, order parameter , chemical potential and pair size. We show\nthat the transition from the normal to the superconducting s tate is controlled by the Berezinskii-\nKosterlitz-Thouless vortex-antivortex pairing mechanis m, and that the evolution from high carrier\ndensity (BCS pairing) to low carrier density (Bose pairing) is just a crossover in s-wave systems. We\ncompare our results to recent experiments on the supercondu ctor Li xZrNCl, a lithium-intercalated\nlayered nitride, and obtain very good quantitative agreeme nt at low and intermediate densities.\nIntroduction: The physics of two-dimensional (2D) su-\nperconductors is a fascinating subject with a long his-\ntory [1]. The theoretical prediction that 2D Fermi sys-\ntems can exhibit superconductivity can be inferred from\nthe seminal papers of Berezinskii [2], Kosterlitz and\nThouless [3], where the mechanism of vortex-antivortex\nunbinding was proposed. The topological nature of\nthe superconducting transition in two dimensions at\nfixed carrier density was unveiled in experimental stud-\nies of thin films [4, 5], which provided strong evidence\nfor the Berezinskii-Kosterlitz-Thouless (BKT) transition\nand vortex-antivortex unbinding [6].\nChemical substitutions or atom deficiencies in 2D ma-\nterials can change carrier density, but very often they\nalso introduce disorder and inhomogenous doping, cre-\nating difficulties in the interpretration of experimental\ndata and phase diagrams. A way to circumvent such dif-\nficulties is to study 2D systems via electrostatic doping,\nwherethe carrierdensity ncanbetuned fromaminimum\nnminto a maximum nmaxvalue. It was theoretically pro-\nposed that topological quantum phase transitions occur\nduring the evolution from BCS to Bose pairing in 2D d-\nandp-wave superfluids and superconductors as a func-\ntion of density or interaction strength [7–10]. However,\nexperiments that used electrostatic doping in 2D d-wave\nsuperconductors (cuprates) [11, 12] did not investigate\nthe existence of a quantum critical point from gapless to\ngapped from high (BCS pairing) to low (Bose pairing)\ndensity of carriers, but indicated that phase fluctuations\nplayanimportantroleindeterminingthecriticaltemper-\natureat lowcarrierdensities. This experimentalobserva-\ntion was in agreement with a theoretical argument that\nclassicalphase fluctuations were very important in estab-\nlishing an upper bound for the critical temperature of 2D\nsuperconductors with small superfluid density [13]. Fur-\nthermore, these early experiments provided support to\nthe viewpoint that the relation between the critical tem-\nperatureTcand the muon depolarization rate in cuprate\nsuperconductors [14, 15] should be reinterpreted as an\nupper bound on Tcgiven by the ordering temperaturefor phase fluctuations [13].\nFor single band 2D superfluids and superconductors\nwith parabolic bands, it was shown explicitly that this\nupper bound for the critical temperature Tc=TBKT\nis exactly one-eighth of the Fermi energy εF, that is,\nTBKT≤εF/8 = 0.125εF[16] in units where /planckover2pi1=kB= 1.\nSuch a general relation is obtained rather trivially from\nthe BKT transition temperature TBKT=πρs(TBKT)/2,\nwhereρsis the superfluid density. It is sufficient to no-\ntice that, at any temperature T,ρs(T) cannot exceed\nn/(4m), wherenis the carrier density and mis the car-\nrier effective mass. Physically, this bound is the ratio\nbetween the maximum fermion pair density ( n/2) and\nthe fermion pair mass 2 mand is reflects the Ferrell-\nGlover-Tinkham [17–19] optical conductivity sum rule.\nThe same bound is valid even when the superfluid den-\nsity tensor is anisotropic or when spin-orbit effects are\nincluded [20, 21].\nExperiments seeking to study the evolution from BCS\nto Bose pairing in essentially 2D superconductors like\nFeSe [22–24], twisted bilayer graphene [25, 26], and lay-\nered nitrides [27] have been attempted, but only very\nrecently it has been shown that in Li xZrNCl, a lithium-\nintercalated layered nitride, the electrolyte LiClO 4can\ndope a 2D layer of ZrNCl with Lithium ions via a gate\nvoltageVG, over a range of nearly two orders of magni-\ntude [28]. Because of this doping tunability, this tech-\nnique potentially alllows for the study of the evolution\nfrom BCS (high carrier density) to the Bose (low carrier\ndensity) superconductivity in 2D systems as discussed\ntheoretically in the context of ultracold fermions [16]. In\nthis paper, we show that a theory involving phase fluc-\ntuations for 2D s-wave superconductors leads to criti-\ncal temperature and order parameter modulus in good\nagreement with experimental results for Li xZrNCl [28]\nas the system evolves from BCS to Bose pairing in the\nrelevant carrier concentration range. Furthermore, we\nextract from experimental data the effective mass of the\ncharge carriers, as well as, the magnitude and range of\nthe effective two-body interaction between them.2\nHamiltonian: We use a two-band continuum model of\nfermion with identical parabolic bands centered at points\nKandK′of the Brillouin zone of Li xZrNCl. Since\nthe two bands are identical, the effective mass of the\ncarriers in each band is mand density of carriers per\nband isn=k2\nF/2π. Furthermore, we take the interac-\ntions to be the same in both bands and assume that the\ntwo bands are independent. In this case, it is sufficient\nto consider the Hamiltonian per band. Thus, we start\nfrom the Hamiltonian density per band ( /planckover2pi1=kB= 1)\nH(r) =HK(r)+HI(r),where the kinetic energy density\nwith respect to the chemical potential µis\nHK(r) =/summationdisplay\nsψ†\ns(r)/bracketleftbigg\n−∇2\n2m−µ/bracketrightbigg\nψs(r),(1)\nand the interaction energy density is\nHI(r) =/integraldisplay\nd2r′V(r,r′)ψ†\n↑(r)ψ†\n↓(r′)ψ↓(r′)ψ↑(r) (2)\nwithV(r,r′) =−Vsg(|r−r′|/R). Here,Vsis the magni-\ntude of the s-wave attractive interaction with dimension\nof energy, and g(|r−r′|/R) is a dimensionless function\nwith range R. The field operator ψ†\ns(r) creates a fermion\nwith spin projection sat position r. The Hamiltonian\nH=/integraltextd2rH(r) has three microscopic parameters: the\neffective mass m, the interaction strength Vsand the in-\nteraction range R. In momentum space, the Hamiltonian\nreads\nH=/summationdisplay\nk,sξkψ†\nk,sψk,s+/summationdisplay\nk,k′,qVkk′b†\nkqbk′q,(3)\nwherebkq=ψ−k+q/2,↓ψk+q/2,↑is the pairing operator\nandξk=εk−µ, with energy dispersion εk=k2/2m.\nFollowing standard procedure [7], we obtain a separable\ninterparticle potential in momentum space\nVkk′=−VsΓs(k)Γs(k′), (4)\nwhere Γ s(k) = (1+k/kR)−1/2,andkR∼R−1plays the\nrole of the interaction range in momentum space [29].\nThe Hamiltonian in Eq. (3) has three microsopic param-\neters: the effective mass m, the interaction strength Vs\nand rangeR∼k−1\nR. As the density nis varied, we com-\npare the interparticle spacing ℓip= 1/√n=√\n2π/kF\ntoRand notice that the interactions are short-ranged\nwhenR/ℓip∼(kF/kR)(1/√\n2π)≪1 and long-ranged\nwhenR/ℓip∼(kF/kR)(1/√\n2π)≫1.\nEffective Action: We consider the order parameter\nmodulus |∆|and its associated phase θ, to write the\nphase fluctuation effective action per band as [16]\nSeff=Ssp(|∆|)+Sph(|∆|,θ)+Szp(|∆|,θ) (5)\nThe first term is the saddle-point action\nSsp(|∆|) =/summationdisplay\nk/bracketleftbigg(ξk−Ek)\nT−2ln/parenleftBig\n1+e−Ek/T/parenrightBig/bracketrightbigg\n+|∆|2\nTVs,\n(6)whereEk=/radicalbig\nξ2\nk+|∆k|2is the quasiparticle excitation\nenergy, and ∆ k= ∆Γ s(k) plays the role of the order\nparameter function for s-wave pairing. The second term\nis the phase fluctuation action\nSph=1\n2/integraldisplay\ndr\n/summationdisplay\nijρij∂iθ(r)∂jθ(r)+κs[∂τθ(r)]2\n,\n(7)\nwherethe integrationis overposition andimaginarytime\nr= (r,τ), with/integraltext\ndr≡/integraltext1/T\n0dτ/integraltext\nd2r. In Eq. (7), the first\nterm contains the superfluid density tensor\nρij(µ,|∆|,T) =1\n4L2/summationdisplay\nk[2nsp(k)∂i∂jξk−Yk∂iξk∂jξk],\n(8)\nwhere∂idenotes the partial derivative with respect to\nmomentum kiwithi={x,y},\nnsp(k) =1\n2/bracketleftbigg\n1−ξk\nEktanh/parenleftbiggEk\n2T/parenrightbigg/bracketrightbigg\n(9)\nis the momentum distribution per spin state, and Yk=\n(2T)−1sech2(Ek/2T) is the Yoshida function. Here, the\nsuperfluid density tensor is diagonal since ρxy=ρyx= 0,\nand its diagonal elements are equal, that is, ρxx=ρyy=\nρs. The second term in Eq. (7) is\nκs(µ,|∆|,T) =1\n4L2/summationdisplay\nk/bracketleftbigg|∆k|2\nE3\nktanh/parenleftbiggEk\n2T/parenrightbigg\n+ξ2\nk\nE2\nkYk/bracketrightbigg\n,\n(10)\nwithκs=κ/4, whereκ=∂n/∂µ|T,Vis related to the\nthermodynamic compressibility K=κ/n2. The zero-\npoint action is Szp=/summationtext\nqωq/2T, whereω(q) =c|q|is\nthe frequency and c=/radicalbig\nρs/κsis the speed of sound.\nTo elucidate the role of phase fluctuations, we write\nθ(r,τ) =θv(r) +θc(r,τ), where the classical ( τ-\nindependent) contribution θv(r) is due to vortices (trans-\nverse velocities), while the quantum ( τ-dependent) con-\ntributionθc(r,τ) is due to collective modes (longitudinal\nvelocities). The resulting phase action per band is\nSph=Sv+Sc, (11)\nsince contributions involving cross-terms in θvandθc\nvanish. The vortex action is\nSv=1\n2T/integraldisplay\nd2rρs[∇θv(r)]2, (12)\nwhile the collective mode action is\nSc=1\n2/integraldisplay\ndr/bracketleftBig\nρs[∇θc(r)]2+κs[∂τθc(r)]2/bracketrightBig\n.(13)\nThe thermodynamic potential is Ω = Ω sp+Ωph, where\nthe first term is the saddle-point Ω sp=TSsp, and the\nsecond is due to phase fluctuations Ω ph= Ωv+Ωc+Ωzp.3\nThe contribution due to vorticesis Ω v=−TlnZv, where\nZv=/integraltext\ndθve−Sv. Integration over phase fluctuations\nθc(r,τ) leads to Ω c=/summationtext\nqTln[1−exp(−ωq/T)],while\nthe zero-point term is Ω zp=TSzp.\nCritical Temperature: The self-consistency relations\nfor|∆|andTcfor fixed chemical potential µare derived\nfrom the effective action Seff=Ssp+Sphas follows. The\norder parameter equation is obtained through the sta-\ntionarity condition δSsp/δ∆∗\n0= 0,leading to\n1\nVs=/summationdisplay\nk|Γs(k)|2\n2Ektanh/parenleftbiggEk\n2T/parenrightbigg\n. (14)\nThe equation for the critical temperature Tc=TBKT\nis determined by the Kosterlitz-Thouless [3] condition\nTBKT=π\n2ρs(µ,|∆|,TBKT), (15)\nobtained by using the vortex action of Eq. (12). For\nfixed interaction Vsand a given chemical potential µ,\nEqs. (14) and (15) determine the order parameter mod-\nulus|∆|and the critical temperature TBKT. These two\nequations arise from the saddle-point and vortex actions.\nTo relateµandn=N/L2, whereNis the total num-\nber ofparticlesper bandand L2is the areaofthe sample,\nwe usen=−∂Ω/∂µ|T,V,withΩ = Ω/L2, leading to\nn=nsp+nph+nzp. (16)\nHere,nj=−∂Ωj/∂µ|T,Vwherej={sp,ph,zp}. The\nexpression of nspis particularly simple, that is, nsp=\n2/summationtext\nknsp(k), while the others can be easily obtained\nfrom their respective Ωj. The self-consistent solutions\nof Eqs. (14), (15), and (16) determine µ,|∆|andTBKT\nas function of density nfor given effective mass m, in-\nteraction strength Vsand momentum-space interaction\nrangekR. A measure of the evolution from BCS to Bose\npairing is provided by the pair size [7]\nξ2\npair=/integraltext\nd2kϕ∗(k)/bracketleftbig\n−∇2\nk/bracketrightbig\nϕ(k)/integraltext\nd2k|ϕ(k)|2(17)\nin terms of the non-normalized two-body wavefunction\nϕ(k) = ∆(k)/2E(k).\nWe use momentum and energy scales kRandεR=\nk2\nR/2m, respectively, and write the dimensionless order\nparameter equation as\n1\n/tildewideVs=/integraldisplay\nd˜k˜k|/tildewideΓs(˜k)|2\n2/tildewideE˜ktanh/parenleftBigg/tildewideE˜k\n2/tildewideT/parenrightBigg\n,(18)\nwhere/tildewideVs=Vsg2D, withg2D=mL2/πbeingthe total2D\ndensity of states per band. The dimensionless functions\nare/tildewideΓs(˜k) = (1 + ˜k)−1/2for the interaction symmetry\nfactor,/tildewideE˜k=/bracketleftBig\n˜ξ2\n˜k+|/tildewide∆˜k|2/bracketrightBig1/2\nfor the quasiparticle exci-\ntation energy, with ˜ξ˜k=˜k2−˜µfor the kinetic energyand/tildewide∆˜k=/tildewide∆/tildewideΓs(˜k) for the order parameter function. The\ndimensionless critical temperature is\n/tildewideTBKT=˜nsp\n8−1\n16/tildewideTBKT/integraldisplay\nd˜k˜k3sech2/parenleftBigg/tildewideE˜k\n2/tildewideTBKT/parenrightBigg\n,(19)\nwhere/tildewideTBKT=TBKT/TR, withTR=εR. The dimension-\nless density ˜ n=n/nR, wherenR=k2\nR/2π, is expressed\nsolely in terms of the saddle-point contribution\n˜n= ˜εF= ˜nsp=/integraldisplay\nd˜k˜k/bracketleftBigg\n1−˜ξ˜k\n/tildewideE˜ktanh/parenleftBigg/tildewideE˜k\n2/tildewideT/parenrightBigg/bracketrightBigg\n,(20)\nsincenphandnzpare small for T≈TBKT, except\nwhennand/orT→0 [30]. For comparison, we also\nobtain the dimensionless mean field (MF) temperature\n/tildewideTMF=TMF/TRby setting |/tildewide∆|= 0 and solving only\nEqs. (18) and (20), because MF neglects all phase fluc-\ntuations including the vortex-antivortex binding mecha-\nnism described by Eq. (19).\nIn Fig. 1, we show temperatures /tildewideTBKT=TBKT/TR,\n/tildewideTMF=TMF/TR, order parameter |/tildewide∆|=|∆|/εR, chem-\nical potential ˜ µ=µ/εRand pair size ˜ξpair=kRξpair\nversus density ˜ n=n/nRfor fixed/tildewideVs= 0.7100. No-\ntice that ˜n= 2π(R/ℓip)2and a crossover from short- to\nlong-ranged interactions occurs at R/ℓip∼1, that is,\nat ˜n∼˜ncr= 2π= 6.283 as ˜ngrows. In Fig. 1(a),\nwe show/tildewideTBKTand/tildewideTMFversus ˜nand note that /tildewideTMF≥\n/tildewideTBKT. The gray dotted line represents the upper bound\n/tildewideTBKT= ˜n/8 = ˜εF/8 discussed in the introduction [16].\nBoth/tildewideTMFand/tildewideTBKTarenon-monotonicandhaveamaxi-\nmum located at ˜ n= 0.3241 and ˜n= 0.5099, respectively.\nThe inclusion of phase fluctuations, via the BKT mecha-\nnism of vortex-antivortex binding, reduces TMFtoTBKT\nthrough Eq. (19). Therefore the range TBKT<T <T MF\ndetermines the region where classical phase fluctuations\nare important. In this region, preformed pairs are guar-\nanteed to exist [31, 32]. However, we cannot interpret\nTMFas the experimental pseudogap temperature T∗de-\nfined asa 1% dropin the zero-biasconductance [28]. The\ntheoretical understanding of T∗requires further analy-\nsis [33].\nIn Fig. 1(b), we show the |/tildewide∆|versus ˜nboth at/tildewideT=\n/tildewideTBKTand/tildewideT= 0, noticing that |/tildewide∆|at/tildewideT=/tildewideTBKThas\na maximum at ˜ n= 0.1939, while |/tildewide∆|at/tildewideT= 0 has a\nmaximum at ˜ n= 0.4356. In Fig. 1(c), we show ˜ µver-\nsus ˜nboth at/tildewideT=/tildewideTBKTand/tildewideT= 0, observing that ˜ µ\nis a monotonically increasing function of ˜ nas expected.\nFor higher densities, where the superconductor is deep\nin the degenerate BCS regime, ˜ µ→˜εF= ˜n, as indi-\ncated by the gray dot-dashed line. For lower densities,\nwhere the superconductor is deep in the non-degenerate\nBose regime, ˜ µ→ −|/tildewideEB|/2 =−0.004309 as indicated by\nthe gray dotted line. Here, /tildewideEB=−0.008619 is the di-\nmensionless binding energy obtained from the two-body4\n0.01 0.1 1\nn/nR00.010.02(a)T/TR\n0.01 0.1 1\nn/nR00.020.04(b)|∆|/εR\n0.01 0.1 1   \nn/nR0  0.20.40.60.81  (c)µ/εR\n0.001 0.01 -0.010    0.01 0.02 \n0.01 0.1 1   \nn/nR100101102103104\n(d)kRξpair\nFIG.1. Thecouplingconstantis /tildewideVs= 0.7100 inall panels. (a)\nTemperatures /tildewideTBKT(red solid line), /tildewideTMF(blue dashed line)\nand upper bound /tildewideTBKT= ˜εF/8 = ˜n/8 (gray dotted line) vs\n˜n. (b) Order parameter |/tildewide∆|vs ˜nat/tildewideT=/tildewideTBKT(red solid line)\nand/tildewideT= 0 (black dashed line). (c) Chemical potential ˜ µvs ˜n\nat/tildewideT=/tildewideTBKT(red solid line) and /tildewideT=TMF(blue dashed line).\nInset: ˜µ= ˜εF= ˜n(gray dot-dashed line) and ˜ µ=−|/tildewideEB|/2\n(gray dotted line). (d) Pair size ˜ξpairvs ˜nat/tildewideT=/tildewideTBKT(red\nsolid line), /tildewideT= 0 (black dashed line), and ˜ξpair= ˜n−1/2(gray\ndotted line), that is, kFξpair= 1 at/tildewideT= 0.\nbound state eigenvalue equation [16]\n1\n/tildewideVs=/integraldisplay\nd˜k˜k|/tildewideΓs(˜k)|2\n2˜k2−/tildewideEB. (21)\nfor/tildewideVs= 0.7100. Spectroscopically, the BCS-Bose\ncrossover occurs at ˜ µ= 0 (˜n= 0.004032), where the gap\nfor quasiparticle excitations changes from /tildewideEg=|/tildewide∆|on\ntheBCS-pairingside(˜ n>0.004032)to/tildewideEg=/radicalBig\n˜µ2+|/tildewide∆|2\non the Bose-pairing side (˜ n<0.004032).\nIn Fig. 1(d), we show that the pair size ˜ξpair=kRξpair\nis not a monotonically increasing function of ˜ n, having a\nminimum at ˜ n= 0.03275. We emphasize that ξpairis not\nthe phase coherence length ξphthat diverges at the BKT\ntransition, but it provides a measure that the BCS-Bose\npairing crossover region occurs when kFξpair∼1 [31].\nThe condition kFξpair= 1 or˜ξpair=˜k−1\nF= ˜n−1/2is\nrepresented by the gray dotted line. The BCS-pairing\nlimit corresponds to ˜ξpair≫˜n−1/2, that is,kFξpair≫1.\nIn contrast, the Bose-pairinglimit correspondsto ˜ξpair≪\n˜n−1/2, that is,kFξpair≪1.\nIn Fig. 2, we compare our theoretical results for\nTBKT,|∆|,µ, andξpairto experimental results for\nLixZrNCl [28]. In Figs. 2(a) and (b), we vary the di-\nmensionless coupling constant /tildewideVsto match the theoreti-0.01 0.1\nx0102030\n(a)T(K)\n0.01 0.1\nx0246\n(b)|∆|(meV)\n0 0.1 0.2 0.3\n|∆|/εF00.10.20.3(c)T/TF0.010.1 0.0010.01 0.1  1    \nx|∆|/εF\n0.0001 0.01\nx10-1100101102\n-1-0.500.51 (d)kFξpair\nµ/εF\nFIG.2. Theoretical parametersare /tildewideVs= 0.7100,TR= 1094K,\nkR= 0.4310/a. Experimental data [28] (red diamonds and\nyellow circles). (a) T(in K) vs x, theoretical TBKT(red solid\nline). (b) |∆|(in meV) at T= 2K, theoretical |∆|(red solid\nline). (c) T/TFvs|∆|/εF,TBKT/TF(red solid line), TMF/TF\n(blue dashed line), and TBKT/TF= 1/8 = 0.125 (gray dotted\nline). Inset: |∆|vsx, theoretical (yellow dot-dashed line).\n(d) Theoretical kFξpair(red solid line), experimental kFξexp\n(red diamonds) both at T= 0;µ/εF(blue dot-dashed line)\natT=TBKT. The vertical gray dotted lines at x={6.549×\n10−5,5.238×10−4,6.564×10−3}givekFξpair={0.5,1.0,5.0}.\ncalTBKTand|∆|to the experimental Tcand|∆|versus\nconcentration x. The best fit is achieved for /tildewideVs= 0.7100,\nand from it we extract the temperature scale TR=εR=\nk2\nR/2mleading toTR= 1094K and the momentum scale\nkRby relating ˜ ntox, that is, by using the carrier den-\nsity per band n=x/Acell, whereAcell=a2sin(π/3)\nis the area of the unit cell of Li xZrNCl. In this case,\n˜n= 2πx/k2\nRAcell, and the momentum scale from the fit\niskR= 0.4310/a= 1.197nm−1, wherea= 0.3601nm is\nthe characteristic length of the unit cell. Note that the\ninteraction range R∼1/kR= 2.320a= 0.8354nm spans\na few lattice spacings, suggesting that the extension to a\nlattice model should require interactions involving only\na few neighbors. The effective mass mis obtained from\nm=k2\nR/2εRleading tom= 0.5791me, wheremeis the\nbare electron mass, in good agreement with band struc-\nture calculations [34, 35]. Since ˜ n= 2π(R/ℓip)2, the\ncrossover from short- to long-ranged interactions occurs\natR/ℓip∼1, that is, ˜n∼˜ncr= 2π(x∼0.1608), mean-\ningthatinteractionsaresufficientlyshort-rangedoverthe\nexperimental range, however a non-zero Ris absolutely\nessential in producing a maximum in TBKTversusx.\nIn Fig. 2(b), we show |∆|(in meV) at T= 2K ver-\nsusx. The red diamonds are the experimental data,\nand the red solid line are theoretical results using the5\nsame values of TR,kRand/tildewideVsgiven in Fig. 2(a). In\nFig. 2(c), we show TBKT/TFandTMF/TFversus|∆|/εF\natT= 2K for the same values of TR,kRand/tildewideVs. The\nupper bound TBKT= 0.125TF[16] is the gray dotted\nline. The range TBKT/TF< T/T F< TMF/TFdeter-\nmines the region where phase fluctuations are impor-\ntant. The inset shows |∆|/εFversusxcomparing ex-\nperiment (yellow circles) and theory (yellow dot-dashed\nline). In Fig. 2(d), we show kFξpairandµ/εFversus\nx. The theoretical kFξpairis compared with the ex-\nperimental kFξexp[36] atT= 0, while the theoretical\nµ/εFis atT=TBKT. The vertical gray dotted lines\natx={6.549×10−5,5.238×10−4,6.564×10−3}give\nkFξpair={0.5,1.0,5.0}. The crossover region from BCS\nto Bose pairing occurs in the range 0 .5< kFξpair<5.0\nor 6.549×10−5<x<6.564×10−3, with corresponding\n−0.6113< µ/ε F<0.9995. At the lowest experimental\nconcentration x= 0.0048, we find kFξpair= 3.826 and\nµ/εF= 0.9979, showing that the crossover region has\njust been entered. Ourresults demonstratethat the Bose\npairing regime has not yet been reached experimentally\nfor gated 2D Li xZrNCl, as it requires x<6.549×10−5,\nkFξpair<0.5 andµ/εF<−0.6113.\nFinal Remarks: Our theoretical work shows good\nagreement with experimental data in the range 0 .0048≤\nx≤0.133 [28]. However, for x≥0.1 there are some\ndeviations which our current model does not capture as\ndescribed next. First, according to Ref. [28], for x≥0.1\nthere is a crossover in behavior from two to three di-\nmensions, which our strictly 2D model does not include.\nSecond, it was suggested [28] that electron-phonon inter-\nactions may play a role specially for x≥0.1, but our\nmodel does not include retardation effects. Third, for\nx≥0.1, our results suggest that it is necessary to in-\nclude band-structure effects (non-parabolicity of bands)\nwithacorrespondingincreaseinthedensityofstatesthat\nleads to an increase in Tc,|∆|and a decrease in kFξpair.\nConclusions: We investigated a density induced evolu-\ntion from BCS to Bose pairing in gated two-dimensional\nsuperconductors with two degenerate parabolic bands\nand showed that there is good agreement with experi-\nmental results [28] of Li xZrNCl forx≤0.1, where the ef-\nfectivemassapproximationisvalid. We concludethat, at\nthe lowest reported concentration x= 0.0048, the chem-\nical potential is still very close to the Fermi energy, and\nthat the pair size is kFξpair= 3.826, indicating that ex-\nperiments havejust entered the crossoverregion from the\nBCS-side. We suggest that it is necessary to reduce the\nlowest concentration by at least one order of magnitude\nto start entering the Bose pairing regime experimentally.\nWe thank Yoshiro Iwasa for discussions, and\nthe National Key R&D Program of China (Grant\n2018YFA0306501),theNationalNaturalScienceFounda-\ntion of China (Grants 11522436& 11774425),the Beijing\nNatural Science Foundation (Grant Z180013), and the\nResearch Funds of Renmin University of China (Grants16XNLQ03 &18XNLQ15) for financial support.\n∗wzhangl@ruc.edu.cn\n†carlos.sademelo@physics.gatech.edu\n[1] A. M. Goldman, The Berezinskii-Kosterlitz-Thouless\ntransition in superconducors, In 40 years of the\nBerezinskii-Kosterlitz-Thouless theory, 135, World Sci-\nentific (2013).\n[2] V. L. Berezinskii, Destruction of long-range order in on e-\ndimensional and two-dimensional systems having a con-\ntinuoussymmetrygroup: I.Classical systems.Sov.Phys.\nJETP32, 493 (1970).\n[3] J. M. Kosterlitz and D. 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B 98,\n064512 (2018).\n[28] Yuji Nakagawa, Yuichi Kasahara, Takuya Nomoto, Ry-otaro Arita, Tsutomu Nojima, Yoshihiro Iwasa, Gate-\ncontrolled BCS-BEC crossover in a two-dimensional su-\nperconductor, Science 372, 190–195 (2021).\n[29] This parametrization is necessary to produce a separa-\nble potential in momentum space that behaves both at\nsmall and large momenta in a form that is compatible to\nan acceptable real space interaction potential V(r,r′), as\npreviously investigated in the literature [7].\n[30] Additional contributions to the number equation beyon d\nphase fluctuations also arise in the Bose limit (low densi-\nties) and lead to logarithmic corrections of TBKTdue to\nresidual boson-boson interactions. See, e.g., D. S. Fisher ,\nand P. C. Hohenberg, Dilute Bose gas in two dimensions,\nPhys. Rev. B 37, 4936 (1988).\n[31] C. A. R. S´ ade Melo, M. Randeria, and J. R. Engelbrecht,\nCrossover from BCS to Bose superconductivity: Transi-\ntion temperature and time-dependent Ginzburg-Landau\ntheory, Phys. Rev. Lett. 71, 3202 (1993).\n[32] C. A. R. S´ a de Melo, When fermions become bosons:\nPairing in ultracold gases, Physics Today, October issue,\n45 (2008).\n[33] For T > T MF, where |∆|= 0, fermions still interact\nand cannot be treated as free particles. Since there is\nalways a two-body bound state in two dimensions for an\narbitrary small attractive interaction, these bound state s\nmustbeincludedfor T > T MFandthedensityoffermions\nneeds to separated at least into bound (preformed pairs)\nand unbound contributions. To recover bound fermions\naboveTMFit is necessary to include amplitude fluctua-\ntions of ∆. Therefore, T∗is related to the Saha disso-\nciation temperature TS, where a fraction of preformed\npairsBdissociates into two fermions F↑andF↓with op-\nposite spins, satisfying the chemical equilibrium equatio n\nB⇋F↑+F↓. Details of this analysis will be investigated\nin a future publication.\n[34] R. Weht, A. Filippetti, W. E. Pickett, Electron doping\nin the honeycomb bilayer superconductors (Zr ,Hf)NCl,\nEurophys. Lett. 48, 320–325 (1999).\n[35] R. Heid, K. P. Bohnen, Ab Initio lattice dynamics and\nelectron-phonon coupling in Li xZrNCl. Phys. Rev. B 72,\n134527 (2005).\n[36] The experimental parameter ξexpis extracted from the\nmeasurement of the temperature dependent of the up-\nper critical field Hc2(T) using the Ginzburg-Landau re-\nlationHc2(T) = Φ 0/2πξ2(T), with ξ(T) =ξexp(1−\nT/Tc)−1/2and extrapolating it to T= 0, from Hc2(T) =/parenleftbig\nΦ0/2πξ2\nexp/parenrightbig\n(1−T/Tc) forT < T c. Here Φ 0is the flux\nquantum /planckover2pi1c/2e, andTcis the zero-field critical temper-\nature. The experimental value ξexp= [2πHc2(0)/Φ0]1/2\nand the theoretical values of ξpairare in good agreement\nin the range 0 ≤x≤0.1, where the system is expected\nto be in the truly 2D regime [28], band structure (lattice)\neffects are not important, and the experimental system\nis closer to the BCS-pairing side of the crossover. At the\nBose-pairing side of the crossover is approached, we ex-\npect substantial deviations between ξpairandξexpas dis-\ncussed in the three-dimensional version of the BCS-BEC\ncrossover [31]."    },    {        "title": "2106.12764v1.Density_Constrained_Reinforcement_Learning.pdf",        "content": "Density Constrained Reinforcement Learning\nZengyi Qin1Yuxiao Chen2Chuchu Fan1\nAbstract\nWe study constrained reinforcement learning\n(CRL) from a novel perspective by setting con-\nstraints directly on state density functions , rather\nthan the value functions considered by previous\nworks. State density has a clear physical and math-\nematical interpretation, and is able to express a\nwide variety of constraints such as resource limits\nand safety requirements. Density constraints can\nalso avoid the time-consuming process of design-\ning and tuning cost functions required by value\nfunction-based constraints to encode system speci-\nfications. We leverage the duality between density\nfunctions and Q functions to develop an effec-\ntive algorithm to solve the density constrained RL\nproblem optimally and the constrains are guaran-\nteed to be satisfied. We prove that the proposed\nalgorithm converges to a near-optimal solution\nwith a bounded error even when the policy update\nis imperfect. We use a set of comprehensive exper-\niments to demonstrate the advantages of our ap-\nproach over state-of-the-art CRL methods, with a\nwide range of density constrained tasks as well as\nstandard CRL benchmarks such as Safety-Gym.\n1. Introduction\nConstrained reinforcement learning (CRL) (Altman, 1999;\nAchiam et al., 2017; Dalal et al., 2018; Paternain et al., 2019;\nTessler et al., 2019; Yang et al., 2020; Zhang et al., 2020;\nStooke et al., 2020) aims to find the optimal policy that\nmaximizes the cumulative reward signal while respecting\ncertain constraints such as safety requirements and resource\nlimits. Existing CRL approaches typically involve construct-\ning suitable cost functions and value functions to take into\naccount the constraints. Then a crucial step is to choose\nappropriate parameters such as thresholds for the cost and\nvalue functions to encode the constraints. However, one\n1Massachusetts Institute of Technology2California In-\nstitute of Technology. Correspondence to: Chuchu Fan\n<chuchu@mit.edu >.\nProceedings of the 38thInternational Conference on Machine\nLearning , PMLR 139, 2021. Copyright 2021 by the author(s).\nGoalCharging stationRoad networkFigure 1. Autonomous electric vehicle routing in Manhattan: Con-\ntrol the electric vehicles to reach the goals and always keep the\nremaining energy above a threshold. Vehicles can be charged at\nthe charging stations. The vehicle density at each charging station\nis constrained by an upper bound due to resource limits. The roads\nand charging stations are from the real-world data (Blahoudek\net al., 2020).\nsignificant gap between the use of such methods and solving\nCRL problems is the correct construction of the cost and\nvalue functions, which is typically not solved systematically\nbut relies on engineering intuitions (Paternain et al., 2019).\nSimple cost functions may not exhibit satisfactory perfor-\nmance, while sophisticated cost functions may not have\nclear physical meanings. When cost functions lack clear\nphysical interpretations, it is difficult to formally guarantee\nthe satisfaction of the performance specifications, even if\nthe constraints on the cost functions are fulfilled. More-\nover, different environments generally need different cost\nfunctions, which makes the tuning process time-consuming.\nIn this work, we study CRL from a novel perspective by im-\nposing constraints on state density functions , which avoids\nthe use of cost or value function-based constraints in pre-\nvious CRL literature. Density is a measurement of state\nconcentration in the state space, and is directly related to\nthe state distribution. A variety of real-world constraints are\nnaturally expressed as density constraints in the state space.\nFor example, safety constraints is a special case of densityarXiv:2106.12764v1  [cs.LG]  24 Jun 2021Density Constrained Reinforcement Learning\nconstraints where we require the entire density distribution\nof states being contained in safe regions. Resource limits\ncan also be encoded by imposing maximum allowed density\non certain states. For instance, consider the electric vehi-\ncle routing problem (Blahoudek et al., 2020) in Figure 1.\nIn order to keep the remaining energy above a predefined\nthreshold, vehicles can be charged at the charging stations.\nDue to resource limits, the vehicle density at each charging\nstation is constrained by an upper bound. In our exper-\niments, we will demonstrate that density constraints can\nbe used in more general examples, which also include the\nstandard value function-based constraints as special cases.\nWe cannot directly apply existing CRL optimization meth-\nods based on constraining cost or value functions to solve\ndensity constrained reinforcement learning (DCRL) prob-\nlems, because the resulting solution cannot guarantee the\nsatisfaction of required density constraints. Instead, we pro-\npose a novel general model-free algorithm (Algorithm 1)\nthat exploits the duality between density functions and Q\nfunctions. Such algorithm enables us to incorporate state\ndensity constraints straightforwardly in CRL. Our proposed\nalgorithm is applicable of handling both discrete and contin-\nuous state and action spaces, and can be flexibly combined\nwith off-the-shelf RL methods to update the policy. We also\nprove that our algorithm can converge to a near-optimal so-\nlution with a bounded error even when the RL-based policy\nupdate is imperfect (Theorem 2).\nComprehensive experiments are conducted on various den-\nsity constrained problems, which demonstrate the advan-\ntages of our approach over state-of-the-art CRL methods in\nterms of satisfying system specifications and improving the\ncumulative reward. A notable achievement is that although\nour method can handle much more general constraints than\nsafety constraints, our DCRL algorithm still shows consis-\ntent improvement over existing CRL approaches on standard\nCRL benchmarks from MuJoCo and Safety-Gym (Ray et al.,\n2019) where constraints are primarily about safety.\nOur main contributions are: 1) We are the first to in-\ntroduce the DCRL problem with density constraints as a\npromising way to encode system specifications, which dif-\nfers from the value function-based constraints in existing\nCRL literature. 2) We propose a general algorithm for solv-\ning DCRL problems by leveraging the duality between den-\nsity functions and Q functions, and prove that our algorithm\nis guaranteed to converge to a near-optimal solution with\na bounded error even when the policy update is imperfect.\n3) We use an extensive set of experiments to examine the\nadvantages of our method over leading CRL approaches,\nin a wide variety of density constrained tasks as well as\nstandard CRL benchmarks.2. Related Work\nConstrained reinforcement learning (Garcıa & Fern ´andez,\n2015) primarily focuses on two approaches: modifying\nthe optimality criteria by combining a risk factor (Heger,\n1994; Nilim & El Ghaoui, 2005; Howard & Matheson, 1972;\nBorkar, 2002; Basu et al., 2008; Sato et al., 2001; Dotan\nDi Castro & Mannor, 2012; Kadota et al., 2006; L ¨otjens\net al., 2019) and incorporating extra knowledge to the ex-\nploration process (Moldovan & Abbeel, 2012; Abbeel et al.,\n2010; Tang et al., 2010; Geramifard et al., 2013; Clouse\n& Utgoff, 1992; Thomaz et al., 2006; Chow et al., 2018).\nOur method falls into the first category by imposing con-\nstraints and is closely related to constrained Markov deci-\nsion processes (Altman, 1999) (CMDPs). CMDPs has been\nextensively studied in robotics (Gu et al., 2017; Pham et al.,\n2018), game theory (Altman & Shwartz, 2000), and commu-\nnication and networks (Hou & Zhao, 2017; Bovopoulos &\nLazar, 1992). Most previous works consider the constraints\non value functions, cost functions and reward functions (Alt-\nman, 1999; Paternain et al., 2019; Altman & Shwartz, 2000;\nDalal et al., 2018; Achiam et al., 2017; Ding et al., 2020).\nInstead, we directly impose constraints on the state density\nfunctions. Chen et al. (2019) and Chen et al. (2019) study\nthe duality between density functions and value functions\nin CMDPs when the full model dynamics are known, while\nwe consider the model-free reinforcement learning and take\na further step to prove the duality of density functions to\nQ functions. In Geibel and Wysotzki (2005) density was\nstudied as the probability of entering error states and thus\nhas fundamentally different physical interpretations from\nus. In Dai et al. (2017) the duality was used to boost the\nactor-critic algorithm. The duality is also used in the policy\nevaluation community (Nachum et al., 2019; Nachum &\nDai, 2020; Tang et al., 2019). The offline policy evaluation\nmethod proposed by Nachum et al. (2019) can also be used\nto estimate the state density in our paper, but their focus is\npolicy evaluation rather than constrained RL. Therefore, we\nclaim that this paper is the first work to consider density\nconstraints and use the duality property to solve CRL.\n3. Preliminaries\nMarkov Decision Processes (MDP). An MDPMis a\ntuplehS;A;P;R;\ri, where (1)Sis the (possibly infinite)\nset of states; (2) Ais the (possibly infinite) set of actions;\n(3)P:S\u0002A\u0002S7![0;1]is the transition probability with\nP(s;a;s0)the probability of transitioning from state stos0\nwhen action a2Ais taken; (4) R:S\u0002A\u0002S7!Ris\nthe reward associated with the transition Punder the action\na2A; (5)\r2[0;1]is a discount factor.\nA policy\u0019maps states to a probability distribution over\nactions where \u0019(ajs)denotes the probability of choos-\ning actionaat states. Let a function \u001e:S7!RDensity Constrained Reinforcement Learning\nspecifie the initial state distribution. The objective of an\nMDP optimization is to find the optimal policy that maxi-\nmizes the overall discounted reward Jp=R\nS\u001e(s)V\u0019(s)ds,\nwhereV\u0019(s)is called the value function and satisfies\nV\u0019(s) =r\u0019(s) +\rR\nA\u0019(ajs)R\nSP(s;a;s0)V\u0019(s0)ds0da,\nandr\u0019(s) =R\nA\u0019(ajs)R\nSP(s;a;s0)R(s;a;s0)ds0dais the\none-step reward from state sfollowing policy \u0019. For every\nstateswith occurring as an initial state with probability\n\u001e(s), it incurs a expected cumulative discounted reward of\nV\u0019(s). Therefore the overall reward isR\nS\u001e(s)V\u0019(s)ds.\nThe formulation of value functions in MDP typically cannot\nhandle constraints on state distribution, which motivates the\ndensity functions.\nDensity Functions. State density functions \u001a:S7!R\u00150\nmeasure the state concentration in the state space (Chen\n& Ames, 2019; Rantzer, 2001).1We will show later that\ngeneric duality relationship exists between density func-\ntions and Q functions, which allows us to directly impose\ndensity constraints in RL problems. For infinite horizon\nMDPs, given a policy \u0019and an initial state distribution \u001e,\nthe stationary density of state sis expressed as:\n\u001a\u0019(s) =1X\nt=0\rtP(st=sj\u0019;s0\u0018\u001e);\nwhich is the discounted sum of the probability of visiting s.\nWe prove in the supplementary material that the density has\nan equivalent expression:\n\u001a\u0019(s) =\u001e(s) +\rZ\nSZ\nA\u0019(ajs0)P(s0;a;s)\u001a\u0019(s0)dads0;\n4. Density Constrained Reinforcement\nLearning\nIn this section, we first formally define the novel DCRL\nproblem and clarify its benefits compared to value function-\nbased constraints. To solve DCRL, we will establish the\nduality between Q functions and density functions, then\npropose a general algorithm for DCRL with convergence\nguarantees.\n4.1. Problem Statement\nGiven an MDPM=hS;A;P;R;\riand an initial state\ndistribution \u001e, DCRL finds the optimal policy \u0019?to the\nfollowing optimization problem:\nmaxZ\nS\u001e(s)V\u0019(s)ds\ns:t:\u001amin(s)\u0014\u001a\u0019(s)\u0014\u001amax(s);8s2S;(1)\n1\u001ais not necessarily a probability density function, which\nmeansR\nS\u001a(s) = 1 is not enforced.whereR\nS\u001e(s)V\u0019(s)dsis the expected cumulative reward.\nThe density constraints \u001amin(s)and\u001amax(s)are functions\nof states, and different states can have different lower and\nupper bound constraints on their densities.\nThe formulation in (1)is different from most of the previous\nwork (Achiam et al., 2017; Tessler et al., 2019; Yang et al.,\n2020), which did not consider the density constraint but\ninstead used the cost (or reward) value as constraint:\nmaxR\nS\u001e(s)V\u0019(s)ds\ns:t:R\nS\u001e(s)V\u0019\nC(s)ds\u0014\u0011;(2)\nwhereV\u0019\nCis the cost value function and \u0011is the threshold\nfor the expected cumulative cost.\nBenefits of the Density Constraint. The density con-\nstraint in (1)can bring at least two benefits. First, density\nhas a clear physical and mathematical interpretation as a\nmeasurement of state concentration in the state space. A\nwide range of real-world constraints can be conveniently\nexpressed as density constraints (e.g., the vehicle density\nin certain areas, and the frequency of agents entering un-\ndesirable states). Second, the value function constraint in\n(2)requires the time-consuming process of designing and\ntuning cost functions, which are completely avoided by the\ndensity constraint since no cost function tuning is needed.\n4.2. Duality of Density Functions and Q Functions\nTo solve the DCRL problem in (1), we cannot directly apply\nexisting RL algorithms (Lillicrap et al., 2016; Schulman\net al., 2015; 2017), which are designed to optimize the\nexpected cumulative reward or cost, but not the state density\nfunction. However, we will show that density functions are\nactually dual to Q functions, and the density constraints can\nbe realized by modifying the formulation of Q functions.\nThen the off-the-shelf RL algorithms can be used to optimize\nthe modified Q functions to enforce the density constraints.\nWe extend the stationary density \u001a\u0019to consider the action\ntaken at each state. Let \u0016\u001a:S\u0002A!R\u00150be a stationary\nstate-action density function, which represents the amount\nof state instances taking action aat states.\u0016\u001ais related to \u001a\nvia marginalization: \u001a(s) =R\nA\u0016\u001a(s;a)da. Under a policy\n\u0019, we also have \u0016\u001a\u0019(s;a) =\u001a\u0019(s)\u0019(ajs). Letr(s;a) =R\nSP(s;a;s0)R(s;a;s0)ds0. Consider the density function\noptimization:\nJd= max\n\u0016\u001a;\u0019Z\nSZ\nA\u0016\u001a\u0019(s;a)r(s;a)dads\ns:t:\u0016\u001a\u0019(s;a) =\u0019(ajs)\u0010\n\u001e(s)+\n\rZ\nSZ\nAP(s0;a0;s)\u0016\u001a\u0019(s0;a0)da0ds0\u0011\n\u001amin(s)\u0014\u001a\u0019(s)\u0014\u001amax(s)(3)Density Constrained Reinforcement Learning\nLemma 1. The optimization problems in (1)and(3)share\nthe same optimal policy \u0019?.\nThe proof of Lemma 1 is provided in the supplementary\nmaterial. It shows that solving (1)is equivalent to solving\n(3). Now we are ready to present the duality between den-\nsity functions and Q functions, which reveals that (3)can\nbe solved via standard RL algorithms by modifying the Q\nfunction. We use the Lagrangian method for (3)and denote\nthe Lagrange multipliers for \u001a\u0019\u0015\u001aminand\u001a\u0019\u0014\u001amax as\n\u001b\u0000:S7!R\u00150and\u001b+:S7!R\u00150. Then we consider the\nfollowing optimization problem:\nJp= max\nQ;\u0019Z\nS\u001e(s)Z\nAQ\u0019(s;a)\u0019(ajs)dads\ns:t:Q\u0019(s;a) =r(s;a) +\u001b\u0000(s)\u0000\u001b+(s)+\n\rZ\nSP(s;a;s0)Z\nA\u0019(a0js0)Q\u0019(s0;a0)da0ds0(4)\nThe difference between the Q function in (4)and the stan-\ndard Q function is that the reward r(s;a)is modified to\nr(s;a) +\u001b\u0000(s)\u0000\u001b+(s).\nTheorem 1. The density constrained optimization objec-\ntivesJdin(3)andJpin(4)are dual to each other. If both\nare feasible and the KKT conditions are satisfied, then they\nshare the same optimal policy \u0019?.\nThe proof of Theorem 1 is provided in the appendix. The-\norem 1 reveals that when the KKT conditions are satisfied,\nthe optimal solution to the adjusted unconstrained primal\nproblem (4) is exactly the optimal solution to the dual prob-\nlem (3) with density constraints. Such an optimal solution\nnot only satisfies the state density constraints, but also maxi-\nmizes the the total reward Jd. Thus it is the optimal solution\nto the DCRL problem. Note that (4)is solvable using stan-\ndard RL algorithms (Lillicrap et al., 2016; Schulman et al.,\n2015; 2017). However, the update of policy \u0019will also\nresult in an update of state density function \u001a\u0019, which will\nchange the Lagrangian multipliers \u001b+and\u001b\u0000to in order to\nsatisfy the KKT conditions of (3). This motivates us to de-\nvelop an algorithm that iteratively update \u0019;\u001a\u0019;\u001b+and\u001b\u0000,\nwhich will be detailed in Section 4.3.\nRemark 1. While the duality between density functions and\nvalue functions has already been studied (Chen & Ames,\n2019), we take a step further to show the duality property be-\ntween the density functions and Q functions . We prove and\nuse the duality between density functions and Q functions\nover continuous state space to solve density constrained RL\nfrom a novel perspective, which has not been explored and\nutilized by published RL literature.\nRemark 2. The dual variables in Linear Programming are\ndifferent from ours and do not have a clear physical interpre-\ntation. Technically, any non-negative dual variable satisfy-\ning the conservation law (Liouville PDE in the continuouscase, see Chen et al. (2019)) is a valid dual. However,\namong all valid dual variables, the state density is associ-\nated with a clear physical interpretation as the concentration\nof states, and we are able to directly apply constraints on\nthe density in RL.\nRemark 3. The value function-based constraint in CMDPs\ncan be viewed as a special case of density constraint.R\nS\u001e(s)V\u0019\nC(s)ds\u0014\u0011is equivalent toR\nS\u001a\u0019(s)rc(s)ds\u0014\u0011.\nThus the value function-based CRL problems can also be\nsolved by the DCRL framework.\nRemark 4. The general DCRL problem cannot be solved by\nvalue function-based RL methods. One may argue that for\na single state sk, its visiting frequency can be constrained\nby defining a cost function ckthat returns 1 when skis\nvisited and 0 otherwise. Then value function-based methods\ncan be used to constrain the expected value VCkof the\ncost, as is inR\nS\u001e(s)VCk(s)ds\u0014\u0011k(\u0003). However, when\nevery statesjhas unique density constraints \u001amin(sj)and\n\u001amax(sj), every state sjneeds a value function VCj(for\nthe cost of visiting sj) and an inequality (\u0003). In order to\ncover all states, it would require an extremely large and even\ninfinite number (when the state space is continuous) of value\nfunctions and inequalities, which can make the problem\nintractable for value function-based methods. Furthermore,\nwe have explicitly mentioned in Remark 3 that the value\nfunction-based constraint in CMDP is equivalent to a special\ncase of density constraint, and our general formulation of\nstate-wise density constraint is more expressive and cannot\nbe trivially converted into value function-based constraints.\n4.3. The DCRL Algorithm\nThe density function optimization in continuous state space\nis an infinite dimensional linear program and there is no\nconvenient approximation method. We thus turn to the\nequivalent Q function optimization where standard RL tools\nare applicable. By utilizing the duality between density func-\ntion and Q function (Theorem 1) in the density constrained\noptimization, the DCRL problem can be solved by alternat-\ning between the primal problem (4)and dual problem (3), as\nis illustrated in Figure 2. In the primal domain, we solve the\nadjusted primal problem (reward adjusted by Lagrange mul-\ntipliers) in (4) using off-the-shelf unconstrained RL methods\nsuch as TRPO (Schulman et al., 2015) and DDPG (Lillicrap\net al., 2016). Note that the density constraints are enforced\nin dual domain and the primal domain is still an uncon-\nstrained problem, which means we can make use of existing\nRL methods to solve the primal problem. In the dual do-\nmain, the policy is used to evaluate the state density function,\nwhich is described in details in Section 4.5. If the KKT con-\nditions\u001b+\u0001(\u001a\u0019\u0000\u001amax) = 0; \u001b\u0000\u0001(\u001amin\u0000\u001a\u0019) = 0 and\n\u001amin\u0014\u001a\u0019\u0014\u001amax are not satisfied, the Lagrange multi-\npliers are updated and enter the next loop. The key insightDensity Constrained Reinforcement Learning\nPrimal domain\nUpdate \n𝜋\nbased on \n𝜎\n+\nand \n𝜎\n−\nDual domain\nUpdate \n𝜎\n+\nand \n𝜎\n−\nbased on \n𝜋\n𝜋\n𝜎\n+\nand \n𝜎\n−\nPrimal \ndomain\nD\nual\ndomain\nUpdate \n𝜋\nbased on \n𝜎\n+\nand \n𝜎\n−\nUpdate \n𝜎\n+\nand \n𝜎\n−\nbased on \n𝜋\n𝜋\n𝜎\n+\nand \n𝜎\n−\nFigure 2. Illustration of the iterative optimization in DCRL. In the primal domain, we solve the adjusted primal problem in (4) to obtain\nthe policy\u0019. Then in the dual domain, the \u0019is used to evaluate the state density. The Lagrange multipliers \u001b+and\u001b\u0000are updated as\n\u001b+ max( 0;\u001b++\u000b(\u001a\u0019\u0000\u001amax))and\u001b\u0000 max( 0;\u001b\u0000+\u000b(\u001amin\u0000\u001a\u0019)). In the next loop, since the reward r(s;a)+\u001b\u0000(s)\u0000\u001b+(s)\nis updated, the primal optimization solves for the new \u0019under the updated reward. The loop stops when the KKT conditions are satisfied.\nis that the density constraints can be enforced in the dual\nproblem, and we can solve the dual problem by solving\nthe equivalent primal problem using existing algorithms.\nAlternating between the primal and dual optimization can\ngradually adjust the Lagrange multipliers until the KKT\nconditions are satisfied.\nAlgorithm 1 Template of the DCRL algorithm\n1:Input MDPM, initial condition distribution \u001e, con-\nstraints on the state density \u001amax and\u001amin\n2:Initialize\u0019randomly,\u001b+ 0,\u001b\u0000 0\n3:Generate experience D\u0019\u001af(s;a;r;s0)js0\u0018\u001e;a\u0018\n\u0019(s)thenrands0are observedg\n4:repeat\n5: (s;a;r;s0) (s;a;r +\u001b\u0000(s)\u0000\u001b+(s);s0)for each\n(s;a;r;s0)inD\u0019\n6: Solve for\u0019of (4) using D\u0019via standard RL\n7: Generate experience D\u0019\u001a f(s;a;r;s0)js0\u0018\n\u001e;a\u0018\u0019(s)thenrands0are observedg\n8: Compute stationary density \u001a\u0019usingD\u0019\n9:\u001b+ max( 0;\u001b++\u000b(\u001a\u0019\u0000\u001amax))\n10:\u001b\u0000 max( 0;\u001b\u0000+\u000b(\u001amin\u0000\u001a\u0019))\n11:until\u001b+\u0001(\u001a\u0019\u0000\u001amax) = 0; \u001b\u0000\u0001(\u001amin\u0000\u001a\u0019) =\n0and\u001amin\u0014\u001a\u0019\u0014\u001amax\n12:Return\u0019; \u001a\u0019\nA general template of the density constraint policy opti-\nmization is provided in Algorithm 1. In Algorithm 1, the\nLagrange multipliers \u001b+and\u001b\u0000are used to adjust rewards,\nwhich lead to an update of the policy \u0019. Then the policy\nis used to evaluate the stationary density, and the Lagrange\nmultipliers are updated following dual ascent. The iteration\nstops when all the KKT conditions are satisfied.\n4.4. Convergence Guarantee under Imperfect Updates\nIf every loop of Algorithm 1 gives the perfect (optimal) pol-\nicy\u0019of(4)under the current \u001b\u0000and\u001b+, then the conver-\ngence can be derived from the subgradient methods (Boyd\net al., 2003) and will not be detailed here. But in reality,\nthe policy update in each loop can be imperfect, which is\nunavoidable for standard RL algorithms.\nIn light of this, we are interested in the case where the policy\nsolved at each iteration is imperfect, and will provide conver-gence guarantees of Algorithm 1. We will also identify how\nthe error in policy updates propagates to the solution found\nby Algorithm 1 and derive an upper bound of its distance to\nthe optimal solution. Denote the negative objective function\nof(3)asf(\u001a\u0019) =\u0000R\nSR\nA\u001a\u0019(s)\u0019(ajs)r(s;a)dads . We\nassumefis strongly convex modulus \u0016. If this is not the\ncase, one can add a strongly convex regularization term to\nthe (already convex) objective function. Then (3)minimizes\nf(\u001a\u0019)subject to the density constraints. For brevity, we\nonly consider the density upper bound in this convergence\nanalysis, and the same result exists for the density lower\nbound. The Lagrangian dual function is formulated as\nd(\u001b+) = min\n\u001a\u0019f(\u001a\u0019) +Z\nS\u001b+(s)(\u001a\u0019(s)\u0000\u001amax(s))ds\n(5)\nDenote the feasible set of Lagrangian multipliers as P. The\ndual ascent is to find max\u001b+2Pd(\u001b+). We assume the pol-\nicy update is imperfect and the suboptimality of the solution\n^\u001a\u0019to (3) is upper bounded as:\nf(^\u001a\u0019) +Z\nS\u001b+(s)(^\u001a\u0019(s)\u0000\u001amax(s))ds\u0000d(\u001b+)\u0014\u000f\nAnd the imperfect dual ascent takes the form \u001b+ \nmax( 0;\u001b++\u000br^d(\u001b+)), wherer^d(\u001b+) = ^\u001a\u0019\u0000\u001amax.\nLet^g(\u001b+;\u000b) =1\n\u000b(max( 0;\u001b++\u000br^d(\u001b+))\u0000\u001b+), which\nis the residual of \u001b+before and after the imperfect update\nin a single loop of Algorithm 1.\nLemma 2. Following the imperfect dual ascent with step\nsize\u000b\u0014\u0016, we have\nd(\u001bk+1\n+)\u0015d(\u001bk\n+) +\u000b\n2jj^g(\u001bk\n+;\u000b)jj2\u0000r2\u000f\n\u0016jj^g(\u001bk\n+;\u000b)jj\nThe superscript kdenotes thekthloop of Algorithm 1. The\nproof of Lemma 2 is provided in the supplementary material.\nLetP?=f\u001b+jd(\u001b+) = max\u001b+2Pd(\u001b+)gbe the set of\noptimal solutions to (5).\nTheorem 2. There exists constants \u0015>0and\u0018 >0such\nthat Algorithm 1 with imperfect policy updates converges to\na dual solution ^\u001b+that satisfies\nmin\n\u001b0\n+2P?jj^\u001b+(s)\u0000\u001b0\n+jj\u0014\u0015r\u000f\n\u0016(6)Density Constrained Reinforcement Learning\nThe dual function also converges to a bounded neighbour-\nhood of its optimal value:\nmin\n\u001b0\n+2P?jjd(^\u001b+)\u0000d(\u001b0\n+)jj\u0014\u0018\u000f\n\u00162(7)\nThe proof of Theorem 2 is provided in the supplementary\nmaterial. Theorem 2 shows that Algorithm 1 is guaranteed\nto converge to a near-optimal solution even under imperfect\npolicy updates.\n4.5. Computational Approaches\nAlgorithm 1 requires computing the policy \u0019, density\u001a\u0019,\nLagrange multipliers \u001b+and\u001b\u0000. For\u0019, there are well-\ndeveloped representations such as neural networks and tabu-\nlar methods. Solving \u0019from experience D\u0019is also straight-\nforward via standard model-free RL. By contrast, the com-\nputation of\u001a\u0019,\u001b+and\u001b\u0000need to be addressed.\nDensity Functions. In the discrete state case, \u001a\u0019is rep-\nresented by a vector where each element corresponds to\na state. To compute \u001a\u0019from experience D\u0019(line 1 of\nAlgorithm 1), let D\u0019containNepisodes where episode\niends at time Ti. Letsijrepresent the state reached at\ntimejin theithepisode. Initialize \u001a\u0019 0. For all\ni2 f1;\u0001\u0001\u0001;Ngandj2 f0;1;\u0001\u0001\u0001;Tig, do the update\n\u001a\u0019(sij) \u001a\u0019(sij) +1\nN\rj. The resulting vector \u001a\u0019ap-\nproximates the stationary state density. In the continuous\nstate space, \u001a\u0019cannot be represented as a vector since\nthere are infinitely many states. We utilize the kernel den-\nsity estimation method (Chen, 2017; Chen & Ames, 2019)\nthat computes \u001a\u0019(s)at statesusing the samples in D\u0019\nwith\u001a\u0019(s) =1\nNPN\ni=1PTi\nj=0\rjKh(s\u0000sij)whereKh\nis the kernel function satisfying 8s2S;Kh(s)\u00150andR\nSKh(s)ds= 1. There are multiple choices of the ker-\nnelKh, e.g. Gaussian, Spheric, and Epanechnikov ker-\nnels (Chen, 2017), and probabilistic guarantee of accuracy\ncan be derived (Wasserman, 2019).\nLagrange Multipliers. If the state space is discrete,\nboth\u001b+and\u001b\u0000are vectors whose length equals to the\nnumber of states. In each loop of Algorithm 1, after\nthe stationary density is computed, \u001b+and\u001b\u0000are up-\ndated following Line 1 and Line 1 respectively in Algo-\nrithm 1. If the state space is continuous, we construct\nLagrange multiplier functions \u001b+and\u001b\u0000from samples\nin the state space leveraging linear interpolation. Let\n\u0016s= [s1;s2;\u0001\u0001\u0001]represent the samples in the state space. In\nevery loop of Algorithm 1, denote the Lagrange function\ncomputed by the previous loop as \u001bo\n+and\u001bo\n\u0000. We com-\npute the updated Lagrange multipliers at states \u0016sas\u0016\u001b+=\n[max(0;\u001bo\n+(s1)+\u000b(\u001a\u0019(s1)\u0000\u001amax(s1));max(0;\u001bo\n+(s2)+\n\u000b(\u001a\u0019(s2)\u0000\u001amax(s2));\u0001\u0001\u0001]and\u0016\u001b\u0000= [max(0;\u001bo\n\u0000(s1) +\n\u000b(\u001amin(s1)\u0000\u001a\u0019(s1));max(0;\u001bo\n\u0000(s2) +\u000b(\u001amin(s2)\u0000\n\u001a\u0019(s2));\u0001\u0001\u0001]. Then the new \u001b+and\u001b\u0000are obtained bylinearly interpolating \u0016\u001b+and\u0016\u001b\u0000respectively.\n5. Experiment\nWe consider a wide variety of CRL benchmarks and demon-\nstrate how they can be effectively solve by the proposed\nDCRL approach. The density constrained benchmarks in-\nclude autonomous electrical vehicle routing (Blahoudek\net al., 2020), safe motor control (Traue et al., 2019) and\nagricultural drone control. The standard CRL benchmarks\nare from MuJoCo and Safety-Gym (Ray et al., 2019). The\ndefinition of reward and constraint vary from task to task\nand will be explained when each task is introduced.\nBaseline Approaches. Three CRL baselines are com-\npared. PCPO (Yang et al., 2020) first performs an uncon-\nstrained policy update then project the action to the con-\nstrained set. CPO (Achiam et al., 2017) maximizes the re-\nward in a small neighbourhood that enforces the constraints.\nRCPO (Tessler et al., 2019) incorporates the cost terms and\nLagrange multipliers with the reward function to encourage\nthe satisfaction of the constraints. We used the original im-\nplementation of CPO and PCPO with KL-projection that\nleads to the best performance. For RCPO, since the official\nimplementation is not available, we re-implemented RCPO\nand made sure it matches the original performance. All\nthe three baseline approaches and our DCRL have the same\nnumber of neural network parameters. Note that the baseline\napproaches enforce the constraints by restricting values func-\ntions while our DCRL restricts the state density functions .\nWhen we train the value function-based CRL methods on\ndensity constrained tasks, we convert the density threshold\nto the value function threshold by duality between density\nfunctions and value functions (Chen & Ames, 2019).\n5.1. Autonomous Electrical Vehicle Routing\nFigure 3. Vehicle densities at the charging stations. All results are\naveraged over 10 independent trials. 16 charging stations with the\nhighest vehicle densities are kept for visualization.Density Constrained Reinforcement Learning\nFigure 4. Average reward and remaining energy. All results are\naveraged over 10 independent trials.\nOur first case study is about controlling autonomous electric\nvehicles (EV) in the middle of Manhattan, New York. It\nis adopted from Blahoudek et al. (2020) and is shown in\nFigure 1. While EVs drive to their destinations, they can\navoid running out of power by recharging at the fast charg-\ning stations along the roads. At the same time, the vehicles\nshould not stay at the charging stations for too long in order\nto save resources and avoid congestion. An road intersec-\ntion is called a node. In each episode, an autonomous EV\nstarts from a random node and drives to the goals. At each\nnode, the EV chooses a direction and reaches the next node\nalong that direction at the next step. The consumed electric\nenergy is assumed to be proportional to traveling distance.\nThere are 1024 nodes and 137 charging stations in total.\nDenote the full energy capacity of the vehicle as cfand the\nremaining energy as cr. When arriving at a charging station,\nthe EV chooses a charging time \u001c2[0;1], then its energy\nincreases to min(cf;cr+\u001ccf). The action space includes\nthe EV direction and the charging time \u001c. The state space\nS\u001aR3is consisted of the current 2D location and the\nremaining energy cr. The agent receives a negative reward\nproportional to its traveling distance to restrict energy con-\nsumption, and a +10 reward when it reaches the goal. In one\nepisode, a single agent starts from a random initial location\nand reaches the goal. The density is accumulated for 2000\nepisodes, which is equivalent to having 2000 agents.\nTwo types of constraints are considered: (1) the minimum\nremaining energy should keep close to a required threshold\nand (2) the vehicle density at charging stations should be\nless than a given threshold. Apparently, if the EV chooses a\nlarger\u001c, then the constraint (1) is more likely to be satisfied,\nwhile (2) is more likely to be violated, since a larger \u001c\nwill increase the vehicle density at the charging station.\nThese contradictory constraints pose a greater challenge\nto the RL algorithms. Both constraints can be naturally\nexpressed as density constraints. For constraint (1), we can\nlimit the density of low-energy states. For constraint (2), it is\nstraightforward to limit the EV density (a function of E[\u001c])\nat charging stations. The threshold of density constraints are\ntransformed to the threshold of value functions to be used by\nthe baseline methods. The conversion is based on the duality\nof density functions and value functions (Chen & Ames,\n2019). Figure 3 demonstrates that our DCRL can avoidthe vehicle densities from exceeding the thresholds, while\nthe baseline methods suffer from constraint violation. This\nis because DCRL allows us to explicitly set state density\nthresholds. Figure 4 shows that all the compared methods\ncan satisfy the constraint on remaining energy, and DCRL\ncan reach the highest reward.\n5.2. MuJoCo Benchmark and Safety-Gym\nExperiments are conducted on three tasks adopted from\nCPO (Achiam et al., 2017) and two tasks from the Safety-\nGym (Ray et al., 2019), built on top of the MuJoCo sim-\nulator. The tasks include Point-Gather, Point-Circle, Ant-\nCircle, Car-Goal and Car-Button. In the Point-Gather task,\na point agent moves on a plane and tries to gather as many\napples as possible while keeping the probability of gather\nbombs below the threshold. In the Point-Circle task, a point\nagent tries to follow a circular path that maximizes the re-\nward, while constraining the probability of exiting an given\narea. The Ant-Circle task is similar to the Point-Circle task\nexcept that the agent is an ant instead of a point. Detailed\nconfigurations of the three benchmarks can be found in\nAchiam et al. (2017). In the Car-Goal task, a car agent\ntries to navigate to a goal while avoiding hazards. In the\nCar-Button task, a car agent tries to press the goal button\nwhile avoiding hazards, and not to press the wrong buttons.\nDetailed descriptions of the Safety-Gym benchmark can be\nfound in Ray et al. (2019). The original value function con-\nstraints for these benchmarks are converted to state density\nconstraints for DCRL by duality between density functions\nand value functions.\nFigure 5 demonstrates the performance of the four methods.\nIn general, DCRL is able to achieve higher reward than\nother methods while satisfying the constraint thresholds. In\nthe Point-Gather and Point-Circle environments, all the four\napproaches exhibit stable performance with relatively small\nvariances. In the Ant-Circle environment, the variances\nof reward and constraint values are significantly greater\nthan that in Point environments, which is mainly due to\nthe complexity of ant dynamics. In Ant-Circle, after 600\niterations of policy updates, the constraint values of the four\napproaches converge to the neighbourhood of the threshold.\nThe reward of DCRL falls behind PCPO in the first 400\niterations of updates but outperforms PCPO thereafter.\n5.3. Safe Motor Control\nThe safe motor control environment is adopted from Traue\net al. (2019) and shown in Figure 6 (a). The objective is to\ncontrol the direct current series motor and ensure its angular\nvelocity follows a random reference trajectory and prevent\nthe motor from overheating. The state space S\u001aR6and\nconsists of six variables: angular velocity, torque, current,\nvoltage, temperature and the reference angular velocity. TheDensity Constrained Reinforcement Learning\nFigure 5. Performance on the constrained reinforcement learning tasks on the MuJoCo (Todorov et al., 2012) simulator. The first three\ntasks are from CPO (Achiam et al., 2017) and the last two tasks are from the Safety-Gym (Ray et al., 2019). All results are averaged over\n10 independent trials. The constraint thresholds are all upper bounds.\n𝑢𝑖𝑛𝑖𝑖𝑛 SensorsAction\nRotorAgent Convertor\n(a) Direct current series motor (b) A farm divided into 5 regions\n0\n31\n42   \nFigure 6. Illustration of the safe motor control and drone applica-\ntion environments. (a) Control the rotor to follow a reference\nangular velocity trajectory while limiting the density of high-\ntemperature states. (b) Control the drones to spray pesticide over a\nfarmland which is divided into five parts and each requires different\ndensities of pesticide.\nFigure 7. Reward and state density constraint in the safe motor\ncontrol task. Results are averaged over 10 independent trials.\naction space A\u001aRis the electrical duty cycle that controls\nthe motor power. The agent outputs a duty cycle at each\ntime step to drive the angular velocity close to the refer-\nence. When the reference angular velocity increases, the\nrequired duty cycle will need to increase. As a result, the\nmotor’s power and angular velocity will increase and cause\nthe motor temperature to grow. The reward is defined as the\nnegative distance between the measured angular velocity\nand the reference angular velocity. We consider the state\ndensity w.r.t. the motor temperature as constraint. High-\ntemperature states should have a low density to protect the\nmotor. When we train the baseline methods, the density con-\nFigure 8. Result of the agricultural spraying problem. (a) Percent-\nage of the entire area that satisfies the pesticide density requirement.\n(b) Time consumption in steps to reach the destination. Whiskers\nin the plots denote confidence intervals. Results are calculated\nwith 10 independent trials after each method converges.\nstraints are converted to equivalent cost value constraints by\nduality between density functions and value functions. Fig-\nure 7 shows the reward and state density for each method.\nWe find that DCRL is successful at reaching the highest\nreward among the compared methods while respecting the\nstate density constraints. More experiments under different\nsettings can be found in the supplementary material.\n5.4. Agricultural Spraying Drone\nWe consider the problem of using drones to spray pesticides\nover a farmland in simulation. The farmland is shown in\nFigure 6 (b), which is divided into 5 regions and different\nregions require different pesticide densities represented by\na lower bound \u001aminand an upper bound \u001amax. The drone\nstarts from the top-left corner, flies over the farmland spray-\ning pesticides, and stops at the bottom-right corner. The\nstate space constrains the position and velocity of the drone,\nas well as the required pesticide density of the current re-\ngion. The action space contains the angular acceleration and\nthe vertical thrust to control the drone. The drone sprays a\nconstant volume of pesticide at each time step, and controls\nthe pesticide density on the land by adjusting its position\nand velocity. A +1 reward will be given when the pesticideDensity Constrained Reinforcement Learning\ndensity of the area below the drone enters the required range,\nand a -1 reward will be given when the density exists the\nrange. A +10 reward will be given when the drone reaches\nthe bottom-right corner that is the destination. Figure 8\nshows the experimental results. With the DCRL method, we\ncan obtain the highest percentage of area that satisfies the\npesticide density constraint. Also, DCRL requires the least\nsteps to reach the destination. Details of the experiment\nconfigurations and results under different settings can be\nfound in the supplementary material.\n6. Conclusion and future works\nWe study a novel class of constrained reinforcement learning\nproblems where the constraints are imposed on state density\nfunctions, rather than value functions considered by previ-\nous literature. State densities have clear physical meanings\nand can express a variety of constraints of the environment\nand the system. We prove the duality between density func-\ntions and Q functions, then leverage the duality to develop\na general algorithm to solve the DCRL problems. We also\nprovide the convergence guarantee of our algorithm even\nunder imperfect policy updates. Note that our algorithm\ndoes not guarantee the satisfaction of density constraints\nduring training, and we do see occasional violations at the\nbeginning iterations. 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Equivalent Expression of the Density Function\nIn Section 3, we point out that the state density function \u001a\u0019\nhas two equivalent expressions, which we prove as follows:\n\u001a\u0019(s) =1X\nt=0\rtP(st=sj\u0019;s0\u0018\u001e)\n=P(s0=sj\u0019;s0\u0018\u001e)+\n1X\nt=1\rtP(st=sj\u0019;s0\u0018\u001e)\n=\u001e(s) +1X\nt=0\rt+1P(st+1=sj\u0019;s0\u0018\u001e)\n=\u001e(s) +\r1X\nt=0\rtP(st+1=sj\u0019;s0\u0018\u001e)\n=\u001e(s) +\rZ\nSZ\nA\u0019(ajs0)Pa(s0;s)\n1X\nt=0\rtP(st=s0j\u0019;s0\u0018\u001e)dads0\n=\u001e(s) +\rZ\nSZ\nA\u0019(ajs0)Pa(s0;s)\u001a\u0019(s0)dads0\nA.2. Proof of Lemma 1\nThe Lagrangian of (3) is\nL=Z\nSZ\nAr(s;a)\u0016\u001a\u0019\ns(s;a)dads\u0000\nZ\nSZ\nA\u0016(s;a)\u0010\n\u0016\u001a\u0019\ns(s;a)\u0000\u0019(ajs)(\u001e(s)+\n\rZ\nSZ\nAPa0(s0;s)\u0016\u001a\u0019\ns(s0;a0)da0ds0)\u0011\ndads (8)\nwhere\u0016:S\u0002A!Ris the Lagrange multiplier. The key\nstep is by noting that\nZ\nSZ\nAZ\nSZ\nA\u0016(s;a)\u0019(ajs)Pa0(s0;s)\u0016\u001a\u0019\ns(s0;a0)da0ds0dads\u0011\nZ\nSZ\nAZ\nSZ\nA\u0016(s0;a0)\u0019(a0js0)Pa(s;s0)\u0016\u001a\u0019\ns(s;a)da0ds0dads\nThen by rearranging terms, the Lagrangian becomes\nL=Z\nS\u001e(s)Z\nA\u0016(s;a)\u0019(ajs)dads\u0000\nZ\nSZ\nA\u0016\u001a\u0019\ns(s;a)\u0010\n\u0016(s;a)\u0000r(s;a)\u0000\n\rZ\nSPa(s;s0)Z\nA\u0019(a0js0)\u0016(s0;a0)da0ds0\u0011\ndads (9)\nBy the KKT condition and taking Q=\u0016, the optimality\ncondition satisfies (1) exactly.A.3. Proof of Theorem 1\nThe solution \u0019to the primal problem is the optimal policy\nfor the modified MDP with reward r+\u001b\u0000\u0000\u001b+, which\nmeans\u0019is the optimal solution to:\nmax\nQ;\u0019Z\nS\u001e(s)Z\nAQ\u0019(s;a)\u0019(ajs)dads\ns:t:Q\u0019(s;a) =r(s;a) +\u001b\u0000(s)\u0000\u001b+(s)+\n\rZ\nSPa(s;s0)Z\nA\u0019(a0js0)Q\u0019(s0;a0)da0ds0\n\u0019is also the optimal solution to:\nmax\n\u0016\u001a;\u0019Z\nSZ\nA\u0016\u001a\u0019(s;a)(r(s;a) +\u001b\u0000(s)\u0000\u001b+(s))dads\ns:t:\u0016\u001a\u0019(s;a) =\u0019(ajs)\u0010\n\u001e(s)+\n\rZ\nSZ\nAPa0(s0;s)\u0016\u001a\u0019(s0;a0)da0ds0\u0011\nTherefore, for any feasible policy \u00190the following inequality\nholds:\nZ\nSZ\nA\u0016\u001a\u0019(s;a)(r(s;a) +\u001b\u0000(s)\u0000\u001b+(s))dads\u0015\nZ\nSZ\nA\u0016\u001a\u00190(s;a)(r(s;a) +\u001b\u0000(s)\u0000\u001b+(s))dads (10)\nBy complementary slackness, if \u001b\u0000(s)>0, then\u001a\u0019(s) =\n\u001amin(s). The same applies to \u001b+and\u001amax. Since\u001a\u00190(s)\u0015\n\u001amin(s)and\u001a\u00190(s)\u0014\u001amax(s), we have:\nZ\nSZ\nA\u0016\u001a\u0019(s;a)(\u001b\u0000(s)\u0000\u001b+(s))dads\u0014\nZ\nSZ\nA\u0016\u001a\u00190(s;a)(\u001b\u0000(s)\u0000\u001b+(s))dads (11)\nThen we use (11) to eliminate the \u001b\u0000(s)\u0000\u001b+(s)in (10)\nand derive:\nZ\nSZ\nA\u0016\u001a\u0019(s;a)r(s;a)dads\u0015Z\nSZ\nA\u0016\u001a\u00190(s;a)r(s;a)dads\nwhich means \u0019is the optimal solution maximizing J?\nd\namong all the solutions satisfying the density constraints.\nAs a result,\u0019is the optimal solution to the DCRL problem.\nA.4. Proof of Lemma 2\nThe proof of Lemma 2 follows from the proof of Theo-\nrem 2.2.7 in Rockafellar (1970). For simplicity, let ^g=Density Constrained Reinforcement Learning\n^g(\u001b+;\u000b). For all\u001b0\n+2P, we have\nd(\u001b0\n+)\u0000\u0016d\n2jj\u001b0\n+\u0000\u001bk\n+jj2\n\u0014d(\u001bk\n+) +hrd(\u001bk\n+);\u001b0\n+\u0000\u001bk\n+i\n=d(\u001bk\n+) +hrd(\u001bk\n+);y+\u0000\u001bk\n+i+\nhrd(\u001b+);\u001b0\n+\u0000\u001bk+1\n+i\n\u0014d(\u001bk\n+) +hrd(\u001bk\n+);\u001bk+1\n+\u0000\u001bk\n+i+\nh^g;\u001b0\n+\u0000\u001bk+1\n+i+r2\u000f\n\u0016jj\u001b0\n+\u0000\u001bk+1\n+jj\n\u0014d(\u001bk+1\n+)\u0000\u000b\n2jj^gjj2+\nh^g;\u001b0\n+\u0000\u001bk\n+i+r2\u000f\n\u0016jj\u001b0\n+\u0000\u001bk+1\n+jj\nTaking\u001b0\n+=\u001bk\n+and utilizing the fact that \u001bk+1\n+\u0000\u001bk\n+=\u000b^g\ngives the result.\nA.5. Proof of Theorem 2\nLet\u001e(\u001b+) = min\u001b0\n+2P?jj\u001b+(s)\u0000\u001b0\n+jj. Based on the The-\norem 4.1 of Luo and Tseng (1993), there exists a constant \u001c\nsuch that\n\u001e(\u001bk\n+) +jj\u001a?\u0000\u001ajj\u0014\u001cjj\u001bk+1\n+\u0000\u001bk\n+jj (12)\nThis shows that the distance to P?decreases linearly with\n^g. From Lemma 2, it is clear that d(\u001b+)monotonically\nincreases for a sequence generated by Algorithm 1 when\njj^gjj>21\n\u000bq\n2\u000f\n\u0016, and the imperfect dual ascent would reach\na^\u001b+satisfyingjj^g(^\u001b+;\u000b)jj<2\n\u000bq\n2\u000f\n\u0016. From the fact that\nprojection does not increase the distance between vectors,\njjg(\u001b+;\u000b)jj<(2\n\u000b+1)q\n2\u000f\n\u0016. Taking\u0015=p\n2\u001c(2+\u000b), then\nit follows that min\u001b0\n+2P?jj^\u001b+\u0000\u001b0\n+jj\u0014\u0015q\n\u000f\n\u0016. Then we\nhave\nd(^\u001b+) =d(\u001b0\n+) +Z1\n0rd(\u001b0\n++t(^\u001b+\u0000\u001b0\n+))dt(^\u001b+\u0000\u001b0\n+)\n\u0014d(\u001b0\n+) +Z1\n0rd(\u001b0\n+) +1\n\u0016(^\u001b+\u0000\u001b0\n+)tdt(^\u001b+\u0000\u001b0\n+)\n=d(\u001b0\n+) +Z1\n0rd(\u001b0\n+)(^\u001b+\u0000\u001b0\n+)dt+1\n2\u0016jj^\u001b+\u0000\u001b0\n+jj2\n=d(\u001b0\n+) +1\n2\u0016jj^\u001b+\u0000\u001b0\n+jj2\nAlso note that d(^\u001b+)\u0000d(\u001b0\n+)\u00150. Thus we have:\nmin\n\u001b0\n+2P?jjd(^\u001b+)\u0000d(\u001b0\n+)jj\u0014\u001521\n2\u0016\u000f\n\u0016=\u00152\n2\u000f\n\u00162(13)\nB. Supplementary Experiments\nIn this section, we provide additional case studies that\nare not covered in the main paper. We mainly comparewith RCPO (Tessler et al., 2019) and the unconstrained\nDDPG (Lillicrap et al., 2016), which serves as the upper\nbound of the reward that can be achieved if the constraints\nare ignored.\nB.1. Express Delivery Service\n(a) (b)\nFigure 9. An example of the express delivery service company’s\ntransportation network with one ship center (red) and 29 service\npoints (gold). (a) The vans start from the service points bounded\nby squares with equal probability, then visit other service points\nfollowing a transition probability (policy), and finally reach the\nship center (goal). (b) The standard Q-Learning method finds a\npolicy that drives the vans directly to the goal without visiting any\nother service points, which minimizes the cost (traveling distance).\nThe sizes of gold nodes represent the state density.\nAn express delivery service company has several service\npoints and a ship center in a city. An example configuration\nis illustrated in Figure 9 (a). The company uses vans to\ntransport the packages from each service point to the ship\ncenter. The vans start from some service points following an\ninitial distribution, travel through some service points and\nfinally reach the ship center. The cost is formulated as the\ntraveling distance. The frequency that each service point is\nvisited by vans should exceed a given threshold in order to\ntransport the packages in the service points to the ship cen-\nter. Such frequency constraints can be naturally viewed as\ndensity constraints. A policy is represented as the transition\nprobability of the vans from one point to surrounding points.\nThe optimal policy should satisfy the density constraints\nand minimize the transportation distance.\nThis case study is proposed to further understand Algo-\nrithm 1 and its key steps. In Algorithm 1, our approach adds\nLagrange multipliers to the original reward in order to com-\npute a policy that satisfies density constraints. The update of\nLagrange multipliers follows the dual ascent, which is key\nto satisfying the KKT conditions. In this experiment, we\ntry to update the Lagrange multipliers using an alternative\napproach and see how the performance changes. We replace\nthe dual ascent with the cross-entropy method, where a set\nof Lagrange multipliers \u0006 = [\u001b1;\u001b2;\u001b3;\u0001\u0001\u0001]are drawn\nfrom an initial distribution \u001b\u0018Z(\u001b)and utilized to ad-\njust the reward respectively, after which a set of policiesDensity Constrained Reinforcement Learning\nTable 1. Results of the express delivery service transportation task. The maximum allowed running time to solve for a feasible policy is\n600s. The cost is the expectation of traveling distance from initial states to the goal.\nDensity space Method\u001amin= 0:1 \u001amin= 0:3 \u001amin= 0:5\nSolved Time (s) Cost Solved Time (s) Cost Solved Time (s) Cost\n\u001as2R10 CERS True 163.69 3.85 True 183.81 5.36 True 466.32 5.12\nDCRL True 1.35 2.91 True 2.05 2.66 True 4.31 4.28\n\u001as2R20 CERS True 227.15 5.80 True 527.26 6.00 False Timeout -\nDCRL True 3.41 5.58 True 3.99 6.16 True 5.28 6.27\n\u001as2R100 CERS True 161.62 10.26 False Timeout - False Timeout -\nDCRL True 3.53 10.23 True 116.24 12.13 True 153.86 14.06\n(a) \nDCRL\n(b) RCPO\n(c) DDPG\nFigure 10. Visualization of the behavior of three methods in the safe electrical motor control task.\n[\u00191;\u00192;\u00193;\u0001\u0001\u0001]are obtained following the same procedure\nin Algorithm 1. A small subset of \u0006whose\u0019has the least\nviolation of the density constraints are chosen to compute a\nnew distribution Z(\u001b), which is utilized to sample a new \u0006.\nThe loop continues until we find a \u001bwhose\u0019completely\nsatisfies the density constraints. We call this cross-entropy\nreward shaping (CERS). We experiment with 10D, 20D and\n100D state spaces (corresponding to 10, 20 and 100 service\npoints in the road network), whose density constraints lie\ninR10,R20andR100respectively. The density constraint\nvector\u001amin:S7!Ris set to identical values for each state\n(service point). For example, \u001amin= 0:1indicates the min-\nimum allowed density at each state is 0:1. In Algorithm 1,\nwe use Q-Learning to update the policy for both DCRL and\nCERS since the state and action space are discrete.\nFrom Table 1, there are two important observations. First,\nour computational time of finding the policy is significantly\nless than that of CERS. When \u001as2R10and\u001amin= 0:1,\nour approach is at least 100 times faster than CERS on the\nsame machine. When \u001as2R100and\u001amin= 0:5, CERS\ncannot solve the problem (no policy found can completely\nsatisfy the constraints) in the maximum allowed time (600s),\nwhile our approach can solve the problem in 153.86s. Sec-\nond, the cost reached by our method is generally lower\nthan that of CERS, which means our method can find better\nsolutions in most cases.B.2. Safe Electrical Motor Control (Section 5.3)\nTo gain more insight on the behavior of our DCRL agent,\nwe visualize the trajectories and actions (duty cycles) taken\nat different temperatures and reference angular velocities in\nFigure 10. In Figure 10 (a), The trajectory using DCRL can\nbe divided into three phases. In Phase 1, as the reference\nangular velocity grows, the duty cycle also increases, so the\nmotor temperature goes up. When the temperature is too\nhigh, the algorithm enters Phase 2 where it reduces the duty\ncycle to control the temperature, even though the reference\nangular velocity remains high. As the temperature goes\ndown, the algorithm enters Phase 3 and increases the duty\ncycle again to drive the motor angular velocity closer to the\nreference. In Figure 10 (b), when the temperature is high,\nthe RCPO algorithm will stop increasing the duty cycle but\nwill not decrease it as Algorithm 1 does. So the temperature\nremains high and thus the density constraints are violated. In\nFigure 10 (c), the unconstrained DDPG algorithm continues\nto increase the duty cycle in spite of the high temperature.\nB.3. Agricultural Spraying Drone (Section 5.4)\nIn the agricultural pesticide spraying problem, we examine\nthe methods with different pesticide density requirements\nand drone configurations to assess their capability of gen-\neralizing to new scenarios. In our main paper, from area\n0 to 4, the minimum and maximum pesticide density areDensity Constrained Reinforcement Learning\nFigure 11. Results of the agricultural spraying problem with minimum pesticide density (0;1;1;0;1)and maximum density (0;2;2;0;2)\nfrom area 0to4. Left: Percentage of the entire area that satisfies the pesticide density requirement. Middle: Time consumption in steps.\nWhiskers in the left and middle plots denote confidence intervals. Right: visualization of the velocity densities using different methods.\nFigure 12. Results of the agricultural spraying problem with minimum pesticide density (0;0;1;1;0)and maximum density (0;0;2;2;0)\nfrom area 0to4. Left: Percentage of the entire area that satisfies the pesticide density requirement. Middle: Time consumption in steps.\nWhiskers in the left and middle plots denote confidence intervals. Right: visualization of the velocity densities using different methods.\n(1;0;0;1;1)and(2;0;0;2;2)respectively. In this supple-\nmentary material, we evaluate with two new configurations.\nIn Figure 11, the minimum and maximum density are set to\n(0;1;1;0;1)and(0;2;2;0;2)from area 0to4. In Figure 12,\nthe minimum and maximum density are set to (0;0;1;1;0)\nand(0;0;2;2;0)from from area 0to4.\nAlthough the settings are different from our main paper, the\nresults convey consistent information. In Figure 11 and 12,\nDCRL and RCPO demonstrates similar performance in con-\ntrolling pesticide densities to be within the minimum and\nmaximum thresholds, while DCRL demands less time to\nfinish the task. DDPG only minimizes the time consumption\nand thus requires the least time among the three methods,\nbut cannot guarantee the pesticide density is satisfied. In\nterms of the velocity control, both DCRL and RCPO can\navoid the high-speed movement. These observations sug-\ngest that when both DCRL and RCPO finds feasible policies\nsatisfying density constraints, the policy found by DCRL\ncan achieve lower cost or higher reward defined by the orig-\ninal unconstrained problem, which is the time consumption\nof executing the task in this case study.\n1\n2\n3Figure 13. The mars rover environment. The agent starts from a\nrandom location in area 1 and is required to reach area 3. Area 2\nis considered dangerous and the constraint is set on the total time\nthat the agent is within area 2.\nB.4. Mars Rover\nWe consider a mars rover task where the agent must con-\nstrain the amount of time within the dangerous region. The\nenvironment is shown in Figure 13, where there are three\nareas marked in yellow. The agent starts from a random lo-\ncation in area 1 and needs to reach area 3. At each timestep,\nthe agent will receive a negative reward proportional to the\nenergy consumption rate, and will receive a +10 reward after\nit reaches area 3. Area 2 is considered dangerous and theDensity Constrained Reinforcement Learning\nFigure 14. Results of the mars rover task.\namount of time that the agent stays in area 2 is constrained\nby an upper bound. In the RCPO method, the agent receives\na negative reward if it is inside area 2, and the magnitude\nof this negative reward is automatically tuned by RCPO\nitself. In our DCRL method, the constraint is converted to\nan equivalent state density constraint in area 2.\nThe objective is to minimize the energy consumption while\nrespecting the time constraint in area 2. Figure 14 shows the\nperformance of the 3 methods. It is shown that DCRL and\nRCPO have similar energy consumption and both satisfy\nthe density constraint. DDPG only minimizes the energy\nconsumption and does not consider the constraint in area 2."    },    {        "title": "2107.04621v1.Majorana_fermion_arcs_and_the_local_density_of_states_of_UTe__2_.pdf",        "content": "Majorana fermion arcs and the local density of states of UTe 2\nYue Yu,1Vidya Madhavan,2and S. Raghu1, 3\n1Department of physics, Stanford University, Stanford, California 94305, USA\n2Department of Physics and Materials Research Laboratory,\nUniversity of Illinois Urbana-Champaign, Urbana, IL, USA\n3SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025, USA\nUTe 2is a leading candidate for chiral p-wave superconductivity, and for hosting exotic Majorana\nfermion quasiparticles. Motivated by recent STM experiments in this system, we study particle-\nhole symmetry breaking in chiral p-wave superconductors. We compute the local density of states\nfrom Majorana fermion surface states in the presence of Rashba surface spin-orbit coupling, which\nis expected to be sizeable in heavy-fermion materials like UTe 2. We show that time-reversal and\nsurface re\rection symmetry breaking lead to a natural pairing tendency towards a triplet pair density\nwave state, which naturally can account for broken particle-hole symmetry.\nIntroduction - Odd parity superconductors are no-\ntable for having a rich pattern of broken symmetries1,2\nand non-trivial topological properties3{6, making them\npromising avenues for the realization of Majorana\nfermion modes. Odd parity superconductivity is in-\nevitably unconventional, arising from electron-electron\nrepulsion rather than from a more conventional electron-\nphonon pairing mechanism7. Thus, they tend to be real-\nized in strongly correlated systems with complex phase\ndiagrams and competing ordering tendencies. While odd\nparity superconductivity has been observed in relatively\nfew systems, UTe 2is a strong candidate: it exhibits\nseveral striking phenomena including an extraordinarily\nlarge upper critical \feld8, reentrant superconductivity9\nand broken time-reversal symmetry10.\nRecent scanning tunneling spectroscopy\nmeasurements11of UTe 2have revealed an unusual\nsignature that is yet to be explained. STM spec-\ntra obtained near step edges in the superconducting\nphase show broken particle-hole symmetry (PHS).\nThis observation is sharply distinct from classical BCS\nsuperconductors, which exhibit an emergent PHS at\nenergies below the gap: the di\u000berence between adding\nand removing an electron is a Cooper pair, which is\nunobservable in a conventional Cooper pair condensate.\nSince STM directly measures local density of states,\nthese unusual edge modes may be signatures of putative\nchiral Majorana modes, and thus, it is of central interest\nto understand these experimental observations.\nA clue to understanding the question of broken PHS\nin a superconductor comes from studies of materials with\nnon-zero Cooper pair momentum12{14which are typically\nassociated with a pair density wave (PDW) modulation.\nIt is well-known that when superconductivity is accompa-\nnied by charge modulation, PHS breaking usually occurs.\nIt is thus possible that the observations of broken PHS in\nUTe 2can be explained by the development of inhomoge-\nneous superconductivity near the surface. In this letter,\nwe show how PHS breaking can emerge from \fnite mo-\nmentum Cooper pairing near the surface of an odd parity\nchiral superconductor.\nSymmetry considerations - Since the PDW state is nota generic weak-coupling instability of a Fermi liquid, one\ncannot predict unambiguously when it might occur. Nev-\nertheless, there are some reliable rules-of-thumb. When\nthe normal state has both time-reversal and inversion\nsymmetry, the spectrum consists of Kramers degenerate\npairs at momenta ( \u0006k) and uniform superconductivity is\noverwhelmingly preferred in the weak-coupling limit. In\nthis case, the BCS superconductor is labeled by a \\pseu-\ndospin\" degree of freedom stemming from the normal\nstate Kramers' degeneracy. Thus, to tilt the balance in\nfavor of the PDW state, either inversion or time-reversal\nsymmetry ought to be broken in the normal state. For\ninstance, a Zeeman \feld can help stabilize a PDW state\nin pseudospin singlet superconductors, as is believed to\nbe the case in CeCoIn 515,16. Similarly, a PDW state in an\nodd parity superconductor can be stabilized by breaking\ninversion (or re\rection) symmetry, as we explain below.\nIn the case of UTe 2, both time-reversal and inversion\nsymmetries occur in the normal state in the bulk. How-\never, spatial re\rection symmetry is broken at the surface .\nThis leads to a Rashba spin-orbit coupling (SOC) which\ndecays in strength away from the surface. If this decay\nlength is su\u000eciently long compared to the superconduct-\ning correlation length, or if the SOC itself is su\u000eciently\nstrong (as is likely the case in UTe 2due to the large\nbulk atomic SOC scales), Majorana surface modes will\nbe strongly a\u000bected by the re\rection symmetry breaking\nat the surface, which results in a PDW component to the\ncondensate. The symmetry considerations therefore sug-\ngest that the STM observations of broken PH in UTe 2\nmay be attributed to non-uniform superconductivity in-\nduced at the surface by sizeable Rashba SOC.\nModel- We \frst provide an explicit example that illus-\ntrates how non-uniform superconductivity is induced by\nlocal inversion symmetry breaking. Since the electronic\nstructure of UTe 2remains poorly understood we instead\nconsider a single band e\u000bective description. Let us sup-\npose that the overwhelming pairing tendency is in the\nodd parity pseudopsin triplet channel without a PDW\ncomponent. Deep in the bulk, the superconductor is de-arXiv:2107.04621v1  [cond-mat.supr-con]  9 Jul 20212\nscribed by mean-\feld Hamiltonian of the following form:\nH=H0+H\u0001+HRashba\nH0=X\nk;\u001b(Ek\u001b\u0000\u0016)cy\nk\u001bck\u001b\nH\u0001=X\nk;\u001b;\u001b0\u0001\u001b\u001b0kck;\u001bc\u0000k;\u001b0+ h:c:: (1)\nWe assume that the system has lattice translation sym-\nmetry. Thus, ck\u001bdestroys an electron with crystal mo-\nmentum k, and pseudospin \u001b. The band energies Ek\u001b\nare time-reversal symmetric and appropriate for an or-\nthorhombic crystal, such as that of UTe 2. Pseudospin-\ntriplet superconducting states are then characterized by\na matrix-valued order parameter \u0001 \u001b\u001b0k.\nNear the surface, superconductivity is subject to a size-\nable Rashba spin-orbit coupling due to the breaking of\nre\rection symmetry. Letting ^ nbe the vector normal to a\nsurface, the Rashba SOC Hamiltonian usually takes the\nform\nH=\u0015X\nk\u001b^n\u0001(k\u0002\u001b): (2)\nIn an orthorhombic crystal, such as UTe 2, the lack of a\nfourfold rotational axis results in a less restrictive form\nof Rashba SOC. Letting ^ n= ^y, the surface Rashba SOC\nof an orthorhombic system has the form\nHRashba =\u0000\u0015kxSz+\u00150kzSx; (3)\nwith generically distinct values of \u0015and\u00150. In what\nfollows, we will neglect spatial decay of \u0015;\u00150into the\nbulk and treat them as constant parameters of the sur-\nface Hamiltonian. In addressing the e\u000bect of these sur-\nface Fermi surface distortions on superconductivity, we\nassume that the superconducting gap scale is small com-\npared to the Rashba coupling, as is usually the case for\nBCS superconductors.\nThe existence of an induced PDW is most clearly\ndemonstrated in the extreme orthorhombic limit where\n\u00150\u001c\u0015. In this limit, the pseudospin Szis well approx-\nimated as a good quantum number. Moreover, \u0001 \"\"and\n\u0001##condensate now \\live\" on separate Fermi surfaces as\nshown in Fig. 1, and therefore prefer non-zero centers\nof mass momenta q\u0006\u0011(qx\u0006;0;0). And it remains en-\nergetically more favorable for the condensate in the \u0001 \"#\nchannel to have zero center of mass momentum. Since\nchiral p-wave state breaks TRS, in general q+6=\u0000q\u0000. If\nall three channels are ordered, there will be three distinct\ncondensate momenta, while two condensate momenta are\nalready su\u000ecient for the PDW13,14. The di\u000berence be-\ntween those two momenta determines the orientation and\nperiodicity of PDW:\n\u0001(x)\u0018cos(1\n2\u0001q\u0001x) (4)\nSince the three pairing channels have di\u000berent conden-\nsate momenta, they will naturally compete with each\n-2 0 2\nkx-4-202kz\nWithout SOC\nspin-up\nspin-downFIG. 1: Projected Fermi surfaces onto ( kx;kz) plane for\n(Black line)Fermi surface in the absence of Rashba SOC, as\nwell as (Red,Blue) pseudospin-up,down Fermi surface under\nRashba SOC. Band dispersion is Ek\u001b=P\nik2\ni\n2mi+P\nik4\ni\n2nifor\nthe plots, with mx= 1,my= 2,mz= 3,nx= 20,ny= 40,\nnz= 60,\u0016= 1,\u0015= 0:8 and\u00150= 0. Blue and red dots are\nthe approximate position for q\u0006.\nother. In this \u00150\u001c\u0015limit, if \u0001\"#channel is suppressed\nby competition, the remaining competition between \u0001 \"\"\nand \u0001##channel will be weak; therefore it is energeti-\ncally easier for these two condensates to coexist. Other\ncoexisting scenarios are more di\u000ecult to happen, but still\npossible under certain pairing interactions.\nThe complementary extreme limit \u00150>> \u0015 , can be\nhandled in the same way upon exchanging the xandz\ncoordinates. In terms of the original coordinate system,\nthe resulting PDW state would now have its center of\nmass momenta along the z-direction.\nAs we deviate from the limit \u00150\u001c\u0015, the \frst cor-\nrections from \u00150would be to alter the shape of Fermi\nsurfaces and the momenta of the PDW phase, but we\nexpect the PDW phases to survive to a \fnite range of\n\u00150=\u0015. Similarly, we expect the PDW in the complemen-\ntary limit to survive up to a \fnite range of \u0015=\u00150. Since\nthe two PDW phases in either extreme have distinct cen-\nter of mass momenta, there are a variety of possibilities\nfor intermediate \u00150=\u0015. There could be a direct \frst-order\ntransition where the PDW momenta jump abruptly from\none phase to the other, or a coexistence phase with both\nsets of PDW momenta present. There could also be an\nintermediate phase without PDW order. The correct sce-\nnario needs to be determined by the detail of the Fermi\nsurface and the pairing interactions, while the \u0015=\u00150ratio\nis determined by the orientation of the measured surface.\nBut we can state with certainty that the PDW phases\nobtained in the limit of extreme orthorhombicity do not\nrequire \fne-tuning.\nChiral edge modes and LDOS - Next we consider the\nquasiparticle spectrum in the strongly orthorhombic limit\nwith\u00156= 0 and\u00150= 0, assuming all three channels are\nordered. Since translational symmetry is broken in the3\nPDW, the Bogoliubov{de Gennes (BdG) Hamiltonian is\nnot block-diagonal, and truncation is needed for concrete\ncomputations. To illustrate the essential idea at a quali-\ntative level, we truncate the Hamiltonian to the following\n4\u00024 form to include just two PDW momenta q+;q\u0000:\nH=X\nk\by\nkhk\bk\n\by\nk= [cy\nk\"cy\nk+q\u0000#c\u0000k+q+\"c\u0000k#]\nhk=2\n664\"k\" 0 \u0001\"\"(k) \u0001\"#(k)\n0\"k+q\u0000# 0 \u0001##(k+q\u0000)\n\u0001\u0003\n\"\"(k) 0\u0000\"\u0000k+q+\" 0\n\u0001\u0003\n\"#(k) \u0001\u0003\n##(k+q\u0000) 0\u0000\"\u0000k#3\n775\n(5)\nwith the following simple normal state dispersion:\n\"k\"=k2\n2m\u0000\u0016\u0000\u0015kx;\"k#=k2\n2m\u0000\u0016+\u0015kx(6)\nThe anti-commutation relationship imposes the following\nconstraint on the pairing function:\n\u0001\"\"(k) =\u0000\u0001\"\"(q+\u0000k)\n\u0001\"#(k) =\u0000\u0001\"#(\u0000k)\n\u0001##(k) =\u0000\u0001##(q\u0000\u0000k)(7)\nIt should be noted that much larger matrices and a real-\nistic band structure are required for future quantitative\nanalysis. The result for 8 \u00028 truncation can be found in\nthe appendix.\nUsing the above Hamiltonian, we compute the surface\nbound state spectrum and the local density of states to\nvalidate the qualitative picture above of broken PHS. We\nwill compare the results in (1) the uniform q=0state\nwithout Rashba SOC, and (2) the non-uniform PDW\nstate with Rashba SOC. For simplicity, we take the same\n\\px+ipy\" pairing state for both cases:\n\u0001\"\"(k) = \u0001 1kz\n\u0001\"#(k) = \u0001#\"(k) = \u0001 2kx+i\u00013ky\n\u0001##(k) = \u0001 4kz;(8)\nwith real coe\u000ecient \u0001 i.\nGiven a BdG Hamiltonian, the existence of chiral\nsurface-bound states is governed by the regions in the\nbulk Fermi surface where the pairing gap closes. In a\nthree-dimensional chiral p-wave state, a Fermi surface\nthat is closed and encloses time-reversal invariant mo-\nmenta will necessarily have point nodes corresponding to\nbulk Majorana fermion excitations. With the parame-\nters chosen in the caption of Fig.2, these point nodes are\nlocated on the ( kx;kz) plane, shown as the black dots in\nthe upper panel of Fig.2. They appear in pairs, with op-\nposite momenta, due to both PH symmetry and inversion\nsymmetry.\nFor surface-bound states on the xzplane,kxandkzare\nstill good quantum numbers and label the eigenenergies,\n-1 0 1\nkx-1-0.500.51kz\n-1-0.500.51\nE/\nFIG. 2: (Top) Arc states without SOC ( \u0015= 0,qx\u0006= 0).\n(bottom)Arc states in the non-uniform state ( \u0015= 0:8,qx+=\n0:8,qx\u0000=\u00000:6). Black dots denote the point nodes, located\non (kx;kz) plane for the chosen parameters. m= 0:5,\u0016= 1,\n\u00011= 0:2, \u0001 2= \u0001 3= \u0001 4= 0:15 are used for both cases.\nDashed lines are the boundary of the projected normal state\nFermi surface. \u0001 = 0 :2 is used for the colorbar.\nwhile the state decays along the y-direction into the bulk.\nFor a \fxed energy E, the states satisfying E(kx;kz) =E\nform arcs in ( kx;kz) plane. If the system has Majorana\npoint nodes, there are zero-energy Majorana arc states,\nconnecting two projected point nodes17.\nFor the uniform q=0state (upper panel of Fig.2),\nzero-energy Majorana arc states (with E(kx;kz) = 0) are\ndenoted in green lines, which are surrounded by non-zero\nenergy arc states in other colors. Particle-hole symmetry\nis preserved. For example, any state on the red arc has\na counterpart on the blue arc, with opposite energy and\nopposite momentum. For the non-uniform PDW state\n(lower panel), PH symmetry is broken, and there is no\ncorrespondence between positive energy states and neg-\native energy states.\nSincekxandkzare now good quantum numbers, num-\nber of states in unit square in ( kx;kz) plane is uniform.\nThis allows us to \fnd the quasi-particle local density of\nstates (LDOS), contributed by arc states:\n\u001aLDOS (E) =X\nkxkz\u000e(E\u0000E(kx;kz)) (9)\nThe results can be found in Fig.3. LDOS in the uniform\nstate is symmetric and peaked at E= 0, while LDOS\nin the PDW state is asymmetric. The shape of LDOS\ndepends on the details of the Fermi surface, the pairing\nfunction, and the orientation of the measured surface.4\n-1 -0.5 0 0.5 1\nE/00.20.40.60.81LDOSWithout SOC\nWith SOC\nFIG. 3: Density of states contributed by arc states. In the\nabsence of Rashba SOC(blue line), system has particle-hole\nsymmetry; and LDOS is symmetric. For non-uniform states\nunder Rashba SOC (Red line), particle hole symmetry is bro-\nken with a peak in DOS at non-zero energy. Same parameters\nare used as in Fig. 2. \u0001 \u00110:2 is used.\nIt should be noted that signal in STM is contributed\nby both surface-bound states and also bulk states, so\nthe above analysis is far from complete. Even among\nthe surface states, states closer to the bulk nodes will\nhave a longer decay lengths, i.e. less localized near the\nsurface. This may a\u000bect sensitivity to STM. Therefore, a\nmore detailed calculation is required for the quantitative\ndescription of the STM experiment.\nFIG. 4: Experimental settings on UTe 2. Measurements are\nperformed on the green surfaces. For system on the left,\nparticle-hole symmetry is broken as it approaches the top sur-\nface. For system on the right, particle-hole symmetry is found\nto be unbroken. Number vectors follow the crystal axes.Discussions on the recent STM experiment - In the\nSTM experiment of Ref. 11, the LDOS near the step\nedge on (0;1;1) and (0;\u00001;\u00001) surfaces (shown in green\nin Fig.4) broke PHS. In the present context, the observa-\ntions can be understood by postulating that the strength\nof the Rashba SOC is stronger near the step edge. This\nis a reasonable hypothesis, since the con\fning potential\nis stronger, and the chiral surface modes are therefore\nmore localized near the step edge. Thus, we expect the\nPDW fraction to be higher in such regions.\nFor LDOS on (0 ;1;1) surface, there is a dip and peak\nat around opposite energies. When comparing results\non (0;1;1) with (0;\u00001;\u00001) surface, the position of dips\nand peaks is reversed. Analysis on these two surfaces\ncan be found in the appendix, but we did not \fnd any\nsymmetry explanations for these \fndings, since the only\nsymmetry relating positive and negative energies is PH\nsymmetry, which is broken. On the right \fgure of Fig.4,\nPH symmetry is found to be preserved on this \\45\u000e\" step\nedge. One possible explanation is the absence of PDW\nsince here the Rashba SOC is di\u000berent from the previous\nsurfaces. A more quantitative analysis based on realistic\nFermi surface and pairing functions may explain these\nobservations.\nSummary - We study the e\u000bect of Rashba spin-orbit\ncoupling on the local density of states of chiral p-wave su-\nperconductors. We point to a natural pairing tendency\ntowards the triplet-PDW state, in the absence of an ex-\nternal magnetic \feld and without bulk inversion symme-\ntry breaking. We \fnd that the LDOS near step edges\nshows broken PHS. Our methods are readily applicable\nto various experiments, including the recent STM experi-\nment on UTe 2, and lend increasing support to the notion\nthat this system hosts chiral p-wave superconductivity\nwith Majorana fermion quasiparticle states.\nAcknowledgments - We thank D. Agterberg, B. Brad-\nlyn, S.-B. Chung, S. Kivelson and F. Zhou for helpful\ndiscussions. SR is supported by the Department of En-\nergy, O\u000ece of Basic Energy Sciences, Division of Mate-\nrials Sciences and Engineering, under contract No. DE-\nAC02-76SF00515.\n1A. J. Leggett, Rev. Mod. Phys. 47, 331 (1975),\nURL https://link.aps.org/doi/10.1103/RevModPhys.\n47.331 .\n2D. Vollhardt and P. Wol\re, The super\ruid phases of helium\n3(Courier Corporation, 2013).\n3X.-L. Qi and S.-C. Zhang, Reviews of Modern Physics 83,\n1057 (2011).\n4M. Leijnse and K. Flensberg, Semiconductor Science and\nTechnology 27, 124003 (2012).\n5M. Sato and Y. Ando, Reports on Progress in Physics\n80, 076501 (2017), URL https://doi.org/10.1088/\n1361-6633/aa6ac7 .6A. K. C. Cheung and S. Raghu, Phys. Rev. B\n93, 134516 (2016), URL https://link.aps.org/doi/10.\n1103/PhysRevB.93.134516 .\n7M. Sigrist and K. Ueda, Rev. Mod. Phys. 63, 239 (1991),\nURL https://link.aps.org/doi/10.1103/RevModPhys.\n63.239 .\n8D. Aoki, A. Nakamura, F. Honda, D. Li, Y. Homma,\nY. Shimizu, Y. J. Sato, G. Knebel, J.-P. Brison, A. Pour-\nret, et al., Journal of the Physical Society of Japan 88,\n043702 (2019).\n9S. Ran, I.-L. Liu, Y. S. Eo, D. J. Campbell, P. M. Neves,\nW. T. Fuhrman, S. R. Saha, C. Eckberg, H. Kim, D. Graf,5\net al., Nature Physics 15, 1250 (2019).\n10I. M. Hayes, D. S. Wei, T. Metz, J. Zhang, Y. S. Eo, S. Ran,\nS. R. Saha, J. Collini, N. P. Butch, D. F. Agterberg, et al.,\narXiv preprint arXiv:2002.02539 (2020).\n11L. Jiao, S. Howard, S. Ran, Z. Wang, J. O. Rodriguez,\nM. Sigrist, Z. Wang, N. P. Butch, and V. Madhavan, Na-\nture579, 523 (2020).\n12P. Fulde and R. A. Ferrell, Phys. Rev. 135, A550 (1964),\nURL https://link.aps.org/doi/10.1103/PhysRev.\n135.A550 .\n13A. Larkin and I. Ovchinnikov, Soviet Physics-JETP 20,\n762 (1965).\n14D. F. Agterberg, J. S. Davis, S. D. Edkins, E. Frad-\nkin, D. J. Van Harlingen, S. A. Kivelson, P. A. Lee,\nL. Radzihovsky, J. M. Tranquada, and Y. Wang, An-\nnual Review of Condensed Matter Physics 11, 231{270\n(2020), ISSN 1947-5462, URL http://dx.doi.org/10.1146/annurev-conmatphys-031119-050711 .\n15C. Martin, C. Agosta, S. Tozer, H. Radovan, E. Palm,\nT. Murphy, and J. Sarrao, Physical Review B 71, 020503\n(2005).\n16M. Kenzelmann, T. Str assle, C. Niedermayer,\nM. Sigrist, B. Padmanabhan, M. Zolliker, A. D.\nBianchi, R. Movshovich, E. D. Bauer, J. L. Sar-\nrao, et al., Science 321, 1652 (2008), ISSN 0036-8075,\nhttps://science.sciencemag.org/content/321/5896/1652.full.pdf,\nURL https://science.sciencemag.org/content/321/\n5896/1652 .\n17V. Kozii, J. W. F. Venderbos, and\nL. Fu, Science Advances 2 (2016),\nhttps://advances.sciencemag.org/content/2/12/e1601835.full.pdf,\nURL https://advances.sciencemag.org/content/2/\n12/e1601835 .6\nAppendix A: perfect nesting for spherical Fermi surface\nA special example for PDW state is the elliptical Fermi surface with \"k=P\nik2\ni\n2mi. The spin-up and spin-down Fermi\nsurfaces are shifted by \u0006mx\u0015, as shown in Fig.A.5. Due to the quadratic dispersion, the \fnal Fermi surfaces are not\ndistorted, which leads to a new Fermi surface perfect nesting. Pairing susceptibility diverges at qx+=\u0000qx\u0000= 2mx\u0015\nandqx0= 0. In the calculations for surface-bound states, we used a spherical Fermi surface. But we did not focus on\nFermi surface perfect nesting, i.e. q\u0006;0are taken to be independent parameters in the model.\nFIG. A.5: Spherical Fermi surface with \u00150= 0, projected to ( kx;kz) plane. Perfect nesting with qx+=\u0000qx\u0000= 2m\u0015is shown\nby the dotted line.\n0 0.5 1 1.5 2\nqx30405060708090=0.8\n'=0\n'=0.15\n'=0.3\n'=0.55\n'=0.8\nFIG. A.6: Pairing susceptibility for \fxed \u0015with various \u00150\nAppendix B: pairing susceptibility\nFor simplicity, we calculated pairing susceptibilities for state \u0001q\n\"\"(k) =kz, with Cooper pair momentum q=\n(qx;0;0), while neglecting all other states. The same Hamiltonian and parameters are used as in the caption of Fig.1.\nThe results are shown in Fig.A.6. We found a critical \u00150around 0.3. For \u00150<0:3, pairing susceptibility is maximized\nat non-zero momentum. For \u00150>0:3, it is maximized at zero momentum. It is worth noting that, other states\ncertainly play a more important role for larger \u00150, and more quantitative analysis including pairing interactions is\nneeded to get a complete phase diagram.\nAppendix C: Translational symmetry&coupling to charge-density-wave\nCondensate with a single center of mass momentum preserves particle-hole symmetry, as one can always shift the\norigin of momentum space and let q=0. To break the translational symmetry and particle-hole symmetry, it is7\nnecessary to have at least two condensate momenta. The di\u000berence in these momenta determines the orientation and\nperiodicity of PDW:\n\u0001(x)\u0018cos(1\n2\u0001q\u0001x) (C1)\nThe model generically has more than two condensate momenta. For instance, in the limit of \u00150\u001c\u0015, PDW has one\nCooper pair momentum in each pairing channel (if ordered). Therefore, we will require at least two pairing channels\nto be ordered for PH symmetry breaking. An easier possibility is to have ordered \u0001 \"\"and \u0001##channels since the\ncompetition between them is weak. Other coexisting scenarios will require stronger pairing interactions.\nNext, we discuss a consequence of non-uniform superconductivity: the development of charge modulation. The cou-\npling between the charge density wave (CDW) and superconducting order paramaters can be deduced from Ginzburg-\nLandau theory. Gauge invariance constrains the lowest order coupling to be quadratic in \u0001:\nfint=X\nijab\u000bij\nab\u001aqj\u0000qih\n\u0001qia(\u0001qj\nb)\u0003+ (\u0001\u0000qi\na0)\u0003\u0001\u0000qj\nb0i\n(C2)\n, with proper coe\u000ecients \u000bij\nabdetermined by the normal state dispersion. a;bdenotes di\u000berent pairing channels:\na;b=\"\";\"#;##, while their time-reversal pairs are a0;b0=##;\"#;\"\".\n1. special case at \u00150= 0\nBut right at \u00150= 0, translational symmetry is more subtle. Since the normal state Hamiltonian does not couple\nspin-up and spin-down states, di\u000berent channels do not couple in the above equation; i.e. \u000bij\nab6= 0 only for a=b.\nSince there is a unique qin each pairing channel, the above quadratic terms only couple to uniform charge density,\nrather than CDW. The actual leading order contribution to CDW is then quartic in \u0001:\n\u001a\"\"\nq;\u001a##\nq/h\n\u0001\"\"\nq+\u0001\"#\u0003\nq0\u0001##\nq\u0000\u0001\"#\u0003\nq0+ \u0001\"\"\u0003\n\u0000q+\u0001\"#\n\u0000q0\u0001##\u0003\n\u0000q\u0000\u0001\"#\n\u0000q0i\nq=q++q\u0000\u00002q0;(C3)\nA non-zero wavevector qrequires q++q\u00006=2q0, which is the condition of translational symmetry breaking at \u00150= 0.\nCDW is expected to be weak at \u00150= 0 since it is only contributed by quartic terms in \u0001. CDW should be still small\nin the limit of \u00150<<\u0015 .\nAppendix D: (0;\u00001;\u00001)and (0;1;1)surfaces\nIn this section, ( x;y;z ) axes will denote the crystal axes, while ( x0;y0;z0) axes are local coordinate set up in Fig.4.\nThe pairing state is taken to be kx+ikystate, satisfying the Kerr e\u000bect measurement, with the following gap function\n~d0= (\u0001 3kz;i\u00014kz;\u00011kx+i\u00012ky) (D1)\nAs we consider surface bound state on (0 ;\u00001;\u00001) and (0;1;1) surfaces, we need to go to the local basis, where\ny'-axis is normal vector of the top surface, while z' is the normal vector of the side (0 ;1;1)/(0;\u00001;\u00001) surface. x'-axis\nis the same as crystal x-axis. Basis transformation leads to a vector rotation on both kand a vector rotation of the\nd-vector:\n~d(k0) =M~d0(Mk) (D2)\n, where matrix Mis the standard rotation matrix in 3D.\nNow we can calculate surface-bound states on the two side surfaces, and the results is shown below:\nWe observed that the peak in LDOS on the two surfaces should be located at di\u000berent energies. But LDOS pro\fle\nis not the same.8\nFIG. D.1: (left) (0,1,1) surface. Same parameters \u0001 ifor pairing states were used as in main text.\nAppendix E: Truncation to larger matrix\nHere we would like to show the surface bound states with truncation to 8 \u00028 Hamiltonian:\nH=X\nk\by\nkhk\bk\n\by\nk= [cy\nk\"cy\nk\u0000q+#cy\nk+q\u0000#cy\nk+q++q\u0000\"c\u0000k+q+\"c\u0000k#c\u0000k+q++q\u0000#c\u0000k\u0000q\u0000\"]\nhk=2\n6666666664\"k\" \u0001\"\"(k) \u0001\"#(k)\n\"k\u0000q+# \u0001\"#(k\u0000q+) \u0001 ##(k\u0000q+)\n\"k+q\u0000# \u0001##(k+q\u0000) \u0001 \"#(k+q\u0000)\n\"k+q++q\u0000\" \u0001\"#(k+q++q\u0000)\n\u0001\u0003\n\"\"(k) \u0001\u0003\n\"#(k\u0000q+) \u0000\"\u0000k+q+\"\n\u0001\u0003\n\"#(k) \u0001\u0003\n##(k+q\u0000) \u0000\"\u0000k#\n\u0001\u0003\n##(k\u0000q+) \u0000\"\u0000k+q++q\u0000#\n\u0001\u0003\n\"#(k+q\u0000) \u0001\u0003\n\"#(k+q++q\u0000) \u0000\"\u0000k\u0000q\u0000\"3\n7777777775\n(E1)\nThe result is shown below.\n-0.5 0 0.5 1\nkx-1-0.500.51kz\n-1-0.500.51\nE/\nFIG. E.2: surface bound states of 8 \u00028 truncation. Same parameters are used as in main text."    },    {        "title": "2107.06728v1.Multifractal_correlations_of_the_local_density_of_states_in_dirty_superconducting_films.pdf",        "content": "Multifractal correlations of the local density of states in dirty superconducting \flms\nM. Stosiek,1F. Evers,2and I. S. Burmistrov3, 4\n1Physics Division, Sophia University, Chiyoda-ku, Tokyo 102-8554, Japan\n2Institute of Theoretical Physics, University of Regensburg, D-93040 Regensburg, Germany\n3L.D. Landau Institute for Theoretical Physics, acad. Semenova av. 1-a, 142432 Chernogolovka, Russia\n4Laboratory for Condensed Matter Physics, HSE University, 101000 Moscow, Russia\n(Dated: July 15, 2021, v.4)\nMesoscopic \ructuations of the local density of states encode multifractal correlations in disordered\nelectron systems. We study \ructuations of the local density of states in a superconducting state of\nweakly disordered \flms. We perform numerical computations in the framework of the disordered\nattractive Hubbard model on two-dimensional square lattices. Our numerical results are explained\nby an analytical theory. The numerical data and the theory together form a coherent picture of\nmultifractal correlations of local density of states in weakly disordered superconducting \flms.\nSuperconductivity and Anderson localization are two\nphysical e\u000bects that result in a rich variety of quan-\ntum phenomena. Initially, it was believed that non-\nmagnetic disorder does not a\u000bect superconductivity (so-\ncalled \\Anderson theorem\") [1{3]. Later it became clear\nthat Anderson localization in the presence of disorder\ncan not only suppress superconductivity [4{7] but lead\nto superconductor{to{insulator transition [8] (see Refs.\n[9{11] for a review). In the presence of Coulomb interac-\ntion even a weak disorder was predicted to be su\u000ecients\nto destroy a superconducting state [12{21].\nIn the case of a short-ranged electron-electron interac-\ntion (e.g., if Coulomb interaction is screened), Anderson\nlocalization can enhance the superconducting transition\ntemperature, Tc[22{25]. This surprising phenomenon\noriginates from the multifractality of electron wave func-\ntions in disordered media [26]. Recently, theoretical pre-\ndictions of Refs. [22{25] have been corroborated by nu-\nmerical solutions of the disordered attractive Hubbard\nmodel [27, 28] and experimental observations of enhance-\nment ofTcin disordered niobium dichalcogenide mono-\nlayers [29, 30]. The multifractally{enhanced supercon-\nductivity is predicted to be accompanied by strong meso-\nscopic \ructuations of the local order parameter and the\nlocal density of states (LDoS) [23, 28, 31{35].\nPoint-to-point \ructuations of the tunneling LDoS, {\nfrequently termed as an emergent electronic granularity,\n{ have been observed experimentally in many studies of\ndisordered superconducting \flms [29, 30, 36{42]. The\nmeasurements yield LDoS maps that are used to extract\nthe statistics of the energy gap and the LDoS maximum.\nRecently, the spatial correlations of the energy gaps have\nbeen analysed, and the emergence of a well-de\fned spa-\ntial scale has been demonstrated [41].\nMultifractally-enhanced superconductivity was pre-\ndicted to occur in a relatively narrow region in the\ndisorder|interaction plane [24, 25]. However, signi\fcant\npoint-to-point \ructuations of LDoS has been reported in\nmany experiments on superconducting \flms that do not\ndemonstrate enhancement of Tcwith disorder. There-\nfore, there is a question about the origin of an emergentelectronic granularity, especially in a weakly disordered\nsuperconducting \flms.\nIn this Letter we investigate numerically and with ana-\nlytical means, the \ructuations of the LDoS in the super-\nconducting state of weakly disordered \flms. We perform\nnumerical computations of disordered attractive Hub-\nbard model on a two-dimensional square lattice. We\nfocus on the energy dependence of the variance of the\nLDoS. Even at very low level of disorder we observe pro-\nnounced \ructuations of the LDoS. Our numerical \fnd-\nings are complemented with the analytical theory for the\n\ructuations of the LDoS. The most striking observation\nboth in the numerics and in the theory is the logarith-\nmic divergence of the LDoS-variance (at energies higher\nthan the energy gap) with a system size L. Such di-\nvergence is a direct signature of multifractal behavior of\nthe LDoS, originating from mesoscopic \ructuations. We\ndemonstrate that the numerical \fndings and the analyti-\ncal results form a coherent picture of multifractal correla-\ntions of the LDoS in weakly disordered superconducting\n\flms.\nNumerics. We consider the attractive \u0000UHubbard\nmodel [43] on the square lattice in two-dimensions with\ndouble-periodic boundary conditions. Within the mean-\n\feld approximation the Hamiltonian reads ( U > 0)\n^H=\u0000tX\nhi;ji;\u001b^cy\ni;\u001b^cj;\u001b+X\ni;\u001b\u0000\nVi\u0000\u0016\u0000Un(ri)=2\u0001\n^ni;\u001b\n+X\ni\u0001(ri)^ci;\"^ci;#+ h.c. (1)\nHere ^cy\nj;\u001band ^cj;\u001bstand for the creation and annihilation\noperators of a fermion with spin projection \u001b=\u00061=2 at\na sitej. An on-site random potential is drawn from the\nbox distribution, Vi2[\u0000W;W ]. The chemical potential\n\u0016\fxes the \flling factor to 0 :3; throughout this work the\ninteraction is taken as U= 2:2t. The local occupation\nnumbern(ri) and the pairing amplitude \u0001( ri) are deter-\nmined self-consistently,\nn(ri) =X\n\u001bh^ni;\u001bi; \u0001(ri) =Uh^cy\ni;#^cy\ni;\"i; (2)arXiv:2107.06728v1  [cond-mat.dis-nn]  14 Jul 20212\n-0.4 0.0 0.4\nE[t]0\n50\n100\n150y[a]\n0.20.40.6\n-0.4 0.0 0.4\nE[t]\n123\n0 100\nx[a]\n0.20.40.60.8\n0 100\nx[a]\n12\n(r,E)[1/t]\nFIG. 1. Left Panels: LDoS-linecuts of typical samples along axial y-direction of the lattice and x= 0 for W=0.5 (outer left\npanel) and W=1.25 (middle left). Right panels: LDoS-maps of the same samples as in the left panels for W= 0:5 (middle\nright) andW= 1:25 (outer right) on the entire lattice for E=Emaxwith the peak energy of the disorder-averaged DoS Emax.\n(Parameters: zero temperature, 0.3 \flling, U= 2:2t,L= 192, and NC= 8192.)\nwhere ^ni;\u001b= ^cy\ni;\u001b^ci;\u001band the average is taken with re-\nspect to the equilibrium density matrix corresponding to\nthe Hamiltonian (1) at zero temperature. We solve Eqs.\n(1){(2) iteratively and terminate the self-consistency cy-\ncle when\u000b(n), the relative change in \u0001( ri) in iteration\nn, is at each site rismaller than some \u000b:\n\u000b(n)= max\nri\f\f\f\f\u0001(n)(ri)\u0000\u0001(n\u00001)(ri)\n\u0001(n\u00001)(ri)\f\f\f\f<\u000b: (3)\nFor system size L= 192 we chose \u000b= 10\u00003and for the\nsmaller system sizes \u000b= 10\u00004. We employ the Kernel\nPolynomial Method [46] to compute n(ri), \u0001( ri) and the\nLDoS. The underlying expansion of the time-evolution\noperator in Chebyshev polynomials to order NCcauses\nGibbs oscillations that we deal with employing the Jack-\nson kernel, see Ref. [34] for further computational de-\ntails. The ensemble averaging over observables involves,\ntypically, several hundred samples.\nThe dependence of the LDoS on energy, E, and the\nspatial coordinate, y, along a cut through the sample is\nshown in Fig. 1 for two values of disorder W. At weak\ndisorder,W= 0:5, (left panel in Fig. 1) \ructuations of\nthe local gap are almost absent and only after an increase\nuptoW= 1:25 the local gap \ructuations become more\nvisible albeit being still small (middle panel in Fig. 1).\nThis observation is qualitatively consistent with an an-\nalytical result for the relative \ructuations of the local\norder parameter, h(\u000e\u0001(r))2i=h\u0001(r)i2= [4=(\u0019g)] ln(\u00180=`)\n[35], where grepresents the dimensionless conductance\nin the normal state, \u00180the coherence length at zero tem-\nperature, and `the mean free path. At weak disorder,\ni.e. largeg, \ructuations of the local order parameter are\nsmall. With increase of disorder, i.e. decreasing g, the\n\ructuations of the local order parameter becomes more\npronounced.\nThe dependence of the average LDoS and its variance\nonEare presented in Fig. 2 for two values of disorder.\nWith increase of disorder the maximum in average LDoS\nbecomes less pronounced as expected. The energy de-\npendence of the LDoS variance has a form similar to theaverage LDoS, in particular, the variance has the maxi-\nmum. We note that this maximum is situated at energy\n~Emaxwhich is smaller than Emax. With increase of dis-\norder the LDoS variance increases.\nTheory. In order to understand salient features of the\nnumerical data, we consider 2D fermions with a BSC-\ntype attraction in the presence of a white-noise random\npotential, i.e. the continuum limit of the Hamiltonian\n(1) without the Hartree term. In the regime of a weak\ndisorder, one can neglect the spatial dependence of the\norder parameter \u0001. Then the LDoS at a given realization\nof disorder can be written as\n\u001a(E;r) =X\na;s=\u0006'2\na(r)\u0000\n1+\"a=E\u0001\n\u000e\u0000\nE\u0000sp\n\"2a+ \u00012\u0001\n;(4)\nwhere\"aand'\u000b(r) are eigen energies and eigen func-\ntions of the single particle Hamiltonian in the absence of\n\u0001. Using the well-known results for statistics of eigen\nenergies and eigen functions of weakly disordered non-\ninteracting Hamiltonian [44], we compute the mean and\nvariance of the local density of states from Eq. (4) (see\nSupplemental Material).\nThe disorder-average density of states is given as\nh\u001a(E)i=\u001a0ReXE, whereXE=E=p\nE2\u0000\u00012\nE. Here \u0001 E\n******************************************************************************************************************************************\n*\n*\n*********************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳ ⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳ ⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳ ⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳ ⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳ ⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳ ⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳ ⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳ ⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳ ⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳ ⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳ ⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳ ⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳ ⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳ ⊳⊳⊳⊳⊳⊳ ⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳∘∘∘∘∘∘ ∘∘∘∘∘∘∘∘ ∘∘ ∘∘∘∘∘∘ ∘∘∘∘∘∘∘∘∘∘ ∘∘∘∘∘∘∘∘∘∘∘∘ ∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘ ∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘∘⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯ ⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯ ⨯ ⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯\n⨯⨯\n⨯⨯⨯\n⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯ ⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯ ⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯ ⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯ ⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯ ⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯ ⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯ ⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯ ⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯ ⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯ ⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯ ⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯ ⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯ ⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯ ⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯ ⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯* -ρ� �=���\n∘-δρ� �=���\n⊳-ρ� �=����\n⨯-δρ� �=����\n-�-���-� ���� ���� �����������������\n�/����〈ρ(�)〉/ρ(����)�δρ�(�)/ρ(����)\nFIG. 2. The numerical data for the average LDoS and the\nLDoS variance for W= 0:5 andW= 1:25.3\n�����+����������������������������\n�/��Γ/Δ\n�=��Γ/Δ=����\n�=���Γ/Δ=����\n� � � � � �����������������\n�/����ρ(�)/ρ(����)\nFIG. 3. The average DOS. Comparison between numerics and\nDynes expression. The position of the maximum is Emax\u00191\nforL= 192 and Emax\u00191:08 forL= 96. (Parameters:\nW= 0:5t,U= 2:2tandNC= 2048 (L= 96), 8192 ( L=\n192).) Inset: Dynes parameter \u0000 versus system size for L=\n48;96;192.\nis the energy dependent gap function. We emphasize\nthat such an energy dependence appears naturally in a\nmore accurate treatment of the disordered electrons in\nthe presence of attraction [35]. Although the energy de-\npendence of \u0001 Ecan be derived microscopically [35], for\na sake of simplicity, we shall use the phenomenological\nDynes ansatz, \u0001 E=\u0001E=(E+i\u0000) with \u0000\u001c\u0001 [45]. We\nnote that the maximum of the average DOS is of the or-\nder of\u001a0p\n\u0001=\u0000 and is situated at Emax'\u0001+\u0000=p\n3. It is\nnatural to expect that \u0000 is enhanced by increasing disor-\nder, so that the peak value reduces and the peak position\nshifts to larger values. Our numerical data, Fig. 2 are in\nqualitative agreement with this.\nIn order to compute the LDoS variance, we use the\nfollowing well-known result for the irreducible part of the\ndynamical structure factor of non-interacting electrons in\nthe di\u000busive regime (see e.g. Ref. [47]),\nX\na;b\n'2\na(r)\u000e(E\u0000\"a)'2\nb(r)\u000e(E0\u0000\"b)\u000b\nirr'\u001a0\n2\u0019\n\u0002Zd2q\n(2\u0019)2Dq2\n(Dq2)2+ (E\u0000E0)2: (5)\nHereD=g=(4\u0019\u001a0) denotes the di\u000busion coe\u000ecient. Using\nEqs. (4) and (5), we \fnd the variance of the normalized\nlocal DOS at T= 0 to the lowest order in 1 =g,\n\u001b2\u0011h[\u000e\u001a(E;r)]2i\nh\u001a(E)i2=4\n\u0019gRe\"\nlnLE\n`\u00001 +jXEj2\n4(ReXE)2ln\u0010ETh\nE\n+ Im1\nXE\u0011\n+1\u0000X2\nE\n4(ReXE)2ln\u0010ETh\nE\u0000i\nXE\u0011#\n:(6)\nHereETh=D=(2L2) stands for the Thouless energy and\nLE=p\nD=(2E). Also we extend the result for the vari-\nance to the energy dependent gap function ( see Ref. [35]\n-�����+�����������������������\n�� �〈δρ�(�∞)〉/〈ρ(�∞)〉�\n�=��Γ/Δ=����\n�=���Γ/Δ=����\n� ��� � ��� ���������������������\n�/����δρ�(�)/ δρ�(����)FIG. 4. The LDoS variance. Comparison between numerics\nand theory. The parameters are the same as in Fig. 3. Fitting\nto Eq. 6, we extract g= 35:4 and`= 6 for the \ft. The\nposition of the maximum is ~Emax\u00190:97 forL= 192 and\n~Emax\u00191:03 forL= 96. Inset: The LDoS variance at large\nenergies,E1'3\u0001, versus system size for L= 48;96;192.\nfor details). Below we assume that the following condi-\ntion holds \u0001\u001dp\n\u0000\u0001\u001dETh\u001d\u0000. This corresponds to the\nparameters of our numerical analysis.\nSpeci\fcally, Eq. (6) predicts that at energies outside\nthe gap-region, ( E\u0000\u0001)&\u0001\u001d\u0000, the normalized variance\nbecomes almost independent of E, and, in particular, is\nlogarithmically divergent with the system size,\nh[\u000e\u001a(E;r)]2i=h\u001a(E)i2'[4=(\u0019g)] ln(L=`): (7)\nThe prefactor in front of ln Lin Eq. (7), coincides\nwith the known result for the multifractal exponent for\na weakly disordered metal, \u0001 2=\u00004=(\u0019g) [26]. The re-\nsult (7) implies that the spatial correlation function of\nLDoS at (E\u0000\u0001)&\u0001\u001d\u0000, behaves ash[\u001a(E;r)\u001a(E;r0)i/\n(L=jr\u0000r0j)\u0000\u00012. Such power-law behavior is surprising\nfor the system with the spectral gap and the \fnite coher-\nence length \u00180.\nThis logarithmic characteristics is a manifestation of\nmultifractality in the superconducting state. It can be\nunderstood on the basis of Eq. (4). To obtain the vari-\nance, each side of (4) needs to be squared; two di\u000berent\ncontributions to the LDoS-variance arise: The \frst one\ncorrelates the electron-like part of the LDoS ( s= + in\nEq. (4)) with the hole-like part ( s=\u0000). This contri-\nbution is sensitive to the gap since the energy di\u000berence\nbetween electron- and hole-like states cannot fall below\n2\u0001. The second contribution resembles correlations be-\ntween purely electron-like or hole-like states. The corre-\nsponding energy di\u000berence can be arbitrarily small even\nin the presence of the gap and therefore fourth-order mo-\nments of wavefunction amplitudes can exhibit signi\fcant\ncorrelations that are the characteristic precursors of mul-\ntifractality.\nMultifractality of the LDoS can be seen in the average\nvalues of the higher moments of \u001a(E;r). In the regime4\nof weak disorder, g\u001d1, the scaling of higher moments\nof the LDoS is fully determined by the second moment,\nh\u001aq(E;r)i=h\u001a(E)iq(h\u001a2(E;r)i=h\u001a(E)i2)q(q\u00001)=2[32, 35,\n48]. Therefore, using the result (6) we can \fnd the distri-\nbution function for the LDoS. After introducing the log-\narithm of the normalized LDoS, x= ln[\u001a(E;r)=h\u001a(E)i],\nits distribution function acquires the log-normal form,\nf(x)\u0019exp[\u0000(x+\u001b2=2)2=(2\u001b2)]=p\n2\u0019\u001b2; (8)\nwhere\u001b2is given by Eq. (6).\nDiscussions. In Fig. 3 we compare the average DOS\nobtained from numerical solution of Hamiltonian (1) with\nthe Dynes ansatz. We observe a reasonable agreement at\nweak disorder W= 0:5 and not too small energies. We\nnote that the Dynes parameter \u0000 depends linearly on\n1=L2(see inset in Fig. 3). This behavior is caused by\nour choice of the number of Chebyshev polynomials NC\nused in the expansion of the LDoS. [50]\nWe continue the discussion with the LDoS variance. A\ndetailed comparison between the numerical data at weak\ndisorderW= 0:5 and the analytical prediction, Eq. (6),\nis presented in Fig. 4. The logarithmic growth of the\nLDoS variance with the system size agrees with numerical\ndata as shown in the inset to Fig. 4. Incidentally, the\nlogarithmic dependence of the LDoS variance on Lallows\nus to extract values of gand`.\nIn agreement with numerical data, see Fig. 2, Eq. (6)\npredicts that the DOS variance has the maximum situ-\nated at energy ~Emaxwhich is smaller than the energy\nEmaxof the LDoS maximum. In particular, one can \fnd\nEmax\u0000~Emax/\u0000=ln(Lp\n\u0000\u0001=`)\u001c\u0000. We note that the dif-\nferenceEmax\u0000~Emaxis enlarged with increase of \u0000, i.e.\nof disorder. In accordance with Eq. (6), the height of\nthe maximum in h[\u000e\u001a(E;r)]2ibecomes of the order of\n[4\u001a2\n0\u0001=(\u0019g\u0000)] ln(Lp\n\u0000\u0001=`). Again, this result is in agree-\nment with the numerical data in Fig. 2 in which the\nheight of the maximum of the DOS variance is enhanced\nwith increasing disorder. Therefore, the expression (6)\nprovides reasonable description of the LDoS variance for\na weak disorder.\nIn Fig. 5 we present comparison of the distribution\nfunction for the logarithm of the normalized LDoS taken\nat two energies, E=EmaxandE= 3Emax, and obtained\nfrom numerical solution of Hamiltonian (1) against the\ntheoretical prediction of weak multifractality theory, Eq.\n(8). There is reasonable agreement between numerics\nand the theory. Also in 5 we plot the curves which cor-\nresponds to normal (rather log-normal) distribution of\nthe LDoS (with the same variance). As one can see, the\nlog-normal distribution is much more in agreement with\nnumerical data for the distribution function than the nor-\nmal distribution. This supports that the \ructuations of\nLDoS seen in numerics are of multifractal origin.\nFinally, we mention that our numerical and analyt-\nical results are also in qualitative agreement with the\n1.5\n 1.0\n 0.5\n 0.0 0.5 1.0 1.5\nln((E,r)/<(E,r)>)\n105\n104\n103\n102\n101\n100f(ln[(E,r)/<(E,r)>])\nE=Emax\nE=Emax analytics\nE=Emax normal\nE=3Emax\nE=3Emax analytics\nE=3Emax normalFIG. 5. The distribution of the logarithm of the nor-\nmalized LDoS, x= ln[\u001a(E;r)=h\u001a(E)i] at two di\u000berent ener-\ngiesE=EmaxandE=3Emax. The normal distribution for\nLDoS in terms of the normalized LDoS has the follow-\ning expression, fn(x)= exp[x\u0000(ex\u00001)2=(2\u001b2)]=(p\n2\u0019\u001b2) with\n\u001b2=h\u000e\u001a2(E;r)i=h\u001a(E)i2taken from numerics, see Fig. 4.\n(Parameters: W=0:5t,U=2:2t,L=192, andNC=8192.)\nexperimental data [49]. While the average and variance\nof the local DOS computed numerically as well as ana-\nlytically demonstrate all features observed in the experi-\nments, a more quantitative comparison is not indicated at\nthis point; it would require to include, e.g., also repulsive\nterms into our model, which goes beyond the scope of our\npresent work. We note that in experimental samples the\ninfra-red logarithmic divergence of the variance should\nbe cut o\u000b by the energy-dependent dephasing length (in-\nstead of the system size) [35].\nSummary. To summarize we report the results of numer-\nical and theoretical analysis of the energy dependence of\n\ructuations of the local DOS in weakly disordered su-\nperconducting \flms. We found that the local DOS has\npronounced \ructuations those variance has the energy\ndependence similar to the one for the average density of\nstates. 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Tamir and Prof. W.\nWulfhekel.\n[50] We choose NCsuch that the number of states within the\nresulting energetic broadening is constant for all system\nsizes. Thus the 1 =L2behavior is a re\rection of the de-\npendence of the mean level spacing on system size. While\nthere is an intrinsic broadening caused by disorder, it is\nnegligible with respect to the broadening brought about\nby our numerical procedure."    },    {        "title": "2107.12508v1.Active_Matter_Shepherding_and_Clustering_in_Inhomogeneous_Environments.pdf",        "content": "Active Matter Shepherding and Clustering in Inhomogeneous Environments\nP. Forg\u0013 acs1, A. Lib\u0013 al1, C. Reichhardt2, and C. J. O. Reichhardt2\n1Mathematics and Computer Science Department, Babe\u0018 s-Bolya University, Cluj 400084, Romania\n2Theoretical Division and Center for Nonlinear Studies,\nLos Alamos National Laboratory, Los Alamos, New Mexico 87545, USA\n(Dated: July 28, 2021)\nWe consider a mixture of active and passive run-and-tumble disks in an inhomogeneous environ-\nment where only half of the sample contains quenched disorder or pinning. The disks are initialized\nin a fully mixed state of uniform density. We identify several distinct dynamical phases as a function\nof motor force and pinning density. At high pinning densities and high motor forces, there is a two\nstep process initiated by a rapid accumulation of both active and passive disks in the pinned region,\nwhich produces a large density gradient in the system. This is followed by a slower species phase sep-\naration process where the inactive disks are shepherded by the active disks into the pin-free region,\nforming a non-clustered \ruid and producing a more uniform density with species phase separation.\nFor higher pinning densities and low motor forces, the dynamics becomes very slow and the system\nmaintains a strong density gradient. For weaker pinning and large motor forces, a \roating clustered\nstate appears and the time averaged density of the system is uniform. We illustrate the appearance\nof these phases in a dynamic phase diagram.\nActive matter, or self-motile particles or agents, arise\nin a variety of contexts including biological as well as\ndesigned systems such as self-propelled colloids and ar-\nti\fcial swimmers [1{3]. Due to their intrinsic non-\nequilibrium nature, such systems exhibit collective be-\nhaviors which are absent in systems with only thermal\nor Brownian motion. One of the best known examples\nof such an e\u000bect is the motility induced phase separation\nfound for a collection of interacting disks. In the Brow-\nnian limit, at lower densities the disks form a uniform\nliquid phase; however, if the disks are active, there can\nbe a transition to a self-clustered or phase separated state\ncomposed of regions of high density or solid clusters co-\nexisting with a low density active gas [2, 4{8]. This phase\nseparation can occur even when all the pairwise interac-\ntions between particles are repulsive.\nActive matter can also exhibit a variety of other ef-\nfects when it is coupled to complex environments [2],\nsuch as clustering along walls [9{13], activity induced\ndepletion forces [14{18], ratchet behavior or spontaneous\ndirected motion through funnels or asymmetric shapes\n[19{23], novel transport in maze geometries [24, 25], and\navalanches or activity induced clogging phenomena in\nconstricted geometries [26{28]. Active matter systems\nwith random quenched disorder can undergo activity in-\nduced jamming [29], trapping [30{32], non-monotonic\nmobility [29, 33{37], and di\u000berent types of motion un-\nder external drift forces [38{42].\nThe quenched disorder can take the form of obstacles\nwhich act as repulsive barriers or pinning sites which be-\nhave as localized traps. For dense pinning sites, the ac-\ntivity induced clustering e\u000bect can be suppressed when\nthe active particles remain spread out due to trapping\nin the pinning sites [31]. In contrast, obstacles can pro-\nmote activity induced clustering [29] by acting as nucle-\nation sites [43]. The enhancement or reduction of activ-\nity included clustering by random disorder depends on\nthe density and strength of the disorder. In addition torandom disorder, there have been a variety of studies\nexamining active matter coupled to periodic substrates\nwhich reveal directional locking e\u000bects [44{46], nonlinear\ntransport [47{50], and commensuration e\u000bects [51, 52].\nIn non-active matter systems with quenched disorder,\nsuch as vortices in superconductors [53, 54], colloidal\nparticles [55, 56], and skyrmions [57], it is known that\nthe dynamics of the system can be modi\fed strongly if\nthe quenched disorder is inhomogeneous. An example\nof such disorder is a sample containing extended regions\nof strong pinning coexisting with regions where the pin-\nning is weak or absent. Here, under an applied drive or\nincreasing temperature, the system exhibits high mobil-\nity in the unpinned regions and reduced mobility in the\npinned regions [53, 55, 57]. Also, as the temperature is\ndecreased from a high value, a glass or solid state \frst\nbegins to form in the pinned region, with glassy behavior\ngradually spreading into the non-pinned regions [54, 55].\nIn inhomogeneous pinning environments, application of\nan external drive can produce an accumulation of parti-\ncles along the interface between the pinned and pin-free\nregions [57]. These behaviors suggest that active mat-\nter moving in a sample containing coexisting pinned and\nnon-pinned regions should also show signi\fcantly di\u000ber-\nent behavior from active systems in uniform pinning.\nIn this work, we numerically examine a bidisperse as-\nsembly of run-and-tumble disks interacting with a sub-\nstrate that is populated on only one half by quenched\ndisorder in the form of pinning sites or obstacles. Half\nof the disks are active while the other half are passive.\nWe vary the number of pinning sites and the active mo-\ntor force, and initialize the system in a fully mixed state\nwith a uniform density across the sample. For high mo-\ntor forces and low pinning densities, the system forms an\nactive mobile cluster phase in which the clusters contain\nboth active and passive disks. These clusters spend equal\namounts of time in the pinned and non-pinned regions,\nand the clusters act as large scale objects which over-arXiv:2107.12508v1  [cond-mat.soft]  26 Jul 20212\n0 50 100 150 200 250 300 350 400\nx0255075100125150175200y\nFIG. 1. Image of the substrate showing that the right half\nof the sample is empty and the left half contains Npnonover-\nlapping pinning sites (dots).\ncome the pinning e\u000bects to form what we call a \roating\ncluster state with a uniform time-averaged density. At\nhigh motor forces and high pinning densities, we observe\na two step dynamical process involving a transition from\nthe uniform state to a density phase separated state fol-\nlowed by a coarsening process in which the di\u000berent disk\nspecies segregate. In the \frst stage, a large build up of\nboth species of disks into low mobility clusters occurs\nin the pinned region, producing a large density gradient\nbetween the pinned and non-pinned regions. The second\nstage is a slower species phase separation process in which\nactive disks gradually shepherd the non-active disks out\nof the pinned regions, producing a phase separated state\nwith a more uniform density in the long time limit. For\nother parameters, clusters can start to form along the\nedge of the pinned region, followed by the slow motion of\na well de\fned density front into the pinned region. This\nprocess becomes slower as the motor force decreases. For\nthe case of obstacles instead of pinning sites, we observe\nsimilar dynamics; however, the \roating clustered state is\nlost and the shepherding behavior appears at much lower\nobstacle densities. When all the disks are active and the\npinning is strong, the system only forms a low mobility\nclustered state in the pinned region and maintains a low\ndensity of active disks in the unpinned region.\nWe note that Chepizhko and Peruani [58] have also\nconsidered individual active particles interacting with\nquenched disorder in the form of obstacles placed in only\nhalf of the sample. They found that for su\u000eciently high\nobstacle densities, the active particle could become lo-\ncally trapped. This model di\u000bers signi\fcantly from the\npresent work, where we focus on strong collective inter-\nactions between the active particles.\nI. SIMULATION\nWe consider a two-dimensional system of size Lx= 400\nandLy= 200 with periodic boundary conditions in the\nxandy-directions. As illustrated in Fig. 1, the system\nis divided into two regions. The right region has no sub-\nstrate, while the left region contains Npnonoverlappingpinning sites that are modeled as attractive wells. We ini-\ntialize the system with a random uniform distribution of\nNddisks with radius ra. The disk-disk interactions are\nrepresented by a sti\u000b harmonic repulsion and the disk\ndensity is\u001e=Nd\u0019r2\na=LxLy. The disks are divided into\ntwo populations, A and B, where NAof the disks are ac-\ntive and experience self propulsion, while the remaining\nNB=Nd\u0000NAdisks are passive and can move only in\nresponse to other disks or the pinning sites. The equation\nof motion of disk iis\n\u000bdvi=Fdd\ni+Fm\ni+Fobs\ni: (1)\nThe disk velocity is vi=dri=dt, where riis the disk\nposition and the damping constant \u000bd= 1:0. The disk-\ndisk force is given by the harmonic repulsive potential\nFdd=PNd\ni 6=jk(2ra\u0000jrijj)\u0002(2ra\u0000jrijj)^rij, where \u0002 is the\nHeaviside step function, rij=ri\u0000rj, and ^rij=rij=jrijj.\nWe setk= 20 and ra= 1:0 for both species Aand\nB. The active disks obey run-and-tumble dynamics in\nwhich a motor force FMis exerted on the disk in a ran-\ndomly chosen direction during a run time of \u001clbefore\ninstantaneously changing to a new randomly chosen di-\nrection during the next run time. We de\fne the run\nlengthlras the distance a disk would travel in the ab-\nsence of pinning sites or disk-disk collisions, lr=FM\u001cl\u000et,\nwhere\u000et= 0:005 is the simulation time step. We vary \u001cl\nover the range \u001cl= 40;000 to 80;000 time steps and the\nmotor force FMover the range FM= 0:175 to 2:0. For\nthe non-active disks, the equation of motion is the same\nexceptFM= 0. We use a total simulation time of 2 \u0002107.\nWe \fx the total number of disks to give a system density\nof\u001e= 0:55. The pinning sites are modeled as harmonic\ntraps with strength Fp= 2:5 and radius rp= 0:5, such\nthat a single pinning site can capture at most one disk.\nAfter initialization, we measure the time evolution of the\nlocal disk density \u001elas a function of xaveraged over y\nfor all disks ( \u001et\nl), only the active disks ( \u001ea\nl), and only\nthe passive disks ( \u001ep\nl). We also measure the ratio of the\naverage disk density in the unpinned, \u001eu, and pinned,\n\u001ep, portions of the sample, giving the density ratio of all\ndisksRt=\u001et\nu=\u001et\np, of the active disks Ra=\u001ea\nu=\u001ea\np, and\nof the passive disks Rp=\u001ep\nu=\u001ep\np.\nII. RESULTS\nIn Fig. 2 we plot the positions of the active and pas-\nsive disks for a system with FM= 1:5,Np= 9000, and\nlr= 600. The initial disk con\fguration in Fig. 2(a) has a\nuniform density with a uniform distribution of both disk\nspecies. At time t= 3:9\u0002104in Fig. 2(b), a large density\nbuild up of both active and passive disks appears at the\npinning interface and begins to move into the pinned re-\ngion. The active and passive disks form patches of denser\nclusters with six-fold ordering in the pinned area, while in\nthe non-pinned region a more uniform \ruid like state ap-\npears. In Fig. 2(c) at t= 5\u0002105, there is a strong density3\n0 50 100 150 200 250 300 350 4000255075100125150175200y(a)\n0 50 100 150 200 250 300 350 4000255075100125150175200y(b)\n0 50 100 150 200 250 300 350 4000255075100125150175200y(c)\n0 50 100 150 200 250 300 350 400\nx0255075100125150175200y(d)\nFIG. 2. Images of the active disks (blue) and passive disks\n(red) for a system with FM= 1:5,Np= 9000, and lr= 600.\n(a) The initial con\fguration in which the disk species are well\nmixed and have a uniform density. (b) At time t= 3:9\u0002104,\na dense front of mixed species moves into the pinned region\nfrom both sides. (c) At t= 5\u0002105, there is a strong density\nimbalance and the unpinned region contains a low density of\nmostly passive disks while the pinned region contains a dense\nmixed state. (d) At t= 2\u0002107, the phase separation is more\npronounced but the density di\u000berence between the pinned and\nunpinned regions is diminished.\nbuildup in the pinned region while the non-pinned region\ncontains a small density of mostly passive disks, with a\ndensity ratio of close to 2 : 1 in the pinned and unpinned\nregions. At longer times, a coarsening process occurs in\nwhich the passive disks are gradually shepherded out of\nthe pinned region by the active disks. At t= 2\u0002107in\nFig. 2(d), the majority of the active disks have collected\n0.20.40.60.8t\nl\n(a)\n0.10.20.30.40.5a\nl\n(b) t=0.0\nt=5.0×105\nt=5.0×106\nt=2.0×107\n0 50 100 150 200 250 300 350\nx0.20.30.4p\nl\n(c)FIG. 3. Local densities \u001elas a function of xfor the sample\nin Fig. 2 with FM= 1:5,Np= 9000, and lr= 600 at times\nt= 0:0 (blue) where all the densities are uniform, t= 5\u0002105\n(orange),t= 5\u0002106(green), and t= 2\u0002107(red). (a) The\nlocal density of all disks \u001et\nlversusx. (b) The local density\nof active disks \u001ea\nlversusx. (c) The local density of passive\ndisks\u001ep\nlversusx.\nin the pinned region and show pronounced clustering,\nwhile the unpinned region contains mostly passive disks\nbeing pushed around by a small number of active disks.\nThe overall density of the system is more uniform com-\npared to Fig. 2(c). The clusters in the pinned region\ngenerally have low mobility, while weaker clustering of\nthe passive disks in the unpinned region occurs when the\nactive disks push the passive disks together.\nTo get a better picture of the time evolution, in\nFig. 3(a) we plot the local density of all disks \u001et\nlversusx\nfor the system in Fig. 2 at three di\u000berent times, while in\nFig. 3(b,c) we show the local densities \u001ea\nland\u001ep\nlof the\nactive and passive disks, respectively, versus x. Pinning\nis present in the region with x<200. Att= 0, the den-\nsity is uniform throughout the system. For t= 5\u0002105,\n\u001et\nlincreases in the pinned region until the ratio of den-\nsities in the pinned and unpinned regions is 2 : 1. The\nlocal density of active disks \u001ea\nlhas a much stronger in-\ncrease in the pinned region, with a ratio of close to 15 : 1\nfor the density of active particles in the pinned and un-\npinned regions. At t= 5\u0002106andt= 2\u0002107,\u001ea\nlin\nthe pinned region remains \fxed while the local density of\npassive disks \u001ep\nlin the pinned region drops. This occurs\nwhen the active particles shepherd the passive particles\nout of the pinned area, and causes the local density of all\ndisks\u001et\nlto become more uniform across the system.\nAs shown in Fig. 2, the initial build up of active disks in4\n0.05.0×1041.0×1051.5×1052.0×105\nt0.0100.0200.0300.0400.0x\n0.1 0.2 0.3 0.4 0.5 0.6\na\nl\nFIG. 4. Heat map of the local active disk density \u001ea\nlas a\nfunction of position xversus time for the system in Fig. 3\nwithFM= 1:5,Np= 9000, and lr= 600. The active disks\nmove into the pinned regime, which is the region from x= 0\ntox= 200, in the form of a front of higher density.\nthe pinned region occurs through the formation of a dense\nfront that moves into the pinned portion of the sample.\nThis is more clearly illustrated in Fig. 4 where we plot a\nheat map of \u001ea\nlas a function of xposition versus time.\nAtt= 0,\u001ea\nlis initially uniform, and then local maxima\ndevelop at the edges of the pinned region. These high\ndensity areas become fronts of active disks which move\ninto the pinned region from either side and gradually\nmerge at the center of the pinned region over time. For\nt<5\u0002104, in the pinned region there are bands of higher\ndensity which appear as horizontal lines, indicating that\nthe active disks have formed immobilized dense clusters\nin these locations, while in the unpinned region, the lines\ndenoting high density areas run at angles, indicating that\nthe active disks are in mobile clusters.\nIn Fig. 5(a) we plot the ratio Rt=\u001et\nu=\u001et\npof the den-\nsity\u001et\nuof all disks in the unpinned region to the density\n\u001et\npof all disks in the pinned region for the system in\nFig. 3 at di\u000berent values of FM. Figure 5(b) shows the\nratioRa=\u001ea\nu=\u001ea\npof the density of active disks in the un-\npinned and pinned regions, while in Fig. 5(c) we plot the\ncorresponding ratio Rp=\u001ep\nu=\u001ep\npfor the passive disks.\nAt initialization, we have Rt=Ra=Rp= 1:0. We\n\frst focus on the case of FM= 1:5, which matches the\nsystem shown in Fig. 2. There is an initial rapid drop\ninRtfromRt= 1:0 toRt= 0:4 which is accompanied\nby an even more rapid drop in the active particle ratio\nfromRa= 1:0 toRa\u00190:1. As the active disks en-\nter the pinned region, they drag along a portion of the\npassive disks, as indicated by the small drop in Rpfor\nt<0:05\u0002107in Fig. 5(c). In Fig. 6 we show a blow up\nof Fig. 5 over the range t<0:1\u0002107in order to illustrate\nmore clearly the initial invasion of the active and passive\ndisks into the pinned region. A portion of the passive\ndisks are dragged into the pinned region by the active\ndisks, causing Rpto decrease. For t>0:05\u0002107, there\nis a crossover to the second stage in which Raremains\n\fxed butRpgradually increases due to the shepherd-\nFIG. 5. Ratios Rof the average density \u001euin the unpinned\nregion to the average density \u001epin the pinned region versus\ntime for the system in Fig. 3 with Np= 9000 and lr= 600 at\nvariedFM= 0:5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, and\n1.5, from top left to bottom left. (a) The ratio Rt=\u001et\nu=\u001et\np\nof the density of all disks. (b) The ratio Ra=\u001ea\nu=\u001ea\npof the\nactive disk density. (c) The ratio Rp=\u001ep\nu=\u001ep\npof the passive\ndisk density. We \fnd a two stage evolution. In the \frst stage,\nthe active disks invade the pinned region, while in the second\nstage, the active particles shepherd the passive particles out of\nthe pinned region and into the unpinned region. The duration\nof the \frst stage increases as FMdecreases.\ning process in which the active disks eject the passive\ndisks from the pinned region. Figure 5 shows that Rp\nincreases from Rp= 0:9 toRp\u00192:5. At the same time,\nthe density ratio for all disks Rtincreases and approaches\nRt= 0:75 at the longest times. This suggests that for\neven longer times, the system should gradually approach\na state with nearly uniform total density in which the ac-\ntive and passive disks are phase separated in the pinned\nand unpinned regions. In Fig. 5 we \fnd that the time\nneeded to reach a species separated state increases with\ndecreasing FM. The duration of the initial invasion of\nthe active particles into the pinned region also increases\nwith decreasing FM, and forFM<0:9 we resolve only\nthe initial invasion stage.\nIn Fig. 7 we plot Raversus time on a linear-log scale\nto show more clearly the slowing of the initial invasion\nstage with decreasing motor force. The curves for FM<\n1:0 can be described as having three parts. In the \frst\npart fort <2\u0002105, there is an initial sharp decrease\ninRacorresponding to the moving front of active disks.\nThis is followed by a slower relaxation of the form Ra/\nA\u0000Blog(t), and \fnally by an eventual saturation close\nto a ratioRa= 0:1. Logarithmic relaxation has been5\n0.40.60.81.0Rt(a)\n0.20.40.60.81.0Ra(b)\n0.02.5×1055.0×1057.5×1051.0×106\nt0.70.80.91.0Rp(c)\nFIG. 6. A blow up of the density ratio versus time curves from\nFig. 5 over the range t<1:0\u0002106. HereNp= 9000,lr= 600,\nandFM= 0:5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, and 1.5,\nfrom top left to bottom left. (a) Density ratio for all disks\nRt=\u001et\nu=\u001et\np. (b) Active disk density ratio Ra=\u001ea\nu=\u001ea\np. (c)\nPassive disk density ratio Rp=\u001ep\nu=\u001ep\np. The invasion of active\ndisks into the pinned region is accompanied by the dragging\nof some of the passive disks into the pinned region. This is\nfollowed by a crossover to passive particle shepherding out of\nthe pinned region, indicated by an increase in Rp.\nFIG. 7. Linear-log plot of the active disk density ratio Ra\nversus time for the system in Fig. 5 with Np= 9000,lr=\n600, and varied FM. For lower motor forces the curves decay\napproximately as Ra/A\u0000Blog(t), as indicated by the upper\nsolid line.\nFIG. 8. Density ratios for the unpinned to pinned regions\nfor a system with Np= 6000 and lr= 600 at varied FM. (a)\nThe density ratio of all disks Rt=\u001et\nu=\u001et\np. (b) The active disk\ndensity ratio Ra=\u001ea\nu=\u001ea\np. (c) The passive disk density ratio\nRp=\u001ep\nu=\u001ep\np. At highFM, the system saturates to a state\nwithRt= 0:75.\nobserved in vibrated granular matter [59] where there is\na gradual compaction to a denser state. In our system\nthe equivalent of the compaction dynamics is the build\nup of the active disks in the pinned region.\nIn Fig. 8 we plot Rt,Ra, andRpversus time for a\nsystem with a lower number of pinning sites Np= 6000\nat di\u000berent values of FM. The overall behavior is similar\nto that found for the Np= 9000 sample. When FM>\n0:8 we \fnd the two step process of an initial active disk\ninvasion of the pinned region and the later passive disk\nshepherding out of the pinned region. For FM>1:2, the\nsystem reaches a steady state with Rt= 0:75, indicating\na higher overall density in the pinned region compared\nto the unpinned region.\nIn Fig. 9 we plot Rt,Ra, andRpversus time for\na system with an even lower number of pinning sites\nNp= 2000. For FM<0:8 we \fnd the usual two stage\nbehavior of a rapid advancement of active disks into the\npinned region followed by a slow species phase separation\nat longer times. For FM>0:8, the behavior changes and\nthere are strong oscillations in Rt,Ra, andRp. These os-\ncillations result when a large scale motility induced clus-\ntering state forms, as shown in Fig. 10(a) at t= 2\u0002107\nfor the system in Fig. 9 at FM= 1:5. In the pinned\nregion, there is a background of pinned disks coexisting\nwith a much larger cluster composed of both active and\ninactive disks. The density ratio oscillations in Fig. 9\nare correlated with each other, indicating that there is6\nFIG. 9. Density ratios for the unpinned to pinned regions\nversus time for a system with Np= 2000 and lr= 600 at\nFM= 0:5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, and 1.5,\nfrom top left to bottom left. (a) The density ratio of all disks\nRt=\u001et\nu=\u001et\np. (b) The active disk density ratio Ra=\u001ea\nu=\u001ea\np.\n(c) The passive disk density ratio Rp=\u001ep\nu=\u001ep\np. ForFM>0:8,\nwe \fnd strong density oscillations due to motility induced\nclustering, as illustrated in Fig. 10(a).\n0 50 100 150 200 250 300 350 4000255075100125150175200y(a)\n0 50 100 150 200 250 300 350 400\nx0255075100125150175200y(b)\nFIG. 10. Images of the active disks (blue) and passive disks\n(red) for the system in Fig. 9 with Np= 2000 and lr= 600\natt= 2\u0002107. (a) ForFM= 1:5, the system forms a motility\ninduced cluster state. (b) At FM= 0:5, there is a liquid like\nstate with some species phase separation.\n2000 4000 6000 8000 10000\nN\np0.000.200.400.600.801.001.201.401.601.802.00F\nM\n0.500.751.001.251.501.752.002.252.502.75\nR\npFIG. 11. Dynamic phase diagram as a function of motor\nforceFMversus number of pinning sites Npobtained from the\nvalue of the passive disk density ratio Rpat a timet= 2\u0002107.\nGreen: strong shepherding with species separation. Yellow:\nWeak or no shepherding. Red: large density oscillations occur\ndue to the formation of the motility induced cluster state\nillustrated in Figs. 9 and 10(a).\neither no species phase separation or at most only very\nweak separation. In the snapshot, the large cluster is lo-\ncated in the pinned region, but the cluster is mobile and\nspends an equal amount of time in the pinned and un-\npinned regions, producing the density ratio oscillations\nand giving a time averaged density that is close to uni-\nform. The motility induced cluster in Fig. 10(a) di\u000bers\nfrom the clustering found in the pinned region at higher\nNp, since the latter clusters are mostly frozen in time and\nhave little to no mobility. In contrast, the cluster shown\nin Fig. 10(a) gradually drifts through the entire sample\nover time. In Fig. 10(b), we illustrate the disk con\fgura-\ntions for the system in Fig. 9 for FM= 0:5 att= 2\u0002107,\nwhere the clustering is lost but there is still some phase\nseparation. We have constructed a series of plots similar\nto those shown in Fig. 9 and \fnd that motility induced\nclustering always occurs in samples with Np<4500 at\nthe larger motor forces.\nBased on the behavior of the density ratios and our\ndetermination of whether density oscillations of the type\nshown in Fig. 9 are present, we can construct a dynamic\nphase diagram of the di\u000berent behaviors as a function of\nFMversusNp, as shown in Fig. 11 using the value of\nthe passive disk density ratio Rpat a timet= 2\u0002107.\nIn the green region, we \fnd strong shepherding e\u000bects\nand the sample reaches a saturated state, while in the\nyellow region, the shepherding is weak or absent. The\nmotility induced cluster state of the type illustrated in\nFig. 10(a) appears in the red region. We \fnd that shep-\nherding can only occur for a su\u000eciently large number of\npinning sites and a su\u000eciently high motor force. The7\n0 50 100 150 200 250 300 350 4000255075100125150175200y(a)\n0 50 100 150 200 250 300 350 400\nx0255075100125150175200y(b)\nFIG. 12. Images of the active disks (blue) and passive disks\n(red) for a system with Np= 8500,lr= 600, and FM=\n0:7, near the transition between the shepherding and non-\nshepherding states in Fig. 11. (a) At t= 5\u0002105, a dense\ninterface of active disks has formed at the edge of the pinned\nregion. (b) The image at a much later time of t= 2\u0002107\nshows that this interface is e\u000bectively pinned.\nmotility induced cluster state grows in extent as FMin-\ncreases. When the number of pins is large but FMis\nsmall, the pinned portion of the sample is in a disor-\ndered or glassy state. In this study we \fx Fp= 2:5 and\nconsider only the regime FM< Fp, but we expect that\nwhenFM> F p, the motility induced cluster state will\ndominate the phase diagram.\nAt intermediate motor forces and larger Np, near the\nphase boundary between shepherding (green) and ab-\nsence of shepherding (yellow) in Fig. 11, a frozen in-\nvasion front can appear, as illustrated in Fig. 12 for a\nsample with Np= 8500 and FM= 0:7. Here the initial\ninvasion of the active front occurs slowly enough that\nas the front starts to penetrate the pinned region, the\nfront itself becomes pinned, forming a crystallized state\nnear the edge of the pinned region. This pinned dense\nfront makes it di\u000ecult for the passive disks to escape\ninto the unpinned region. Figure 12(b) shows that at a\nmuch later time of t= 2\u0002107, the dense front has be-\ncome better de\fned and has failed to penetrate further\ninto the pinned region. The e\u000bective drive experienced\nby the dense front is proportional to FM. In other sys-\ntems where an interface moves over quenched disorder,\nit is known that there is is a minimum critical force Fc\nwhich must be applied in order for the interface to depin\n[60], so the crossover from the shepherding regime to the\nnon-shepherding glassy state can be viewed as occurring\nFIG. 13. Density ratios for the unpinned to pinned regions\nversus time for a system with Nobs= 1500 repulsive obstacles\nandlr= 600. (a) The density ratio of all disks Rt=\u001et\nu=\u001et\np.\n(b) The active disk density ratio Ra=\u001ea\nu=\u001ea\np. (c) The passive\ndisk density ratio Rp=\u001ep\nu=\u001ep\np. Here we \fnd that the active\ndisks move rapidly into the obstacle-\flled region and then\ngradually shepherd the passive particles into the obstacle-free\nregion.\nat the point where FMdrops below the e\u000bective Fcfor\ninterface depinning.\nThere have been several studies examining bidisperse\nmixtures of passive and active particles which have shown\nthat both motility induced phase separation and species\nseparation can occur [61{64]. It has also been demon-\nstrated that active particles can move passive particles\nand organize them into particular states even when there\nare only a small number of active particles present, which\nis known as active doping [65{68]. This is similar to\nthe shepherding phenomenon we observe. Our results\nshow that quenched disorder could be used to achieve\nboth density and species phase separation. For exam-\nple, it should be possible to create a patterned state with\npinning where the active particles could accumulate and\ngradually move the inactivate particles in order to cre-\nate a tailored self-assembled arrangement of active and\npassive particles.\nIII. OBSTACLE ARRAYS\nWe have also considered samples in which the attrac-\ntive pinning sites are replaced by repulsive obstacles in\nthe form of immobile disks. In Fig. 13(a,b,c) we plot\nRt,Ra, andRpas a function of time for a system with8\n0 50 100 150 200 250 300 350 4000255075100125150175200y(a)\n0 50 100 150 200 250 300 350 400\nx0255075100125150175200y(b)\nFIG. 14. Images of the active disks (blue), passive disks\n(red), and obstacle locations (black) for the system in Fig. 13\nwith repulsive obstacles instead of attractive pinning sites at\nNobs= 1500,lr= 600, and FM= 1:5. (a) Att= 0:05\u0002107,\nthere is an initial invasion into the region with obstacles. (b)\nAtt= 2\u0002107, there is a stronger species separation.\nNobs= 1500 obstacles and FM= 1:5. The active disks\nrapidly move into the obstacle region, followed by the\nlong time shepherding of the inactive disks into the un-\npinned region, similar to what we \fnd for samples with\nattractive pinning sites at much higher pinning densities\nofNp>5000.\nIn Fig. 14(a) we show the positions of the active and\npassive disks along with the obstacle locations for the\nsystem in Fig. 13 at t= 0:05\u0002107, where the active\nand passive disks begin to cluster in the obstacle-\flled\nregion. In Fig. 14(b), the same system at a later time of\nt= 2:0\u0002107has a stronger species separation. In the\nobstacle-\flled region, the active disk density is higher\nthan in samples containing pinning sites. Previous work\nfor active matter on random pinning arrays showed that\nthere can be a wetting transition in which mobility in-\nduced clusters break apart as the active particles spread\nout and attempt to occupy as many pinning sites as pos-\nsible [31]. On the other hand, it has also been shown\nthat repulsive obstacles can act as nucleation sites pro-\nmoting the clustering of active particles [43]. The clus-\nters which are induced by the presence of the obstacles\nremain pinned at the locations of the obstacles and are\nnot mobile. In our system with obstacles, we do not ob-\nserve drifting or \roating clusters of the type illustrated in\nFig. 10(a) for pinning site systems with low Npand high\nFM, since even a small number of obstacles can pin down\nany cluster that forms, while the cluster can e\u000bectively\n0 50 100 150 200 250 300 350 4000255075100125150175200y(a)\n0 50 100 150 200 250 300 350 4000255075100125150175200y(b)\n0 50 100 150 200 250 300 350 400\nx0255075100125150175200y(c)FIG. 15. Images of the active disks (blue) for a system with\nNp= 9000 pinning sites and no passive disks at lr= 600\nandFM= 1:5. (a) The t= 0 initial uniform con\fguration.\n(b) Att= 0:02\u0002107, clusters form on both sides of the\npinned region. (c) The fully density phase separated state at\nt= 2:0\u0002107.\n\roat above the pinning sites if FMis su\u000eciently large. In\nthe obstacle-free region we observe some weak clustering\ndue to the shepherding of the passive particles by a small\nnumber of active particles, as shown in Fig. 14(b). We\n\fnd behavior similar to that illustrated in Figs. 13 and\n14 for other obstacle densities.\nIV. MONODISPERSE ACTIVE DISKS\nWe have also considered monodisperse systems in\nwhich all of the disks are active and there are no pas-\nsive disks. Here, we \fnd only two generic phases. The\n\frst is the accumulation of active disks in the pinned re-\ngion, forming a large density gradient, and the second is\nthe formation of a large scale drifting cluster that \roats\nover the random pinning. In Fig. 15(a) we show the\ninitial uniform con\fguration for a monodisperse system9\ncontaining only active disks with Np= 9000 pinning sites\nandFM= 1:5. Att= 0:01\u0002107in Fig. 15(b), mobile\nclusters have formed in the unpinned region while pinned\nclusters appear in the pinned region. In Fig. 15(c) at\nt= 2\u0002107, most of the disks are in the pinned region\nwith a low density gas of active disks present in the un-\npinned region. For smaller FM, the same behavior occurs\nbut the time required for the active disks to penetrate the\npinned region fully increases.\nIf we place monodisperse active disks in a sample con-\ntaining obstacles instead of pinning sites, we \fnd behav-\nior similar to that shown in Fig. 14, but the mobile clus-\nter phase is absent since the clusters become pinned in\nthe obstacle-\flled region. In general, one might expect\nthe active disks to accumulate in the unpinned region\nwhere they have more space to move; however, this is\nnot what occurs due to a combination of e\u000bects. The\n\frst is the reduction of the mobility through direct in-\nteractions between the active disks and the pinning sites\nor obstacles, and the second is the motility induced clus-\ntering e\u000bect which slows down the disks and is enhanced\nby the presence of pinning sites or obstacles. In a system\nwhere one region has a low di\u000busion coe\u000ecient and an-\nother region has a high di\u000busion coe\u000ecient, particles in\nthe high di\u000busion region can rapidly explore space and\nreach the low di\u000busion region, whereas particles in the\nlow di\u000busion region have reduced mobility and cannot\nreach the high di\u000busion region easily. The resulting \rux\nimbalance causes the low di\u000busion region to accumulate\na higher concentration of particles. In the case of active\ndisks, the motility induced clustering state also produces\na lower e\u000bective di\u000busion coe\u000ecient compared to freely\nmoving non-interacting active disks. If a landscape struc-\nture could be identi\fed which breaks apart the motility\ninduced clusters only in the disordered region but not in\nthe disorder-free region, it would be possible to achieve\na higher e\u000bective di\u000busion coe\u000ecient in the disordered\nregion, causing an accumulation of particles in the un-\npinned region, as opposed to what we observe, which is\nthe opposite e\u000bect where the active disks accumulate in\nthe pinned or obstacle-\flled region.\nV. SUMMARY\nWe have examined a bidisperse mixture of active and\npassive disks interacting with inhomogeneous disorder inwhich half of the sample contains randomly distributed\npinning sites and the other half is free of disorder. We\nconsider active disks obeying run-and-tumble dynamics,\nand vary the magnitude of the motor force and the den-\nsity of pinning sites. For dense pinning and high mo-\ntor forces, we observe a two step process in which active\nand passive particles accumulate in the pinned region,\nproducing a large density gradient, followed by a slower\nshepherding process in which the passive disks are pushed\ninto the unpinned region, producing a state that is more\nuniform in density but is phase separated. For lower mo-\ntor forces, the initial invasion process becomes slower,\nand if the motor force is below a critical value, the in-\nvasion front entering the pinned region becomes pinned.\nFor larger motor forces and lower pinning density, we \fnd\nlarge scale drifting clusters containing a mixture of active\nand passive disks which e\u000bectively \roat over the pinning\nsites, leading to large oscillations in the density on either\nside of the sample but giving a uniform time averaged\ndensity across the entire sample. If we replace the at-\ntractive pinning sites by repulsive obstacles, the drifting\ncluster phase is lost and we observe an expanded shep-\nherding region with the formation of much more compact\nclusters in the disordered portion of the sample. For a\nmonodisperse system containing only active disks and no\npassive disks, the disks accumulate in the pinned region\nfor higher pinning density and form a drifting motility\ninduced cluster state at low pinning densities. Our re-\nsults indicate that active particles could be used to move\npassive particles though complex landscapes or to control\nthe invasion of active \ruids into disordered media.\nACKNOWLEDGMENTS\nThis work was supported by the US Department of\nEnergy through the Los Alamos National Laboratory.\nLos Alamos National Laboratory is operated by Triad\nNational Security, LLC, for the National Nuclear Secu-\nrity Administration of the U. S. Department of Energy\n(Contract No. 892333218NCA000001). 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Sansone 1, 50019 Sesto Fiorentino, Italy\nWe analyze the change in the spontaneous decay rate, or Purcell e\u000bect, of an extended quantum\nemitter in a structured photonic environment. Based on a simple theory, we show that the cross-\ndensity of states is the central quantity driving interferences in the emission process. Using numerical\nsimulations in realistic photonic cavity geometries, we demonstrate that a structured cross-density\nof states can induce subradiance or superradiance, and change substantially the emission spectrum.\nInterestingly, the spectral lineshape of the Purcell e\u000bect of an extended source cannot be predicted\nfrom the sole knowledge of the spectral dependence of the local density of states.\nI. INTRODUCTION\nIn the last two decades, the ability to tailor the spon-\ntaneous decay rate of a quantum emitter, by molding\nthe dielectric environment at the nanoscale, has been\nthe driving force of a large variety of studies in cav-\nity quantum electrodynamics [1], semiconductor exciton\nphysics [2], and nanophotonics [3{5]. The engineering\nof the electromagnetic local density of states (LDOS), in\norder to tune the spontaneous decay rate by the Purcell\ne\u000bect [6], has become a major branch of nanophotonics\nand plasmonics. The standard formulation of the Pur-\ncell e\u000bect is based on the electric dipole approximation,\nwhich requires the electromagnetic \feld to be uniform\nover the spatial extent of the emitter. The dipole ap-\nproximation is expected to breakdown when extended\nsources are used (for example extended excitons in semi-\nconductors [7]), or when the external \feld is con\fned at\nthe nanometer scale, or beyond [8]. A theory govern-\ning the spontaneous emission by extended excitons has\nbeen proposed in Ref. [9]. The theory predicts large os-\ncillator strengths, exceeding 103for quantum-dots (QDs)\nwith diameters on the order of 100 nm. These ideas led\nto the concept of single-photon superradiance, which has\nbeen observed in natural QDs arising in monolayer \ruc-\ntuation of GaAs quantum wells, with radiative lifetime\nbelow 100 ps. On a di\u000berent scheme, Rydberg excitons\nhave been observed in CuO 2bulk samples, with a Bohr\nradius extending over micrometers [10, 11]. Extended\nsources are also at play in microwave cavity quantum\nelectrodynamics with Rydberg atoms [12], and in elec-\ntron spectroscopy where the interaction with light de-\npends on non-local excitations [13]. In a slightly di\u000berent\ncontext, understanding and controlling the emission of a\nset of single-photon sources connected through their elec-\ntromagnetic environments has become a key issue for the\ntreatment of quantum information in photonics [14, 15],\nor to favor collective emission [16, 17]. This has stimu-\nlated the development of new theoretical approaches to\n\u0003remi.carminati@espci.psl.eu\nymassimo.gurioli@uni\f.itdescribe photonic states in ensembles of quantum emit-\nters [18, 19]. In the simple case of two (point) quantum\nemitters, the role of the cross density of states (CDOS),\nthat describes electromagnetic spatial coherence between\ndi\u000berent points in an arbitrary environment [20], has\nbeen put forward [21, 22]. It was shown that the degree\nof quantum coherence in the emitted light is controlled\nby a combination of the LDOS and CDOS, the latter de-\nscribing the coupling between the sources in the emission\nprocess.\nIn the case of extended quantum sources that cannot\nbe reduced to a single point dipole or an ensemble of inde-\npendent (mutually incoherent) point sources, it has been\nshown that spontaneous decay results from a non-local\ninteraction with the environment [9]. Nevertheless, the\nexact role of the CDOS on the Purcell e\u000bect has not been\nexplored in its full generality. In particular, beyond the\ngeneration of large e\u000bective oscillator strengths, one can\nexpect an engineering of the CDOS to a\u000bect the emis-\nsion substantially, e.g. by producing subradiance or su-\nperradiance, or by changing the emission spectrum. The\npurpose of this paper is to analyze precisely the Pur-\ncell e\u000bect for an extended quantum source. Based on\nsimple theoretical arguments, we show that the CDOS\nis the essential ingredient to understand the change in\nthe spontaneous emission decay rate, beyond the dipole\napproximation. The theoretical results are highlighted\nby numerical simulations, in a series of simple examples\ninvolving either a single high- Qcavity or coupled cavi-\nties with non Lorentzian spectra. Interestingly, the spec-\ntral emission lineshape of an extended source covering\na nanostructured environment cannot be predicted from\nthe spectral dependence of the LDOS alone.\nII. SPONTANEOUS DECAY RATE IN\nSTRUCTURED ENVIRONMENTS\nIn this section we introduce the expressions of the\nspontaneous decay rate of point and extended dipole\nemitters. We assume a weak-coupling regime, in which\nthe population of the excited state exhibits exponential\ndecay, and the transition dipole is not modi\fed by thearXiv:2107.13980v3  [quant-ph]  26 Apr 20222\ninteraction with the environment. We also restrict the\ndiscussion to electric-dipole transitions, and the theory\ninvolves the electric part of the density of states of the\nelectromagnetic \feld. For magnetic-dipole transitions,\nsimilar expressions could be easily deduced by using the\nmagnetic contribution to the density of states [5, 23, 24].\nA. Dipole emitter\nIn the weak-coupling regime, the spontaneous decay\nrate \u0000 of a two-level dipole emitter can be calculated\nfrom perturbation theory. For an emitter with transition\ndipole p, emission frequency !, and located at the point\nrs, it can be shown that [25, 26]\n\u0000(rs;!) =2\u00160!2\n~jpj2Im [^u\u0001G(rs;rs;!)^u]:(1)\nIn this expression Gis the electric Green's function de-\nscribing the electromagnetic response of the environment,\nand ^u=p=jpjis the unit vector de\fning the orienta-\ntion of the transition dipole. We note that the inter-\nnal dynamics of the emitter, described by the transition\ndipole pthat depends on the excited and ground state\nwavefunctions, and the response of the environment fac-\ntorize. In free space, the Green's function G0is such\nthat Im [ ^u\u0001G0(rs;rs;!)^u] =!=(6\u0019c), withcthe speed\nof light in vacuum, and the spontaneous decay rate is\n\u00000(!) =!3jpj2=(3\u0019~\u000f0c3). Normalizing, we \fnd that\n\u0000(rs;!)\n\u00000(!)=6\u0019c\n!Im [^u\u0001G(rs;rs;!)^u]: (2)\nIt will prove useful to recall a well-established corre-\nspondence between classical and quantum calculations.\nConsidering a classical electric dipole poscillating at fre-\nquency!, the time-averaged power released in the envi-\nronment is P(rs;!) = (\u00160!3=2)jpj2Im [^u\u0001G(rs;rs;!)^u].\nNormalizing by the power P0radiated in free space, we\nobtain\nP(rs;!)\nP0(!)=6\u0019c\n!Im [^u\u0001G(rs;rs;!)^u]: (3)\nThe right-hand side is the same as in Eq. (2), showing\nthat the change in the spontaneous decay rate of a quan-\ntum emitter due to the electromagnetic interaction with\nits environment, and the change in the power emitted by\na classical dipole, are identical [4, 5, 27].\nIn the particular case of a single mode cavity, the right-\nhand side in Eqs. (2) and (3) de\fnes the Purcell fac-\ntor. For an emitter on resonance, it is usually written as\n3=(4\u00192)\u00153\nmQ=V, withQthe mode quality factor, Vthe\nmode volume and \u0015mthe resonance wavelength [5, 6].\nEquations (2) and (3) generalize the Purcell e\u000bect to a\nmore complex environment, and the right-hand side can\nbe seen as a generalized Purcell factor. This factor actu-\nally describes the change in the projected LDOS\n\u001a(rs;!) =2!\n\u0019c2Im [^u\u0001G(rs;rs;!)^u]; (4)which measures the coupling strength of a unit dipole ^u\nto the electromagnetic environment. The LDOS actually\ncounts the relative contribution of the electromagnetic\nmodes at position rand frequency ![4, 5, 28]. In terms\nof the LDOS we \fnd that\n\u0000(rs;!)\n\u00000(!)=P(rs;!)\nP0(!)=\u001a(rs;!)\n\u001a0(!); (5)\nwhere\u001a0(!) =!2=(3\u00192c3) is the projected LDOS in free\nspace.\nB. Purcell e\u000bect for an extended source\nA full quantum theory of spontaneous emission by an\nextended exciton has been developed in Ref.[9], empha-\nsizing the role of the internal dynamics of the source.\nIn the present work, we rather focus on the role of the\nenvironment, and take advantage of the equivalence of\nthe classical and quantum descriptions to build the ex-\npression of the decay rate starting from the expression of\nthe emitted power. For a monochromatic classical source\nwith current density j(r) con\fned within a volume V,\nand oscillating at frequency !, the emitted power can be\nwritten as\nP=\u00001\n2ReZ\nVj\u0003(r)\u0001E(r)d3r; (6)\nwhere the superscript \u0003stands for complex conjugate. In\nterms of the Green's function Gthis becomes\nP=\u00160!\n2Z\nIm [j\u0003(r)\u0001G(r;r0;!)j(r0)]d3rd3r0;(7)\nwhich, by making use of the reciprocity relation\nG(r;r0;!) =GT(r0;r;!), can be rewritten as\nP=\u00160!\n2Z\nj\u0003(r)\u0001ImG(r;r0;!)j(r0)d3rd3r0:(8)\nNormalizing by the power emitted in free space, we \fnd\nthat\nP\nP0=R\nj\u0003(r)\u0001ImG(r;r0;!)j(r0)d3rd3r0\nR\nj\u0003(r)\u0001ImG0(r;r0;!)j(r0)d3rd3r0:(9)\nIn the classical picture, the current density corresponding\nto an extended exciton can be taken of the form\nj(r) =\u0000i!^u\u0011(r;rs); (10)\nwhere\u0011(r;rs) is the volume density of transition dipole\nat the position r,^ugives the local orientation of the tran-\nsition dipole and rsde\fnes the position of the source (the\nreference point rscan be chosen arbitrarily, for example\nas the center of the source spatial distribution). This\nmodel allows to take into account a possible heterogene-\nity of the distribution of elementary dipoles, in ampli-\ntude and orientation, inside the extended emitter. Also3\nnote that a single extended exciton corresponds to spa-\ntially coherent source, that can be seen as a continuous\ndistribution of mutually coherent elementary transition\ndipoles. Inserting this expression into Eq. (9), and in-\nvoking again the classical to quantum correspondence,\nwe obtain\n\u0000(rs;!)\n\u00000(!)=R\n\u0011\u0003(r;rs)\u0011(r0;rs) Im[ ^u\u0001G(r;r0;!)^u0]d3rd3r0\nR\n\u0011\u0003(r;rs)\u0011(r0;rs) Im[ ^u\u0001G0(r;r0;!)^u0]d3rd3r0:\n(11)\nThis expression of the normalized decay rate of an ex-\ntended source accounts for the non-local interaction with\nthe environment, as already put forward in Ref. [9]. The\nsimilarity with Eq. (5) can be made more explicit by in-\ntroducing the projected CDOS [20], de\fned as\n\u001a(r;r0;!) =2!\n\u0019c2Im [^u\u0001G(r;r0;!)^u0]: (12)\nThe projected CDOS measures the connection between\ntwo unit dipoles ^uand^u0located at the points randr0.\nMore precisely, the CDOS counts the contribution of the\nelectromagnetic modes connecting randr0at frequency\n![5, 20, 29]. It can also be understood as a measure of\nthe intrinsic spatial coherence between the two points r\nandr0. We also note that for r=r0the projected CDOS\ncoincides with the projected LDOS evaluated at point r.\nIn terms of the CDOS, we \fnd that\n\u0000(rs;!)\n\u00000(!)=R\n\u0011\u0003(r;rs)\u0011(r0;rs)\u001a(r;r0;!)d3rd3r0\nR\n\u0011\u0003(r;rs)\u0011(r0;rs)\u001a0(r;r0;!)d3rd3r0;\n(13)\nwhere\u001a0(r;r0;!) is the CDOS in free space. This expres-\nsion of the normalized decay rate (or Purcell enhance-\nment) generalizes Eq. (5) to extended emitters beyond\nthe dipole approximation. Indeed, it can be seen imme-\ndiately that by using \u0011(r;rs) =p\u000e(r\u0000rs) in Eq. (13),\nwhich corresponds to a point dipole source with ampli-\ntudeplocated at the point rs, we recover Eq. (5).\nIII. EXTENDED SOURCES IN PHOTONIC\nCAVITIES\nIt is instructive to start with the simplest model of a\nspatially coherent extended source, made of two mutually\ncoherent point dipoles with amplitude p1=p1^u1and\np2=p2^u2, and located at positions r1andr2, that is\nj(r) =\u0000i!^u1p1\u000e(r\u0000r1)\u0000i!^u2p2\u000e(r\u0000r2):(14)\nAccording to Eq. (13), the spontaneous decay rate is\n\u0000\u0018jp1j2\u001a11+jp2j2\u001a22+ 2Re(p1p\u0003\n2)\u001a12; (15)\nup to a constant prefactor (independent of the environ-\nment) that we do not specify. Here \u001aij= 2!=(\u0019c2)Im[^ui\u0001\nG(ri;rj;!)^uj] is the LDOS for i=jand the CDOS for\ni6=j. If the two dipoles have the same amplitude jpjand\na phase di\u000berence \u001e, we simply have\n\u0000\u0018jpj2[\u001a11+\u001a22+ 2\u001a12cos\u001e]: (16)The mutual coherence between the two sources is essen-\ntial for the appearance of the interference term involving\nthe CDOS. This situation is di\u000berent from that analyzed\nin Refs. [21, 22, 30], where spontaneous emission by two\nindependent sources ( e.g., two \ruorescent molecules) in\na structured environment was considered. In this case,\nit was shown that intensity \ructuations are in\ruenced\nby a cross term involving the CDOS. In the situation\nconsidered here, interferences directly modify the spon-\ntaneous decay rate. The state of interference not only\ndepends on the relative phase of the dipole sources, but\nalso on the environment through the CDOS that can take\npositive or negative values. The increase or decrease of\nthe spontaneous decay rate of an extended source due\nto interferences corresponds to superradiance and sub-\nradiance [31, 32]. In the following subsections, we use\nnumerical simulations to illustrate the substantial role of\nthe CDOS on the spontaneous decay rate of extended\nsources in photonic cavities.\nA. High- Qcavity\nLet us \frst consider a source immersed in a discrete\nset of high- Qelectromagnetic modes, whose resonances\ndo not overlap spectrally. The Green's function in this\ncase takes the following form [5]\nG(r;r0;!) =X\nmc2em(r)\ne\u0003\nm(r0)\n!2m\u0000!2\u0000i!\rm; (17)\nwhere emare eigenmodes of the vector Helmholtz equa-\ntion in absence of losses, !mare eigenfrequencies, and\n\rmis a phenomenological mode damping rate. This ex-\npression can also be seen as low-loss limit ( \rm\u001c!m) of\nan expansion in quasi-normal modes (QNMs) [28]. For\nthe CDOS we \fnd that [5, 29]\n\u001a12=X\nm\rm\n2\u0019Re[(^u1\u0001em(r1)) (^u2\u0001e\u0003\nm(r2))]\n(!\u0000!m)2+\r2m=4:(18)\nThis expression shows explicitly that the CDOS accounts\nfor the phase di\u000berence of each \feld mode embetween\nthe points r1andr2. In the particular case of a single-\nmode high- Qcavity, the CDOS factorizes in the form\n\u001a2\n12=\u001a11\u001a22[22]. For a source made of two coherent\ndipoles located at positions r1andr2in a single mode\ncavity, we \fnd from Eq. (16) that the spontaneous decay\nrate is\n\u0000\u0018jpj2[\u001a11+\u001a22\u00062p\u001a11\u001a22cos\u001e]: (19)\nThe \feld mode in this case is a standing wave with real\namplitude, and the plus or minus sign in the interference\nterm corresponds to ^u1\u0001em(r1) and ^u2\u0001em(r2) having\nidentical or opposite signs (which corresponds to a posi-\ntive or negative CDOS \u001a12).\nIn order to illustrate the relevance of Eq. (19) in a\nrealistic situation, we consider a single mode contribu-\ntion in the well-known L3 photonic crystal cavity [33].4\nThe cavity is formed by a line of three missing holes in\na triangular-lattice pattern of air holes on a slab with\nrefractive index n= 3:48 and thickness L= 200 nm.\nCalculated maps of the amplitudes Ey(r) = Re[ ^y\u0001em(r)]\nandEx(r) = Re[ ^x\u0001em(r)] of the fundamental mode\nm, obtained from a FDTD simulation, are displayed in\nFig. 1(a) and 1(b).The positions of the photonic crystal\nholes, designing the L3 cavity, are indicated as circles.\nThe fundamental mode mis characterized by a LDOS\nwith a Lorentzian lineshape (not shown), with a reso-\nnance wavelength \u0015m= 1270 nm and a quality factor\nQm=!m=\rm'2000. We focus on the emission by\ntwo coherent electric dipoles placed either at the points\n\u000bindicated in Fig. 1(a), or at the points \findicated\nin Fig. 1(b). We show in Fig. 1(c) [resp. Fig. 1(d)] the\nPurcell enhancement \u0000 =\u00000for two dipoles with identi-\ncal amplitude and phase di\u000berence \u001e, and oriented along\nthexdirection [resp. ydirection]. The black curve\ncorresponds to two dipoles in phase ( \u001e= 0), and the\nred curve to two dipoles with opposite signs ( \u001e=\u0019).\nThe decay rate is obtained from a FDTD simulation, in\nwhich the emitted power is calculated from the net \rux\nof the Poynting vector through a box containing the two\ndipoles, and having spectral linewidth much larger than\nthe linewidth of the cavity mode. For two dipoles placed\nat positions \u000b[Fig. 1(c)], that are connected by a nega-\ntive CDOS since the \feld amplitudes have opposite signs,\nwe observe a subradiant emission for \u001e= 0 and perfect\nsuperradiance for \u001e=\u0019(we have veri\fed that the emis-\nsion rate in this case is twice that obtained for a single\ndipole). For two dipoles placed at positions \f[Fig. 1(d)],\nconnected by a positive CDOS, we observe superradi-\nance for\u001e= 0 and subradiance for \u001e=\u0019, as expected.\nNote that the distances between the two points \u000band\f\nare \u0001r= 320 nm and \u0001 r= 480 nm, respectively, that is\ncomparable or larger than \u0015=n, with\u0015the emission wave-\nlength in vacuum. In bulk materials, super or subradiant\ne\u000bects would be observable only for \u0001 r\u001c\u0015=n. The sim-\nulations clearly con\frm that the photonic mode imposes\nthe constructive or destructive interference, through the\nsign of the CDOS, in agreement with Eq. (19).\nNext, in order to mimic the emission by a one-\ndimensional extended exciton (as found for example in\na carbon nanotube or a quantum wire), we consider a\nline source with length d(hereafter denoted by emitting\nnanowire). In the FDTD simulation, the source model\ncorresponding to the emission by a single exciton is a\nlinear cluster of Nelementary and mutually coherent\ndipoles with amplitude p=p\nN(as already pointed out\nthe source is spatially coherent for a single exciton). In\norder to focus on the in\ruence of the environment, we\nconsider the situation where all elementary dipoles are\nin phase (but the approach allows one to use a more re-\n\fned source model). The emitting nanowire is embedded\nin the L3 cavity, and we can expect very di\u000berent radia-\ntive rates depending on its location and orientation. As\na case study, we consider a nanowire oriented along the\nxdirection, polarized along the ydirection, and located\nFigure 1. Emission by two dipoles in a L3 photonic crystal\ncavity on a slab. The slab has a refractive index n= 3:48\nand a thickness L= 200 nm. The photonic crystal is made of\nholes with a diameter d= 180 nm and a lattice constant a=\n320 nm. (a) and (b): Field amplitudes Ex(r) = Re[ ^x\u0001em(r)]\nandEy(r) = Re[ ^y\u0001em(r)] of the cavity mode em(r). The\n\feld is calculated in a plane coinciding with the middle of the\nslab. The points \u000band\findicate the positions of the two\nemitting dipoles. The locations of the photonic crystal holes\nare also highlighted. (c) and (d): Spectrum of the normalized\nspontaneous decay rate \u0000 =\u00000(or Purcell enhancement) of two\ncoherent dipoles with identical amplitudes, phase di\u000berence\n\u001e, located at positions \u000band oriented along the xdirection\n(panel c), or located at positions \fand oriented along the y\ndirection (panel d). Black line: \u001e= 0. Red line: \u001e=\u0019.\nThe curves corresponding to subradiant emission have been\nmultiplied by a factor of 103(panel c) and 104(panel d) for\nthe sake of visibility.\nat the center of the L3 cavity. The emission wavelength\n\u0015= 1270 nm is on resonance with the cavity mode, and\nthe nanowire can be seen as immersed in the \feld dis-\ntribution shown in Fig. 1(b). We show in Fig. 2(a) the\ncalculated radiative rate versus the nanowire length d\nin the L3 cavity (green circles), and in a homogeneous\nmedium with the same refractive index n= 3:48 as the\ncavity slab (blue triangles). The radiative rate in the\nhomogeneous medium increases with the exciton size, in\nagreement with the result already found in Ref. [9], up to\na maximum reached for a size d=\u0015=2n. Interestingly, a\nvery similar trend is observed for the nanowire in the L3\ncavity.\nTo address this length dependence in more details, we\nshow in Fig. 2(b) the Purcell enhancement \u0000 =\u00000in the\nL3 cavity (red dots) versus the emitting nanowire length.\nIn the same \fgure we also display the \feld amplitude of\nthe cavity mode at the extremities of the nanowire (blue\ndashed line). We see that the Purcell enhancement is\nalmost constant for d\u0014300 nm, i.e. as long as the\nnanowire is inside the central hot spot of the \feld ampli-\ntude seen in Fig. 1(b). In this case, the CDOS remains\npositive over the emitting nanowire extent (the \feld am-\nplitude of the mode does not change sign from the cen-5\nFigure 2. (a) Comparison of the spontaneous decay rate of\nan emitting nanowire polarized along the ydirection and ex-\ntended along the xdirection in the L3 cavity (green circles),\nand in a homogeneous medium with refractive index n= 3:48\n(blue triangles), versus the nanowire length d. In the case of\nthe L3 cavity, the center of the nanowire coincides with the\ncenter of the cavity. (b) Purcell enhancement \u0000 =\u00000versus\nthe emitting nanowire length (red dots). The \feld amplitude\nEy(r) = Re[ ^y\u0001em(r)] of the cavity mode at the extremities\nof the emitting nanowire is also indicated (blue dashed line).\nUp to a length d'300 nm, the center and the extremities of\nthe nanowire are connected by a positive CDOS (the \feld am-\nplitude is positive at the center and at the extremities). For\nd&300 nm, the center and the extremities of the emitting\nnanowire are connected by a negative CDOS, and destructive\ninterferences reduce the spontaneous decay rate. The mini-\nmum is observed for d'560 nm.\nter to the extremities of the nanowire). Constructive in-\nterferences between the emissions from di\u000berent parts of\nthe nanowire lead to a substantial Purcell enhancement.\nHowever, for d&300 nm, the center and the extremities\nof the emitting nanowire are connected by a negative\nCDOS (the \feld amplitude of the mode is positive at the\ncenter of the nanowire and negative at the extremities,\nsince they overlap with the negative spots indicated as \f\nin Fig. 1(b)). Destructive interferences reduce the spon-\ntaneous decay rate, leading to a minimum of the Purcell\nenhancement for d'560 nm. This subradiance of the\nemitting nanowire arises from the dominance of negative\nCDOS contributions, induced by the spatial structure of\nthe underlying cavity mode.\nB. Coupled cavities with non-Lorentzian spectrum\nIn addition to generating subradiance or superradi-\nance, the CDOS contribution in the interferometric emis-\nsion process may also in\ruence the spectral lineshape.\nAn interesting situation is that of two coupled photonic\ncavities generating a LDOS with a non-Lorentzian spec-\ntrum, which can be achieved by selecting two L3 cavities\nwith unbalanced losses [34]. This situation goes beyond\nthe high-Qmodel used above, and the expansion (18) for\nthe LDOS and CDOS has to be replaced by a more gen-\neral expansion in QNMs (for an introduction to QNMs in\nphotonics, see for example [32]). Considering that eachcavity can be described by a single QNM, the Green's\nfunction takes the form [28, 29]\nG(r;r0;!)'c2\n2!\"~Ea(r)\n~Ea(r0)\n~!a\u0000!+~Eb(r)\n~Eb(r0)\n~!b\u0000!#\n:\n(20)\nHere ~Eaand~Ebare QNMs corresponding to cavity aand\nb, respectively, with ~ !aand ~!bthe associated complex\nresonance frequencies. A major di\u000berence with Eq. (17)\nis that the expression above is not limited to the low-loss\n(or high-Q) regime.\nComing back to the simple source model described\nby Eq. (14), with two coherent dipoles at positions r1\nandr2, we can derive the expressions of the LDOS and\nCDOS entering Eq. (16). De\fning !m= Re~!mand\n\rm=\u00002Im~!m, withm2fa;bg, the result can be cast\nin a form involving Fano pro\fles of the form\nF(m;q;! ) =\rm=2\nq2+ 1\u0014(q2\u00001)\rm=2 + 2q(!\u0000!m)\n(!\u0000!m)2+\r2m=4\u0015\n;\n(21)\nwhereqis a parameter de\fning the resonance line-\nshape [35]. Introducing the phases of the QNMs at the\nsource points 'm\ni= arg[ ^ui\u0001~Em(ri)], withi2 f1;2g,\nand the parameters qm\ni= tan('m\ni),qm\n12= [tan('m\n1) +\ntan('m\n2)]=2, a simple calculation leads to\n\u001a11\u0018F(a;qa\n1;!) +F(b;qb\n1;!) (22)\n\u001a22\u0018F(a;qa\n2;!) +F(b;qb\n2;!) (23)\n\u001a12\u0018F(a;qa\n12;!) +F(b;qb\n12;!): (24)\nWe \fnd that the spectrum of the LDOS and CDOS is a\nsum of Fano lineshapes, each arising from one QNM ( a\nandb), withqparameters depending on the location of\nthe source points (1 and 2). These expressions reveal the\ncomplexity of the prediction of the spontaneous emission\ndynamics of an extended source in a structured environ-\nment, that depends on the LDOS and the CDOS, and\non the spatial distribution of the phase of the QNMs. In\nparticular, it is interesting to note that the spectrum of\nthe decay rate cannot be predicted solely from the knowl-\nedge of the spectrum of the LDOS at each point. The\nrelative phases 'm\nibetween di\u000berent source points, im-\nposed by the electromagnetic QNMs, play a crucial role\non the lineshape of the emission spectrum.\nTo illustrate this result, we have carried out numeri-\ncal simulations in the system depicted in Fig. 3(a), made\nof two coupled photonic-crystal cavities with unbalanced\nlosses. The holes with di\u000berent colors have di\u000berent radii,\nthe green holes being designed to tailor the losses, and the\npale blue hole being designed to reduce the frequency de-\ntuning between the two cavities. Unbalanced losses lead\nto highly non-Lorentzian LDOS and CDOS spectra (not\nshown), resulting from the superposition of two strongly\nasymmetric Fano contributions according to Eqs. (22-24).\nAs a result, we can expect the spectral pro\fle of the spon-\ntaneous decay rate of a point dipole source to be very dif-\nferent from that of an extended exciton. This is clearly6\nseen in Fig. 3(b), where we show the calculated spectrum\nof the spontaneous decay rate \u0000 of a point dipole (black\ndashed line) and an emitting nanowire (red dashed line).\nBoth sources are polarized in the xdirection, and cen-\ntered on the position indicated by the red dot in Fig. 3(a).\nThe nanowire has a length d= 300 nm and is oriented\nalong thexdirection. We observe that the CDOS con-\ntribution not only produces a subradiant e\u000bect reducing\nthe decay rate to very small values, but also induces a\nsubstantial reshaping of the spectral pro\fle. This exam-\nple clearly demonstrates that the emission lineshape in\nthe Purcell e\u000bect of an extended source cannot be pre-\ndicted from the sole knowledge of the LDOS spectrum at\nall points.\nFigure 3. (a): Sketch of the system made of two coupled\nphotonic-crystal cavities on a slab, with the same thickness\nand refractive index as in Fig. 1. The white holes have a\ndiameter of 180 nm, the green holes of 140 nm and the blue\nhole of 150 nm. (b): Spectrum of the decay rate for a point\ndipole source (black dashed line) and an emitting nanowire\n(red dashed line). The red cross in the left cavity in panel\n(a) indicates the location of the center of the source, that\ncoincides with one of the \u000bpoints indicated in Fig. 1(a). Both\nsources are polarized along the xdirection. The emitting\nnanowire has a length d= 300 nm and is oriented along the\nxdirection.IV. CONCLUSION\nIn summary, we have studied theoretically the change\nin the spontaneous decay rate (or Purcell e\u000bect) of an\nextended emitter in a structured environment, empha-\nsizing the role of the CDOS in driving interferences in\nthe emission process. Beyond the generation of large ef-\nfective oscillator strengths, that had been put forward\nand exploited in previous studies, we have demonstrated\nthat a structured CDOS (in space and/or spectrum) may\nproduce subradiance or superradiance, and change sub-\nstantially the emission spectrum. These results, based\non a simple but robust theory, have been highlighted by\nnumerical simulations in realistic photonic-crystal cavity\ngeometries. The study clearly shows that a knowledge of\nthe LDOS spectrum at all points does not permit to pre-\ndict the change in the decay rate, neither in value nor in\nlineshape. It also emphasizes the importance of the engi-\nneering of both the LDOS and the CDOS for the control\nof spontaneous emission by extended sources.\nACKNOWLEDGMENTS\nWe thank P. Lalanne and K. Vynck for helpful discus-\nsions at the initial stage of this study. 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Department of Physics  \n     Arizona State University  \n     Tempe  \n      AZ 85287 -1504  \n2.   School For Engineering of Matter Transport and Energy  \n     Arizona State University  \n     Tempe  \n      AZ 85287 -6106  \n3.  Fraunhofer Institute for Mechanics of Materials IWM  \n     Wöhlerstraße 11 \n     79108 Freiburg  \n     Germany  \n \nAbstract  \n \nSince it is now possible to record vibrational spectr a at nanometer scales in the electron \nmicroscope it is of interest to explore whether defects such as dislocations or grain \nboundaries will result in measurable changes  of the spectra .  Phonon densities of st ates \nwere calculated for a set of high angle gra in boundaries in silicon. Since these boundaries \nare mode led by supercells with up to 160 atoms , the density of states was calculated by \ntaking the Fourier transform of the velocity -velocity autocorrelation function from  \nmolecular dynamics simulations  based on  new supercells  doubled in each direction . In \nselect cases t he results were checked  on the original super cells with fewer atoms by \ncomparison with  the densities of states obtained by diagonalizing  the dynamical matrix \ncalculated using  densi ty funct ional theory. Near the core of the grain boundary the height \nof the optic phonon peak in the density of states at 60 me V wa s suppressed relative to \nfeatures due to acoustic phonons that are largely unchanged  relative to their bulk values .  \nThis can be attributed to the variation in the strength of bonds in grain boundary core \nregions where there is  a range of bond lengths.  It also means that  changes in the density \nof states intrinsic to  grain boundaries are unlikely to affect thermal con ductivity at \nambient temperatures, which are most likely dominated by the scattering of acoustic \nphonons.  \n \n \nIntroduction  \n \nRecent developments in monochromators and electron energy loss spectrometers for \ntransmission electron microscopy have made it possib le to record vibrational spectra, \npotentially at nanometer resolution, in the electron microscope1.  Atomic resolution  2 \nimages at energies corresponding to vibrational modes or phonons have been \ndemonstrated in BN2 and Si3, by selecting phonon wave vectors corresponding to \nlocalized or impact scattering.  This raises the intriguing possibility of experimentally \nmeas uring localized phonon modes at extended structural defects in crystals and using \nmeasured local densities of states to investigate how defects may change important \nmacroscopic thermal transport properties such as thermal conductivity.  Localized modes \nassociated with a single s ilicon atom in a graphene sheet have been observed by Hage et \nal4, and a change in phonon de nsity of states corresponding to a stacking fault in 3C SiC \nwas reported by Yan et al5.   \n \nLocalized modes associated with a point defect, namely an isotope atom with a differen t \nmass, are to be expected, though  the spectacular result of Yan et al 5 showing an \nobservable change at a stacking fault in 3C SiC could be as well a consequence of the \nstacking fault being a 2 layer region of 2H SiC,  a different polytype.  Although grain \nboundaries or dislocations should give rise to differences in the phonon  density of states \n(DOS ), it is still not certain whether they would be detectable  unequivocally .  The energy \nresolution of the monochromator spe ctrometer combination in the electron microscope is \nabout 5 meV or 40 cm-1; this is a significant advance on what as been  achieved in the \npast, but still very much inferior  to what is possible with infra -red absorption or Raman \nspectroscopy on bulk specime ns.   \n \nZiebart h et al6 and St offers et al7 performed first -principles  calculations, using density \nfunctional theory  (DFT) , to determine the atomic  positions in supercells representing \nvarious gra in boundaries in Silicon. For a 9 boundary Stoff ers et al7 also compared the  \nDFT  results with high resolution scanning transmission electron microcopy images. Since  \nVenkatraman et al3  have shown that it is possible to mea sure the phonon scattering at the \natomic scale in Silicon us ing an on -axis detector, any local changes at grain boundaries \nmay be experimentally detectable.  \n \nWe used the atomic coordinates of the structurally relaxed grain -boundary models of \nZiebart h et al6 and Stoffers et al7 to calculate the phonon DOS  in two ways: by \nenumerating the modes obtained by diagonalizing the dynamical matrix , calculated \ndirectly by DFT ; by taking the Fourier t ransform of the velocity -velocity correlation \nfunction of the  atoms in an extended molecular dynamics (MD) simulation.  In \ncalculating the DOS  we made a distinction between those atoms at the grain boundary \ncore and those  that were in regions that can  be considered as similar to the bulk Si crystal .  \nIn all cases except th e  5 (130) boundary  the peak in the DOS at 60 meV was depressed \nfor atoms in the core region.  We attribute the anomalous result for the  5 (130) \nboundary  to the small s ize of the supercell which means  that the environment of all the \natom s is more like t hat of the grain boundary core  than that of the bulk crystal.  \n \nTheory  \n \nPhonons are the quantized vibrations of the atoms in the crystal lattice;  they are uniquely \ndetermined  by their wave  vector in the 1st Brillouin Zone and their polarization  vector .  \nSince the grain boundary systems that we are studying are represented by large  3 \nsupercells , the Brillouin Zone is relatively small, and as a first approximation the density \nof states need s only to be evaluated by enumerating the frequencies at the  point ( or \ntypically only  a small number of k points)  8.  The frequencies are obtained by \ndiagonalizing the dynamical matr ix D \n \n𝐷(𝑘\n𝑏𝑏′\n𝛼𝛼′)=𝑒−𝑖𝐤.𝐑𝐛′\n√𝑀𝑏′𝜙(𝑏𝑏′\n𝛼𝛼′)𝑒𝑖𝐤.𝐑𝐛\n√𝑀𝑏      1 \n \nwhere b labels atoms of mass Mb in the simulation cell with periodic boundary conditions , \n labels the polarization and  is the force -constant matrix , and k is the wave  vector.  \nThe number of modes is 3 N where N is the number of atoms in the unit cell.  The original \nsupercells that represent our grain boundaries contain up to 160  atoms, thus the \ndynamical matrix can be of size  up to  480 x 480 .  All DFT  calculations were done using \nthe projector -augmented -wave method as implemented in the plane -wave code VASP 9-11 \nwithin the gener alized gradient approximation of Perdew , Burke  and Ernzerhof for \nexchange -correlation .12 A plane -wave energy cutoff of 400 eV was used. A k-point \nsampling mesh with at least 10 points per Å-1 was used for the supercells together with a \n0.01 eV smearing width of the Gaussian smearing scheme. The grain -boundary (GB) \nmodels of Ziebarth et al6 and Stoffers et al7 were re -relaxed using the aforementioned \nparameters. These settings were sufficient to converge all five GB cells to a total force \nper atom of less than 0.05 eV/ Å or better and a n energy tolerance of 10-5 eV. A \ncompilation of the lattice parameters of the cells and the number s of atoms is given in \nTable 1.  Phonon calculations were performed using the direct finite -difference \nmethod13,14 implemented  within the Phonopy package 15. To obtain the projected phon on \ndensity of states, a 1x1x1  supercell approach was used fo r all GB cells with a 10x10x10 \nk-point sampling mesh density. Finite ato mic displacements of 0.01 Å were applied to \ncalculate the force -constant matrix . The atomic displacements of the phonon  modes, \nobtained from the eigenvectors of the modes at the Gamma point, were simulated using \nthe v_sim code  (https://gitlab.com/l_sim/v_sim ) \n \n \nTable 1      Compilation of number s of atoms and unit cell parameters of the  supercells  for \nthe five investigated grain boundaries . The “large” MD supercells contain 23=8 times as \nmany atoms and have unit-cell parameters twice as long.  \n \n 3 (112) \nMirror -\nsymmetric  3 (112)  \nNon- \nsymmetric  5 (130)  5 (120)  9 (221)  \n# of atoms  in \nDFT supercell  144 136 40 160 68 \n# of atoms in \nMD supercell  1152  1088  320 1280  544 \na (Å)  9.479 0 9.4336  5.3335  12.3261  11.6531  \nb (Å)  40.5757  38.7540  17.6788  24.5705  31.0746  \nc (Å)  7.6715  7.6715  8.7036  10.8160  3.8576  \n  4 \n \n \n \n \n \n \nFigure 1   The original DFT s upercells for the various grain boundaries.  The atoms at the \ngrain boundary core are shown as open red circles  and the “bulk” atoms are shown as \nfilled blue circles . \n \n \nAn alternative method to calculate the phonon DOS is to take the F ourier transform of the \nvelocity -velocity corre lation function from MD trajectories  of the atoms16,17 ,18.  The \ninstantaneous velocities are determined by the difference s in the atom positions in the \ntime evolution of the MD trajectories.   In practice it was found to be beneficial to break \nthe trajectory  into L time slices  of M steps and average over the resulting spectra.   The \nphonon density of states  I(k) is  \n \n \n𝐼(𝜔𝑘)=1\n𝐿∑∑ 𝑒𝑥𝑝(𝑖𝜔𝑘𝑡𝑗)∑∑𝑣𝑖𝑛𝑛=𝑁\n𝑛=1𝑖=𝑝\n𝑖=1 (𝑡𝑗)𝑣𝑖𝑛(𝑡2)𝑒𝑥𝑝(−𝜎2𝑡𝑗2\n4)𝑗=(𝑙+1)𝑀\n𝑗=2+𝑙𝑀𝑙=𝐿−1\n𝑙=0  \n \n          2 \nwhere  vik(tj) is proportional to the velocity at time tj \n 5 \n \n𝑣𝑖𝑘(𝑡𝑗)=𝑢𝑖𝑘(𝑡𝑗)−𝑢𝑖𝑘(𝑡𝑗−1) \n           3 \nand M is the number of time steps  in each time slice l,  tj is the jth time step, N the number \nof atoms, k is a frequency,  ui is the coordinate in the i’th direction (x,y,z) for atom  k.  \nFor the total density of states it is summed over dimensions x,y and z (p=3).  The \nparameter  is related to the instrumental broadening, E by \n \n𝜎=Δ𝐸𝑒\n2ℏ√𝑙𝑛2          4 \n \nwhen E is in eV.  \n \n \nThis approach was proposed by Beeman and Alben 19 and has been used to calculate the \nthermal conductivity of MgO by de Koker20 , phonon dispersions and lifetimes in carbon \nnanotubes by Thomas et al,  21 and infra red spectra of large proteins by Mott et al  22. \n \nScattering wave vectors i n electron  energy loss  spectroscopy (EELS) in the electron \nmicroscope are predominately in the plane of the specimen, normal to the incident beam.  \nFor densities of states that would be observed by EELS in the electron microscope the \ndisplacement direction, i, is only summed over the 2 dimensions i n the plane of the \nspecimen (p=2).   The specimen ’s surface plane  normal for the  3 and 9 boundaries in \nFig 1 is (110 ) , which is the most frequently used orientation for  high-resolution electron \nmicroscopy of silic on23,24.  There have been fewer high -resolution electron microscopy \nresults for the (100) orientation which is the specimen normal for the  5 boundaries25. \n \nMolecular dynamics simulations were performed with the LAMMPS 26 package using the \nphonon fix command that measures the elastic Green’s function  by making use of the \nfluctuation -dissipation theorem implemented by Kong  27 to obtain the phonon spectrum .  \nThe positions of atoms in the supercells of grain boundaries determined by Ziebarth et al \n6 and Stoffers et al 7  were used as the initial atom coordinates.  In practice it was found \nthat their supercells were too small  for calculating the phonon DOS .  Various multiples of \nthe supercell were tried, and it was found that convergence was achieved with a la rger \nsupercell containing the original supercells 2 x 2 x 2 times .  The number s of atoms in \nthese larger supercells are given in the 2nd line of Table 1.  These simulations were \nperformed with periodic boundary conditions applied in all directions. The interaction \namong atoms was described using the Tersoff potential  28. The Nosé–Hoover style \nthermostat and barostat 29-32 were employed to maintain the system at 300 K and 1 \natmosphere. A time step of 1 fs was used with a total simulatio n time of 0.05 ns . The \natom positions were saved at each time step. These computational settings were sufficient \nto obtain a stable spectrum. Additional convergence checks were performed for \nsimulation times of 0.1 ns .  For calculating th e spectra as given  by equations 2 and 3 , the \ntrajectories were divided into 24 blocks of 2048 time steps and the results were averaged .  \nAll spectra were normalized using the constraint that there are 3 vibrational modes per \natom.   6 \n \nCalculating phonon density of states by DFT and Phonopy for the supercells shown in \nFig 1 is computer resource  and time  intensive . This was only done for select ed cases as a \nway of checking and validating the results from the analysis of molecular dynamics \ntrajectories  by direct comparison with  the calculation of the phonon spectrum from the \ndynamic al matrix . \n \n \nResults  and Discussion  \n \nTo validate the molecular dynamics approach , in particular the Tersoff potential  we used, \nwe compared the phonon density of states calculated b y both molecular dynamics and by \ndiagonalizing the dynamical matrix using VASP in association with Phon opy.  The result \nfor a perfect crystal is shown as Fig 2.  The most prominent feature is the peak in the \nDOS at 60 meV. Silicon has an optic mode since there are two atoms in each pri mitive \nunit cell.  Th ere is little dispersion with wave  vector for optic modes, which means they \nmake a large contribution to the density of states.  The peaks and their relative heights are \nthe same , though the peaks in the D OS calculated using molecular dynamics trajectories \nwith the Tersoff potential are shifted to slightly higher energies. This is not unexpected, a \nsimilar shift was reported by Rohskopf et al33. However t he strong similarity  between the \nresult s of the two different approaches suggests that taking the Fo urier transform of the \nvelocity -velocity correlation function of instantaneous velocities from MD trajectories is \na valid approach, though it should be remembered that the Tersoff potential  is constructed \noriginally for the perfect crystal structure and might be of limited valid ity for the \ndistorted  structures at grain boundaries.    \n \n   \n \nFig 2  Phonon density of states for perfect Si : (a) calcu lated  with DFT (using VASP  and  \nPhonopy ) (b) calculated from  molecular dynamics trajectories  and the Tersoff potential.  \n \n  \n 7 \n \n  \n \n   \n \n \na 3 (112) mirror -symmetr ic GB,  MD  (b)   3 (112) non-symmetr ic GB,   MD  \n          total DOS      total DOS  \n \n              \n \nc 3 (112) mirror -symmetr ic GB,   DFT total DOS  \n \n 8 \n \nd 3 (112) mirror -symmetric GB, MD   (e)   3 (112) non -symmetric GB,  MD  \n          EELS  DOS       EELS  DOS  \n \n \nFigure 3    Phonon densities of states for   3 (112)  grain b oundaries . Graphs (a), (b) and \n(c) display the total DOS, graphs (d) and (e) illustrate  the partial DOS corresponding to \nEELS s pectra.  \n \n \n  \n 9 \n   \n     \n         (a)5 (130) GB,  MD total DOS                        (b) 5 (12 0) GB, MD total DOS  \n \n \n             (c)  5 (13 0) GB, DFT total DOS  \n 10 \n \n         (d)5 (130) GB,  MD EELS DOS                (e) 5 (120) GB, MD EELS DOS  \n \n \nFigure 4       Phonon densities of states for   5 grain b oundaries . Graphs (a), (b) and (c) \ndisplay the total DOS, graphs (d) and (e) illustrate the partial DOS corresponding to \nEELS spectra.  \n \n \n \n  \n 11 \n \n \n(a)  9 (221) GB, MD total DOS  \n \n \n(b)  9 (221) GB, DFT total DOS  \n 12 \n \n \n(c)  9 (221) GB, MD EELS DOS  \n \nFigure 5  Phonon DOS results for  9 (221) grain b oundary .  Graphs (a) and (b) display \nthe total DOS, graph (c) illustrates a partial DOS corresponding to an EELS spectrum.  \n \n \n \nFigures 3 -5 display comparisons between the  total phonon density of states and the \npartial density of states that would be observed by EELS for atoms that are part of a grain \nboundary ( GB core atoms ) and those that are more distant from the boundary (bulk \natoms ).  The core  and bulk atoms are marked as open red and filled blue circles \nrespectively in Figure 1.  In all the graphs the density of states is broadened by a \nGaussian of 5 meV half width to represent the  instrumental  energy resolution that can \nnow be achieved  in a transmission electron micros cope .  Given that it is possible to \nrecord spectra from regions as small as 0.5  nm, it should be possible to acquire distinct \nspectra representative of the core and bulk regions.  \n \nIn all cases the most significant and observable change is the reduction of the he ight of \nthe optical -phonon peak in the DOS  at 60 meV.  The magnitude of the reduction in peak \nheight is different for the different boundaries.  It is greatest for the  3 (112)  mirror -\nsymmetr ic boundary, and hardly noticeable for the 5 (13 0) bounda ry. The effect for the \nboundary with the lowest tilt angle (or highest sigma value) considered, the  9 (221) \nboundary , is somewhere in between that for the  3 (112) mirror-symmetr ic boundary \nand the 5 (13 0) boundary.  It is interestin g to note that the  projected DOS  that would be \nobserved by EELS in the electron microscope is very similar to the total DOS .  The \nmagnit ude of the DOS in  states/atom/meV is 2/3 of the 3D DOS , as EELS selects a 2D \nprojection.  We perform ed DFT ( VASP & Phonopy ) calculations for the  3 (112)  \nmirror -symmetr ic boundary , the 5 (130) boundary and the 9 (221) boundary.  These \nwere intentionally chosen to span  the range of changes in peak heights obtained by the \n 13 \nMD calculations. Although the exact value s of the he ight reduction in the 60 meV peak \ndiffer between the DFT  and MD calculations, most noticeably in the case of the 5 (13 0) \nboundary, the same trend is apparent.  \n \nThe magnitude of the peak at 60 meV is controlled by the angle of the boundary and the \nCoincidence Site Lattice  misorientation , as measured by the parameter .  The nature of \nthe boundary plane can have a significant effect as shown by the difference between the \npeak height for the  5 (130) and (120) boundaries and between the  3 (112) boundaries \nwith and without mirror symmetry.   However for the  5 (130) boundary the separation \nbetween the boundaries in the supercell was so small that most of the atoms are part of \nthe boundary core s, as can be seen in Fig 1.  Interatomic distances and angles vary \nconsiderably in the core reg ions of the grain boundary , and this is reflected in the \ninteratomic potential and the magnitude of the interatomic forces.  This is very evident \nfrom the map of displacements for the mirror symmetric  3 (112) and the 9 (221) \nboundaries shown as Figs 6a and b, where the displacements are much smaller for atoms \nnear the boundary core.    \n \n \n(a) Displace ments for the 59  meV mode f or the mirror symmetric  3 (112) boundary  \n \n(b) Displacements for the 58  meV mode for the 9 (221) boundary  \n \n \n 14 \n \n \n(c)     Displacements for the 63 meV mode for the 9 (221) boundary  \n \n \nFigure  6 Atomic displacements for modes predominately affecting bulk like atoms for \n(a) the  3 (112) boundary and (b) the  9 (221) boundary,  (c) is a mode localized on \nboundary core atoms of the 9 (221) boundary.  The blue lines mark the locations of \nthe boundary planes  in the supercells . \n \n \nIt is even more striki ng in the animations shown as movies S1 and S2 in Supplementary \nInformation.  Since the magnitude of the DOS is related to the size of the atomic \ndisplacements  as can be seen from equations 2 and 3  it is not surprising that the 60 meV \noptic phonon peak is reduced in both the total DOS and the projected DOS observed by \nEELS.  There are relatively few modes localized to atoms at the core of the boundaries.  \nThere is a  single mode at 63 meV for the 9 (221) boundary with the displacements \nshown as Fig 6c. The  animation is shown as movie S3 in the Supplementary Information.  \nHowever it is unlikely that this single mode would be separately detected when the \ninstrumental resolution is 5 meV.  \n \nConclusions  \n \nPhonon densities of states in silicon calculated from velocity-velocity correlation \nfunctions of molecular dynamics trajectories (using a Tersoff potential ) were validated by \ncomparison  with those calculated directly from  the dynamical matrix  by density \nfunctional theory (using VASP and Phonopy) .  This agreement is not only true for perfect \ncrystals but also for extended structural defects , namely grain boundaries,  where there \nmight be concerns about the applicability of simple force fields.  The phonon density of \nstates due to optic phonon s at 60 meV for 3 to 9 high-angle tilt boundaries is reduced \ncompared to the  optic phonon  density of states in regions of bulk crystal structure .  This \n 15 \nchange should be detectable  by high resolution EELS in electron microscope s in a \nspecimen where the boundary  plane  is perpendi cular to the specimen surface , and the \nelectron beam runs parallel to the boundary core .  The similarity between the densities of \nstates from acoustic phonons at the boundary core and in bulk regions indicates that local \nchanges of densities of states  at the grain boundaries cores should have minimal effect on \nthe thermal conductivity  of a polycrystalline microstructure of Si at ambient temperature .  \nAny changes in thermal conductivity will be a consequence of phonon scattering at the \ngrain boundary lea ding to a reduction in phonon mean free paths.  \n \n \n \nAcknowledgements  \n \nT. B. and A. S.  are supported  in part  by the Arizona State University start -up funds . A. S.  \nis also funded by the NSF DMR grant  #1906030  and as part of ULTRA, an Energy \nFrontier Research Center funded by the U.S. Department of Energy (DOE), Office of \nScience, Basic Energy Sciences (BES), under Award # DE -SC002123 0. 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Mater.  3, 27, \ndoi:10.1038/s41524 -017-0026 -y (2017).  \n "    },    {        "title": "2109.06728v1.Learning_Density_Distribution_of_Reachable_States_for_Autonomous_Systems.pdf",        "content": "Learning Density Distribution of Reachable States for\nAutonomous Systems\nYue Meng\nMIT\nUnited States\nmengyue@mit.eduDawei Sun\nUIUC\nUnited States\ndaweis2@illinois.eduZeng Qiu\nFord\nUnited States\ncqiu1@ford.com\nMd Tawhid Bin Waez\nFord\nUnited States\nmwaez@ford.comChuchu Fan\nMIT\nUnited States\nchuchu@mit.edu\nAbstract: State density distribution, in contrast to worst-case reachability, can\nbe leveraged for safety-related problems to better quantify the likelihood of the\nrisk for potentially hazardous situations. In this work, we propose a data-driven\nmethod to compute the density distribution of reachable states for nonlinear and\neven black-box systems. Our semi-supervised approach learns system dynamics\nand the state density jointly from trajectory data, guided by the fact that the state\ndensity evolution follows the Liouville partial differential equation. With the help\nof neural network reachability tools, our approach can estimate the set of all pos-\nsible future states as well as their density. Moreover, we could perform online\nsafety verification with probability ranges for unsafe behaviors to occur. We use\nan extensive set of experiments to show that our learned solution can produce a\nmuch more accurate estimate on density distribution, and can quantify risks less\nconservatively and flexibly comparing with worst-case analysis.\nKeywords: Reachability Density Distribution, Learning Density Distribution, Li-\nouville Theorem\n1 Introduction\n(a)t= 0s.\n (b)t= 0:35s.\n (c)t= 0:80s.\nFigure 1: Reachability density of Van der Pol.Reachability analysis has been a central topic\nand the key for the verification of safety-critical\nautonomous systems. The majority of existing\nreachability approaches either compute worst-\ncase reachable sets [1, 2, 3, 4, 5, 6], or use\nMonte Carlo simulation to estimate the reach-\nable states with probabilistic guarantees [7, 8,\n9, 2]. There are several obvious disadvantages\nof such methods. 1). Worst-case methods, especially those for nonlinear systems, often over-\napproximate the reachable sets and produce very conservative results (due to convex set representa-\ntions [10, 11, 12, 13], wrap effects [14, 15], low-order approximation [3, 16], etc.), not to say they\nare usually very computationally expensive (also known as the curse of dimensionality). 2). Exist-\ning methods do not care about reachable state concentration and produce the same reachable state\nestimations for different distributions of initial states if those distributions share the same support1.\nThat is, current approaches do not compute which states are more likely to be reached. For example,\nin Fig. 1, the state density distribution of a Van der Pol oscillator evolves over time and concentrates\non certain states (the highlighted part in the figure), even if the initial states are uniformly distributed.\n1The probabilistic guarantees of the sampling-based methods do rely on the form of the initial states distri-\nbution. However, the final reachable sets estimate is the same for different distributions with the same support.\n5th Conference on Robot Learning (CoRL 2021), London, UK.arXiv:2109.06728v1  [cs.AI]  14 Sep 2021If one uses an existing worst-case reachability algorithm, most likely the results in Fig. 1 (b)(c) will\nshow almost the entire black region inside the highlighted part is reachable, as those methods often\nuse convex sets to represent the reachable sets.\nIn this paper, we aim to tackle the above problems and propose a learning-based method that can\ncompute the density distribution of the reachable states from given initial distributions. The state\ndensity is much more powerful than worst-case reachability and can better quantify risks. Our\nproposed method is based on the Liouville theorem [17, 18, 19], which is from classical Hamiltonian\nmechanics and asserts that the state density distribution function (which is the measurement of state\nconcentration) is constant along the trajectories of the system. Given an autonomous system _x=\nf(x)that is locally Lipschitz continuous, the evolution of the state density \u001a(x;t)(i.e. the density\nof statexat timet) is characterized by a Liouville partial differential equation (PDE). We learn\nthe state density as a neural network (NN) while respecting the laws of physics described by the\nnonlinear Liouville PDE, in a way that is similar to the physics-informed NN [20].\nFurthermore, we make two major improvements to make the learned density NN suitable for ver-\nification. 1). Instead of learning \u001afor a fixed initial distribution \u001a(x0;0), we learn the density\nconcentration function which specifies the multiplicative change of density from any \u001a(x0;0). 2).\nWe use a single NN to jointly learn both the reachable state \b(x0;t)and its corresponding density.\nMoreover, we use Reachable Polyhedral Marching (RPM) [21]—an exact ReLU NN reachability\ntool—to parse our learned NN as linear mappings from input polyhedrons to output polyhedrons.\nUsing such parsed polyhedrons, we can perform online forward and backward reachability analysis\nand get the range of density bounds [\u001amin;\u001amax]for each output polyhedron. Together, our method\ncan perform online safety verification by computing the probability of safety (instead of a single yes\nor no answer) under various initial conditions, even with unknown system dynamics f(i.e. a black-\nbox system where we only have access to a simulator of x(t)). In this way, it also has the potential\nto collect environmental data in run-time and update its distribution for online safety verification.\nWe conduct experiments on 10 different benchmarks covering systems from low dimension aca-\ndemic examples to high dimension black-box simulators equipped with either hand-crafted or NN\ncontrollers. Surprisingly, without using any ground-truth density data in the learning process, our\napproach can achieve up to 99.89% of error reduction in KL divergence with respect to the ground-\ntruth value, when compared to sampling-based methods like kernel density estimation and Gaussian\nProcesses. Moreover, our learned density concentration function can also be used for reachability\ndistribution analysis. We show that in several systems, more than 90% of the states can actually\njust reside in a small region (less than 10% of the volume of the convex hull for those states) in\nthe state space, which also points out the conservativeness of the worst-case reachability analysis\nmethods in terms of quantifying risks. We also show that different initial distributions can lead to a\ndrastic change in the safety probability, which can help in cases when unsafe is inevitable but can be\ndesigned to happen with a very low probability.\nOur major contributions are: (1) we are the first to provide an explicit probability density function\nfor reachable states of dynamical systems characterized as ordinary differential equations; this den-\nsity function can be used for online safety analysis, (2) we propose the first data-driven method to\nlearn the state density evolution and give accurate state density estimation for arbitrary (bounded\nsupport) initial density conditions without retraining or fine-tuning, and (3) we use a variety of ex-\namples to show the necessity to perform reachability distribution analysis instead of pure worst-case\nreachability, to flexibly and less conservatively quantify safety risks.\nRelated work Reachability analysis, especially worst-case reachability using sampling-based\nmethods or set propagation, has been studied for decades. The literature on reachability has been\nextensively studied in many surveys [1, 6, 22, 23]. Here we only discuss a few closely related works.\nHamilton Jacobian PDE has been used to derive the exact reachable sets in [24, 1, 5]. However,\nthe HJ-PDE does not provide the density information. Many data-driven approaches can com-\npute probabilistic reachable sets using scenario optimization [7, 25], convex shapes [8, 9, 26],\nsupport vector machines [27, 28], kernel embedding [29], active learning [30], and Gaussian\nprocess [2]. However, the probabilistic guarantees they provide are usually in the form that\nProb (x(t)2estimated set )>1\u0000\u000fwith enough samples, instead of the state density dis-\ntribution. [31] estimates human state distribution using a probabilistic model but requires state\ndiscretization. [32] uses the Liouville equation to maximize the backward reachable set for only\n2polynomial system dynamics. In [33] the authors compute the stochastic reachability by discretiz-\ning stochastic hybrid systems to Markov Chains (MC), then perform probabilistic analysis on the\ndiscretized MC. The closed-form expression of the probability requires integrals over the whole state\nspace hence is computation-heavy and cannot be used for online safety check. The closest to ours\nis [34] where Perron-Frobenius and Koopman operators are learned from samples of trajectories.\nThen, the learned operators can be used to transform the moments of the distribution over time. The\ndistribution at time tis then recovered from the transformed moments. However, as they use mo-\nments up to a specific order to represent a distribution, even if the learned operators are perfect, the\nestimation error of the distribution might not be zero. Also, such a moment-based method is hard to\nscale to large-dimensional systems. Recently, there is also a growing interest in studying the (worst-\ncase) reachability of NN [35, 36, 37, 38, 21] or systems with NN controllers [39, 4, 40, 41, 42]. In\nthis paper, we use [21] to parse our learned NN as a set of linear mappings between polyhedrons for\nonline reachability computation, but this step can be replaced with other NN reachability tools.\nTo measure the probability distribution in the reachable sets, the most na ¨ıve approach is to use his-\ntograms or kernel density estimation [19], similar to the Monte-Carlo method used in dispersion\nanalysis [43]. But this could lead to poor accuracy and computational scalability [44]. Another ap-\nproach propagates the uncertainty by approximating the solution of the probability density function\n(PDF) transport equation [45], which could still be time-consuming due to the optimization process\nperformed at each time step. Our approach finds the PDF transport equation by solving the Liou-\nville PDE2using NN. Similar ideas have been explored in [18]. However, the density NN in [18]\nwas learned solely for a fixed initial states distribution and therefore, cannot be used for online pre-\ndiction. Instead, we jointly learn the reachable states and density changes, and can perform online\nreachability density computation for any initial states distribution.\nThe idea of using deep learning to solve PDEs can be traced back to the 1990s [46, 47, 48],\nwhere the solutions of the PDE on a priori fixed mesh are approximated by NN. Recently,\nthere is a growing interest in the related sub-fields including: mesh-free methods in strong\nform [20, 49, 50] and weak form [51, 52], solving high-dimension PDEs via BSDE method [53],\nsolving parametric PDE [54, 55] and stochastic differential equations [56, 57], learning differential\noperators[58, 59, 60], and developing more advanced toolboxes [61, 62, 63]. Our idea for solving\nLiouville PDE along trajectories is similar to [53] but without the stochastic term.\n2 Preliminaries\nWe denote by R,R\u00150,Rn,Rn\u0002nthe sets of real numbers, non-negative real numbers, n-dimensional\nreal vectors, and n\u0002nreal matrices. We consider autonomous dynamical systems of the form\n_x(t) =f(x(t)), where for all t2R,x(t)2X \u0012 Rdis the state andXis a compact set that\nrepresents the state space. We assume that f:Rd7!Rdis locally Lipschitz continuous. The\nsolution of the above differential equation exists and is unique for a given initial condition x0. We\ndefine the flow map \b :X\u0002R7!X such that \b(x0;t)(also written as x(t)for brevity) is the\nstate at time tstarting from x0at time 0. Note that system parameters can be easily incorporated as\nadditional state variables with time derivative to be 0.\nWe analyze the evolution of the dynamical system by equipping it with a density function \u001a:X\u0002\nR!R\u00150, which measures how states distribute in the state space at a specific time instant. A\nlarger density \u001a(x;t)means the state is more likely to reside around xat timet, and vice versa.\nThe density function is completely determined by the underlying dynamics (i.e., function f) and\nthe initial density map \u001a0:X 7! R\u00150. Specifically, given a \u001a0, the density function \u001asolves the\nfollowing boundary value problem of the Liouville PDE [17]:\n@\u001a\n@t+r\u0001(\u001a\u0001f) = 0; \u001a(x;0) =\u001a0(x); (1)\nwherer\u0001(\u001a\u0001f) =Pd\ni=1@[\u001a\u0001f]i\n@[x]iis the divergence for the vector field \u001a\u0001f, and [\u0001]itakes the\ni-th coordinate of a vector.3Intuitively, as shown in [17], Liouville PDE is analogous to the mass\nconservation in fluid mechanics, where the change of density@\u001a\n@tat one point is balanced by the\n2In the absence of process noise, this PDF transport equation reduces to stochastic Liouville equation [17]\n3A general form of Liouville PDE can have a non-zero term on the right-hand side indicating how many\n(new) states appear or exit from the system during the run time [19]. In all the systems we discuss here, there\n3total flux traversing the surface of a small volume surrounding that point. It is hard to solve the\nLiouville PDE for a closed from of the density function \u001a. However, it is relatively easy to evaluate\nthe density along a trajectory of the system. To do that, we first convert the PDE into an ODE as\nfollows. Considering a trajectory \b(x0;t), the density along this trajectory is an univariate function\noft, i.e.,\u001a(t) =\u001a(\b(x0;t);t). If we consider the augmented system with states [x;\u001a], from Eq. (1)\nwe can easily get the dynamics of the augmented system [19]:\n\u0014\n_x\n_\u001a\u0015\n=\u0014\nf(x)\n\u0000r\u0001f(x)\u001a\u0015\n: (2)\nTherefore, to compute the density at an arbitrary point xT2X at timeT, one can simply proceed\nas follows: First, find the initial state x0= \b(xT;\u0000T)using the inverse dynamics \u0000f. Then,\nsolve the Eq. (2) with initial condition [x0;\u001a0(x0)]. The solution at time Tjust gives the desired\ndensity value. However, such a procedure only gives the density at a single point, therefore, cannot\nbe used in reachability analysis. Instead, we need to compute the solution of Eq. (2) for a set of\ninitial conditions and use that to compute the reachable sets. To achieve this, we use an NN with\nReLU (Rectified Linear Unit) activation functions [64] to jointly approximate the flow map \band\nthedensity concentration function , as will be shown in the next section.\n3 Density learning and online reachability density computation\nLet us take a closer look at Eq. (2). For an initial condition [x0;\u001a0(x0)], the closed form\nof the solution \u001ais\u001a(\b(x0;t);t) =\u001a0(x0) exp\u0010\n\u0000Rt\n0r\u0001f(\b(x0;\u001c))d\u001c\u0011\n:Interestingly, the\nsolution of \u001ais linear in the initial condition \u001a0. The gained part denoted by G(x0;t) :=\nexp\u0010\n\u0000Rt\n0r\u0001f(\b(x0;\u001c))d\u001c\u0011\nis a function of the initial state x0and timet, but is independent\nof\u001a0. This mapping Gis completely determined by the underlying dynamics and we call it the\ndensity concentration function . WithGin hand, the density at an arbitrary time and state can be\nquickly computed from any \u001a0. However,Gis obviously hard to compute. Therefore, we use NN to\napproximate the density concentration function . In addition to G, we also use NN to learn the flow\nmap\b, which will be a necessity for computing the reachable set distribution shown in Sec. 3.2.\n3.1 System dynamics and density learning framework\nLet the parameterized versions of the flow map and the density concentration function be\b!and\nG\u0012respectively, where !and\u0012are parameters. To train the neural network, we construct a dataset\nby randomly sampling Ntrajectories of the system: Dtrain =f\u0018igN\u00001\ni=0inTtime steps (with time\ninterval \u0001t):\u0018i=f(xi\n0;0);(xi\n1;\u0001t);:::;(xi\nT\u00001;(T\u00001)\u0001t)gwherexi\nj= \b(xi\n0;j\u0001t). Then, the\ngoal of the learning is to find parameters !and\u0012satisfying\n(\n\b!(xi\n0;k\u0001t)\u0000xi\nk= 0;8i;k;\n@G\u0012(xi\n0;k\u0001t)\n@t+G\u0012(xi\n0;k\u0001t)\u0001(r\u0001f(xi\nk)) = 0;8i;k;(3)\nwhere the first constraint is for the flow map estimation, and the second constraint enforces the\nLiouville equation for all the data points.\nAs for the implementation, we model \b!andG\u0012jointly as a fully-connected neural network NN (\u0001;\u0001)\nwith ReLU activations. To ensure numerical stability as Gis an exponential function, we add a\nnonlinear transform from the NN output to the density concentration function :\n\u001aG\u0012(x0;t) = exp(t\u0001NN[0:1](x0;t)) = exp(t\u0001z(x0;t));\n\b!(x0;t) =NN[1:n+1](x0;t);(4)\nwhere NN [i:j+1]is to choose the i;i+ 1;:::;j -th dimensions from the output of the NN, and zis the\nintermediate density estimation from the NN. In this way, we guarantee that the density concentra-\ntion function att= 0is always 1. We optimize our NN via back propagation with the loss function:\nis no state entering (other than the initial states) or leaving the system, so the right-hand side of Eq. (1) is zero.\nThe total density of the systems we consider is invariant over time.\n4L=\u0015\u0001X\ni;k\u0002\n\b!(xi\n0;k\u0001t)\u0000xi\nk\u00032+X\ni;kh\n_G\u0012(xi\n0;k\u0001t) +G\u0012(xi\n0;k\u0001t)\u0000\nr\u0001f(xi\nk)\u0001i2\n; (5)\nwhere the first term denotes the state estimator square error [65], the second term indicates how\nfar (in the sense of L2-norm) the solution deviates from the Liouville Equation, and \u0015balances\nthese two loss terms. We approximate the time derivative of the density concentration function by\n_G\u0012(xi\n0;k\u0001t) =\u0002\nG\u0012(xi\n0;(k+ 1)\u0001t)\u0000G\u0012(xi\n0;k\u0001t)\u0003\n=\u0001t. Our method can also work for black-\nbox systems if we approximate r\u0001f(\u0001)numerically. With some tools from statistical learning\ntheory, we can show that with a large enough number Nof samples, the learned flow map and the\ndensity concentration function can be arbitrarily accurate. A formal proof is shown in Appendix A.\n3.2 System reachable set distribution computation via NN Reachability Analaysis\nIn the last section, we have learned an NN that can estimate the density at a single point given the\ninitial density. In this section, we will boost this single-point estimation to set-based estimation\nby analyzing the reachability of the learned NN. To compute the set of all reachable states and the\ncorresponding density from a set of initial conditions, we use the Reachable Polyhedron Marching\n(RPM) method [21] to further process the learned NN for Gand\b. RPM is a polyhedron-based\napproach for exact reachability analysis for ReLU NNs. It partitions the input space into polyhedral\ncells so that in each cell the ReLU activation map does not change and the NN becomes a fixed affine\nmapping. With the cells and the corresponding affine mapping on each cell, the exact reachable set\nfor a set of input can be quickly evaluated. Backward reachability analysis can also be performed by\ncomputing the intersection of the pre-image of a query output set with the input polyhedron cells.\nSystem forward reachable set with density. Recall that the input of the NN in Sec. 3.1 consists of\nthe initial state x0and timet. For the simplicity of comparison with other methods, we only estimate\nthe reachable set and the density at given fixed time instances tand fix the last element of the input\nof the NN. Thus for each time step t, the input polyhedral cells generated from the RPM will be\na set of linear inequality constraints on x, and those input cells together with the set of the affine\nmappings and output polyhedral cells can be represented as: f(Ak;bk;Ck;dk;Ek;fk)gN\nk=1, where\nin each input cell Hk:=fv2RdjAkv\u0014bkg, the NN becomes an affine mapping y=Ckv+dk,\nand thus the image of the input cell is also a polyhedron Mk=fy2Rd+1jEky\u0014fkg.\nRecall that the first dimension of our NN output estimates the density concentration function\nz, and the rest dimensions estimate the state x, thus the output cell can be written as Mk=\nf(z;x)jEk[z; x]T\u0014fkg. By projecting it to the state space, we get the reachable set of the system,\ni.e.,Ro\nk:=fx2Xj (z;x)2Mkg. Then, in each cell Mk, we evaluate the lower and upper bounds\nofz, and denote them by zk;min andzk;max . The density bound for cell Mkis then computed as\n\u001a\u001ak;min =\u001a0(\u0016xk)\u0001exp(t\u0001zk;min );\n\u001ak;max =\u001a0(\u0016xk)\u0001exp(t\u0001zk;max );(6)\nwhere \u0016xkis the center of Hk. Finally the system forward reachable set is a union of projected output\npolyhedral cells:NS\nk=1Ro\nkwhere each cell Ro\nkis associated with a density bound [\u001ak;min;\u001ak;max ].\nSystem reachable set probability computation. Given an initial state distribution, we want to\nfigure out the probability distribution of the system forward reachable sets, as well as the probability\nfor the states land into a query set (e.g., the query set could be the unsafe region).\nFor an arbitrary initial probability density function (whose support is bounded), we can apply RPM\nto partition its support into cells4as in the last section. Finally, we obtain the reachable sets with\nbounded state densities f(Hk;\u001amin\nk;\u001amax\nk)gN\nk=1and the probability bound in each cell is:\n\u001aPmin\nk=V ol(Hk)\u001amin\nk;\nPmax\nk=V ol(Hk)\u001amax\nk;(7)\n4We can always further divide those input cells to make the bound tighter/ more precise, while still guaran-\nteeing the Neural Network on each cell can be seen as an affine transformation.\n5System Dim. Control\nVan der Pol Oscillator (vdp) 2 -\nDouble integrator (dint) [42] 2 NN\nKraichnan-Orszag system (kop) [66] 3 -\nInverted pendulum (pend) [67] 4 LQR\nGround robot navigation (rpbot) 4 NN\nFACTEST car tracking system (car) [68] 5 Tracking\nQuadrotor control system (quad) [42] 6 NN\nAdaptive cruise control system (acc) [4] 7 NN\nF-16 Ground collision avoidance (gcas) [69] 13 Hybrid\n8-Car platoon system (toon) [70] 16 NN\nTable 1: Benchmarks from low dimen-\nsion academic models to complex systems\nwith handcrafted or NN controllers.\nFigure 2: KL divergence with ODE-simulated density. Our\nmethod consistently outperforms other baselines. The histogram\nmethod cannot estimate the density for “acc”, “gcas” and “toon”.\nwhere V ol (\u0001)computes the volume for a polyhedron. By computing for all input cells fHkgN\nk=1,\nwe can derive the system forward reachable set and the corresponding probability bound as\b\n(Hk;Pmin\nk;Pmax\nk)\t\nk=1:N. The backward reachable set probability can be computed in a similar\nfashion, by checking the intersection between the query output region and the output cells derived by\nRPM, computing each intersection’s probability range by its volume and density bound and finally\naggregating the probability of all intersections. Detailed computation is shown in Appendix B.5\n4 Experimental evaluation in simulation\nHere we show the benefits of learning density concentration function from Liouville PDE in state\ndensity and reachable set distribution estimation. More experiments are provided in Appendix C \u0018H.\n4.1 Implementation details\nDatasets: We investigate 10 benchmark dynamical systems as reported in Table 1. These bench-\nmark systems range from low dimension academic models (2 \u00183 dimensions) to complex and even\nblack-box systems (13 \u001816 dimensions) controlled by handcrafted or NN controllers. All controllers\n(except for the ground robot navigation example) are from the original references. More details (the\nmodel description and initial distributions) will be provided in Appendix C.\nTraining: For each system, we generate 10k trajectories through simulation with varied trajectory\nlengths from 10 to 100 time-steps, depending on different configurations of the simulation envi-\nronment. We use 80% of the samples to train the NN as described in Sec. 3.1 and use the rest for\nvalidation. For the trajectory data, we collect the system states and compute for the system diver-\ngence term. For black-box systems, we use the gradient perturbation method to approximate the\nderivatives. We use feed-forward NN which has 3 hidden layers with 64 hidden units in each layer.\nWe use PyTorch [71] to train the NN and the training takes 1 \u00182 hours on an RTX2080 Ti GPU.\n4.2 Density estimation verification\nWe first test the density estimation accuracy of our learned NN. We compare our approach with\nother baselines including kernel density estimation (KDE), Sigmoidal Gaussian Process Density\n(SGPD) [72] and the histogram approach. For each simulation scenario, we first solve Eq. (2) to\ngenerate 20k\u0018100k trajectories of (state, density) pairs, and treat this density value as the ground\ntruth. For the KDE method, we choose an Epanechnikov kernel. We then measure the KL divergence\nbetween the density estimate of each method and the ground truth. As shown in Fig. 2, our approach\nhas consistently outperformed KDE, SGPD and histogram approaches, with the largest reduction of\n99.69% in KL divergence when compared with the histogram approach for the Kraichnan-Orszag\nsystem, while our method doesn’t use any ODE generated density values during training. Also in\nhigh-dimension systems (dimension \u00157), the histogram approach fails to predict the density due to\nthe curse of the dimensionality, whereas our approach can always predict the density, with a 30.13%\nto 99.87% decrease in KL divergence comparing to KDE. More plots will be given in Appendix D.\n6(a) The system forward reach set computed by the ConvexHull method(blue) [9], DryVR(red) [68] and\nGSC(green) [73]. We further show the relative volume of each kind of reachable set, comparing to the vol-\nume of the convex hull of the sampled points (gray). As time evolves, the conventional reachability methods\noften result in a larger over-approximation of the reachable sets with an increasing estimation error.\n(b) The system forward reach set derived by RPM. The reachable sets are represented in polyhedral cells. The\ncolor ranged from dark purple to light yellow indicates the density inside the polyhedral cells. The edges\ncolored in green indicate the boundaries of the RPM polyhedral cells with density below a threshold.\nFigure 3: The Van der Pol oscillator forward reachable set comparison between (a) the worst-case reachability\nmethods [68, 73] and (b) our probabilistic approach. Our approach clearly identifies reachable sets in high and\nlow densities, and shows that as time evolves, the states will concentrate on a limit cycle, which is as expected.\n(a) Van der Pol system\n (b) Double integrator\n (c) Ground robot model\n (d) FACTEST car model\nFigure 4: The (relative) volume of reachable set with different probability level using our method, and com-\nparison to other methods. The top part of each figure is in logarithm scale.\n4.3 Reachable set distribution analysis\nForward reachable set distribution analysis. Being confident that our approach is able to pro-\nvide an accurate state density estimation, we extend our NN to do distribution analysis, which is a\nvaluable technique in safety-related applications like autonomous driving. Here we use an existing\nreachability tool RPM [21] to compute the forward reachable sets with probability bounds. Details\nabout how to derive the density and probability bound for the reachable sets are presented in Sec. 3.2\nand in Appendix B. Also, RPM was only able to parse the NN for Van der Pol, Double integrator,\nground robot navigation, and the FACTEST car model. It fails in handling other high-dimension\ncomplex systems due to numerical issues when partitioning for the input set. Thus, we only report\nthe results on those 4 models in this section. The main purpose is to show that for some systems the\ndensity tends to concentrate on certain states, where a small portion of the reachable sets contains the\nmajority of states that are more likely to be reached. Therefore, our method can better quantify risks\nthan worst-case reachability, by providing a flexible threshold for the probability of reachability.\nWe start with the Van der Pol oscillator. The initial states are uniformly sampled from a square\nregion: [\u00002:5;2:5]\u0002[\u00002:5;2:5]. As illustrated in Fig. 1, the system states will gradually converge to\na limit cycle. Using worst-case reachability analysis for this system will result in a very conservative\n5The RPM method cannot handle systems with higher than 4 dimensional state space in our experiments.\nBut the technique discussed in Sec. 3.2 can work with any set-based NN reachability tools with slight modifica-\ntion based on the set presentation of the tool. Developing a better NN reachability tool that can scale to higher\ndimensional systems is another topic and is out of the scope of our paper.\n7over-approximation, and this over-approximation will propagate over time and lead to increasing\nconservativeness of the reachable set estimation. As shown in Fig. 3(a), for the worst-case methods\nlike DryVR(red) [68, 74] and GSG(Green) [73], the volume of their estimated reachable set relative\nto the volume of the convex hull of the system states keeps increasing over time, from 1.9670X to\n5.3027X for the DryVR [68] approach, and from 1.7636X to 2.8086X for GSG [73]. However, our\nmethod can give the probability bound for every reachable set in the state space as shown in the\nheatmap in Fig. 3(b), clearly identifying the region around the limit cycle in high density (bright\ncolor), and the rest space in low density (dark color).\nWe can also compute the relative volume of the reachable sets (comparing to the convex hull) pre-\nserving different levels of reachable probability, whose evolution over time reflects the system’s\ntendency for concentration. As shown in Fig. 4, we use the above 4 systems and study the volume\nof the reachable set with probability threshold 0.50, 0.70, 0.80, 0.90, and 0.99. As expected, the rel-\native volume will increase as the probability threshold increases. In all cases, there exist some time\ninstances where a small volume of the reachable set actually preserves high probability, which shows\nthe state concentration exists in many existing systems. While as shown in Fig. 4, the worst-case\nreachability tools can only generate one curve which presents the (relative) volume of the reach-\nable set that covers all possible states. Not surprisingly, the worst-case reachability tools give very\nconservative results. We believe using our proposed method to do reachable set distribution analy-\nsis will benefit future study for systems with uncertainty, and for systems where the failure case is\ninevitable but happens with a low probability. More comparisons with state-of-the-art reachability\nmethods (Verisig [39], Sherlock [75] and ReachNN [40]) are shown in Appendix H.\n(a)\n (b)\nFigure 5: (a) A robot tries to reach the goal while\navoiding the obstacle. (b) Probabilistic safety verifica-\ntion under different initial distributions. As the robot\nhas less uncertainty in its initial position and velocity, it\nwill have lower probability to collide.Online safety verification under different ini-\ntial state distributions. Since our approach\nlearns the density concentration function in-\nstead of absolute density value, it has the flex-\nibility to estimate online reach sets distribution\nwith any possible (bounded support) initial dis-\ntributions. Consider the safety verification for\nthe ground robot navigation problem (shown in\nFig. 5(a)): The probability of colliding with\na obstacle in the center of the map is deter-\nmined by the initial state distribution6, which\nis parametrized as a truncated Gaussian distri-\nbutionN(\u0016;\u001b2I)where\u0016and\u001bmeasure the\nexpectation and uncertainty of the initial robot state. Our method can estimate the upper and lower\nbounds for the probability of colliding with the obstacle, and this safety evaluation process can run\nin faster than 50Hz with parallel computation and heuristic searching used (see details in Appendix\nE). As shown in Fig. 5(b), when the initial state uncertainty \u001bdecreases from 1:0to0:02, the up-\nper and lower bounds for the probability of collision decrease to close to zero (from 0:04236 to\n3:5344\u000210\u000006, where other worst-case reachability analysis methods can only report a collision\nis inevitable, without quantifying the corresponding risks. This also shows the advantage of our ap-\nproach in adapting to different density conditions in computation without retraining or fine-tuning.\nLimitations and trade-offs of performing reachable set distribution analysis. Experimental\nresults show that our method can compute much less conservative probabilistic reachable sets than\nmost worst-case reachability methods. This less conservative result benefits from RPM which can\nprovide exact NN reachability analysis by sacrificing scalability. Therefore RPM also constrains us\nfrom performing online reachability analysis for high-dimensional systems or systems with large ini-\ntial sets. Technically, we are solving a harder problem than worst-case reachability approximation,\nas we need not only the reachable set, but also the density over those reachable states. Like worst-\ncase analysis, this is the reason why it can only scale to lower-dimensional systems and smaller\ninitial sets when we want to perform accurate reachable computation. We believe that our approach\ncan better quantify risks under different conditions, especially when unsafe is inevitable (similar to\nFig. 5(a)). Our method can also give worst-case reachability by taking all output reachable cells\nproduced by RPM regardless of their density. This worst-case reachability using RPM is less con-\nservative than other NN reachability tools, at the cost of not scaling to high-dimensional systems.\n6How to compute the probability of a reachable set is discussed in Sec.3.2\n85 Conclusion and discussion\nIn this paper, we propose a Neural Network (NN)-based probabilistic safety verification frame-\nwork that can estimate state density, compute reachable sets and corresponding probability. Our\nLiouville-based NN can accurately estimate the state density even for high-dimension systems. Our\nprobabilistic reachable set framework can handle nonlinear (and potentially black-box) systems with\nvarying initial state distributions and can be used for fast online safety verification. We recognize\nthat the task of computing probabilistic reachable sets is very useful, and our method is more help-\nful than worst-case reachability particularly when the system states are more likely to concentrate.\nOne limitation of our approach is that the NN reachability tool we used (RPM) cannot handle high-\ndimension systems or cases where the initial set is very large, due to the numerical issues when\npartitioning for polyhedral cells. This limitation is due to the scalability and accuracy trade-off of\nNN reachability, which is an independent problem from our paper. We plan to explore other NN\nreachability methods and more complicated hybrid systems in real-world applications.\nAcknowledgments\nThe NASA University Leadership initiative (grant #80NSSC20M0163) and Ford Motor Company\nprovided funds to assist the authors with their research, but this article solely reflects the opinions\nand conclusions of its authors and not any NASA or Ford entity.\nReferences\n[1] M. 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Pritzel, N. Heess, T. Erez, Y . Tassa, D. Silver, and D. Wierstra.\nContinuous control with deep reinforcement learning. arXiv preprint arXiv:1509.02971 , 2015.\n13Learning Density Distribution of Reachable States for\nAutonomous Systems (Supplementary Material)\nA Generalization error bound for the learning framework\nWith sufficient amount of data and a large enough neural network, we can approximate the state\nand density estimation at arbitrary small errors [76]. In the language of statistical learning theory,\nthe neural network generating functions (\b!;G\u0012)is called a hypothesis and denoted by h. The set\ncontaining all the possible hypotheses is called the hypothesis class H. For a hypothesis hgenerating\n(\b!;G\u0012), we denotel(h;\u0018i) =P\n(xk\ni;k\u0001T)2\u0018i(\b!(xi\n0;k\u0001t)\u0000xi\nk)2+(@G\u0012(xi\n0;k\u0001t)\n@t+G\u0012(xi\n0;k\u0001t)\u0001(r\u0001\nf(xk\ni)))2\u0000\rwhere\r\u00150is an error tolerance term which is further used to derive the probabilistic\nguarantee. Assume that the optimization problem in Eq.(3) is feasible, and ^!and^\u0012solve Eq. (3).\nLet^hNbe the hypothesis that generates (\b^!;G^\u0012). Furthermore, assume that jl(\u0001;\u0001)j\u0014Bl, and\ndenote the sample distribution D(where the training sample trajectories are sampled from). Then\naccording to Theorem 5 in [77], the following statement holds with probability at least 1\u0000\u000eover a\ntraining data set consisting of Ni.i.d. random trajectories:\nP(E\n\u0018\u0018Dl(^hN;\u0018)>0)\u0014K\u0012log3N\n\r2R2\nN(H) +2 log(log(4Bl=\r)=\u000e)\nN\u0013\n(8)\nwhereKis a universal constant, and RN(H)is the Rademacher complexity for Hdefined as:\nRN(H) = sup\n\u00181;\u00182;\u0001\u0001\u0001;\u0018N\"\nE\n\u001b\"\nsup\nh2H1\nNNX\ni=1\u001bil(h;\u0018i)##\n(9)\nwhere\u001b= [\u001b1;\u001b2;\u0001\u0001\u0001;\u001bN]are i.i.d. random variables with P(\u001bi= 1) = P(\u001bi=\u00001) = 0:5.\nRemarks: Here we reduce bounding the generalization error to bounding the Rademacher com-\nplexityRN(H), where RN(H)can be further bounded as RN(H)\u0014o(k\nN)for Lipschitz paramet-\nric function classes (including neural networks) where kdenotes the number of learnable parame-\nters [78][Theorem 4.2.]. In this way, we show that for a fixed error threshold \r, as the number of\ntraining samples Nincreases, the probability that our learning framework fails to satisfy the Liou-\nville equation or fails to estimate the system dynamics will gradually decrease to zero. We show an\nempirical result to support this in Figure 1. For the Van der Pol Oscillator benchmark example, we\ntrain the neural network with different numbers of training samples (from 8\u0002100\u00188\u0002104) and\nreport the testing error (mean square error for the state estimation and density concentration function\ncomparing to the groundtruth) for a fixed testing set. As the number of training samples increases,\nthe testing error gradually converges to zero.\nAssume the functions on the right hand side of Eq. (2) are uniformly Lipschitz continuous in (x;\u001a),\nthen the function will have a unique solution according to Picard-Lindel ¨of theorem[79][Theorem\nI.3.1]. Then if our estimator satisfies the Liouville equation everywhere, we can recover the\ngroundtruth density concentration function as well as the system dynamics.\nB Implementation details for system reachable set probability computation\nusing RPM\nB.1 Online query set probability bound computation under different initial state\ndistributions\nThe problem formulation is: given a query set Rqwith density concentration function constraints\n[zmin;zmax](the range that the density concentration function can change from the initial condition\nto the terminal condition; if this constraint is not specified, the default value is \u00001\u0014z\u00141 ),\ncompute the probability that the system will reach this query set (with optional density constraints).\nIn our case, when using RPM to compute the reachable sets, we represent Rqas a polyhedron, and\nsince [zmin\u0014z\u0014zmax]is a set of linear inequality constraints, the set Mq=f(z;v)jzmin\u0014z\u0014\n1Figure 1: The testing error decreases as more training samples are used.\nFigure 2: Illustration for the online query set probability bound computation (for an output cell Mk).\nzmax;v2Rqgis also a polyhedron. At each time step t, from Sec. 3.2 we can represent the NN\ninput cells, affine mapping and output cells at this time step as f(Ak;bk;Ck;dk;Ek;fk)gN\nk=1(here\nwe omit the subscript for tfor the brevity in the notation) where each input cell is a polyhedron\nHk=fx2Rd+1jAkx\u0014bkg, with an affine mapping y=Ckx+dkand the resulting output cell\nis also polyhedron Mk=fy2Rd+1jEky\u0014fkg. Then for each output cell Mk, we check for the\nintersection between the query cell and the output cell Mq\nk=fyjy2Mq; Eky\u0014fkg. Next we can\nderive the intermediate density concentration function bound [zk;min;zk;max ]onMq\nkby solving the\nfollowing Linear Programming problem (here taking zk;min as an example; to solve zk;max we just\nneed to change the “min” to “max” in the objective function in Eq. 10; and here [y]0denotes the first\ncoordinate of y, thus [y]0=zas we denote y= (z;x)T):\n\u001amin [y]0\ns.t.y2Mq\nk(10)\nAfter we derive the bound for zonMq\nk, the density bound for Mq\nkis computed as (similar to Eq.(6)):\n\u001a\u001ak;min =\u001a0(~xk)et\u0001zk;min\n\u001ak;max =\u001a0(~xk)et\u0001zk;max(11)\nwhere ~xkis the center of Hk. And the probability bound can be computed by pk;min =\n\u001ak;min\u0001V ol(Mq\nk)andpk;max =\u001ak;max\u0001V ol(Mq\nk)where the V ol (Mq\nk)is the volume for the\nintersection. Finally the probability of the system reach this query set at time tis bounded by\u0014\nPmin=NP\nk=1pk;min;Pmax=NP\nk=1pk;max\u0015\n. An illustrative figure is shown in Fig. 2\nRemarks: This algorithm can be used for online safety verification under different initial state\ndistributions by just representing the dangerous set in Rq, and changing the \u001a0(\u0001)function in (11) on\n2the fly. Here we approximate the density distribution in Hkusing the density evaluated at ~xkwhich\nis the center of Hk- the accuracy of this approximation will converge to 1 as the partition on \u001a0gets\nfiner.\nB.2 Backward reachable set probability computation\nThe problem formulation is: given a query set Rqwith density concentration function constraints\n[zmin;zmax](the range that the density concentration function can change from the initial condition\nto the terminal condition; if this constraint is not specified, the default value is \u00001\u0014z\u00141 ),\ncompute for all possible initial conditions as well as probabilities that lead the system to reach the\nquery set (with optional density constraints).\nSimilar to Sec. B.1, we can denote this query set as Mq=f(z;v)jzmin\u0014z\u0014\nzmax;v2Rqg. At each time step t, the NN input cells, affine mapping and output cells are\nf(Ak;bk;Ck;dk;Ek;fk)gN\nk=1where each input cell is a polyhedron Hk=fx2Rd+1jAkx\u0014bkg,\nwith an affine mapping y=Ckx+dkand the resulting output cell is also polyhedron Mk=fy2\nRd+1jEky\u0014fkg. Then for each output cell Mk, we check for the intersection between the query\ncell and the output cell Mq\nk=fyjy2Mq; Eky\u0014fkg. Using the affine mapping with invertible\nCk7, we can derive the pre-image of this intersection to be Hq\nk=fxjx=C\u00001\nky\u0000C\u00001\nkdk; y2Mq\nkg.\nThus the reachable set can be computed using projection: Ri;q\nk=fx2Xj (x;t)2Hq\nkgand the\ncorresponding probability is pi;q\nk=V ol(Ri;q\nk)\u001a0(~xi;q\nk)where\u001a0(\u0001)is the initial state distribution\nfunction and~xi;q\nkis the center of Ri;q\nk. By performing this for all output cells and for all time steps\nt, we derive the backward reachable set ff(Ri;q\nt;k;pi;q\nt;k)gN\nk=1gT\u00001\nt=0.\nB.3 Speed up the probability computation by using hyper-rectangle heuristic\nThe computation in both Sec. B.1 and Sec. B.2 requires checking the intersection between polyhe-\ndralHiandHj, where one approach is to check whether a feasible solution exists for the linear\nprogramming problem : min 0Tx;s.t.x2Hi\\Hj. Solving this for x2RnrequiresO(n2:5)time\nwhen the interior method is used. To speed up the intersection checking process, we introduce a\nhyper-rectangle heuristic: at the pre-processing stage, we over-approximate each polyhedron Hiby\nits outer hyper-rectangle ~Hi(derived by computing the range for the vertices of Hiin each dimen-\nsion). When checking for the polyhedron intersection between HiandHj, we first check whether\ntheir corresponding hyper-rectangles ~Hiand~Hjwill intersect. If ~Hiand~Hjdo not intersect, then it\nis guaranteed that the polyhedra HiandHjwon’t intersect. Otherwise, we further check the inter-\nsection ofHiandHjby using the interior method. Checking hyper-rectangles’ intersection can be\nimplemented in O(n), hence greatly accelerates the computation process. A detailed computation\ntime comparison will be presented in Sec. E.\nC Simulation environments\nIn this section, we present the implementation details for all 10 simulation environments used in our\nmain paper, sorted in the same order as shown in Table. 2.\nC.1 Van der Pol Oscillator\nConsider the Van der Pol Oscillator problem:d2x\ndt2\u0000\u0016(1\u0000x2)dx\ndt+x= 0where the position variable\nxis a function of tand the scalar parameter \u0016indicates the strength of the system damping effect.\nBy doing a transformation: y= _x, the original problem can be shaped to the following 2d system\ndynamics:\u001a_x=y\n_y=\u0016(1\u0000x2)y\u0000x(12)\n7In practice, Ckis in high probability to be invertible. This is because the set of all non-invertible random\nmatrices forms a hyper-surface with Lebesque measure zero. When Ckis singular, we can use elimination\nmethod like Fourier-Motzkin elimination as in [21] to derive the set representation in the input side.\n3where the divergence term r\u0001fused in (2) can be computed as: r\u0001f=\u0016(1\u0000x2). In the simulation,\nwe set\u0016= 1:0, the initial state distribution as an uniform distribution U[\u00002:5;2:5]\u0002[\u00002:5;2:5]and the\ntime step duration \u0001t= 0:05s. We run each simulation for 50 time steps to collect the trajectories.\nC.2 Double Integrator with an NN controller\nWe consider a discrete double integrator system introduced in [41]:\n\u0012\nxt+1\nyt+1\u0013\n=\u0014\n1 1\n0 1\u0015\u0012\nxt\nyt\u0013\n+\u0014\n0:5\n1\u0015\nut (13)\nwhere (xt;yt)Tdenotes the 2d state variable, and utis the output of a neural network controller\nwhich is trained to mimic the behavior of an MPC controller [41, 42]. We convert the system to the\ncontinuous system with state (x;y)Tand time step duration \u0001t= 1:0sas :\n_\u0012\nx\ny\u0013\n=\u0014\n0 1\n0 0\u0015\u0012\nx\ny\u0013\n+\u0014\n0:5\n1\u0015\nu (14)\nand here the divergence term r\u0001fused in (2) can be computed as: r\u0001f= 0:5@u\n@x+@u\n@y, where the\n@u\n@xis the gradient of the neural network controller output uwith respect to the input x(and similar\nfor@u\n@yandy) and can be calculated using automatic differentiation engine in PyTorch [71]. We set\nthe initial state distribution as an uniform distribution U[\u00000:5;4:0]\u0002[\u00001:0;1:0]. Similar to [41, 42], we\nrun each simulation for 10 time steps to collect the trajectories.\nC.3 Kraichnan-Orszag system\nThe system dynamics of the Kraichnan-Orszag problem [66, 18] is defined as:\n8\n<\n:_x1=x1x3\n_x2=\u0000x2x3\n_x3=\u0000x2\n1+x2\n2(15)\nand here an interesting fact is that the divergence term r\u0001fused in (2) is just:r\u0001f=x3\u0000x3+0 = 0 ,\nwhich means the density along each trajectory won’t change over time, and only depends on the\ninitial state distribution. Similar to [18], we set the initial state x(0) = (x1(0);x2(0);x3(0))T\ndistribution as an Gaussian distribution with:8\n<\n:x1(0)\u0018N(1;1=42)\nx2(0)\u0018N(0;1=22)\nx3(0)\u0018N(0;1=22)(16)\nwhere we further truncate the initial state within the range f0\u0014x1(0)\u00142;\u00002\u0014x2(0)\u00142;\u00002\u0014\nx3(0)\u00142g. We set the time step duration \u0001t= 0:125sand run each simulation for 80 time steps\nto collect the trajectories.\nC.4 Inverted pendulum\nThe inverted pendulum problem [67] is defined as \u0012+b\nmL2_\u0012\u0000g\nLsin\u0012\u00001\nmL2uLQR = 0, where\u0012\ndenotes the pendulum’s relative angle to the the up-right position, m;L;g;b are pre-defined param-\neters anduLQR denotes the output of an LQR controller [67] u=K1\u0012+K2_\u0012whereK1andK2are\nscalar-valued coefficients. To test for the system performance under different coefficient settings for\nthe LQR controller, we include k1,k2into the system state variable and study the following system\ndynamics:8\n>><\n>>:_\u0012=!\n_!=1\nm\u0001L2(mgL sin\u0012\u0000b!+u)\n_k1= 0\n_k2= 0(17)\nwhereu=K1\n50ek1\u0012+K2\n50ek2!. Now the divergence term r\u0001fused in (2) can be computed as:\nr\u0001f=\u0000b\nmL2+1\nmL2@u\n@w=\u0000b\nmL2+K2ek2\n50mL2. Based on [67], we set g= 9:80;L= 0:50;m=\n40:15;b= 0:00;K1=\u000023:59;K2=\u00005:31, the time step duration \u0001t= 0:02s. We set the\ninitial state distribution as a uniform distribution U[\u00002:1;2:1]\u0002[\u00005:5;5:5]\u0002[\u00002:0;2:0]\u0002[\u00002:0;2:0]and run\neach simulation for 50 time steps to collect the trajectories.\nC.5 Ground robot navigation with an NN controller\nFigure 3: The screenshot for robot navigation problem.\nWe design a ground robot navigation experiment (as shown in Fig. 3), where the objective is to reach\nthe green regionf(x;y)j(x\u0000xgoal)2+ (y\u0000ygoal)2\u0014r2\ngoalgwhile avoiding to enter the red region\nf(x;y)j(x\u0000xobs)2+ (y\u0000yobs)2\u0014r2\nobsg. The robot is following an Dubins car model:\n8\n>><\n>>:_x=vcos\u0012\n_y=vsin\u0012\n_\u0012=uw\n_v=ua(18)\nwherex;y;\u0012;v represent robot’s x and y position, heading angle and velocity respectively. We use\nan NN controller to output control signals uw;uv. The NN controller is a feedforward NN with\n2 hidden layers and 32 hidden units in each layer. We use ReLU for the intermediate activation\nfunctions and use Tanh as the activation function for the last layer to make sure the control output is\nalways bounded. During training, we use this NN controller to collect trajectory data and do back-\npropagation with the loss function: L=N\u00001P\ni=0T\u00001P\nk=0\u000b\u0002\n(xi\nk\u0000xgoal)2+ (yi\nk\u0000ygoal)2\u0003\n+ 1fdobs<\nrobsg(dobs\u0000robs)wheredobs=p\n(xi\nk\u0000xobs)2+ (yi\nk\u0000yobs)2. Here the divergence term r\u0001f\nused in (2) can be computed as: r\u0001f=@uw\n@\u0012+@ua\n@v. We set the initial state distribution as an\nuniform distribution U[\u00001:8;\u00001:2]\u0002[\u00001:8;\u00001:2]\u0002[0;\u0019=2]\u0002[1:0;1:5]. We run each simulation for 50 time\nsteps with time duration \u0001t= 0:05sto collect the trajectories.\nC.6 FACTEST car tracking system\nConsider a rearwheel kinematic car in 2D scenarios where the dynamics is:\n2\n4_x\n_y\n_\u00123\n5=\"cos(\u0012) 0\nsin(\u0012) 0\n0 1#\u0014\nv\n!\u0015\n(19)\nand the corresponding errors are measured by:\n\"ex\ney\ne\u0012#\n=\"cos(\u0012) sin(\u0012) 0\n\u0000sin(\u0012) cos(\u0012) 0\n0 0 1#\"xref\u0000x\nyref\u0000y\n\u0012ref\u0000\u0012#\n(20)\n5withxref;yref;\u0012refbeing some predefined tracking points (in this experiment, we assume the\ntracking points are not changing over time). With the following tracking controller defined in ( wref\nandvrefare referenced angular velocity and velocity respectively, k1;k2;k3are the parameters\ncontrolling how fast the system will converge to the reference point) [68]:\u001av=vrefcos(e\u0012) +k1ex\n!=!ref+vref(k2ey+k3sin(e\u0012))(21)\nand with an uncertainty error in the dynamics of exandey(denoted as a), the error dynamics\nbecome:2\n64_ex\n_ey\n_e\u0012\n_a3\n75=2\n64(!ref+vref(k2ey+k3sin(e\u0012)))ey\u0000k1ex+aex\n\u0000(!ref+vref(k2ey+k3sin(e\u0012)))ex+vrefsin(e\u0012)+aey\n\u0000vref(k2ey+k3sin(e\u0012))\n03\n75 (22)\nThe uncertain parameter a2[0;1]. We will show that althought now the reachable set will be much\nlarger than the case when a= 0, the probability that the system does not converge to the origin\n(zero-error) is very low.\nHere the divergence term r\u0001fused in (2) can be computed as: r\u0001f= 2a\u0000k1\u0000k2vrefex\u0000\nvrefk3cose\u0012. In our experiment, we set xref=yref=\u0012ref= 0; k1=k2= 0:5; k3=\n1:0; wref= 0; vref= 1 . We set the initial state distribution as an uniform distribution\nU[\u00002:1;2:1]\u0002[\u00002:1;2:1]\u0002[0;0:1]\u0002[0:0;1:0]. We run each simulation for 50 time steps with time duration\n\u0001t= 0:10sto collect the trajectories.\nC.7 6D Quadrotor with an NN controller\nConsider a 6D quadrotor [42]:\n_x=\u0014\n03\u00023I3\n03\u0002303\u00023\u0015\nx+\"g0 0\n03\u000230\u0000g0\n0 0 1#T\nu+\u0014\n05\u00021\n\u0000g\u0015\n(23)\nwhere the state vector xcontains 3D positions and velocities [px;py;pz;vx;vy;vz],gis the gravity\n(set to 9:8m=s2), and the control u= (u1;u2;u3)Tis from the output of an NN controller taking\nthe state vector as the input [42]. Here the divergence term r\u0001fused in (2) can be computed\nas:r\u0001f=g\u0001@u1\n@vx\u0000g\u0001@u2\n@vy+@u3\n@vz. Similar to [42], we set the initial state distribution as an\nuniform distribution U[4:65;4:75]\u0002[4:65;4:75]\u0002[2:95;3:05]\u0002[0:94;0:96]\u0002[\u00000:05;0:05]\u0002[\u00000:5;0:5]. We run each\nsimulation for 12 time steps with time duration \u0001t= 0:10sto collect the trajectories.\nC.8 Adaptive cruise control system\nConsider a learning-based adaptive cruise control (ACC) problem with plant dynamics [4]:8\n>>>>>>>>><\n>>>>>>>>>:_xrel=vlead\u0000vego\n_vlead=\rlead\n_\rlead=alead\n_vego=\rego\n_\rego=\u00002\rego+ 2u(xrel;vlead\u0000vego\u0000\rego\u001c;vego+\rego\u001c)\n_alead=\u00002\rlead\n_\u001c= 0(24)\nherexreldenotes the relative distance from the leading vehicle to the ego vehicle, vlead and\nvegodenote the velocity of leading and ego vehicles and \rlead and\regodenote the correspond-\ning acceleration rates of the two vehicles ( alead models the change in the leading vehicle’s ac-\nceleration rate, similar to the MATLAB implementation in [4]). And the controller uis taking\nthe relative distance, velocity, and ego vehicle’s velocity as input and outputs the change in the\nego vehicle’s acceleration rate. We model the velocity perception uncertainty as \u001cand pass it\nthrough the neural network. Here the divergence term r\u0001fused in (2) can be computed as:\nr\u0001f=\u00002\u0000@u\n@(vlead\u0000vego\u0000\rego\u001c)\u001c+@u\n@(vego+\rego\u001c)\u001c. We set the initial state distribution as an\nuniform distribution U[59:0;62:0]\u0002[26:0;30:0]\u0002[\u00000:01;0:01]\u0002[30:0;30:5]\u0002[\u00000:01;0:01]\u0002[\u000010:1;\u00009:9]\u0002[\u00002:0;2:0]\nand run each simulation for 50 time steps with time duration \u0001t= 0:10sto collect the trajectories.\n6C.9 F-16 ground-collision avoidance system\nThis F-16 Ground-Collision Avoidance System (GCAS) performs a recovery maneuver for the F-16\naircraft when a ground collision is detected. The F-16 aircraft is modelled with 6 degrees of freedom\n(DoF) associated with 13 nonlinear equations (three equations each for forces, kinematics, moments\nand position of the aircraft, and one extra to capture the F-16 turbojet engine). The hierarchical\ncontrol system has an outer-loop autopilot controller and an inner loop tracking and stabilizing\ncontroller (ILC). More details can be found in [69]. Specifically in this experiment, the GCAS\ndrives the roll angle and its rate to 0 and then accelerates upwards to avoid ground collision. The\nsafety specification is to make sure the altitude is always non-negative (not hitting the ground).\nWe collect the trajectories using the F-16 simulator provided in [69]. The trajectories has a time\nstep duration as 0:0333sand has 106 time steps in total. The hierarchical controller made the\nclosed-loop F-16 system a black-box system without a clean ODE expression. Therefore, there is\nno analytical way to compute for the system dynamics. As we discussed in the main paper, we could\napproximate the divergence of the system dynamics by using gradient perturbation. Recall that for\nsystem _x= (f1(x);f2(x);:::;fd(x))T, the system divergence is r\u0001f=dP\ni=1@fi\n@xi, so we approximate\nthe gradient for@fi(x)\nxiby(fi(x1;:::;xi+\u000f;:::;xn)\u0000fi(x1;:::;xi\u0000\u000f;:::;xn))=(2\u000f)where\u000fis a\nvery small number and we set \u000f= 10\u00008in our experiments.\nC.10 8-car platooning with model error\nIn this experiment we consider a 8-car platoon model [80, 70]. The state variable is x2R15, where\nx1represents the first vehicle’s (which is also the leading vehicle in the platoon) velocity, x2k\u00001\n(k=2,3...,8) represents the relative velocity of the k\u00001-th vehicle comparing to the k-th vehicle, and\nx2k\u00002(k=2,3...,8) represents the relative longitudinal offset of the k\u00001-th vehicle comparing to the\nk-th vehicle. The dynamics of the system hence is given by:\n_x2k\u00001=\u001au1; k = 1\nuk\u00001\u0000uk; k = 2;3;:::8\n_x2k\u00002=x2k\u00001+w; k = 2;3;:::8(25)\nwhereu= (u1;:::;u 8)Tis the NN controller’s output (for changing the vehicles’ acceleration rates)\nandwmodels the noise in the vehicles’ velocity dynamics. Here the neural network controller is\ntrained via RL [81]. Here the divergence term r\u0001fused in (2) can be computed as: r\u0001f=\n@u1\n@x1+8P\nk=2(@uk\u00001\n@x2k\u00001\u0000@uk\n@x2k\u00001). We set the initial state distribution as an uniform distribution Ufor\n19:9\u0014x1\u001420:1,0:9\u0014x2k\u00003\u00141:1;k= 2;3:::;8and\u00000:1\u0014x2k\u00002\u00140:1;k= 2;3:::;8and\n\u00000:01\u0014w\u00140:01. We run each simulation for 50 time steps with time duration \u0001t= 0:15sto\ncollect the trajectories.\nD Forward reachable set distribution under different probability thresholds\nInstead of over-approximating the reachable sets like traditional methods, our approach can ren-\nder varied sizes of reachable sets under different probability thresholds and under different initial\nstate distributions. We compute for the varied reachable sets at t=1.0s for ground robot navigation\nexperiment under three different (truncated) multivariate Gaussian distributions: N1=N(\u0016=\n(\u00001:7;\u00001:7;0:2;1:4)T; \u0006 = 0:02I),N2=N(\u0016= (\u00001:7;\u00001:7;1:3;1:4)T; \u0006 = 0:02I), and\nN3=N(\u0016= (\u00001:7;\u00001:7;0:2;1:4)T; \u0006 = 0:1I). The difference between N1andN2is the\nchange of the mean vector, and the difference between N1andN3is the change in the covariance\nmatrix. As shown in Fig. 4 \u0018Fig. 6, as the probability threshold decreases, the relative volume of the\nreachable set (comparing to the volume in Fig.3(a)) decreases drastically. And our approach shows\nthat under the initial distribution N1, a large portion of the states (p >=0.8980) actually only reside\nin a small region (vol=0.03X) in the state space (as shown in Fig. 4(e)). Whereas under different ini-\ntial state distributions, the concentration region might be different (comparing Fig. 4(f) and Fig. 5(f))\nor the degree of concentration is different (comparing Fig. 4(f) and Fig. 6(e)).\n7(a) vol=1.00X; p=1.0000\n (b) vol=0.24X; p >=0.9999\n (c) vol=0.13X; p >=0.9995\n(d) vol=0.05X; p >=0.9813\n (e) vol=0.03X; p >=0.8980\n (f) vol=0.02X; p >=0.5697\nFigure 4: The system forward reach set distribution under different probability thresholds at t=1.0s\nwith initial condition N1=N(\u0016= (\u00001:7;\u00001:7;0:2;1:4)T; \u0006 = 0:02I). The color ranged from\ndark purple to light yellow indicates the density inside the polyhedral cells. The density is shown\nin logarithm magnitude. The edges colored in green indicate the boundaries of the RPM polyhedral\ncells with density below a threshold. As the probability thresholds (p) decreases, the relative volume\n(vol) of the reachable set decreases drastically. Our approach indicates under this distribution, the\nsystem state has large probability concentrating in the right bottom curve as shown in Fig. 4(f).\nE Runtime for fast safety checking\nLow density Medium density High density\ne\u000010\u0014\u001a\u0014e2e2\u0014\u001a\u0014e3\u001a\u0015e3\nVanilla Heuristic Vanilla Heuristic Vanilla Heuristic\nTime (sec) 3.1594 0.8425 3.0854 0.8122 3.0585 0.7644\n#(Rect) - 391 - 31 - 4\n#(Poly) 303 303 2 2 0 0\nIs safe? No No No No Yes Yes\nTable 2: Online safe verification comparison under different density conditions (Low / Medium /\nHigh). We measure the computation time (“Time””), number of rectangle intersections (“#(Rect)”),\nnumber of polyhedral intersections (“#(Poly)”) and whether the initial condition will avoid to drive to\nunsafe region (“Is safe?”) under each density condition with and without using the hyper-rectangle\nheuristics (“Heuristic”/“Vanilla”). As shown in Table. 2, the trajectories sampled from the initial\nstate will only reach the unsafe region under low and medium densities while won’t reach the unsafe\nregion in high density. Using heuristics can reduce the computation time in all conditions by 70%.\nWe also perform the system safety verification for the ground robot task. Specifically, we want\nto verify whether the trajectories starting from the initial condition Sinitwill drive to the unsafe\nregionSunsafe under different density conditions. We set Sinit=f\u00001:8\u0014x\u0014\u00001:2;\u00001:8\u0014\ny\u0014 \u0000 1:2;0:0\u0014\u0012\u0014\u0019=2;0:0\u0014v\u00141:0g,Sunsafe =f\u00000:5\u0014x\u00140:0;\u00000:5\u0014y\u0014\n0:0gand try three different density constraints: which are low density ( e\u000010\u0014\u001a\u0014e2), medium\n8(a) vol=1.00X; p=1.0000\n (b) vol=0.97X; p >=0.9999\n (c) vol=0.83X; p >=0.9995\n(d) vol=0.48X; p >=0.9851\n (e) vol=0.11X; p >=0.8749\n (f) vol=0.06X; p >=0.6139\nFigure 5: The system forward reach set distribution under different probability thresholds at t=1.0s\nwith initial condition N2=N(\u0016= (\u00001:7;\u00001:7;1:3;1:4)T; \u0006 = 0:02I). The color ranged from\ndark purple to light yellow indicates the density inside the polyhedral cells. The density is shown\nin logarithm magnitude. The edges colored in green indicate the boundaries of the RPM polyhedral\ncells with density below a threshold. As the probability thresholds (p) decreases, the relative volume\n(vol) of the reachable set decreases drastically. Our approach indicates under this distribution, the\nsystem state has large probability concentrating in the top left curve as shown in Fig. 5(f).\ndensity (e2\u0014\u001a\u0014e3) and high density ( \u001a\u0015e3). We measure whether the initial condition\nwill avoid to lead the system to reach the unsafe region under each density condition (“Is safe?”).\nTo illustrate how the heuristic method introduced in Sec. B.3 accelerates the computation process,\nwe also measure the computation time (“Time””), number of rectangle intersections (“#(Rect)”),\nnumber of polyhedral intersections (“#(Poly)”) and , with and without using the hyper-rectangle\nheuristics(“Heuristic”/“Vanilla”). Our program is implemented in Python with parallel computation\ndeployed on a 12-core CPU.\nAs shown in Table. 2, the trajectories sampled from the initial state will only reach the unsafe region\nunder low and medium densities, and won’t reach the unsafe region in high density. This can be\nhelpful when we are considering planning problems with density constraints. Besides, our approach\nwith hyper-rectangle heuristic can finish the online safety verification for 50 time steps in only 0.8\nseconds, which reduces 70% of the computation time comparing to the vanilla algorithm. Doing\nsafety verification for each time step only needs 0:016s, which is much smaller than the actual \u0001t\nused for the ground robot navigation benchmark ( \u0001t= 0:05s). With code-level optimization (e.g.\nwrite the program in C++ or Julia) and more CPU cores being used in parallel, our approach can\nfurther benefit for real-time applications.\nF Density (Ours, KDE, histogram, groundtruth) and reachability\nvisualizations\nHere we compare the density prediction results on all 10 benchmark examples mentioned in Table 1,\nand compare our reachable set result with other worst-case reachability tools (Convex Hull [9],\nGSG [73] and DryVR [68]) on 4 of the benchmark examples. As shown in figures in F.1, our\n9(a) vol=1.00X; p=1.0000\n (b) vol=0.99X; p >=0.9951\n (c) vol=0.96X; p >=0.9420\n(d) vol=0.91X; p >=0.8625\n (e) vol=0.77X; p >=0.5513\n (f) vol=0.46X; p >=0.2223\nFigure 6: The system forward reach set distribution under different probability thresholds at t=1.0s\nwith initial condition N3=N(\u0016= (\u00001:7;\u00001:7;0:2;1:4)T; \u0006 = 0:1I). The color ranged from\ndark purple to light yellow indicates the density inside the polyhedral cells. The density is shown\nin logarithm magnitude. The edges colored in green indicate the boundaries of the RPM polyhedral\ncells with density below a threshold. As the probability thresholds (p) decreases, the relative volume\n(vol) of the reachable set decreases drastically. Our approach indicates under this distribution, the\nsystem state has large probability concentrating in the right bottom curve as shown in Fig. 6(f), but\nis not as concentrated as shown in Fig. 4(f).\napproach can consistently achieve the closest state density distribution among other approaches\n(Kernel density, histogram), and doesn’t have a restriction for high-dimension systems (whereas the\nhistogram method cannot estimate the density for high-dimension systems like in Fig. 28 \u0018Fig. 36).\nFor the reachability comparison, different from the worst-case reachability analysis tools (Convex\nHull [9], GSG [73] and DryVR [68]), our approach can compute the density and probability for each\nof the reachable set, hence is able to tell where do states concentrate (a high probability of states\nonly reside in a small region in the state space, as shown in Fig. 39, Fig. 42, Fig. 44, Fig. 48, etc).\nOur method is more precise and informative than those worst-case reachability analysis approaches.\nMore figures can be found out in the supplementary video.\nF.1 Comparison of density prediction accuracies\n(a) Groundtruth\n (b) Kernel density\n (c) Histogram\n (d) Ours\n (e) SGPD\nFigure 7: Comparison of density prediction accuracies (Van der Pol, t=0)\n10(a) Groundtruth\n (b) Kernel density\n (c) Histogram\n (d) Ours\n (e) SGPD\nFigure 8: Comparison of density prediction accuracies (Van der Pol, t=20)\n(a) Groundtruth\n (b) Kernel density\n (c) Histogram\n (d) Ours\n (e) SGPD\nFigure 9: Comparison of density prediction accuracies (Van der Pol, t=49)\n(a) Groundtruth\n (b) Kernel density\n (c) Histogram\n (d) Ours\n (e) SGPD\nFigure 10: Comparison of density prediction accuracies (Double integrator, t=0)\n(a) Groundtruth\n (b) Kernel density\n (c) Histogram\n (d) Ours\n (e) SGPD\nFigure 11: Comparison of density prediction accuracies (Double integrator, t=3)\n(a) Groundtruth\n (b) Kernel density\n (c) Histogram\n (d) Ours\n (e) SGPD\nFigure 12: Comparison of density prediction accuracies (Double integrator, t=9)\n11(a) Groundtruth\n (b) Kernel density\n (c) Histogram\n (d) Ours\n (e) SGPD\nFigure 13: Comparison of density prediction accuracies (Kraichnan-Orszag system, t=0)\n(a) Groundtruth\n (b) Kernel density\n (c) Histogram\n (d) Ours\n (e) SGPD\nFigure 14: Comparison of density prediction accuracies (Kraichnan-Orszag system, t=20)\n(a) Groundtruth\n (b) Kernel density\n (c) Histogram\n (d) Ours\n (e) SGPD\nFigure 15: Comparison of density prediction accuracies (Kraichnan-Orszag system, t=79)\n(a) Groundtruth\n (b) Kernel density\n (c) Histogram\n (d) Ours\n (e) SGPD\nFigure 16: Comparison of density prediction accuracies (Inverted pendulum, t=0)\n(a) Groundtruth\n (b) Kernel density\n (c) Histogram\n (d) Ours\n (e) SGPD\nFigure 17: Comparison of density prediction accuracies (Inverted pendulum, t=20)\n12(a) Groundtruth\n (b) Kernel density\n (c) Histogram\n (d) Ours\n (e) SGPD\nFigure 18: Comparison of density prediction accuracies (Inverted pendulum, t=49)\n(a) Groundtruth\n (b) Kernel density\n (c) Histogram\n (d) Ours\n (e) SGPD\nFigure 19: Comparison of density prediction accuracies (Ground robot navigation, t=0)\n(a) Groundtruth\n (b) Kernel density\n (c) Histogram\n (d) Ours\n (e) SGPD\nFigure 20: Comparison of density prediction accuracies (Ground robot navigation, t=20)\n(a) Groundtruth\n (b) Kernel density\n (c) Histogram\n (d) Ours\n (e) SGPD\nFigure 21: Comparison of density prediction accuracies (Ground robot navigation, t=49)\n(a) Groundtruth\n (b) Kernel density\n (c) Histogram\n (d) Ours\n (e) SGPD\nFigure 22: Comparison of density prediction accuracies (FACTEST car model, t=0)\n13(a) Groundtruth\n (b) Kernel density\n (c) Histogram\n (d) Ours\n (e) SGPD\nFigure 23: Comparison of density prediction accuracies (FACTEST car model, t=20)\n(a) Groundtruth\n (b) Kernel density\n (c) Histogram\n (d) Ours\n (e) SGPD\nFigure 24: Comparison of density prediction accuracies (FACTEST car model, t=49)\n(a) Groundtruth\n (b) Kernel density\n (c) Histogram\n (d) Ours\n (e) SGPD\nFigure 25: Comparison of density prediction accuracies (Quadrotor control system, t=0)\n(a) Groundtruth\n (b) Kernel density\n (c) Histogram\n (d) Ours\n (e) SGPD\nFigure 26: Comparison of density prediction accuracies (Quadrotor control system, t=4)\n(a) Groundtruth\n (b) Kernel density\n (c) Histogram\n (d) Ours\n (e) SGPD\nFigure 27: Comparison of density prediction accuracies (Quadrotor control system, t=11)\n14(a) Groundtruth\n (b) Kernel density\n (c) Histogram\n (d) Ours\n (e) SGPD\nFigure 28: Comparison of density prediction accuracies (Adaptive cruise control system, t=0)\n(a) Groundtruth\n (b) Kernel density\n (c) Histogram\n (d) Ours\n (e) SGPD\nFigure 29: Comparison of density prediction accuracies (Adaptive cruise control system, t=20)\n(a) Groundtruth\n (b) Kernel density\n (c) Histogram\n (d) Ours\n (e) SGPD\nFigure 30: Comparison of density prediction accuracies (Adaptive cruise control system, t=49)\n(a) Groundtruth\n (b) Kernel density\n (c) Histogram\n (d) Ours\n (e) SGPD\nFigure 31: Comparison of density prediction accuracies (Ground collision avoidance system, t=0)\n(a) Groundtruth\n (b) Kernel density\n (c) Histogram\n (d) Ours\n (e) SGPD\nFigure 32: Comparison of density prediction accuracies (Ground collision avoidance system, t=20)\n15(a) Groundtruth\n (b) Kernel density\n (c) Histogram\n (d) Ours\n (e) SGPD\nFigure 33: Comparison of density prediction accuracies (Ground collision avoidance system, t=59)\n(a) Groundtruth\n (b) Kernel density\n (c) Histogram\n (d) Ours\n (e) SGPD\nFigure 34: Comparison of density prediction accuracies (8-Car platoon system, t=0)\n(a) Groundtruth\n (b) Kernel density\n (c) Histogram\n (d) Ours\n (e) SGPD\nFigure 35: Comparison of density prediction accuracies (8-Car platoon system, t=20)\n(a) Groundtruth\n (b) Kernel density\n (c) Histogram\n (d) Ours\n (e) SGPD\nFigure 36: Comparison of density prediction accuracies (8-Car platoon system, t=49)\nF.2 Comparison of reachable set computation among different tools\n(a) Convex Hull\n (b) GSG\n (c) DryVR\n (d) Ours\nFigure 37: Comparison of reachable set computation among different tools (Van der Pol, t=0). The\ngray dots are sampled points and blue / green / red / colored regions are reachability results\n16(a) Convex Hull\n (b) GSG\n (c) DryVR\n (d) Ours\nFigure 38: Comparison of reachable set computation among different tools (Van der Pol, t=40). The\ngray dots are sampled points and blue / green / red / colored regions are reachability results\n(a) Convex Hull\n (b) GSG\n (c) DryVR\n (d) Ours\nFigure 39: Comparison of reachable set computation among different tools (Van der Pol, t=49). The\ngray dots are sampled points and blue / green / red / colored regions are reachability results\n(a) Convex Hull\n (b) GSG\n (c) DryVR\n (d) Ours\nFigure 40: Comparison of reachable set computation among different tools (Double integrator, t=0).\nThe gray dots are sampled points and blue / green / red / colored regions are reachability results\n(a) Convex Hull\n (b) GSG\n (c) DryVR\n (d) Ours\nFigure 41: Comparison of reachable set computation among different tools (Double integrator, t=3).\nThe gray dots are sampled points and blue / green / red / colored regions are reachability results\n17(a) Convex Hull\n (b) GSG\n (c) DryVR\n (d) Ours\nFigure 42: Comparison of reachable set computation among different tools (Double integrator, t=7).\nThe gray dots are sampled points and blue / green / red / colored regions are reachability results\n(a) Convex Hull\n (b) GSG\n (c) DryVR\n (d) Ours\nFigure 43: Comparison of reachable set computation among different tools (Ground robot naviga-\ntion, t=0). The gray dots are sampled points and blue / green / red / colored regions are reachability\nresults\n(a) Convex Hull\n (b) GSG\n (c) DryVR\n (d) Ours\nFigure 44: Comparison of reachable set computation among different tools (Ground robot naviga-\ntion, t=20). The gray dots are sampled points and blue / green / red / colored regions are reachability\nresults\n(a) Convex Hull\n (b) GSG\n (c) DryVR\n (d) Ours\nFigure 45: Comparison of reachable set computation among different tools (Ground robot naviga-\ntion, t=40). The gray dots are sampled points and blue / green / red / colored regions are reachability\nresults\n18(a) Convex Hull\n (b) GSG\n (c) DryVR\n (d) Ours\nFigure 46: Comparison of reachable set computation among different tools (FACTEST car model,\nt=0). The gray dots are sampled points and blue / green / red / colored regions are reachability results\n(a) Convex Hull\n (b) GSG\n (c) DryVR\n (d) Ours\nFigure 47: Comparison of reachable set computation among different tools (FACTEST car model,\nt=20). The gray dots are sampled points and blue / green / red / colored regions are reachability\nresults\n(a) Convex Hull\n (b) GSG\n (c) DryVR\n (d) Ours\nFigure 48: Comparison of reachable set computation among different tools (FACTEST car model,\nt=49). The gray dots are sampled points and blue / green / red / colored regions are reachability\nresults\n19G Comparison between histogram-based and Liouville-based approaches\nThe advantage of learning density distribution by solving Liouville ODE is that it requires less\ntraining samples than histogram-based approaches, hence has the potential to generalize to high-\ndimension cases. To show its advantages in training efficiency and testing accuracy, we compare the\nhistogram-based approach and Liouville-based approach’s density estimation for the following sys-\ntem, under different number of training samples. To make sure we can compare to the “groundtruth”\ndensity, we manually design the system such that the state density distribution at each time step has\na closed form solution.\nConsider a 1-d system: _x=\u0000x2with initial states ranged from [0;1]. Under uniformly distributed\ninitialization, the system dynamics x(t)and density distribution \f(x;t)(here\f(x;t)denotes the\ndensity at time tat locationx) can be directly written out in the closed form:\n\u001ax(t) = 1=(C+t)\n\f(x;t) = 1=(1\u0000x\u0001t)2(26)\nwhere the parameter Ccan be derived from the initial condition x(0) =x0= 1=C.\nFigure 49: KL Divergence (comparing to groundtruth state density) for histogram-based approach\nand Liouville-based approach in different numbers of training samples. The groundtruth state den-\nsity is computed analytically in closed form. We use 10000, 100000, 1000000 training samples\nfor the histogram-based approach and use only 10000 training samples for the Liouville-based ap-\nproach. The KL divergence is computed on a separate testing set (200 samples). Comparing to the\nhistogram-based approach, the Liouville-based approach can achieve a smaller KL divergence while\nusing 0.01X number of training samples.\nWe then use histogram-based and Liouville-based approach to estimate the state density for this\nsystem. We uniformly sample initial states and generate 1000000 trajectories using ODE45 solver.\nWe use 10000, 100000 and 1000000 training samples for the histogram-based approach and use only\n10000 training samples for the Liouville-based approach, then we estimate the density on a separate\ntesting set of trajectories using nearest neighbor interpolation. At each time step, we measure the\nestimation accuracy on the test set by computing the KL divergence to the groundtruth density.\nAs shown in Fig. 49, histogram-based approach needs lots of samples to accurately approximate a\ngood distribution (the KL divergence converges to zero at each time step as the number of samples\nincreases), where our approach can learn the density distribution with the lowest KL divergence\n20using just 0.01X of the sampled trajectories. This shows the advantage of solving Liouville ODE to\nestimate the state density.\nH Comparison with state-of-the-art worst-case reachability approaches\nWe compare our approach with three state-of-the-art worst-case reachability methods: Sher-\nlock [75], Verisig [39] and ReachNN [40]. We use the official implementation of Verisig and\nReachNN which focus on reachable set computation for neural network control systems (NNCS),\nand use the re-implementation of Sherlock from [6], which is for neural network verification.\nTo make a fair comparison, we set a timeout limit of six hours for all approaches. Among all the\nfour datasets that our method has computed, Sherlock can solve for the reachable sets for the datasets\n“Double integrator”, “Ground robot navigation” and “FACTEST car tracking system”, and Verisig\nand ReachNN can only calculate for the “Double integrator” dataset - Verisig encounters numerical\nissue on this dataset at first due to the large initial set, and we have to divide the initial set to smaller\nsets and run the program multiple times in parallel to compute for the reachable sets. Similar in\nSec. 4.3, we measure the reachable sets by computing the volume of the reachable sets relative to\nthe volume of the convex hull of the sampled points.\nWe use different networks when doing reachability analysis, because all those methods have differ-\nent requirements for the analyzed system:\n(a) The RPM used in our approach is doing reachability analysis for ReLU-based NNs. For\nthe “Double Integrator” system, the controller is another ReLU-based NN that has a clip\nfunction at the output (to rectify the control output between [\u00001;1])\n(b) The Sherlock approach we used in [6] can only work with ReLU-based NN (not NNCS).\nThus we used the same NN used in (a) and conducted the experiments. Since we only\ncompute for the reachable set, we just collect the flow map estimation \b!part of this NN\n(i.e. we did not need to use the density estimator part of the NN).\n(c) Verisig can only work with a Neural Network Controlled System (NNCS) with\nSigmoid/Tanh-based NN controllers. Thus we re-trained a Tanh-based NN controller (us-\ning the same number of hidden layers and hidden units) to reproduce the output of the\noriginal controller in (a) and use this new controller to do reachability analysis. We verified\nthat the L2 error between the Tanh-based NN controller and the original controller is less\nthan 0.001 on the testing set, and we also inspected the trajectories generated using these\ntwo controllers and cannot find a substantial difference.\n(d) ReachNN can work with NNCS that has Sigmoid/Tanh/ReLU-based NN controllers. How-\never, it cannot directly process the controller we had in (a) because the controller in (a) has\na clip function at the output to rectify the control output between [\u00001;1]. Therefore, we\ntrained another ReLU-based NN controller that does not have that clip function to repro-\nduce the output of the original controller in (a). We use this newly trained controller to do\nreachability analysis in ReachNN.\nAs shown in Fig. 50 \u0018Fig. 53, in the “Double integrator” experiment, all of the three worst-case\nreachability analysis methods can only over-approximate the reachable sets of the system, with the\nreachable volume increasing over time. The approximation error for Versig and ReachNN will\nseverely accumulate, hence the corresponding reachable sets gradually occupy the whole figure\n(where the growths is 32.24X for Verisig and 67.96X for ReachNN respectively), whereas our ap-\nproach estimated reachable sets have volume less than the convex hull volume, and can reflect the\nconvergence of the majority of the system states owing to the ability to predict the state density. For\nhigher dimension benchmarks like “Ground robot navigation” and “FACTEST car tracking system”\n(as shown in Fig. 54 \u0018Fig. 57), only our approach and Sherlock are able to compute the reachable\nset under the timeout limit. Due to the high dimensionality, Sherlock’s estimated volume grows\ndramatically over time (16.51X for the “Ground robot navigation”, and 42.60X for the “FACTEST\ncar model”), while our approach still gives more compact reachable sets. These observations il-\nlustrate the advantages of our approach in precisely estimating the system reachable sets as well\nas the state density distribution. One advantage of Sherlock over ours is that it can also solve for\nother benchmarks listed in Table. 2, where our approach cannot solve due to the numerical issues in\n21RPM. Another limitation is that our approach only solves for NN with ReLU activations, which is\nagain a restriction inherited from RPM. We believe combining our learning framework with a more\nadvanced exact reachability tool will resolve this issue in the future.\n(a) Sherlock\n (b) Verisig\n (c) ReachNN\n (d) Ours\nFigure 50: Comparison of reachable sets (Double integrator, t=0)\n(a) Sherlock\n (b) Verisig\n (c) ReachNN\n (d) Ours\nFigure 51: Comparison of reachable sets (Double integrator, t=3)\n(a) Sherlock\n (b) Verisig\n (c) ReachNN\n (d) Ours\nFigure 52: Comparison of reachable sets (Double integrator, t=7)\n(a) Sherlock\n (b) Verisig\n (c) ReachNN\n (d) Ours\nFigure 53: Comparison of reachable sets (Double integrator, t=9)\n22(a) Sherlock, t=0\n (b) Sherlock, t=20\n (c) Sherlock, t=30\n (d) Sherlock, t=40\nFigure 54: Sherlock results (Ground robot navigation, t=0, 20, 30, 40)\n(a) Ours, t=0\n (b) Ours, t=20\n (c) Ours, t=30\n (d) Ours, t=40\nFigure 55: Our results (Ground robot navigation, t=0, 20, 30, 40)\n(a) Sherlock, t=0\n (b) Sherlock, t=20\n (c) Sherlock, t=30\n (d) Sherlock, t=40\nFigure 56: Sherlock results (FACTEST car model, t=0, 20, 30, 40)\n(a) Ours, t=0\n (b) Ours, t=20\n (c) Ours, t=30\n (d) Ours, t=40\nFigure 57: Our results (FACTEST car model, t=0, 20, 30, 40)\n23"    },    {        "title": "2109.09895v1.Ultrafast_excitation_and_topological_soliton_formation_in_incommensurate_charge_density_wave_states.pdf",        "content": "Ultrafast excitation and topological soliton formation in incommensurate charge\ndensity wave states\nXiao-Xiao Zhang,1Dirk Manske,2and Naoto Nagaosa3, 4\n1Department of Physics and Astronomy & Stewart Blusson Quantum Matter Institute,\nUniversity of British Columbia, Vancouver, BC, V6T 1Z4 Canada\n2Max Planck Institute for Solid State Research, 70569 Stuttgart, Germany\n3Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan\n4RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan\nTopological soliton is a nonperturbative excitation in commensurate density wave states and con-\nnects degenerate ground states. In incommensurate density wave states, ground states are continu-\nously degenerate and topological soliton is reckoned to be smoothly connected to the perturbative\nphason excitation. We study the ultrafast nonequilibrium dynamics due to photoexcited electron-\nhole pair in a one-dimensional chain with an incommensurate charge density wave ground state.\nTime-resolved evolution reveals both perturbative excitation of collective modes and nonperturba-\ntive topological phase transition due to creating novel topological solitons, where the continuous\ncomplex order parameter with amplitude and phase is essential. We identify the nontrivial phase-\nwinding solitons in the complex plane unique to this nonequilibrium state and capture it by a low-\nenergy e\u000bective model. The perturbative temporal gap oscillation and the solitonic in-gap states\nenter the optical conductivity absorption edge and the spectral density related to spectroscopic\nmeasurement, providing concrete connections to real experiments.\nI. INTRODUCTION\nTopological soliton is a subject of intense interest, gov-\nerning various types of novel phenomena such as charge\nfractionalization and spin-charge separation[1, 2]. It is re-\ngarded as the excited state connecting degenerate ground\nstates, e.g., two di\u000berent phases of bond dimerization in\nthe representative half-\flled polyacetylene where it can\nbe detected by induced optical absorption[3]. Commen-\nsurate density waves in general have the similar situation\nwith discrete multiple ground states and associated soli-\ntonic structures. In the incommensurate charge-density-\nwave (ICDW) state, however, the broken translational\nsymmetry renders the ground states continuously degen-\nerate: the phase Goldstone boson constitutes the low-\nlying excitations[4, 5], and soliton is usually not well-\nde\fned[6].\nOn the other hand, signi\fcant progress in the pre-\ncise access to ultrafast light-induced dynamics[7{13] {\nespecially time-resolved spectroscopic techniques includ-\ning angle-resolved photoemission spectroscopy (ARPES),\noptical bulk re\rection or thin-\flm transmission mea-\nsurements, electron di\u000buse scattering[14, 15], terahertz\npump-probe scanning-tunneling microscopy (STM)[16,\n17], resonant inelastic X-ray scattering[18, 19], etc. {\nbrings to the forefront nontrivial systems with broken\nsymmetries via the nonlinear phenomena naturally aris-\ning from intense pulses and the subsequent nonequilib-\nrium evolution. An intriguing question arises: What\nhappens to the ICDW and if any soliton in the nonequi-\nlibrium? A closely related and immediate need to the\ncommunity is elucidating the real-time dynamics and\npossible nonlinear responses against optical disturbance\nin some prototypical low-dimensional quantum systems\nwith especially broken continuous symmetries and or-\ndered phases. This is in contrast to the \feld of Floquetengineering of topological phases developed in the past\ndecade, where the photon's periodic driving in\ruence is\ntaken into account in another limit of steady states[20].\nTo address these and motivated by the recent ex-\nperimental interest in bulk and low-dimensional charge-\nordered systems, especially in relation to photoinduced\nphenomena[21{37], we study the photoexcited dynam-\nics of an epitome, the one-dimensional ICDW[6, 38, 39].\nThe spatiotemporally varying complex order parame-\nter and the transport and spectroscopic implications re-\nveal exceptionally rich photoinduced nonequilibrium dy-\nnamics { the perturbative excitation and propagation\nof collective amplitude and phase modes and the topo-\nlogical phase transitions due to highly nontrivial phase-\nwinding solitons even for ICDW. The latter emerges as\na novel photoinduced topological phenomenon unique to\nthe nonequilibrium and nonlinear system evolution.\nII. MODEL SYSTEM WITH BROKEN\nCONTINUOUS SYMMETRY\n!\"#2∆\nFIG. 1. Optical irradiation locally onto the yellow region of a\nchain with ICDW order close to one-third-\flling will induce\nultrafast nonequilibrium and nonlinear dynamics. Inset: we\nmainly focus on the consequence of photoinduced electron-\nhole excitations across the spectral gap 2\u0001.\nFig. 1 schematically illustrates our basic setting: anarXiv:2109.09895v1  [cond-mat.mes-hall]  21 Sep 20212\nICDW chain is formed as the equilibrium ground state\nfrom the electron-phonon interaction, which originally\nhasNions uniformly spaced at lattice constant a. To be\nspeci\fc, we study the Peierls-type electron-phonon cou-\npling Hamiltonian[3, 39, 40]\nH=Hel(~ u) +Hphn(~ u): (1)\nThe electronic part Hel(~ u) =PN\ni;\u001b(\u0000tic†\ni;\u001bci+1;\u001b+ h:c:)\nincludes spin- \u001bdegeneracy and electron-phonon coupling\n\u000bthrough the ion-dependent hopping ti=t[1\u0000\u000b(ui+1\u0000\nui)]; the kinetic and potential energies of the phonon dis-\nplacement \feld ~ ugiveHphn(~ u) =P\niM\n2_u2\ni+\u0014\n2(ui\u0000ui+1)2\nwith the ion mass Mand elastic sti\u000bness \u0014. Coprime\n\flling fraction representatively asp\nq=67\n199\u00191\n3with a\nlarge denominator is used for a long chain N= 10q,\napproaching the incommensurate limit. In equilibrium,\nthe phonons condense at the Kohn anomaly momentum\nQ= 2kFand form a CDW with ionic displacement\nu(0)(x) =Usin (Qx+\u001e0). Note that the phenomena\nof our interest are generic to any ICDW systems, be-\ncause it is the incommensurability, rather than any spe-\nci\fc nonuniversal \flling, that is essential to the physical\nmechanism. We provide extensive examples of other \fll-\ning fractions in Supplemental Material (SM)[41].\nThis incommensurate system with broken translational\nsymmetry and the associated complex and continuous or-\nder parameter Ois fundamentally di\u000berent from any dis-\ncrete order like in polyacetylene[3]. In Fig. 2, we schemat-\nically describe the signi\fcant distinction of ICDW from\nany half-\flled or general commensurate situation, which\nfollows from the spontaneous U(1) symmetry breaking.\nNote that only with the incommensurability, which is the\nkey ingredient in our study, the freely varying phase de-\ngree of freedom becomes possible. And it, cooperatively\ntogether with the amplitude, plays the key role to gener-\nate the novel nonequilibrium topological phase.\nIII. NONEQUILIBRIUM TIME EVOLUTION\nExperimentally an intense light pulse is exerted to a\nrelatively local region at time T < 0 and drives the sys-\ntem via light-matter interaction; notwithstanding an in-\ncomplete knowledge of this highly nonlinear and nonequi-\nlibrium optical disturbance, we directly focus on the most\nprobable and common photoexcitation { an electron-hole\npair is created at time T= 0 via absorbing the pho-\nton across the spectrum with a CDW gap 2\u0001[42, 43].\nThe phonon dynamical period \u0018T0= 2\u0019=! 0within\nan excited energy surface is typically more than one or-\nders of magnitude shorter than the interband sponta-\nneous decay ranging from several hundred femtoseconds\nto picoseconds[23, 42, 44, 45]. Encoding the pumping\ne\u000bect, we thus take this T= 0 state as the starting\npoint as well as a given nonequilibrium occupation dur-\ning the system's subsequent evolution before the possi-\nble (non)radiative recombination decay; alternatively, apossibly persistent photoirradiation can maintain the ex-\ncited state beyond free decay time[46]. Justi\fed by the\nhierarchy and separation of two distinct time scales {\nelectronic and ionic (phonon), such an adiabatic approx-\nimation and associated instantaneous electronic states\nare physically the most relevant since we are interested\nin the characteristic dynamics or time scale directly re-\nlated to the phonons' nonequilibrium behavior. In par-\nticular, electrons will in general rapidly relax and adjust\nto the instantaneous phonon coordinates at an orders-of-\nmagnitude faster time scale due to the mass di\u000berence;\nas the typical phonon frequency !0=p\u0014\nM\u001c2\u0001, o\u000b-\ndiagonal electronic density-matrix elements \u001amnoscillate\nmuch faster than the lattice and hence negligible electron-\nphonon energy transfer can interfere the phonon dynam-\nics. Indeed, in the following the collective excitations\nand the topological transition with soliton formation are\ninherently in terms of the phonon coordinate dynamics,\nupon which we de\fne and study the complex order pa-\nrameter.\nIn Appendix A, we outline the simulation of this time-\ndependent system. Due to the scaling property, the\nnonequilibrium system possesses two primary parame-\nters: the coherence length \u0018=vF\n\u0019\u0001speci\fes the electron-\nphonon system and can be varied from the coupling \u000bin\nEq. (1) and the variationally determined CDW strength\nU; the ratio\u001f=\u001c=T0between minimal time step \u001cand\nphonon period for a total evolution time T= 1000\u001c.\nTo faithfully simulate the evolution, we set typically\n\u001f= 1=20\u0019, i.e., phonons complete a full oscillation cycle\nafter more than 60 steps. Thus, as T0\u001860fs in realistic\nsystems, we have \u001c\u00181fs andT \u0018 1ps within or compa-\nrable to the foregoing electronic decay time. Henceforth,\ntimeTwill be measured in units of \u001c. Lastly, pinning\nthe outset of traveling collective modes and solitons, we\nput an initial tiny perturbation to the ionic displacement\n\feld at a few points around the chain center, which as\nwell mimics the local photoirradiation to the chain in\nFig. 1.\nThe most important feature of the nonequilibrium sys-\ntem consists in the instantaneous complex order param-\neterO. As the CDW condensate at momentum Qis\nstirred by the photoexcitation both perturbatively and\nnonperturbatively, it in general becomes highly inhomo-\ngeneous\nO(x) =m(x)ei\u001e(x)=\u0002q0\n\u0000q0dquq+Qei(qx+\u001e0)(2)\nwithuqthe momentum-space transform of the instan-\ntaneous phonon \feld u(x) andq0= 10\u0019=qa a cuto\u000b\nmomentum that stably captures the long-wavelength be-\nhavior. Shown in Fig. 3, time-dependent real \felds\nm(x);\u001e(x), which are constant m=U=2;\u001e= 0 at equi-\nlibrium, become respectively the mass (amplitude) and\nthe phase. While they are decoupled at linear order,\nnonlinear and mutual excitations are naturally possible\nand incorporated in the simulation, thereby fully encod-\ning both the photoinduced perturbative and nonpertur-3\n𝑂0(a) Half-filled case(c) Incommensurate caseamplitudon\n(b) General commensurate caseIm𝑂Re𝑂𝑚𝜙amplitudonIm𝑂Re𝑂𝑚𝜙phason00Real-axis of 𝑂Complex 𝑂-planeComplex 𝑂-plane\nFIG. 2. Conceptual illustration in the complex order parameter O-plane of the CDW symmetry breaking pro\fles and collective\ndegrees of freedom. (a) Half-\flled case leads to a real order parameter of only the amplitude that can take two values denoted\nby the orange dots. (b) In the general commensurate case, illustrated as a 1 =3-\flled case for presentational brevity, Otakes\nthree discrete possible values in the complex plane. Only amplitude can \ructuate as a collective excitation. (c) For an\nincommensurate case, Ocan continuously take any value on the circle of a given amplitude. Both amplitude and phase can\n\ructuate as collective excitations.\nbative e\u000bects. We inspect below these two major aspects\nspeci\fc to ICDW: collective excitation of intrinsic gap-\nless phason and gapped amplitudon exhibits rich physical\nevolution; formation of exceptional \u0019-winding phase soli-\ntons, partially captured by low-energy models, realizes\nnew nonequilibrium topological-winding phases.\nIV. COLLECTIVE EXCITATIONS\nIn Figs. 3(a), we plot the spatiotemporal pro\fle of am-\nplitudemand phase\u001eafter the initial photoexcitation.\nInstead of obvious wave propagation, Fig. 3(a1) shows a\nrobust fast temporal oscillation. This should owe to the\ngapped amplitudon that gives a vanishing group veloc-\nity d!amp=dkjk=0and determines the observed standing-\nwave oscillation frequency !amp(k= 0). The spacetime\nV-shape indicated by the green arrows in Fig. 3(a2) ex-\nactly depicts a phason 'light cone' emanating from the\nchain center where solitons germinate and one can read\nthe gapless phason velocity vphsfrom the envelope shape.\nThe phason wavefront initially splits the spatial pattern\nand then pushes the phase from the center towards both\nedges as time elapses. Note that in the polyacetylene\ncase only the amplitude mode exists and the continuous\nphason degree of freedom associated with the U(1) order\nparameter is completely absent. The rich collective ex-\ncitation behavior in Fig. 3 is only made possible by the\nincommensurate nature.\nFrom the mean-\feld and random-phase-approximation\ncalculation of the collective excitations for ICDW[4], we\nobtain the phason velocity vphs=vF\r1=2!Q=2\u0001 and the\namplitude frequency !amp(0) =\r1=2!Qwith the dimen-\nsionless electron-phonon coupling \r= (ln 2\u0019\u0018\"F=vF)\u00001\nand the uncoupled acoustic phonon frequency !k=\n2!0jsin1\n2kajevaluated at k=Q. From another view-\npoint of practically making a CDW approach incommen-\nsurability with large \flling denominators, the phason and\namplitudon originate from the rapidly softened lowest\ntwo optical phonons in the Kohn anomaly on the insu-\nlating side of the CDW transition[47, 48], which simi-larly suggests constant vphs=!0aand!amp(0)=!0for a\n\fxed coherence length \u0018as the ICDW expression im-\nplies. We con\frm this general expectation by varying\nelastic sti\u000bness in our simulation. Also note that the\nobserved!amp\u0018!0for our incommensurate system\nclose to1\n3-\flling can be understood via the simpler com-\nmensurate1\n3-\flling case, where the amplitudon, origi-\nnally as the second longitudinal optical phonon branch\nof frequencyp\n3!0atq= 0 for a triatomic chain, is\nlowered around !0[48, 49]. On the other hand, we\ncan further tune the condensate at di\u000berent coherence\nlengths, e.g., \u00181;2=a\u00193;9, for which the theory predicts\nvphs2=vphs1\u00192:5 and!amp2(0)=!amp1(0)\u00190:86. As this\nmean-\feld prediction can underestimate \ructuations and\nis valid as to order of magnitude in one dimension, it\nis in reasonable agreement with vphs2=vphs1\u00192:1 and\n!amp2(0)=!amp1(0)\u00190:87 measured directly from our\nsimulation (see SM[41]).\nV. TOPOLOGICAL PHASE TRANSITION AND\nSOLITON\nApart from the collective excitations as a background,\nthere lie more drastic nonperturbative features at T&80:\nthe topological soliton-pair formation and its subsequent\nmovement are consistently visible in Figs. 3(a1,a2) re-\nspectively as considerable amplitude m-reduction and\nrapid phase \u001e-jump. After the initial period when m(x)\nsoftens till zero at the chain center, the soliton pair is cre-\nated and accompanied by a topological transition of the\ntotal phase winding along the chain from zero to \u00062\u0019.\nPlotted in Figs. 3(b,c), in the complex order parameter\nO=mei\u001e-plane, this transition means that a photoin-\nduced initially small closed O(x)-trajectory deforms to\neventually encircle the origin. See also SM[41] for ex-\ntended examples, supporting the topological robustness\nof this origin-winding transition and the insigni\fcance of\nthe speci\fc nonuniversal trajectory shape. We will later\nrevisit the process with spectral and transport probes,\nwhere the associated solitonic states formed inside the4\n0.10.30.50.7m1 500 1500 19900300700999\n(a1)xT\n-1.0-0.500.51.0ϕ/2π1 500 1500 19900300700999\n(a2)xT\n-0.5 0.5-0.010.01\n(b1)O-plane\n-0.5 0.5-0.010.01\n(b2)O-plane\n-0.5 0.5-0.040.04\n(b3)O-plane\n05001500199000.5\n(c1)\nx-0.0300.03\nm ϕ05001500199000.5\n(c2)\nx-0.300.3\nm ϕ05001500199000.5\n(c3)\nx0π2π\nm ϕ\nFIG. 3. (a) Amplitude and phase of the nonequilibrium spatiotemporal ICDW complex order parameter O(x;T) =mei\u001eat\ncoherence length \u0018\u00199a. Horizontal (vertical) axis corresponds to the whole chain (evolution time). Orange vertical lines\natx= 333;801 in (a1) are used in Fig. 5(c). (b1-3) Complex O(x)-trajectories as xtraverses the whole chain respectively\natT= 71;79;185 illustrate the topological transition near T= 80 and the later stabilized solitons. Note the di\u000berence in\nscales between real and imaginary axes. (c1-3) m(x)- and\u001e(x)-pro\fles in one-to-one correspondence to (b1-3). Parameters are\n~=a=t= 1;!0= 0:006;\u000b= 0:55.\nCDW gap become clear. After the initial topological\ntransition to a nontrivial phase, the horizontal stripes in\nFig. 3(a2) indicate the general possibility of occasional\ntopological switchings between di\u000berent-winding topo-\nlogical phases. Note that because of the nonintegrable\nnature of the system, the particular topological-switching\npattern and long-time soliton behavior are nonuniversal\nand dependent on the model details. In reality, solitons\nwith opposite charge can possibly attract each other and\neventually get damped or annihilated; here in the ab-\nsence of an explicit decay channel, they persist and can\ncontingently collide.\nThe exact topological winding quantization mostly\nconsists of two separate \u0019-solitons [Figs. 3(b3,c3)], where\namplitudemis reduced but remains \fnite at the soliton\nposition, distinct from the amplitude/mass sign-change\nsolitons or alike edge states. Instead, the novel nonequi-\nlibrium topological soliton herein is stabilized by taking\nadvantage of the phase freedom and its winding quan-\ntization. Only right at a topological phase transition,\nincluding the initial one [Figs. 3(b2,c2)] and later topo-\nlogical switchings between \u00062\u0019[see Figs. S11,S12 in SM]\ncan the solitons become singularly sharp and m!0.\nWe also show in SM higher-winding transitions as a gen-\neral feature of more than one electron-hole photoexcita-\ntions. The emergence of (multiples of) \u0019-solitons needs\nto be understood as a nontrvial feature of the ICDW,because it naturally holds the continuous phase freedom\naforementioned in contrast to polyacetylene's staggered\norder or any commensurate case. Instead, the incom-\nmensurability eventually suppresses all the possible com-\nmensurate multiples of 2 p\u0019=q -windings[50, 51] but singles\nout\u0019-phase-kink solitons to compose the total winding\nquantization for the nonequilibrium ICDW, which is fun-\ndamentally di\u000berent from polyacetylene's amplitude soli-\nton.\nm\n0m0\n(a)\nx0 -x0m0(1-δ)FWHMϕ\n0Φ\nFIG. 4. (a) Soliton pro\fle centered at x= 0 of amplitude m\nand phase\u001ewith relative weakening \u000e, phase-winding \b, and\nfull-width-half-maximum (FWHM) 2 x0= 2\u0015sech\u000011\n2. (b)\nSoliton energy \u000f(\u0015;\u000e;\b) solved from the continuum model.\nGreen line: Jackiw-Rebbi zero mode.\nWe can derive from Eq. (1) a low-energy continuum\nmodel to partially capture the in-gap soliton state (see\nAppendix B)\nh=vF(\u0000i@)sz+\f[O(x)s++O\u0003(x)s\u0000]; (3)\nwhere\f= 2\u000bvF=a. The pseudospin sz;2s\u0006=sx\u0006isy5\ncombines the electrons around two Fermi points and real\nspin degeneracy is assumed. Resembling the realistic pro-\n\fles like Fig. 3(c), the complex order parameter O(x)\nenters for a soliton as m(x) =m0(1\u0000\u000esechx\n\u0015);\u001e(x) =\n\b\n2tanh(x\n\u0015+ 1), where \u0015;\u000e;\b control respectively the soli-\nton width, amplitude weakening and phase winding in\nFig. 4(a). Scaling properties of hmakes it su\u000ece to set\nvF=m0=\f= 1 and have x;\u0015measured in units of\n\u0015\u0003=vF=m0\f=a=2\u000bm0\u001960afor Fig. 3. This soliton\ncharacteristic scale \u0015\u0003, in comparison to the condensa-\ntion momentum Q, signi\fes the length scale separation\nbetween the slowly varying envelope and the rapidly os-\ncillating CDW pattern and hence justi\fes the continuum\ntheory and the cuto\u000b-independent long-wavelength de-\nscription of the inhomogeneous O. Although this low-\nenergyhsu\u000bers spectral pollution[52], we are able to\nsolve its bound-state energies \u000f2(\u0000m0;m0) by the\ncompound-matrix method relying on a topological invari-\nant of the di\u000berential system[53].\nIn Fig. 4(b), we plot energy \u000flowest in absolute\nvalue and \fnd it monotonically increasing with \u000e;\b,\nwhich is the low-energy excitation around Fermi points\nand enables energy gain from soliton formation. As\n\u0015shrinks, the variation against \u000e(\b) becomes \ratter\n(steeper) and the \u000f= 0 solutions move more aligned\nto \b =\u0019. This is asymptotically consistent with the\nJackiw-Rebbi zero mode, because solutions become in-\nsusceptible to \u000ewhen\u0015!0. For typical smooth soli-\ntons\u0015.1 [Fig. 3(c3)], though \fnite \u000ecosts elastic en-\nergy due to the amplitude deformation, it helps to form\nnear-zero-energy solitons: phase-winding and amplitude-\nreduction together assist in stabilizing the photoexcita-\ntion. Single 2 \u0019-solitons can have nearly zero energy only\nwhen\u0015is long enough, which indeed is the rare case\nat nonequilibrium because of the higher elastic-energy\npenalty accumulated along the wider distortion. The to-\ntal 2\u0019-winding is more commonly achieved by two sep-\narate sharper kinks of \u0019-winding [Fig. 3(c3)], because,\nexcept the Jackiw-Rebbi resemblance for sharper soli-\ntons, the most energy gain from bound-state formation,\njm0\u0000\u000f(2\u0019\u0000\b)j+j\u0000m0\u0000\u000f(\b)j, is maximized around\n\b =\u0019following the monotonicity of \u000f. Therefore, the\nrobust nonequilibrium topological soliton in ICDW, for\nenergetic and topological reasons, nontrivially embodies\na\u0019-winding spinon soliton with neutral charge accumu-\nlation as per the single occupation of electron-hole pair.\nVI. TRANSPORT AND SPECTRAL PROBES\nTime-resolved optical conductivity \u001band spectral\nfunctionAconstitute the comprehensive experimental\naccess towards dynamical properties of this nonequi-\nlibrium system. Conductivity can be extracted from\ntransport and terahertz pump-probe optical re\rection\nor transmission measurement[8, 54{59]. Spectral weight,\nwhile directly corresponding to local or integrated den-\nsity of states and hence STM tunneling I-V spectra thatcan resolve in-gap solitons[16, 17, 60{62], is also a tar-\nget of spatially-resolved nano-ARPES using synchrotron\nlights[7, 63, 64]. We derive in Appendix C\n\u001b(\";T) =\u0000ie2\nLX\nmn(fn\u0000fm)(\"n\u0000\"m)jxmnj2\n(\"+ i\u0011) +\"n\u0000\"m(4)\nwith the instantaneous electronic spectrum and states\nf\"i;jiigand the nonequilibrium occupation fidi-\nrectly from the simulation and the position operator\nxmn=hmjxjnifor the aperiodicity after photoexcitation.\nAlongside, we calculate the time- and space-dependent\nA(\";x;T ) =\u00001\n\u0019=Gr(\";x;x;T ) from the retarded elec-\ntronic Green's function Gr.\nWe observe in Fig. 5(a1) that below the dark-red con-\nductivity continuum due to across-gap excitations, there\nlies the region ( \".2\u0001) exhibiting reddish nonzero sig-\nnals only present due to the photoexcitation and in-gap\ntopological states. As the evolution starts, the excited\nelectron and hole at the conduction/valence band edges,\nabsorb photon and transition to many nearby empty\nor \flled states [earlier than but similar to Fig. 5(b1)];\nhence the dense reddish signal starting from \"= 0 in\nFigs. 5(a1,a2) within the conductivity gap for a short\ntimeT.50, prior to the soliton maturation or topo-\nlogical transition T\u001980 in Fig. 3. Within this period,\naround the chain center the originally uniform band edges\ndeform towards the gap and prepare the soliton germi-\nnation [Figs. 5(b1-2)]. Later, the electron-hole states\ngradually split o\u000b the band continuum and move deep\ninside the gap, developing the spatially separated dou-\nble\u0019-winding structure and \fnally transforming into the\nstabler solitons around the gap middle [Figs. 5(b2-4)];\nconductivity now mainly originates from the partially\n\flled soliton states and those unstable states around band\nedges { for instance, the upward thin-line (50 <T < 120)\nemanating from the \"= 0 bottom in Fig. 5(a1) is pri-\nmarily attributed to the increasing energy separation be-\ntween solitons and band edges.\nDuring the later time evolution, while stabilized soli-\ntons could still \ructuate, band-edge states constantly\nbear variation in energy and spectral weight [Figs. 5(b2-\n6)], which can even slightly detach transient states tem-\nporarily into the gap, complicating the conductivity sig-\nnals. However, major in-gap signals roughly around the\nconductivity gap midpoint \"\u0019\u0001 in Fig. 5(a1) clearly re-\n\rect the contribution from the solitons around the mid-\ndle of the CDW gap and the apparently oscillating band\nedges seen through Figs. 5(b). This regular gap oscilla-\ntion becomes conspicuous in the time-dependent spectral\nfunction at a site away from the soliton trajectory, oth-\nerwise obscured by the soliton contribution at a site tra-\nversed by the soliton [Figs. 5(c)]. Correspondingly, it is\nclearly (vaguely) picked up in Fig. 5(a1) by the dark-red\ncontinuum (the reddish in-gap signals). This temporal\noscillation in both \u001bandAis the direct consequence of\nthe photoexcited amplitudon and perfectly matches the\noscillation in Fig. 3(a1). Furthermore, Fig. 5(c2) pre-\nvails in excluding the soliton spectral weight whereas6\n0 300 700 99900.09\n(a1)\nTε\n0.10.50.9T=35T=80T=2550.03 0.07\n-1\n-3\n-5(a2)ε\n1500 1500 19900.961.08(b1)\nxε\n1500 1500 19900.961.08(b2)\nxε\n1500 1500 19900.961.08(b3)\nxε\n1500 1500 19900.961.08(b4)\nxε\n1500 1500 19900.961.08(b5)\nxε\n1500 1500 19900.961.08(b6)\nxε\n0 300 700 9990.961.08\n(c1)Tε\n0 300 700 9990.961.08\n(c2)Tε\n0.10.50.9\nFIG. 5. (a1) Real part of the time-dependent optical conductivity <\u001b(\";T). (a2) Snapshots of log <\u001b(\";T) atT= 35;90;255.\n(b1-6) Space-dependent spectral function A(\";x) respectively at times T= 65;75;90;100;200;700. (c1-2) Time-dependent\nspectral function A(\";T) at sitesx= 801;333 corresponding to the orange lines in Fig. 3(a1). All quantities, calculated from\nthe case of Fig. 3 with \u0011= 10\u00003tcomparable to the level spacing, are scaled with respect to their maxima. See SM for extended\nearly-time data.\nFig. 5(c1) recapitulates the soliton energetic evolution\nthat \frst leaves the band edges and eventually stabilizes\ninside the gap. The temporally light-heavy-light-heavy\nin-gap weight in Fig. 5(c1) exactly records the 8-shaped\nsoliton trajectory crossing the orange line in Fig. 3(a1).\nVII. DISCUSSION\nHere, we comment on the physical validity of the ap-\nproach we adopt in relation to the pump-probe-like spec-\ntroscopy, admittedly one of the relevant and practical\nexperiment for realizing the proposal of nonequilibrium\nphenomenon. We make use of the adiabatic approxi-\nmation, which has been widely used and is physically\njusti\fed for the ion/phonon nonequilibrium dynamics of\nour interest. In this regard, at the time scale or resolu-\ntion for experimentally monitoring the leading nonequi-\nlibrium dynamics of interest, the system has very rapid\nelectronic transition: the electronic states e\u000bectively ad-\njust to the phonon coordinates instantaneously, during\nwhich phonons mainly remain intact. Explicitly incor-\nporating the electromagnetic pump-probe \feld does not\nalter the physical content of the optical conductivity and\nspectral function at this time scale of practical and the-\noretical interest. This justi\fes that instantaneous eigen-\nstates su\u000ece to capture the experimentally relevant time-\ndependent signals.As for the initial optical disturbance or pumping that\ngives rise to the electron-hole excitation, the optical ab-\nsorption especially happens at a much higher energy\nscale with the lower bound given by the large CDW\ngap, where the transition time is correspondingly much\nshorter than the interested phonon dynamics. There-\nfore, again at the time scale relevant to the nonequi-\nlibrium dynamics of our interest, encoding the optical\npumping e\u000bect by the electron-hole excitation is su\u000ecient\nand physically most relevant. This is also in the same\nspirit as the established quenched dynamics in, e.g., su-\nperconducting systems[65, 66]. The emphasis is directly\nput on the instant change of the pairing instead of an\nall-encompassing time-dependent electromagnetic pump-\nprobe description, although experimentally it is realized\nin pump-probe spectroscopy. Such theories capture the\nexperimentally most relevant collective excitations, such\nas the twice-gap oscillations in both conventional and\nunconventional superconductors. We hence expect our\ntheory to play a similar role in capturing the important\nnonequilibrium phenomena in a straightforward manner\nand to stimulate further interest and study.\nOn the other hand, the explicit pump-probe approach\nwould bring about much theoretical and technical dif-\n\fculty. In the present case, as incommensurability is of\ncentral signi\fcance to the phason phenomena, we are con-\nfronted by a nonequilibrium correlated CDW system of\nthree main di\u000eculties: the lack of momentum as a good7\nquantum number, an inevitably large enough inhomoge-\nneous system with many degrees of freedom of the full\nphonon and electron spectra, and the essential nonper-\nturbative e\u000bect. This is in stark contrast to the usual\nincorporation of a pump-probe process: it is more feasi-\nble to simpler and smaller systems such as a single band\n(and typically single-frequency phonon) with a quench or\nuniform pump-probe \feld and often limited to perturba-\ntion or other approximation, which cannot capture the\nnovel real-space nonperturbative topological phenomena\nin our system[67, 68]. This type of explicit pump-probe\ntreatment has a di\u000berent focus and advantage.\nInstead, we in this study succeed in providing the way\nto overcome these di\u000eculties and tackle interesting sys-\ntems hitherto inaccessible from other methods. And in\nthe spirit aforementioned, we extract the most relevant\nnonequilibrium phenomena by directly targeting at our\nlarge system right after some pumping transition that\nabsorbs the photon and creates the particle-hole exci-\ntation, which is the most ordinary e\u000bect of the optical\ndisturbance in this system gapped by the CDW order.\nIn summary, we combine ICDW as a prototypical quan-\ntum phase bearing broken continuous symmetry with\nthe facet of optically induced nonequilibrium and nonlin-\nearity. Real-time ultrafast evolution highlights the spa-\ntiotemporally varying complex order parameter and the\nexperimental implication, revealing unique features from\nthe continuous amplitude and especially the phase varia-\ntion that leads to rich collective dynamics and highly non-\ntrivial solitonic topological phases and transitions unique\nto nonequilibrium.\nAppendix A: Simulation of time evolution\nInstead of the involved full coupled quantum dynam-\nics in a large phase space, we treat the system semi-\nclassically: electrons e\u000bectively integrated out at each\ninstant generate adiabatic forces dominating short-time\ndynamics, whereby we evolve the ionic con\fguration.\nThe instantaneous electronic spectrum and eigenstates\nf\"i;jiigsolved fromHel(~ u) lead to the electronic en-\nergyEel[~ u] =P\nifi\"iwhere the nonequilibrium occu-\npationfiincorporates spin degeneracy and keeps the\nelectron-hole pair excited at the band edge. Variation\nof the total energy Etot[~ u], combining Eeland the poten-\ntial part ofHphn(~ u), determines the conservative force\nFi[~ u] =\u0000\u000eEtot\n\u000euifelt by theith ion. The phonon dynamics\ncan thus be performed using the velocity Verlet method,\npreserving the symplectic structure and is energy-drift-\nfree.\nBased on Eq. (1), the electronic part Helcan be solved\nat any given displacement con\fguration ~ u, leading to en-\nergy levels \"i=1;\u0001\u0001\u0001;N. The electronic energy then reads\nEel[u] =P\nifi\"i. In the ground state, the occupation\nnumberfi= 2 for the two spins when i= 1;\u0001\u0001\u0001;Ne=2\nandfi= 0 otherwise. If the light excites an electron\nfrom an fully occupied level jto an unoccupied levelk, i.e., an electron-hole pair, it di\u000bers from the ground\nstate infj=fk= 1. In general, we mostly focus on\nthe scenario of exciting an electron-hole pair right across\nthe CDW gap, i.e., j=p\nqN=Ne=2;k=j+ 1. On\nthe other hand, the phonon potential energy Vphn[~ u] =P\ni\u0014\n2(ui\u0000ui+1)2. We thus de\fne a total 'potential' en-\nergyEtot[~ u] =Eel[~ u] +Vphn[~ u] that directly determines\nthe conservative force Fi[~ u] =\u0000\u000eEtot\n\u000euifelt by the ion\ni. Any ground state must be determined for a \fxed\n\u000bby the condition\u000eEtot\n\u000eU= 0 that energetically mini-\nmizes the system for the given equilibrium CDW pat-\nternu(0)\nn=Usin(Qna +\u001e0), where we set the sponta-\nneously broken phase \u001e0= 0 without loss of generality.\nWhen other control parameters are \fxed, \u000bwill deter-\nmineU;\u0001;\u0018. As concrete values of \u000b;U; \u0001 are not of\ninterest, we primarily tune \u000bto vary at will \u0018in units\nofa, because it fully represents the CDW state's fea-\nture. Without loss of generality, we impose the periodic\nboundary condition in the calculation. We also con\frm\nall essential results with the open boundary condition.\nRegarding the nonequilibrium time evolution, we inte-\ngrate the phonon dynamics M~ u=~F[~ u] in time steps of \u001c\nusing the velocity Verlet method in molecular dynamics\nthat updates both ~ uand its velocity consecutively\n~ u(i+1)=~ u(i)+~ v(i)\u001c+~F(i)\n2M\u001c2\n~ v(i+1)=~ v(i)+~F(i)+~F(i+1)\n2M\u001c(A1)\nwhere the force \feld ~F(i)at timeT=i\u001cis solved from\nvariating the instantaneous electron system H(i)\nelagainst\nthe ion \feld ~ u(i). This method is of second-order global\nerror. As mentioned in the main text, the whole time-\ndependent system possesses two independent dimension-\nless parameters: \u0018=aand\u001f=\u001c=T0for a \fxed number of\ntime steps. To see this, we look at a few scaling proper-\nties below. Derived from Eq. (1), when \u0018=ais invariant\none has\nEtot(r;s) =rEtot(1;1);~F(r;s) =r\ns~F(1;1); (A2)\nwhereEtot(r;s)\u0011Etot(rt;r\ns2\u0014;s~ u;\u000b\ns) and similarly\n~F(r;s) are de\fned for a system with rescaled parame-\nters. Note that \u0001 scales the same as Etotdoes. Alter-\nnatively, this can be seen as altering the units of energy\nand length. On the other hand, when one integrates the\nphonon dynamics via Eq. (A1),\n~ u(n)=~ u(0)+n~ v(0)\u001c+\u001c2\n2M[n~F(0)+ 2n\u00001X\ni=1(n\u0000i)~F(i)]\n~ v(n)=~ v(0)+\u001c\n2M[~F(0)+ 2n\u00001X\ni=1~F(i)+~F(n)]:\n(A3)\nThe force \feld ~F[~ u] along the chain is purely a func-\ntional of the ion displacement \feld ~ u. Therefore, when8\nthe ions are initially at rest ~ v(0)= 0, which is the present\ncase, either ~F(n)[~ u] or the ion displacement from its ini-\ntial value~ u(n)\u0000~ u(0)(implicitly) depends on the com-\nbination\u001c2\nM=(2\u0019\u001f)2\n\u0014. We immediately observe from\nEqs. (A2)(A3) that as long as \u001fis \fxed, i.e.,\u001c2\nM!\u001c2\nMs2\nr\nunder the scaling, the evolution of ~ uis merely rescaled\nto bes~ uand hence no essential di\u000berence although one\nuses completely di\u000berent t;\u0014;\u000b . Besides, given other pa-\nrameters, the time evolution is invariant when one tune\nMand\u001csuch that\u001c2\nMis \fxed, which also means that !0\ncan be tuned to ful\fll the adiabatic condition !0\u001c2\u0001.\nReadily seen, the relevance of a concrete choice of M;\u001c\nenters when we calculate the velocities of collective exci-\ntations, which involves the time lapse in units of \u001cand\nthe space traversed in units of a. One has to \fx a real-\nisticMand hence\u001cin order to compare these velocities\nwith, say, a realistic Fermi velocity vF. See SM for other\nsimulation details.\nAppendix B: Low-energy continuum model\nThe complete second-quantization form of Eq. (1) is\nH=X\nk;\u001b\"kc†\nk\u001bck\u001b+X\nq!qb†\nqbq\n+X\nk;q;\u001b(gk;qc†\nk+q;\u001bck\u001b(bq+b†\n\u0000q) + h:c:)(B1)\nwhereinck\u001b =1p\nNP\nne\u0000iknacn\u001b,un =\nP\nqq\n~\n2NM!q(bq+b†\n\u0000q)eiqna, and the electron-phonon\nvertexgkq=t\u000bq\n~\n2NM!qeika(eiqa\u00001). Once the phonons\nin Eq. (B1) are integrated out, the electron-phonon\ncoupling will e\u000bectively induce attractive four-fermion\ninteractions and hence a BCS-type theory of the CDW\npairing. More explicitly, Eq. (B1) becomes at the\nmean-\feld level\nH=X\nk;\u001b\"kc†\nk\u001bck\u001b+X\nq!qhb†\nqbqi+X\nk;q;\u001b(fk;qc†\nk+q;\u001bck\u001b+h:c:)\n(B2)\nwhere we de\fne fk;q=gk;qhbq+b†\n\u0000qi. Taking into ac-\ncount the phonon condensation at q=\u0006Q, the linearized\nlow-energy theory around the Fermi level \"Fis given by\nsettingk!\u0006kF+k;q!\u0006 (Q+q) and then summing\noverk(q) that is henceforth small and measured relativeto\u0006kF(\u0007Q)\nh=X\ns=\u0006;k!skF+k[\"kc†\nkck+X\nq(fk;\u0000s(Q+q)c†\nk\u0000s(Q+q)ck\n+f\u0003\nk;s(Q+q)c†\nkck+s(Q+q))]\n=X\ns=\u0006;k!skF+k[\"kc†\nkck\n+X\nq(fk;\u0000s(Q+q)+f\u0003\nk\u0000s(Q+q);s(Q+q))c†\nk\u0000s(Q+q)ck]\n=X\nk[X\ns=\u0006\"s;kc†\ns;kcs;k+X\nq(Ok;qc†\n+;k+c\u0000;k\u0000+ h:c:)]\n(B3)\nwhere we drop the constant term and the spin indices\nfor simplicity, shift the reference energy to \"Fby\"\u0006;k=\n\"\u0006kF+k\u0000\"F=\u0006vFk, shiftk!k+sq\n2respectively for k\nins=\u0006branches to univocally de\fne ks=k+sq\n2, and\nOk;q=f\u0000kF+k\u0000;Q+q+f\u0003\nkF+k+;\u0000Q\u0000q. Theks-dependence,\ne\u0006iksa, inOk;qat most generates elements tridiagonal\nin the real-space lattice representation. The continuum\nlimit near\u0006kFand the long wavelength limit of q(a!0)\nassure of dropping this ksaandqadependence and thus\nwe replaceOk;qby a complex mass\nOq=f\u0000kF;Q+q+f\u0003\nkF;\u0000Q\u0000q= i4t\u000bsin (kFa)1p\nNuQ+q:\n(B4)\nFor the equilibrium state of a constant mass O=Oq=0\nonly, from Eq. (B3), we thus have in the real space\nh(0)=vF(\u0000i@)sz+O0s++O\u0003\n0s\u0000with the pseudospin\nsz;s\u0006=1\n2(sx\u0006isy) for the two Fermi points. For the\nmore general case with perturbations and soliton forma-\ntion in nonequilibrium, we obtain from Eq. (B3)\nh\u0019X\nk[X\ns=\u0006\"s;kc†\ns;kcs;k+X\nq(Oqc†\n+;k+c\u0000;k\u0000+ h:c:)]\n=\u0002\ndx[X\ns=\u0006svF(\u0000i@)c†\ns;xcs;x+ (i\fO(x)c†\n+;xc\u0000;x+ h:c:)]\n(B5)\nwherein\f= 4t\u000bsin (kFa), we have usedP\nkeik(x\u0000x0)=\nN\u000ex;x0!\u000e(x\u0000x) for transitioning to the continuum\nlimit, and we can now de\fne the complex mass or order\nparameter\nO(x) =m(x)ei\u001e(x)=X\nq1p\nNuQ+qeiqx(B6)\nthat no longer remains a constant. We use the Fourier\ntransform uk=1p\nNP\nxe\u0000ikxu(x). Real quantities\nm(x);\u001e(x) take values m=U=2;\u001e=\u001e0at equilibrium.\nTherefore, we \fnally have the Hamiltonian\nh=vF(\u0000i@)sz+ (i\fO(x)s++ h:c:) (B7)9\nappllied on a wavefunction  (x) = ( +(x); \u0000(x))T.\nUsing the concrete form of O(x) =m0~O(x\n\u0015) with a\ncharacteristic length scale \u0015, we \frst nondimensionalize\nEq. (B7)\n~h=\u0000i@~xsz+ 2~\u000b~m0(i~O(~x\n~\u0015)s++ h:c:) (B8)\nwhere all tilde quantities ~ x=x=a; ~\u0015=\u0015=a; ~\u000b=\na\u000b;~m0=m0=a,~h=h=\u000f0with\u000f0= 2tsinkFaand\n~Oare dimensionless. An important scaling property\nof Eq. (B7) is that h(rO(rx)) (rx) =r\u000f (rx) follows\nfromh(O(x)) (x) =\u000f (x), i.e., another state  (rx) is\nan eigenstate of h(rO(rx)) with energy r\u000fwhereris a\ndimensionaless scaling factor. This scaling property im-\nmediately suggests that it su\u000eces to consider only the\nfollowing dimensionless Hamiltonian\nh=\u0000i@xsz+ (O(x\n\u0015)s++ h:c:) (B9)\nwhere we drop all tildes for brevity, r= 1=(2~\u000b~m0), and\n\u0015is now measured in units of \u0015\u0003=ra=a=(2~\u000b~m0).\nNote that we also drop the imaginary factor of Osimply\nbecause a global phase does not a\u000bect the result. The\ntypical dimensionless value \u0015= 1 therefore corresponds\nto the dimensionful \u0015=\u0015\u0003\u001920a;60afor\u0018=a\u00193;9.\nAppendix C: Spectral function and optical\nconductivity\nAt each time slice, we diagnolize the instantaneous\nelectronic Hamiltonian S†HelS= diag(\"1;\"2;\u0001\u0001\u0001;\"N).\nThe retarded Green's function in the real space is given\nby\nGr(x;y;! ) =\u0000i\u00021\n\u00001dtei!t\u0012(t)hfcx(t);c†\ny(0)gi\n=X\nlSxlS\u0003\nyl\n!\u0000\"l+ i\u0011(C1)\nThe spectral function reads A(x;y;! ) =\u00001\n\u0019=Gr(x;y;! )\nand we will mainly focus on its diagonal components\nA(x;!) =A(x;x;! ) that re\rects the spectral weight on\na certain lattice site x. For our \fnite-size system one\nhas to set\u0011larger than or comparable to the smallest\nlevel splitting, which is estimated to be 4 t=2N\u00190:001t,\notherwise the \fnite-size structure becomes visible in the\nspectral function and the optical conductivity.\nThe retarded current-current correlator and its spec-\ntral expansion is given by\n\u0005\u0016\u0017(!) =\u0000i\n~V\u00021\n0dtei!th[j\u0016(t);j\u0017(0)]i\n=\u0000i\n~V\u00021\n0dtei!tTr (^\u001a[j\u0016(t);j\u0017(0)])\n=1\nVX\nmne\u0000\f\"n\u0000e\u0000\f\"m\nZhnjj\u0016jmihmjj\u0017jni\n~(!+ i\u0011) +\"n\u0000\"m;\n(C2)where we use eigenstates Heljni=\"njni, the parti-\ntion function Z= Tr(e\u0000\fH), the density matrix ^ \u001a=\ne\u0000\fH=Z, and the time dependence of the current operator\nhnjj\u0016(t)jmi=hnjeiHtj\u0016e\u0000iHtjmi= ei(\"n\u0000\"m)thnjj\u0016jmi.\nThen, we have the conductivity tensor\n\u001b\u0016\u0017(!) =i\n!(\u0005\u0016\u0017(!)\u0000\u0005\u0016\u0017(0))\n=\u0005\u0016\u0017(0)\ni!+i~\nVX\nmne\u0000\f\"n\u0000e\u0000\f\"m\nZ\u0002\nhnjj\u0016jmihmjj\u0017jni\n~i\u0011+\"n\u0000\"m(1\n~!\u00001\n~(!+ i\u0011) +\"n\u0000\"m)\n=i~\nVX\nmne\u0000\f\"n\u0000e\u0000\f\"m\nZ(\"m\u0000\"n)hnjj\u0016jmihmjj\u0017jni\n~(!+ i\u0011) +\"n\u0000\"m\n=i~\nVX\nmnfn\u0000fm\n\"m\u0000\"nhnjj\u0016jmihmjj\u0017jni\n~(!+ i\u0011) +\"n\u0000\"m\n(C3)\nwhere in the last line we identify the Fermi distribution\nnF(\"n) =fn=e\u0000\f\"n\nZfor this e\u000bectively noninteracting\nsystem. In the second line, writing the diamagnetic term\n\u0005\u0016\u0017(0)\ni!with the expansion Eq. (C2), it exactly cancels\nthe \frst term in the parentheses. The electric current\noperatorj=evassumes velocity operator v=i\n~[H;x]\nand\nvmn=hmji\n~[H;x]jni=i\n~(\"m\u0000\"n)xmn: (C4)\nEq. (C3) in the 1D case then gives\n\u001b(!) =i~\nVX\nmnfn\u0000fm\n\"m\u0000\"njjmnj2\n~(!+ i\u0011) +\"n\u0000\"m\n=\u0000ie2\n~VX\nmn(fn\u0000fm)(\"n\u0000\"m)jxmnj2\n~(!+ i\u0011) +\"n\u0000\"m;(C5)\nwhich is our \fnal formula of the optical conductivity.\nFrom the last line of Eq. (C3), we have the symmetry\nproperty for the general conductivity tensor \u001b\u0016\u0017(!) =\n\u001b\u0003\n\u0016\u0017(\u0000!) and hence\n<\u001b\u0016\u0017(!) =<\u001b\u0016\u0017(\u0000!);=\u001b\u0016\u0017(!) =\u0000=\u001b\u0016\u0017(\u0000!);(C6)\nwhich certainly holds for the longitudinal conductivity\n\u001b(!) =\u001b\u0016\u0016(!) in Eq. (C5) as well. Therefore, with the\nsymmetry\u001b(\") =\u001b\u0003(\u0000\"), we consider only \"\u00150 in\nFigs. 5(a1-2) in the main text.\nACKNOWLEDGMENTS\nWe thank the Max Planck-UBC-UTokyo Center for\nQuantum Materials for fruitful collaborations and \fnan-\ncial support. This work was supported by CFREF,\nNSERC and CIfAR, JSPS KAKENHI (No. 18H03676),\nand JST CREST (Nos. JPMJCR16F1 & JPMJCR1874).\nX.-X.Z was also partially supported by the Riken Special\nPostdoctoral Researcher Program.10\n[1] R. Jackiw and C. Rebbi, Physical Review D 13, 3398\n(1976).\n[2] R. Rajaraman, Solitons and Instantons, Volume 15:\nAn Introduction to Solitons and Instantons in Quantum\nField Theory (North-Holland Personal Library) , 1st ed.\n(North Holland, Amsterdam, 1987).\n[3] W. P. Su, J. R. Schrie\u000ber, and A. J. Heeger, Physical\nReview Letters 42, 1698 (1979); A. J. Heeger, S. Kivel-\nson, J. R. Schrie\u000ber, and W. P. Su, Reviews of Modern\nPhysics 60, 781 (1988).\n[4] P. Lee, T. Rice, and P. 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Taking a meta-modelling approach to\ngenerate a huge set of equation of state that satisfy nuclear matter properties close to n0and\nthat do not contain a \frst order phase transition, the possibility of constraining the high density\nequation of state was investigated. The entire information obtained from the GW170817 event for\nthe probability distribution of ~\u0003 was used to make a probabilistic inference of the EOS, which goes\nbeyond the constraints imposed by nuclear matter properties. Nuclear matter properties close to\nsaturation, below 2 n0, do not allow us to distinguish between equations of state that predict di\u000berent\nneutron star (NS) maximum masses. This is, however, not true if the equation of state is constrained\nat low densities by the tidal deformability of the NS merger associated to GW170817. Above 3 n0,\ndi\u000berences may be large, for both approaches, and, in particular, the pressure and speed of sound\nof the sets studied do not overlap, showing that the knowledge of the NS maximum mass may give\nimportant information on the high density EOS. Narrowing the maximum mass uncertainty interval\nwill have a sizeable e\u000bect on constraining the high density EOS.\nI. INTRODUCTION\nNeutron stars (NS) are special astrophysical ob-\njects through which the properties of cold super-dense\nneutron-rich nuclear matter may be investigated. The\nmassive NSs observed, e.g., PSR J1614 \u00002230 withM=\n1:908\u00060:016M\f[1{3], have established strong con-\nstraints on the equation of state (EOS) of nuclear mat-\nter. Further pulsars' observations, PSR J0348+0432 with\nM= 2:01\u00060:04M\f[4] and MSP J0740+6620, with a\nmass 2:08+0:07\n\u00000:07M\f[5, 6], and radius 12 :39+1:30\n\u00000:98km ob-\ntained from NICER and XMM-Newton data [7] (together\nwith an updated mass 2 :072+0:067\n\u00000:066M\f), have strength-\nened the already sti\u000b constraints on the EOS. The\nNICER mission [8] has estimated the mass and radius of\nthe pulsar PSR J0030-0451: respectively, 1 :34+0:15\n\u00000:16M\f,\n12:71+1:14\n\u00001:19km [9], and 1 :44+0:15\n\u00000:15M\f, 13:02+1:24\n\u00001:06km [10].\nGravitational waves (GWs) are another crucial source\nof information about NS matter. GWs are emitted dur-\ning the coalescence of binary NS systems and carry im-\nportant information on the high density properties of the\nEOS. The analysis of the event GW170817 has settled an\nupper bound on the e\u000bective tidal deformability of the bi-\nnary ~\u0003 [11]. Using a low-spin prior, which is consistent\nwith the observed NS population, the value ~\u0003\u0014800 (90%\ncon\fdence) was determined from the GW170817 event.\nTighter constraints were found in a follow up reanalysis\n[12], with ~\u0003 = 300+420\n\u0000230(90% con\fdence), under minimal\nassumptions about the nature of the compact objects.\nThe two NS radii for the GW170817 event were estimated\nin [13], under the hypothesis that both NS are described\n\u0003marcio.ferreira@uc.pt\nycp@uc.ptby the same EOS and have spins within the range ob-\nserved in Galactic binary NSs, to be R1= 11:9+1:4\n\u00001:4km\n(heavier star) and R2= 11:9+1:4\n\u00001:4km (lighter star). These\nconstraints on R1;2were obtained requiring that the EOS\nsupports NS with masses larger than 1 :97M\f.\nThe detection of GWs from the GW170817 event was\nfollowed by electromagnetic counterparts, the gamma-ray\nburst (GRB) GRB170817A [14], and the electromagnetic\ntransient AT2017gfo [15], that set extra constraints on\nthe lower limit of the tidal deformability [16{20]. This\nlast constraint seems to rule out very soft EOS: the lower\nlimit of the tidal deformability of a 1.37 M\fstar set by the\nabove studies limits the tidal deformability to \u0003 1:37M\f>\n210 [18], 300 [17], 279 [19], and 309 [20].\nThe LIGO/Virgo collaboration has recently reported\nthe gravitational-wave observation of a compact binary\ncoalescence (GW190814) [21]. While the primary\ncomponent of GW190814 is conclusively a black hole\nof mass 22 :2\u000024:3M\f, the secondary component of\nthe binary with a mass 2 :50\u00002:67M\fremains yet\ninconclusive, which might be either the heaviest neutron\nstar or the lightest black hole ever discovered in a double\ncompact-object system [21]. The absence of measurable\ntidal deformations and the absence of an electromagnetic\ncounterpart from the GW190814 event are consistent\nboth scenarios [21]. The uncertainty on the nature\nof the second component have prompted an ongoing\ndebate on whether the EOS of nuclear matter is able to\naccommodate such a massive NS.\nIn the present study, we will analyse which informa-\ntion on the high density EOS can be drawn from the\nknowledge of the NS maximum mass. Several works\nhave estimated NS maximum mass using di\u000berent ap-\nproaches taking into account the GW170817 observa-\ntion, the electromagnetic follow-up or/and NICER PSRarXiv:2110.00305v1  [nucl-th]  1 Oct 20212\nJ0030+0451 data [22{27]: the upper limit was \fxed at\n\u00192:15\u00002:25M\fin [22], and updated to Mmax<2:3M\f\nin [26]; by the interval 2 :01+0:04\n\u00000:04\u0014Mmax=M\f\u00142:26+0:17\n\u00000:15\nwas obtained for a nonrotating NS in [25]; [23] places the\nupper limit Mmax.2:17M\f(90%); in [24] the limits\n2:0< M max=M\f<2:2 (68%) and 2 :0< M max=M\f<\n2:6M\f(90%) have been obtained; in [28] the maxi-\nmum mass 2 :16\u00002:28M\fwas calculated; in a recent\nstudy considering hybrid stars and both the GW170817\nand the NICER observations the limits 2 :36+0:49\n\u00000:26M\f\nand 2:39+0:47\n\u00000:28M\f, taking two di\u000berent hadronic models,\nwhere obtained at 90% credible interval [27]. Considering\nthat a black-hole was formed in GW170817 and taking\nconservative assumptions the authors of [29] concluded\nthat the maximum mass should be <2:53M\f.\nWe assume that no \frst order phase transition occurs.\nAlthough restrictive, this hypothesis is supported by re-\ncent studies. In [30], a one branch EOS was favored con-\nsidering a non-parametric EOS model based on Gaussian\nprocesses and taking as constraints radio, x-ray and GW\nNS data, including the mass and radius measurement of\nthe pulsar J0740+6620. A similar result was obtained in\n[31], where a \frst-order phase transition inside NSs was\nshown to be not favored, although not ruled out.\nNuclear matter properties are reasonably well con-\nstrained for densities below twice the saturation density\nby nuclear experiments. Therefore, as a second as-\nsumption, we take this information into account by\nconsidering a set of meta-EOS that satisfy experimental\nnuclear matter constraints below \u00192n0and give rise\nto thermodynamically consistent EOSs that describe\n1.97M\fNSs. Applying a probabilistic approach, we\nanalyze how the information contained in the entire\nposterior probability distribution (pdf) for the binary\ntidal deformability of the GW170817 event [32] a\u000bects\nthe EOS pressure and speed of sound at high densities.\nIt will be shown that non-overlapping 90% credible in-\ntervals are obtained when di\u000berent maximum masses are\nconsidered, indicating that a maximum mass constraint\ngives information on the star EOS. However, we should\nkeep in mind that there is always the possibility that\nthe \"real\" nuclear matter EOS consists in an unlikely\nrealization for any probabilistic analysis approach, and\nthus the \"real\" EOS might actually be realized in a low\nprobability region. It is the nuclear force that determines\nthe characteristics of the EOS. Having this in mind\nin most of our plots we represent the extremes of our sets.\nThe paper is organized as follows. The EOS\nparametrization and the method for generating the EOSs\nare presented in Sec. II. The results are discussed in Sec.\nIII, where we analyze the properties of the di\u000berent EOS\ndata sets and perform a probabilistic inference on the\nhigh density region of the EOS. Finally, the conclusions\nare drawn in Sec. IV.II. EOS MODELLING\nWe assume the generic functional form for the energy\nper particle of homogeneous nuclear matter\nE(n;\u000e) =esat(n) +esym(n)\u000e2(1)\nwith\nesat(n) =Esat+1\n2Ksatx2+1\n6Qsatx3+1\n24Zsatx4(2)\nesym(n) =Esym+Lsymx+1\n2Ksymx2+1\n6Qsymx3\n+1\n24Zsymx4(3)\nwherex= (n\u0000n0)=(3n0). The baryon density is given by\nn=nn+npand\u000e= (nn\u0000np)=nis the asymmetry, with\nnnandnpdenoting the neutron and proton densities,\nrespectively. This approach of Taylor expanding the en-\nergy functional up to fourth order around the saturation\ndensity,n0, has been applied recently in several works\n[33{36]. The empirical parameters can be identi\fed as\nthe coe\u000ecients of the expansion. The isoscalar empirical\nparameters are de\fned as successive density derivatives\nofesat,Pk\nis= (3n0)k@kesat=@nkfor symmetrical nuclear\nmatter at saturation, i.e., at ( \u000e= 0;n=n0), while\nthe isovector parameters are connected to the density\nderivatives of esym,Pk\niv= (3n0)k@kesym=@nkalso at\n(\u000e= 0;n=n0).\nEach EOS, denoted by index i, is represented by a\npoint in the 8-dimensional space of parameters\nEOS i= (Esym;Lsym;Ksat;Ksym;Qsat;Qsym;Zsat;Zsym)i:\n(4)\nA set of EOS is generated by drawing random samples\nthrough a multivariate Gaussian distribution\nEOS i\u0018N(\u0016;\u0006); (5)\nwhere the mean vector and (diagonal) covariance matrix\nare, respectively,\n\u0016T= (Esym;Lsym;Ksat;Ksym;Qsat;Qsym;Zsat;Zsym)\nand\n\u0006= diag(\u0006 Esym;\u0006Lsym;\u0006Ksat;\u0006Ksym;\u0006Qsat;\u0006Qsym;\u0006Zsat;\u0006Zsym):\n(6)\nWhile the lower order coe\u000ecients are quite well\nconstrained experimentally [37{42], the higher order\nQsat; ZsatandKsym; Qsym; Zsym are poorly known\n[34, 43{48]. In the present work, we \fx Esat=\u000015:8\nMeV andn0= 0:155 fm\u00003as these quantities are rather\nwell constrained. Table I shows the parameters means\nand standard deviations, \u001ba=p\u0006a, we use [34].3\nPiEsymLsymKsatKsymQsatQsymZsatZsym\nPi32 60 230 \u0000100 300 0\u0000500\u0000500\n\u001bPi2 15 20 100 400 400 1000 1000\nTable I. The means Piand standard deviations \u001bPithat char-\nacterize the multivariate Gaussian distribution [Eq. (5)]. All\nthe quantities are in units of MeV.\nEach valid EOS sampled describes npe\u0016 matter in\n\f-equilibrium and satisfy the following conditions i) the\npressure is an increasing function of density (thermody-\nnamic stability); ii) the speed of sound is smaller than\nthe speed of light (causality); iii) the EOS supports\na maximum mass at least as high as 1 :97M\f[1{4].\nThis EOS sampling approach introduces no a priori\ncorrelations between the parameters (zero covariances),\nand the correlations present in \fnal set of valid EOS\nare solely induced from the above conditions. We use\nthe nuclear SLy4 EOS [49] for the low density region\nofn < 0:5n0, and a power-law interpolation between\n0:5n0<n<n 0is performed to ensure a matching at n0\nwith the sampled EOS (di\u000berent matching procedures\nwere analyzed in [50]).\nEmploying the EOS Taylor expansion around sat-\nuration for symmetric nuclear matter as the EOS\nparametrization itself is widely used [33{35, 47, 51{57].\nThere are no convergence issues when it is considered\nas an EOS parametrization, and this is a reliable\napproximation for realistic EOS around the saturation\ndensity for symmetric nuclear matter [33]. In [33]\nthe authors have also shown that they could \ft any\nEOS to a quite good accuracy using the EOS forth\norder Taylor expansion. However, the higher-order\nparameters must be seen as e\u000bective terms, which may\nsubstantially deviate from the actual nuclear matter\nexpansion coe\u000ecients. Nevertheless, the present EOS\nparametrization allows to directly constrain the EOS\nfrom the properties of symmetric nuclear matter and the\nsymmetry energy near saturation density, while di\u000berent\nEOS parametrizations impose such properties in a indi-\nrect way [47]. A recent work [58] has pointed out that\nthe use of such Taylor expansion parametrization may\nbe an inadequate approach. The authors have compared\nthe results from two parametrization procedures i) a\nthird order Taylor expansion; ii) an hybrid formulation\nwhere a second order Taylor expansion describes matter\nbelow 2n0and three polytropes are used to describe\nmatter above that density. While we agree that the\npresent EOS parametrization has its limitations, it\nshould not be compared with a hybrid procedure where\npolytropes are randomly added to the low density Taylor\nexpansion, as it brakes the correlations between the\ninformation around the saturation density, from the\nTaylor expansion, with the high-density region de\fned\nby the polytropes. In this approach, the correlationsamong the Taylor coe\u000ecients and the polytropes indices\nare close to zero (which is visible in the posterior\nprobabilities in [58]), and both low- and high-density\nregions are disconnected. Since we are only interested in\nconstraining the thermodynamic properties of the EOS\nand not the nuclear matter properties, and we consider\na one branch EOS, the Taylor expansion is as reliable as\nother parametrizations, describing smooth equations of\nstate.\nFor each EOS, we compute the dimensionless tidal de-\nformability \u0003 =2\n3k2C\u00005, wherek2is the quadrupole\ntidal Love number and C=GM= (c2R) is the star's com-\npactness [59]. We also determine the e\u000bective tidal de-\nformability of binary systems,\n~\u0003 =16\n13(12q+ 1)\u0003 1+ (12 +q)q4\u00032\n(1 +q)5; (7)\nwhereq=M2=M1<1 is the binary mass ratio and\n\u00031(M1) and \u0003 2(M2) represent the tidal deformability\n(mass) of the primary and the secondary NS in the bi-\nnary, respectively. ~\u0003 is the leading tidal parameter in\ngravitational-wave signal from a NS merger. The con-\nstraint ~\u0003 = 300+500\n\u0000190(symmetric 90% credible interval)\nwas estimated from the GW170817 with a mass ra-\ntio bounded as 0 :73\u0014q\u00141 [12]. The binary chirp\nmass,Mchirp =M1q3=5=(1 +q)1=5, was measured as\nMchirp = 1:186+0:001\n\u00000:001M\ffor the GW170817 event [12].\nIn the present work, we \fx the Mchirp to 1:186M\f.\nIII. RESULTS\nBy applying the statistical procedure described in the\nlast section, we have generated a dataset containing\n150000 EOS that describe neutron star matter in \f-\nequilibrium.\nA. Subsets de\fned by Mmax\nWe are interested in studying the e\u000bect of the maxi-\nmum NS mass, Mmax, on the NS EOS properties. There-\nfore, we split our data set into three subsets depending\nonMmaxreached by each EOS: 1 :97<M max=M\f\u00142:20\n(set 1), 2:20< M max=M\f\u00142:40 (set 2), and\n2:40< M max=M\f\u00142:66 (set 3). The highest value\nreached in our data-set is Mmax = 2:66M\f. The\nhistogram of the number of EOS as a function of Mmax\nis shown in Fig. 1, displaying an unimodal distribution\nwith a peak around 2 :25M\f, being the most likely out-\ncome forMmaxwithin the present EOS parametrization\napproach and sampling procedure. Only 3162 EOS were\nable to sustain a NS with Mmax>2:5M\f, 2.11% of our\ndataset.4\n0500100015002000\n2.0 2.2 2.4 2.6\nMmax[M⊙]Number of EoS\nFigure 1. Number of EOS as a function of Mmax. The colors\nidentify the subsets: 1 :97<M max=M\f\u00142:20 (blue), 2 :20<\nMmax=M\f\u00142:40 (green), and 2 :40< M max=M\f\u00142:66\n(red).\nLet us recall that di\u000berent experiments have recently\nconcentrated on getting more information on the nuclear\nsymmetry energy at saturation. In [60], the authors have\nconstrained the symmetry energy and its slope at satura-\ntion to, respectively, 32 :5<E sym<38:1 MeV and 42 <\nLsym<110 MeV using spectra of charged pions pro-\nduced in intermediate energy collisions. These values are\nin accordance with the ones obtained in [61] from charge\nexchange and elastic scattering reactions, respectively,\n33:5<E sym<36:4 MeV and 70 <L sym<101 MeV, but\nsmaller than the ones determined from the measurement\nof the neutron radius of208Pb by the PREX-2 collabo-\nration [62, 63], which have obtained for the slope of the\nsymmetry energy Lsym= 106\u000637 MeV. These estimates\nare somehow larger than the ones proposed in [40], which\ntake into account experimental, theoretical and observa-\ntional constraints, e. g. 29 :0< E sym<32:7 MeV and\n44< L sym<66 MeV ([40] and updated in [64]). The\nsymmetry energy properties of our EOS, see Table II,\nessentially reproduce at one standard deviation the con-\nstraints of [64], which are strongly in\ruenced by chiral ef-\nfective \feld theory results for neutron matter [65]. How-\never, the three sets also include EOS that fall within the\nrecent experimental intervals within two to three stan-\ndard deviations. Moreover, the extremes are well out of\nthese ranges.\nTable II shows the mean and standard deviations of\neach set. We see that, except for Zsym, isovector prop-\nerties decrease from set 1 to set 3. Larger Mmaxvalues\nrequire smaller isovector properties, while the isoscalar\nproperties show the opposite behaviour, e.g., Ksatin-\ncreases slightly and there is a considerable increase of\nQsat, butZsatshows a di\u000berent behavior. Two comments\nare in order: i) both EsymandLsymare very similar in\nthe three sets indicating that the maximum mass star is\nnot sensitive to the symmetry energy close to saturation\ndensity; ii) since Zsatis the last term of the expansionconsidered, it has an e\u000bective character taking into ac-\ncount all the missed higher order terms. As we mentioned\nin Sec. II, the high-order terms should be seen as e\u000bective\nones, and their values might deviate from the true nu-\nclear matter properties. A recent review [56] discussing\nthe constraints set on the nuclear symmetry energy by\nNS observations since the detection of GW170817, pro-\nposes for the LsymandKsymvalues and uncertainties of\nthe order of the ones presented in Table II.\nSet 1 Set 2 Set 3\nMmax=M\f]1:97;2:20] ]2:20;2:40] ]2:40;2:66]\n# EOSs 61660 69789 18551\nmean std mean std mean std\nKsat 229.28 19.83 233.17 19.63 238.74 19.75\nQsat -93.44 65.23 18.11 71.09 148.11 77.45\nZsat -3.47 67.31 -109.11 81.75 -237.37 96.68\nEsym 32.01 1.98 32.01 2.00 31.90 2.01\nLsym 61.21 14.85 60.97 14.89 58.19 14.89\nKsym -64.06 90.62 -72.54 89.88 -107.78 78.87\nQsym 300.44 328.93 227.27 320.31 112.70 277.83\nZsym 392.77 684.98 292.95 598.48 433.28 509.64\nTable II. The mean [MeV] and standard deviation [MeV] (std)\nof the EOS parameters for each set. In the following \fgures\nthe di\u000berent sets are represented by the colors blue (Set 1),\ngreen (Set 2) and red (Set 3).\n1. Neutron stars properties\nIn order to study NS properties, we integrate the\nTolmann-Oppenheimer-Volko\u000b (TOV) equations [66, 67],\ntogether with the di\u000berential equations that determine\nthe tidal deformability [68]. Figure 2 shows the radius\n(left panel) and tidal deformability (right panel) as a\nfunction of the NS mass for each set. Statistics for some\nspeci\fc NS masses are given in Table III. As expected,\nlargerMmaxvalues correspond to larger NS radii and\ntidal deformabilities. The lower bound on Ris more sen-\nsitive toMmaxthan the upper bound. However, we are\nunable to constrain the Mmaxvalue from the analysis of\nthe millisecond pulsar PSR J0030+0451 (obtained from\nthe NICER x-ray data) [69, 70] since all sets describe the\nNICER data. Next, we will also conclude that we cannot\nconstrain the Mmaxfrom the tidal deformability inferred\nfrom the GW170817 event [12].\nFigure 3 shows the extreme values (maximum and min-\nimum) for the tidal deformability from our entire data set\n(left panel) and the e\u000bective tidal deformability of the\nbinary (Mchirp = 1:186M\f) as a function of the binary\nmass ratioq=M2=M1. Two main conclusion can be ex-\ntracted: i) the range of ~\u0003 shifts to higher values as Mmax\nincreases, but all sets are compatible with ~\u0003 = 300+500\n\u0000190\n(symmetric 90% credible interval) from the GW170817\nevent [12]; ii) furthermore, the NS tidal deformability5\n9.510.511.512.513.5\n1.00 1.25 1.50 1.75\nM [M ⊙]R [km]\n101102103\n1.00 1.25 1.50 1.75\nM [M ⊙]Λ\nFigure 2. The minimum/maximum (dashed lines) and mean (dotted lines) values for the radius (left panel) and tidal deforma-\nbility (right panel) as a function of NS mass. The color indicates the set: set 1 (blue), set 2 (green), and set 3 (red). The\nregions enclosed by solid lines display ( M;R ) constraints obtained by two independent analysis using the NICER x-ray data\nfrom the millisecond pulsar PSR J0030+0451 [69, 70], while the orange bar shows the recent constraint, R1:4M\f= 12:45\u00060:65\nkm, extracted from a combination of NICER and XMM-Newton data [71].\nrange of our data set is consistent with the 90% pos-\nterior credible level obtained in [32], when the analysis\nassumes that the EOS should describe 1.97 M\fNS. Some\nstatistics for the di\u000berent sets are given in Table III.\nNotice that in [72] the authors have obtained for the\nradius of a 1.4 M\fstar 12.33+0:76\n\u00000:81km and 12.18+0:56\n\u00000:79km\nwithin two di\u000berent models. The radii obtained within\nour sets, see Table III, are in accordance with these pre-\ndictions. However, the prediction obtained for the ra-\ndius of the PSR J0740+6620 in [71], 12 :39+1:30\n\u00000:98km with\nthe mass 2:072+0:067\n\u00000:066M\f[7] seems to be more compatible\nwith the two sets with larger maximum masses, although\nset 1 is not excluded.\nWe also give the prediction for the radius of a 1.6 M\f\nstar that has been constrained in [73] to be larger than\n10.68+0:15\n\u00000:04km considering that there was no prompt col-\nlapse following GW170817 as suggested by the associated\nelectromagnetic emission. All our three sets satisfy this\nconstraint. As expected, the predicted central density of\nmaximum mass stars decreases from the less massive set\nto the most massive. For the last set we have obtained\na central density of 5 :05< n max=n0<6:19 compatible\nwith the values proposed in [30] for cold non-spinning\nNSs.\n2. EOS thermodynamics\nLet us now analyze the thermodynamics properties of\neach EOS subset. The pressure and speed of sound are\nshown in Fig. 4, where the shaded colored regions repre-\nsent the 90% credible (equal-tailed) interval regions (sets\n1, 2 and 3 are respectively, blue, green and red). For the\npressure (left panel), the results of each set begins to de-\nviate from each other only at moderate densities, n\u00190:3\nfm\u00003. Although the results are within the range of the\nLIGO/Virgo analysis (gray region), there is a consider-able discrepancy for densities n < 0:55 fm\u00003, although\nthe extreme values cover a larger region. This is expected\nsince the EOS parametrization we use is constrained at\nlow densities by the low order nuclear matter parameters,\nsuch asLsymandKsat, while a polytropic approach as the\none used in the analysis of GW170917, being solely con-\nstrained by thermodynamic stability conditions, is able\nto reproduce extreme behaviours at any density region\n(even close to the saturation density).\nThe speed of sound (right panel of Fig. 4) also shows\na similar dependence at low densities for all sets, but a\ndi\u000berent prediction at higher densities. If we compare\nset 1 (blue), 1 :97< M max=M\f<2:20, with set 3 (red),\n2:40<M max=M\f<2:66, we see that the regions do not\noverlap for n > 0:4 fm\u00003, showing that the high den-\nsity dependence of v2\nsis considerable bound to the Mmax\nvalue. Moreover, the extremes show the possibility of oc-\ncurring a large softening of the EOS, as would be present\nfor a phase transition. These are the EOS that in the\npresent approach cover the lower range of the P(n) band\ndetermined by LIGO/Virgo collaboration. On the other\nhand, the upper limit of the speed of sound is clearly\nde\fned by the causality constraint at high densities.\nB. Constraining the EOS from ~\u0003\nThe analysis of the GW170817 event by the\nLIGO/Virgo collaboration has provided a constraint on\nthe e\u000bective tidal deformability of the binary system,\n~\u0003 = 300+500\n\u0000190(symmetric 90% credible interval) [12].\nA possible way of learning how such a credible region\nconstrains the properties of the nuclear matter consists\nin \fltering out all EOS that do not ful\fll such interval\nand analyze the properties of the remaining EOS.\nHowever, the set of EOS used in the present study\npredicts ~\u0003 values that range between 168 and 7906\n101102103\n1.00 1.25 1.50 1.75\nM [M ⊙]Λ\n200400600800\n0.8 0.9 1.0\nqΛ~\nq\nFigure 3. Left panel: the tidal deformability minimum/maximum values (dashed lines) for the entire data set, i.e., 1 :97<\nMmax=M\f\u00142:66. The red region corresponds to the 90% posterior credible level obtained in [32], when it is imposed that\nthe EOS should describe 1.97 M\fNS.; Right panel: the minimum/maximum (dashed lines) and mean (dotted lines) values for\nthe e\u000bective tidal deformability of the binary, ~\u0003, as a function of qwithMchirp = 1:186M\f. The color indicates the set: set 1\n(blue), set 2 (green), and set 3 (red).\nSet 1 Set 2 Set 3\n1:97<M max=M\f\u00142:20 2:20<M max=M\f\u00142:40 2:40<M max=M\f\u00142:66\nmean std min max mean std min max mean std min max\nR1:4M\f[km] 12:34 0.22 10.42 12.99 12.51 0.22 11.19 13.25 12.61 0.20 11.70 13.25\nR1:6M\f[km] 12:22 0.24 10.34 12.89 12.47 0.21 11.24 13.20 12.63 0.20 11.84 13.28\nR2:08M\f[km] 11:37 0.36 9.76 12.36 12.05 0.26 10.89 12.95 12.48 0.19 11.76 13.29\n\u00031:4M\f 427:36 50.37 139.55 598.21 477.83 50.23 245.1 660.47 514.68 47.29 346.4 688.17\n\u00031:6M\f 260:01 25.05 54.45 260.01 205.37 23.56 105.18 296.23 229.95 21.95 153.81 319.37\n~\u0003q=0:9 506:41 57.53 167.28 700.38 561.13 58.07 286.23 772.35 600.42 54.83 403.99 797.54\nnmax=n0 7.09 0.42 5.72 8.65 6.35 0.27 5.51 7.32 5.76 0.16 5.05 6.19\n(vcen\ns)2[c2] 0:75 0.17 0.03 1 0.86 0.12 0.02 1 0.91 0.08 0.36 1\nTable III. Sample statistics for RMi, \u0003Mi,~\u0003q=0:9, and the NS' central density nmax=n0and speed of sound squared ( vcen\ns)2for\nthe three sets considered.\n100101102\n0.2 0.3 0.4 0.5 0.6\nn [fm-3]P [MeV/fm3]\n0.000.250.500.751.00\n0.2 0.3 0.4 0.5 0.6\nn [fm-3]vs2 [c2]\nFigure 4. The 90% credible levels (dashed lines) and maximum/minimum values (dotted lines) for the pressure (left) and speed\nof sound squared (right) as a function of baryonic density. The color identi\fes the set: set 1 (blue), set 2 (green), and set 3\n(red). The gray region in p(n) corresponds to the 90% posterior credible level from [13].\n(sample minimum/maximum values), not reproducing neither 110 <~\u0003<168 nor ~\u0003>790, i.e., the smallest7\nand largest values obtained from the GW170817 analysis\nare not represented in our set. Notice, however, that the\nkilonova/macronova AT2017gfo sets a lower bound on\nbinary tidal deformability, which should satisfy, for the\nGW170817 event, ~\u0003>200 according to [74] or ~\u0003>300\naccording to [16, 17], and, therefore, our lower bound\nis reasonable. Even though our set statistics presents\na good agreement with ~\u0003 = 300+500\n\u0000190(see Table III),\nin the following we explore a probabilistic inference of\nthe EOS using the entire information encoded in the\nprobability distribution of ~\u0003 for the GW170817 event\n[12] and a multivariate Gaussian distribution of the EOS\nbased on the properties of our EOS sets. Note that the\nformalism developed in the present section considers the\nentire dataset, without any class subset related with the\nmaximum mass Mmax.\n1. Probability distributions for the pressure and speed of\nsound\nWe use a joint multivariate Gaussian distribution\nand its conditioning properties to constrain the EOS\nthermodynamics, namely p(n) andv2\ns(n). This allows\nus to consider the complete information encoded in\nthe posterior probability density function (pdf) of ~\u0003,\nPLV(~\u0003) [12].\nLet us consider a N-multivariate Gaussian probabil-\nity distribution P(z1;z2;:::;z N) =N(z1;z2;:::zN;\u0016;\u0006),\nwhere \u0016= (E[z1];E[z2];:::E[zN])Trepresents the mean\nvector and \u0006 ij= E[(zi\u0000\u0016i)(zj\u0000\u0016j)] are the covariance\nmatrix entries ( Erepresents the expected value opera-\ntor). To simplify the notation, let us rewrite the above\nprobability distribution as\nN(z1;z2;:::zN;\u0016;\u0006) =N(X;Y;\u0016;\u0006); (8)\nwhere X= (z1;:::;z q)TandY= (zq+1;:::;z N)Tare, re-\nspectively, two subsets with dimension qandN\u0000qof the\noriginalNrandom variables. This way, the mean vector\nand covariance matrix are partitioned into blocks,\n\u0016= \nE[X]\nE[Y]!\n\u0011 \n\u0016X\n\u0016Y!\nand \u0006= \n\u0006XX\u0006XY\n\u0006YX\u0006YY!\n;\n(9)\nrespectively. For instance, the block \u0006XY is a\nq\u0002(N\u0000q) matrix with the (1 ;1) entry being\nE[(z1\u0000\u00161)(zq+1\u0000\u0016q+1)].\nA crucial operation for Gaussian distributions is the\nconditional probability. It allows us to determine the\nprobability of a set of variables depending (conditioning)\non the other set variables, yielding a modi\fed - but still -\nGaussian probability distribution. Conditioning the pdf\nP(X;Y) =N(X;Y;\u0016;\u0006) upon the set variables Ygives\nP(XjY) =N(XjY;\u0016\u0016;\u0016\u0006); (10)where\n\u0016\u0016=\u0016X+\u0006XY\u0006\u00001\nYY(Y\u0000\u0016Y) (11)\n\u0006=\u0006XX\u0000\u0006XY\u0006\u00001\nYY\u0006YX: (12)\nThe dimensions of the mean vector, \u0016\u0016, and of the\ncovariance matrix, \u0006, areq\u00021 andq\u0002q, respectively.\nOur goal is to infer the thermodynamic properties of\nthe EOS from the knowledge of ~\u0003, extracted from the\nGW170817 event. Given the current uncertainty on the\nMmax, we want to analyze how the EOS inference de-\npends on the Mmaxvalue. In the following, we con-\nsider the EOS pressure, p(n), and speed of sound, v2\ns(n).\nFor each density n, we model the probability distribu-\ntionsP(p(n);Mmax;~\u0003) andP(v2\ns(n);Mmax;~\u0003) as multi-\nvariate Gaussian distributions (Eq. 8), where both the\nmean vector and covariance matrix have a density depen-\ndence. Then, we condition upon ~\u0003, i.e.,P(p(n);Mmaxj~\u0003)\nandP(v2\ns(n);Mmaxj~\u0003) by applying Eq. (10), where X\ncorresponds to ( p(n);Mmax)Tand (v2\ns(n);Mmax)T, re-\nspectively, whereas Y= (~\u0003). Next, we determine\nthe joint probability distributions P(p(n);Mmax) and\nP(v2\ns(n);Mmax), by marginalizing over ~\u0003, weighting by\nthe prior pdf PLV(~\u0003) [12],\nP(p(n);Mmax) =Z\nP(p(n);Mmaxj~\u0003)PLV(~\u0003)d~\u0003:(13)\nP(v2\ns(n);Mmax) =Z\nP(v2\ns(n);Mmaxj~\u0003)PLV(~\u0003)d~\u0003:\n(14)\nThe probability distribution P(p(n);Mmax) is no\nlonger Gaussian, due to the non-Gaussian char-\nacteristic of the prior PLV(~\u0003). To obtain the\nprobability distribution for the pressure, given a\nspeci\fc value of Mmax, sayMmax =a, we calculate\nP(p(n)jMmax=a) =P(p(n);Mmax=a)=P(Mmax=a).\nFinally, we characterize this pdf using symmetric 90%\ncredible intervals. The same procedure is applied for the\nspeed of sound. Equations (13)-(14) can be interpreted\nas compound pdfs. Note that any pdf can be thought\nas a marginal of some (higher dimension) joint pdf, e.g.,\nP(p(n);Mmax) =R\nP(p(n);Mmax;~\u0003)d~\u0003, and, further-\nmore, any joint pdf can be written as a conditional pdf,\ne.g.,P(p(n);Mmax;~\u0003) =P(p(n);Mmaxj~\u0003)P(~\u0003). That\nis, Eqs. (13)-(14) become trivial identities if we change\nPLV(~\u0003) by the (dataset) marginal distribution P(~\u0003).\nCompound pdfs arise when one uses an \"external\" pdf\nfor some of the probabilistic model variables, i.e., we\nare integrating out ~\u0003 from our probabilistic models\nP(p(n);Mmaxj~\u0003) andP(v2\ns(n);Mmaxj~\u0003) by weighting\neach value of ~\u0003 by a speci\fc (prior) pdf, which has a\ndi\u000berent structure than the datatset P(~\u0003).\nFigure 5 shows the results for P(p(n)jMmax) (left) and\nP(v2\ns(n)jMmax) (right), for three values of Mmax=M\f: 2:08\n100101102\n0.2 0.3 0.4 0.5 0.6\nn [fm-3]p [MeV/fm3]\n0.000.250.500.751.00\n0.2 0.3 0.4 0.5 0.6\nn [fm-3]vs2 [c2]\nFigure 5. Inference results for the pressure (left) and speed of sound squared (right) as a function of baryonic density. The\ncolor regions show the 90% credible intervals for the pdfs P(p(n)jMmax) (left) and P(v2\ns(n)jMmax) (right) for three values of\nMmax=M\f: 2:0 (blue), 2:3 (green), and 2 :6 (red).The gray region in the p(n) plot corresponds to the 90% posterior credible\nlevel from [13].\n(red), 2:3 (green), and 2 :6 (blue). The gray band on the\nP(n) plot indicates the prediction region determined by\nthe LIGO/Virgo analysis [13]. It is interesting to con-\nclude that: i) the bands do not overlap, i.e. knowing the\nmaximum star mass it is possible to extract quite con-\nstrained information on the high density density EOS; ii)\nthe three bands obtained with di\u000berent maximum masses\nlie almost inside the 90% credible interval predicted from\nthe GW170817 by LIGO/Virgo, obtained imposing that\na maximum star mass equal to 1.97 M\f, the main dif-\nference lying on the upper boundary corresponding to\nthe more massive stars, iii) the probabilistic approach\nwhich allows to take into account the whole GW170817\npdf has given more freedom to the low density EOS, giv-\ning rise to distinct bands also at low densities contrary\nto the previous conclusions. The p(n) band obtained by\nthe LIGO/Virgo collaboration was obtained from a set of\nEOS not constrained by nuclear matter properties and,\ntherefore, may cover a region that would be excluded by\nnuclear properties.\nUsing the probabilistic model, it was possible to de-\ntermine the expected pressure band for a maximum star\nmass within a well de\fned mass range. For mass intervals\nthat take as the lower limit observed 2 solar mass stars\nthe lower limit of the LIGO/Virgo collaboration pressure\nband is not populated at 90% credible probability. For\nthe speed of sound, a monotonic increasing function of\nthe density, values quite far from the conformal limit\nwere obtained. The maximum mass clearly constrains\nthe speed of sound, with values close to one being ob-\ntained only if a maximum mass of 2.66 M\fis imposed.\nThe main e\u000bect of the probabilistic approach that takes\ninto account the full pdf obtained by the LIGO/Virgo\ncollaboration for the tidal deformability with respectto the study done in the previous section is to enlarge\nthe speed of sound range at low densities, allowing\nfor quite low values of the speed of sound, and, there-\nfore giving rise to a harder EOS at intermediate densities.\nInstead of a \fxed value for Mmax, we can\nconsider an interval by determining P(p(n)) =Rb\naP(p(n);Mmax)dMmax. The results are in Fig. 6. Two\nmain conclusions can be drawn: i) the detection of NS\nwith a mass above two solar masses that decreases the\nNS maximum mass uncertainty interval from below will\nhelp constraining the high density EOS. Future observa-\ntion from Square Kilometer Array (SKA) will certainly\nbring information on this lower limit; ii) taking as hy-\npothesis a one branch mass-radius NS curve and impos-\ning maximum masses above 2 solar masses does not allow\nto cover the whole pressure-density range de\fned by the\nLIGO/Virgo collaboration for the GW170817 binary NS\nmerger.\nIV. CONCLUSIONS\nIn the present study we have studied the possibility of\nconstraining the EOS from the knowledge of the NS max-\nimum mass, Mmax, starting from the hypothesis of a EOS\nwith no \frst order phase transition and, as a second ob-\njective, we have shown how to use the entire information\nobtained from the GW170817 event for the probability\ndistribution of ~\u0003 to make a probabilistic inference of the\nEOS. We have generated a set of 150000 EOS that are\nconstrained at low densities, saturation density and be-\nlow, by nuclear properties. Otherwise, these EOS were\nrequired to be thermodynamically consistent, casual and9\n100101102\n0.2 0.3 0.4 0.5 0.6\nn [fm-3]p [MeV/fm3]\n0.000.250.500.751.00\n0.2 0.3 0.4 0.5 0.6\nn [fm-3]vs2 [c2]\nFigure 6. Inference results for the pressure (left) and speed of sound squared (right) as a function of baryonic density. The\nred, blue, and green regions indicate the 90% credible intervals for Mmax=M\f2[2:0;2:66],2[2:2;2:66], and2[2:5;2:66],\nrespectively. The gray region corresponds to the 90% posterior credible level from [13].\nto describe NS with maximum masses above 1.97 M\f.\nIn a \frst step the EOS set was divided into three subsets\ncorresponding to three intervals for the maximum mass,\ne.g. [1.97, 2.2] M\f, [2.2,2.4]M\fand above 2.4 M\fand\nthe pressure of catalysed beta-equilibrium neutral mat-\nter was obtained as a function of the baryonic density. It\nwas shown that, while at low densities the pressure band\nobtained for each set coincide below two times saturation\ndensity, above these density the di\u000berences are large and\nthe pressure and speed of sound of the extreme sets (the\nlowest and largest mass sets) do not overlap, showing\nthat the knowledge of the NS maximum mass may give\nsome information on the high density EOS. These sets,\nhowever, do not cover the whole EOS band predicted by\nthe GW170817 event [32]. The set we are using is more\nrestrictive at low densities, since the EOS set used in\nthe GW170817 analysis does not take into account nu-\nclear properties. Recently some tension between obser-\nvational data and the nuclear physics inputs, and on the\ndeformability probability distribution depending on the\nthe inclusion or not of multimessenger information was\ndiscussed in [55]. In particular, this analysis discusses\nthe implications on the nuclear matter EOS properties.\nWe have next used a probabilistic approach based\non our complete EOS dataset in order to use all the\ninformation the GW170817 gives us. This has allowed\nus to explore a region of the EOS phase space that\nwas not accessible if nuclear matter properties close\nto saturation density are imposed. It is shown that it\nis possible to extract constrained information on the\nhigh density density EOS from the knowledge of themaximum mass. In particular, it was shown that the\n90% credible intervals of the pressure as a function of the\nbaryonic density obtained considering a maximum mass\nof 2M\f, 2:3M\fand 2:6M\fdo not overlap for densities\nabove 0.45 fm\u00003. Moreover, neither the pressure nor the\nspeed of sound 90% credible intervals considering the\ntwo maximum masses, 2 M\fand 2:6M\foverlap for any\ndensity above saturation density. This seems to indicate\nthat determining the NS maximum mass will give us\nstrong constraints on the EOS at high densities. In\nparticular, the detection of more massive NS will narrow\nthe uncertainty on the high density EOS. It was also\nshown that the speed of sound increases monotonically\nwell above the conformal limit, and that a maximum\nmass of the order of 2.6 M\fmay push the upper limit of\nthe speed of sound to \u00181 at 4n0. However, there is also\na \fnite probability of obtaining an EOS that satis\fes\nv2\ns.1=3 if the maximum mass is close to 2 M\f.\nV. ACKNOWLEDGMENTS\nThis work was partially supported by national funds\nfrom FCT (Funda\u0018 c~ ao para a Ci^ encia e a Tecnologia, I.P,\nPortugal) under the Projects No. UID/FIS/04564/2019,\nNo. UID/04564/2020, and No. POCI-01-0145-FEDER-\n029912 with \fnancial support from Science, Technology\nand Innovation, in its FEDER component, and by the\nFCT/MCTES budget through national funds (OE).\n[1] Z. Arzoumanian et al. 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Col\u0012 o,1, 2X. Roca-Maza,1, 2and E. Vigezzi2\n1Dipartimento di Fisica \\Aldo Pontremoli\", Universit\u0012 a degli Studi di Milano, 20133 Milano, Italy\n2INFN, Sezione di Milano, 20133 Milano, Italy\nA complete solution to the inverse problem of Kohn-Sham (KS) density functional theory is\nproposed. Our method consists of two steps. First, the e\u000bective KS potential is determined from\nthe ground state density of a given system. Then, the knowledge of the potentials along a path\nin the space of densities is exploited in a line integration formula to determine numerically the KS\nenergy of that system. A possible choice for the density path is proposed. A benchmark in the\ncase of a simpli\fed yet realistic nuclear system is shown to be successful, so that the method seems\npromising for future applications.\nPACS numbers: 21.60.Jz, 21.60.-n\nI. INTRODUCTION\nThe Inverse Kohn-Sham (IKS) problem [1, 2] aims at\nreverse engineering the two steps that characterise Den-\nsity Functional Theory in the Kohn-Sham direct scheme\n(KS-DFT) [3, 4]. In the direct DFT problem, one usually\nassumes an Energy Density Functional, or EDF, E[\u001a].\nMinimisation of this functional, in the form \u000eE[\u001a] = 0,\nprovides a formulation of the ground state (g.s.) prob-\nlem, while the Runge-Gross theorem sets the formal basis\nfor the study of excited states within a time dependent\nDFT framework [5].\nSpeci\fcally, in the KS-DFT scheme a single-particle\n(s.p.) representation is employed. Indeed, an auxiliary\nsystem of independent particles with the same g.s. den-\nsity as the true interacting system is introduced. The KS\ndensity, then, is a function of the s.p. orbitals \u001e\u000b(r) and\nreads\n\u001a(r) =X\n\u000b\u001e\u0003\n\u000b(r)\u001e\u000b(r): (1)\nThe EDF is customarily written as\nE[\u001a] =T+F+Vext; (2)\nwhereTis the kinetic energy of a system of indepen-\ndent particles, Fis the internal potential energy of the\nauxiliary KS system, and Vextis the contribution to the\nenergy due to an external potential vext(r). A potential\nv[\u001a] is associated to Fby means of\nv[\u001a]\u0011\u000eF\n\u000e\u001a: (3)\nThen, the variational equation \u000eE[\u001a] = 0, with the con-\nstraint that the orbitals form an orthonormal set, gives\nrise to the so-called KS equations, that are a set of one-\nparticle Schr odinger-like equations. Hence, the \frst step\nof the direct KS problem consists in calculating the ef-\nfective potential v[\u001a] from an assumed form for F. The\n\u0003francesco.marino@unimi.itsecond step consists in solving the KS equations in or-\nder to determine the s.p. orbitals and the corresponding\ndensity (1). In short,\nE[\u001a]!v[\u001a]!\u001a (4)\nis a view of the direct KS problem.\nThe IKS problem, that is\n\u001a!v[\u001a]!E[\u001a]; (5)\nis far less simple. Inverse problems are in general dif-\n\fcult to tackle from a mathematical and computational\nviewpoint. First, we should stress that there are many\nmore attempts and results concerning the \frst branch\n(density-to-potential, hereafter D2P) than for the second\none (potential-to-energy, hereafter denoted by P2E). As\nfor the D2P step, the \frst techniques to solve the IKS\nproblem have been proposed in the context of electronic\nDFT in Refs. [6{9]. This topic has recently encountered\na renewed interest both in quantum chemistry [10{14]\nand in nuclear physics [15, 16] (our group had undertaken\npreliminary steps before Ref. [15], cf. e.g. [17]). Sev-\neral inversion techniques that are relevant in this context\nare reviewed in the useful Refs. [1, 2, 18], and inversion\ncodes either have been developed ex novo [13] or have\nbeen integrated into existing libraries, such as Octopus\n[19]. Complementary to these practical implementations,\nwe should also mention a recent attempt to delve more\ninto the mathematical aspects of the IKS problem [20].\nHowever, in spite of the indubitable technical progress\n[10], the \frst branch of the IKS problem has found per\nseonly a limited number of applications. So far, indeed,\nthe reverse-engineered KS potentials have been mostly\nemployed to benchmark existing models [12, 21]. In our\nview, the topic of utmost interest is the determination of\nthe EDF itself. To this purpose, the P2E step is necessar-\nily involved. Among the few attempts in this direction,\nwe mention that machine learning approaches to DFT\nmay bene\ft from the knowledge of the exact KS poten-\ntial [10]. For example, in Ref. [22] it has been shown that\ntraining a neural network EDF not only on energies, but\nalso on potentials, greatly improves its performances, atarXiv:2110.11193v1  [nucl-th]  21 Oct 20212\nleast in a simpli\fed one-dimensional framework. A dif-\nferent line of research combines density functional per-\nturbation theory (DFPT), where an ansatz for the EDF\nfunctional form must be assumed, and IKS to improve\nthe initial model towards the exact EDF [11, 16]. Illus-\ntrative calculations have been performed in the case of\ncovariant nuclear DFT [16].\nA more direct approach to \fnd the EDF from a so-\nlution of the whole IKS problem, as sketched in (5), is\nour main interest in this work. The second branch, P2E,\nhas been the subject of some theoretical investigation,\nalthough not of a full-\redged solution. A line integra-\ntion formula has been proposed by R. Van Leeuwen and\nE.J. Baerends in [23] as a mean to perform the functional\nintegration and obtain the total energy Eof a given sys-\ntem with density \u001afromv. Such formula requires the\nknowledge of the system densities not only in the ground\nstate but also along a given path, together with the corre-\nsponding potentials. The relation between the potential\nvand the EDF has been thoroughly investigated by A.P.\nGaiduk et al. in [24{26]. The line integration formula\nhas been also discussed by the authors of Ref. [27], as we\nshall mention below.\nThe novel contribution of this paper consists in outlin-\ning a possible path towards the full solution of the IKS\nproblem, and of its implementation for a simple and yet\nrealistic case, in nuclear physics. In particular, we have\nmerged the solution of the \frst branch of the IKS prob-\nlem (as already obtained in [15]) with the line integra-\ntion formula proposed in Ref. [23]. We will demonstrate\nthat, by choosing a path of conveniently scaled densities\n\u001at(r), parametrized by a real number t, and by inserting\ndensities and associated potentials in the line integra-\ntion formula, we can reconstruct the total energy of the\nsystem, relative to the g.s. energy, as a function of the\nscaling parametertwith very good accuracy. This will\nallow to gain some insights, as we shall discuss, about\nthe functional form of the underlying EDF.\nOur method shows some resemblance with the scheme\nproposed by Dobaczewski et al. in Refs. [28, 29]. In-\ndeed, both approaches share the idea that information\nabout nuclei driven outside their ground state should\nbe used in order to probe the exact EDF. To do that,\nsome external perturbation must be introduced. In Ref.\n[29] extensive ab initio calculations have been performed,\nwhere both one- and two-body operators were added to\nthe Hamiltonian. Our approach di\u000bers in that we gener-\nate a set of benchmark densities by means of a perturb-\ning one-particle external potential Vext(r), simplifying in\nthat way the required benchmark calculations for the IKS\nmethod.\nIn order to appreciate the relevance of our work, a few\nwords about the status of nuclear DFT are in order. Nu-\nclear DFT is a more challenging problem than electronic\nDFT. In the case of electronic systems, the Coulomb in-\nteraction is well-known and when building an EDF it\nis natural to single out the contribution from the direct\nHartree term. This, together with the kinetic energy,provides the largest contribution to the EDF and leaves\nout only the so-called exchange-correlation part to be\ndetermined. In contrast, the e\u000bective Hamiltonian that\ndescribes the nucleon-nucleon interaction is not so accu-\nrately known (an overview of ab initio nuclear physics\ncan be found in Refs. [30{32]). A further issue is that\nthree nucleon forces are relevant and cannot be ignored.\nIn spite of all these di\u000eculties, nuclear DFT has\nachieved a considerable number of successes. These are\ndescribed in the recent textbook [33]. A shorter review\ncan be found in Ref. [4]. Throughout the isotope chart,\nstate-of-the-art nuclear EDFs can describe the overall\ntrend of masses (with about 1 MeV average accuracy)\nand nuclear radii (with about few times 10\u00002fm average\naccuracy), as well as some other properties like the elec-\ntric quadrupole moments or the main features of giant\nresonances. Hundreds of EDFs exist because many possi-\nble functional forms can be assumed, and many protocols\nto determine the EDF parameters can be designed. It is\nfar from obvious that new attempts to extend the EDF\nforms or to employ more re\fned \ftting protocols may\nlead to substantial improvements [34]. Current EDFs\nare frequently based on nuclear phenomenology and suf-\nfer from large extrapolation errors [35], which are very\ndi\u000ecult to quantify. In other words, a well established\ntheoretical scheme for a systematic improvement does not\nseem to exist at present. For those reasons, some prac-\ntitioners believe that a re-thinking of the nuclear DFT\nstrategy would be timely [28, 34, 36] (see also the work\nof some of us in Ref. [37]).\nIn this respect, non-empirical EDFs, or even strategies\nto better constrain some terms of the EDFs, would be\nextremely welcome. Deriving the nuclear EDF from ab\ninitio is a possible avenue and is the subject of intense\ninvestigations [29, 37, 38]. Solving the IKS problem with\nexperimental or ab initio densities as an input is a com-\nplementary possibility. Our work has the goal of showing\nthat, at least in a simple situation, the IKS problem can\nindeed be solved.\nIn Sec. II, the KS framework is reviewed and the line\nintegration formula is presented. Then, in Sec. III, a\nfew key aspects related to the use of this formula are\ndiscussed in detail. Sec. IV provides a description of the\nmethod we employ to perform the D2P inversion, that is,\nthe constrained variational (CV) method. In Sec. V, our\nmethod is applied to a case study. Lastly, conclusions\nand perspectives of our work are outlined in Sec. VI.\nII. THE LINE INTEGRATION FORMULA\nWe \frst remind the essential notions of DFT and clar-\nify our notation. Useful references are [3, 4, 33].\nIn the KS scheme, the EDF is written as in Eq. (2)\nas a functional of the auxiliary s.p. orbitals. Tis the\nnon-interacting kinetic energy.\nFrom the universal part F, that ultimately stems from\nall the inter-particle interactions, we de\fne the potential3\nv(also named self-consistent potential in what follows),\nv([\u001a];r) =\u000eF\n\u000e\u001a(r): (6)\nOn the other hand, the external potential vext(r) and the\nexternal contribution to the total energy are related by\nVext=Z\nd3r vext(r)\u001a(r) (7)\nvext(r) =\u000eVext\n\u000e\u001a(r): (8)\nApplying the variational principle to Eq. (2) leads to\nthe following KS equations for the s.p. orbitals:\n\u0012\n\u0000~2\n2mr2+vKS\u0013\n\u001ei=\u000fi\u001ei; (9)\nvKS=v[\u001a] +vext: (10)\nThe KS potential vKSis the sum of the self-consistent\npotential and the external term. The latter is a function\nof the position only, while the former also depends on the\ndensity.\nAfter setting the framework, we can now focus on\nthe relation between the self-consistent potential and the\nEDF, namely the aforementioned line integration for-\nmula. In Ref. [23] (cf. also [24]), it is shown that\nif one knows the e\u000bective potential v[\u001a] along a path\nof densities, then the corresponding change in the en-\nergy functional can be reconstructed. In particular, a\none-parameter family of densities is considered. Ac-\ncordingly, we shall write the densities as \u001at(r), with\nA\u0014t\u0014B. The reconstruction formula has been dis-\ncussed in [23, 24], which focus on electronic DFT, for the\nexchange-correlation part of the functional only. On the\nsame grounds, we can write the following formula for F:\nF[\u001aB]\u0000F[\u001aA] =ZB\nAdtZ\nd3rv([\u001at(r)];r)d\u001at(r)\ndt:(11)\nThis formula is actually holds for any functional of the\ndensity and of its gradients. We also remind that EDFs\nare generally written as the integral of an energy density\nf\nF[\u001a] =Z\nd3r f(r;\u001a;r\u001a;::: ):\nwhich is unique up to a gauge transformation, i.e. a term\n\u0012(r) whose volume integral vanishes.\nFor later convenience, we also de\fne It(R) by means\nof\nIt(R) =ZR\n0drr2Z\nd\nv([\u001at(r)];r)d\u001at(r)\ndt; (12)\nwhere \n is the solid angle and Ris the radial upper bound\nfor the numerical integration. We expect It(R) to be aconvergent function as Rgoes to in\fnity. Then, (11) can\nalso be written as\nF[\u001aB]\u0000F[\u001aA] =ZB\nAdt It(R\u0000!+1): (13)\nIn principle, any reasonable density path could be chosen\nin order to perform the line integral. Three speci\fc paths\nwere discussed in Ref. [24]. Our actual choice of the\ndensity path shall be discussed below.\nThe line integration formula, so far, has been mostly\nemployed in cases where an analytical dependence of the\ne\u000bective potential with respect to the density is given, see\n[24]. The same perspective is taken by the Levy-Perdew\nvirial relation [39, 40] between exchange potential and\nexchange energy, where the energy can be deduced from\nthe potential evaluated for just one density. In passing,\nwe mention that line integration techniques have found\napplications also in constructing approximation to the\nnon-local kinetic energy in orbital free DFT [41, 42].\nOur idea is to move one step forward: the formula\n(11) shall be applied to cases where such prior analytic\nknowledgev[\u001a] is not available. Speci\fcally, the relation\nbetween densities and potentials shall be de\fned by a\nnumerical inversion procedure, as detailed in Sec. IV.\nIII. CONCEPTUAL REMARKS\nWe present here some remarks that concern the valid-\nity of the line integration formula.\nA. v-representable solutions\nThe \frst question concerns the choice of the density\npath. While it is simple to design a path \u001at(r) that in-\ncludes only N-representable densities, it is not possible\nto guarantee, in general, that the path avoids densities\nthat are not v-representable [43]. From a physical view-\npoint, a reasonable and somehow conservative choice may\nbe that of spanning a path of densities that are close to\nthe actual ground state density of a given system or to a\nrealistic approximation thereof, e.g., as it may be deter-\nmined by an ab initio calculation. A theorem by Kohn\n[44] gives theoretical support to our argument.\nAs an example, in Sec. V results will be presented\nfor densities originating from monopole deformations of\na nucleus [45]. In this case, the existence of a wide lit-\nerature on the e\u000bects of small variations of the shape of\na nucleus around its equilibrium state can be exploited\nto design clever and physically motivated density paths.\nTo sum up, we are con\fdent that a sound physical in-\ntuition of the systems at hand allows to avoid the v-\nrepresentability problem.4\nB. Self-consistent potential\nThe second question is a key issue for our discussion.\nThe D2P solution allows us to determine the KS potential\nof (10), that is,\nvKS=v[\u001a] +vext:\nHowever, Eq. (11) speci\fcally requires the knowledge of\nv[\u001a] alone. While we cannot suggest a general solution,\nwe propose a way to circumvent this di\u000eculty in speci\fc\ncases. Indeed, the freedom in the choice of the density\npath has to be exploited. Theoretical calculations of the\nsystems under study, subject to a known external po-\ntentialvext;t, can be performed and the corresponding\ndensities\u001at(r) can be taken as input. In Sec. V, we will\nshow a demonstration of the feasibility of this approach\nwhere constrained Hartree-Fock (CHF) calculations of an\natomic nucleus [46] are employed to produce the bench-\nmark\u001at(r) from a given vext;t. In the longer term, it\nwould be interesting to employ ab initio simulations as a\nbenchmark.\nC. Energy conservation\nIn this context, an interesting identity that has been\n\frst introduced in Ref. [27], is worth discussing, because\nit is an useful test of our numerical procedure. Let us\nassume that \u001at(r) is a number-preserving path, so that all\ndensities are normalized to the same number of particles.\nLet us consider the fundamental DFT Euler equation [3]\n\u000eT[\u001a]\n\u000e\u001a(r)+vKS([\u001a];r) =\u0016[\u001a]; (14)\nwhere\u0016is the chemical potential. Then, we can multiply\nEq. (14) byd\u001at\ndtand integrate over both tand r. Since\n\u0016is independent from randR\ndr\u001at(r) is a constant with\nrespect tot, the r.h.s. vanishes. It follows immediately\nthat\n\u0001T=\u0000ZB\nAdtZ\ndrvKS([\u001at(r)];r)d\u001at(r)\ndt: (15)\nNote that the right hand side of this equation is di\u000ber-\nent from that of Eq. (11) and it is equal to \u0000\u0001UKS=\n\u0000[\u0001F+ \u0001Uext]. The total variation of the energy\n\u0001T+ \u0001UKS= 0 is conserved as it should since the cho-\nsen path comes from a minimization of Eq. (2) for each\nvalue oft. Therefore, this relation is a conservation law\nthat holds on density and potential paths \u001atandvKS[\u001at].\nIt is potentially well suited to test the correctness of the\nimplementation of both the D2P step and Eq. (11). In-\ndeed, as detailed in Sec. IV below, from an IKS calcu-\nlation both the KS potential and the kinetic energy are\nobtained. One can then compare \u0001 T=T[\u001aB]\u0000T[\u001aA]\nto the integral on the r.h.s. of Eq. (15), thus gaining in-\nformation on the relative numerical accuracy of the two\nprocedures (Sec. V).IV. CONSTRAINED-VARIATIONAL METHOD\nIn this Section, the formalism and the implementation\nof the CV method [1, 15] are reviewed. In the KS scheme\n[3], a system of Nindependent particles is \frst consid-\nered, which is described by a set of single-particle (s.p.)\norbitals\u001e\u000b(r). The independent-particle kinetic energy\nis then a functional of the orbitals and reads\nT[f\u001e\u000b(r)g] =\u0000~2\n2mNX\n\u000b=1Z\ndr\u001e\u0003\n\u000b(r)r2\u001e\u000b(r):(16)\nSuppose that the ground state (g.s.) density of the sys-\ntem\u001a(r) (1) be equal to a given target density ~ \u001a(r). Such\ndensity may be known from experiment or an ab initio\ncalculation. Under these assumptions, the g.s. is char-\nacterized by the condition that the kinetic energy (as a\nfunctional of the orbitals) is minimized under the con-\nstraint\u001a(r) = ~\u001a(r). Moreover, the additional orthonor-\nmality constraintsR\ndr\u001e\u0003\n\u000b(r)\u001e\f(r) =\u000e\u000b\fare imposed on\nthe orbitals. The problem has the form of a constrained\nvariation of an integral functional. The Lagrange multi-\npliers method allows to convert it into the free optimiza-\ntion of the objective functional\nJ[f\u001e\u000b(r)g;vKS(r);f\u000f\u000b\fg] =T[f\u001e\u000b(r)g] +\n+Z\ndrvKS(r) [\u001a(r)\u0000~\u001a(r)]\n\u0000X\n\u000b<\f\u000f\u000b\f\u0012Z\ndr\u001e\u0003\n\u000b(r)\u001e\f(r)\u0000\u000e\u000b\f\u0013\n: (17)\nWe stress that Jis a functional of the additional vari-\nablesvKS(r) and of\u000f\u000b\f. We have readily identi\fed the\nLagrange multiplier associated to the density constraint\nwith the KS potential vKS. These two quantities coincide\nonly if the minimum of Jhas been obtained.\nBy solving \u000eJ= 0 plus the subsidiary conditions, the\nvalue of the orbitals and of the multipliers can be deter-\nmined. The Euler-Lagrange equations for Jread\n\u0000~2\n2mr2\u001e\u000b(r) +vKS(r)\u001e\u000b(r) =X\n\f\u000f\u000b\f\u001e\f(r):(18)\nThese are s.p. Schr odinger equations in a non-canonical\nform [27]. After these have been solved, a diagonaliza-\ntion of the \u000fmatrix allows to determine the canonical\neigenfunctions and the s.p. energies, which satisfy\n\u0000~2\n2mr2\u001e\u000b(r) +vKS(r)\u001e\u000b(r) =\u0015\u000b\u001e\u000b(r): (19)\nThus, Eq. (19) con\frms the physical interpretation of\nthe Lagarnge multipliers.\nIn practice, works on the IKS problem have focused\non spherically symmetric systems [15, 16]. As a conse-\nquence, the problem simpli\fes to a one-dimensional ra-\ndial equation, and the labels \u000b;\fwill correspond to the5\nusual quantum numbers ( nlj) with the standard shell-\nmodel ordering. The angular and spin parts of the wave\nfunctions are then known, and only the radial functions\nu\u000b(r) have to be determined. The orthonormality condi-\ntions have to be imposed only if l\u000b=l\fandj\u000b=j\fand\nreadR+1\n0dr u\u000b(r)u\f(r) =\u000en\u000bn\f. We note here that the\ninclusion of a spin-orbit potential is left for future devel-\nopments. The KS potential is then orbital-independent.\nThe numerical implementation is based on a discretiza-\ntion scheme, where the radial functions are evaluated on\na radial mesh and the \fnite di\u000berence method is used to\napproximate the derivatives [1]. The solution proceeds\nas follows. First, the constrained optimization library\nIPOPT [47, 48] is used to \fnd the orbitals u\u000b(r) that\nminimize the kinetic energy and at the same time sat-\nisfy the aforementioned constraints within a given toler-\nance (see also Ref. [15]). Next, we aim at determining\nthe KS potential and the energy eigenvalues. The pre-\nviously determined radial functions are plugged into the\nnon-canonical Schr odinger equations (18) that now turn\ninto a linear system, whose unknowns are vKS(r) on each\npoint of the mesh and the \u000f\u000b\f's [1]. This is an overdeter-\nmined algebraic system, with one equation for each point\nof the mesh and each di\u000berent radial function. A least-\nsquares algorithm is employed to solve it. Lastly, the \u000f\nmatrix is diagonalized and the canonical solutions of (19)\nare thus found. The outcome of the CV method, there-\nfore, consists in the orbitals u\u000b(r), the non-interacting\nkinetic energy T, the KS potential vKS(r) and the eigen-\nvalues\u0015\u000bof the e\u000bective s.p. description of the system.\nA new code has been developed in Python which al-\nlows to perform the D2P inversion once a density is given\nas input. A technical note on the numerical inputs of the\nCV calculations is here in order. We have found that the\nconvergence of the method is mostly a\u000bected by the fol-\nlowing IPOPT parameters: the relative tolerance on the\nvalue of the objective function and the absolute tolerance\non the ful\fllment of the constraints [47]. The choice of\nthe upper bound Rof the radial mesh is also important.\nTo be physically meaningful, a radial potential should\nvanish asr\u0000!+1. It is well-known that the asymp-\ntotic behaviour of the target density directly impacts that\nof the corresponding potential. For example, Gaussian\ntails are notoriously problematic, cf. [10, 15]. The densi-\nties employed in this work, however, are all well-behaved,\nas far as their asymptotic properties are concerned. A\nfurther caveat relates to the fact that a potential is de-\n\fned only up to a constant. Thus both v(r) and\u0015\u000b\nmust be shifted, so as to ensure that v(r) goes to zero\nat larger. Actually, the correct asymptotic behaviour of\nthe potential is ensured by a correct reproduction of the\nionization energy in atomic physics or neutron/proton\nseparation energy in nuclear physics [49, 50]. This is a\nfurther check or recipe that should be applied whenever\nthis information is available.\nLastly, we mention that some inaccuracies may occa-\nsionally show up near the border of the radial mesh. For\nexample, the potential may take unnatural values in alimited number of points in such region. However, we\nhave veri\fed this has no impact on the energy integrals\ndue to the exponentially decreasing behaviour of the den-\nsities.\nV. RESULTS\nA. Physical case and numerical details\nIn this Section, we will implement the strategy outlined\nin the previous Section and solve the two-step IKS prob-\nlem. Namely, we will try to reconstruct the functional\nstarting from a set of densities \u001a\u0016, with\u0016A< \u0016 < \u0016B.\nThe densities will be generated by means of a set of KS\ncalculations carried out with a Skyrme EDF, to which a\nharmonic external potential vext;\u0016 =\u0016r2is added. Note\nthat in the nuclear physics context these are often called\nself-consistent (constrained) Hartree-Fock calculations.\nFor each value of \u0016, we will use the procedure of\nSec. IV to obtain the KS potential v([\u001a\u0016];r) by inver-\nsion (\frst step, D2P). We will then insert these poten-\ntials in Eq. (11) to reconstruct the energy di\u000berence\n\u0001F=F[\u001a\u0016B]\u0000F[\u001a\u0016A] (second step, P2E). At the end,\nwe will assess to what accuracy the procedure reproduces\nthe benchmark energy di\u000berence associated with the orig-\ninal Skyrme EDF.\nWe highlight that the same procedure can be followed\nstarting from a set of densities generated by ab initio cal-\nculations, without making reference to any pre-existing\nEDF. The two-step solution of the IKS problem can then\nlead to values of \u0001 Fthat will provide precise information\non the underlying EDF.\nFor this \frst application of the method, we will present\nresults obtained with a simpli\fed EDF. In particular, we\nbenchmark our method to a t0\u0000t3Skyrme EDF (see e.g.\nRef. [51]) with which we generate the target densities.\nThis EDF is characterized by the energy density\nF=Z\nd3rF=Z\nd3r(F0+F1); (20)\nwhere\nF0=1\n4t0(2\u001a2\u0000(\u001a2\np+\u001a2\nn)); (21)\nF1=1\n24t3\u001a\u000b(2\u001a2\u0000(\u001a2\np+\u001a2\nn)); (22)\n\u001a=\u001ap+\u001an: (23)\nSinceFdepends on the densities, but not on their gra-\ndients, the potential takes the form\nvq([\u001a(r)];r) =\u000eF\n\u000e\u001aq=@F\n@\u001aq; (24)\nwhereq=n;p. The EDF parameters are reported in\nTab. I for a few choices of the parameter \u000b. Once\u000bhas\nbeen chosen, the coe\u000ecients t0andt3are determined6\n\u000b t 0(MeV fm3)t3(MeV fm3+3\u000b)\n0.16 -3068.63 19578.08\n0.2 -2581.88 16853.70\n0.32 -1851.75 13124.44\n1 -1024.27 14602.57\nTable I: Parameters of the t0\u0000t3EDFs.\nin such a way that the equation of state of symmetric\nnuclear matter predicted by the EDF has a minimum at\nthe empirical saturation point, \u001a=0.16 fm\u00003andE=\n\u000016 MeV. In Sec. V B and V C, we test our method by\nemploying the EDF with \u000b=0.32. To further simplify\nthe problem, we neglect Coulomb and spin-orbit and we\npick up the representative case of the16O densities ( \u001an=\n\u001ap=\u001a=2).\nA family of scaled densities are generated by means of\nconstrained Hartree-Fock (CHF) calculations [46, 52]. As\na perturbation, a harmonic external potential vext;\u0016(r) =\n\u0016r2, with\u0016in the range [-0.25,0.25] MeV \u0001fm\u00002is em-\nployed. The physical meaning is that of driving a scaling\nof the radius (and of the whole nuclear density) with re-\nspect to the unperturbed case. This approach has its\noriginal motivation in the study of monopole deforma-\ntions [45]. The true ground state of the system is given\nby\u0016= 0 and is an absolute minimum of the energy T+F.\nThe D2P inversion is performed by means of the CV\nmethod described in Sec. IV, which provides both the\ne\u000bective KS potential vKSand the kinetic energy associ-\nated to the neutron density.\nB. Potentials\nTo exemplify the subtle point concerning the di\u000ber-\nence between vKSandv[\u001a] (Sec. III), in Fig. 1 and\n2 the CHF potential for either \u0016=\u00000:2 or\u0016= 0:2\nMev fm\u00002is compared to the IKS potential (KS). The\ntwo functions are clearly di\u000berent, while the third curve\nv[\u001a\u0016] =vKS;\u0016\u0000vext;\u0016 =vKS;\u0016\u0000\u0016r2, de\fned by subtract-\ning the harmonic term from the IKS potential, matches\nrather well the CHF potential. This proves the accuracy\nof the CV method.\nFrom now on, only the self-consistent part of the po-\ntential,v[\u001a\u0016], shall be displayed. In Fig. 3, three den-\nsities (top) and the corresponding potentials (bottom),\ndetermined by means of the CV inversion, are compared\nfor three di\u000berent values of \u0016. A positive value \u0016 > 0\nacts as a con\fning potential and as a consequence leads\nto a more compact density pro\fle than in the unper-\nturbed case. Conversely, a repulsive external potential\n(\u0016 < 0) leads to systems which are more spread out.\nConsequently, in the latter case the density peak in the\ninterior of the nucleus is less pronounced.\n0 1 2 3 4 5 6\nr (fm)60\n50\n40\n30\n20\n10\n01020v(r) (MeV)vKS\nv\nvCHFFigure 1: The potential derived from \u001a\u0016by the IKS\nprocedure ( vKS), the CHF potential vCHF and\nv=vKS;\u0016\u0000\u0016r2are compared for \u0016=\u00000:2 MeV\u0001fm\u00002.\n0 1 2 3 4 5 6\nr (fm)60\n50\n40\n30\n20\n10\n01020v(r) (MeV)vKS\nv\nvCHF\nFigure 2: Same as Fig. 1, but for \u0016= 0:2 MeV\u0001fm\u00002.\nC. Line integration\nWe can now move on to the study of the P2E and the\nline integration formula. A preliminary test is presented\nin Fig. 4. There, the function I\u0016(R) (12) is plotted as\na function of the radius Rin the cases \u0016=\u00000.2, 0, 0.2\nMeV\u0001fm\u00002. Our concern here is that of verifying the7\n/uni00000013/uni00000011/uni00000013/uni00000013/uni00000013/uni00000011/uni00000013/uni00000015/uni00000013/uni00000011/uni00000013/uni00000017/uni00000013/uni00000011/uni00000013/uni00000019/uni00000013/uni00000011/uni00000013/uni0000001b/uni00000013/uni00000011/uni00000014/uni00000013/uni00000013/uni00000011/uni00000014/uni00000015/uni0000000b/uni00000055/uni0000000c/uni00000003/uni0000000b/uni00000049/uni000000503/uni0000000c\n/uni00000013 /uni00000014 /uni00000015 /uni00000016 /uni00000017 /uni00000018 /uni00000019/uni00000010/uni00000019/uni00000013/uni00000010/uni00000017/uni00000018/uni00000010/uni00000016/uni00000013/uni00000010/uni00000014/uni00000018/uni00000013v([]/uni0000000f/uni00000055/uni0000000c/uni00000003/uni0000000b/uni00000030/uni00000048/uni00000039/uni0000000c\n=/uni00000010/uni00000013/uni00000011/uni00000015\n=/uni00000013/uni00000011/uni00000013\n=/uni00000013/uni00000011/uni00000015\nFigure 3:16O neutron densities (top) and corresponding\nself-consistent potentials (bottom) for three values of\nthe perturbation strength \u0016(in MeV fm\u00002).\nasymptotic convergence of I\u0016to a constant for large R.\nIndeed, the convergence is quite fast and a stable result\nis reached already for R= 4 fm. We remind here that\n16O has a root mean square radius of about 3 fm and,\nthus,I\u0016is expected to converge for Rvalues that scale\nwithA1=3.\nNext, we can turn to the study of the kinetic energy\nand the potential energy contributions separately. In Fig.\n5 di\u000berent behaviours of the kinetic energy di\u000berences,\n\u0001T(\u0016) =T(\u0016)\u0000T(\u0016= 0), are shown as functions of \u0016.\nWe consider two independent ways of evaluating \u0001 T(\u0016).\nIndeed, one can simply subtract the kinetic energies pro-\nvided by the CV method when performing inversion at\n\u0016= 0 and at \fnite \u0016. Otherwise, the formula by Karasiev\net al. [27], Eq. (15), can be exploited, where the kinetic\nenergy is computed from the knowledge of the poten-\ntialvKS;\u0016 obtained in the CV framework on a grid of\nintermediate values of \u0016. The two estimates of the ki-\nnetic energy di\u000berence, labelled as IKS and Eq. (15) in\nthe \fgure, are compared to the CHF results. The top\npanel highlights that there is a very good agreement be-\ntween the three curves for all values of \u0016. The deviations\nfrom the CHF results are plotted in the bottom panel. It\ncan be appreciated that the discrepancies are quite small,\nbeing of the order of 10\u00002MeV. In comparison, the neu-\ntron kinetic energy of the unperturbed system amounts\ntoT\u0016=0= 148:37 MeV (including the center of mass cor-\nrection). We observe, however, that Eq. (15) guarantees\nan overall better accuracy. Therefore, it will be used in\n0 1 2 3 4 5 6\nR (fm)-35-30-25-20-15-10-50I=0(R) (fm2)\n=-0.2\n=0.0\n=0.2\nFigure 4:I\u0016(R) (de\fned in (12)) as a function of the\nintegration limit Rfor three representitative values of\n\u0016. The integral function converges asymptotically to a\nconstant for large enough R.\nthe following to estimate \u0001 T.\nWe now compute the universal term For, more pre-\ncisely, \u0001F(\u0016) =F(\u0016)\u0000F(\u0016= 0), by the line integration\nmethod introduced in Sec. II according to Eq. (11). Fig.\n6 shows that the reconstructed IKS curve is in excellent\nagreement with the CHF values obtained with the origi-\nnal Skyrme EDF, with discrepancies of about 10\u00003MeV.\nThis is rather encouraging, since it implies that the line\nintegration has been implemented to a high numerical\nprecision. At the same time, it is a strong support to\nthe reliability of the IKS machinery. This is con\frmed\nby Fig. 7, where we display the total energy di\u000berence\n\u0001F(\u0016) + \u0001T(\u0016), accurate up to a fraction of keV.\nA question now arises: how much information have we\nactually gained about the original EDF, by solving the\ntwo-step inverse IKS problem? We will limit ourselves\nto consider the family of t0\u0000t3Skyrme EDFs, obtained\nby varying the value of the exponent \u000b(cf. Tab. I) and\nby assuming that they are determined in symmetric nu-\nclear matter, as discussed above (cf. Tab. I). In other\nwords, we will not discuss how to determine the absolute\nstrengths (t0andt3) of the di\u000berent terms in the under-\nlingF[\u001a], but only if we can be sensitive to the exponent\nof the density dependence. To this aim, we compare in\nFig. 8 the values obtained for \u0001 T+ \u0001Fas a function of\n\u0016for a few di\u000berent values of \u000b. The resolving power of\nthe method will depend on the range of \u0016taken into ac-\ncount in the calculation. Nonetheless, the \fgure already\nshows that, for a modest change in \u0016, our benchmark\nvalue\u000b= 0:32 can be rather clearly distinguished from8\n6\n4\n2\n024T (MeV)\nIKS\nEq.(15)\nCHF\n0.2\n 0.1\n 0.0 0.1 0.2\n (MeV fm2)\n1.2\n0.9\n0.6\n0.3\n0.00.3(TTCHF)102 (MeV)\nIKS-CHF\nEq.(15)-CHF\nFigure 5: Top: kinetic energy di\u000berences\n\u0001T(\u0016) =T(\u0016)\u0000T(\u0016= 0) obtained with CHF\ncalculations, by inversion of the densities \u001a\u0016(r) (IKS),\nand by employing Eq. (15). Bottom: discrepancy\nbetween the CHF and both IKS and Eq. (15) results.\n\u000b=0.16 and\u000b=1. This type of information may become\ninstrumental in building new EDFs based on ab inito cal-\nculations.\nVI. CONCLUSIONS AND PERSPECTIVES\nA complete solution to the inverse Kohn-Sham prob-\nlem of Density Functional Theory has been proposed.\nFirst, the e\u000bective KS potential associated to a given\nground state density is determined. Second, a path of\nperturbed densities is chosen, and the knowledge of the\nassociated KS potentials is exploited to compute the dif-\nference between the energies of the perturbed and unper-\nturbed states.\nBenchmark calculations have been performed in a case\nrelevant for nuclear physics. In particular, we have shown\nthat, for a simple t0\u0000t3Skyrme EDF, and perturbing the\nsystem with an external harmonic potential, this method\nallows to reconstruct the energies with a rather good nu-\nmerical accuracy.\nThese promising results open up a number of perspec-\ntives. First, one should apply our method using ab initio\ncalculations as an input. The feasibility of these calcula-\ntions when the nuclear systems are perturbed by external\n4\n2\n0246F (MeV)\nIKS\nCHF\n0.2\n 0.1\n 0.0 0.1 0.2\n (MeV fm2)\n3.0\n1.5\n0.01.53.0(FIKS FCHF)103 (MeV)\nIKS-CHFFigure 6: Top: universal energy di\u000berences\n\u0001F(\u0016) =F(\u0016)\u0000F(\u0016= 0) obtained with CHF\ncalculations, by inversion of the densities \u001a\u0016(r) (IKS).\nBottom: discrepancy between the IKS and CHF results.\npotentials, and the assessment of the associated theoreti-\ncal uncertainties, are issues that are mandatory to clarify.\nIn principle, accurate ab initio calculations of nuclear\ndensities perturbed by a variety of external potentials\ncontain a lot of useful information to improve existing\nEDFs. In practice, one could apply the IKS methodology\noutlined in the present work to produce a set of ab initio\nmetadata on how energies change under the action of\nperturbations.\nIn our work, we have not tackled the problem of how\nto fully determine the EDF from these metadata. When\nab initio inputs are available, we envisage to use also the\nresulting absolute values of the energy. Implementing\nappropriate \ftting algorithms on these absolute energies\nmay allow obtaining a set of novel constraints on the\ndi\u000berent terms of the EDF. As the convergence of this\nprocedure is not guaranteed, because the choice of the\ndi\u000berent terms is the result of an ansatz, and the EDF\nwhich is compatible with the ab initio input may not\ncorrespond to that ansatz, one could devise as an alter-\nnative a machine learning algorithm (see, e.g., [22]). This\nshould be able to select the terms in the EDF that are\nmore likely to appear, and quantitatively estimate their\nrelevance.\nIn a later stage, further studies could be envisaged by\nconsidering a set of di\u000berent external potentials that are9\n0.000.050.100.150.200.25(T+F) (MeV)\nIKS\nCHF\n0.2\n 0.1\n 0.0 0.1 0.2\n (MeV fm2)\n4\n3\n2\n1\n01((T+F)IKS (T+F)CHF)\n104 (MeV)\nIKS-CHF\nFigure 7: Top: total energy \u0001 F(\u0016) + \u0001T(\u0016) obtained\nwith IKS and CHF calculations. Bottom: discrepancy\nbetween the IKS and CHF results.\n0.2\n 0.1\n 0.0 0.1 0.2\n (MeV fm2)\n0.00.10.20.30.40.5(T+F) (MeV)\n=0.16\n=0.2\n=0.32\n=1.0\nFigure 8: The total energy \u0001 F(\u0016) + \u0001T(\u0016)\nreconstructed with the two-step IKS method and\nassociated with the value \u000b= 0:32 is compared to the\nvalues obtained with t0\u0000t3EDFs characterized by\ndi\u000berent values of \u000b.\ncoupled to spin densities, isospin densities, or other types\nof densities. 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R. de Baar1;2\n1DIFFER - Dutch Institute for Fundamental Energy Research, PO Box 6336,\n5600HH Eindhoven, The Netherlands\n2Eindhoven University of Technology, Dept. of Mechanical Engineering, Control\nSystems Technology Group, PO Box 513, 5600MB Eindhoven, The Netherlands\nAbstract. In contemporary magnetic confinement devices, the density\ndistribution is sensed with interferometers and actuated with feedback controlled\ngas injection and open-loop pellet injection. This is at variance with the density\ncontrol for ITER and DEMO, that will depend mainly on pellet injection as an\nactuator in feed-back control. This paper presents recent developments in state\nestimation and control of the electron density profile for ITER using relevant\nsensors and actuators. As a first step, Thomson scattering is included in an\nexisting dynamic state observer. Second, model predictive control is developed as\na strategy to regulate the density profile while avoiding limits associated with the\ntotal density (Greenwald limit) or gradients in the density distribution (e.g. neo-\nclassical impurity transport). Simulations show that high quality density profile\nestimation can be achieved with Thomson Scattering and that the controller is\ncapable of regulating the distribution as desired.\nSubmitted to: Journal of Physics CommunicationsarXiv:2110.14975v1  [physics.plasm-ph]  28 Oct 2021Model-based electron density profile estimation and control, applied to ITER 2\n1. Introduction\nFuture reactors, such as ITER and DEMO, aim to\nhave a net energy gain [1]. This requires these\nreactors to operate close to physical limits and to\noptimize quantities such as temperature, density, and\nconfinement. In practice, shot-to-shot differences\nin reactor conditions will always be present and\nthe optimal control action will vary subject to the\nplasma scenario and plasma state. Hence, active\nplasma feedback control is essential in achieving these\nrequirements [2].\nIn a tokamak, the produced fusion power directly\ncorrelates to the density in the hot core of the\nreactor [3], particle transport and non-inductive\ncurrent drive depend on the gradient of the spatial\nparticle distribution [4,5], a.k.a the (particle) density\nprofile. Furthermore, turbulence and impurity\ntransport depend on the logarithmic gradient of the\ndensity profile [6–8] and the density is subject to limits\nthat can lead to detrimental plasma instabilities when\nviolated [9–11]. Consequently, for optimal and reliable\nhigh-performance operation of fusion reactors, particle\ndensity profile estimation and control are mandatory\n[12–15].\nTo synthesize such controllers, model-based con-\ntrol design is required as experimental time for tuning\nand validation on reactors such as ITER will be scarce\nand expensive [16,17]. Model-based design has already\nsuccessfully been applied to synthesize controllers for\nplasma shape [18] and profiles in a tokamak [19–22].\nEspecially, in [23], a controller for the current density\nprofile is presented that can handle plasma limits and\nactuator constraints simultaneously. The design pro-\ncedure requires a control-oriented model of the input-\noutput dynamics, i.e., a model that captures the main\ndynamic relationships between inputs (actuators) and\noutputs (plasma quantities) to be controlled.\nRecently, the control-oriented model RAPDENS\nwas developed for the particle transport dynamics in a\ntokamak [24]. This model has been used to perform\ndensity profile estimation in TCV and AUG with\ninterferometry and spectroscopy [24,25], to design a\ncontroller for the volume-averaged density during the\nITER ramp-up with pellets and gas [26], to estimate\nthe core density in density control experiments with\npellets in AUG [27], and to derive a controller for the\nvolume-averagedensityinTCVusinggasinjection[25].\nHowever, a controller for the particle densitydistribution,: a) that is capable of handling shot-to-\nshot differences; b) can handle state physical limits;\nc) and uses multiple actuators is one key research\nand development aspect still to be solved for a fusion\nreactor.\nIn this work, a two-step model-based approach is\nproposedtodevelopsuchadensityprofilecontrollerfor\nITER. The first is the estimation (or reconstruction)\nof the particle density profile with an observer and\nrelevant sensors. The second is the design and\nvalidation of a model-predictive controller for the\ndensity profile using two pellet injectors.\nFor the first step, RAPDENS [24] is applied to\nITER and extended with a Thomson scattering (TS)\noutput model. Subsequently, the model and synthetic\nTS measurements are used together in an extended\nversion of the dynamic state observer (DSO) proposed\nin [24] to reconstruct the density profile. Simulations\nof the observer are used to determine the quality of the\nachieved profile estimation for an H-mode, 15 MA, 5.3\nT, DT ITER baseline discharge.\nFor the second step, model-predictive control\n(MPC) is applied to synthesize a controller for the\ndensity distribution in ITER using pellet injection as\nmain actuator. For the first time, constraints on the\ndensity profile due to impurity transport are taken\ninto consideration in the control problem. Simulations\nusingthelightweightcontrol-orientedplasmasimulator\nRAPDENS are used to assess the effectiveness of the\ncontroller.\nThe remainder of this paper is structured as\nfollows. Section 2 briefly summarizes the control-\noriented model used for design and simulation. The\nforward TS model and the DSO are presented in\nSection 3. The density reconstruction results are\npresented in Section 4. The controller design\nmethodologyisdescribedinSection5andtheresultsof\ncontrol simulations are shown in Section 6. This paper\nis concluded by a discussion of the work in Section 7.\n2. Control-oriented plasma simulator\nIn this work, we have used and extended the RAP-\nDENS model (originally proposed in [24]) for the DSO\nand the synthesis of the MPC controller. A summary\nof the model is given here.\nThe RAPDENS model is a lightweight, control-\noriented model of the electron particle transport in aModel-based electron density profile estimation and control, applied to ITER 3\ntokamak [24]. It is a multi-inventory model in which\nthe particles inside the vessel are attributed to one\nof three inventories: the plasma, the wall, or the\nvacuum. Note that the vacuum denotes the region\nsurrounding the plasma that is filled with neutrals.\nHowever, the name \"vacuum\" is kept for continuity\nand conherence with previous papers on RAPDENS.\nThe model consists of a 1D-PDE for the evolution of\nthe flux-surface averaged electron density and two OD-\nODE’sfortheevolutionofthewallandneutralvacuum\ninventories. Radial particle transport is modeled with\na drift-diffusion model with ad-hoc chosen transport\ncoefficients. The processes that drive the particle\nexchange between these inventories are modeled in\na heuristic fashion. An overview is given in figure\n1. Additionally, it contains representations of the\nfueling with neutral beam, pellet and gas injection. A\nsummary of the model equations is given is Appendix\nA. The full details and derivations of the model\nequations are outside the scope of this paper and can\nbe found in [24].\nrz\nNBI/Pellet\nWallVacuum\nIonizationRecombination\nPump\nValve Plasma\nRecyclingSOLwall\nSOLe\nFigure 1. Overview of the modeled processes in the RAPDENS\nmodel. The particles inside the tokamak are attributed either to\nthe plasma, the wall or the vacuum. Green arrows represent\nthe modeled particle fluxes between the inventories. Orange\narrows represent particle fluxes crossing the system boundaries.\nAdapted from [28]\n.\nFor the applications reported in this paper,\nRAPDENS is used to formulate a system of nonlinear\ndiscrete-time ODEs that describes the evolution of the\nplasma density due to relevant actuator inputs. The\ninternal variables of the model (states), the external\nparameters, the considered actuators inputs, and the\nsystem of ODEs are briefly discussed next.States: In RAPDENS, the flux-surface averaged elec-\ntron density distribution ne(\u001a;t)is discretized in space\nusing finite elements (see e.g. [29]), i.e, the electron\ndensity is approximated by\nne(\u001a;t) =mX\n\u000b=1\u0003\u000b(\u001a)b\u000b(t); (1)\nwhere\u001ais the normalized toroidal flux. The basis\nfunction \u0003\u000b: [0;\u001ae]![0;1];\u000b= 1;2;:::;mare\ndefined as cubic B-splines with finite support [30]\n(\u001ae>1being the artificially prolonged flux label to\nencompass the scrape-off layer. This is discussed in\nmore details in [24]). The variables b\u000b(t)2Rmare\ntime dependent spline coefficients. These coefficients\ntogether with the time dependent wall and vacuum\ninventories, respectively denoted Nw(t)andNv(t),\ncomprise the state vector x(t)2Rnxwithnx=m+ 2,\ni.e., the state vector is given by\nx(t) =\u0002b(t)Nw(t)Nv(t)\u0003>: (2)\nExternal parameters: RAPDENS requires external\nparameters to be provided in an external parameter\nvectorp(t)2Pdefined as\np= [cDcHTe;bIpV V0\u0014  axis LCFSG1G0];\n(3)\nwherecDindicates limited ( cD= 0) or diverted ( cD=\n1) plasma,cHimplies low confinement ( cH= 0) or\nhigh confinement ( cH= 1) regime,Te;bis the electron\ntemperature at the last-closed flux surface (LCFS),\nIpis the plasma current, Vis the plasma volume,\nV0=dV=dt, and\u0014is the plasma elongation,  axis\nand LCFSare the equilibrium  (R;Z)evaluated at\nthe magnetic axis and LCFS, G1=h(r\u001a)2)iand\nG0=hjr\u001ajiwithhjr\u001aji=hjr ji(@ \n@\u001a)\u00001(whereh\u0001i\ndenote the flux-surface average).\nInputs: The particle fueling rate by pellet injection\n\u0000i\npellet(t)8i= 1;:::;nu, wherenuis the number of\nused pellet injectors, comprise the inputs u(t)2Rnu\nto the modelz. The input vector is defined as\nu(t) =\u0002\u00001\npellet(t);:::; \u0000nu\npellet(t)\u0003>:(4)\nNonlinear system of ODEs: Thespatialdiscretization\nof the density profile (1) is inserted in (A.1)-(A.5)\nand discretized in time using an equidistant temporal\ndiscretization tk=t0+kTsto form the nonlinear set\nof ODEs\nx(tk+1) =f(p(tk);x(tk);u(tk)); (5)\nzNote that RAPDENS also contains models for gas fueling and\nNBI but these are not considered in this work.Model-based electron density profile estimation and control, applied to ITER 4\nwheref:P\u0002Rnx\u0002Rnu!Rnxisanonlinearfunctionx\nof the state that represents the state evolution due to\nphysical phenomena and the influence of the actuators.\nThis formulation is the foundation of the work\npresented in the coming sections. In sections 3 and 4,\nit is used to perform state estimation and in sections\n5 and 6, it is used to derive and test a density profile\ncontroller.\n3. Dynamic state observer for the density\nprofile\nA requirement to perform profile control is the\nestimation of the profile from available measurements.\nFor this purpose, a dynamic state observer (DSO) was\nproposed in [24,25]. In this section, we summarize the\nDSO and we present a forward diagnostic model for TS\nthat will allow us to perform profile estimations using\nTS measurements.\nNotation 1: Let a generic variable e(t)be a\nphysical quantity of a real system. The estimate of\nthis quantity made by a DSO at time t=tk1with\ndiagnosticsignalsfromtime t=tk2isdenotedas ^ek1jk2.\n3.1. Forward Thomson scattering model\nThe DSO is comprised of a prediction model, a\ndiagnostic model, and an observer gain (see figure\n4). The original observer, developed in [24, 25],\nincluded diagnostic models for interferometry and\nbremmstrahlung but not for TS. Here, we discuss the\ninclusion of a TS model that will allow us to perform\ndensity estimation with TS measurements.\nTS systems are used in tokamaks to measure the\nradially resolved distributions of electron temperature\nand density [31]. In combination with knowledge of the\n2D equilibrium, the spatial measurements ne(R;Z;t )\ncan be transformed into equilibrium measurements of\nthe density ne(\u001a;t)at known equilibrium locations \u0016\u001ai\nwithi= 1;:::;nTS.nTSdenotes the total number of\nradial measurement locations. The spatial resolution\nof the ITER core TS system is 67mm [32], which,\nin a typical ITER plasma with a radius of rp\u00192\nm, provides\u001929measurement locations distributed\nover the high and low field side. In this work, to be\nconservative, ny=nTS= 11is chosen.\nThe TS output vector y2RnTSshould contain\nthe plasma density evaluated at the corresponding flux\nxNote that for the modeling is this paper, the function f\nis linear with respect to u. Hence, (5) can be rewritten as\nx(tk+1) = f(p(tk); x(tk)) + Bd(p(tk))u(tk), where Bdis a\nparameter varying matrix.labels \u0016\u001ai, i.e.,\ny(t) =\u0014\nne(\u001a;t)\f\f\f\n\u0016\u001a1\u0001\u0001\u0001ne(\u001a;t)\f\f\f\n\u0016\u001anTS\u0015>\n:(6)\nThe relation between (2) and (6) is obtained\nby evaluating the basis functions used for spatial\ndiscretization in (1) at \u001a= \u0016\u001ai8i= 1;:::;nTS.\nInserting the spatial discretization (1) in (6) results\nin the numerical expression of the output vector as\nfunction of the b-spline coefficients:\ny(t) =2\n66664Pm\n\u000b=1\u0003\u000b(\u001a)\f\f\f\n\u0016\u001a1b\u000b(t)\n...Pm\n\u000b=1\u0003\u000b(\u001a)\f\f\f\n\u0016\u001anTSb\u000b(t)3\n77775;(7)\nwhere \u0003(\u001a)j\u0016\u001aiare the basis function evaluated at\nthe corresponding radial locations. By defining\n\u0003TS2RnTS\u0002mthe matrix containing the basis\nfunction evaluated at all the nTSradial locations,\nwe can formulate an output mapping C(p(t)) =\u0002\n\u0003TS 0nTS\u00022\u0003\nsuch that the forward outputs are\ngiven by\ny(t) =C\u0000\np(t)\u0001\nx(t): (8)\nThe output mapping Cchanges as function of the\nequilibrium, hence, it depends on the parameter vector\np(t)introduced in a previous paragraph. For the\nsimulations presented in this work however, it is\nassumed for simplicity that the measured locations are\nconstant.\n3.2. Working principles of the observer\nA dynamic state observer (DSO) is an algorithm that\nuses measurements to constrain prediction made by\na dynamic model to iteratively estimate the internal\nstate of a system. A brief overview of the working\nprinciples of the observer is given here, more details\ncan be found in Appendix B.\nIn the case of the density distribution, we are\ndealing with a nonlinear system. Hence, we are using\nthe extended Kalman filter (EKF) framework [33] (see\nAppendix B.2) to estimate the density. The algorithm\nworks as follows. At each time step, the latest\nmeasurements ~ykare compared with the predicted\nmeasurements ^ykjk\u00001to form the residual zk(B.2).\nThe residual is multiplied by the Kalman gain (B.4)\nand used to updated the predicted state ^xkjk\u00001to\nobtain the state estimate ^xkjk. Finally, the predicted\nstate ^xk+1jkand measurement ^yk+1jkat the next\ntime step are computed using RAPDENS (5) and the\nforward diagnostic model (8).\nThe performance, stability, and robustness of the\nobserver depends on the choice of the measurementModel-based electron density profile estimation and control, applied to ITER 5\ncovarianceRk, the process covariance Qx\nk, and the dis-\nturbance covariance Q\u0010\nk. This is detailed in section 4.2.\nIn this section, we included a forward TS model\nin a DSO. With this addition, TS measurements can\nbe used to perform estimations of the density profile.\nIn the next section, the performances of the observer\nare assessed for ITER.\n4. Density profile estimation in ITER baseline\nsimulations using Thomson scattering\nmeasurements\nIn this section, density estimation simulations are\ndiscussed where we demonstrate the effectiveness\nof using TS measurements to update the state\npredictions in the DSO. First, we introduce the\nsimulated plasma scenario and argue the choice of\nthe covariance matrices, i.e. the tuning knobs of an\nEKF. Subsequently, we discuss the simulation set-up\nand present the observer’s performance in a simulation\nwith plant-observer model mismatch and realistic noise\nlevels.\n4.1. Plasma scenario\nThe observer is used to estimate the density profile\nwith TS measurements for an ITER H-mode, 15 MA,\n-5.3 T, baseline discharge. ITER shot 134173 is chosen\nas reference scenario as it is up to date the most\ncomplete prediction of an ITER baseline discharge\navailable.\nThe scenario is obtained with an integrated model\n(IM) simulation [34]. The simulation was performed\nas an open-loop coupling within IMAS [35] of the\nfree boundary equilibrium code DINA [36–38] with the\nJINTRAC [39] suite of codes. The baseline scenario\nwas assessed from the formation of the X-point to the\ntransition from X-point to limiter. The time evolution\nof relevant plasma parameters are given in figure 2.(a)-\n(c).\nIn the reference scenario, the ramp-up is modeled\nto last for 75 s with the L-H transition being triggered\nby the increase in auxiliary power. The flat-top is\nmodeled to last from t= 75s tot= 730s. During\nthat time the particle source is pellet injection with a\npellet composition of 50% deuterium and 50% tritium\natoms. It is represented as a continuous source of\n2:3\u00021022atoms/s deposited at \u001a= 0:85with Gaussian\ndeposition profile with normalized deposition width of\n0.15.\n4.2. Design of covariance matrices\nThe accuracy, convergence speed, and robustness of\nthe state estimation depends on the choice of the\n0 200 400 600 800050100150PfusPecPNBI\nt[s]Power [MW] (a)\n0 200 400 600 800051015\nIp [MA]Bt [T]\nt[s](b)\n0 200 400 600 800051015ne [#1e19/m3]Te [keV]Ti [keV]\nt[s](c)\n0 0.5 1510151019\ntorne [m-3]\nni [m-3]\n(d)\n0 0.5 101020\ntorTe [keV]\nTi [keV]\n(e)Figure 2. Reference scenario ITER #134173. Evolution of\nthe heating power (a); plasma current and toroidal magnetic\nfield (b); central plasma density and temperature (c); and\ndistributions of density and temperature for t= 436s (d-e). The\nvertical dashed line indicates the time for which the distributions\nare shown.\ncovariance matrices Rk,Qx\nk, andQ\u0010\nk[33]. The choice\nof these matrices is discussed here.\nThe measurement covariance matrix Rkis chosen\nas a diagonal matrix with the square root of the\nmodeled noise covariance on the diagonal (the modeled\nnoise is discussed in section 4.3). Note that this\nassumes that each TS measurement location is an\nuncorrelated individual output.\nThe process covariance matrix Qx\nkis constructed\nas a symmetric Toeplitz matrixkwith descending first\nrow. The values on the first row of the Toeplitz\nmatrices determine the spatial correlation between the\nestimates and is chosen as exponentially decaying. The\ndisturbance covariance matrix Q\u0010\nkis constructed as a\ndiagonal matrix.\nThe magnitude of the entries of Qx\nkandQ\u0010\nkreflect\nthe amount of trust in the model predictions and\ninfluence the Kalman gain (B.4) [33]. A low observer\ngain reflects small trust in measurements and high\nkAToeplitz matrix is a matrix of which the values on the\ndiagonals are constant.Model-based electron density profile estimation and control, applied to ITER 6\ntrust in the model predictions. Hence, the EKF will\nreact slowly to the differences between measurements\nand model predictions. A low gain avoids fitting the\nnoise but relies more heavily on the model predictions,\nmaking the reconstruction more sensitive to modeling\nerrors. On the contrary, a high Kalman gain reflects\nhigh trust in the measurements and large model\nuncertainty. The DSO will react fast to discrepancies\nbetween model and measurements. The reconstruction\nwill be more robust to modeling error but sensitive to\ndiagnostic noise and errors.\nThe magnitude of the covariance matrices’ entries\nare chosen based on a preliminary analysis of\nRAPDENS for ITER. The heuristic parameters of\nRAPDENS are tuned to approximate the flat-top\nplasma behavior of the reference discharge scenario\ndescribed in section 4.1. From the analysis, it\nis concluded that the model had good predictive\ncapabilitiesinflat-topbutthatmodelingoftheL-mode\nand L-H transition is not fully accurate. Hence, a high\nKalman gain is chosen in the ramp-up and ramp-down\nphases and a low Kalman gain is chosen during flat-\ntop. Furthermore, it was observed that the density\nprediction mismatched at the edge of the plasma.\nHence, the diagonal entries of Qx\nkcorresponding to the\nedge of the plasma were increased (see figure 3).\n24681012142\n4\n6\n8\n10\n12\n14\n00.050.10.150.2\nFigure 3. Graphical representation of the matrix Qx\nkfor the\nlow Kalman gain.\n4.3. Simulation set-up\nBefore proceeding to the estimation results, we briefly\ndescribe the set-up of the simulation used to evaluate\nthe observer’s performances. A visual depiction of\nthe simulation set-up is given in figure 4. The\nEKF is simulated for the entire discharge presented\nin section 4.1. The external parameters required to\nrun RAPDENS (see sec 2), the particle inputs, the\ninitial conditions, and the ’real’ (to be estimated)\ndensity profile are taken from the database of the IM\nsimulation.\nThe heuristic parameters of RAPDENS that were\nn\ne\nITER \nDatabase\nRAPDENS\nSynthetic \nTS\nObserver \ngain\n+\n-\n+\n+\nTime \nshift\np\nk\n?\npellet\ny\nk|k-1\n^\nz\nk\nx\nk|k\n^\ny\nk\n~\nTS \nmodel\nx\nk|k-1\n^\nDSO\nx\n0Figure 4. Block diagram of the DSO and the set-up used\nfor the density estimation simulations for ITER. The external\nparameters, the particle fueling rates, and the initial conditions\nare taken from the ITER database and used to simulate\nRAPDENS. The ’real’ density distribution is used to synthesize\nsynthetic TS measurements.\ntuned to approximate the flat-top plasma behavior in\nthepreliminaryanalysisareusedintheestimationsim-\nulation. However, to represent inaccurate knowledge\nof the transport and investigate the performance of\nthe observer with model uncertainty, systematic plant-\nobserver model mismatch is manually introduced by\nchanging the tuned transport coefficients and the pel-\nlet spatial deposition function.\nFurthermore, measurement inaccuracies and noise\nare included in the simulation with synthetic TS\nmeasurements ~yk. They created by adding artificial\nnoise to the density profile computed by the IM. While\nthe signal-to-noise ratio (SNR) for TS depends on\nPoisson statistics [40] and thus scales with 1=pne, we\nchoose to model the noise as a Gaussian with expected\nvalueE[\u0001TS] = 0and covariance \u001b2\nk= 0:05\u0003max(ne).\nThis way, we approximate the required accuracy of\nthe TS system [32] and account for sources of noise\nsuch as shot noise, thermal noise in the detection\ncircuit, noise due to plasma light, and noise due to\nneutron damage on optics installation [41,42]. It is\nimportant to note that Thomson scattering can also be\naffected by systematic errors, such as misalignment or\ndue to inaccuracies in the equilibrium reconstruction.\nThese are not accounted for in this work. When using\n\"real\" Thomson scattering measurements, detection\nand correction for these errors can be performed by\nintegrating the Thomson scattering data with other\ndiagnostics in the observer. This is elaborated further\nin Section 7.Model-based electron density profile estimation and control, applied to ITER 7\nFigure 5. Results of density estimation simulations for ITER\n#134173usingsyntheticTSdatatoupdatethestatepredictions.\nPlant-observer mismatch is present. Frame a) The estimated\ndensity (faded) is compared with the IM density (solid) for four\nflux labels. Dashed vertical lines show the time at which the\nprofiles are shown. Frame c-e) The estimated density profile\n(red) is compared with the real density profile (black). The TS\nmeasurementpointsareshown(orange). Theobserveriscapable\nof accurately estimating the density profile. The estimation is of\nbest quality during the flat-top part of the discharge.\n4.4. Density profile estimation results\nThe simulation results are shown in figure 5. The\nestimation error, depicted in figure 5.b, is expressed\nas the normalized 2-norm of the error vector: jj^ykjk\u0000\nnejj2=jjnejj2, where ^ykjk=C(pk)^xkjkwithC(pk)\ndefined in (8) and nethe density profile of the IM\nsimulation. Only the part of the profile measured by\nthe core TS system [32], i.e., 0\u0014\u001a\u00140:9, are used in\nthe error computation.\nOverall, gooddensityreconstructionperformances\ncan be seen for the entire profile, i.e., the estimated\ndensities are shown to correctly track the true densities\n(see figure 5.a). The normalized estimation error\naverages around\u001910%during the ramp-up and \u00192%for the flat-top (see figure 5.b).\nDuring the ramp-up and the L-H transition, from\nt= 0s tot= 75s the effect of the high Kalman gain\ncan clearly be distinguished. During that time, the\nartificialmeasurementnoiseisvisibleintheestimation,\nas can be seen in figure 5.b. Decreasing the Kalman\ngain during this phase would reduce the influence\nof noise but increase the overall estimation error as\nthe model is not accurate for the early stages of the\ndischarge.\nAn increase in the estimation error can be seen\nbetweent= 75s andt= 90s. Att= 75s, the plasma\nenters H-mode. As was discussed in section 4.2, the\nobserver gain is changed from a high to a low Kalman\ngain at that time. This means that the observer relies\nmore on the model predictions. The modeled density\nincrease in early H-mode is steeper in RAPDENS,\nhence, the density profile is overestimated (see figure\n5.d). This could be solved by slowly transitioning from\na high to a low observer gain once the plasma enters\nH-mode. After t= 90s, the estimation error decreases\nrapidly. A good agreement can be seen between the\nestimated and real density profile for the flat-top (see\nfigure 5.e).\nNote that the transport dynamics change after the\nsudden decrease in the density at t= 634s. This\nchange in dynamics is not modeled in RAPDENS and\nresults in an increase of the estimation error to \u00195%.\nThissectionshowsthatusingTSmeasurementsto\nupdate the predictions made by the RAPDENS model\nin an EKF results in accurate density profile estima-\ntions. Furthermore, the quality of the estimation is\ndeemed sufficient to be used for profile control, espe-\ncially during the flat-top.\n5. Design of an MPC controller\nIn this section, the design of an MPC controller for the\ndensity distribution is presented. We first introduce\nthe concepts of MPC and motivate the use of this\nstrategy for the control of the density distribution.\nSubsequently, we discuss in detail each component of\nthe controller.\n5.1. Model predictive control\nModel-predictive control (MPC) is widely adopted as\nan effective control strategy to deal with multivariate\nconstrained control problems [43,44]. A block diagram\nof an MPC controller is given in figure 6.a.\nIn such a controller, an explicit model of the\nto be controlled system is used to predict future\nstates/outputs over a prediction horizon (denoted N2\nN). These predictions are used to formulate an optimal\ncontrol problem where the future tracking error, i.e.Model-based electron density profile estimation and control, applied to ITER 8\nFigure 6. a) Block diagram of the main components of a\nMPC controller. B) Illustration of MPC for a discrete-time\nsingle-input single-output system with constant state and input\nconstraints.\nthe difference between the desired and predicted future\noutputs is minimized. Limits on the output, states,\nand inputs can be taken into account as constraints in\nthe optimal control problem. This is illustrated for a\ndiscrete-time, single-state single-input system in figure\n6.b.\nThe design of an MPC controller consists in:\nchoosing a prediction model, formulating a cost func-\ntion that encompasses the control objectives, formulat-\ning constraint functions that translate physical limits\nof the plasma and actuators as state and input con-\nstraints, and implementing and solving the optimal\ncontrol problem in a numerical optimization solver.\nThese steps are discussed in the coming sub-sections.\n5.2. Control objectives and motivation for MPC\nWe aim to synthesize a controller that satisfies the\nfollowing control objectives:\n(i) Track a high-performance reference density profile\nin a region of interest (ROI).\n(ii) Guaranteeing that the Greenwald density limit isnot being violated.\n(iii) Optimizing impurity transport in the ROI.\nNote that these objectives are generic, in section\n6.1 the objectives are refined for the specific control\nsimulations.\nTo control the density profile, the controller\nwill have access to multiple actuators, e.g., multiple\npellet injectors [45], and it will have access to\nthe reconstructed density profile. The control of\nthe density profile is thus a multivariate control\nproblem. Additionally, the Greenwald density and the\noptimization of impurity transport can be formulated\nas state and inputs constraints. Hence, the control\nproblem at hand can be classified as multivariate and\nsubject to state and actutator constraints, which is\nexactly the type of problem for which MPC is suited.\n5.3. Prediction model\nThe first component of a MPC controller is an explicit\nmodel of the systems dynamics. In this work, we use\na linear disturbance-augmented model derived from\nRAPDENS for this purpose.\nThe prediction model is derived as follows. First,\nthe nonlinear state-space (5) is linearized around the\ndesired flat-top conditions fxft;pftgto obtain a linear\napproximation of the flat-top transport dynamics.\nNext, as the MPC controller runs in discrete-time,\nthe continuous time model is discretized temporally\nby exact discretization using an equidistant temporal\ndiscretization tk=t0+kTs, withTs= 3ms. This\nresults in a discrete-time linear state-space given by\n(\nx(tk+1) =Ax(tk) +Bu(tk)\ny(tk) =Cx(tk):(9)\nNext, to incorporate integral action and enable\nthe controller to compensate for steady-state offsets\ninduced by model mismatch and slow changing\ndisturbances, a disturbance model as proposed in\n[46,47] is used. To do so, (9) is augmented to\n8\n><\n>:x(tk+1) =Ax(tk) +Bu(tk) +Bdd(tk)\nd(tk+1) =d(tk)\ny(tk) =Cx(tk) +Cdd(tk);(10)\nsuch that it includes the disturbance d(t)2Rnd=ny\nwith disturbance model matrices BdandCd. These\nmatrices are chosen to guarantee the observability of\nthe augmented system (10) [47].\nThe first condition for the augmented system to\nbe observable is that the non-augmented system is\nobservable. The local linear model (9) (with state\nvector (2)) contains two unobservable states. These\nare the particle inventories of the wall Nwand vacuumModel-based electron density profile estimation and control, applied to ITER 9\nNv. Hence, for the controller we use the minimal\nrealization of input-output dynamics. Next, we choose\nBd=Inx;ndandCd=Ind;nd, such that\nrank\u0014A\u0000I Bd\nC Cd\u0015\n=nx+nd: (11)\nguaranteeing that the augmented prediction model is\nobservable [47].\nNotation 2: For a system with state variable\nxand input variable u, we distinguish between the\nsystem’s state at time t=tkdenotedx(tk)and\nthe predicted state at time t=tk+idenotedxi;k.\nAnalogously, u(tk)denotes the input to the system at\ntimet=tkandui;kdenotes the input that would be\napplied at time t=tk+i.\nUsing notation 2, the discrete-time linear distur-\nbance augmented state-space prediction model is then\ngiven by\n8\n><\n>:xi+1;k=Axi;k+Bui;k+Bddi;k\ndi+1;k=di;k\nyi;k =Cxi;k+Cddi;k;(12)\nwherexi;k,di;k,yi;k, andui;kdenoted the predicted\nstate, disturbance, output and control input at i2N\ntime steps ahead of the starting time t=tk.\nBy introducing, for a given prediction horizon N,\nthe stacked notations\nXk=2\n666664x1;k\nx2;k\n...\nxN\u00001;k\nxN;k3\n777775\n|{z}\nFuture states;Uk=2\n666664u0;k\nu1;k\n...\nuN\u00002;k\nuN\u00001;k3\n777775\n|{z}\nFuture inputs;(13)\nthe compact formulation of the entire prediction\nsequence is given by\nXk=Axx(tk) +BuUk+Bdd(tk);(14)\nthe stacked prediction matrices Ax,Bu, andBdare de-\nrived in Appendix C.1.\nThe prediction model allows us to relate an\nunknown input sequence Ukto the current state x(tk)\nand current disturbance d(tk). In the coming sections,\nan optimal control problem is formulated using this\nmodel that can be used to compute an optimal input\nsequence.\n5.4. Disturbance estimation\nThe disturbance d(tk)introduced in (10) is estimated\nat each time instance using a Kalman filter (KF).The KF equations can be found in Appendix B. The\nperformanceandstabilityoftheobserver(andpartially\nthat of the controller) are determined by the choice of\nthe measurements covariance Rand the disturbance\ncovarianceQd. They have been tuned manually and\nchosen as diagonal matrices with R= 5\u000210\u00003Inyand\nQd= 1\u000210\u00006IndWith this design, the estimate of the\ndisturbancesconvergesin \u00192s. Afterconvergence, the\nprediction model is capable of dealing with systematic\nand slow-varying plant-model mismatches.\n5.5. State and actuator constraints\nNext, the construction of the constraints functions\nis explained. The constraint functions are used\nto constraint the future states and inputs in the\noptimization problem.\nThree different type of constraints are imposed on\nthe system in this work: linear input constraints to\nrepresent the minimal and maximal fueling rates of the\npelletinjectors, linearstateconstraintstorepresentthe\nGreenwald density limit, and nonlinear state and input\nconstraints to optimize impurity transport.\nFirst, linear constraints are defined to represent\nactuator limits and plasma density limit. As shown in\nAppendix C.2, these constraints can be formulated as\nlinear inequality constraints on the input sequence Uk\nas\nAineqUk\u0014bineq: (15)\nSecond, a nonlinear constraint is imposed to\nthe states and inputs that translate the desire\nto optimize impurity transport. A condition for\noutward neo-classical impurity transport can be\nderivedasafavorableratiobetweenthelogarithmicion\ntemperature gradient LTiand the logarithmic density\ngradientLne, i.e.,LTi=Lne'\u0011ic\u00191[6]. Assuming\ntypical peaked temperature and density profiles in\nITER flat-top, the ratio can be used to formulate an\ninequality constraint on the logarithmic gradient of the\ndensity profile:\njLnej\njLTij\u00140:95 +\u000f\n\u0011j@ne(\u001a)\n@\u001a1\nne(\u001a)j\u0014jLTij(0:95 +\u000f):(16)\nNote that a soft constraint formulation is used to avoid\ninfeasibility of the optimization problem [48]. In such\na formulation, violation of the constraint is allowed\nbut penalized via the parameter \u000fand the cost matrix\nW\u000fin the cost function (see (18)). In Appendix C.3,\nthe constraint is formulated as a nonlinear function\nof the state, control input, disturbance estimate, and\nsoft constraint parameter. The nonlinear constraint\nfunctiong1(xk;uk;dk;\u000f)isdefinedas(timedependenceModel-based electron density profile estimation and control, applied to ITER 10\ndenoted by subscript kfor readability):\ng1(xk;uk;dk;\u000f)\u0011jWyC0(Axk+Buk+Bddk)j\u0000\njWy\u0010\nC(Axk+Buk+Bddk)+Cddk\u0011\n(0:95WyLTi+\u000f)j\u00140:\n(17)\nThe linear and nonlinear constraints are used to\nconstrain the optimization problem. With their\ninclusion the controller accounts for physical limits\nof the system (actuator and plasma) and the density\nprofile - inward impurity transport interaction.\n5.6. Cost function\nThe objective of the controller is to track the high-\nperformance reference rkwhile avoiding aggressive\ncontrol and violating the constraints. This is expressed\nin the cost function Jkas:\nJk=kxN;k\u0000\u0016xNk2\nP+N\u00001X\nj=0(kxj;k\u0000\u0016xjk2\nWx+\nkuj;k\u0000\u0016ujk2\nWu) +W\u000f\u000f2;(18)\nwhere for a given matrix Tand vectorxof appropriate\ndimensions, we define kxk2\nT,x>Tx.\u0016xiand \u0016uiare\nthe desired steady-state states and control inputs, i.e.,\nthe states and inputs that are to be reached for the\nreference to be tracked. They are computed using\n\u0014\u0016xi\n\u0016ui\u0015\n=\u0014A\u0000I B\nHC 0\u0015\u00001\u0014\u0000Bd^d(tk)\nr(tk)\u0000Cd^d(tk)\u0015\n;(19)\nwhere (A;B;Bd;C;Cd)are the matrices of (10) and\nH2f0;1gnz;nxis the controlled variable matrix.\nIn (18), the stage state cost matrixWxis used\nto penalize deviations between the desired steady-\nstate state and the future predicted states, up to but\nexcluding the terminal state ( xN;k). Thestage input\ncostmatrixWuis used to weigh the inputs and avoid\naggressive control actions. The terminal state cost\nmatrixPis used to penalize the deviation of the\nterminal state. Finally, soft constraint violation cost\nmatrixW\u000fis used to penalize the violation of the\nnonlinear state constraint (see section 5.5).\nAn iterative procedure is used to choose the\nweights. The matrix Wxis chosen as a block-diagonal\nmatrix of two matrices: one being a weighted unity,\nthe other being a zero matrix. The dimensions of these\nmatrices ensure that only the states that parametrize\nthe ROI are penalized. The matrix Wuis chosen as a\nweighted unity matrix, i.e., Wu= 5\u000110\u00003I, and matrix\nW\u000fis chosen as positive definite diagonal matrix with\nW\u000f\u001fQ. Finally,Pis chosen as the solution of the\ndiscrete algebraic Riccati equation\nP=A>PA\u0000(A>PB)(B>PB+Wu)\u00001(B>PA)+Wx:\n(20)By defining these weights, the cost function Jis convex\nand quadratic in Uk[49].\n5.7. Optimization problem\nThe future inputs Ukare computed by minimizing the\ncost function (18) subject to the constraints (15) and\n(17). Using the prediction model, the nonlinear online\noptimization is defined as:\nmin\nUk;\u000fU>\nkWu;uUk+U>\nk(Wu;x^xk;k+Wu;d^dkjk+\nWu;r) +W\u000f\u000f2\nsubj.toAineqUk\u0014bineq; (21)\ng1(xk;uk;dk;\u000f)\u00140:\nThe matrices Wu;u,Wu;x,Wu;d, andWu;rare defined\nin Appendix C.4. The optimization problem is\nnonlinear due to the presence of the impurity transport\nconstraint (17). Therefore, sequential quadratic\nprogramming [50] is used to solve the problem. Due\nto the nonlinear constraint, no guarantee on convexity\n(and global optimality) can be given.\nOncean\"optimal\"solution U\u0003\nkof(21)isobtained,\nthe firstnuentries of the optimal sequence are applied\nas control inputs to the system. In the next section,\nthe results of control simulations are shown to test the\nperformance of the controller.\n6. Simulation results of closed-loop density\ndistribution control\nInthissection,wepresentcontrolsimulationsthatwere\nused to test the MPC controller. Note that it is not the\nobjective of these simulations to provide quantitative\nestimates of the ITER performance or controllability\nof a particular scenario, but to illustrate the potential\nof MPC control for the density profile and help identify\nchallenges that remain to be solved.\nFirst, we discuss the simulation set-up and refine\nthe control requirements. Next, we demonstrate\ntracking of a high-performance density profile while\nminimizing inward impurity transport. This is done\nin simulations with plant-controller model mismatch\nand a continuous pellet representation.\n6.1. Simulation set-up and control requirements\nThe considered actuators to perform control are the\npellet injection systems that have their injection\nlocation situated at the high field side (HFS). During\nITER baseline operation, two HFS PIS are planned\nwith the possibility to expand to four [45]. The control\ninputs (signals changed by the controller) u(tk)are\nchosen as the fueling rates of the PIS and defined as\nu(tk) =\u0002\u0000p;1(tk) \u0000p;2(tk)\u0003>:(22)Model-based electron density profile estimation and control, applied to ITER 11\n0 5 10020406080\npellet(=0.7) [1020/s]\nt[s](a)\n0 5 100100200300\npellet(=0.85) [1020/s]\nt[s](b)\n0 5 1000.10.20.30.4\n||n-nref||2/||nref||2 [A.U]\nt[s](c)\n0 5 1011.21.41.6Line-average density vs\nGreenwald limit\nt[s](d)\n0 0.5 100.511.52\nLog. gradient [m-1]\nt=1s\ntor(e)\n0 0.5 100.511.52\nLog. gradient [m-1]\nt=2s\ntor(f)Limit (d-f) / Reference (g-i) Closed-loop Open-loop\n0 0.5 100.511.52ne() [1020/m3]\nt=1s\ntor(g)\n0 0.5 100.511.52ne() [1020/m3]\nt=2s\ntor(h)\n0 0.5 100.511.52ne() [1020/m3]\nt=5s\ntor(i)\nFigure 7. Tracking of a high-performance control reference in flat-top phase. The case with feedforward only (blue) is compared\nwith the case with feedback (red). Frame a-b) Required particle flow per pellet injector; Frame c) Relative tracking error for the\ncontrolled part of the profile; Frame d) The approximated line average density is compared with the Greenwald density limit (dotted\nblack); Frame e-f) The logarithmic gradient profiles of the ion temperature (dotted black) are shown together with the logarithmic\ngradients of the density profiles; Frame g-i) Reference density profile (dotted black) is shown together with the controlled profiles for\ndifferent time instances. The time instances are shown with the dotted vertical lines in c). The MPC controller reduces the tracking\nerror significantly while keeping a favorable ratio between the logarithmic gradient profiles hence minimizing the inward impurity\ntransport.\nThe deposition function are modeled as parabolic\nfunctions with normalized radius \u001ap;1= 0:7and\u001ap;2=\n0:85and normalized widths of wp;1= 0:3andwp;2=\n0:15. These represent obtainable fueling profiles for 3\nand 5 mm pellets in ITER [51].\nThecontrolledvariables z(t)arethedensityprofile\nevaluated on an equidistant equilibrium grid for 0\u0014\n\u001a\u00140:9. Similarly, the control reference r(t)is\nthe desired density profile evaluated on the same\nequilibrium grid. The controller is synthesized using\nthis nominal configuration.\nTo assess the performance of the controller in\nsimulation, the control objectives discussed in section\n5.2 are named and specified as follows:\n(i)Error (Err) requirement: The controller\ncan maintain the average tracking error below\n5%. The tracking error is expressed in thenormalized 2-norm of the error vector: jjz(tk)\u0000\nr(tk)jj2=jjr(tk)jj2.\n(ii)Greenwald (Gw) requirement: The controller\ncan guarantee that the line-average density will\nnot exceed the Greenwald density nGW[9] for any\ntimet\u0015t0.\n(iii)Transport(Trsp)requirement: Thecontroller\ncan optimize neoclassical impurity transport\nby maintaining a favourable ratio between the\nlogarithmic gradients of the ion temperature and\ndensity profiles for the domain \nc=f(t;\u001a)2\nRjt\u0015t0;0\u0014\u001a\u00140:9g.Model-based electron density profile estimation and control, applied to ITER 12\n0 5 10050100150\npellet(=0.7) [1020/s]\nt[s](a)\n0 5 100100200300\npellet(=0.85) [1020/s]\nt[s](b)\n0 5 1000.10.20.30.4\n||n-nref||2/||nref||2 [A.U]\nt[s](c)\n0 5 1011.21.41.6Line-average density vs\nGreenwald limit\nt[s](d)\n0 0.5 100.511.52\nLog. gradient [m-1]\nt=1s\ntor(e)\n0 0.5 100.511.52\nLog. gradient [m-1]\nt=2s\ntor(f)Limit (d-f) / Reference (g-i) Closed-loop Open-loop\n0 0.5 100.511.52ne() [1020/m3]\nt=1s\ntor(g)\n0 0.5 100.511.52ne() [1020/m3]\nt=2s\ntor(h)\n0 0.5 100.511.52ne() [1020/m3]\nt=5s\ntor(i)\nFigure 8. Tracking of a high-performance control reference in flat-top phase with uncertain pellet deposition location. The case\nwith feedforward only (blue) is compared with the case with feedback (red). Frame a-b) Required particle flow per pellet injector;\nFrame c) Relative tracking error for the controlled part of the profile; Frame d) The approximated line average density is compared\nwith the Greenwald density limit (dotted black); Frame e-f) The logarithmic gradient profiles of the ion temperature (dotted black)\nare shown together with the logarithmic gradients of the density profiles; Frame g-i) Reference density profile (dotted black) is shown\ntogether with the controlled profiles for different time instances. The time instances are shown with the dotted vertical lines in c).\nThe MPC controller reduces the tracking error significantly while keeping a favorable ratio between the logarithmic gradient profiles\nhence minimizing the inward impurity transport.\n6.2. Tracking of a reference in flat-top with\nplant-controller model mismatch\nThe goal of the controller is to compensate for\nmismatches between the real system and the model\nin the controller. Therefore, mismatch is introduced\nin the plant with respect to the controller model.\nThe results of two simulations are presented here. In\nthe first (Figure 7), transport mismatch is introduced\nby decreasing the diffusion coefficient and increasing\nthe drift coefficient. Furthermore, a mismatch is\nintroduced by changing the width of the pellet\ndeposition profile. These changes increase inward\ntransport significantly resulting in higher densities at\nnominal particle inputs. In the second (Figure 8),\ntransport mismatch is introduced by increasing the\ndiffusion coefficient and decreasing the drift coefficient\nintheplant. Furthermore, amismatchisintroducedbychanging the deposition radii of the pellet deposition\nprofile, respectively to [0.8,0.9]. These changes\ndecrease inward transport significantly resulting in\nlower densities at nominal particle inputs.\nThe open-loop control inputs are computed based\non the nominal configuration of the system. For\nboth simulations, the uncertainty in transport results\nin a large tracking error (figure 7.c & 8.c). In the\nfirst simulation, the uncertainty causes a violation\nof the Greenwald density limit (figure 7.d) and of\nthe logarithmic gradient constraint (figure 7.e-f). It\nis shown in figure 7.g-i & 8.g-i that the open-loop\ndensity profiles (blue) do not match the reference\nprofile (black).\nThe MPC controller is used to control the density\nprofile in closed-loop. The control inputs differ from\nthe open-loop (figure 7.a-b & 8.a-b). It can beModel-based electron density profile estimation and control, applied to ITER 13\nseen that in the presented simulations, the controller\nachieves the control objectives: the tracking error is\nbelow 5%(figure 7.c & 8.c); the Greenwald density\nlimit is respected (figure 7.d); and the favorable\nratio between ion-temperature and density logarithmic\ngradients is maintained (figure 7.e-f). Furthermore,\na good agreement between the controlled (red) and\nreference (black) profiles can be seen (figure 7.g-i &\n8.g-i).\nThese results show that with continuous actua-\ntors, the MPC controller is capable of tracking a high-\nperformance reference density profile in the presence of\nplant-controller model mismatch while simultaneously\naccounting for inward impurity transport.\nNote that the relation between the density and\nthe logarithmic gradient profile depend on a lot of\nfactors, e.g. ratio between transport coefficients, pellet\ndeposition location and deposition profiles, and the\nfueling rates (control inputs). For the controller to\naccount for inward impurity transport, the control\ninputs must give enough freedom to find a feasible\ninput sequence that minimizes the control error while\nrespecting the constraint. It is however possible\nthat the control reference and logarithmic gradient\nconstraint become unfeasible (e.g. if the transport\ncoefficients profiles changes drastically), i.e., there does\nnot exist an input sequence for which the logarithmic\ngradient constraint is respected (this depends greatly\non the degrees of freedom of the controller). In\nthis case, the controller cannot optimize the impurity\ntransport. However, it can give a warning to the\noverarching supervisory controller that the constraint\nis violated. Subsequently, fast update methods of\nthe transport coefficients [52] can be used to update\nthe transport model in the controller and determine a\nbetter suited control reference.\n7. Discussion and outlook\nIn this work, we extend a dynamic state observer\nto estimate the density distribution in ITER using\nThomson scattering (TS) measurements. For this,\nwe combine the control-oriented particle transport\nmodel RAPDENS with a forward TS model and\nsynthetic TS measurements in an extended Kalman\nfilter (EKF) framework. We have shown that reliable,\nhigh-quality density profile estimates can be obtained\nin simulation with realistic measurement noise levels.\nHowever, we use a simple model for the synthetic\nTS measurements (Gaussian distribution) and thus\ndo not account for systematic errors (e.g., due to\nequilibrium reconstruction errors or misalignment) or\ndensity dependent errors. These error can be present\nwhen using real TS measurements. Their presencewill reduce the performance of the observer and might\nrequire a specific error handling procedure to ensure\nreliable reconstruction of the density profile. As a\nnext step, we propose to test the extended observer\non an experimental reactor with real diagnostic\ndata, preferably AUG or TCV as it was already\nimplemented there [24]. On these reactors, the\nTS measurements will be used together with other\ndiagnostics (interferometers on TCV; interferometers\nand radiation measurements on AUG) to reconstruct\nthe density profile. It can then be investigated if the\nexpected density estimation performances are achieved\nand the increase in reconstruction performance by\nincluding TS can be quantified. Testing the observer\nwith experimental data will help identify if additional\nhandling is required to deal with systematic errors and\ndesign it if need be.\nFurthermore in this work, we have synthesized a\nmodel-predictive controller for the density distribution\nin a tokamak. The controller uses the pellet fueling\nrate as control input to track a high-performance\nreference profile, avoid density limits, and optimize\nfor impurity transport. We show that for continuous\nactuators, the controller is capable of achieving\nthese control requirements in simulations with plant-\ncontroller model mismatch.\nEven so, we use the heuristic control-oriented\nmodel RAPDENS to synthesize and validate the\ncontroller. For further validation, it will be relevant as\na next step to couple the controller to more complex\nfirst-principles transport codes, e.g., to JETTO as\npart of JINTRAC [39] to perform closed-loop control\nsimulations, and investigate the achieved control\nperformances. Additionally, pellets are inherently\ndiscrete events. However since this works presents a\nfor the first time a controller for the density profile,\nthis discrete nature is not included. Knowing that in a\nreal reactor, the pellet size cannot be changed for each\ncontrol action and is subject to constraints linked to\nfor example plasma penetration, it will be relevant to\ninvestigate the performance of the controller when the\ndiscrete pellet nature is introduced in the control loop\nand when only pellets of a fixed size can be used.\nFurthermore, we wouldlike to address the possible\nextensions of the choice of control inputs. In this work\nand in literature, e.g. [26,27], the pellet fueling rate is\nconsidered as control input. This is a straightforward\nchoice as it enables the use of linear controllers. In\npractice, however, the actuation with pellet injection\nis a complex function of pellet size, pellet velocity,\ninjection frequency, and plasma temperature. We\nthink it is relevant to study the possibility of using\na more complete input space, e.g., by also taking\ninto account pellet velocity, and analyze the effect on\ncontrollability and control performances.Model-based electron density profile estimation and control, applied to ITER 14\nAcknowledgments This work has been carried\nout within the framework of the EUROfusion Consor-\ntium and has received funding from the Euratom re-\nsearch and training programme 2014-2018 and 2019-\n2020 under grant agreement No 633053. The views\nand opinions expressed herein do not necessarily re-\nflect those of the European Commission. 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Strand\nP and Szepesi G 2014 Plasma and Fusion Research 9\n1–4 ISSN 18806821\n[40] Stoneking M R and Den Hartog D J 1997 Review of\nScientific Instruments 68914–917 ISSN 00346748\n[41] Togashi H, Ejiri A, Hiratsuka J, Nakamura K, Takase Y,\nYamaguchi T, Furui H, Imamura K, Inada T, Kakuda H,\nNakanishi A, Oosako T, Shinya T, Sonehara M, Tsuda\nS, Tsujii N, Wakatsuki T, Hasegawa M, Nagashima\nY, Narihara K, Yamada I and Tojo H 2014 Review\nof Scientific Instruments 852–5 ISSN 10897623 URL\nhttp://dx.doi.org/10.1063/1.4891707\n[42] Tanimura Y and Iida T 1998 Journal of Nuclear Materials\n258-263 1812–1816 ISSN 00223115\n[43] Qin J and Badgwell T 1997 AIChE Symposium Series 93\n[44] LeeJH2011 9415–424ISSN15986446URL https://link.\nspringer.com/article/10.1007/s12555-011-0300-6\n[45] Maruyama S, Yang Y, Sugihara M, Pitts R A, Baylor L,\nCombs S, Meitner S, Li B and Li W 2009 Proceedings -\nSymposium on Fusion Engineering\n[46] Pannocchia G 2003 Journal of Process Control 13693–701\nISSN 09591524\n[47] Maeder U, Borrelli F and Morari M 2009 Automatica 45\n2214–2222 ISSN 00051098 URL http://dx.doi.org/10.\n1016/j.automatica.2009.06.005\n[48] Vada J, Slupphaug O and Johansen T 1999 Efficient\ninfeasibilityhandlinginlinearMPCsubjecttoprioritized\nconstraints 1999 European Control Conference (ECC)\n(IEEE) pp 1166–1171 ISBN 978-3-9524173-5-5 URL\nhttps://ieeexplore.ieee.org/document/7099467/\n[49] Rawlings J, Mayne D and Diehl M 2017 Model Predictive\nControl: Theory, Computation, and Design\n[50] Nocedal J and Wright S 2006 Numerical OptimizationSpringer Series in Operations Research and Finan-\ncial Engineering (Springer New York) ISBN 978-0-\n387-30303-1 URL http://link.springer.com/10.1007/\n978-0-387-40065-5\n[51] Baylor L R, Parks P B, Jernigan T C, Caughman J B,\nCombs S K, Foust C R, Houlberg W A, Maruyama S\nand Rasmussen D A 2007 Nuclear Fusion 47443–448\nISSN 00295515\n[52] van de Plassche K L, Citrin J, Bourdelle C, Camenen Y,\nCassonFJ,DagnelieVI,FeliciF,HoAandVanMulders\nS 2020Physics of Plasmas 27022310 ISSN 1070-664X\nURL https://doi.org/10.1063/1.5134126http://aip.\nscitation.org/doi/10.1063/1.5134126\n[53] Hinton L and Hazeltine R D 1976 Reviews of Modern\nPhysics 48239\nAppendix A. Summary of RAPDENS\nequations\nIn this appendix, a summary of the RAPDENS\nequations is given. Mass conservation and radial\ntransport [53] are used to model the evolution of the\nradial plasma density ne(\u001a;t)as\n\u0012@V\n@\u001a\u0013\u00001@\n@t\u0012\nne@V\n@\u001a\u0013\n+\u0012@V\n@\u001a\u0013\u00001@\u0000e\n@\u001a=Se(A.1)\nwhere\u001ais the normalized toroidal magnetic flux, Vis\nthe plasma volume, \u0000e(\u001a;t)is the radial transport flux\nmodeled as\n\u0000e=\u0000@V\n@\u001a\u0012\n(r\u001a)2\u000b\n\u001f@ne\n@\u001a+hjr\u001aji\u0017ne\u0013\n(A.2)\nwith\u001f(\u001a;cHL)the diffusivity and \u0017(\u001a;cLH;Ip)the\npinch(ordrift)velocitywhere cLHindicatestheregime\nof the plasma (L or H-mode). In A.2, Se(\u001a;t)is the net\nelectron source modeled as\nSe=h\u001bviiznnne\u0000h\u001bvirecneni\u0000nej\u001a>1\n\u001cSOL+Spellet+SNBI\n(A.3)\nwithh\u001bviiz(Te)andh\u001bvirecthe temperature dependent\ncross-sections for ionization and recombination [3],\n\u001cSOLthe time constant for particle loss in the scrape-\noff layer,Spelletthe net particle source due to pellet\ninjection, and SNBIthe net particle source due to\nneutral beam injection.\nThe evolution of the particle inventories Nw(t)in\nthe wall and neutral vacuum density nn(t)are modeled\nby the 0D ODE’s\ndNw\ndt=\u0000Nw\u0000cwVv;0nn\n\u001crelease+\u0012\n1\u0000Nw\nNsat\u0013Z\nVSOLne\n\u001cSOLdV;\n(A.4)Model-based electron density profile estimation and control, applied to ITER 16\nVvdnn\ndt=Nw\u0000cwVv;0nn\n\u001crelease+Nw\nNsatZ\nVSOLne\n\u001cSOLdV\n+Z\nVp(h\u001bvirecneni\u0000h\u001bviiznnne) dV\n\u0000nnVv;0\n\u001cpump+ \u0000j\u001ae+ \u0000valve;\n(A.5)\nwith\u001creleasethe particle release rate, Nsatthe wall\nsaturation inventory, \u001cpumpthe pumping time scale, cw\na dimensionless constant that determines the steady-\nstate balance between the wall and vacuum inventory,\nandVv;0the nominal vacuum volume.\nAppendix B. Dynamic state observer: theory\nand application to density profile estimation\nAppendix B.1. Kalman filter equations\nGiven a system Gwith state vector x, output vector\ny, and input vector u, the Kalman filter (KF)\nassumes that the dynamics of Gevolve in discrete-time\nfollowing a linear state-space as\n(\nx(tk+1) =F(tk)x(tk) +B(tk)u(tk) +w(tk)\ny(tk) =H(tk)x(tk) +D(tk)u(tk) +v(tk);\n(B.1)\nwhere the state transition dynamics are captured\nin matrix Fand the input dynamics in B. The\noutput model and influence of the inputs on the\noutputs are captured in matrices HandD. The\nstochastic behavior of measurement and process noises\nare modeled in vandwas zero-mean Gaussian\nwhite noises with respective covariance matrices R\nandQsuch thatv(tk)\u0018 N (0;R(tk))andw(tk)\u0018\nN(0;Q(tk)). The tuning knobs of the KF are the\ncovariance matrices RandQ.\nThealgorithmisinitializedwithaninitialstatees-\ntimate ^x1j0(with corresponding output estimate ^y1j0)\nand an initial a priori estimate covariance \u00061j0. For\neach discrete time instance t=tk, the estimate of the\nstates are obtained as follows (for readability the time\ndependence is denoted by the subscript k).\nUpdate step:\nUsing the latest available measurements ~yk, the\nresidualzkand innovation covariance \nkare computed\nby\nzk= ^ykjk\u00001\u0000~yk (B.2)\n\nk=Hk\u0006kjk\u00001H>\nk+Rk; (B.3)\nwhereH(tk)is the output matrix and \u0006kjk\u00001thea\nprioriprocess covariance. Next, the optimalKalmangainLkis computed with\nLk= \u0006kjk\u00001H>\nk\n\u00001\nk; (B.4)\nand is used to obtain the updated state estimate ^xkjk\nand thea posteriori state estimate covariance \u0006kjkas\n^xkjk= ^xkjk\u00001+Lkzk; (B.5)\n\u0006kjk= (I\u0000\u0006kjk\u00001H>\nk\n\u00001\nkHk)\u0006kjk\u00001:(B.6)\nPrediction step:\nBasedonthemostlikelystate ^xkjk, theactuatorinputs\nat the present time uk, and the process covariance Q,\nthe one step ahead predicted state ^xk+1jkand outputs\n^yk+1jkandthea prioricovarianceattimestep t=tk+1\nare given by\n^xk+1jk=Fk^xkjk+Bkuk (B.7)\n^yk+1jk=Hk^xkjk+Dkuk (B.8)\n\u0006k+1jk=Fk\u0006kjkF>\nk+Qk: (B.9)\nAppendix B.2. Extended Kalman filter equations\nIn case a system cannot be represented accurately by\na linear model, the EKF equations are used. The\nobserver assumes that the system’s dynamics evolve\nfollowing\n(\nx(tk+1) =f(x(tk);u(tk)) +w(tk)\ny(tk) =h(x(tk);u(tk)) +v(tk);(B.10)\nwheref:Rnx\u0002Rnu!Rnxandh:Rnx\u0002Rnu!Rny\nare non-linear functions of the state and input. The\nstochastic variables vandware similar as defined in\nAppendix B.1.\nThere are two main differences between the KF\nand EKF equations:\n(i) The matrices FandH(used in the computation\nthe measurement covariance (B.3), the Kalman\ngain (B.4), and the state covariances (B.9) and\n(B.6)) are the jacobians of fandh, i.e.,\nF(tk) =@f\n@x\f\f\f\f\nxk\u00001jk\u00001H(tk) =@h\n@x\f\f\f\f\nxk\u00001jk\u00001:\n(B.11)\n(ii) The prediction step is performed using the\nnonlinear model (B.10), i.e., (B.7) and (B.8)\nbecome\n^xk+1jk=f(^xkjk;uk) (B.12)\n^yk+1jk=h(^xkjk;uk) (B.13)\nIt is important to note that the EKF is a\nlinearized version of the Kalman filter for nonlinear\ndynamicalsystems, hence, noguaranteeaboutstability\nor estimation accuracy can be given [33]. These\nproperties are to be checked experimentally for the\nspecific implementations.Model-based electron density profile estimation and control, applied to ITER 17\nAppendix B.3. Application of the in the DSO for the\ndensity profile estimation\nFor the density reconstruction, the RAPDENS\nplasma simulator is used as the one step ahead\nprediction model. The physical state vector x(tk)\n(2) is augmented with additive unknown disturbances\n\u0010(tk)2Rnx. These disturbances are co-estimated\nby the observer and give a measure of the modeling\nerrors, unmodeled processes, unaccounted particles\nsources, and errors in the diagnostics models. The\nstochastic behavior of the state and disturbances is\nmodeled by additive zero-mean white noises wx\nkandw\u0010\nk\nwith respective covariance matrices Qx\nkandQ\u0010\nk. The\naugmented nonlinear state-space model is given by\n\u0014xk+1\n\u0010k+1\u0015\n=\u0014fd(xk;pk)B\u0010\u0010k\n0\u0010k\u0015\n+\u0014Bd(pk)\n0\u0015\nuk+\u0014wx\nk\nw\u0010\nk\u0015\n(B.14)\nwhereB\u0010=\u0002Im\u0002m0m\u00022\u0003\n.\nAn augmented state is defined as\n\u0018k=\u0002xk\u0010k\u0003>: (B.15)\nThe stochastic measurement noise is represented by\nan additive zero-mean white noise vkwith associated\ncovariance matrix Rk. The outputs are assumed to\nevolve as:\nyk=C(pk)xk+vk: (B.16)\nThe final step consists in defining the matrices\nFk=\"@f\n@xk\f\f\n^xkjk;pkB\u0010\n0Im\u0002m#\nBk=\u0014\nBd\n0\u0015\nHk=\u0002C(pk) 0\u0003\n: (B.17)\nThe augmented state \u0018(tk)is then estimated by using\n(B.14) and (B.17) in (B.2)-(B.9).\nAppendix C. Controller: definitions and\nderivations\nIn this appendix we provide the definitions and\nderivations of the controller matrices described in\nsection 5.\nAppendix C.1. Prediciton model matrices\nFor a system described by a linear time-invariant\ndisturbance augmented model such as (10), the\npredicted future states xi;kfori= 0 :Ncan be related\nto the current state xk, current disturbance estimate\n^dk, and input sequence uk;:::;uk+N\u00001by:\nxi;k=Aix0;k+i\u00001X\nj=0AjBui\u00001\u0000j;k+i\u00001X\nj=0AjBd^dk:(C.1)We define matrices Ax,Bu, andBdas\nAx=2\n666664A\nA2\n...\nAN\u00001\nAN3\n777775; (C.2)\nBu=2\n6664B 0\u0001\u0001\u0001 0\nAB B \u0001\u0001\u0001 0\n............\nAN\u00001B AN\u00002B\u0001\u0001\u0001B3\n7775;(C.3)\nBd=2\n6664Bd 0\u0001\u0001\u0001 0\nABdBd\u0001\u0001\u0001 0\n............\nAN\u00001BdAN\u00002Bd\u0001\u0001\u0001Bd3\n77752\n66664I\n...\n...\nI3\n77775:(C.4)\nUsing (C.2), (C.3), and (C.4), we rewrite (C.1) for the\nstackedpredictedstatesandinputsequenceintroduced\nin notation 2:\nXk=Ax^xk+BuUk+Bd^dk:(C.5)\nAppendix C.2. Linear state and actuator constraints\nHere the linear and nonlinear constraints are derived.\nThe pellet particle input is constrained as 0\u0014\u0000pellet\u0014\n\u0000max\npellet. The pellet injectors are designed to provide a\nmass flow rate _m= 0:23 + 30% = 0 :33gs\u00001of solid\nDT to the plasma [51]. The maximum electron fueling\nrate is than given by\n_\u0000max\npellet = _m\u0003(fDMD+fTMT)\u0003Navg;(C.6)\nwherefDandfTare the fractions of deuterium and\ntritium atoms in the pellet, MDandMTthe molecular\nmasses of the two isotopes and Navgrepresenting the\nAvogadro constant. Assuming a pellet composed at\n50% of deuterium and 50% of tritium, the maximal\nfueling rate is equal to _\u0000max\npellet = 4:115e22 [#=s].\nHence, the number of injected particles is constraint\nby\u0000max\npellet = 2:57e21 [#].\nNo lower constraint exist for the states hence\nxmin=\u00001. The Greenwald density limit [9] sets an\nupper bound on the line-average electron density in a\nspecific device configuration. This upper limit is given\nby\nngw=Ip\n\u0019a2(C.7)\nwhereIpis the plasma current and athe minor radius\nof the tokamak. By constructing a matrix Z2R1\u0002nx\nsuch thatZx\u0019\u0016nwhere \u0016nis the line-average density,\nthislimitcanbeformulatedasalinearstateconstraint.Model-based electron density profile estimation and control, applied to ITER 18\nThe linear constraints are given by:\numin\u0014ui;k\u0014umax;8i= 0;1;\u0001\u0001\u0001;N\u00001;\nxmin\u0014xi;k8i= 0;1;\u0001\u0001\u0001;N;\nZxi;k\u0014ngw8i= 1;2;\u0001\u0001\u0001;N; (C.8)\nBy defining\nMi=0\nBB@0nu\u0002nx\n0nu\u0002nx\n\u0000Inx\nZ1\nCCA; Ei=0\nBB@\u0000Inu\nInu\n0nx\u0002nu\n01\u0002nu1\nCCA; bi=0\nBB@\u0000umin\numax\n\u0000xmin\nngw1\nCCA\nMN=\u0012\u0000Inx\nZ\u0013\n;andbN=\u0012\u0000xmin\nngw\u0013\n; (C.9)\nwe rewrite (C.8) as\nMixi;k+Eiui;k\u0014bi8i= 0;1;:::;N\u00001\nMNxN;k\u0014bN: (C.10)\nUsing the stacked notation (13) and defining M0,Mi,\nEiandbas\nM0=0\nBBB@M0\n0\n...\n01\nCCCA; (C.11)\nMi=0\nBBB@0\u0001\u0001\u0001 0\nM1\u0001\u0001\u0001 0\n.........\n0\u0001\u0001\u0001MN1\nCCCA;(C.12)\nEi=0\nBBB@E0\u0001\u0001\u0001 0\n.........\n0\u0001\u0001\u0001EN\u00001\n0\u0001\u0001\u0001 01\nCCCA;(C.13)\nb=0\nBBB@b0\nb1\n...\nbn1\nCCCA; (C.14)\nthe constraints in (C.10) are grouped as\nM0x(k) +MiXk+EiUk\u0014b: (C.15)\nFinally, the matrices of (15) are obtained by inserting\n(C.5) in (C.15) and are defined as:\nAineq=Mi\u0000u+Ei; (C.16)\nbineq=b\u0000(M0+Mi\u0000x)x(k)\u0000Mi\u0000dd(k):(C.17)Appendix C.3. Nonlinear state constraints\nHere we formulate the nonlinear constraint function g1\nintroduced in section 5.5. Recalling that y(tk+1) =\nne(\u0016\u001a;tk+1), with \u0016\u001adefined in section 3.1, the\naugmentedmodel(10)canbeusedtorelatethedensity\nprofile at time t=tk+1can be related to the state and\ninputs at time t=tk. The relation is given by:\nne(\u0016\u001a;tk+1) =y(tk+1)\n=Cx(tk+1) +Cdd(tk+1)\n=C(Ax(tk) +Bu(tk) +Bdd(tk))+\nCdd(tk+1): (C.18)\nUnder the assumption of slow varying disturbances,\nsuch thatd(tk+1)\u0019d(tk), we can write\nne(\u0016\u001a;tk+1) =C(Ax(tk) +Bu(tk) +Bdd(tk)) +Cdd(tk):\n(C.19)\nFinally, we define C0as\nC0=@\u0010Pm\n\u000b=1\u0003\u000b(\u0016\u001a)\u0011\n@\u001a; (C.20)\nsuch thatC0x(tk) =@ne(\u001a;tk)\n@\u001aand we construct an\noutput weight matrix Wyto weight the importance of\ntheoutputsinthenonlinearconstraint. Inthisworkwe\nchose this matrix such that the weight of the outputs\nfor which\u001a<0:9are unity and the outputs for which\n\u001a>0:9are weighted 0.\nInserting (C.19), (C.20), and Wyin (16) and\nrewriting the equation in negative null form, we can\nformulate the nonlinear constraint function g1as (time\ndependence denoted by subscript kfor readability):\ng1(xk;uk;dk;\u000f)\u0011jWyC0(Axk+Buk+Bddk)j\u0000\njWy\u0010\nC(Axk+Buk+Bddk)+Cddk\u0011\n(0:95WyLTi+\u000f)j\u00140:\n(C.21)\nAppendix C.4. Cost function matrices\nHere we derive the matrices of the numerical\nimplementation of the cost function (18) as presented\nin (21).\nFirst, similaras(13), wedefinethestackedsteady-\nstate states and inputs as\n\u0016Xk=2\n666664\u0016x1jk\n\u0016x2jk\n...\n\u0016xN\u00001jk\n\u0016xNjk3\n777775\u0016Uk=2\n666664\u0016u0jk\n\u0016u1jk\n...\n\u0016uN\u00002jk\n\u0016uN\u00001jk3\n777775:(C.22)\nSubsequently, the cost matrices Wx,Wu, andWNfirst\nintroducedin(18)areusedtodefinetheblock-diagonalModel-based electron density profile estimation and control, applied to ITER 19\n0 0.5 1\nFlux label 00.511.5[#1020/m3]\nN=5(a)\n0 0.5 1\nFlux label N=15(b)\n0 0.5 1\nFlux label N=35(c)\n0 0.5 1\nFlux label N=50(d)\n05101520253035404550\nPrediction horizon N0123||e||2(e)\nFigure C1. Validation of the LDAM and KF in the MPC set-up at different times of the open-loop simulation, i.e., different values\nof the disturbance estimate ^dk. In red the value of the disturbance after t= 0:25sis used. In blue, the disturbance estimate after\nt= 1:25sis used and in pink the disturbance estimate at t= 2:5s. Frame a-d): The \"real\" density profile computed by the nonlinear\nRAPDENS model (black) is compared with the predicted density profile by the LDAM for different values of the prediction horizon\nN. Frame(e): The prediction error is given as a function of the prediction horizon for different initialization times.\nmatrices WxandWuas\nWx=2\n66664Wx0\u0001\u0001\u0001 0\n0Wx......\n............\n0 0\u0001\u0001\u0001WN3\n77775; (C.23)\nWu=2\n66664Wu0\u0001\u0001\u0001 0\n0Wu......\n............\n0 0\u0001\u0001\u0001Wu3\n77775: (C.24)\nUsing(13), (C.22), (C.23), and(C.24)thecostfunction\n(18) can be rewritten as\nJk=x(k)>Qx(k) +X>\nkWxXk\u00002X>\nkWx\u0016Xk\n+\u0016X>\nkWx\u0016Xk+U>\nkWuUk\u00002U>\nkWu\u0016Uk\n+\u0016U>\nkWu\u0016Uk:(C.25)Inserting (C.5) in (C.25) we obtain:\nJk=x(k)>Qx(k) +x(k)>\u0000>\nxWx\u0000xx(k)\n+ 2U>\nk\u0000>Wx\u0000xx(k) +U>\nk\u0000>\nuWx\u0000uUk\n+ 2x>\nk\u0000>\nxWx\u0000dd(k) + 2U>\nk\u0000>\nuWx\u0000dd(k)\n+d(k)>\u0000>\ndWx\u0000dd(k)\u00002x(k)>\u0000>\nxWx\u0016Xk\n\u00002U>\nk\u0000>\nuWx\u0016Xk\u00002d(k)>\u0000>\ndWx\u0016Xk\n+\u0016X>\nkWx\u0016Xk+U>\nkWuUk\u00002U>\nkWu\u0016Uk\n+\u0016U>\nkWu\u0016Uk: (C.26)\nFinally, by regrouping the quadratic terms in Uk,\nthe terms that depend on fUk;x(k)g,fUk;d(k)g, and\nfUk;\u0016Xk;\u0016Ukgwe can define the matrices Wu;u,Wu;x,\nWu;d, andWu;rfrom (21) as\nWu;u=Wu+ \u0000>\nuWx\u0000u; Wu;x= 2\u0000>\nuWx\u0000x;\nWu;d= 2\u0000>\nuWx\u0000d; Wu;r=\u00002\u0000uWx\u0016Xk\u00002Wu\u0016Uk;\nAppendix C.5. Choice of prediction horizon\nThe predictive capacities of the LDAM depend greatly\non the estimated disturbance ^dk. The choice of the\nprediction horizon Nis made to trade off control\nperformances and prediction accuracy. In figure C1(a-\nd) the predicted density profile is shown for differentModel-based electron density profile estimation and control, applied to ITER 20\nprediction horizons and initialization times. The red\nlines are obtained with the disturbance estimate ^dkat\ntimek= 0:25s. The blue lines are obtained with the\ndisturbanceestimate ^dkattimek= 1:25s. Finally, the\npink lines are obtained with the disturbance estimate\n^dkat timek= 2:5s. In (e), the two-norm of the\npredictionerrorisshownasafunctionoftheprediction\nhorizon for the three initialization times.\nIt can be seen in figure C1(e) that the prediction\nerror increases with the prediction horizon but that is\nincrease is very small when the disturbance estimate\nhas reached the steady-state values (see pink line in\nC1(e)) and in that case large prediction horizons can\nbe used accurately. In this work, we have chosen to\nwork with N = 20."    },    {        "title": "2111.00564v1.Density_of_Bloch_states_inside_a_one_dimensional_photonic_crystal.pdf",        "content": "arXiv:2111.00564v1  [physics.optics]  31 Oct 20211\nDensity of Bloch states inside a one dimensional\nphotonic crystal\nEbrahim Forati, Member, IEEE\nAbstract —The density of Bloch electromagnetic states inside a\none dimensional photonic crystal (1D PC) is formulated base d\non its dispersion relations. The formulation applied to any\nanisotropic medium with known dispersion relations and iso -\nfrequency surfaces. Using a practical 1D PC parameters in th e\nvisible range, the density of Bloch states for different mod es are\ncalculated.\nINTRODUCTION\nThe density of electromagnetic modes/states (DOS) is an\nimportant quantity in statistical physics. It is often used as the\ndegeneracy function for the energy levels in thermal radiat ion\nstudies such as Planck’s blackbody radiation [1], [2]. Cons ider\nthe distribution of photons inside a large cavity in thermal\nequilibrium with a solid matter at temperature T (a.k.a. clo ud\nof photons). The solid matter provides the mechanism to\nconvert photons energies (i.e. annihilate and create them)\naccording to the temperature. Since photons are bosons, the\nnumber of photons at each energy level ( εi=/planckover2pi1ω) follows\nBose-Einstein distribution as [3]\nni=gi\ne−α−βεi−1(1)\nwheregiandniare the degeneracy and photon number of\neach energy level ( εi), respectively. Equation (1) is obtained\nby maximizing the number of micro-states in the system (see\n[3]). The constants αandβare determined by enforcing the\nconditions/summationtextni=N,and/summationtextεini=UwhereNandU\nare the total number and energy of bosons, respectively. The\nlater condition leads to β=1\nkBTwherekBis Boltzmann’s\nconstant, and the former does not apply to photons, as they\ncan be created and annihilated bythe solid matter. This lead s to\nsettingα= 0. The modes degeneracy, gi,is where the DOS is\nrequired. If there is no photons reflection at the surface of t he\nthermal radiator (i.e. an ideal black body,) the DOS inside t he\nthermal radiator is essentially the same as its ambient medi um.\nAlternatively, we could explain thermal emission by\nconsidering the distribution of emitters inside a matter at\ntemperature T, and relate it to the radiational modes. An\nexample of such methods is briefly reviewed in the appendix,\nfor self-consistency.\nIf the emitting body is a 1D PC, the DOS consists of two\ncontributions: Bloch states which can propagate and depart\na finite-size 1D PC, and the wave-guided states which trap\nphotons inside the 1D PC (mostly within higher permittivity\nlayers). A complete analysis of these states is performed in\n[4] based on Green’s tensor approach. However, the density\nPO box 3043, Santa Clara, CA 95051, USA\nemail: forati@ieee.orgof Bloch states ( DOSBloch ) can be calculated, exactly, using\ntheir dispersion relations in the 1D PC. This is the paramete r\nwhich, depending on the geometry, can be used directly or\nindirectly as the degeneracy function in thermal emission f rom\n1D PCs.\nDOSBloch(ω)INSIDE A 1D PC\nWe start by formulating the DOS inside a medium with\nknown dispersion relations, and general non-spherical iso -\nfrequency surfaces. The conventional method of calculatin g\nthe DOS is to consider a large rectangular cavity with size L\nin all dimensions, filled with the medium, and with periodic\nboundary conditions on its walls. Eigen solutions of the\nharmonic electromagnetic wave equation, at frequency ω,in\nsuch geometry are modes in the form\nFmnl=Cp\nmnle−iωtei(2πm\nLx+2πn\nLy+2πl\nLz)(2)\nwhereFmnl is an electric (magnetic) field component of\nthe mode and Cp\nmnlis its associated coefficient (which has\nboth frequency and geometry dependences.) The superscript\npinCp\nmnldenotes the mode’s polarization and can take\ntwo values (any two orthogonal polarizations is correct, bu t\ntransverse electric and transverse magnetic modes along a\ncartesian coordinate axis are commonly chosen.) The values in\nthe triplet (m,n,l)can be any positive integer number subject\nto an equation obtained from the eigen value problem\nf/parenleftbigg2πm\nL,2πn\nL,2πl\nL,ω/parenrightbigg\n= 0, (3)\nEquation (3) is known as the dispersion equation after\ndefining and replacing the wave-vectors as\n(kx,ky,kz) =/parenleftbigg2πm\nL,2πn\nL,2πl\nL/parenrightbigg\n. (4)\nIn case of an isotropic homogenous material, equation (3)\nbecomes/parenleftbig2πm\nL/parenrightbig2+/parenleftbig2πn\nL/parenrightbig2+/parenleftbig2πl\nL/parenrightbig2=k2\n0εrµr. The density\nof electromagnetic modes, DOS(ω),is defined as the spectral\ndensity of the modes per cavity volume. That is, DOS(ω)dω\nis the number of supported modes per volume in the interval\n[ω,ω+dω],\nDOS(ω)dω=/summationdisplay2\nL3\n{(m,n,l):f(2πm\nL,2πn\nL,2πl\nL,ω<ω′≤ω+dω)=0}(5)\nwhere the factor of 2 accounts for the two possible orthogona l\npolarizations per mode. We used the notation {x:p(x)}\nwhich refers to the set of xfor which p(x)is true. We may\nwrite (5) as2\n1.91.81.71.6kx/k11.51.41.3-10ky/k11\n0.5\n0\n-0.5\n-1\n1kz/k1Blue: ω1=350 Trad/s\nRed:  ω2=340 Trad/s\nFig. 1. The iso-frequency surfaces of a 1D PC (TE modes) at two different\nfrequencies normalized to k1=ω√µ0ε0ε1.Parameters of the 1D PC are\nT1= 300nm, T 2= 700nm, ε1= 2.1, ε2= 11.9.\nDOS(ω)dω=/summationdisplay2\nL3△m△n△l\n{(m,n,l):f(2πm\nL,2πn\nL,2πl\nL,ω<ω′≤ω+dω)=0}(6)\nsincem,n, andlare all integers, that is △m=△n=△l=\n1.Replacing (4) in (6) and converting the summation to the\nintegration (since L is large) gives\nDOS(ω)dω=2\n(2π)3˚\ndVk\n{(kx,ky,kz):f(kx,ky,kz,,ω<ω′≤ω+dω)=0}\n(7)\nwheredVk=dkxdkydkzis the volume differential in k- space.\nThe integration in (7) is a volume (three-fold) integration\nbetween the two 3D surfaces in k- space, satisfying dispersi on\nequation (3) at ωandω+dω, respectively. These surfaces are\nalso known as the iso-frequency surfaces. Solution of equat ion\n(7) gives the DOS. Equation (7) holds regardless of the\nshape of the iso-frequency surfaces, and can be used directl y\nfor simple surfaces. For example, since the iso-frequency\nsurfaces of a homogeneous isotropic space are spherical, us ing\nk2\nx+k2\ny+k2\nz=k2in (7) gives\nDOS(ω)dω=2\n(2π)3(ω+dω)\nc√εrµrˆ\nω\nc√εrµrπˆ\n02πˆ\n0k2sinθdϕdθdk (8)\nwhich after simple manipulation leads to the known equation\nDOS(ω) =ω2(µrεr)3/2\nπ2c3.\nWe may prepare (7) further for an arbitrary iso-frequency\nsurface in k-space. This preparation becomes useful in findi ng\nDOSBloch(ω)in photonic crystals and other non-trivial and\nanisotropic materials such as hyperbolic metamaterials wh ere\niso-frequency surfaces’ shapes depend on the frequency. Fo r\nexample, Fig. 1 shows the iso-frequency surfaces of a 1D PC\nat two nearby frequencies.\nEach of the iso-frequency surfaces in (7) (and in Fig. 1) can\nbe parametrized using two independent parameters θandϕas\nS(ω) : (kx(θ,ϕ),ky(θ,ϕ),kz(θ,ϕ)) (9)\nwhere(kx,ky,kz)represents a vector in k-space. In general,\nthe choice of parameters θandϕis arbitrary and depends\non the shape of the iso-frequency surface. However, it isconvenient to choose θandϕas spherical coordinate’s polar\nand azimuthal angles for 1D PCs. The surface differential fo r\nsuch arbitrary surface is\ndA=/bardbltθ×tϕ/bardbldθdϕ (10)\nwheretθ,tϕare tangential vectors to the surface as\ntθ=/parenleftbigg∂kx(θ,ϕ)\n∂θ,∂ky(θ,ϕ)\n∂θ,∂kz(θ,ϕ)\n∂θ/parenrightbigg\n, (11)\ntϕ=/parenleftbigg∂kx(θ,ϕ)\n∂ϕ,∂ky(θ,ϕ)\n∂ϕ,∂kz(θ,ϕ)\n∂ϕ/parenrightbigg\n, (12)\nand/bardbl./bardbland×are the vector L2norm the external vector\nproduct, respectively. Since tθ×tϕis a vector normal to the\nsurface, the volume differential between the two iso-frequ ency\nsurfaces at ωandω+dωis\ndV=|(S(ω+dω)−S(ω)).(tθ×tϕ)dθdϕ|, (13)\nwhich simplifies to\ndV=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg∂kx\n∂ω,∂ky\n∂ω,∂kz\n∂ω/parenrightbigg\n.(tθ×tϕ)dθdϕdω/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (14)\nNote the|.|operator is necessary in the volume calculations\n(since the volume between the two surfaces is independent of\ntheir order). Replacing (14) into (7) gives\nDOS(ω) =2\n(2π)3\"\nS/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg∂kx\n∂ω,∂ky\n∂ω,∂kz\n∂ω/parenrightbigg\n.(tθ×tϕ)/vextendsingle/vextendsingle/vextendsingle/vextendsingledθdϕ\n(15)\nwhich can be used for any iso-frequency surface provided tha t\nit can be parametrized. As a simple example, the spherical is o-\nfrequency surfaces of a non-magnetic ( µr= 1) homogeneous\nisotropic medium can be parametrized as\nS:/parenleftbiggω√εr\ncsinθcosϕ,ω√εr\ncsinθsinϕ,ω√εr\nccosθ/parenrightbigg\n(16)\nwhich easily leads to the expected result DOS(ω) =ω2ε3/2\nr\nπ2c3.\nIn a non-magnetic 1D PC, the dispersion relation is [5], [6]\nkx=1\nT1+T2cos−1(cos(kx1T1)cos(kx2T2)− (17)\n0.5/parenleftbiggp2\np1+p1\np2/parenrightbigg\nsin(kx2d2)sin(kx1d1)/parenrightbigg\n,\nwhere\nkxi=/radicalBig\nk2\ni−k2y−k2z;pi=/braceleftBigg\nkxi\nωµ0\nωε0εi\nkxiTE\nTM,(18)\nand transverse electric (TE) and magnetic (TM) modes are\ndefined with respect to the y-z plane (interface plane).\nThe iso-frequency surface of the 1D PC can be parametrized\nas3\nω [Trad/s]0 250 500 750 1000 1250DOSBloch(ω) [m-3 (rad/s)-1]×104\n00.511.52\nTE\nTM\nTE+TM\nunbounded ǫ1\nFig. 2. The Bloch density of states in the 1D PC. The DOS of an is otropic\nhomogenous ε1= 2.1medium is also shown as a reference.\nS(θ,ϕ) :kx(ω,θ),ω\nc√ε1cosθsinϕ,ω\nc√ε1cosθcosϕ (19)\nwhere(θ,ϕ)are defined inside the material with the\nlower permittivity ( ε1) because the tangential wave-vector, /radicalBig\nk2y+k2z, inside the PC cannot exceedω\nc√εmin.The\ntangential vectors to the iso-frequency surface are\ntθ:/parenleftbiggdkx(ω,θ)\ndθ,−ω\nc√ε1sinθsinϕ, −ω\nc√ε1sinθcosϕ/parenrightbigg\n,\n(20)\ntϕ:/parenleftBig\n0,ω\nc√ε1cosθcosϕ, −ω\nc√ε1cosθsinϕ/parenrightBig\n. (21)\nUsing (15), (20), and (21), it is straight forward to show the\ndensity of Bloch states for TE and TM modes are\nDOSi\nBloch(ω) =ωε1\n4c2π2πˆ\n0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftBigg\ncosθ∂ki\nx(ω,θ)\n∂θ+ωsinθ∂ki\nx(ω,θ)\n∂ω/parenrightBigg\ncosθ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingledθ,\n(22)\nwherei= TE or TM . Equation (22) can be solved, as is,\nusing commercially available solvers and no further analyt ical\nexpansion is necessary. However, special care should be giv en\nto the calculation of kxusing (17) as it passes through different\nRiemann sheets, as will be further discussed later.\nAs an example, Fig. 2 shows the density of Bloch states\ninside a 1D PC with parameters ε1= 11.9,ε2= 2.1,\nT1= 300nm , andT2= 700nm . It also includes DOS of\nan isotropic homogeneous material with ε1= 2.1. Note that\nBloch wave’s tangential wave-vector inside a 1D PC is always\nlimited by the material with the lower permittivity (i.e. ε1).\nTheDOSBloch in Fig. 2 shows some dirivative discontinuities\nwhich are at frequencies near the band edges of TM or TE\nmodes. Some of these local peaks exceed the DOS of an\nunbounded ε1region.\nNote that the inverse cosine function in (17) is a multi-\nvalued function with branch cuts on positive and negative\nsides of the real axis ( [−∞,−1]and[1,+∞]segments).\nEvery pass of the inverse cosine argument across a branch cut\nrequires considering a different (appropriate) Riemann sh eet.\nThe argument of the cos−1function in (17) for TE modes of\nthe 1D PC example discussed here is shown in Fig. (3). The\ncrossing points of this function through branch cuts are whe re\n(a)\n(b)\n(c)\nFig. 3. a) argument of cos−1in 17 for TE modes. Flat surfaces (red) at ±1\nare added as a reference, b) regions with different Riemann s heets, and c)\nkTE\nxafter removing regions with imaginary kTE\nx(band-gaps). The 1D PC\nparameters are d1= 300nm, d2= 700nm, ε1= 11.9, ε2= 2.1.\nits∂\n∂ω(.) = 0.This leads to the Riemann sheet assignment\nshown in Fig. 5 (b) so that\ncos−1(x) =/braceleftBigg\n(n−1)π+P.V.\nnπ−P.V.@Riemann n,foroddn\n@Riemann n,forevenn\n(23)\nwhere P.V . is the principal value of cos−1(x)as0≤P.V.≤π.\nThe resulting kTE\nxis also shown in Fig. 3.\nCONCLUSION\nThe density of Bloch states inside a 1D PC was formulated\nbased on its dispersion relations for both TE and TM modes.\nThe quantities were calculated for a practical 1D PC in the\nvisible range.4\nAPPENDIX\nThe ratio of the radiation to absorption, F(ω,T),is\nconsidered to be a universal function for all solid matters b ased\non Kirchhoff’s law (controversies regarding Kirchhoff’s l aw\nare beyond the purpose of this paper, see [7], [8] for details .)\nThere are several methods to obtain F(ω,T),a.k.a. the black\nbody radiation spectrum, inside a black body cavity. They al l\nlead to\nF(ω,T) =ρ(ω)U(ω,T) (24)\nwhereF(ω,T)is considered to be a continuous function of\nωwith the unit of joules per frequency per volume (J\nm3Hz),\nandUis the average total energy of the oscillators inside the\nblack body (or photons in the cavity). In the state of thermal\nequilibrium, Ucan be obtained either using the equipartition\ntheorem in classical statistical mechanics ( U=kT,wherekis\nBoltzmann’s constant) leading to Rayleigh-Jeans distribu tion,\nor using Plank’s energy quantization arguments leading to\nU=/planckover2pi1ω//parenleftbigg\nexp(/planckover2pi1ω\nkT)−1/parenrightbigg\n. (25)\nOne of the most intuitive methods to obtain ρ(ω)in\n(24) (for isotropic homogeneous materials) is to count the\nsupported electromagnetic modes inside the resonator in th e\ninterval[ω,ω+dω], as we did in the main text. It can also\nbe obtained from the emission by classical resonators into a n\nunbounded isotropic medium (modeling a very large cavity)\nas follows [9]. Consider a particle with mass mand charge\neacted upon by an elastic restoring force −mω2\n0zand an\nexternal electric field of Ez(t).For simplicity, assume the\nparticle only moves in one dimension ( z). Newton’s equation\nof motion for such particle is\n¨z+ω2\n0z=e\nmEz(t)+e\nmERR(t) (26)\nwhereERR(t)is the reaction electric field originated from\nthe moving particle itself, and can be shown to be ERR(t) =\n2e\n3c3...z(t).If the particle is inside a large rectangular cavity\n(with size L), the energy absorption rate by the particle\noscillating with frequency ω0from a cavity resonant mode\nat frequency ωis\n˙WA=e2\n2mγω4E2\nz(ω)\n(ω2−ω2\n0)2+γ2ω6;γ=2e2\n3mc3(27)\nwhereEz(ω)is thez−directed electric field associated with\nthe electromagnetic mode, and cis the speed of ( z−polarized)\nlight in the medium. If the radiation has a continuous\nbroadband spectrum, with the (z-directed electric field) en ergy\ndensity of Fz(ω,T)dω=E2\nz(ω)/(8π)in the interval\n[ω,ω+dω], it can be shown that [9]\n˙WA=4πe2\nmˆ\nωγFz(ω,T)dω\n(1−ω2\n0/ω2)2+γ2ω2. (28)For the frequencies of interest in thermal emissions, γω≪\n1. IfF(ω,T)is not sharply peaked, we may simplify (28) as\n˙WA=4πe2γFz(ω0,T)\nmˆ\nωdω\n4ω−2\n0(ω−ω0)2+γ2ω2o(29)\n=4πe2γFz(ω0,T)\nmω2\n0\n42π\nγω2\n0.\nThe energy emission rate of the oscillating charge is\n˙WE=2e2ω2\n0\n3mc3U (30)\nwherecis the speed of light in vacuum, and Uis the average\ntotal oscillator energy. In the state of thermal equilibriu m\nbetween the radiational energy and the matter, (30) and (29)\nshould be equal, leading to\nF(ω0,T) =ω2\n0\nπ2c3U, (31)\nwhereFx(ω0,T) =Fy(ω0,T) =Fz(ω0,T) =F(ω0,T)/3\nis used. Similar to the resonator mode counting method, (31)\ngives\nρ(ω) =ω2\nπ2c3. (32)\nAlso, equations (24), (25), and (32) provide the well-known\nPlank’s emission spectrum inside a blackbody cavity filled\nwith an isotropic homogeneous material.\nThis method of obtaining ρ(ω)provides insight into how\nradiators inside the matter (e.g. blackbody) couple to the\nelectromagnetic modes in common situations (leading to\nPlank’s radiation spectrum). Two critical assumptions whi ch\nsimplified (28) to (29) are 1) radiation spectrum, Fz(ω,T),\ndoes not have any sharp peaks, and 2) the resonators’\ncoupling to the radiation is sharply peaked around their\nnatural frequency, ω0.This assumption also implicitly requires\nlinearity of the material. Only with these assumptions we\nobtained the same ρ(ω)as using the other, more fundamental,\nmethods such as BE distribution of photons in the cavity in\nthermal equilibrium. It appears that (28) is a good starting\npoint to study the thermal emission from non-linear (or any\nother uncommon) material.\nREFERENCES\n[1] G. Kirchhoff, “On the relation between the radiating and absorbing\npowers of different bodies for light and heat,” The London, Edinburgh,\nand Dublin Philosophical Magazine and Journal of Science , vol. 20, no.\n130, pp. 1–21, 1860.\n[2] M. Planck, The theory of heat radiation . Courier Corporation, 2013.\n[3] Reichl, Linda E. \"A modern course in statistical physics .\" (1999): 1285-\n1287.\n[4] E. Forati, “Spontaneous emission rate and the density of states inside a\none dimensional photonic crystal,” xx-arxiv , 2021.\n[5] L. Qi and C. Liu, “Complex band structures of 1d anisotrop ic graphene\nphotonic crystal,” Photonics Research , vol. 5, no. 6, pp. 543–551, 2017.\n[6] J. Joannopoulos, R. Meade, and J. Winn, “Photonic crysta ls,”Molding\nthe flow of light , 1995.\n[7] P.-M. Robitaille, “Kirchhoff’s law of thermal emission : 150 years,” Progr.\nPhys , vol. 4, pp. 3–13, 2009.\n[8] P.-M. Robitaille, “On the validity of Kirchhoff’s law of thermal emission,”\nIEEE transactions on plasma science , vol. 31, no. 6, pp. 1263–1267,\n2003.\n[9] P. W. Milonni, The quantum vacuum: an introduction to quantum\nelectrodynamics . Academic press, 2013."    },    {        "title": "2112.09108v6.On_the_Normalization_and_Density_of_1D_Scattering_States.pdf",        "content": "arXiv:2112.09108v6  [quant-ph]  11 Dec 2023On the Normalization and Density of 1D Scattering States\nChris L. Lin\nDepartment of Physics, University of Houston, Houston, TX 7 7204-5005\n(Dated: December 12, 2023)\n1Abstract\nThenormalization of scattering states is more than a rote st ep necessary to calculate expectation\nvalues. This normalization actually contains important in formation regarding the density of the\nscattering spectrum (along with useful details on the bound states). For many applications, this\ninformation is more useful than the wavefunctions themselv es. In this paper we show that this\ncorrespondence between scattering state normalization an d the density of states is a consequence\nof the completeness relation, and we present formulas for ca lculating the density of states which\nare applicable to certain potentials. We then apply these fo rmulas to the delta function potential\nand the square well. We then illustrate how the density of sta tes can be used to calculate the\npartition function for a system of two particles with a point -like (delta potential) interaction.\nI. INTRODUCTION\nIn nonrelativistic quantum mechanics, eigenstates of the free Ham iltonian ˆH0=ˆP2\n2mare also\nmomentum eigenstates |k/an}⌊ra⌋ketri}ht. These states satisfy the completeness relation\nˆI=/integraldisplay∞\n−∞dk\n2π|k/an}⌊ra⌋ketri}ht/an}⌊ra⌋ketle{tk|, (1)\nwhereˆIis the identity operator, and the normalization is /an}⌊ra⌋ketle{tq|k/an}⌊ra⌋ketri}ht= 2πδ(q−k) assuming\nthe convention /an}⌊ra⌋ketle{tx|k/an}⌊ra⌋ketri}ht ≡φk(x) =eikx. In the presence of a potential ˆV, the Hamiltonian\nˆH=ˆH0+ˆVmay possess bound states |ψbi/an}⌊ra⌋ketri}htas well as scattering states |ψk/an}⌊ra⌋ketri}ht, which are\nsolutions to ˆH|ψk/an}⌊ra⌋ketri}ht=/planckover2pi12k2\n2m|ψk/an}⌊ra⌋ketri}ht. These scattering states are the sum of the free wave and a\nscattered part, i.e., |ψk/an}⌊ra⌋ketri}ht=|k/an}⌊ra⌋ketri}ht+|ψscat\nk/an}⌊ra⌋ketri}ht. The completeness relation then takes the form [1]:\nˆI=/summationdisplay\ni|ψbi/an}⌊ra⌋ketri}ht/an}⌊ra⌋ketle{tψbi|+/integraldisplay∞\n−∞dk\n2π|ψk/an}⌊ra⌋ketri}ht/an}⌊ra⌋ketle{tψk|. (2)\nSince the trace of the identity matrix gives the dimension of the unde rlying vector space,\nwe can set the trace of Eq. (1) equal to the trace of Eq. (2) giving :\nNb=−/integraldisplay∞\n−∞dk∆ρ(k) (3)\n∆ρ(k) =1\n2π/integraldisplay∞\n−∞dx/parenleftbig\n|ψk(x)|2−|φk(x)|2/parenrightbig\n,\n2whereNbis the number of bound states and we have defined ∆ ρ(k) as the change in the\nmomentum density of scattering states from that of the free sta tes. Intuitively, Eq. (3)\nreflects the fact that that an increase in the number of bound sta tes necessitates a decrease\nin the number of scattering states [2]. We will sometimes evaluate ∆ ρ(k) by calculating a\nmore general quantity\n∆ρ(k,q) =1\n2π/integraldisplay∞\n−∞dx/parenleftbig\nψ∗\nq(x)ψk(x)−φ∗\nq(x)φk(x)/parenrightbig\n, (4)\nfrom which ∆ ρ(k) can be obtained by setting q=k.\nThefactthat ψk(x)andφk(x)havedifferentnormalizationsmaycomeasasurprise, although\nthis has been observed in other papers [3–6]. We will explicitly derive t he difference in their\nnormalizations in multiple ways and for multiple systems. Due to this cor respondence with\nthe density of states, the normalization of the scattering states is not merely an academic\nexercise.\nIt is instructive to briefly outline how Eq. (3) can be obtained startin g from a system with a\ncountable discrete spectrum where both the bound and scatterin g states can be normalized\nto unity. This can be accomplished for instance by confining the inter acting system to a box\nof lengthLwith periodic boundary conditions. The allowed energies and momenta can then\nbe labeled by a quantum number nand are therefore discrete satisfying E(n,L) =/planckover2pi12k2\nn\n2m. For\nlargeL, increasing nby 1 only causes an infinitesimal increment in k, so a sum over states\ncan be turned into an integral by the sequence/summationtext\nn...→/integraltext\ndn...→/integraltext\ndkdn\ndk...=/integraltext\ndkρ(k)....\nIn particular, the completeness relation can be written [7]:\nˆI=/summationdisplay\ni|ψbi/an}⌊ra⌋ketri}ht/an}⌊ra⌋ketle{tψbi|+/summationdisplay\nn|ψkn/an}⌊ra⌋ketri}ht/an}⌊ra⌋ketle{tψkn| (5)\n=/summationdisplay\ni|ψbi/an}⌊ra⌋ketri}ht/an}⌊ra⌋ketle{tψbi|+/integraldisplay\ndkρ(k)|ψkn/an}⌊ra⌋ketri}ht/an}⌊ra⌋ketle{tψkn|.\nBy defining the continuum spectrum solution as |ψk/an}⌊ra⌋ketri}ht=/radicalbig\n2πρ(k)|ψkn/an}⌊ra⌋ketri}htwe accomplish three\nthings: Eq. (5)goesintoitscontinuumversionEq. (2); /an}⌊ra⌋ketle{tψk|ψk/an}⌊ra⌋ketri}ht= 2πρ(k)/an}⌊ra⌋ketle{tψkn|ψkn/an}⌊ra⌋ketri}ht= 2πρ(k)\nwhich gives Eq. (3); |ψk/an}⌊ra⌋ketri}htreceives the correct units of square root of inverse momentum as\nthose are the units of/radicalbig\n2πρ(k) (by contrast, |ψkn/an}⌊ra⌋ketri}htis dimensionless). Although in princi-\nple one can calculate the density of states ρ(k) by putting the system in a box, imposing\n3boundary conditions, solving the Schr¨ odinger equation for k=k(n,L) and then inverting\n(n=n(k,L)) to calculate ρ(k) =dn\ndk, in this paper we will not discretize the system and\ninstead directly use Eq. (3) to calculate ∆ ρ(k).\nIn section II, we derive formulas for the normalization of the state s in 1D, finite-range\nsymmetric potentials. In section III we apply these formulas to the delta function potential\nand the square well, and plot the density of states. In section IV we use the density of\nstates to calculate the partition function for a system of two part icles with a delta potential\ninteraction.\nII. DERIVATION OF THE CHANGE IN THE DENSITY OF STATES\nWe consider a symmetric potential V(x) =V(−x) that is zero outside a region |x|> a.\nSymmetry allows us to consider only right-moving waves ( k >0) since ∆ρ(k) = ∆ρ(−k).\nThe scattering state |ψk/an}⌊ra⌋ketri}htis then given by\nψk(x) =\n\neikx+R(k)e−ikxx<−a\nψkI(x) −a≤x≤a\nT(k)eikxx>a,(6)\nwhereψkI(x) is the wavefunction inside the range of the potential. First, we will p lace the\nsystem in a box of length Land then consider the L→ ∞limit [8]. Plugging Eq. (6)\ninto Eq. (4) directly taking q=kin the integrand, and using |R(k)|2+|T(k)|2= 1, it is\nstraightforward to show that\n∆ρ(k) =−a\nπ+1\n2π/integraldisplaya\n−adxψ∗\nkI(x)ψkI(x)\n−1\n2πk/parenleftbig\nRe[R(k)]sin(2ka)+Im[R(k)]cos(2ka)/parenrightbig\n+sin(kL)\n2πkRe[R(k)]+cos(kL)\n2πkIm[R(k)].(7)\nBoth sin(kL) and cos(kL) oscillate very fast when L→ ∞, so that when ∆ ρ(k) is integrated\noverk, these terms will produce zero unless their coefficients are non-an alytic. The coeffi-\ncients are analytic except at k= 0. However, it is a general feature of scattering in 1D that\n4R(0) =−1, i.e. complete inversion with no imaginary part [9, 10], and reflects th e fact that\nk= 0 constitutes the opening of the continuum channel [11]. Using tha tsin(kL)\nk→πδ(k) as\nL→ ∞we get:\n∆ρ(k) =−a\nπ−Re[R(k)]sin(2ka)+Im[R(k)]cos(2ka)\n2πk−1\n2δ(k)+1\n2π/integraldisplaya\n−a|ψkI(x)|2dx.(8)\nWe will comment on the −1\n2δ(k) term in the next section; such a term is also found in\nstudies of Levinson’s theorem in 1D [12, 13].\nThe integral in Eq. (8) contains the wavefunction ψkI(x) in the interaction region. An\nalternate formula for ∆ ρ(k) that does not require knowledge of this wave function is:\n∆ρ(k) =−i\n2π/parenleftbigg\nR∗(k)dR(k)\ndk+T∗(k)dT(k)\ndk/parenrightbigg\n−1\n2δ(k). (9)\nWe briefly outline the derivation of Eq. (9) in appendix A, while in append ix B we will\nderive both Eq. (8) and Eq. (9) using a convergence factor in place of a box.\nIII. EXAMPLES\nA.δ(x)Potential\nFor the potential V(x) =−gδ(x) the reflection coefficient is R(k) =−(1+ik/κ)−1, where\nκ=mg//planckover2pi12. This is valid for both the delta function well ( g >0) and the delta function\nbarrier (g <0). Wheng >0,κis equal to/radicalbig\n2mEb//planckover2pi12for the single bound state energy\nE=−Eb=−mg2/2/planckover2pi12. By setting a→0 in Eq. (8) and plugging in R(k), we get:\n∆ρ(k) =−1\n2πκ\nκ2+k2−1\n2δ(k), (10)\nwhich is plotted in Fig. 1. For the case of a barrier ( g <0),κ<0, so the graph in Fig. 1\nwould be inverted, implying all states except the k= 0 state would experience an increase\nin density.\nAsk→ ∞, ∆ρ(k)→ −κ/2πk2. It is interesting to note that κ, which is a measure of the\nbound state depth when g >0, is the coefficient of the k−2decaying tail, which suggests a\ndeeper bound state influences the spectrum at large kmore than a shallower bound state.\n5-6-4-2 246k[]\n-0.15-0.10-0.05Δρ[κ-1]\nFIG. 1. The change in the density of states for the potential V(x) =−gδ(x), assuming g>0. Not\nshown is a −1\n2δ(k) term in ∆ ρ(k) that only affects the k= 0 state.\nFinally we verify the preservation of the size of the vector space:\nNb=−/integraldisplay∞\n−∞dk/parenleftbigg\n−1\n2πκ\nκ2+k2−1\n2δ(k)/parenrightbigg\n=\n\n1κ>0\n0κ<0.(11)\nB. Square Well\nFor a right-moving wave in a square well of length 2 aand depthV0we express the results\nin terms of the transmission coefficient\nT(k) =e−2ika\ncos(2ℓa)−ik2+ℓ2\n2kℓsin(2ℓa), (12)\nwhereℓ=/radicalbig\nk2+q2,q2= 2mV0//planckover2pi12[14]. The reflection coefficient and the wavefunction in\nthe interacting region can then be expressed as\nR(k) =isin(2ℓa)(ℓ2−k2)\n2kℓT(k) (13)\nψkI=Csin(ℓx)+Dcos(ℓx)\nC= [sin(ℓa)+ik\nℓcos(ℓa)]eikaT(k)\nD= [cos(ℓa)−ik\nℓsin(ℓa)]eikaT(k).\nPlugging these into Eq. (8) we get:\n6-10 -5 510k[1/a]\n-0.3-0.2-0.1\n\u0001 \u0000[a]\nqa=1\nqa=3\nqa=10\nFIG. 2. The change in the density of states for the square well , plotted for various values of\nqa=/radicalig\n2mV0\n/planckover2pi12a. Not shown is a −1\n2δ(k) term in ∆ ρ(k) that only affects the k= 0 state.\n∆ρ(k) =1\n2π\n−2a+8ak2(2k2+q2)−2q4sin/parenleftBig\n4a√\nk2+q2/parenrightBig\n√\nk2+q2\n−q4cos/parenleftig\n4a/radicalbig\nk2+q2/parenrightig\n+8k4+8k2q2+q4\n−1\n2δ(k),(14)\nwhich is plotted in Fig. 2.\nA largeqasignifies a deep and wide well which allows for many bound states – corr e-\nspondingly, the area above the curves increases with qa. It is straightforward to show that\nasymptotically ∆ ρ(k)→ −aq2/(2πk2) ask→ ∞. Once again the coefficient of the k−2\ndecaying tail is related to the depth of the well.\nFinally, using the substitution u=ka, we calculate the number of bound states by integrat-\ning the density of states:\nNb=1\nπ/integraldisplay∞\n0du\n2−8u2(2u2+(qa)2)−2(qa)4sin/parenleftBig\n4√\nu2+(qa)2/parenrightBig\n√\nu2+(qa)2\n8u4+8u2(qa)2+(qa)4−(qa)4cos/parenleftig\n4/radicalbig\nu2+(qa)2/parenrightig\n+1\n2.(15)\nThe result is plotted in Fig. 3 for various values of qa. An analysis of the bound state sector\n[15] gives the number of bound states as Nb= 1+⌊2\nπ(qa)⌋, where the floor function ⌊2\nπ(qa)⌋\nreturns the greatest integer that is smaller than2\nπ(qa). It is remarkable that a numerical\nintegration of Eq. (15) can produce the sharp steps seen in Fig. 3.\n72 468102mVo\nℏ2a123\n4567Nb\nFIG. 3. The number of bound states in the square well, found by numerical integration of the\nchange of the density of states, ∆ ρ(k). The well always has one bound state no matter how small\nthe attraction, and gains an additional bound state wheneve r/radicalig\n2mV0\n/planckover2pi12aincreases byπ\n2.\nIV. APPLICATION TO A THERMODYNAMIC SYSTEM\nIn this section we illustrate how this technique of calculating the dens ity of states can be\napplied to a physical system in thermal equilibrium. Consider a system of two particles of\nmassmconfinedtoalineoflength Landinthermalcontact withaheatbathattemperature\nT. The particles interact with each other through the contact inter actionV(x1,x2) =\n−gδ(x1−x2). The Hamiltonian for such a system is H=p2\n1\n2m+p2\n2\n2m−gδ(x1−x2). Changing to\ncenterofmassandrelativecoordinatestheHamiltonianbecomesse parable(H=Hcom+Hrel)\nwith\nHcom=P2\n4m(16)\nHrel=p2\nm−gδ(x),\nwhereP=p1+p2,p=1\n2(p1−p2), andx=x1−x2are canonical variables. The partition\nfunction for this system can then be factorized as Q2=Qrel\n2Qcom\n2[16].Hcomhas the form of\na Hamiltonian for a free particle of mass 2 mso its partition function is\nQcom\n2=/integraldisplay∞\n−∞dkρfreee−β/planckover2pi12k2\n4m=L/radicalbiggm\nπβ/planckover2pi12, (17)\nwhere we inserted ρfree=L\n2π, the density of states for the free particle [17]. The relative\ntwo-particle partition function can be calculated as\n8Qrel\n2=/summationdisplay\nie−βEbi+/integraldisplay∞\n−∞dk(ρfree+∆ρ(k))e−β/planckover2pi12k2\n2µ, (18)\nwhereµis the reduced mass, β= (kBT)−1(wherekBis Boltzmann’s constant), and Ebi\nare bound state energies (if they exist for the system). We calcula teQrel\n2by integrating the\nscattering spectrum over the density of states. Inserting Eq. ( 10) into Eq. (18) yields the\ntwo-particle relative partition function:\nQrel\n2=/bracketleftbigg\ne−β/parenleftBig\n−/planckover2pi12κ2\nm/parenrightBig\n+/integraldisplay∞\n−∞dk/parenleftbiggL\n2π−1\n2πκ\nκ2+k2−1\n2δ(k)/parenrightbigg\ne−β/planckover2pi12k2\nm/bracketrightbigg\n(19)\n=1\n2/braceleftigg\nL/radicalbiggm\nπβ/planckover2pi12+eβ/planckover2pi12κ2\nm/bracketleftigg\nerf/parenleftigg\n/planckover2pi1κ/radicalbigg\nβ\nm/parenrightigg\n+1/bracketrightigg\n−1/bracerightigg\n,\nwhere erf(x) is the error function. If the interaction is very weak ( κ→0),Qrel\n2approaches\nthe partition function of a free particle of mass m/2: the bound state term would exactly\ncancel the scattering term in Eq. (19). This provides a nontrivial c heck of Eq. (8) which\nwas used to derive the scattering term.\nV. CONCLUSIONS\nWe have shown that the density of scattering states lies hidden in th e normalization of the\nscattering wave functions. Moreover, the scattering state sec tor contains information about\ntheboundstatesector. Theconnection between boundstatesa ndscattering statesisusually\npresented to students through either Levinson’s theorem, [18] o r through a discussion of the\nanalytic propertiesof thescattering matrix[19]. Inthis paper we h ighlighted thisconnection\nusing an alternative approach which also makes contact with the den sity of states. Because\nthe density of states is an important concept in statistical mechan ics, we find this approach\nto be pedagogically beneficial and have used it to calculate the exact partition function of\nan interacting system.\nVI. CONFLICT OF INTERESTS\nThe author has no conflicts to disclose.\n9ACKNOWLEDGEMENTS\nThe author would like to acknowledge the reviewers for their helpful suggestions.\nAppendix A: Asymptotic Form\nThe derivation of Eq. (9) is challenging. Consider the Schr¨ odinger e quation and its conju-\ngate:\n−/planckover2pi12\n2md2\ndx2ψk+Vψk=/planckover2pi12k2\n2mψk (A1)\n−/planckover2pi12\n2md2\ndx2ψ∗\nq+Vψ∗\nq=/planckover2pi12q2\n2mψ∗\nq.\nMultiplying the top line by ψ∗\nqand the bottom line by ψkand subtracting removes reference\nto the potential:\nψ∗\nqψk=1\nq2−k2/parenleftbigg\nψ∗\nqd2ψk\ndx2−ψkd2ψ∗\nq\ndx2/parenrightbigg\n. (A2)\nWe can use integration by parts to get an expression involving the wa vefunction only in the\nasymptotic region:\n/integraldisplayL/2\n−L/2dxψ∗\nqψk=1\nq2−k2/parenleftbigg\nψ∗\nqdψk\ndx−ψkdψ∗\nq\ndx/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleL/2\n−L/2≡N(k,q)\nq2−k2. (A3)\nSinceLis large, the asymptotic forms ψk(L/2) =T(k)eik(L/2),ψ′\nk(L/2) =ikT(k)eik(L/2),\nψk(−L/2) =eik(−L/2)+R(k)e−ik(−L/2), etc. can be used to find the numerator N(k,q).\nPlugging Eq. (A3) into Eq. (4) and taking q→kgives ∆ρ(k). L’Hˆ opital’s rule can be used\nto evaluate the limit as\nlim\nq→kN(k,q)\nq2−k2=∂qN(k,q)|q=k\n2k, (A4)\nwhich creates the derivatives seen in Eq. (9). After the limit is taken , the conservation of\nprobability and its derivative\n10R∗(k)R(k)+T∗(k)T(k) = 1 (A5)\ndR∗\ndkR+dT∗\ndkT=−/parenleftbiggdR\ndkR∗+dT\ndkT∗/parenrightbigg\ncan be used to further simplify the expression to get the final form in Eq. (9).\nAppendix B: Convergence Factor Derivation\nWithout a box, we will utilize the help of a convergence factor ( ǫ>0) that converts oscil-\nlatory expressions into decaying ones:\n∆ρ(k,q) =1\n2πlim\nǫ→0+/integraldisplay∞\n−∞dx/parenleftbig\nψ∗\nq(x)ψk(x)−φ∗\nq(x)φk(x)/parenrightbig\ne−ǫ|x|. (B1)\nInserting Eq. (6) into Eq. (B1) all integrals in the asymptotic region |x|>aare of the form\n/integraltext∞\nadxeisxe−ǫx=−eisae−ǫa\nis−ǫand/integraltext−a\n−∞dxeisxeǫx=e−isae−ǫa\nis+ǫ. Performing these integrals we get:\n/integraldisplay∞\n−∞dxψ∗\nq(x)ψk(x)e−ǫ|x|= (T∗(q)T(k)+R∗(q)R(k))/parenleftbigg1\ni(k−q)+ǫ−1\ni(k−q)−ǫ/parenrightbigg\nei(k−q)ae−ǫa\n+/parenleftbig\n1−(T∗(q)T(k)+R∗(q)R(k))e2ia(k−q)/parenrightbige−ia(k−q)e−ǫa\ni(k−q)+ǫ\n+/parenleftbigg\nR(k)eia(k+q)\n−i(k+q)+ǫ+R∗(q)e−ia(k+q)\ni(k+q)+ǫ/parenrightbigg\ne−ǫa\n+/integraldisplaya\n−adxψ∗\nqI(x)ψkI(x)e−ǫ|x|.(B2)\nSettingq=k, the first term on the right-hand side produces 2 πδ(0)−2a, the second\nterm is 0 from |R(k)|2+|T(k)|2= 1, and the third term produces −1\nk/parenleftbig\nRe[R(k)]sin(2ka)+\nIm[R(k)]cos(2ka)/parenrightbig\n−πδ(k). This is in agreement with Eq. (8) once the free normalization\nis subtracted, which gets rid of the 2 πδ(0) term.\nOn the other hand, if in Eq. (B2) we take kclose to, but not equal to q, then we can set\n11ǫ= 0 to get a useful relation:\n0 = 0+ lim\nk→q/parenleftbig\n1−(T∗(q)T(k)+R∗(q)R(k))e2ia(k−q)/parenrightbige−ia(k−q)\ni(k−q)\n−1\nq/parenleftbig\nRe[R(q)]sin(2qa)+Im[R(q)]cos(2qa)/parenrightbig\n+/integraldisplaya\n−adxψ∗\nqI(x)ψqI(x).(B3)\nUsing L’Hˆ opital’s rule on the first term on the right-hand of Eq. (B3) gives−2a+\niT∗(q)dT(q)\ndq+iR∗(q)dR(q)\ndq. This allows us to solve for/integraltexta\n−adxψ∗\nqI(x)ψqI(x) in terms of asymp-\ntotic quantities:\n/integraldisplaya\n−adxψ∗\nqI(x)ψqI(x) = 2a−iT∗(q)dT(q)\ndq−iR∗(q)dR(q)\ndq\n+1\nq/parenleftbig\nRe[R(q)]sin(2qa)+Im[R(q)]cos(2qa)/parenrightbig\n,(B4)\nwhich we then subsequently plug into Eq. (8) to get Eq. (9).\n[1] S. H. Patil, American Journal of Physics 68, 712 (2000), https://doi.org/10.1119/1.19532.\n[2] In linear algebra, two different basis {|k/an}⌊ra⌋ketri}ht}and{|φbi/an}⌊ra⌋ketri}ht,|ψk/an}⌊ra⌋ketri}ht}should have the same size, so the\nmore states {|φbi/an}⌊ra⌋ketri}ht}that exist the less the density of the {|ψk/an}⌊ra⌋ketri}ht}states compared to the density\nof the{|k/an}⌊ra⌋ketri}ht}states .\n[3] M. Stone and P. Goldbart, “Mathematics for Physics: A Gui ded Tour for Graduate Students,”\n(Cambridge University Press, 2009) p. 130.\n[4] N. Poliatzky, Helvetica Physica Acta 66, 241 (1993).\n[5] R. G. Newton, Helv. Phys. Acta 67, 20 (1994).\n[6] N. Poliatzky, Helv. Phys. Acta 67, 683 (1994), arXiv:hep-th/9411197\n.\n[7] An interval of size ∆ kcentered around the momentum k, may contain several discrete\nmomenta {...,kn−1,kn,kn+1,...}. Instead of adding them ...+|ψkn−1/an}⌊ra⌋ketri}ht/an}⌊ra⌋ketle{tψkn−1|+|ψkn/an}⌊ra⌋ketri}ht/an}⌊ra⌋ketle{tψkn|+\n|ψkn+1/an}⌊ra⌋ketri}ht/an}⌊ra⌋ketle{tψkn+1|+..., we take any one of the discrete momenta as representative an d multiply\nby the number of elements in the set:\n12...+|ψkn−1/an}⌊ra⌋ketri}ht/an}⌊ra⌋ketle{tψkn−1|+|ψkn/an}⌊ra⌋ketri}ht/an}⌊ra⌋ketle{tψkn|+|ψkn+1/an}⌊ra⌋ketri}ht/an}⌊ra⌋ketle{tψkn+1|+...=ρ(k)∆k|ψkn/an}⌊ra⌋ketri}ht/an}⌊ra⌋ketle{tψkn|\nTherefore the |ψkn/an}⌊ra⌋ketri}htin the last term of Eq. (5) represents any of the discrete stat es that fall\nwithin the interval dkcentered around the momentum k.\n[8] We will not impose any boundary conditions at the endpoin ts of the box as we are not\nattempting to discretize the spectrum, but rather regulate the infinite volume limit .\n[9] P. Senn, American Journal of Physics 56, 916 (1988), https://doi.org/10.1119/1.15359.\n[10] Y. Nogami and C. K. Ross, American Journal of Physics 64, 923 (1996),\nhttps://doi.org/10.1119/1.18123.\n[11] For exotic cases where there exists a zero-energy bound state, we may have R(0)/ne}ationslash=−1, which\nis known as threshold anomaly. We do not consider such cases h ere .\n[12] M. Sassoli de Bianchi, Journal of Mathematical Physics 35, 2719 (1994),\nhttps://doi.org/10.1063/1.530481.\n[13] G. Barton, Journal of Physics A: Mathematical and Gener al18, 479 (1985).\n[14] D. J. Griffiths and D. F. Schroeter, “Introduction to Quan tum Mechanics; Third Edition,”\n(Cambridge University Press, 2018) pp. 96–97.\n[15] F. Williams, “Topics in Quantum Mechanics,” (Springer Science, New York, 2003) p. 57.\n[16] When the energy of a system of two particles decomposes i ndependently into Ecom\ni+Erel\ni, then\nthe partition function factorizes: Q2=/summationtext\nI,ie−β(Ecom\nI+Erel\ni)=/parenleftbig/summationtext\nIe−βEcom\nI/parenrightbig/parenleftig/summationtext\nie−βErel\ni/parenrightig\n=\nQcom\n2Qrel\n2.\n[17] If we impose periodic boundary conditions on a free part icle in a box of length L, we get the\nconditionk(n) =2πn\nL. The density is then ρ(k) =dn\ndk=L\n2π.\n[18] M. Wellner, American Journal of Physics 32, 787 (1964), https://doi.org/10.1119/1.1969857.\n[19] J. J. Sakurai and J. Napolitano, “Modern Quantum Mechan ics; Second Edition,” (Addison-\nWesley, San Francisco, CA, 1994) pp. 429–430.\n13"    },    {        "title": "2201.11723v2.Broadened_Yu_Shiba_Rusinov_states_in_dirty_superconducting_films_and_heterostructures.pdf",        "content": "Broadened Yu-Shiba-Rusinov states in dirty superconducting films and\nheterostructures\nS. S. Babkin,1, 2A. A. Lyublinskaya,1, 2and I. S. Burmistrov2, 3\n1Moscow Institute of Physics and Technology, 141700 Dolgoprudnyi, Moscow Region, Russia\n2L. D. Landau Institute for Theoretical Physics, acad. Semenova av. 1-a, 142432 Chernogolovka, Russia\n3Laboratory for Condensed Matter Physics, HSE University, 101000 Moscow, Russia\n(Dated: May 24, 2022, v.5 - resubmitted)\nThe interplay of a potential and magnetic disorder in superconductors remains an active field of\nresearch for decades. Within the framework of the Usadel equation, we study the local density of\nstates near a solitary classical magnetic impurity in a dirty superconducting film. We find that a\npotential disorder results in broadening of the delta-function peak in the local density of states at the\nYu-Shiba-Rusinov (YSR) energy. This broadening is proportional to the square root of a normal-\nstate spreading resistance of the film. We demonstrate that modification of multiple scattering\non the magnetic impurity due to intermediate scattering on surrounding potential disorder affects\ncrucially a profile of the local density of states in the vicinity of the YSR energy. In addition, we find\nthat a scanning-tunneling-microscopy tip can mask an YSR feature in the local density of states.\nAlso, we study the local density of states near a chain of magnetic impurities situated in the normal\nregion of a dirty superconductor/normal-metal junction. We find a resonance in the local density of\nstates near the YSR energy. The energy scale of the resonant peak is controlled by the square root\nof the film resistance per square in the normal state.\nI. INTRODUCTION\nStudies of the e\u000bect of imperfections on superconduct-\ning properties have been remaining an active \feld of re-\nsearch since the middle of the last century. Initially, it\nwas believed that the potential scattering in 𝑠-wave su-\nperconductors does not a\u000bect superconducting properties\n(so-called, Anderson theorem) [1{3]. Later it was under-\nstood that signi\fcant amount of potential disorder re-\nsults in superconductor to insulator transition [4] which\nis manifestation of competition between Anderson local-\nization and Cooper-channel attraction (see Refs. [5{7]\nand references therein).\nClassical magnetic impurities being a source for time-\nreversal symmetry violation cause much severe e\u000bect on\ns-wave superconductivity than potential imperfections.\nWithout any quantum interference e\u000bects taken into ac-\ncount (mean-\feld approximation), magnetic impurities\nsuppress the superconducting state provided their con-\ncentration is high enough [8, 9]. Beyond the Born approx-\nimation, the scattering of quasiparticles by a magnetic\nimpurity leads to the appearance of subgap Yu-Shiba-\nRusinov (YSR) states in a superconductor [10{13]. At a\n\fnite concentration of magnetic impurities, YSR states\nare hybridized and can form energy bands with hard gaps\nin the averaged density of states. Depending on the con-\ncentration of magnetic impurities and their strength, a\nrich phase diagram arises (see Ref. [14] for a review).\nVarious inhomogeneity e\u000bects, such as rare \ructua-\ntions of a random potential [15{18], \ructuations in con-\ncentration of magnetic impurities [19], \ructuations of su-\nperconducting order parameter [20], etc. lead to smear-\ning of hard gaps in the density of states (see Refs. [21, 22]\nfor a review). Recently, it has been shown [23] that\nmesoscopic (point-to-point) \ructuations of e\u000bective ex-\nchange interaction between spins of magnetic impurityand quasiparticles caused by non-magnetic disorder re-\nsult in strong modi\fcation of the YSR bands in the av-\nerage density of states in comparison with the mean-\feld\nanalysis.\nFor a long time, modi\fcation of the superconducting\nstate by a single magnetic impurity has been remaining\ntheoretical concept only [24, 25]. Progress in scanning\ntunneling microscopy (STM) makes possible to resolve\nspatial and energy dependence of YSR states [26{33]. Re-\ncent STM experimental studies have revealed rich physics\nof YSR states in superconductors (see Ref. [34] for a re-\nview).\nCurrently, experimental studies of solitary YSR states\nare limited to relatively clean superconductors (typically,\nMn or Cr atoms in Pb \flm or monolayer). Nevertheless,\nthere is an intriguing and yet unresolved question of how\nnon-magnetic disorder a\u000bects spatial and energy depen-\ndence of YSR states. This question can be of additional\nimportance due to the presence of intrinsic magnetic im-\nperfections in nominally non-magnetic disordered super-\nconducting \flms [35].\nRecently, the e\u000bect of a random potential on the YSR\nstate has been theoretically studied in Ref. [36]. The au-\nthors extended the scattering approach used in Ref. [13]\nto incorporate additional scattering on the nonmagnetic\nimpurities. The broadening of YSR state has been es-\ntimated within the lowest order perturbation theory in\npotential disorder. However, behavior of the local den-\nsity of states (LDoS) near a magnetic impurity has not\nbeen addressed.\nAnother interesting question is the fate of YSR\nstates in superconducting heterostructures, e.g.,\nsuperconductor/normal-metal/superconductor (SNS)\nor superconductor/normal-metal (SN) junctions. In\na clean SNS junction, a magnetic impurity situated\nin the normal region leads to interesting interplay ofarXiv:2201.11723v2  [cond-mat.mes-hall]  22 May 20222\nFIG. 1. Left: A schematic view of a dirty superconducting film with a solitary magnetic impurity with spin 𝑆. Potential impu-\nrities are shown as yellow circles. A singlet Cooper pair is depicted as a wavy green line. The mean free path is assumed to be\nmuch shorter than clean superconducting coherence length, ℓ≪𝜉cl. Right: A schematic view of a dirty superconductor/normal-\nmetal junction, with a chain of magnetic impurities with one-dimensional concentration 𝑛𝑠situated in the normal region at a\ndistance 𝑏from the boundary of the materials. Potential impurities are shown as yellow circles.\nYSR states and Andreev levels (see Refs. [37, 38] and\nreferences therein). We are not aware of similar studies\nof the LDoS near a magnetic impurity in the normal\nregion of superconducting heterostructures in the dirty\nregime.\nIn this paper, we study the LDoS near a solitary clas-\nsical magnetic impurity in a dirty superconducting \flm\nwith elastic mean free path, ℓ, being shorter than the su-\nperconducting coherence length, 𝜉cl≫ℓ(see Fig. 1, left\npanel). Our theoretical analysis is based on the Usadel\nequation. We investigate the energy and spatial pro\fles\nof the LDoS. In the absence of potential disorder, the\nYSR state due to a single magnetic impurity yields the\ndelta-function contribution to energy dependence of the\nLDoS. We demonstrate that a potential disorder results\nin broadening of the delta-function into a peak. Its en-\nergy width is controlled by the square root of the spread-\ning resistance of the \flm in the normal state, see Eq. (8)\nfor the precise de\fnition. We \fnd that the pro\fle of\nthe LDoS near the YSR energy is signi\fcantly a\u000bected\nby modi\fcation of multiple scattering on the magnetic\nimpurity due to intermediate scattering on surrounding\npotential disorder (treated in the Born approximation).\nSurprisingly, the corresponding term in the Usadel equa-\ntion seems to be analogous to the term which takes into\naccount the e\u000bect of the mesoscopic \ructuations of the ef-\nfective magnetic scattering amplitude in the case of \fnite\nconcentration of magnetic impurities [23]. Unexpectedly,\nwe \fnd that the potential-disorder-induced broadening\nof the YSR state at a solitary magnetic impurity seems\nto be of the order of the variance for the YSR energy\nwhich is caused by point-to-point \ructuations of the di-\nmensionless strength of the magnetic impurity due to the\npotential disorder found in Refs. [23, 36]. Additionally,\nwe study how the STM tip applied in vicinity of the mag-\nnetic impurity masks the YSR feature in the LDoS.\nAlso, we investigate the LDoS near a chain of mag-\nnetic atoms situated in the normal region of a dirty SN\njunction (see Fig. 1, right panel). We \fnd that mag-netic impurities increase the LDoS in the vicinity of the\nYSR energy. However, on the contrary to a homoge-\nneous superconductor, the energy controlling the position\nof the LDoS peak acquires an imaginary part. The latter\nmeans that magnetic impurities in the normal region of\nthe SN junction result in quasibound states rather than\nthe bound ones.\nThe outline of the paper is as follows. In Sec. II we\ncalculate the LDoS in a dirty superconducting \flm with\na solitary magnetic impurity. The LDoS in the SN junc-\ntion with a chain of magnetic atoms is analyzed in Sec.\nIII. The discussion of the obtained results as well as con-\nclusions are given in Sec. IV. Some technical details are\npresent in Appendices.\nII. A DIRTY SUPERCONDUCTING FILM\nWITH MAGNETIC IMPURITY\nIn this section, we consider a dirty superconducting\n\flm with a single classical magnetic impurity. We assume\nthat the elastic mean free path ℓis much shorter than the\nclean superconducting coherence length 𝜉cl=𝑣𝐹/\u0001. Here\n𝑣𝐹and \u0001 denote the Fermi velocity and the supercon-\nducting gap, respectively. We shall treat the problem in\nthe framework of Usadel equation [39] which is a stan-\ndard approach for description of superconductors in dirty\nlimit, ℓ≪𝜉cl.\nA. Standard Usadel equation\nIn the presence of a solitary magnetic impurity situ-\nated at the origin of the coordinate system the standard\nUsadel equation acquires the following form [40],\n𝐷\n2∇2𝜃𝜎+𝑖𝐸sin𝜃𝜎+\u0001 cos 𝜃𝜎=[𝑖𝜎√𝛼/(𝜋𝜈)] sin 𝜃𝜎\n1−𝛼+2𝑖𝜎√𝛼cos𝜃𝜎𝛿(𝑟).\n(1)3\nHere 𝐷is the di\u000busion coe\u000ecient in the normal phase,\n𝜎=±stands for the projection of an electron spin onto\nthe direction of the impurity spin, and 𝛿(𝑟) is the two-\ndimensional Dirac delta-function. The dimensionless pa-\nrameter 𝛼=(𝜋𝜈𝐽𝑆 )2is the e\u000bective strength of the mag-\nnetic impurity expressed in terms of the impurity spin, 𝑆,\nthe exchange interaction constant, 𝐽, and the density of\nstates at the Fermi level in the normal state (per one spin\nprojection), 𝜈. The spectral angle 𝜃𝜎(𝐸,𝑟) parametrizes\nthe quasiclassical Green's function (see Appendix B). In\nparticular, the spin resolved LDoS is given as\n𝜌𝜎(𝐸,𝑟) =𝜈Re cos 𝜃𝜎(𝐸,𝑟). (2)\nWe note that the right-hand side of Eq. (1) is essen-\ntially a T-matrix describing the multiple scattering on\nthe magnetic impurity. Also, we mention that the stan-\ndard Usadel equation has the following symmetry: the\nsolution 𝜃𝜎for the impurity strength 𝛼coincides with\nthe solution 𝜃−𝜎for 1/𝛼. Therefore, below, when dis-\ncussing the solution of the standard Usadel equation, we\nshall consider the case 𝛼61. The opposite case, 𝛼>1, can\nbe restored by changing 𝜎to−𝜎.\nIn Eq. (1) we approximate an exchange potential of the\nmagnetic impurity by the delta-function 𝛿(𝑟). In fact, the\npotential has some radius 𝜆. Since the Usadel equation\ndescribes physics at length scales larger than the mean\nfree path, an impurity with 𝜆.ℓcan be described by the\ndelta-function potential.\nIt is worthwhile to mention that we neglect the spin-\nindependent part of the potential of the magnetic impu-\nrity in Eq. (1). We shall discuss its e\u000bect in Sec. IV.\n1. The LDoS inside the gap, |𝐸|<Δ\nIn order to study the LDoS inside the superconducting\ngap, 𝐸<\u0001, it is convenient to parameterize the spectral\nangle as 𝜃𝜎=𝜋/2+𝑖𝜓𝜎so that\n𝜌𝜎(𝐸,𝑟) =𝜈Im sinh 𝜓𝜎. (3)\nIn terms of 𝜓𝜎, the Usadel equation (1) becomes\n𝐷\n2∇2𝜓𝜎+𝐸cosh𝜓𝜎−\u0001 sinh 𝜓𝜎=(cosh 𝜓𝜎)/(2𝜋𝜈)\n𝜎√𝛽+ sinh 𝜓𝜎𝛿(𝑟),\n𝛽=(1−𝛼)2/(4𝛼). (4)\nIn the absence of a magnetic impurity, Eq. (4) has the ho-\nmogeneous solution, 𝜓∞= arcsinh( 𝐸/√\n\u00012−𝐸2), corre-\nsponding to the density of states in the Bardeen-Cooper-\nSchrie\u000ber (BCS) theory. The magnetic impurity disturbs\nthe homogeneous solution, 𝜓𝜎=𝜓∞+𝛿𝜓𝜎, where 𝛿𝜓𝜎sat-\nis\fes the two-dimensional sinh-Gordon equation,\n𝜉2∇2𝛿𝜓𝜎−sinh𝛿𝜓𝜎=[𝜉2/(𝜋𝜈𝐷)] cosh 𝜓𝜎\n𝜎√𝛽+ sinh 𝜓𝜎𝛿(𝑟).(5)\nHere the length 𝜉≡𝜉(𝐸)=√︁\n𝐷/(2√︀\n|\u00012−𝐸2|) controls\nthe spatial extent of the perturbation of the homogeneous\nsolution.As we shall see below, the perturbation 𝛿𝜓𝜎occurs\nto be small,|𝛿𝜓𝜎|≪1. Then we can approximate the\nfunction sinh 𝛿𝜓𝜎by its argument in such a way that Eq.\n(5) reduces to the quantum mechanical problem of a two-\ndimensional particle in the presence of a delta-function\npotential (see Appendix A). Therefore, we \fnd\n𝛿𝜓𝜎(𝑟) =~𝜓𝜎−𝜓∞\nln(𝜉/ℓ)𝐾0(𝑟/𝜉), 𝑟>ℓ. (6)\nHere 𝐾0(𝑥) denotes the modi\fed Bessel function. We\nnote that the mean free path ℓappeared under the log-\narithm in Eq. (6) as a short-distance regularization for\nthe delta-function. The quantity ~𝜓𝜎satis\fes the follow-\ning nonlinear algebraic equation,\n~𝜓𝜎=𝜓∞−𝑡𝜎cosh ~𝜓𝜎√𝛽+𝜎sinh ~𝜓𝜎. (7)\nThe perturbation of the homogeneous solution by the\nimpurity is controlled by the parameter\n𝑡=2\n𝜋𝑔ln𝜉\nℓ, (8)\nwhere 𝑔=4𝜋𝜈𝐷≡ℎ/(𝑒2𝑅\u0003) is the bare dimensionless nor-\nmal state conductance of the \flm. Here 𝑅\u0003is the resis-\ntance per square in the normal phase. We emphasize that\nthe parameter 𝑡is energy-dependent, since 𝜉depends on\nenergy.\nOur approach based on the Usadel equation does not\ntake into account the localization e\u000bects. Therefore, our\nresults are limited to the range of energies such that the\n𝜉≪𝜉locwhere 𝜉loc≃ℓexp(𝜋𝑔/2) is the localization length\nin two dimensions. This condition is equivalent to the\nfollowing inequality,\n𝑡≪1. (9)\nIn view of the relation (9) one could try to solve Eq.\n(7) iteratively, substituting 𝜓∞for~𝜓𝜎in the right-hand\nside. However, there are two energies, 𝐸=±𝐸YSR, where\n𝐸YSR= \u0001√︃\n𝛽\n1 +𝛽= \u00011−𝛼\n1 +𝛼, (10)\nat which the denominator√𝛽+𝜎sinh𝜓∞in the right-\nhand side of Eq. (7) diverges. We note that ±𝐸YSRare\njust the energies of the localized YSR states in a clean su-\nperconductor. The divergence of the denominator in Eq.\n(7) indicates that near the energy −𝜎𝐸YSR, the spec-\ntral angle ~𝜓𝜎can be perturbed from the homogeneous\nsolution 𝜓∞parametrically larger than by the term ∼𝑡.\nAlso, the zero in the denominator implies the existence\nof complex solution for the spectral angle ~𝜓𝜎near the en-\nergy−𝜎𝐸YSR, as illustrated in Fig. 2. As it follows from\nEq. (3), the complex solution for ~𝜓𝜎implies the nonzero\ndensity of states in some interval of energies around the\nenergy−𝜎𝐸YSR. As shown in Fig. 2, the boundaries of4\n0.8\nFIG. 2. Graphical illustration for Eqs. (7) and (11).\nThe function−𝑡𝜎cosh ˜𝜓𝜎/(√𝛽+𝜎sinh ˜𝜓𝜎) is shown in orange\ncolor and the function ˜𝜓𝜎−𝜓∞is shown in green lines that\ncorrespond to different values of energy, 𝐸=𝐸<and𝐸=𝐸>.\nThe shaded bright-green area corresponds to straight lines\n˜𝜓𝜎−𝜓∞that do not cross the orange curve and, therefore, cor-\nrespond to complex solution for ˜𝜓𝜎. So, the non-zero density\nof states appears at energies 𝐸<<𝐸<𝐸 >. We use the follow-\ning values: 𝛼=0.3,𝑡=0.0025, and 𝜎=−1. For such parame-\nters, the boundary energies are 𝐸</Δ≃0.46 and 𝐸>/Δ≃0.60.\nthis interval can be found from the combined solution of\nEq. (7) and the following equation,\n𝑡(1−𝜎√𝛽sinh ~𝜓𝜎)\n(√𝛽+𝜎sinh ~𝜓𝜎)2= 1. (11)\nAlthough, Eq. (7) can be easily solved numerically, it\nis instructive to discuss its analytical solution using the\ncondition (9).\nExpanding sinh ~𝜓𝜎(cosh ~𝜓𝜎) in the denominator (enu-\nmerator) of the fraction in the right-hand side of Eq. (7)\nto the \frst (zeroth) order in deviation ~𝜓𝜎−𝜓∞, we \fnd\n~𝜓𝜎≃𝜓∞−𝜎√𝛽+ sinh 𝜓∞\n2 cosh 𝜓∞+𝑖√︃\n𝑡−(𝜎√𝛽+ sinh 𝜓∞)2\n4 cosh2𝜓∞.\n(12)\nWe note that the choice of the sign in front of the\nsquare root corresponds to Im ~𝜓𝜎>0, that guarantees\nnon-negativity of the density of states. Using the ex-\nplicit solution (12), one can check that the assumption\n|~𝜓𝜎−𝜓∞|≪1 is justi\fed in virtue of the inequality (9).\nNow we can compute the LDoS, see Eq. (3). The\nresult reads,\n𝜌𝜎(𝐸,𝑟) =𝛿𝜌0(−𝜎𝐸)\nln(𝜉𝛽/ℓ)𝐾0(︀\n𝑟/𝜉𝛽)︀\n, 𝑟>ℓ,\n𝛿𝜌0(𝐸)≃𝜈(1 +𝛽)3/2\n2\u0001Re[︁\n\u00002−(𝐸YSR−𝐸)2]︁1/2\n,(13)\n\u0000 =2√𝑡𝛽\n1 +𝛽\u0001, 𝑡 𝛽=2\n𝜋𝑔ln𝜉𝛽\nℓ,\nwhere 𝜉𝛽=𝜉(0)(1 + 𝛽)1/4. Therefore, the energy depen-\ndence of the density of states has the semicircle shape\nwith the width 2\u0000 around the YSR energy 𝐸YSR. Wemention that the square-root energy dependence of the\ndensity of states corresponds to the treatment of the\nnon-magnetic random potential within the self-consistent\nBorn approximation (see Refs. [36, 41] for details). Un-\nexpectedly, \u0000 /2 coincides with the variance of the YSR\nenergy due to the presence of potential disorder calcu-\nlated in Ref. [36]. This indicates that the nonzero LDoS\naround 𝐸YSR caused by di\u000busive motion of quasiparti-\ncles around the magnetic impurity can be thought, phys-\nically, as a result of \ructuations of the YSR energy (more\nprecisely, of 𝛼, see Ref. [23]) due to dependence on a re-\nalization of potential disorder.\nA few remarks are in order here. At \frst, we note that\nfor𝛼→0 the condition 𝑡𝛽≪1 implies that 𝛼≫(𝜉(0)/𝜉loc)4,\ni.e., the result (13) is not applicable for extremely weak\nimpurity strengths. Secondly, we mention that the per-\nturbation of the LDoS around the YSR energy contains\nexactly one fermion state,\nΓ∫︁\n−Γ𝑑𝐸∫︁\n𝑑2𝑟𝜌𝜎(𝐸,𝑟) = 1 /2. (14)\nAt third, there is the critical impurity strength, 𝛼𝑐, such\nthat the density of states at the Fermi energy becomes\nnonzero for 𝛼>𝛼 𝑐. Using Eq. (13), one \fnds the critical\nstrength as\n𝛼𝑐= 1−4√𝑡0. (15)\nAt fourth, we note that the LDoS per spin (13) is asym-\nmetric with respect to the chemical potential.\nIn Fig. 4 (left panel) we plot the LDoS obtained from\nthe numerical solution of Eq. (5) and compare it with\nthe analytic solution (13). As one can notice, there is\nhardly any di\u000berence between the numerical and ana-\nlytical solutions. In accordance with the analytical re-\nsult (13), the nonzero LDoS region around 𝐸YSRbroad-\nens with an increase in 𝛼. We note an interesting non-\nmonotonous behavior of the total LDoS with energy. For\n𝛼𝑐<𝛼< (1+3𝛼𝑐)/4, the total LDoS has three local max-\nima: at 𝐸=±𝐸YSRand at 𝐸=0. At 𝛼>(1+3𝛼𝑐)/4 only\na single maximum at the Fermi energy, 𝐸=0, remains.\n2. The LDoS outside the gap, |𝐸|>Δ\nFor description of the e\u000bect of the magnetic impurity\non the LDoS outside the superconducting gap, 𝐸>\u0001, it is\nconvenient to parameterize the spectral angle as 𝜃𝜎=𝑖𝜒𝜎.\nThen the LDoS becomes\n𝜌𝜎(𝐸,𝑟) =𝜈Re cosh 𝜒𝜎. (16)\nIn terms of 𝜒𝜎, the Usadel equation (1) reads,\n𝐷\n2𝑖∇2𝜒𝜎+𝐸sinh𝜒𝜎−\u0001 cosh 𝜒𝜎=[1/(𝜋𝜈)] sinh 𝜒𝜎\n𝜎√𝛽+𝑖cosh𝜒𝜎𝛿(𝑟).\n(17)5\nWithout the right-hand side, Eq. (4) has the homoge-\nneous solution, 𝜒∞= sgn 𝐸arcsinh(\u0001 /√\n𝐸2−\u00012) that re-\nproduces the BCS density of states,\n𝜌0(𝐸) =𝜈|𝐸|√\n𝐸2−\u00012,|𝐸|>\u0001. (18)\nThe magnetic impurity perturbs the homogeneous solu-\ntion: 𝜒𝜎=𝜒∞+𝛿𝜒𝜎, where 𝛿𝜒𝜎solves the following equa-\ntion,\n𝑖𝜉2∇2𝛿𝜒𝜎−sgn𝐸sinh𝛿𝜒𝜎=−(4𝜎𝜉2/𝑔) sinh 𝜒𝜎√𝛽+𝑖𝜎cosh𝜒𝜎𝛿(𝑟).\n(19)\nAs we shall see below, the correction 𝛿𝜒𝜎occurs to be\nsmall,|𝛿𝜒𝜎|≪1. Then Eq. (19) can be easily solved,\n𝛿𝜒𝜎(𝑟) =~𝜒𝜎−𝜒∞\nln(𝜉/ℓ)𝐾0(︁\n𝑒−𝑖𝜋sgn𝐸/4𝑟/𝜉)︁\n, 𝑟>ℓ.(20)\nThe quantity ~ 𝜒𝜎satis\fes the following nonlinear alge-\nbraic equation,\n~𝜒𝜎=𝜒∞−𝑖𝑡𝜎sinh ~𝜒𝜎√𝛽+𝑖𝜎cosh ~𝜒𝜎. (21)\nIt is worthwhile to mention the di\u000berence between the\nequations (7) and (21). In the latter case, we seek the\nsolution ~ 𝜒𝜎with nonzero real part. Due to the imaginary\nunity in the denominator of the fraction in the right-hand\nside of Eq. (21), the denominator does not vanish for any\nreal ~𝜒𝜎. Considering the inequality (9), we can solve Eq.\n(21) iteratively. Substituting 𝜒∞for ~𝜒𝜎in its right-hand\nside, we obtain\n~𝜒𝜎≃𝜒∞−𝑖𝑡𝜎sinh𝜒∞√𝛽+𝑖𝜎cosh𝜒∞. (22)\nAs one can check, the condition (9) guarantees the in-\nequality|~𝜒𝜎−𝜒∞|≪1. Now, using Eq. (16), we \fnd the\nLDoS at 𝐸>\u0001 and 𝑟>ℓ,\n𝜌𝜎(𝐸,𝑟) =𝜌0(𝐸) + Re𝛿𝜌0(𝐸)\nln(𝜉/ℓ)(︃\n1 +𝑖√︀\n𝛽√\n𝐸2−\u00012\n|𝐸|)︃\n×𝐾0(︁\n𝑒−𝑖𝜋𝜎sgn𝐸/4𝑟/𝜉)︁\n, (23)\nwhere\n𝛿𝜌0(𝐸)≃−𝜈𝑡\n1 +𝛽\u00012|𝐸|\n(𝐸2−𝐸2\nYSR)√\n𝐸2−\u00012. (24)\nWe note that 𝛿𝜌0(𝐸) is the change of the LDoS at\nthe position of the magnetic impurity. The condition\n𝑡≪1 implies that the result (23) is not applicable for\n|𝐸|−\u0001≪(𝜉(0)/𝜉loc)4\u0001. As we shall see below in Sec.\nII A 3, the region in which the result (23) is not applica-\nble turns to be wider.Using Eq. (23), we \fnd\n∫︁\n𝑑2𝑟𝛿𝜌𝜎(𝐸,𝑟) =𝜎𝜈√𝛽\n2𝜋(1 +𝛽)sgn𝐸√\n𝐸2−\u00012\u00012\n𝐸2−𝐸2\nYSR.\n(25)\nWe note the appearance of sgn 𝐸in Eq. (25). There-\nfore, integrating the above expression over |𝐸|>\u0001, we\n\fnd that there is no change in the number of states for a\ngiven spin projection 𝜎at|𝐸|>\u0001.\n3. Suppression of the order parameter near the magnetic\nimpurity\nTo study the suppression of the order parameter near\nthe magnetic impurity, it is convenient to rewrite the\nUsadel equation (1) for the imaginary (Matsubara) 𝜀=\n𝜋𝑇(2𝑛+ 1), rather than real, energies,\n𝐷\n2∇2𝜃𝜎−|𝜀|sin𝜃𝜎+ \u0001 cos 𝜃𝜎=[𝑖/(2𝜋𝜈)] sin 𝜃𝜎\n𝜎√𝛽+𝑖cos𝜃𝜎𝛿(𝑟).\n(26)\nThe superconducting order parameter satis\fes the self-\nconsistent equation,\n\u0001(𝑟) =𝜋𝑇|𝛾𝑐0|∑︁\n𝜎=±∑︁\n𝜀>0sin𝜃𝜎(𝑟), (27)\nwhere 𝛾𝑐0<0 is the bare dimensionless attraction inter-\naction in the Cooper channel.\nIn the absence of the magnetic impurity, Eqs. (26)\nand (27) reduce to the standard self-consistent equation\nof the BCS theory for the homogeneous superconducting\norder parameter \u0001 0,\n\u00010= 2𝜋𝑇|𝛾𝑐0|∑︁\n𝜀>0sin𝜃∞,sin𝜃∞=\u00010√︀\n𝜀2+ \u00012\n0.(28)\nIt is convenient to introduce 𝛿𝜃𝜎=𝜃𝜎−𝜃∞and\n𝛿\u0001=\u0001−\u00010. They describe the deviations of 𝜃𝜎\nand \u0001 from the homogeneous solutions. As we shall see\nbelow, these deviations are small. Therefore, we can\nlinearize Eqs. (26) and (27) as follows:\n𝜉2\n𝜀∇2𝛿𝜃𝜎−𝛿𝜃𝜎+|𝜀|𝛿\u0001\n𝜀2+ \u00012\n0=(4𝑖𝜎𝜉2\n𝜀/𝑔) sin𝜃∞√𝛽+𝑖𝜎cos𝜃∞𝛿(𝑟),\n𝛿\u0001(𝑟) =𝜋𝑇|𝛾𝑐0|∑︁\n𝜀>0𝜀𝛿𝜃𝜎(𝑟)√︀\n𝜀2+ \u00012\n0, (29)\nwhere 𝜉2\n𝜀=𝐷/(2√︀\n𝜀2+\u00012\n0). We note that the denomina-\ntor in the right-hand side of the linearized Usadel equa-\ntion does not turn into zero. Making the Fourier trans-\nform from the spatial coordinate 𝑟to the momentum 𝑞,\n𝛿\u0001𝑞=∫︁\n𝑑2𝑟𝛿\u0001(𝑟)𝑒−𝑖𝑞𝑟, 𝛿𝜃𝜎,𝑞=∫︁\n𝑑2𝑟𝛿𝜃𝜎(𝑟)𝑒−𝑖𝑞𝑟,\n(30)6\nwe \fnd\n𝛿\u0001𝑞\n\u00010=−4𝜉(0)2\n𝑔ℱ𝛽(𝑞𝜉(0), 𝑇/\u00010)\nℱ(𝑞𝜉(0), 𝑇/\u00010). (31)\nHere the functions ℱ𝛽andℱare de\fned as follows:\nℱ𝛽(𝑞𝜉(0), 𝑇/\u00010) =𝑇∑︁\n𝜀>01\n1 +𝑞2𝜉2𝜀𝜀2\u00010/(𝜀2+ \u00012\n0)\n((1 + 𝛽)𝜀2+𝛽\u00012\n0),\nℱ(𝑞𝜉(0), 𝑇/\u00010) =𝑇∑︁\n𝜀>0𝑞2𝜉2\n𝜀+ \u00012\n0/(𝜀2+ \u00012\n0)\n(1 +𝑞2𝜉2𝜀)√︀\n𝜀2+ \u00012\n0.(32)\nWe note that this result coincide with the expression de-\nrived previously (see Eq. (B11) in Ref. [22]).\nTo estimate the e\u000bect of the magnetic impurity on the\nsuperconductor order parameter, we consider the case of\nzero temperature. Then, at 𝑇=0 the summation over\n𝜀in expressions for the functions ℱ𝛽(𝑞𝜉(0), 𝑇/\u00010) and\nℱ(𝑞𝜉(0), 𝑇/\u00010) can be performed exactly, and we obtain\n𝛿\u0001𝑞\n\u00010=−4𝜉(0)2\n𝑔𝐹𝛽(𝑞2𝜉(0)2)\n𝐹(𝑞2𝜉(0)2), (33)\nwhere\n𝐹(𝑧) =1\n4𝑧−√︀\n|1−𝑧2|\n2𝜋𝑧{︃\narccos 𝑧, 𝑧 61,\n(−) arccosh 𝑧, 𝑧 > 1,(34)\nand\n𝐹𝛽(𝑧) =1\n(1 +𝛽)𝑧2−1{︃\n𝑧𝐹(𝑧)−√𝛽+ 1−√𝛽\n4√𝛽+ 1\n−𝑧√𝛽\n2𝜋arctan(︁\n1/√︀\n𝛽)︁}︃\n. (35)\nThe functions 𝐹(𝑧) and 𝐹𝛽(𝑧) have the following asymp-\ntotic behavior at 𝑧≫1:\n𝐹(𝑧)≃ln(2𝑧)\n2𝜋, 𝐹 𝛽(𝑧)≃𝐹(𝑧)\n𝑧(1 +𝛽). (36)\nHence, we \fnd with logarithmic accuracy the modi\fca-\ntion of the superconducting order parameter at the loca-\ntion of the magnetic impurity,\n𝛿\u0001(0)\n\u00010≃−𝑡0\n1 +𝛽. (37)\nFor|𝐸|<\u00010the change in the superconducting order\nparameter produces the purely real correction to the 𝛿𝜓𝜎,\n𝛿𝜓(Δ)\n𝜎,𝑞=4𝜉(0)2\n𝑔\u00010𝐸/√︀\n\u00012\n0−𝐸2\n√︀\n\u00012\n0−𝐸2+𝑞2𝜉(0)2\u00010𝐹𝛽(𝑞2𝜉(0)2)\n𝐹(𝑞2𝜉(0)2).\n(38)\nTherefore, suppression of the superconducting order pa-\nrameter does not a\u000bect the average LDoS at |𝐸|<\u00010.For𝐸>\u00010we compute the Fourier transform of the\ncorrection 𝛿𝜒(Δ)\n𝜎(𝑟) as\n𝛿𝜒(Δ)\n𝜎,𝑞=−4\n𝑔𝜉(0)2\nsgn𝐸+𝑖𝑞2𝜉2\u00010|𝐸|\n𝐸2−\u00012\n0𝐹𝛽(𝑞2𝜉(0)2)\n𝐹(𝑞2𝜉(0)2).(39)\nNext, we estimate the correction 𝛿𝜒(Δ)\n𝜎at the spatial\npoint where the magnetic impurity is situated,\nRe𝛿𝜒(Δ)\n𝜎(0) =−1\n𝜋𝑔∞∫︁\n0𝑑𝑧\u00010𝐸\n𝐸2−\u00012\n0+𝑧2\u00012\n0𝐹𝛽(𝑧)\n𝐹(𝑧).(40)\nWe note that the integral over 𝑧=𝑞2𝜉(0)2is convergent\nin the ultraviolet. Performing the integration over 𝑧, we\n\fnd\nRe𝛿𝜒(Δ)\n𝜎(0)≃−\u00010sgn𝐸\n𝜋𝑔(1 +𝛽)√︀\n𝐸2−\u00012\n0\n×{︃\n𝜋2(1 +𝛽)𝐹𝛽(0),|𝐸|−\u00010≪\u00010,\nln(|𝐸|/\u00010),|𝐸|≫\u00010.(41)\nIn derivation of the above result we used expansion in\n𝛿𝜒(Δ)\n𝜎. Therefore, Eq. (41) is valid for |Re𝛿𝜒(Δ)\n𝜎(0)|≪1,\ni.e., for energies not too close to the unrenormalized gap,\n(|𝐸|−\u00010)/\u00010≫𝐹2\n𝛽(0)/𝑔2.\nUsing Eqs. (16) and (41), we obtain the following cor-\nrection to the LDoS due to renormalization of the super-\nconducting order parameter,\n𝛿𝜌(Δ)\n0(0)≃−𝜈\u00012\n0\n𝜋𝑔(1 +𝛽)(𝐸2−\u00012\n0)\n×{︃\n𝜋2(1 +𝛽)𝐹𝛽(0),|𝐸|−\u00010≪\u00010,\nln(|𝐸|/\u00010),|𝐸|≫\u00010.(42)\nComparing Eqs. (42) and (24), one can check\nthat for|𝐸|≫\u0001 the suppression of the superconduct-\ning order parameter results in the substitution of 𝑡\nin Eq. (24) by 𝑡0. Next, near the band edge,\n|𝐸|−\u00010≪\u00010, one can neglect the renormalization of \u0001\nfor (|𝐸|−\u00010)/\u00010≫𝜋4𝐹2\n𝛽(0)\nln2(𝜉(0)/ℓ)only. In the opposite case,\n𝜋4𝐹2\n𝛽(0)\nln2(𝜉(0)/ℓ)≫(|𝐸|−\u00010)/\u00010≫(𝐹𝛽(0)/𝑔)2, the correction to\nthe LDoS is dominated by the renormalization of the su-\nperconducting order parameter. Since in this work we\nare interested in the behavior of the density of states at\nenergies|𝐸|<\u00010, we shall not study that regime in de-\ntails.\nB. Renormalized Usadel equation and the LDoS at\n|𝐸|<Δ\nThe solution of the standard Usadel equation (1) re-\nsults in the broadening of the YSR state due to potential\ndisorder. However, Eq. (1) produces the LDoS with7\nFIG. 3. Sketch of quasiparticle scattering on potential disor-\nder between rescattering on the magnetic impurity. Potential\nimpurities are shown as yellow circles, a solitary magnetic im-\npurity with spin 𝑆is shown as a purple circle. The schematic\ntrajectory of a quasiparticle is depicted as a dotted green line.\nsharp edges, cf. Eq. (13). As we discussed above, physi-\ncally, the broadening of the YSR state can be understood\nas the result of \ructuations of the impurity strength 𝛼.\nTherefore, one expects smooth energy dependence of the\nLDoS around the YSR energy. This indicates that the\nstandard Usadel equation (1) is not suited for calculation\nof the LDoS at|𝐸|−𝐸YSR&\u0000.\nOne way to improve the standard Usadel equation is\nto consider non-symmetric in replica space solutions for\nthe spectral angle, as it was done in Ref. [22]. Here we\nemploy an alternative idea introduced in Ref. [23] for the\ncase of dilute concentration of magnetic impurities, 𝑛(2)\n𝑠,\ndistributed in the \flm according to the Poisson distribu-\ntion. The standard Usadel equation (1) can be derived as\nthe saddle-point of the nonlinear sigma model (NLSM)\n[18]. However, as it was shown in Ref. [23], since we\nare interested in physics at the length scale 𝜉(alterna-\ntively, at the energy scale 𝐸), we need to renormalize\nthe NLSM action from the mean free path up to 𝜉(or\nfrom elastic scattering rate 1 /𝜏down to 𝐸). Upon this\nrenormalization, the term describing scattering by mag-\nnetic impurities is strongly renormalized. In the case of\na single magnetic impurity, there exists similar renormal-\nization of the NLSM such that the renormalized Usadelequation acquires the following form (see Appendix B),\n𝐷\n2∇2𝜃𝜎+𝑖𝐸sin𝜃𝜎+ \u0001 cos 𝜃𝜎\n=⟨[𝑖𝜎√a/(𝜋𝜈)] sin 𝜃𝜎\n1−a+2𝑖𝜎√a cos𝜃𝜎⟩\na𝛿(𝑟). (43)\nHere⟨. . .⟩ais the average with respect to the following\nlog-normal distribution,\n𝒫𝛼(a, 𝑡) =1\n4a√\n𝜋𝑡exp[︃\n−1\n4𝑡(︂1\n2lna\n𝛼+𝑡)︂2]︃\n(44)\nSince𝒫𝛼(a, 𝑡→0)→𝛿(a−𝛼), Eq. (43) transforms into Eq.\n(1) as 𝑡→0.\nWe note that the right hand side of Eq. (43) can be\nthought as the T-matrix renormalized by scattering of a\nquasiparticle on potential disorder between rescattering\non the magnetic impurity, this is illustrated schematically\nin Fig. 3. Surprisingly, the result (43) can be obtained\nfrom the renormalized Usadel equation of Ref. [23] upon\nsubstitution of 𝑛(2)\n𝑠by𝛿(𝑟).\nIn order to \fnd the LDoS at |𝐸|<\u0001, we follow the\nsame approach as in Sec. II A 1. Parametrizing the spec-\ntral angle as 𝜃𝜎=𝜋/2+𝑖𝜓𝜎with 𝜓𝜎=𝜓∞+𝛿𝜓𝜎, we \fnd\nthat 𝛿𝜓𝜎(𝑟) is given by Eq. (6), where ~𝜓𝜎satis\fes the\nfollowing nonlinear equation (cf. Eq. (7)),\n~𝜓𝜎=𝜓∞−⟨\n2𝑡𝜎√a cosh ~𝜓𝜎\n1−a + 2 𝜎√a sinh ~𝜓𝜎⟩\na. (45)\nUsing the condition (9), we rewrite Eq. (45) as\n~𝜓𝜎=𝜓∞−√\n𝑡𝜎cosh ~𝜓𝜎√1 +𝛽𝑡ℋ(︃√𝛽𝑡+𝜎sinh ~𝜓𝜎\n2√\n𝑡√1 +𝛽𝑡)︃\n,(46)\nwhereℋ(𝑧)=√𝜋𝑒−𝑧2[er\f(𝑧)−𝑖sgn(Im 𝑧)]/2 and\n𝛽𝑡=(1−𝛼𝑡)2/(4𝛼𝑡) with 𝛼𝑡=𝛼𝑒−2𝑡. Although alge-\nbraic Eq. (46) can be solved numerically, it is instructive\nto discuss its analytic solution in limiting cases.\nWe start from the case of the vicinity of the YSR en-\nergy,⃒⃒|𝐸|−𝐸YSR⃒⃒≪\u0000. In this regime, the argument of\nthe functionℋin Eq. (46) is small. Performing series\nexpansion ofℋin its argument and using |~𝜓𝜎−𝜓∞|≪1,\nwe \fnd a simple but lengthy result,\n~𝜓𝜎≃𝜓∞+ 2𝑖𝑧0√\n𝑡[︃\n1−(︂8(1−ℎ1)√𝑡𝛽𝑡\n(2 +ℎ1)2√1 +𝛽𝑡−𝑖𝜎ℎ1/𝑧0\n2 +ℎ1)︂\n×√𝛽𝑡+𝜎sinh𝜓∞\n2√\n𝑡√1 +𝛽𝑡+2𝑖ℎ2/𝑧0\n(2 +ℎ1)3(︂√𝛽𝑡+𝜎sinh𝜓∞\n2√\n𝑡√1 +𝛽𝑡)︂2]︃\n.\n(47)\nHere ℎ𝑘≡ℋ(𝑘)(𝑖𝑧0) denotes the 𝑘-th derivative ofℋ(𝑧) at\nthe point 𝑧0≈0.32. The latter is the positive solution of\nthe equation 𝑧0=𝑖ℋ(𝑖𝑧0)/2. We note that ℎ1≃0.59 and8\nFIG. 4. Dependence of the LDoS on energy at the position of the magnetic impurity obtained from standard (left panel) and\nrenormalized (right panel) Usadel equations for different values of 𝛼. The total LDoS is shown in green, whereas 𝜌𝜎(𝐸,0) for\n𝜎=−1(+1) is shown in orange (blue) color. The analytical (numerical) result is depicted as a solid (dotted) line. We choose\n𝑡=0.0025 and neglect its weak energy dependence.\nℎ2≃0.90𝑖. Hence, we obtain the average LDoS in the form\nof Eq. (13) but with 𝛿𝜌0(𝐸) given as (⃒⃒|𝐸|−~𝐸YSR⃒⃒≪~\u0000)1\n𝛿𝜌0(𝐸)≃[︃\n1 +𝑡𝛽1−𝛼\n1 +𝛼(︂\n1 + 4 𝑐1𝑐2\n21−𝛼\n1 +𝛼)︂]︃\n×𝑧0𝜈(1 +𝛽)3/2\u0000\n\u0001[︃\n1−(𝐸−~𝐸YSR)2\n~\u00002]︃\n.(48)\nWe note that the typical broadening of the YSR state is\nenhanced,\n~\u0000 = \u0000 /√𝑐1, (49)\nwhere 𝑐1=2|ℎ2|/[𝑧0(2+ℎ1)3]≃0.32. Also, there is a non-\nzero shift of the energy at which the LDoS has maximum,\n~𝐸YSR≃\u0001[︂1−𝛼\n1 +𝛼+4𝛼𝑡𝛽\n(1 +𝛼)2(︂\n1 + 4 𝑐21−𝛼\n1 +𝛼)︂]︂\n,(50)\nwhere 𝑐2=𝑧0(4−ℎ1)(2+ℎ1)/(2|ℎ2|)≃1.57. We emphasize\nthat Eq. (48) predicts a dramatic reduction (by a factor\nof 2𝑧0≈0.64) of the maximal magnitude of the LDoS in\ncomparison with the result (13). We reiterate that the\nbroadening ~\u0000 (as well as \u0000) is of the order of variance of\nthe YSR energy (Eq. (10) with 𝛼substituted by a) due\nto log-normal distribution (44) of the impurity strength.\nNow we turn our attention to the study of energy tails\nin the LDoS. We consider the energy interval in which\n1In order to derive this result, we expanded sinh 𝜓𝜎in Eq. (3) to\nthe second order in difference 𝛿𝜓𝜎=˜𝜓𝜎−𝜓∞. We note that still\nit is enough to use the Usadel equation Eq. (5) with a linearized\nleft-hand side.there are no states within the standard Usadel equa-\ntion (5). In this regime, the argument of the function\nℋin Eq. (46) is so large that we can use its asymptotic\nexpression,ℋ(𝑧)≃1/(2𝑧)−𝑖[sgn(Im 𝑧)]√𝜋exp(︀\n−𝑧2)︀\n/2 at\n|𝑧|≫1. Then, assuming that Im ~𝜓𝜎≪1, we \fnd,\nIm~𝜓𝜎=√\n𝜋𝑡cosh ~𝜓′\n𝜎\n2√1 +𝛽𝑡(︃\n1 +𝑡1−𝜎√𝛽𝑡sinh ~𝜓′\n𝜎\n(√𝛽𝑡+𝜎sinh ~𝜓′𝜎)2)︃\n×exp[︃\n−(√𝛽𝑡+𝜎sinh ~𝜓′\n𝜎)2\n4𝑡(1 +𝛽𝑡)]︃\n. (51)\nHere the quantity ~𝜓′\n𝜎is the real part of ~𝜓𝜎,~𝜓′\n𝜎≡Re~𝜓𝜎,\nand satis\fes Eq. (7). Hence, we obtain the LDoS in the\nparametric form,\n𝛿𝜌0(𝐸) =𝜈√\n𝜋𝑡cosh2𝑥\n2√1 +𝛽𝑡(︂\n1 +𝑡1 +√𝛽𝑡sinh𝑥\n(√𝛽𝑡−sinh𝑥)2)︂\n(52)\n×exp[︂\n−(√𝛽𝑡−sinh𝑥)2\n4𝑡(1 +𝛽𝑡)]︂\n,\n𝐸\n\u0001= tanh[︁\n𝑥−𝑡cosh𝑥√𝛽𝑡−sinh𝑥]︁\n.\nThis expression holds for a real variable 𝑥that satis-\n\fes|√𝛽𝑡−sinh𝑥|≫2√︀\n𝑡(1+𝛽𝑡). This implies that the\nLDoS is exponentially small away from 𝐸YSR. In\nparticular, there is \fnite, albeit exponentially small,\n∼exp[−𝛽𝑡/(4𝑡(1 +𝛽𝑡))], LDoS at the Fermi energy, 𝐸=0,\nfor𝛼&𝛼𝑐.\nIn Fig. 4 (right panel) we plot the LDoS obtained from\nthe numerical solution of Eq. (43) and compare it against\nthe analytic asymptotes (48) and (52). As can be seen,\nanalytics and numerics are in full agreement. The renor-\nmalized Usadel equation results not only in suppression\nof the magnitude of the LDoS near the YSR energy but9\nFIG. 5. Dependence of the LDoS on energy and distance\nto the magnetic impurity ( 𝑟>𝑙) obtained numerically from\nrenormalized Usadel equations for different values of 𝛼. We\nchoose 𝑡=0.0025 and 𝑙/𝜉(0)=0 .1.\nmakes the LDoS to be asymmetric. We note that this\nasymmetry disappears in the total LDoS after merging\npeaks around±𝐸YSR. Also, our numerical analysis re-\nveals smaller value of 𝛼𝑐in comparison with Eq. (15),\nalthough we \fnd 1 −𝛼𝑐∼√𝑡0still.\nIn Fig. 5 we plot the dependence of the total LDoS\non energy and distance to the magnetic impurity for dif-\nferent values of 𝛼. Here, the LDoS is obtained by the\nnumerical solution of Eq. (43). For convenience, we nor-\nmalize the LDoS by its maximal magnitude for each 𝛼.\nAs can be seen, the LDoS decays with distance on the\nscale 𝜉𝛽, in full agreement with Eq. (6).\nC. The effect of a tip\nThe LDoS in the superconductor can be a\u000bected by\nan STM tip. In this section, we study this e\u000bect. We\nassume that the tip (either superconducting or metallic)\nis placed near the magnetic impurity. Possibility of tun-\nneling from/to the superconducting \flm to/from the tip\nresults in modi\fcation of the Usadel equation [42, 43],\n𝐷\n2∇2𝜃𝜎+𝑖𝐸sin𝜃𝜎+ \u0001 cos 𝜃𝜎=[𝑖𝜎√𝛼/(𝜋𝜈)] sin 𝜃𝜎\n1−𝛼+2𝑖𝜎√𝛼cos𝜃𝜎𝛿(𝑟)\n−1\n2𝜋𝜈𝑁∑︁\n𝑘=1√𝒯𝑘sin(𝜃tip−𝜃𝜎)\n1 +𝒯𝑘+ 2√𝒯𝑘cos(𝜃tip−𝜃𝜎)𝛿(𝑟).(53)\nHere we assume 𝑁tunneling channels with the tunneling\nprobability 𝑇𝑘each. The quantity 𝒯𝑘=𝑇2\n𝑘/(2−𝑇𝑘)2is the\nAndreev conductance of the 𝑘-th channel. For a sake\nof simplicity, we study the e\u000bect of the tip within the\nstandard Usadel equation. As above, we are interested\nin the modi\fcation of the LDoS near the YSR energy\nalone.Following the same steps as in Sec. II A, we \fnd the\nfollowing equation for the spectral angle at the position\nof the magnetic impurity, cf. Eq. (7),\n~𝜓𝜎=𝜓∞−𝑡𝜎cosh ~𝜓𝜎√𝛽+𝜎sinh ~𝜓𝜎+𝐶(~𝜓𝜎),\n𝐶(~𝜓𝜎) =𝑁∑︁\n𝑘=1𝑖𝑡√𝒯𝑘cos(︁\n𝜃tip−𝑖~𝜓𝜎)︁\n1 +𝒯𝑘+ 2√𝒯𝑘sin(︁\n𝜃tip−𝑖~𝜓𝜎)︁.(54)\nThe term 𝐶(~𝜓𝜎) has no singularity in its denominator at\nthe YSR energy. So, the di\u000berence between ~𝜓𝜎and𝜓∞\ncan be neglected there. Solving Eq. (54) in the same way\nas Eq. (7), we \fnd\n~𝜓𝜎=𝜓∞+1\n2𝐶(𝜓∞)−𝜎√𝛽+ sinh 𝜓∞\n2 cosh 𝜓∞\n+𝑖√︃\n𝑡−1\n4[︂𝜎√𝛽+ sinh 𝜓∞\ncosh𝜓∞+𝐶(𝜓∞)]︂2\n.(55)\nThe above result allows us to compute the LDoS near\nthe energy 𝐸YSR for arbitrary magnitude of 𝜃tip. For\nconcreteness, we consider the cases of normal metal and\nsuperconducting tips only.\n1. Normal-metal tip\nFor a normal-metal tip, the spectral angle is zero,\n𝜃tip=0. Then, using Eq. (55), we \fnd that the LDoS\nis given by the expressions (13) but with\n𝛿𝜌0(𝐸) =𝜈(1 +𝛽)3/2\n2\u0001[︃\nIm~𝐸YSR\n+ Re(︁\n\u00002−(𝐸−~𝐸YSR)2)︁1/2]︃\n. (56)\nHere we introduced the complex YSR energy,\n~𝐸YSR=𝐸YSR(︃\n1−𝑁∑︁\n𝑘=1𝑖𝑡√︀\n𝒯𝑘/𝛽\n1 +𝒯𝑘+ 2𝑖√𝛽𝒯𝑘)︃\n. (57)\nThe normal-metal tip results not only in a shift ∼𝑡of\nthe YSR energy, but also the appearance of an imaginary\npart∼𝑡. The latter signals that the YSR state becomes\na quasibound one, since it can decay into the normal tip.\nThe existence of the imaginary part in ~𝐸YSRsmears the\nsharp edges of the LDoS.\n2. Superconducting tip\nIn the case of a superconducting tip with large super-\nconducting order parameter, we can neglect the energy\ndependence of the spectral angle and use the following10\napproximation: 𝜃tip=𝜋/2. Then, using Eq. (55), we ob-\ntain the LDoS given by Eq. (56) with\n~𝐸YSR=𝐸YSR(︃\n1 +𝑁∑︁\n𝑘=1𝑡√︀\n𝒯𝑘/(1 +𝛽)\n1 +𝒯𝑘+ 2√︀\n(1 +𝛽)𝒯𝑘)︃\n.(58)\nWe mention that the superconducting tip results in the\nshift of the YSR energy. The imaginary part of ~𝐸YSRis\nzero due to the absence of quasiparticle tunneling into\nthe superconducting tip. Therefore, sharp edges of the\nLDoS near the YSR energy remain.\nIII. YSR RESONANCE IN A DIRTY SN\nJUNCTION\nNow we discuss how magnetic impurities situated in\na dirty normal metal near a superconductor's boundary\na\u000bect the LDoS. We consider a dirty two-dimensional SN\njunction with a rare chain of magnetic atoms with one-\ndimensional concentration 𝑛𝑠. For a sake of simplicity,\nwe assume that both the SN boundary situated at 𝑥=0\nand the chain of magnetic atoms situated at 𝑥=−𝑏are\nstraight and parallel to each other (see Fig. 1b). Also, we\nsuppose that the spins of magnetic atoms are classical,\nstatistically independent vectors of the length 𝑆with the\n\rat distribution over their orientations.\nThe e\u000bect of impurities will be estimated based on the\nchange in the LDoS in comparison with the one without\nmagnetic atoms. We expect that in the presence of a\nnormal metal, the localized state at the YSR energy is\nsmeared out, forming a peak with a \fnite width. Thus,\nour goal is to determine the conditions under which the\npresence of impurities a\u000bects the LDoS near the YSR\nenergy most pronounced.\nA. Standard Usadel equation\nTo describe the LDoS in a dirty SN junction with a\nchain of magnetic impurities, we employ the standard\nUsadel equation. Contrary to Eq. (1), the spectral angle\nis now independent of the spin projection. Due to the\nheterogeneity of our model, the Usadel equation should\nbe written separately in the regions of the superconductor\n(𝑥>0) and the normal metal ( 𝑥<0). Under the assump-\ntion of an in\fnite system size in the 𝑦direction (see Fig.\n1b), the spectral angle depends solely on the 𝑥coordi-\nnate. Then the standard Usadel equation becomes\n𝐷\n2𝜕2\n𝑥𝜃s+𝑖𝐸sin𝜃s+ \u0001 cos 𝜃s= 0 (59)\nfor𝑥>0 (the superconductor), and\n𝐷n\n2𝜕2\n𝑥𝜃n+𝑖𝐸sin𝜃n=[𝑛𝑠𝛼/(𝜋𝜈n)] sin 2 𝜃n\n1 +𝛼2+ 2𝛼cos 2𝜃n𝛿(𝑥+𝑏) (60)\nfor𝑥<0 (the normal metal). Here 𝐷ndenotes the di\u000bu-\nsion coe\u000ecient of the normal metal, 𝜈nis the density ofstates per one spin projection in the normal metal, and\n𝜃s(𝐸, 𝑥) (𝜃n(𝐸, 𝑥)) stands for the spectral angle in the\nsuperconductor (the normal metal).\nThe Usadel Eqs. (59){(60) needs to be supplemented\nwith boundary conditions. We assume that away from\nthe SN boundary the superconductor and the normal\nmetal behave as in\fnite bulk materials, i.e.,\n𝜃s(𝐸, 𝑥→+∞) =𝜋\n2+𝑖𝜓∞, 𝜃 n(𝐸, 𝑥→−∞ ) = 0 .\n(61)\nAt the SN boundary we employ the following boundary\nconditions [44, 45],\n𝜃s(𝐸,0) = 𝜃n(𝐸,0),\n𝑔𝜕𝑥𝜃s(𝐸, 𝑥)⃒⃒⃒\n𝑥=0=𝑔n𝜕𝑥𝜃n(𝐸, 𝑥)⃒⃒⃒\n𝑥=0,(62)\nwhere 𝑔n=4𝜋𝜈n𝐷nis the conductance of the normal\nmetal.\nThe LDoS reads, cf. Eq. (2),\n𝜌(𝐸, 𝑥) ={︃\n2𝜈Re cos 𝜃s(𝐸, 𝑥), 𝑥 >0,\n2𝜈nRe cos 𝜃n(𝐸, 𝑥), 𝑥 < 0.(63)\nThe Usadel equation (60) is justi\fed, provided that the\nmagnetic impurities are rare enough, so the scattering of\nelectrons by them can be considered independently. To\nguarantee such situation, we assume that the following\nconditions are satis\fed,\n𝑛𝑠≪⎧\n⎪⎨\n⎪⎩𝑔n/𝜉n, 𝑏≫𝜉n,\n𝑔/𝜉, 𝑏 ≪min{𝜉n, 𝜉}, 𝑔/𝜉≫𝑔n/𝜉n,\nmax{𝑔, 𝑔n}/𝜉n, 𝑏≪min{𝜉n, 𝜉}, 𝑔/𝜉≪𝑔n/𝜉n.\n(64)\nHere the length 𝜉n=√︀\n𝐷n/(2|𝐸|) for the normal metal\nis introduced. We note that this inequality is analogous\nto the corresponding condition in the case of impurities\nscattered in the whole two-dimensional plane (see Ref.\n[23] and Appendix C).\nSolving the Usadel equations (59){(60) consists in \fnd-\ning solutions in three domains ( 𝑥<−𝑏,−𝑏<𝑥< 0, and\n𝑥>0) and stitching these solutions at the points 𝑥=−𝑏\nand𝑥=0, applying the boundary conditions (62). Using\nthe boundary conditions (61) at spatial in\fnity, 𝑥→∞,\nwe can immediately write the solutions for the spectral\nangle in the region 𝑥<−𝑏,\n𝜃n(𝐸, 𝑥) = 4 arctan[︂\nexp(︂𝑥+𝑏\n𝜉n𝑒−𝑖𝜋sgn𝐸/4)︂\ntan𝜃b\n4]︂\n,\n(65)\nand in the domain 𝑥>0,\n𝜃s(𝐸, 𝑥) =𝜋\n2+𝑖𝜓∞+4𝑖arctanh(︂\n𝑒−𝑥/𝜉tanh𝜓0\n4)︂\n.(66)\nThe constants 𝜃band𝜓0have to be determined from the\nboundary conditions (62).11\nIn order to \fnd the solution for the spectral angle on\nthe interval−𝑏6𝑥60, we need to write out the \frst inte-\ngral of the equation (60),\n𝜉2\nn\n2(𝜕𝑥𝜃n)2−𝑖sgn𝐸cos𝜃n=𝐶, (67)\nwhere 𝐶is the constant. This allows us to reduce the\nsolution of Eq. (60) to the inversion of the incomplete\nelliptic integral,\n𝑥+𝑏\n𝜉n=𝜃n(𝐸,𝑥)∫︁\n𝜃b𝑑𝜃√2𝐶+ 2𝑖sgn𝐸cos𝜃. (68)\nThus, we have the explicit solutions (65){(66) in the re-\ngions 𝑥>0,𝑥6−𝑏and the implicit solution (68) in the\ninterval−𝑏<𝑥< 0. To obtain the \fnal result for the spec-\ntral angle, it remains to determine the values of the con-\nstants 𝜃b,𝜓0and𝐶using the boundary conditions (62).\nHence, we can derive a system of algebraic equations for\nthese coe\u000ecients. Boundary conditions at the point 𝑥=0\nyield\n𝐶=−2𝑔2𝜉2\nn\n𝑔2n𝜉2sinh2𝜓0\n2−sgn𝐸sinh ( 𝜓∞+𝜓0).(69)\nBoundary conditions at the point 𝑥=−𝑏lead to\n𝐶= 2(︂\n𝑒−𝑖𝜋sgn𝐸/4sin𝜃b\n2+(4𝑛𝑠𝜉n𝛼/𝑔n) sin 2 𝜃b\n1 +𝛼2+ 2𝛼cos 2𝜃b)︂2\n−𝑖sgn𝐸cos𝜃b. (70)\nFinally, from Eq. (68) with 𝑥=0 and the boundary con-\nditions at 𝑥=0, we \fnd the third relation:\n𝑏\n𝜉n=∫︁𝜋\n2+𝑖𝜓∞+𝑖𝜓0\n𝜃b𝑑𝜃√2𝐶+ 2𝑖sgn𝐸cos𝜃. (71)\nThereby, the solution of the Usadel equation (59){(60)\nis fully determined by the algebraic system of equations,\n(69){(71), and by the functions (65), (66), and (68) in\nthe domains 𝑥<−𝑏,𝑥>0, and−𝑏<𝑥< 0, respectively. Al-\nthough Eqs. (69){(71) can be solved numerically, at \frst,\nit is instructive to discuss their analytic solutions in some\nlimiting cases.\nB. The LDoS in the case of 𝑏=0\nThe algebraic system of equations, (69){(71), can be\nsolved analytically in the case of magnetic impurities sit-\nuated exactly at the SN boundary, i.e., at 𝑏=0. In the\nsuperconductor, 𝑥>0, the spectral angle is given by Eq.\n(66). At 𝑥<0 the spectral angle is described by Eq. (65)\nwith 𝑏=0 and\n𝜃b=𝜋/2 +𝑖𝜓∞+𝑖𝜓0. (72)Next, using Eqs. (69) and (70), we \fnd the following\nclosed equation for 𝜓0,\n𝑖𝛾sinh𝜓0\n2+𝑒−𝑖𝜋sgn𝐸/4sin(︂𝜋\n4+𝑖𝜓∞+𝜓0\n2)︂\n=(4𝑖𝑛𝑠𝜉n𝛼/𝑔n) sinh[2( 𝜓∞+𝜓0)]\n1 +𝛼2−2𝛼cosh[2( 𝜓∞+𝜓0)]. (73)\nHere the energy function 𝛾=𝑔𝜉n/(𝑔n𝜉) is introduced. We\nnote that we choose the sign in front of the term propor-\ntional to 𝛾in Eq. (73) in such a way that the equation\nreproduces the known solution for 𝜓0in the absence of\nmagnetic impurities, i.e., at 𝛼=0,\n𝜓0,0≡𝜓0⃒⃒⃒\n𝛼=0= ln𝛾+𝑒𝑖𝜋(1−sgn𝐸)/4𝑒−𝜓∞/2\n𝛾+𝑒−𝑖𝜋(1+sgn 𝐸)/4𝑒𝜓∞/2.(74)\nWe mention that the solution of Eq. (73) for 𝐸<0 can be\nobtained from the solution for 𝐸>0 through the transfor-\nmation 𝜓0→−𝜓*\n0. It guarantees that the LDoS is sym-\nmetric with respect to 𝐸=0. Therefore, below we shall\nfocus on the case 𝐸>0.\nFor a su\u000eciently low concentration of magnetic im-\npurities, 𝑛𝑠, we can assume that the solution to Eq.\n(73) is close to the function (74) due to the smallness\nof the term proportional to 𝑛𝑠. A noticeable e\u000bect of\nthat term in Eq. (73) can be expected if its denomina-\ntor, 1+ 𝛼2−2𝛼cosh[2( 𝜓∞+𝜓0)], becomes close to zero.\nAs we discussed above, this expression vanishes when\n𝜓0=0 and 𝜓∞= arcsinh(√𝛽), corresponding to the YSR\nenergy, 𝐸YSR, are substituted into it. Combining these\nideas, it becomes clear that a peak in the LDoS near the\nYSR energy is possible if, at \frst, the concentration 𝑛𝑠\nis su\u000eciently low and, secondly, the function (74) for the\nenergy 𝐸YSRis close to zero. Perturbation of the LDoS\naway from the YSR energy is weak, provided\n𝑛𝑠𝜉/𝑔≪1. (75)\nWe emphasize that this inequality coincides with the in-\nequality given by Eq. (64) for 𝑏≪min{𝜉n, 𝜉}. The second\ncondition (smallness of 𝜓0,0at the YSR energy) means\nthat√︀\n𝐷n/𝐷≫(𝑔n/𝑔)√︀\n(1−𝛼)/(2𝛼). (76)\nThus, we assume that, when the conditions (75) and\n(76) are satis\fed, the LDoS determined by Eq. (73) will\ncoincide with the one given by the solution (74) every-\nwhere outside the vicinity of the YSR energy, where the\npeak is expected. This means that to complete the ana-\nlytical description of the considered case, 𝑏=0, it is su\u000e-\ncient to solve Eq. (73) near the YSR energy. Expanding\nthe left-hand side and denominator of the right-hand side\nin Eq. (73) in small parameter 𝜓0−𝜓0,0up to the \frst or-\nder, we \fnd\n𝜓0=𝜓0,0\n2+𝛽−sinh2𝜓∞\n2 sinh(2 𝜓∞)\n+𝑖[︃\n2𝑛𝑠𝜉\n𝑔−(︂𝛽−sinh2𝜓∞\n2 sinh(2 𝜓∞)−𝜓0,0\n2)︂2]︃1/2\n.(77)12\nFIG. 6. Dependence of the LDoS on the energy at the position of the magnetic impurities in the case of the impurity located\nat the SN boundary, i.e., at 𝑏=0. Vertical dashed lines denote the position of the YSR energy at a given value of 𝛼, which\nis𝛼=0.3 on the left and 𝛼=0.9 on the right. Blue, orange and green curves show the LDoS for 𝑛𝑠𝜉(0)/𝑔=0.001,0.0025,and\n0.005, respectively. The dashed curves denote, for comparison, the LDoS without magnetic impurities (i.e., for 𝑛𝑠=0). We use√︀\n𝐷n/𝐷=20 and 𝑔=𝑔n.\nWe note that this result holds for |𝜓∞−arcsinh 𝛽|≪1.\nNext, using Eq. (77), we extract the LDoS for⃒⃒|𝐸|−𝐸YSR⃒⃒≪\u0001/(1+𝛽). In the superconducting region,\n𝑥>0, we \fnd\n𝜌(𝐸, 𝑥) =𝜌0(𝐸, 𝑥) +𝛿𝜌(𝐸, 𝑥). (78)\nHere the \frst term in the right-hand side is the LDoS in\nthe absence of magnetic impurities,\n𝜌0(𝐸, 𝑥)≃−2𝜈(1 +𝛽)3/2\n\u0001𝑒−𝑥/𝜉𝛽Im^𝐸YSR. (79)\nThe second term describes the contribution of magnetic\nimpurities,\n𝛿𝜌(𝐸, 𝑥)≃𝜈(1 +𝛽)3/2\n\u0001𝑒−𝑥/𝜉𝛽[︃\nIm^𝐸YSR\n+ Re√︁\n\u00002ns−(^𝐸YSR−|𝐸|)2]︃\n. (80)\nHere the energy parameter\n^𝐸YSR= \u00011−𝛼\n1 +𝛼[︃\n1−23/2𝛼+𝑖√𝛼\n(1 +𝛼)√1−𝛼𝑔n√\n𝐷\n𝑔√𝐷n]︃\n(81)\ndescribes the YSR energy modi\fed by the presence of the\nnormal region. We note that, in addition to some shift of\nthe YSR energy in comparison with the case of a homo-\ngeneous superconductor, ^𝐸YSRhas a negative imaginary\npart. It indicates that the YSR state becomes the quasi-\nbound state rather than the bound one [37]. We mention\nthat for 𝛼→1 the shift of the YSR energy due to the\npresence of the SN boundary can become dominant.\nThe energy scale\n\u0000ns=2\u0001√︀\n2𝑛𝑠𝜉𝛽/𝑔\n1 +𝛽(82)determines the e\u000bective width of the YSR resonance.\nWe mention that at 𝑛𝑠∼1/𝜉𝛽, \u0000nsmatches the disorder\nbroadening \u0000 for a single magnetic impurity problem,\ncf. Eq. (13). Also, we note that Eq. (80) resembles\nthe result for the LDoS on the magnetic impurity in the\npresence of the STM tip, cf. Eq. (56).\nIn the normal region, 𝑥60, the LDoS can be written\nin the form of Eq. (78) with\n𝜌0(𝐸, 𝑥)≃𝜈nRe cos(︃\n4 arctan(︃\n𝑒−|𝑥|𝛽1/4(1−𝑖)/(√\n2𝜉𝛽)\n×tan(︂𝜋−𝑖ln𝛼\n8)︂)︃)︃\n+𝜈n𝑔n√\n𝐷\n𝑔√𝐷n1 +𝛼√︀\n2𝛼(1−𝛼)𝑒−|𝑥|𝛽1/4/(√\n2𝜉𝛽)\n×(︃\n√𝛼cos(︃\n𝑥𝛽1/4\n√\n2𝜉𝛽)︃\n+ sin(︃\n𝑥𝛽1/4\n√\n2𝜉𝛽)︃)︃]︃\n(83)\nand\n𝛿𝜌(𝐸, 𝑥)≃𝜈n(1 +𝛽)3/2\n\u0001𝑒−|𝑥|𝛽1/4/(√\n2𝜉𝛽)[︃\ncos(︃\n𝑥𝛽1/4\n√\n2𝜉𝛽)︃\n×(︁\nIm^𝐸YSR+ Re√︁\n\u00002ns−(^𝐸YSR−|𝐸|)2)︁\n,\n−sin(︃\n𝑥𝛽1/4\n√\n2𝜉𝛽)︃(︁\nRe^𝐸YSR−|𝐸|\n−Im√︁\n\u00002ns−(^𝐸YSR−|𝐸|)2)︁]︃\n. (84)\nWe note that according to Eqs. (83) and (84), the LDoS\nin the normal metal oscillates with the distance from the\nSN boundary (and the magnetic impurities). However,13\nthe period of these oscillations coincide with the decay\nlength such that they are not visible.\nThe obtained analytical results are con\frmed by a nu-\nmerical solution. The black dashed curves on graphs in\nFig. 6 shows the density of states in the absence of mag-\nnetic impurities. We note that V-shape form of the den-\nsity of states appearing due to inverse proximity e\u000bect is\nreminiscence of the density of states with Thouless en-\nergy minigap in the SNS junction. The solid curves in\nFig. 6 show the sought-for peak in the energy dependence\nof the LDoS near 𝐸YSRfor di\u000berent values of 𝛼and𝑛𝑠.\nThe position of the peak is shifted relative to the YSR\nenergy, as predicted in Eq. (84). This picture displays\nhow the peak grows in height and width as the concen-\ntration of magnetic impurities 𝑛𝑠increases. We note that\ngrowth of the peak height and width with increasing 𝑛𝑠is\nlimited by the inequality (75). Additionally, in the right\npanel of Fig. 6, we present how the peaks for positive\nand negative energies merge when 𝛼approaches unity.\nFig. 7 shows the decrease in the relative size of the\npeak (as before, shifted from 𝐸YSR) with distance from\nthe impurity and the superconductor. As can be seen,\nthe e\u000bect of magnetic impurities extends over distances\nof the order of the length 𝜉ninside the normal metal.\nInterestingly, there is no decay with the distance of the\nrelative correction to the LDoS, 𝛿𝜌/𝜌 0, in the supercon-\nductor. This phenomenon can be explained as follows.\nIn the superconducting part of NS junction ( 𝑥 >0) any\nspatially dependent perturbation decays on the scale of\nsuperconducting coherence length 𝜉(𝐸) for a given en-\nergy𝐸. There is no other spatial scale within the Usadel\nequation. In particular, the perturbations of the density\nof states due to proximity e\u000bect in the absence of mag-\nFIG. 7. Dependence of the relative change in the LDoS, 𝛿𝜌/𝜌 0,\non energy and 𝑥-coordinate as a result of the magnetic im-\npurities located at the SN boundary (that is, 𝑏=0). Black\nvertical lines denote the position of the YSR energy ( 𝛼=0.3).\nWe use 𝑛𝑠𝜉(0)/𝑔= 0.0025,√︀\n𝐷n/𝐷=20 and 𝑔=𝑔n.netic impurities at the YSR energy, Eq. (79), and due\nto magnetic impurities, Eq. (80), decay with the same\nspatial scale 𝜉𝛽. That is why the ratio 𝛿𝜌/𝜌 0does not\ndepend on the coordinate, although the magnitude of 𝛿𝜌\ntends to zero with increasing 𝑥.\nC. The LDoS in the general case, 𝑏>0\nHere we return to the system of Eqs. (69){(71), the\nsolution of which, together with the expressions (63){\n(66), describes the LDoS for 𝑏>0. As in the previous\nsection, we investigate the region of parameters in which\nthe LDoS has a peak near the YSR energy, and is oth-\nerwise close to the impurity-free solution determined by\nthe expressions (72) and (74).\nIt is easy to see that the presence of impurities in the\nsystem (69){(71) is re\rected in only one term from the\nsecond equation. This term completely coincides with\nthe one investigated in the equation (73). Repeating the\nprevious reasoning, we again come to the necessity of ful-\n\flling the inequalities (75){(76). However, these condi-\ntions are not enough. If the impurities are moved to the\ndepth of the normal metal, the proximity e\u000bect ceases\nto work, and the peculiarity in the YSR energy region\ndisappears. For the impurities to remain in the super-\nconductor region of in\ruence, the condition\n𝑏≪𝜉n (85)\nis necessary. It means that the spectral angle 𝜃nat\nthe point 𝑥=−𝑏where the magnetic impurities are lo-\ncated should not be close to zero { its magnitude in\na homogeneous normal metal (see (65)). We mention\nthat condition (85) becomes more relax with increase of\n𝛼towards unity. Indeed, at the YSR energy, we \fnd\n𝜉n(𝐸YSR)=𝜉(0)√︀\n(1 +𝛼)/(1−𝛼).\nFig. 8 shows the energy dependence of the LDoS for\ndi\u000berent values of 𝑏. Again, the peak is displaced from\nthe YSR energy (see Eq. (84)). As can be seen, when the\nimpurity is removed farther away from the SN boundary,\nthe peak is blurred, becoming lower and wider. This\nbehavior is in agreement with the inequality (85).\nFig. 9 displays the energy and coordinate dependence\nof the relative size of the shifted peak with distance from\nthe impurity. As in Fig. 7, in the superconductor region,\nthe ratio 𝛿𝜌/𝜌 0does not depend on the coordinate.\nAs one can see from Figs. 8 and 9, the LDoS for mag-\nnetic impurities situated in the normal metal within the\ndistance 𝑏≪𝜉nis qualitatively the same as the one in the\ncase𝑏=0.\nFor completeness, in Appendix D we present the den-\nsity of states in the case of SN junction with a chain\nof magnetic impurities situated in the superconducting\nregion.14\nFIG. 8. Dependence of the LDoS on the energy at the SN boundary (left) and at the position of the magnetic impurities\n(right). Vertical dashed lines denote the position of the YSR energy at a given value of 𝛼=0.3. Blue, orange and green curves\nshow the LDoS for 𝑏/𝜉n(Δ)=0 .01,0.05,and 0 .15, respectively. The dashed curves denote, for comparison, the LDoS without\nmagnetic impurities, i.e., for 𝑛𝑠=0. We use 𝑛𝑠𝜉(0)/𝑔= 0.002,√︀\n𝐷n/𝐷=20 and 𝑔=𝑔n.\nIV. DISCUSSIONS AND CONCLUSIONS\nIn our paper, we do not take into account the spin-\nindependent part of the magnetic impurity strength, 𝛼0.\nHowever, its e\u000bect can be incorporated into rede\fnition\nof the parameter 𝛽→(1−𝛼+𝛼0)2/(4𝛼). Therefore, all our\nresults can be easily applied to that more general case.\nThe results of Sec. II A for a solitary magnetic impu-\nrity are related with the case of \fnite impurity concen-\ntration 𝑛(2)\n𝑠. We remind that a single magnetic impurity\nproduces a perturbation of the LDoS with spatial extent\nof the order of the superconducting coherence length at\nFIG. 9. Dependence of the relative change in the LDoS 𝛿𝜌/𝜌 0\non energy and 𝑥-coordinate as a result of the magnetic im-\npurities located at 𝑥=−𝑏=−0.1𝜉n(Δ). Black vertical lines\ndenote the position of the YSR energy ( 𝛼=0.3). We use\n𝑛𝑠𝜉(0)/𝑔= 0.002,√︀\n𝐷n/𝐷=20 and 𝑔=𝑔n.the YSR energy, 𝜉𝛽. Therefore, we can expect our re-\nsults to be applicable for a \fnite impurity concentration\n𝑛(2)\n𝑠∼1/𝜉2\n𝛽. In this case, using Eq. (96) of Ref. [22], we\n\fnd the width of the impurity band to be of the order of\n\u0001/[(1 + 𝛽)√𝑔]. The latter estimate coincides with \u0000 up\nto a logarithm in the de\fnition of spreading resistance\n𝑡, cf. Eq. (8). We emphasize that two seemingly di\u000ber-\nent problems { spatially inhomogeneous one for a solitary\nmagnetic impurity and homogeneous one for a \fnite con-\ncentration of magnetic impurities { occur to be related.\nAs mentioned above, the broadening of the LDoS near\nthe YSR energy is caused by the \ructuations of the di-\nmensionless e\u000bective strength 𝛼of a magnetic impurity.\nTherefore, it would be tempting to say that the distribu-\ntion of the YSR energy (de\fned as the energy at which\nthe peak in the LDoS has the maximum) can be directly\nread from the log-normal distribution (44) and Eq. (10).\nHowever, Eq. (50) demonstrates clearly that this is not\nthe case. In fact, the problem of computation of the YSR\nenergy distribution in a dirty superconducting \flm is\nmore complicated and goes far beyond the present work.\nWe emphasize that in the case of a single magnetic im-\npurity randomness of the YSR state is introduced due to\ndi\u000berent realizations of potential disorder. It should be\ncontrasted with the case of rare magnetic impurities con-\nsidered in Ref. [23] where \ructuations of YSR states at\ndi\u000berent magnetic impurities were related to the point-\nto-point \ructuations of the local density of states due to\npotential disorder. Surprisingly, on the level of the Us-\nadel equation both e\u000bects can be described by the very\nsame log-normal distribution of the dimensionless e\u000bec-\ntive strength 𝛼, cf. Eq. (44).\nIn our work, in order to \fnd the LDoS, we solve the\nUsadel equation for a spatially dependent spectral an-\ngle. We remind that the Usadel equation corresponds\nto the saddle-point treatment of the NLSM (see Refs.15\n[18, 22, 23] for details). Renormalization of the NLSM\naction between the length scales ℓand𝜉should be taken\ninto account. It leads to the renormalized Usadel equa-\ntion. However, one can treat the renormalized NLSM\nbeyond the saddle-point approximation. This results in\nadditional \ructuation corrections to the LDoS. For a \f-\nnite impurity concentration in a dirty superconducting\n\flm, one can estimate the relative \ructuation correction\nto the LDoS to be of the order of ∼(𝑛(2)\n𝑠𝜉2)/𝑔2[46]. Ap-\nplying this estimate with 𝑛(2)\n𝑠∼1/𝜉2for a solitary mag-\nnetic impurity, we \fnd that the \ructuation corrections\nto the LDoS are negligible in comparison with the results\nderived from the Usadel equation. We expect a similar\nconclusion in the case of a magnetic impurity chain near\nthe SN boundary.\nIn our paper, we treat the magnetic impurity spin fully\nclassically. Such approximation is formally justi\fed by\nthe limit 𝑆≫1. Since in reality the magnetic impurity\nspin is not that large, 𝑆.5/2, it would be interesting\nto investigate the e\u000bect of potential disorder on the YSR\nstate treating the spin quantum mechanically (for a clean\ncase see Ref. [47] and references therein).\nFor a chain of magnetic impurities, the quantum dy-\nnamics of their spins leads to an intriguing competition\nbetween the Kondo e\u000bect and the indirect exchange inter-\naction that can be probed by STM measurements [48, 49].\nTaking into account potential disorder is likely to be im-\nportant for interpretation of the STM data.\nFor a chain of rare magnetic impurities near the SN\nboundary, we limit our consideration by simplest geom-\netry when the chain is parallel to the SN interface. Re-\ncently, YSR-type features in the LDoS at grain bound-\naries in graphene with Pb islands have been measured\n[50]. In view of these experimental \fndings, it would\nbe worthwhile to study more complicated geometries of\nmagnetic impurity chains near SN interfaces.\nTo summarize, we reported the results of detailed stud-\nies of the e\u000bects of potential disorder on YSR states in\nsuperconducting \flms. We focus on two setups: (i) a soli-\ntary magnetic impurity in a dirty superconducting \flm\nand (ii) a chain of magnetic impurities situated in a nor-\nmal region of an SN junction. Solving the Usadel equa-\ntion for a spatially dependent spectral angle, we found\nthat potential disorder broadens the YSR state. This\nmanifests as the peak in LDoS at energies near 𝐸YSR.\nThe broadening of the peak is proportional to the\nsquare root of resistance per square of the \flm. Thus,\nit is larger than one could naively expect. The physical\nmechanism for appearance of broadening is \ructuations\nof the LDoS in the normal state. The latter results in\n\ructuations of dimensionless impurity strength 𝛼and,\nconsequently, to \ructuations of an energy of the YSR\nstate. In the case of a single magnetic impurity in a\ndirty superconducting \flm, we demonstrate that\nmodi\fcation of multiple scattering on the magnetic im-\npurity due to intermediate scattering on surrounding po-\ntential disorder is of crucial importance for correct de-\nscription of the LDoS pro\fle near the YSR energy. Inparticular, the account of this modi\fcation allowed us\nto remove unphysical abrupt vanishing of the LDoS ob-\ntained within standard Usadel equation. We are not\naware of any systematic experimental studies of the de-\npendence of the YSR peak width in the LDoS on the\nsheet resistance of a \flm.\nWe demonstrated that existence of a normal metal\nmakes the YSR state to be the quasibound state rather\nthan the bound one. For a solitary magnetic impurity\nin a dirty superconducting \flm, such an e\u000bect is caused\nby the normal-metal tip used for STM measurements. In\nthe case of a magnetic impurity chain, the normal region\nof the SN heterostructure provides a channel for decay of\nthe YSR state.\nFinally, we mention that it would be interesting to ex-\ntend our study to superconducting systems with spin-\norbit coupling in which a magnetic impurity chain can\nhost Majorana bound states together with YSR states.\nACKNOWLEDGMENTS\nThe authors are grateful to Ya. Fominov, A. Melnikov,\nM. Skvortsov for very useful discussions. We are espe-\ncially grateful to I. Tamir for providing us the experimen-\ntal data on the LDoS. The research was partially sup-\nported by the Russian Ministry of Science and Higher\nEducation, the Russian Foundation for Basic Research\n(grant No. 20-52-12013) - Deutsche Forschungsgemein-\nschaft (grant No. EV 30/14-1) cooperation, and by the\nBasic Research Program of HSE. A. Lyublinskaya is also\ngrateful to JetBrains Co. Ltd. for a personal scholar-\nship through the program to support women and girls in\nSTEM.\nAppendix A: Derivation of Eq. (7)from the\nstandard Usadel equation (1)\nIn this Appendix we present brief derivation of Eq. (7)\nfrom the standard Usadel equation (1). It is expressed\nas follows\n𝜉2∇2𝛿𝜓𝜎−sinh𝛿𝜓𝜎=[︀\n𝜉2/(𝜋𝜈𝐷)]︀\ncosh𝜓𝜎\n𝜎√𝛽+ sinh 𝜓𝜎𝛿(𝑟).(A1)\nTaking into account the smallness of deviation from the\nhomogeneous solution, |𝛿𝜓𝜎|≪1, we can treat the lin-\nearized equation,\n𝜉2∇2𝛿𝜓𝜎−𝛿𝜓𝜎=[︀\n𝜉2/(𝜋𝜈𝐷)]︀\ncosh𝜓𝜎\n𝜎√𝛽+ sinh 𝜓𝜎𝛿(𝑟). (A2)\nThis equation for 𝛿𝜓𝜎is similar to the 2D Schr odinger\nequation with 𝛿(𝑟) potential. For 𝑟>𝑙the solution of Eq.\n(A2) can be written as\n𝛿𝜓𝜎(𝑟) =(︁\n~𝜓𝜎−𝜓∞)︁\n·𝐾0(𝑟/𝜉)\nln𝜉/𝑙. (A3)16\nHere we have introduced the notation ~𝜓𝜎=𝛿𝜓𝜎(𝑙)+𝜓∞\nand have taken into account the small parameter 𝑙/𝜉≪1.\nIn order to treat the delta-functional potential we ap-\nply the Fourier transformation to the equation (A2),\n−(︀\n𝑞2𝜉2+ 1)︀\n𝛿𝜓𝜎(𝑞) =[︀\n𝜉2/(𝜋𝜈𝐷)]︀\ncosh ~𝜓𝜎\n𝜎√𝛽+ sinh ~𝜓𝜎.(A4)\nHere 𝛿𝜓𝜎(𝑞) is the Fourier transform of 𝛿𝜓𝜎(𝑟). Thus,\nwe derive the self-consistent equation (7) for ~𝜓𝜎.\nAppendix B: Derivation of the Usadel equation for a\nsolitary magnetic impurity\nThe NLSM action for a dirty superconducting \flm with\na solitary magnetic impurity can be written as (see Ref.\n[7] for details):\n𝑆=𝑆𝜎+𝑆Δ+𝑆mag. (B1)\nHere the \frst term in the right-hand side of Eq. (B1) is\ngiven by\n𝑆𝜎=𝑔\n32∫︁\n𝑑𝑟Tr(∇𝑄)2−2𝑍𝜔∫︁\n𝑑𝑟Tr[︁\n^𝜀+^\u0001]︁\n𝑄.(B2)\nThe \feld 𝑄(𝑟) is a matrix in the replica, spin, Matsubara,\nand particle-hole spaces. The trace Tr acts in the same\nspaces. The matrix \feld 𝑄obeys the nonlinear constraint\nand charge-conjugation symmetry relation,\n𝑄2(𝑟) = 1 ,Tr𝑄= 0, 𝑄 =𝑄†=−𝐶𝑄𝑇𝐶, (B3)\nwhere 𝐶=𝑖𝑡12. The action (B2) involves two constant\nmatrices:\n^𝜀𝛼𝛽\n𝑛𝑚=𝜀𝑛𝛿𝜀𝑛,𝜀𝑚𝛿𝛼𝛽𝑡00,^\u0001𝛼𝛽\n𝑛𝑚= \u0001𝛿𝜀𝑛,−𝜀𝑚𝛿𝛼𝛽𝑡10.\n(B4)\nHere 𝛼, 𝛽=1, . . . , 𝑁 𝑟stand for replica indices, while in-\ntegers 𝑛, 𝑚 correspond to the Matsubara fermionic fre-\nquencies 𝜀𝑛=𝜋𝑇(2𝑛+1). The superconducting order pa-\nrameter \u0001 is assumed to be a real scalar. The sixteen\nmatrices,\n𝑡𝑟𝑗=𝜏𝑟⊗𝑠𝑗, 𝑟, 𝑗 = 0,1,2,3, (B5)\noperate in the spin (subscript 𝑗) and particle-hole (sub-\nscript 𝑟) spaces. The matrices 𝜏𝑟and𝑠𝑟are the standard\nPauli matrices. We note that the parameter 𝑍𝜔describes\nthe frequency renormalization upon the renormalization\ngroup \row (see Ref. [51] for details). The bare value of\n𝑍𝜔is equal to 𝜋𝜈/4. The second term of the action (B1)\nreads\n𝑆Δ=−4𝑍𝜔𝑁𝑟\n𝜋𝑇𝛾 𝑐0∫︁\n𝑑𝑟\u00012. (B6)\nThe last term of 𝑆describes the action of the solitary\nmagnetic impurity,\n𝑆mag=−1\n2Tr ln(︀\n1 +𝑖√𝛼0𝑄(0) + 𝑖√𝛼𝑄(0)𝑡33)︀\n.(B7)We choose the following form of the saddle-point 𝑄-\nmatrix,\n𝑄𝛼𝛽\n𝑛𝑚=1\n2∑︁\n𝜎=±(︁\ncos𝜃𝜎(𝑡00sgn𝜀𝑛+𝜎𝑡33)𝛿𝜀𝑛,𝜀𝑚\n+ sin 𝜃𝜎(𝑡10−𝑖𝜎𝑡23sgn𝜀𝑛)𝛿𝜀𝑛,−𝜀𝑚)︁\n𝛿𝛼𝛽.(B8)\nHere we assume that the spectral angle 𝜃𝜎≡𝜃𝜎(𝜀𝑛) is an\neven function of 𝜀𝑛. Then variation of the saddle-point\naction 𝑆[𝑄] with respect to the spectral angle 𝜃𝜎(𝜀𝑛) re-\nsults in the standard Usadel equation (1). Varying 𝑆[𝑄]\nover \u0001 yields a self-consistent equation for the supercon-\nducting order parameter, Eq. (28).\nIn order to derive the renormalized Usadel equation,\nwe need to consider the renormalization of the NLSM\naction. Let us split the matrix \feld 𝑄into the fast 𝑞\nand slow 𝑄0=𝑇−1\u0003𝑇components. Here we introduce\nthe matrix\n\u0003𝛼𝛽\n𝑛𝑚= sgn 𝜀𝑛𝛿𝜀𝑛,𝜀𝑚𝛿𝛼𝛽𝑡00. (B9)\nThe renormalized action for a magnetic impurity is de-\ntermined as follows,\n𝑆(ren)\nmag[𝑄0] =−ln⟨\n𝑒−𝑆mag[𝑇−1𝑞𝑇]⟩\n𝑞. (B10)\nHere the averaging ⟨. . .⟩𝑞is with respect to the NLSM ac-\ntion𝑆𝜎for the fast modes 𝑞. As was derived in Ref. [23],\nthe term exp(︀\n−𝑆mag[𝑇−1𝑞𝑇])︀\ntransforms upon renormal-\nization as follows,\n⟨\n𝑒1\n2Tr ln(1+𝑖√𝛼𝑇−1𝑞(0)𝑇𝑡33)⟩\n𝑞→⟨\n𝑒1\n2Tr ln(1+𝑖√a𝑄0(0)𝑡33)⟩\na.\n(B11)\nHere we set 𝛼0=0 for a sake of simplicity. The average\n⟨. . .⟩ais de\fned with respect to the distribution function\n(44). For the derivation of the Usadel equation we need to\nknow the saddle-point action in the replica limit, 𝑁𝑟→0,\nalone. Therefore, we \fnd\n𝑆(ren)\nmag[𝑄]≃−1\n2⟨\ntr ln(︀\n1 +𝑖√a𝑄(0)𝑡33)︀⟩\na. (B12)\nVarying 𝑆𝜎[𝑄]+𝑆(ren)\nmag[𝑄] over the spectral angle 𝜃𝜎(𝜀𝑛)\nyields the renormalized Usadel equation (43).\nWe note that for a nonzero 𝛼0, we would obtain\nthe renormalized Usadel equation with the distribu-\ntion functions (44) for quantities corresponding to both\n(√𝛼+√𝛼0)2and (√𝛼−√𝛼0)2.\nAppendix C: Condition for rareness of magnetic\nimpurities in the case of SN junction\nThe MLSM approach allows us to establish the con-\ndition of rareness of magnetic impurities. In the case of17\na superconducting \flm, the corresponding condition can\nbe formulated as [23]\n𝑛𝑠\n𝜈n|𝒟(𝑟,𝑟)|≫𝑛2\n𝑠\n𝜈2n∫︁\n𝑑2𝑟′|𝒟(𝑟,𝑟′)𝒟(𝑟′,𝑟)|, (C1)\nwhere𝒟(𝑟,𝑟′) stands for the di\u000busion propagator. In the\ncase of a homogeneous 2D superconductor, the di\u000busion\npropagator can be written as\n𝒟(𝑟,𝑟′) =∫︁𝑑2𝑞\n(2𝜋)2𝑒𝑖𝑞(𝑟−𝑟′)\n𝐷(𝑞2+𝜉−2). (C2)\nHence we \fnd the inequality (C1) reduces to the condi-\ntion, 𝑛𝑠𝜉2/𝑔≪1. We note that we neglect a logarithmic\nfactor.\nIn the case of a chain of magnetic impurities paral-\nlel to the SN boundary, one needs to \fnd the di\u000busive\npropagator. It satis\fes the following equations:\n𝐷n[−𝜕2\n𝑥+𝑞2\n𝑦−𝑖𝜉−2\nn]𝒟(𝑞𝑦;𝑥, 𝑥′) =𝛿(𝑥−𝑥′), 𝑥 < 0\n𝐷[−𝜕2\n𝑥+𝑞2\n𝑦+𝜉−2]𝒟(𝑞𝑦;𝑥, 𝑥′) =𝛿(𝑥−𝑥′), 𝑥 > 0.\n(C3)\nThe boundary condition at 𝑥=0 reads\n𝒟(𝑞𝑦;−0+, 𝑥′) =𝒟(𝑞𝑦; 0+, 𝑥′),\n𝑔n𝜕𝑥𝒟(𝑞𝑦;𝑥, 𝑥′)|𝑥=−0+=𝑔𝜕𝑥𝒟(𝑞𝑦;𝑥, 𝑥′)|𝑥=0+.(C4)\nHere we perform the Fourier transform with respect to\n𝑦coordinate (which is parallel to the SN boundary).\nHence, in the case of a magnetic impurities chain, the\ncondition (C1) becomes\n∫︁𝑑𝑞𝑦\n2𝜋|𝒟(𝑞𝑦;−𝑏,−𝑏)|≫𝑛𝑠\n𝜈n∫︁𝑑𝑞𝑦\n2𝜋|𝒟(𝑞𝑦;−𝑏,−𝑏)|2,\n(C5)\nSolving Eqs. (C3), we \fnd the following expression for\nthe di\u000busive propagator,\n𝒟(𝑞𝑦;−𝑏,−𝑏) =1\n2𝐷n√︁\n𝑞2𝑦−𝑖𝜉−2n[︃\n1−𝛾√\n1+𝑞2𝑦𝜉2√\n1+𝑖𝑞2𝑦𝜉2n−𝑒−𝑖𝜋\n4\n𝛾√\n1+𝑞2𝑦𝜉2√\n1+𝑖𝑞2𝑦𝜉2n+𝑒−𝑖𝜋\n4\n×𝑒−2𝑏√︁\n𝑞2𝑦−𝑖𝜉−2\nn]︃\n. (C6)\nIn the case 𝑏≫𝜉n, the inequality (C5) reduces to the fol-\nlowing condition,\n𝑛𝑠𝜉n/𝑔n≪1. (C7)\nIn the opposite case, 𝑏≪min{𝜉n, 𝜉}, we \fnd from Eq.\n(C5), the following inequalities\n𝑛𝑠𝜉/𝑔≪1, 𝛾≫1,\n𝑛𝑠𝜉n/max{𝑔, 𝑔n}≪1, 𝛾≪1.(C8)\nAs one can see, Eqs. (C7) and (C8) are equivalent to\nEq. (64).\nAppendix D: YSR resonance in a dirty SN junction\nwith magnetic impurities situated inside the\nsuperconductor.\nIn this Appendix we consider how a chain of magnetic\nimpurities situated inside the superconducting region inSN junction a\u000bects the density of states. We shall per-\nform calculations in a way similar to the one described in\nSec. III A. We assume that the chain is parallel to the SN\ninterface and is situated at the point 𝑥=𝑏. By analogy\nwith Eqs. (65)-(66), we write out the spectral angle in\nthe region 𝑥<0,\n𝜃n(𝐸, 𝑥) = 4 arctan[︂\nexp(︂𝑥\n𝜉n𝑒−𝑖𝜋sgn𝐸/4)︂\ntan𝜃0\n4]︂\n,\n(D1)\nand the region 𝑥>𝑏,\n𝜃s(𝐸, 𝑥) =𝜋\n2+𝑖𝜓∞+ 4𝑖arctanh(︂\n𝑒−(𝑥−𝑏)/𝜉tanh𝜓b\n4)︂\n.\n(D2)\nNext, to \fnd a solution on the interval 0 <𝑥<𝑏 , we use\nthe \frst integral of the Usadel equation:\n𝜉2\n2(𝜕𝑥𝜃s)2+ sin ( 𝜃s−𝑖𝜓∞) =𝐶, (D3)\nwhere 𝐶is the constant. This allows us to reduce the so-\nlution to the inversion of the incomplete elliptic integral,\n𝑏−𝑥\n𝜉=𝜃s(𝐸,𝑥)∫︁\n𝜋\n2+𝑖𝜓∞+𝑖𝜓b𝑑𝜃√︀\n2𝐶−2 sin ( 𝜃−𝑖𝜓∞).(D4)\nTo fully determine the spectral angle, we need to \fnd\nconstants 𝜃0,𝜓b, and 𝐶. Boundary conditions (62) at\nthe point 𝑥=0 yield\n𝐶=2𝑔2\nn𝜉2\n𝑔2𝜉2n𝑒−𝑖𝜋sgn𝐸/2sin2𝜃0\n2+ sin ( 𝜃0−𝑖𝜓∞) (D5)\nand at the point 𝑥=𝑏lead to\n𝐶=−2(︂\nsinh𝜓b\n2−(4𝑛𝑠𝜉𝛼/𝑔 ) sinh(2 𝜓∞+ 2𝜓b)\n1 +𝛼2−2𝛼cosh(2 𝜓∞+ 2𝜓b))︂2\n+ cosh 𝜓b. (D6)\nThe last equation is obtained from the Eq. (D4) by\nsubstituting 𝑥=0:\n𝑏\n𝜉=𝜃0∫︁\n𝜋\n2+𝑖𝜓∞+𝑖𝜓b𝑑𝜃√︀\n2𝐶−2 sin ( 𝜃−𝑖𝜓∞). 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Fern´ andez-Rossier, and I. Brihuega, Observation of\nYu–Shiba–Rusinov States in superconducting graphene,\nAdv. Mat. 33, 2008113 (2021).\n[51] A. M. Finkelstein, Electron liquid in disordered conduc-\ntors, in Soviet scientific reviews, Vol. 14, edited by I. M.\nKhalatnikov (Harwood Academic Publishers, 1990)."    },    {        "title": "2202.02605v1.Many_body_density_of_states_in_the_edge_of_the_spectrum__non_interacting_limit.pdf",        "content": "arXiv:2202.02605v1  [cond-mat.stat-mech]  5 Feb 2022Many body density of states in the edge of the spectrum:\nnon-interacting limit\nPragya Shukla\nDepartment of Physics, Indian Institute of Technology, Kha ragpur-721302, India\n(Dated: February 8, 2022)\nAbstract\nIn noninteracting limit, the density of states of a many body system can be expressed as the\nconvolution of single body density of states of its subunits . Here we use the formulation to derive\nthe ensemble averaged many body density of states for the cas es in which subunits can be modelled\nby Gaussian or Wishart random matrix ensembles.\n1I. INTRODUCTION\nI would like to dedicate this work to the meomory of Fritz Haake, to ce lebrate his rich\nand diverse contributions to Mathematical Physics, especially Quan tum Chaos. It was his\nbook [1] on the latter topic, an important resource for my graduat e studies and I am deeply\nindebted to him for writing it.\nConsider a many body system composed of independent or weakly int eracting particles\nwith mutual interaction not strong enough to perturb the single pa rticle spectrum. The\nlatter situation may arise, for example, for the systems in which the interactions are local\nwithin a short spatial scale, say Lsand negligible at large scales. The system can then\nbe described as a collection of non-interacting/ weakly interacting s ubunits of linear size\nLsand its Hamiltonian can be expressed as a tensor sum of the local Ham iltonians (of the\nsubunits) anditsphysical properties canthenbedetermined from those ofsubunits (referred\nas ”particles” hereafter). The many body density of states ( mdos) can then be obtained by\na convolution of the single particle states ( sdos) [2].\nFor many body systems with non-local interactions, the bulk densit y of many body states\nis very large and even a small non-local interaction can in general mix the single particle\nstates, wiping out their details (due to small level spacing). The con volution route to derive\nmdosis therefore not valid in the bulk of the spectrum of such a system. T he situation\nhowever is different in their edge: due to vanishing dosin the edge, the separation between\nlevels is large and the many body interactions may not be strong enou gh to mix the single\nparticle energy states. The mdosin the edge of the spectrum can then be derived by the\nconvolution of single particle spectrum.\nFor a system consisting of a large number of particles and with a very weak restriction\nonsdos, the convolution route predicts, based on central limit theorem (C LT), a Gaussian\nbehaviour of the mdosin the bulk of the spectrum [2]. The CLT however is not applicable\nin the regime where the single particle spectrum is singular e.g. the spe ctral-edge. With\ngrowing interest in the low temperature physics, it becomes now impo rtant to know the\nbehaviour of the mdosnear the edge i.e the region where the ground state of the many bod y\nsystem lies. This motivates us to consider following questions:\n(i) assuming the theoretical form for the sdosin the single body spectral edge is known,\nwhat would be the behaviour of mdosin the same energy range,\n2(ii) for what cases of sdos, themdoswill retain the same functional form albeit approxi-\nmately?\nThe standard route to determine the energy states of a system is based on a matrix\nrepresentation of the Hamiltonian in a physically motivated basis. Bas ed on the nature of\ninteractions between different subunits, referred here as ”part icles”, the Hamiltonian matrix\ncan be of various types e.g. dense (full), sparse, banded and subj ected to additional matrix\nconstrains due to symmetry and conservation laws [1, 3, 4]. Besides , the complexity of inter-\nactions often renders an exact calculation of the matrix elements a technically challenging\ntask. The deterministic error appears as a distribution of the matr ix element around its\nmost probable value and the Hamiltonian matrix behaves as a random m atrix with some\nor all random elements. The physical properties of the Hamiltonian m atrix are then best\ndescribed by an appropriate ensemble of its replicas, with ensemble p arameters determined\nfrom the system conditions. With density of states as a backbone o f many physical prop-\nerties, it is therefore necessary to derive its general formulation applicable for a wide range\nof systems. This is however technically very complicated due to syst em dependence of the\nrandomization of the many body Hamiltonian. The mdosalso varies from one replica to\nthe other, thus making it necessary to consider its average as well fluctuations across the\nensemble. However for cases in which sdosis known apriori, the mdoscan be determined\nthrough convolution route.\nIt is now well-established, theoretically as well as experimentally, tha t the fluctuation\nproperties of a one body (single particle) operator in the ergodic re gime of its wavefunction\ndynamics can be well described by the Wigner-Dyson ensembles of He rmitian matrices if the\noperator is Hamiltonian and by stationary Wishart ensembles if it is a se mi-positive definite\n(can be described by a full Wishart random matrix) [1, 3, 4]. In case o f the localized/\npartially localized wavefunctions or weakly violated exact symmetries of one body operator,\nhowever, the statistics is described by generalized ensembles e.g. m ulti-parameter dependent\nsparse random matrix ensembles, brownian ensembles etc [4–7]. The present study analyses\nthe ensemble averaged mdosin the spectrum edge of a many particle system that consists\nof many non-interacting particles, with their fluctuation propertie s described by Gaussian\nor Wishart ensembles of both stationary/ non-stationary types [8 , 9].\nThe paper is organized as follows. Our approach is based on first writ ing the many body\ndensity of states mdosas the convolution of single body densities sdos. For the systems con-\n3sisting of smaller subunits e.g. a 2-body system with known sdos, themdoscan be obtained\ndirectly from the convolution integral. However the integration rou te becomes increasingly\ncomplicated as the number of subunits increase, motivating us ti co nsider an alternative\nroute based on the differential equation satisfied by the sdos. A repeated application of\nthe latter to convolution integral leads to a non-homogeneous diffe rential equation for mdos\nwhich can then be solved to obtain the mdos, subjected to the boundary conditions derived\nfrom a smooth connection of the edge mdoswith that of bulk. Under some approximations,\nmdoscan also be given by the homogeneous solution of the related differen tial equation.\nII. AVERAGE MANY BODY DENSITY OF STATES\nConsider the case in which the system Hamiltonian Hwithginteracting subunits can be\nwritten as\nH=H0+V (1)\nwithH0as a sum over non-interacting Hamiltonians H(s),s= 1→g,\nH0=g/summationdisplay\ns=1H(s)(2)\nandVas the many body interaction among subunits. Here we consider the energy limit in\nwhichVcan be ignored e.g. a system generated by distributing gnon-interacting particles\noverNsingle particle states Ekwithk= 1→N(ignoring the occupancy constraints due to\nPauli principle for cases g≪N, referred as dilute limit). Another energy range, where V\noften plays a negligible role, is the lower edge region of the many body s pectrum and is the\nfocus of current study.\nAssuming E1,E2,...,E Mas the eigenvalues of H, withM=Ngas the size of H-matrix,\nthe many body dosρg(e) can be expressed as\nρg(e) =M/summationdisplay\nn=1δ(e−En) (3)\nUsing the same formulation for the single particle dosρ1(E), (with E1,E2,...,EMas the\neigenvalues of H), we have\nρ1(E) =N/summationdisplay\nk=1δ(E −Ek) (4)\nIn non-interacting limit, the energy Enof a many body levels Ψ ncan further be expressed\nin terms of the single single particle state as follows. Using ˆ nkas the number operator for\nthekthsingle particle state Ek(counting the number of particles with single particle energy\n4Ekin the manybody state Ψ n),Hrepresented in the Ng-dimensional product basis of single\nparticle states becomes\nH=/summationdisplay\nkˆnkEk (5)\nThis in turn leads to En=∝an}bracketle{tΨn|HΨn∝an}bracketri}ht=/summationtext\nknkEkwherenk=∝an}bracketle{tΨn|ˆnk|Ψn∝an}bracketri}ht\nFollowing from the above, ρg(e) can then be expressed as the covolution of many body\ndosof (g−1) particles and a single particle dos\nρg(e) =/integraldisplayeb\neaρg−1(x)ρ1(e−x) dx (6)\nwith notation ρmdenoting the dosofmparticles and ea,ebas the many body energy range\nof interest.\nAn averaging of ρg(e) over all replicas of Hin the ensemble at a fixed ethen leads to its\nensemble average at e:\n∝an}bracketle{tρg(e)∝an}bracketri}ht=M/summationdisplay\nn=1∝an}bracketle{tδ(e−En)∝an}bracketri}ht. (7)\nAs the particles are almost mutually independent, an ensemble avera ging of eq.(6) can be\nexpressed as\n∝an}bracketle{tρg(e)∝an}bracketri}ht=/integraldisplayeb\nea∝an}bracketle{tρg−1(x)∝an}bracketri}ht ∝an}bracketle{tρ1(e−x)∝an}bracketri}htdx (8)\nFor later purposes, we also consider an alternative formulation for evenge.g.g= 2n,\nn= 1,2,...;ρ2n(e) can be rewritten as\n∝an}bracketle{tρ2n(e)∝an}bracketri}ht=/integraldisplayeb\nea∝an}bracketle{tρ2n−1(x)∝an}bracketri}ht ∝an}bracketle{tρ2n−1(e−x)∝an}bracketri}htdx. (9)\nThe above in turn leads to\n∝an}bracketle{tρ2n+1(e)∝an}bracketri}ht=/integraldisplayeb\nea∝an}bracketle{tρ2n(x)∝an}bracketri}ht ∝an}bracketle{tρ1(e−x)∝an}bracketri}htdx. (10)\nHereafter our analysis is confined to the edge region of the single bo dy spectrum. Using\nthe notation ∝an}bracketle{tρm,edge(e)∝an}bracketri}htfor the ensemble averaged m-bodydosin the edge region of the\nspectrum, we have, from eq.(8),\n∝an}bracketle{tρg,edge(e)∝an}bracketri}ht=/integraldisplaye0\n−∞∝an}bracketle{tρg−1,edge(x)∝an}bracketri}ht ∝an}bracketle{tρ1,edge(e−x)∝an}bracketri}htdx (11)\nwith−∞< e < e 0as the single body spectral edge. Here we note that due to rapidly\nincreasing mdos, the many body spectral edge indeed lies below single body spectral edge\ne0. Although the sum over contributions of single particle spectral ed ges ofgbodies is g e0,\nthe number of many body levels near this energy is quite large (e.g ∼g), thus making it\npart of the bulk spectrum instead of the edge.\nFurther mathematical steps depend on the behaviour of ∝an}bracketle{tρ1,edge(e−x)∝an}bracketri}htwhich can vary\nbased on the complexity of the system. As mentioned in previous sec tion, here we confine\n5the analysis to the many body systems consisting of single body Hamilt onians which can\nbe modelled by Gaussian and Wishart ensembles. Moreover, with pres ent study focussed\non the edge behaviour of the average spectral density ρg,edge, hereafter the subscript′′edge′′\nwill be suppressed unless needed for clarity of presentation.\nIII. STATIONARY GAUSSIAN ENSEMBLES\nA stationary Gaussian ensemble, also known as Wigner-Dyson ensem ble, consists of Her-\nmitian matrices (denoted here by symbol H) with basis-invariant Gaussian ensemble den-\nsityρH(H)∝exp[−TrH2]. The underlying symmetry constraints lead to three universality\nclasses of WDEs, labelled by the symmetry parameter β: (i) Gaussian orthogonal ensem-\nbles (GOE) of real-symmetric matrices ( β= 1), (ii) Gaussian unitary ensembles (GUE) of\ncomplex Hermitian matrices ( β= 2), (iii) Gaussian symplectic ensembles (GSE) of real-\nquaternion matrices ( β= 4). The average dosfor a Wigner-Dyson ensemble follows a\nsemicircle behaviour in the bulk of the spectrum and an Airy function t ype behaviour in its\nedge.\nFollowing standard notation, the ensemble averaged sdosis given as ∝an}bracketle{tρ1(e)∝an}bracketri}ht=KN(e,e)\nwhere the kernel Khas different scaling behaviour in the bulk and edge of the spec-\ntrum. For a stationary Gaussian ensemble, it has been shown to beh ave as follows [10]:\nlimN→∞1√\n2NKN(e√\n2N,e√\n2N) =1\nπ√\n1−e2in bulk (e∼0) and\nlim\nN→∞1\n21/2N1/6KN/parenleftbigg\n−√\n2N+e\n21/2N1/6,−√\n2N+e\n21/2N1/6/parenrightbigg\n=f1(e)\nin the lower edge ( e∼ −√\n2N) wheref1(e) depends on the symmetry class of the ensemble.\nShifting the origin of spectrum to e=−√\n2N, thesdos∝an}bracketle{tρ1(e)∝an}bracketri}htin the lower edge region,\nfor a single particle Hamiltonian modelled by a stationary Gaussian ense mble, can then be\ngiven as\n∝an}bracketle{tρ1(e)∝an}bracketri}ht=γ N f 1(γ e)−∞< e < e 0. (12)\nwithγ= 21/2N1/6.\nAs our approach is based on the convolution of sdos, it is useful to define the mdosalso\nin a rescaled form\n∝an}bracketle{tρg(e)∝an}bracketri}ht=γ Ngfg(γe). (13)\n6From eq.(11), we then have\nfg(e) =/integraldisplaye0\n−∞fg−1(x)f1(e−x)∝an}bracketri}htdx (14)\nSimilarly eq.(9) gives\nf2n(e) =/integraldisplaye0\n−∞f2(n−1)(x)f2(n−1)(e−x)∝an}bracketri}htdx (15)\nThe fom of f1(e) depends on the symmetry class of the ensemble and can thereby le ad to\ndifferent fg(e). Here we consider the cases with f1given by a GOE or GUE only. The steps\ncan however be extended directly to GSE too.\nA. Gaussian Unitary ensemble (GUE)\nFor a GUE, f1(e) in the lower edge is\nf1(e)≈eAi2(−e)+(Ai′(−e))2(16)\nwith Ai(e) and Ai′(e) as the Airy function of the first kind and its derivative, respective ly.\nWe note that the exact differential equation satisfied by f(y) is [10]/parenleftBiggd3\ndy3+4yd\ndy−2/parenrightBigg\nf1(y) = 0 (17)\nTo derive ∝an}bracketle{tρg(e)∝an}bracketri}htin the regime e < e0, we proceed iteratively and start with g= 2. Using\nthe definition\n∝an}bracketle{tρg(e)∝an}bracketri}ht=γ Ngfg(γe) (18)\nwithf1(x) given by eq.(16), a substitution of eq.(12) in eq.(8) gives f2(the rescaled 2-body\ndos) in the edge as\nf2(y) =/integraldisplaye0\n−∞f1(x)f1(y−x) dx. (19)\nIt is easy to check that f2(y) satisfies/parenleftBiggd3\ndy3+4yd\ndy−2/parenrightBigg\nf2(y) =−4I2(y). (20)\nwith\nI2(y) =/integraldisplaye0\n−∞x f1(x)∂f1(y−x)\n∂xdx. (21)\nwhere we have used the relation∂f1(y−x)\n∂(y−x)=−∂f1(y−x)\n∂x=∂f1(y−x)\n∂y. Applying partial integra-\ntion,I2can now be rewritten as\nI2(y) =−3\n2f2(y)+1\n4d2f2(y)\ndy2+Q2(y,e0) (22)\nwith\nQ2(y,x) =1\n4/bracketleftBig\nf1(y−x) D2\nxf1(x)−(Dxf1(y−x))(Dxf1(x))+(D2\nxf1(y−x))f1(x)/bracketrightBig\n+\n+x f1(x)f1(y−x) (23)\nwhereDk\nx≡dk\ndxk.\n7Substitution of the above in eq.(20) gives/parenleftBigg\n2d3\ndy3+4yd\ndy−8/parenrightBigg\nf2(y) =−4Q2(y,e0) (24)\nFurtherusingeq.(15)with n= 2andapplyingthedifferentialoperator/parenleftBig\n2d3\ndy3+4yd\ndy−8/parenrightBig\nto both sides of the integral then leads to a differential equation fo rf4(e). Repeating the\nsteps multiple times with n= 3,...mthen leads to the differential equation for fg(e) with\ng= 2n,\n/parenleftBigg\nagd3\ndy3+4yd\ndy−bg/parenrightBigg\nfg(y) =−4Qg(y,e0)−4Gg(y,e0) (25)\nwherea2n= 2n,b2n= 2(2+b2(n−1)) witha1= 1,b1=−1 forn= 0) and\nG2n(y,e0) =/integraldisplay\n[Q2n−1(y−x)+G2n−1(y−x)]f2n−1(x)rdx (26)\nand\nQ2n(y,x) =2(n−1)\n4/bracketleftBig\nf2(n−1)(y−x) D2\nxf2(n−1)(x)−(Dxf2(n−1)(y−x))(Dxf2(n−1)(x)) +\n+ (D2\nxf2(n−1)(y−x))f2(n−1)(x)/bracketrightBig\n+x f2(n−1)(x)f2(n−1)(y−x) (27)\nUsing eq.(25) with g= 2nand substitution in the definition f2n+1(y) =/integraltexte0\n−∞f2n(x)f1(y−\nx) dxfollowed by partial integration and rearrangements, we can derive the differential\nequation for f2n+1(e); the latter turns out to be same as eq.(25) but now g= 2n+ 1,\na2n+1= 2n+1,b2n+1= 2(5+b2n) and\nG2n+1(y,e0) =/integraldisplay\n[Q2n(y−x)+G2n(y−x)]f1(x) dx (28)\nand\nQ2n+1(y,x) =2n\n4/bracketleftBig\nf2n(x) D2\nxf1(y−x)−(Dxf2n(y−x))(Dxf1(x)) +\n+ (D2\nxf2n(y−x))f1(x)/bracketrightBig\n+x f2n(x)f1(y−x) (29)\nEq.(25) is a linear nonhomogeneous differential equation of the 3rdorder. Writing its\ngeneral solution as a sum of homogeneous and nonhomogeneous pa rts\nfg(y) =fg,h(y)+fg,ih(y), (30)\nthe three independent solutions of the homogeneous part fg,h(y) can be given as\nfg,h(y) =c1F/parenleftBigg\n−bg\n12,{1\n3,2\n3},−4y3\n9ag/parenrightBigg\n+c2/parenleftBigg4y3\n9ag/parenrightBigg1/3\nF/parenleftBigg\n{1\n3−bg\n12},{2\n3,4\n3},−4y3\n9ag/parenrightBigg\n+c3/parenleftBigg4y3\n9ag/parenrightBigg2/3\nF/parenleftBigg2\n3−bg\n12,{4\n3,5\n3},−4y3\n9ag/parenrightBigg\n(31)\nAs a check, we note that, for g= 1, eq.(25) reduces to linear homogeneous differential\nequation given by eq.(17) and the solution given by eq.(31) reduces t o eq.(16) for c1=\n80.066987,c2= 0.091888095 ,c3= 0.12604490. Clearly f1,ih= 0. For arbitrary g >1 case,\nhowever, it is apriori not obvious whether fg,ihcan be ignored.\nFor large gcases, which are of wider physical interests, here we consider an a ltrenative\nalthough approximate route: it is based on neglecting the first orde r derivative in eq.(25),\nits coefficient being much smaller than the coefficients of the other tw o terms on the left of\neq.(25). The approximation reduces the latter equation to a linear d ifferential equation with\nconstant coefficients:/parenleftBigg\nagd3\ndy3−bg/parenrightBigg\nfg(y) =−4Qg(y,e0)−4Gg(y,e0) (32)\nand its solution can then be given as\nfg(y) =3/summationdisplay\nk=1cketky/bracketleftbigg\n1+Ak/integraldisplay\netkx(Qg(x)+Gg(x))/bracketrightbigg\n(33)\nwithtkas the roots of equation ( D3−bg/ag) = 0 and Ak=1\n3t2\nk. Here two of the arbitrary\nconstants c1,c2,c3can be determined by imposing the requirement that fg(−∞)→0 and\nfg(e0) approaches correct bulk density smoothly. Substitution of eq.(3 1) in eq.(18) then\nleads to mdos. The remaining constant can be determined by invoking the normaliza tion\ncondition on full mdos. Here we note that, for a sdosmodeled by a GUE, the /tieaccentlowercasemdos in\nthe bulk turns out to be a Gaussian or semicircle based on whether th e nature of partice\ninteractions [2].\nB. Gaussian orthogonal ensemble (GOE)\nAs in the previous case, the sdos∝an}bracketle{tρ1(e)∝an}bracketri}htin the lower edge region, for a single particle\nHamiltonian modeled by a GOE, can again be given by eq.(12) but now\nf1(x)≈xAi2(−x)+(Ai′(−x))2+1\n2Ai(−x)/integraldisplay−x\n−∞Ai(y) dy. (34)\nwith Ai(x) same as in eq.(16). We note that the origin of spectrum here again is shifted to\ne=−√\n2N. It is easy to check that f1(y) satisfies/parenleftBiggd2\ndy2+y/parenrightBigg\nf1(y) =Q1(y) (35)\nwith\nQ1(y) =y2Ai2(−y)+y(Ai′(−y))2−1\n2Ai(−y)Ai′(−y) (36)\nTo derive ∝an}bracketle{tρg(e)∝an}bracketri}htin the edge regime e < e0, we again use relation (18), proceed iteratively\nand use eq.(8) with g= 2, equivalently eq.(19). A double differentiation of the latter along\nwith rearrangement of the terms then leads to following differential equation for f2\n9/parenleftBiggd2\ndy2+y/parenrightBigg\nf2(y) =I2(y)+Q2(y). (37)\nwhere\nI2(y) =/integraldisplaye0\n−∞d2f1(x)\ndx2f1(y−x) dx (38)\nQ2(y) =/integraldisplaye0\n−∞(Q1(x)f1(y−x)+f1(x)Q1(y−x)) dx. (39)\nApplying repeated partial integration, we have\nI2(y) =u2(y)−d2f2(x)\ndy2(40)\nwhereu2(y) =f1(e0)df1(y−e0)\nde0−df1(e0)\nde0f1(y−e0). Substitution of the above in eq.(37) reduces\nit as/parenleftBigg\n2d2\ndy2+y/parenrightBigg\nf2=u2(y)+Q2(y) (41)\nFurther using eq.(9) and repeating the above steps then leads to t he differential equation\nforf2n(y). The latter along with eq.(10) can subsequently be used to derive t he differential\nequation for f2n+1. For both cases ( g= 2norg= 2n+1), we have/bracketleftBigg\ngd2\ndy2+y/bracketrightBigg\nfg(y) =ug(y)+Qg(y). (42)\nwhere\nug(y) =g/2/summationdisplay\nm=1/bracketleftBiggdfg−m(e0)\nde0fm(y−e0)−fg−m(e0)dfm(y−e0)\nde0/bracketrightBigg\n(43)\nand\nQ2n(y) =/integraldisplaye0\n−∞[Q2n−1(x)f2n−1(y−x)+f2n−1(x)Q2n−1(y−x)] dx, (44)\nQ2n+1(y) =/integraldisplaye0\n−∞[Q2n(x)f1(y−x)+f2n(x)Q1(y−x)] dx (45)\nEq.(42) is exact and its solution can be written as fg(y) =fg,h(y) +fg,ih(y) withfg,h\nandfg,ihas its homogeneous and nonhomogeneous solutions. Here the two in dependent\nhomogeneous solutions are\nfg,h(y) =c1Ai(−g−1/3y)+c2Bi(−g−1/3y) (46)\nwith Ai(y) same as in eq.(16) and Bi(y) as the Airy functions of the second kind. Further,\nas in the GUE case, the constants c1,c2can be determined from the boundary conditions\nonfg(y), namely, fg(−∞)→0 andfg(e0) approaches correct bulk density smoothly. Using\nthe standard approach (based on the wronskin of two independen t homogeneous solutions\nof a second order differential equation), the inhomogeneous solut ion of eq.(42) is then\nfg,ih(y) =A(y) Ai(−g−1/3y)+B(y) Bi(−g−1/3y). (47)\n10with\nA(y) =/integraldisplayy\n(w(x))−1Bi(−g−1/3x) [ug(x)+Qg(x)] dx, (48)\nB(y) =/integraldisplayy\n(w(x))−1Ai(−g−1/3x) [ug(x)+Qg(x)] dx. (49)\nThe above integrals can be caculated by a substitution of eq.(43) an d eq.(45). Here the\nwronskin w(x) = Ai(−g−1/3x)Bi′(−g−1/3x)−Bi(−g−1/3x)Ai′(−g−1/3x) turns out to be a\nconstant. Further addition of eq.(46) and eq.(47) and subsequen tly imposing the boundary\ncondition fg(−∞) = 0 then leads to fg(y)≈(c1+A(y)) Ai(−g−1/3y). The latter along\nwith eq.(18) then leads to ensemble averaged mdos\n∝an}bracketle{tρg(e)∝an}bracketri}ht ≈(c1+A(y))γ NgAi(−g−1/3γ e) (50)\nIV. STATIONARY WISHART ENSEMBLES (WE)\nA stationary Wishart ensemble consists of Hermitian matrices L≡C†.CwithCas\na (N+α)×Ncomplex matrix. The ensemble averaged sdosfor this case can again\nbe given as ∝an}bracketle{tρ1(e)∝an}bracketri}ht=KN(e,e) where the kernel Khas different scaling behavior in the\nbulk and edge of the spectrum. The kernal Kfor average single particle dosin this case\nhas following scaling behavior: lim N→∞1√\n2NKN(e/2N,e/2N) =1\n2π√e√1−ein bulk and\nlimN→∞1\n4NKN/parenleftBig\ne\n4N,e\n4N/parenrightBig\n=f1(e) in the lower edge ( e∼0) where f1(e) again depends on the\nsymmetry class of the Wishart ensemble [10–12].\nFollowing from the above, the sdos∝an}bracketle{tρ1(e)∝an}bracketri}htin the lower edge region, for a single particle\noperator modeled by a stationary Wishart esemble, can then be give n as\n∝an}bracketle{tρ1(e)∝an}bracketri}ht=γ N f 1(γ e) 0< e < e 0. (51)\nwithγ= 4N. Here again the scaled form of mdosis given by eq.(18) but with γas in\neq.(51).\nA. Wishart Unitray ensemble (WUE)\nFor a single particle operator modeled by a WUE, f1(x) in the edge region can be given\nas (eq.(2.10) of [10])\nf1(x) =1\n4/bracketleftBigg\n(J′\na(√x)2+Ja(√x)J′\na+1(√x)−a√xJa(√x)J′\na(√x)+1√xJa(√x)Ja+1(√x)/bracketrightBigg\n(52)\n11As discussed in [10], f1(y) in this case satisfies/parenleftBigg\nyd2\ndy2−αd\ndy−1/parenrightBigg\nf1(y) =α\nyf1(y) (53)\nAgain using eq.(8) and proceeding as in previous cases, it can be show n that∝an}bracketle{tρ2(y)∝an}bracketri}htnow\nsatisfies/parenleftBigg\nyd2\ndy2−αd\ndy−1/parenrightBigg\nf2(y) =I2(y)−α h2(y). (54)\nwhere now\nI2(y) =/integraldisplaye0\n0x f1(x)∂2f1(y−x)\n∂x2dx. (55)\nand\nh2(y) =/integraldisplaye0\n0f1(x)f1(y−x)\ny−xdx. (56)\nApplying partial integration, I2can be rewritten as\nI2(y) =Q2\n2+y\n2d2\ndy2f2(y) (57)\nSubstitution of the above in eq.(54) leads to/parenleftBiggy\n2d2\ndy2−αd\ndy−1/parenrightBigg\nf2(y) =Q2\n2−α h2(y) (58)\nwhere\nQ2(y) =b/parenleftBigg\nf1(b)∂f1(y−b)\n∂b−f1(y−b)∂f1(b)\n∂b/parenrightBigg\n−f1(b)f1(y−b) (59)\nAgain using eq.(18) for g= 4,8,...,2n,narbitrary, and iterating the above steps repeat-\nedly (as in previous cases) leads to/parenleftBiggy\n2nd2\ndy2−αd\ndy−1/parenrightBigg\nf2n(y) =R2n(y) (60)\nwhereR2n(y) =1\n2Q2n(y)+f2n(y)−α h2n(y) and\nQ2n(y) =b/parenleftBigg\nf2n−1(b)∂f2n−1(y−b)\n∂b−f2n−1(y−b)∂f2n−1(b)\n∂b/parenrightBigg\n−f2n−1(b)f2n−1(y−b)(61)\nf2n(y) =/integraldisplaye0\n0f2n−1(x)Q2n−1(y−x) dx (62)\nh2n(y) =/integraldisplaye0\n0f2n−1(x)h2n−1(y−x) dx (63)\nAs in the previous cases, the differential equation for the present case with g= 2n+1 can\nnow be derived from eq.(60) along with eq.(14).\nThe two independent solutions of the homogeneous part of eq.(60) can be given as, with\nγ= 1+αgand again using subscript ginstead of 2n,\nfg,h(y) =c1Γ(1−γ) (gy)γ/2I−γ(2√gy)+c2(−1)γΓ(1+γ) (gy)γ/2Iγ(2√gy) (64)\nwithIγas the modified Bessel function and c1,c2as the constants of integration, to be\ndetermined from the boundary conditions. The above in turn gives t he wronskin w(x) of\nthe two homogeneous solutions as w(x) = (−g)(1+ag)(1+ag)xag. Following standard route,\n12the solution fg,ihof the inhomogeneous part of eq.(60) can be written as\nfg,ih(y) =−γ0(gy)γ/2[A(g)I−γ(2√gy)+B(g)Iγ(2√gy) ] (65)\nwithγ0=−(−1)γΓ(1+γ)Γ(1−γ) and\nAg(y) =/integraldisplay\n(gx)γ/2Iγ(2√gx)Rg(x) (w(x))−1dx (66)\nBg(y) =/integraldisplay\n(gx)γ/2I−γ(2√gx)Rg(x) (w(x))−1dx. (67)\nUsingfg(y) =fg,h(y)+fg,ih(y), the general solution of eq.(60) can now be written as\nfg(y) = (c1Γ(1−γ)−γ0Ag(y))(gy)γ/2I−γ(2√gy)+\n+(c2(−1)γΓ(1+γ)−γ0Bg(y)) (gy)γ/2Iγ(2√gy) (68)\nhere again the constants c1,c2can be detremined by imposing the boundary conditions at\ny= 0 and y=e0. The substitution of fg(y) in eq.(18) then leads to mdos∝an}bracketle{tρg∝an}bracketri}ht.\nB. Wishart Orthogonal ensemble (WOE)\nAs in the previous case, the sdos∝an}bracketle{tρ1(e)∝an}bracketri}htin the edge region for a WOE, can still be\ndescribed by eq.(51) with f1(x) given as [10]\nf1(x) =f1,U(x)−Ja+1(√x)\n4√x/parenleftBigg/integraldisplay√x\n0Ja+1(v) dv−1/parenrightBigg\n(69)\nwheref1,U(x) is same as f1(x) of WUE case, given by eq.(52)\nIn principle, proceeding as in previous cases, the exact differential equation for fg(y) for\nthis case can again be derived. The derivation can however be simplifie d by noting that ,\nnear the lower edge x∼0, the 2ndterm in eq.(69) is negligible with respect to first term\nand one can approximate\nf1(x)≈f1,U(x) (70)\nThe above approximation in turn would again lead to eq.(60) for mdosin WOE case. This\nindicatestheinsensitiivty ofthebehaviour intheedgeofthespectr um ofaWishart ensemble\nto exact symmetry conditions.\nV. BROWNIAN ENSEMBLES\nA Brownian ensemble (BE) in general refers to an intermediate stat e of perturbation\nof a stationary random matrix ensemble by another one of a differen t universality class\n[1, 3, 9, 13]. The type of a BE, appearing during the cross-over, de pends on the nature of\nthe stationary ensembles and their different pairs may give rise to diff erent BEs [4, 9]. Here\n13we consider the BEs appearing between stationary ensembles of Ga ussian and Wishart type\noly.\nA. Gaussian Brownian Ensembles\nA Brownian ensemble of Hermitian matrices Hcan be described as H=√f(H0+tV)\nwithV(t)asarandomperturbationofstrength t, takenfromastationaryGaussianensemble\ncharacterized by symmetry parameter β, and applied to an initial stationary state H0(see\nalso [5]). Here f= (1+γ t2)−1withγas an arbitrary positive constant.\nThesdosfor the above ensemble is described by a diffusion equation, referre d as Dyson-\nPastur equation, [5, 9]:\n1\n2β∂∝an}bracketle{tρ1∝an}bracketri}ht\n∂Y=∂\n∂e[γ e−α(e) ]∝an}bracketle{tρ1∝an}bracketri}ht (71)\nwithY∝1\n2γln(1+γ;t2) and\nα(e)≡αe≡P/integraldisplay∞\n−∞∝an}bracketle{tρ1(e′)∝an}bracketri}ht\ne−e′. (72)\nWe note that here e0=−√\n2N, the standard edge limit for Gaussian stationary ensembles\n(as eq.(71) is derived for origin of the spectrum at e= 0).\nTo derive ∝an}bracketle{tρg(e)∝an}bracketri}htin the edge regime for the present case, we again proceed iterative ly and\nstart with g= 2. Using the definition in eq.(12) for ∝an}bracketle{tρ2∝an}bracketri}htand differentiating it with respect\ntoY, we have\n∂∝an}bracketle{tρ2∝an}bracketri}ht\n∂Y=J1+J2 (73)\nwith\nJ1=/integraldisplaye0\n−∞∂∝an}bracketle{tρ1(x)∝an}bracketri}ht\n∂Y∝an}bracketle{tρ1(e−x)∝an}bracketri}ht;dx (74)\nJ2=/integraldisplaye0\n−∞∝an}bracketle{tρ1(x)∝an}bracketri}ht∂∝an}bracketle{tρ1(e−x)∝an}bracketri}ht\n∂Ydx. (75)\nSubstitution of eq.(71) followed by partial integration reduces J1as\nJ1=S2(e,e0)−/integraldisplaye0\n−∞(γx−α(x))∝an}bracketle{tρ1(x)∝an}bracketri}ht∂x∝an}bracketle{tρ1(e−x)∝an}bracketri}htdx (76)\n=S2(e,e0)+∂e/integraldisplaye0\n−∞(γx−α(x))∝an}bracketle{tρ1(x)∝an}bracketri}ht ∝an}bracketle{tρ1(e−x)∝an}bracketri}htdx (77)\nwithS2(e,e0) = (γe0−α(e0))ρ1(e−e0)ρ1(e0) and∂x≡∂\n∂x. Similarly, following substitution\nof eq.(71) and writing∂/angbracketleftρ1(e−x)/angbracketright\n∂(e−x)=∂/angbracketleftρ1(e−x)/angbracketright\n∂e,J2can be rewritten as\nJ2=/integraldisplaye0\n−∞∝an}bracketle{tρ1(x)∝an}bracketri}ht∂e−x(γ(e−x)−αe−x)∝an}bracketle{tρ1(e−x)∝an}bracketri}htdx (78)\n14=∂e/integraldisplay\n(γ(e−x)−αe−x)∝an}bracketle{tρ1(x)∝an}bracketri}ht ∝an}bracketle{tρ1(e−x)∝an}bracketri}htdx (79)\nSubstitution of the above in eq.(73) gives\n1\n2β∂∝an}bracketle{tρ2∝an}bracketri}ht\n∂Y=S2(e,e0)+∂\n∂e(γe∝an}bracketle{tρ2∝an}bracketri}ht−G2(e)) (80)\nwithG2(e) =/integraltexte0\n−∞(αx+αe−x)∝an}bracketle{tρ1(x)∝an}bracketri}ht ∝an}bracketle{tρ1(e−x)∝an}bracketri}htdx. As the maximum contribu-\ntion toG2comes from the neighborhood of e0, it can be approximated as G2(e)≈\n(αe0+αe−e0)/integraltexte0\n−∞∝an}bracketle{tρ1(x)∝an}bracketri}ht ∝an}bracketle{tρ1(e−x)∝an}bracketri}htdx= (αe0+αe−e0)∝an}bracketle{tρ2∝an}bracketri}ht. This reduces eq.(80) as\n1\n2β∂∝an}bracketle{tρ2∝an}bracketri}ht\n∂Y=S2(e,e0)+∂\n∂e[γe−α2,e]∝an}bracketle{tρ2∝an}bracketri}ht (81)\nwithα2,e≡α2(e) =α(e0) +α(e−e0). Proceeding iteratively with eq.(8) again, one can\nsimilarly derive the diffusion equation for ∝an}bracketle{tρ2n∝an}bracketri}ht:\n1\n2β∂∝an}bracketle{tρ2n∝an}bracketri}ht\n∂Y=n/summationdisplay\nk=12n−kS2k−1,2n−2k−1+∂\n∂e[γe−α2n,e]∝an}bracketle{tρ2n∝an}bracketri}ht (82)\nwithSab= (γe0−α(e0))ρa(e0)ρb(e−e0) andα2n,e≡α2n(e) =α2n−1(e0)+α2n−1(e−e0).\nWe note that the case α(e0) = 0 implies Sab= 0 and eq.(82) for mdosreduces to the\nsame form as that of sdosi.e eq.(71).\nB. Wishart Brownian ensembles\nConsider an ensemble of Na×Nrectangular matrices A(t) =√f(A0+tV(t)) with\nf= (1 +γ t2)−1[8, 9] with A0as a fixed matrix and γas an arbitrary positive constant.\nThe matrices L=A†Acorrespond to Wishart Brownian ensembles (WBE) if the matrices\nV†Vare taken from stationary Wishart ensembles e.g WOE or WUE. As clea r,A=A0for\nt→0,A→V/√γfort→ ∞.\nA variation of strength tof the random perturbation Vleads to diffusion of the matrix\nelements Akl(t) =√f(A0;kl+tVkl(t)). Forρv(V) =/parenleftBig\n1\n2πv2/parenrightBigβNaN/2e−1\n2v2Tr(VV†), the diffusion\nequation for the sdosfor theL-ensemble is described by a diffusion equation [5]:\n1\n2β∂∝an}bracketle{tρ1∝an}bracketri}ht\n∂Y=∂\n∂e(γ e−αw(e))∝an}bracketle{tρ1∝an}bracketri}ht (83)\nwithY=−1\n2γlnf=1\n2γln(1+γ t2) andαw(e) =e α(e) same as in eq.(71).\nWenote that theformof the eq.(83) is same asthat of eq.(71) exce pt forα(e) in thelatter\nreplaced by αw(e) in the former. Thus proceeding again as in previous case, the evolu tion\nof∝an}bracketle{tρ2n∝an}bracketri}htcan again be described by eq.(82) but with α(e)→αw(e):\n1\n2β∂∝an}bracketle{tρ2n∝an}bracketri}ht\n∂Y=n/summationdisplay\nk=12n−kS2k−1,2n−2k−1+∂\n∂e[e(γ−α2n,e)]∝an}bracketle{tρ2n∝an}bracketri}ht (84)\nwithSab=e0(γ−α(e0))ρa(e0)ρb(e−e0) andα2n,e≡α2n(e) =α2n−1(e0)+α2n−1(e−e0).\n15C. Multiparametric Ensembles\nConsider an operator of a complex system modeled by a multiparamet ric Gaussian en-\nsemble of Hermitian matrices with probability density\nρH(H,v,b) =Cexp[−/summationdisplay\nk≤l1\n2vkl(Hkl−bkl)2]; (85)\nhere the variances vkland mean values bklcan take arbitrary values (e.g. vkl→0 for the\nnon-random elements). As discussed in a series of studies [4, 5, 7], t he evolution of the sdos\nof the above ensemble is again described by eq.(71) but now Yis given by a combination of\nensemble parameters, Y=−1\nγ Mln/bracketleftBig/producttext\nk≤l/producttextβ\nq=1|xkl| |bkl+b0|2/bracketrightBig\n+constant and is referred\nas the ”complexity parameter”. Here b0= 1 or 0 if bkl= 0 or∝ne}ationslash= 0 respectively.\nTheequivalenceofthedifferentialequationfor sdosfurtherimpliesthat mdosforasystem,\nconsiting of subunits statistically described by eq.(85), is also given b y eq.(82) however Yis\nnow the complexity parameter mentione above.\nVI. CONCLUSION\nIn the end, we summarize with our main results and open questions.\nBasedontheinformationabouttheensemble averageddensityofs tatesofasinglesubunit\nof a many body system in its spectral edge, we derive the ensemble a veraged many body\ndensity of states. Our analysis clearly indicates that the latter cha nges with increasing\nnumber of subunits and differs from that of single body density of st ates even in non-\ninteracting limit.\nWhile the analysis in the present study is confined to the edge of the s pectrum, the\nderivation can directly be extended to any other energy range inclu ding bulk, within nonin-\nteracting approximation. Although the convolution route however is not applicable for the\ninteracting many body systems, it would be interesting to compare s ome available results\nfor the latter with the results derived in the present study and stu dy the extent of their\ndifferences/ deviation and sensitivity to interaction parameters.\n16ACKNOWLEDGMENTS\nI am grateful to Professor Michael Berry for some technical help withmathematica code\nused for solving the differential equations. I also thank science and educational research\nborad (SERB), department of science (DST), India for the financial support provided for\nthe research under MATRICES grant scheme.\n17REFERENCES\n[1] F. Haake, Quantum Signatures of Chaos , Springer (Berlin), (1991).\n[2] T.A.Brody, J.Flores, J.B.French, P.A.Mello, A.Pandey and S.S.M.Wong, Rev. Mod. Phys.,\n53, 385, (1981).\n[3] M. L. Mehta, Random Matrices , Academic Press, (1991).\n[4] P. Shukla, Int. J. Mod. Phys. B (WSPC) 26, 12300008, (2012 ).\n[5] P.Shukla, Phys. Rev. E, 62, 2098, (2000); Phys Rev. E 71, 0 26226 (2005); Phys. Rev. B 98,\n184202 (2018); R. Dutta and P. Shukla, Phys. Rev. E 76, 051124 (2007).\n[6] P. Shukla, Phys. Rev. E 75, 051113 (2007).\n[7] P. Shukla, J.Phys.: Condens. Matter 17, 1653, (2005);\n[8] P. Shukla, J. Phys. A: Math. Theor 50, 435003 (2017).\n[9] S. Kumar and A. Pandey, Ann. Phys. 326, 1877, (2011); Phys . Rev. E, 79, 026211, (2009).\n[10] P. J. Forrester, Nuc. Phys. 402, 709, (1993).\n[11] P. J. Forrester, T. Nagao and G. Honner, Nuc. Phys. B, 553 , 601, (1999).\n[12] T. Nagao and P. J. Forrester, Nuc. Phys. B, 435, 401, (199 5).\n[13] F.Dyson, J. Math. Phys. 3, 1191 (1962).\n[14] S. Sadhukhan and P. Shukla, Physical Review E 96, 012109 (2017).\n18"    },    {        "title": "2202.02761v4.The_two_dimensional_density_of_states_in_normal_and_superconducting_compounds.pdf",        "content": "Canadian Journal of Pure and Applied Sciences  \nVol. 1 7, No. 2, pp. 5659 -5671 , June 2023 \nOnline ISSN: 1920 -3853; Print ISSN : 1715 -9997  \nAvailable online at www.cjpas.net  \n \n \n \nTHE TWO -DIMENSIONAL DENSITY OF STATES IN  NORMAL AND \nSUPERCONDUCTING COMP OUNDS  \n \n*P. Contreras1, A. Devi2, D. Uzcátegui1 and E. Ochoa1 \nPhysics Department, University of Los Andes, Mérida, 5001, Ve nezuela  \nDepartment of Physics, Government College Baroh, Kan gra, Himachal Pradesh, India  \n \n \nABSTRACT  \n \nThe present  work represents a review for the numerical calculation of the density of states (DoS) for two -dimensional \ntight-binding models with first neighbors in its normal state and for two superconducting order p arameters. One is a \nsinglet scalar state and the other is a triplet vector state. At the beginning an emphasis is given to the general expression s \ncommonly used to the calculation of the density of states as the number of partial and total number of states , the degrees \nof freedom and the ab-initio methods most commonly used to its calculation. Then, the transition happening to the DoS \nnormal states by varying the Fermi energy and the hopping parameter is investigated. After that, the numerical \ncalculation o f the superconducting density of states using the zero -temperature scattering cross -section is performed for \nthe two order parameters. Finally, the residual density of states depending on disorder and the scattering potential \nstrength using the Larkin equa tion are calculated for the two order parameter symmetries different in nature .  \n \nKeywords:  Normal density of states, superconducting and residual density of states, Larkin equation, degrees of \nfreedom, reduced phase space .  \n \n \nINTRODUCTION  \n \nThe density of states (DoS) is a quantum mechanical \nconcept derived from the total number of quantum states \n(Φ) with an energy less than the value of E in a \nmicroscopic system, where Φ provides the total number \nof states and increases in energy with the DoS defined as  \nρ(E) = dΦ(E)/dE. Also, the DoS can be derived within a \nsmall interval of energies δE as function of a reduced \nnumber of microscopic states (Ώ) with the DoS being the \nproportionality coefficient in the relationship Ώ( E) = \nρ(E)δE, where Ώ is constant in ene rgy (Reif, 1965). On \nthe other hand, the relation between the DoS and the \ndegrees of freedom ( f) states is the microscopic motion of \na physical system that follows the Gibbs distribution (with \na constant energy) and the behavior of some particles is \nquasi -classical and happens only for some degrees of \nfreedom. However, for the rest of degrees of freedom the \nmotion is quantized and those degrees of freedom can be \nwritten as function of a quantum number ( n), where the \nenergy is quantized as En(q, p) (Landau a nd Lifshitz, \n1969).   \n \nAdditionally, the DoS can be function of external \nparameters such as those of extensive type defined in \nstatistical thermodynamics (Reif, 1965) and in that case, \ndifferent degrees of freedom can be included in a \nmacroscopic system. Thus, another way to derive the DoS comes from comparing the Gibbs distribution and the \nmicrocanonical ensemble in Statistical Mechanics \n(Landau and Lifshitz, 1969). It is more complicated to \nunderstand the derivation that includes the Planck \nconstant (2 πħ) and the relation with a volume in a \nhyperspace with one Planck constant for each pair of the \nconjugate variables q and p in the phase space, where \neach microstate belongs to a 2fN-dimensional \n“hypercube” with a length such as 2 πħ and a volume such \nas (2πħ)fN (Landau and Lifshitz, 1969).  \n \nIn general, the DoS it is a measure of how many \nmicroscopic states are available to a system in a particular \nrange of values of the energy. If the ground state energy \nfor a physical system with N particles is given by E0 = \nfNϵ0, the energy difference from of an excited state from \nthe ground state is represented by a general expression \nlinking several parameters. Thus, E – E0 = fN(〈ϵ〉 – ϵ0), \nwhere the notation indicates that 〈ϵ〉 is the mean quantum \nenergy per particle, ϵ0 is the ground state energy of each \nparticle, N is the number of particles, E and E0 are the \ntotal energy and ground state energy of the N particles, \nand f are the degrees of freedom. This mean that the total \nnumber of states Φ( E – E0) ~ Φ(〈ϵ – ϵ0〉)N ~ (〈ϵ〉 – ϵ0)N is a \nbig number even for only one kind ( f = 1) of degrees of \nfreedom.  \n \n_____________________________________________________________________  \n*Corresponding author e-mail: pcontreras@ula.ve  Canadian Journal of Pure and Applied Sciences  5660  \nIn solid state physics, the DoS is expressed in terms of the \nsystem´s energy. As some authors point out (Mulhall and \nMoelter, 2014), each element of volume/area in the phase \nspace of position q and momenta p is replaced by a \nweighting factor in an energy integral, which is easier to \nwork with at the quantum level. We make use of the \nrationalized Planck units ( ħ = kB = c = 1) to have a single \nunit of measurement because it cond ucts to the conceptual \nframework of the reduced phase space for the zero -\ntemperature elastic scattering cross -section, which we \nhave used previously in the following work (Contreras, et \nal., 2023 ). A tale of the scattering lifetime and the mean \nfree path. arXiv:2301.05322 [cond -mat.supr -con] DOI: \n10.48550/arXiv.2301.05322).  \n \nThus, a DoS equation with equal number of spin up and \ndown particles can be written as: 1) the sum of infinite \ndelta functions; 2) as the derivative of the total number of \nstates; 3) a s a proportionality coefficient of the partial \nnumber of states, i.e.  \n \n𝑁 𝜔 =  2𝑉  𝛿(𝜔− 𝜔𝑖)∞\n𝑖= 𝑑 Φ𝑑 𝜔 ≈ Ω𝛿 𝜔  \n \n \nThe last two expressions relate the DoS and the number of \ntotal or partial states. An instructive interpretation of the \nDoS from a geometrical perspective is given in (Mulhall \nand Mo elter, 2014) where the DoS is defined as the slope \nbetween the number of macroscopically allowed quantum \npartial number of states ( Ω) and an infinitesimal energy \ninterval from ω to ω + δω in a two -dimensional space \nwith variables ( ω, Ω) (Mulhall and Moelter, 2014).  \n \nThe sum inside the delta function in momentum space is \nadequate for the normal state DoS since disorder only \nchanges  the DoS value by a constant quantity. But in the \nother cases such as unconventional superconductors, it is \neasy to replace the sum by a weighting factor into an \nenergy integral, which is easier to deal with. \nSummarizing, the DoS is extensively used in applic ations \nto Statistical Mechanics, Solid State Physics, and \nQuantum Chemistry. Ab-initio  routines include the \ncalculation of the DoS, and more important is that the \nDoS can be calculated not only for systems with N \nparticles but also for: 1) single molecules  at the Hartree -\nFock HF/6 -311G* level ( Contreras et al., 2021 ) using the \nTDoS formalism (Lu and Chen, 2012); 2) dimer or trimer \nmolecular systems with lack of inversion symmetry at the \nUDFT/B3LYP level ( Burgos et al., 2017 ), or in one and \ntwo-dimensional m onolayers such as nanowires and \nnanoflakes, where it is clearly observed from several DoS \ncalculations and their visualizations show that the \nmaterials symmetrical for its up and down channels are \nnonmagnetic and asymmetrical materials are magnetic in \nnature (Devi et al., 2019, 2021a, 2021b, 2022).  \n The structure of this work is as follows: In the following \nsection, some details of the computational approach are \noutlined. In the third section, a detailed DoS calculation \nof the normal state with a tight bind ing model is \nperformed. In the fourth section, the calculation of the \nsuperconducting DoS is performed for singlet and a triplet \norder parameters (OP). In the final section, the behavior \nof the residual density of states is addressed for both \nmodels using the formalism following the Larkin equation \n(Larkin, 1965).  \n \nComputational details for the density of states with \nsums and Fermi averages  \nFor the numerical calculation of the DoS in the normal \nstate, we make use an approximation of the Delta function \nusing  a 2D sum for momentum dependency with a δ-\nfunction approximated by   \n \n𝑁 𝐸 ≅ 𝛿 𝐸−𝜉𝑖,𝒌 \n𝑘𝑥,𝑘𝑦=1\n𝜋 𝑛\n 𝐸−𝜉𝑖,𝒌 2+𝑛2\n𝑘𝑥,𝑘𝑦 \n  \n(1) \n \nwhere Σ kx,kyδ(E – ξi,k) is approximated by a Lorentzian 2D \nfunction. For the calculation of equation (1), it has been \nused a mesh of ( N × N) k-points with N = ± 400 points. \nThe other parameters in equation (1) are n = 0.005 that \ngives a well -defined delta function, and a dispersion TB \nlaw with first neighbors where two terms are responsible \nfor the behavior of the DoS, i.e.  \n \n𝜉𝑖,𝒌 𝑘𝑥,𝑘𝑦 = 𝜀𝐹+ 𝜉ℎ𝑜𝑝 𝑘𝑥,𝑘𝑦  \n (2) \n \nwhere the k dependence is carried in the hoping term \nξhop(kx, ky). Using the first neig hbors’ coefficient t, this \nfunction is given by  \n \n𝜉ℎ𝑜𝑝 𝑘𝑥,𝑘𝑦 =2𝑡  cos 𝑘𝑥𝜋 \n𝑁  + cos 𝑘𝑦𝜋 \n𝑁    \n \n \nOn the other hand, for a superconductor with \nnonmagnetic impurity scattering it is used the equation \nwhich is derived from the Green function formalism (also \nknown as T matrix formalism) (Mineev and Samo khin, \n1999; Hussey, 2002) N(͠ω) = NFℜ[g(͠ω)] where ℜ means \nthe real part of the function g(͠ω). The Fermi level DoS is \nNF and the function containing the impurity effects is  \n \n𝑔 𝜔  =   \n  2− Δ02 𝑘𝑥,𝑘𝑦    𝐹𝑆 \n \n \nand has a zero superconducting energy gap parameter \ndependence. The function g(͠ω) has the Fermi average \n<…> FS. This part requires a calculation that implies \nuncommon numerical routines to find from the zero self -\nconsistent elastic scattering cross -section, the real and \nimaginary parts. The study of the zero -temperature elastic Contreras  et al. 5661  \n𝑣𝑘     = grad (𝐸) \nscattering cross -section was fi rstly proposed in (Pethick \nand Pines, 1986) and used with a specific disorder \nparametrization, i.e. the inverse dimensionless strength c \nand the impurity density Γ+ in (Schachinger and Carbotte, \n2003) and references therein for isotropic Fermi surfaces.  \n \nAdditionally, extended studies of the zero elastic \nscattering cross -section were recently performed in \n(Contreras and Moreno, 2019) to calculate  ͠ω using two \ndifferent numerical self -consistent routines for isotropic \nFS and a linear nodes OP with different c and Γ+ in order \nto establish differences in numerical routines. The work \nby Contreras and Osorio (2021) was performed for a \nlinear nodes HTSC model using a ti ght binding \nparametrization for three different collisional regimes. For \nthe for the Miyake Narikiyo quasi -nodes triplet OP \n(Miyake and Narikiyo, 1999), the tight -binding \ncalculation of  ͠ω was performed for ten values of the \ninverse strength parameter c showing that the cross -\nsection is mostly in the unitary limit and few times in the \nintermedium limit ( Contreras et al., 2022a ).  \n \nIn (Contreras et al ., 2022b), the calculation was \nperformed as function of the Fermi energy and it was \ndistinguished the point nodes model from the quasi -nodal \noriginal model in the elastic scattering cross -section. In \n(Contreras et al ., 2022c), the dependence on the zero \ntemperature Δ0 was modeled self -consistently for a triplet \nOP finding that the imaginary elastic scattering cross -\nsection is always positive and fits well in the unitary limit. \nFinally, the quasi -nodal model was contrasted with the \nlinear OP behavior by fixing the Fer mi energy and the \nzero-temperature superconducting gap in other to see the \ninterplay between different kind of quasiparticles \n(Kaganov and Lifshits, 1989) in the elastic scattering \ncross -section (Contreras et al., 2022d, 2022e).  \n \nIf we are dealing with mo re than one Fermi surface sheet, \nthe DoS is calculated according to equations such as  \n \n𝑁()\n𝑁𝐹=𝑝𝛾𝑁𝛾 \n∆0𝛾 +𝑝𝛼,𝛽𝑁𝛼,𝛽 \n∆0𝛼,𝛽  \n \n \nwhich is suitable for strontium ruthenate in a non -self-\nconsistent way using a TB parametrization aiming at \nfitting experimental low -temperature data such as \nultrasou nd attenuation in the superconducting state (T) \n(Lupien et al., 2001; Contreras et al., 2004), the electronic \nsuperconducting thermal conductivity (T) (Tanatar et al., \n2001; Contreras, 2011) and the electronic \nsuperconducting specific heat C(T) (Nishizak i et al ., \n1998; Contreras et al ., 2014) with relatively clean \nsamples. Details of the original use of pγ and pα/β are \nfound for the Sr 2RuO 4 normal state viscosity calculation \nin (Walker et al., 2001). The other work of relevance for \nexperimental fittings in strontium ruthenate can be found in (Nomura, 2005;  Taniguchi et al., 2015;  Wu and Joynt, \n2001; Zhitom irsky and Rice, 2001).  \n \nThe equation used to calculate the DoS in dirty \nsuperconductors when taking into account the reduced \nphase space is the following one (Devi et al., 2021b):  \n \n𝑁 𝜔  \n𝑁𝐹= ℜ(𝜔 )\n 2 𝜌𝑘  1+ 𝑎𝑘\n𝜌𝑘 𝐹𝑆+  ℑ 𝜔  \n 2 𝜌𝑘  1− 𝑎𝑘\n𝜌𝑘 𝐹𝑆 \n  \n(3) \n \nwhere ℜ(͠ω) and ℑ(͠ω) are the coordinates in the reduced \nphase s pace, and the other symbols are ak = ℜ(͠ω)2 – ℑ(͠ω)2 \n– Δk2, b = 2ℜ (͠ω)ℑ(͠ω), and ρk = (ak2+ b2)1/2. Equation (3) is \na very suitable because it is directed related to the reduced \nphase space.  \n \nThe tight -binding Fermi averages replacing the sum \n“Σkx,ky(…)” are performed using a weight in energy instead \nof the sum, i.e.  \n \n  … ∞\n𝑘= 1\n4 𝜋2  𝑑 𝑆𝐹\n|𝑣𝑘     |  𝑑 𝐸 … =  ⋯ 𝐹𝑆  \n,  \n \nwhere E is the energy of the normal state, the Fermi \nvelocity is given by the gradient , the \nsurface k-space element is expressed as dSF = (dkx2 + \ndky2)1/2, and the normal state density  of states at the Fermi \nlevel is calculated by Ashcroft and Mermin (1976) as \nfollows:  \n \n𝑁𝐹= 1\n4 𝜋2  𝑑 𝑆𝐹\n|𝑣𝑘     |  \n  \n \nNormal state DoS and its evolution according to the \nFermi energy  \nIn this section, we demonstrate how the evolution of the \ndensity of states (DoS) can be numerically  modeled by \nvarying the Fermi energy parameter in equation (2). \nUsing the results shown in Figure 1, it is found that the \nimplicit Fermi surface with ξi,k(kx, ky) = 0 evolves from \nhaving a behavior with a mesh centered at (0, 0) \ncoordinates when the Fermi energy has a negative sign \nand the hoping coefficient has a positive sign, to a \ndifferent behavior when both the Fermi energy and the \nhoping parameter have positive values, and the Fermi \nsurface for this case is centered in four pockets at the ± \n(N, N) cor ners. The values used for the model are listed in \nTable 1. The implicit Fermi surface evolution in the N × N \nmesh is sketched in Figure 1, where two well defined \nbehaviors are seen.  \n \nFor the values of the parameters listed in Table 1, two of \nthe four impl icit plots are centered at zero point (when the \nhoping parameter value is t = 0.20 meV, i.e. the green and Canadian Journal of Pure and Applied Sciences  5662  \nred implicit plots). Meanwhile, the other two are centered \nat the corners of the square (when the hoping parameter \nvalue is t = + 0.40 meV, i.e. the  yellow and blue implicit \nplots). In addition, in this section, the normal density of \nstates is calculated using equation (1) with the same \nparameters listed in Table 1. The results are shown in \nFigure 2.  \n \nFig. 1. The evolution o f ξi,k(kx, ky) = 0 in a 400 × 400 \npoints mesh. The TB values for each color are listed in \nTable 1.  \n \n \nFig. 2. The density of states N(E) for 4 values of EF in a \n400 × 400 points mesh. The TB values for each color are \nfrom Table 1 and Figure 1.  \n As it can b e seen in Figure 2, the density of states with \nFermi energy values close to zero (blue and green colors) \nhave almost an electron -hole symmetry behavior. Also, \nthose where the two tight binding coefficients have \nsimilar order of magnitude are less symmetric  (yellow and \nred colors). If the hoping parameter is smaller as happen \nfor the red and green cases, both centered at the corners, \nthe DoS have more available quantum states than when \nthe Fermi surface is centered at zero points. This is \nimportant and shows  how the quantum behavior of those \nceramics having tight -binding parametrization (the red \nand green cases) (for instance, some HTSC in its normal \nstate) represent a more difficult quantum interpretation \ncompared with the blue and yellow cases that mostly \nrepresent metallic alloys.  \n \nThe drop to zero of the normal DoS can be understood in \nterms of a number of partial available quantum states \nconstant as it has been explained in the introduction. \nTherefore, the DoS being a coefficient of the partial \nnumber of  states Ω (see the introduction of this work) \nbecomes negligible and probably other degrees of \nfreedom start to play a more important role at those \nenergies where N(E) drops to a zero value, meaning a \nconstant number of partial states Ω. \n \nImpurity supercon ducting DoS and its evolution as \nfunction of the scattering strength and disorder  \nIn the superconducting state, we can compare two OP \nmodels. The 2D TB line nodes used to model the \nstrontium doped lanthanum copper oxide superconductor \nwith a Tc  44.35 K f or a polycrystalline sample (Bednorz \nand Müller, 1986; Kastner et al., 1998; Xiao et al., 1987) \nmodeled  with the parameters t = 0.2 meV, εF = –0.4 meV \nand Δ0 = 33.9 meV (Yoshida et al., 2012) shadowed gray \nin the 1st line of Table 1. The linear nodes OP has even \nparity i that belongs to the irreducible representation B 1g \nof the D 4h point symmetry group (Scalapino, 1995; Tsuei \nand Kirtley, 2 000).  \n \nThe triplet case is represented in this section for the 2D γ-\nsheet Miyake Narikiyo quasi -point nodes model (Larkin, \n1965) for strontium ruthenate, and Tc  1.5 K for a bulk \nclean sample (Maeno et al., 1994; Rice and Sigrist, 1995) \nwith the followin g values: t = 0.4 meV, εF = –0.4 meV, \nand Δ0 = 1.0 meV shadowed gray in the 4th line of Table \n1. In this case, the OP has odd parity i that belongs to the \nirreducible representation E 2u of the D 4h point symmetry \ngroup with GL coefficients (1, i) and 2D basis (sin( kx), \nTable 1. The evolution of the implicit Fermi surface for different sets of TB parameters.  \nFermi energy εF Hoping parameter t Centered at N × N mesh points  ξi,k(kx, ky) = 0  \n– 0.40 meV  + 0.20 meV  ± (400, 400)  Red color  \n– 0.04 meV  + 0.20 meV  ± (400, 400)  Green color  \n+ 0.04 meV  + 0.40 meV  (0, 0)  Blue color  \n+ 0.40 meV  + 0.40 meV  (0, 0)  Yellow color  \n Contreras  et al. 5663  \nsin(ky)) (Miyake and Narikiyo, 1999; Walker and \nContreras, 2002; Sigrist, 2002; Contreras et al ., 2016). \nWe would like to point out the intriguing 2D electronic \nnature of this material as is pointed out in ( Maeno et al., \n1997 ).  \n \nIn addition, to take into account the nonmagnetic disorder \neffects in both order parameter models inside the \nsuperconducting DoS and the residual DoS, we use:  \n \n The Born limit given by l kF >> 1 or l a–1 >> 1, where \nl is the mean free path, a is the lattice parameter, and \nkF is the Fermi momentum.  \n   The intermediate scattering regime with l kF ~ l a–1 > 1. \n The unitary limit where holds that l kF ~ l a–1 ~ 1.  \n \nWe use the parameter  c which is inverse to the scattering \nstrength U 0 to describe the dispersion lim its for both OP \nnumerical models. During the 1970s and the 1980s, the \nformalism and some phenomenology of the physics for \nnonmagnetic impurity scattering in normal metals and \nalloys were described in (Ziman, 1979) from a work \nfirstly proposed by Edwards (1 958). In monography \n(Lifshits et al ., 1988), it was noticed that in studying \nmetallic alloys, there are singularities in the DoS for \ndisordered systems such as those involving nonmagnetic \nimpurities, or stoichiometric nonmagnetic atomic \npotentials U 0, they  pointed out that keeping a 1st power in \nimpurity concentration suffixes the calculation. We \nconsider also instructive to mention that the ARPES \ntechnique is related to a fundamental equation “the Fermi \nGolden Rule” . A robust introduction to ARPES can be \nfound in (Palczewski, 2010).  \n \nNumerical results for the self -consistent DoS with \nnonmagnetic disorder  \nFigure 3 shows the density of states (DoS) calculated \nnumerically using equation (3) that includes the zero \nelastic scattering cross -section reduced phase  space. The \nresulting plot is the density of states N(͠ω)/NF as a function \nof the normalized frequency  ω/Δ0. In the clean limit \n(without impurities), the parameter for the concentration \nof disorder Γ+ is normalized by the zero -energy gap and \ndenoted as ζ = Γ+/Δ0. It is observed that, there are no \ndressed fer mionic quasiparticles if there is no disorder. \nThus, the number of occupied states starts to grow \nlinearly, since it represents a nodal line OP (Scalapino, \n1995) until an energy value equal to that of the zero \nsuperconducting gap  is reached. At ω = Δ0 there is a \ndrastic change in slope that resembles on the right side the \nshape of a BCS superconductor (Bardeen et al ., 1957), \nwhere the DoS represents a singularity for ω = Δ0 and is \nequal to zero below the gap, the other type of suppression \nin superconducting  states (when there are magnetic \nimpurities) is explained following a physical kinetic \nanalysis in (Ambegaokar and Griffin, 1965).  \n \nFig. 3. The superconducting density of states (DoS) for a \ntight binding line nodes OP if ζ = 0. There aren’t residual \nstates at zero energy.  \n \nFigure 4 shows five curves calculated for the values of \ndisorder with ζ = 0.001, 0.005, 0.010, 0.015, and 0.020 in \nthe unitary limit when c = 0. However, at zero frequency, \nit is observed that there is a residual density of states for \nall 5 values of ζ, which is a consequence of the presence \nof nonmagnetic impurities in the reduced phase space for \nthe line nodes model, which behavior is modified by the \ninverse scattering lifetime. It is noticed that the higher \nconcentration of impurities , the number of states at the \nfrequency value of ω = 0.0 meV is also higher. The \nresidual normalized DoS is bigger for the thinner line ( ζ = \n0.020) compared to other four curves, the thicker line ( ζ = \n0.001) has still a considerable number of residual DoS.  \n \n \nFig. 4. The superconducting DoS for the line nodes OP. \nThe residual DoS shows significant dressed quantum \nlevels at zero energy due to the strong scattering potential.  \n Canadian Journal of Pure and Applied Sciences  5664  \nFigure 5 shows the superconducting DoS for a weaker \nscattering potential with c = 0.4 and the disorder \nparameter ζ = 0.001, 0.005, 0.010, 0.015, and 0.020. The \nstrength when c = 0.4 was established numerically from \nthe analysis of the zero -temperature elastic scattering \ncross -section as the Born limit for a linear OP (Contreras \nand Osorio, 2021). However, in this cas e the residual \nN(0)/NF weakly increases as ζ increases and it is \nnoticeably smaller compared to the N(0)/NF values shown \nin Figure 2 that corresponds to the unitary limit, where \nthere is a strong elastic scattering, suggesting that a strong \nnonmagnetic dis persion potential produces more occupied \nstates than weaker scattering potentials at zero energy. \nThis can be defined as the signature for the unitary state \nin the residual DoS analysis compare with the nonlocal \nminimum observed in the analysis of the imag inary part \nof the elastic cross -section ℑ[͠ω](ω + i0+) (Contreras and \nOsorio, 2021).  \n \n \nFig. 5. The DoS for the nodal line OP in strontium doped \nlanthanum compound  with 5 disorder values. The residual \ndensity is small when compared to the results shown in \nFigure 4. There are only a few quantum s tates available in \nthe hydrodynamic limit.  \n \nThe next calculation is performed for the γ sheet of the \ntriplet superconductor strontium ruthenate with an OP that \nbelongs to the irrep E 2u of the D 4h point group. Figure 6 \nshows six curves calculated for differ ent values of the \nnormalized disorder ζ = Γ+/Δ0. The simulation was \nperformed for ζ = 0.00, 0.01, 0.02, 0.03, 0.10, 0.20 in the \nunitary limit when the inverse elastic scattering parameter \nc = 0, according to the analysis (Miyake and Narikiyo, \n1999; Maeno et al., 1994; Rice and Sigrist, 1995; Walker \nand Contreras, 2002; Sigrist, 2002; Contreras et al., 2016; \nMaeno et al., 1997 ). Triplet OP are suitable to analyze in \nthis limit due to strong nonmagnetic strontium potential. \nA contrast concerning the previous case shown in Figures 3, 4, and 5 is that the dimensionless disorder parameter \nfor the triplet case is one order of magnitude bigger than \nthe dimensionless singlet OP.  \n \nThis is partially explained because the γ-sheet uses an \nexperimental zero gap value of Δ0 = 1.0 meV that is an \norder of magnitude smaller that the lines ’ nodes OP with a \nzero gap Δ0 = 33.9 meV and therefore, the triplet case has \na smaller reduced phase space for scattering events. \nAdditionally, in the triplet compound,  Sr atoms are \nlocated in the lattice with an additional nonmagnetic \nimpurity level in the energy zone. Thus, Sr atoms are part \nof the D 4h tetragonal structure and also are the scattering \ncenters that explains the stronger pair break ing \nmechanism and the additional impurity level.  \n \nFigure 6 shows that in the absence of impurity levels \n(black line where ζ = 0), we do not observe nonzero \nvalues for the density N(͠ω)/NF as happens for BCS \nsuperconductors (Bardeen et al ., 1957). It means that \nwhen doing calculations that involved triplet states and \nscattering is excluded, there is no pair breaking effects \nand bosonic quasiparticles dominate the behavior below \nthe transition temperature. It occurs numerically below \n0.83 meV in this calculation. Generally speaking, this \nvalue depends on the choice of the TB parameters, i.e. \nhow close will be the Fermi surface t o the zero gap value \nof Δ0 in the MN model. We have used parameters from \nthe 4th column in Table 1 (shadowed gray) to  calculate the \nDoS.  \n \nFor the parameter value of ζ = 0.01 , we observe still an \nintermedia well -formed BCS gap from frequencies in the \ninterval (0.4, 0.83) meV and w ith a small quantity of \nquantum states at low frequencies due to strong scattering \nthat happens in the unitary regime when the reduced \nphase space is activated with a nonzero imaginary part of \nthe cross -section, and the mean free path l is comparable \nto th e magnitude of the inverse Fermi length | kF|-1, or to \nthe value of the lattice parameter a. To illustrate what \nhappens numerically we show in the insert on the upper \nleft side of Figure 6, the reduced phase space calculation, \nand it is observed the followi ng: for the case of ζ = 0.01 , \nthe imaginary part of the cross -section dies inside of the \nsuperconducting phase. Therefore, it does not become a \nnormal metal and could be a signature of an \nantiferromagnetic state as happens to the \nantiferromagnetic insulato r LaCuO (Kastner et al., 1998).  \n \nFor the impurity value of ζ = 0.05,  it agrees with an \ninhomogeneous phase that is the Miyake -Narikiyo tiny \ngap inside of which there are not fermionic quantum \nlevels. Henceforth, the tiny Miyake -Narikiyo tiny gap \npredicted and used to explain microscopically the \nbehavior of triplet pairing superconductors such as \nstrontium ruthenate (Miyake and Narikiyo, 1999), is \nshown in Figure 6 for an impurity value of ζ = 0.05  as Contreras  et al. 5665  \nwas also observed in ( Contreras et al ., 2022a, 2022b, \n2022c) using the imaginary part analysis of the scattering \ncross -section ℑ[͠ω](ω + i0+). The DoS calculation also agrees with the one by Contreras et al . (2022a)  in the \nsense that only the unitary limit persists in Sr 2RuO 4. It is \nnoticed that the higher conc entration of impurity levels \n \nFig. 6. The DoS for the quasi -point nodes  with 6 values of nonmagnetic impurity doping including the clean case. The \nresidual N(0) is found for 5 nonzero ζ-values. The tiny MN gap is found for ζ = 0.05.  \n \n \nTable 2. The C0 parameter and the residual DoS. It is summarized for both irrep,  B1g and E2u and the scattering limits.  \n \nResidual \ndensity of \nstates RDoS \nTB formalism  Expressions for C0 in the \nBorn and interme dia  limits  Expressions for C0 in the \nunitary limit  Residual DoS \nfor the Born and \nintermedia \nregimes  Residual DoS for \nthe unitary limit  \nSinglet parity \nlinear nodes \nOP \n𝐶0=𝜋𝜂𝑐   𝐶02+ 𝜙𝑘2 \n𝐶0 𝐹𝑆−1\n  \n𝐶0=𝜋𝜂𝑐  𝐶02+ 𝜙𝑘2 \n𝐶0 𝐹𝑆  \n𝑁 0 \n𝑁𝐹= 𝐶0\n𝜋𝜂𝑐  \n𝑁 0 \n𝑁𝐹=(𝐶0\n𝜋𝜂𝑐)−1  \nTriplet parity \nquasi -point \nnodes OP  \n𝐶0=𝜋𝜂𝑐   𝐶02+ |𝑑𝑘2| \n𝐶0 𝐹𝑆−1\n  \n𝐶0=𝜋𝜂𝑐  𝐶02+ |𝑑𝑘2| \n𝐶0 𝐹𝑆  \n𝑁 0 \n𝑁𝐹= 𝐶0\n𝜋𝜂𝑐  \n𝑁 0 \n𝑁𝐹=(𝐶0\n𝜋𝜂𝑐)−1  \n \n \n Canadian Journal of Pure and Applied Sciences  5666  \ngiven by the values of ζ = 0.10, 0.15, 0.20 , the larger \nnumber of occupied quantum states exists at both  the low \nand high frequencies . This is a consequence of having an \nincreasing reduced phase space and therefore more \npossibi lities for scattering events . However,  it is still \nsmall compare d with the HTSC nodal lines ’ OP case \n(Contreras et al., 2022d).  \n \nTherefore, for the disorder parameter values of ζ = 0.10, \n0.15, 0.20 , there are normal state DoS levels available, \nand the peak  at ω = 1.4 meV considerable  reduces with a \ntendency where N(ω) ~ NF above Tc. Therefore, in the \nsuperconducting triplet model, we observe two phases, \none tiny phase (the MN gap) without electronic levels \n(BCS type) and another with normal -state dressed \nquantum levels, contrasting with the strontium substitute \nlanthanum cuprate calculation, where dressed fermionic \nlevels are found.  \n \n \nNumerical results for the residual density of states and \nthe pair breaking  \nThe residual equation for the density of states is  \ntheoretically obtained by setting up the real frequency as \nan imaginary number, i.e. ω = iα in equation (4). Thus, \nwe have   ͠ω = ω + iα with α being a new disorder \nparameter (0 ≤ α ≤ 1). This does not require a self -\nconsistent routine but it needs a fixed -point numerical \ncalculation. In such a case, we get the following general \nequation fo r the residual DoS ( N(0)/NF) with a new \ndimensionless disorder parameter C0 = α/Δ0:  \n \n𝑁 0 =𝑁𝐹  𝐶0\n 𝐶02+ Δ02 𝑘𝑥,𝑘𝑦    𝐹𝑆 \n  \n(4) \n \nwhere equation (4) depends on the symmetry of the OP, \nthe elastic scattering regime of the imaginary part of the \nzero-temperature elastic scattering cross -sectio n, i.e. the \nunitary, intermedia, and Born cases; and finally also \ndepends on the Fermi surface average, so we can control \nthe residual DoS the same way as we did for the zero -\ntemperature elastic scattering cross -section and the \nsuperconducting DoS.  \n \nThere  is the functional dependence Tc/Tc0 = f(N(0)/NF), \nwhere Tc0 indicates the transition temperature without \ndisorder and Tc is the transition temperature including \ndisorder in the Larkin equation (Larkin, 1965) for \nsuppression of impurity states in the case of nonmagnetic \ndisorder when the critical temperature Tc decreases as a \nfunction of the pair breaking parameter ηc. So, we have  \n \nln𝑇𝑐\n𝑇𝑐0=𝜓 1\n2 − 𝜓 1\n2+ 𝜂𝑐 =𝜓´ 1\n2 Γ+\n2𝜋𝑇𝑐 \n  \n(5) \n In equation (5), the superconducting pair breaking \nparameter is defined as ηc = –(4π/1.764)  × ln(Tc/Tc0), \nwhere Tc0 is the transition temperature for a clean \nsuperconductor ( α = 0), Tc represents the transition \ntemperature  for dirty superconductors, i.e. α ≠ 0, ψ(x) is \nthe digamma function, and ψ′(x) is the derivative of the \ndigamma function. Table 2 lists the  expressions that can \nbe obtained if the tight binding approximation is \naccounted for and they do not differentiate from the \nisotropic case with angular dependence of the Fermi \nsurface.  \n \nThe analytical expressions for the calculation of the \nrelationship Tc/Tc0 = f(N(0)/NF) for the cases plotted in \nthis section can be obtained after some long algebraic \nmanipulations using equations (7) and (8). The difference \nfrom the previous work by Sun and Maki (1995) and \nMomono and Ido (1996) is that the Fermi surface a verage \ndepends on more parameters <…> FS, the basis function ϕk \nin the case  of a scalar line nodes OP, and the complex \ntriplet vector OP dk are the same used for the density of \nstates calculation in the previous paragraph, the tight \nbinding parameters are t hose shadowed gray in Table 1. A \ndiscussion with second and third harmonics for the triple \nOP is given in Miyake -Narikiyo original work (Miyake \nand Narikiyo, 1999).  \n \nTheoretically it is known that nonmagnetic impurities \ndestroy superconductivity in unconve ntional \nsuperconductors (Larkin, 1965; Sun and Maki, 1995) and \nreduce the value of the transition temperature Tc0. The \nresidual DoS changes as a function of Tc, as it is shown \naccordingly to the data listed in Table 2. Noticeable in \nthis work is that the p arameter C0 depends on the Fermi \nsurface averages. In order to numerically evaluate the \npolynomic expressions involved, it is more convenient to \nsimplify the analysis to the three cases: The Born, \nintermedia and unitary scattering regimes (see in Table 2 \nfor a summary of the equations involved).  \n \nThe results of the numerical calculation using equations \ntaken from Table 1 are shown in Figure 7 for the case of \nlinear nodes OP. At Tc/Tc0 = 0 the largest residual DoS \nvalue for both the cases is obtained (for u nitary and Born \nlimits). On the other hand, as the value of N(0)/NF \nincreases, the ratio Tc/Tc0 can rapidly reach a zero value, \nfaster for a weak scattering Born nonmagnetic potential \nlimit than for the unitary case. In addition, we see in \nFigure 5 that th e unitary limit presents a curve that always \nhas the same sign in slope; meanwhile the Born limit \nchanges it signs and even has a linear behavior \ndependence for N(0) ~ 0.5 NF.  \n \nFigure 8 compares the unitary limit of the OP used with \nmeasurements taken from  specific heat in the compound \nstrontium doped lanthanum ceramic for different \nexperimental values of strontium doped obtained Contreras  et al. 5667  \nexperimentally (Momono and Ido, 1996). Strontium \ndoping has been extensively studied in the cuprate La 2-\nxSrxCuO 4 for smaller orde rs of strontium concentration \n(Yoshida et al., 2012; Momono and Ido, 1996). The color \npoints with the impurity atoms are from values of specific \nelectronic heat capacity C(T) in the superconducting state, \nwhere in Figure 6 the gray color corresponds to x = 0.10, \nthe green color to x = 0.18, the blue color to x = 0.20, and \nthe red color to x = 0.22 (Momono and Ido, 1996). We see \na tendency for the experimental red points with x = 0.22 \ncorresponding to our reduced value of ζ = 0.02 in \ncorrespondence with both  the unitary theoretical residual \ndensity of states and the self -consistent unitary case  of the \nprevious section (Fig.  4).  \n \nIn Figure 9, the value of Tc/Tc0 approaches zero slowly if \nthe fixed -point calculation is done for the unitary limit, as \nin the cas e of line nodes. Meanwhile, the unitary triplet \nOP presents a curve that always has the same shape and \nslope; the intermediate limit changes its slope weaker, \nslightly contrasting with the OP nodal line situation in \nFigure 7, where there are intermediate s cattering events. \nAlso, Figure 9 shows the comparison of the unitary limit \nof the triplet OP with the experimental values of the \ncompound strontium ruthenate (Miyake and Narikiyo, \n1999; Nishizaki  et al., 1998). In this case, we recall that \nstrontium atoms add an additional impurity level since \nthey are part of the crystal structure. The tight binding \ncalculation here confirms the MN original results (Miyake \nand Narikiyo, 1999).  \n \n \n \nFig. 7. The numerical calculation of the residual DoS in \nthe case of the sin glet line nodes in the Born and the \nunitary limits, following equations listed in Table 1.  \n \n \nFig. 8. The fits of the residual DoS for the unitary limit \nwith data from La 2-xSrxCuO 4 (Rice and Sigrist, 1995). \nDifferent colors correspond to different hopping \nparameters.  \n \n \nFig. 9. The numerical calculation of the residual DoS for \nthe triplet OP in intermedia and unitary limits. The blue \ncolor shows the experimental fits corresponding to \nSr2CuO 4 (Zhitomirsky and Rice, 2001; Walker and \nContreras, 2002).  \n \nCONCLUS ION  \n \nThis work was aimed at revisiting the calculation of the \ndensity of states (DoS) and the residual DoS with \nfrequency values taken from a self -consistent calculation \nof the real and imaginary parts of the elastic scattering \ncross -section with a tight b inding framework and for two \norder parameter models. The strontium -substituted \nlanthanum cuprate line nodes case (Scalapino, 1995), and Canadian Journal of Pure and Applied Sciences  5668  \nstrontium ruthenate symmetry breaking triplet model \n(Miyake and Narikiyo, 1999), where the DoS levels come \nfrom calculat ions in the reduced phase space when \nnonmagnetic pair breaking disorder destroys \nsuperconductivity. Self -consistent calculations are in \ngeneral very computing demanding as stated in (Jansen et \nal., 1991).  \n \nIt was concisely reviewed main concepts and the \nimportance of the DoS in ab-initio calculations for novel \nmaterials and also outlined the details of the \ncomputational approach. Also, a detailed DoS calculation \nof the normal state with a tight binding model was \nperformed and interpreted in the terms of the  degrees of \nfreedom. In addition, the calculation of the \nsuperconducting DoS was self -consistently performed for \nsinglet and a triplet OP using the zero -temperature elastic \ncross -section for three scattering regimes. Finally, the \nbehavior of the residual D oS was addressed for both \nmodels using the formalism following the Larkin equation \n(Larkin, 1965).  \n \nRECOMMENDATIONS  \n \nIt is recommended to use the tight -binding and other ab-\ninitio frameworks to the study of numerical simulations as \nthe density of states an d other physical properties in \ndoped  unconventional superconductors as recently it has \nbeen done in the following work (Contreras and Osorio, \n2021; Contreras et al ., 2022b, 2022c; Käser, 2021; \nPhotopoulos and Frésard, 2019 ; Kang et al., 2022 ; Yu et \nal., 2023). \n \nConcerning the  approach developed  in this work  to obtain \nthe imaginary part of the self -energy used to calculate the \nDoS, it  is worthy to mention  that several  interesting  \npioneer studies about the decay of single -particle \nexcitations in weak -coupling  superconductors  and the \ncalculation of the conductivity scattering rate were \nconducted  in the following work ( Prabhu , 1978;  Marsiglio \nand Carbotte, 1997 ). Referring  to the unitary limit , \ninteresting results ha ve been  obtained  in the work  by \nDomański  (2011 , 2016 ). We suggest that it would be \ninteresting to  compare  the Domański results with the \nmore recently work by Contreras  et al. (2023)  in or der to \nsee how the supercoducting zero energy gap might \ninfluence the DoS.  \n \nACKNOWLEDGEMENT  \n \nThis research did not receive any specific grant from \nfunding agencies in the public, commercial, or not -for-\nprofit sectors. The authors acknowledge discussions about \nthe pre sentation of the results with Dianela  Osorio.  \n \n \n REFERENCES  \n \nAmbegaokar, V. and Griffin, A. 1965. Theor y of the \nthermal conductivity of superconducting alloys with \nparamagnetic impurities. Physical Review. \n137(4A):A1151. DOI: \nhttps://doi.org/10.1103/PhysRev.137.A1151  \nAshcroft, NW. and Mermin, ND. 1976. Holt, Rinehart \nand Winston. New York, USA. pp.826. 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Cambridge University Press, Cambridge -\nLondon -New York -Melbourne, England, UK. pp.525.  \n \nReceived: April 25, 2023; Accepted: June 1, 2023  \n \nCopyright©2023, Contreras  et al. This is an open access article distributed under the Creative \nCommons Attribution Non Commercial License, which permits unrestricted use, distribution, \nand reproduction in any medium, provided the original wor k is properly cited.  \n \n \n \n \n \n \n "    },    {        "title": "2202.13963v2.Detection_of__d__1__otimes_d__2___Dimensional_Bipartite_Entangled_State__A_Graph_Theoretical_Approach.pdf",        "content": "Detection of d1\nd2Dimensional Bipartite Entangled State: A Graph Theoretical Approach\nRohit Kumar, Satyabrata Adhikari1,\u0003\n1Delhi Technological University, Delhi-110042, Delhi, India\nBraunstein et. al. have started the study of entanglement properties of the quantum states through graph\ntheoretical approach. Their idea was to start from a simple unweighted graph Gand then they have defined the\nquantum state from the Laplacian of the graph G. A lot of research had already been done using the similar\nidea. We ask here the opposite one i.e can we generate a graph from the density matrix? To investigate this\nquestion, we have constructed a unital map \u001esuch that\u001e(\u001a) =L\u001a+\u001a, where the quantum state is described\nby the density operator \u001a. The entries of L\u001adepends on the entries of the quantum state \u001aand the entries are\ntaken in such a way that L\u001asatisfies all the properties of the Laplacian. This make possible to design a simple\nconnected weighted graph from the Laplacian L\u001a. We show that the constructed unital map \u001echaracterize the\nquantum state with respect to its purity by showing that if the determinant of the matrix \u001e(\u001a)\u0000Iis positive then\nthe quantum state \u001arepresent a mixed state. Moreover, we study the positive partial transpose (PPT) criterion\nin terms of the spectrum of the density matrix under investigation and the spectrum of the Laplacian associated\nwith the given density matrix. Furthermore, we derive the inequality between the minimum eigenvalue of the\ndensity matrix and the weight of the edges of the connected subgraph of a simple weighted graph to detect the\nentanglement of d1\nd2dimensional bipartite quantum states. Lastly, we have illustrated our results with few\nexamples.\nPACS numbers: 03.67.Hk, 03.67.-a\nI. INTRODUCTION\nEntanglement [1] is one of the key topic in quantum informa-\ntion theory that lies at the heart of quantum mechanics. This\nfeature of quantum mechanics has no classical analogue and\nplay a vital role in enhancing the power of quantum computa-\ntion [2]. It also acts as a useful resource in many quantum in-\nformation processing tasks such as quantum teleportation [3],\nquantum superdense coding [4], quantum cryptography [5, 6]\netc. Thus, it is important to generate the entangled state in the\nlaboratory for processing the quantum information tasks. But\nthe experimentalist may face the problem in identifying the\ngenerated state: whether the state at the output of the experi-\nment is entangled or not? This type of problem is known as\nentanglement detection problem.\nIt is one of the prime task for the researcher to develop the\ncriterion for the detection of entanglement. A lot of research\nhave already been carried out in this direction and as a con-\nsequence different criterions such as partial transposition cri-\nterion [7, 8], computable cross norm or realignment criterion\n[9, 10], reduction criterion [11] have been emerged to detect\nthe entangled state. Other criterion for the detection of entan-\nglement can be found in [12].\nThere exist another approach to study the problems in quan-\ntum information theory - the graph theoretical approach.\nAlong this line of research, Braunstein et.al. [13] have started\nwith any graph Gand then associate a specific mixed quantum\nstate with it. Mixed state is described by the density matrix\nand therefore they called it as the density matrix of graph G.\nIn this way, they have introduced the concept of density matri-\nces of graph to study the graphical representation of quantum\nstates and their properties. The entanglement properties of\n\u0003Electronic address: rohitkumar@dtu.ac.in, satyabrata@dtu.ac.inthe mixed density matrices obtained from the combinatorial\nLaplacians has been studied in [14]. A. Cabello et.al. [15]\nused graph to characterize the correlations with respect to dif-\nferent sets of probabilities obtained through non-contextual\ntheories, quantum theory and more general probabilistic theo-\nries. M. Ray et.al. [16] have taken graph theoretical approach\nto find the minimum quantum dimension required for per-\nforming given quantum task by identifying quantum dimen-\nsion witnesses. The graphical characterization of the entan-\nglement properties of the grid states has been studied in [17].\nThe separability problem of bipartite quantum states generat-\ning from graphs has been studied in [18].\nIn this work, we will study the entanglement detection prob-\nlem using the graph theoretical approach. But our approach\nis different from the other graph-theoretical approaches that\nare existing in the literature. In the existing graph theoreti-\ncal approaches, the density matrices are generated from the\nLaplacian of the given graph but in our approach, we have\nconstructed an unital map that take the density matrices as the\ninput and at the output, it provides the input density matrix to-\ngether with other matrix L. Later, we prove that the matrix L\nsatisfies all the properties of Laplacian. Then we can construct\na graph corresponding to each Laplacian at the output of the\nmap. Further, we will show that the entanglement property of\nthe quantum states can be studied using the eigenvalues of the\ngenerated Laplacian.\nThe work can be distributed in different sections in the follow-\ning way: In section-II, we discuss about the basics of graph\ntheory and its terminology. In particular, we will talk about a\nsimple weighted graph that may be associated with the quan-\ntum state described by the density matrix. In section-III, we\ndescribe briefly the tensor formalism in quantum information\ntheory. In section-IV , we revisit few results that have been\nobtained earlier in the literature. In section-V , we construct\na unital map and study the properties of the map. Also, we\nhave discussed the physical interpretation of the total degreearXiv:2202.13963v2  [quant-ph]  8 Feb 20232\nof the graph corresponding to the Laplacian generated at the\noutput of the map. In section-VI, we derive the necessary con-\ndition to test whether a given quantum state described by the\ndensity operator \u001ais either a pure state or mixed state. In\nsection-VII, we study the PPT criterion in terms of the eigen-\nvalues of the given d1\nd2dimensional bipartite state and the\ncorresponding Laplacian. In section-VIII, we study the entan-\nglement properties using the graph terminology.\nII. BASICS OF GRAPH THEORY AND THE ASSOCIATED\nTERMINOLOGY\nLetG= (V;E)be a simple weighted graph (may or may not\nbe connected) with vertex set V=fv1;v2;:::::;vngand edge\nsetE. If the two vertices viandvjare connected by an edge\neijthen we write i\u0018j. A graph is said to be a simple graph\nif it has no loops and multiple edges. If we assigned a weight\nwi;jto each edge ei;jin a graphGthen the graph is known\nas a weighted graph. The weight wi;jassociated to each edge\nis usually a positive number. If widenote the weight of the\nvertexvithen it can be defined as wi=P\nk\u0018iwk;i. Ifwi;j=\n1for all edges ei;j, then the graph will become an unweighted\ngraph. Let us now consider an unweighted graph G0which\nhasnverticesui; i= 1;:::;n . IfA(G0)denote the adjacency\nmatrix of the graph G0then adjacency matrix can be defined\nasn\u0002nmatrix [aij]whereaijis given by\naij= 1;ifi\u0018j\n= 0;ifi\u001cj (1)\nThe Laplacian matrix L(G0)of the graph G0may be defined as\nL(G0) = \u0001(G0)\u0000A(G0), where \u0001(G0)denote the diagonal\nmatrix whose diagonal entries represent the degree diof each\nvertexuiof the graph G0. Therefore, the Laplacian matrix\nL(G0) = [l0\nij]of an unweighted graph G0is given by\nl0\nij=di;ifi=j\n=\u00001;ifi\u0018j\n= 0;ifi\u001cj;i6=j (2)\nAnalogously, The Laplacian matrix L(G)of the weighted\ngraphGis defined as the n\u0002nmatrixL(G) = [lij], where\nlij=wi;ifi=j\n=\u0000wij;ifi\u0018j\n= 0;ifi\u001cj;i6=j (3)\nThe Laplacian matrix L(G)is a real symmetric matrix.\nUsing the fact that L(G)is a real symmetric matrix and\nGershgorin’s theorem, it can be shown that the eigenvalues\nofL(G)are non-negative real numbers [19]. Thus L(G)\nrepresent a positive semi-definite matrix.\nThere exist a vast literature [20–25] on Laplacian eigen-\nvalues and their relation to various properties of the simple\nunweighted graph but there exist few literatures [26–29]\nthat studied the properties of the Laplacian matrix of the\nsimple weighted graph. In this work, we will design thesimple weighted graph corresponding to the Laplacian gen-\nerated as a output of the constructed linear positive unital map.\nIII. TENSOR FORMALISM\nThe state space of a composite quantum system can be rep-\nresented as the tensor product of the state spaces of component\nquantum systems. For instance, if a composite quantum sys-\ntem contain two components and if the 1st and 2nd component\nare described by the Hilbert space HAandHBrespectively\nthen the composite quantum system is described by HA\nHB.\nGeneralising to n components j 1i,j 2i,j 3i,....,j ni, the\nstatej iof the total system is given by\nj i=nO\ni=1j ii=j 1i\nj 2i\n:::::\nj ni (4)\nFor a bipartite system, let us assume iA,i= 0;1;2;:::::;d 1\u00001\nbe a basis for the d1\u0000dimensional Hilbert space HAandjB,\nj= 0;1;2;:::::;d 2\u00001be a basis for the d2\u0000dimensional\nHilbert space HB. Then thed1d2statesjiAi\njjBiform a\nbasis for the composite space HAB=HA\nHB. Therefore,\nthe dimension of the Hilbert space HABisd1d2. For a two-\nqubit system, the four basis states are given by\nB1\u0011j00i\u0011j0i\nj0i=\u0012\n1\n0\u0013\n\n\u0012\n1\n0\u0013\n=0\nB@1\n0\n0\n01\nCA\nB2\u0011j01i\u0011j0i\nj1i=\u0012\n1\n0\u0013\n\n\u0012\n0\n1\u0013\n=0\nB@0\n1\n0\n01\nCA\nB3\u0011j10i\u0011j1i\nj0i=\u0012\n0\n1\u0013\n\n\u0012\n1\n0\u0013\n=0\nB@0\n0\n1\n01\nCA\nB4\u0011j11i\u0011j1i\nj1i=\u0012\n0\n1\u0013\n\n\u0012\n0\n1\u0013\n=0\nB@0\n0\n0\n11\nCA (5)\nAn arbitrary dimensional bipartite state j iAB2HABcan be\nexpressed as\nj iAB=X\ni;j\u000bi;jjiAi\njjBi;X\ni;jj\u000bi;jj2= 1 (6)\nThe concept of bases for tensor product spaces can be ex-\ntended from the arbitrary dimensional bipartite system to the\nmultipartite system.\nThe tensor product structure is linear and it also satisfies the\nassociative and distributive properties. But the commutative\nproperty is not satisfied by the tensor product.3\nIV . PRELIMINARY RESULTS\nIn this section, we recapitulate the important results obtained\nin the previous works and then we will use these results in the\nforthcoming sections.\nResult-1 [30, 31]: IfXandYdenote any two n\u0002nHermitian\nmatrices then we have\n\u0015min[X]tr[Y]\u0014tr[XY]\u0014\u0015max[X]tr[Y] (7)\nwhere\u0015min[X]and\u0015max[X]denote the minimum and max-\nimum eigenvalue of X.tr[:]denote the trace of the matrix [:].\nResult-2: LetX;Y2Mnbe Hermitian matrices and let the\nitheigenvalues of X,YandX+Ybe denoted by \u0015i[X],\n\u0015i[Y]and\u0015i[X+Y]. Fork= 1;2;3;::::;n , the Weyl’s in-\nequality [32] is given by\n\u0015k[X] +\u0015min[Y]\u0014\u0015k[X+Y]\u0014\u0015k[X] +\u0015max[Y](8)\nwhereMndenote the set of n\u0002nHermitian matrices and\n\u0015min(Y),\u0015max(Y)denotes the minimum and maximum\neigenvalues of Y , respectively.\nResult-3: Let\b :Mn!Mkbe a positive unital linear map.\nMnandMkdenote the set of all matrices of order nandkre-\nspectively. For every n\u0002nHermitian matrix A, the inequality\ngiven below follows:\n[\b(A)]2\u0014\b(A2) (9)\nThe inequality (9) is known as Kadison’s inequality [33].\nResult-4 [34]: LetCbe ann\u0002ncomplex matrix with real\neigenvalues \u0015[C]. Then\nm\u0000sp\nn\u00001\u0014\u0015min[C]\u0014m\u0000spn\u00001(10)\nwherem=tr[C]\nnands2=tr[C2]\nn\u0000m2.\nResult-5 [19]: If a simple connected weighted graph of order\nnrepresented by Gthen\n\u0015max[L]\u00141\n2maxi\u0018jW[i;j] (11)\nwhereLdenote the Laplacian corresponding to the graph G\nandW[i;j]is given by [19]\nW[i;j] =wi+wj+X\nk\u0018i;k\u001cjwik+X\nk\u0018j;k\u001ciwjk\n+X\nk\u0018i;k\u0018jjwik\u0000wjkj (12)\nV . CONSTRUCTION OF A UNITAL MAP\nIn this section, we construct a map that may provide Laplacian\nat the output and also study the properties of the constructed\nmap. Then we provide the physical interpretation of the total\ndegree of the graph, which is constructed from the generated\nLaplacian from the map.LetMn(R+)denote the set of all n\u0002nmatrices over the set\nR+of all positive real numbers. The map \u001e:Mn(R+)!\nMn(R+)may be defined as\n\u001e(A) =LA+A (13)\nwhereA2Mn(R+)andLAdenote a matrix of order n,\nwhich is constructed using the entries of the matrix A.\nLet us consider the n\u0002nreal matrixA, which is given by\nA=0\nB@a1;1a1;2:::::: a 1;n\u00001a1;n\na2;1a2;2:::::: a 2;n\u00001a2;n\n: : :::::: : :\nan;1an;2:::::: an;n\u00001an;n1\nCA;nX\ni=1ai;i= 1:(14)\nThe matrixLAcan be constructed as\nLA=0\nBBB@d1l1;2:::::: l 1;n\u00001l1;n\nl1;2d2:::::: l 2;n\u00001l2;n\n: : :::::: : :\nl1;n\u00001l2;n\u00001:::::: dn\u00001ln\u00001;n\nl1;nl2;n:::::: ln\u00001;ndn1\nCCCA(15)\nwhereli;j=\u00001\n2(ai;j+aj;i); i;j = 1:::n\u00001; i6=j:,di=Pn\nj6=i;j=1jli;jj.\nIn particular, we can consider the domain D\u001aMn(R+)as\nthe set of all d1\nd2dimensional bipartite quantum states\ndescribed by the density matrices of order d1d2\u0002d1d2. For\nthe domainD, the map\u001ecan be re-expressed as\n\u001e(\u001a) =L\u001a+\u001a (16)\nwhere the input state described by the density matrix \u001a2D\nis given by (14) with\u001a= [\u001ai;j];i;j = 1;2;::::;n and at the\noutputL\u001ais given by (15) withli;j=\u0000\u001ai;j; i6=j;i;j=\n1;2:::n,di=Pn\nj6=i;j=1jli;jj.\nIt can be easily seen that the matrix L\u001asatisfies the following:\n(i)L\u001ais symmetric and positive semi-definite.\n(i) The smallest eigenvalue of L\u001ais zero and (1,1,1,...,1) repre-\nsent an eigenvector corresponding to the smallest eigenvalue.\nThus, the matrix L\u001aact as a Laplacian corresponding to the\nmatrix\u001a.\nA. Properties of the map \u001e\nWe are now in a position to discuss the properties of the map\n\u001edefined in (13) .\nP-1:\u001eis a linear map.\nProof: LetA1andA2be any two matrices of same order from\nMn(R+)and let\u000b1;\u000b2be any two real scalars. Let ai;jand\nbi;jbe the (i;j)thentries ofA1andA2respectively. Then, it\nis clear that\u000b1ai;j+\u000b2bi;jis the (i;j)thentry ofL\u000b1A1+\u000b2A2\nAccording to the definition of the Laplacian, we have\nL\u000b1A1+\u000b2A2=\u000b1LA1+\u000b2LA2 (17)\nTherefore, using (17), \u001e(\u000b1A1+\u000b2A2)can be written as\n\u001e(\u000b1A1+\u000b2A2) =L\u000b1A1+\u000b2A2+\u000b1A1+\u000b2A2\n=\u000b1LA1+\u000b2LA2+\u000b1A1+\u000b2A2\n=\u000b1(LA1+A1) +\u000b2(LA2+A2)\n=\u000b1\u001e(A1) +\u000b2\u001e(A2) (18)4\nHence, the map \u001eis linear.\nP-2: The map\u001eis unital i.e. \u001e(I) =I.\nProof: It follows from the definition of the map \u001e.\nP-3: If the input matrix Ais Hermitian and positive\nsemi-definite then \u001e(A)is also positive-semidefinite.\nProof: Let\u0015min(:)be minimum eigenvalue of (:). SinceA\nandLAare Hermitian matrices so from Weyl’s inequality, we\nhave\n\u0015min(A) +\u0015min(LA)\u0014\u0015min(A+LA) =\u0015min(\u001e(A))(19)\nSinceLAdenote the Laplacian corresponding to the density\nmatrixAso\u0015min(LA) = 0 . Therefore, (19) reduces to\n\u0015min(A+LA) =\u0015min(\u001e(A))\u0015\u0015min(A)\u00150 (20)\nThe last inequality follows from the positive semi-definiteness\nofA. Hence proved.\nB. Physical interpretation of the total degree of the graph\ncorresponding to the Laplacian L\u001a\nTo start with, let us recall the quantum state in d1\nd2\ndimensional space described by the density matrix \u001a. The\nLaplacian corresponding to \u001ais denoted by L\u001a. LetGbe\nthe simple weighted graph for the Laplacian L\u001aand letdGbe\ntotal degree of the graph G. ThendGcan be expressed as\ndG=Tr[L\u001a] =d1+d2+:::::+dn\n=X\ni;j;i6=jj\u001ai;jj\n=Cl1(\u001a) (21)\nwhereCl1(\u001a)denote the l1-norm of quantum coherence,\nwhich is defined as the summation of modulus of the off-\ndiagonal terms of given quantum state \u001a. Thus, the total de-\ngree of the graph Gcorresponding to the Laplacian of the\ndensity matrix \u001acan be interpreted as the l1\u0000norm of the\ncoherence of the state \u001a.\nVI. NECESSARY CONDITION FOR THE\nDETERMINATION OF THE PURITY OF A QUANTUM\nSTATE\nA quantum system can exist in two forms: a pure state or a\nmixed state. A pure state is a projector while the mixed state\ncan be expressed as a convex combination of pure states. It\nis not always possible to prepare a pure state in the laboratory\ndue to noisy environment. Thus, it is an important issue for the\nexperimentalist to ascertain whether the state prepared in the\nlaboratory is the pure state or the mixed state. To probe this,\nsome method is needed by which pure state and mixed state\ncan be identified. The oldest and easiest method that can be\nadopted to discriminate pure and mixed state is the following:\n(i)Tr(\u001a2) = 1 , if the state is pure.(ii)Tr(\u001a2)<1, if the state \u001ais a mixed state.\nWe need two copies of the state to implement this method.\nLinear entropy is another possible way to distinguish pure and\nmixed state. It is a quantity that can quantify the amount of\nmixedness in the quantum state. The linear entropy SLfor the\nd\u0002ddensity matrix \u001acan be defined as\nSL=d2\nd2\u00001(1\u0000Tr(\u001a2)) (22)\nIn case of pure state, SL= 0 while 0< SL\u00141holds for\nmixed state.\nThe linear entropy involves non-linear functional of the quan-\ntum state and thus the value of the linear entropy depends on\nthed2\u00001parameter of the quantum state. All the unknown pa-\nrameters of the quantum state can be determined by tomogra-\nphy. The method of tomography needs lot of measurement to\nget the information about the state parameter and the number\nof measurement increases as the dimension of the system in-\ncreases. Additionally, tomography is very expensive in terms\nof resources also. Thus, in order to bypass the procedure of\ntomography, Ekert et.al. [35] have devised the quantum net-\nwork, which is controlled by input data. This method require\nonly to estimate d\u00001parameters to extract the information\naboutd2\u00001parameters of d\u0002ddensity matrix \u001a. Gener-\nalised uncertainty relation can be used to discriminate pure\nand mixed bipartite qutrit system [36].\nWe are now in a position to discuss the detection pure and\nmixed quantum system using graph theoretical approach. Our\ncriterion depends on the linear function of \u001a. Thus, our\nmethod requires single copy of the quantum state to test\nwhether the state is pure or mixed.\nTheorem-1: If the density operator \u001arepresent a pure quan-\ntum state then\nDet(L\u001a+\u001a\u0000I)\u00140 (23)\nwhereDet(:)denote the determinant.\nProof: Since the density operator \u001ais Hermitian so using\nResult-3, we can re-express Kadison’s inequality in terms of\n\u001aas\n[\b(\u001a)]2\u0014\b(\u001a2) (24)\nIf the quantum state \u001ais pure then we have\n\u001a2=\u001a (25)\nCombining (24) and (25), we get\n[\b(\u001a)]2\u0014\b(\u001a)\n)(L\u001a+\u001a)2\u0014L\u001a+\u001a\n)(L\u001a+\u001a)(L\u001a+\u001a\u0000I)\u00140\n)\b(\u001a)(L\u001a+\u001a\u0000I)\u00140 (26)\nTaking determinant both sides, we get\nDet[\b(\u001a)(L\u001a+\u001a\u0000I)]\u00140\n)Det[\b(\u001a)]:Det[L\u001a+\u001a\u0000I]\u00140 (27)5\nSince \b(\u001a)is positive so we have\nDet(L\u001a+\u001a\u0000I)\u00140 (28)\nHence proved.\nCorollary-1: If the state\u001arepresent a pure state then the op-\neratorL\u001a+\u001a\u0000Ihas odd number of negative eigenvalues.\nIllustration-1: To illustrate Theorem-1, let us consider a two-\nqubit pure statej iof the form\nj i=1\n2j00i+1\n2j01i+1\n4j10i+r\n7\n16j11i (29)\nThe density operator \u001aj iof the statej iis given by\n\u001aj i=0\nBBB@1\n41\n41\n8p\n7\n8\n1\n41\n41\n8p\n7\n8\n1\n81\n81\n16p\n7\n16p\n7\n8p\n7\n8p\n7\n167\n161\nCCCA(30)\nThe Laplacian associated with the density matrix \u001aj iis given\nby\nL\u001aj i=0\nBBB@3\n8+p\n7\n8\u00001\n4\u00001\n8\u0000p\n7\n8\n\u00001\n43\n8+p\n7\n8\u00001\n8\u0000p\n7\n8\n\u00001\n8\u00001\n81\n4+p\n7\n16\u0000p\n7\n16\n\u0000p\n7\n8\u0000p\n7\n8\u0000p\n7\n165p\n7\n161\nCCCA(31)\nIfI4denote the 4\u00024identity matrix then the value of the\ndeterminant of the operator L\u001aj i+\u001aj i\u0000I4is given by\nDet[L\u001aj i+\u001aj i\u0000I4] =\u00000:00027059<0 (32)\nThus, Theorem-1 is verified. Also, one can easily verify that\nthere are three negative eigenvalues and one positive eigen-\nvalue of the operator L\u001aj i+\u001aj i\u0000I4. This verify the\ncorollary-1.\nTheorem-2: IfDet(L\u001a+\u001a\u0000I)>0then the state described\nby the density operator \u001ais a mixed state.\nCorollary-2: If the operator L\u001a+\u001a\u0000Ihas either all eigen-\nvalues positive or even number of negative eigenvalues then\nthe density operator \u001arepresent a mixed state.\nIllustration-2: Let us take a 2\n4dimensional bipartite quan-\ntum state described by the density operator \u001a1, which is given\nby\n\u001a1=0\nBBBBBBBBBB@1\n80 0 01\n810 01\n81\n01\n80 0 0 01\n810\n0 01\n80 01\n80 0\n0 0 01\n81\n810 01\n811\n810 01\n811\n80 0 0\n0 01\n80 01\n80 0\n01\n810 0 0 01\n80\n1\n810 01\n810 0 01\n81\nCCCCCCCCCCA(33)The Laplacian associated with the density matrix \u001a1is given\nby\nL\u001a1=0\nBBBBBBBBBB@2\n810 0 0\u00001\n810 0\u00001\n81\n01\n810 0 0 0 \u00001\n810\n0 01\n80 0\u00001\n80 0\n0 0 02\n81\u00001\n810 0\u00001\n81\n\u00001\n810 0\u00001\n812\n80 0 0\n0 0\u00001\n80 01\n80 0\n0\u00001\n810 0 0 01\n810\n\u00001\n810 0\u00001\n810 0 02\n811\nCCCCCCCCCCA(34)\nIfI8denote the 8\u00028identity matrix then the value of the\ndeterminant of the operator L\u001a1+\u001a1\u0000I8is given by\nDet[L\u001a1+\u001a1\u0000I8]w0:2188278>0 (35)\nFrom (35), we can conclude that the state described by the\ndensity matrix \u001a1is a mixed state. Moreover, we find that all\neigenvalues of the operator L\u001a1+\u001a1\u0000I8are negative. Thus,\nthe number of negative eigenvalues are even and hence this\nverify the corollary-2.\nVII. PPT CRITERION IN TERMS OF LAPLACIAN FOR\nd1\nd2DIMENSIONAL SYSTEM\nA bipartite entangled state in d1\nd2dimensional system can\nbe divided into two categories: (i) Negative partial transpose\nentangled states (NPTES) and (ii) Positive partial transpose\nentangled states (PPTES) or bound entangled states. The first\ncriterion for the entanglement detection problem was given\nby Peres and Horodecki [7, 8] and it may be called as PH\ncriterion. The PH criteria states that a quantum state is sep-\narable if and only if the eigenspectrum of partial transposed\nstate contain positive eigenvalues. The criterion is necessary\nand sufficient for 2\n2and2\n3system but in higher di-\nmensional systems, there exist entangled states which satisfy\nthe PH criterion. This means that the eigenspectrum of par-\ntial transposition of the density matrix that represent the en-\ntangled state contain positive eigenvalues. The states which\npossesses this type of properties are known as positive partial\ntranspose entangled states (PPTES) or bound entangled states\n(BES). Since PPTES are not detected by partial transposition\nmethod so other criterion such as the computable cross norm\nand re-alignment Criterion [9, 10], range criterion [37], ma-\njorization criterion [38] developed in detecting the bound en-\ntangled states. D. P. DiVincenzo et.al. [39] provided an exam-\nple of a class of bipartite bound entangled state in d\ndsystem\nwhich is also negative partial transpose entangled state.\nIn this section, we derive few criterion based on (i) the spec-\ntrum of the density matrix of the state under investigation and\n(ii) the spectrum of the Laplacian corresponding to the density\nmatrix. The criterion may serve to identify PPT states and also\ntake part in detecting the negative partial transpose entangled\nstates (NPTES). We then illustrate our criterion by taking few\nexamples of quantum states in higher dimensional system.6\nA. Few PPT Criterion\nFirstly, we will derive the separability criterion for a bipartite\nquantum state which is proved to be necessary and sufficient\nin2\n2dimensional system. Then we will show that the con-\ndition is only sufficient for the d1\nd2dimensional bipartite\nPPT state where either d1;d2\u00153ord1= 2;d2>3. Sec-\nondly, we deduce PPT criterion for d1\nd2dimensional bi-\npartite state in terms of the minimum eigenvalue of the probe\nstate and its corresponding Laplacian.\nTheorem-3: The state described by the density operator \u001a2\n2\nAB\nin2\n2dimensional system is a separable state if and only if\n\u0015min[L\u001a2\n2\nAB+ (\u001a2\n2\nAB)TB]\u00150 (36)\nwhereTBdenote the partial transposition with respect to the\nsystemBandL\u001a2\n2\nABrepresent the laplacian corresponding to\nthe density operator \u001a2\n2\nAB.\nProof: For2\n2dimensional system, the state \u001a2\n2\nABis sepa-\nrable if and only if (\u001a2\n2\nAB)TB\u00150. This implies\n\u0015min[(\u001a2\n2\nAB)TB]\u00150 (37)\nUsing the Result-2 given in (8) on two Hermitian operators\n\u001aABandL\u001aAB, we get\n\u0015min[L\u001a2\n2\nAB+ (\u001a2\n2\nAB)TB]\u0015\u0015min[L\u001a2\n2\nAB] +\u0015min[(\u001a2\n2\nAB]TB](38)\nSinceL\u001a2\n2\nABis the Laplacian so \u0015min[L\u001a2\n2\nAB] = 0 . Therefore,\nthe inequality (38) reduces to\n\u0015min[L\u001a2\n2\nAB+ (\u001a2\n2\nAB)TB]\u0015\u0015min[(\u001a2\n2\nAB)TB] (39)\nUsing (37) and (39), we get the required result.\nOne may note that the necessary and sufficient condition given\nin Theorem-3 also holds for 2\n3dimensional bipartite system\nbut the condition is only sufficient in the higher dimensional\nsystem because of the existence of positive partial transpose\nentangled states (PPTES). Thus, theorem-3 may be modified\nfor the higher dimensional system in the following way:\nTheorem-4: If any bipartite state in d1\nd2(eitherd1;d2\u00153\nord1= 2;d2>3), dimensional system described by the\ndensity operator \u001ad1\nd2\nAB represent a positive partial transpose\nstate then it satisfies the inequality\n\u0015min(L\u001ad1\nd2\nAB+ (\u001ad1\nd2\nAB )TB)\u00150 (40)\nWe note here that the condition (40) is only sufficient for d1\nd2dimensional system while the condition (36) is necessary\nand sufficient for 2\n2and2\n3dimensional system.\nCorollary-4: If any arbitrary d1\nd2dimensional bipartite\nstate\u001ad1\nd2\nAB satisfies the inequality\n\u0015min(L\u001ad1\nd2\nAB+ (\u001ad1\nd2\nAB )TB)<0 (41)\nthen the state \u001ad1\nd2\nAB is negative partially transposed entan-\ngled state (NPTES).\nNext our task is to derive few other criterion based on the max-\nimum and minimum eigenvalues of the Laplacian of the givenstate that may identify whether the given state is a PPT state\nor not? These conditions are necessary conditions and hence\nthey are of particular importance.\nTheorem-5: If any arbitrary full rank d1\nd2dimensional\nstate described by the density operator \u001asatisfies the inequal-\nity\n\u0015min(\u001a)\u0015\u0015max(LTB\u001a)\u0000\u0015min(LTB\u001a) (42)\nthen the state \u001ais either separable state or PPTES.\nProof: To start with, let us consider the functional Tr[(L\u001a+\n\u001a)\u001aTB].\nTr[(L\u001a+\u001a)\u001aTB] =Tr[L\u001a\u001aTB] +Tr[\u001a\u001aTB]\n=Tr[LTB\u001a\u001a] +Tr[\u001a\u001aTB]\n\u0015\u0015min[LTB] +\u0015min[\u001a] (43)\nWe may now proceed further with the inequality (43) that may\nbe re-expressed as\n\u0015min[LTB] +\u0015min[\u001a]\u0014Tr[(L\u001a+\u001a)\u001aTB]\n=Tr[(L\u001a+\u001a)TB\u001a]\n=Tr[(LTB\u001a+\u001aTB)\u001a]\n\u0014\u0015max[LTB+\u001aTB]\n\u0014\u0015max[LTB] +\u0015min(\u001aTB)(44)\nRearranging the terms of the inequality (44), we get\n\u0015min[\u001aTB]\u0015\u0015min[\u001a]\u0000[\u0015max[LTB]\u0000\u0015min[LTB]](45)\nTherefore, if a quantum state \u001asatisfies the inequality\n\u0015min[\u001a]\u0015\u0015max[LTB]\u0000\u0015min[LTB], then\u0015min[\u001aTB]\u00150.\nThus, the state \u001arepresent either a separable state or bound\nentangled state. Hence proved.\nCorollary-5: If the inequality (42) is violated by any quantum\nstate\u001athen the state may or may not be NPTES.\nNow it can be seen that the criterion given in (42) depends\non the partial transposition of the Laplacian of the given state\nbut since partial transposition is not a physical operation so\nit cannot be implemented in the laboratory. Therefore, the\nabove theorem and corollary that involve partial transposition\noperation, cannot be used in an experiment for the detection\nof positive partial transpose states. Thus, we need to modify\nTheorem-5 in such a way so that we can avoid partial transpo-\nsition operation in deducing the condition for the detection of\nstates with positive partial transpose. In doing this, we will de-\nduce another criterion which is free from partial transposition\noperation but can discriminate between positive partial trans-\npose states and negative partial transpose states. The modified\ncriterion can be expressed via the following theorem:\nTheorem-6: If a full rank state \u001athat exist in d1\nd2dimen-\nsional system satisfies the inequality\n\u0015min[\u001a]\u0015\u0015max[L\u001a] (46)\nthen the state \u001ais either separable state or PPTES.\nProof: Let us begin with the functional Tr[(L\u001a+\u001aTB)\u001a]. The\nlower bound of Tr[(L\u001a+\u001aTB)\u001a]can be derived as\nTr[(L\u001a+\u001aTB)\u001a] =Tr[L\u001a\u001a] +Tr[\u001aTB\u001a]\n\u0015\u0015min[L\u001a] +\u0015min(\u001a) (47)7\nThe inequality (47) may be written as\n\u0015min[L\u001a] +\u0015min[\u001a]\u0014Tr[(L\u001a+\u001aTB)\u001a]\n\u0014\u0015max[L\u001a+\u001aTB]\n\u0014\u0015max[L\u001a] +\u0015min[\u001aTB](48)\nConsidering the fact that \u0015min[L\u001a] = 0 and then rearranging\nthe terms of the inequality (48), we get\n\u0015min[\u001aTB]\u0015\u0015min[\u001a]\u0000\u0015max[L\u001a] (49)\nIf we now further consider that \u0015min[\u001a]\u0015\u0015max[L\u001a]then\n\u0015min[\u001aTB]\u00150and thus we can conclude that the state under\ninvestigation will be either separable state or bound entangled\nstate, that is, a positive partial transpose state.\nB. Examples\nExample-1: Let us consider a 2\n2dimensional bipartite\nsystemABdescribed by the density operator \u001aAB. It is given\nby\n\u001aAB=0\nB@0:1 0 0 0\n0 0:2x0\n0x0:4 0\n0 0 0 0 :31\nCA;0\u0014x\u00140:283 (50)\nIf\u001aTB\nABdenote the partial transposition of the state \u001aABthen\nthe eigenvalues of \u001aTB\nABare given by\n\u00151[\u001aTB\nAB] = 0:4;\u00152[\u001aTB\nAB] = 0:2;\n\u00153[\u001aTB\nAB] =1\n5+1\n10p\n1 + 100x2;\n\u00154[\u001aTB\nAB] =1\n5\u00001\n10p\n1 + 100x2 (51)\nThe minimum eigenvalue of \u001aTB\nABis given by\n\u0015min[\u001aTB\nAB] =1\n5\u00001\n10p\n1 + 100x2\u00150;for0\u0014x\u00140:173\n\u00140;for0:173<x\u00140:283\n(52)\nTherefore, the state \u001aABis separable for 0\u0014x\u00140:173\nwhile it is entangled for 0:173<x\u00140:283. IfL\u001aABdenote\nthe Laplacian associated with the density matrix \u001aABthen it\nis given by\nL\u001aAB=0\nB@0 0 0 0\n0x\u0000x0\n0\u0000x x 0\n0 0 0 01\nCA;0\u0014x\u00140:283 (53)\nThe eigenvalues of L\u001aABare given by 0;0;0;2x. The eigen-\nvalues of the partial transposition of L\u001aAB, which is denoted\nbyLTB\u001aABare given by\u0000x;x;x;x .\nNow we are in a position to make the following observation:\nObservation-1: The minimum eigenvalue of L\u001aAB+\u001aTB\nABisgiven by\u0015min[L\u001aAB+\u001aTB\nAB] =1\n5\u00001\n10p\n1 + 100x2, which is\nnon-negative for 0\u0014x\u00140:173. The region 0\u0014x\u00140:173\nrepresent the separability region and thus Theorem-3 is satis-\nfied for the state \u001aAB.\nObservation-2: \u0015min[L\u001aAB+\u001aTB\nAB] =1\n5\u00001\n10p\n1 + 100x2<\n0for0:173<x\u00140:283. Therefore, the region 0:173<x\u0014\n0:283represent the entanglement region and thus Corollary-4\nis satisfied for the state \u001aAB.\nObservation-3: The rank of \u001aABgiven in (50) is 4. Thus,\nthe state\u001aABis a full rank state. It can be easily verified that\nTheorem-5 and Theorem-6 are satisfied for 0\u0014x\u00140:05.\nExample-2: Let us now consider a 2\n4dimensional bipar-\ntite system described by the density operator \u001a2. It is given by\n[40]\n\u001a2=0\nBBBBBBBBBB@1\n80 0 01\n810 01\n81\n01\n80 0 0 0 0 0\n0 01\n80 0 0 0 0\n0 0 01\n81\n810 01\n811\n80 01\n811\n80 0 0\n0 0 0 0 01\n80 0\n0 0 0 0 0 01\n80\n1\n810 01\n810 0 01\n81\nCCCCCCCCCCA; (54)\nThe eigenvalues of \u001a2are given by\n\u00161=97\n648>\u00162=\u00163=\u00164=\u00165=\u00166=\u00167=1\n8\u0015\n\u00168=65\n648(55)\nIn this example, we can find that the partial transposed state\n\u001aTB\n2is identical with the state \u001a2. Thus, we have\n\u001a2=\u001aTB\n2 (56)\nIt has been shown that the state \u001a2is a separable state [40].\nIfL\u001a2andLTB\u001a2denote the Laplacian and partial transposition\nof the Laplacian associated with the density matrix \u001a2then\nthey are given by\nL\u001a2=LTB\u001a2=0\nBBBBBBBBB@2\n810 0 0\u00001\n810 0\u00001\n81\n0 0 0 0 0 0 0 0\n0 0 0 0 0 0 0 0\n0 0 02\n81\u00001\n810 0\u00001\n81\u00001\n810 0\u00001\n812\n810 0 0\n0 0 0 0 0 0 0 0\n0 0 0 0 0 0 0 0\n\u00001\n810 0\u00001\n810 0 02\n811\nCCCCCCCCCA;(57)\nTherefore, the eigenvalues of L\u001a2=LTB\u001a2are given by\n0;0;0;0;0;2\n81;2\n81;4\n81.\nNow we observe the following facts:\nObservation-1: It can be easily verified that the minimum\neigenvalue\u0015min[L\u001a2+\u001aTB\n2]is greater than zero. Thus ac-\ncording to the Theorem-4, we can infer that the state \u001a2is a\nPPT state. But using Theorem-4, we are unable to tell that\nwhether the state \u001a2is separable state or PPTES.\nObservation-2: The state\u001a2represent a full rank state. Now,8\n1 2 3\n4 5 6\n7 81\n81\n1\n81\n1\n81\n1\n81\nFIG. 1: Graph corresponding to L\u001a2\ngoing through the eigenvalues of \u001a2given in (55), we can de-\ntermine that \u0015min[\u001a2] =65\n648. Thus, we have\n\u0015min[\u001a2] =65\n648>\u0015max[LTB\u001a2]\u0000\u0015min[LTB\u001a2] =4\n81(58)\nHence, Theorem-5 is satisfied for the state \u001a2.\nObservation-3: Since the state \u001a2is a full rank state so we\ncan apply Theorem-6. We can then easily verify Theorem-6\nfor the state \u001a2.\nVIII. GRAPHICAL INTERPRETATION\nHere, we will discuss the graphical interpretation of results\ngiven in section-VI A. We will show that PPT criterion of a\nquantum state can be interpreted through the properties of a\ngraph associated with the quantum states.\nTheorem-3A: The state described by the density operator\n\u001a2\n2in2\n2dimensional system is a separable state if and\nonly if\n0\u0014\u0015min[L\u001a2\n2+ (\u001a2\n2)TB]\u00141 +dG (59)\nwheredGdenote the total degree of the graph Gassociated\nwith the density operator \u001a2\n2.\nProof: The lower bound of \u0015min[L\u001a2\n2+ (\u001a2\n2)TB]may be\nverified from the inequality (36) given in Theorem-3. Now\nour task is to derive the upper bound. The state \u001a2\n2in2\n2\ndimensional system is a separable state if and only if its par-\ntial transposed form denoted by (\u001a2\n2)TBis a positive semi-\ndefinite operator. Further, since L\u001a2\n2is also a positive semi-\ndefinite operator so we can write\nTr[L\u001a2\n2+ (\u001a2\n2)TB]\u0015\u0015min[L\u001a2\n2+ (\u001a2\n2)TB](60)\nThe inequality (60) can be further re-expressed as\nTr[L\u001a2\n2] +Tr[(\u001a2\n2)TB]\u0015\u0015min[L\u001a2\n2+ (\u001a2\n2)TB]\n)dG+ 1\u0015\u0015min[L\u001a2\n2+ (\u001a2\n2)TB] (61)\nwhere we used the fact that Tr[(\u001a2\n2)TB] = 1 and\nTr[L\u001a2\n2] =dG.Combining Theorem-3 and the inequality (61), we get the re-\nquired result.\nTheorem-3B: If\u001adenoted1\nd2dimensional NPTES and if\nthe graphGcorresponding to the density matrix \u001ais simple\nconnected weighted graph then\n\u0015min[L\u001a+\u001aTB]\u00141\n2maxi\u0018jW[i;j] (62)\nwhereW[i;j]is given by Result-5.\nProof: Let us start with \u0015min[L\u001a+\u001aTB]. Using Result-2, we\nget\n\u0015min[L\u001a+\u001aTB]\u0014\u0015max[L\u001a] +\u0015min[\u001aTB]\n\u0014\u0015max[L\u001a]\n\u00141\n2maxi\u0018jW[i;j] (63)\nThe second inequality follows from the fact that the state \u001ais\na NPTES and the last inequality follows from Result-5. Hence\nproved.\nIllustration-3: Let us consider a state \u001a3, which is given by\n\u001a3=0\nB@0:4 0:2 0:1 0:1\n0:2 0:3 0:2 0:1\n0:1 0:2 0:2 0:1\n0:1 0:1 0:1 0:11\nCA (64)\nThe partial transposed state \u001aTB\n3can be expressed as\n\u001aTB\n3=0\nB@0:4 0:2 0:1 0:2\n0:2 0:3 0:1 0:1\n0:1 0:1 0:2 0:1\n0:2 0:1 0:1 0:11\nCA (65)\nThe eigenvalues of \u001aTB\n3are: 0:7;0:16;0:14;\u00000:01. This\nshows that the state \u001a3is a NPTES.\nThe Laplacian of the state described by the density operator\n\u001a3is given by\nL\u001a3=0\nB@0:4\u00000:2\u00000:1\u00000:1\n\u00000:2 0:5\u00000:2\u00000:1\n\u00000:1\u00000:2 0:4\u00000:1\n\u00000:1\u00000:1\u00000:1 0:31\nCA (66)\nIf\u0015min(L\u001a3+\u001aTB\n3)denote the minimum eigenvalue of L\u001a3+\n\u001aTB\n3then\n\u0015min(L\u001a3+\u001aTB\n3) = 0:3763 (67)\nUsing (12), we can calculate maxi\u0018jW[i;j]for the graph\nshown in figure 2 and it is given by\nmaxi\u0018jW[i;j] = 0:9 (68)\nTherefore, the inequality (62) can be easily verified using (67)\nand (68).\nTheorem-4A: For anyd1\nd2dimensional positive partial\ntransposed bipartite state \u001ad1\nd2, the inequality\n1 +dG\u0015\u0015min[L\u001ad1\nd2+ (\u001ad1\nd2)TB] (69)9\n1 2\n3 40.2\n0.10.1\n0.20.1\n0.1\nFIG. 2: Graph corresponding to L\u001a3\nholds.\nCorollary-4A: LetdGdenote the total degree of the simple\nconnected weighted graph Gassociated with any arbitrary\nd1\nd2dimensional bipartite state %d1\nd2. If the state %d1\nd2\nsatisfies the inequality\n1 +dG<(d1d2\u00001)(1\n2maxi\u0018jW[i;j] +\u0015max[(%d1\nd2)TB])\n(70)\nthen the state %d1\nd2is negative partially transposed entangled\nstate (NPTES).\nProof: Let us recall corollary-4. It implies that if\n\u0015min[L%d1\nd2+ (%d1\nd2)TB]<0 (71)\nthen the state %d1\nd2is NPTES.\nUsing Result-4, the inequality (71) can be re-expressed as\nm<sp\nd1d2\u00001 (72)\nwherem =Tr[L%d1\nd2+(%d1\nd2)TB]\nd1d2ands2=\nTr[(L%d1\nd2+(%d1\nd2)TB)2]\u0000(Tr[L%d1\nd2+(%d1\nd2)TB]\nd1d2)2\nd1d2.\nSimplifying the inequality (72), we get\n(1 +dG)2<(d1d2\u00001)Tr[(L%d1\nd2+ (%d1\nd2)TB)2]\n)(1 +dG)2<(d1d2\u00001)\u0015max[L%d1\nd2+ (%d1\nd2)TB](1 +dG)\n)1 +dG<(d1d2\u00001)\u0015max[L%d1\nd2+ (%d1\nd2)TB] (73)\nThe second line follows from Result-1. Moreover, using\nResult-2 on\u0015max[L%d1\nd2+ (%d1\nd2)TB], the inequality (73)\nfurther reduces to\n1 +dG<(d1d2\u00001)(\u0015max[L%d1\nd2] +\u0015max[(%d1\nd2)TB])\n\u0014(d1d2\u00001)(1\n2maxi\u0018jW[i;j] +\u0015max[(%d1\nd2)TB])\n(74)\nThe last inequality follows from Result-5.\nTheorem-7: If anyd1\nd2dimensional bipartite NPTES\nstate described by the density operator \u001athen\n\u0015min[\u001a]\u00141\n2maxi\u0018jW[i;j] (75)whereW[i;j]is given by Result-5 for the simple connected\nweighted graph Gassociated with the density operator \u001a.\nProof: Let us recall the inequality (49), which can be re-\nexpressed in the form as\n\u0015min[\u001a]\u0014\u0015max[L\u001a] +\u0015min[\u001aTB] (76)\nSince the state \u001arepresent a NPTES so \u0015min[\u001aTB]<0. Thus,\nwe have\n\u0015min[\u001a]\u0014\u0015max[L\u001a] (77)\nThe non-trivial upper bound of \u0015max[L\u001a]is given by the\nResult-5. Using Result-5, the inequality (77) reduces to\n\u0015min[\u001a]\u00141\n2maxi\u0018jW[i;j] (78)\nHence proved.\nCorollary-6: If anyd1\nd2dimensional full rank bipartite\nstate described by the density operator \u001asatisfies the inequal-\nity\n\u0015min[\u001a]>1\n2maxi\u0018jW[i;j] (79)\nthen the full rank state \u001amust be a PPT state.\nIllustration-4: The quantum state described by the density\noperator\u001a5is given by\n\u001a5=0\nBB@1\n41\n2001\n201\n201\n40 0\n0 01\n41\n201\n2001\n201\n41\nCCA(80)\nThe eigenvalues of \u001a5 are as follows:\n0:3309;0:2809;0:2191;0:1691 . The Laplacian corre-\nsponding to the density matrix is given by\nL\u001a5=0\nBB@1\n10\u00001\n200\u00001\n20\n\u00001\n201\n200 0\n0 01\n20\u00001\n20\n\u00001\n200\u00001\n201\n101\nCCA(81)\n1 2\n3 41\n20\n1\n20\n1\n20\nFIG. 3: Graph corresponding to L\u001a5\nFrom the graph given in Fig. 3, it is clear that there are three\nedges namely e1;2;e1;4ande3;4. Thus, we need to calculate10\nW[1;2];W[1;4]andW[3;4]as per the prescription given in\n(12). Therefore, we have\nW[1;2] =w1+w2+X\nk\u00181;k\u001c2w1k+X\nk\u00182;k\u001c1w2k\n+X\nk\u00181;k\u00182jw1k\u0000w2kj=3\n10(82)\nW[1;4] =w1+w4+X\nk\u00181;k\u001c4w1k+X\nk\u00184;k\u001c1w4k\n+X\nk\u00181;k\u00184jw1k\u0000w4kj=1\n5(83)\nW[3;4] =w3+w4+X\nk\u00183;k\u001c4w3k+X\nk\u00184;k\u001c3w4k\n+X\nk\u00183;k\u00184jw3k\u0000w4kj=1\n5(84)\nIt can be easily shown that for the state \u001a5, the inequality (79)\nis verified. Thus, we can conclude that the two-qubit state \u001a5\nis a separable state.\nIllustration-5: Let us consider a 3\n3dimensional bipartite\nstate\u001a6, which is given by\n\u001a6=1\nN0\nBBBBBBBBBBB@x0:01 0 0 0 0 0 0 a\nz x 0 0z0 0 0 0\n0 0x z 0 0 0 0 0\n0 0z x 0 0 0z0\n0z0 0x z 0 0a\n0 0 0 0 z x z 0 0\n0 0 0 0 0 z y z 0\n0 0 0z0 0z x 0\na0 0 0a0 0 0y1\nCCCCCCCCCCCA(85)\nwhereN= 400a+ 1,x= 50a,y=50a+1\n2,z= 0:01and\n0:01\u0014a\u00141.\nIt can be easily checked that the eigenvalues of \u001a6are posi-\ntive for 0:01\u0014a\u00141. The Laplacian corresponding to the\ndensity matrix \u001a6is given by\nL\u001a6=1\nN0\nBBBBBBBBBBB@u\u0000z0 0 0 0 0 0 \u0000a\n\u0000z w 0 0\u0000z0 0 0 0\n0 0z\u0000z0 0 0 0 0\n0 0\u0000z w 0 0 0\u0000z0\n0\u0000z0 0v\u0000z0 0\u0000a\n0 0 0 0\u0000z w\u0000z0 0\n0 0 0 0 0 \u0000z w\u0000z0\n0 0 0\u0000z0 0\u0000z w 0\n\u0000a0 0 0\u0000a0 0 0 2 a1\nCCCCCCCCCCCA(86)\nwhereu=a+ 0:01,v=a+ 0:02andw= 0:02.\nThe graph can be constructed from the Laplacian L\u001a6, which\nis a connected graph shown in figure 5. Thus, we calcu-\nlateW[1;2];W[1;9];W[2;5];W[3;4];W[4;8];W[5;6],W[6;7];W[7;8];W[5;9]as per the prescription given in\n(12). Therefore, we have\nW[1;2] =2a+ 0:04\nN;W[1;9] =4a+ 0:02\nN;\nW[2;5] =2a+ 0:06\nN;W[3;4] =0:03\nN;\nW[4;8] =0:04\nN;W[5;6] =2a+ 0:06\nN;\nW[6;7] =0:06\nN;W[7;8] =0:06\nN;\nW[5;9] =4a+ 0:04\nN(87)\nThe quantity maxi\u0018jW[i;j]is given by\nmaxi\u0018jW[i;j] =4a+ 0:04\n400a+ 1;when 0:01\u0014a\u00141(88)\nFrom figure 4, it is clear that for the state \u001a6, the inequality\nFIG. 4: Minimum eigenvalue of \u001a6andmaxi\u0018jW[i;j]\n(79) is satisfied. Thus, we can conclude that the two-qutrit\nstate\u001a6is either a separable state or PPTES i.e. a PPT state.\n1 2 3\n4 5 6\n7 8 9z\nN\na\nNz\nN z\nN\nz\nNz\nN\na\nNz\nN\nz\nN\nFIG. 5: Graph corresponding to L\u001a611\nIX. CONCLUSION\nTo summarize, we have constructed a mapping in which\nthe domain set contain any n\u0002nmatrix over the field of real\nnumbers where all off diagonal entries are either positive or\nnegative. In particular, We can consider the domain set as the\nset of density matrices. The map takes each density matrix\ninto the sum of the input density matrix and another matrix\nL. The matrix Lis constructed from the elements of the input\ndensity matrix in such a way that it satisfies all the properties\nof the Laplacian. The constructed map is shown to be unital.\nUsing the unital property and the structure of the map, we are\nable to derive a criterion that characterize the quantum state\neither as a pure state or a mixed state. We also use the con-\nstructed Laplacian and its partial transpose to derive the PPT\ncriterion. Further, we have derived the inequality between the\nminimum eigenvalue of the full rank bipartite NPTES and thefunctionW[i;j]of the weights of the edges of a simple con-\nnected weighted subgraph of a graph Gconstructed from the\nLaplacian. The violation of the derived inequality prove the\nfact that the given state is a PPT state. Thus, we have studied\nthe entanglement properties of d1\nd2dimensional bipartite\nquantum system through graph theoretical approach. In this\nwork, we restrict ourselves only to the bipartite system but\nour result is more general in the sense that it can be applied\nto multipartite system also. Specifically, the result obtained in\nthis work may be useful in studying the three-qubit states in\nnon-inertial frame [41–44].\nX. DATA A VAILABILITY STATEMENT\nData sharing not applicable to this article as no datasets\nwere generated or analysed during the current study.\n[1] R. Horodecki, P. Horodecki, M. Horodecki and K. Horodecki,\nRev. Mod. Phys. 81, 865 (2009).\n[2] R. Jozsa and N. Linden, Proc. R. Soc. Lond. A 459, 2011\n(2003).\n[3] C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres and\nW. K. Wootters, Phys. Rev. Lett. 70, 1895 (1993).\n[4] C. H. Bennett and S. Wiesner, Phys. Rev. Lett. 69, 2881 (1992).\n[5] A. K. Ekert, Phys. Rev. Lett. 67, 661 (1991).\n[6] N. Gisin, G. 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Leon-Montiel, G-H Sun and S-H Dong, En-\ntropy 24, 1011 (2022)."    },    {        "title": "2203.03465v2.A_Time_Dependent_Random_State_Approach_for_Large_scale_Density_Functional_Calculations.pdf",        "content": "A Time-Dependent Random State Approach for Large-scale Density Functional\nCalculations\nWeiqing Zhou1and Shengjun Yuan1,∗\n1Key Laboratory of Artificial Micro- and Nano-structures of Ministry of Education\nand School of Physics and Technology, Wuhan University, Wuhan 430072, China\n(Dated: January 25, 2023)\nWe develop a self-consistent first-principle method based on the density functional theory. Physi-\ncal quantities, such as the density of states, Fermi energy and electron density are obtained using a\ntime-dependent random state method without diagonalization. The numerical error for calculating\neither global or local variables always scales as 1 /√SN e, where Neis the number of electrons and S\nis the number of random states, leading to a sublinear computational cost with the system size. In\nthe limit of large systems, one random state could be enough to achieve reasonable accuracy. The\nmethod’s accuracy and scaling properties are derived analytically and verified numerically in dif-\nferent condensed matter systems. Our time-dependent random state approach provides a powerful\nstrategy for large-scale density functional calculations.\nPACS numbers: 31.15.E-; 71.15.-m; 71.15.Mb\nKeywords: Density Functional Theory, Random State, Time-dependent Schr¨ odinger Equation, First-principle\nCalculation\nFirst-principles calculation using the Density Func-\ntional Theory (DFT) is one of the most powerful com-\nputational methods for multi-electron systems and con-\ntributes extensively to physics, chemistry, and material\nscience. DFT theorems prove that there is a one-to-one\nmapping between the ground-state wave function and\nthe ground-state electron density [1]. In the mid-1960s,\nKohn and Sham showed that the finding of the ground-\nstate density could be determined by a set of single-\nelectron equations (Kohn-Sham equations) [2], which is\nalso known as KS-DFT. However, KS-DFT suffers from\na size limitation caused by diagonalization, in which the\ncomputational cost exhibits a cubic scaling with the sys-\ntem size. Although many efforts, such as iterative diago-\nnalization schemes [3], preconditioned conjugate-gradient\nminimizations [4–6], and the Car-Parrinello method [7],\nhave improved the scaling behaviour for a relatively small\nsystem, it is still hard to handle systems of more than a\nfew hundred or thousand atoms.\nThis size limitation has stimulated the development\nof linear-scaling DFT [8–21]. The first attempt can\nbe traced back to the ’divide and conquer’ method of\nYang [11]. In 1992, Baroni and Giannozzi also pro-\nposed an algorithm that determines the electron den-\nsity directly by using Green’s function [12]. In 1993, the\ndensity-matrix minimization approach was proposed by\nLi, Numes, and Vanderbilt [13]. Following these strate-\ngies, many linear-scaling DFT codes have been developed\n[10, 15–20]. Chebyshev filter method is another success-\nful attempt to reduce the size of the effective dimension\nof Hilbert space, but there are other non-linear factors\ndominated in large systems[22]. A linear-scaling algo-\nrithm using atomic orbitals (LCAO) basis sets [14] can be\napplied to suitable systems with clearly separated occu-\npied and empty states [21]. Furthermore, the orbital-free\nDFT (OF-DFT) [23, 24] is a linear-scaling approach thatavoids complete diagonalization, but the kinetic energy\ndensity functionals have not yet reached a good accu-\nracy for many elements [25, 26]. Recently, a linear-scaled\nDFT is realized by using a stochastic technique in a trace\nformula [9, 27–30], in which the statistical error of cal-\nculating a global variable, such as the total energy (per\nelectron), is reduced by the sample average from differ-\nent random orbitals. For local quantities, such as the\nelectron density, many stochastic samples are required\nto reach a reasonable accuracy.\nIn this letter, we develop a self-consistent first-principle\ncalculation method based on DFT without any diagonal-\nization of the Hamiltonian matrix. The physical quan-\ntities, such as the density of states (DOS), Fermi en-\nergy and the real-space distribution of electron density,\nare calculated using the so-called time-dependent random\nstate (TDRS) method. We show that the numerical er-\nror of a global or local variable always scales as 1 /√SNe,\nwhereNeis the number of electrons and Sis the num-\nber of random states. It leads to an overall sublinear\nscaling of the computational costs, and one needs fewer\nrandom states for larger systems. The method becomes\nextremely powerful for massive quantum systems, and a\ncalculation using one random state is enough to achieve\nreasonable accuracy when Ne→∞ . Our time-dependent\nrandom-states DFT (rsDFT) originates from the real-\nspace TDRS method developed in the tight-binding cal-\nculations, with an extension from globe variables (such\nas density of states [31], electronic and optical conductiv-\nities [32], polarization and screening functions [33], etc.),\nto a local variable of electron density. It is a general strat-\negy for the local variable calculations and can be applied\nin the tight-binding model or other physical models as\nwell.\nDensity of states and Fermi energy. — In KS-DFT, the\nFermi energy is determined by counting the number ofarXiv:2203.03465v2  [cond-mat.mes-hall]  24 Jan 20232\n(c) (d)(a) (b)\nFIG. 1: (a) The DOS of a graphite nanocrystal with 42\natoms, calculated by exact diagonalization or using the\nTDRS method averaged with a different number ( S) of\nrandom samples. (b) The DOS of a graphite\nnanocrystal with 11440 atoms was calculated using the\nTDRS method without random sample averaging. The\ndifferent colours represent different initial random\nstates. (c) The standard deviations (SD) of µand\nEocc(µ) for graphite nanocrystals with different size\n(Ne), obtained from the statisicial analysis of results\nfrom 500 individual random states. (d) The error of\nEocc(µ), with respect to the exact diagonalization, as a\nfunction of the number of random states ( S) used in\nTDRS for fullerenes C 60and C 540. Here, each point in\n(d) is averaged from 100 groups of Srandom states.\noccupied eigenstates, which are obtained from the diag-\nonalization of the Hamiltonian. In rsDFT, the Fermi\nenergy is extracted by the integration of the DOS, which\nis calculated with the TDRS method without diagonal-\nization [31, 34]:\nD(ε) =1\n2π/integraldisplay∞\n−∞eiεt/angbracketleftϕ|e−iHt|ϕ/angbracketrightdt, (1)\nhere,|ϕ/angbracketright=/summationtext\nici|ri/angbracketrightis a random state in the real-space\nand{ci}are normalized random complex numbers. The\nstate|ϕ/angbracketrightis also a random superposition state in the\nenergy-space and can be expressed as |ϕ/angbracketright=/summationtext\nnbn|En/angbracketright,\nwherebn=/summationtext\nicia∗\ni(En) andai(En) =/angbracketleftri|En/angbracketright. Thus\nEq. 1, becomes\nD(ε) =N/summationdisplay\nn=1|bn|2δ(ε−En), (2)\nwhereNis the dimension of the Hamiltonian. For a\nlarge but finite N,|bn|→1/√\nN, the error of using D(ε)\nto approximate the DOS vanish with 1 /√\nN[31, 34]. Aswe normally use the same grid density, the dimension\nof the Hamiltonian N, determined by the number of\ngrids, is linearly proportional to the number of atoms\nand the number of electrons. Thus the numerical error\nof calculating D(ε) scales also with 1 /√Ne. The Fermi\nenergyµis determined by Ne=/integraltextµ\n−∞D(ε)dε. In the\ncase thatNeis not enough to provide a desired accu-\nracy, additional average of D(ε) from different random\nstates (|ϕp/angbracketright=/summationtext\nici,p|ri/angbracketright, wherep= 1,2,...,S ) can be\nintroduced to reduce the statistical error. Then, accord-\ning to the central limit theory, the overall error of cal-\nculatingD(ε) scales as 1 /√SNe[31, 34]. Numerically,\nthe time-evolution operator e−iHtcan be decomposed\nusing the Chebyshev polynomial method as discussed in\nRef. [31, 34], which is unconditionally stable and leads to\na linear scaling on the system size as the Hamiltonian H\nis a spare matrix in DFT. The energy resolution is deter-\nmined by 1 /Ntτ, whereNtis the number of time steps\nandτis the time interval ( dt).\nAs a numerical check, we calculated the DOS of\ngraphite nanocrystals and fullerene with a different num-\nber of carbon atoms. Here, the Kohn-Sham Hamiltonian\nis constructed with a given initial electron density ρ(r)\nas\nH=−∇2\n2+Vext[ρ(r)] +VH[ρ(r)] +Vxc[ρ(r)],(3)\nwhere−∇2/2 is the kinetic energy, Vextis the external\npotential,VHis the Hartree potential, and Vxcis the\nexchange and correlation potential. The kinetic energy\nin the KS-Hamiltonian (Eq. (3)) is approximated by us-\ning the higher-order finite-difference expansion for the\nLaplacian operator in a uniform real-space grid [35, 36].\nThe Hartree potential VHis derived by solving the Pois-\nson equation [36]. For exchange and correlation poten-\ntialVxc, we use the local density approximation (LDA)\n[37]. The full ionic potential Vextis effectively replaced\nby pseudo-potential in Kleinman-Bylander forms [38].\nIn Fig. 1(a), we plot the DOS of a graphite nanocrystal\nwith 42 carbon atoms, showing that the TDRS results\nwith more random samples match better to the value\nfrom the diagonalization. For large graphite nanocrys-\ntals, such as the one with 11440 atoms shown in Fig. 1(b),\nthe TDRS results obtained from two individual random\nstates are quite close to each other. As a quantitative\nmeasurement of the statistical error, the standard devi-\nation (SD) of results using only one random state are\ncollected in Fig. 1(c). Here, in each case we consid-\nered 500 different random states and plotted the SDs\nofµandEocc(µ) for graphite nanocrystals with differ-\nent sizes, where Eocc(µ) is the occupied energy defined\nasEocc(µ) =/integraltextµ\n−∞εD(ε)dε. It is clear that the SDs of\nµandEocc(µ) reduce significantly when there are more\nelectrons and approach to zero with an error scales as\n1/√Ne. This indicates that for very large systems, it is\nnot necessary to have additional averages with different3\nrandom states, and thus using one random initial state\nis enough to provide a converged result in TDRS. On the\nother hand, for finite systems such as the fullerenes C 60\nand C 540shown in Fig. 1(d), the accuracy of using TDRS\ncan be improved by the average using more random states\nand the result converges to the exact value (from diag-\nonalization) with an error scale as 1 /√\nS. In total, our\nnumerical results prove that the statistical error of using\nthe TDRS method to calculate DOS or related variables\nscales as 1/√SNe, just as we expected (see also the error\nanalysis of DOS in the Supplementary Materials).\nElectron density. — In KS-DFT, the electron density\nis constructed by the superposition of occupied KS or-\nbitals|En/angbracketright, namelyρ(r) =/summationtext\nnf(En)||En(r)/angbracketright|2wheref\nis the Fermi-Dirac function. In rsDFT, the knowledge\nof|En/angbracketrightis absent as we do not perform any diagonaliza-\ntion, butρ(r) will be obtained in a different way. In\nthe basis of real-space grid, {|ri/angbracketright}, the wave functions of\nKS orbitals can be expressed as |En/angbracketright=/summationtextN\ni=1ai(En)|ri/angbracketright,\nwhereNis the total number of grid points, i.e., the di-\nmension of the Hamiltonian. We consider a random state\n|ϕ/angbracketrightin the real space, as the one used in Eq. 1, which is\nalso a random superposition state in the energy space.\nThus, a superposition of all occupied states, with a given\nFermi energy µand temperature Tcan be constructed\nby applying a Fermi-Dirac (FD) filter on the random\nstate as|ϕ/angbracketrightFD≡/radicalbig\nf(H)|ϕ/angbracketright=/summationtextbn/radicalbig\nf(En)|En/angbracketright, where\nf(H) = 1/(e(H−µ)/kBT+ 1) is the Fermi-Dirac opera-\ntor. The intensity of state vector |ϕ/angbracketrightFDat grid rjcan be\nexpressed as\n/bardblϕ/angbracketrightFD(rj)|2=/summationdisplay\nn|bn|2f(En)|aj(En)|2+\n/summationdisplay\nm/negationslash=nb∗\nmbnf1\n2(Em)f1\n2(En)a∗\nj(Em)aj(En).(4)\nAs we see in the calculation of DOS, for a large but finite\nN,|bn|→1/√\nN[39], thus the first term in Eq. 4 con-\nverges toρ(rj)/N, whereρ(rj) =/summationtext\nnf(En)|aj(En)|2is\nexactly the electron density at grid rj. However, the\nvalue of this term is of the order of O(Ne/N2),which\nis aboutNetimes smaller than the second term ∼\nO(N2\ne/N2). This means that the second term in Eq. 4\nbecomes dominant when increasing the system size. To\napproximate ρ(rj) by using/bardblϕ/angbracketrightFD(rj)|2, one needs to re-\nduce the second term significantly. This can be realized\nby using a time-dependent approach in the following way.\nIntroduce the electron density ρRS(r) using a time-\ndependent RS approach as\nρRS(rj)≡N\n2π/integraldisplay∞\n−∞/vextendsingle/vextendsinglee−iHt|ϕ/angbracketrightFD(rj)/vextendsingle/vextendsingle2dt, (5)\nand one can prove that, ρRS(r) converges to ρ(r) in the\nlimit ofN→∞ . Defineδρ≡/summationtext\nj|ρRS(rj)−ρ(rj)|/Neas\na measurement error of electron density, and by using the\n(a) (b)\n(c) (d)FIG. 2: (a) The statistical error δρas a function of the\nevolution time Ntfor C 20, C60, C180, C240and C 540,\nwhereρRefis the charge density obtained from\ndiagonalization. The time step is τ= 64πand the\nnumber of random samples S= 20. (b) The values of δρ\nin (a) in the limit of Nt→∞ , plotted as a function of\nNe. The error bars indicate the corresponding standard\ndeviations. (c) is the same as (b) but with finite Nt,\nand the reference charge density is the mean value\naveraged from 20 random samples with Nt= 144.\nNt= 0 means no time-dependent approach is involved\nin calculating charge density. (d) The value of δρ,\nplotted as a function of sample number Sfor C 60,\nwithout or with time-dependent approach ( τ= 64π).\nproperty/integraltext∞\n−∞e−i(En−Em)tdt= 2πδ(En−Em), we have\nδρ=1\nNe/summationdisplay\nj|/summationdisplay\nn(N|bn|2−1)f(En)|aj(En)|2+\n2πN/summationdisplay\nm/negationslash=nb∗\nmbnf1\n2(Em)f1\n2(En)a∗\nj(Em)aj(En)δ(En−Em)|.\n(6)\nFor a large but finite N,|bn|→ 1/√\nN, both terms in\nEq. 6 converge to zero, but the statistical error of the\nfirst and second terms is O(1/√\nN) andO(1/N), respec-\ntively. This indicates that in the limit of N→∞ ,ρRS(r)\nconverges to ρ(r), andδρis dominated by the first term\nand reduces to zero with a statistical error ∼O(1/√\nN).\nThe accuracy of using Eq. 5 to obtain the electron den-\nsity can be further improved by averaging ρRS(r) from\ndifferent initial random states. Similar to the calcula-\ntion of DOS, consider a set of random states |ϕp/angbracketright=/summationtext\nici,p|ri/angbracketright, according to the central limit theorem, for\na large but finite S,/summationtextS\np=1ci,pci/prime,p/S=E(|c|2)δi,i/prime+\nO(1/√\nS)[34], where E(|c|2)∼1/Nis the expectation\nvalue of|ci|2. Asbn=/summationtextN\ni=1cia∗\ni(En), using the normal-4\nization property/summationtextN\ni=1|ai(En)|2= 1 and the orthogonal\nproperty/summationtextN\ni=1ai(En)a∗\ni(Em) = 0 forn/negationslash=m, we have/summationtext\npb∗\nm,pbn,p/S=δm,n/N+O(1/√\nS). Thus, together\nwith extra random states average, ρRS(r) is an accurate\napproximation of ρ(r) with a statistical error δρscales as\n1/√SNe. This scaling behaviour is indeed the same as\nthe calculations of DOS given in Eq. 1.\nHere, the time-evolution operator e−iHtand the Fermi-\nDirac filter/radicalbig\nf(H) are carried out numerically using the\nChebyshev polynomials[31], which is very efficient and\naccurate for sparse matrix Has we discussed in the part\nof DOS. In the Chebyshev decomposition of the Fermi-\nDirac filter, the temperature has to be finite, and we\nuseT= 10Kin all the calculations. The method intro-\nduced here becomes more efficient at high temperatures,\nin which the ground state calculations also involve many\nstates above the Fermi energy due to a nonzero occupa-\ntion probability given by the Fermi-Dirac distribution.\nIn the following, we show several examples and check the\naccuracy of obtaining electron density using the TDRS\nmethod introduced in Eq. 5.\nWe first consider fullerene with different numbers of\ncarbon atoms and calculate the charge density for a given\nHamiltonian using either the TDRS method or the stan-\ndard diagonalization. In Fig. 2(a), we plot the statistical\nerrorδρas a function of the number of time steps Nt\nfor C 20, C 60, C 180, C 240, and C 540, where the reference\ncharge density ρin each case is the one obtained from\nthe diagonalization. We see that in all cases, δρdrops\nrapidly when introducing the time-evolution, and for a\ngiven evolution period (same Nt),δρis always smaller in\na larger sample. In Fig. 2(b), we plot the minimum value\nofδρas a function of the number of electrons Nein the\nlimit ofNt→∞ . It shows clearly that δρscales exactly\nasO(1/√Ne),same as we predicted from Eq. 6. The er-\nror bars are the standard deviations of electron density\nobtained from each individual random initial state, i.e.,\nwithout any random sample averaging. In practice, it\nmight be numerically expensive to perform a very long\nevolution, and thus one needs to consider using only finite\nor relatively small Nt. Thus we plot in Fig. 2(c) similar\nresults as (b) but with finite Nt. We see that: (1) in the\nabsence of the time-dependent approach ( Nt= 0), the\nvalue ofδρkeeps the same amplitude, independent of the\nsystem size; (2) when the time-evolution is introduced,\nthe value of δρstarts to decrease when increasing the size\nof the sample ( Ne). In Fig. 2(d), we consider the influ-\nence of sample average on the statistical error by plotting\nthe value of δρas a function of sample number S. Here\nthe reference charge density is the one obtained from C 60\nwith exact diagonalization, and we see that the scaling\nof the error follows as 1 /√\nS, for both cases with or with-\nout the time-dependent approach. The results presented\nin Fig. 2 verified numerically that the statistical error of\ncalculating the charge density using the TDRS method\n(a) (b)FIG. 3: (a) For C 60, the eigenvalue distribution of the\nground state calculated from VASP, diagonalization and\nrsDFT, respectively. (b)The convergence of the total\nenergy during the iteration for C 20, C60, and C 240. Here\nthe total energy of the ground state ( EGS) is obtained\nfrom diagonalization.\nscales as 1/√SNe, with the same scaling behaviour as\nthe calculation of DOS.\nSelf-consistent iteration. — Now, we consider the de-\ntailed self-consistent iterations in rsDFT. Here, the nu-\nmerical results obtained from the widely used commer-\ncial KS-DFT package VASP (Vienna Ab initio Simula-\ntion Package) [40] are also presented as references.\nIn Fig. 3(a), we show the ground state DOS of C 60\ncalculated using rsDFT, standard KS-DFT (with diago-\nnalization) and VASP, respectively. Pulay mix is adopted\nin the iteration to optimize the input density and accel-\nerate the convergence [41]. In rsDFT, we use 10 random\nsamples in DOS calculations and 36 random samples with\nNt= 36,τ= 64πin electron density calculations. The\nDOS obtained from different approaches agree well, indi-\ncating that (1) our DFT code based on diagonalization\ncorrectly reproduces the ground state from VASP, (2)\nthe newly proposed rsDFT provides accurate results as\nthese can be obtained from standard KS-DFT and the\ndiagonalization can be completely ruled out in the entire\niteration process. In Fig. 3(b), we present more results of\nconverged rsDFT calculations of fullerenes with different\nsizes, and show the total energy difference compared with\nthe result from diagonalization in each iteration step. In\ngeneral, the convergence to the ground state is more dif-\nficult in the larger system in KS-DFT, requiring more\niteration steps to reach the same accuracy for the total\nenergy. In rsDFT, however, the accuracy is increased au-\ntomatically in larger systems due to the 1 /√SNedepen-\ndence of the statistical error, as shown clearly in converge\nof the total energy during the iterations in Fig. 3(b).\nScaling Behavior. — At last, we check the scaling be-\nhaviour of rsDFT. The non-diagonal elements of KS-\nHamiltonian are from the kinetic energy and non-local\npseudopotentials in Eq. (3), leading to a highly sparse\nHamiltonian matrix due to the locality. The number\nof the non-zero elements in the matrix scales linearly\nwith the number of atoms in the system. In rsDFT,\nthe basic and dominant calculations are the multiplica-5\n(a) (b)\n(c) (d)\n 5040 atoms 11440 atoms\nFIG. 4: (a,b) The cost of memory and CPU time in\nrsDFT for A-B stacked graphite with different numbers\nof carbon atoms. The black and red curves in (a) are\nthe total memory cost and the memory used to store\nthe sparse Hamiltonian matrix. The black, red and blue\ncurves in (b) correspond to the time cost of one\niteration for the calculations of DOS, Fermi-Dirac filter\nand time-evolution averaging, respectively. (c,d) The\ndifference of input and output electron density as a\nfunction of iteration steps for A-B stacked graphite with\n5040 and 11,440 atoms, respectively. The insets indicate\nthe converged ground state density.\ntions between the Hamiltonian matrix and a state vec-\ntor, in which the number of operations scales linearly\nwith the nonzero elements in the matrix. Therefore,\nthe total computational load (CPU time) and memory\ncost of rsDFT are linearly dependent on the dimension\nof the Hamiltonian, which is proportional to the num-\nber of atoms in the system. These linear-scaling be-\nhaviours are verified using graphite crystal, with up to\n11440 carbon atoms (45760 electrons) in Fig. 4 (a-b). As\nbenchmark tests, we perform complete ground state cal-\nculations of graphite with 5040 and 11440 carbon atoms,\nrespectively. The difference of input and output electron\ndensity (∆ ρ≡/summationtext\ni|ρout(ri)−ρin(ri)|/N) as a function of\niterative steps is plotted in Fig. 4 (c-d). In all cases, we\nused the same calculation parameters, such as the same\nreal-space grid density, and the same accuracy for the\nChebyshev decompositions of the time-evolution opera-\ntor and the Fermi-Dirac filter. One should notice that,\nfor 11440 carbon atoms, the total memory cost is only 16\nGB. Due to the linear-scaling behaviour, a self-consistent\ncalculation of millions of atoms should be possible for\ncomputer clusters with Terabytes (TB) memory.\nConclusion and Discussion. — We developed a\nsublinear-scaled self-consistent first-principle calculationmethod rsDFT. We use a TDRS method to calculate\nthe density of states and determine the Fermi energy. A\nFermi-Dirac filter on a random state and, subsequently,\na wave propagation according to the time-dependent\nSchr¨ odinger equation are introduced to approximate the\nspatial distribution of the electron density. The accu-\nracy can be improved by the average using different ini-\ntial random states. The overall numerical error of either\na global quantity or a local variable scales as 1 /√SNe,\nwhereNeis the number of electrons and Sis the number\nof random states. It leads to an overall sublinear scal-\ning of the computational costs, as for larger systems, one\nneeds fewer random states for the sample average. The\nmethod becomes extremely powerful when Ne→∞ , and\na calculation using one random state is enough to achieve\nreasonable accuracy.\nIn the recently developed stochastic DFT (sDFT)\n[9, 27–30], the physical quantities, such as the Fermi en-\nergy and electron density, are calculated using the trace\nformula of the stochastic technique without diagonaliza-\ntion, i.e., the trace of a variable is approximated by the\naverage of its expectation values in stochastic orbitals.\nOne of the main differences between rsDFT and sDFT\nis that all the variables in rsDFT are obtained in a more\ndeterministic way based on numerical solutions of the\ntime-dependent Schr¨ odinger equation, which is absent in\nsDFT. In particular, the dominant noise in the electron\ndensity induced by occupied states after the Fermi-Dirac\nfilter is dramatically reduced by the time-dependent ap-\nproach in rsDFT. A large number of stochastic orbitals\n(random samples) is required to reduce the statistical er-\nror of the electron density in sDFT. And to keep the same\naccuracy of calculating the local variables such as elec-\ntron density, one can not use fewer stochastic orbitals for\nlarger systems, even in the limit of Ne→∞ .\nIn our earlier works, we have developed a so-called\ntight-binding propagation method (TBPM) for large-\nscale modelling of complex quantum systems. A direct\nextension of TBPM in rsDFT is straightforward. For\nexample, by using the TDRS-based method without di-\nagonalization, one can calculate the electronic and op-\ntical conductivities [42, 43], polarization and screening\nfunctions [32, 44], diffusion coefficient and localization\nlength [33, 44], quasieigenstate [45], and many other ap-\nplications as implemented in our homemade simulation\npackage, TBPLaS [46]. The main advantage of rsDFT\nand the other time-dependent methods mentioned above\nis that there is no diagonalization of the Hamiltonian ma-\ntrix in the whole process, and the errors of these calcu-\nlated variables all scale as 1 /√SNe, leading to an overall\nsublinear scaling on the computational costs. Further-\nmore, the atomic force can be calculated the same way\nas the traditional DFT or OF-DFT by using the formu-\nlation based on the electron density in real space [47–50].\nIt is also possible to extract the atomic force using the\nwave function of approximated ground states, which will6\nbe discussed in future work. 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Baer,\nEquilibrium configurations of large nanostructures using\nthe embedded saturated-fragments stochastic density func-\ntional theory , J. Chem. Phys. 146, 224111 (2017).Supplementary Materials: A Time-Dependent Random State Approach for\nLarge-scale Density Functional Calculations\nWeiqing Zhou1and Shengjun Yuan1,∗\n1Key Laboratory of Artificial Micro- and Nano-structures of Ministry of Education\nand School of Physics and Technology, Wuhan University, Wuhan 430072, China\n1. Higher-order Finite-difference Pseudopotential Method\nWithin the non-relativistic Kohn −Sham DFT, the ground state of a system of Neelectrons subject to an external\npotential can be obtained by solving a set of one-particle equations, the Kohn −Sham equations (atomic units will be\nused throughout):\n[−∇2\n2+VKS[ρ(r)]]ϕi(r) =εiϕi(r) (1)\nwhere Kohn-Sham potential VKS[ρ(r)] is usually divided as:\nVKS[ρ(r)] =Vext[ρ(r)] +VH[ρ(r)] +Vxc[ρ(r)] (2)\nwhereVextis the external potential, VHis the Hartree potential, and Vxcis the exchange and correlation potential.\nIn this paper, we implement real-space finite-element methods, resulting in VKS[ρ(r)] =VKS(r).\nIn our letter, we impose a simple, uniform orthogonal three-dimensional (3D) grid where the points are described\nin a finite domain by ( xi,yj,zk) [1]. Kinetic-energy operator can be described by high-order finite-element difference\nmethod [2],\n∂2ϕ\n∂x2=Nh/summationdisplay\nn=−NhCnϕ(xi+nh,yj,zk) +O/parenleftbig\nh2Nh+2/parenrightbig\n(3)\nwherehis the grid spacing and Nhis the order of finite-element difference. Expansion coefficients Cnfor a uniform\ngrid are given in Table. S1 [2].\nTABLE S1: Expansion coefficients Cnfor higher-order finite-difference expressions of the second derivative.\nCiCi±1Ci±2Ci±3Ci±4Ci±5Ci±6\nNh= 1 -2 1\nNh= 2 −5\n24\n3−1\n12\nNh= 3 −49\n183\n2−3\n201\n90\nNh= 4 −205\n728\n5−1\n58\n315−1\n560\nNh= 5 −5269\n18005\n3−5\n215\n126−5\n10081\n3150\nNh= 6 −5369\n180012\n7−15\n5610\n189−1\n1122\n1925−1\n16632\nThe Hartree energy density and potential are given by:\nεH(r) =1\n2/integraldisplay\ndr/primen(r/prime)\n|r−r/prime|(4)\nVH(r) =/integraldisplay\ndr/primen(r/prime)\n|r−r/prime|(5)\nThe Hartree potential VHcould be obtained by solving Poisson’s equation.\nFor the exchange-correlation part, we use local-density approximation (LDA):\nExc[n] =/integraldisplay\ndrn(r)εxc(n(r)) (6)arXiv:2203.03465v2  [cond-mat.mes-hall]  24 Jan 20232\n(a) (b)\nFIG. S1: (a) pseudopotential of carbon atom generated by program ATOM [5]. (b) atomic wavefunction (Red\ndashed curve) and pseudo-wavefunction (black solid curve) of carbon.\nThe most accurate formulae for the exchange-correlation functional were obtained by fitting the QMC results for\nthe Jellium model. Various parameterizations are available. We use one of the most popular choices proposed by\nVosko-Wilk [3]. From the Jellium model, the local part of the exchange is given by :\nεx=−3\n4(3\n2π)2/31\nrs,\nVx= (3\n2π)2/31\nrs.(7)\nForVext(r), we use full ionic potential −Z\nrfor the cases of single atoms. For other systems, we implement a\npseudopotential operator to reduce the computational demand. We use the projection scheme of the pseudopotential\noperator suggested by Kleinman and Bylander [4]:\nVps(r) =N/summationdisplay\na=1/bracketleftBigg\nVloc\na,ps(r) +/vextendsingle/vextendsingle∆Vl\na,ps(r)ϕa\nlm/angbracketrightbig/angbracketleftbig\n∆Vl\na,ps(r)ϕa\nlm(r)/vextendsingle/vextendsingle\n/angbracketleftbig\nϕa\nlm(r)/vextendsingle/vextendsingle∆Vla,ps(r)/vextendsingle/vextendsingleϕa\nlm(r)/angbracketrightbig/bracketrightBigg\n(8)\nwhere the total pseudopotential can be divided into non-local and local part ∆ Vl\na,ps(r)≡Vl\na,ps(r)−Vloc\na,ps(r).Vloc\na,ps\nis the local part with specific angular momentum lcomponent of atom a, which differs from zero only in the region\nsmaller than the cutoff radius r<rc.ϕa\nlmis the atomic pseudo wave function with lmquantum angular momentum\nnumbers. It is worth noticing that the pseudopotential operator only needs to be calculated once at the very beginning\nsince it only depends on the atomic configuration. Taking the carbon atom as an example, we use the program ATOM\n[5] to generate its pseudopotential. The type of pseudopotential is chosen as local density approximation (LDA) [6]\nand plotted in Fig. S1. In Fig. S1 (b), we show the atomic pseudo-wave-function, and indeed it is the same as a\nfull-potential wave-function in the range of r > rswherersis the cutoff radius. The construction of charge density\nhas been described in the main context.\n2. Another Fermi-Dirac Filter\nIn this part, we add some detailed discussion of the methods used in rsDFT. First, we construct a random super-\nposition state in a uniform real-space grid as an initial state,\n|ϕ0/angbracketright=N/summationdisplay\ni=1ci|ri/angbracketright, (9)3\n(a) (b)\nFIG. S2: The statistical error of calculated DOS as a function of the number of electrons for graphite nanocrystals\n(a) or the number of random states for C 540(b). In (a), the standard deviation of the DOS spectrum/integraltext∞\n−∞|D(ε)−<D(ε)>|dεis calculated based on the results from 500 individual random states, where <D(ε)>is\nthe mean value of {Dk(ε)}withk= 1,2,...,500.In (b), the error is defined as ∆ D≡/integraltext∞\n−∞|D(ε)−DDiag(ε)|dε, where\nDDiag(ε) is the result obtained from the diagonalization, and each point is averaged from 100 groups of Srandom\nstates.\nwhereNis the number of grid, {ri}are the real space basis states, and {ci}are random complex numbers. Assuming\nthat\n|En/angbracketright=N/summationdisplay\niai(En)|ri/angbracketright, (10)\nwe have\n|ϕ0/angbracketright=N/summationdisplay\ni=1ciN/summationdisplay\nn=1|En/angbracketright/angbracketleftEn|ri/angbracketright\n=N/summationdisplay\ni=1N/summationdisplay\nn=1cia∗\ni(En)|En/angbracketright\n=N/summationdisplay\ni=1N/summationdisplay\nj=1N/summationdisplay\nn=1cia∗\ni(En)|rj/angbracketright/angbracketleftrj|En/angbracketright\n=N/summationdisplay\ni=1N/summationdisplay\nj=1N/summationdisplay\nn=1cia∗\ni(En)aj(En)|rj/angbracketright.(11)\nNow we consider another type of Dirac-Fermi filter different from the one introduced in the main text:\n|ϕ/angbracketrightfd≡f(H)|ϕ0/angbracketright\n=N/summationdisplay\ni=1N/summationdisplay\nj=1N/summationdisplay\nn=1cia∗\ni(En)f(En)aj(En)|rj/angbracketright(12)4\nIn the inner product of /angbracketleftϕ0|ϕ/angbracketrightfdat grid rjcan be calculated by using Eq. (11) and Eq. (12),\nρfd(rj) =0/angbracketleftϕ|rj/angbracketright/angbracketleftrj|ϕ/angbracketrightfd\n=N/summationdisplay\ni,i/prime=1N/summationdisplay\nn,m=1c∗\niai(En)a∗\nj(En)ci/primea∗\ni/prime(Em)f(Em)aj(Em)\n=N/summationdisplay\ni,i/prime=1N/summationdisplay\nn=mc∗\niai(En)ci/primea∗\ni/prime(En)f(En)|aj(En)|2\n+N/summationdisplay\ni,i/prime=1N/summationdisplay\nn/negationslash=mc∗\niai(En)a∗\nj(En)ci/primea∗\ni/prime(Em)f(Em)aj(Em)(13)\nAccording to the central limit theorem, for a large but finite number ( S) of the random states |ϕp/angbracketright=/summationtext\nici,p|ri/angbracketright, we\nhave\n1\nSS/summationdisplay\np=1ci,pci/prime,p=E(c2)δi,i/prime+O(1√\nS). (14)\nTherefore, one proves that\nlim\nS→∞1\nSS/summationdisplay\np=1/angbracketleftϕp|rj/angbracketright/angbracketleftrj|ϕp/angbracketrightfd\n=N/summationdisplay\ni=1N/summationdisplay\nn=1E(|c|2)f(En)|ai(En)|2|aj(En)|2\n+N/summationdisplay\ni=1N/summationdisplay\nn/negationslash=mE(|c|2)f(Em)ai(En)a∗\ni(Em)a∗\nj(En)aj(Em)\n=N/summationdisplay\ni=1|ai(En)|2N/summationdisplay\nn=1E(|c|2)f(En)|aj(En)|2\n=1\nNN/summationdisplay\nn=1f(En)|aj(En)|2\n=1\nNρdiag(xj)(15)\nhere we used the normalization property of KS orbitals\nN/summationdisplay\ni=1|ai(En)|2= 1 (16)\nand the orthogonal property\nN/summationdisplay\ni=1ai(En)a∗\ni(Em) = 0 (17)\nform/negationslash=n. Eq. (15) indicates that\nρfd(rj)≡N\nSS/summationdisplay\np=1/angbracketleftϕp|rj/angbracketright/angbracketleftrj|ϕp/angbracketrightfd (18)\nis an approximation of the charge density at rjwith an error vanishes as 1 /√\nS, which can be verified in the zoom-in\nfigure of Fig. S3 .5\nFIG. S3: The difference between the electron density obtained from KS-DFT ρdiagand rsDFT as a function of the\nnumber of random samples S. In rsDFT, the results are obtained by using ρFD(Eq. (4) in the main context) or ρfd\n(Eq. (13)). Using/radicalbig\nf(H) instead of f(H) in the Fermi-Dirac filter will significantly reduce the statistical error.\nHowever, the converge of ρfd(rj) using the Fermi-Dirac filter f(H) is slower than the one using double/radicalbig\nf(H)\nintroduced in the main text. The reason is that, in Eq. (13), the sum in the second term ( n/negationslash=m) involves all\nunoccupied states associated with the index n, and their number is several orders larger than the number of occupied\nstates because N/greatermuchNe. To overcome this difficulty, we introduce |ϕ/angbracketrightFD=/radicalbig\nf(H)|ϕ/angbracketright0, the one used in the main\ncontext. The main advantage of using ρFD(rj) is that the sums in the second term ( n/negationslash=m) of Eq. (4) of the main\ncontext includes only occupied states, leading to a much faster convergence compared with Eq. (13) (see Fig. S3).\n3. Chebyshev Polynomials Method\nIn the numerical calculation, the operators1√\neβ(H−µ)+1ande−iHtare approximated by using the Chebyshev poly-\nnomial method. In general, a function f(x) whose values are in the range [-1,1] can be expressed as,\nf(x) =1\n2c0T0(x) +∞/summationdisplay\nk=1ckTk(x) (19)\nwhereTk(x) = cos(karccosx) and the coefficients ckare\nck=2\nπ/integraldisplay1\n−1dx√\n1−x2f(x)Tk(x) (20)\nif we letx=cosθ, thenTk(x) =Tk(cosθ) = coskθ, and\nck=2\nπ/integraldisplayπ\n0f(cosθ) coskθdθ\n= Re/bracketleftBigg\n2\nNN−1/summationdisplay\nn=0f/parenleftbigg\ncos2πn\nN/parenrightbigg\ne2πink/N/bracketrightBigg\n,(21)\nwhich can be calculated by the fast Fourier transform (FFT). We normalize Hsuch that/tildewideH=H//bardblH/bardblhas eigenvalues\nin the range [-1,1] and put /tildewideβ=β//bardblβ/bardbl. Then\nf(˜H) =∞/summationdisplay\nk=0ckTk(˜H) (22)6\n(a) (b)\n(c) (d)\nFIG. S4: (a) DOS and carrier density of one He atom calculated using the time-dependent random state method of\none He atom. The vertical dashed lines indicate the energies obtained from the diagonalization. (b) The output\nelectron density after one iteration using diagonalization (black) and Fermi-Dirac filter on random states (red) from\nthe same input electron density (blue). For the He atom, due to the spherical symmetry, the electron density is only\na function of distance ( r) from the center of the atom. (c) The converged ground-state electron density of the He\natom was obtained from KS-DFT and rsDFT. (d) The total energy as a function of bond length of molecular H 2\nfrom obtained from KS-DFT and rsDFT, respectively.\nwhere the Chebyshev polynomial Tk(x) is the Chebyshev polynomial of the first kind. Tk(x) obeys the following\nrecurrence relation:\nTk+1(x) +Tk−1(x) = 2xTk(x)\nwith\nT0(x) = 1,T1(x) =x.\nIn Table. S2, we present the number of nonzero Bessel function ( NBessel ) as a function of time step τ, and the\ncorresponding number of total matrix-vector operations Noperations for the same propagation time T= 1024π. We\ncan see a larger τleads to fewer operations with the same total propagation time.\n4. SINGLE ATOM\nLet us consider a single Helium atom. In step (I), we construct a KS-Hamiltonian based on an initial electron\ndensity. In step (II), we obtain DOS D(ε) by using the time-evolution method without the diagonalization of the\nHamiltonian matrix and subsequently determine the Fermi level µ(see Fig. S4(a)).\nAs a comparison, the energies of KS orbitals from the diagonalization are also shown in Fig. S4(a), which agree very\nwell with our results. In step (III), as there is only one occupied state, one can just use ρFDto obtain the electron7\nFIG. S5: Comparisons of ground-state charge density calculated by KS-DFT based on diagonalization and rsDFT of\nsingle atoms.\n(a) (b)\nFIG. S6: (a) The statistical error δρas a function of total operations. Different colour indicates different time step τ\nin time evolution. (b) The statistical error δρas a function of 1 /√Ntfor different fullerenes, where τ= 64π.\nTABLE S2: The number of nonzero Bessel function ( NBessel ) as a function of time step τ, and the corresponding\nnumber of total matrix-vector operations Noperations for the same certain propagation time T= 1024π.\nτ N Bessel Nt Noperations\nπ 20 1024 20480\n2π 27 512 13824\n4π 38 256 9728\n8π 56 128 7168\n16π 89 64 5695\n32π 149 32 4768\n64π 261 16 4176\n128π 478 8 3824\n256π 899 4 3596\n512π 1727 2 34548\n1 8 16 24\n32\n 48 64\n 128\nFIG. S7: The charge density of C 60calculated from average over a different number of random samples without time\nevolution.\n32 64 96 128\n32 64 96 128(a) time intervals dt = 𝜋\n(b) time intervals dt = 32 𝜋\nFIG. S8: The charge density of C 60calculated from one random state averaging over from time evolution for 32, 64,\n96, 128 steps with dt= π(a) and dt=32 π(b).\ndensity without time evolution. As a comparison, we also calculate the electron density ρdiagbased on the occupied\nKS orbital obtained from the diagonalization of KS-Hamiltonian, and plot together with ρFDin Fig. S4(b). We see\nthat with only S=10 random samples, ρFDconverges to ρdiagwith an error of ∆( ρFD−ρdiag) = 8.06×10−5, where\n∆(ρFD−ρdiag)≡/summationtextN\nj=1|ρFD(rj)−ρdiag(rj)|/N. In step (IV), we use ρFDas the new input electron density and\nperform the next iteration. The self-consistent iterations, including steps (I) to (IV), are continued until a threshold\nis reached. In our approach, since the space resolution (determined by N) is much larger than the energy resolution\n(determined by Nt), it is more accurate to use the electron density instead of the total energy to define the convergence9\nVASP PM method\nRS-DFT\nC60 C60V ASP\nrsDFTrsDFT\nC540 C540VASPVASP\nFIG. S9: The ground-state charge density calculated by rsDFT and VASP, respectively.\ncriterion. The ground-state electron density obtained from rsDFT without any diagonalization agrees well with the\none from the common KS-DFT with diagonalization. Both are plotted in Fig. S4(c) for comparison. More examples\nof other single atoms can be found in Fig. S5.\n4. MOLECULES\nFor molecular systems, we consider a diatomic model H 2. The iterative calculations are similar to those of a single\natom. Here we verify our approach by calculating the total energies for different H-H bond lengths and compare\nthe results from our rsDFT approach and the common KS-DFT in Fig. S4(d). The two methods yield similar total\nenergies for a given H-H bond length. The bond lengths in the ground state obtained from both ways are the same\n(74 pm), which agrees with the well-known result [7].\n5. CLUSTERS\nWe extend our calculations to large atomic clusters of fullerenes C 60and C 540. In Fig. S7, we plot the ρFD\naveraging from up to 128 random states. As a comparison, we also present ρRSusing only one random state, but\ndifferent propagation time in Fig. S8(a) with τ=πand Fig. S8(b) with τ= 32π. The real-space distribution of electron\ndensity in the ground state is visualized by VESTA [8] in Fig. S9. We use VASP (Vienna Ab initio Simulation Package)\n[9] to represent the standard KS-DFT method. VASP is a very efficient and widely used commercial KS-DFT package.\nThe electron density distributions obtained from rsDFT and VASP are very similar.10\n42 atoms 264 atoms 2202 atoms\nFIG. S10: ∆( ρin−ρout) as a function of iterative steps for A-B stacked graphite calculated by rsDFT. The insets\nindicate the converged ground state densities.\n(a) (b)\nFIG. S11: Time cost of per iteration (a) and memory cost (b) during self-consistent calculations for fullerenes with\ndifferent numbers of electrons. Black points refer to the traditional KS-DFT method with diagonalization, and red\npoints refer to the rsDFT method without diagonalization. In rsDFT, we used 36 random samples for the average in\nthe DOS and charge density calculations, the time step is τ= 64πand the number of time steps is Nt= 36.\n6. CRYSTALS\nMore rsDFT calculations of graphite nanocrystals with different numbers of carbon atoms are plotted in Fig. S10.\n7. CPU TIME AND MEMORY COST\nTo have a direct comparison of the CPU time and memory cost between the traditional KS-DFT (with diagonal-\nization) and rsDFT (without diagonalization), we performed calculations for the fullerenes with different numbers of\natoms (electrons) on a server with 40 CPU cores (2*Intel(R) Xeon(R) CPU Gold 6248). As shown in Fig. S11(a), if\nthe system has less than ∼1000 electrons, the traditional KS-DFT is much faster, but when the system size reaches\n∼1000 electrons, the rsDFT method becomes more efficient. The results in Fig. S11(a) also indicates that the time\ncost of rsDFT scales linearly with the system size, whereas the traditional KS-DFT scales approximately as O(N3\ne).\nAlthough the accuracy of rsDFT and KS-DFT are not exactly the same, we estimate that rsDFT becomes more\nefficient when the system contains a few thousand or more electrons.11\n∗s.yuan@whu.edu.cn\n[1] J. R. Chelikowsky, N. Troullier, and Y. Saad, Phys. Rev. Lett. 72, 1240 (1994).\n[2] J. R. Chelikowsky, N. Troullier, K. Wu, and Y. Saad, Higher-order finite-difference pseudopotential method: An application\nto diatomic molecules , Phys. Rev. B 50, 11355 (1994).\n[3] S. H. Vosko, L. Wilk, and M. Nusair, Accurate spin-dependent electron liquid correlation energies for local spin density\ncalculations: a critical analysis , Can. J. Phys. 58, 1200 (1980).\n[4] L. Kleinman and D. M. Bylander, Efficacious Form for Model Pseudopotentials , Phys. Rev. Lett. 48, 1425 (1982).\n[5] ATOM, a program for DFT calculations in atoms and pseudopotential generation, distributed as part of the SIESTA\nsoftware package. See http://www.icmab.es/siesta/atom.\n[6] D. R. Hamann, M. Schl¨ uter, and C. Chiang, Norm-Conserving Pseudopotentials , Phys. Rev. Lett. 43, 1494 (1979).\n[7] K.-P. Huber, Molecular spectra and molecular structure: IV. Constants of diatomic molecules (Springer Science & Business\nMedia, 2013).\n[8] K. Momma and F. Izumi, VESTA: a three-dimensional visualization system for electronic and structural analysis , J. Appl.\nCrystallogr. 41, 653 (2008).\n[9] G. Kresse and J. Furthm¨ uller, Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-\nwave basis set , Comput. Mater. Sci. 6, 15 (1996)."    },    {        "title": "2203.08317v2.TAKDE__Temporal_Adaptive_Kernel_Density_Estimator_for_Real_Time_Dynamic_Density_Estimation.pdf",        "content": "1\nTAKDE: Temporal Adaptive Kernel Density\nEstimator for Real-Time Dynamic Density\nEstimation\nYinsong Wang, Yu Ding, Senior Member, IEEE, and Shahin Shahrampour, Senior Member, IEEE\nAbstract —Real-time density estimation is ubiquitous in many applications, including computer vision and signal processing. Kernel\ndensity estimation is arguably one of the most commonly used density estimation techniques, and the use of “sliding window” mechanism\nadapts kernel density estimators to dynamic processes. In this paper, we derive the asymptotic mean integrated squared error (AMISE)\nupper bound for the “sliding window” kernel density estimator. This upper bound provides a principled guide to devise a novel estimator,\nwhich we name the temporal adaptive kernel density estimator (TAKDE). Compared to heuristic approaches for “sliding window” kernel\ndensity estimator, TAKDE is theoretically optimal in terms of the worst-case AMISE. We provide numerical experiments using synthetic\nand real-world datasets, showing that TAKDE outperforms other state-of-the-art dynamic density estimators (including those outside of\nkernel family). In particular, TAKDE achieves a superior test log-likelihood with a smaller run-time.\nIndex Terms —Real-time Density Estimation, Kernel Density Estimation, Adaptive Estimation, Asymptotic Mean Integrated Squared Error.\n✦\n1 I NTRODUCTION\nTHis work is concerned with estimation and tracking\nof dynamic probability density functions in real time,\nmotivated by a nanoscience application. The introduction of\nin situ transmission electron microscope (TEM) technology\n[1] allows the growth of nanoparticles to be captured in\nreal time and has the potential to enable precise control in\nnanoparticle self-assembly processes. Part of the underlying\nnanoscience problem is framed into a learning problem\nwith the following characteristics [2]: (1) Estimation and\ntracking of a time-varying probability density function that\nreflects the collective changes across ensembles of the nano\nobjects. (2) It seems inevitable to adopt a non-parametric\napproach in the density tracking, because there is no settled\nparametric density function that can adequately describe\ngrowth mechanisms in a multi-stage nanoparticle growth\nprocess [1], [3]. (3) In order to be useful for in-process decision\nmaking, the density estimation and tracking needs to be\nconducted in real time. By \"real-time\" we mean that the\nlearning and computation speed ought to be fast enough\nrelative to the imaging rate (or the data arrival rate in\ngeneral), which is 15frames per second (fps) in [1]. While\nthe research is motivated by the dynamic nano imaging, we\nbelieve that the aforementioned characteristics are rather\ncommon in many types of dynamic streaming data, brought\nforth in various applications by fast-pace data collection\ncapability. The objective of this research is to present one\n•Y. Wang and S. Shahrampour are with the Department of Mechanical and\nIndustrial Engineering at Northeastern University, Boston, MA 02115\nUSA.\nE-mail: wang.yinso@northeastern.edu\nE-mail: s.shahrampour@northeastern.edu\n•Y. Ding is with the Wm Michael Barnes ’64 Department of Industrial\nand Systems Engineering at Texas A&M University, College Station, TX\n77843 USA.\nE-mail: yuding@tamu.educompetitive solution for dynamic density estimation and\ntracking.\nOn the subject of density estimation, kernel density\nestimator has had great success (in terms of accuracy) for\nstatic datasets [4]. The direct adaptation of kernel density\nestimator to dynamic density estimation [5] is infeasible as\nthe memory and computation cost constantly scale with the\ntotal number of incoming data points. [6] further shows that\neven with unlimited computation and storage resources, a\ntraditional kernel density estimator will only be a consistent\nestimator for a few specific dynamic systems. [2] also shows\nthat traditional kernel density estimation falls short in\npractice in dynamic density estimation due to limited data\navailability.\nTo address the disadvantages of traditional kernel density\nestimator in dynamic density estimation, most researchers\nresort to the \"sliding window\" mechanism [7], [8], [9]. For\nexample, [7] proposed the M-kernel algorithm, where the\ncontribution of each data point in the \"sliding window\" is\napproximated as an additional weight added to the kernel\ndensity at the closest grid point. This approach manages\nto keep the memory and computation costs within budget\ndespite the growth of the total number of data points.\nHowever, with a poor choice of grid points, it can suffer\nfrom either over-fitting or under-fitting. [8] employed cluster\nkernel and resampling technique to improve the merger\nperformance. This approach uses the exponentially decaying\nweight scheme to capture the dynamic of the true density.\n[10] proposed the local region kernel density estimator\n(LRKDE), where the kernel bandwidth varies in different\nregions. The regions are divided such that the sum of data\nvariances in each region is minimized. LRKDE also uses a\n\"sliding window\" to capture the dynamic of the true density.\n[9] further improved upon the previous works by using\nlinear interpolation with kernel densities at grid points toarXiv:2203.08317v2  [stat.ML]  8 Nov 20232\napproximate the kernel density estimator and then updating\nthe kernel densities at the grid points with data points within\na \"sliding window\".\nThe \"sliding window\" kernel density estimators do not\nonly use the data points at the current time stamp, and\nthey take into account older data points for inferring the\ncurrent distribution. Intuitively, this mechanism provides\ntwo improvements that allow the kernel density estimator\nto work well in dynamic density estimation. First, defining\na window size according to the computation and memory\nlimit of the learning machine can alleviate the scalability\nissue of the kernel density estimator as old data points that\nare irrelevant to the current distribution can be discarded.\nSecond, including older data points in the window can\nhelp alleviate the low data volume issue for most streaming\ndata applications. However, to the best of our knowledge,\nall \"sliding window\" kernel density estimators proposed\nso far focus on modifying the kernel density estimator\nitself, and less attention has been given to the \"sliding\nwindow\" mechanism. As the only component that addresses\nthe \"dynamic\" part of dynamic density estimation, there\nis no answer regarding how this mechanism affects the\nperformance of the estimation.\nWe note that there also exists another line of works that\nmodel the dynamic density transition using a dynamical\nsystem with a fixed number of parameters. One class of\nframeworks is based on Bayesian learning [11], [12], [13],\nwhich models the prior with an evolving Dirichlet process\ncalled dependent Dirichlet process, where the dependence be-\ntween a class of Dirichlet processes is defined by a covariate.\nWhen using the covariate to describe time, the dependent\nDirichlet process can be used to model the evolution of\nthe dynamic distribution. The computation and memory\ncosts are also maintained at a constant level. Another\napproach [2] couples B-spline with Kalman filter to capture\nthe density evolution with a state space model. It imposes\nspace continuity with B-spline smoothing and time continuity\nwith Kalman filter to develop a fast density estimator for real-\ntime process control. However, these estimators always need\na normalization process with numerical operations to return\na proper density function. For real-time density estimation\ntasks that require a model update cycle in the order of sub-\nsecond, these methods may not be ideal as we will later show\nin simulations.\nIn this paper, we propose the temporal adaptive kernel\ndensity estimator (TAKDE), a novel kernel density estimator\nfor real-time dynamic density estimation that is theoreti-\ncally optimal in terms of the worst-case asymptotic mean\nintegrated squared error (AMISE). For the first time, we\nderive the AMISE upper bound for the \"sliding window\"\nkernel density estimator in a dynamic density estimation\ncontext. The minimizer of the upper bound entails a novel\nsequence for bandwidth selection and data weighting, which\nforms the basis of TAKDE. We provide numerical exper-\niments on synthetic datasets to support our theoretical\nclaim, and we then use several real-world datasets to show\nthat TAKDE outperforms other state-of-the-art fast dynamic\ndensity estimators, such as the B-spline Kalman Filter [2]\nand KDEtrack [9] in terms of mean test log-likelihood metric.\nInterestingly, TAKDE also dominates these algorithms in\nterms of achieving a smaller run-time.The organization of the paper is as follows. We present\nin Section 2 the preliminaries, including definitions and\nnotations used throughout the paper. In Section 3, we\npresent the details for TAKDE design, which addresses\nthree important questions, i.e., the selection of window size,\nbandwidth and the data weights. We provide in Section 4\nnumerical experiments with synthetic and real datasets to\ndemonstrate the performance of TAKDE. Finally, we draw\nconclusions and discuss the potential and limitations of\nTAKDE in Section 5.\n2 P RELIMINARIES\n2.1 Kernel Density Estimation: A Brief Overview\nThe kernel density estimator for a given set of data points\n{xi}n\ni=1is as follows\nˆp(x;σ) =1\nnnX\ni=1Kσ(x−xi), (1)\nwhere Kσ(·)is the kernel function with the bandwidth\nσ. Throughout this paper, K(·)denotes a standard ker-\nnel function with a unit kernel bandwidth. We have that\nKσ(x) =1\nσK(x\nσ). We further impose the following mild\nassumptions on the kernel function K(·).\nAssumption 1. [14] The bandwidth sequence σn(the subscript\nn shows the dependence of σto the number of data points) has the\nfollowing properties\nlim\nn→∞σn= 0\nlim\nn→∞nσn=∞,(2)\nwhich implies that the bandwidth σndecays slower than n−1and\nconverges to 0. The standard kernel function K(·)is a bounded,\nsymmetric probability density function with a zero first moment\nand a finite second moment. That is, the following properties hold\nZ\nK(x)dx= 1\nZ\nxK(x)dx= 0\nZ\nx2K(x)dx <∞.(3)\nThe convergence to 0for bandwidth is rather intuitive,\nin that when we have infinitely many data points at hand,\nour estimator can be as flexible as possible without having\nto be concerned about over-fitting. It is also easy to verify\nthat many commonly used kernels (e.g., the Gaussian kernel\nK(x) =1√\n2πe−x2) satisfy (3).\n2.2 Problem Formulation\nIn dynamic estimation, the density evolves over time. The\nevolution might be continuous in nature, but we only observe\nsamples from time to time. Here, we consider the case where\nthe streaming data comes in batches. We first define the\ndynamic streaming dataset, where we observe one new\nbatch of data points x(t)={x(t)\ni∈R}nt\ni=1at a new time\nstamp t. This data structure applies to most real-world\nstreaming datasets. An important example is estimating\ndensity information in video datasets [2] like the dynamic3\nnano imaging problem mentioned in the introduction. An\nimage processing tool extracts the sizes of nanoparticles as\nthe sample points for estimating the normalized particle size\ndistribution (NSPD), which is an indicator to anticipate and\ndetect phase changes in the nanoparticle growth. This data\nstructure further applies to many time-series datasets [15].\nFor the cases where streaming data comes in on a per point\nbasis, one can convert those types of data into our defined\nstructure through combining consecutive data points into\nbatches.\nWe assume data points x(t)are generated independently\nfrom pt(x), the true density at time stamp t. Also, the data\npoints x(t)andx(t′)in different time stamps ( t̸=t′) are\nindependent from each other. We impose the following\nassumption on the true density function.\nAssumption 2. The true density function pt(x)at any time\nstamp tis twice differentiable, and its second derivative p′′\nt(x)is\ncontinuous and square integrable.\nAssumption 2 is commonly used for continuous density\nfunctions [14]. The square integrable condition is necessary\nas the integrated second order Taylor expansion appears later\nin the error bound derivation.\nFollowing (1), we write the traditional kernel density\nestimator of the density pt(x)as follows\nˆpt(x;σ) =1\nntntX\ni=1Kσ(x−x(t)\ni). (4)\nThe \"sliding window\" kernel density estimator, popularly\nused in dynamic density estimation [7], [8], [9], takes the\nfollowing form\nˆht(x) =X\nj∈Ttα(t)\njˆpj(x;σ(t)\nj), (5)\nwhere Ttrepresents the set of batches within the moving\nwindow (memory), ˆpjis defined following (4), and α(t)\njis a\nnon-negative weight sequence that satisfiesP\nj∈Ttα(t)\nj= 1,\nto ensure that the output is a proper density function. The\nwindow size is Tt, i.e.|Tt|=Tt, so that Ttcan be naturally\nwritten as Tt={t−Tt+ 1, . . . , t}. The superscripts (t)onα\nandσare omitted hereafter for the presentation clarity.\nIn order to develop a fast real-time estimator, we need to\naddress the following three problems.\nProblem 1. How do we choose the set Ttto have a good enough\n\"memory\" for estimating the density at time twhile maintaining\nreal-time processing?\nProblem 2. How do we design the weight sequence in (5)?\nProblem 3. How do we devise a kernel bandwidth selector in (4)?\n3 A LGORITHM DESIGN\nIn this section, we derive the AMISE upper bound for the\ngeneral \"sliding window\" kernel density estimator in (5).\nWe then present a novel weight and bandwidth sequence,\nentailed by the upper bound minimizer (Problems 2-3). We\nuse these sequences to design the TAKDE algorithm.3.1 Asymptotic Mean Integrated Squared Error Upper\nBound\nAMISE is a popular metric used to theoretically evaluate\nthe performance of a density estimator [14]. For a given\ndensity estimator ˆh(x)of a density function p(x), the mean\nintegrated squared error (MISE) is defined as follows\nMISE (ˆh, p)≜Z\nE[(ˆh(x)−p(x))2]dx\n=Z\nMSE (ˆh, p)dx,(6)\nwhere the expectation is taken with respect to the distri-\nbutions of data points involved in estimator ˆh. MISE is the\nintegration of the mean squared error of the density estimator\nover the support. [14] shows that the asymptotic expression\n(with respect to the sample size n) of the MISE for a standard\nkernel density estimator ˆp(x;σn)with kernel bandwidth σn\nis\nAMISE (ˆp, p) =R(K)\nnσn+1\n4σ4\nnµ2\n2(K)R(p′′), (7)\nwhere\nR(f) =Z\nf2(x)dx,\nµ2(f) =Z\nx2f(x)dx.(8)\nWe can see that the conditions in (2)guarantee that AMISE\nconverges to zero as n→ ∞ . The MISE and AMISE have\nbeen popular measures for characterizing non-parametric\ndensity estimators, including binned density estimator [16],\nkernel density estimator [14], wavelet density estimator [17],\nand diffusion estimator with a static limit [18]. The exact\nexpression for kernel density estimator can also be derived\nin the case of specific distributions like Gaussian distribution\n[14]. However, all these derivations assume that data points\nin the non-parametric density estimator are samples from a\nstatic target density function.\nIn the following theorem, we derive the theoretical upper\nbound of AMISE for the \"sliding window\" kernel density\nestimator given in (5)in the context of dynamic density\nestimation. To the best of our knowledge, this is the first\nAMISE bound for \"sliding window\" kernel density estimator\nin estimating the evolving true density pt(x).\nTheorem 1. Let Assumptions 1-2 hold. The AMISE of a \"sliding\nwindow\" kernel density estimator ˆhtat time twith window size\n|Tt|=Tt, weight sequence {αi}Tt\ni=1, and bandwidth sequence\n{σi}Tt\ni=1has the following upper bound\nAMISE (ˆht, pt)≤X\ni∈Ttα2\ni\nniσiR(K)\n+ (2Tt−1)X\ni∈Ttα2\niR(b(t)\ni)\n+2Tt−1\n4µ2\n2(K)X\ni∈Ttα2\niσ4\niR(p′′\ni),(9)\nwhere b(j)\ni(x)defines the difference between density functions\npi(x), pj(x)(j≥i)\nb(j)\ni(x)≜pi(x)−pj(x). (10)4\nProof. We omit the superscript (t)for weight αand band-\nwidth σfor the presentation clarity. First, recall the definition\nofˆht(x)from (4)-(5), where we have\nˆht(x) =X\ni∈Ttαiˆpi(x;σi) =X\ni∈Ttαi\nniniX\nj=1Kσi(x−x(i)\nj).(11)\nThe bias of the estimator can be written as\nB(ˆht(x))≜E[ˆht(x)−pt(x)]\n=EhX\ni∈Ttαi\nniniX\nj=1Kσi(x−x(i)\nj)−pt(x)i\n=X\ni∈TtαiZ\nKσi(x−y)pi(y)dy−pt(x)\n=X\ni∈Ttαi(Kσi∗pi)(x)−pt(x),(12)\nwhere ∗denotes the convolution, and pi(·)is the true density\nof batch i.\nUsing V(·)to denote the variance operator, the estimator\nvariance can be calculated as\nV(ˆht(x)) =X\ni∈Ttα2\niV(ˆpi(x;σi)),(13)\ndue to the independence of batches, where\nV(ˆpi(x;σi)) =1\nni\u0010\n(K2\nσi∗pi)(x)−(Kσi∗pi)2(x)\u0011\n.(14)\nThe decomposition of the MSE of the \"sliding window\"\nestimator ˆhtis as follows\nMSE (ˆht, pt) =E[(ˆht(x)−pt(x))2]\n=V(ˆht(x)) +B2(ˆht(x)).(15)\nIntegrating above over x, we have\nMISE (ˆht, pt) =Z\nMSE (ˆht, pt)dx. (16)\nGiven the expressions of bias (12) and variance (14), to calcu-\nlate AMISE, we need to derive the Taylor approximations of\nthe following quantities\n(K2\nσi∗pi)(x)\n(Kσi∗pi)(x).(17)\nFirst, we have\n(K2\nσi∗pi)(x) =Z\nK2\nσi(x−y)pi(y)dy\n=1\nσiZ\nK2(z)pi(x−σiz)dz\n=pi(x)\nσiR(K) +o(1),(18)\nwhere we note that pi(x−σiz) =pi(x)+o(1)holds, because\nσi→0asni→ ∞ . We also have that\n(Kσi∗pi)(x) =Z\nKσi(x−y)pi(y)dy\n=Z\nK(z)pi(x−σiz)dz\n=Z\nK(z)(pi(x)−σizp′\ni(x)\n+1\n2σ2\niz2p′′\ni(x) +o(σ2\ni))dz\n=pi(x) +1\n2σ2\nip′′\ni(x)µ2(K) +o(σ2\ni).(19)where we used the assumptions thatRK(z)dz= 1 andRzK(z)dz= 0. Given the above asymptotic characterization\nof the quantities, we can rewrite the bias term (12) as\nB(ˆht(x)) =X\ni∈Tt\u0010\nαib(t)\ni(x) +1\n2αiσ2\nip′′\ni(x)µ2(K) +o(σ2\ni)\u0011\n.\n(20)\nWe can also write the variance (14) as\nV(ˆht(x)) =X\ni∈Tt\u0010α2\ni\nniσiR(K)pi(x) +o(1\nniσi)\u0011\n. (21)\nWe can now simplify the MSE (15) as\nMSE (ˆht, pt) =X\ni∈Tt\u0010α2\ni\nniσiR(K)pi(x) +o(1\nniσi)\u0011\n+\u0012X\ni∈Ttαib(t)\ni(x) +X\ni∈Tt1\n2σ2\niαip′′\ni(x)µ2(K) +X\ni∈Tto(σ2\ni)\u00132\n.\n(22)\nDisregarding the terms that converge to zero and taking\nintegral over x, we can derive an upper bound for AMISE as\nAMISE (ˆht, pt)≤X\ni∈Ttα2\ni\nniσiR(K)\n+ (2|Tt| −1)X\ni∈Ttα2\niR(b(t)\ni)\n+2|Tt| −1\n4µ2\n2(K)X\ni∈Ttα2\niσ4\niR(p′′\ni),(23)\nwhere the last two lines follow from the Cauchy-Schwarz\ninequality for the 2|Tt| −1terms in the square. Note that\nb(t)\nt= 0 by definition. Observing that |Tt|=Ttcompletes\nthe proof of Theorem 1.\nLet us call the three lines in the right hand side of (9)as\nterm 1, term 2, and term 3, respectively. Term 1is due to\nthe variance of the estimator, and terms 2and3are the\nbias terms. Terms 1and 3are asymptotically vanishing\nin the sense that when ni→ ∞ , they both go to zero\nper condition (2). We can make several observations about\nthe upper bound expression (9). First, the dynamic density\nestimation with \"sliding window\" kernel density estimators\nwill have a non-vanishing error term 2, induced by keeping\ndensities of various time stamps in the memory. We will later\nsee in Corollary 3 that under optimal weight design, this\nterm can also go to zero when ni→ ∞ . Second, when the\ndistribution evolution is mild (i.e., R(b(t)\ni)is small), there can\nbe a theoretical advantage in including previous samples in\nthe memory to reduce the variance term 1. Later simulations\nwill show this advantage can be significant in practice. Third,\nwhen the previous distributions are very different from the\ncurrent distribution, it is desirable to only keep one batch\n(the current batch) in the memory, i.e., Tt={t}andTt= 1.\nIn this case, R(b(t)\nt) = 0 by definition (10) and the upper\nbound (9)exactly recovers the AMISE for the traditional\nkernel density estimator in (7).\n3.2 Window Generator\nIn the existing literature, kernel density estimators are\nmodified using arbitrary \"sliding windows\" to adapt to the5\ndynamic estimation. This approach performs better than the\ntraditional kernel density estimator, as a static kernel density\nestimator works poorly for dynamic density estimation\n[2]. However, this heuristic approach lacks a theoretical\njustification. In fact, based on the theoretical upper bound\nof AMISE (9), it is intuitive that the window size should\ndepend on the density evolution to keep the AMISE small.\nFor example, when the true density changes drastically, it is\nideal to decrease the window size to adapt to the fast density\nchange. Therefore, we propose a histogram-based window\nsize generator that will allow the kernel density estimator to\nbe adaptive to dynamic changes.\nWe observe in (9)that compared to the static AMISE, the\nworst-case AMISE for dynamic density estimation depends\non one more quantity, namely the difference function b(t)\ni. In\nprinciple, we can use this quantity as an indicator to adapt\nthe dynamic kernel density estimator to the changes in the\nunderlying density function.\nWe define a cutoff threshold to determine the number of\nbatches (sliding window size) to be kept in the memory of the\ndynamic kernel density estimator. In doing so, we first define\nthe temporal adaptive (TA) distance between two density\nfunctions. Here, we use histograms to approximate the\ndensity functions as true density functions are unavailable.\nWe denote the number of bins in the histograms by m, set\nusing the Sturges’ rule [19]\nm= 1 + 3 .322 log n, (24)\nwhere nis the smallest batch size among all batches in the\ncurrent memory. Sturges’ rule is a widely adopted, simple\nbinning algorithm in the literature. It is derived for normally\ndistributed data. The user can choose other binning rules,\nsuch as Doane’s rule [20], Scott’s rule [21], or Freedman and\nDiaconis’s rule [22] as appropriate. However, we note that all\nexisting binning guidelines provide bins similar to Sturges’\nrule under low data volume (less than 200) [4].\nThe temporal adaptive distance between two histograms\nhistiandhistjis expressed as\n∥histi, hist j∥TA≜∥yi−yj∥2\n2, (25)\nwhere ∥ · ∥ 2denotes the ℓ2norm and yiis the probability\nmass vector on bins in batch i, i.e.,∥yi∥1= 1. This TA\ndistance serves as a measure proportional to ˆR(b(t)\ni), the\napproximation of R(b(t)\ni)in (9), i.e.,\nˆR(b(t)\ni)∝ ∥histi, hist t∥TA. (26)\nTo control the bias, one can set a cutoff threshold sfor the\nTA distance. Upon receiving batch t, the number of batches\nto be kept in the memory can be set as Ttthat satisfies the\nfollowing two inequalities\nt−1X\nj=t−Tt∥histj, hist t∥TA> s,t−1X\nj=t−Tt+1∥histj, hist t∥TA≤s.\n(27)\nNote that from a practical standpoint, the cutoff threshold\nsshould not be the only criterion for window selection,\nbecause when the true density goes through a long static\nperiod, it is possible that (27) will induce a large memory\nwindow that exceeds the computational limit for real-time\ndensity estimation. Therefore, there should exist a hardcapwto account for computational limits. Combining both\nconsiderations, the actual number of batches in the memory\nshould be set as min(Tt, w).\nRemark 1. Note that the main purpose of cutoff value sis to\nreduce the window size (and computation cost) when dealing\nwith rapidly changing densities. The bias-variance decomposition\nsuggests that including more batches in TAKDE can induce a\nlower variance (first term in equation (21)) at the cost of increasing\nthe bias (first term in equation (20)). Moreover, we will show in\nCorollary 3 that TAKDE is consistent regardless of window size\nTt. Later, synthetic data simulation also suggests the empirical\nperformance difference is not too sensitive to the cutoff value, so\none can heuristically choose it in favor of fast processing rather\nthan through intensive cross-validation.\n3.3 Bandwidth and Weight Generator\nThe dynamic nature of the underlying true density makes\nit practically impossible to understand the actual difference\nfunctions and the second derivative of the true densities.\nHowever, using the AMISE upper bound in Theorem 1, we\ncan find theoretically optimal sequences for kernel band-\nwidths and weights, which in turn helps in the algorithm\ndesign. In view of Theorem 1, we present the following\ncorollary.\nCorollary 2. The optimal sequences of weights and bandwidths\nthat minimize the AMISE upper bound of the dynamic kernel\ndensity estimator are as follows\nσi=\u0014R(K)\nniµ2\n2(K)R(p′′\ni)(2Tt−1)\u00151\n5\n,\nαi=1/SiP\nj∈Tt1/Sj,(28)\nwhere the sequence Si(with superscript (t)omitted) is such that\nSi=5R(K)\n4niσi+ (2Tt−1)R(b(t)\ni). (29)\nProof. Equation (23) shows that the upper bound on AMISE\ndepends on the weight sequence αiand the bandwidth\nsequence σi. Therefore, we can minimize the upper bound\nwith respect to both of these parameters.\nDifferentiating with respect to σiyields the following\n(optimal) sequence\nσi=\u0014R(K)\nniµ2\n2(K)R(p′′\ni)(2Tt−1)\u00151\n5\n. (30)\nWe can find the optimal sequence of weights by simply\nsolving the minimization of Lagrangian of (23) with the\nconstraintPαi= 1 and incorporating (30). This will result\nin the following expression for the sequence αi\nSi=5R(K)\n4niσi+ (2Tt−1)R(b(t)\ni)\nαi=1/SiP\nj∈Tt1/Sj,(31)\nwhich completes the proof of Corollary 2.\nRemark 2. Corollary 2 provides some insights concerning the\nbandwidth and weight choices.6\nWindow Generator\nWidth Generator\nWeight Generator\nTAKDE\n𝜎1𝜎i𝜎T𝜎i\nT batches…………⍺1⍺i⍺TR(bi)\nIncoming Batch\nFig. 1. TAKDE framework.\n1) The bandwidth sequence suggests that we should make\nthe kernel more flexible as more batches of data points are\nincluded in the estimation. This aligns with the intuition\nfrom the traditional kernel density estimator, where the\nestimator can be more flexible with more sample points.\n2) The weight sequence provides the following insights. First,\nwhen the number of data points at a particular batch is\nconsiderably large, we should assign more weight to that\nbatch with the hope of extracting more information to\ninfer the current density. Second, the R(b(t)\ni)quantity\nprovides a countermeasure to prevent us from assigning a\nlarge weight to data points coming from a very different\ndistribution compared to the current batch. Third, we\nshould assign more weights to the batches with larger\nkernel bandwidths, which means we are favoring smoother\nestimators in principle.\nCorollary 3. Under Assumptions 1-2, the optimal weight se-\nquence and kernel bandwidth sequence in Corollary 2 will ensure\nthat for any ϵ >0,\nPr(|ˆht−pt|2> ϵ)→0, (32)\nasni→ ∞ .\nProof. First, notice that following Corollary 2, we have σi→\n0andαi→0for every batch except the last batch where\nαt→1(since R(b(t)\nt) = 0 ) asni→ ∞ . It is easy to verify\nthatE[|ˆht−pt|2]→0under this bandwidth and weight\nsequence, based on the expression of the mean squared error\nin (22). Then, by Markov inequality, we have\nPr(|ˆht−pt|2> ϵ)≤E[|ˆht−pt|2]\nϵ→0. (33)The proof is complete.\nCorollary 3 shows that TAKDE is weakly consistent as\nni→ ∞ regardless of Tt. This is rather intuitive as TAKDE\ncan precisely recover the traditional KDE in this extreme case.\n3.4 Kernel Bandwidth Selector\nThe bandwidth sequence in Corollary 2 presents a principle\nfor choosing the kernel bandwidth. However, the quantity\nR(p′′\ni)is unknown in practice, and we still need to find\na kernel bandwidth selector to calculate the actual kernel\nbandwidth values. There exist extensive studies for the choice\nof bandwidth in traditional kernel density estimation. One\npopular choice is the cross-validation approach [23], [24], [25],\n[26]. However, the computational cost of cross-validation\nprohibits its application in high-frequency density estimation\nas every new batch of data points needs to be cross-validated\nfor a new kernel bandwidth.\nMinimizing AMISE in (7)reveals a simple expression for\nthe optimal kernel bandwidth. [14] characterized the optimal\nkernel bandwidth based on (7) as follows\nσAMISE =\u0014R(K)\nnµ2\n2(K)R(p′′)\u00151\n5\n. (34)\nWe notice that (34) coincides with the optimal kernel band-\nwidth sequence we derived in Corollary 2 except for a factor\nof(2Tt−1)1/5. This relationship allows us to directly adopt\nexisting kernel bandwidth selection methods for optimal\nAMISE.\nExpression (34) is still dependent on the unknown R(p′′),\nbut there exist a number of studies that explore different7\nmethods for estimating R(p′′). For example, [27] approxi-\nmates the AMISE objective function assuming the density\nis Poisson and then proceeds to find the minimizer as the\noptimal kernel bandwidth. However, this method is not\napplicable in real-time dynamic density estimation as the\noptimization process is expensive. [9] provides an iterative\nupdate framework by estimating R(p′′)through R(ˆp′′),\nwhich is the numerical square integration of the second\nderivative of the density estimator. This approach does not\nimpose any strict assumption on the underlying distribution,\nwhich offers a robust estimation of R(p′′). However, the\niterative algorithm still requires numerical operations like\nnumerical derivatives and numerical integration, which may\nnot be efficient enough for real-time density estimation.\nIn TAKDE, we adopt the normal rule introduced in [28].\nAssuming the true density is Gaussian, the optimal kernel\nbandwidth can be approximated as follows\nσAMISE ≈cˆσn−1\n5, (35)\nwhere cis the smoothness parameter depending on the kernel\nfunction and the underlying true density, and ˆσis the sample\nstandard deviation of the data points. The normal rule is\nparticularly appealing for the design of TAKDE due to its\nsimple structure, which allows a direct plug-in of smoothness\nparameter cand enables fast real-time processing.\nThere are two commonly used recommendations for the\nsmoothness parameter cin(35). The first choice given in [14]\nis as follows\nσAMISE ≈\"\n8π1/2R(K)\n3µ2\n2(K)n#1\n5\nˆσ, (36)\nwhere ˆσis the estimated standard deviation assuming the\ntrue density is normal. The smoothness parameter cof\nGaussian Kernel in this setting is (32/3)1/5.\nThe second recommendation [29] comes from the up-\nper bound of the AMISE-optimal kernel bandwidth using\nbeta(4,4)or triweight density function, that is,\nσAMISE ≤\u0014243R(K)\n35µ2\n2(K)n\u00151\n5\nˆσ. (44)\nThis bandwidth provides an oversmoothed density estimator\nthat might not perform well with respect to metrics like log-\nlikelihood or MSE. However, an oversmoothed density esti-\nmator is often preferred for real-world applications, because\nthe results are visually plausible. In this case, the smoothness\nparameter cof Gaussian Kernel is (972/35√π)1/5.\nRemark 3. The only reason for adopting the normal rule in\nTAKDE is its computation simplicity. We must note that the\nweight sequence given in Corollary 2 is compatible with any\nexisting R(p′′)approximation method.\n3.5 Algorithm Design\nIn this subsection, we present the final form of TAKDE.\nThe algorithm requires as input a cutoff value s, a hard\ncapw, a smoothness parameter c, and a kernel function\nK. Upon receiving the batch of data points at time t, the\nwindow generator decides the set of batches Ttto be used\nfor the density estimation. The window generator will also\nreturn the sequence of approximated ˆR(b(t)\nj)as in (26) for allAlgorithm 1 Temporal Adaptive Kernel Density Estimator\n(TAKDE)\nInput: Kernel function K(·), cutoff value s, hard cap w,\nsmoothness parameter c.\nFort= 1,2, . . .\n1:Receive new batch of data x(t)at time t.\n2:Window Generator: Generate and record histtand\nforget histt−w. SetDistance = 0, Tt= 0,Tt=∅.\nWhile Tt< w:\nDistance =Distance +∥histt, hist t−Tt∥TA (37)\nBreak If:\nDistance > s, (38)\nElse:\nTt=Tt∪x(t−Tt)Tt=Tt+ 1. (39)\nReturn: TtandTtand the sequence {ˆR(b(t)\nj)}j∈Ttwhere\nˆR(b(t)\nj) =m∥histj, hist t∥TA. (40)\n3:Bandwidth Generator: Receive the batch set Tt.\nForj∈ {t−Tt+ 1, . . . , t}:\nσj=cˆσj\n((2Tt−1)nj)1/5, (41)\nwhere cis defined by the kernel bandwidth selector,\nnj=|x(j)|, and ˆσjis the sample standard deviation of\ndata in batch j.\nReturn: Bandwidth sequence σj.\n4:Weight Generator: Receive bandwidth sequence σjand\nthe approximated ˆR(b(t)\nj)sequence. Let\nαj=1/SjP\ni∈Tt1/Si,\nSj=5R(K)\n4njσj+ (2Tt−1)ˆR(b(t)\nj).(42)\nReturn: Weight sequence αj.\nOutput: The Temporal Adaptive Kernel Density Estimator\ngiven as\nˆht(x) =X\nj∈Ttαjˆpj(x;σj),\nˆpj(x;σj) =1\nnjnjX\ni=1Kσj(x−x(j)\ni).(43)\nbatches in the memory. Then, all batches within the memory\nwill be fed into the bandwidth generator to generate the\nsequence of kernel bandwidths σjas in Corollary 2. Then,\nthe approximated ˆR(b(t)\nj)and bandwidth sequence σjwill\nbe fed into the weight generator to generate the sequence αj\nas in Corollary 2. Finally, all parameters will be put together\nto generate a proper kernel density estimator for estimating\nthe density at time t. Fig. 1 illustrates the workflow of\nTAKDE. The algorithmic presentation of TAKDE is outlined\nin Algorithm 1.8\n4 E XPERIMENT\nWe now present numerical experiments to verify the effi-\nciency of TAKDE both on synthetic data and real-world data.\nAll experimental results established in this section are based\non Gaussian kernel function.\n4.1 Algorithm Design Evaluation\nBefore we compare TAKDE with other established bench-\nmark algorithms, we evaluate the design of TAKDE on\nsynthetic data. The specific question that we aim to address\nis that whether our proposed weighting scheme, derived\nfrom the AMISE upper bound, outperforms other heuristic\nweight sequences such as uniform (or average) weighting\nand exponentially decaying weighting.\n4.1.1 Synthetic Dataset Design\nWe create a synthetic dataset to test the performance of\nTAKDE in dynamic density estimation. We design the\nsynthetic dataset following some general principles.\n1) The true densities involved in the generation of\nthe dataset need to have analytical forms and have\nalready been established in the literature.\n2) Each batch of data points has a size in the range of\n[5,20], so that the batches do not differ too drastically\nin terms of the data amount.\n3) The number of testing points for all batches should\nbe the same for comparison purposes.\n4) The dynamics of the underlying densities varies for\ndifferent batches.\nFollowing the above principles, we adopt the 15Gaussian\nmixture densities, recommended by [30], as the baseline\ndensities for our synthetic dataset design. The 15densities\nare shown in Fig. 2.\nTo design the true density, we first consider 14sections,\nwhere each section consists of multiple batches. Let us denote\nthe15Gaussian mixtures with g1(x), . . . , g 15(x)and repre-\nsent the 14sections with S1, . . . ,S14, where |S1|+. . .+|S14|\nequals to the total number of batches in the dataset. To be\nspecific, section Sihas|Si|consecutive batches of data points\nin it, and the first batch of data in section Si+1will start\nafter the last batch in section Si. For batch iin section j,\nwhere 1≤i≤ |S j|, the density function h(j)\ni(x)is defined\nas follows\nh(j)\ni(x) =|Sj| −i+ 1\n|Sj|gj(x) +i−1\n|Sj|gj+1(x). (45)\nTo be consistent with our previous notation, h(j)\ni(x) =ptij(x)\nfortij=|S1|+. . .+|Sj−1|+i. Notice that in section j, thej-th\nGaussian mixture linearly transforms to the j+1-th Gaussian\nmixture. After we move on to section j+ 1, none of previous\nGaussian mixtures g1(x), . . . , g j(x)will appear in the section.\nGiven the density of batch iin section j, we sample a random\nnumber between 5to20as the number of training points\nand500 for testing points to perform the comparison. To\naccount for the randomness in partitioning the batches into\n14sections and the randomness in samples, we generate 300\nsynthetic datasets for Monte-Carlo simulations.4.1.2 TAKDE Evaluation\nWe now compare the weight generator in TAKDE with\ntwo heuristic approaches in the literature. One approach\nis to assign uniform weights to the batches, assuming\nolder data points are of the same importance as the new\ndata points, and the other one is to assign exponentially\ndecaying weights, assuming the new points are much more\nimportant [8], [9]. To ensure a fair comparison, we only\nchange the weight generator of TAKDE to uniform and\nexponential weighting, and we keep the other components\nof the algorithm unchanged. The uniform weight sequence\nis set as follows\nαj=1\nTt,∀j∈ {t−Tt+ 1, . . . , t}. (46)\nThe exponential weight sequence is set as follows\nαj= (1−e)et−j,∀j∈ {t−Tt+ 2, . . . , t}, (47)\nandαt−Tt+1=eTt−1, where eis the decay ratio. We compare\nthe above to αjcorresponding to the expression in (42). In\nour simulation, e= 0.9in general yields the best result under\ndifferent settings; therefore, the decay ratio for exponential\nweight sequence is set to e= 0.9.\nOur comparison is performed under several kernel\nbandwidth selectors, including the normal selector and\noversmooth selector mentioned in Section 3.4 and under\nvarious cutoff values.\nFirst, we consider normal bandwidth selector (36) and\noversmooth bandwidth selector (44). For each bandwidth\nselector, we conduct the comparison with datasets having\nfrom 100to500batches of data to reflect different underlying\ndynamics. Notice that for the data with 100 batches, the\ndynamic change is more drastic than that of the data with\n500batches.\nThe simulation result is shown in Fig. 3. We can observe\nthat TAKDE with AMISE-based weight sequence dominates\nthe uniform and exponential weight sequences in terms of\nthe test log-likelihood. We also see that when using the\nheuristic weight sequences, increasing the memory (i.e.,\nlarger cutoff value) mostly exacerbates the density estima-\ntion performance. The results show that the performance\ndifference between TAKDE and other two methods is larger\nwhen the total number of batches is smaller. This suggests\nthat TAKDE with AMISE-based weight sequence is better\nat adapting to more drastic dynamic changes. The smaller\ndifferences in 500-batch simulations are consistent with our\ntheoretical results, where the weighting sequence in Corollary\n2 gets closer to uniform weighting as R(b(t)\ni)converges to\n0, equivalent to a static density estimation. We observe that\nchanges in the cutoff value do not have a significant effect on\nTAKDE performance compared to others. This verifies our\ndiscussion in Remark 1.\nSecond, we conduct the comparison using a synthetic\ndataset with 100 batches of data for different bandwidth\nselectors, i.e., varying the smoothness parameter cin(35).\nThe simulation results are shown in Fig. 4. Again, we\nobserve the same performance trend for the algorithms.\nThese simulations empirically verify that the performance\nadvantage of our proposed weight sequence against the\nheuristic weight sequences is robust to different kernel\nbandwidths and different window sizes.9\n/uni0000002a/uni00000044/uni00000058/uni00000056/uni00000056/uni0000004c/uni00000044/uni00000051 /uni00000036/uni0000004e/uni00000048/uni0000005a/uni00000048/uni00000047/uni00000003/uni00000038/uni00000051/uni0000004c/uni00000050/uni00000052/uni00000047/uni00000044/uni0000004f /uni00000036/uni00000057/uni00000055/uni00000052/uni00000051/uni0000004a/uni0000004f/uni0000005c/uni00000003/uni00000036/uni0000004e/uni00000048/uni0000005a/uni00000048/uni00000047 /uni0000002e/uni00000058/uni00000055/uni00000057/uni00000052/uni00000057/uni0000004c/uni00000046/uni00000003/uni00000038/uni00000051/uni0000004c/uni00000050/uni00000052/uni00000047/uni00000044/uni0000004f /uni00000032/uni00000058/uni00000057/uni0000004f/uni0000004c/uni00000048/uni00000055\n/uni00000025/uni0000004c/uni00000050/uni00000052/uni00000047/uni00000044/uni0000004f /uni00000036/uni00000048/uni00000053/uni00000044/uni00000055/uni00000044/uni00000057/uni00000048/uni00000047/uni00000003/uni00000025/uni0000004c/uni00000050/uni00000052/uni00000047/uni00000044/uni0000004f /uni00000036/uni0000004e/uni00000048/uni0000005a/uni00000048/uni00000047/uni00000003/uni00000025/uni0000004c/uni00000050/uni00000052/uni00000047/uni00000044/uni0000004f /uni00000037/uni00000055/uni0000004c/uni00000050/uni00000052/uni00000047/uni00000044/uni0000004f /uni00000026/uni0000004f/uni00000044/uni0000005a\n/uni00000027/uni00000052/uni00000058/uni00000045/uni0000004f/uni00000048/uni00000003/uni00000026/uni0000004f/uni00000044/uni0000005a /uni00000024/uni00000056/uni0000005c/uni00000050/uni00000050/uni00000048/uni00000057/uni00000055/uni0000004c/uni00000046/uni00000003/uni00000026/uni0000004f/uni00000044/uni0000005a /uni00000024/uni00000056/uni0000005c/uni00000050/uni00000050/uni00000048/uni00000057/uni00000055/uni0000004c/uni00000046/uni00000003/uni00000027/uni00000052/uni00000058/uni00000045/uni0000004f/uni00000048/uni00000003/uni00000026/uni0000004f/uni00000044/uni0000005a /uni00000036/uni00000050/uni00000052/uni00000052/uni00000057/uni0000004b/uni00000003/uni00000026/uni00000052/uni00000050/uni00000045 /uni00000027/uni0000004c/uni00000056/uni00000046/uni00000055/uni00000048/uni00000057/uni00000048/uni00000003/uni00000026/uni00000052/uni00000050/uni00000045\nFig. 2. The 15Gaussian mixture densities used in the synthetic dataset design.\nFig. 3. The test log-likelihood comparison between TAKDE vs. the heuristic approaches. The x-axis represents the cutoff value and the y-\naxis represents the test log-likelihood. The first row shows the result under normal bandwidth selector and the second row shows the result\nunder oversmooth bandwidth selector. In each row, the plots from left to right represent the simulation results using synthetic datasets with\n100,200,300,400,500batches of data.\n4.2 Comparison with Benchmark Algorithms\nNext, we compare TAKDE with three density estimation\nmethods on real-world datasets. We consider both the mean\ntest log-likelihood and the run-time to show the advantages\nof TAKDE.4.2.1 Benchmark Algorithms\n1)Kernel Density Estimator (KDE) : The first bench-\nmark algorithm is the traditional kernel density\nestimator. The main reason to include kernel density\nestimator in the comparison is to show why a\ntraditional density estimator is not ideal for dynamic10\nFig. 4. The test log-likelihood comparison between TAKDE vs. the heuristic approaches over different bandwidth selectors. The x-axis represents the\nvalue of the smoothness parameter c. The y-axis represents the test log-likelihood. Each plot from left to right represents the simulation conducted\nwith cutoff values from 1to5.\nTABLE 1\nThe best experimental settings for different benchmark algorithms in different datasets.\nAlgorithm Noise Parameter α1 Noise Parameter α2 Smoothness Parameter c Cutoff Value (Window Size)\nTEM\nKDE - - 1.34 -\nB-spline 0.66 0.04 - -\nKDEtrack - - 0.45 1(16)\nTAKDE - - 0.15 1(16)\nECG\nKDE - - 0.98 -\nB-spline 0.82 0.05 - -\nKDEtrack - - 0.1 1(60)\nTAKDE - - 0.7 1(60)\nWafer\nKDE - - 0.22 -\nB-spline 0.96 0.06 - -\nKDEtrack - - 1.05 1(20)\nTAKDE - - 0.15 1(20)\nEarth\nKDE - - 0.4 -\nB-spline 0.81 0.05 - -\nKDEtrack - - 0.8 0.2(15)\nTAKDE - - 0.05 1.4(90)\nStar\nKDE - - 0.9 -\nB-spline 0.30 0.02 - -\nKDEtrack - - 1 0.3(20)\nTAKDE - - 0.35 1.8(38)\ndensity estimation. The kernel density estimator is\nformulated as (1). The bandwidth selector is\nσ=cˆσn−1\n5, (48)\nwhere we use cross-validation to choose c(rather\nthan the actual bandwidth) for easy comparison with\nTAKDE.\n2)B-spline Kalman Filter (BKF) [2] : B-spline Kalman\nfilter models the underlying density function as\na count measure defined on the partitions of the\ndensity support. The density estimator is defined as\nˆp(x) =1\nCexpmX\ni=1βiBi(x), (49)where Cis the normalization constant calculated\nwith numerical integration, mis the number of\npartitions, and Bi(x)are the B-spline bases. The\nalgorithm updates its states βiusing a B-spline ma-\ntrix evaluated on the centers of the density support\npartitions and the count vector at each batch.\n3)KDEtrack [9] : KDEtrack partitions the support of the\ndensity using a collection of grid points. The set of\ngrid points and the density values at the grid points\nare updated after each new batch of data points is\nreceived and evaluated. The density evaluation at a\ntest point will be the linear interpolation at the test\npoint using the closest grid points.\nRemark 4. We do not include the M-kernel and LRKDE methods11\nTABLE 2\nMean test log-likelihood on five real datasets.\nAlgorithm TEM ECG Wafer Earth Star\nKDE −0.026±0.0001 0.060±0.00002 0.0229±0.0007 0.048±0.0002 0.0078±0.00002\nB-spline Kalman Filter 0.171±0.0062 1.580±0.0011 1.204±0.0034 1.324±0.0051 0.685±0.0024\nKDEtrack 0.245±0.0057 1.095±0.0009 0.866±0.0018 0.915±0.0012 0.640±0.0007\nTAKDE(normal) 0.130±0.0016 1.639±0.0004 1.530±0.0015 1.247±0.0017 0.696±0.0007\nTAKDE(cor) 0.246±0.0022 1.625±0.0010 1.627±0.0017 1.331±0.0026 0.705±0.0008\nTAKDE 0.362±0.0036 1.648±0.0009 1.848±0.0025 1.504±0.0026 0.710±0.0012\nTABLE 3\nRun-time comparison (seconds) on five real datasets.\nAlgorithm TEM ECG Wafer Earth Star\nB-spline Kalman Filter 7.08 4.099 0.379 0.907 1.752\nKDEtrack 5.461 4.712 1.542 1.569 14.85\nTAKDE 0.378 0.557 0.114 0.704 0.851\nsince [9] has showed that KDEtrack is superior to these two\nmethods.\n4.2.2 Datasets\n•In situ TEM video data : The first dataset we use is\nin situ TEM dataset introduced in Section 1. It is the\n76.6second in situ TEM video published in [1]. It has\na total of 1149 frames of images and 5−20particle\ncounts in each frame.\n•CinCECGTorso (ECG) data : CinCECGTorso dataset is\nan ECG dataset taken from multiple torso surface sites\nof four patients from the Computers in Cardiology\nchallenges. This dataset is available on UCR time-\nseries data archive [15].\nThe dataset consists of ECG measurements of four pa-\ntients. We use the ECG signal sequence of one person\nto highlight the density dynamics over time. Note that\nsimulations on all four patients yield similar results.\nThere are 342ECG signals (data points) available at\neach batch, and there are a total of 1639 batches of\ndata points over time. The batches are collected at\n2-kHz frequency, which requires the density estimator\nto be updated 2000 times per second. For each batch\nof data points at a certain time stamp, we randomly\nsample 5to20data points to train and use the\nrest of the data points to evaluate the algorithms.\nThe number of training data points at each batch is\ndetermined only once throughout all the Monte-Carlo\nsimulations. However, the set of training points are\nsampled randomly in each Monte-Carlo simulation.\n•Wafer data : Wafer dataset is a collection of sensor\nreadings in a semiconductor wafer manufacturing\nprocess over time, available on UCR time-series data\narchive [15]. Unlike the previous two datasets, a wafer\nmanufacturing process is a rather slow process that\ncould span over 10weeks. However, this dataset\nis still illustrative for evaluating the accuracy of\nTAKDE. We use the readings in the normal state\nwafer manufacturing process to conduct our analysis.\nThere are 600readings (data points) available at each\nbatch, and there are a total of 152 batches of data\npoints over time. Again, we adopt the same train-testsplit approach as in the ECG dataset.\n•Earthquakes (Earth) data : The earthquake dataset is\na sensor reading dataset from Northern California\nEarthquake Data Center available on UCR time-series\ndata archive [15]. It consists of 461readings at each\nbatch with a total of 512batches.\n•StarLight Curves (Star) data : The starlight curves\ndataset consists of time-series sensor readings on the\nbrightness of a collection of celestial objects. It is also\navailable on UCR time-series data archive [15]. This\ndataset includes the readings of 1000 celestial objects\nat each batch with a total of 1024 batches.\n4.2.3 Experimental Settings\nIn comparing across different density estimators, we only\npresent the best performance of B-spline Kalman filter, where\nthe noise prior parameters are cross-validated using a grid\nsearch with an interval size of 0.01. For the traditional\nkernel density estimator, we report its best performance, but\neven that is significantly inferior to other density estimators.\nFor KDEtrack and TAKDE, we report the best settings\nperformances (in terms of smoothness parameter cand cutoff\nvalue s). Notice we do not adopt the iterative bandwidth\nupdate in KDEtrack for the computation reason explained in\nSection 3.4, but instead we use the same bandwidth generator\nas in TAKDE. All the simulations are conducted over 100\nMonte-Carlo simulations for random training-testing splits\nto generate the standard errors of the performance. The\nperformance metric is the mean test log-likelihood of the test\npoints.\n4.2.4 Performance\nThe parameter settings leading to respective best perfor-\nmance for all benchmark algorithms are shown in Table 1.\nThese settings are cross-validated using the first 10% batches\nof each dataset ( 20% for Wafer and Earth dataset).\nThe results are tabulated in Table 2. TAKDE tagged\nwith \"(normal)\" represents the performance achieved with\nsmoothness parameter recommended in equation (36) (nor-\nmal bandwidth selector) and the optimal cutoff in Table\n1. TAKDE tagged with \"(cor)\" represents the performance\nachieved by TAKDE under KDEtrack best settings in terms of12\nFig. 5. Visualization of the density estimators on the TEM dataset. The first row shows TAKDE at its normal setting and optimal setting. The second\nrow shows B-spline Kalman Filter at its optimal setting. The third row shows KDEtrack at its optimal setting. Figures from left to right represent the\nestimation at time stamps 225,450,675, and 900, respectively.\ncutoff value and smoothness parameter. As we can observe,\nTAKDE dominates all other benchmark algorithms in terms\nof test log-likelihood by a large margin. TAKDE is also\nrobust with respect to different cutoff values and different\nsmoothness parameters, as it dominates all other benchmark\nalgorithms even under the best settings for KDEtrack. The\nonly exception is TAKDE with normal bandwidth selector on\nthe TEM dataset. The underlying reason is that the low data\nvolume available at different batches (training and testing\ncombined) forces the \"true\" density distribution at each time\nstamp to an average of Dirac measures, which is far from the\nnormal assumption of the normal bandwidth selector.\nThe run-time comparisons are shown in Table 3. The\nvalues represent the time used for executing the density\nestimation for all test data points in all batches. We can\nobserve that in addition to being more accurate than the\nbenchmark algorithms, TAKDE is also much faster in speed\nas it requires negligible calculations in addition to kernel\ndensity evaluation. The computation advantage makes a\nhuge difference for the ECG dataset in particular, as the other\ntwo benchmark algorithms do not run nearly fast enough to\ncatch up with the 2kHz data collection rate.\n4.3 Visual Examination\nIn this subsection, we visualize the previously compared den-\nsity estimators. We pick the time stamps {225,450,675,900}\nin1150 batches of data in the TEM dataset for visualization.\nThe results are shown in Fig. 5. As we can observe, TAKDE\nat its optimal setting (for test log-likelihood) yields a more\nflexible model compared to other algorithms. TAKDE withnormal smoothness parameter yields the smoothest model\namong all. Our results in Table 2 also show that the normal\nsmoothness parameter can achieve estimation performance\nclose to the optimal setting while yielding smooth density\nfunctions that facilitate easy interpretation. For this reason,\nin most real-world applications that do not place estimation\naccuracy as their first priority, we do recommend using the\nnormal smoothness parameter (36) to avoid cross-validation.\n5 C ONCLUSION\nIn this paper, we established a theoretical AMISE upper\nbound expression for the \"sliding window\" kernel density\nestimator in dynamic density estimation. We proposed the\ntemporal adaptive kernel density estimator that maintains\nthe fast processing advantage of the \"sliding window\" kernel\ndensity estimator, while being theoretically optimal under\nthe worst-case AMISE. We provided extensive numerical\nsimulations to verify that TAKDE is superior to state-of-the-\nart real-time dynamic density estimators in terms of the mean\ntest log-likelihood. TAKDE also dominated these algorithms\nin terms of achieving smaller run-times.\nThe proposed weight sequence is reminiscent of the\nattention mechanism in a transformer neural network for\nsequence re-weighting [31]. 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Marron, “On optimal data-\nbased bandwidth selection in kernel density estimation,” Biometrika ,\nvol. 78, no. 2, pp. 263–269, 1991.\n[25] P . Robert, “On the choice of smoothing parameters for parzen\nestimators of probability density functions,” IEEE Transactions on\nComputers , vol. 25, no. 11, pp. 1175–1179, 1976.\n[26] M. Rudemo, “Empirical choice of histograms and kernel density\nestimators,” Scandinavian Journal of Statistics , vol. 9, pp. 65–78, 1982.\n[27] H. Shimazaki and S. Shinomoto, “Kernel bandwidth optimization\nin spike rate estimation,” Journal of Computational Neuroscience ,\nvol. 29, no. 1, pp. 171–182, 2010.\n[28] B. W. Silverman, Density Estimation for Statistics and Data Analysis .\nRoutledge, 2018.\n[29] G. R. Terrell, “The maximal smoothing principle in density estima-\ntion,” Journal of the American Statistical Association , vol. 85, no. 410,\npp. 470–477, 1990.\n[30] J. S. Marron and M. P . Wand, “Exact mean integrated squared error,”\nThe Annals of Statistics , vol. 20, no. 2, pp. 712–736, 1992.\n[31] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N.\nGomez, Ł. Kaiser, and I. Polosukhin, “Attention is all you need,”\nAdvances in neural information processing systems , vol. 30, 2017.\nYinsong Wang received his B.S. degree in\nMechanical Engineering from Shandong Univer-\nsity, China, in 2017, and his M.S. degree in\nManufacturing System Engineering and Manage-\nment from The Hong Kong Polytechnic University,\nHong Kong, in 2018. He is currently working\ntoward a Ph.D. degree in Industrial Engineering\nat Northeastern University. His research interests\ninclude machine learning, data science, and ker-\nnel methods.\nYu Ding (M’01, SM’11) received B.S. from Uni-\nversity of Science & Technology of China (1993);\nM.S. from Tsinghua University, China (1996); M.S.\nfrom Penn State University (1998); received Ph.D.\nin Mechanical Engineering from University of\nMichigan (2001). He is currently the Mike and\nSugar Barnes Professor of Industrial & Systems\nEngineering and a Professor of Electrical & Com-\nputer Engineering at Texas A&M University. His\nresearch interests are in data and quality science.\nDr. Ding is the Editor-in-Chief of IISE Transactions\nfor the term of 2021–2024. Dr. Ding is a fellow of IIE, a fellow of ASME, a\nsenior member of IEEE, and a member of INFORMS.\nShahin Shahrampour received the Ph.D. degree\nin Electrical and Systems Engineering, the M.A.\ndegree in Statistics (The Wharton School), and\nthe M.S.E. degree in Electrical Engineering, all\nfrom the University of Pennsylvania, in 2015,\n2014, and 2012, respectively. He is currently an\nAssistant Professor in the Department of Mechan-\nical and Industrial Engineering at Northeastern\nUniversity. His research interests include ma-\nchine learning, optimization, sequential decision-\nmaking, and distributed learning, with a focus on\ndeveloping computationally efficient methods for data analytics. He is a\nSenior Member of the IEEE."    },    {        "title": "2203.14635v1.Embedding_short_range_correlations_in_relativistic_density_functionals_through_quasi_deuterons.pdf",        "content": "arXiv:2203.14635v1  [nucl-th]  28 Mar 2022Embedding short-range correlations in relativistic densi ty functionals through\nquasi-deuterons\nS. Burrello1,∗and S. Typel1,2,†\n1Technische Universität Darmstadt, Fachbereich Physik,\nInstitut für Kernphysik, Schlossgartenstraße 9, D-64289 D armstadt, Germany\n2GSI Helmholtzzentrum für Schwerionenforschung GmbH,\nTheorie, Planckstraße 1, D-64291 Darmstadt, Germany\n(Dated: March 29, 2022)\nBackground: The formation of clusters at sub-saturation densities, as a result of many-body correlations, con-\nstitutes an essential feature for a reliable modelization o f the nuclear matter equation of state (EoS). Phenomeno-\nlogical models that make use of energy density functionals ( EDFs) offer a convenient approach to account for\nthe presence of these bound states of nucleons when introduc ed as additional degrees of freedom. However, in\nthese models clusters dissolve, by construction, when the n uclear saturation density is approached from below,\nrevealing inconsistencies with recent findings that eviden ce the existence of short-range correlations (SRCs) even\nat larger densities.\nPurpose: The idea of this work is to incorporate SRCs in established mo dels for the EoS, in light of the im-\nportance of these features for the description of heavy-ion collisions, nuclear structure and in the astrophysical\ncontext. Our aim is to describe SRCs at supra-saturation den sities by using effective quasi-clusters immersed\nin dense matter as a surrogate for correlations, in a regime w here cluster dissolution is usually predicted in\nphenomenological models.\nMethod: Within the EDF framework, we explore a novel approach to embe d SRCs within a relativistic mean-\nfield model with density dependent couplings through the int roduction of suitable in-medium modifications of\nthe cluster properties, in particular their binding energy shifts, which are responsible for describing the cluster\ndissolution. As a first exploratory step, the example of a qua si-deuteron within the generalized relativistic density\nfunctional approach is investigated. The zero temperature case is examined, where the deuteron fraction is given\nby the density of a boson condensate.\nResults: For the first time, suitable parameterizations of the cluste r mass shift at zero temperature are derived\nfor all baryon densities. They are constrained by experimen tal results for the effective deuteron fraction in nuclear\nmatter near saturation and by microscopic many-body calcul ations in the low-density limit. A proper description\nof well-constrained nuclear matter quantities at saturati on is kept through a refit of the nucleon meson coupling\nstrengths. The proposed parameterizations allow to also de termine the density dependence of the quasi-deuteron\nmass fraction at arbitrary isospin asymmetries. The streng th of the deuteron-meson couplings is assessed to be\nof crucial importance. Novel effects on some thermodynamic q uantities, such as the matter incompressibility, the\nsymmetry energy and its slope, are finally discerned and disc ussed.\nConclusions: The findings of the present study represent a first step to impr ove the description of nuclear matter\nand its EoS at supra-saturation densities in EDFs by conside ring correlations in an effective way. In a next step,\nthe single-particle momentum distributions in nuclear mat ter can be explored using proper wave functions of the\nquasi-deuteron in the medium. The momentum distributions i s expected to exhibit a high-momentum tail, as\nobserved in the experimental study of SRCs by nucleon knocko ut with high-energy electrons.\nI. INTRODUCTION\nThe Equation of State (EoS) of strongly interacting\nmatter is a fundamental ingredient in the theoretical de-\nscription of compact stars. It plays also a crucial role\nin many astrophysical simulations of, e.g., core-collapse\nsupernovae and neutron star mergers [ 1–4]. Tight links\nhave been also established between the properties of the\nEoS and those of finite nuclei, concerning both their\nstructure and reaction dynamics [ 5–7]. A large vari-\nety of approaches has been employed therefore in the\nlast decades to develop reliable models for the EoS,\n∗burrello@lns.infn.it, ORCID: 0000-0002-1132-4073\n†stypel@ikp.tu-darmstadt.de, ORCID: 0000-0003-3238-997 3which are constrained by both nuclear physics experi-\nments and astronomical observations. For astrophysical\nsimulations, tables of global, multi-purpose EoSs are in\nparticular needed, being provided by sophisticated theo-\nretical methods available in literature [ 8].\nOne class is given by microscopic ab-initio models,\nwhich try to solve the nuclear many-body problem using\nadvanced methods and realistic interactions that are con-\nstrained by scattering data and properties of few-nucleon\nsystems, see, e.g., Refs. [ 9,10] Other approaches based on\nthe effective-field theory exploit systematic developments\nfrom quantum chromodynamics and symmetry concepts,\ne.g. in chiral perturbation theory, providing also uncer-\ntainty estimates [ 11–13]. However, these models fail in\nproperly describing the formation of clusters at densities\nbelow nuclear saturation, where these bound states of\nnucleons emerge as many-body correlations generated by2\nthe short-range nucleon-nucleon interaction. Such low-\ndensity conditions are encountered in various systems:\nthe debris of heavy-ion collisions (HIC), the post-bounce\nevolution of core-collapse supernovae, and the surface of\nnuclei [ 14,15] where the formation of clusters is a prereq-\nuisite for cluster radioactivity and, in particular, α-decay\nof heavy nuclei. The emergence of clusters is definitely\nan essential feature for the modelization of a realistic EoS\n[16].\nPhenomenological models, whose parameters are con-\nstrained from experimental results of HIC and astronom-\nical observations and/or by directly fitting properties of\nfinite nuclei and nuclear matter near saturation, offer a\nconvenient alternative to microscopic models to approach\nthis problem. A widely used class of phenomenolog-\nical approaches is based on energy density functionals\n(EDFs), which are usually derived in the self-consistent\nmean-field approximation with an effective in-medium\ninteraction without a direct connection to the nucleon-\nnucleon interaction in free space [ 17]. There are various\nversions of this approach, e.g., non-relativistic models u s-\ning Skyrme or Gogny type interactions [ 18,19] or rela-\ntivistic models based on the exchange of mesons [ 20]. In\nrecent years, several attempts have been made to also di-\nrectly link EDFs to microscopic ingredients [ 21–23]. For\nexample, a special class of functionals inspired by effec-\ntive field theories (EFTs) and bench-marked on ab-initio\npredictions have been designed [ 24,25]. They were ap-\nplied to finite nuclei [ 26] and finite temperature EoS cal-\nculations of pure neutron matter (PNM) [ 27]. Moreover,\nfew steps were made towards the construction of a power\ncounting in EDF [ 28,29].\nHowever, EDFs derived from phenomenological mean-\nfield models fail as well, when only nucleons are con-\nsidered as basic constituents. Further progress is only\nachieved if clusters are introduced as additional explicit\ndegrees of freedom at low densities [ 30]. The dilute mat-\nter is then depicted as an ideal mixture of nucleons and\nall nuclei from the table of isotopes in thermodynamic\nequilibrium. Such a model is called nuclear statistical\nequilibrium (NSE) and is widely used in the astrophysi-\ncal context, where it leads to a reliable description for the\nchemical composition of stellar matter at sub-saturation\ndensities [ 16,31]. However, models like NSE are consid-\nered valid only as long as the interaction between the con-\nstituents can be neglected. Thus they fail at higher den-\nsities where in-medium effects become important, lead-\ning to the dissolution of clusters and the transition to\ncluster-free nuclear matter. The dissolution of a cluster,\ni.e., the so-called Mott effect, which is expected when\napproaching nuclear saturation density from below, can\nbe produced through the introduction of an excluded-\nvolume mechanism, which is just a simple geometric con-\ncept [ 16,32–34]. More microscopically, the formation\nand dissolution of light clusters in nuclear matter can be\ntreated using a quantum statistical approach with ther-\nmodynamic Green’s functions, see, e.g., Refs. [ 35,36].\nAn alternative scenario was proposed in the lastdecade, when interacting clusters were introduced as ex-\nplicit degrees in relativistic density functionals, firstl y\nconcentrating on light hydrogen and helium isotopes,\nwhose properties are modified in the medium [ 30]. Con-\ntrary to the chemical picture traditionally adopted in\nNSE-like models, where the properties of the correlated\nstates of nucleons are assumed to be independent of the\nmedium, in such a physical picture, the in-medium effects\nare addressed by introducing a proper modification of the\nmasses of the clusters, as inspired by the quantum statis-\ntical approach. Following this idea, the effective binding\nenergies are expected to increase with density [ 37].\nMany-nucleon correlations in the continuum, which\nstill survive above the Mott density, are then described in\nterms of effective resonances or quasi-clusters. In current\nphenomenological approaches with the excluded-volume\nmechanism or with medium-dependent mass shifts, these\nstates are statistically suppressed by construction beyon d\nsaturation, so that only nucleons should remain as inde-\npendent quasi-particles [ 38]. Mean-field type descriptions\nof nuclear matter above saturation consider thus the sys-\ntem as a free Fermi gas, with the usual step function in\nthe single-particle momentum distribution at zero tem-\nperature. Such a picture is however inconsistent with\nrecent experimental results from nucleon knock-out reac-\ntions on nuclei using inelastic electron scattering [ 39,40].\nThese studies clearly evidence the smearing of the nu-\ncleon Fermi surface and the emergence a high-momentum\ntail (HMT) in the single-nucleon momentum distribution\nof cold nucleonic matter, ascribable to the existence of\nsizeable nucleon-nucleon short-range correlations (SRCs )\neven at saturation density [ 40–42].\nExperimental investigations assessed that SRCs pairs\nare formed by approximately 20% of nucleons in vari-\nous measured nuclei [ 43–45]. They are characterized by\nlarge relative and small center-of-mass (c.m.) momenta.\nMoreover, some results concluded that their magnitude\nis spin- and isospin-dependent, with a clear dominance in\nthe neutron-proton channel [ 44–47], that affects the ratio\nof minority and majority species in asymmetric nuclear\nmatter, in the bulk part and tail of the single-particle\nmomentum distributions [ 46,48]. Extrapolating these\nexperimental results from finite nuclei to infinite nuclear\nmatter, useful information around saturation density was\nthen deduced. At higher densities, only numerical inves-\ntigations exist to predict the density dependence of these\nSRC pairs. From the analysis, the probability for nucle-\nons to form SRC pairs seems to have a minimum in the\nneighbourhood of the saturation density, owing to the in-\nterplay between the tensor component and the repulsive\ncore of the nuclear force [ 49–51].\nSRCs may also change the balance between kinetic\nand potential contributions to the energy of the system.\nThus their introduction is expected to significantly af-\nfect the controversial density dependence, in particular\nat supra-saturation densities, of the nuclear symmetry\nenergy [ 52,53], which quantifies the difference between\nthe total energy of PNM and symmetric nuclear mat-3\nter (SNM). Many theoretical and experimental investiga-\ntions are currently investigating the density dependence\nof this quantity [ 54–58].\nThe purpose of the present work is to incorporate\nSRCs in established models for the EoS, in light of the\nkey-importance of these features for the description of\nHIC, nuclear structure and in the astrophysical context\n[59–62]. Our aim is to explicitly treat SRCs at supra-\nsaturation densities by using effective quasi-clusters im-\nmersed in dense matter as a surrogate for correlations,\nthrough the introduction of proper in-medium modifica-\ntions of the cluster mass shifts. In this regime cluster\ndissolution is usually predicted in actual realizations of\nphenomenological models. Within the EDF framework,\nwe propose thus a novel approach to embed the SRCs\nwithin a relativistic mean-field model (RMF) with den-\nsity dependent coupling [ 63] through a substantial mod-\nification of the cluster mass shift at high densities.\nGiven its phenomenological nature, the adopted ap-\nproach does not allow to investigate the origin of the\nSRCs. Several effects coexist in the high-density regime\nthat can lead to a smearing of the single-particle distri-\nbution functions and the appearance of HMTs as known\nfrom the description of heavy-ion collisions in transport\ntheory [ 64]. They are due to different (repulsive) com-\nponents of the nucleon-nucleon interaction and can be\nhardly disentangled. A deeper insight on these features\nwould indeed require a more microscopic treatment of\nthese SRCs, going beyond the scope of the present work.\nAs a first exploratory step, the example of a quasi-\ndeuteron within this generalized relativistic density fun c-\ntional (GRDF) is currently explored, since two-body\nSRCs in the neutron-proton3S1channel are much more\nimportant than other many-body correlations. The zero\ntemperature case is examined, where the deuteron is rep-\nresented by a boson condensate that determines the mass\nfraction and leads to specific conditions to the parame-\nterization of the mass shift. The purpose of this work\nis then to propose possible mass shift parameterizations\nthat will be employed to determine the density depen-\ndence of the quasi-deuteron mass fraction at arbitrary\nisospin asymmetries. The final ambitious goal is to inves-\ntigate the effect of accounting for the SRCs in an effective\nway on the EoS, and some related thermodynamic quan-\ntities, at supra-saturation densities. In this context, it is\nworthwhile to mention that a recent study employed the\nconcept of a mass shift in a similar approach to model\nthe effective interaction of a possible heavy particle with\nbaryon number B= 2, the so-called sexaquark , in nu-\nclear matter. It is treated as a boson condensate like the\ndeuteron and affects the EoS of compact-star matter and\nthus the properties of neutron stars [ 65].\nThe manuscript is structured as follows. In Section\nIIthe theoretical formalism is illustrated and the funda-\nmental principles and basic formulas of the GRDF are\ngiven. The case of zero temperature is studied, where\nquasi-deuteron condensation is expected, and the role of\nthe deuteron-meson coupling strength is discussed. Sec-tionIIIexplores the different constraints for the deuteron\nmass shifts. Section IVconcentrates on the mass shift pa-\nrameterization, as suitably derived for nuclear matter at\nzero temperature. Then, the corresponding density de-\npendence of the quasi-deuteron mass fraction is obtained.\nThe impact on the EoS and on some general properties of\nnuclear matter at arbitrary neutron-proton asymmetries\nis shown in Section V. Conclusions and an outlook are\nfinally given in Section VI. Details on the formal deriva-\ntion of some quantities, on the conversion of parameters\nand analytical expressions for the mass shift parameters\nare furthermore collected in four appendices.\nII. THEORETICAL FORMALISM\nA. Generalized Relativistic Density Functional\nThe GRDF is a density functional derived from a RMF\nwith nucleons and further degrees of freedom. Their ef-\nfective interaction in the medium is described by the\nexchange of mesons with density dependent couplings\n[30,63]. In such a model, light clusters are explicitly\nintroduced, allowing for a unified treatment from matter\nwith bound nucleons at low densities to matter possibly\nmade of only neutrons ( n) and protons ( p) at high den-\nsities. All degrees of freedom are represented by quasi-\nparticles with self-energies that incorporate the effects o f\nthe interaction. For sake of simplicity, only2H nuclei\n(labeled as din the following) are added to nucleons as\ndegrees of freedom in light of their expected prevalent\nimportance discussed in Section I.\nLet us thus consider the general case of asymmetric\nnuclear matter (ANM) composed of neutrons, protons\nand deuterons with particle number densities ni(i=\nn,p,d ), baryon numbers Aiand charge numbers Zi. The\nsystem is usually characterized by specifying the baryon\ndensitynb=nn+np+2nd, the isospin asymmetry β=\n(nn−np)/nband the temperature T. In this section, the\nbasic formulas of the theoretical formalism are given for\nthe most general case of finite temperature. However, the\nzero temperature case will be considered in the analysis\nperformed in the following sections, leaving the analysis\nat finite Tfor future work. In this work, the masses\nof neutrons and protons will be taken as equal to the\naverage nucleon mass mnucso thatmn=mp=mnuc.\nThe same values as given in Ref. [ 30] are considered.\nFollowing the framework illustrated in Ref. [ 38,66],\nthe thermodynamic properties of nuclear matter are com-\npletely determined once the grand canonical potential\ndensity˜ω(T,{µi})is specified. It depends, apart from\nthe temperature, on the chemical potentials µiof all con-\nstituents i. In the present work, three types of mesons are\nconsidered: an isoscalar scalar σmeson to describe the\nattraction between nucleons, an isoscalar vector ωmeson\nfor their repulsion and an isovector vector ρmeson for the\nisospin dependence of the strong force. The interaction\nbetween a baryon iand a meson j(j=σ,ω,ρ ) is realized4\nby a minimal coupling with a strength that is given by\nthe product of a scaling factor χij, the mass number Ai,\nand a coupling Γj. The latter quantity depends on the\nbaryon density nbto describe the medium dependence of\nthe effective interaction. Different prescriptions might be\nadopted as recently discussed in Ref. [ 67]. In this work\nthe functional form of the couplings as introduced in Ref.\n[63] is used. In the application of the GRDF to homoge-\nneous nuclear matter, only the ratio Γj/mjof coupling\nand mass mjof the mesons jis relevant. Hence it is\nconvenient to introduce the coefficients\nCj=Γ2\nj\nm2\nj(1)\nand their derivatives\nC′\nj=dCj\ndnb= 2Γj\nm2\njdΓj\ndnb(2)\nwith respect to the baryon density.\nThe total grand canonical potential density of the sys-\ntem can be written as\n˜ω(T,{µi}) = (3)/summationdisplay\ni˜ωi+ ˜ω(cond)\nd+ ˜ωmeson−˜ω(r)\nmeson−˜ω(r)\nmass\ncontaining the standard expression for the single quasi-\nparticle (non-mesonic) contribution\n˜ωi=−Tgi\nσi/integraldisplayd3k\n(2π)3ln/bracketleftbigg\n1+σiexp/parenleftbigg\n−Ei−µ∗\ni\nT/parenrightbigg/bracketrightbigg\n(4)\nwith the well-known integral over the momentum kthat\nappears in the quasi-particle energy\nEi=/radicalBig\nk2+(m∗\ni)2(5)\nassuming natural units such that /planckover2pi1=c= 1. In Eq.\n(4), the sign factor σidistinguishes the particle statistics\n(σi= 1for fermions and σi=−1for bosons, respec-\ntively) and giis the spin-degeneracy factor. The boson\ncondensate term\n˜ω(cond)\nd=1−σi\n2n(cond)\nd(m∗\nd−µ∗\nd) (6)\nin Eq. ( 3) with the density of the condensate, n(cond)\nd, is\nonly relevant for deuterons.\nThe effective chemical potential\nµ∗\ni=µi−Vi. (7)\nand the effective mass\nm∗\ni=mi+∆mi−Si (8)depend on scalar\nSi=χiσAiCσnσ (9)\nand vector\nVi=χiωAiCωnω+χiρAiCρnρ+AiV(r)+W(r)\ni(10)\npotentials, respectively. They are defined in terms of the\ndifferent source densities\nnσ=/summationdisplay\niχiσAin(s)\ni (11)\nnω=/summationdisplay\niχiωAin(v)\ni (12)\nnρ=/summationdisplay\niχiρAin(v)\ni (13)\nwhere the single-particle scalar densities ( n(s)\ni) and vector\ndensities ( n(v)\ni) appear.\nThe mass shift ∆miin the effective mass ( 8) appears\nonly for the deuteron and is assumed to depend on the\nbaryon density nb. This dependence leads to rearrange-\nment contribution\nW(r)\ni=n(s)\nd∂∆md\n∂n(v)\ni(14)\nin the vector potentials ( 10) of nucleons and the deuteron.\nThe further rearrangement contribution\nV(r)=1\n2/parenleftbig\nC′\nωn2\nω+C′\nρn2\nρ−C′\nσn2\nσ/parenrightbig\n(15)\nin the vector potential ( 10) is due to the density depen-\ndence of the couplings Γjand can be expressed with the\ncoefficients Cjand source densities njof the three mesons\nconsidered here.\nCorresponding to the two rearrangement contribution\nin (10), there are also two such terms in the total grand\npotential density ( 3): the meson term\n˜ω(r)\nmeson=V(r)nb (16)\nand the mass shift term\n˜ω(r)\nmass=/summationdisplay\nin(v)\niW(r)\ni (17)\nFurthermore, the meson contribution in ( 3) is given by\n˜ωmeson=−1\n2/parenleftbig\nCωn2\nω+Cρn2\nρ−Cσn2\nσ/parenrightbig\n(18)\nsimilar in structure to ( 15).\nThe single-particle number densities can be derived5\nfrom ( 3) using the thermodynamic definitions\nn(v)\ni=−∂˜ω\n∂µi/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nT,{µj}j/negationslash=i(19)\nn(s)\ni=∂˜ω\n∂mi/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nT,{µj}(20)\nthat give\nn(v)\ni=gi/integraldisplayd3k\n(2π)3di+1−σi\n2n(cond)\ni (21)\nn(s)\ni=gi/integraldisplayd3k\n(2π)3m∗\ni\nEidi+1−σi\n2n(cond)\ni (22)\nwith a thermal and a condensate contribution, once the\ndistribution function\ndi(T,k,m∗\ni,µ∗\ni) =/bracketleftbigg\nexp/parenleftbiggEi−µ∗\ni\nT/parenrightbigg\n+σi/bracketrightbigg−1\n(23)\nis defined. These expressions are consistent with the\nusual definitions of the densities. The vector densities\ncan be expressed as\nn(v)\nn=1+β−Xd\n2nb (24)\nn(v)\np=1−β−Xd\n2nb (25)\nn(v)\nd=Xd\n2nb (26)\nusing the baryon density nb, the asymmetry β, and the\ndeuteron fraction Xd.\nThe deuteron fraction has to stay below a maximum\nvalue of\nX(max)\nd=mnuc\nχCσnb(27)\nin order to ensure positive effective masses of the nucle-\nons. This limit is reached when the effective masses of\nthe nucleons are zero, i.e., Sn=Sp=Cσnσ=mnuc\nwith the source density nσ= 2χn(s)\nd=χX(max)\ndnbof\ntheσmeson. It only has a contribution from the quasi-\ndeuterons because the scalar densities of the nucleons\nvanish for m∗\nnuc= 0. A second limitation of the deuteron\nfraction arises from the fact that for every neutron a\nproton is needed, or vice versa, to form the cluster.\nThis translates to the condition Xd≤1− |β|depend-\ning on the isospin asymmetry β. So, in total, one has\n0≤Xd≤min/braceleftBig\nX(max)\nd,1−|β|/bracerightBig\n.\nAt high densities or temperatures, a mixture of\ndeuterons, neutrons and protons might be expected. If\nall three particle species have non-zero densities, the con -\ndition of chemical equilibrium\nµd=µn+µp, (28)\napplies between the chemical potentials of the degrees offreedom involved. Using the definitions ( 7) and ( 8) with\nthe potentials ( 9) and ( 10), this relation can be written\nas\n∆md=µ∗\nn+µ∗\np−md+Sd+Vn+Vp−Vd,(29)\nand thus an expression for the deuteron mass shift is\nobtained.\nAll thermodynamic quantities of the system can be\neasily obtained from the grand canonical thermodynamic\npotential ( 3). For instance, the pressure is given by\nP=−ω(T,{µi}) (30)\nand the free energy density Fcan be expressed as\nF=/summationdisplay\ni=n,p,dµin(v)\ni+ω=/parenleftbigg\nµb+1−β\n2µc/parenrightbigg\nnb−P(31)\nwith the baryon chemical potential µb=µnand the\ncharge chemical potential µc=µp−µn. The entropy\ndensity finally can be written as\nS=−∂ω\n∂T/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n{µi}+Scond (32)\n=−/summationdisplay\nigi/integraldisplayd3k\n(2π)3/bracketleftbigg\ndilndi+1−σidi\nσiln(1−σidi)/bracketrightbigg\n+n(cond)\ndlngd\nwith the distribution functions ( 23) and a contribution of\nthe condensed deuterons. The latter contribution arises\nbecause deuterons have spin 1and thus the ground state\nof nuclear matter at T= 0contains a mixture of the\ndifferent spin substates.\nB. Zero temperature and quasi-deuteron\ncondensation\nIf the temperature vanishes, the vector and scalar den-\nsities of the the nucleons q=n,pcan be expressed in\nanalytical form as\nn(v)\nq=gq\n6π2k3\nq (33)\nwithgq= 2and\nn(s)\nq=gqm∗\nq\n4π2/bracketleftbigg\nkqµ∗\nq−(m∗\nq)2lnkq+µ∗\nq\nm∗q/bracketrightbigg\n(34)\nwith the Fermi momentum kqand the effective chemical\npotential\nµ∗\nq=/radicalBig\nk2q+/parenleftbig\nm∗q/parenrightbig2. (35)\nSince quasi-deuterons are bosons, they can exist only as a\ncondensate in the zero temperature case we are focusing6\non. For the deuteron there is no thermal contribution to\nthe density but the condensate term\nn(v)\nd=n(s)\nd=n(cond)\nd(36)\nwith equal vector and scalar densities and the effective\nchemical potential becomes\nµ∗\nd=m∗\nd. (37)\nIn nuclear matter at very low densities, it is advanta-\ngeous to form quasi-deuterons, which still have a positive\nbinding energy, to gain energy as compared to a system\ncomposed of neutrons and protons only. In SNM, all pro-\ntons and neutrons will be bound in quasi-deuterons in the\nlow-density limit and no free nucleons remain. The bind-\ning energy per nucleon will approach half of the deuteron\nbinding energy for nb→0and not zero as for homoge-\nneous nucleonic matter without clusters. For ANM, only\nquasi-deuterons and neutrons (protons) will be the active\nconstituents in case of positive (negative) isospin asym-\nmetryβ.\nWith the help of the nucleon and deuteron densities, a\nsimple expression for the pressure Pis found. Since the\nthe momentum integral ( 4) can be calculated explicitly\nafter partial integration, one obtains\nP=/summationdisplay\nq=n,p1\n4/parenleftBig\nµ∗\nqn(v)\nq−m∗\nqn(s)\nq/parenrightBig\n(38)\n+1\n2/parenleftbig\nDωn2\nω+Dρn2\nρ−Dσn2\nσ/parenrightbig\n+/summationdisplay\ni=n,p,dn(v)\niW(r)\ni\nwith\nDi=Ci+C′\ninb (39)\nfor the meson couplings ( i=σ,ω, andρ). Because the\nentropy density ( 32) vanishes for T= 0, the internal\nenergy density Eis identical to the free energy density\n(31).\nC. Coupling strength scaling factors\nIn Eqs. ( 9) - (13), the scaling factors χijappear. These\nfactors are always unitary for nucleons and, in particular,\nχnσ=χnω=χnρ= 1\nχpσ=χpω=−χpρ= 1 (40)\nfor the three mesons. For the nucleons bound in clusters,\nthe choice of the scaling factors is instead widely debated\n[68].\nIt is a natural choice to assume that the nucleons inside\nthe deuteron couple to the mesons with the same strength\nas the unbound nucleons. Nevertheless, previous studies\nhave already shown that, to take into account in-medium\neffects in calculation of the EoS for astrophysical applica-\ntions, a universal scaling factor smaller than 1should beassumed for the cluster-meson coupling strength [ 69,70].\nA reduced value for the coupling of the σ-meson to differ-\nent light clusters, including deuterons, allows also a good\ndescription of the chemical equilibrium constants deter-\nmined from the NIMROD data [ 71]. However, recent\nBayesian analysis [ 72,73] have highlighted that a larger\nvalue should be taken for the cluster σ-meson coupling,\nto describe recent results from INDRA collaboration as\nwell [74]. Moreover, a possible model-dependence of this\nresult was recently assessed [ 75], calling for further micro-\nscopic analysis to get more stringent constraints. On the\nother hand, the possible choice of different values for the\nσandωcoupling strength factors is known to produce a\nstrong imbalance between the corresponding scalar and\nthe vector potentials. This would, in turn, reflect itself\nin an unrealistic behavior of the nuclear EoS, owing to\na corresponding change of the central potential given by\nthe difference Vi−Siof very large potentials in lowest or-\nder non-relativistic approximation. A reasonable choice\nwould be therefore to assume the same scaling factor for\nbothσandωmeson (χdσ=χdω=χ) and explore the\nsensitivity of our results to this ingredient ( χdρ= 0be-\ncause the deuteron has zero isospin.) This is thus the\nstrategy that will be adopted in the following.\nIII. QUASI-DEUTERON MASS SHIFT\nCONSTRAINTS\nIn the GRDF, a mass shift ∆mdis introduced in the\neffective mass defined in Eq. ( 8) in order to suppress\nthe cluster formation at supra-saturation densities. This\nmass shift may generally receive several contributions.\nFor example, in compact star matter, where electrons\nare included to fulfill the requirement of charge neutral-\nity, a possible contribution comes from the screening of\nthe Coulomb potential produced by the electronic back-\nground.\nIn the nuclear matter case, the mass shift is usually\nprovided just from Pauli blocking, which plays a cru-\ncial role in suppressing the cluster formation. Indeed,\nthe Pauli exclusion principle implies that a single-partic le\nstate in momentum space would not be longer available\nfor formation of a cluster when it is already occupied by\nnucleons of the medium. The Pauli blocking of states\nstrongly reduces at high temperatures with increasing\ndiffuseness of the Fermi sphere or when the c.m. mo-\nmentum of the cluster is much larger than the typical ra-\ndius of the Fermi sphere. This effect can be represented\neffectively as a repulsive, medium-dependent potential\nor a change of the cluster binding energy. A quantita-\ntive value can be calculated by solving the many-body\nSchrödinger equation when proper potentials for the nu-\ncleons in matter are introduced. The corresponding re-\nsults, which are explicitly calculated for various con-\nditions of temperature, density and isospin asymmetry\nof the medium, are then usually approximated by suit-\nable parameterizations in a wide range of thermodynamic7\nvariables [ 30,76–78].\nThe change of the cluster binding energy, calculated\nin this approach, is valid only at sub-saturation densities\nand has to be extrapolated to higher densities, in partic-\nular above the cluster dissociation (Mott) density, where\nthe binding energy vanishes, and only a many-body cor-\nrelation in the continuum remains. For this purpose, suit-\nable heuristic parameterizations were introduced within\nthe GRDF [ 66,79]. Focusing on deuteron-like correla-\ntions, the aim of our work is to provide a unified param-\neterization of the quasi-deuteron binding energy shifts,\nso that it can be used as effective means to treat SRCs\nat supra-saturation densities. In this context, the pio-\nneering analysis developed in Ref. [ 37] already suggests\nthat a substantial modification of the cluster mass shift\nhas to be expected in the regime beyond the deuteron\ndissociation density, in comparison to the traditionally\nadopted form.\nSuch a parameterization should be appropriately cho-\nsen to interpolate between the low-density limit con-\nstrained by microscopic many-body calculations and the\nhigh-density behavior postulated under the assumed bo-\nson condensation condition. In the following subsections\nthe different constraints to be imposed to the deuteron\nbinding-energy shifts will be discussed. As will be seen,\nvarious scenarios will emerge depending also on the value\nof the deuteron-meson scaling factors χ=χdσ=χdωin-\ntroduced in Eqs. ( 9) - (13).\nA. Low-density constraint\nSeveral parameterizations of the binding energy or\nmass shift of the deuteron at sub-saturation densities\nhave been developed to account for the Pauli blocking\neffects. In particular, when neglecting the dependence of\nthe mass shift on the momentum of the deuteron with\nrespect to the medium and limiting to the zero tempera-\nture case, the simplified functional form\n∆m(low)\nd=δBd(0)n(eff)\nd(41)\ncan be assumed, where the effective vector density n(eff)\nd,\nas defined in Ref. [ 37]\nn(eff)\nd=2\nAd/bracketleftBig\nNd/parenleftBig\nn(v)\nn+n(v)\nd/parenrightBig\n+Zd/parenleftBig\nn(v)\np+n(v)\nd/parenrightBig/bracketrightBig\n(42)\nwithNd=Zd= 1andAd= 2, turns out to be equal\nto the total baryon density nb, independent of the global\nisospin asymmetry βof the system. The quantity δBd\ngenerally regulates the temperature dependence; its zero\ntemperature limit which appears in Eq. ( 41) isδBd(0) =\n3634.16MeV fm3, as given in Ref. [ 30], where explicit\nexpressions for finite temperature calculations are also\nprovided.\nA linear increase of the mass shift proportional to nb\nat low baryon densities is consistent with the results ob-tained in [ 30,76–78], at least for density values lying\nbelow the dissociation or Mott density defined as\nn(diss)\nd=Bd\nδBd(0), (43)\nwhereBd=mn+mp−md= 2.225MeV [ 80] is the\ndeuteron binding energy in vacuum.\nB. High-density limit\nIn the traditional treatment of cluster dissolution us-\ning the concept of mass shifts, a heuristic density depen-\ndence stronger than linear in nbis customarily assumed\nat baryon densities above the dissociation density to pre-\nvent the clusters to reappear. A divergence of the mass\nshift ensures in particular the deuteron removal from the\nsystem when approaching saturation density, resulting\nin pure nucleonic matter above saturation. The possi-\nbility of using quasi-deuterons to effectively embed nu-\nclear SRCs at supra-saturation densities requires thus a\nproper change of the usual parameterization adopted in\nthe GRDF, as first noticed in Ref. [ 37].\n1. Deuteron mass shift\nAssuming a dependence on the effective density\nn(eff)\nd=nb, as defined in Eq. ( 42), the quasi-deuteron\nmass shift expression derived in Eq. ( 29) simplifies to\n∆m(high)\nd(44)\n=µ∗\nn+µ∗\np−md+2χCσnσ+2(1−χ)Cωnω\nbecause\nW(r)\nn+W(r)\np−W(r)\nd=n(s)\nd∂∆md\n∂nb(1+1−2) = 0.(45)\nEq. (44) can be used to calculate the deuteron mass shift\nas a function of the baryon density nb, the deuteron frac-\ntionXdand the isospin asymmetry β.\nAn impression of the density dependence of the quasi-\ndeuteron mass shift is depicted in Fig. 1for given, con-\nstant mass fractions Xdand two values of the scaling\nfactorχof the deuteron. A unitary value of χis con-\nsidered in panel (a), whereas a reduced scaling factor of\nχ= 1/√\n2is assumed in panel (b). The choice of the lat-\nterχvalue will be explained below. In both panels, for\nsake of simplicity, the SNM case is considered. Moreover,\nthe DD2 parameterization [ 30] of the nucleon-meson ef-\nfective interaction is adopted.\nFig.1shows that the deuteron mass shift ∆mddepends\nquite strongly on the value assumed for the scaling factor\nχ. This is deduced by comparing the results of panel (a)\nwith the corresponding ones of panel (b), especially in the\nhigh-density regime, we are focusing on. Indeed, for large8\n0 0.1 0.2 0.3 0.4 0.50.6 0.7 0.8 0.9 1\nbaryon number density nb [fm-3]-200020040060080010001200deuteron mass shift ∆md [MeV]\n0.45 0.5 0.55 0.6400450500\n(a)\n0 0.1 0.2 0.3 0.4 0.50.6 0.7 0.8 0.9 1\nbaryon number density nb [fm-3]-200020040060080010001200deuteron mass shift ∆md [MeV]Xd = 0.0\nXd = 0.2\nXd = 0.5\nXd = 0.8\nXd = 1.0\n(b)\nFigure 1. Panel (a): Deuteron mass shift as function of the ba ryon density, as determined according to Eq. ( 44) in the SNM\ncase, by assuming a unitary scaling factor χfor the deuteron-meson coupling strengths. Panel (b): the s ame as in panel (a),\nbut assuming a reduced scaling factor χ= 1/√\n2. In both panels, the DD2 parameterization [ 30] of the nucleon-meson effective\ninteraction is adopted. The inset in panel (a) shows a zoom ar ound the density values where the curves cross each other. Th e\nthin blue line indicates in both panels the binding energy of the deuteron in vacuum Bd.\ndensities, the effective chemical potentials of the nucle-\nons in Eq. ( 44) are dominated by the Fermi momenta kq\nas defined in Eq. ( 33) and thus exhibit a n1/3\nbbehavior.\nSince the meson couplings CσandCωapproach constants\nat high densities, the mesonic contributions to the mass\nshift are determined by the behavior of the source den-\nsitiesnσandnω. Forχ/ne}ationslash= 1, the source density nωrules\nthe dominating term. Since the deuteron fraction has\nto vanish for nb→ ∞ due the constraint ( 27), a linear\nasymptotic increase of ∆mdwith the baryon density is\nexpected. Indeed, such a dependence is observed in panel\n(b) of Fig. 1forχ= 1/√\n2. A softer increase of the mass\nshift with the baryon density is seen in panel (a) because\nthe last term in Eq. ( 44) does not contribute for χ= 1\nand the high-density behavior is driven by the σmeson\nterm. However, the source density nσmay asymptoti-\ncally receive contributions only from the scalar densities\nof the nucleons because the deuteron fraction asymptot-\nically approaches zero. An asymptotic dependence pro-\nportional to n2/3\nbis then expected for the σmeson term\nand the deuteron mass shift of Eq. ( 44).\nIn the (unrealistic) case when all nucleons are bound\ninside the deuteron ( Xd= 1), independent of the den-\nsity, both the scalar and vector densities of the nucleons\nvanish. The mass shift defined in Eq. ( 44) assumes the\nfollowing form\n∆m(high)\nd(Xd= 1) =Bd(0)+2χ(1−χ)(Cω−Cσ)nb,(46)\ni.e., a linear dependence on nbis generally predicted, for\nany finite value of the scaling factor χ, as shown by the\ndotted line of panel (b). An exception is the case with\nχ= 1, when the deuteron mass-shift coincides with the\ndeuteron binding energy, see dotted line in panel (a).\nFurthermore, Fig. 1shows that the largest deuteronfraction generally corresponds to the lowest mass shift,\nwith a clear ordering in panel (b). However, in the case\nwhen a unitary scaling factor is adopted, such a state-\nment actually holds only below the region where the lines\ncross each other, which is observed above 0.45fm−3. This\nregion is also emphasized in the inset of panel (a) of Fig. 1\nto evidence the fact that there is no single crossing point\nbetween the different curves.\nThe emergence of this crossing can be easily under-\nstood when looking at the explicit form of Eq. ( 44). In-\ndeed, for χ= 1, the vector contribution vanishes, and\nthere exists a delicate interplay between the remaining\nterms. At low densities, the leading role is played by\nthe effective chemical potentials, which reduce when the\ndeuteron mass fraction increases. However, at larger den-\nsities, the importance of the term involving the source\ndensity of the scalar meson is enhanced. Since the corre-\nsponding contribution increases with the deuteron frac-\ntion, a crossing among the curves is observed at a certain\nbaryon density ncross\nband an inversion of their previous\nordering is expected at higher densities. On the other\nhand, for smaller values of the scaling factor, as the one\nconsidered in panel (b) of Fig. 1, the vector term plays\nalso a role. The source density of the ωmeson field does\nnot change with the deuteron mass fraction at constant\nnb. Since the vector term dominates asymptotically over\nthe scalar one, in light of the power of density involved\nthere, for scaling factor values small enough, such as the\none considered in panel (b) of Fig. 1, no crossing is ob-\nserved and the same ordering for all allowed densities is\npreserved.\nMoreover, it is worthwhile to notice in panel (a)\nthat the dotted-dashed line, which corresponds to the\ndeuteron mass shift obtained for a fixed deuteron mass\nfractionXd= 0.5, ends at a density around 0.84fm−3.9\nBeyond this value, the deuteron mass fraction would ex-\nceed its maximum allowed value, see Eq. ( 27). The same\noccurs also for the two largest deuteron mass fraction\nvalues considered in both panels of Fig. 1, although at\nsmaller densities. It is also worth noticing that Eq. ( 27)\nimplies that any finite asymptotic value for the deuteron\nmass fraction is not allowed. As a result, the cluster is\nforced to dissolve asymptotically, at least as n−1\nb, for any\nfinite value of χ.\nThe observation of the line crossing in panel (a) of Fig.\n1motivates to study the dependence of the deuteron mass\nshift on the deuteron mass fraction for constant baryon\ndensitynband constant asymmetry β. A deeper insight\ninto this behavior can be achieved, by looking at Fig.\n2, where∆mdis plotted as a function of the deuteron\nmass fraction for different baryon density values. In order\nto facilitate the comparison among the curves, in Fig. 2\nthe differences with respect to the deuteron mass shift\nevaluated in the limiting case without deuteron (denoted\nas∆m(0)\nd) are actually considered. These differences turn\nout to be systematically lower than zero, at least for the\nreduced value of the scaling factor considered in panel\n(b). A change of sign is instead observed for χ= 1at\nlarger densities, i.e., beyond the crossing point observed\nin the panel (a) of Fig. 1and already discussed above.\nQuite interestingly, one observes in both panels that the\n(negative) slope of the curves strongly increases for those\nlines that approach Xd= 1. This represents the ideal,\nbut unrealistic, case where no free nucleons exist in the\nsystem also at supra-saturation densities. The reason\nbehind this behavior will be clarified in the following,\nwhen the mass shift derivatives are investigated in detail.\n2. Mass shift derivatives\nIn Appendix A, an explicit, general expression for the\nmass fraction derivative of the mass shift is derived. For\nsake of simplicity, the case of SNM is considered here,\nwhere the Fermi momenta, effective masses and effec-\ntive chemical potentials of the nucleons are identical, i.e .,\nknuc=kn=kp,m∗\nnuc=m∗\nn=m∗\np,µ∗\nnuc=µ∗\nn=µ∗\np.\nThen the simple form\n∂∆m(high)\nd\n∂Xd/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nnb,β=0= (47)\n/bracketleftBigg\n2Cσ\n1+fnucCσ/parenleftbigg\nχ−m∗\nnuc\nµ∗nuc/parenrightbigg2\n−π2\nknuc1\nµ∗nuc/bracketrightBigg\nnb\nwith the factor\nfnuc= 3/parenleftBigg\nn(s)\nnuc\nm∗nuc−n(v)\nnuc\nµ∗nuc/parenrightBigg\n(48)\nand the total nucleon densities n(v)\nnuc=n(v)\nn+n(v)\npand\nn(s)\nnuc=n(s)\nn+n(s)\npis obtained. The derivative ( 47) isthe difference of two positive contributions and depend-\ning on the choice of χthere can be a zero at a certain\nbaryon density. The dependence of the derivative ( 47)\non the baryon density nbis depicted in Fig. 3panel (a)\nforχ= 1and in panel (b) for χ= 1/√\n2. The same\nconstant values of Xdas in Fig. 1are considered. Only\nthe limit case with Xd= 1is not shown because for this\ndeuteron mass fraction, knucvanishes and a (negative)\ndivergent mass shift derivative is obtained. Moreover, as\nalready observed in Fig. 2, this result is independent of\nthe adopted value for the scaling factor.\nEven though we are interested here in the investiga-\ntion of the high-density limit, it is instructive to study\nboth the low- and high-density behaviors of the deriva-\ntive (47). One observes that, for knuc→0, the deriva-\ntive is dominated by the negative, diverging contribu-\ntion inside the brackets and the derivative approaches\nthus zero from negative values, as shown in both pan-\nels of Fig. 3. Forknuc→ ∞, the total nucleonic vec-\ntor and scalar densities scale as n(v)\nnuc∼[2/(3π2)]k3\nnuc\nandn(s)\nnuc∼(1/π2)m∗\nnuck2\nnuc, respectively. Furthermore,\nnb∼n(v)\nnuc,m∗\nnuc∼π2/(Cσk2\nnuc), andµ∗\nnuc∼knuc, so that\nfnuc∼k2\nnuc/π2and the simple asymptotic form\n∂∆m(high)\nd\n∂Xd/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nnb,β=0∼/parenleftbig\n2χ2−1/parenrightbig2\n3knuc (49)\nremains. For χ <1/√\n2the derivative in the high-density\nlimit is negative as for knuc→0and no zero at finite\nbaryon densities is expected. In contrast, for χ= 1, the\nderivative approaches asymptotically a positive value and\na zero at a certain baryon value, ncross\nb, appears, as de-\npicted in Fig. 3, panel (a). The curves cross the zero line\nat similar values above 0.45fm−3with a weak depen-\ndence on the deuteron fraction, highlighted in the inset.\nBelow the crossing an ordering of the lines as observed in\npanel (b) of Fig. 1follows, whereas an inversion will ap-\npear above ncross\nb. Forχ= 1/√\n2, however, the derivative\nof the mass shift with respect to the deuteron mass frac-\ntion of Eq. ( 47) is always non-positive, as evidenced in the\npanel (b) of Fig. 3. Then, the zero line is only asymptot-\nically approached and χ= 1/√\n2constitutes the largest\nvalue for which the ordering of the lines with respect to\nXd, depicted in Fig. 1, panel (b) persists, for all baryon\ndensities.\nIn the following, also the derivative of the mass shift\nwith respect to the baryon density will be studied. It can\nbe calculated explicitly from Eq. ( 44). The general case\nfor arbitrary values of βis treated in Appendix B. Again,\nthe result for the simplified case of SNM is given here. It\ncan be written for χ=χdω=χdσas\n∂∆m(high)\nd\n∂nb/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nβ=0=WSNM\nd−ZSNM\nd∂(nbXd)\n∂nb/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nβ=0(50)10\n0 0.1 0.2 0.3 0.4 0.50.6 0.7 0.8 0.9 1\ndeuteron mass fraction Xd-350-300-250-200-150-100-50050100∆md - ∆md(0) [MeV]\nnb = n0\nnb = 0.35 fm-3\nnb = 0.55 fm-3\nnb = 0.75 fm-3(a)\n0 0.1 0.2 0.3 0.4 0.50.6 0.7 0.8 0.9 1\ndeuteron mass fraction Xd-350-300-250-200-150-100-50050100∆md - ∆md(0) [MeV](b)\nFigure 2. Panel (a): Difference between the deuteron mass shi ft∆md, as determined according to Eq. ( 44) and its value in the\ncase with zero deuteron mass fraction ∆m(0)\nd, as function of the deuteron mass fraction Xdfor different, but constant, values\nof the baryon number density nb. The SNM case is considered and a unitary scaling factor χfor the deuteron-meson coupling\nstrengths are assumed. Panel (b): the same as in panel (a), bu t assuming a reduced scaling factor χ= 1/√\n2. In both panels,\nthe DD2 parameterization [ 30] of the nucleon-meson effective interaction is adopted.\n0 0.1 0.2 0.3 0.4 0.50.6 0.7 0.8 0.9 1\nbaryon number density nb [fm-3]-600-500-400-300-200-1000100200300∂∆md/∂Xd [MeV]\n0.45 0.5 0.55 0.6-25025\n(a)\n0 0.1 0.2 0.3 0.4 0.50.6 0.7 0.8 0.9 1\nbaryon number density nb [fm-3]-600-500-400-300-200-1000100200300∂∆md/∂Xd [MeV]Xd = 0.0\nXd = 0.2\nXd = 0.5\nXd = 0.8\n(b)\nFigure 3. Panel (a): Mass fraction derivative of the deutero n mass shift as function of the baryon density, as determined\naccording to Eq. ( 47) for different, but constant values of the deuteron mass frac tionXd. The SNM case is considered and\na unitary scaling factor χfor the deuteron-meson coupling strengths is assumed. Pane l (b): The same as in panel (a), but\nassuming a reduced scaling factor χ= 1/√\n2. In both panels, the DD2 parameterization [ 30] of the nucleon-meson effective\ninteraction is adopted. The thin blue line indicates the zer o of the deuteron mass fraction derivative of the mass shift. The\ninset in panel (a) shows a zoom around the density where the de rivative vanishes.\nwith the quantities\nZSNM\nd=π2\nµ∗nucknuc+2(1−χ)2Cω (51)\n−2Cσ\n1+fnucCσ/parenleftbigg\nχ−m∗\nnuc\nµ∗nuc/parenrightbigg2and\nWSNM\nd=π2\nµ∗nucknuc+2(1−χ)(Cω+C′\nωnω)(52)\n+2\n1+fnucCσ/parenleftbigg\nC′\nσnσ+Cσm∗\nnuc\nµ∗nuc/parenrightbigg/parenleftbigg\nχ−m∗\nnuc\nµ∗nuc/parenrightbigg\nthat contain again the factor ( 48). The dependence of\nthe derivative ( 50) for the SNM case is depicted in the\ntwo panels of Fig. 4for the selected scaling factors of\nχ= 1(left) and χ= 1/√\n2(right). In the zero-density\nlimit, a divergent behavior is observed, owing to the11\n0 0.1 0.2 0.3 0.4 0.50.6 0.7 0.8 0.9 1\nbaryon number density nb [fm-3]-5000500100015002000∂∆md/∂nb [MeV fm3](a)\n0 0.1 0.2 0.3 0.4 0.50.6 0.7 0.8 0.9 1\nbaryon number density nb [fm-3]-5000500100015002000∂∆md/∂nb [MeV fm3]\nXd = 0.0\nXd = 0.2\nXd = 0.5\nXd = 0.8\nXd = 1.0(b)\nFigure 4. Panel (a): Baryon density derivative of the deuter on mass shift as function of the baryon density, as determine d\naccording to Eq. ( 50) for different, but constant values of the deuteron mass frac tionXd. The SNM case is considered and\na unitary scaling factor χfor the deuteron-meson coupling strengths is assumed. Pane l (b): The same as in panel (a), but\nassuming a reduced scaling factor χ= 1/√\n2. In both panels, the DD2 parameterization [ 30] of the nucleon-meson effective\ninteraction is adopted.\ncontribution originating from the density derivatives of\nthe effective chemical potentials that lead to the terms\nproportional to k−1\nnucinZSNM\ndandWSNM\nd. The diver-\ngence of ( 50) fornb→0will disappear only for constant\nXd= 1. The relative importance of the terms propor-\ntional to k−1\nnucstrongly reduces with increasing density,\nso that a rise driven by the σmeson contribution is ob-\nserved at larger nbvalues, until a maximum is reached.\nThen, a continuous decrease of the density derivative of\nthe deuteron mass shift emerges, which asymptotically\napproaches zero or a constant value for the unitary or the\nreduced value of χ, respectively. An interplay among the\ndifferent involved terms takes place, analogously to the\none illustrated to describe the results of Fig. 1. As a con-\nsequence, the ordering of the curves in Fig. 1is reflected\nin Fig. 4. As discussed before, for any finite deuteron\nfraction value Xd, the corresponding curve ends again at\nthe baryon density where X(max)\nd, as a function of nb,\nattains this value.\nIV. MASS SHIFT PARAMETERIZATION\nThe condensation condition allows to calculate, for a\ngiven mass-fraction function, the quasi-deuteron mass\nshift and its derivatives at high-densities through Eqs.\n(44), (47) and ( 50), respectively. However, the density\ndependence of the mass fraction Xdis not known a pri-\nori, in particular at supra-saturation densities. It shoul d\noriginate from microscopic calculations in a similar man-\nner as the fractions of light clusters are determined by\ntheir mass shifts in the low-density domain, see Ref. [ 30].\nSince calculations of the deuteron mass shift using proper\ninteractions and many-body methods are only available\nat very low densities, it is necessary to resort to exem-plary forms of the mass shift in the full range of baryon\ndensities to study the properties of the system. Instead of\ncalculating the mass shift and its derivatives for a given\nmass fraction function, the main aim is then to choose a\ndensity dependent parameterization of the mass shift and\nto determine the deuteron fraction, not only for symmet-\nric but also asymmetric matter. Although such a function\nis not yet available from microscopic models, the pro-\nposed form should comply with the available constraints.\nA. Piecewise mass-shift parameterization\nThe most simple choice for the deuteron mass shift\nfunction is the piecewise parameterization\n∆md(nb) = (53)\nmin/braceleftBig\n∆m(low)\nd(nb),∆m(high)\nd(nb,Xd)/bracerightBig\nthat combines the low-density form ( 41) with the high-\ndensity function ( 44) assuming, e.g., a constant deuteron\nfraction in the high-density region as discussed in the pre-\nvious subsection. Although Eq. ( 53) provides a continu-\nous function, the same does not apply for its derivatives.\nIn addition, a constant deuteron fraction at high den-\nsities is not compatible with the asymptotic constraint\nwhich imposes a vanishing mass fraction Xdfornb→ ∞.\nDespite these shortcomings, it is informative to inves-\ntigate the effect of finite deuteron fractions on the energy\nper nucleon E/A. Fig. 5depicts the density behavior of\nE/Ain SNM as obtained with the piecewise mass shift\nparameterization ( 53) for different, but constant mass\nfractions in the high-density regime. For comparison,\nthe energy per nucleon obtained from the deuteron free\ncase of the standard DD2 parameterization is also shown.12\n0 0.1 0.2 0.3 0.4 0.50.6 0.7 0.8 0.9 1\nbaryon number density nb [fm-3]-1000100200300400energy per nucleon E / A [MeV]DD2\nXd(high) = 0.0\nXd(high) = 0.2\nXd(high) = 0.5\nXd(high) = 0.8\nXd(high) = 1.010-510-410-310-210-1-16-80\n(a)\n0 0.1 0.2 0.3 0.4 0.50.6 0.7 0.8 0.9 1\nbaryon number density nb [fm-3]-1000100200300400energy per nucleon E / A [MeV]10-510-410-310-210-1-16-80\n(b)\nFigure 5. Panel (a): Energy per nucleon as function of the bar yon density as determined in the SNM case, by assuming a\nunitary scaling factor χfor the deuteron-meson coupling strengths. A simple piecew ise parameterization, as given by Eq. ( 53)\nand interpolating between the low- and high-density constr aints for the mass shift is adopted. Different, but constant, values\nof the deuteron mass fraction X(high)\nd at high density are considered. Panel (b): The same as in pane l (a), but assuming a\nreduced scaling factor χ= 1/√\n2. In both panels, the DD2 parameterization [ 30] of the nucleon-meson effective interaction is\nadopted.\nOne observes that an extra binding is predicted in the en-\nergy per nucleon around saturation density, with respect\nto the case without deuterons. This result is indepen-\ndent of the adopted choice of the scaling factor χof the\ndeuteron-meson couplings. Moreover, Fig. 5shows that\nthe overbinding persists much beyond the saturation den-\nsity, especially in panel (b), where a reduced value of χ\nis considered. However, the stiffness of these curves in\nthe high-density regime increases with the deuteron mass\nfraction. As a result, a crossing among the curves is gen-\nerally observed at larger densities in case when χ≥1/√\n2,\nexcept for deuteron fraction values for which the corre-\nsponding curves end at lower densities.\nBoth insets of Fig. 5highlight that the correct low-\ndensity limit is reproduced because the low-density con-\nstraint is properly taken into account. As a result, dif-\nferently from the standard DD2 model without clusters,\nthe curves tend to half the deuteron binding energy in\nthe limit nb→0.\nThe overbinding observed in Fig. 5in a broad range of\ndensities around saturation seems to be surprising in light\nof the large positive mass shift of the deuteron depicted in\nFig.1. The increase of the binding energy per nucleon is\na result of two main effects. When nucleons are replaced\nby quasi-deuterons at a given baryon density, the value\nofnωdoes not change when χ= 1. At the same time,\nthe source density nσincreases because the scalar and\nvector densities of the condensed deuterons are identical\nwhereas n(s)\nq/n(v)\nq<1for nucleons. Hence, on the one\nhand, a stronger attraction from the σmeson is induced.\nThis effect will be smaller for χ <1when nucleons are\nreplaced by deuterons. Here, the source density of the ω\nmeson will reduce but the corresponding decrease of the\nσmeson source density is also less strong. On the otherhand, nucleons at the Fermi surface with energies close\nto the nucleon chemical potential\nµ(0)\nq=/radicalbigg/parenleftBig\nk(0)\nq/parenrightBig2\n+/parenleftBig\nmq−S(0)\nq/parenrightBig2\n+V(0)\nq (54)\nof the deuteron-free system, indicated by a superscript\n(0), are replaced by deuterons with energy\nEd=µd=md−Sd+Vd+∆md=µn+µp.(55)\nSincekq< k(0)\nq,Sq> S(0)\nq, andVq≈V(0)\nq, the chemical\npotentials of the nucleons µqare lowered, i.e., µq< µ(0)\nq,\neven for a positive deuteron mass shift ∆md. This cor-\nresponds to a stronger binding of the system.\nB. Specific features in the determination of\ndeuteron mass fractions from mass shifts\nIn principle, it would be sufficient to use the conden-\nsation condition ( 44) to find Xdfor a given ∆m(high)\nd. In\npractice, however, it is found that, in the high-density\nregime, Eq. ( 44) can have multiple solutions when a me-\nson coupling scaling factor χ≥1/√\n2is considered. As a\nconsequence, Eq. ( 44) can not be inverted uniquely in all\ncases. The correct solution has to be selected such that\na continuous function of the density is obtained for the\ndeuteron mass fraction.\nIn this context, further insight can be obtained from\nthe explicit functional form of the mass shift derivative\nwith respect to the density, given by Eq. ( 50). It is,\nin fact, a first-order differential equation for Xd. The\nexpression can be analyzed most easily for SNM. It is13\nconvenient to write Eq. ( 50) in the form\n∂(nbXd)\n∂nb/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nβ=0= (56)\n1\nZSNM\nd/parenleftBigg\nWSNM\nd−∂∆md\n∂nb/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nβ=0/parenrightBigg\nwith the functions ZSNM\ndandWSNM\ndas defined in Eqs.\n(51) and ( 52), respectively. A special role is played by\nZSNM\ndthat can be written as\nZSNM\nd= 2(1−χ)2Cω−1\nnb∂∆md\n∂Xd/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nnb,β=0(57)\nwith the help of the mass fraction derivative defined in\nEq. (47). The superscript (high) is no longer added to\nthe mass shift because a mass shift parameterization is\nconsidered now in the whole density range. For χ= 1,\nthe term from the ωmeson does not contribute. The\nremaining term in Eq. ( 57) develops a zero at a certain\ndensityncross\nbas discussed in section III Bclose to Eq.\n(49). Thus the density derivative ( 56) develops a pole at\nxcross\nband a continuous solution of the differential equa-\ntion can only be obtained if the term in parentheses in\nEq. (56) vanishes at the same density. For χ≤1/√\n2,\nhowever, the mass-fraction derivative ( 47) is negative,\nimplying a positive ZSNM\ndbecause also the contribution\nof theωmeson is positive. Then, a continuous solution of\nthe differential equation ( 56) can be found for all densi-\nties. The discussion developed above justifies the choice\nof the reduced scaling factor. Thus, two different val-\nues of the scaling factor, namely χ= 1andχ= 1/√\n2,\nwill be considered in the following analysis. It is worth-\nwhile to notice that the value of χ= 1/√\n2is significantly\nsmaller than the universal scaling factor for the cluster-\nmeson coupling strength proposed in some recent works\n[69,70]. Our choice complies, however, with the aim to\nconsider two extreme values of χwith two distinct paths,\nbearing in mind that any intermediate behavior may also\noccur.\nC. Saturation constraints\nThe overbinding observed around saturation density\nin both panels of Fig. 5implies that a proper refit of\nthe nucleon-meson couplings to the saturation proper-\nties of SNM is mandatory if one wants to keep nuclear\nmatter quantities around the saturation point n0well\nconstrained. The actual deuteron fraction at saturation,\nXd,0, which has to be specified to fix the couplings, can\nbe imposed from recent experimental investigations of\nSRCs by extrapolating results of nuclei to infinite nu-\nclear matter. They assess that SRCs pairs amount to\napproximately 20%of the nucleon density [ 43–45]. Here,\nas in the following, the index 0on the quantities indicates\nthe values at saturation.In order to reproduce the properties at saturation in\nSNM, e.g., of the DD2 model, the binding energy per\nnucleonB0and the effective nucleon mass m∗\nnuc,0at the\nsaturation density should be obtained also in the model\nwith a finite deuteron fraction Xd,0. This will be real-\nized by rescaling the meson-nucleon couplings assuming\nno change in their density dependence as given in a ref-\nerence parameterization. The values of n0,B0,mnuc,0\nandXd,0together with the pressure P0= 0MeV fm−3\ngive four conditions that allow to determine the rescaled\ncouplings Γσ,0,Γω,0, the deuteron mass shift ∆md,0and\nits derivative d∆md/dnb|n0at saturation. Owing to the\nrescaling of the σandωcoupling strengths, also the en-\nergy per nucleon of PNM would be modified. A proper\nrescaling also of the ρ-nucleon coupling strength is hence\nin order if one wants to keep the symmetry energy at\nsaturation J0unaltered, with respect to the selected ref-\nerence parameterization. Here, the symmetry energy is\ncalculated in the parabolic approximation as the differ-\nence between the energies per nucleon in PNM and SNM.\nThis recipe gives finite values at sub-saturation densities\nin models with clusters or liquid-gas phase transition dif-\nferently than the original definition using second deriva-\ntives of the energy per nucleon in SNM with respect to\nthe isospin asymmetry, see, e.g., Ref. [ 81]. The full con-\nversion procedure is illustrated in detail in Appendix C.\nThe actual values of these quantities are given in Table\nIfor the two considered values of the deuteron coupling\nscaling factor χand a deuteron fraction Xd,0= 0.2. The\nstandard DD2 model is chosen as reference parameter-\nization with n0= 0.149065 fm−3,B0= 16.0224MeV,\nm∗\nnuc,0= 0.562544mnucandJ0= 32.73MeV. The σ,\nω, andρnucleon-meson couplings are given by Γσ,0=\n10.686681 ,Γω,0= 13.342362 andΓρ,0= 3.626940 in this\ncase.\nQuite interestingly, one observes that, in the case with\na unitary scaling factor χ, a small reduction of the three\nnucleon-meson coupling strengths is obtained at satura-\ntion, with respect to the original DD2 parameterization.\nOn the other hand, these quantities turn out to be larger\nfor both σ- andω-meson, when the smaller χvalue is\nconsidered. This change reflects the balance between the\ncouplings and scaling factors to achieve the same strength\nof the effective interaction at saturation density. Con-\ncerning the mass shifts, larger values and stronger den-\nsity slopes are predicted at saturation when assuming\nthat the nucleons bound inside the deuterons couple to\nthe mesons with the same strength as the unbound nu-\ncleons, see also Fig. 1.\nA possible mass shift parameterization will be pro-\nposed in the following section. It will be constrained\nat saturation density, in the low-density limit by micro-\nscopic many-body calculations and at high-density by an\nassumed mass-fraction behavior that respects the con-\ndensation condition and the constraint on the maximum\ndeuteron fraction.14\nTable I. Values at saturation density of nucleon-meson coup ling strengths Γj,0(j=σ,ω,ρ)), deuteron mass shift ∆md,0and\nits slope, for the two different scaling factors χconsidered in this work. The mass shift and its slope are expr essed in MeV and\nMeV fm3, respectively, whereas all other quantities are dimension less.\nχΓσ,0Γω,0Γρ,0∆md,0[MeV]d∆md\ndn/vextendsingle/vextendsingle\nn0,β=0[MeV fm3]\n110.580042 13 .217226 3 .556424 104 .92 813 .98\n1/√\n2 10.919963 13 .719324 3 .400187 58 .23 570 .80\nD. Unified mass shift parameterization\nDifferent forms of the mass shift density parameteri-\nzation might be employed to interpolate among the low-\nand the high-density constraints discussed in the previous\nsection, while keeping the required saturation properties .\nA unified form, adopted for the SNM case, might be then\nemployed also for ANM to obtain predictions at arbitrary\nisospin asymmetries. The mass shift parameterization\nintroduced in this work combines two limiting dependen-\ncies on the density that reproduce by construction the\nlinear increase in the zero-density limit of the dilute re-\ngion and the high-density asymptotic behavior. Taking\ninto account also the two constraints imposed at satura-\ntion, such a parameterization has to depend on at least\nfour parameters and satisfy ∆md(0) = 0 . However, some\nadditional parameters should enter in the proposed pa-\nrameterization to guarantee a smooth transition between\nthe different density regimes and leave some freedom in\nthe high-density behavior of the deuteron fraction.\nA possible choice, among others, is the function\n∆md(x) = ∆md,1(x)+∆md,2(x)+∆md,3(x)(58)\ndepending on x=nb/n0with three contributions\n∆md,1(x) =ax\n1+bx(59)\n∆md,2(x) =cxη+1[1−tanh(ex)] (60)\n∆md,3(x) =fxγtanh(gx) (61)\nwithγ= 1or2/3forχ= 1/√\n2or1, respectively, and\nseven coefficients a,b,c,η,e,f,g, which allow to comply\nwith the constraints by a proper choice.\nThe different terms in Eqs. ( 58) - (61) are chosen so\nthat the density derivative of the mass shift is given by\nd∆md\ndnb/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nnb=0=a\nn0(62)\nat vanishing nband the high-density behavior is domi-\nnated by the third term ∆md,3. The second contribu-\ntion acts mainly in an intermediate density range. The\ncoefficient ais determined as δBd(0)n0by the limiting\nform of the deuteron mass shift parameterization from\nmicroscopic calculations, c.f., Eq. ( 41). There is no a\npriori constraint for the parameter b. Here, it is set to\nb=a/Bd=n0/n(diss)\ndso thatlimnb→∞∆md,1=Bdwith\nthe deuteron binding energy Bdand the dissociation den-sityn(diss)\nddefined in Eq. ( 43). In the asymptotic limit,\nforχ= 1/√\n2, the mass shift approaches a linear func-\ntion in the baryon density and a slope determined by the\nratio offand the saturation density n0. On the other\nhand, the asymptotic form\n∆md(nb)∼f/parenleftbiggnb\nn0/parenrightbigg2/3\n(63)\nis expected for the deuteron mass shift, in the case when\nχ= 1.\nThe coefficients candη, whose analytical expressions\nare given in Appendix D, are finally determined by the\nconstraints introduced on the mass shift and its deriva-\ntive at saturation. The remaining parameters eandg\nare free and allow to tune the relative role of the differ-\nent contributions ∆md,2and∆md,3in Eq. ( 58). Only a\ntiny sensitivity of the results was assessed by varying the\nparameter e. Thus, this parameter was kept fixed to 1\nand only different values for the parameter gwere con-\nsidered. Different choices of gpermit indeed to produce\nalternative supra-saturation scenarios while keeping the\nsame asymptotic behavior.\n1. Deuteron mass shift and mass fraction for χ= 1\nLet us consider first the case with χ= 1. For such\na scaling factor, three different sets of parameters will\nbe considered. They are determined according to the\nparameterization proposed in Eqs. ( 58) - (61) withγ=\n2/3and are labeled in the following as DD2-d1, DD2-\nd2 and DD2-d3. The values of the parameters for these\nmass shift parameterizations are listed in the first three\nlines of Table II. They were obtained by employing the\nDD2 nucleon-meson effective interaction with properly\nrescaled meson coupling strengths at saturation as given\nin Table I. Panel (a) of Fig. 6shows the deuteron mass\nshift as function of the baryon density for these three\ndifferent sets. The red shaded area corresponds to the\nrange of possible mass shift values that are explored by\nassuming, for each density, deuteron fractions within the\nrange[0,min{1,X(max)\nd}]. First of all, Fig. 6highlights\nthe validity of the proposed parameterization to comply\nwith the constraints imposed on the deuteron mass-shift.\nThe three lines lie indeed within the red area for the\nwhole range of displayed baryon densities.\nHowever, as shown in Table II, a rather small range of15\nTable II. Values of the parameters in the deuteron mass shift parameterization defined by Eqs. ( 58) - (61) for six different sets.\nThey are obtained by employing the DD2 nucleon-meson effecti ve interaction, with properly rescaled meson coupling stre ngths\nat saturation. The parameters a,candfare expressed in MeV, b,η,e,gandγare dimensionless. The first three sets refer to\nthe case with deuteron-meson coupling scaling factor χ= 1, the others to χ= 1/√\n2.\na b c η e f g γ\nDD2 - d1 541.726060 243 .472387−83.230901 3 .491787 1 .0 214.368137 0 .65 2/3\nDD2 - d2 541.726060 243 .472387−98.923123 3 .200967 1 .0 214.368137 0 .67632 2/3\nDD2 - d3 541.726060 243 .472387−140.309501 2 .715545 1 .0 214.368137 0 .75 2/3\nDD2 -χd1541.726060 243 .472387 99 .677247 1 .656159 1 .0 181.113975 0 .18 1\nDD2 -χd2541.726060 243 .472387 70 .476986 1 .230947 1 .0 181.113975 0 .22 1\nDD2 -χd3541.726060 243 .472387 41 .777908 0 .257252 1 .0 181.113975 0 .26 1\n0 0.1 0.2 0.3 0.4 0.50.6 0.7 0.8 0.9 1\nbaryon number density nb [fm-3]0200400600800deuteron mass shift ∆md [MeV]DD2 - d1\nDD2 - d2\nDD2 - d3\n(a)\n0 0.1 0.2 0.3 0.4 0.50.6 0.7 0.8 0.9 1\nbaryon number density nb [fm-3]0.10.20.30.40.50.60.70.80.91deuteron mass fraction Xd\n(b)\nFigure 6. Panel (a): Deuteron mass shift as function of the ba ryon density, as determined according to the parameterizat ion\nproposed in Eqs. ( 58) - (61). The SNM case is considered and a scaling factor χ= 1is assumed for the deuteron-meson coupling\nstrengths. Three different set of parameters were employed. The red shaded areas evidence the region of allowed mass shif t\nor deuteron mass fraction values. Panel (b): Deuteron mass f ractionXdas function of the baryon density, as determined by\nemploying the same sets of parameters considered in panel (a ). In both panels, the DD2 nucleon-meson effective interacti on,\nwith properly rescaled meson coupling strengths at saturat ion, is adopted. Two curves with symbols (not shown in panel ( a))\nare considered in panel (b), as a result of slightly varying t hegparameter of the set labeled as DD2-d2 (see text for more\ndetails).\nthe parameter gmay be actually explored, owing to the\nshrinkage of the red area in the density region around the\ncrossing points, which were observed in Fig. 1, panel (a).\nDespite their proximity, when these parameterizations\nare employed in Eq. ( 56) to determine the density be-\nhavior of the deuteron fraction, extremely different out-\ncomes are obtained, at least when approaching the region\nof the crossings. As clearly depicted in Fig. 6, panel (b),\nthe three adopted parameterizations correctly reproduce\nthe low-density limit, corresponding to the situation in\nwhich the matter is entirely clusterized ( Xd= 1), and\naccount also for the constraints at saturation.\nNonetheless, highly diverse scenarios are predicted at\nhigh-densities. The dashed curve, which is the lowest\none in the region around the shrinkage observed in panel\n(a), corresponds to a deuteron mass fraction which tends\nto exceed, at a certain density, the maximum allowed\nvalue for X(max)\nd. We remind that the density regionbeyond this point would be characterized by a negative\nvalue of the Dirac effective mass of the nucleons. To ex-\nclude this opportunity, the curve in Fig. 6, panel (b) is\nthus stopped beyond that density. Owing to the unfea-\nsible high-density scenario, this set of parameter will not\nbe further employed in the following. Conversely, the\nDD2-d3 parameterization, plotted as the dotted line, ly-\ning above the other curves around 0.5fm−3in panel (a),\npredicts a sudden disappearance of the deuterons, when\napproaching the pole of Eq. ( 56). Actually, even in this\ncase, the curve is not plotted beyond the region of the\ncrossing, to exclude the unrealistic situation in which the\nclusters reappear at very high densities. As already an-\nticipated in Section IVB, a quasi-continuous solution of\nEq. (56) might be arranged for all densities with a fine\ntuning of the parameters. The corresponding curve is\nplotted in both panels of Fig. 6as the full line. In this\ncase, a smooth behavior is apparently recovered for the16\ndensity behavior of the deuteron mass fraction. However,\na strong sensitivity persists in correspondence of the pole ,\nas manifested by the two thin lines with symbols plotted\nin panel (b). These curves are obtained with mass shifts\nfunctions that are found by varying the gparameter only\nby 1% of the set labeled as DD2-d2. Owing to the pres-\nence of the pole, the DD2-d2 set of parameters will be\nset aside hereafter too. A more refined method to find\na continuous function would be to consider a more gen-\neral form of the mass shift parameterization with respect\nto the one proposed in Eqs. ( 58)-(61). By enlarging the\nnumber of the involved parameters, it would be possi-\nble to constrain the mass-shift and its slope values such\nthat the pole will be definitely washed out. Nonetheless,\nnone of such possible parameterizations would allow to\naccomplish our aim to extend the predictions to ANM.\nSince the position of the pole evolves with the asym-\nmetry, a divergence of the mass fraction would emerge\nonce again as soon as the asymmetry of the matter is\nchanged, despite its removal in the SNM case. The same\nholds, obviously, for any deuteron-meson coupling scal-\ning factor χ >1/√\n2. However, one should bear in mind\nthat, for values of the scaling factor smaller than 1, the\npole is expected to appear at higher densities. Since for\nχ= 1the pole emerges already at rather large densities\n(around 3n0), one expects that for more realistic values\nofχ, its position will be located much beyond the range\nthat is relevant in the applications of the model. Then\nits emergence could be neglected in practice. For χ= 1,\nonly the DD2-d3 parameterization will be employed in\nthe following, when the general properties of both SNM\nand ANM matter will be investigated.\n2. Deuteron mass shift and mass fraction for χ= 1/√\n2\nAs discussed in the previous sections, a possible way\nout to overcome the issue of the pole might be to assume\na smaller scaling factor which is, at maximum, equal to\n1/√\n2. For such a scaling factor, three different sets of\nparameters, labeled as DD2- χd1, DD2- χd2, and DD2-\nχd3, are proposed here. The values of the parameters in\nEqs. ( 58) - (61) for these sets are listed again in Table II.\nOne observes a strong sensitivity to the gparameter\nof the deduced values for candη. Moreover, when in-\ncreasing g, both parameters to account for the saturation\nconstraints decrease. A further increase of gbeyond a\nmaximum value gmax≈0.26is excluded, since it would\nimply a negative value for dand thus a dominant role of\n∆md,2in the zero-density limit, where a pole could even\nemerge. However ∆md,1returns already, by construc-\ntion, the correct low-density trend.\nFurther insights may be achieved by looking at both\npanels of Fig. 7. The density behaviors of these mass\nshifts are shown in Fig. 7panel (a). The same pa-\nrameterizations are then employed in panel (b) of Fig.\n7to determine the corresponding density behavior of\nthe deuteron mass fraction Xd. In the two panels, thered shaded area evidences the allowed region between\nthe (upper) curve, related to the deuteron-free case, i.e.\nXd= 0, and the (lower) one, obtained by assuming\nXd=min/braceleftBig\n1,X(max)\nd/bracerightBig\n.\nFirst of all, looking at panel (a), one notices that the\nblack curves lie always within the red shaded area up to\nvery large baryon densities, so validating the choice of\nthe adopted parameter sets. In light of the constraints\nimposed in the extremely dilute regime and at saturation,\nthe full, dashed and dottes black curves remain rather\nclose up to n0. All the curves also converge to the line\ncharacterized by Xd= 0, in the asymptotic limit. Some\ndifferences emerge instead in the high-density behavior,\naround and beyond 3n0.\nThe observed differences in the mass shifts are then re-\nflected in the density behavior of the deuteron mass frac-\ntion, which is plotted in panel (b). Independent on the\nparameterization, in the zero-density limit, the matter\nis completely clusterized and the deuteron mass fraction\nXdis equal to 1. With increasing density, a consider-\nable reduction of Xdis observed and a local minimum\nemerges in a density region around the saturation den-\nsity. There the value Xd= 0.2is reached, as required\nin agreement with the experimental evidences concern-\ning the emergence of SRCs pairs. At supra-saturation\ndensities, several scenarios take place. The curves never\novershoot the line indicating the maximum allowed value\nforXd, which ensures a non-negative value for the Dirac\neffective mass of the nucleons. Then, at higher densi-\nties, a decreasing trend is observed for all curves, which\nconverge each other, approaching zero asymptotically.\nIt is worthwhile to notice that alternative scenarios,\nsimilar to the ones displayed by the dotted and the\ndashed lines in panel (b) of Fig. 6, would be possible\nalso in the case with χ= 1/√\n2. Solutions with deuteron\nmass fraction values which tend to exceed the maximum\nallowed at a certain density or abruptly vanishing might\naccidentally occur, when considering mass shift parame-\nterizations which cross the lowest or the highest border,\nrespectively, of the red shaded area shown in panel (a)\nof Fig. 7. Differently than the case with χ= 1, these so-\nlutions are not connected to the emergence of any pole.\nThe three sets of parameters proposed in panel (b) of Fig.\n7avoid these scenarios. Solutions with the disappear-\nance of the clusters at a certain baryon density would be\nlikewise acceptable, even though none of the three cho-\nsen sets of parameters for χ= 1/√\n2provides a similar\nresult. Indeed, for this class of solutions, the density be-\nhavior of the deuteron mass fraction would closely resem-\nble the one obtained with the DD2-d3 parameterization.\nThe main difference will be only the possible wider den-\nsity range with a non-vanishing deuteron mass fraction\nvalues before the cluster is suppressed. For χ= 1/√\n2,\nwe will consider three set of parameters, such that the\nmass shifts are characterized by similar smooth trends as\nfunctions of the baryon density, but different sizes of the\ndeuteron mass fractions in the supra-saturation density17\n0 0.5 1 1.5 2 2.5 3\nbaryon number density nb [fm-3]01000200030004000deuteron mass shift ∆md [MeV]DD2 - χd1\nDD2 - χd2\nDD2 - χd3\n(a)\n0 0.5 1 1.5 2 2.5 3\nbaryon number density nb [fm-3]0.10.20.30.40.50.60.70.80.91deuteron mass fraction Xd\n(b)\nFigure 7. Panel (a): Deuteron mass shift as function of the ba ryon density, as determined according to the parameterizat ion\nproposed in Eqs. ( 58) - (61). The SNM case is considered and a scaling factor χ= 1/√\n2is assumed for the deuteron-meson\ncoupling strengths. Three different set of parameters were e mployed. The red shaded area evidences the region between th e\n(upper) curve, related to the deuteron-free case, i.e.Xd= 0, and the (lower) one, obtained by assuming Xd=min/braceleftBig\n1,X(max)\nd/bracerightBig\n.\nPanel (b): Deuteron mass fraction Xdas function of the baryon density as determined by employing the same sets of parameters\nof panel (a). In both panels, the DD2 nucleon-meson effective interaction, with properly rescaled meson coupling streng ths at\nsaturation, is adopted.\nregime. In such a way, we could assess and isolate the\nrole of this ingredient.\nThe sets of parameters DD2-d3, DD2- χd1, DD2- χd2,\nand DD2- χd3 listed in Table IIreturn a smooth behavior\nof the mass fraction for the whole range of baryon densi-\nties and for any asymmetry. Thus, they will be employed\nin the next section to study various properties, both for\nSNM and ANM.\nV. THERMODYNAMIC QUANTITIES\nOnce the density behavior of the mass shift and of the\ncorresponding deuteron mass fraction is determined, it is\ninteresting to see how the embedding of quasi-clusters at\nsupra-saturation densities affects some general thermo-\ndynamic quantities.\nA. SNM: EoS and incompressibility\nLet us focus in this section on the results for SNM.\nThe density dependence of the energy per nucleon E/A\nis plotted in Fig. 8. The sets of parameters DD2-d3,\nDD2-χd1, DD2- χd2, DD2- χd3 listed in Table IIare em-\nployed. The standard DD2 parameterization, describ-\ning the deuteron free case, is also shown as reference\nfor comparison. The inset of Fig. 8depicts in particu-\nlar the differences between the energy per nucleon de-\nrived with each set and the chosen reference. First of all,\none notices that, in light of the fit performed at satura-\ntion, all the curves remain rather close in the low-density0 0.1 0.2 0.3 0.4 0.50.6 0.7 0.8 0.9 1\nbaryon number density nb [fm-3]0100200300400energy per nucleon E / A [MeV]DD2\nDD2 - d3\nDD2 - χd1\nDD2 - χd2\nDD2 - χd30.00.51.01.52.0-20020406080\n(E- EDD2)/A [MeV] \nFigure 8. Energy per nucleon as a function of the baryon\ndensity, as determined by employing four selected set of pa-\nrameters listed in Table II, for the mass shift parameterization\ngiven in Eqs. ( 58)-(61). For comparison, the curve obtained\nfor the DD2 parameterization in the deuteron-free case is al so\nshown, as the red full line. The inset shows the differences\nwith respect to the predictions of the standard DD2 parame-\nterization in a larger baryon density range.\nregime. However, the parameterizations with deuterons\ndiffer from the standard DD2 result in the zero-density\nlimit, approaching one half of the deuteron binding en-\nergy in vacuum. Moreover, remarkable differences also\nemerge in the high-density behavior of the energy per\nnucleon. Despite the fit performed at saturation, as in\nFig.5, a stronger binding is generally observed in the18\n0 0.1 0.2 0.3 0.4 0.50.6 0.7 0.8 0.9 1\nbaryon number density nb [fm-3]04080120160200incompressibility K [102 MeV]\nDD2\nDD2 - d3\nDD2 - χd1\nDD2 - χd2\nDD2 - χd300.511.522.53-1001020\nK - KDD2 [102 MeV]\nFigure 9. Incompressibility Kas a function of the baryon\ndensity as determined by employing four selected set of pa-\nrameters listed in Table IIand for the mass shift parameter-\nization given in Eqs. ( 58)-(61). For comparison, the curve\nobtained for the DD2 parameterization in the deuteron-free\ncase is also shown as the red full line. The inset shows the dif -\nferences with respect to the predictions of the standard DD2\nparameterization in a larger baryon density range.\nneighbourhood of n0in the parameterizations accounting\nfor the presence of deuterons. This stronger attraction\npersists also to all densities in the case of the DD2-d3\nset, despite the disappearance of the deuterons which is\nexpected to occur at nbaround 0.45 fm−3with this pa-\nrameterization (see Fig. 6). The reduction in the energy\nper nucleon observed at higher densities with the DD2-\nd3 is thus only driven by the changed balance between\nthe scalar and vector components, which is a result of the\nrescaling of the meson coupling strengths at saturation.\nA different scenario manifests itself with the param-\neterizations derived with a reduced scaling factor χ.\nThere, no systematic increased binding, as compared to\nthe DD2 case, is observed. A delicate interplay takes\nplace between the stronger attraction, which is produced\nby the presence of the deuterons, and the repulsion de-\ntermined by the increased stiffness of the EoS. This is a\nresult of the modification introduced in the strengths of\ntheσandωmeson couplings. A global repulsive contri-\nbution is seen at large baryon densities, while a significant\nreduction of the energy per nucleon might be observed up\nto approximately 3n0, depending on the value reached by\nthe deuteron mass fraction in correspondence of the local\nmaximum observed in Fig. 7. However, as highlighted in\nthe inset of Fig. 8, the three black curves converge in the\nasymptotic limit, where a smooth disappearance of the\nclusters was depicted in panel (b) of Fig. 6.\nThe incompressibility characterizes the curvature of\nthe energy per nucleon. It is defined here as\nK(nb) = 9n2\nb∂2(E/A)\n∂n2\nb(64)Table III. Values of the incompressibility K0, in MeV, at sat-\nuration density as derived according to Eq. ( 64), for four se-\nlected parameterizations employed in this work. The result\nfor the DD2 is also given for comparison.\nDD2 DD2-d3 DD2- χd1 DD2- χd2 DD2- χd3\nK0[MeV]242.7 199.6 185 .3 207 .3 240 .3\nthrough a second derivative with respect to the baryon\ndensity.1We recall that the incompressibility was not\nconstrained within the approach adopted in this work.\nA constraint on the incompressibility would translate to\na constraint on the second density derivative of the mass\nshift at saturation. In addition, it would require, as a\nfurther input at saturation, the knowledge of the density\nderivative of the deuteron mass fraction. Although some\nnumerical analyses have suggested that SRC pairs have a\nminimum in the neighbourhood of the saturation, owing\nto the interplay between the tensor component and the\nrepulsive core of the nuclear force [ 49–51], we preferred\nto prescind from applying such a constraint. Instead, the\npredictions for the density behavior of Kare numerically\nextracted from the energy per nucleon. They are plot-\nted in Fig. 9for the four selected parameterizations of\nTable IIconsidered before. The related inset shows the\ndifferences with respect to the standard DD2 reference,\nwhich is also shown in the main plot for comparison. As\na general feature, one observes that a softening of the\nEoS is recovered in the region beyond saturation, up to a\ndensity around 3n0. Furthermore, the size of this effect\ndepends on the magnitude of the deuteron mass fraction.\nAs clearly evidenced in Table III, the predictions for\nthe incompressibility at saturation lie within or below the\nrange of values generally assumed for this quantity [ 82–\n85] for all the parameterizations here employed. Strong\ndifferences are observed in the high-density region, de-\npending on the value for the scaling factor χ. A much\nstiffer EoS is envisaged in particular at very large baryon\ndensities, when a scaling factor χ= 1/√\n2is assumed, as\na consequence of the significant change introduced in the\nbalance between the scalar and the vector components in\nthis case.\nFig.9also exhibits another aspect to be discussed. The\nblue curve, which corresponds to the DD2-d3 parame-\nterization, reveals the emergence of a discontinuity at a\nbaryon density around 0.45 fm−3. This striking feature\nsignals the abrupt disappearance of the cluster, which\nwas observed in Fig. 6, panel (b) and already discussed\nbefore. It is worthwhile to mention that a discontinuity\nin the matter incompressibility or in any other quantity\nrelated to the second derivative of a thermodynamic po-\n1The original definition of Kuses a second derivative with respect\nto the Fermi momentum and gives different results as compared\nto the definition used here but can not be used at finite temper-\natures. However, both definitions coincide at saturation.19\ntential is the signature for the possible emergence of a\nsecond-order phase transition. One notices, by the way,\nthat this feature is in complete analogy to the disappear-\nance of pairing correlations that was observed in previ-\nous works at low density [ 86,87]. Another discountinuity\nwould emerge moreover at larger densities, if the calcu-\nlation with the DD2-d3 parameterization is not stopped\nwhen the cluster dissolves, so that a further reappear-\nance of the deuterons at higher density is allowed. The\ninset of Fig. 9shows that no discontinuity is instead\nobserved when the deuteron mass fraction smoothly de-\ncreases with the density. The latter situation may how-\never occur only for the parameterizations characterized\nby a reduced value of the scaling factor χ.\nB. Predictions for ANM\n1. Deuteron mass fraction and EoS\nLet us finally concentrate on the predictions for ANM.\nIn Fig. 10, panel (a), the density dependence of the\ndeuteron mass fraction is plotted for different values of\nthe isospin asymmetry |β|. The DD2- χd1 parameteriza-\ntion is employed for sake of illustration. The adopted pa-\nrameterization allows one to get the largest value for the\ndeuteron mass fraction around the local maximum, which\nwas observed beyond 0.5 fm−3in panel (b) of Fig. 7. In\nsuch a way, the effect of embedding the quasi-deuterons\nat supra-saturation densities is better emphasized. How-\never, similar results, at least from a qualitative point of\nview, would be obtained with the other sets of parameters\naccounting for the presence of the deuterons. The red line\nindicates the maximum allowed deuteron mass fraction\nvalues, which are compatible with a non negative value of\nthe Dirac effective mass of the nucleons. Let us recall that\nthis curve corresponds to Xd=min{1−|β|,X(max)\nd}, so\nthat it evolves with |β|. Then only the border for SNM\n(β= 0) is plotted in panel (a) of Fig. 10to avoid to\noverload the figure.\nAs a quite interesting result, Fig. 10highlights that, al-\nthough the mass shift function defined in Eqs. ( 58)-(61)\nhas no explicit dependence on the isospin asymmetry,\nthe corresponding deuteron mass fraction evolves with\n|β|, giving rise to a continuous overall reduction when\nincreasing the neutron-proton asymmetry of the matter.\nMoreover, the smooth transition to the cluster-free mat-\nter realized in SNM with the DD2- χd1 is preserved also\nin the ANM case.\nThe density behavior of the energy per nucleon is de-\npicted in panel (b) of Fig. 10, for the same parameteriza-\ntion and the same asymmetry values considered in panel\n(a). The inset of panel (b) displays moreover the differ-\nence in the energy per nucleon, as determined with the\nDD2-χd1 and the DD2 parameterization. Although not\nclearly visible, differently than in the deuteron-free DD2\ncase, the zero-density limit does not approach zero, ex-\ncept for the PNM case ( β= 1) in which deuterons areobviously not formed. This feature will be more visible\nbelow, when studying the symmetry energy.\nThe results shown in the inset of panel (b) help to dis-\nentangle the effect induced on the stiffness of the EoS,\nowing to the rescaling of the meson couplings at satu-\nration and the changes ascribable to the presence of the\ndeuterons. The curve related to |β|= 1, being character-\nized byXd= 0, demonstrates that, apart from a tiny en-\nhancement of the attraction below saturation (not clearly\nvisible in the figure), a much more repulsive PNM EoS\nis produced for DD2- χd1 beyond saturation as compared\nto DD2. This is the result of the modification induced\non the effective interaction by changing the meson cou-\npling strengths. However, for the curves characterized\nby smaller asymmetry values, such a repulsion is coun-\nterbalanced by the attraction produced by the deuterons.\nAn interplay analogous to the one discussed in the SNM\ncase takes place. In such a way, a reduction of the energy\nper nucleon of ANM might be observed with the DD2-\nχd1 parameterization, in the intermediate density region\nbeyond saturation. This region may actually extend up\nto very large densities, close to nb= 0.9fm−3in SNM.\nOn the other hand, in the asymptotic limit the quasi-\ndeuterons tend to dissolve, so that their extra-binding\nvanishes and the black curves depicted in the inset con-\nverge to the |β|= 1one.\nA further insight into the dependence of the deuteron\nmass fraction and the energy per nucleon on the isospin\nasymmetry might be achieved by looking at Fig. 11.\nThree different values of the total baryon density are\nconsidered: n(low)\nb= 10−4fm−3(panels (a) and (d)),\nn0(panels (b) and (e)) and n(high)\nb= 100fm−3(panels\n(c) and (f)). The standard DD2 parameterization is also\nplotted in the right panels.\nFirst of all, one observes that all the quantities are\nsymmetric with respect to the SNM ( β= 0) case. At\nthe lowest density value considered in Fig. 11,n(low)\nb, the\ndeuteron fraction Xdequals the maximum allowed frac-\ntion and behaves thus like Xd= 1−|β|, for the parame-\nterizations accounting for the presence of the deuterons.\nAs a result, a characteristic triangular shape of Xdand\nE/Ais observed in panels (a) and (d). The energy per\nnucleon does not follow the standard parabolic law which\nis predicted in the deuteron-free DD2 case and reaches\nsmaller values in SNM. It approaches half of the deuteron\nbinding energy in vacuum in the zero-density limit. The\nenergy per nucleon determined with the parameteriza-\ntions accounting for the deuterons coincides with the\nDD2 result only for matter composed exclusively of neu-\ntrons or protons.\nSecondly, a different picture is observed at the satura-\ntion density n0. There, by varying the isospin asymme-\ntry, the deuteron mass fraction departs from the value\nXd= 0.2imposed for β= 0(see panel (b)). A mild de-\npendence of the deuteron mass fraction on βis assessed\nfor the parameterization characterized by a reduced scal-\ning factor χ. A larger sensitivity exists in the case of20\n0 0.5 1 1.5 2 2.5 3\nbaryon number density nb [fm-3]00.10.20.30.40.50.60.70.80.91deuteron mass fraction Xd(a)\n0 0.1 0.2 0.3 0.4 0.50.6 0.7 0.8 0.9 1\nbaryon number density nb [fm-3]0100200300400500energy per nucleon E / A [MeV]β = 0\n|β| = 0.2\n|β| = 0.5\n|β| = 0.7\n|β| = 10.00.51.01.52.0-20020406080\n(E - EDD2)/A [MeV](b)\nFigure 10. Panel(a): Deuteron mass fraction Xdas function of the baryon density, as determined by employin g the DD2- χd1\nparameterization. The results for SNM ( β= 0) are compared with the corresponding ones deduced with diffe rent values of\nthe asymmetry |β|. Panel (b): Energy per nucleon as a function of the baryon den sity, as determined by employing the same\nparameterization and the same asymmetry values as in panel ( a). The inset shows the difference of the energy per nucleon\nbetween the DD2- χd1 and DD2 parameterizations for the same asymmetry values, in a wider range of baryon densities.\n00.20.40.60.81\n00.20.40.60.8deuteron mass fraction Xd\n-1 -0.5 0 0.5 1\nisospin asymmetry β00.20.40.60.8(a)\n(b)\n(c)-1-0.50\n-16-80816energy per nucleon E / A [MeV]DD2\nDD2 - d3\nDD2 - χd1\nDD2 - χd2\nDD2 - χd3\n-1 -0.5 0 0.5 1\nisospin asymmetry β400450500(d)\n(e)\n(f)\nFigure 11. Left panels: Deuteron mass fraction as function o f the isospin asymmetry β, for four selected parameterizations\naccounting for the presence of the deuterons considered in t his work. Right panels: Energy per nucleon as a function of βas\ndetermined by employing the same parameterizations as in th e left panels. For comparison, the curve obtained for the DD2\nparameterization in the deuteron-free case is also shown. T hree different values of the total baryon density are conside red:\nn(low)\nb= 10−4fm−3(panels (a) and (d)), n0(panels (b) and (e)) and n(high)\nb= 100fm−3(panels (c) and (f)).\nthe DD2-d3 parameterization. Let us recall that, for\nthis parameterization, the clusters disappear for SNM\nat density around 0.45 fm−3. Panel (b) shows that the\nclusters may dissolve already at saturation density for\nfiniteβvalues, at least in the case of the DD2-d3 pa-\nrameterization. On the other hand, the presence of the\ndeuteron persists at n0for all asymmetry values except|β|= 1, for parameterizations with χ= 1/√\n2. However,\nthe parabolic dependence of the energy per nucleon on\nthe isospin asymmetry of the DD2 parameterization is\nperfectly reproduced with all the considered parameter-\nizations. This result is clearly shown in panel (e). It\noriginates from the requirement to keep the energy per\nnucleon at saturation constrained, both for SNM and for21\nmatter composed exclusively of neutrons or protons.\nThirdly, it is interesting to discuss what happens at the\nhighest density value, n(high)\nb, considered in Fig. 11, pan-\nels (c) and (f). Here, different results are obtained among\nthe parameterizations characterized by a scaling factor\nχ= 1/√\n2. However, for this density value, the deuterons\nsurvive at all asymmetries only with the DD2- χd1 pa-\nrameterization, for which the largest value was already\npredicted in the SNM case. For the other two parame-\nterizations with χ= 1/√\n2, the clusters dissolve already\nfor|β|values smaller than 1. On the other hand, since\nn(high)\nblies beyond the density at which the cluster dis-\nsolution is predicted in SNM, the deuteron mass fraction\nidentically vanishes in case of the DD2-d3 parameteriza-\ntion. The corresponding asymmetry dependence of the\nenergy per nucleon, which is depicted in panel (f), is then\ndriven only by the modification in the coupling strengths\nwhich was needed to keep the saturation properties well\nconstrained. As in Fig. 8, a slightly larger attraction is\nforeseen with the DD2-d3 parameterization, with respect\nto the DD2 reference case. The opposite happens instead\nwhen the parameterizations with a reduced value of the\nscaling factor are considered. In this case, a stronger\nrepulsion is envisaged, partially mitigated, at least for\nsmall asymmetry values, by the stronger binding pro-\nvided by the presence of deuterons. Quite interestingly,\none observes that the parameterizations plotted by black\ncurves always converge when approaching |β|= 1, where\nXd= 0. A change of the parameterization, which im-\nplies an according change of the deuteron fraction at\nsupra-saturation densities, affects the curvature of the\ndependence of the energy per nucleon on βand thus the\nsymmetry energy. The latter quantity will be studied in\ndetail below.\n2. Symmetry energy and its slope\nIn the present work, the symmetry energy Jis calcu-\nlated as the difference between the energies per nucleon\nin PNM and SNM\nJ(nb) =E\nA/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nβ=1(nb)−E\nA/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nβ=0(nb). (65)\nThe quantity obtained through this equation is identical\nto the symmetry energy calculated from the usual defi-\nnition\nJ(nb) =1\n2∂2(E/A)\n∂β2/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nβ=0(66)\nusing a second derivative of the energy per nucleon with\nrespect to the asymmetry, if E/Afollows a quadratic de-\npendence on β. The density dependence of Jis plotted\nin Fig. 12, panel (a). The inset of panel (a) shows a\nzoom at sub-saturation densities. Once again, the in-\nset highlights the dissimilar behavior in the zero densitylimit, when the presence of the clusters is taken into ac-\ncount or not. It reflects the differences existing in the\nvery dilute regime of the SNM EoS with respect to the\ndeuteron-free case. If clustering is taken into account, th e\nsymmetry energy approaches indeed half of the deuteron\nbinding energy in the zero-density limit in contrast to the\nsimple description without explicit two-particle correla -\ntions. Some differences emerge among the parameteri-\nzations which account for the presence of the deuterons\nbelown0. Apart from the constraint at saturation, no\nrestrictions have been imposed on the density behavior\nofJ.\nConcerning the behavior beyond saturation density,\ndespite the presence of the deuterons, at least up to\n0.45fm−3, the blue curve remains close to the result of\nthe standard DD2 parameterization. Huge differences\nare instead observed in the supra-saturation density re-\ngion when the parameterizations with a reduced value of\nthe deuteron-meson coupling scaling factor is considered.\nFurthermore, the size of the effect depends quite strongly\non the mass fraction of the deuterons. The differences\nvanish in the asymptotic limit, where the deuterons dis-\nappear. The black and blue curves approach the result of\nthe standard DD2 parameterization as the contribution\nof theρmeson to the symmetry energy vanishes due to\nthe suppression of its coupling, leaving the imbalance of\nthe Fermi momenta of the nucleons as the main contri-\nbution to J.\nFinally, it is interesting to look at the density derivative\nof the curves plotted in panel (a). The slope Lof the\nsymmetry energy is numerically calculated here as\nL(nb) = 3nbdJ\ndnb(67)\nand the result is shown in Fig. 12, panel (b). Except for\nthe different behavior in the zero-density limit, the blue\ncurve roughly coincides with the standard DD2 in the\nwhole range of densities. A small kink is only observed\nfor the DD2-d3 parameterization around 0.45 fm−3. This\nkink is related to the disappearance of the deuterons in\nthe SNM case. We recall that this feature was also re-\nsponsible for the emergence of the discontinuity in the\nmatter incompressibility discussed before. A huge double\noscillation around the result obtained with the standard\nDD2 parameterization is observed for the parameteriza-\ntions with a reduced scaling factor χ. The magnitude\nof this oscillation depends again on the deuteron mass\nfraction predicted at supra-saturation densities, thus re -\nflecting the result shown in panel (a) of Fig. 12. The inset\nof panel (b) displays the predictions for the symmetry en-\nergy slope around saturation. In spite of the extremely\nlarge differences in the high-density regime, reasonable\nvalues are obtained at saturation density with all the pa-\nrameterizations considered in this work. These values,\nwhich are collected in Table IV, lie within the range usu-\nally assumed for the slope of the symmetry energy, see,\ne.g., [88] and references therein. The three black curves,22\n0 0.5 1 1.5 2 2.5 3\nbaryon number density nb [fm-3]050100150symmetry energy J [MeV]DD2\nDD2 - d3\nDD2 - χd1\nDD2 - χd2\nDD2 - χd3\n10-410-310-210-110-1100101(a)\n0 0.5 1 1.5 2 2.5 3\nbaryon number density nb [fm-3]-50050100150200250300350symmetry energy slope L [MeV]0.14 0.1650607080 (b)\nFigure 12. Panel (a): Symmetry energy Jas function of the baryon density, as determined through Eq. (65) for four selected\nparameterizations employed in this work. The inset shows a z oom at sub-saturation densities. Panel (b): Slope Lof the\nsymmetry energy as function of the baryon density, as determ ined through Eq. ( 67), for the same parameterizations as in panel\n(a). The inset shows a zoom around the saturation density n0.\nTable IV. Values of the slope of the symmetry energy L0, in\nMeV, at saturation density as derived according to Eq. ( 67),\nfor four selected parameterizations employed in this work.\nThe result for the DD2 is also given for comparison.\nDD2 DD2-d3 DD2- χd1 DD2- χd2 DD2- χd3\nL0[MeV]57.94 56.49 67 .50 67 .50 67 .50\ncrossing each other at saturation, naturally provide the\nsame value. Alternative scenarios manifest for the high-\ndensity behavior of the symmetry energy and its slope.\nThe stiffness of the EoS in the supra-saturation density\nregime turns out to be strongly affected by the value\nof the scaling factor. It is less dependent on the mere\npresence of the deuterons. Thus, as a general feature,\none concludes that smaller values of the scaling factor\ncorrespond to higher stiffness values in the density re-\ngion beyond saturation relevant for the applications of\nthe model.\nVI. CONCLUSIONS AND OUTLOOK\nIn this paper, we have proposed and explored a novel\napproach to embed SRCs within the GRDF, a well-\nestablished phenomenological EDF based on nucleon and\ncluster degrees of freedom. In such a way, we aim to\novercome the inconsistencies between recent experimen-\ntal evidences, which brought to light the existence of\nSRCs, and the predictions of phenomenological models\nderived from mean-field approaches without explicit cor-\nrelations at densities around saturation. Previous gen-\neralisations of these EDFs represent many-body corre-\nlation by cluster which dissolve, by construction, when\nthe nuclear saturation density is approached from below.Within an extended relativistic mean-field model with\ndensity dependent couplings, the idea of this work was\nto effectively account for the existence of SRCs through\nproper in-medium modifications of the cluster properties\naround saturation and above. They are considered as\nquasi-particles with density-dependent binding energies .\nQuasi-deuterons immersed in dense matter are used as\nsurrogate for correlations in this first exploratory step.\nFor the time being, the zero temperature case, where the\ndeuteron fraction is determined by the density of a boson\ncondensate, was addressed.\nSuitable parameterizations of the cluster mass shift\nwere derived, for the first time, for all baryon densi-\nties. The proposed mass shift functions comply with the\navailable constraints and were employed to determine the\ndensity dependence of the quasi-deuteron mass fraction\nat arbitrary isospin asymmetries, thus for symmetric as\nwell as for asymmetric nuclear matter. They were con-\nstrained by microscopic many-body calculations in the\nlow-density limit, by specifying the actual deuteron frac-\ntion at saturation and by assuming a deuteron mass frac-\ntion behavior that respects the boson condensation con-\ndition at higher densities. The effective deuteron fraction\naround saturation was specified by extrapolating the ex-\nperimental results on SRC pairs in nuclei to infinite nu-\nclear matter. Further constraints were moreover imposed\nat supra-saturation densities by the maximum allowed\ndeuteron fraction, compatible with a non-negative value\nof the Dirac effective mass of the nucleons.\nA proper description of well-constrained nuclear mat-\nter quantities at saturation required a refit of the nucleon-\nmeson coupling strengths. An important role of the cou-\npling scaling factor χwas revealed. It rules the coupling\nstrength of the mesons with nucleons bound in the clus-\nters. Such a scaling factor plays actually a primary role\nin the whole analysis developed in this work.23\nThe natural choice was supposed to be that the nucle-\nons inside the deuterons couple to the mesons with the\nsame strength as the unbound nucleons. However, with\nthis choice, the deuteron mass shift is not a monotonic\nfunction of the deuteron mass fraction for all baryon den-\nsities. As a result, the relation between the mass shift and\nthe deuteron mass fraction can not be inverted uniquely\nin all cases. The same holds for any scaling factor χ\nlarger than 1/√\n2. Thus, as possible extreme values of\ntwo well distinct behaviors, two different values of the\nscaling factor, namely χ= 1andχ= 1/√\n2were chosen,\nin the calculations performed in this work. The latter\nvalue is however significantly smaller than the universal\nscaling factor for the cluster-meson coupling strength. A\nvalue smaller than χ= 1was proposed in previous calcu-\nlations of the EoS to take into account in-medium effects\nand to get a good description of the chemical equilibrium\nconstants determined from recent experimental data.\nAs a general feature, our analysis shows that, for χ= 1,\nthe only possible smooth solution for the density depen-\ndence of the deuteron mass fraction implies a sudden dis-\nappearance of the clusters at a density below the one\ncorresponding to the emergence of a pole. In correspon-\ndence of this density, a discontinuity in the matter in-\ncompressibility emerges, analogous to the one observed\nat low density, owing to the disappearance of the pair-\ning correlations and indicating the emergence of a sec-\nond order phase transition. The analogy between the\nbehavior of pairing and SRCs deserves however further\ninvestigation. For χ= 1, the density where the pole\nemerges is located around three times the saturation den-\nsity. However, for more realistic values of the scaling\nfactor, the pole is expected to appear at much higher\ndensities, thus much beyond the range that is relevant in\napplications of the model. When the scaling factor value\nχ= 1/√\n2is considered, alternative solutions exist, per-\nmitting smooth functions of the density dependence of\nthe deuteron mass fraction for all densities. Three differ-\nent parameterizations, providing such a smooth behavior\nand characterized by different maximum deuteron frac-\ntion values at supra-saturation densities, were proposed.\nStriking effects on some thermodynamic quantities\nare recognized, owing to the presence of the quasi-\ndeuterons in the neighbourhood of saturation and at\nsupra-saturation densities. In particular, a softening of\nthe SNM EoS is systematically observed with respect to\nthe standard DD2 parameterization, which does not in-\nclude deuteron-like correlations. However, the stronger\nattraction, which is produced by the presence of the\ndeuterons, might be counterbalanced by the repulsion\ndriven by the modified coupling strengths. This delicate\ninterplay is additionally tuned by the value of the scaling\nfactor, which determines then alternative scenarios for\nthe high-density behavior of the symmetry energy and\nits slope. In general, one concludes that smaller values\nof the scaling factor correspond to higher stiffness of the\nEoS in the supra-saturation density regime.\nLast, but not least, it is worthwhile to recall that ouranalysis permits to also recover the correct low-density\nlimit of the EoS. Indeed, at zero-density, both the energy\nper nucleon of SNM and the symmetry energy tend to\nbe equal to one half of the deuteron binding energy in\nvacuum, in contrast to the predictions of standard mean-\nfield models without cluster correlations.\nThe findings of the present study represent a first\nstep to improve the description of nuclear matter and\nits EoS at supra-saturation densities in EDFs by consid-\nering correlations in an effective way. In a next step,\nthe single-particle momentum distributions can be ex-\nplored using proper wave functions of the quasi-deuteron\nin the medium. They have to be derived consistently\nwith the interaction used in the model and will lead to\nprediction of the cluster mass shifts and fractions that\ncan be compared to the suggested forms of the present\nwork. The many-body wave function of a cluster contains\ncorrelated nucleons with a specific momentum distribu-\ntion. Then an imprint on the single-nucleon momentum\ndistribution in nuclear matter is expected, such that a\nhigh-momentum tail develops even at zero temperature,\nas observed in the experimental study of SRCs by nucleon\nknockout with high-energy electrons.\nThe present approach can be generalized to finite tem-\nperatures, where a further change of the single-nucleon\nmomentum distribution arises owing to the thermal\nchange in the distribution functions. Also a momentum\ndependence of the mass shift and a more involved depen-\ndence on the isospin asymmetry might be considered in a\nfuture work, together with the effect of including heavier\nclusters and to investigate their relative importance.\nAs a perspective, we finally aim at investigating the\neffect of SRCs on neutron stars in the EDF framework,\nsimilarly to what was done in some prior studies, see,\ne.g., [89], but our approach is to replace heuristic param-\neterizations of the momentum distributions with more\nmicroscopically founded descriptions.\nMore in general, we aim at achieving a more compre-\nhensive description of correlations and clustering phe-\nnomena, which represents still a challenge from a the-\noretical point of view, despite the importance of these\nfeatures in the widest scope of astrophysical applications\nand for general aspects of reactions dynamics in heavy-\nion collisions.\nACKNOWLEDGMENTS\nThe authors thank Maria Colonna and Gerd Röpke\nfor their comments and suggestions on this work. S. B.\nacknowledges support from the Alexander von Humboldt\nfoundation.24\nAppendix A: Mass fraction derivative of the\ndeuteron mass shift\nThe deuteron mass shift ( 44) is in general a function\nof the baryon density nb, the asymmetry β, the deuteron\nfractionXd, and the temperature T. In this section, the\nderivative of ∆m(high)\ndwith respect to Xdis derived for\nconstant nbandβatT= 0. In a first step, the derivative\nof the nucleon effective chemical potential with respect\ntoXdis expressed as\n∂µ∗\nq\n∂Xd/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nnb,β=1\nµ∗q/parenleftBigg\nkq∂kq\n∂Xd/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nnb,β+m∗\nq∂m∗\nq\n∂Xd/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nnb,β/parenrightBigg\n(A1)\nusing Eq. ( 35). The derivative of the Fermi momentum\nof a nucleon q=n,pis found with help of the relation\n∂n(v)\nq\n∂Xd/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nnb,β=−nb\n2=3n(v)\nq\nkq∂kq\n∂Xd/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nnb,β(A2)\nfor the vector density ( 19). Using Eqs. ( A1) and ( A2),\nthe derivative of the scalar density of the nucleons can\nbe written as\n∂n(s)\nq\n∂Xd/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nnb,β=−nb\n2m∗\nq\nµ∗q+fq∂m∗\nq\n∂Xd/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nnb,β(A3)\nwith the factor\nfq= 3/parenleftBigg\nn(s)\nq\nm∗q−n(v)\nq\nµ∗q/parenrightBigg\n(A4)\nafter several steps of recasting the individual contribu-\ntions. Eq. ( A3) contains again the derivative of the ef-\nfective mass that assumes the simple form\n∂m∗\nq\n∂Xd/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nnb,β=−Cσ∂nσ\n∂Xd/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nnb,β(A5)\nbecauseCσdepends only on nb. With the derivatives\n∂n(v)\nd\n∂Xd/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nnb,β=∂n(s)\nd\n∂Xd/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nnb,β=nb\n2(A6)\nof the deuteron densities, the derivative of the σmeson\nsource density ( 11) is found as\n∂nσ\n∂Xd/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nnb,β=nb\n1+(fn+fp)CσUd (A7)\nwith\nUd=χdσ−m∗\nnuc\n2/parenleftbigg1\nµ∗n+1\nµ∗p/parenrightbigg\n(A8)whereas\n∂nω\n∂Xd/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nnb,β=−nb(1−χdω) (A9)\nfor the source density of the ωmeson. Finally, the deriva-\ntive of the mass shift with respect to the mass fraction is\nobtained in the compact form\n∂∆m(high)\nd\n∂Xd/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nnb,β(A10)\n=/bracketleftbigg2Cσ\n1+(fn+fp)CσU2\nd−π2\n2µ∗nkn−π2\n2µ∗pkp/bracketrightbigg\nnb\nwith a contribution from the σmeson and kinetic terms.\nAppendix B: Density derivative of deuteron mass\nshift\nIn this section, the derivative of ∆m(high)\ndwith respect\ntonbis derived for an arbitrary function Xd(nb)and\nconstant βatT= 0. It requires again several steps.\nFirst, the derivatives of the source densities have to be\ndetermined. For the ωmeson one finds\n∂nω\n∂nb/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nβ= 1−(1−χdω)Yd (B1)\nwith the quantity\nYd=∂(nbXd)\n∂nb/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nβ=Xd+nb∂Xd\n∂nb/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nβ(B2)\nthat contains the derivative of the deuteron mass frac-\ntion. For the σmeson, the calculation is more involved.\nHere, the relations\n∂nσ\n∂nb/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nβ= (fn+fp)∂m∗\nnuc\n∂nb/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nβ+χdσYd (B3)\n+m∗\nnuc\n2µ∗n(1+β−Yd)+m∗\nnuc\n2µ∗p(1−β−Yd)\nwith the factor fqdefined in Eq. ( A4) and\n∂m∗\nnuc\n∂nb/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nβ=−Cσ∂nσ\n∂nb/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nβ−C′\nσnσ (B4)\nfor the derivative of the effective nucleon mass can be\ncombined to obtain the form\n∂nσ\n∂nb/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nβ= [1+(fn+fp)Cσ]−1(B5)\n[−(fn+fp)C′\nσnσ+χdσYd\n+m∗\nnuc\n2µ∗n(1+β−Yd)+m∗\nnuc\n2µ∗p(1−β−Yd)/bracketrightbigg25\nwith an explicit dependence on Yd. In the next step, the\nderivative of the mass shift assumes the form\n∂∆m(high)\nd\n∂nb/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nβ(B6)\n=kn\nµ∗n∂kn\n∂nb/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nβ+kp\nµ∗p∂kp\n∂nb/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nβ+/parenleftbigg1\nµ∗n+1\nµ∗p/parenrightbigg\nm∗\nnuc∂m∗\nnuc\n∂nb/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nβ\n+2(1−χdω)/parenleftBigg\nCω∂nω\n∂nb/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nβ+C′\nωnω/parenrightBigg\n+2χdσ/parenleftBigg\nCσ∂nσ\n∂nb/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nβ+C′\nσnσ/parenrightBigg\nwith\n∂kn\n∂nb/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nβ=kn\n3n(v)\nn/parenleftbigg1+β−Yd\n2/parenrightbigg\n(B7)\nand\n∂kp\n∂nb/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nβ=kp\n3n(v)\np/parenleftbigg1−β−Yd\n2/parenrightbigg\n. (B8)\nUsing the expressions ( B1) and ( B5), the final result can\nbe expressed in compact form as\n∂∆m(high)\nd\n∂nb/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nβ=Wd−ZdYd (B9)\nwith the auxiliary quantities\nZd=π2\n2µ∗nkn+π2\n2µ∗pkp+2(1−χdω)2Cω(B10)\n−2Cσ\n1+(fn+fp)CσU2\nd\nWd=π2\n2µ∗nkn(1+β)+π2\n2µ∗pkp(1−β) (B11)\n+2(1−χdω)(Cω+C′\nωnω)\n+2\n1+(fn+fp)CσUd\n/bracketleftbigg\nC′\nσnσ+Cσm∗\nnuc\n2/parenleftbigg1+β\nµ∗n+1−β\nµ∗p/parenrightbigg/bracketrightbigg\nandUdas given in ( A8).\nAppendix C: Conversion of parameters at saturation\nIn order to find the coupling strengths Γσ,0andΓω,0as\nwell as the deuteron mass shift and its density derivative\nat saturation, a step-by-step procedure can be followed.\nThese quantities are determined as soon as the satura-\ntion density n0, the binding energy per nucleon B0, the\neffective nucleon mass m∗\nnuc,0and deuteron fraction Xd,0\nof SNM are specified.In a first step, the scalar and vector densities\nn(s)\nd,0=n(v)\nd,0=n0Xd,0\n2(C1)\nof the deuteron and the total vector density\nn(v)\nnuc,0=n0(1−Xd,0) (C2)\nof the nucleons are immediately obtained from n0and\nXd,0in SNM. Then the Fermi momentum\nknuc,0=/bracketleftbigg6π2\ngnucn(v)\nnuc,0/bracketrightbigg1/3\n(C3)\nwith degeneracy factor gnuc= 4and the effective chemi-\ncal potential\nµ∗\nnuc,0=/radicalBig\nk2\nnuc,0+/parenleftbig\nm∗\nnuc,0/parenrightbig2(C4)\nallow to calculate the scalar density\nn(s)\nnuc,0 (C5)\n=gnucm∗\nnuc,0\n4π2/bracketleftBigg\nknuc,0µ∗\nnuc,0−/parenleftbig\nm∗\nnuc,0/parenrightbig2lnknuc,0+µ∗\nnuc,0\nm∗\nnuc,0/bracketrightBigg\nusing the effective nucleon mass m∗\nnuc,0. Then the source\ndensities\nnσ,0=n(s)\nnuc+2χn(s)\nd,0(C6)\nand\nnω,0=n(v)\nnuc+2χn(v)\nd,0(C7)\nwith the deuteron-meson coupling scaling factor χare\nfound and the pressure contribution\npnuc,0=1\n4/bracketleftBig\nµ∗\nnuc,0n(v)\nnuc,0−m∗\nnuc,0n(s)\nnuc,0/bracketrightBig\n(C8)\nof the nucleons can be calculated immediately.\nIn the next step, the effective nucleon mass determines\nthe scalar potential\nSnuc,0=mnuc−m∗\nnuc,0 (C9)\nof the nucleons and thus the scalar coupling\nCσ,0=Snuc,0\nnσ,0(C10)\nand finally the coupling strength\nΓσ,0=mσ/radicalbig\nCσ,0 (C11)\nof theσmeson. The binding energy per nucleon B0gives\nthe chemical potential\nµnuc,0=mnuc−B0 (C12)26\nat saturation and then the vector potential\nVnuc,0=µnuc,0−µ∗\nnuc,0 (C13)\nof the nucleons. The latter quantity can be expressed in\ngeneral as\nVnuc,0=Cω,0nω,0+Cρ,0nρ,0+U(r)\n0 (C14)\nwith the auxiliary quantity\nU(r)\n0=V(r)\n0+W(r)\n0 (C15)\nthat also appears in the total pressure\nP0=pnuc,0 (C16)\n+1\n2/parenleftbig\nCω,0n2\nω,0+Cρ,0n2\nρ,0−Cσ,0n2\nσ,0/parenrightbig\n+U(r)\n0n0.\nFor SNM, however, the ρ-meson contribution does not\nappear, since nρis identically zero. The two equations\n(C14) and ( C16) allow to solve for the ωcoupling\nCω,0=/parenleftbig\n2nω,0n0−n2\nω,0/parenrightbig−1(C17)\n/parenleftbig\n2pnuc,0+2Vnuc,0n0−Cσ,0n2\nσ,0/parenrightbig\nand further the coupling strength\nΓω,0=mω/radicalbig\nCω,0 (C18)\nusingP0= 0. With known Cσ,0andCω,0, their deriva-\ntivesC′\nσ,0andC′\nω,0can be determined using the same\nfunctional density dependence of the couplings as in the\nreference parameterization. Thus also the rearrangement\ncontribution\nV(r)\n0=1\n2/parenleftbig\nC′\nω,0n2\nω,0+C′\nρ,0n2\nρ,0−C′\nσ,0n2\nσ,0/parenrightbig\n(C19)\nis given.\nFinally, from Eqs. ( C14) and ( C15) one finds\nW(r)\n0=Vnuc,0−Cω,0nω,0−V(r)\n0 (C20)\nand the deuteron mass shift derivative\nd∆md\ndnb/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nn0=W(r)\n0\nn(s)\nd,0(C21)\nat saturation. The deuteron mass shift itself is deter-\nmined as\n∆md,0=Bd+2/parenleftbig\nµ∗\nnuc,0−m∗\nnuc,0/parenrightbig\n(C22)\n+2(1−χdω)Cωnω−2(1−χdσ)Cσnσ\nfrom the condensation condition with the binding energy\nof the deuteron in vacuum Bd.\nThe rescaling of the σandωcoupling strengths induces\na modification of the energy per nucleon in PNM at sat-\nuration and thus of the symmetry energy. Within theparabolic approximation, the symmetry energy at satu-\nration is indeed given by\nJ0=E\nA/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nn0,β=1−E\nA/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nn0,β=0=E\nA/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nn0,β=1+B0.(C23)\nThen, constraining the value of J0implies a constraint\nonE/Aof PNM at n0. Taking into account Eq. ( 31), the\ncondition above writes\nµ∗\nn,0+Vn,0−Pn\nn0/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nn0,β=1=mnuc+J0−B0(C24)\nwith the effective chemical potential\nµ∗\nn,0=/radicalBig\nk2\nn,0+/parenleftbig\nm∗\nn,0/parenrightbig2(C25)\nof the neutron at the saturation density n0. The effective\nmass of the neutron m∗\nn,0=mnuc−Γσn(s)\nn,0has to be\ndetermined self-consistently with the scalar density n(s)\nn,0,\ndefined in Eq. ( 34), using the Fermi momentum kn,0=/parenleftbig\n3π2n0/parenrightbig1/3of the neutron. Since there are no deuterons\nin PNM, the vector potential of the neutron is given by\nVn,0=Cω,0nω+Cρ,0nρ (C26)\n+1\n2/parenleftbig\nC′\nω,0n2\nω+C′\nρ,0n2\nρ−C′\nσ,0n2\nσ/parenrightbig\nand the pressure assumes the simple form\nPn(n0) =1\n4/parenleftBig\nµ∗\nn,0n(v)\nn,0−m∗\nn,0n(s)\nn,0/parenrightBig\n(C27)\n+1\n2/bracketleftbig\nDω,0n2\nω+Dρ,0n2\nρ−Dσ,0n2\nσ/bracketrightbig\nwithnω=nρ=n(v)\nn,0=n0andnσ=n(s)\nn,0. The rescaled\ncouplings Cω,0,Cσ,0,C′\nω,0,C′\nσ,0,Dω,0, andDσ,0, c.f., Eq.\n(39), at saturation are already known and thus Cρ,0can\nbe deduced from\nCρ,0=2\nn0/bracketleftbigg\nmnuc+J0−B0−3\n4µ∗\nn,0 (C28)\n−1\n4m∗\nn,0n(s)\nn,0\nn0−1\n2Cω,0nω−1\n2n0Cσ,0n2\nσ/bracketrightBigg\nand, finally,\nΓρ,0=mρ/radicalbig\nCρ,0 (C29)\nfor the coupling of the ρmeson at saturation.27\nAppendix D: Analytical expressions for the mass\nshift parameters\nThe following analytical expressions permit to calcu-\nlate the parameters canddappearing in Eq. 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Indeed, for instance the ground state of a system can be extracted\nfrom the minimization of the energy over the set of two-parti cle density ma-\ntrices coming from fermionic wavefunctions [2,3,16], and k nowing a simple\ncharacterization of this set would enable to reduce the exac t many-body\nproblem to a low-dimensional problem. Those issues are call ed the pure-\nstate representability problems. For one-body densities i t was solved by\nHarriman in [7], using determinantal functions, and was lat er put on rigor-\nous grounds by Lieb in [13, Theorem 1.2]. Then the problem of p ure-state\nrepresentability of current densities was addressed by Lie b and Schrader\nin [14], solved with explicit bounds when the velocity field i s curl-free. The\ncase of pure-state representability of magnetizations was solved by Gontier\nin [5,6], also with explicit bounds. Finally, see [2] and [15 , Theorem 3.2]\nfor a solution of the mixed state representability of one-bo dy density matri-\nces. On the contrary, representing pair densities [1,19,20 ] and two-body (or\nq-body) density matrices [2,9,17,23] with N-body fermionic states are im-\nportant but still open problems, simple necessary and suffici ent conditions\nare not known.\nIn this document, we address the corresponding problem of st ate rep-\nresentability in the quantum thermodynamic ensembles. Mor e precisely,\nwe build mixed states having prescribed density and entropy (ρ,S) in the\ncanonical case, and having prescribed density, mean number of particles and\nentropy ( p,N,S) in the grand-canonical case, for both bosons and fermions,\nprovidingboundsonthekineticenergyofthosemixedstates . Finally, wecan\napply those results to define internal Levy-Lieb functional s in the Thermal\nDFT [21] configuration, and show some basic properties of tho se functionals.\nDate: March 31, 2022.\n12 L. GARRIGUE\n2.Main result\n2.1.Definitions. We useN=N∪{0}as theset of integers. For any Hilbert\nspaceH, we denote by L(H) the set of bounded operators of H, the first\nSchatten space is defined as the set S1(H) ={Γ∈ L(H),Tr|Γ|<+∞}\nendowed with the trace norm, and it is a Banach space. We consi der the\nentropyS(Γ) :=−TrΓlnΓ, wherethetrace is taken on thenatural operator\nHilbert space on which Γ is defined. The set of canonical state s is\nSN\ncan:=/braceleftbig\nΓ = Γ∗/greaterorequalslant0/vextendsingle/vextendsingleΓ∈S1/parenleftbig\nL2\na/s/parenleftbig\nRdN/parenrightbig/parenrightbig\n,\nTrΓ = 1,Tr/summationtextN\ni=1(−∆i)Γ<+∞/bracerightbig\n,\nwhereL2\na(RdN) :=∧NL2(Rd) is the set of fermionic N-body states and\nL2\ns(RdN) :=∨NL2(Rd) is the set of bosonic N-body state. The set of grand-\ncanonical states is\nSN\ngc:=/braceleftig\nΓ = Γ∗/greaterorequalslant0/vextendsingle/vextendsingleΓ∈S1/parenleftbig\n⊕+∞\nn=0L2\na/s/parenleftbig\nRdn/parenrightbig/parenrightbig\n,TrΓ = 1,TrKΓ<+∞/bracerightig\nwhere⊕+∞\nn=0L2\na/s/parenleftbig\nRdn/parenrightbig\nis the fermionic/bosonic Fock space, and where\nK:=+∞/circleplusdisplay\nn=0n/summationdisplay\nj=1(−∆j)\nisthegrand-canonical kineticenergyoperator. Alsodenot ingbyΓthekernel\nof the mixed states Γ, we define the one-body density of a canon ical state\nΓNby\nρΓ(x) :=N/integraldisplay\nRd(N−1)ΓN(x,x2,...,x N;x,x2,...,x N)dx2···dxN.\nThe density ρΓof a grand-canonical state Γ = ⊕+∞\nn=0Γn, and its normalized\ndensitypΓare defined by\nρΓ:=+∞/summationdisplay\nn=1ρΓn, p Γ:=ρΓ/integraltext\nRdρΓ.\nA classical construction from Harriman and Lieb [7,13] show s that a den-\nsityρ∈L1(Rd,R+) is the one-body density of a physical pure wavefunction\nΨ∈H1\na(RdN) :=/parenleftbig\nL2\na∩H1/parenrightbig\n(RdN),/integraltext\nRdN|Ψ|2= 1 if and only if ρ∈ IN,\nwhere\nIN:=/braceleftbigg\nρ∈L1(Rd,R)/vextendsingle/vextendsingle/vextendsingleρ/greaterorequalslant0,√ρ∈H1(Rd),/integraldisplay\nRdρ=N/bracerightbigg\n.\nMoreover, there is an explicit kinetic energy bound for Ψ, be ing\n/integraldisplay\nRdN|∇Ψ|2/lessorequalslant(4π)2N2/integraldisplay\nRd|∇√ρ|2.\nIn Thermal DFT [21], which is an approach of quantum thermody namics,\nthe corresponding representability problem consists in ch aracterizing the set\nof pairs ( ρ,S)∈L1(Rd)×R+coming from mixed states. In this case, theREPRESENTABILITY OF ENTROPY-DENSITY 3\nset of admissible density-entropy pairs is\nIN\ncan:={(ρ,S)∈L1(Rd)×R+/vextendsingle/vextendsingle∃ΓN∈ SN\ncan,ρΓN=ρ,\nS(ΓN) =S,Tr/summationtextN\ni=1(−∆i)ΓN<+∞},\nfor the canonical setting, and\nIgc:={(p,N,S)∈/parenleftbig\nL1(Rd)∩{∫p= 1}/parenrightbig\n×(R+\\{0})×R+/vextendsingle/vextendsingle∃Γ∈ Sgc,\npΓ=p,TrNΓ =N,S(Γ) =S,TrKΓ<+∞},\nfor the grand-canonical setting, where N=⊕+∞\nn=0nis the particle number\noperator. The mean number of particles is\nN(Γ) := Tr NΓ =/integraldisplay\nRdρΓ.\nFinally we define the N-body Hamiltonian\nHN(v) :=N/summationdisplay\ni=1(−∆i+v(xi))+/summationdisplay\n1/lessorequalslanti<j/lessorequalslantNw(xi−xj),\nacting on canonical states of the N-particle sector, and\nH(v) :=+∞/circleplusdisplay\nn=0Hn(v)\nacting on the Fock space. The function vis the electric potential while w\nis the interaction potential. We will always assume that v,w∈Lq\nloc(Rd),\nwhere\nq= 1 ford= 1, q > 1 ford= 2, q =d\n2ford/greaterorequalslant3.(1)\n2.2.Representability of density-entropy pairs.\nTheorem 1 (Representability of ( ρ,S) with canonical states) .Takeρ∈ IN\nandS∈R+. There exists ΓN∈ SN\ncansuch that ρΓN=ρ,S(ΓN) =S, and\nTr/summationtextN\ni=1(−∆i)ΓN/lessorequalslant/parenleftbigg\n1+4π2N2/parenleftig\neS\nN+21\nN+2\nN+ξ/parenrightig2/parenrightbigg/integraltext\nRd/vextendsingle/vextendsingle∇√ρ/vextendsingle/vextendsingle2,\nwhereξ= 1for fermions and ξ= 0for bosons. We have thus an explicit\ncharacterization of representable densities,\nIN\ncan=IN×R+.\nWe then treat the grand-canonical case where first the mean nu mber of\nparticles is not controled.\nTheorem 2 (Representability of ( p,S) with grand-canonical states) .Take\np∈ I1andS∈R+, there exists Γ∈ Sgcsuch that pΓ=p, and\nTrKΓ/lessorequalslant5×212(S+3)2/integraldisplay\nRd|∇√p|2,N(Γ)/lessorequalslant24(S+3),\nboth for fermions and bosons.4 L. GARRIGUE\nFinally, we treat the grand-canonical case where we control S,pandN\nat the same time. Let us define the entropic function\ns(x) :=−xlnx−(1−x)ln(1−x),\non [0,1\n2], which is strictly increasing and spans [0 ,ln2], let us denote by s−1\nthe reciprocal function of s, which is hence defined on [0 ,ln2] and strictly\nincreasing, and for any S∈[0,+∞), define\nξS:=/braceleftigg\ns−1(S) ifS/greaterorequalslantln2\n1 otherwise .(2)\nFor any ε∈(0,1), there is xε∈(0,1) such that x/lessorequalslants(x)/lessorequalslantx1\n1+εfor all\nx∈[0,xε], hencey1+ε/lessorequalslants−1(y)/lessorequalslantyin a right neighborhood of {0}.\nTheorem 3 (Representability of ( p,N,S) with grand-canonical states) .\nTakep∈ I1,S∈R+\\{0}andN∈R+\\{0}. There exists Γ∈ Sgcsuch\nthatN(Γ) =N,pΓ=p,S(Γ) =S, and their kinetic energies are bounded\nby\nTrKΓ/lessorequalslant5π2/parenleftbiggN\nξS+1/parenrightbigg3\neS1\nξS+1\nN\nmax/parenleftbigg\n1,N\nξS/parenrightbigg\n+5\n2\n/integraldisplay\nRd|∇√p|2.(3)\nboth for fermions and bosons, where ξSis defined in (2). We have\nIgc=I1×/parenleftbig\n((R+\\{N∪{0}})×(R+\\{0}))∪((N\\{0})×R+)/parenrightbig\n.(4)\n•WhenN→0 andS/greaterorequalslantln2, the bound (3) diverges like a constant\ntimese2S\nNat leading order.\n•WhenS→0,ξS=s−1(S)→0 and (3) diverges like a constant times/parenleftbig\ns−1(S)/parenrightbig−3, which diverges more slowly than S−1−εfor anyε >0. This is\nnot surprisingsince when N /∈N∪{0}, it is not possibleto build a state with\nthis mean number of particles and with vanishing entropy. Wh enN∈N\nandS= 0, we can take a pure state as in the standard Harriman-Lieb\nconstruction [7,13].\n•WhenS→+∞, (3) diverges like a constant times e2S1+1\nN\nmax(1,N)\n•WhenN→+∞, (3) diverges like a constant times N3.\n2.3.Free energies. To prepare the application of our representability the-\norems, we define the free energies and give some of their prope rties.\nThe energy functionals are\nEN\nv,T(Γ) := Tr HN(v)Γ−TS(Γ),Ev,T(Γ) := Tr H(v)Γ−TS(Γ),\nand the free energies are\nEN\ncan(v,T) := inf\nΓ∈SN\ncanEN\nv,T(Γ), E gc(v,T,µ) := inf\nΓ∈SgcEv−µ,T(Γ).\nFirst,EN\ncan(v,T)isjointlyconcave in( v,T)andEgc(v,T,µ)isjointlyconcave\nin (v,T,µ). They are increasing in vand decreasing in T.\nWe want to manipulate finite energies, so we add a confining pot ential\nv0∈Lq\nloc(Rd) such that ( v0)−∈(Lq+L∞)(Rd) and/integraltext\nRde−(v0)+/T0<+∞REPRESENTABILITY OF ENTROPY-DENSITY 5\nfor some T0∈R+\\{0},qas in (1), but we also could have restricted the\nphysical space to a cube. Now those quantities are finite and c ontinuous.\nProposition 1. Takeqas in(1). For a confining potential v0∈Lq\nloc(Rd)\nandT0∈R+\\{0}suchthat (v0)−∈(Lq+L∞)(Rd)and/integraltext\nRde−(v0)+/T0<+∞,\nthe functionals (v,T)/ma√∫to→EN\ncan(v0+v,T)and(v,T)/ma√∫to→Egc(v0+v,T,µ)\nare locally Lipschitz in vin theLq+L∞norm, and globally Lipschitz in\nT∈[0,T0]\n2.4.Free Levy-Lieb functionals. Forρ∈ IN,p∈ I1,N∈R+, the free\nLevy-Lieb functionals are\nFN,T\ncan(ρ) := inf\nΓ∈SN\ncanρΓ=ρ/parenleftbig\nTrHN\n0Γ−TS(Γ)/parenrightbig\n, FT\ngc(Np) := inf\nΓ∈Sgc\nρΓ=Np(TrH0Γ−TS(Γ)),\n(5)\nwhich elable to recover the free energies by Legendre transf orms. On IN,\nwe haveFN,T\ncan=FT\ngc. Some properties of those Levy-Lieb functionals at zero\ntemperature were recently studied in [11,12].\n2.5.Internal Levy-Lieb functionals. There is a second way of defining\nLevy-Liebfunctionals, intermsininternalquantities onl y, thatisnoexternal\nfield is involved. Their definition is an application of our re presentability\ntheorems.\nFor (ρ,S)∈ IN×R+, and (p,S,N)∈ Igcin the grand-canonical case, we\ndefine\nFN\ncan(ρ,S) := inf\nΓ∈SN\ncan\n(ρΓ,S(Γ))=(ρ,S)TrHN\n0Γ, F gc(Np,S) := inf\nΓ∈Sgc\n(ρΓ,S(Γ))=(Np,S)TrH0Γ,\n(6)\nand call them internal Levy-Lieb functionals. They are boun ded from below\n(uniformly in ρ) by a constant depending only on w. The internal Levy-Lieb\nfunctionals (6) enable to recover the free energies by Legen dre transforms,\nEN\ncan(v,T) = inf\n(ρ,S)∈IN×R+/parenleftbigg\nFN\ncan(ρ,S)+/integraldisplay\nRdρv−TS/parenrightbigg\n,\nEgc(v,T,µ) = inf\n(p,S,N)∈Igc/parenleftbigg\nFgc(Np,S)+N/integraldisplay\nRdp(v−µ)−TS/parenrightbigg\n.\nThey are finite by the representability results Theorem 1 and Theorem 3,\nand the next result shows that they have a minimum.\nProposition 2. If there are constants c,ε,R > 0such that1BR(x)ρ(x)/lessorequalslant\nc|x|−d−ε,1BR(x)p(x)/lessorequalslantc|x|−d−ε, then the infima FN\ncan(ρ,S)andFgc(Np,S)\nare minima.\nWe remark that we could also define Lieb functionals, similar ly as in [13,\n(3.17)], they are convex and weakly lower-semicontinuous, and a reconstruc-\ntion formula similar as [13, (3.20)] also holds.\nWe recall that the map ( v,T)/ma√∫to→(ρΓ,SΓ), where Γ is the Gibbs state\nof (v,T), is injective, this is the Hohenberg-Kohn theorem for syst ems at\npositive temperature, see [18] and [4, Theorem 4.1], and thi s exposes the6 L. GARRIGUE\n(v,T)−(ρ,S) duality. The information of the internal quantities ( ρ,S)\nenables to recover the information of the external fields ( v,T).\n3.Proofs\nWe define the kinetic energy T(Ψ) of pure states Ψ, the kinetic en-\nergyT(ΓN) of canonical states Γ N, and the kinetic energy T(Γ) of grand-\ncanonical states Γ as\nT(Ψ) :=/integraldisplay\nRdN|∇Ψ|2, T(ΓN) := Tr/summationtextN\ni=1(−∆i)ΓN, T(Γ) := Tr KΓ.\n3.1.Proof of Theorem 1.\nFermions in the canonical case.\nWe use the construction of Harriman and Lieb [7,13], which co rresponds to\nthe pure case S= 0. We recall Harriman’s function\nf(x1) :=2π\nN/integraldisplayx1\n−∞ds/integraldisplay\nRd−1dx2···dxdρ(s,x2,...,xd),\nwherex= (x1,...,xd)∈Rd. The one-particle orbitals at stake are\nφk(x) :=/radicalbigg\nρ(x)\nNeikf(x1), (7)\nfork∈Z, which is an orthonormal familly, /a\\}b∇ack⌉tl⌉{tφℓ,φm/a\\}b∇ack⌉t∇i}ht=δℓm, and we will call\nkthe momentum of φk. Then it is proved in [13, Proof of Theorem 1.2] that\n/integraldisplay\nRd|∇φk|2/lessorequalslant1\nN/parenleftbig\n1+16π2k2/parenrightbig/integraldisplay\nRd|∇√ρ|2,\nandthat/integraltext\nRd|∇Ψk|2increaseswith k2. Wedefinetheorderedset(Ψ k)k∈N\\{0}\nsuch that\n{Ψk}k∈N\\{0}=\n\n/logicalanddisplay\nj1<···<jNφji/vextendsingle/vextendsingle/vextendsinglej1,...,jN∈Z\n\n, (8)\nand such that/integraltext\nRdN|∇Ψk|2/lessorequalslant/integraltext\nRdN|∇Ψℓ|2fork < ℓ. The Ψ k’s form an\northonormal familly. We call\nJα(k) := max\n1/lessorequalslanti/lessorequalslantN\nΨk=∧N\ni=1φji|ji|\nthe maximum absolute value momentum of Ψ k, where α∈ {even,odd}.\nJevendenotes the case when Nis even, and Jodddenotes the case when N\nis odd, and we write Jα. We have\n/integraldisplay\nRdN|∇Ψk|2=/summationdisplay\na∈{j1,...,jN}\nΨk=∧N\ni=1φji/integraldisplay\nRd|∇φa|2/lessorequalslantN/integraldisplay\nRd/vextendsingle/vextendsingle∇φJα(k)/vextendsingle/vextendsingle2\n/lessorequalslant/parenleftbig\n1+16π2Jα(k)2/parenrightbig/integraldisplay\nRd|∇√ρ|2,\nand since we will use the pure states Ψ k, we will need an upper bound on\ntheir kinetic energy, hence we now search for an upper bound o nJα.REPRESENTABILITY OF ENTROPY-DENSITY 7\nForN= 2neven,n∈N, we can for instance take\nΨ1:=φ−(n−1)∧···∧φn\nΨ2:=φ−n∧φ−(n−2)∧···∧φn\nΨ3:=φ−n∧φ−(n−1)∧φ−(n−3)∧···∧φn\n...\nΨN:=φ−n∧···∧φn−2∧φn\nΨN+1:=φ−n∧···∧φn−1,\nwhich amounts to choose 2 nnumbers in a list of 2 n+ 1 elements, to keep\nJeven(k) =n. Hence\nJeven(1) =Jeven(2) =···=Jeven(N+1) =n.\nThen we can use orbitals of higher momentum and take for insta nce\nΨN+2:=φ−(n+1)∧···∧φn−2\nandJeven(N+2) =n+1. For ℓandnfixed, choosing all the combinations\nofji’s such that Jeven(k)/lessorequalslantn+ℓamounts to choose 2 nnumbers among\n2(n+ℓ) + 1 = N+ 2ℓ+ 1 ones. For p∈N∪ {0}andq∈Z, we use the\nnotation/parenleftbigg\nq\np/parenrightbigg\n=q!\np!(q−p)!whenp/lessorequalslantq,/parenleftbigg\nq\np/parenrightbigg\n= 0 when q < porq <0.(9)\nSince we sorted the combinations such that Jevenis increasing, we have\nJeven/parenleftbigg/parenleftbigg\nN+2ℓ−1\nN/parenrightbigg\n+1/parenrightbigg\n=···=Jeven/parenleftbigg/parenleftbigg\nN+2ℓ+1\nN/parenrightbigg/parenrightbigg\n=n+ℓ=/floorleftbiggN\n2/floorrightbigg\n+ℓ,\nfor anyℓ∈N∪{0}. Moreover, for any N,ℓ∈N,\n(2ℓ)N\nN!/lessorequalslant/parenleftbigg\nN+2ℓ\nN/parenrightbigg\n/lessorequalslant/parenleftbigg\nN+2ℓ+1\nN/parenrightbigg\n. (10)\nWe defined JαonN\\{0}but we extend it to all of R+such that it remains\nincreasing, hence for y∈R+,\nJeven/parenleftigg\n(2y)N\nN!/parenrightigg\n/lessorequalslantJeven/parenleftigg\n(2⌈y⌉)N\nN!/parenrightigg\n/lessorequalslantJeven/parenleftbigg/parenleftbigg\nN+2⌈y⌉+1\nN/parenrightbigg/parenrightbigg\n/lessorequalslant/floorleftbiggN\n2/floorrightbigg\n+⌈y⌉/lessorequalslantN\n2+y+1.\nAs forN= 2n+1 odd, we can take\nΨ1:=φ−n∧···∧φn\nsoJodd(1) =n. As in the even case, choosing all the combinations of\nji’s such that Jodd(k)/lessorequalslantn+ℓamounts to choose 2 n+ 1 numbers among\n2(n+ℓ)+1 =N+2ℓones, hence\nJodd/parenleftbigg/parenleftbigg\nN+2ℓ−2\nN/parenrightbigg\n+1/parenrightbigg\n=···=Jodd/parenleftbigg/parenleftbigg\nN+2ℓ\nN/parenrightbigg/parenrightbigg\n=n+ℓ=/floorleftbiggN\n2/floorrightbigg\n+ℓ,8 L. GARRIGUE\nfor anyℓ∈N∪{0}. By the first inequality of (10), and following the same\nreasoning as in the even case, we also have\nJodd/parenleftigg\n(2y)N\nN!/parenrightigg\n/lessorequalslantN\n2+y+1,\nfor anyy∈R+. We now invert the formula\nt=(2y)N\nN!⇐⇒y=1\n2t1\nNN!1\nN,\nto deduce that for any t∈R+andα∈ {even,odd},\nJα(t)/lessorequalslant1\n2t1\nNN!1\nN+N\n2+1/lessorequalslant1\n2N/parenleftbigg\nt1\nN+1+2\nN/parenrightbigg\n,\nand in particular it holds for t∈N\\{0}. Remark that ( N!)1\nN∼N→+∞\ne−1N. This enables us to get an upper bound on the kinetic energy of the\nprevious antisymmetric states\nT(Ψk)/lessorequalslant/parenleftbig\n1+16π2Jα(k)2/parenrightbig/integraldisplay\nRd|∇√ρ|2(11)\n/lessorequalslant/parenleftbigg\n1+4π2N2/parenleftig\nk1\nN+1+2\nN/parenrightig2/parenrightbigg/integraldisplay\nRd|∇√ρ|2.(12)\nForθ∈/bracketleftbig\n0,π\n2/bracketrightbig\nandM∈N\\{0}, we define the mixed state\nΓN(θ,M) :=(cosθ)2\nMM/summationdisplay\nk=1|Ψk/a\\}b∇ack⌉t∇i}ht/a\\}b∇ack⌉tl⌉{tΨk|+(sinθ)2|ΨM+1/a\\}b∇ack⌉t∇i}ht/a\\}b∇ack⌉tl⌉{tΨM+1|,(13)\nwhichbelongsto SN\ncan, hasdensity ρΓN(θ,M)=ρandhasfinitekinetic energy.\nWe choose M=/floorleftbig\neS/floorrightbig\n+1. The entropy of the state is\nS(ΓN(θ,M)) =−cos2θlncos2θ\nM−sin2θlnsin2θ.\nIt satisfies S(ΓN(0,M)) = lnM > S andS/parenleftbig\nΓN/parenleftbigπ\n2,M/parenrightbig/parenrightbig\n= 0. Since the\nmapθ/ma√∫to→S(ΓN(θ,M)) is continuous, there is some value θ0∈/bracketleftbig\n0,π\n2/bracketrightbig\nfor\nwhich\nS(ΓN(θ0,M)) =S.\nEventually, we have a bound on the kinetic energy\nT(ΓN(θ,M)) =(cosθ)2\nMM/summationdisplay\nk=1T(Ψk)+(sinθ)2T(ΨM+1)\n/lessorequalslantT(ΨM+1)\n/lessorequalslant/parenleftigg\n1+4π2N2/parenleftbigg/parenleftbig/floorleftbig\neS/floorrightbig\n+2/parenrightbig1\nN+1+2\nN/parenrightbigg2/parenrightigg/integraldisplay\nRd|∇√ρ|2\n/lessorequalslant/parenleftbigg\n1+4π2N2/parenleftig\neS\nN+1+21\nN+2\nN/parenrightig2/parenrightbigg/integraldisplay\nRd|∇√ρ|2.(14)\nBosons in the canonical case.\nFor bosons, the construction that we give does not depend on t he parity ofREPRESENTABILITY OF ENTROPY-DENSITY 9\nN. We consider the same one-particle orbitals as previously b ut we define\nnewN-particle functions using the symmetrized states\nΨ1:=N/logicalordisplay\ni=1φ0,Ψ2:=φ1N−1/logicalordisplay\ni=1φ0,Ψ3:=φ−1N−1/logicalordisplay\ni=1φ0,\nΨ4:=φ1∨φ1N−2/logicalordisplay\ni=1φ0,Ψ5:=φ−1∨φ−1N−2/logicalordisplay\ni=1φ0,Ψ6:=φ−1∨φ1N−2/logicalordisplay\ni=1φ0,\netc, and we fill the Norbitals by φi’s such that, as in the case of fermions,\nT(Ψk)/lessorequalslantT(Ψm) fork < m.\nWe denote by s(m,N) the number of ways to fill Nindistinguishable\nboxes with mtypes of indistinguishable balls. For instance when the thr ee\ntypes of balls are represented by 1, 2 and 3 and when there are 3 boxes, the\nconfigurations 221 ,212,122 are the same, and one can compute s(3,3) = 10.\nA configuration is exactly given by its numbers of balls of typ e 1, of balls\nof type 2, etc, and its number of balls of type m. We can thus order a\nconfiguration by placing all its 1’s on the left, then all its 2 ’s on the right of\nthe 1’s, etc, ending by all its m’s. To count s(m,N) we thus only need to\nplacem−1 separators between the different types, so between the Nboxes,\nwhich is the same as choosing m−1 balls among N+m−1 boxes. We\ndeduce that\ns(m,N) =/parenleftbigg\nN+m−1\nm−1/parenrightbigg\n=/parenleftbigg\nN+m−1\nN/parenrightbigg\n.\nIfJ(k) is defined as the maximum absolute value momentum of orbital s\ncomposing Ψ k, as previously, then J(1) = 0, and for ℓ∈N\\{0}, we have\nJ(s(2ℓ−1,N)+1) = ···=J(s(2ℓ+1,N)) =ℓ\nor equivalently,\nJ/parenleftbigg/parenleftbigg\nN+2ℓ−1\nN/parenrightbigg\n+1/parenrightbigg\n=···=J/parenleftbigg/parenleftbigg\nN+2ℓ\nN/parenrightbigg/parenrightbigg\n=ℓ (15)\nWe choose any extension of JtoR+such that it remains increasing, and\nsimilarly as previously, we find that\nJ/parenleftbigg(2y)N\nN!/parenrightbigg\n/lessorequalslanty+1,\nthus for any k∈N\\{0},\nJ(k)/lessorequalslant1\n2k1\nNN!1\nN+1/lessorequalslant1\n2N/parenleftbigg\nk1\nN+2\nN/parenrightbigg\n,\nWe take the same mixed state Γ N(θ,M) as in (13), and the same Mand\nθ0as previously, so the only difference with fermions is that for the kinetic\nenergy bound we obtain\nT(ΓN)/lessorequalslant/parenleftbigg\n1+4π2N2/parenleftig\neS\nN+21\nN+2\nN/parenrightig2/parenrightbigg/integraldisplay\nRd|∇√ρ|2,(16)\nwhich is slightly better than the fermionic case.10 L. GARRIGUE\n3.2.Proof of Theorem 2.\nFermions in the grand-canonical case.\nWe fix some normalized density p∈ I1, that is/integraltext\nRdp= 1. From (7), we have\n/integraldisplay\nRd|∇φk|2/lessorequalslant/parenleftbig\n1+16π2k2/parenrightbig/integraldisplay\nRd|∇√p|2.\nIn the case of fermions in the grand-canonical ensemble, we c an fill the\nN-particle sectors for N∈ {1,...,k}. We give a first solution but better\nfillings can be done. In a general way, we define the ordered set (Ψj)j∈N\\{0}\nsuch that the states composed by orbitals having maximum abs olute value\nof momentum lower or equal to ℓare written\nΨk=Nmax(k)/circleplusdisplay\nn=1Ψ(n)\nm,\nwhere Ψ(n)\nmis ann-particle antisymmetric state of the form\nΨ(n)\nm=/logicalanddisplay\ns∈I(n)\nmφs, I(n)\nm⊂ {s∈Z,|s|/lessorequalslantℓ},/vextendsingle/vextendsingle/vextendsingleI(n)\nm/vextendsingle/vextendsingle/vextendsingle=n.\nThe number Nmax(k) is the highest particle sector number on which Ψ khas\na non vanishing component. We remark that the Ψ k’s form an orthonormal\nfamilly.\nWe define the states Ψ krecursively as follows. We start by defining the\nvaccuum\nΨ0:= Ω\nthen for the step ℓ= 0,\nΨ1:=φ0.\nFor the step ℓ= 1,\nΨ2:=φ1, Ψ3:=φ−1,\nΨ4:=φ0∧φ1, Ψ5:=φ−1∧φ0, Ψ6:=φ−1∧φ1,\nΨ7:=φ0⊕φ0∧φ1,Ψ8:=φ0⊕φ−1∧φ0,Ψ9:=φ0⊕φ−1∧φ1,\nΨ10:=φ1⊕φ0∧φ1,Ψ11:=φ1⊕φ−1∧φ0,Ψ12:=φ1⊕φ−1∧φ1,\nΨ13:=φ−1⊕φ0∧φ1,Ψ14:=φ−1⊕φ−1∧φ0,Ψ15:=φ−1⊕φ−1∧φ1,\nΨ16+j:= Ψj⊕φ−1∧φ0∧φ1,forj∈ {0,...,15}.\nThen for the step ℓ= 2, we use φ2and continue to define the following Ψ j’s,\nAt stepℓ, we define the corresponding Ψ k’s by taking their definition for\nℓ−1, and we define new Ψ k’s by putting one (or zero) Slater state in each\nn-particle sector, from n= 1 ton=Nmax(k)/lessorequalslant2ℓ+ 1. At step ℓ, we\nthus have defined all the possible ways of filling the 2 ℓ+1 first sectors with\nSlater states formed by orbitals having maximum absolute va lue label ℓ. For\nk/greaterorequalslant1, all those grand-canonical states have the same normalize d density\npΨk=ρΨk/N=p.REPRESENTABILITY OF ENTROPY-DENSITY 11\nAs in the canonical case, we define\nJ(k) := max\nj∈I(m)\nn\nΨk=/circleplustextNmax(k)\nn=1/logicalandtext\ns∈I(n)\nmφs|j|\nas the maximum of the absolute values of momentum of orbitals composing\nΨk. By construction, the last particle sector of Ψ kcannot be composed\nof orbitals with maximum absolute value momentum larger tha nJ(k), so\nwe have Nmax(k)/lessorequalslant2J(k) + 1, and the mean number of particles of Ψ kis\nbounded by\nN(Ψk)/lessorequalslantNmax(k)/summationdisplay\nN=1N/lessorequalslant(J(k)+1)(2J(k) +1). (17)\nWe note that the Ψ k’s do not necessarily have increasing kinetic energy, but\nwe can still bound\nT(Ψk)/lessorequalslantN(Ψk)T(φJ(k))/lessorequalslant(J(k)+1)(2J(k) +1)T(φJ(k)) (18)\nFor each N∈ {0,...,2ℓ+1}, and recalling the definition of binomial coef-\nficients (9), we have\n1+/parenleftbigg\n2ℓ+1\nN/parenrightbigg\nways to build an N-particle Slater state containing maximum absolute value\nmomentum lower or equal to ℓ, or take no state, the 1 summed represents\nthis choice of taking no particle. So placing one Slater stat e, or nothing, in\neach of those sectors, we have for ℓ∈N∪{0}\nJ/parenleftigg2ℓ−1/productdisplay\nN=1/parenleftbigg\n1+/parenleftbigg\n2ℓ−1\nN/parenrightbigg/parenrightbigg/parenrightigg\n=···=J/parenleftigg\n−1+2ℓ+1/productdisplay\nN=1/parenleftbigg\n1+/parenleftbigg\n2ℓ+1\nN/parenrightbigg/parenrightbigg/parenrightigg\n=ℓ,\nwhere/producttextk\nN=1:= 1 when k/lessorequalslant0. We have the lower bound\n2ℓ+1/productdisplay\nN=1/parenleftbigg\n1+/parenleftbigg\n2ℓ+1\nN/parenrightbigg/parenrightbigg\n/greaterorequalslant1+2ℓ+1/productdisplay\nN=1/parenleftbigg\n2ℓ+1\nN/parenrightbigg\n= 1+g(2ℓ+1),\nwhereg(n) :=/producttextn\nk=0/parenleftbigg\nn\nk/parenrightbigg\n, soJ(g(2ℓ+1))/lessorequalslantℓ. Moreover, since t/ma√∫to→tlntis\nnon-decreasing on [1 ,+∞),\nn/summationdisplay\nk=0klnk/lessorequalslantnlnn+/integraldisplayn\n1xlnxdx=nlnn+1\n2n2lnn−n2\n4+1\n4\nhence for n∈N\\{0},\nlng(n)/greaterorequalslantlnn/productdisplay\nk=0/parenleftign\nk/parenrightigk\n=1\n2n(n+1)lnn−n/summationdisplay\nk=0klnk/greaterorequalslantn2\n4−1\n2nlnn−1\n4\n/greaterorequalslantn2\n8−1,12 L. GARRIGUE\nandJ/parenleftbigg\ne(2ℓ+1)2\n8−1/parenrightbigg\n/lessorequalslantℓ. As before, we extend JonR+such that it remains\nincreasing, and we get\nJ/parenleftbigg\ne(2y+1)2\n8−1/parenrightbigg\n/lessorequalslanty+1 fory∈[0,+∞),\nJ(t)/lessorequalslant√\n2√\n1+lnt+1\n2fort∈[1,+∞).\nThe kinetic energy can be estimated by (18),\nT(Ψk)/lessorequalslant(J(k)+1)(2J(k) +1)/parenleftbig\n1+16π2J(k)2/parenrightbig/integraldisplay\nRd|∇√p|2\nand we can show that ( x+ 1)(2x+ 1)(1 + 16 π2x2)/lessorequalslant20(2x+1)4for any\nx∈R+, hence\nT(Ψk)/lessorequalslant20(2J(k)+1)4/integraldisplay\nRd|∇√p|2/lessorequalslant320/parenleftig√\n2√\n1+lnk+1/parenrightig4/integraldisplay\nRd|∇√p|2.\nWe now use the same mixed states construction as in (13), with M=/floorleftbig\neS/floorrightbig\n+\n1.\nT(ΓN)/lessorequalslantT(ΨM+1)/lessorequalslant320/parenleftbigg√\n2/radicalig\n1+ln(eS+2)+1/parenrightbigg4/integraldisplay\nRd|∇√p|2\n/lessorequalslant5×28/parenleftig√\nS+3/parenrightig4/integraldisplay\nRd|∇√p|2/lessorequalslant5×212(S+3)2/integraldisplay\nRd|∇√p|2,\nwhere we used that 4( x+ 3)/greaterorequalslant(√x+ 3)2for anyx/greaterorequalslant0. Eventually, and\nusing (17), we have\nN(Γ(θ,M))/lessorequalslantN(ΨM+1)/lessorequalslant(J(M+1)+1)(2 J(M+1)+1)\n/lessorequalslant2(J(M+1)+1)2/lessorequalslant2/parenleftbigg√\n2/radicalig\n1+ln(eS+2)+3\n2/parenrightbigg2\n/lessorequalslant24(S+3).\nBosons in the grand-canonical case.\nAs explained in (15), there are\n1+/parenleftbigg\n2ℓ+N\n2ℓ/parenrightbigg\n= 1+/parenleftbigg\n2ℓ+N\n2ℓ/parenrightbigg\nways of buildingan N-particles state composed by orbitals having maximum\nabsolute value momentum lower or equal to ℓ, or choosing an empty state.\nHence filling the N-particles sectors from N= 1 toN=ℓ, which is an\narbitrary choice, we have\nh(ℓ) :=/parenleftbigg\n1+/parenleftbigg\n2ℓ+1\n2ℓ/parenrightbigg/parenrightbigg/parenleftbigg\n1+/parenleftbigg\n2ℓ+2\n2ℓ/parenrightbigg/parenrightbigg\n···/parenleftbigg\n1+/parenleftbigg\n3ℓ\n2ℓ/parenrightbigg/parenrightbigg\n,\nways to build states containing maximal absolute value mome ntum lower or\nequal to ℓ, soJ(h(ℓ))/lessorequalslantℓ. We have\nlnh(ℓ)/greaterorequalslantln/parenleftbigg(2ℓ+1)(2ℓ+2)···3ℓ\n2ℓ···2ℓ/parenrightbigg2ℓ\n/greaterorequalslant−2ℓln(2ℓ)+2ℓℓ/summationdisplay\ni=1ln(2ℓ+i),REPRESENTABILITY OF ENTROPY-DENSITY 13\nbut\nℓ/summationdisplay\ni=1ln(2ℓ+i)/greaterorequalslantln(2ℓ+1)+/integraldisplayℓ\n1ln(2ℓ+t)dt,\nand hence\nlnh(ℓ)/greaterorequalslant6ℓ2ln(3ℓ)−2ℓ(2ℓln(2ℓ+1)+ln(2 ℓ))+2ℓ(1−ℓ)/greaterorequalslant2ℓ2lnℓ,\nand therefore h(ℓ)/greaterorequalslante2ℓ2lnℓ. We define f(x) :=x2lnxforx∈[1,+∞),\nwhich is strictly increasing, and we call ξthe reciprocal function of f, which\nis defined on [0 ,+∞), we have thus ξ(x2lnx) =x. We extend Jto [1,+∞)\nand have J(x)/lessorequalslantξ/parenleftbiglnx\n2/parenrightbig\n. We can continue and obtain a bound similar to\nthe fermionic case, but with better constants.\nThe characterization of the set of representable densities follows from the\nprevious results and the Hoffmann-Ostenhof inequalities der ived from [8],\n/integraldisplay\nRd/vextendsingle/vextendsingle∇√ρΓN/vextendsingle/vextendsingle2/lessorequalslantTr/summationtextN\ni=1(−∆i)ΓN,/integraltext\nRd/vextendsingle/vextendsingle∇√ρΓ/vextendsingle/vextendsingle2/lessorequalslantTrKΓ.(19)\n3.3.Proof of Theorem 3. The inclusion Igc⊂ ···in (4) is justified by\nusing the second Hoffmann-Ostenhof inequality of (19). We the n show the\nreciprocal inclusion.\nWe denote by Ω the vaccum in the Fock space, and we define the mix ed\nstate\nΓ(λ,α) :=λ|ΨM+1/a\\}b∇ack⌉t∇i}ht/a\\}b∇ack⌉tl⌉{tΨM+1|+α\nMM/summationdisplay\nk=1|Ψk/a\\}b∇ack⌉t∇i}ht/a\\}b∇ack⌉tl⌉{tΨk|+(1−λ−α)|Ω/a\\}b∇ack⌉t∇i}ht/a\\}b∇ack⌉tl⌉{tΩ|,\nwhere Ψ kare the same N-particle wavefunctions as in the proof of Theorem\n1, (8) for fermions. We will choose N∈NandM∈Nlater, and we have\nΓ(λ,α)∈ Sgc. Coefficients of projectors need to be positive so we also\nimposeλ,α∈[0,1] such that λ+α/lessorequalslant1.\nWe compute that\nN(Γ(λ,α)) = (λ+α)N\nS(Γ(λ,α)) =αlnM−λlnλ−αlnα−(1−λ−α)ln(1−λ−α)\nand\npΓ(λ,α)=ρΓ(λ,α)\nN(Γ(λ,α))=p.\nHence to satisfy N(Γ(λ,α)) =N, we impose\nλ+α=N\nN,\nand we consider now that αis fixed by the value of λ, and that the only\nremaining degree of freedom is λ. We remark that we need N/lessorequalslantNto have\nλ+α/lessorequalslant1. Then we define\nS(λ) :=S/parenleftig\nΓ/parenleftig\nλ,N\nN−λ/parenrightig/parenrightig\n, λ∈/bracketleftig\n0,N\nN/bracketrightig\n,\nwhich is continuous. we have\nS(0) =N\nNlnM+s/parenleftig\nN\nN/parenrightig\n, S/parenleftig\nN\nN/parenrightig\n=s/parenleftig\nN\nN/parenrightig\n.14 L. GARRIGUE\nWith an adapted choice of M, we now impose that S/lessorequalslantN\nNlnM, which is\nequivalent to\nM/greaterorequalslanteN\nNS. (20)\nWe have max x∈[0,1]s(x) = ln2, hence if S/greaterorequalslantln2,S(λ) is a continuous\nfunction such that\nS/parenleftig\nN\nN/parenrightig\n/lessorequalslantS/lessorequalslantS(0), (21)\nand then there exists λ∗∈/bracketleftig\n0,N\nN/bracketrightig\nsuch that S(λ) =S. Then we can choose\nN=/ceilingleftbig\nN/ceilingrightbig\n, M =/ceilingleftbigg\neN+1\nNS/ceilingrightbigg\n=/ceilingleftbigg\ne/parenleftBig\n1+1\nN/parenrightBig\nS/ceilingrightbigg\n.\nIfS/lessorequalslantln2, we need to impose s/parenleftig\nN\nN/parenrightig\n/lessorequalslantS, which is equivalent to\nN/greaterorequalslantN\ns−1(S).\nIn this case we choose\nN=/ceilingleftbiggN\ns−1(S)/ceilingrightbigg\n, M =/ceilingleftbigg\neS\nN/parenleftBig\nN\ns−1(S)+1/parenrightBig/ceilingrightbigg\n=/ceilingleftbigg\neS/parenleftBig\n1\ns−1(S)+1\nN/parenrightBig/ceilingrightbigg\n.\nso that we also have (20). Then we again have (21), and as befor e, since\nS(λ) is continuous, there exists λ∗such that S(λ∗) =S.\nFinally, the kinetic energy of our mixed state is bounded by\nT(Γ(λ,α)) =λT(ΨM+1)+α\nMM/summationdisplay\nk=1T(Ψk)/lessorequalslantT(ΨM+1)\nwe use the same bound as in (11) and with α∗:=N\nN−λ∗, we finally obtain\nT(Γ(λ∗,α∗))/lessorequalslant/parenleftbigg\n1+4π2N2/parenleftig\n(M+1)1\nN+1+2\nN/parenrightig2/parenrightbigg/integraldisplay\nRd|∇√ρ|2\n/lessorequalslant5π2N3/parenleftig\n(M−1)1\nN+5/parenrightig2/integraldisplay\nRd|∇√p|2\n/lessorequalslant5π2/parenleftbiggN\nξS+1/parenrightbigg3\neS1\nξS+1\nN\nmax/parenleftbigg\n1,N\nξS/parenrightbigg\n+5\n2\n/integraldisplay\nRd|∇√p|2.\n3.4.Proof of Proposition 1.\nThe continuity in vcan be proved by the same method as in [13, Theorem\n3.1]. For the continuity in T, we use that concave functions on convex sets\nare continuous.REPRESENTABILITY OF ENTROPY-DENSITY 15\n3.5.Proof of Proposition 2.\nCanonical case. We adapt the argument of the proof for the pure states\ncase, found in [13, Proof of Theorem 3.3].\nThe geometric convergence in truncated Fock spaces S/lessorequalslantN\ncanis denoted by\n⇀gand defined in [10]. Let (Γ j)j∈Nbe a minimizing sequence, so verifying\nρΓj=ρ. This minimizing sequence is bounded in\n/braceleftig\nΓ∈S1/parenleftbig\nL2\na(RdN)/parenrightbig/vextendsingle/vextendsingleTrΓ = 1/bracerightig\n,\nthus by a Banach-Alaoglu-type theorem [10, Lemma 2.2], and b y weak lower\nsemi-continuity of Γ /ma√∫to→TrHN\n0Γ showed in [22, 2.7.d], we have Γ j⇀gΓ with\nTrHN\n0Γ/lessorequalslantFN\ncan(ρ,S).\nNow we want to show that ρΓ=ρ. LetBR⊂Rddenote the ball of\nradiusRcentered at the origin. The kinetic energy is uniformly boun ded,\nthat is Tr/summationtextN\ni=1(−∆i)Γj/lessorequalslantc, so by a Rellich-Kondrachov-type theorem for\nmixed states [10, Lemma 3.2], the localizations Γ j,BR→S1ΓBR, defined\nin [10, Proposition 3.1], converge strongly. Also, we have\nTrΓj,BR=/integraldisplay\nRdNΓj(x1,...,x N;x1,...,x N)N/productdisplay\ni=11BR(xi)dx1···dxN.\nTakeε >0, there exists R >0 large enough so that\nε >/integraldisplay\nRdρ(1−1BR)\n=/integraldisplay\nRdNΓj(x1,...,x N;x1,...,x N)N/summationdisplay\ni=1(1−1BR(xi))dx1···dxN\n/greaterorequalslant/integraldisplay\nRdNΓj(x1,...,x N;x1,...,x N)/parenleftigg\n1−N/productdisplay\ni=11BR(xi)/parenrightigg\ndx1···dxN\n= 1−TrΓj,BR.\nThus by strong convergence of Γ j’s kernels on B⊗N\nR, we get\nTrΓ/greaterorequalslantTrΓBR= lim\nj→+∞TrΓj,BR/greaterorequalslant1−ε.\nHence TrΓ = 1 so by [10, Lemma 2.3], Γ j→Γ strongly in S1. Since the\nmap Γ/ma√∫to→ρΓis globally Lipschitz from S1→L1, we deduce that ρΓ=ρ.\nLet us now show that S(Γ) =S. With ρ(x)/lessorequalslantc|x|−d−εfar enough\nfrom the origin, and taking v(x) =|x|ε/2, we have/vextendsingle/vextendsingle/integraltext\nRdvρ/vextendsingle/vextendsingle<+∞, so\nTre−HN(v)/T<+∞for anyT∈R+and from this last property, as ex-\nplained in [24, II.D], we can conclude that S(Γj)→S(Γ). Since Γ belongs\nto the minimizing set of FN\ncan, thenFN\ncan(ρ,S) = TrHN\n0Γ.\nGrand-canonical case. The lemmas [10, Lemma 2.2, 2.3 and 3.2] hold\nwhen states belong to the untruncated Fock space if we add the uniform\nboundN(Γj)/lessorequalslantc. We can thus proceed as in the canonical case.16 L. GARRIGUE\nAcknowledgement. I warmly thank Mathieu Lewin for his advices. This\nprojecthasreceived fundingfromtheEuropeanResearchCou ncil(ERC)un-\nder the European Union’s Horizon 2020 research and innovati on programme\n(grant agreements MDFT No 725528 and EMC2 No 810367).\nReferences\n[1]P. W. Ayers and E. R. 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Phys, 50 (1978), pp. 221–260."    },    {        "title": "2204.09925v2.Honing_in_on_a_topological_zero_bias_conductance_peak.pdf",        "content": "Honing in on a topological zero-bias conductance peak\nSubhajit Pal and Colin Benjamin∗\nSchool of Physical Sciences, National Institute of Science Education & Research, Jatni-752050, India and\nHomi Bhabha National Institute, Training School Complex, AnushaktiNagar, Mumbai, 400094, India\nA popular signature of Majorana bound states in topological superconductors is the quantized\nzero-energy conductance peak. However, a similar zero energy conductance peak can also arise due\nto non-topological reasons. Here we show that these trivial and topological zero energy conductance\npeaks can be distinguished via the zero energy local density of states and local magnetization density\nof states. We find that the zero-energy local density of states and the local magnetization density\nof states exhibit periodic oscillations for a trivial zero-bias conductance peak. In contrast, these\noscillations disappear for the topological zero-bias conductance peak because of perfect Andreev\nreflection at zero energy in topological superconductor junctions. Our results suggest that the\nzero-energy local density of states and the local magnetization density of states can be used as\nan experimental probe to distinguish a trivial zero-energy conductance peak from a topological\nzero-energy conductance peak.\nI. INTRODUCTION\nOne of the most significant signatures of Majorana zero modes (MZMs) is the observation of zero-bias conductance\npeak (ZBCP), which is reported in many experiments[1–5]. ZBCP with heights near the theoretically predicted\nquantized value of 2 e2/h[6] leads to optimism regarding the real sighting of a Majorana bound state (MBS). However,\nthe zero-energy behaviour of MBS has been questioned recently because many papers have reported ZBCP due to\nnon-topological reasons[7–22]. Many thought experiments have shown that ZBCP height can be tuned to be near\n2e2/hvalues. For example, Andreev bound states induced by inhomogeneous chemical potential or quantum dots can\ngenerate quantized ZBCPs[23–25]. Further, disorder-induced random potential can generate ZBCP (quantized almost\nat 2e2/hvalues) with sufficient fine-tuning[26–29] due to non-topological reasons. Sometimes, these trivial ZBCP can\neven accidentally show stable quantization[30–33] to some extent via fine-tuning in the system, leading to confusion\nbetween trivial and topological ZBCP. Further, in Ref. [10], the authors show that spin splitting, spin-orbit coupling,\nand ordinary s-wave superconductivity sometimes produce a situation where trivial ZBCP mimics the topological\nZBCP. Some theoretical papers even indicate that any smooth change in the chemical potential may give rise to\ntrivial ZPCP[7, 11, 34]. Ref. [10] also shows through detailed calculation of the differential conductance that the\ntrivial and non-trivial ZBCP may behave essentially similarly in some situations. In Ref. [35], the authors observe\nlarge ZBCPs near 2 e2/hin a thin InAs-Al hybrid nanowire device, which forms a plateau within 5% tolerance by\nsweeping magnetic field and gate voltage. Further, in this device, both tunnelling and Coulomb spectroscopy can be\nperformed to identify Majorana bound states, as shown in Ref. [36]. Ref. [37] shows experimentally the existence of\na trivial ZBCP arising from Andreev-bound states which is analogous to the observed non-trivial ZBCP arising from\nMBS. Furthermore, a trivial ZBCP is reported in a normal metal-insulator- d-wave superconductor junction[38]. In\nthe case of a d-wave superconductor-insulator- d-wave superconductor Josephson junction, zero-energy states emerge\naround the surface of the d-wave superconductor[39]. This occurs because the quasiparticle inside the superconductor\nencounters varying signs of the pair potential depending on its direction of motion. Thus, it is a central challenge to\ndistinguish topological ZBCP from trivial ZBCP based on current experimental techniques. Just by noticing, ZBCP\nis not a sufficient condition for MBS.\nIn Ref. [40], authors show that trivial and non-trivial ZBCP can be distinguished using interferometry experiments\nthat have been proposed recently[41, 42]. In Ref. [43], authors distinguish trivial zero energy states from topological\nzero energy states via Josephson current, where trivial zero energy states generate a πshift in the Josephson current.\nIn contrast, topological zero energy states do not. Further, in Ref. [44], authors propose a method to distinguish\nMajorana bound states from Andreev bound states in a 1D semiconductor-superconductor nanowire junction by\nusing a sharp local potential barrier, whose position and strength could be controlled by applying a gate voltage\non the nanowire. In addition, Ref. [45] provides some specific experimental tools through numerical simulations\nfor differential tunnelling conductance measurements to distinguish between topological MBS and non-topological\nAndreev bound states[46].\n∗colin.nano@gmail.comarXiv:2204.09925v2  [cond-mat.mes-hall]  19 Oct 20232\nThe main aim of this article is to distinguish trivial ZBCP from topological ZBCP via local density of states (LDOS)\nand local magnetization density of states (LMDOS). We have two setups shown in Figs. 1(a), 1(b). Fig. 1(a) consists\nof a spin-flipper at x=−a, and at x= 0, there is a delta potential barrier. There are two metallic regions for x <−a\nand−a < x < 0. There is a superconductor for x >0, either a non-topological s-wave or topological p-wave. For\nhigh values of the exchange interaction and low values of the spin flipper’s spin-flip probability, a zero-energy peak\nappears in the conductance spectra due to the merger of two bound state energies, which is very similar to that of the\ntopological zero-energy peak when a p-wave superconductor replaces s-wave superconductor. In Fig. 1(b), we depict\nthe second setup wherein both trivial and topological phases are seen via a change in chemical potential.\nTo distinguish trivial ZBCP from topological ZBCP, we examine zero energy LDOS and LMDOS for parameters\nwhen both trivial and topological zero energy peaks appear in the conductance spectra. Zero-energy LDOS shows\noscillations in the left normal metal region for trivial junctions, while it is constant for topological junctions. In the\nsuperconducting region, the zero-energy LDOS decays exponentially for trivial junctions. In contrast, the zero-energy\nLDOS shows an oscillatory decay in topological junctions. The zero-energy LMDOS in the metallic region exhibits a\nnice oscillation for trivial cases, while it vanishes in topological cases. The zero-energy LMDOS exhibits an oscillatory\ndecay for both trivial and topological junctions in the superconducting region. However, the oscillation periods are\ndifferent in the two cases. Thus, by measuring LDOS and LMDOS in tandem, one can distinguish trivial ZBCP from\ntopological ZBCP.\nThe rest of the paper is organized as follows: in the next section, we discuss our two setups, the first setup (see\nFig. 1(a)) and then provide the theory to compute differential charge conductance and Green’s functions. Following\nthis, we discuss the second setup (see Fig. 1(b)). In the same section, we also provide the method to compute differential\ncharge conductance and LDOS, LMDOS from retarded Green’s function for both of our setups. In sections III, IV\nand, V, we show our results for these two setups. In section VI, we analyze our results via a table. Finally, we\nconclude with an experimental realization in section VII. The detailed wavefunctions of our two setups and explicit\nexpressions for Green’s functions are given in the Appendix.\nII. THEORY\nA. Hamiltonian, wavefunctions and boundary conditions in the first setup: normal metal-normal\nmetal-superconductor junction in the presence of a spin flipper\n1. Hamiltonian\nWe consider a one-dimensional metal (N 1)-Normal metal (N 2)-Superconductor (S) junction as shown in Fig. 1(a),\nwhere the superconductor is either conventional s-wave or topological p-wave. There is a spin flipper (SF) between two\nmetallic regions at x=−a. A delta potential barrier models the Normal metal-Superconductor (N 2S) interface. Using\nBTK approach[47], we solve the problem. In our problem, the spin flipper is a delta potential magnetic impurity, and\nits Hamiltonian from Refs. [13, 48–51] is,\nHSpin flipper=−J0δ(x)⃗ s.⃗S, (1)\nThe Bogoliubov-de Gennes (BdG) Hamiltonian of our system (see Fig. 1(a)) with s-wave superconductor is\nHBdG(x) =\u0014\nHˆI i ∆θ(x)ˆσy\n−i∆∗θ(x)ˆσy−HˆI\u0015\n, (2)\nwhile the BdG Hamiltonian of our system with p-wave superconductor is\nHBdG(x) =\u0014\nHˆI ∆′θ(x)ˆσx\n∆′∗θ(x)ˆσx−HˆI\u0015\n, (3)\nwhere H=p2\n2m∗+V δ(x)−J0δ(x+a)⃗ s.⃗S −EF,θ(x) represents Heaviside step function, ∆ is superconducting gap\nfor an s-wave superconductor and in case of p-wave superconductor ∆ is replaced by ∆′=p∆/kF, where kFis the\nFermi wave-vector.p2\n2m∗is the electron’s kinetic energy with momentum pand effective mass m∗,Vis the delta\npotential barrier strength at the N 2S interface, J0is the exchange interaction strength between the spin of electron ⃗ s\nand the spin of spin flipper ⃗S,ˆIrepresents unit matrix, ˆ σrepresents Pauli spin matrix, and EFrepresents the Fermi\nenergy. The spin flipper’s spin can exist in various states. For instance, if the spin flipper’s spin S= 1/2, then it\ncan have two distinct states: m′=−1/2 and m′= 1/2. Similarly, when S= 3/2, then spin flipper’s spin can have3\n \n  I                                             II                                                III\nx = -a                                                  x = 0                     e                                e\n     \n                                                                  \n                                                \n    e\n e\n e\n h\n h e                                e\n h                                 h\n h                                 h e \n  e \n h\n   hV\nN1                                    N2                                        S \n  SF(a)\n \n  I                                             II                                     III\nx = -a                                                  x = 0                     e                                \n \n                                                             Z  \n    e\n e\n e\n h\n h e                               \n h                         e       h\n h                                  e \n  e                                                e       h\n h\n   h                                                                                 N1                                    N2                                        pSc  \n SF (b)\nFIG. 1: (a) N 1N2S junction with spin flipper at N 1N2interface and a delta potential barrier at N 2S interface, (b) N 1N2pSc\njunction with spin flipper at N 1N2interface and a delta potential barrier at N 2pSc interface. In both (a) and (b), the\nscattering of an up-spin electron incident is represented. When a spin-up electron is an incident at x=−ainterface from the\nleft metallic region, it interacts with the spin flipper through an exchange interaction, which may induce a mutual spin-flip.\nAs a result, an electron or hole with spin up or down is reflected into the left normal metal region. There is also the\ntransmission of electronlike and hole-like quasiparticles with spin-up or spin-down into the superconducting region for energies\nabove the gap. In (a) and (b) the spin flipper’s spin is oriented along any arbitrary direction.\nfour distinct states: m′= 3/2,1/2,−1/2,−3/2. We calculate conductance/LDOS/LMDOS for each of the 2 S+ 1\npossible values of m′for spin flipper’s spin Sand finally take an average over all m′values. When a spin-up electron\ninteracts with the spin flipper whose spin S= 1/2, a product statem′\n2\u0000\n|↑⟩e⊗ |↑⟩ SF\u0001\nis formed if the spin flipper’s\nspin is in the up state ( m′= 1/2). However, an entangled state\u0010With mutual spin flipz }| {\nf\n2(|↓⟩e⊗ |↑⟩ SF) +Without spin flipz }| {\nm′\n2(|↑⟩e⊗ |↓⟩ SF)\u0011\nmay emerge\nafter scattering, if the spin flipper’s spin is in the spin-down state ( m′=−1/2). In the subsequent scattering event at\nthe spin flipper, when a spin-down electron interacts, it encounters the spin flipper in either the spin-up or spin-down\nstate. If the spin flipper is in the spin-up state, a spin flip occurs, leading again to the formation of an entangled state.\nConsequently, measurable quantities like charge conductance, LDOS, and LMDOS are determined as the averages of\nthese two processes: with and without spin flips. A similar scenario happens for S= 3/2 also. This approach applies\nanalogously to higher spin states of the spin flipper’s spin S, and measurable quantities are calculated by averaging\nover all possible values of m′. We will later use the dimensionless parameter J=m∗J0\nkFfor exchange coupling[48]\nstrength and Z=m∗V\n¯h2kFfor interface transparency[47].\n2. Wavefunctions\nAfter diagonalizing the BdG Hamiltonians (2),(3), the wavefunctions in metallic and superconducting regions for\ndifferent kinds of scattering processes are obtained. The wavefunctions in different regions of our system with sand\np-wave superconductors are shown in Appendix. When a spin-up electron is incident at Metal- s-wave superconductor\ninterface, it is Andreev reflected as a spin-down hole; however, when a spin-up electron is incident at Metal- p-wave\nsuperconductor (equal spin-triplet) interface, it is Andreev reflected as a spin up hole. It is the main difference in\nthe Andreev reflection process between trivial ( s-wave) and topological ( p-wave) superconductor junctions. Wave-\nfunctions in different regions for trivial and topological superconductor junctions are written keeping this in mind\nso that in the absence of spin flipper, only spin-up electron and spin-down hole contribute to the wavefunction for\ntrivial superconductor junction, while spin-up electron and spin-up hole contribute to wavefunction of topological\nsuperconductor junction. Both are for a spin-up electron incident at the interface. In wavefunction for trivial super-\nconductor junctions, ψ1(x) is the wavefunction when spin-up electron is incident, while Ψ 1(x) replaces ψ1(x) for the\ntopological case. In ψ1(x) or Ψ 1(x), in the region x <−a, the first term represents the incident spin-up electron,\nthe second and third terms represent the Andreev reflected spin up and spin down hole respectively, the fourth and\nfifth terms represent the normal reflected spin up and spin down electron respectively. In ψ1(x) or Ψ 1(x), in the\nregion, x >0, the first two terms represent the spin up and spin down electron-like quasiparticles, respectively. The\nlast two terms represent the spin-up and spin-down hole-like quasiparticles, respectively. Similarly, in ψ2(x) (Ψ 2(x)),4\nψ3(x) (Ψ 3(x)),..., ψ8(x) (Ψ 8(x)), the different terms represent the normal and Andreev reflection processes as well as\ntransmission process for trivial (topological) superconductor junction.\n3. Boundary conditions\nWe obtain the amplitudes in the scattering states from the boundary conditions. The boundary conditions (at\nx=−a) are-\nχi|x<−a=χi|−a<x< 0, (4)\ndχi|−a<x< 0\ndx−dχi|x<−a\ndx=−2m∗J0⃗ s.⃗S\n¯h2χi|x=−a, (5)\nwhere χi≡ψifor trivial superconductor junction, while χi≡Ψifor topological superconductor junction and ⃗ s.⃗S=\nszSz+(s+S−+s−S+)\n2represents the exchange operator in spin flipper’s Hamiltonian[48] with s±=sx±isyfor electron\nandS±=Sx±iSyfor spin flipper. At x= 0, the boundary conditions are-\nχi|−a<x< 0=χi|x>0, (6)\ndχi|x>0\ndx−dχi|−a<x< 0\ndx=2m∗V\n¯h2χi|x=0. (7)\nThe spin-flipper and electron spin interaction term: ⃗ s.⃗Sgive\n⃗ s.⃗SφN\n1ϕS\nm′=¯h2m′\n2φN\n1ϕS\nm′+¯h2f\n2φN\n2ϕS\nm′+1,and, ⃗ s.⃗SφN\n2ϕS\nm′=−¯h2m′\n2φN\n2ϕS\nm′+¯h2f′\n2φN\n1ϕS\nm′−1. (8)\nFurther, for wave-functions involving up spin hole for s-wave (down spin hole for p-wave) and down spin hole for\ns-wave (up spin hole for p-wave), operations of ⃗ s.⃗Sare\n⃗ s.⃗SφN\n3ϕS\nm′=−¯h2m′\n2φN\n3ϕS\nm′+¯h2f′\n2φN\n4ϕS\nm′−1,and, ⃗ s.⃗SφN\n4ϕS\nm′=¯h2m′\n2φN\n4ϕS\nm′+¯h2f\n2φN\n3ϕS\nm′+1. (9)\nIn Eqs. (8),(9), f=p\n(S −m′)(S+m′+ 1) represents spin flipper’s flip probability for spin-up electron and spin-\ndown/spin-up hole incident process, and f′=p\n(S+m′)(S −m′+ 1) represents spin flipper’s flip probability for\nspin-down electron and spin-up/ spin-down hole incident process in case of s-wave/ p-wave superconductor junction.\nWe use the above equations and solve for the boundary conditions to obtain 16 equations for each kind of incident\nprocess, see Eqs. (A1),(A2). Different scattering amplitudes aij,bij,cij,dij,eij,fij,gij,hijfor each type of incident\nprocess are obtained from these 16 equations. By using these scattering amplitudes, we calculate differential charge\nconductance as well as retarded Green’s function in metallic and superconducting regions of our setup. LDOS and\nLMDOS in metallic and superconducting regions are found from retarded Green’s function. In the next subsection,\nwe consider another setup with a p-wave superconductor junction wherein one can tune the system from trivial to\ntopological regimes via changing the chemical potential[52].\nB. Hamiltonian, wavefunctions and boundary conditions in the second setup: normal metal-normal\nmetal- p-wave superconductor junction in the presence of a spin flipper\n1. Hamiltonian\nWe consider a 1D Normal metal (N 1)-Normal metal (N 2)-p-wave superconductor (pSc) junction as shown in\nFig. 1(b), where there is a spin flipper between two metallic regions at x=−a. A δ-like potential barrier with\nstrength Zmodels the Normal metal-pSc interface at x= 0. For such a pSc junction, one can tune the system from\na trivial to a topological regime by changing the chemical potential. Thus, for this second setup, one can get both\ntrivial and topological regimes in the same device. The metallic region is spinful, and spin-flip processes are confined\nto the metallic region. When a spin up or spin down electron is incident at x=−ainterface from the normal metal\nregion, it interacts with the spin flipper through an exchange interaction which may induce a mutual spin flip. The\nincident electron can be reflected or transmitted to the middle normal metal region, with spin-up or spin-down. When\nthis transmitted spin-up electron is incident at metal- p-wave superconductor interface ( x= 0), it can be normally\nreflected as a spin-up electron from the interface or could be Andreev reflected only as a spin-up hole back to the5\nmiddle metallic region. In the p-wave superconductor, the spin-up electron and spin-up hole form one transport\nchannel, while the spin-down electron and spin-down hole form another transport channel for energies above the gap,\nsee Fig. 1(b). In the p-wave superconductor, there is no spin mixing. These are two different channels. So, in our\ncase, the p-wave superconductor is still spinless regardless of the presence of a spin flipper in its vicinity. The BdG\nHamiltonians for normal metal (N) and p-wave superconductor (pSc) are[52]\nHN=\u0010\n−¯h2∂2\nx\n2m∗−µN\u0011\nτz, (10a)\nHpSc=\u0010\n−¯h2∂2\nx\n2m∗−µpSc\u0011\nτz−i∆pSc∂xτx, (10b)\nwhere µNandµpScare the chemical potentials for normal metal and pSc, respectively. ∆ pSc≥0 is the pairing\npotential for pSc. τρ=σρ⊗I, where Iis the 2 ×2 identity matrix and σρ(ρ=x, y, z ) are Pauli matrices. For\nsimplicity, we take ¯ h= 2m∗=µN= 1.\n2. Wavefunctions\nIf we diagonalize the BdG Hamiltonian (10), we will get the wavefunctions in metallic and superconducting regions\nfor different scattering processes. The wavefunctions in different regions of our system are shown in the Appendix.\n3. Boundary conditions\nWe get the amplitudes in the scattering states from the boundary conditions. The boundary conditions (at x=−a)\nare-\nΦi|x<−a= Φ i|−a<x< 0, (11)\n2i∂xτzΦi|x<−a−2i∂xτzΦi|−a<x< 0= 2iJ⃗ s.⃗SτzΦi|x=−a. (12)\nAtx= 0, the boundary conditions are-\nΦi|−a<x< 0= Φ i|x>0, (13)\n(−2i∂xτz+ ∆ pScτx)Φi|x>0+ 2i∂xτzΦi|−a<x< 0=−2iZτzΦi|x=0. (14)\nThe spin-flipper and electron spin interaction ⃗ s.⃗Sgive\n⃗ s.⃗SτzφN\n1ϕS\nm′=¯h2m′\n2φN\n1ϕS\nm′+¯h2f\n2φN\n2ϕS\nm′+1,and, ⃗ s.⃗SτzφN\n2ϕS\nm′=−¯h2m′\n2φN\n2ϕS\nm′+¯h2f′\n2φN\n1ϕS\nm′−1. (15)\nFurther, hole spin and spin flipper interaction ⃗ s.⃗Sgive\n⃗ s.⃗SτzφN\n3ϕS\nm′=¯h2m′\n2φN\n3ϕS\nm′−¯h2f′\n2φN\n4ϕS\nm′−1,and, ⃗ s.⃗SτzφN\n4ϕS\nm′=−¯h2m′\n2φN\n4ϕS\nm′−¯h2f\n2φN\n3ϕS\nm′+1. (16)\nUsing the above equations and solving for the boundary conditions, we obtain 16 equations for each kind of incident\nprocess, see Eq. (B1). Different scattering amplitudes a′\nij,b′\nij,c′\nij,d′\nij,e′\nij,f′\nij,g′\nij,h′\nijfor each type of incident\nprocess are obtained from these 16 equations. We use these scattering amplitudes to calculate retarded Green’s\nfunction in metallic and superconducting regions of our second setup. LDOS, LMDOS and SPLDOS in normal metal\nand superconducting regions are obtained from retarded Green’s function.\nC. Differential charge conductance\nThe differential charge conductance is given as[53, 54]\nGc=GN(2 +A11+A12−B11−B12+A21+A22−B21−B22), (17)\nwhere GN=e2/his the charge conductance in the normal state with ∆ = 0. A11(=qh\nqe|a11|2) orA12(=qh\nqe|a12|2)\nare the Andreev reflection probabilities when a spin-up electron is reflected as a spin-up hole or spin-down hole6\nrespectively. A21(=qh\nqe|a21|2) or, A22(=qh\nqe|a22|2) are the Andreev reflection probabilities for a spin-down electron\nreflected as a spin-up hole or spin-down hole respectively. Similarly, B11(=|b11|2) orB12(=|b12|2) are the normal\nreflection probabilities for a spin-up electron reflected as spin-up electron or spin-down electron respectively. In\ncontrast, B21(=|b21|2), or B22(=|b22|2) are the normal reflection probabilities when a spin-down electron is reflected\nas spin-up electron or spin-down electron respectively. Real part of the complex poles of Gcare energy bound states\nE±.\nD. Retarded Green’s functions\nThis paper distinguishes trivial ZBCP from topological ZBCP via LDOS and LMDOS. For this reason, we follow\nRefs. [55, 56] and form the retarded Green’s function Gr(x, x′, ω) in our setup because of interface scattering[57].\nFrom retarded Green’s function, one can calculate LDOS, LMDOS and SPLDOS. The retarded Green’s function is\nGr(x, x′, ω) =\n\nχ1(x)[α11˜χT\n5(x′) +α12˜χT\n6(x′) +α13˜χT\n7(x′) +α14˜χT\n8(x′)]\n+χ2(x)[α21˜χT\n5(x′) +α22˜χT\n6(x′) +α23˜χT\n7(x′) +α24˜χT\n8(x′)]\n+χ3(x)[α31˜χT\n5(x′) +α32˜χT\n6(x′) +α33˜χT\n7(x′) +α34˜χT\n8(x′)]\n+χ4(x)[α41˜χT\n5(x′) +α42˜χT\n6(x′) +α43˜χT\n7(x′) +α44˜χT\n8(x′)], x > x′\nχ5(x)[β11˜χT\n1(x′) +β12˜χT\n2(x′) +β13˜χT\n3(x′) +β14˜χT\n4(x′)]\n+χ6(x)[β21˜χT\n1(x′) +β22˜χT\n2(x′) +β23˜χT\n3(x′) +β24˜χT\n4(x′)]\n+χ7(x)[β31˜χT\n1(x′) +β32˜χT\n2(x′) +β33˜χT\n3(x′) +β34˜χT\n4(x′)]\n+ψ8(x)[β41˜χT\n1(x′) +β42˜χT\n2(x′) +β43˜χT\n3(x′) +β44˜χT\n4(x′)], x < x′,(18)\nwhere χi≡ψiin our first setup for Normal metal-Spin flipper-Normal metal- s-wave superconductor junction, while Ψ i\nreplaces ψiin our first setup for Normal metal-Spin flipper-Normal metal-topological p-wave superconductor junction.\nFurther, χi≡Φiin our second setup for Normal metal-Spin flipper-Normal metal-pSc junction, where pSc can be\ntuned from trivial to topological regime via changing the chemical potential. The coefficients αijandβmnin Eq. (18)\nare determined from the continuity of the Green’s function\n[ω−HBdG(x)]Gr(x, x′, ω) =δ(x−x′). (19)\nBy integrating Eq. (19) around x=x′we get,\n[Gr(x > x′)]x=x′= [Gr(x < x′)]x=x′,\n[d\ndxGr(x > x′)]x=x′−[d\ndxGr(x < x′)]x=x′=ητzσ0,(20)\nwhere η=2m∗\n¯h2andτidenote Pauli matrices in particle-hole space, while σidenote Pauli matrices in spin space. σ0\nis unit matrix. Generally, in particle-hole space Grbeing a 2 ×2 matrix,\nGr(x, x′, ω) =\u0014\nGr\neeGr\neh\nGr\nheGr\nhh\u0015\n, (21)\nwhere Gr\nee,Gr\neh,Gr\nheand,Gr\nhhare matrices. For spin-flip process, each element of Gris given as\nGr\nij(x, x′, ω) =\u0012[Gr\nij]↑↑[Gr\nij]↑↓\n[Gr\nij]↓↑[Gr\nij]↓↓\u0013\n,with i, j∈ {e, h}. (22)\nIn the Appendix, we give an explicit form of Green’s functions.\n1. LDOS, SPLDOS and LMDOS\nFrom retarded Green’s function one can obtain LDOS ν(x, ω) and LMDOS m(x, ω)[58],\nν(x, ω) =−1\nπlim\nε→0Im[Tr{Gr\nee(x, x, ω +iε)}],\nm(x, ω) =−1\nπlim\nε→0Im[Tr{⃗ σi.Gr\nee(x, x, ω +iε)}].(23)\nUsing Eq. (23), the up spin and down spin components of the spin-polarized local density of states (SPLDOS) are\ncomputed as[58] νσ=ν+σ|m|\n2, where σ= 1 for up spin and σ=−1 for a down spin.7\nIII. RESULTS FOR THE FIRST SETUP\nA. Differential charge conductance and Energy bound states\nThe differential charge conductance for N 1-SF-N 2-s-wave superconductor junction at zero energy and kFa= 0,\nfrom Eq. (17) is given as\nGc=GNh\n8(J2f2(4+(2 Jm′+J)2)+(2+ J2((1+m′)2−f2)+4J(m′+1)Z+4Z2)2)\n((f2+m′+m′2)2J4−4(f2+m′+m′2)J3Z+4(2Z2+1)2+8J(2Z3+Z)+J2(2+(4−8f2)Z2+m′(1+m′)(4−8Z2)))2\n+8(J2f′2(4+(2 J(m′−1)+J)2)+(2+ J2((m′2−f′2)+4Jm′Z+4Z2)2)\n((f′2−m′+m′2)2J4−4(f′2−m′+m′2)J3Z+4(2Z2+1)2+8J(2Z3+Z)+J2(2+(4−8f′2)Z2+m′(m′−1)(4−8Z2)))2i\n,\n(24)\nwhile for N 1-SF-N 2-p-wave superconductor junction Gc= 4GNindependent of all parameters at zero energy. In\nFig. 2 normalized charge conductance and energy bound states are plotted for high values of the exchange interaction\nJand low values of spin flipper’s spin S. From Fig. 2(a), we notice that a ZBCP appears for N 1-SF-N 2-s-wave\nsuperconductor junction. This ZBCP is almost quantized ≈2e2/h, but it arises due to the merger of two bound\nstate energies as shown in Fig. 2(e), where bound state energies vs Zis plotted. Thus, the reason this zero energy\nconductance peak arises is non-topological. In Figs. 2(a), (b), (c) and (d), we calculate average charge conductance,\nsummed over all m′values for spin flipper’s spin S, the reason being the normal metal is unpolarized and both spin\nup as well as spin down electrons/holes may be incident on the spin flipper. From Fig. 2(c), we see that normalized\ncharge conductance at zero energy first increases and attains its maximum value around Z= 0.77 and then decreases\nwith Z. Differential charge conductance vanishes in the tunnelling regime ( Z >> 1). What happens if we replace\nthes-wave superconductor with a topological p-wave superconductor? We plot normalized charge conductance as a\nfunction of ωfor N 1-SF-N 2-p-wave superconductor junction in Fig. 2(b). We find that a ZBCP appears, which is\nquantized at 4 e2/h. In the case of topological superconductor junction, there is a perfect Andreev reflection at zero\nenergy. Thus, A11+A12=A21+A22= 1 and B11+B12=B21+B22= 0 in Eq. (17) at zero energy. Therefore, at zero\nenergy, Gc= 4e2/hfor topological superconductor junction. Hence, for a topological superconductor junction, zero\nenergy conductance peak is quantized at 4 e2/h. This ZBCP does not depend on the system parameters. Fig. 2(d)\nshows that zero-energy conductance is constant at 4 e2/hand does not change with interface transparency Z. This\nzero energy conductance peak is topological. However, for trivial superconductor junction, there is no perfect Andreev\nreflection at zero energy, and zero energy conductance peak is almost quantized at 2 e2/hfor some particular values\nof system parameters as shown in Fig. 2(a). In Fig. 2(f), we also plot bound state energies vs Zand see that in the\ncase of N 1-SF-N 2-p-wave superconductor junction energy bound states merge at zero energy for all values of interface\ntransparency Z. In the case of topological superconductor junction, E±is zero regardless of different parameters like\nS,f,J,kFaetc. Energy bound states for spin up electron incident with S= 1/2,m′=−1/2 are same as energy\nbound states for a spin down electron incident with S= 1/2,m′= 1/2. For S= 1/2,m′= 1/2 (m′=−1/2), there\nare no energy bound states below the gap for a spin up (down) electron incident in case of trivial superconductor\njunction.\nB. Local density of states (LDOS)\nHerein we discuss our proposed recipe for distinguishing non-topological zero energy conductance peaks from topo-\nlogical zero energy conductance peaks via LDOS, LMDOS, and SPLDOS. For LDOS, in case of N 1-SF-N 2-s-wave\nsuperconductor junction we get from Eq. (23)\nν(x, ω) =\n\n1\nπIm\"\niη(2+b11e−i2qe(x+a)+b22e−i2qe(x+a))\n2qe#\n,(N1region)\ne−2µx×\"\n1\nπIm[γ1]#\n,(S region)(25a)\nwhere µ=√\n∆2−ω2[kF/(2EF)] and,\nγ1=iη((b51+b62)ei2kFx(kF−iµ)u2+ 2(a72+a81)kFuv+ (b71+b82)e−i2kFx(kF+iµ)v2+ 2e2µx(kF−iµ(u2−v2)))\n2(u2−v2)(k2\nF+µ2).\n(25b)8\n-2 -1 0 1 2ω/Δ0.51.01.52.0Gc/GN(a)\n-2 -1 0 1 2ω/Δ1234Gc/GN (b)\n0.0 0.5 1.0 1.5 2.0 2.5 3.0Z0.51.01.52.0Gc/GN\n(c)\n0.0 0.5 1.0 1.5 2.0 2.5 3.0Z1234Gc/GN (d)\n0.5 1.0 1.5 2.0 2.5 3.0Z\n-1.0-0.50.51.0E±/△\n0.7 0.9 1.1-0.0800.08\n(e)\n0.5 1.0 1.5 2.0 2.5 3.0Z\n-1.0-0.50.51.0E±/△ (f)\nFIG. 2: Charge conductance vs ωfor (a) N 1-SF-N 2-s-wave superconductor junction and (b) N 1-SF-N 2-p-wave\nsuperconductor junction. Charge conductance vs Zfor (c) N 1-SF-N 2-s-wave superconductor junction and (d)\nN1-SF-N 2-p-wave superconductor junction. Bound state energies vs Zfor (e) N 1-SF-N 2-s-wave superconductor junction and\n(f) N 1-SF-N 2-p-wave superconductor junction. Parameters: S= 1/2,J= 3.9,Z= 0.7(for (a) and (b)), ω= 0(for (c) and\n(d)), EF= 10∆ ,kFa= 0.84π. In (e) and (f) we consider the sepcific case of S=−m′= 1/2.\nHowever, for N 1-SF-N 2-p-wave superconductor (topological) junction we get from Eq. (23)\nν(x, ω) =\n\n1\nπIm\"\niη(2+b11e−i2qe(x+a)+b22e−i2qe(x+a))\n2qe#\n,(N1region)\ne−2µx×\"\n1\nπIm[ϱ1]#\n,(S region)(26a)\nwhere, ϱ1=iη((b51+b62)ei2kFx(kF−iµ)u2+ 2(a71+a82)kFuv−(b71+b82)e−i2kFx(kF+iµ)v2+ 2e2µx(kF(u2−v2)−iµ))\n2(k2\nF+µ2).\n(26b)\nAtω= 0, there is a perfect Andreev reflection in case of N 1-SF-N 2-p-wave superconductor junction, therefore\nb11=b22= 0 in Eq. (26a) and zero energy LDOS is constant atη\nqeπin the left metallic region. However, in the\ncase of N 1-SF-N 2-s-wave superconductor junction, there is no perfect Andreev reflection at ω= 0, therefore b119\n-30 -20 -10 0 100.00.51.01.52.0\nx/ξν(a)\n-30 -20 -10 0 100.00.51.01.52.0\nx/ξν0 100.00.20.40.60.81.01.2 (b)\nFIG. 3: Spatial dependence of the zero-energy LDOS in metal and superconducting region in the presence of spin-flip\nscattering for (a) N 1-SF-N 2-s-wave superconductor junction, (b) N 1-SF-N 2-p-wave superconductor junction. Parameters:\nS= 1/2,J= 3.9,Z= 0.7,kFa= 0.84π,ω= 0,EF= 10∆ .\nandb22are finite in Eq. (25a) and zero energy LDOS exhibits a nice oscillation in the left metallic region. In the\nsuperconducting region b82=b∗\n51andb71=b∗\n62, therefore only the term 2( a72+a81)kFuv, which is independent\nof position ( x), contributes. Thus the zero energy LDOS exhibits an exponential decay due to the factor e−2µxin\nEq. (25a) for the case of a trivial superconductor. For N 1-SF-N 2-p-wave superconductor junction, the first three\nterms in Eq. (26b) contribute to zero energy LDOS. Thus the zero energy LDOS shows an oscillatory decay in the\nS region. Since our interest is in zero energy states, see Fig. 3, wherein we show the spatial dependence of the\nzero-energy LDOS in metallic and superconducting regions for (a) N 1-SF-N 2-s-wave superconductor junction and\n(b) N 1-SF-N 2-p-wave superconductor junction for parameters when both trivial and topological zero energy peaks\nappear in the conductance spectra. For trivial superconductor junction, zero energy LDOS shows a nice oscillation\nin the metallic region with periodπ\nqe. However, the topological superconductor junction is independent of xin the\nmetallic region. The reason is that there is perfect Andreev reflection at zero energy in the topological superconductor\njunction. Therefore, zero energy LDOS has a constant value ofη\nqeπin the normal metal region. However, in the case\nof trivial superconductor junction, due to the presence of the term e−i2qe(x+a), zero energy LDOS show an oscillatory\nbehaviour in the normal metal region. In the superconducting region, zero-energy LDOS show an exponential decay\nin the case of trivial superconductor junction, while in the case of topological superconductor junction zero-energy,\nLDOS exhibits an oscillatory decay with periodπ\nkF. Since zero-energy LDOS shows different behaviour for trivial and\ntopological junctions, thus via LDOS, one can distinguish a trivial zero-energy peak from a topological zero-energy\npeak.\nC. Local magnetization density of states (LMDOS)\nFor LMDOS, in case of N 1-SF-N 2-s-wave superconductor junction using Eq. (23) we obtain,\nm(x, ω) =\n\n1\nπIm\"\niη(b12+b21)e−i2qe(x+a)\n2qe#\nˆx,(N1region)\ne−2µx×1\nπIm[γ2]ˆx,(S region)(27a)\nwhere, γ2=−iη((b72+b81)e−i2kFx(kF+iµ)v2+ ((a51+a62+a71+a82)kF−iκ(a51+a62−a71−a82))uv+ (b52+b61)ei2kFx(kF−iµ)u2)\n2(u2−v2)(k2\nF+µ2).\n(27b)10\nFor N 1-SF-N 2-p-wave superconductor junction using Eq. (23) we obtain,\nm(x, ω) =\n\n1\nπIm\"\niη(b12+b21)e−i2qe(x+a)\n2qe#\nˆx,(N1region)\ne−2µx×1\nπIm[ϱ2]ˆx,(S region)(28a)\nwhere, ϱ2=iη((b52+b61)ei2kFx(kF−iµ)u2−((a52+a61−a72−a81)kF−iκ(a52+a61+a72+a81))uv−(b72+b81)e−i2kFx(kF+iµ)v2)\n2(k2\nF+µ2).\n(28b)\nAt zero energy ( ω= 0), there is perfect Andreev reflection in the case of topological superconductor junction. Therefore\nb12=b21= 0 in Eq. (28a) and zero energy LMDOS vanishes in the left metallic region. In Fig. 4, zero-energy LMDOS is\nplotted as a function of position in both metallic and superconducting regions for (a) N 1-SF-N 2-s-wave superconductor\njunction and (b) N 1-SF-N 2-p-wave superconductor junction for parameters when both trivial and topological zero\nenergy peaks appear in the conductance spectra. For N 1-SF-N 2-s-wave superconductor junction zero energy LMDOS\n-30 -20 -10 0 100.00.20.40.60.81.01.21.4\nx/ξ|m|-20 -1000.030.06(a)\n-30 -20 -10 0 100.00.10.20.30.40.50.60.7\nx/ξ|m|0 100.00.10.20.30.40.50.60.7 (b)\nFIG. 4: Spatial dependence of the zero-energy LMDOS in metal and superconducting region for (a) N 1-SF-N 2-s-wave\nsuperconductor junction, (b) N 1-SF-N 2-p-wave superconductor junction. Parameters: S= 1/2,J= 3.9,Z= 0.7,\nkFa= 0.84π,ω= 0,EF= 10∆ .\nshows a nice periodic oscillation in the normal metal region; however, for N 1-SF-N 2-p-wave superconductor junction\nLMDOS vanishes in the metallic region. Zero-energy LMDOS shows an oscillatory decay for trivial and topological\nsuperconductor junctions in the superconducting region. However, in the topological superconductor, LMDOS exhibits\na superposition of two different periodic oscillations to that seen in a trivial superconductor junction where LMDOS\nexhibits no such superposition. Thus, measuring LMDOS at zero energy distinguishes a trivial zero energy conductance\npeak from a topological zero energy conductance peak. In the absence of a spin flipper, LMDOS vanishes in both\ntrivial and topological superconductor junctions. Further, there isn’t any trivial zero energy conductance peak since\nbound state energies do not merge. Thus, in our work, the role of the spin flipper is crucial to distinguish trivial zero\nenergy conductance peaks from topological zero energy conductance peaks.\nD. Spin polarized local density of states (SPLDOS)\nFinally, from LDOS and LMDOS, we can calculate SPLDOS; see below Eq. (23). In case of N 1-SF-N 2-s-wave\nsuperconductor junction we obtain,\nνσ(x, ω) =\n\n1\n2πIm\"\niη(2+b11e−i2qe(x+a)+b22e−i2qe(x+a))\n2qe#\n+σ\n2πvuutIm\"\niη(b12+b21)e−i2qe(x+a)\n2qe#2\n,(N1region)\ne−2µx×\"\n1\n2πIm[γ1] +σ\n2πp\nIm[γ2]2#\n.(S region)(29)11\nFor N 1-SF-N 2-p-wave superconductor junction we obtain,\nνσ(x, ω) =\n\n1\n2πIm\"\niη(2+b11e−i2qe(x+a)+b22e−i2qe(x+a))\n2qe#\n+σ\n2πvuutIm\"\niη(b12+b21)e−i2qe(x+a)\n2qe#2\n,(N1region)\ne−2µx×\"\n1\n2πIm[ϱ1] +σ\n2πp\nIm[ϱ2]2#\n.(S region)(30)\nEqs. (29),(30) reveal that SPLDOS includes contributions from both the bulk and surface in both trivial and topological\nsuperconductor junctions. LMDOS, on the other hand, exclusively consists of surface contributions, distinguishing\nit from SPLDOS. However, the spatial characteristics of both LMDOS and SPLDOS are nearly identical, as they\nboth depend on the exponential factors e−i2qe(x+a)in the metallic region and e−2µxin the superconducting region,\nand these factors are independent of spin flip scattering. In Fig. 5, we show the spatial dependence of the zero-\nenergy SPLDOS in metallic and superconducting regions for (a) trivial superconductor junction and (b) topological\nsuperconductor junction for parameters when both trivial and topological zero energy peaks appear in the conductance\nspectra. In the left normal metal region, zero-energy SPLDOS shows a periodic oscillation with periodπ\nqefor N 1-SF-\nν↑\nν↓\n-30 -20 -10 0100.00.51.01.5\nx/ξ-20 -100.050.150.250 100.00.10.20.30.40.50.6(a)\nν↑\nν↓\n-30 -20 -10 0100.00.20.40.60.81.01.2\nx/ξ0 100.00.20.40.60.8 (b)\nFIG. 5: Spatial dependence of the zero-energy SPLDOS in metal and superconducting region for (a) N 1-SF-N 2-s-wave\nsuperconductor junction, (b) N 1-SF-N 2-p-wave superconductor junction. Parameters: S= 1/2,J= 3.9,Z= 0.7,\nkFa= 0.84π,ω= 0,EF= 10∆ .\nN2-s-wave superconductor junction, while for N 1-SF-N 2-p-wave superconductor junction zero-energy SPLDOS is at a\nconstant valueη\n2qeπ. In the superconducting region, zero-energy SPLDOS exhibits an oscillatory decay with period\nπ\nkFin both trivial and topological superconductor wave junctions. However, for N 1-SF-N 2-p-wave superconductor\njunction zero-energy SPLDOS shows the superposition of two different periodic oscillations in contrast to N 1-SF-N 2-\ns-wave superconductor junction. Since zero-energy SPLDOS exhibits distinct behaviours for trivial and topological\nsuperconductor junctions, thus, via measuring zero-energy SPLDOS, one can also distinguish the trivial zero-energy\npeak from the topological zero-energy peak. In this section, we have discussed the results for our first setup (Fig. 1(a)),\nand in the next section, we show the results for our second setup (Fig. 1(b)).\nIV. RESULTS FOR THE SECOND SETUP\nA. Local density of states (LDOS)\nFor LDOS, when our system as shown in Fig. 1(b) is in trivial regime we get from Eq. (23)\nν(x, ω) =\n\n1\nπIm\"\niη(2+b′\n11e−2i(x+a)+b′\n22e−2i(x+a))\n2#\n,(N1region)\n1\nπIm[α1],(S region)(31a)12\nwhere, α1=iη\n2(q+κ+−q−κ−)q\n(1−κ2\n+)(1−κ2\n−)\u0010\nei(q++q−)x((a′\n51+a′\n62)κ+−(a′\n71+a′\n82)κ−) +\u0000\n2κ+−2κ−−e2iq+x\n(b′\n71+b′\n82)κ++e2iq−x(b′\n51+b′\n62)κ−\u0001q\n(1−κ2\n+)(1−κ2\n−) +κ+κ−ei(q++q−)x((a′\n71+a′\n82)κ+−(a′\n51+a′\n62)κ−)\u0011\n.\n(31b)\nHowever, in the topological regime, we get from Eq. (23)\nν(x, ω) =\n\n1\nπIm\"\niη(2+b′\n11e−2i(x+a)+b′\n22e−2i(x+a))\n2#\n(N1region)\n−1\nπIm[β1],(S region)(32a)\nwhere, β1=iη(2κ−−2κ++e2iq+x(b′\n82+b′\n71)κ+−e2iq−x(b′\n51+b′\n62)κ−+ei(q++q−)x((a′\n82+a′\n71)κ−−(a′\n51+a′\n62)κ+))\n2(q+κ+−q−κ−).\n(32b)\nAt zero energy, there is a perfect Andreev reflection in the topological regime, thus b′\n11=b′\n22= 0 in Eq. (32a).\nTherefore, zero energy LDOS is constant atη\nπin the left normal metal region. However, in the trivial regime, there\nis no perfect Andreev reflection at zero energy, thus b′\n11andb′\n22are finite in Eq. (31a) and zero energy LDOS shows\na nice periodic oscillation with period π. In the superconducting region, we find that κ±=±i,a′\n51=a′\n82(both are\nreal), a′\n62=a′\n71(both are real), b′\n51=b′\n82(both are imaginary), b′\n62=b′\n71(both are imaginary) and q±are imaginary\nin trivial regime at zero energy. Therefore only the fifth and sixth terms in Eq. (31a) contribute to zero energy LDOS,\nand zero energy LDOS shows a rapid decay without any oscillation in the S region. For the topological regime, we\n-30 -20 -10 0 100.00.10.20.30.40.50.60.7\nx/ξν(a)\n-30 -20 -10 0 100.00.20.40.60.81.01.2\nx/ξν (b)\nFIG. 6: Spatial dependence of the zero-energy LDOS in the metal and superconducting region for second setup in (a) trivial\nregime and (b) topological regime. Parameters: S= 1/2,J= 3.9,Z= 0.7,kFa= 0.84π,µpSc=−3(for (a)), µpSc= 1(for\n(b)), ∆pSc= 0.05,ω= 0,EF= 10∆ .\nfind that κ+=κ−=i,b′\n51=−b′∗\n82,b′\n62=−b′∗\n71,a′\n51=−a′\n82(both are real), a′\n62=−a′\n71(both are real), q∗\n−=−q+at\nzero energy and thus only the third and fourth terms in Eq. (32a) contribute to zero energy LDOS and zero energy\nLDOS show an oscillatory decay. Since we concentrate on zero energy states, in Fig. 6, we plot zero-energy LDOS as\na function of position xin both metallic and superconducting regions for a second setup. We see that in the trivial\nregime (Fig. 6(a)), zero-energy LDOS shows a nice periodic oscillation with period πin the metallic region. In the\nsuperconducting region, it rapidly decays without any oscillation. However, in the topological regime (Fig. 6(b)),\nzero-energy LDOS is constant with value η/πin the metallic region. In the superconducting region, it shows an\noscillatory decay with periodπ\n(µpSc−delta2\npSc/2).13\nB. Local magnetization density of states (LMDOS)\nFor LMDOS, when our system (Fig. 1(b)) is in trivial regime, using Eq. (23) we obtain\nm(x, ω) =\n\n1\nπIm\"\niη(b′\n12+b′\n21)e−2i(x+a)\n2#\nˆx,(N1region)\n−1\nπIm[α2]ˆx,(S region)(33a)\nwhere, α2=iη\n2(q+κ+−q−κ−)q\n(1−κ2\n+)(1−κ2\n−)\u0010\ne2iq+x(b′\n81+b′\n72)κ+q\n(1−κ2\n+)(1−κ2\n−)−e2iq−x(b′\n52+b′\n61)κ−\nq\n(1−κ2\n+)(1−κ2\n−) +ei(q++q−)x((a′\n81+a′\n72)κ−(1−κ2\n+)−(a′\n52+a′\n61)κ+(1−κ2\n−))\u0011\n.\n(33b)\nFor topological regime using Eq. (23), we obtain\nm(x, ω) =\n\n1\nπIm\"\niη(b′\n12+b′\n21)e−2i(x+a)\n2#\nˆx,(N1region)\n−1\nπIm[β2]ˆx,(S region)(34a)\nwhere, β2=iη(e2iq+x(b′\n72+b′\n81)κ+−e2iq−x(b′\n52+b′\n61)κ−+ei(q++q−)x((a′\n72+a′\n81)κ−−(a′\n52+a′\n61)κ+))\n2(q+κ+−q−κ−).(34b)\nAtω= 0, there is a perfect Andreev reflection in the topological regime. Thus, b′\n12=b′\n21= 0 in Eq. (34a) and zero\nenergy LMDOS becomes zero in the left metallic region. In Fig. 7, we show the spatial dependence of the zero-energy\nLMDOS in metallic and superconducting regions for the second setup. Zero-energy LMDOS is computed by taking an\naverage over all possible values of m′for spin flipper’s spin S. We find that in the trivial regime (Fig. 7(a)), zero-energy\nLMDOS exhibits a nice periodic oscillation in the metallic region; however, in the superconducting region, LMDOS\nalmost vanishes. In the topological regime (Fig. 7(b)), zero-energy LMDOS vanishes in the metallic region and in the\nsuperconducting region it exhibits an oscillatory decay with superposition of two different periodic oscillations.\n-30 -20 -10 0 100.000.050.100.150.20\nx/ξ|m|-20 -1000.0050.01(a)\n-30 -20 -10 0 100.00.10.20.30.40.50.60.7\nx/ξ|m|0 1000.250.5 (b)\nFIG. 7: Spatial dependence of the zero-energy LMDOS in the metal and superconducting region for second setup in (a)\ntrivial regime and (b) topological regime. Parameters: S= 1/2,J= 3.9,Z= 0.7,kFa= 0.84π,µpSc=−3(for (a)), µpSc= 1\n(for (b)), ∆pSc= 0.05,ω= 0,EF= 10∆ .14\nC. Spin polarized local density of states (SPLDOS)\nFinally, from LDOS and LMDOS, we can get SPLDOS; see below Eq. (23).For SPLDOS, when our system (Fig. 1(b))\nis in trivial regime we obtain,\nνσ(x, ω) =\n\n1\n2πIm\"\niη(2+b′\n11e−2i(x+a)+b′\n22e−2i(x+a))\n2#\n+σ\n2πvuutIm\"\niη(b′\n12+b′\n21)e−2i(x+a)\n2#2\n,(N1region)\n×\"\n1\n2πIm[α1] +σ\n2πp\nIm[α2]2#\n.(S region)(35)\nFor topological regime, we obtain,\nνσ(x, ω) =\n\n1\n2πIm\"\niη(2+b′\n11e−2i(x+a)+b′\n22e−2i(x+a))\n2#\n+σ\n2πvuutIm\"\niη(b′\n12+b′\n21)e−2i(x+a)\n2#2\n,(N1region)\n×\"\n1\n2πIm[β1] +σ\n2πp\nIm[β2]2#\n.(S region)(36)\nFrom Eqs. (35),(36), it is evident that SPLDOS has bulk and surface contributions in both trivial and topological\nregimes. However, LMDOS is solely composed of surface contributions, distinguishing it from SPLDOS. In Fig. 8, we\npresent the spatial dependence of the zero-energy SPLDOS in metallic and superconducting regions for the second\nsetup. We see that in the trivial regime (Fig. 8(a)), zero-energy SPLDOS exhibits a periodic oscillation with period\nπin the metallic region. In the superconducting region, it first shows rapid decay, then vanishes. However, in\nthe topological regime (Fig. 8(b)), zero-energy SPLDOS is constant with valueη\n2π, in the metallic region. For the\nsuperconducting region, it shows an oscillatory decay with superposition of two different periodic oscillations with\nperiodπ\n(µpSc−delta2\npSc/2). Thus, the obtained results for LDOS, LMDOS and SPLDOS in the second setup match\nquite well with the first setup.\nν↑\nν↓\n-30 -20 -10 0 100.00.10.20.30.4\nx/ξ(a)\nν↑ν↓\n-30 -20 -10 0 100.00.20.40.60.8\nx/ξ0 1000.30.6 (b)\nFIG. 8: Spatial dependence of the zero-energy SPLDOS in the metal and superconducting region for second setup in (a)\ntrivial regime and (b) topological regime. Parameters: S= 1/2,J= 3.9,Z= 0.7,kFa= 0.84π,µpSc=−3(for (a)), µpSc= 1\n(for (b)), ∆pSc= 0.05,ω= 0,EF= 10∆ .\nV. FINITE TEMPERATURE EFFECT\nIn sections III and IV, we have discussed our results at zero temperature. This section discusses the effect of\nfinite temperature on our results. To calculate LDOS at the finite temperature, we take the temperature-dependent15\nsuperconducting gap, which follows that ∆( T) = ∆(0) tanh\u0010\n1.74q\nTc\nT−1\u0011\n, where T cis the critical temperature[59].\nIn Fig. 9, we plot the zero energy LDOS at finite temperature as a function of position in both metallic and supercon-\nducting regions for (a) N 1-SF-N 2-s-wave superconductor junction and (b) N 1-SF-N 2-p-wave superconductor junction.\nWe see that zero energy LDOS exhibits a nice periodic oscillation in the metallic region for trivial superconductor\njunctions. However, LDOS is independent of position in the metallic region for topological superconductor junction.\nIn the superconducting region, zero-energy LDOS exhibit an exponential decay in the case of N 1-SF-N 2-s-wave su-\nperconductor junction, while in the case of N 1-SF-N 2-p-wave superconductor junction zero-energy LDOS show an\noscillatory decay. At finite temperatures, the nature of oscillations does not change; only magnitude changes. Thus,\nthe main conclusion of our work remains unchanged even at finite temperatures.\n-30 -20 -10 0 100.00.51.01.52.02.5\nx/ξν(a)\n-30 -20 -10 0 100.00.51.01.52.0\nx/ξν0 100.00.20.40.60.81.01.2 (b)\nFIG. 9: Spatial dependence of the zero-energy LDOS in metal and superconducting region in the presence of spin-flip\nscattering at finite temperature for (a) N 1-SF-N 2-s-wave superconductor junction, (b) N 1-SF-N 2-p-wave superconductor\njunction. Parameters: S= 1/2,J= 3.9,Z= 0.7,T= 0.5Tc,kFa= 0.84π,ω= 0,EF= 10∆ .\nVI. ANALYSIS\nIn this section, we analyze our obtained results for both setups. Table I compares zero-energy LDOS, LMDOS, and\nSPLDOS between trivial and topological cases for our two setups. For the first setup, we see that zero-energy LDOS\nand SPLDOS exhibit oscillations in the metallic region in the case of a trivial superconductor junction. However,\nzero energy LDOS and SPLDOS do not show any oscillations for topological superconductor junctions. In the\nsuperconducting region, zero energy LDOS shows an exponential decay for trivial superconductor junction, but it\nshows an oscillatory decay for topological superconductor junction. Zero energy SPLDOS and LMDOS show an\noscillatory decay in the superconducting region for both trivial and topological superconductor junctions. However,\nfor topological superconductor junctions, zero energy SPLDOS and LMDOS exhibit the superposition of two different\nperiodic oscillations compared to trivial superconductor junctions. For the second setup, we notice that in the trivial\nregime, zero-energy LDOS and SPLDOS show periodic oscillations in the metallic region. However, in the topological\nregime, they do not exhibit any oscillations. Zero-energy LDOS in the trivial regime exhibits a rapid decay in the\nsuperconducting region; however, it shows an oscillatory decay in the topological regime. Zero-energy SPLDOS and\nLMDOS exhibit the superposition of two different periodic oscillations in the topological regime. To conclude our\nanalysis, the results obtained for the second setup are very similar to those for the first setup in the topological regime\nmarking the universal nature of the probes.\nVII. EXPERIMENTAL REALIZATION AND CONCLUSION\nThe setups, as seen in Figs. 1(a), 1(b), can be realized in a lab. The Metal-Insulator-Superconductor junctions\nhave been realized experimentally for more than 40 years[60]. Replacing a spin flipper at the Metal-Superconductor\ninterface should not be difficult, especially with a s-wave or p-wave superconductor. It should be perfectly possible.16\nTABLE I: Comparison of LDOS, LMDOS and SPLDOS between trivial and topological superconductor junction at\nzero energy\nTrivial junction Topological junction\nMetallic region Superconducting region Metallic region Superconducting region\nFirst setup Second setup First setup Second setup First setup Second setup First setup Second setup\nLDOS at\nω= 0Periodic oscil-\nlation with pe-\nriodπ\nqePeriodic oscilla-\ntion with period\nπExponential\ndecayRapid decay Constant Constant Oscillatory de-\ncay with period\nπ\nkFOscillatory de-\ncay with period\nπ\n(µpSc−∆2\npSc/2)\nLMDOS\natω= 0Periodic oscil-\nlation with pe-\nriodπ\nqePeriodic oscilla-\ntion with period\nπOscillatory de-\ncay with period\nπ\nkFRapid decay Zero Zero Superposition\nof two different\nperiodic oscil-\nlations with\nperiodπ\nkFSuperposition\nof two differ-\nent periodic\noscillations\nwith period\nπ\n(µpSc−∆2\npSc/2)\nSPLDOS\natω= 0Periodic oscil-\nlation with pe-\nriodπ\nqePeriodic oscilla-\ntion with period\nπOscillatory de-\ncay with period\nπ\nkFRapid decay Constant Constant Superposition\nof two different\nperiodic oscil-\nlations with\nperiodπ\nkFSuperposition\nof two differ-\nent periodic\noscillations\nwith period\nπ\n(µpSc−∆2\npSc/2)\nExperimentally, high spin molecules like Fe 19-complex with a spin of S= 33/2[61] or Mn 4O3complex with a spin of\nS= 9/2[62] can to a certain extent be a model for the spin flipper. The internal dynamics of such a high-spin molecule\nare perhaps quite distinct from the spin-flipper. However, the spin flipper can mimic the half-integer spin states up\nto any arbitrary high value, the associated spin magnetic moment of the high spin molecule, and the consequence of\nan electron interacting with such, to a large extent. LDOS and LMDOS/SPLDOS can be measured experimentally\nusing a scanning tunnelling microscope and spin-polarized scanning tunnelling microscope, respectively[63, 64]. STM\nmeasurements on a nanowire covered by a superconductor are done in Ref. [65], using InAs nanowires covered by an\nAl superconductor.\nTo conclude, in this work, we have shown a way to distinguish trivial zero energy conductance peaks from topological\nzero energy conductance peaks via LDOS and LMDOS. For high values of the exchange interaction and low values of\nspin flipper’s spin, a ZBCP arises for trivial superconductor junction, which is similar to that of ZBCP for topological\nsuperconductor junction. We examine zero energy LDOS, LMDOS, and SPLDOS for parameters when both trivial\nand topological zero energy peaks appear in the conductance spectra. Table I compares zero energy LDOS, LMDOS,\nand SPLDOS between trivial and topological superconductor junctions. We find that zero-energy LDOS and SPLDOS\nshow a periodic oscillation in the metallic region in the case of trivial superconductor junction, but for topological\nsuperconductor junction, they do not exhibit any oscillations. Zero-energy LDOS exhibits an exponential decay in the\nsuperconducting region for trivial superconductor junction. However, zero-energy LDOS exhibits an oscillatory decay\nfor topological superconductor junctions. Our work may help experimentalists in distinguishing trivial zero-energy\nconductance peaks from topological zero energy conductance peaks by measuring LDOS, LMDOS, and SPLDOS.\nACKNOWLEDGMENTS\nThe grants which supported this work: 1. Josephson junctions with strained Dirac materials and their application\nin quantum information processing, SERB Grant No. CRG/20l9/006258, and 2. Nash equilibrium versus Pareto\noptimality in N-Player games, SERB MATRICS Grant No. MTR/2018/000070.17\nAppendix A: Wavefunctions in the first setup\nThe wavefunctions in different regions of Normal metal-Spin flipper-Normal metal-Superconductor junction with\ns-wave superconductor are given as-\nψ1(x) =\n\nφN\n1eiqe(x+a)ϕS\nm′+a11φN\n3eiqh(x+a)ϕS\nm′+1+a12φN\n4eiqh(x+a)ϕS\nm′+b11φN\n1e−iqe(x+a)ϕS\nm′+b12φN\n2e−iqe(x+a)ϕS\nm′+1, x < −a\nc11φN\n1eiqe(x+a)ϕS\nm′+c12φN\n2eiqe(x+a)ϕS\nm′+1+d11φN\n1e−iqexϕS\nm′+d12φN\n2e−iqexϕS\nm′+1+e11φN\n3eiqhxϕS\nm′+1+e12φN\n4eiqhxϕS\nm′\n+f11φN\n3e−iqh(x+a)ϕS\nm′+1+f12φN\n4e−iqh(x+a)ϕS\nm′, −a < x < 0\ng11φS\n1eiqS\nexϕS\nm′+g12φS\n2eiqS\nexϕS\nm′+1+h11φS\n3e−iqS\nhxϕS\nm′+1+h12φS\n4e−iqS\nhxϕS\nm′, x > 0\nψ2(x) =\n\nφN\n2eiqe(x+a)ϕS\nm′+a21φN\n3eiqh(x+a)ϕS\nm′+a22φN\n4eiqh(x+a)ϕS\nm′−1+b21φN\n1e−iqe(x+a)ϕS\nm′−1+b22φN\n2e−iqe(x+a)ϕS\nm′, x < −a\nc21φN\n1eiqe(x+a)ϕS\nm′−1+c22φN\n2eiqe(x+a)ϕS\nm′+d21φN\n1e−iqexϕS\nm′−1+d22φN\n2e−iqexϕS\nm′+e21φN\n3eiqhxϕS\nm′+e22φN\n4eiqhxϕS\nm′−1\n+f21φN\n3e−iqh(x+a)ϕS\nm′+f22φN\n4e−iqh(x+a)ϕS\nm′−1, −a < x < 0\ng21φS\n1eiqS\nexϕS\nm′−1+g22φS\n2eiqS\nexϕS\nm′+h21φS\n3e−iqS\nhxϕS\nm′+h22φS\n4e−iqS\nhxϕS\nm′−1, x > 0\nψ3(x) =\n\nφN\n3e−iqh(x+a)ϕS\nm′+a31φN\n1eiqe(x+a)ϕS\nm′−1+a32φN\n2eiqe(x+a)ϕS\nm′+b31φN\n3e−iqh(x+a)ϕS\nm′+b32φN\n4e−iqh(x+a)ϕS\nm′−1, x < −a\nc31φN\n1eiqe(x+a)ϕS\nm′−1+c32φN\n2eiqe(x+a)ϕS\nm′+d31φN\n1e−iqexϕS\nm′−1+d32φN\n2e−iqexϕS\nm′+e31φN\n3eiqhxϕS\nm′+e32φN\n4eiqhxϕS\nm′−1\n+f31φN\n3e−iqh(x+a)ϕS\nm′+f32φN\n4e−iqh(x+a)ϕS\nm′−1, −a < x < 0\ng31φS\n1eiqS\nexϕS\nm′−1+g32φS\n2eiqS\nexϕS\nm′+h31φS\n3e−iqS\nhxϕS\nm′+h32φS\n4e−iqS\nhxϕS\nm′−1, x > 0\nψ4(x) =\n\nφN\n4e−iqh(x+a)ϕS\nm′+a41φN\n1eiqe(x+a)ϕS\nm′+a42φN\n2eiqe(x+a)ϕS\nm′+1+b41φN\n3e−iqh(x+a)ϕS\nm′+1+b42φN\n4e−iqh(x+a)ϕS\nm′, x < −a\nc41φN\n1eiqe(x+a)ϕS\nm′+c42φN\n2eiqe(x+a)ϕS\nm′+1+d41φN\n1e−iqexϕS\nm′+d42φN\n2e−iqexϕS\nm′+1+e41φN\n3eiqhxϕS\nm′+1+e42φN\n4eiqhxϕS\nm′\n+f41φN\n3e−iqh(x+a)ϕS\nm′+1+f42φN\n4e−iqh(x+a)ϕS\nm′, −a < x < 0\ng41φS\n1eiqS\nexϕS\nm′+g42φS\n2eiqS\nexϕS\nm′+1+h41φS\n3e−iqS\nhxϕS\nm′+1+h42φS\n4e−iqS\nhxϕS\nm′, x > 0\nψ5(x) =\n\ng51φN\n1eiqe(x+a)ϕS\nm′+g52φN\n2eiqe(x+a)ϕS\nm′+1+h51φN\n3e−iqh(x+a)ϕS\nm′+1+h52φN\n4e−iqh(x+a)ϕS\nm′, x < −a\nc51φN\n1eiqe(x+a)ϕS\nm′+c52φN\n2eiqe(x+a)ϕS\nm′+1+d51φN\n1e−iqexϕS\nm′+d52φN\n2e−iqexϕS\nm′+1+e51φN\n3eiqhxϕS\nm′+1+e52φN\n4eiqhxϕS\nm′\n+f51φN\n3e−iqh(x+a)ϕS\nm′+1+f52φN\n4e−iqh(x+a)ϕS\nm′, −a < x < 0\nφS\n1e−iqS\nexϕS\nm′+a51φS\n3e−iqS\nhxϕS\nm′+1+a52φS\n4e−iqS\nhxϕS\nm′+b51φS\n1eiqS\nexϕS\nm′+b52φS\n2eiqS\nexϕS\nm′+1, x > 0\nψ6(x) =\n\ng61φN\n1eiqe(x+a)ϕS\nm′−1+g62φN\n2eiqe(x+a)ϕS\nm′+h61φN\n3e−iqh(x+a)ϕS\nm′+h62φN\n4e−iqh(x+a)ϕS\nm′−1, x < −a\nc61φN\n1eiqe(x+a)ϕS\nm′−1+c62φN\n2eiqe(x+a)ϕS\nm′+d61φN\n1e−iqexϕS\nm′−1+d62φN\n2e−iqexϕS\nm′+e61φN\n3eiqhxϕS\nm′+e62φN\n4eiqhxϕS\nm′−1\n+f61φN\n3e−iqh(x+a)ϕS\nm′+f62φN\n4e−iqh(x+a)ϕS\nm′−1, −a < x < 0\nφS\n2e−iqS\nexϕS\nm′+a61φS\n3e−iqS\nhxϕS\nm′+a62φS\n4e−iqS\nhxϕS\nm′−1+b61φS\n1eiqS\nexϕS\nm′−1+b62φS\n2eiqS\nexϕS\nm′, x > 0\nψ7(x) =\n\ng71φN\n1eiqe(x+a)ϕS\nm′−1+g72φN\n2eiqe(x+a)ϕS\nm′+h71φN\n3e−iqh(x+a)ϕS\nm′+h72φN\n4e−iqh(x+a)ϕS\nm′−1, x < −a\nc71φN\n1eiqe(x+a)ϕS\nm′−1+c72φN\n2eiqe(x+a)ϕS\nm′+d71φN\n1e−iqexϕS\nm′−1+d72φN\n2e−iqexϕS\nm′+e71φN\n3eiqhxϕS\nm′+e72φN\n4eiqhxϕS\nm′−1\n+f71φN\n3e−iqh(x+a)ϕS\nm′+f72φN\n4e−iqh(x+a)ϕS\nm′−1, −a < x < 0\nφS\n3eiqS\nhxϕS\nm′+a71φS\n1e−iqS\nexϕS\nm′−1+a72φS\n2e−iqS\nexϕS\nm′+b71φS\n3eiqS\nhxϕS\nm′+b72φS\n4eiqS\nhxϕS\nm′−1, x > 0\nψ8(x) =\n\ng81φN\n1eiqe(x+a)ϕS\nm′+g82φN\n2eiqe(x+a)ϕS\nm′+1+h81φN\n3e−iqh(x+a)ϕS\nm′+1+h82φN\n4e−iqh(x+a)ϕS\nm′, x < −a\nc81φN\n1eiqe(x+a)ϕS\nm′+c82φN\n2eiqe(x+a)ϕS\nm′+1+d81φN\n1e−iqexϕS\nm′+d82φN\n2e−iqexϕS\nm′+1+e81φN\n3eiqhxϕS\nm′+1+e82φN\n4eiqhxϕS\nm′\n+f81φN\n3e−iqh(x+a)ϕS\nm′+1+f82φN\n4e−iqh(x+a)ϕS\nm′, −a < x < 0\nφS\n4eiqS\nhxϕS\nm′+a81φS\n1e−iqS\nexϕS\nm′+a82φS\n2e−iqS\nexϕS\nm′+1+b81φS\n3eiqS\nhxϕS\nm′+1+b82φS\n4eiqS\nhxϕS\nm′. x > 0(A1)\nSimilarly, the wavefunctions in different regions of our system with p-wave superconductor are given as-\nΨ1(x) =\n\nφN\n1eiqe(x+a)ϕS\nm′+a11φN\n4eiqh(x+a)ϕS\nm′+a12φN\n3eiqh(x+a)ϕS\nm′+1+b11φN\n1e−iqe(x+a)ϕS\nm′+b12φN\n2e−iqe(x+a)ϕS\nm′+1, x < −a\nc11φN\n1eiqe(x+a)ϕS\nm′+c12φN\n2eiqe(x+a)ϕS\nm′+1+d11φN\n1e−iqexϕS\nm′+d12φN\n2e−iqexϕS\nm′+1+e11φN\n4eiqhxϕS\nm′+e12φN\n3eiqhxϕS\nm′+1\n+f11φN\n4e−iqh(x+a)ϕS\nm′+f12φN\n3e−iqh(x+a)ϕS\nm′+1, −a < x < 0\ng11φS\n1eiqS\nexϕS\nm′+g12φS\n2eiqS\nexϕS\nm′+1+h11φS\n4e−iqS\nhxϕS\nm′+h12φS\n3e−iqS\nhxϕS\nm′+1, x > 0\nΨ2(x) =\n\nφN\n2eiqe(x+a)ϕS\nm′+a21φN\n4eiqh(x+a)ϕS\nm′−1+a22φN\n3eiqh(x+a)ϕS\nm′+b21φN\n1e−iqe(x+a)ϕS\nm′−1+b22φN\n2e−iqe(x+a)ϕS\nm′, x < −a\nc21φN\n1eiqe(x+a)ϕS\nm′−1+c22φN\n2eiqe(x+a)ϕS\nm′+d21φN\n1e−iqexϕS\nm′−1+d22φN\n2e−iqexϕS\nm′+e21φN\n4eiqhxϕS\nm′−1+e22φN\n3eiqhxϕS\nm′\n+f21φN\n4e−iqh(x+a)ϕS\nm′−1+f22φN\n3e−iqh(x+a)ϕS\nm′, −a < x < 0\ng21φS\n1eiqS\nexϕS\nm′−1+g22φS\n2eiqS\nexϕS\nm′+h21φS\n4e−iqS\nhxϕS\nm′−1+h22φS\n3e−iqS\nhxϕS\nm′, x > 018\nΨ3(x) =\n\nφN\n4e−iqh(x+a)ϕS\nm′+a31φN\n1eiqe(x+a)ϕS\nm′+a32φN\n2eiqe(x+a)ϕS\nm′+1+b31φN\n4e−iqh(x+a)ϕS\nm′+b32φN\n3e−iqh(x+a)ϕS\nm′+1, x < −a\nc31φN\n1eiqe(x+a)ϕS\nm′+c32φN\n2eiqe(x+a)ϕS\nm′+1+d31φN\n1e−iqexϕS\nm′+d32φN\n2e−iqexϕS\nm′+1+e31φN\n4eiqhxϕS\nm′+e32φN\n3eiqhxϕS\nm′+1\n+f31φN\n4e−iqh(x+a)ϕS\nm′+f32φN\n3e−iqh(x+a)ϕS\nm′+1, −a < x < 0\ng31φS\n1eiqS\nexϕS\nm′+g32φS\n2eiqS\nexϕS\nm′+1+h31φS\n4e−iqS\nhxϕS\nm′+h32φS\n3e−iqS\nhxϕS\nm′+1, x > 0\nΨ4(x) =\n\nφN\n3e−iqh(x+a)ϕS\nm′+a41φN\n1eiqe(x+a)ϕS\nm′−1+a42φN\n2eiqe(x+a)ϕS\nm′+b41φN\n4e−iqh(x+a)ϕS\nm′−1+b42φN\n3e−iqh(x+a)ϕS\nm′, x < −a\nc41φN\n1eiqe(x+a)ϕS\nm′−1+c42φN\n2eiqe(x+a)ϕS\nm′+d41φN\n1e−iqexϕS\nm′−1+d42φN\n2e−iqexϕS\nm′+e41φN\n4eiqhxϕS\nm′−1+e42φN\n3eiqhxϕS\nm′\n+f41φN\n4e−iqh(x+a)ϕS\nm′−1+f42φN\n3e−iqh(x+a)ϕS\nm′, −a < x < 0\ng41φS\n1eiqS\nexϕS\nm′−1+g42φS\n2eiqS\nexϕS\nm′+h41φS\n4e−iqS\nhxϕS\nm′−1+h42φS\n3e−iqS\nhxϕS\nm′, x > 0\nΨ5(x) =\n\ng51φN\n1eiqe(x+a)ϕS\nm′+g52φN\n2eiqe(x+a)ϕS\nm′+1+h51φN\n4e−iqh(x+a)ϕS\nm′+h52φN\n3e−iqh(x+a)ϕS\nm′+1, x < −a\nc51φN\n1eiqe(x+a)ϕS\nm′+c52φN\n2eiqe(x+a)ϕS\nm′+1+d51φN\n1e−iqexϕS\nm′+d52φN\n2e−iqexϕS\nm′+1+e51φN\n4eiqhxϕS\nm′+e52φN\n3eiqhxϕS\nm′+1\n+f51φN\n4e−iqh(x+a)ϕS\nm′+f52φN\n3e−iqh(x+a)ϕS\nm′+1, −a < x < 0\nφS\n1e−iqS\nexϕS\nm′+a51φS\n4e−iqS\nhxϕS\nm′+a52φS\n3e−iqS\nhxϕS\nm′+1+b51φS\n1eiqS\nexϕS\nm′+b52φS\n2eiqS\nexϕS\nm′+1, x > 0\nΨ6(x) =\n\ng61φN\n1eiqe(x+a)ϕS\nm′−1+g62φN\n2eiqe(x+a)ϕS\nm′+h61φN\n4e−iqh(x+a)ϕS\nm′−1+h62φN\n3e−iqh(x+a)ϕS\nm′, x < −a\nc61φN\n1eiqe(x+a)ϕS\nm′−1+c62φN\n2eiqe(x+a)ϕS\nm′+d61φN\n1e−iqexϕS\nm′−1+d62φN\n2e−iqexϕS\nm′+e61φN\n4eiqhxϕS\nm′−1+e62φN\n3eiqhxϕS\nm′\n+f61φN\n4e−iqh(x+a)ϕS\nm′−1+f62φN\n3e−iqh(x+a)ϕS\nm′, −a < x < 0\nφS\n2e−iqS\nexϕS\nm′+a61φS\n4e−iqS\nhxϕS\nm′−1+a62φS\n3e−iqS\nhxϕS\nm′+b61φS\n1eiqS\nexϕS\nm′−1+b62φS\n2eiqS\nexϕS\nm′, x > 0\nΨ7(x) =\n\ng71φN\n1eiqe(x+a)ϕS\nm′+g72φN\n2eiqe(x+a)ϕS\nm′+1+h71φN\n4e−iqh(x+a)ϕS\nm′+h72φN\n3e−iqh(x+a)ϕS\nm′+1, x < −a\nc71φN\n1eiqe(x+a)ϕS\nm′+c72φN\n2eiqe(x+a)ϕS\nm′+1+d71φN\n1e−iqexϕS\nm′+d72φN\n2e−iqexϕS\nm′+1+e71φN\n4eiqhxϕS\nm′+e72φN\n3eiqhxϕS\nm′+1\n+f71φN\n4e−iqh(x+a)ϕS\nm′+f72φN\n3e−iqh(x+a)ϕS\nm′+1, −a < x < 0\nφS\n4eiqS\nhxϕS\nm′+a71φS\n1e−iqS\nexϕS\nm′+a72φS\n2e−iqS\nexϕS\nm′+1+b71φS\n4eiqS\nhxϕS\nm′+b72φS\n3eiqS\nhxϕS\nm′+1, x > 0\nΨ8(x) =\n\ng81φN\n1eiqe(x+a)ϕS\nm′−1+g82φN\n2eiqe(x+a)ϕS\nm′+h81φN\n4e−iqh(x+a)ϕS\nm′−1+h82φN\n3e−iqh(x+a)ϕS\nm′, x < −a\nc81φN\n1eiqe(x+a)ϕS\nm′−1+c82φN\n2eiqe(x+a)ϕS\nm′+d81φN\n1e−iqexϕS\nm′−1+d82φN\n2e−iqexϕS\nm′+e81φN\n4eiqhxϕS\nm′−1+e82φN\n3eiqhxϕS\nm′\n+f81φN\n4e−iqh(x+a)ϕS\nm′−1+f82φN\n3e−iqh(x+a)ϕS\nm′, −a < x < 0\nφS\n3eiqS\nhxϕS\nm′+a81φS\n1e−iqS\nexϕS\nm′−1+a82φS\n2e−iqS\nexϕS\nm′+b81φS\n4eiqS\nhxϕS\nm′−1+b82φS\n3eiqS\nhxϕS\nm′, x > 0.(A2)\nIn Eqs. (A1),(A2), φN\n1=\n1\n0\n0\n0\n,φN\n2=\n0\n1\n0\n0\n,φN\n3=\n0\n0\n1\n0\n,φN\n4=\n0\n0\n0\n1\n,φS\n1=\nu\n0\n0\nv\n,φS\n2=\n0\n−λu\nv\n0\n,φS\n3=\n0\n−v\nu\n0\nand\nφS\n4=\nλv\n0\n0\nu\nwith λ= 1 for an s-wave superconductor and λ=−1 for an p-wave superconductor. In Eqs. (A1),(A2),\nψ1,ψ2,ψ3andψ4are the wavefunctions when electron with up spin, an electron with down spin, hole with up spin\nor hole with down spin is incident from left normal metal (N 1) respectively. In contrast, ψ5,ψ6,ψ7andψ8are the\nwavefunctions for an electron with up spin, an electron with down spin, hole with up spin or hole with down spin are\nincident from superconductor (S) respectively. aijandbijare the Andreev reflection amplitudes and normal reflection\namplitudes, respectively, while gijandhijare the transmission amplitudes for electronlike and holelike quasiparticles\nrespectively. We multiplyq\nqh\nqe\u0010q\nqe\nqh\u0011\nandq\nqS\nh\nqSe\u0010q\nqSe\nqS\nh\u0011\nwith the Andreev reflection amplitudes aijfor the electron\n(hole) and electronlike (holelike) quasiparticle incident from metallic and superconducting region respectively so that\nthe square of the absolute values of the amplitudes gives the Andreev reflection probability. Further,q\nqSe\nqe(qh)(u2−v2)\nandq\nqS\nh\nqe(qh)(u2−v2) are multiplied with the transmission amplitudes gijandhijfor electron (hole) incident from\nmetallic region respectively, whileqqe\nqSe(qS\nh)(u2−v2)andqqh\nqSe(qS\nh)(u2−v2)are multiplied with the transmission amplitudes\ngijandhijfor electronlike (holelike) quasiparticle incident from superconducting region respectively. ϕS\nm′is the\neigenfunction of spin flipper with its spin Sand spin magnetic moment m′. The Szoperator of spin flipper’s spin\nacts as- SzϕS\nm′= ¯hm′ϕS\nm′. In Eqs. (A1),(A2), u=q\n1\n2(1 +√\nω2−∆2\nω) and v=q\n1\n2(1−√\nω2−∆2\nω). In normal metal,\nthe electron, hole wavevectors are qe,h=q\n2m∗\n¯h2(EF±ω), while in superconductor the quasiparticle wavevectors are19\nqS\ne,h=q\n2m∗\n¯h2(EF±√\nω2−∆2). The conjugated processes ˜ψinecessary to form retarded Green’s functions in section\nIII are found by diagonalizing Hamiltonian H∗\nBdG(−k) instead of HBdG(k). For N 1-SF-N 2-S junction we note that\n˜φiN=φN\ni, however ˜ φ1S=\nλu\n0\n0\nv\n, ˜φ2S=\n0\n−u\nv\n0\n, ˜φ3S=\n0\n−λv\nu\n0\nand, ˜ φ4S=\nv\n0\n0\nu\n. In our work, qe,h≈kF(1±ω\n2EF)\nandqS\ne,h≈kF±iµwhere kF=q\n2m∗EF/¯h2andµ=√\n∆2−ω2[kF/(2EF)] when EF≫∆, ω. The superconducting\ncoherence length is denoted as[66] ξ=¯h\nm∗∆.\nAppendix B: Wavefunctions in the second setup\nThe wavefunctions in different regions of Normal metal-Spin flipper-Normal metal-pSc junction are given as-\nΦ1(x) =\n\nφN\n1ei(x+a)ϕS\nm′+a′\n11φN\n4ei(x+a)ϕS\nm′+a′\n12φN\n3ei(x+a)ϕS\nm′+1+b′\n11φN\n1e−i(x+a)ϕS\nm′+b′\n12φN\n2e−i(x+a)ϕS\nm′+1, x < −a\nc′\n11φN\n1ei(x+a)ϕS\nm′+c′\n12φN\n2ei(x+a)ϕS\nm′+1+d′\n11φN\n1e−ixϕS\nm′+d′\n12φN\n2e−ixϕS\nm′+1+e′\n11φN\n4eixϕS\nm′+e′\n12φN\n3eixϕS\nm′+1\n+f′\n11φN\n4e−i(x+a)ϕS\nm′+f′\n12φN\n3e−i(x+a)ϕS\nm′+1, −a < x < 0\ng′\n11φSR\n1eiq−xϕS\nm′+g′\n12φSR\n2eiq−xϕS\nm′+1+h′\n11φSR\n4eiq+xϕS\nm′+h′\n12φSR\n3eiq+xϕS\nm′+1, x > 0\nΦ2(x) =\n\nφN\n2ei(x+a)ϕS\nm′+a′\n21φN\n4ei(x+a)ϕS\nm′−1+a′\n22φN\n3ei(x+a)ϕS\nm′+b′\n21φN\n1e−i(x+a)ϕS\nm′−1+b′\n22φN\n2e−i(x+a)ϕS\nm′, x < −a\nc′\n21φN\n1ei(x+a)ϕS\nm′−1+c′\n22φN\n2ei(x+a)ϕS\nm′+d′\n21φN\n1e−ixϕS\nm′−1+d′\n22φN\n2e−ixϕS\nm′+e′\n21φN\n4eixϕS\nm′−1+e′\n22φN\n3eixϕS\nm′\n+f′\n21φN\n4e−i(x+a)ϕS\nm′−1+f′\n22φN\n3e−i(x+a)ϕS\nm′, −a < x < 0\ng′\n21φSR\n1eiq−xϕS\nm′−1+g′\n22φSR\n2eiq−xϕS\nm′+h′\n21φSR\n4eiq+xϕS\nm′−1+h′\n22φSR\n3eiq+xϕS\nm′, x > 0\nΦ3(x) =\n\nφN\n4e−i(x+a)ϕS\nm′+a′\n31φN\n1ei(x+a)ϕS\nm′+a′\n32φN\n2ei(x+a)ϕS\nm′+1+b′\n31φN\n4e−i(x+a)ϕS\nm′+b′\n32φN\n3e−i(x+a)ϕS\nm′+1, x < −a\nc′\n31φN\n1ei(x+a)ϕS\nm′+c′\n32φN\n2ei(x+a)ϕS\nm′+1+d′\n31φN\n1e−ixϕS\nm′+d′\n32φN\n2e−ixϕS\nm′+1+e′\n31φN\n4eixϕS\nm′+e′\n32φN\n3eixϕS\nm′+1\n+f′\n31φN\n4e−i(x+a)ϕS\nm′+f′\n32φN\n3e−i(x+a)ϕS\nm′+1, −a < x < 0\ng′\n31φSR\n1eiq−xϕS\nm′+g′\n32φSR\n2eiq−xϕS\nm′+1+h′\n31φSR\n4eiq+xϕS\nm′+h′\n32φSR\n3eiq+xϕS\nm′+1, x > 0\nΦ4(x) =\n\nφN\n3e−i(x+a)ϕS\nm′+a′\n41φN\n1ei(x+a)ϕS\nm′−1+a′\n42φN\n2ei(x+a)ϕS\nm′+b′\n41φN\n4e−i(x+a)ϕS\nm′−1+b′\n42φN\n3e−i(x+a)ϕS\nm′, x < −a\nc′\n41φN\n1ei(x+a)ϕS\nm′−1+c′\n42φN\n2ei(x+a)ϕS\nm′+d′\n41φN\n1e−ixϕS\nm′−1+d′\n42φN\n2e−ixϕS\nm′+e′\n41φN\n4eixϕS\nm′−1+e′\n42φN\n3eixϕS\nm′\n+f′\n41φN\n4e−i(x+a)ϕS\nm′−1+f′\n42φN\n3e−i(x+a)ϕS\nm′, −a < x < 0\ng′\n41φSR\n1eiq−xϕS\nm′−1+g′\n42φSR\n2eiq−xϕS\nm′+h′\n41φSR\n4eiq+xϕS\nm′−1+h′\n42φSR\n3eiq+xϕS\nm′, x > 0\nΦ5(x) =\n\ng′\n51φN\n1ei(x+a)ϕS\nm′+g′\n52φN\n2ei(x+a)ϕS\nm′+1+h′\n51φN\n4e−i(x+a)ϕS\nm′+h′\n52φN\n3e−i(x+a)ϕS\nm′+1, x < −a\nc′\n51φN\n1ei(x+a)ϕS\nm′+c′\n52φN\n2ei(x+a)ϕS\nm′+1+d′\n51φN\n1e−ixϕS\nm′+d′\n52φN\n2e−ixϕS\nm′+1+e′\n51φN\n4eixϕS\nm′+e′\n52φN\n3eixϕS\nm′+1\n+f′\n51φN\n4e−i(x+a)ϕS\nm′+f′\n52φN\n3e−i(x+a)ϕS\nm′+1, −a < x < 0\nφ′SR\n1e−iq−xϕS\nm′+a′\n51φSR\n4eiq+xϕS\nm′+a′\n52φSR\n3eiq+xϕS\nm′+1+b′\n51φSR\n1eiq−xϕS\nm′+b′\n52φSR\n2eiq−xϕS\nm′+1, x > 0\nΦ6(x) =\n\ng′\n61φN\n1ei(x+a)ϕS\nm′−1+g′\n62φN\n2ei(x+a)ϕS\nm′+h′\n61φN\n4e−i(x+a)ϕS\nm′−1+h′\n62φN\n3e−i(x+a)ϕS\nm′, x < −a\nc′\n61φN\n1ei(x+a)ϕS\nm′−1+c′\n62φN\n2ei(x+a)ϕS\nm′+d′\n61φN\n1e−ixϕS\nm′−1+d′\n62φN\n2e−ixϕS\nm′+e′\n61φN\n4eixϕS\nm′−1+e′\n62φN\n3eixϕS\nm′\n+f′\n61φN\n4e−i(x+a)ϕS\nm′−1+f′\n62φN\n3e−i(x+a)ϕS\nm′, −a < x < 0\nφ′SR\n2e−iq−xϕS\nm′+a′\n61φSR\n4eiq+xϕS\nm′−1+a′\n62φSR\n3eiq+xϕS\nm′+b′\n61φSR\n1eiq−xϕS\nm′−1+b′\n62φSR\n2eiq−xϕS\nm′, x > 0\nΦ7(x) =\n\ng′\n71φN\n1ei(x+a)ϕS\nm′+g′\n72φN\n2ei(x+a)ϕS\nm′+1+h′\n71φN\n4e−i(x+a)ϕS\nm′+h′\n72φN\n3e−i(x+a)ϕS\nm′+1, x < −a\nc′\n71φN\n1ei(x+a)ϕS\nm′+c′\n72φN\n2ei(x+a)ϕS\nm′+1+d′\n71φN\n1e−ixϕS\nm′+d′\n72φN\n2e−ixϕS\nm′+1+e′\n71φN\n4eixϕS\nm′+e′\n72φN\n3eixϕS\nm′+1\n+f′\n71φN\n4e−i(x+a)ϕS\nm′+f′\n72φN\n3e−i(x+a)ϕS\nm′+1, −a < x < 0\nφ′SR\n4e−iq+xϕS\nm′+a′\n71φSR\n1eiq−xϕS\nm′+a′\n72φSR\n2eiq−xϕS\nm′+1+b′\n71φSR\n4eiq+xϕS\nm′+b′\n72φSR\n3eiq+xϕS\nm′+1, x > 0\nΦ8(x) =\n\ng′\n81φN\n1ei(x+a)ϕS\nm′−1+g′\n82φN\n2ei(x+a)ϕS\nm′+h′\n81φN\n4e−i(x+a)ϕS\nm′−1+h′\n82φN\n3e−i(x+a)ϕS\nm′, x < −a\nc′\n81φN\n1ei(x+a)ϕS\nm′−1+c′\n82φN\n2ei(x+a)ϕS\nm′+d′\n81φN\n1e−ixϕS\nm′−1+d′\n82φN\n2e−ixϕS\nm′+e′\n81φN\n4eixϕS\nm′−1+e′\n82φN\n3eixϕS\nm′\n+f′\n81φN\n4e−i(x+a)ϕS\nm′−1+f′\n82φN\n3e−i(x+a)ϕS\nm′, −a < x < 0\nφ′SR\n3e−iq+xϕS\nm′+a′\n81φSR\n1eiq−xϕS\nm′−1+a′\n82φSR\n2eiq−xϕS\nm′+b′\n81φSR\n4eiq+xϕS\nm′−1+b′\n82φSR\n3eiq+xϕS\nm′, x > 0(B1)20\nwhere φSR\n1=1√\n|κ−|2+1\nκ−\n0\n0\n1\n,φSR\n2=1√\n|κ−|2+1\n0\nκ−\n1\n0\n,φSR\n3=1√\n|κ+|2+1\n0\nκ+\n1\n0\n,φSR\n4=1√\n|κ+|2+1\nκ+\n0\n0\n1\n,φ′SR\n1=\n1√\n|κ−|2+1\n−κ−\n0\n0\n1\n,φ′SR\n2=1√\n|κ−|2+1\n0\n−κ−\n1\n0\n,φ′SR\n3=1√\n|κ+|2+1\n0\n−κ+\n1\n0\n,φ′SR\n4=1√\n|κ+|2+1\n−κ+\n0\n0\n1\nandκ±= (E+q2\n±−\nµpSc)/(∆pScq±).a′\nmnandb′\nmnare the Andreev reflection amplitudes and normal reflection amplitudes, respectively.\nIn contrast, g′\nmnandh′\nmnare the transmission amplitudes for electron/electronlike quasiparticles and hole/holelike\nquasiparticles, respectively. In Eq. (B1) wavevector in metallic region is approximated by the Fermi wavevector\nkF=√2m∗µN/¯h= 1 (since ¯ h=µN= 2m∗= 1) with E≪EF. In pSc wave vector’s q±are solutions of-\nE2= (q2−µpSc)2+ (∆ pScq)2. (B2)\nSolutions of Eq. (B2) for different values of chemical potential µpSc>0 (topological regime) and µpSc<0 (trivial\nregime) with energy Eare given in Table I of Ref. [52]. The conjugated processes ˜ψinecessary to form retarded Green’s\nfunctions in section II.D are found by diagonalizing Hamiltonian H∗\nN(−k) instead of HN(k) for normal metal and\nH∗\npSc(−k) instead of HpSc(k) for pSc. For our second setup we note that ˜ φiN=φN\ni, however ˜ φ1SR=1√\n|κ−|2+1\n−κ−\n0\n0\n1\n,\n˜φ2SR=1√\n|κ−|2+1\n0\n−κ−\n1\n0\n, ˜φ3SR=1√\n|κ+|2+1\n0\n−κ+\n1\n0\n, ˜φ4SR=1√\n|κ+|2+1\n−κ+\n0\n0\n1\n, ˜φ1′SR=1√\n|κ−|2+1\nκ−\n0\n0\n1\n, ˜φ2′SR=\n1√\n|κ−|2+1\n0\nκ−\n1\n0\n, ˜φ3′SR=1√\n|κ+|2+1\n0\nκ+\n1\n0\nand, ˜ φ4′SR=1√\n|κ+|2+1\nκ+\n0\n0\n1\n.\nAppendix C: Explicit form of expressions for normal Green’s functions for the first setup\nIn this section, we provide an explicit form of expressions for normal Green’s functions ( Gr\nee) for the first setup. Gr\nee\nare utilized to compute LDOS and LMDOS in section III.\n1. Green’s function in N 1region\nGrin N 1region are calculated by putting wavefunctions from Eqs. (A1),(A2) into Eq. (18) with aijandbijobtained\nfrom Eqs. (4)-(7). For Gr\neewe find\n[Gr\nee]↑↑=−iη\n2qe[b11e−iqe(x+2a+x′)+eiqe|x−x′|],[Gr\nee]↓↓=−iη\n2qe[b22e−iqe(x+2a+x′)+eiqe|x−x′|],\n[Gr\nee]↑↓=−iη\n2qeb21e−iqe(x+2a+x′),and [Gr\nee]↓↑=−iη\n2qeb12e−iqe(x+2a+x′).(C1)21\n2. Green’s function in S region\nWe use a similar method in the superconducting region as for the metallic region and finally obtain normal com-\nponents of Green’s function for s-wave superconductor as-\n[Gr\nee]↑↑=η\n2i(u2−v2)\"\neiqS\ne|x−x′|u2+b51eiqS\ne(x+x′)u2+a52ei(qS\nex′−qS\nhx)uv\nqSe+a81ei(qS\nex−qS\nhx′)uv+b82e−iqS\nh(x+x′)v2+v2e−iqS\nh|x−x′|\nqS\nh#\n,\n[Gr\nee]↓↓=η\n2i(u2−v2)\"\neiqS\ne|x−x′|u2+b62eiqS\ne(x+x′)u2+a61ei(qS\nex′−qS\nhx)uv\nqSe+a72ei(qS\nex−qS\nhx′)uv+b71e−iqS\nh(x+x′)v2+v2e−iqS\nh|x−x′|\nqS\nh#\n,\n[Gr\nee]↑↓=−η\n2i(u2−v2)\"\nb61eiqS\ne(x+x′)u2+a62ei(qS\nex′−qS\nhx)uv\nqSe+b72e−iqS\nh(x+x′)v2+a71ei(qS\nex−qS\nhx′)uv\nqS\nh#\n,\n[Gr\nee]↓↑=−η\n2i(u2−v2)\"\nb52eiqS\ne(x+x′)u2+a51ei(qS\nex′−qS\nhx)uv\nqSe+b81e−iqS\nh(x+x′)v2+a82ei(qS\nex−qS\nhx′)uv\nqS\nh#\n.\n(C2)\nForp-wave superconductor, we obtain normal components of Green’s function as-\n[Gr\nee]↑↑=η\n2i\"\neiqS\ne|x−x′|u2+b51eiqS\ne(x+x′)u2−a51ei(qS\nex′−qS\nhx)uv\nqSe+a82ei(qS\nex−qS\nhx′)uv−b82e−iqS\nh(x+x′)v2−v2e−iqS\nh|x−x′|\nqS\nh#\n,\n[Gr\nee]↓↓=η\n2i\"\neiqS\ne|x−x′|u2+b62eiqS\ne(x+x′)u2−a62ei(qS\nex′−qS\nhx)uv\nqSe+a71ei(qS\nex−qS\nhx′)uv−b71e−iqS\nh(x+x′)v2−v2e−iqS\nh|x−x′|\nqS\nh#\n,\n[Gr\nee]↑↓=η\n2i\"\nb61eiqS\ne(x+x′)u2−a61ei(qS\nex′−qS\nhx)uv\nqSe−b72e−iqS\nh(x+x′)v2−a72ei(qS\nex−qS\nhx′)uv\nqS\nh#\n,\n[Gr\nee]↓↑=η\n2i\"\nb52eiqS\ne(x+x′)u2−a52ei(qS\nex′−qS\nhx)uv\nqSe−b81e−iqS\nh(x+x′)v2−a81ei(qS\nex−qS\nhx′)uv\nqS\nh#\n.\n(C3)\nAppendix D: Explicit form of expressions for normal Green’s functions for the second setup\nIn this section, we mention the explicit form of expressions for normal Green’s functions ( Gr\nee) for the second setup.\nGr\neeare utilized to compute LDOS and LMDOS in section IV.\n1. Green’s function in N 1region\nGrin N 1region are determined by putting wavefunctions from Eq. (B1) into Eq. (18) with a′\nijandb′\nijobtained\nfrom Eqs. (11)-(14). For Gr\neewe find\n[Gr\nee]↑↑=−iη\n2[b′\n11e−i(x+2a+x′)+ei|x−x′|],[Gr\nee]↓↓=−iη\n2[b′\n22e−i(x+2a+x′)+ei|x−x′|],\n[Gr\nee]↑↓=−iη\n2b′\n21e−i(x+2a+x′),and [Gr\nee]↓↑=−iη\n2b′\n12e−i(x+2a+x′).(D1)22\n2. Green’s function in S region\nWe use a similar procedure in the superconducting region as for the metallic region and finally get normal compo-\nnents of Green’s function in the trivial regime for our second setup as-\n[Gr\nee]↑↑=iη\n2(q+κ+−q−κ−)q\n(1−κ2\n+)(1−κ2\n−)\u0010\nei(q−x+q+x′)a′\n82κ−(1−κ2\n+)−ei(q+x+q−x′)a′\n51κ+(1−κ2\n−)\n+\u0000\nκ+b′\n82eiq+(x+x′)−κ+eiq+|x−x′|+κ−eiq−|x−x′|−κ−b′\n51eiq−(x+x′)\u0001q\n(1−κ2\n+)(1−κ2\n−)\u0011\n,\n[Gr\nee]↓↓=iη\n2(q+κ+−q−κ−)q\n(1−κ2\n+)(1−κ2\n−)\u0010\nei(q−x+q+x′)a′\n71κ−(1−κ2\n+)−ei(q+x+q−x′)a′\n62κ+(1−κ2\n−)\n+\u0000\nκ+b′\n71eiq+(x+x′)−κ+eiq+|x−x′|+κ−eiq−|x−x′|−κ−b′\n62eiq−(x+x′)\u0001q\n(1−κ2\n+)(1−κ2\n−)\u0011\n,\n[Gr\nee]↑↓=iη\n2(q+κ+−q−κ−)q\n(1−κ2\n+)(1−κ2\n−)\u0010\nei(q−x+q+x′)a′\n72κ−(1−κ2\n+)−ei(q+x+q−x′)a′\n61κ+(1−κ2\n−)\n+\u0000\nκ+b′\n72eiq+(x+x′)−κ−b′\n61eiq−(x+x′)\u0001q\n(1−κ2\n+)(1−κ2\n−)\u0011\n,\n[Gr\nee]↓↑=iη\n2(q+κ+−q−κ−)q\n(1−κ2\n+)(1−κ2\n−)\u0010\nei(q−x+q+x′)a′\n81κ−(1−κ2\n+)−ei(q+x+q−x′)a′\n52κ+(1−κ2\n−)\n+\u0000\nκ+b′\n81eiq+(x+x′)−κ−b′\n52eiq−(x+x′)\u0001q\n(1−κ2\n+)(1−κ2\n−)\u0011\n.(D2)\nIn the topological regime, we get normal components of Green’s function as-\n[Gr\nee]↑↑=iη\n2(q+κ+−q−κ−)\u0010\nei(q−x+q+x′)a′\n82κ−−ei(q+x+q−x′)a′\n51κ++κ+b′\n82eiq+(x+x′)−κ+eiq+|x−x′|\n+κ−eiq−|x−x′|−κ−b′\n51eiq−(x+x′)\u0011\n,\n[Gr\nee]↓↓=iη\n2(q+κ+−q−κ−)\u0010\nei(q−x+q+x′)a′\n71κ−−ei(q+x+q−x′)a′\n62κ++κ+b′\n71eiq+(x+x′)−κ+eiq+|x−x′|\n+κ−eiq−|x−x′|−κ−b′\n62eiq−(x+x′)\u0011\n,\n[Gr\nee]↑↓=iη\n2(q+κ+−q−κ−)\u0010\nei(q−x+q+x′)a′\n72κ−−ei(q+x+q−x′)a′\n61κ++κ+b′\n72eiq+(x+x′)−κ−b′\n61eiq−(x+x′)\u0011\n,\n[Gr\nee]↓↑=iη\n2(q+κ+−q−κ−)\u0010\nei(q−x+q+x′)a′\n81κ−−ei(q+x+q−x′)a′\n52κ++κ+b′\n81eiq+(x+x′)−κ−b′\n52eiq−(x+x′)\u0011\n.(D3)\n[1] M. 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B 100, 115433 (2019)."    },    {        "title": "2205.07136v3.Electronic_excited_states_in_extreme_limits_via_ensemble_density_functionals.pdf",        "content": "Electronic excited states in extreme limits via ensemble density functionals\nTim Gould\nQueensland Micro- and Nanotechnology Centre, Gri\u000eth University, Nathan, Qld 4111, Australia\u0003\nDerk P. Kooi and Paola Gori-Giorgi\nDepartment of Chemistry & Pharmaceutical Sciences and Amsterdam Institute of Molecular and Life Sciences (AIMMS),\nFaculty of Science, Vrije Universiteit, De Boelelaan 1083, 1081HV Amsterdam, The Netherlands\nStefano Pittalis\nCNR-Istituto Nanoscienze, Via Campi 213A, I-41125 Modena, Italy\nDensity functional theory (DFT) has greatly expanded our ability to a\u000bordably compute and\nunderstand electronic ground states, by replacing intractable ab initio calculations by models based\non paradigmatic physics from high- and low-density limits. But, a comparable treatment of excited\nstates lags behind. Here, we solve this outstanding problem by employing a generalization of density\nfunctional theory to ensemble states (EDFT). We thus address important paradigmatic cases of all\nelectronic systems in strongly (low-density) and weakly (high-density) correlated regimes. We show\nthat the high-density limit connects to recent, exactly-solvable EDFT results. The low-density limit\nreveals an unnoticed and most unexpected result { density functionals for strictly correlated ground\nstates can be reused directly for excited states. Non-trivial dependence on excitation structure only\nshows up at third leading order. Overall, our results provide foundations for e\u000bective models of\nexcited states that interpolate between exact low- and high-density limits, which we illustrate on\nthe cases of singlet-singlet excitations in H 2and a ring of quantum wells.\nPreamble. Density functional theory (DFT) [1, 2] is\nbest known as a computational modelling tool used in\ntens of thousands of applicative scienti\fc papers every\nyear. What is less widely known is that DFT o\u000bers\na natural connection between quantum mechanics and\nparadigmatic physical conditions (high- and low-density\nlimits) of matter, in which electronic correlations attain\ntwo quantitatively (weak and strong, respectively) and\nqualitatively di\u000berent fundamental ends. In this context,\nDFT serves as a formal tool to understand the behaviour\nof ground state electronic structure via a rigorous con-\nstrained variational approach to the electronic structure\nproblem. Understanding of paradigmatic conditions then\ninforms model development, e.g. the popular \\PBE\" [3]\napproximation, and computational studies therefrom.\nUnfortunately, DFT is only de\fned for ground states,\nso cannot elucidate the structure of excited states. This\nLetter will demonstrate that ensemble density functional\ntheory (EDFT) for excited states [4, 5] can tackle this\noutstanding problem. We shall show that recently de-\nrived Hartree and exchange physics [6, 7] become exact\nin the high density (weak interaction); so high-density\nexcited electronic states may be solved using these tools.\nMore importantly, we shall show that the low density\n(strong interaction) limit of excited states behaves exactly\nlike a ground state . Therefore, the full suite of ground\nstate strictly correlated electron (SCE) tools and approx-\nimations [8{13] may be used to solve both ground and\nexcited states of low-density many-electron systems.\nOur work thereby improves understanding of excited\nstates in paradigmatic limits and connects their be-\nhaviour to well-de\fned density functionals for which ex-act forms and approximations are available. It presents a\ncrucial step toward e\u000ecient excited state approximations\nthat capture important limits; and promises to acceler-\nate and generalize recent progress on low cost modelling\nof single [14{19] and double excitations [16, 20{23] that\nmay range from weakly to strongly correlated regimes.\nThe rest of this Letter proceeds as follows: First, we\nintroduce EDFT and show how it can be used to under-\nstand the high- and low-density limits of interacting elec-\ntrons in realistic inhomogeneous systems. Then, using as\nan illustration the strong interaction limit of electrons\nin an harmonic well, we derive the asymptotic proper-\nties of the density functionals for describing excitations\nin Wigner-like systems via EDFT. We then reveal that\nthesecond leading term in the low-density limit is also\nthe same in ground and excited states, and that a non-\ntrivial dependence only appears in the third leading term\n{ which therefore describe more realistic correlated exci-\ntations. We then illustrate the importance of our \fndings\nfor applications by studying excitations in two examples.\nFinally, we conclude.\nTheoretical framework. Excited state EDFT is con-\ncerned with the behaviour of countable sets of excited\nstates. In practice, a \fnite set of low-lying solutions of\n^Hj\u0014i=E\u0014j\u0014i. These are grouped in an ensemble state\n^\u0000w=P\n\u0014w\u0014j\u0014ih\u0014jusing some prescribed weights [24]\nsuch thatw\u0014\u00150 andP\n\u0014w\u0014= 1 (collectively, w). The\naverage of an operator, ^O, over ^\u0000wis given byOw:=\nTr[^\u0000w^O]. Crucially, choosing w\u0014\u0014w\u00140forE\u0014\u0015E\u00140,\nensures that ^\u0000wful\flls an extended variational princi-\nple [4] according to which Ew= inf ^\u0000w\ntrialTr[^\u0000w\ntrial^H] where\nthe argument for the in\fmum (usually a minimum),arXiv:2205.07136v3  [physics.chem-ph]  15 Oct 20222\n^\u0000w\ntrial=P\n\u0014w\u0014j\u0014tih\u0014tj, involves prescribed weights, w,\nand mutually orthonormal trial wavefunctions j\u0014ti.\nDensity functionalizing the above variational principle\nin terms of the ensemble particle density, n, yields, [5, 6]\nEw= min\nn\u001a\nTw\ns[n] +Ew\nHxc[n] +Z\nn(r)vext(r)dr\u001b\n:(1)\nHere,Tw\ns[n] = min ^\u0000w\ntrial!nTr[^\u0000w\ntrial^T] is the kinetic en-\nergy of the Kohn-Sham (KS) system { i.e., an auxiliary\nsystems reproducing the particle density of the ensem-\nble; the minimum is attained at ^\u0000s\u0011P\n\u0014w\u0014j\u0014sih\u0014sj. [25]\nEw\nHxc[n] takes care of the remaining Hartree (H), exchange\n(x), and correlation (c) energies. Together, they yield the\n\\universal\" functional for ensembles: Fw=Tw\ns+Ew\nHxc.\nIn fact, eq. (1) describes di\u000berent functionals for ev-\nery choice of w. To stress this important point we use\ncapital calligraphic letters to refer to energies of mixed\nstates; and superscripts, w, to indicate quantities that\nexplicitly depend on their weights. Pure ground states\ninvolve setting w0= 1 ( ^\u00000=j0ih0j) for which eq. (1) at-\ntains usual DFT forms. Spin and spatial symmetries are\npreserved at the Kohn-Sham level [26] (see high-density\nlimit discussion for further details) by equally weight-\ning degenerate states. Varying weights (e.g. via partial\nderivatives) lets us address individual excited states. [27]\nTherefore, the weight-dependence of ensemble function-\nals is directly related to the structure and behaviour of\nground and excited electronic states.\nThe universal energy functional may be generalized to,\nF\u0015;w[n] = inf\n^\u0000w!nTr\u0002^\u0000w(^T+\u0015^Vee)\u0003\n; (2)\nwhere ^Tis the kinetic energy operator and ^Veeis the\nCoulombic interaction operator. \u0015sets the strength of\nthe interaction. This functional is referred to as \\uni-\nversal\" because the external potential, which speci\fes\nthe molecule or solid we wish to treat, does not ap-\npear explicitly in its de\fnition [the density being given\nand \fxed in eq. (2)]. Matching terms from above yields\nFw\u0011F\u0015=1;w,Tw\ns\u0011F0;wandEw\nHxc:=Fw\u0000F0;w. In\nfact, we stress that F,TsandEHxcaremulti -universal be-\ncause each set of weights, w, de\fnes a di\u000berent excitation\nstructure and, thus, a di\u000berent universal functional. As\nwe shall show below, this multi -universality evolves into\nasimple universality in the low-density limit of matter.\nIn what follows, our main objective is to determine\nthe salient behavior of key ensemble density functionals\nin the high- (weakly interacting) and low-density (strictly\ninteracting) limits. In doing so, we shall extend to excited\nstates concepts and core results which have previously\nbeen worked out for pure ground states only. [8{13] These\nworks can be understood as providing a generalization of\nthe seminal work of Wigner [28, 29] to inhomogeneous\nsystems within DFT. Our current work completes the\ngeneralization to include excited inhomogeneous systems\n100120140V/Vcl [%]\n Triplet\n3S03S13S23S33S43S5\n1 2 3 5 10 15 25 50\nInteraction strength  [unitless]\n6080100V/Vcl [%]\nSinglet\n1S01S11S21S31S41S5FIG. 1. Ratio of quantum and classical interaction energies\nfor two electrons in an Harmonic well. Shows six triplet (top)\nand singlet (bottom) energies. Scaled densities, 4 \u0019r2n(r), of\nthe states are also shown for the case \u0015= 50.\nwithin EDFT. It thus provides a complete treatment of\nelectronic structure of two important paradigmatic and\nfundamentally di\u000berent regimes, within a consistent and\nversatile approach.\nHigh-density limit. In the parlance of modern density\nfunctional, the high- and low-density limits entail uni-\nform scaling of the coordinates of the electrons, say, by\n\r >0 in such a way n(r)!\r3n(\rr) =:n\r(r). To keep\nthe discussion simple, we may think of a \fnite system\nlike an atom, molecule or quantum dot. Scaling gives\n^T!\r\u00002^T\rand ^Vee!\r\u00001^Vee;\r; so,\nF\u0015;w[n\r] =\r2F\u0015=\r;w[n]: (3)\nBecause the scaled ensemble density is the density of a\nstationary ensemble of the Hamiltonian with interaction\n\u0015= 1=\r, we see that the high- and low-density limits\nare related to the weak- and strong-interaction limits,\nrespectively. [30]\nLet us \frst consider the high-density (i.e., weak inter-\naction,\r!1 ) limit. Scaling yields,\nlim\n\r!1Fw[n\r]\n\r2=Tw\ns[n];lim\n\r!1Ew\nHxc[n\r]\n\r=Ew\nHx[n];(4)\nwhere the second result follows from the de\fnition [6] of\nEHxas a gradient ofF. The high-density limit thus in-\nherits good properties of EHx: [6, 7, 23, 26] i) it preserves\nspin and spatial symmetries of the system; ii) the relevant\nKS states can be linear combinations of Slater determi-\nnants (SD) that are eigenstates of spin and proper gen-\nerators of point groups, unlike the single SD of conven-\ntional spin-DFT treatments; iii) yet, it enable e\u000bective\nreuse of conventional spin-density functional approxima-\ntions for exchange, via combination rules or on-top pair\ndensities. [7, 17, 19, 23]\nNext, consider the adiabatic connection formula,\nEw\nHxc[n] =Fw[n]\u0000F0;w[n] =Z1\n0V\u0015;w\nee[n]d\u0015; (5)\nV\u0015;w\nee[n] = lim\n\u0011!0+F\u0015+\u0011;w\u0000F\u0015;w\n\u0011:= Tr[ ^Vee^\u0000\u0015+;w]:(6)3\nThe `Hx' component is recovered as Ew\nHx[n] =V0;w\nee[n].\nScaling givesEw\nHxc[n\r] =\r2R1=\r\n0V\u0015;w\nee[n]d\u0015, from which\n(for \fnite systems) we get, [31]\nFw[n\r]\u0000!\n\r!1\r2Tw\ns[n] +\rEw\nHx[n] +EGL2;w\nc [n] +:::(7)\nThe \frst correlation contribution follows from G orling-\nLevy [32] perturbation theory for ensembles, [33] which\nmust also be adapted for KS states in the form of linear\ncombinations of SDs. Correlations may alternatively be\ncaptured by employing expressions previously reported\nin Ref. 7, 34, and 35; as in (e.g.) Ref. 23.\nLow-density limit. Approaching a most striking, and\npreviously unnoticed fact, let us turn to the the low-\ndensity (i.e. strong interaction, \r!0+) limit,\nlim\n\r!0+F1;w[n\r] = lim\n\r!0+V1;w\nee[n\r]: (8)\nCrucially,\nlim\n\r!0+V1;w\nee[n\r]\n\r=VSCE;w\nee [n]\u0011VSCE\nee[n] (9)\nwhereVSCE\nee[n] = inf \t!nh\tj^Veej\tiis the known in-\nteraction energy functional of strictly correlated elec-\ntrons in a ground state , but here evaluated at the en-\nsemble particle density. This result says that, in the\nlow-density limit, the functional dependence on weights\ndisappears from both F1;w[n\r] andV1;w\nee[n\r]. Depen-\ndence on the weights enters only via the particle density,\nn:= Tr[ ^\u0000w^n] =P\n\u0014w\u0014n\u0014, of the ensemble. Eq. (8), and\nits extension to higher orders in \rdiscussed later, are the\ncentral result of this work. In this context, SCE results,\nanalysis and understanding for ground states [8{11] be-\ncome special cases of the above more general result.\nProof of Eqs (8)and (9). Here, we shall guide the\nreader through the main steps and key physics. A full\nproof is reported in Section 1 of the Supplementary Ma-\nterial (Supp. Mat. Sec. I).\nThe salient features can be already grasped by observ-\ning the behaviour of an interacting system as interactions\nare increased in a model system. We choose two-electron\nHarmonium in which two electrons interact in an exter-\nnal potential vext=1\n2r2with an interaction strength \u0015.\nThe scaled classical interaction energy of this system is\nVcl= 0:7937\u00152=3. [36] Quantum solutions may be found\nnumerically. Details are in Supp. Mat. Sec. II. Figure 1\nshows the interaction energies, V\u0015\n\u0014:=h\u0014j^Veej\u0014i, of six\nlow-lying spherically symmetric triplet (3S) and singlet\n(1S) states. It is clear that quantum and classical inter-\naction energies all become the same as \u0015is increased {\ni.e., all excitations tend toward the same classical limit.\nTo prove our result for EDFT we need to consider a\nsimilar physical setting ( \u0015!1 ), in which instead of\n\fxing the external potential we \fx the ensemble den-\nsity, containing the excited states we want to treat.Proving eq. (9) then entails showing that the degen-\neracy behaviour carries through to systems in which\nthe density is kept \fxed. Our argument involves the\nexpansion of wave functions for large but \fnite inter-\naction strengths, [9, 37] around the strictly correlated\nlimit. In this e\u000bectively classical limit, which yields\nthe leading term as \u0015! 1 of the ground-state uni-\nversal functional F\u0015[n] [10, 11], the N-body distribu-\ntion of an N-electron system is PN[n](r1\u0001\u0001\u0001rN) =Rn(s)\nNQN\ni=1\u000e(ri\u0000fi(s))dswhich leads to F\u0015!1[n]!\n\u0015VSCE\nee[n] =\u0015PN\ni=2R\nn(r)dr\n2jr\u0000fi(r)j. Here, fi(r) are co-\nmotion maps that preserve the density and the indistin-\nguishability of electrons. [8, 12]\nAt large but \fnite \u0015we construct orthonormal wave\nfunctions,j\u0014\u0015i, based on quantum harmonic oscilla-\ntions (QHO) around the strictly-correlated distribution,\nPN[n]. [9, 38] The QHOs act on curvilinear coordi-\nnates orthogonal to the manifold parametrized by the co-\nmotion functions; and contribute at O(p\n\u0015) in the kinetic\nand potential energies. Prima facie , the wave functions\nj\u0014\u0015ihave di\u000berent densities. However, it is also possible\nto quantize along the manifold, which contributes only\natO(1) in kinetic energy and is amenable to the Har-\nriman construction [39] of orthogonal orbitals yielding\ndensityn. We thereby obtain a countable number of or-\nthornomal wavefunctions that all have the same density\nn, and the same energy up to O(1). Thus,VSCE;w\nee [n]\n:=P\n\u0014w\u0014VSCE\nee[n] =VSCE\nee[n] and eqs (8) and (9) follow\nfrom the equivalence of \r!0+and\u0015!1 in eq. (3).\nNext-leading terms in the low-density limit. To an-\nalyze the next leading terms, it useful to \frst rewrite\nFw[n] :=TSCE;w[n] +VSCE\nee[n], taking SCE as the refer-\nence system and letting TSCE;wcapture all the ensemble\ne\u000bects. Then an alternative adiadatic connection yields,\nTSCE;w[n] =Z1\n1T\u0015;w[n]\n\u00152d\u0015: (10)\nHere, we introduced, T\u0015;w=\u0000\u00152@\u0015F\u0015;w\n\u0015:= Tr[ ^T^\u0000\u0015+;w],\nwhere the derivative and trace must be treated with cau-\ntion, like in eq. (6). Eq. (10) generalizes known results\nfor the ground state-only case [40{42] to the ensembles\nconsidered in this work.\nNext, we show that T\u0015!1;wis independent of weights\nto leading order, which leads to TSCE;w[n\r!0+] also inde-\npendent of weights. Ensemblization of known results [9,\n38, 43] givesT\u0015!1;w!p\n\u0015\n2FZPE;wwhereFZPE;w\ninvolves\u0015-normalized zero point energy (ZPE) of the\nQHOs,j\u0014\u0015i, introduced earlier. Hence, T\u0015;wbecomes\nindependent of weights if we can show that FZPE;w[n]\u0011\nlim\u0015!12p\n\u0015T\u0015;w[n] is independent of weights. Supp.\nMat. Sec. I naturally covers this case { for guidance,\nbelow, we touch on essential steps and consequences.\nThe Harriman construction introduced earlier (also,\nSupp. Mat. Sec. IB) yields h\u0014\u0015j^Tj\u0014\u0015i=h0\u0015j^Tj0\u0015i+\nO(1) for\u0015! 1 . Thus,T\u0015;w=P\n\u0014w\u0014h\u0014\u0015j^Tj\u0014\u0015i=4\nh0\u0015j^Tj0\u0015i+O(1) is independent of weights to lead-\ning order, giving weight-independent, FZPE;w[n]\u0011\nlim\u0015!12T\u0015;w[n]p\n\u0015=FZPE[n]. Here,FZPE[n] is the well-\nstudied ground state functional [9, 37, 38], but eval-\nuated on the ensemble density. Using the latter re-\nsult in (10), and applying scaling laws, \fnally yields,\nlim\r!0+TSCE;w[n\r] =FZPE[n\r] =\r3=2FZPE[n]. Thus\nwe conclude that T\u0015;w[n] andTSCE;w[n] are indepen-\ndent of the ensemble weights in the low-density limit.\nFw[n\r!0+] =\rVSCE\nee[n] +\r3=2FZPE[n] is therefore also\nindependent of weights to second leading order. Details\nof scaling are in Supp. Mat. Sec. III.\nWhere and how does the weight dependence appears\nin the low-density limit? Eqs (S31){(S37) of Supp. Mat.\nSec. IB reveal that it appears in the third leading term,\nFw[n\r]\u0000!\n\r!0+\rVSCE\nee[n] +\r3=2FZPE[n]\n+\r2\u0001T(2);w[n] +::: : (11)\nTheO(\r2) term, \u0001T(2);w=P\n\u0014w\u0014\u0001T(2)\n\u0014, has an explicit\nweight dependence on each excited state. It captures the\nenergy of oscillations \\perpendicular\" to the collective\nZPE modes, according to the metric dictated by the SCE\nmanifold { see eq. (S36) and discussion for details. Note,\na similar result was previously observed in the special\ncase of Hubbard dimers. [44]\n2 4 6 8 10\nDistance, D [Bohr]51015ESS [eV]\nH2\nRun1 Run2 Run3 Run4 Run5 Run6 Run7 Mean0246ESS [eV]\nRing\nFIG. 2. Excitation energies, for dissociating H 2(top);\nand seven random runs and the mean of 25 runs for a ring\nof wells with disorder (bottom). Shows ensemblized EXX\n(dots/green), GL2 (dashes/red), ISI (dash-dot/blue) and ex-\nact (lines/maroon) energies. GL2 errors in the ring are so\nlarge they cannot be shown in the lower panel.\nUse of high- and low-density limits in approxima-\ntions. We have so far derived series expansions in the\nhigh-density [eq. (7)] and low-density [eq. (11)] limits.\nNext, we shall illustrate their relevance in applications.\nFirst, we remark that eqs (5), (7) and (11) imply that,\nlim\n\r!+1Ew\nHxc[n\r] =\rEw\nHx[n] +EGL2;w\nc [n] +::: ; (12)\nlim\n\r!0+Ew\nHxc[n\r] =\rVSCE\nee[n] +\r3=2FZPE[n] +::: : (13)Especially note that the low-density (strictly correlated)\nlimit of bothFw[n] [see (8)] andEw\nHxc[n] depend on the\nexcitation structure only trivially, via the ensemble parti-\ncle density. Weight dependence appears at higher order.\nTo illustrate the usefulness of the limits, we \frst con-\nsider the lowest singlet-singlet excitation in dissociating\nH2. This problem is a rather stringent test of density\nfunctionals { failed by time-dependent DFT approxima-\ntions [45, 46] { because: i) the ground state is dynam-\nically correlated near its minima but becomes strongly\ncorrelated when dissociated; ii) the \frst excited state is\nalways dynamically correlated, thus cancellation of errors\nin the approximate excitation energy from the ground\nstate may be unreliable during dissociation; iii) the \frst\nexcited state in the KS ensemble involves a superposition\nof two SDs, and its symmetry and related properties are\nirreproducible by an adiabatic single-SD approach.\nFigure 2 (top) reports the excitation energy \u0001 ESS=\nES1\u0000ES0of the lowest singlet states ( S0andS1) of H 2us-\ning: EXact eXchange (EXX) energies { the leading term\nin the high-density series of eq. (12); G orling-Levy [32]\n(GL2) perturbation theory { the next leading term of\n(12); and the Interaction Strength Interpolation (ISI) ap-\nproximation [47, 48] that uses high- and low-density lim-\nits { the latter via the harmonium point charge plus con-\ntinuum (hPC) approximation [48]. Note, all approxima-\ntions are ensemblized versions of ground states analogs,\ni.e.Eapprox\nHxc!Eapprox;w\nHxc is adapted for excited states.\nAll relevant energy expressions and technical details on\nthe calculations are in Supp. Mat. Sec. IV.\nOnly ISI performs well across the whole H 2dissocia-\ntion curve, which unambiguously highlights the bene\ft\nof using both eq. (12) and eq. (13) to construct approx-\nimations that capture di\u000berent correlation regimes. In\nfact, using only eq. (12) leads to very poor results for the\nground state energy: EXX overestimates and GL2 dras-\ntically underestimates as correlations become stronger.\nNext, we carry out similar calculations for four elec-\ntrons in a ring of four quantum wells; see Supp. Mat.\nSec. IV for further details. Lattice disorder in this sys-\ntem yields \u0001 ESS= 2:15 eV on average, versus \u0001 ESS=\n0:003 eV of the ordered lattice. Results are shown in\nFigure 2 (bottom). Again, we see that the low-density\nbehaviour included in ISI reduces errors: from 100%\n(EXX) down to -12% (ISI). GL2 energies (not shown)\nhave orders of magnitude worse errors. This example\n(also, Ref. 49) thus suggests that seamless interpolation\nbetween high- and low-density limits may be crucial for\npredicting optical gaps in disordered nanostructures.\nSummary and conclusions. The results presented\nin this work describe, via ensemble density function-\nals, the behaviour of excited many-electron states in the\nparadigmatic high- (weakly-) and low-density (strictly-\ncorrelated) limits (regimes) { summarized for the impor-\ntantEHxcfunctional in eqs (12) and (13), respectively.\nThe high-density limit follows intuition and connects5\ndirectly to previous results which use the ensemble Kohn-\nSham system as reference system. The corresponding\nauxiliary pure states have the form of symmetry adapted\ncombination of Slater determinants; and EHxchas strong\nweight-dependence. Approximations based on this limit\nhave already successfully described weakly to moderately\ncorrelated excitations.\nThe low-density limit, in contrast, revealed an unex-\npected fact: the \frst two leading order terms of excited\nstates may be described by existing tools used for strictly\ncorrelated ground states. Therefore the ensemblization\nof ground-state approximations is, for once, straightfor-\nward. Dependence of EHxcon the weights (and therefore\nexcitation structure) only shows up in the third lead-\ning order term. The provided model applications illus-\ntrate that generally correlated regimes of excited states\nrequires seamless treatment of both density regimes.\nOne immediate consequence of the present work is\nthat electronic interaction models must interpolate be-\ntween Fermionic mean-\feld like excitation-structure de-\npendence at high-densities, and no excitation-structure\ndependence at low-densities. Not only is this of direct im-\nportance for traditional analytic-driven approximations,\nas seen in the examples reported here, it also provides\nconstraints for data-driven methodologies based on ma-\nchine learning. Ensemble-derived constraints were used\nto great success in the machine-learned \\Deep Mind 21\"\nground state approximation [50] { our work promises to\nextend this success to excited states.\nNatural next steps from the present results are to con-\nsider \fnite-temperature ensembles and magnetic inter-\nactions. Finite temperature imposes a \u0015-dependence\non the weights. Prior work [51, 52] showed that the\nhigh/low-density limit may be more relevant to the be-\nhavior of density functionals at low/high temperatures.\nMagnetic interactions require extra basic densities (e.g.\nspin-densities and currents) and related Hxc quanti-\nties; [53, 54] and must consistently ful\fll gauge symme-\ntries. [55] Further work along both lines is being pursued.\nTG was supported by an Australian Research Coun-\ncil (ARC) Discovery Project (DP200100033) and Fu-\nture Fellowship (FT210100663). 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Our\nmethod, called Marginal Contrastive Discrimination, MCD , reformulates the\nconditional density function into two factors, the margina l density function of the\ntarget variable and a ratio of density functions which can be estimated through\nbinary classification. Like noise-contrastive methods, MCD can leverage state-\nof-the-art supervised learning techniques to perform CDE , including neural net-\nworks. Our benchmark reveals that our method significantly o utperforms in prac-\ntice existing methods on most density models and regression datasets.\n1 Introduction\nWe consider the problem of conditional density estimation ( CDE ), which is a major topic of\ninterest in the fields of statistical and machine learning.\nWe consider a couple of random variables (X,Y)taking values inX×Y such thatX⊆ /CAp\nandY⊆ /CAk. We assume in this article that all random variables admit a d ensity function\nwith respect to a dominant measure. We also assume all these d ensities are proper i.e. they\nintegrate to 1. We denote fXand fYthe marginal densities of XandYwith respect to a\ndominant measure. Our goal is to estimate the conditional de nsity function:\nX×Y−→ /CA+\n(x,y)/mapsto−→fY|X=x(y)\nThis problem is at the root of the majority of machine learnin g tasks, including supervised\nand unsupervised learning or generative modelling. Superv ised learning techniques aims at\nestimating the conditional mean. Meanwhile, in the binary c lassification setting, these two\ntasks are equivalent, since E[Y|X=x] =P[Y=1|X=x]. In the regression setting where\nYis a continuous variable, the conditional density is far mor e informative than the mean value.\nThis is especially true when the conditional distribution i s multi-modal, heteroscedastic\nor heavy tailed. Moreover, in many fields such as actuarial sc ience, asset management,\nclimatology, econometrics, medicine or astronomy, one is i nterested in quantities other than\nexpectation, such as higher order moments (variance, skewn ess kurtosis), prediction intervals,\nquantile regression, outlier boundaries, etc.\nMeanwhile, most unsupervised learning techniques aim to di scover relationships and\npatterns between random variables. This corresponds to the joint density probability function\nestimation subtask, itself a subtask of CDE . Similarly, the field of generative modeling,\n1whose goal is to generate synthetic data by expressing the jo int distribution as a product\nof univariate conditional distributions with respect to la tent variables, can also be consid-\nered a CDE subtask, as realistic images or sounds correspond to the mod es of the distribution.\n1.1 Related work\nHistorically, the first attempts were based on the use of Baye s’ formula (1) which transforms\ntheCDE into the estimation of two density functions, thus allowing the use of techniques\ndedicated to density estimation.\nfY|X=x(y) =fX,Y(x,y)\nfX(x),fX(x)/nequal0 (1)\nIn this article , we address the many real-world application s of supervised learning regres-\nsions where we have a one-dimensional target but a large numb er of features. A major flaw\nof reformulating a conditional density as the ratio of two de nsities, is that even if Yis low di-\nmensional, we will still incur the curse of dimensionality i fXis high-dimensional. Although\ntechniques exist to alleviate the curse to some extent by lev eraging sparsity or other properties\nofX, it would be far easier to reformulate the CDE so that only the estimate of the marginal\ndensity fYis required instead of fX.\nNonparametric density estimation. Nonparametric estimation is a powerful tool for density\nestimation as it does not require any prior knowledge of the u nderlying density. One of\nthe first intuitions of kernel estimators was proposed by Ros enblatt (1956) and later by\nParzen (1962). Kernel estimators were then widely studied r anging from bandwidth selection\n(Goldenshluger and Lepski (2011)), non linear aggregation (Rigollet and Tsybakov (2006)),\ncomputational optimisation (Langrené and Warin (2020)) to the extension to conditional\ndensities (Bertin et al. (2014)). We refer to Silverman (201 7) and references therein for inter-\nested reader. A very popular and e ffective nonparametric method is the k-Nearest Neighbors\n(Fix and Hodges (1989)), but like other nonparametric metho ds (kernel estimators, histogram\nPearson (1895),...), a main limitation is the curse of dimen sionality (Nagler and Czado (2016),\nScott (1991)). Silverman (2017) showed that, in a density es timation setting, the number of\npoints n, needed to obtain equivalent results when fitting a d-dimensional random variable,\ngrows at a rate of n4\n4+d. Meanwhile, the impact of dimensionality on the computatio nal time\nalso scales exponentially. To the best of our knowledge, the best performing method for\nkernel estimation, based on a divide-and-conquer algorith m (Langrené and Warin (2020)),\nhas a complexity of O(nlog(n)max(d−1,1)). Both of these relationships are compounded, as\nhigher dimensionality means that exponentially more data i s required and the computational\ntime relative to the size of the dataset will also grow expone ntially.\nNoise Contrastive methods. Another important family of techniques for density estimat ion\nis noise-contrastive learning (Gutmann and Hyvärinen (201 0)). These techniques reformulate\nthe estimation of the density into a binary classification pr oblem (up to a constant). It consists\nin introducing a known probability density g(·)and sampling from it a synthetic dataset. The\nlatter is concatenated with the original dataset, then a tar get value Ziis associated such that Zi\nis equal to 1 if the observation comes from the original datas et and 0 otherwise. A contrast\nq(·), which can be, under certain conditions on g, directly related to the density of the original\n2observations is introduced to obtain an estimation of the de nsity function :\nq(x):=P[Z=1|X=x] =fX(x)\nfX(x)+g(x)and fX(x) =g(x)q(x)\n1−q(x). (2)\nA binary classifier is then trained to predict this target val ue conditionally on the associated\nobservation.\nContrast learning has been successfully applied in the area of self-supervision learning\n(He et al. (2020), Jaiswal et al. (2020)), especially on comp uter vision tasks (Bachman et al.\n(2019)). Many extensions and improvements have been made, i ncluding w.r.t. the learning\nloss function (Khosla et al. (2021)), data augmentation tec hniques (Chen et al. (2020)), net-\nwork architectures (He et al. (2020)) and computational e fficiency (Yeh et al. (2021)). Our\nproposed method is partially based on this technique, with t wo major differences. First, our\ntechnique addresses CDE problem and not density estimation or self-supervised lear ning. Sec-\nond, the distribution we choose is unknown, potentially int ractable, and/or with a large set\nof highly dependent components, which violates the usual re strictions for performing noise\ncontrastive density estimation but allows us to taylor the n oise distribution precisely to the\nestimated distribution. This allows us to fit the noise distr ibution precisely to the estimated\ndistribution. To the best of our knowledge, the technique cl osest to our method is designed to\nevaluate the deviation from the independence setting, i.e. when all random features {Xj}j=1,···,p\nare independent, in an unsupervised framework. It has been b riefly described in the second\nedition of Tibshirani et al. (1960) (pages 495-497) where, b ased on the noise contrastive re-\nformulation given above with g(·) = Πp\nj=1fXj(·). Note that gis unknown and corresponds\nto what we will call later the noise distribution. Neverthel ess, noise samples can be generated\nby applying a random permutation on the feature columns of th e dataset, which is su fficient\nto discover association rules between the Xfeatures but not to estimate the density function\nfX. On the other hand, although each component of Xis one-dimensional by definition, the er-\nrors made when estimating the marginal densities pwill compound when estimating g, which\nmeans that the dimensionality of Xis again a limiting factor.\nCDE reformulation. There are other ways to reformulate the CDE , for example\nSugiyama et al. (2010) propose a reformulation into a Least- Squares Density Ratio\n(LSCond ) and Meinshausen (1960) into a quantile regression task. St ill others take advan-\ntage of a reformulation of the CDE into a supervised learning regression task. Meanwhile,\nRFCDE (Pospisil and Lee (2018)) and NNKCDE (Pospisil (2020)) are techniques that adapt\nmethods that have proven to be e ffective to the CDE task. Other methods, such as Deep-\ncde(D’Isanto and Polsterer (2018)) and FlexCode (Izbicki and B. Lee (2017)), go further\nand design a surrogate regression task that can be run by an o ff-the-shelf supervised learning\nmethod taking advantage of many mature and well-supported o pen-source projects, e.g. scikit-\nlearn (Pedregosa et al. (2011)), pytorch (Paszke et al. (2019)), fastai (Howard and Gugger\n(2020)) , keras (Chollet and team (2022)), tensorflow (Martin Abadi et al. (2015)), Cat-\nBoost (Prokhorenkova et al. (2018)), XGboost (Chen and Guestrin (2016)), etc. Our\nmethod implementation also has this advantage, since it can take as arguments any\nclass that follows the scikit-learn init/f it/predict _proba API, which includes our\noff-the-shelf MLP implementations, linear grid units ( GLU ,Gorishniy et al. (2021)),\nResBlock (Gorishniy et al. (2021)) and self-normalizing networks ( SNN ,Klambauer et al.\n(2017)).\n3Neural networks and variational inference. Long before the current resurgence of in-\nterest in deep learning, there were already neural networks designed specifically to han-\ndleCDE ,e.g. Bishop (1994) adressed the problem of estimating a probabi lity density us-\ning Mixture Density Networks ( MDN ). More recently, Kernel Mixing Networks ( KMN ,\nAmbrogioni et al. (2017)) are networks trained to estimate a family of kernels to perform the\nCDE task. Another famous variational inference technique is Fl ow Normalization ( N.Flow ,\nRezende and Mohamed (2016)), designed to solve the problem o f finding the appropriate ap-\nproximation of the posterior distribution. A common challe nge with most of these methods is\nscalability in terms of computation resources, which our ex perimental benchmark confirmed.\nApython implementation of MDN ,KMN andN.Flow is provided by the f reelunchtheorem\npackage (freelunchtheoremDoc (2022)), which we included i n our benchmark.\n1.2 Our contributions\nOur method, called contrastive marginal discrimination, MCD , combines several charac-\nteristics of the above methods but without their respective limitations. At a basic level,\nMCD begins by reformulating the conditional density function i nto two factors, the marginal\ndensity function of Yand a ratio of density functions as follows\nfY|X=x(y) =fY(y)fX,Y(x,y)\nfX(x)fY(y),∀(x,y)∈X×Y s.t.fX(x)/nequal0,fY(y)/nequal0\nWe propose to estimate these two quantities separately. In m ost real applications of CDE ,Y\nis univariate or low-dimensional, while Xis not. In these cases, it is much easier to estimate\nfYrather than fXand fX,Y. In this article , we do not focus on how to choose the estimati on\nmethod for fYor introduce new techniques to estimate fY. By contrast, our experimental\nresults show that out-of-shelf kernel estimators with defa ult parameters perform very well on\nboth simulated density models and real datasets, as expecte d for univariate distributions.\nThe core of our method is the reformulation of the ratio of the density functions\nfX,Y//parenleftBig\nfXfY/parenrightBig\ninto a contrast. In our method, the introduced noise in equat ion 2 always corre-\nsponds to the density function g(·) = fXfY. This is akin to the reformulation proposed by\nTibshirani et al. (1960), except that we only break the relat ionship between the two elements\nof the pair (X,Y)but not between each component of XandY:fX,Y(·) = fX(·)fY(·)q(·)\n1−q(·).\nTo estimate the joint density fX,Yit would be necessary to estimate both fXand fY. But\nwhen we apply Bayes’ formula (1), we divide by fX, which disappears from the expression\noffY|X, meaning we only need to estimate fYandq.\nLike noise-contrastive methods, MCD can leverage state-of-the-art supervised learning\ntechniques to perform CDE , especially neural networks. Our numerical experiments re veal\nMCD performances are far superior when using neural networks co mpared to other popular\nclassifiers like CatBoost ,XGboost or Random Forest. Our benchmark also reveals that our\nmethod significantly outperforms in practice RFCDE ,NNKCDE ,MDN ,KMN ,N.Flow ,\nDeepcde ,FlexCode and LSCond on most the density models and regression datasets\nincluded in our benchmark. Moreover, the MCD reformulation enables us to train the binary\nclassifier on a contrast training set much larger than the ori ginal dataset. Evermore, MCD can\neasily take advantage of additional data. Unlabeled observ ations can be directly used to\nincrease the size of the training set, without any drawbacks . Similarly, in the case where each\nobservation is associated with more than one target value, t hey can all be included in the\ntraining dataset.\n4Our main contributions are as follows:\n•We introduce a reformulation of the CDE problem into a contrastive learning task which\ncombines a binary classification task and a marginal density estimation task, which are\nboth much easier than CDE .\n•We prove that given a training set of size nit is always possible to generate a i.i.d.\ntraining set of size/floorleftBign\n2/floorrightBig\ncorresponding to the contrast learning task. We also prove t hat\nit is always possible to generate a training set of size at mos tn2in the non i.i.d. case.\n•We provide the corresponding construction procedures and t hepython implementation.\nWe also provide construction procedures to leverage additi onal marginal data and mul-\ntiple targets per observations, which can improve performa nces significantly.\n•We produce a benchmark of 9 density models and 12 datasets. We combine our method\nwith a large set of classifiers and neural networks architect ures, and compare ourselves\nagainst a large set of CDE methods. Our benchmark reveals that MCD outperforms\nall the existing methods on the majority of density models an d datasets, sometimes\nby a very significant margin .\n•We provide a python implementation of our method compatible with any py-\ntorch module or scikit-learn classifier, and the complete code to replicate our experi-\nments.\nThe rest of this article is organised as follows: section 2 pr ovides a theoretical background\nfor the reformulation. Section 4 provides the implementati on details and evaluates our\nmethod on density models and regression datasets, comparin g results with the methods\nimplemented in the python frameworks CDEtools Dalmasso et al. (2020), Pospisil (2022)\nandfreelunchtheorem freelunchtheorem (2022), Rothfuss et al. (2019). Section 5 provides an\nablation study of our method. Section 6 provides the proofs f or theoretical results and the\nalgorithms to construct the training dataset.\n2 Marginal Contrastive Discrimination\n2.1 Setting\nIn this article , we consider three frameworks correspondin g to three different situation in\npractice.\nFramework 1 [Independent Identically Distributed Samples]\nIn this most classic setting, we consider DX,Y\nn={(Xi,Yi)}i=1,···,na training dataset of\nsize n∈ /C6∗such that∀i=1,···,n, the(Xi,Yi)are i.i.d.of density fX,Y.\nIn practice, it is often the case that additional observatio ns are available but without the\nassociated target values and vice versa (Framework 2). In th is article we show that it is\npossible to take advantage of these additional samples to in crease the size of the training\ndataset without using any additional unsupervised learnin g techniques.\n5Framework 2 [Additional Marginal Data]\nIn this framework, we still consider a i .i.d.training dataset of size n ∈ /C6∗of density fX,Y\ndenotedDX,Y\nn={(Xi,Yi)}i=1,···,n. Moreover we assume we have one or two additional\ndatasets.\n•LetDX\nnx={/tildewideXi}i=1,···,nxbe i.i.d.an additional dataset of size n x∈ /C6of density fX.\n•LetDY\nny={/tildewideYi}i=1,···,nybe i.i.d.an additional dataset of size n y∈ /C6of density fY.\nWe assume thatDX,Y\nn,DX\nnxandDY\nnyare independent.\nLet us introduce a last framework, the one where more than one target value is associated\nto the same observation. To our knowledge, the article of (Bo tt and Kohler (2017)) is the\nonly attempt to deal with this case. We show in this article th at our method can exploit and\ntake advantage of these additional targets, again, without requiring an additional learning\nscheme. This can be the case for example in mechanics when per forming fatigue analysis\n(Bott and Kohler (2017), Manson (1965)).\nFramework 3 [Multiple Target per Sample]\nLet(X,Y)be a couple of random variables taking values in X×Ymof density fX,Y. Let\nDX,Y\nn,m={(Xi,Yi)}n\ni=1a training dataset of size n ∈ /C6∗such that\n•The{Xi}i=1,···,nare sampled such that the X iare i.i.d.of density fX.\n•The{Yi}i=1,···,nare sampled such that the Yi|Xiare i.i.d.of density fY|X.\n2.2 Contrast function\nOur method, MCD , is grounded on a trivial approached based on the successive application\nof the Bayes’ formula (1).\nWe reformulate the problem di fferently from the existing noise contrastive methods, focus -\ning on the contrast between the joint law fX,Yand the marginal laws fXand fY. To do this,\nwe define (Definition 1) a new contrast q(·,·), called the marginal contrast function.\nDefinition 1 [Marginal Contrast function MCF (r)]\nLet r∈(0, 1)be a real number. Consider (X,Y)a couple of random variables taking\nvalues inX×Y . The Marginal Contrast Function with ratio r of the couple (X,Y),\ndenoted q (·,·), is defined as:\nX×Y−→ [0, 1)\n(x,y)/mapsto−→q(x,y):=r fX,Y(x,y)\nr fX,Y(x,y)+( 1−r)fX(x)fY(y).\nThis new contrast is motivated by the Fact 1: CDE is equivalent to the marginal density of\nYand the marginal contrast function q. The CDE task is therefore reduced to the estimation\nof the constrast and a marginal density.\n6Fact 1 Let r∈(0, 1)be a real number. Consider (X,Y)a couple of random variables\ntaking values inX×Y . For all(x,y)∈X×Y , we have\nfY|X=x(y) =fY(y)q(x,y)\n1−q(x,y)1−r\nr\nwhere q(·,·)denotes the MCF(r).\nIn the next section, we determine the conditions (Marginal D iscrimination Conditions) un-\nder which the contrast function estimation can be transform ed into an easy supervised learning\nestimation problem.\n2.3 Marginal Discrimination Conditions\nTo transform the problem of estimating qinto a problem of supervised learning, we first need\nto introduce a couple of random variables (W,Z)satisfying the Marginal Discrimination\nCondition (MDcond) of the couple (X,Y)with ratio r∈(0, 1).\nDefinition 2 [Marginal Discrimination Condition MDcond (r)]\nLet r∈(0, 1)be a real number. Consider (X,Y)two random variables taking values in\nX×Y . A couple of random variables (W,Z)is said to satisfy the Marginal Discrimina-\ntion Condition (MDcond) of the couple (X,Y)with ratio r if\n(Cd 1 ) The random variable Z follows a Bernoulli law of parameter r (Z∼B(r)).\n(Cd 2 ) The support of W is W=X×Y .\n(Cd 3 ) For all(x,y)∈X×Y ,we have fW(x,y) =r fX,Y(x,y)+( 1−r)fX(x)fY(y).\n(Cd 4 ) For all(x,y,z)∈X×Y×{ 0; 1}, we have\nfW|Z=z(x,y) =1z=1fX,Y(x,y)+1z=0fX(x)fY(y).\nRemark that condition ( Cd 3 ) is satisfied if conditions ( Cd 1 ),(Cd 2 ) and ( Cd 4 ) are veri-\nfied. These conditions are su fficient to characterise both the joint and marginal laws of the\ncouple(W,Z).\nEstimation of the contrast function through supervised lea rning The Proposition 1\nspecifies how the marginal contrast function qproblem can be estimated by a supervised\nlearning task, using either a regressor or a binary classifie r, provided that we have access to a\nsample of identically distributed ( i.d.) of random variables that satisfies the MDcond (r).\nProposition 1 [Constrast Estimation]\nLet r∈(0, 1)be a real number. Consider (X,Y)a couple of random variables taking\nvalue inX×Y . For all(x,y)∈X×Y , the Marginal Contrast Function of couple (X,Y)\nwith ratio r denoted by q satisfies the following property:\nq(x,y) =E[Z|W= (x,y)] = P[Z=1|W= (x,y)].\nThe proof is given in section 6.2. It remains to prove that it i s possible to construct a training\nset of identically distributed (i.d.) samples of (W,Z)using the elements of the original dataset.\n73 Contrast datasets construction\n3.1 Classical Dataset (Framework 1)\nTheorem 1 establishes the existence of an i.i.d. sample of (W,Z)satisfying the MDcond (r)\nin Framework 1. In its proof (section 6.3), such of construct ion based on the original data\nset is derived. From now and for all α∈ /CA, the quantity⌊α⌋denote the largest integer value\nsmaller or equal to α.\nTheorem 1 [Construction of an i.i.d.training Set]\nLet r∈(0, 1)be a real number. Consider the dataset DX,Y\nndefined in Framework 1.\nThen, we can construct a dataset DW,Z\nN={(Wi,Zi)}i=1,···,Nof size N =⌊n/2⌋of i.i.d.\nobservations satisfying the MDcond (r).\nNote that, in practice, having access to a larger data set imp roves the results considerably.\nBy dispensing with the independence property, it is possibl e to construct a much larger data\nset without deteriorating the results (see numerical exper iments). This is the purpose of\nTheorem 2\nTheorem 2 [Construction of a larger i.d.training Set]\nConsider the dataset DX,Y\nndefined in Framework 1. Moreover, assume that DX,Y\nnis such\nthat∀(i,j)∈{1,···,n}2with i/nequalj\nXi/nequalXjand Y i/nequalYj\nThen, for any couple of integers (nJ,nM)such that\n/braceleftBigg1≤nJ≤n\n1≤nM≤n(n−1)\nwe can construct a dataset DW,Z\nN={(Wi,Zi)}i=1,···,Nof size N=nJ+nMof i.d.random\nobservations satisfying the MDcond/parenleftBignJ\nN/parenrightBig\n.\nNote first that we can at most construct a dataset of size N=n2, with r=1\nn. On the\nother hand, if we want to have r=1\n2, we can generate a dataset of size N=2n. Second, the\nadditional conditions on the dataset are introduced to excl ude the trivial case where a larger\ndataset is constructed by simply repeating the existing sam ples. In practice, we can always\navoid this case by removing redundant samples. Note, howeve r, that the repetition of values\noccurs with probability 0, almost surely, because we consid er continuous densities. Finally,\nthe complete construction of such a dataset is described in s ection 6.4\n3.2 Additional Marginal Data (Framework 2)\nNow consider that we have additional features and /or targets for which the target or associated\nfeature is not available. We include this additional data in our training process without using\na semi-supervised scheme.\n8Theorem 3 [Construction of an i.i.d.training set]\nLet r∈(0, 1)be a real number. Consider the datasets defined in Framework 2 . Set\nN=min/parenleftBigg\nn,/floorleftBiggn+nx+ny\n2/floorrightBigg/parenrightBigg\n,\nthen, we can construct DW,Z\nN={(Wi,Zi)}i=1,···,Na dataset of size N of i .i.d.observations\nsatisfying the MDcond (r).\nThis theorem implies that as soon as we have nx+ny≥n, we can generate a training set\nfor the discriminator as large as the original set, i.e. N=n. In practice, this can happen\nin many cases. For example, when data annotation is expensiv e or difficult, we often have\nnx>> n. At the same time, to use contrastive marginal discriminati on to estimate the\nconditional density, we need to know or estimate the margina l density fY. The proof of this\ntheorem is done in section 6.5\nTheorem 4 [Construction of a larger i.d.training set]\nConsider the dataset defined in Framework 3. Moreover assume\n•The datasetDX,Y\nnis such that∀(i,j)∈{1,···,n}2s.t.i/nequalj\nXi/nequalXjand Y i/nequalYj.\n•The datasetsDX\nnxnxandDY\nnynyare s .t.∀(i,j)∈{1,···,nx}2and∀(i′,j′)∈{1,···,ny}2\n/tildewideXi/nequal/tildewideXjand/tildewideYi′/nequal/tildewideYj′\n•Moreover, we assume that DX,Y\nn,DX\nnxandDY\nnyare such that∀(i,i′)∈{1,···,n}×\n{1,···,nx}and∀(j,j′)∈{1,···,n}×{1,···,ny}\nXi/nequal/tildewideXi′and Y j/nequal/tildewideYj′\nThen, for any couple of integers (nJ,nM)such that\n/braceleftBigg1≤nJ≤n\n1≤nM≤(n+nx)(n+ny)−n\nwe can a datasetDW,Z\nN={(Wi,Zi)}i=1,···,Nof size N=nJ+nMof i.d.random observa-\ntions satisfying the MDcond (nJ\nN).\nHere again, it is possible to build a much larger i.d. training dataset under some weak assump-\ntion. Indeed, the repetition of values occurs with probabil ity 0, almost surely. The complete\nconstruction of such a dataset is described in section 6.6\n3.2.1 Multiple targets per observations (Framework 3)\nIn Framework 3, to each of the nobservations Xi, there exists a m-associated target\nYi=/parenleftBig\nYi,1,···,Yi,m/parenrightBig\nofi.i.d. components such that the (Yi,j|Xi)j=1,···,marei.i.d.. In this\nsetting, it is still possible to construct, under some weak a ssumption, a larger i.d. training\nset satisfying the MDcond. Recall, the repetition of values occurs with probability 0,\n9almost surely. Note that we can construct, at most, a data set of size n2×m, with r=1\nn. On\nthe other hand, if we want to have a ratio of r=1\n2, the generated dataset will be of size 2 n×m.\nTheorem 5 [Construction of a larger i.d.training set]\nConsider the dataset DX,Y\nn,m={(Xi,Yi)}n\ni=1a training dataset of size n ∈ /C6∗defined in\nFramework 3. Assume that\n•For all(i,i′)∈{1,···,n}2such that i/nequali′\nXi/nequalXi′.\n•For all(i,j),(i′,j′)∈{1,···,n}×{1,···,m}such that i/nequali′or j/nequalj′\nYi,j/nequalYi′,j′\nThen, for any couple of integers (nJ,nM)such that\n/braceleftBigg1≤nJ≤n×m\n1≤nM≤n(n−1)×m\nwe can construct a dataset DW,Z\nN={(Wi,Zi)}i=1,···,Nof size N=nJ+nMof i.d.random\nobservations satisfying the MDcond/parenleftBignJ\nN/parenrightBig\n.\nThe complete construction of such a dataset is described in s ection 6.7\n4 Experiments\nIn this section we detail the implementation of MCD in section 4.1 and provide a benchmark\nto compare MCD to other available methods. All our experiments are done in python and the\nrandom seed is always set such that the results of our method a re fully reproducible.\nWe compare our method MCD with other well-known methods presented in section 4.2 on\nboth density models including two new ones, described in sec tion 4.3 and real dataset. Our\nresults are displayed in sections 4.4 and 4.5.\n4.1 Method implementation\nTraining. First describe the procedure used for both parametric and no nparametric estima-\ntion, to train our estimator MCD on dataset corresponding respectively to Framework 1, 2\nor 3. Table 4.1 details which Construction to use in each Fram ework.\nTraining (Step1). Set r.\n(Step2). Estimate fYusing{Yi}i=1,···,nfromDX,Y\nn.\n(Step3). GenerateDW,Z\nN.\n(Step4). Train either a regressor or a binary classifier on DW,Z\nN.\nThe estimators obtained in steps 2 and 4 are called respectiv ely the marginal estimator /hatwidefY\nand the discriminator /tildewideq. Note that for any (x,y)∈X×Y , we have /tildewideq(x,y)∈[0, 1]while\nq(x,y)∈[0, 1). To obtain an appropriate prediction, we introduce a thresh olding constant\nǫ=10−6and set\n/hatwideq(x,y) =min(/tildewideq(x,y), 1−ǫ)∈[0, 1−ǫ]⊂[0, 1).\n10Framework Available datasets i.i.d. samples i.d. samples\nFramework 1DX,Y\nn Construction 1 Construction 2\nFramework 2DX,Y\nn,DX\nnx,DY\nnyConstruction 3 Construction 4\nFramework 3DX,Y\nn,m Not applicable Construction 5\nTable 1: Look-up table to determine the appropriate constru ction corresponding to each framework.\nNext, underline that MCD can be used to perform two di fferent tasks:\n•Nonparametric estimation of the conditional density fY|X=x(·)for all x∈X.\n•Pointwise estimation of the conditional density fY|X=x(y)for any(x,y)∈X×Y .\nIndeed, using Fact 1 we have: ∀(x,y)∈X×Y ,\n/hatwidefY|X=x(y) =/hatwidefY(y)/hatwideq(x,y)\n1−/hatwideq(x,y)1−r\nr. (3)\nThis implies that it is su fficient to have estimators of both fYandqto have a point estimate\noffY|X=x(y). We may also deduce the literal expression of /hatwidefY|X=x(·), the nonparametric\nestimate of the conditional density, provided we know the li teral expressions of /hatwideq(x,·)and\n/hatwidefY(·). It is the case for /hatwideqin Deep Learning, as we can write the literal expression of /hatwideq(x,·)\nfrom the NNparameters learned in the learning step and a value x. Under certain assumptions\non/hatwideq,/hatwidefY|X=x(·)is a true density (Gutmann and Hyvärinen (2010)).\nPrediction. For parametric pointwise estimation, at test time, given an y new observation\nx∈ X, for any chosen target value y∈ Y , we can estimate fY|X=x(y)the value of the\nprobability density function evaluated on (x,y):\nPrediction (Step1). Evaluate /tildewideq(x,y).\n(Step2). Apply thresholding: /hatwideq(x,y) =min(/tildewideq(x,y), 1−ǫ).\n(Step3). Evaluate /hatwidefY(y).\n(Step4).Plug in fY|X=x(y)by applying equation 3 given above.\nRemark that if the discriminator is a classifier, the predict ed value should be the probability\nof class 1, i.e.P[Z=1|W= (x,y)].\nChoice of parameters. There are 4 major choices to make when implementing our metho d:\nthe marginal density estimator method, the discriminator m ethod, the construction and the\ncontrast ratio r.\n•Marginal density estimator. Since the point of our method is to provide an estimation\nof the conditional density in cases where the marginal densi ty is relatively easy to estimate\n(meaningY⊆ /CA), we pick a simple yet e ffective technique, the univariate kernel density\nestimation KDEUnivariate provided in the statsmodels package. We always keep the default\nparameters, i.e. gaussian kernels, bandwidth set using the normal referenc e, and the fast\nFourier transform algorithm to fit the kernels.\n11Neural Networks architectures Other supervised learning classifiers\nMultiLayerPerceptron ( MLP ) Random Forests ( RF)\nMLP w/o Drop-Out nor Batch-Norm ( MLP :no-D.O. )Elastic-net\nResNet ( ResBlock ) Gorishniy et al. (2021) XGboost Chen and Guestrin (2016)\nGated Linear Unit ( GLU ) Gorishniy et al. (2021) CatBoost Prokhorenkova et al. (2018)\nSelf Normalizing Networks ( SNN ) LGBM Ke et al. (2017))\nKlambauer et al. (2017)\nTable 2: List of discriminator methods evaluated with MCD .\n•Marginal contrast discriminator. We evaluate the performance of MCD combined with\nNeural Networks, Decision Tree based classifiers and Logist ic Elastic-net (see table 4.1 for\nthe exhaustive list).\n•Dataset construction and ratio: We compare in our ablation study (Section 5) the Con-\nstructions 1, 2, 3, 4 and 5 in Frameworks 1, 2 and 3. Following t he findings of the ablation\nstudy, we use Construction 2 and set r=0.05 in other experiments.\n4.2 Other benchmarked methods and Application Programming Interface\n(API)\nThere is a small number of CDE methods for which a readily available open source imple-\nmentation in python exists. One notable di fficulty when introducing a new CDE package is\nthat there is no gold standard API in python on top of which new packages can build upon.\nMost existing implementations are standalone, with their o wn unique syntax for common\nfunctions. We choose to include in our benchmark the methods provided by two of the most\nmature python projects, the freelunchtheorem github repository by freelunchtheorem (2022),\nRothfuss et al. (2019), and a network of packages created by D almasso et al. (2020) and\nPospisil (2022).\nThe freelunchtheorem github provides an implementation of KMN ,N.Flow ,MDN and\nLSCond . The freelunchtheorem github has a quite consistent API across all provided meth-\nods. Notably, freelunchtheorem (2022), Rothfuss et al. (20 19), also provide implementations\nfor several statistical models ( EconDensity ,ARMAJump ,JumpDiffusion ,LinearGauss ,\nLinearStudentT ,SkewNormal andGaussianMixt ), which we include in our benchmark\n(see the density model section 4.3).\nMeanwhile, the project of Dalmasso et al. (2020) and Pospisi l (2022) is built around the\nCDEtools github repository and consists of several repository of varying ma turity and ease\nof use corresponding to each method they implemented ( RFCDE , f-RFCDE ,FlexCode ,\nDeepcde andNNKCDE ). Notably, they also provide implementations in Java and R f or some\nof these methods. Their implementation of Deepcde allows custom pytorch andtensor f low\narchitectures to be plugged-in, which gave us the opportuni ty to adapt the architectures and\ntraining schemes used with MCD toDeepcde . As such, the comparison between MCD and\nDeepcde is done on equal ground.\nIn total, we include in our benchmark 10 other CDE methods: NNKCDE ,N.Flow ,LSCond ,\nMDN ,KMN ,Deepcde ,RFCDE ,f-RFCDE ,FlexCode :NNandFlexCode :XGboost .\nAlthough it is not the main goal of this work, we provide an ove rhead over these two\nprojects and our method to facilitate the comparison betwee n them. Each evaluated\n12method is encapsulated in a class which inherits the same uni que API from the parent class\nConditionalEstimator , with same input and output format and global behavior. Simi lar to\nscikit-learn , the estimator is an instance of the class, with hyper-param eters provided during\ninitialisation (__ init__), and the observation matrix and target vector provided (as numpy\narrays) when calling the f itfunction. At predict time however, the estimator predictio n is\na function called pdf_from_X which predicts the probability density function on a grid of\ntarget values. This package overhead allows us to compare al l methods on equal ground:\nFor density models all methods are trained on the same sample d training set. We also use\nthe same grid of target values and test set of observations to evaluate the probability density\nfunction of all compared methods. Likewise, for real-world datasets, we use the same dataset\ntrain-test splits for all compared methods.\n4.3 Estimation of theoretical models\nWe first evaluate our method on the core task it aims to handle o n theoretical models: numeri-\ncally estimating a conditional density function with respe ct to a new observation. We choose\nto evaluate the quality of the prediction empirically: for e ach predicted and target functions,\nwe evaluate for a grid of target values the empirical Kullback-Leibler divergence (KL).\nDensity models Thefreelunchtheorem package provides 7 conditional densities implementa-\ntions for which we have a function to generate a training data set and a function to evaluate\nthe theoretical density function on a grid of target values g iven an observation. These 7\nmodels are EconDensity ,ARMAJump ,JumpDiffusion ,LinearGauss ,LinearStudentT ,\nSkewNormal andGaussianMixt . We refer to the freelunchtheorem github documentation\nfreelunchtheorem (2022) for a detailed description of each model. Although these conditional\ndensity models cover a diverse set of cases, we do however int roduce two other density\nmodels to illustrate the specific drawbacks of some benchmar ked methods.\nModel 1 [BasicLinear ]: Let p =10and fix p coefficientsβ={βj}j=1,···,pdrawn\nindependently at random, such that ∀j=1,···,p,βjis uniformly distributed over\n(0, 1), i.e.βj∼U(0, 1).\nConstruct now our first density model\n•Let X∼N(0p,Ip)be a gaussian vector.\n•Let Y∈ /CAbe a random variable such that Y =X⊤β+σǫwhereǫ∼N(0, 1)and\nǫand X independent.\nBasicLinear is a very simple linear model included to check that sophisti cated methods\nwhich can estimate complex models are not outperformed in si mple cases. We also add\na second model, AsymmetricLinear , which corresponds to BasicLinear with a simple\nmodification: we use asymmetric noise ( |ǫ|instead ofǫ).\nModel 2 [AsymmetricLinear ]: Letβ, X andǫbe as in BasicLinear 1. In our second\ndensity model, Y is as follows:\nY=X⊤β+σ|ǫ|.\nHere,|·|denotes the absolute value.\n13//\nThe major difficulty is that the support of the conditional density of Ywith respect to X\ndiffers from the support of the marginal density of Y(which isY= /CA) and depends on the\nobservation X.\n4.4 Results on density models\nEvaluation protocol. We use 9 density models (section 4.3) as ground truth, on whic h we\nevaluate the MCD with various discriminators and the methods presented in se ction 4.2.To\ngenerateDX,Y\nnfor each density model, we sample n=100 observations and for each\nobservation we sample one target value using the conditiona l law of the density model.\nWe train all benchmarked methods on this same dataset. Next, we sample ntest=100\nobservations from the density model to generate a test set. W e also generate a unique grid of\n10000 target values, spread uniformly on Y. For each observation of the test set, we evaluate\nthe true conditional density function on the grid of target v alues. Then, for all benchmarked\nmethod, we estimate the conditional density for each observ ation on that same grid of target\nvalues and evaluate the empirical Kullback-Leibler ( KL) divergence defined below.\nEmpirical Kullback-Leibler divergence .\nSetδ=10−6a numerical stability constant. Let x∈ /CApbe an observation of the test\nsetDtestandy∈ /CAbe a point onG, a grid of target values to be estimated. Then, for\nthe evaluation of the target value fx(y) = max(fY|X=x(y),δ)and the predicted value\ngx(y) =max(/hatwidefY|X=x(y),δ), we define the empirical KLδdivergence as follows:\nKL :=KLδ(f/⌊ard⌊lg) =/summationdisplay\nx∈D test/summationdisplay\ny∈Gfx(y)×ln/parenleftBiggfx(y)\ngx(y)/parenrightBigg\n. (4)\nBenchmark results.\n➢We first combine the MCD method with a classic Multi Layer Perceptron ( MLP ) as dis-\ncriminator and a kernel estimator as marginal density estim ator, named MCD :MLP . Ta-\nble 3 depicts the global performance in terms of empirical KL divergence over 9 models of\nMCD :MLP compared to 10 others methods, described in section 4.2.\n•The main take away is that in 6 out of 9 cases, MCD :MLP outperforms all others.\n•On 3 density models, BasicLinear ,LinearGauss and LinearStudentT , the\nKLempirical divergence is less than half of the second best met hod. Meanwhile, on\nEconDensity ,N.Flow andMCD :MLP share the first place.\n•When MCD :MLP is outperformed, which corresponds to JumpDiffusion and\nSkewNormal , the best performing methods are N.Flow andMDN . Otherwise, the sec-\nond best performing method is either N.Flow orNNKCDE .\n➢We also assess the performance of MCD combined with other popular supervised learning\nmethods. Table 4 depicts the performance of MCD with various discriminators on the same\nbenchmark of density models. The classifiers included are li sted in Table 2 and described in\nsection 4.1.\n14Empirical KL\nMCD :MLP\nNNKCDE N.Flow LSCond MDN KMN Deepcde RFCDE f-RFCDE\nFlexCode :NN\nFlexCode :XGboostARMAJump 0.573 0.929 0.196 0.408 0.29 0.312 1.754 0.529 0.526 1.201 2.226\nAsymmetricLinear 0.158 0.245 0.498 0.882 0.437 0.338 0.666 0.324 0.328 0.359 0.483\nBasicLinear 0.009 0.087 0.313 0.473 0.195 0.167 0.317 0.139 0.139 0.174 0.116\nEconDensity 0.006 0.01 0.006 0.021 0.013 0.022 0.068 0.049 0.045 0.034 0.048\nGaussianMixt 0.005 0.008 0.012 0.023 0.016 0.018 0.127 0.026 0.028 0.048 0.023\nJumpDiffusion 1.352 1.632 4.481 9.371 4.576 5.568 22.45 10.45 10.45 6.347 4.363\nLinearGauss 0.189 1.742 0.868 3.15 2.318 2.892 14.71 15.88 15.88 13.75 3.571\nLinearStudentT 0.141 0.301 6.238 9.09 3.109 3.136 7.583 2.363 2.363 1.686 0.821\nSkewNormal 0.722 0.089 0.019 0.1 0.014 0.036 18.94 0.551 0.551 0.636 2.255\nTable 3: Evaluation of the empirical KLdivergence of different CDE methods, for 9 density models with n=\n100. Best performance is in bold print, second best performance is underlined. Lower values are better.\nColumn 2 corresponds to the performance of MCD combined with the classic MLP . Columns 3 to 12\nshow the results of the 10 other benchmarked methods.\n•In density model ARMAJump where MCD :MLP is outperformed by other methods,\nsimply removing the Batch-Normalization and Drop-Out is su fficient to obtain the best\nperformance.\n•In density models BasicLinear ,EconDensity ,GaussianMixt ,JumpDiffusion ,Lin-\nearGauss andLinearStudentT , the results for MCD :MLP are very close to other\nNNarchitectures performances. This means that in most cases, MCD does not require\nheavy tuning to perform well.\n•The best discriminator besides NNis always either CatBoost orE.Net .\n•NNdiscriminators are outperformed by other classifiers in onl y one case, the SkewNor-\nmaldensity model. This corresponds to the density model includ ed in our benchmark\nwhere all versions of MCD are outperformed by another method.\nEmpirical KL\nMLP SNN GLU ResBlock no-D.O. CatBoost E.Net XGboost LGBM RF XRFARMAJump 0.573 1.047 0.731 0.224 0.1 0.241 1.09 0.598 0.953 2.265 4.469\nAsymmetricLinear 0.158 0.05 00.05 0.055 0.318 0.2 0.262 0.274 0.28 0.279 0.319\nBasicLinear 0.009 0.008 0.01 0.009 0.108 0.059 0.083 0.113 0.098 0.094 0.106\nEconDensity 0.006 0.007 0.005 0.005 0.006 0.012 0.016 0.06 0.07 0.305 0.444\nGaussianMixt 0.005 0.006 0.006 0.005 0.007 0.01 0.007 0.057 0.061 0.297 0.422\nJumpDiffusion 1.352 1.353 1.352 1.352 1.352 1.485 1.352 5.694 5.11 10.17 19.99\nLinearGauss 0.189 0.189 0.19 0.19 0.19 1.274 0.19 9.33 10.99 67.46 94.76\nLinearStudentT 0.141 0.141 0.141 0.141 0.141 0.236 0.141 1.58 1.027 0.912 1.617\nSkewNormal 0.722 0.796 0.669 0.356 0.155 0.083 0.833 0.223 0.12 5.036 7.289\nTable 4: Evaluation of the empirical KLdivergence of different MCD discriminators, for 9 density models with\nn=100. Best performance is in bold print, second best performance is underlined. Lower values\nare better. Columns 2 to 6 correspond to the performance of MCD combined with NNarchitectures.\nColumns 7 to 12 show the results of MCD combined with other popular classifiers.\nImpact of dimensionality.\n➢Then, we check if in cases where MCD outperforms all others, our method maintains its\n15good performances when p, the number of features, changes. Table 5 depicts the impact of the\ndimensionality of Xon the performances of each method in BasicLinear , the density model\nwhere the performance gap in our benchmark between MCD and other methods is the largest.\n•Performances decrease across the board when pincreases. Although BasicLinear is a\nsetting where a larger pcorresponds to a higher signal to noise ratio, this is not eno ugh\nto counter the curse of dimensionality.\n•MCD :MLP outperforms all others at all ranges of p. Note that at p=300, the em-\npirical KLofMCD :MLP is equivalent to that of fY, the marginal density of the target\nvalue. This means that in this high-dimensional case, where MCD :MLP is not able to\ncapture the link between XandY, it does not overfit the training set, and instead takes\na conservative approach.\n•NNKCDE maintains good results across the board. Meanwhile, MCD combined with\nCatBoost performs almost as well as MCD :MLP when p=3, but its relative perfor-\nmances are average at best when p=300.\nEmpirical KLMCD MCDNNKCDEFlexCodeKMN RFCDEFlexCode\nMLP CatBoost XGboost NN\np=3 0.008 0.009 *0.022* 0.093 0.067 0.037 0.094\np=10 0.036 0.042 *0.063* 0.076 0.096 0.105 0.106\np=30 0.115 0.202 0.154 0.196 *0.181* 0.238 0.22\np=100 0.162 *0.224* 0.173 0.264 0.401 0.238 0.234\np=300 0.244 0.308 *0.282* 0.302 0.304 0.507 0.253\nTable 5: Evaluation of the KLdivergence values for various feature sizes p, on the BasicLinear density model,\nwith n=100. Best performance is in bold print, second best performance is underlined, *third best*\nperformance is between asterisks. Lower is better. Methods depicted achieve top 4 performances for at\nleast one feature size regime.\nExecution time and scalability.\n➢Next, we evaluate the scability with respect to the size of th e dataset of MCD combined\nwith either MLP orCatBoost , two classifiers known to be e fficient but slow. We include\nas reference other methods belonging to the same categories , meaning those based on su-\npervised learning NNand Decision Trees respectively. We also include NNKCDE and the\nmethods based on variational inference, as they perform wel l in our benchmark. Table 6 de-\npicts the computation time of MCD and other CDE methods for n=30, 100, 300 and 1000,\nonBasicLinear , the density model where it is most beneficial to use MCD instead of another\nmethod. The reported computation times include all steps (i nitialization, training, prediction).\nForMCD , this includes both the time taken by the discriminator and t he time taken by the\nestimator. Note also that for MCD , we use the ratio r=0.05, meaning the actual training set\nsize is 20 times larger.\n•The main take away is that MCD mostly scales like its discriminator would in a super-\nvised learning setting.\n•Regarding methods based on supervised learning with neural networks, we compare\nMCD :MLP with FlexCode :NNand Deepcde .FlexCode :NNis much faster than\nDeepcde andMCD :MLP . When using equivalent architectures, Deepcde is slower\nthan MCD when n=1000 (which corresponds to N=n\nr=20000 for MCD ). This\ncan be partly explained by the fact that Deepcde uses a transformation of the target\n16which corresponds to an output layer width of 30, while MCD increases the input size\nof the network by 1, since observations and target values are concatenated.\n•Regarding methods based on supervised learning with decisi on trees, we compare\nMCD :CatBoost with RFCDE andFlexCode :XGboost .RFCDE is the only method\nwhich does not leverage GPU acceleration, yet, it is faster than MCD :CatBoost and\nFlexCode :XGboost .\n•Among the best performing methods in our benchmark, NNKCDE is by far the fastest.\nMeanwhile, the 3 variational inference methods included, MDN ,KMN andN.Flow ,\nare much slower, which is to be expected.\nCategory Decision Tree Based Methods NNbased Methods Kernel Variational inference\nMethod RFCDE MCD FlexCode FlexCode MCD Deepcde NNKCDE MDN N.Flow KMN\nBased on: RF CatBoost XGboost NN MLP MLP N.Neighbor NN NN NN\nn=30 0.044 0.46 9.574 0.018 3.133 1.943 0.01 35.46 50.74 47.92\nn=100 0.149 0.369 9.57 0.017 3.149 2.11 0.022 35.45 50.72 102.7\nn=300 0.448 0.446 13.19 0.017 3.132 3.225 0.04 35.65 50.62 150.6\nn=1000 0.952 0.689 25.24 0.02 3.296 6.388 0.043 35.73 51.12 143.6\nTable 6: Training Time in seconds for various training set si zesn, on the BasicLinear density model. Row 1\ncorresponds to the method category, row 2 corresponds to the benchmarked CDE method and row 3\ncorresponds to the underlying supervised learning method u sed. Column 1 corresponds to the training set\nsize. Columns 2, 3 and 4 correspond to methods which leverage a supervised learning method based on\nDecision trees. Columns 5, 6 and 7 correspond to methods whic h leverage a supervised learning method\nbased on NN. Column 8 corresponds to nearest neighbors kernels. Column s 9, 10 and 11 correspond to\nthree variational inference methods.\n4.5 Real-world datasets\nDataset origins and methodology.\n➢We include in our benchmark 12 datasets, taken from two sourc es:\n•The CDEtools framework provides the dataset \"Teddy\" (see Beck et al. (201 7) for a\ndescription). We follow the pre-processing used in the give n packages.\n•The freelunchtheorem framework provides 2 toy datasets, 7 datasets from the UCI\n(Asuncion and Newman (2007) repository, and one dataset fro m the kaggle plateform\nKaggle (2000) which includes 2 targets, for a total of 11 data sets. Here again, we follow\nthe pre-processing used in the given package.\nEvaluation protocol.\n➢For all datasets, we standardize the observations and targe t values, then we perform the\nsame train-test split for all methods benchmarked (manuall y setting the random seed for re-\nproductibility). Because datasets are of varying sizes and as some methods are extremely\nslow for large datasets, we only take a subset of observation s to include in the training set,\ndoing the split as such. Let nmaxbe the original size of the dataset, the trainset is of size\nn=min(300,⌊nmax×0.8⌋)and the test set is of size ntest=min(300, nmax−n).\nOne challenge when benchmarking CDE methods is that real-world datasets almost never\nprovide the conditional density function associated to an o bservation, but instead only one\nrealisation. This means that the usual metrics for CDE (eg.: KL) cannot be evaluated on\nthese datasets. We nonetheless evaluate our method on real- world datasets, using the negative\nlog-likelihood metric, denoted NLL , to compare the estimat ed probability density function\nagainst the target value.\n17Empirical Negative Log-likelihood .\nSetδ=10−6a numerical stability constant. Let (x,y)∈ /CAp× /CAbe a sample from\nthe test setDtest, and gx(y) = max(/hatwidefY|X=x(y),δ)be the predicted value, we define the\nNLL metric as follows:\nNLL :=NLLδ(g) =−/summationdisplay\n(x,y)∈D testln(gx(y)). (5)\nResults.\n➢Table 7 depicts the global performance in terms of negative l og-likelihood for the 12\ndatasets.\n•The main take away is that this time, MCD :MLP outperforms existing methods in 7\nout of 12 cases, including the popular datasets BostonHousi ng and Concrete.\n•On the WineRed and WineWhite datasets, MCD lags far behind NNKCDE andFlex-\nCode . One possible cause is that for these two datasets, the targe t variable Ytakes\ndiscrete values:Y={3, 4, 5, 6, 7, 8}, which does not correspond to the regression task\nfor which MCD was designed.\n•Besides MCD , the top performing methods are NNKCDE ,N.Flow ,FlexCode (both\nversions) and KMN .\n•Regarding the choice of the discriminator, MCD :MLP is not outperforming\nMCD :CatBoost to the same extent it does on density models. The di fference in nega-\ntive log-likelihood between MCD :MLP andMCD :CatBoost is below 0.1 in 7 out of\n12 cases.\n•Besides, MCD :CatBoost obtains Top 2 performances in 3 of the 5 cases where\nMCD :MLP is outperformed by other methods. Notably, on the Yacht data set where\nMCD :MLP performs poorly, MCD :CatBoost outperforms all others by a wide mar-\ngin. This indicates that this time, MLP andCatBoost are complementary, as together\nthey can obtain at least Top 2 performances in all cases besid es the WineRed and\nWineWhite datasets.\nEmpirical NLLMCD MCDNNKCDEFlexCode FlexCodeKMN N.FlowMLP CatBoost NN XGboost\nBostonHousing -0.64 -0.59 -0.81 -1.99 -1.84 -1.22 -1.63\nConcrete -0.86 -1.02 -1.23 -2.26 -1.25 -2.30 -2.13\nNCYTaxiDropoff:lon. -1.30 -1.28 -1.68 -2.05 -2.17 -2.51 -2.85\nNCYTaxiDropoff:lat. -1.31 -1.31 -1.44 -1.67 -6.18 -2.45 -2.96\nPower -0.06 -0.36 -0.73 -1.09 -0.75 -0.35 -0.39\nProtein -0.09 -0.42 -0.77 -0.83 -1.38 -0.68 -0.54\nWineRed -0.89 -0.89 3.486 1.062 0.965 -0.90 -2.43\nWineWhite -1.18 -1.13 2.99 -0.73 -0.63 -1.7 -4.18\nYacht 0.14 0.822 -0.46 -1.23 0.144 0.025 0.401\nteddy -0.47 -0.51 -0.83 -0.83 -1.34 -0.76 -0.94\ntoy dataset 1 -0.99 -0.47 -0.63 -0.88 -1.46 -0.35 -0.71\ntoy dataset 2 -1.40 -1.33 -1.31 -1.54 -1.43 -1.33 -1.39\nTable 7: Evaluation of the negative log-likelihood (NLL ) fo r 12 datasets. Best performance is in bold print,\nsecond best performance is underlined. Higher values are better. Methods included o utperform all others\nbesides MCD on at least one dataset.\n185 Ablation\nWe now present an ablation study of the impact of the construc tion strategy used to build\nDW,Z\nNand the chosen value for ratio r. We also assess the impact of additional data on perfor-\nmances in Framework 2 and 3. Our experiments are done on the AsymmetricLinear density\nmodel, which corresponds to a case where MCD is performing well but there is still room\nfor improvement.\n•The main take away is that in Framework 1, i.d.-Construction 2 is far better than i.i.d.-\nConstruction 1.\n•The appropriate ratio rfor the i.d.-Construction 2 should be around 0.05, meaning\nN=20×n.\n•In Framework 3, as soon as two target values are associated to each observation, the i.d.-\nConstruction 5 can massively improve the performances of MCD , which are already\nvery good when using i.d.-Construction 2.\nConstruction strategy and ratio r\n➢Table 8 depicts the impact of the construction strategy and r atioron the performance in\nterms of empirical KLdivergence in the classical Framework 1 on the AsymmetricLinear\ndensity model.\n•The appropriate ratio rfor the i.i.d.-Construction 1 is 0.5 which corresponds to a bal-\nanced distribution between the two classes.\n•It seems clear that the i.d.-Construction 2 produces much better results than the i.i.d.-\nConstruction 1. For the latter, the performances are worse t han those obtained with\nthe concurrent method NNKCDE (see Table 3: KL=0.245). On the other hand, the\ni.d.-Construction 2 obtains Top 1 performances on our benchmar k, by a wide margin.\n•For the i.d.-Construction 2, it seems preferable to choose a ratio of 0. 15 or 0.05, which\ncorresponds respectively to a 6 or 20 times larger dataset. I ndeed, in that case since\nN=n\nr, we have to make a trade-o ffbetween the size of the training dataset and the\nimbalance between the classes.\n•Following these findings, in our benchmark, we choose to use t hei.d.-Construction 2\nwith a ratio of 0.05.\nConstruction Ratio r N KL\ni.i.d.-Construction 10.05 50 0.3284\n0.15 50 0.2865\n0.5 50 0.2771\n0.85 50 0.4211\ni.d.-Construction 20.01 10000 0.0563\n0.015 6666 0.0546\n0.05 2000 0.0550\n0.15 666 0.0551\n0.5 200 0.0996\nTable 8: Evaluation of the KLdivergence of the MCD :MLP method on the AsymmetricLinear density model in\nFramework 1 with various values for ratio r, with i.i.d. and i.d. constructions. Best performance for each\nconstruction is in bold print.\n19Additional marginal data\n➢Table 9 depicts the impact on KL performance when using Constructions 3 and 4\n(corresponding to the i.i.d. and i.d. case respectively) in Framework 2 where additional\nmarginal data is available. We compare the respective benefi t of adding only marginal\nobservations ( nx>0,ny=0), only marginal target values ( nx=0,ny>0), and both\nmarginal observation and marginal target values ( nx>0,ny>0).\n•In the i.i.d. case, using i.i.d.-Construction 3 instead of i.i.d.-Construction 1 produces\nsubstantial improvements in terms of performances, but not enough to outperform\nthei.d.-Construction 2. When using the i.d.-Construction 4 instead of the i.d.-\nConstruction 2, the performance gain is smaller, but bear in mind that performances\nare already very satisfying at that point.\n•In the i.i.d. case, having access to ny=nmarginal target values also allows the marginal\nestimator to be trained on a sample size of ny+n=2n, which may explain why the per-\nformance gain is larger with marginal target values DY\nnythan with marginal observations\nDX\nnxwhen using the i.i.d.-Constructions 3.\n•Meanwhile, in the i.d.case, the size of the training dataset DW,Z\nNwhen max (nx,ny)>0\nis at most (n+nx)(n+ny)>n2, which allows to build an even larger data set ( N>n2).\nHowever, in that case, the choice of Nis constrained by the choice of ratio r:N=n\nr.\nHere the appropriate ratio is r=0.05 and n=100, meaning N=2000≤n2. As such,\nusing Construction 4 to increase Nbeyond n2is not useful in that setting.\n•The Construction 4 does, however, increase the amount of inf ormation present in the\ndatasetDW,Z\nN. Here, the performance gain is higher in the presence of marg inal obser-\nvationsDX\nnxthan marginal target values DY\nny. This may be partly due to the fact that in\ntheAsymmetricLinear model, X∈ /CA10andY∈ /CA, meaning the observations contain\nmore information than the target values.\nSetting Construction Results\nAvailable datasets n nx ny Strategy Construction N KL\nDX,Y\nn 100 0 0 Construction 1 50 0.3456\nDX,Y\nn,DXnx100 100 0 i.i.d. Construction 3 100 0.1204\nDX,Y\nn,DYny100 0 100 r=0.5 Construction 3 100 0.1203\nDX,Y\nn,DXnx,DYny100 25 25 Construction 3 75 0.2058\nDX,Y\nn 100 0 0 Construction 2 2000 0.0551\nDX,Y\nn,DXnx100 500 0 i.d. Construction 4 2000 0.0541\nDX,Y\nn,DYny100 0 500 r=0.05 Construction 4 2000 0.0546\nDX,Y\nn,DXnx,DYny100 150 150 Construction 4 2000 0.0639\nTable 9: Evaluation of the KLdivergence values of the MCD :MLP method on the AsymmetricLinear density\nmodel in Framework 1 and 2 with various values for n,nxandny, with i.i.d. and i.d. constructions. Best\nperformance for i.i.d. and i.d. are in bold print.\nMultiple target values per observations\n➢Table 10 compares the KL performances of i.d.-Construction 5 in Framework 3, when\nmore than one target is associated to each observation in the datasetDX,Y\nn,m. Here we\n20denote mthe number of observations associated to each target. Remar k that when m=1,\ni.d.-Construction 5 is strictly equivalent to i.d.-Construction 2.\n•In the presence of multiple target values per observation, i.d.-Construction 5 allows for\nunparalleled performances. This can probably be explained by the fact that the goal of\ntheCDE task is to determine the relationship between XandYbeyond the conditional\nexpectation, and thus multiple realizations for a single ob servation better quantify the\nvariance.\n•Besides, it seems that when m>1, the appropriate ratio is higher, since the perfor-\nmances for r=0.15 are better than with r=0.05, which is not the case when m=1.\nConstruction m Ratio r N KL\n0.5 200 0.0620\ni.d.-Construction 2 1 0.15 666 0.0502\n0.05 2000 0.0501\n0.5 400 0.0520\ni.d.-Construction 5 2 0.15 1333 0.0438\n0.05 4000 0.0461\n0.5 2000 0.0179\ni.d.-Construction 5 10 0.15 6666 0.0167\n0.05 20000 0.0191\nTable 10: Evaluation of the KLdivergence values of the MCD :MLP method on the AsymmetricLinear density\nmodel in Framework 1 and 3 with various values for ratio randm.Best performance is in bold print.\nBest performance ratio for each value of mis underlined. Column 2 corresponds to the number of target\nvalues associated to each observation.\n6 Proofs and dataset constructions\n6.1 Proof Fact 1\nLet(x,y)∈X×Y be any couple of values for which we need to prove Fact 1. We hav e by\ntheBayes ’s formula:\nfY|X=x(y)\nfY(y)=fX,Y(x,y)\nfX(x)fY(y),fX(x)>0,fY(y)>0 (6)\nThen, for any r∈(0, 1):\nfX,Y(x,y)\nfX(x)fY(y)=r fX,Y(x,y)\nr fX,Y(x,y)+( 1−r)fX(x)fY(y)×r fX,Y(x,y)+( 1−r)fX(x)fY(y)\nr fX(x)fY(y)\nLetq(x,y):=rfX,Y(x,y)\nrfX,Y(x,y)+(1−r)fX(x)fY(y)the marginal constrast function with ratio rdefined on\nDefinition 1\nfX,Y(x,y)\nfX(x)fY(y)=q(x,y)×/parenleftBiggfX,Y(x,y)\nfX(x)fY(y)+1−r\nr/parenrightBigg\n⇔(1−q(x,y))fX,Y(x,y)\nfX(x)fY(y)=q(x,y)1−r\nr\n⇔fY|X=x(y)\nfY(y)=1−r\nrq(x,y)\n1−q(x,y)\n21The last equation is deduced using equation (6).\n6.2 Proof Proposition 1\nBy condition ( Cd 1 ) of Definition 2 we have Z∼B(r), then\nP[Z=1] =randfZ(z) = pz(1−p)1−z.\nMoreover, by condition ( Cd 3 ) of Definition 2, we have ∀(x,y)∈ X×Y ,fW(x,y) =\nr fX,Y(x,y)+( 1−r)fX(x)fY(y). By condition ( Cd 4 ) of Definition 2 we have\nfW|Z=1(x,y) =fX,Y(x,y)\nfW|Z=0(x,y) =fX(x)fY(y)\nBy Definition 1 we have\nq(x,y) =r×fX,Y(x,y)\nr fX,Y(x,y)+( 1−r)fX(x)fY(y)=P[Z=1]×fW|Z=1(x,y)\nr fW|Z=1(x,y)+( 1−r)fW|Z=0(x,y)\n=P[Z=1]×fW|Z=1(x,y)\nfW(x,y)=EZ[1Z=1]×/bracketleftBig\n1×fW|Z=1(x,y)+0×fW|Z=0(x,y)/bracketrightBig\nfW(x,y)\n=EZ[1Z=1]×EZ[fW|Z=z]\nfW(x,y)=EZ[1Z=1fW|Z=z(x,y)]\nfW(x,y)=EZ[1Z=1fW,Z(x,y,z)]\nfW(x,y)fZ(z)\n=EZ1Z=1×fZ|W=(x,y)(z)\nfZ(z)=EZ/bracketleftBigg\n1Z=1×P[Z=z|W= (x,y)]\nP[Z=z]/bracketrightBigg\n=EZ/bracketleftBigg\n1Z=1×P[Z=1|W= (x,y)]\nP[Z=1]/bracketrightBigg\n=EZ[1Z=1]P[Z=1|W= (x,y)]\nP[Z=1]\n=P[Z=1|W= (x,y)] = E[Z|W= (x,y)]\n6.3 Proof Theorem 1 and Construction in the i.i.d. case\nFirst construct a random vector (W,Z)satisfying the MDcond (r)with r∈(0, 1).\n•Consider the random vector (X,Y)admitting fX,Yas density of probability.\n•Let/tildewideYbe a random variable independent of XandYand of same law fYofY(/tildewideYD=Y).\n•Letrbe a real number in (0, 1)andZ∼B(r).\n•SetW= (X,Y1Z=1+/tildewideY1Z=0).\nThen, for all (x,y,/tildewidey)∈X×Y×Y we have\n∀z∈{0; 1}fW|Z=z(x,y,/tildewidey) =/braceleftBiggfX,Y(x,y) ifz=1\nfX(x)fY(/tildewidey)ifz=0\nTherefore,∀(x,y,z)∈X×Y×{ 0; 1},fW|Z=z(x,y) = fX,Y(x,y)1z=1+fX(x)fY(y)1z=0).\nMoreover,∀(x,y)∈X×Y\nfW(x,y) =/integraldisplay\nfW|Z=z(x,y)fZ(z)dz\n=fW|Z=1(x,y)P[Z=1]+fW|Z=0(x,y)P[Z=0]\n=fW|Z=1(x,y)r+fW|Z=0(x,y)(1−r)\n=rfX,Y(x,y)+( 1−r)fX(x)fY(y).\n22Therefore Z,Wsatisfies Definition 2. Now, consider the original n-sampleDX,Y\nnand set N=\n⌊n/2⌋. We can now construct the DW,Z\nNsample.\nConstruction 1/bracketleftBig\ni.i.d.DW,Z\nN/bracketrightBig\nConsider the original n-sampleDX,Y\nnand set N=⌊n/2⌋. We can now construct the DW,Z\nN\nsample:\nStep (1) : First sample Nindependent observations Z1,···,ZNwith respect\ntoB(r).\nStep (2) : Next,∀i=1,···,N, setWi= (Xi,Yi1Zi=1+Yi+N1Zi=0).\nSince N>0 and the Ncouples(Wi,Zi)iare constructed from the N-i.i.d. quadruplets\n(Xi,Yi,Yi+N,Zi)i, they are i.i.d..\n6.4 Proof Theorem 2 and Construction in the i.d. case\nLetnJandnMtwo integers such that 1 ≤nJ≤nand 1≤nM≤n(n−1).\nConstruction 2/bracketleftBig\ni.d.DW,Z\nN/bracketrightBig\nStep (1) : Construct n2observations{(/tildewideWi,/tildewideZi)}i=1,···,n2such that:∀j=\n0,···,n−1,∀k=1,···,n:\n/braceleftBigg/tildewideWjn+k= (Xj+1,Yk)\n/tildewideZjn+k=1j+1=k(7)\nNote that the n2observations are neither independent nor identically dist ributed and do not\nsatisfy yet the MDcond.\nStep (2) : Split the n2couple of observations {(/tildewideWi,/tildewideZi)}i=1,···,n2into two:\n/braceleftBiggSJ={(/tildewideWi,/tildewideZi)}:/tildewideZi=1∀i\nSM={(/tildewideWi,/tildewideZi)}:/tildewideZi=0∀i.\nStep (3) : Sample at random uniformly without replacement nJobserva-\ntions(/tildewideWi,/tildewideZi)from SJand denote /tildewideSJthisnJsample.\nNote that since /tildewideZi=1 if and only if j+1=kin equation(7), we have\n∀(/tildewideWi,/tildewideZi)∈/tildewideSJ,/tildewideZi=1 and f/tildewideWi|/tildewideZi=1≡fX,Y (8)\nThis means that /tildewideWj(n+1)+1= (Xj+1,Yj+1).\nStep (4) : Sample at random uniformly without replacement nMobserva-\ntions(/tildewideWi,/tildewideZi)from SMand denote /tildewideSMthisnMsample.\nNote that since /tildewideZi=0 if and only if j+1/nequalkin equation(7), we have\n∀(/tildewideWi,/tildewideZi)∈/tildewideSM,/tildewideZi=0 and f/tildewideWi|/tildewideZi=0≡fXfY (9)\n23This means that /tildewideWjn+k= (Xj+1,Yk)(recall Xl⊥⊥Xk∀l/nequalk).\nStep (5) : Concatenate the samples /tildewideSJand/tildewideSMand shuffle uniformly to ob-\ntain a NsampleDW,Z\nN=Unif.Shuffle/parenleftBig\n{/tildewideSJ;/tildewideSM}/parenrightBig\n.\nProve now that∀i=1,···,N, the couple (Wi,Zi)∈DW,Z\nNsatisfies MDcond (nJ\nN).\n((Cd 1 )) As we shuffle uniformly the indices, we have ∀i=1,···,N,Zi∈DW,Z\nN,Zi∼B(nJ\nN).\n((Cd 2 )) By Step (1) , it is obvious that all the WiadmitX×Y as support.\n((Cd 4 )) Let(Wi,Zi)be any element ofDW,Z\nN, then by equation (8)\n/braceleftBiggfWi|Zi=1≡fX,Y ifZi=1\nfWi|Zi=0≡fXfYifZi=0\n((Cd 3 )) Moreover, it comes fWi(x,y) =r fX,Y(x,y)+( 1−r)fXfYwith r=nJ\nN.\n6.5 Proof Theorem 3 and i.i.d. Construction in the Additiona l data setting\nTo constructDW,Z\nN={(Wi,Zi)}i=1,···,Na training set of N i.i.d. observations, we concatenate\n3 datasets denoted DW,Z\n|NX,DW,Z\n|NYandDW,Z\n|NX,Yof respective size NX,NYandNX,Ysuch that\nDW,Z\nN=DW,Z\n|NX∪DW,Z\n|NY∪DW,Z\n|NX,Y\nandNX=min(n,nx),NY=min(ny,n−NX)\nNX,Y=/floorleftBign−NX−NY\n2/floorrightBig\n,N=NX+NY+NX,Y\nTo construct these 3 preliminary datasets, we first split\nDX,Y\nn=DX,Y\n|2NX,Y/bracehtipupleft/bracehext/bracehtipdownright/bracehtipdownleft/bracehext/bracehtipupright\nthe first 2 NX,Yobs.∪ DX,Y\n|NY/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nthe next NYobs.∪ DX,Y\n|NX/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nthe next NXobs.∪ D/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nthe rest.\nWe considerDX\n|NXandDY\n|NYthe observations in datasets DX\nnxandDY\nnyrestricted to the NXand\nNYfirst observations respectively.\nConstruction 3/bracketleftBig\ni.i.d.DW,Z\nNAdditional data/bracketrightBig\nStep (1) : Construction ofDW,Z\n|NX,Y. IfNX,Y=0, thenDW,Z\n|NX,Y=∅, otherwise\nwe constructDW,Z\n|NX,Yfrom the initial dataset DX\n|NXas described in\nConstruction 1.\nStep (2) : Construction ofDW,Z\n|NY.First note that if NY=0, thenDW,Z\n|NY=∅.\nIfNY/nequal0, we proceed as follows:\n(i) Sample NYindependent observations (Zi)i=1,···,NYaccording to a\nBernoulli law of parameter r∈(0, 1).\n(ii) For all i=1,···,NY;∀(Xi,Yi)∈DX,Y\n|NYand∀/tildewideYi∈DY\n|NY, set\nWi= (Xi,Yi1Zi=1+/tildewideYi1Zi=0).\n24Note that by construction, (Wi,Zi)NY\ni=1arei.i.d. and follow fW,Z. Moreover, following the\nsame arguments as in proof of Theorem 1, the {(Wi,Zi)}NY\ni=1satisfy the MDcond (r).\nStep (3) : Construction ofDW,Z\n|NX.\n(i) Sample NXindependent observations (Zi)i=1,···,NXaccording to a\nBernoulli law of parameter r∈(0, 1).\n(ii) For all i=1,···,NX;∀(Xi,Yi)∈DX,Y\n|NXand∀/tildewideXi∈DX\n|NX, set\nWi= (Xi1z=1+/tildewideXi1z=0,Yi)\n.\nNote that, by construction DW,Z\n|NX= (Wi,Zi)NX\ni=1arei.i.d. and follow fW,Z. Indeed, the reason-\ning is similar as in proof of Theorem 1. First construct a rand om vector (W,Z)satisfying the\nMDcond(r)with r∈(0, 1):\n•Let(X,Y)be a random variable admitting fX,Yas probability density function.\n•LetZbe a random variable following a Bernoulli of parameter r∈(0, 1).\n•Let/tildewideXbe a random variable independent of (X,Y)following the law fX.\n•SetW= (Xi1Zi=1+/tildewideXi1Zi=0,Yi).\nThen,∀(x,/tildewidex,y)∈X×Y×Y we have∀z∈{0, 1},\ngW|Z=z(x,/tildewidex,y) =/braceleftBiggfX,Y(x,y)ifz=1\nfX(/tildewidex)fY(y)ifz=0\nThen∀(x,y,z)∈X×Y×{ 0, 1}, it comes\nfW|Z=z(x,y) =gW|Z=z(x,x,y) =fX,Y(x,y)1z=1+fX(x)fY(y)1z=0.\nMoreover,∀(x,y)∈X×Y ; we have by equation (8):\nfW(x,y) =rfX,Y(x,y)+( 1−r)fX(x)fY(y).\nSo(W,Z)satisfies the MDcond (r).\nStep (4) : Concatenation. Concatenate the 3 datasets DW,Z\n|NX,DW,Z\n|NYand\nDW,Z\n|NX,Y\nTo conclude our proof, note that :\n•SinceDX,Y\n|2NX,Y,(DX,Y\n|NX,DX\n|NX)and(DX,Y\n|NY,DY\n|NY)are independent, the datasets DW,Z\n|NX,DW,Z\n|NY\nandDW,Z\n|NX,Yare independent by construction.\n•Moreover, sinceDW,Z\n|NX,DW,Z\n|NYandDW,Z\n|NX,Yare composed by i.i.d. random variables follow-\ning the law fW,Z, the sample of size N=NX,Y+NX+NY,\nDW,Z\nN=DW,Z\n|NX∪DW,Z\n|NY∪DW,Z\n|NX,Y\n25is composed by i.i.d. random variables following the law fW,Z.\nConstruction ofDW,Z\n|NX,Y.\nIfNX,Y=0, thenDW,Z\n|NX,Y=∅, otherwise we construct DW,Z\n|NX,Yfrom the initial dataset DX\n|NXas\ndescribed in Construction 1 (see the proof of Theorem 1).\nConstruction of DW,Z\n|NY.\nFirst note that if NY=0, thenDW,Z\n|NY=∅. IfNY/nequal0, we proceed as follows:\n(i) Sample NYindependent observations (Zi)i=1,···,NYaccording to a\nBernoulli law of parameter r∈(0, 1).\n(ii) For all i=1,···,NY;∀(Xi,Yi)∈DX,Y\n|NYand∀/tildewideYi∈DY\n|NY, set\nWi= (Xi,Yi1Zi=1+/tildewideYi1Zi=0).\nNote that by construction, (Wi,Zi)NY\ni=1arei.i.d. and follow fW,Z. Moreover, following\nthe same arguments as in proof of Theorem 1, the {(Wi,Zi)}NY\ni=1satisfy the MDcond (r).\nConstruction of DW,Z\n|NX.\nFirst note that if NX=0, thenDW,Z\n|NX=∅. IfNX/nequal0, we consider the datasets DX,Y\n|NXand\nDX\n|NX, and proceed as follows:\nFirst construct a random vector (W,Z)satisfying the MDcond (r)with r∈(0, 1)following\na similar reasoning as in proof of Theorem 1:\n•Let(X,Y)be a random variable admitting fX,Yas probability density function.\n•LetZbe a random variable following a Bernoulli of parameter r∈(0, 1).\n•Let/tildewideXbe a random variable independent of (X,Y)following the law fX.\n•SetW= (Xi1Zi=1+/tildewideXi1Zi=0,Yi).\nThen,∀(x,/tildewidex,y)∈X×Y×Y we have∀z∈{0, 1},\ngW|Z=z(x,/tildewidex,y) =/braceleftBiggfX,Y(x,y)ifz=1\nfX(/tildewidex)fY(y)ifz=0\nThen∀(x,y,z)∈X×Y×{ 0, 1}, it comes\nfW|Z=z(x,y) =gW|Z=z(x,x,y) =fX,Y(x,y)1z=1+fX(x)fY(y)1z=0.\nMoreover,∀(x,y)∈X×Y ; we have by equation (8):\nfW(x,y) =rfX,Y(x,y)+( 1−r)fX(x)fY(y).\nSo(W,Z)satisfies the MDcond (r).\n26(i) Sample NXindependent observations (Zi)i=1,···,NXaccording to a\nBernoulli law of parameter r∈(0, 1).\n(ii) For all i=1,···,NX;∀(Xi,Yi)∈DX,Y\n|NXand∀/tildewideXi∈DX\n|NX, set\nWi= (Xi1z=1+/tildewideXi1z=0,Yi)\n.\nNote that, by construction DW,Z\n|NX= (Wi,Zi)NX\ni=1arei.i.d. and follow fW,Z.\nConcatenation.\nNow to conclude our proof, note that DW,Z\n|NX,DW,Z\n|NYandDW,Z\n|NX,Yare independent by con-\nstruction, sinceDX,Y\n|2NX,Y,(DX,Y\n|NX,DX\n|NX)and(DX,Y\n|NY,DY\n|NY)are independent. Moreover, since\nDW,Z\n|NX,DW,Z\n|NYandDW,Z\n|NX,Ysamples are i.i.d. random variables following the law fW,Zwe\nhaveDW,Z\nN=DW,Z\n|NX∪DW,Z\n|NY∪DW,Z\n|NX,Ya training set of N=NX,Y+NX+NYi.i.d. samples\nfollowing the law fW,Z.\n6.6 Proof Theorem 4 and i.d. Construction in the Additional d ata setting\nLetnJandnMtwo integers such that 1 ≤nJ≤nand 1≤nM≤(n+nx)(n+ny)−n.\nConstruction 4/bracketleftBig\ni.d.DW,Z\nNAdditional data/bracketrightBig\nThe construction is the same as in the proof of Theorem 2, exce pt we replace Step (1) in\nConstruction 2 with Step (1-bis) detailed below.\nStep (1-bis) : We first generate (n+nx)(n+ny) observations\n{(Wi,Zi)}i=1,···,(n+nx)(n+ny)by concatenating 4 sets of sam-\nples denotedS,S/tildewideX,S/tildewideYandS/tildewideX,/tildewideYof size n2,n×nx,n×nyand\nnx×nyrespectively.\n(A) Generate the set of samples Sexactly like in Step (1) of the Con-\nstruction 2.\n(B) Construct the set of samples S/tildewideX={(Wi,Zi)}i=1,···,n×nxof size\nn×nxas follows:∀j=0,···,n−1,∀k=1,···,nx,\n/braceleftBigg\nWjnx+k= (/tildewideXk,Yn2+j+1)\nZjnx+k=0\nSinceDX\nnxandDX,Y\nnare independent and their respective elements are i.i.d., we have∀i=\n1,···,n2,fWi≡fXfY.\n(C) Construct the set of samples S/tildewideY={(Wi,Zi)}i=1,···,n×nyof size\nn×nyas follows:∀j=0,···,n−1,∀k=1,···,ny,\n/braceleftBigg\nWjny+k= (Xj+1,/tildewideYk)\nZjny+k=0\n27SinceDY\nnyandDX,Y\nnare independent and their respective elements are i.i.d., we have∀i=\n1,···,n×ny,fWi≡fXfY.\n(D) Construct the set of samples S/tildewideX,/tildewideY={(Wi,Zi)}i=1,···,nxnyof size\nnx×nyas follows:∀j=0,···,nx−1,∀k=1,···,ny,\n/braceleftBigg\nWjny+k= (/tildewideXj+1,/tildewideYk)\nZjny+k=0\nSinceDX\nnxandDY\nnyare independent and their respective elements are i.i.d., we have∀i=\n1,···,nxny,fWi≡fXfY.\n(E) Finally, concatenate S,S/tildewideX,S/tildewideYandS/tildewideX,/tildewideYwhich ends the Step\n(1-bis) .\n(Next-Step) : Next, do Step (2) ,Step (3) ,Step (4) andStep (5) of the Con-\nstruction 2.\nUsing the same arguments as in the proof of Theorem 2, DW,Z\nNsatisfies the MDcond (r). Since\n∀i=n2+1,···,(nx+n)(ny+n), we have fWi≡fXfYandZi=0. Therefore∀i=\n1,···,(nx+n)(ny+n),fWi≡fX,Y1Zi=1+fXfY1Zi=0.\n6.7 Proof Theorem 5 and Construction in the non-independent case\nLetnJandnMtwo integers such that 1 ≤nJ≤n×mand 1≤nM≤n(n−1)m. We construct\nthe final dataset similary to Construction 2, except we repla ceStep (1) with Step (1-ter) :\nConstruction 5/bracketleftBig\ni.d.DW,Z\nNFramework 3/bracketrightBig\nStep (1-ter) : Construct n2mobservations{(Wi,Zi)}i=1,···,n2msuch that:\n∀j=0,···,n−1,∀k=1,···,n∀l=1,···,m,/braceleftBiggW(jn+k)m+l= (Xj+1,Yk\nl)\nZ(jn+k)m+l=1j+1=k\n(Next-Step) : Next, do Step (2) ,Step (3) ,Step (4) andStep (5) of the Con-\nstruction 2.\nSince∀j=0,···,n−1,∀k=1,···,n,∀l=1,···,m,\nXj+1⊥⊥Yk\nli f f j+1/nequalk,\nwe can use the same arguments as in the proof of Theorem 2 to sho w that∀i=1,···,N, the\ncouple(Wi,Zi)∈DW,Z\nNsatisfies the MDcond/parenleftBignJ\nN/parenrightBig\n.\n7 Conclusion\nIn this article , we consider the problem of conditional dens ity estimation. We introduce a\nnew method, MCD inspired by contrastive learning. MCD reformulates the initial task into\na problem of supervised learning. 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URL\nhttp://arxiv.org/abs/2110.06848 . arXiv: 2110.06848.\n8 Appendix\n8.1 Method to rescale estimated densities\nIf we want our estimation to correspond to a proper probabili ty density function, that is/integraldisplay\nY/hatwidefY|X=x(y)dy≈1, we can numerically estimate the integral using the trapez oidal rule on\na grid of mtarget values{yi}i=1,···,msuch that its values are evenly distributed between the\nquantiles 0.001 and 0.999 of fY(which we can always estimate through the marginal density\nestimator) and then rescale the predicted value, doing as fo llows:\nMCD. rescale (pdf)•Generate a grid of mtarget values{yi}i=1,···,m.\n•Estimate{fY|X=x(yi)}i=1,···,m.\n•Use the trapezoidal rule to estimate/integraldisplay\nY/hatwidefY|X=x(y)dy.\n•Divide the predicted value by the estimation of/integraldisplay\nY/hatwidefY|X=x(y)dy.\n.\nNote that this step must be repeated for each observation xfor which we want to esti-\nmate a conditional density function, but given a set observa tion x, we can reuse the computed\nintegral for any number of target values. This technique wil l also be used for the other\nbenchmarked methods which do not yield proper integrals by d efault.\n8.2 Python code for training set construction\nWe provide extracts of the python code used to construct DW,Z, the training set for the dis-\ncriminator.\n331importnumpy as np\n2fromsklearn.model_selection importtrain_test_split as tts\n3fromsklearn.utils importshuffle\n4...\n5n_obs, p = X.shape\n6n_extraobs, n_extrasamples = min(len(extraobs),n_obs), min(len(extrasamples\n),n_obs)\n7n_extra =min(max(n_extraobs, n_extrasamples),n_obs)\n8n_contrast = int((n_obs + n_extra )/2)\n9ifn_extra == 0:\n10X_joint, X_discarded, y_joint, y_marginal = tts(X, y, trai n_size =\nn_contrast, random_state = self._random_state)\n11X_marginal = X_joint\n12elifn_extra == n_obs:\n13ifn_extraobs <= n_extrasamples:\n14X_joint, X_marginal = X, X\n15y_joint, y_marginal = y, extrasamples[:n_obs]\n16else:\n17y_joint, y_marginal = y, y\n18X_joint, X_marginal = X, extraobs[:n_obs]\n19elifn_extraobs <= n_extrasamples:\n20X_joint, X_discarded, y_joint, y_marginal_partial = tts( X,y, train_size\n= n_contrast, random_state = self._random_state)\n21X_marginal = X_joint\n22y_marginal = np.concatenate([y_marginal_partial, extra samples[:2 *\nn_contrast - n_obs]], axis= 0)\n23elifn_extraobs > n_extrasamples:\n24X_joint, X_marginal_partial, y_joint, y_discarded = tts( X, y,\ntrain_size = n_contrast, random_state = self._random_sta te)\n25y_marginal = y_joint\n26X_marginal = np.concatenate([X_marginal_partial, extra obs[:2 *\nn_contrast - n_obs]], axis= 0)\n27z = self._random_state.binomial(1, self.contrast_ratio , size = (n_contrast)\n)\n28X_joint, X_marginal, y_joint, y_marginal = X_joint[:n_co ntrast], X_marginal\n[:n_contrast], y_joint[:n_contrast].reshape((n_contr ast,-1)),\ny_marginal[:n_contrast].reshape((n_contrast,-1))\n29W_X = X_joint * z.reshape((-1,1)) + X_marginal * (1 - z.resha pe((-1,1)))\n30W_y = y_joint * z.reshape((-1,1)) + y_marginal * (1 - z.resha pe((-1,1)))\n31W = self._build_W(W_X, W_y)\n32...\n33def_build_W(self, X, y):\n34if len(X.shape)!=len(y.shape):\n35y = y.reshape(( len(X), -1))\n36returnnp.concatenate([X, y], axis = -1)\nListing 1: Python code extract corresponding to Constructi ons 1 and 3.\n1importnumpy as np\n2fromsklearn.model_selection importtrain_test_split as tts\n3fromsklearn.utils importshuffle\n4...\n5n_obs, p = X.shape\n6n_joint = n_obs\n7ifself.contrast_ratio < 1. / n_obs: self.contrast_ratio = 1. / n_obs#does\nnot consider the case where p < 1 / n <==> n_J < n, as there is no\npractical reason to make this choice\n8n_marginal = int(n_joint / self.contrast_ratio - n_joint)\n9n_marginal_extra = min(len(extraobs) * len(extrasamples), n_marginal)\n10n_marginal_x = min(len(extraobs) * n_obs, n_marginal - n_marginal_extra)\n11n_marginal_y = min(n_obs *len(extrasamples), n_marginal - n_marginal_extra\n- n_marginal_x)\n3412n_marginal_both = min(n_obs * (n_obs - 1), n_marginal - n_marginal_extra -\nn_marginal_y - n_marginal_x)\n13\n14W_joint = self._build_W( X, y)\n15\n16W_marginal = np.zeros((0, W_joint.shape[1]))\n17forn_added_marginal, X_added, y_added, remove_same_index in[(\nn_marginal_extra, extraobs, extrasamples, False),\n18 (n_marginal_x, extraobs, y,\nFalse),\n19 (n_marginal_y, X, extrasamples,\nFalse),\n20 (n_marginal_both, X, y, True)]:\n21ifn_added_marginal:\n22W_marginal_added = self._shuffle_and_sample(X_added, y _added,\nn_added_marginal, remove_same_index = remove_same_inde x)\n23W_marginal = np.concatenate([W_marginal, W_marginal_ad ded], axis =\n0)\n24\n25W = np.concatenate([W_joint, W_marginal], axis = 0)\n26z = (np.arange(n_joint + n_marginal) < n_joint).astype( int)\n27shuffled_indexes = shuffle(np.arange(n_joint + n_margin al), random_state =\nself._random_state)\n28W, z = W[shuffled_indexes], z[shuffled_indexes]\n29#to insure we use the same distribution the discriminator ha s seen during\ntraining\n30true_contrast_ratio = z.mean()\n31...\n32def_shuffle_and_sample(self, X, y, n_samples, remove_same_ index = False):\n33len_x, len_y = len(X),len(y)\n34x_indexes = np.arange(len_x * len_y) % len_x\n35y_indexes = np.arange(len_x * len_y) // len_x\n36ifremove_same_index:\n37isnot_joint = x_indexes != y_indexes\n38x_indexes, y_indexes = x_indexes[isnot_joint], y_indexe s[\nisnot_joint]\n39ifn_samples <len(x_indexes):\n40sampled_indexes = self._random_state.choice(np.arange (len(x_indexes\n)), size = n_samples, replace= False)\n41x_indexes, y_indexes = x_indexes[sampled_indexes], y_in dexes[\nsampled_indexes]\n42returnself._build_W( X[x_indexes], y[y_indexes])\nListing 2: Python code extract corresponding to Constructi ons 2 and 4.\n1importnumpy as np\n2fromsklearn.model_selection importtrain_test_split as tts\n3fromsklearn.utils importshuffle\n4...\n5n_obs, p = X.shape\n6n_joint = n_obs\n7#target values are stored in extrasamples\n8#extrasamples.shape = (n, m)\n9multi_sample_size = extrasamples.shape[-1]\n10ifself.contrast_ratio < 1. / (n_obs * multi_sample_size) : se lf.\ncontrast_ratio = 1. / (n_obs * multi_sample_size) #does not consider\nthe case where r < 1 / n <==> n_J < n, as there is no practical reas on\nto make this choice\n11n_joint = n_obs * multi_sample_size\n12n_total =int(n_joint / self.contrast_ratio)\n13n_marginal = n_total - n_joint\n14\n3515X_joint = X.repeat(multi_sample_size,axis=0)\n16y_joint = extrasamples.reshape((n_obs * multi_sample_si ze,-1))\n17W_joint = self._build_W(X_joint, y_joint)\n18\n19X_marginal_max = []\n20y_marginal_max = []\n21foriin range(n_obs):\n22forjin range(n_obs):\n23ifi != j:\n24 X_marginal_max.append(X[i:i+1].repeat(multi_sample_ size,axis=0)\n)\n25 y_marginal_max.append(extrasamples[j:j+1].reshape((\nmulti_sample_size,-1)))\n26X_marginal_max = np.concatenate(X_marginal_max, axis = 0 )\n27y_marginal_max = np.concatenate(y_marginal_max, axis = 0 )\n28X_marginal, X_discarded, y_marginal, y_discarded = tts(X _marginal_max,\ny_marginal_max, train_size = n_marginal, random_state = s elf.\n_random_state)\n29W_marginal = self._build_W(X_marginal, y_marginal)\n30\n31W = np.concatenate([W_joint, W_marginal], axis = 0)\n32z = (np.arange(n_joint + n_marginal) < n_joint).astype( int)\n33shuffled_indexes = shuffle(np.arange(n_joint + n_margin al), random_state =\nself._random_state)\n34W, z = W[shuffled_indexes], z[shuffled_indexes]\n35#to insure we use the same distribution the discriminator ha s seen during\ntraining\n36true_contrast_ratio = z.mean()\nListing 3: Python code extract corresponding to Constructi on 5.\n8.3 Exhaustive experimental results\nIn this section, we present the exhaustive results correspo nding to Tables 5, 6 and 7.\nEmpirical KL\nMCD :MLP\nMCD :CatBoost\nNNKCDE MDN KMN N.Flow LSCond RFCDE FlexCode :MLP\nFlexCode :XGboost\nDeepcdep=3 0.008 0.009 0.022 0.109 0.067 0.189 0.342 0.037 0.094 0.093 0.152\np=10 0.036 0.042 0.063 0.229 0.096 0.228 0.61 0.105 0.106 0.076 0.205\np=30 0.115 0.202 0.154 0.396 0.181 0.432 0.369 0.238 0.22 0.196 0.385\np=100 0.162 0.224 0.173 0.45 0.401 0.411 1.059 0.238 0.234 0.26 4 0.36\np=300 0.244 0.308 0.282 0.506 0.304 0.67 0.783 0.507 0.253 0.30 2 0.306\nTable 11: Evaluation of the KLdivergence values for various feature sizes p, on the BasicLinear density model,\nwith n=100.\nTime in sec.\nMCD :MLP\nMCD :CatBoost\nNNKCDE MDN KMN N.Flow LSCond RFCDE FlexCode :MLP\nFlexCode :XGboost\nDeepcden=30 3.133 0.46 0.01 35.46 47.92 50.74 12.01 0.044 0.018 9.574 1 .943\nn=100 3.149 0.369 0.022 35.45 102.7 50.72 37.14 0.149 0.017 9.5 7 2.11\nn=300 3.132 0.446 0.04 35.65 150.6 50.62 100.5 0.448 0.017 13.1 9 3.225\nn=1000 3.296 0.689 0.043 35.73 143.6 51.12 167.9 0.952 0.02 25. 24 6.388\nTable 12: Training Time in seconds for various training set s izesn, on the BasicLinear density model.\n36Empirical NLL\nMCD :MLP\nCatBoost NNKCDE MDN KMN N.Flow LSCond FlexCode :MLP\nFlexCode :XGboost\nDeepcdeBostonHousing -0.64 -0.59 -0.81 -1.17 -1.22 -1.63 -2.19 -1. 99 -1.84 -7.09\nConcrete -0.86 -1.02 -1.23 -2.02 -2.30 -2.13 -4.00 -2.26 -1. 25 -10.1\nNCYTaxiDropoff:lon. -1.30 -1.28 -1.68 -2.51 -2.51 -2.85 -3.90 -2.05 -2.17 - 11.2\nNCYTaxiDropoff:lat. -1.31 -1.31 -1.44 -2.51 -2.45 -2.96 -4.72 -1.67 -6.18 - 9.35\nPower -0.06 -0.36 -0.73 -0.55 -0.35 -0.39 -0.70 -1.09 -0.75 - 8.97\nProtein -0.09 -0.42 -0.77 -0.54 -0.68 -0.54 -1.09 -0.83 -1.3 8 -10.5\nWineRed -0.89 -0.89 3.486 -1.27 -0.90 -2.43 -6.25 1.062 0.96 5 -10.1\nWineWhite -1.18 -1.13 2.99 -2.24 -1.7 -4.18 -5.41 -0.73 -0.6 3 -13.2\nYacht 0.14 0.822 -0.46 0.083 0.025 0.401 -2.79 -1.23 0.144 -7 .52\nteddy -0.47 -0.51 -0.83 -0.87 -0.76 -0.94 -0.91 -0.83 -1.34 - 9.59\ntoy dataset 1 -0.99 -0.47 -0.63 -0.40 -0.35 -0.71 -0.70 -0.88 -1.46 -6.10\ntoy dataset 2 -1.40 -1.33 -1.31 -1.40 -1.33 -1.39 -1.35 -1.54 -1.43 -3.53\nTable 13: Evaluation of the negative log-likelihood (NLL ) f or 12 datasets.\n37This figure \"MNIST_pire_diff.png\" is available in \"png\"\n format from:\nhttp://arxiv.org/ps/2206.01592v1"    },    {        "title": "2206.03910v1.Identification_of_a_Nematic_Pair_Density_Wave_State_in_Bi2Sr2CaCu2O8_x.pdf",        "content": " 1 Identification  of a Nematic Pair Density Wave State   \nin Bi2Sr2CaCu 2O8+x \n \nWeijiong Chen1§, Wangping Ren1§, Niall Kennedy1,2, M. H. Hamidian3, S. Uchida4, H. Eisaki4, \nPeter. D. Johnson1,5, Shane O’Mahony2 and J.C. Séamus Davis1,2,3,6  \n \n1. Clarendon Laboratory, U niversity of Oxford, Oxford, OX1 3PU, UK . \n2. Department of Physics, University College Cork, Cork T12 R5C, Ireland  \n3. Department of Physics, Cornell University, Ithaca , NY 14850, USA  \n4. Inst. of Advanced Industrial Science and Tech., Tsukuba, Ibaraki 305 -8568, Jap an. \n5. CMPMS Department, Brookhaven National Laboratory, Upton, NY 11973 , USA  \n6. Max -Planck Institute for Chemical Physics of Solids, D -01187 Dresden, Germany  \n§     These authors contributed equally to this project.  \n \nABSTRACT  Electron -pair density wave (PDW) s tates are now an intense focus of research \nin the field of cuprate correlated superconductivity. PDW’s exhibit periodically modulating \nsuperconductive electron pairing which can be visualized directly using scanned Josephson \ntunneling microscopy (SJTM). A lthough from theory, intertwining the d-wave \nsuperconducting (DSC) and PDW order parameters allows a plethora of global electron -pair \norders to appear, which one actually occurs in the various cuprates is unknown. Here we use \nSJTM to visualize the interpla y of PDW and DSC states in Bi 2Sr2CaCu 2O8+x at a carrier density \nwhere the charge density wave (CDW) modulations are virtually nonexistent. S imultaneous \nvisualization  of their amplitudes reveal s that the intertw ined  PDW and DSC  are mutually  \nattractive  state s. Then, by  separately imaging the electron -pair density modulations of the \ntwo orthogonal PDW s, we discover a robust  nematic PDW state . Its spatial arrangement \nentails  Ising domains of opposite nematicity , each consisting  primarily of unidirectional and \nlattice commensurate electron -pair density modulations . Further, we demonstrate by direct \nimaging that the  scattering resonances identifying Zn impurity atom sites occur \npredominantly within boundaries between these domains . This i mpl ies that the nematic  \nPDW state is pinned by Zn atoms , as was recently proposed (Lozano et al, PHYSICAL REVIEW \nB 103, L020502 (2021)).  Taken i n combination , these data indicate  that the PDW in \nBi2Sr2CaCu 2O8+x is a vestigial nematic  pair density wave state  (J. Wardh and M. Granath  \narXiv:2203.08250v1) .  \n  2  The existence and phenomenology of  PDW  states are fundamental issues  within the \nchallenge to understand cuprate  correlated superconductivity1,2,3,4. Their definitive \ncharacteristic is  a periodically modulating electron -pair  density  which can now be visualized \ndirectly at the atomic -scale, by using scanned Josephson tunneling microscopy5,6,7,8,9 (SJTM). \nThis new capability allows one to explore both  the microscopic electronic structure of the \ncuprate PDW state, and the  interactions between it  and the other electronic orders . \nUnderstanding  the latter is essential, because  intertwining the d-wave superconducting \n(DSC) and PDW order parameters allows a plethora of distinct  electron -pair orders to \nappear1,2,3 . But which one actually occurs in the various cuprates is unknown.  \n \n The simplest  quantum condensate s of electron pairs  are defined by an homogeneous  \nsuperconductive order -parameter   \nΔ0(𝒓)=Δ0𝑒𝑖𝜙(𝑟)           (1) \nfor which 𝜙(𝑟) is the electromagnetic gauge symmetry breaking macroscopic quantum \nphase.  Heuristically, o ne may think of Δ0≡∑𝛾〈𝑐𝑘↑𝑐−𝑘↓〉𝑘 , where  𝑐𝑘↑ ; 𝑐−𝑘↓ are electron \nannihilation  operators of opposite spin and momentum and 𝛾 the pairing strength, as the \namplitude of the electron -pair condensate wavefunction.  A unidirectional PDW state is also \na superconductor, but one that modulates the condensate order parameter  spatially at \nwaveve ctor 𝑃𝑥 such that  \nΔ𝑥(𝒓)=Δ𝑃𝑥(𝒓)𝑒𝑖𝑃𝑥𝑥+Δ−𝑃𝑥∗(𝒓)𝑒−𝑖𝑃𝑥𝑥    (2) \nIn a tetragonal crystal , an orthogonal PDW state can also exist modulating with at \nwavevector 𝑃𝑦 along the y-direction  as \nΔ𝑦(𝒓)=Δ𝑃𝑦(𝒓)𝑒𝑖𝑃𝑦𝑦+Δ−𝑃𝑦∗(𝒓)𝑒−𝑖𝑃𝑦𝑦   (3) \nBecause there are then five complex -valued scalar order parameter  functions , when these \nthree macroscopic quantum phases  are intertwined , a plethora of global electron -pair order \nparameters becomes possible1,2,3,10,11,12.   \n \n     Theoretical a nalysis of how the  DSC and two primary PDW states  become intertwined \nrequires a n intricate Ginzburg -Landau -Wilson (GLW) free energy densit y functional1,2,3,10. \nUsing such an approach, e xcellent success has been achieved  in understanding the induced   3 order parameters,  the most prominent of which  are the induced CDW states with order \nparameter s2,3 𝜌𝑃𝑥,𝑃𝑦(𝒓)∝Δ0∗Δ𝑃𝑥,𝑃𝑦+Δ−𝑃𝑥,−𝑃𝑦∗Δ0 and 𝜌2𝑃𝑥,2𝑃𝑦(𝒓)∝Δ−𝑃𝑥,−𝑃𝑦∗Δ𝑃𝑥,𝑃𝑦. However, \nto fully understand the PDW  of cuprates,  one is not limited to studying  such secondary or \ninduced states , because SJTM imaging can give direct access5-9 to the primary electron -pair \norders . The structure and intertwining between  PDW and DSC  states are  descr ibed  by a \nsubset of terms from the overall  GLW free energy density  \nℱ=𝛽𝑐1|Δ0|2(|ΔP𝑥|2+|ΔP𝑦|2\n+|ΔP−𝑥|2+|ΔP−𝑦|2\n)+𝛽𝑐2[Δ02(ΔP𝑥Δ−P𝑥+ΔP𝑦Δ−P𝑦)∗\n+\n(Δ02)∗(ΔP𝑥Δ−P𝑥+ΔP𝑦Δ−P𝑦)]  (4) \nwhere only the lowest order c oupling terms are considered and the gradient terms are \nignored . Among the possible global electron -pair orders  are2,3 unidirectional or bidirectional \nmodulated PDW phase s 𝑒𝑖𝑃𝑥𝑥;𝑒𝑖𝑃𝑦𝑦 with fixed amplitude,  as in the Fulde -Ferrell13 state ; and \nunidirectional  or bidirectional modulated PDW amplitude s \n|Δ𝑃𝑥|cos (𝑃𝑥𝑥);|Δ𝑃𝑦|cos (𝑃𝑦𝑦) with fixed phase, as  in the Larkin -Ovchinnikov14 state. An \nintriguing hypothetical state is a nematic pair density wave2,3 with  order parameter  \n 𝑁≡(|Δ𝑃𝑥|2+|Δ−𝑃𝑥|2)−(|Δ𝑃𝑦|2\n+|Δ−𝑃𝑦|2\n)       (5) \nand such  state s may also exhibit vestigial nematic phases15,16. But, how the DSC and PDW \norders are intertwined in cuprates, and the ir consequent global electron -pair order \nparameter , are all unknown.   \n \n   Our objective is exploration of  intertwined PDW and DSC states  in cuprates, by using  \nSJTM . In principle, t he total e lectron -pair density  at location r,  𝑛(𝒓), can be  visualized by \nmeasuring electron -pair ( Josephson ) critical -current  𝐼J(𝒓) from the sample to  a \nsuperconducting STM t ip17. This is because   𝑛(𝒓)∝𝐼J2(𝒓)𝑅N2(𝒓) where RN is the normal -state \njunction resistance18,19. But since  typical thermal fluctuations far exceed the Josephson \nenergy 𝐸J between SJTM tip and sample surface,  a phase -diffusive steady -state at voltage \n𝑉𝐽  drives  an electron -pair current 𝐼P(𝑉𝐽)=1\n2𝐼J2𝑍𝑉𝐽(𝑉𝐽2+𝑉c2) ⁄ . Here 𝑉c=2𝑒𝑍𝑘B𝑇ℏ⁄ with Z \nthe high -frequency impedance of the junction20,21. In the theory  of such spectra,  the \nmaximum value of elec tron -pair current  𝐼m=(ℏ/8𝑒𝑘𝐵𝑇)𝐼𝐽2, providing  the basis for  4 atomically resolved SJTM visualization of the electron -pair condensate in superconductors5-\n9 as \n𝑛(𝒓)∝𝐼m(𝒓) 𝑅N2(𝒓)       (6) \n \n   In that context,  we study  single crystals of Bi2Sr2CaCu 2O8+x with CuO 2 plaquette hole -\ndensity 𝑝≈0.17, by u sing  a dilution refrigerator based S JTM5. The cryogen ically cleaved \nsamples terminate at the BiO crystal  layer , and t he d-wave superconducti ng scan -tip is \nprepared  by exfoliating a nanometer scale Bi 2Sr2CaCu 2O8+δ flake from th at sample surface5,9.  \nA typical topographic image at T= 45 m K  then consists of atomically resolved and registered \nsurface  Bi atoms  (Fig. 1 A) with the typical measured 𝐼P(𝑉𝐽) shown as inset . Using such tips  \nand a virtually constant  𝑅N(𝒓)≈20 MΩ as determined f rom analysis of topography at the \nsetup voltage, we image 𝐼P(𝑉𝐽,𝒓) and thus  𝐼m(𝒓), at T=45mK . The Fourier transform of  the \nmeasured  𝐼m(𝒓) image,   𝐼m(𝒒), is shown in Fig. 1 B. In such experiments  on Bi2Sr2CaCu 2O8+δ, \nthe 𝐼m(𝒓) and 𝐼m(𝒒) are both dominated by effects of the crystal supermodulation , a bulk 26Å \nperiodic modulation of unit -cell dimensions22 whose general effects are discussed \nelsewhere9 . The focus of interest here is on the two PDWs  observable  in 𝐼m(𝒒) as four \nrelatively broad peaks surrounding the wavevectors5 𝒒≈(2𝜋\n𝑎)(0,±0.25);(2𝜋\n𝑎)(±0.25,0). \nAdditionally, the somewhat heterogeneous el ectron -pair density  of the DSC state is \nrepresented by  the broad peak at 𝒒=(2𝜋\n𝑎)(0,0). The phenomena detected separately at \nthese five wavevectors can reveal key information on the interplay of the five order \nparameter functions of Eqns. 1 -3. \n \n    To separately visualize the two PDWs we inverse Fourier transform  𝐼m(𝒒) in Fig. 1 B \nfor wavevectors ±𝑷𝒙≈(2𝜋\n𝑎)(0,±0.25) and ±𝑷𝒚≈(2𝜋\n𝑎)(±0.25,0) respectively , using the \n𝒒 -space regions represented inside the colored circles . The result ing  𝐼m𝑥(𝒓), and  𝐼m𝑦(𝒓)  \nimages  shown in Fig . 1C,D respectively,  reveal th e spatial arrangements of the two \northogonal PDW states (in the presence of a predominant DSC .) They are relatively coherent \n4a0-periodioc modulations with amplit udes  differently disordered , and without obvious \ncorrelation  between them . Clearly , there is a proliferation of edge -dislocation 2 topological  5 defects23 in  𝐼m𝑥,𝑦(𝒓), whose ongoing study will be reported elsewhere . Here we concentrate  \non the inte rplay of between the  simultaneously observed  PDW and DSC states.   \n \n   To do so, w e parameterize the two PDW state  signatures  in terms  \n 𝐼𝑚𝑥,𝑦(𝒓)=A𝑥,𝑦(𝒓) 𝑐𝑜𝑠(𝑷𝒙,𝒚∙𝒓+𝛿𝑥,𝑦(𝒓))≡ A𝑥,𝑦(𝒓) 𝑐𝑜𝑠(𝚽𝒙,𝒚(𝒓))      (7) \nFourier component selection  at ±𝑷𝒙=(2𝜋\n𝑎)(0,±0.25) and ±𝑷𝒚=(2𝜋\n𝑎)(±0.25,0) then \nyields  \n𝐼𝑚′(𝒓) = =1\n√2𝜋𝜎∫𝑑𝒓′𝐼m(𝒓′) 𝑒−𝑖𝑷𝒙,𝒚∙𝒓′𝑒|𝒓−𝒓′|2\n2𝜎02    (8) \nsuch that  \n 𝐴𝑥,𝑦(𝒓)≡2√(ℛ𝑒 𝐼𝑚′(𝒓))2+(ℐ𝑚 𝐼𝑚′(𝒓))2      (9) \n(SI Section 1). The DSC state characteristic , A0(𝒓), is defined similarly  by inverse Fourier \ntransform within the yellow circle surrounding wavevector 𝒒=(0,0). Identical value of \nσ0=σ𝑥=σ𝑦=3 𝑛𝑚 are applied throughout.  Using  this approach, Fig s. 2A,B,C respectively \nshow A𝑥(𝒓), A𝑦(𝒓) and A0(𝒓)  visualized simultaneously in the FOV of Fig. 1 A. Thus  it \nbecomes possible  to directly explore  the phenomenology of intertwining  the DSC and PDW \nstates  (SI Section 2). Figure  2D presents measured 〈A𝑥+A𝑦〉  versus A0 , wh ere the average \nis carried out over all locations 𝒓 at which A0(𝒓) has the value  indicated on the abscissa . The \npositive slope 𝑠=0.0389  through zero indi cates that, throughout the variation s in \nA𝑥(𝒓), A𝑦(𝒓) and A0(𝒓) (Figs. 1C,D and Figs. 2A,B,C), the PDW and the DSC states are \nmutually enhancing  on the average . This conclusion is independent of any inadvertent \nheterogeneity  in the normal -state Josephson junction resistance R𝑁(𝐫)  in Eqn. 6, because it \ngets  divided out by the definition of 𝑠. The implication of Fig. 2 it that, within the GLW context  \n(Eqn. 4  and Refs. 1,2,3) the DSC and PDW states  are attractive ( (𝛽𝑐1−|𝛽𝑐2|<0) electronic \nphases  (SI Section 4) at zero magnetic field , although there is evidence of repulsion when \nstrong gradients pr esent as in the vortex core .   \n \n   Next, t o searc h for the hypothetical2,3 nematic PDW state  we define  a nematic order \nparameter   6                 𝒩(𝐫)={Ax(𝐫)− Ay(𝐫)}/{Ax(𝐫)+ Ay(𝐫)}               (10)  \nAnalyzing the data from Figs. 2A,B in this way  generates  𝒩(𝐫) as shown in Fig. 3 A. Again w e \nnote that Eqn. 10  provides a n empirical  definition for a nematic PDW state based directly  on \n𝐼m(𝒓) data , i.e. independent of  any possible variations in R𝑁(𝐫). Figure 3 A then reveals the \nexistence of strong PDW nematicity  in Bi2Sr2CaCu 2O8. Indeed, the h istogram of all 𝒩(𝐫) \nvalues from  Fig. 3A in the inset  demonstrates  that  |𝒩(𝐫)|>0.3 for 45% of the sam ple area \nand thus that these nematic PDW state s can predominate . To exemplify , Fig. 3B shows  \nexamples of measured I𝑚(𝐫) along the x -axis  (y-axis ) in its left (right)  panels, both  within \ndomains where 𝒩(𝐫)≫0. Figure 3 C shows equivalent exemplary data for domains where \n𝒩(r)≪0. Overall , we find that a robust nematic PDW state, consisting of nanoscale domains \nof opposite nematicity in electron -pair density, occurs in Bi 2Sr2CaCu 2O8 at 𝑝≈0.17. \n \n But t his discovery  begs the question of what  it is that  sets the size and location of the  \nnematic PDW domains ?  One  clue come s from recent report s24 that , in La 2-xBaxCuO 4 away \nfrom 𝑝≈0.125 , introducing Zn atom substitution at 1% of Cu sites, leads to a  cascade \ntransition from  2D superconductivity  to 3D superconduct ivity with falling temperature . The  \ninference derived from these studies is that the Zn atoms pins PDW  order locally. In cuprates,  \nat each  Zn impurity atom there is a superconductive  impurity state with energy 𝐸≈−1𝑚𝑒𝑉 , \nand also a powerful suppression o f electron -pair condensate 𝑛(𝒓) as ex emplified by Fig. S2F. \nIn Bi2Sr2CaCu 2O8 the Zn impurity states  can be imaged  directly  in superconduc ting -tip \ndifferential conductance imaging , by finding the local maxim a in 𝑍(𝒓)≡I(20mV ,𝒓)−\nI(−20mV ,𝒓) where I(V) is the superconductor -insulator -superconductor (SIS) single -\nelectron tunnel current.  These maxima occur because the coherence peak s of SJTM tip  \ndensity -of-states near 𝐸≈±20𝑚𝑒𝑉   are convoluted with the Zn impurity state density -of-\nstates peak 𝐸≈−1𝑚𝑒𝑉  to produce a strong jump in tunnel currents I(𝐸≈±20𝑚𝑒𝑉 ,𝒓). \nFigure 4 A show s the resulting  image 𝑍(𝒓)≡I(20mV ,𝒓)−I(−20mV ,𝒓) from which each Zn \nimpurity -state maximum is identified  by blue circles  (SI Section 3). In Figs. 4B we show  the \namplitude of nematic order parameter  |𝒩(𝒓)| with the sites of Zn impurity resonances  \noverlaid as blue  dots. Visually  the Zn sites seem to occur near the domain boundaries  in \n|𝒩(𝒓)|. This can be quantified by plotting the histogram of the distribution of distances  7 between each Zn impurity atoms and its nearest PDW  domain wall  and comparing to the \nexpected average distance of uncorrelated random points . The result shown in Fig. 4 C \nreveals that the Zn sites occur highly preferentially  near the boundaries of the PDW  Ising \ndomain s. This direct visualization i ndicates that the PDW nematic domains are pinned by \ninteractions with Zn impurity atoms at the Cu sites, and is therefore highly consist ent with \ndeductions from the transport studies of  Zn-doped La2-xBaxCuO 4.24  \n \n Spectroscopic imaging STM and resonant  X-ray scattering  have produced a wealth of \nunderstanding of unidirectionality, commensuration and domain formation for CDW \nmodulations in strongly underdoped cuprates . Howeve r, because photon scattering cannot \n(yet) detect PDW states , because the pseudogap in single particle tunneling masks the true \nelectron -pairing energy gap and thus order parameter,  and because SJTM visualization of \nPDW states is recent5-9, equivalent issues for the cuprate  PDW  state are unresolved . Our \nSJTM visualization now demonstrates  that  the Bi2Sr2CaCu 2O8 PDW state s tend strongly to be  \nboth unidirectional and com mensurate , a situation widely predicted25-34 as a consequence of \nstrong -coupling physic s within the CuO 2 Hubbard model . Further, w e note that visualization \nof these  characteristics in a robust PDW state occurs in Bi 2Sr2CaCu 2O8+x at a carrier den sity \nwhere the charge density wave (CDW) modulations are virtually nonexistent35,36, as is the \nempirical case in the samples studied here  (SI Fig. S3) .  The implication, consistent with the  \neight unit -cell periodic energy gap modulations observed in single p article tunneling37,38 , is \nthat the Bi2Sr2CaCu 2O8 PDW states are not induced by the existence of a CDW and instead \nare the primary translation symmetry breaking state  of hole -doped CuO 2. Furthermore, since \nthe spatial configurations of  the PDWs studied here  appear  uninfluenced by a preexistent \nCDW yet are obviously  disordered, effects of  chemical (and possibly dopant -ion) \nrandomness on the PDW are adumbrated.  Indeed , recent transport studies24 provide \nexperimental  evidence that Zn impurity atoms pin the PDW  order in La 2-xBaxCuO 4. For \ncomparison,  our visualization of the Bi2Sr2CaCu 2O8 PDW nematic domains simultaneously  \nwith Zn scattering  resonances demonstrate s directly tha t the Zn impurity -atom sites occur \npredomin antly within boundary regions between these domains . Overall , a vestigial state is  \none that only partially breaks the symmetry of the  true  ordered state. Here  we find a  nematic  8 PDW state breaking  the rotational ( C4) symmetry of the lattice but not  long -\nrange  translational  symmetry,  in what appears to be a disorder -pinned  realization of  a \nglobal  nematic  order . Hence, a plausible context in which to consider the PDW in \nBi2Sr2CaCu 2O8+x is as a new form of vestigial nematic state based on a disordered \nunidir ectional density wave of electron pairs39 rather than of single electrons40. \n \n   Ergo, by using SJTM to simultaneously visualize  the PDW and DSC states of \nBi2Sr2CaCu 2O8 near  𝑝≈0.17 where no CDW phenomena obtrude , we demonstrate t hat the y \nare intertwined in mutually attractive phases  (Fig. 2) . Conversely,  separate imaging of the \nelectron -pair density modulations of the two orthogonal PDWs  reveals a robust nematic \nPDW state , with  primarily unidirectional and lattice -commensurate electron -pair density \nmodulatio ns forming  Ising domains of opposite nematicity  (Fig. 3) . Imaging  the sites of Zn \nimpurity -states finds them occurring preferentially in boundary regions of minimal  nematic \norder  (Fig. 4)  implying that the PDW domains are pinned thereby. Generally , these d ata \nsignify that the PDW state of Bi 2Sr2CaCu 2O8+x is a vestigial nematic phase of electron pairs , \nthat it is locally unidirectional and lattice commensurate,  and that it is not a subordinate but \na primary electronic order  of hole -doped CuO 2. \n   9 FIGURES  \n \nFigure 1: Visualizing Electron Pair Density 𝑛(𝒓) \nA. SJTM topographic image T(𝒓) of BiO termination layer  of Bi 2Sr2CaCu 2O8. Inset shows \naverage  electron -pair current spectrum I𝑃(𝑉𝐽) measure  in this FOV at T=45mK  and \n𝑅𝑁≈20 MOhm , with maxima occurr ing at ±I𝑚. \nB. Power spectral density Fourier transform of I𝑚(𝒓), I𝑚(𝐪), as measured  in FOV of 1a. \nFour broad PDW peaks surround the wavevectors 𝐏=\n(2π 𝑎⁄)(0,±0.25);(2π 𝑎⁄)(±0.25,0) as indicated by pairs of red and blue arrows \nrespectively. The DSC  electron -pair density  is represented by the broad peak at 𝐪=\n(2π 𝑎⁄)(0,0) as indicated by the yellow circle . \nC. Fourier filtration of I𝑚(𝐪) 𝐏=(2π 𝑎⁄)(0,±0.25) as in Fig. 1 B, to visualize  the ±𝑷𝒙 \nPDW modulating along the CuO 2 x-axis. \nD. Fourier filtration of I𝑚(𝐪) 𝐏=(2π 𝑎⁄)(±0.25,0) in Fig. 1 B, to visualize the ±𝑷𝒚 PDW \nmodulating along the CuO 2 y-axis. \n \nFigure 2: Intertwined  DSC and PDW Order Parameters  \nA. Amplitude of I𝑚(𝐫) modulations A𝑃𝑥(𝐫)  for PDW state with ±𝑷𝒙, from 1 C. \nB. Amplitude of I𝑚(𝐫) modulations A𝑃𝑦(𝐫)  for PDW state with ±𝑷𝒚, from 1 D. \nC. Amplitude of DSC electron -pair density A0(𝐫) derived from Eqn. 6 with q=0 as \nindicated by yellow circle in Fig. 1 B. \nD. 〈Ax(𝐫)+ Ay(𝐫)〉 avera ged over all locations r where A0(𝐫) equals the abscissa  value  \nA0. The solid line is a linear fit to these data through (0,0).  Inset shows the 2D \nhistogram of same data.  \n \nFigure  3: Nematic PDW State  of Bi2Sr2CaCu 2O8  \nA. Nemati c order parameter  𝒩(𝐫)={Ax(𝐫)− Ay(𝐫)}/{Ax(𝐫)+ Ay(𝐫)} derived from \n2A,B. Domains  of opposite nematicity occur with  correlation length  ξ≈15nm . `Inset: \nHistogram  of all 𝒩(𝐫) values in A, showing nonzero mean value . The magnitude |   10 𝒩|>0.3 | for approximately 45% of the FOV, indicating a strong nematic  interaction \nbetween the two PDW.  \nB. At left, four examples of measured I𝑚(𝐫) along the x -axis; at right,  four examples of \nmeasured I𝑚(𝐫) along the y -axis : both within domains where 𝒩(𝐫)≫0. \nC. At left, four exam ples of measured I𝑚(𝐫) along the x -axis; at right, four examples of \nmeasured I𝑚(𝐫) along the y -axis : both within domains where 𝒩(𝐫)≪0 \n \nFigure 4:  Pinning of PDW Nematic Domains by Zn Impurity Atoms  \nA. Locations of zinc impurity atoms Z( r) in A, as detected in  𝑍(𝑟)≡I(r,20mV )−\n I(r,−20mV ) are show n as blue circles . The identification and register of Zn impurity \nsites is discussed in SI section 3.  \nB. |𝒩(𝒓)|, the amplitude of  the nematic order parameter from Fig. 3A, with the sites of \nZn impurity  resonances  overlaid as  blue dots.  \nC. The distribution of distances between each Zn impur ity atoms and its nearest PDW  \ndomain walls  (red). This is compared to the expected average distance if there is no \ncorrelation between Zn impurity atoms and the PDW doma in walls (blue).  Zn impurity \natoms are concentrated  near the PDW domain walls . \n \n   11 Acknowledgements : We acknowledge and thank  Owen H.S. Davis  and Shuqiu Wang for key \ndiscussions  and suggestions . M.H.H. and J.C.S.D. acknowledge  support from the Moore \nFoundat ion’s EPiQS Initiative through Grant GBMF9457.  W.R. and J.C.S.D.  acknowledge \nsupport from the European Research Council (ERC) under Award DLV -788932.  W.C. and \nJ.C.S.D. acknowledge support from the Royal Society under Award R64897. N.K., S.O’M and \nJ.C.S.D.  acknowledge support from Science Foundation of Ireland under Award SFI \n17/RP/5445.  H.E. acknowledges support f rom  JSPS KAKENHI (No. JP19H05823). PDJ \nacknowledges support by QuantEmX grant GBMF9616 from ICAM / Moore Foundation and \nby a Visiting Fellowship at Wadham College, Oxford., UK  \n \nAuthor Contributions: W.C. and  J.C.S.D.  conceived the project.  P.D.J. and  J.C.S.D. supervised \nthe research.  H.E. and S.U.  synthesized and characterized the samples; M.H.H. and J.C.S.D. \ncarried out SJTM measurements . Analys is and scholarship was carried out by W.C., W.R., N.K., \nP.D.J. and S.O’M.   J.C.S.D.  and W.C  wrote the paper with key contributions from N.K., W.R. and \nP.D.J.. The manuscript reflects the contributions and ideas of all authors.  \n \nAuthor Information  Correspon dence and requests for materials should be addressed to \njcseamusdavis@gmail.com . \n   12 References  \n1  E. Berg, E. Fradkin, S. A. Kivelson, J. M. 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Quenched disorder and vestigial nematicity in the \npseudogap regime of the cuprates Proc. Natl. A cad. Sci. U.S.A.  111 , 7980 -7985 (2014)  \n Im(q) High Low\n0.4 -0.4\nImy(r) (pA)\n20 nm\nyx\n20 nm\n0.4 -0.4\nImx(r) (pA)yxA B\nC\n-0.4 -0.2 0 0.2 0.4\n VJ(mV)-8-4048 Ip (pA)\nE\nyx20 nmIm \n-Im \nDqyqx (0,π/a0)  (π/a0,0)\nFigure 1Experimental Data\nFitting Curve\n0 2 4 6 8 1000.10.20.30.40.50.6Ax+Ay (pA)\nA0(pA)\n00.6\n0 9.5 A0Ax+Ay D0.3 0\nAx(r) (pA)yx\nyx\n10 3\nA0(r) (pA)yx\n0.3 0\nAy(r) (pA)A B\nC\nFigure 2A\n0.5 0\nN(r)-1 -0.5\n 104008001200 Count1600\n-0.3 0.3\nyx20 nm\na4b4c4\nd4a3\nb3c3\nd3\nB\n0 1 2 3 4-0.500.51\nr (nm)Imx(pA)\n0 1 2 3 4-0.500.51\nr (nm)Imy(pA)\n0 1 2 3 4-0.500.51\nr (nm)Imy(pA)\nr (nm)0 1 2 3 4-0.500.51Imx(pA)Ca1\nb1\nc1\nd1a2\nb2\nc2\nd2\na3\nb3\nc3\nd3a4\nb4\nc4\nd4a2b2\nc2d2\na1b1\nc1d1\nFigure 30 4 8 12\nd(pix)04812 Countsddistirbution\ndexperiment\nC\nyx\n|N(r)|1 010 nm\nyx\nZ(r) (pA)10 1.510 nmA\nB\nFigure 4Supporting Information  for \nIdentification of a Nematic Pair Density Wave State in Bi 2Sr2CaCu 2O8+x \n \nWeijiong Chen, Wangping Ren, Niall Kennedy, M. H. Hamidian, S. Uchida, H. Eisaki, Peter. D. \nJohnson, Shane M. O’Mahony and J.C. Séamus Davis  \n  \n1. Fourier Decomposition of SJTM Data  \nTo separate the components of PDW ( 𝒒=𝑷𝒙,𝒚) in the Im(r) map , we use a two -dimensional \nlock -in method. We parameterize the two PDW state signatures in terms  \n𝐼𝑚𝑥,𝑦(𝒓)=A𝑥,𝑦(𝒓) 𝑐𝑜𝑠(𝑷𝒙,𝒚∙𝒓+𝛿𝑥,𝑦(𝒓))≡ A𝑥,𝑦(𝒓) 𝑐𝑜𝑠(𝚽𝒙,𝒚(𝒓))      (1) \nFourier component selection at ±𝑷𝒙≈(2𝜋\n𝑎)(0,±0.25) and ±𝑷𝒚≈(2𝜋\n𝑎)(±0.25,0). A𝑥,𝑦(𝒓) \nand 𝛿𝑥,𝑦(𝒓) are the local amplitude and phase shift of each PDW component , respectively. To \nseparate these signatures  from the Im(r) map , we carry out a  two -dimensional lock -in \nmethod in which references 𝛼(𝒓)= 𝑐𝑜𝑠(𝑷𝒙,𝒚∙𝒓) and 𝛽(𝒓)= 𝑠𝑖𝑛(𝑷𝒙,𝒚∙𝒓) are multiplied by \nIm(r) \n𝑋(𝑟)= 𝐼𝑚(𝐫)⋅𝛼(𝒓) \n𝑌(𝑟)= 𝐼𝑚(𝐫)⋅𝛽(𝒓) (2) \nIn practice, it is more convenient to calculate 𝑋(𝑟) 𝑎𝑛𝑑  𝑌(𝑟) together in complex number \nwith 𝛼(𝒓)−𝑖⋅𝛽(𝒓)=𝑒−𝑖𝑷𝒙,𝒚∙𝒓. Then , low pass filtering these products:  \n𝐼𝑚′(𝒓) = =1\n√2𝜋𝜎∫𝑑𝒓′𝐼𝑚(𝒓′)𝑒−𝑖𝑷𝒙,𝒚∙𝒓′𝑒|𝒓−𝒓′|2\n2σ2                              (3) \nWe use  the filter width  𝜎=3 𝑛𝑚. The real part of 𝐼𝑚′(𝒓) contains A𝑥,𝑦(𝒓) and cos(𝛿𝑥,𝑦(𝒓)). \nAnd the imaginary part of 𝐼𝑚′(𝒓) contains A𝑥,𝑦(𝒓) and sin(𝛿𝑥,𝑦(𝒓)) \nℛ𝑒 𝐼𝑚′(𝒓)=𝐴𝑥,𝑦(𝒓)cos(𝛿𝑥,𝑦(𝒓))\n2(4) ℐ𝑚 𝐼𝑚′(𝒓)=𝐴𝑥,𝑦(𝒓)sin(𝛿𝑥,𝑦(𝒓))\n2(5) \nFinally, we achieve  the amplitude and phase shift of 𝑷𝒙,𝒚PDW state by equation  \n𝐴𝑥,𝑦(𝒓)≡2√(ℛ𝑒 𝐼𝑚′(𝒓))2+(ℐ𝑚 𝐼𝑚′(𝒓))2(6) \n𝛿𝑥,𝑦(𝒓)=arctan (ℐ𝑚 𝐼𝑚′(𝒓)\nℛ𝑒 𝐼𝑚′(𝒓)) (7) \n𝐴0(𝒓) (fig. 2C) is calculated by the same algorithm with q=0, which is equivalent to low pass \nfilter ing Im(r) map  with 𝜎=3 𝑛𝑚. \n \n2. Bi2Sr2CaCu 2O8+𝛿 tip segregation of Pair Tunneling to  DSC and PDW Orders    \nHere, we discuss the link between the Px,y≈(0,±0.25) 𝑎𝑛𝑑  (±0.25,0)2𝜋\n𝑎0 observed in \nelectron pair density modulation s in the present study and the  Qx,y≈\n(0,±0.125) 𝑎𝑛𝑑  (±0.125 ,0)2𝜋\n𝑎0 as reported in energy gap  Δ(r) imaging of  Refs. 1,2 . It has \nbeen proposed that this distinction  in wavevectors  is due to the segregation of pair tunneling \nto DSC and PDW orders  when using a Bi2Sr2CaCu 2O8+𝛿 nanoflake tip .  \n A heuristic  model for pair tunneling from a Bi2Sr2CaCu 2O8+𝛿 nanoflake tip to a \nBi2Sr2CaCu 2O8+𝛿 bulk crystal , both of which have coexisting DSC order and PDW order \nparallel to the sample surface  is shown in Fig.  S1. The perfect pair tunneling can only occur \nbetween the states with the same momentum of the center of mass because of the \nconservation of momentum. Such that, the ck†c−k† pairs of the homogenous DSC state at one \nside can only tunnel into the same DSC pairi ng states at another side. And, the ck†c−k+Q† pair \nof the modulating PDW state at one side can only tunnel into the same PDW pairing states at \nanother side. Th ese are  marked as orange  and blue arrows in Fig. S1. \n Consider ing the pair tunneling in the PDW c hannel, the tip’s PDW order parameter \n𝛹T and the sample’s PDW order parameter 𝛹S are written as  𝛹T=Δ1𝑒𝑖𝜃1exp[𝑖𝑄(𝑥+𝛿)] and 𝛹S=Δ2𝑒𝑖𝜃2exp(𝑖𝑄𝑥) (8) \nwhere x is the position and 𝜃 is the phase of the order parameter. In this case, the inter -PDW \nJosephson current takes the form  \n𝐼J𝑃∝𝛹T𝛹S∗−𝛹T∗𝛹S∝sin(𝑄𝛿+𝜙) (9) \nOur present study using SJTM measures the local cooper pair density 𝑛(𝑟)∝𝐼𝐽2. If the \nJosephson tunneling of PDW and DSC channels are independent, then,  \n𝑛(𝑟)∝𝐼𝐽𝑆2+𝐼J𝑃2(10) \nThe pair density of DSC part 𝐼𝐽𝑆2 is locally approximately  constant  while the  pair density of \nPDW part modulates in the form:  \n𝐼J𝑃2∝sin2(𝑄𝛿+𝜙) (11) \nHere 𝛿 is the transverse displacement between the Bi2Sr2CaCu 2O8+𝛿 nano flake tip and the \nBi2Sr2CaCu 2O8+𝛿 sample. Therefore  in such model s2, if the modulat ion of the order parameter  \nhas a wave vector Q ≈(0,0.125 )2𝜋\n𝑎0 or wave length λ≈8𝑎0, the modula tion of our measured \npair density would have  a primary periodicity of 4𝑎0. \n \n3. Identification and Register of Zn Impurity Sites  \n In Bi 2Sr2CaCu 2O8+𝛿, the scanning tunneling spectrum on the Zn site  show s the intense \nscattering resonance peak centered  at E ≈− 1 𝑚𝑒𝑉  (3). This resonance peak is detected at \nhigher energy ( Fig. S2A ) in our experiments, due to the superconductor -insulator -\nsuperconductor (SIS) tunnelling junction. The superconducting tip with coherence peaks \nnear  𝐸≈±20𝑚𝑒𝑉  result s in the convoluted tunnelling current I shows a strong jump near \n𝐸≈±20𝑚𝑒𝑉 , comparing to the I -V curve away from the Zn sites ( Fig. S2B ).  \n Therefore,  we can identify  the Zn impurities by finding the local maxima in 𝑍(𝒓)≡\nI(20mV ,𝒓)−I(−20mV ,𝒓) map . Fig. S2E shows the 𝑍(𝒓) map with identified Zn impurities \nmarked by blue  circles .   Because 𝑍(𝒓) map and I m(r) map s are measured using independent experiments  but  \nin the same FOV, there is usually a small displacement of two sets of data. So, the positions \nof these Zn impurities in 𝑍(𝒓) map must be transformed to their correct positions in I m(r) \nmap. We establish a  transformation matrix . We firstly manually choos e 3 Zn impurities in \nboth 𝑍(𝒓) map  (𝑎1=(x𝑎1,y𝑎1)𝑇,𝑎2=(x𝑎2,y𝑎2)𝑇,𝑎3=(x𝑎3,y𝑎3)𝑇) (blue  circles  in Fig. S2C ) \nand Im(r) map  𝑏1=(x𝑏1,y𝑏1)𝑇,𝑏2=(x𝑏2,y𝑏2)𝑇,𝑏3=(x𝑏3,y𝑏3)𝑇 (cyan  crosses in Fig. S2D) . \nThen, w e generate the transformation matrix  M by equation:  \n𝑀=[𝑏2−𝑏1𝑏3−𝑏1]×[𝑎2−𝑎1𝑎3−𝑎1]−1(12) \nThus,  finally, we can transform the position of each Zn impurity  in Z(r) map  𝑎Zn to its \nposition in Im(r) map 𝑏Zn by \n𝑏Zn=𝑀×(𝑎Zn−𝑎1)+𝑏1 (13) \nFig. S2F  show s the final Zn impurities in I m(r) map which are transformed from Zn impurities \nin Z(r) map  (Fig. S2E ) . \n \n4. Intertwining PDW with DSC order s in GLW free energy density  \n              The structure and intertwining between  the PDW orders \n(Δ𝑃𝑥(𝒓),Δ𝑃𝑦(𝒓),Δ−𝑃𝑥(𝒓),Δ𝑃𝑦(𝒓)) and  DSC order ( Δ0) is described by a subset of terms from \nthe overall GLW free energy density, as mentioned in the main text and ref. 4,5,6. \nℱ=𝛽𝑐1|Δ0|2(|ΔP𝑥|2+|ΔP𝑦|2\n+|ΔP−𝑥|2+|ΔP−𝑦|2\n)\n+𝛽𝑐2[Δ02(ΔP𝑥Δ−P𝑥+ΔP𝑦Δ−P𝑦)∗\n+(Δ02)∗(ΔP𝑥Δ−P𝑥+ΔP𝑦Δ−P𝑦)] (14) \n \nTo minimize the free energy and find the ground  state , one must insert the phase terms of \norder parameters . \nΔ0=|Δ0|𝑒𝑖𝜙0,Δ𝑃𝑥(𝒓)=|Δ𝑥|𝑒𝑖𝜙1,Δ𝑃𝑦(𝒓)=|Δ𝑦|𝑒𝑖𝜙2,\nΔ−𝑃𝑥(𝒓)=|Δ𝑥|𝑒𝑖𝜙3,Δ𝑃𝑦(𝒓)=|Δ𝑦|𝑒𝑖𝜙4 (15) Here, for simplicity, we ignore the  case of the  loop current order. Such that  \nℱ=2|Δ0|2|Δ𝑥|2[𝛽𝑐1+𝛽𝑐2cos(2𝜙0−𝜙1−𝜙3)]\n          +2|Δ0|2|Δ𝑦|2[𝛽𝑐1+𝛽𝑐2cos(2𝜙0−𝜙2−𝜙4)] (16) \nThe coupling term 𝛽𝑐2 locks the phase of the uniform DSC order the PDW orders6. Obviously , \nthe minimum free energy is:  \nℱ=2|Δ0|2(|Δ𝑥|2+|Δ𝑦|2)[𝛽𝑐1−|𝛽𝑐2|] (17) \nTherefore, the positive correlation between |Δ0|2 and |Δ𝑥|2+|Δ𝑦|2 in Fig. 2D implies  𝛽𝑐1−\n|𝛽𝑐2|<0 under the zero magnetic field circumstances of that experiment.  \n \n \n  \n \n  \nFigure S1.  Schematic of possible Josephson tunneling (JT) in the presence of d -wave \nsuperconductivity (DSC) and pair density wave (PDW) order parameters (OPs).  \n \n  \n  \nFigure S2. Identification and transformation of Zn impurity sites. (A). Scanning tunnelling \nspectra  on Zn impurity site measured by  a superconducting tip with coherence peaks near \n𝐸≈±20𝑚𝑒𝑉 . The resonance peak of Zn impurity moves to energy slightly larger than \n±20𝑚𝑒𝑉 . (B). I -V curves measured on Zn impurity site and defect free region. (C). 𝑍(𝒓)≡\nI(20mV ,𝒓)−I(−20mV ,𝒓) map. Three Zn impurities used  to generate the transformation \nmatrix are marked by blue circles . (D). Measured  I𝑚(𝐫) map. Three Zn impurities used to \ngenerate the transformation matrix are marked by cyan crosses. (E). Z(𝒓)≡I(20mV ,𝒓)−\nI(−20mV ,𝒓) map. Identified Zn impurities are marked by blue  circles . (F). Measured  I𝑚(𝐫) \nmap. Red crosses mark the Zn impurities transformed from cyan  crosses in (E).  \n   \nFigure S3. Absence of  charge density modulation  signature in these samples 𝑝≈0.17 (A,C). \nScanning differential tunnelling spectra g( r) map at 𝐸=70 𝑚𝑒𝑉 ~20𝑚𝑒𝑉  (A)  and  𝐸=\n−70 𝑚𝑒𝑉 ~−20𝑚𝑒𝑉  (C). (B,D). Power spectral density Fourier transform  of g( r), g( q) at \n𝐸=70 𝑚𝑒𝑉 ~20𝑚𝑒𝑉  (B)  and  𝐸=−70 𝑚𝑒𝑉 ~−20𝑚𝑒𝑉  (D). Bragg peaks (0,±2𝜋\n𝑎0), \n(±2𝜋\n𝑎0,0) are marked by yellow crosses. The pair density wave peaks Q= (0,±2𝜋\n4𝑎0), (±2𝜋\n4𝑎0,0) \nseen in 𝐼𝑚(𝒒) map of the main paper cannot be detected  throughout g(q) map B and D, \nwhich implies that there is little or no pre-existent  charge density modulation  in our \nsample , consistent with previous studies for this doping range7 in Bi 2Sr2CaCu 2O8. \n \nReferences  \n1 Edkins SD , et al . (2019) Magnetic field -induced pair density wave state in the cuprate \nvortex halo. Science  364 (6444):976 -980.  \n2 Du Z , et al . (2020) Imaging the energy gap modulations of the cuprate pair -density -wave \nstate. Nature  580 (7801):65 -70. \n3 Pan SH , et al . (2000) Imaging the effects of individual zinc impurity atoms on \nsuperconductivity in Bi 2Sr2CaCu 2O8+δ. Nature  403 (6771):746 -750.  \n4 Berg, E., et al . (2009). \"Striped superconductors: how spin, charge and superconducting \norders intertwine in the cuprates.\" New Journal of Physics  11(11): 115004.  \n5 Fradkin, E., et al . (2015). \"Colloquium: Theory of intertwined orders in high temperature \nsuperconductors.\" Reviews of Modern Physics  87(2): 457 -482.  \n6 Agterberg, D. F., et al . (2020). \"The Physics of Pair -Density Waves: Cuprate \nSuperconductors and Beyond.\" Annual Review of Condensed Matter Physics , Vol 11, 2020 \n11(1): 231 -270.  \n7 Fujita, K., et al . (2014). \"Simultaneous Transitions in Cuprate Momentum -Space Topology \nand Electro nic Symmetry Breaking.\" Science  344 (6184): 612 -616.  \n  \n                                                            "    },    {        "title": "2206.14877v1.Configurational_density_of_states_and_melting_of_simple_solids.pdf",        "content": "Configurational density of states and melting of simple solids\nSergio Davisa,b,\u0003, Claudia Loyolab, Joaqu ´ın Peraltab\naResearch Center on the Intersection in Plasma Physics, Matter and Complexity, P2MC,\nComisi´ on Chilena de Energ´ ıa Nuclear, Casilla 188-D, Santiago, Chile\nbDepartamento de F´ ısica, Facultad de Ciencias Exactas, Universidad Andres Bello. Sazi´ e 2212, piso 7, Santiago, 8370136, Chile.\nAbstract\nWe analyze the behavior of the microcanonical and canonical caloric curves for a piecewise model of the configura-\ntional density of states of simple solids, in the context of melting from the superheated state, as realized numerically in\nthe Z-method via atomistic molecular dynamics. A first-order phase transition with metastable regions is reproduced\nby the model, being therefore useful to describe aspects of the melting transition. Within this model, transcendental\nequations connecting the superheating limit, the melting point, and the specific heat of each phase are presented and\nnumerically solved. Our results suggest that the essential elements of the microcanonical Z curves can be extracted\nfrom simple modeling of the configurational density of states.\nKeywords: Density of states, Phase transitions, Melting\n1. Introduction\nOne of the widely used approaches to determine the melting point of materials via atomistic computer simulation\nis the so-called Z-method [1–11], which is based on the empirical observation that the superheated solid at the limit of\nsuperheating temperature TLShas the same internal energy as the liquid at the melting temperature Tm. This observa-\ntion has potential implications for understanding the atomistic melting mechanism from the superheated state [12, 13]\nbut has been left mostly unexplored, disconnected from thermodynamical models of solids.\nIn the Z-method, the isochoric curve T(E) is computed from simulations at di \u000berent total energies E1;E2;:::and\nthe minimum temperature of the liquid branch of this isochoric curve is identified with the melting temperature Tm.\nThis key assumption still lacks a proper explanation in terms of microcanonical thermodynamics of finite systems.\nIn this work, we study the properties of a recently proposed model [14] for the configurational density of states\n(CDOS) of systems with piecewise constant heat capacity by calculating its canonical and microcanonical caloric\ncurves in terms of special functions. The model presents a first-order phase transition with metastable regions where\nthe microcanonical curve T(E) shows a so-called van der Waals loop [15]. The inflection points found in this loop\ncan be associated to TLSandTmof the Z-method description of the superheated solid.\nThis paper is organized as follows. In Section 2 we briefly review the definition of the CDOS and the formalism\nused to compute thermodynamic properties from it. In Section 3 we revisit the model in Ref. [14] and provide some\ninterpretation of its parameters. Sections 4 and 5 show the computation of the caloric curves of the solid model in\nthe canonical and microcanonical ensemble, respectively, and we present some concluding remarks in Section 6.\n2. Configurational density of states and thermodynamics\nFor a classical system with Hamiltonian\nH(r1;:::; rN;p1;:::; pN)=NX\ni=1pi2\n2mi+ \b(r1;:::; rN); (1)\n\u0003Corresponding author\nPreprint submitted to Physica A July 1, 2022arXiv:2206.14877v1  [cond-mat.stat-mech]  29 Jun 2022we will define the configurational density of states (CDOS) as the multidimensional integral\nD(\u001e):=Z\ndr1:::drN\u000e(\b(r1;:::; rN)\u0000\u001e); (2)\nwhere \b(r1;:::; rN) is the potential energy describing the interaction between particles. Using the definition of D(\u001e)\nin (2) is possible to rewrite any configurational integral of the form\nI=Z\ndr1:::drNG(\b(r1;:::; rN)) (3)\nas a one-dimensional integral over \u001e. In fact, taking Iand introducing a factor of 1 as an integral over a Dirac delta\nfunction, we have\nI=Z\ndr1:::drNG(\b(r1;:::; rN))\n=Z\ndr1:::drN\"Z1\n\u00001d\u001e\u000e(\u001e\u0000\b(r1;:::; rN))#\nG(\b(r1;:::; rN))\n=Z1\n0d\u001e(Z\ndr1:::drN\u000e(\u001e\u0000\b(r1;:::; rN)))\nG(\u001e)\n=Z1\n0d\u001eD(\u001e)G(\u001e):(4)\nwhere we have assumed that \bhas its global minimum at \b = 0. One such integral of particular importance is the\ncanonical partition function Z(\f), defined as\nZ(\f):=Z\nd\u0000exp\u0000\u0000\fH(\u0000)\u0001; (5)\nwhere \u0000=(r1;:::; rN;p1;:::; pN), and which can be computed, following (4), as\nZ(\f)=Z\ndp1:::dpNexp\u0010\n\u0000\fPN\ni=1p2\ni\n2mi\u0011\n\u0002R\ndr1:::drNexp\u0010\n\u0000\f\b(r1;:::; rN)\u0011\n=Z0(N)\f\u00003N=2Z1\n0d\u001eD(\u001e) exp(\u0000\f\u001e)\n=Z0(N)\f\u00003N=2Zc(\f);(6)\nwith Z0(N):=QN\ni=1(p2\u0019mi)3a constant only dependent on Nand the masses of the particles, and where\nZc(\f):=Z1\n0d\u001eD(\u001e) exp(\u0000\f\u001e) (7)\nis the configurational partition function. Similarly, the full density of states,\n\n(E):=Z\nd\u0000\u000e(E\u0000H(\u0000)) (8)\ncan also be computed from the CDOS, by a convolution with the density of states \nKof the ideal gas [16], that is,\n\n(E)=Z\nd\u001eD(\u001e)\nK(E\u0000\u001e); (9)\nwhere\n\nK(E):=Z\ndp1:::dpN\u000e\u0010PN\ni=1p2\ni\n2mi\u0000E\u0011\n= \n 0(N)\u0002(E)E3N\n2\u00001: (10)\n2In order to obtain the result in (9), we replace the integral over the momenta in \n(E) by using (10),\n\n(E):=Z\ndr1:::drN\"Z\ndp1:::dpN\u000e\u0010PN\ni=1p2\ni\n2mi+ \b(r1;:::; rN)\u0000E\u0011#\n=Z\ndr1:::drN\nK(E\u0000\b(r1;:::; rN))\n=Z1\n0d\u001eD(\u001e)\nK(E\u0000\u001e);(11)\nwhere in the last equality we have used (4). Finally we have\n\n(E)= \n 0(N)ZE\n0d\u001eD(\u001e)(E\u0000\u001e)3N\n2\u00001= \n 0(N)\u0011(E) (12)\nwhere \n0(N):=Z0(N)=\u0000\u00003N=2\u0001is a function only of the size Nof the system, and\n\u0011(E):=ZE\n0d\u001eD(\u001e)\u0000E\u0000\u001e\u00013N\n2\u00001: (13)\nNow we have all the elements needed for the computation of the caloric curves and the transition energy (or\ntemperature) for a given model of CDOS. In the canonical ensemble, the probability of observing a value \u001eof potential\nenergy at inverse temperature \fis given by\nP(\u001ej\f)=1\nZc(\f)exp(\u0000\f\u001e)D(\u001e) (14)\nand, using this, we can determine the caloric curve (internal energy as a function of inverse temperature) as\n\nH\u000b\n\f=\u0000@\n@\flnZ(\f)=3N\n2\f+\n\u001e\u000b\n\f; (15)\nwhere\n\n\u001e\u000b\n\f=1\nZc(\f)Z1\n0d\u001eD(\u001e) exp(\u0000\f\u001e)\u001e=\u0000@\n@\flnZc(\f): (16)\nOn the other hand, in the microcanonical ensemble the probability of having potential energy \u001eat total energy E\nis given by [17–20]\nP(\u001ejE)=1\n\u0011(E)(E\u0000\u001e)3N\n2\u00001D(\u001e): (17)\nThe microcanonical caloric curve (inverse temperature as a function of internal energy) is given by\n\f(E):=@\n@Eln\n(E); (18)\nwhich can also be rewritten as an expectation as follows. Replacing \n(E) in terms of \u0011(E) in (18) and using (17), we\ncan write\n\f(E)=1\n\u0011(E)@\n@EZE\n0d\u001eD(\u001e)(E\u0000\u001e)3N\n2\u00001\n=1\n\u0011(E)ZE\n0d\u001eD(\u001e)\u0000E\u0000\u001e)3N\n2\u00001\"3N\u00002\n2(E\u0000\u001e)#\n=\nˆ\fK\u000b\nE:(19)\nwhere ˆ\fKis the kinetic inverse temperature estimator\nˆ\fK(\u001e;E):=3N\u00002\n2(E\u0000\u001e): (20)\n33. Model for the CDOS\nIn the following sections we will use, for the configurational density of states, the model presented in Ref. [14]\nwhich is defined piecewise, as\nD(\u001e)=8>>>>><>>>>>:dS(\u001e\u0000\u001eS)\u000bS for\u001e<\u001e c;\ndL(\u001e\u0000\u001eL)\u000bL for\u001e\u0015\u001ec:(21)\nThis model represents two segments, one for the solid phase which, by definition, will have potential energies\n\u001e<\u001e c, and one for the liquid phase where \u001e\u0015\u001ec. That is, our main assumption is that the potential energy landscape\nis e\u000bectively divided into solid and liquid states by a surface\n\b(r1;:::; rN)=\u001ec\nin configurational space. By imposing continuity of the CDOS at \u001e=\u001ec, we must have\ndS(\u001ec\u0000\u001eS)\u000bS=dL(\u001ec\u0000\u001eL)\u000bL: (22)\nHere\u001eSrepresents the potential energy minimum of the ideal solid, that can be set to zero without loss of generality\nprovided that all energies are measured with respect to this value. We can also set dS=1 and express the potential\nenergy in units of \u001ec, and then we have\nD(\u001e)=8>>>>>><>>>>>>:\u001e\u000bS for\u001e<1;\n\u0010\u001e\u0000\r\n1\u0000\r\u0011\u000bLfor\u001e\u00151;(23)\nwhere we have defined the dimensionless parameter\n\r:=\u001eL\n\u001ec:\nIn this way, the model for the CDOS has only three free parameters, namely \u000bS,\u000bLand\r.\n4. The caloric curve in the canonical ensemble\nThe configurational partition function Zc(\f) associated to the model for the CDOS in (23) can be obtained by\npiecewise integration as\nZc(\f)=Z1\n0d\u001eD(\u001e) exp(\u0000\f\u001e)\n=Z1\n0d\u001e\u001e\u000bSexp(\u0000\f\u001e)+1\n(1\u0000\r)\u000bLZ1\n1d\u001e(\u001e\u0000\r)\u000bLexp(\u0000\f\u001e)\n=\f\u0000(\u000bS+1)Z\f\n0duu\u000bSexp(\u0000u)+\f\u0000(\u000bL+1)\n(1\u0000\r)\u000bLZ1\n\fdu(u\u0000\f\r)\u000bLexp(\u0000u)\n=\f\u0000(\u000bS+1)Z\f\n0duu\u000bSexp(\u0000u)+\f\u0000(\u000bL+1)\n(1\u0000\r)\u000bLexp(\u0000\f\r)Z1\n\f(1\u0000\r)dww\u000bLexp(\u0000w):(24)\nFinally we obtain\nZc(\f)=\f\u0000(\u000bS+1)GS(0!\f)+\f\u0000(\u000bL+1)\n(1\u0000\r)\u000bLexp(\u0000\f\r)GL(\f(1\u0000\r)!1 ) (25)\nwhere we have defined, for convenience, the auxiliary functions\nG\u0017(a!b):=Zb\nadtexp(\u0000t)t\u000b\u0017= \u0000(\u000b\u0017+1;b)\u0000\u0000(\u000b\u0017+1;a) (26)\nfor\u0017=S;L, where \u0000(k;x):=Rx\n0dtexp(\u0000t)tk\u00001is the lower incomplete Gamma function.\n44.1. Low and high-temperature limits\nBy taking the limit \f!1 of (25) we see that the second term vanishes, and also GS(0!1 )= \u0000(\u000bS+1) so for\nlow temperatures we can approximate\nZc(\f)\u0019\f\u0000(\u000bS+1)\u0000(\u000bS+1): (27)\nBy replacing (27) in (16) we have\n\n\u001e\u000b\n\f=\u0000@\n@\flnZc(\f)\u0019\u000bS+1\n\f(28)\nand then the solid branch of the canonical caloric curve is given by a straight line,\nES(T):=\nH\u000b\nT;S=3N\n2kBT+\n\b\u000b\nT=\u00003N\n2+\u000bS+1\u0001kBT (29)\nas expected. Here we can verify that ES(T)!0 as T!0, because we have fixed \u001eS=0, and moreover, we learn\nthat the value of the specific heat for the solid phase is\nCS=d\ndTES(T)=\u00003N\n2+\u000bS+1\u0001kB: (30)\nOn the other hand, for high temperatures (i.e. in the limit \f!0) we can approximate\nZc(\f)\u0019\"\f\u0000(\u000bL+1)exp(\u0000\f\r)\n(1\u0000\r)\u000bL#\n\u0000(\u000bL+1); (31)\nobtaining from (16) that\n\n\u001e\u000b\n\f=\u0000@\n@\flnZc(\f)\u0019\u000bL+1\n\f+\r (32)\nhence the liquid branch is also a straight line, given by\nEL(T):=\nH\u000b\nT;L=3N\n2kBT+\n\b\u000b\nT\u0019\r+\u00003N\n2+\u000bL+1\u0001kBT: (33)\nThis allows us to interpret \ras the extrapolation of the liquid branch towards T=0, that is,\u001eLin units of\u001ecis the\npotential energy of a perfectly frozen liquid at T=0,\nEL(T=0)=\r=\u001eL\n\u001ec: (34)\nMoreover, we obtain that the specific heat of the liquid phase is\nCL=d\ndTEL(T)=\u00003N\n2+\u000bL+1\u0001kB: (35)\nIn order for the energy to be extensive in both branches as N!1 , it must hold true that the parameters \u000b\u0017are\nproportional to N, and it follows that\nC\u0017\nkB=\u000b\u0017+3N\n2(36)\nwith\u0017=S;L. Using these definitions we can write ESandELin terms of\fmore compactly, as\nES(\f)=CS\n\f; (37a)\nEL(\f)=\r+CL\n\f: (37b)\n54.2. Melting temperature\nOn account of the assumption that all solid states have \u001e < \u001e cand for the liquid states \u001e > \u001e c, we will define the\nprobabilities P(Sj\f) of being in the solid phase, and P(Lj\f) of being in the liquid phase as\nP(Sj\f):=P(\u001e<1j\f)=1\nZc(\f)Z1\n0d\u001eexp(\u0000\f\u001e)\u001e\u000bS=\f\u0000(\u000bS+1)GS(0!\f)\nZc(\f); (38)\nP(Lj\f):=P(\u001e\u00151j\f)=1\nZc(\f)Z1\n1d\u001eexp(\u0000\f\u001e)(\u001e\u0000\r)\u000bL\n(1\u0000\r)\u000bL=\f\u0000(\u000bL+1)exp(\u0000\f\r)\nZc(\f)(1\u0000\r)\u000bLGL(\f(1\u0000\r)!1 ) (39)\nrespectively, such that\nP(Sj\f)+P(Lj\f)=1\nfor all values of \f. The probability of solid is shown as a function of Tin Fig. 1. The melting temperature Tmis such\nthat both probabilities are equal [21], that is,\nP(Sj\fm)=P(Lj\fm)=1\n2; (40)\nwith\fm=1=(kBTm), which is then the solution of the transcendental equation\n(\fm)\u000bL\u0000\u000bSGS(0!\fm)=exp(\u0000\fm\r)\n(1\u0000\r)\u000bLGL(\fm(1\u0000\r)!1 ): (41)\nUsing P(Sj\f) and P(Lj\f) we can write the canonical caloric curve for any \fas the sum of three contributions, namely\n\nH\u000b\n\f=3N\n2\f\u0000@\n@\flnZc(\f)=ES(\f)P(Sj\f)+EL(\f)\u00001\u0000P(Sj\f)\u0001+Eres(\f); (42)\nwhere the first and second terms account for the solid and liquid branches, respectively, and Eresis equal to\nEres(\f):=\u0000\r\f\u00001exp(\u0000\f)\nZc(\f): (43)\nBy comparing (42) with the low and high temperature limits, namely ESandEL, and noting that P(Sj\f)!1 for\nlow temperatures and P(Sj\f)!0 for high temperatures, we see that Eresmust vanish on both limits, as can be verified\nby using (27) and (31). Therefore, this term is only relevant near the transition region. If we replace (41) into the\nconfigurational partition function in (25) we obtain\nZc(\fm)=2(\fm)\u0000(\u000bS+1)GS(0!\fm) (44)\nand we can determine the melting energy E\u0003as\nE\u0003=3N\n2\fm+1\n2\"\nN\r+\u000bS+\u000bL+2\n\fm#\n\u0000\rexp(\u0000\fm)\n2GS(0!\fm)(\fm)\u000bS: (45)\nBecause\u000bS/N, in the thermodynamic limit we can approximate\nGS(0!\fm)=Z\fm\n0dtexp(\u0000t)t\u000bS\n=Z\fm\n0dtexp(\u0000t+\u000bSlnt)\n\u0019Z\fm\n0dtexp(\u000bSlnt)=\f\u000bS+1\nm\n\u000bS+1;(46)\nand then we have, for the melting energy per particle \"\u0003:=E\u0003=Nin the original energy units, that\nlim\nN!1\"\u0003='S+\r\n2+1\n2 cS+cL\nkB\u0000\raSexp\u0010\n\u0000\u001ec\nkBTm\u0011!\nkBTm; (47)\n6with'S:=\u001eS=N,c\u0017:=C\u0017=Nanda\u0017:=\u000b\u0017=N. This becomes a linear relation between E\u0003andTmfor\r\u00190, because\nlim\n\r!0\"\u0003='s+\u0012cs+cL\n2\u0013\nTm; (48)\nand also for kBTm\u001c\u001ec. Fig. 2 shows the dimensionless intensive quantity\n\u0010:=\"\u0003\u0000'S\nkBTm\nas a function of \r, using the approximation in (47) and the exact value in (45).\n5. The caloric curve in the microcanonical ensemble\nJust as we used the configurational partition function in Section 4 to compute the canonical caloric curve, we can\nuse the full density of states \n(E) to obtain the microcanonical caloric curve, being given in terms of the CDOS as\n\u0011(E)=ZE\n0d\u001eD(\u001e)\u0000E\u0000\u001e\u00013N\n2\u00001=E3N\n2\u00001ZE\n0d\u001eD(\u001e)\u0010\n1\u0000\u001e\nE\u00113N\n2\u00001\n=E3N\n2\u00001\"Zmin( E;1)\n0d\u001e\u0010\n1\u0000\u001e\nE\u00113N\n2\u00001\u001e\u000bS+\u0002(E\u00001)\n(1\u0000\r)\u000bLZE\n1d\u001e\u0010\n1\u0000\u001e\nE\u00113N\n2\u00001(\u001e\u0000\r)\u000bL#\n=ECSBS\u0010\n0!min(1;1\nE)\u0011\n+\u0002(E\u00001)\n(1\u0000\r)\u000bL(E\u0000\r)CLBL\u0010\n\u0015(E)!1\u0011(49)\nwhere we have defined the auxiliary functions\n\u0015(E):=1\u0000\r\nE\u0000\r(50)\nand\nB\u0017(a!b):=Zb\nadt t\u000b\u0017+1(1\u0000t)3N\n2\u00001(51)\nfor convenience of notation, where \u0017=S;Landa;b2[0;1]. Replacing in (49) we obtain \u0011(E) as a piecewise function,\n\u0011(E)=8>>>>>>><>>>>>>>:ECSB(\u000bS+1;3N\n2) for E\u00141\nECSBS(0!1\nE)+(E\u0000\r)CL\n(1\u0000\r)\u000bLBL(\u0015(E)!1) for E>1:(52)\nWe can see that the microcanonical inverse temperature for the branch with E\u00141 is simply given by\n\flow(E)=@\n@Eln(ECS)=CS\nE; (53)\nthat is Eas a function of T(E) is a straight line with slope CS, in agreement with the low temperature approximation\nES(T) of the canonical caloric curve in (29). This means the non-monotonic behavior, i.e. the van der Waals loop and\nthe microcanonical melting energy Emmust ocurr above E=1. Just as we did for the canonical ensemble, we will\ndefine the probabilities of solid and liquid, P(SjE) and P(LjE) respectively, at an energy E\u00151, according to\nP(SjE):=1\n\u0011(E)Z1\n0d\u001eD(\u001e)(E\u0000\u001e)3N\n2\u00001=ECSBS(0!1=E)\n\u0011(E); (54)\nP(LjE):=1\n\u0011(E)ZE\n1d\u001eD(\u001e)(E\u0000\u001e)3N\n2\u00001=(E\u0000\r)CL\n(1\u0000\r)\u000bLBL(\u0015(E)!1)\n\u0011(E): (55)\n7The probability of solid is shown, as a function of energy, in Fig. 3. We will define the microcanonical melting\nenergy Emas the value of Esuch that P(SjEm)=P(LjEm), being then the solution of the transcendental equation\n(Em)CS(1\u0000\r)\u000bLBS(0!1=Em)=(Em\u0000\r)CLBL(\u0015(Em)!1): (56)\nUsing the derivatives\n@\n@EBS(0!1=E)=@\n@EZ1\n0dt t\u000bS+1(1\u0000t)3N\n2\u00001\u0002(1=E\u0000t)=\u0000E\u0000(CS+2)(E\u00001)3N\n2\u00001(57)\n@\n@EBL(\u0015(E)!1)=@\n@EZ1\n0dt t\u000bL+1(1\u0000t)3N\n2\u00001\u0002(t\u0000\u0015(E))=1\n1\u0000\rh\n\u0015(E)\u000bL+3\u0000\u0015(E)CL+2i\n; (58)\nwe can write\n\u00110(E)=\u0011(E) CS\nEP(SjE)+CL\nE\u0000\rP(LjE)!\n+\u0015(E)2h\n(E\u0000\r)3N\n2\u0000(1\u0000\r)3N\n2i\n\u0000E\u00002(E\u00001)3N\n2\u00001(59)\nand then\n\fhigh(E)=\u0012CS\nE\u0013\nP(SjE)+ CL\nE\u0000\r!\u00001\u0000P(SjE)\u0001+\fres(E) (60)\nwhere\n\fres(E):=1\n\u0011(E) \n\u0015(E)2h\n(E\u0000\r)3N\n2\u0000(1\u0000\r)3N\n2i\n\u00001\nE2(E\u00001)3N\n2\u00001!\n(61)\nfrom which we see that the (inverse) temperature is also continuous at E=1. Similarly to Eres(\f) in the canonical\nensemble, we can also see that \fres(E) vanishes for both E!1 and E!1 . By using the Stirling approximation on\nthe representation of the Beta function written in terms of \u0000-functions,\nB(x;y)=\u0000(x)\u0000(y)\n\u0000(x+y)\u0019p\n2\u0019\"xx\u00001=2yy\u00001=2\n(x+y)x+y\u00001=2#\n(62)\nfor large xandywe have\nB\u0017(a!b)\u00198>>>><>>>>:K\u0017if\u000b\u0017=C\u00172[a;b]\n0 otherwise ;(63)\nwhere\nK\u0017:=p\n2\u001926666666664(\u000b\u0017+2)\u000b\u0017+3=2\u00103N\n2\u00113N=2\u00001=2\n(C\u0017+1)C\u0017\u00001=237777777775: (64)\nThe probability of solid phase P(SjE) is then approximated by\nP(SjE)\u0019ECS(1\u0000\r)\u000bLKSQ[E\u0014cS=aS]\nECS(1\u0000\r)\u000bLKSQ[E\u0014cS=aS]+(E\u0000\r)CLKLQ[E\u0015\r+(cL=aL)(1\u0000\r)](65)\nwhere Q( A) is the indicator function [22] of the proposition A, defined as\nQ(A)=8>><>>:1 if Ais true;\n0 otherwise :(66)\nTherefore, in this approximation the transition energy Em, such that P(SjEm)=1=2, must be the solution of\nECSm(1\u0000\r)\u000bLKS=(Em\u0000\r)CLKL (67)\n8provided that no indicator function vanishes, that is, it must hold that\n\r+cL\naL(1\u0000\r)\u0014Em\u0014cS\naS: (68)\nThese inequalities impose a lower limit on \r, namely\n\r\u0015aL\naS\u00001; (69)\nwhich however is only relevant if aL>aS, as it prevents \rfor reaching zero. In order to determine the solutions E\u0003\ncorresponding to the maximum and minimum of microcanonical temperature, we impose\n@\n@E\f(E)\f\f\f\fE=E\u0003=@2\n@E2ln\u0011(E)\f\f\f\fE=E\u0003=\u001100(E\u0003)\n\u0011(E\u0003)\u0000 \u00110(E\u0003)\n\u0011(E\u0003)!2\n=0; (70)\ntherefore\n\u001100(E\u0003)\u0001\u0011(E\u0003)=\u00110(E\u0003)2: (71)\nThe exact conditions under which (71) has exactly two solutions remain to be explored. Nevertheless, we have\nverified this fact numerically, and in Fig. 4 the green diamonds show the numerical solutions of (71) using the\nexpression (52) for \u0011(E) in the case E>1, together with the canonical and microcanonical caloric curves. The\nensemble inequivalence expected in small systems [23] is clearly seen, and a rather remarkable agreement of the\nmicrocanonical curves with the usual shape of the Z curves in the literature is found. 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Negative heat capacity for a cluster of 147 sodium\natoms. Physical Review Letters , 86:1191, 2001.\n[26] H. Behringer, M. Pleimling, and A. Huller. Finite-size behaviour of the microcanonical specific heat. J. Phys. A: Math. Theor. , 38:973–985,\n2005.\n[27] H. Behringer and M. Pleimling. Continuous phase transitions with a complex dip in the microcanonical entropy. Phys. Rev. E , 74:011108,\n2006.\n[28] M. Ery ¨urek and M. H. G ¨uven. Negative heat capacity of Ar55 cluster. Physica A , 377:514–522, 2007.\n[29] M. Ery ¨urek and M. H. G ¨uven. Peculiar thermodynamic properties of LJ N(N=39-55) clusters. European Physical Journal D , 48:221–228,\n2008.\n[30] M. A. Carignano and I. Gladich. Negative heat capacity of small systems in the microcanonical ensemble. EPL, 90:63001, 2010.\n10Figure 1: Canonical probability of solid phase P(SjT) as a function of temperature, for di \u000berent values of \r.\nFigure 2: Ratio between the melting energy and temperature, according to (45) (solid blue line) and its approximation (47) (orange dashed line).\nRed circles indicate the values \r=0.2, 0.3, 0.5 and 0.8.\n11Figure 3: Microcanonical probability of solid phase P(SjE) as a function of energy, for di \u000berent values of \r.\nFigure 4: Canonical and microcanonical isochoric curves for N=128,\u000bS=N=1.53812 and \u000bL=N=1.57241, with (a) \r=0.2, (b)\r=0.3, (c)\r=0.5\nand (d)\r=0.8.\n12"    },    {        "title": "2207.02730v1.On_atomic_state_purity_operator__degree_of_state_purity_and_concurrence_in_the_JC_and_anti_JC_models.pdf",        "content": "arXiv:2207.02730v1  [quant-ph]  6 Jul 2022On atomic state purity operator, degree of state purity and\nconcurrence in the JC and anti-JC models\nJoseph Akeyo Omolo\nDepartment of Physics, Maseno University, P.O. Private Bag, Mase no, Kenya\ne-mail: ojakeyo04@yahoo.co.uk\n10 April 2022\nAbstract\nThe state of an atom in a bipartite qubit, Jaynes-Cummings (J C) or anti-Jaynes-Cummings (aJC)\ninteraction is described by a reduced density operator. The purity of the state has been measured by\ntaking the trace of the square of the reduced density operato r. In this article, we define the square of\nthe reduced density operator as the state purity operator, c omposed of a completely pure state part and\na completely mixed state part. The coefficient of the complete ly mixed state part is the mixed state\nmeasure, formally obtained as the determinant of the reduce d density operator and it is therefore directly\nrelated to tangle, the square of concurrence of the bipartit e system. Expressed in various equivalent\nforms, the mixed state measure provides all the characteris tic elements of state purity or entanglement,\nsuch as eigenvalues of the reduced density operator, noncla ssicality measures and a state purity complex\namplitude. The argument of the state purity complex amplitu de in polar form is the phase of the state\npurity measure, which defines the degree of purity of the stat e. We find that the degree of purity and\nconcurrence are complementary quantifiers satisfying a com plementarity relation. The general form of\nthe mixed state measure provides an interpretation that con currence is fully defined by the Bloch radius\nfour-vector in a spacetime frame. Plots of the degree of puri ty, concurrence and spin excitation number\nat resonance reveal that the atomic state collapses rapidly to a momentary totally mixed or maximally\nentangled state over a very short time after the beginning of the interaction, then evolves gently to a pure\ndisentangled state in the middle of the collapse region of th e spin excitation number. In off-resonance\ndynamics with large values of the frequency detuning parame ters, the degree of purity, concurrence and\nspin excitation number develop periodic evolution at the re spective maximum (1) or minimum (0) values,\nsimultaneously signifying that at very large detuning, the atom evolves to a perfectly pure disentangled\nstate where the spin excitation number is effectively zero.\n1 Introduction\nAtomic state purity or entanglement dynamics in the Jaynes-Cummin gs (JC) model is an old, comprehen-\nsively researched topic. The original studies initiated by Gea-Banac loche [1-3] and comprehensively elabo-\nrated by P L Knight and collaborators [4-6] revealed the important u nderlying property that the atom-field\nstate evolves to a pure disentangled state in the middle of the initial c ollapse region of the atomic state\npopulation inversion. Introducing the reduced density operator ρto describe the state of the atom subsys-\ntem, Gea-Banacloche used the atomic state purity measure Trρ2, while P L Knight and collaborators, used\nthe von Newmann entropy defined in terms of the eigenvalues of ρto study the dynamical evolution of the\natomic state purity or entanglement. These were essentially qualita tive studies demonstrating the nature of\nthe dynamical evolution between the pure disentangled and the max imally entangled states of the atom. It\nwas established that the atomic state evolved rapidly from the initial pure state to a momentary maximally\nentangled state followed by a gentle evolution back to a pure disenta ngled state in the middle of the region of\ncollapse of the state population inversion. The approximate forms o f the emergent pure disentangled states\nof the atom and field mode were determined at half the revival time of the population inversion.\nGeneralizations beginning with the classic works of Bennette, et al [7 ], Hill and Wootters [8 , 9] shifted\nattention from the qualitative features of state purity or entang lement dynamics to a practical interpretation\nofentanglementofabipartitesystemasaquantifiablequantumres ource. Entanglementofformationofapure\nstate bipartite system wasintroduced asa quantifierdefined asth e vonNewmman entropydefined in terms of\n1the eigenvalues of the reduced density operator of either of the t wo subsystems or the binary entropy function\ndefined in terms of a quantifier identified as the concurrence of the bipartite system. Hill and Wootters [8\n, 9] provided an exact formula for the entanglement of formation b y introducing concurrence as the basic\nquantifier of the entanglement of pure or mixed states of bipartite systems. The Hill-Wootters definition of\nconcurrence,basedonaspin-flipsymmetryoperation,wasfurth erdevelopedingeneralformsbyUhlmann[10]\nthrough an arbitrary state conjugation symmetry operation and Rungta, et al [11] through a universal state\ninversion symmetry operation, while Albeverio and Fei [12] derived a g eneral form of concurrence based on\ninvariant traces of moments of the reduced density operator of a subsystem. In generalizations to distributed\nentanglement in multipartite systems, Coffman, Kundu and Wootter s [13] introduced tangle as the natural\nquantifier, being the square of concurrence for a pure state bipa rtite system. In [14], Wootters presented\nan excellent review of the definition of entanglement of formation an d the derivation of exact formulas of\nconcurrence based on the state conjugation symmetry operatio ns introduced in [8-11]. Concurrence has\ntaken a central position in studies of entanglement of multipartite s ystems and current research interest is\nnow focussed on lower bounds and monogamy of concurrence of mix ed state multipartite systems [15-19].\nIn the midst of the current intensive investigations into the detailed features of concurrence, a recent\ninformation-theoretic approach based on the Wigner-Yanase ske w information yielded yet another quantifier\nof entanglement, identified as the atomic nonclassicality quantifier [2 0]. The atomic nonclassicality quantifier\nhas been used to study the evolution of the atomic state in the JC mo del [21], thereby revealing the develop-\nment of a pure disentangled state in the middle of the collapse region o f the spin state population inversion\ndiscovered earlier in [1 , 4 , 6]. In the present article, we have establis hed that this nonclassicality quantifier\nis one of a pair of quantifiers of entanglement which can be interpret ed as a redefinition of the concurrence\nof the atomic state.\nIn this article, we present a simplified approach to an effective theor etical study of the characteristic\nfeatures of atomic state purity or entanglement dynamics in the JC , aJC and two-qubit interaction models.\nThe underlying principle is that in an interacting bipartite system, the dynamics of a subsystem is effectively\ndescribed by the reduced density operator of the subsystem. He nce, in contrast to the well established\ngeneral approaches based on the bipartite system state conjug ation symmetry operations developed in [8-11],\nwe apply an interpretation that the dynamical evolution of the stat e purity or entanglement of the atom\nas a subsystem is fully described by a state purity operator defined as the square of the reduced density\noperator of the atom. Denoting the reduced density operator by ρ, applying the standard property that ρis\nhermitian ( ρ†=ρ) and positive semi-definite ( ρ≥0) provides the state purity operator as ρ2. If the atom\nis in a normalized pure state, then the state purity operator satisfi es the basic property ρ2=ρ. In general,\nthe state purity operator is governed by an operator inequality ρ2≤ρ, where the equality applies to a pure\nstate, while for a mixed state, ρ2< ρ. Indeed, using the reduced density operator, it is established belo w\nthat the state purity operator of the atom in the JC and aJC models satisfies the expected mixed state\ninequality. The state purity operator ρ2is composed of a completely pure state part and a completely mixed\nstate part. We interpret the coefficient of the completely mixed sta te part as the mixed state measure which\nquantifiesthe purityorentanglementofthe atomic state. The mixe dstate measuredirectly definesthe square\nof the concurrence of the bipartite system, while expressing it in va rious equivalent forms provides all the\ncharacteristic elements of state purity or entanglement, namely, the reduced density operator eigenvalues,\nnonclassicality quantifiers and the complex amplitude of the state pu rity measure. The argument of the\ncomplex amplitude in polar form is the phase of the state purity measu re, which defines the degree of purity\nof the state of the atom. A complementarity relation unifying the de gree of purity and concurrence as\ncomplementary quantifiers of state purity or entanglement easily e merges from the form of the mixed state\nmeasure.\nWe begin with a brief description of the JC and aJC dynamics in section 2 , where we derive the general\nform of the reduced density operator of the atom in each model. In section 3, we derive the state purity\noperatoranddetermine the mixed statemeasure, which providesa llthe characteristicelementsofstatepurity\nor entanglement ; we derive a direct relation between the mixed stat e measure and concurrence, then derive\nthe degree of purity of the atomic state and the complementarity r elation between the degree of purity and\nconcurrence. We provide plots of the degree of purity, concurre nce and the atomic spin excitation number\nto demonstrate the dynamical evolution of the state of the atom o ver time in the JC and aJC models.\nThe Conclusion is presented in section 4. Eigenvectors and eigenvalu es of the reduced density operator are\ncalculated in the Appendix.\n22 Atomic state in the JC and aJC models\nThe JC and aJC models are the rotating and antirotating component s of the quantum Rabi model of a two-\nlevel atom interacting with a single mode of quantized electromagnet ic field. The JC and aJC Hamiltonians\nHJC,HaJCexpressed in terms of the respective conserved excitation numbe r and qubit state transition\noperators ( ˆN ,ˆR) , (ˆN ,ˆR) take the form\nHJC= ¯hωˆN+¯hˆR;ˆN= ˆa†ˆa+s+s−;ˆR=δsz+g(ˆas++ˆa†s−) ;δ=ω0−ω (1)\nHaJC= ¯hωˆN+¯hˆR;ˆN= ˆaˆa†+s−s+;ˆR=δsz+g(ˆas−+ˆa†s+) ;δ=ω0+ω(2)\nwheresz,s−,s+,σx=s−+s+,ω0and ˆa,ˆa†,ωare the respective atomic spin-1\n2and field mode state transition\noperators and angular frequencies in standard definition, while δ,δare the red-sideband and blue-sideband\nfrequency detunings. Noting the commutation relations [ ˆN ,ˆR] = 0 , [ˆN ,ˆR] = 0, the time evolution\noperatorsUJC(t) ,UaJC(t) generated by the JC and aJC Hamiltonians HJC,HaJCare expressed in the\nappropriate factorized form\nUJC(t) =e−iˆRte−iωˆNt;UaJC(t) =e−iˆRte−iωˆNt(3)\nConsidering the atom initially in the ground state |g/an}bracketri}htand the field mode initially in a coherent state |α/an}bracketri}ht,\nthe composite atom-field initial state |ψgα/an}bracketri}htand the general time evolving states in the JC , aJC models are\nobtained as\n|ψgα/an}bracketri}ht=∞/summationdisplay\nn=0e−1\n2α2αn\n√\nn!|gn/an}bracketri}ht\n|Ψgα(t)/an}bracketri}ht=∞/summationdisplay\nn=0e−1\n2α2αn\n√\nn!UJC(t)|gn/an}bracketri}ht;|Ψgα(t)/an}bracketri}ht=∞/summationdisplay\nn=0e−1\n2α2αn\n√\nn!UaJC(t)|gn/an}bracketri}ht (4)\nafter expressing |α/an}bracketri}htas a superposition of the field mode number (Fock) states |n/an}bracketri}ht. In this work, we take the\ncoherent state eigenvalue αconstant and real ( α∗=α).\nThe JC , aJC excitation number and qubit state transition operator s (ˆN ,ˆR) , (ˆN ,ˆR) acting on |gn/an}bracketri}ht\ngenerate the respective basic qubit states {|gn/an}bracketri}ht,|φgn/an}bracketri}ht},{|gn/an}bracketri}ht,|φgn/an}bracketri}ht}satisfying transition operations\nˆN|gn/an}bracketri}ht=n|gn/an}bracketri}ht;ˆR|gn/an}bracketri}ht=Rn|φgn/an}bracketri}ht;ˆR|φgn/an}bracketri}ht=Rn|gn/an}bracketri}ht;|φgn/an}bracketri}ht=−cn|gn/an}bracketri}ht+sn|en−1/an}bracketri}ht\nRn=g/radicalbigg\nn+1\n4β2;cn=δ\n2Rn;sn=g√n\nRn;β=δ\ng(5)\nˆN|gn/an}bracketri}ht= (n+1)|gn/an}bracketri}ht;ˆR|gn/an}bracketri}ht=Rn+1|φgn/an}bracketri}ht;ˆR|φgn/an}bracketri}ht=Rn+1|gn/an}bracketri}ht;|φgn/an}bracketri}ht=−cn+1|gn/an}bracketri}ht+sn+1|en+1/an}bracketri}ht\nRn+1=g/radicalbigg\nn+1+1\n4(β+2f)2;cn+1=δ\n2Rn+1;sn+1=g√n+1\nRn+1;f=ω\ng(6)\nIn defining the respective Rabi frequencies Rn,Rn+1of qubit oscillations in equations (5) , (6), we have\nconveniently redefined the red-sideband and blue-sideband frequ ency detunings δ,δ=δ+2ωto introduce\nthe dimensionless coupling parameters β,fas defined in equations (5) , (6), noting the general property\nω/ne}ationslash= 0 ,g/ne}ationslash= 0 ,f >0.\nSubstituting UJC(t) ,UaJC(t) from equation (3) into equation (4) as appropriate and applying th e qubit\nstate transition operations from equations (5) , (6), the respec tive JC , aJC general time evolving state\nvectors|Ψgα(t)/an}bracketri}ht,|Ψgα(t)/an}bracketri}htare obtained and reorganized (Schmidt decomposition) in the form\n|Ψgα(t)/an}bracketri}ht=∞/summationdisplay\nn=0(An(t)ξn(t)|g/an}bracketri}ht−i An+1(t)ηn+1(t)|e/an}bracketri}ht)|n/an}bracketri}ht;ξn(t) = cos(Rnt)+icnsin(Rnt)\nηn+1(t) =sn+1sin(Rn+1t) ;An(t) =/radicalbig\nPne−iωnt;An+1(t) =/radicalbig\nPn+1e−iω(n+1)t(7)\n3|Ψgα(t)/an}bracketri}ht=∞/summationdisplay\nn=0(An+1(t)ξn+1(t)|g/an}bracketri}ht−iAn(t)ηn(t)|e/an}bracketri}ht)|n/an}bracketri}ht;ξn+1(t) = cos(Rn+1t)+icn+1sin(Rn+1t)\nηn(t) =snsin(Rnt) ;An+1(t) =/radicalbig\nPne−iω(n+1)t;An(t) =/radicalbig\nPn−1e−iωnt(8)\nThe parameters and Rabi frequencies ( cn, sn+1, Rn+j) , (cn+1,sn,Rn+j),j= 0,1, are defined according\nto equations (5) , (6), while the photon number state probabilities Pn+jin the coherent state are defined by\nPn+j=e−1\n2α2αn+j\n/radicalbig\n(n+j)!, j= 0,1,−1 (9)\nNote that for completeness, the general solutions include the time evolving global phase factors e−iωnt,\ne−iω(n+1)twhich appear in the respective probability amplitudes ( An(t), An+1(t) ) , (An(t),An+1(t) ).\nThe reduced density operators ρagα(t) ,ρagα(t) of the atom in the JC , aJC dynamics are obtained by\ntracing out the field mode states ( Trf) from the respective bipartite atom-field density operators ρgα(t) ,\nρgα(t) as\nρgα(t) =|Ψgα(t)/an}bracketri}ht/an}bracketle{tΨgα(t)|;ρagα(t) =Trfρgα(t)\nρagα(t) =|Ψgα(t)/an}bracketri}ht/an}bracketle{tΨgα(t)|;ρagα(t) =Trfρgα(t) (10)\nSubstituting |Ψgα(t)/an}bracketri}ht,|Ψgα(t)/an}bracketri}htfrom equations (7) , (8) and reorganizing using symmetrization rela tion\n2(aA+bB) = (a+b)(A+B)+(a−b)(A−B), provides the reduced density operators in the general form\nρ=1\n2(r0σ0+r·/vector σ) ;R= (r0, r1, r2, r3) ; Σ = ( σ0, σ1, σ2, σ3) ;σ0=I(11)\nwhereR= (r0,r) , Σ = (σ0, /vector σ) are the Bloch radius and Pauli spin matrix four-vectors, noting th atσ0=I\nis the 2×2 identity matrix. In the reorganization of the reduced atomic stat e density operators in the above\nform, the Pauli matrices have been defined in terms of the basic ato mic spin states |g/an}bracketri}ht,|e/an}bracketri}htas usual\nσ0=|e/an}bracketri}ht/an}bracketle{te|+|g/an}bracketri}ht/an}bracketle{tg|;σ3=|e/an}bracketri}ht/an}bracketle{te|−|g/an}bracketri}ht/an}bracketle{tg|;σ1=|e/an}bracketri}ht/an}bracketle{tg|+|g/an}bracketri}ht/an}bracketle{te|;σ2=−i(|e/an}bracketri}ht/an}bracketle{tg|−|g/an}bracketri}ht/an}bracketle{te|) (12)\nIn the JC dynamics described by the state vector |Ψgα(t)/an}bracketri}htin equation (7), the components of the Block\nradius four-vector have been obtained in the general form\nr0=∞/summationdisplay\nn=0|An+1|2|ηn+1|2+∞/summationdisplay\nn=0|An|2|ξn|2;r3=∞/summationdisplay\nn=0|An+1|2|ηn+1|2−∞/summationdisplay\nn=0|An|2|ξn|2\nr1=−i∞/summationdisplay\nn=0(An+1A∗\nnηn+1ξ∗\nn−A∗\nn+1Anη∗\nn+1ξn) ;r2=∞/summationdisplay\nn=0(An+1A∗\nnηn+1ξ∗\nn+A∗\nn+1Anη∗\nn+1ξn) (13)\nIn the aJC dynamics described by the state vector |Ψgα(t)/an}bracketri}htin equation (8), the components of the Bloch\nradius four-vector have been obtained in the general form\nr0=∞/summationdisplay\nn=0|An|2|ηn|2+∞/summationdisplay\nn=0|An+1|2|ξn+1|2;r3=∞/summationdisplay\nn=0|An|2|ηn|2−∞/summationdisplay\nn=0|An+1|2|ξn+1|2\nr1=−i∞/summationdisplay\nn=0(AnA∗\nn+1ηnξ∗\nn+1−A∗\nnAn+1η∗\nnξn+1);r2=∞/summationdisplay\nn=0(AnA∗\nn+1ηnξ∗\nn+1+A∗\nnAn+1η∗\nnξn+1) (14)\nWe observe that, in general, the Bloch radius four-vector compon entsrj,j= 0,1,2,3 are obtained as\nthe mean values of the corresponding Pauli matrices σjwith respect to the reduced density operators\nρ=ρagα(t),ρagα(t) of the atom in the respective JC , aJC general time evolving states (|Ψ(t)/an}bracketri}ht=\n|Ψgα(t)/an}bracketri}ht,|Ψgα(t)/an}bracketri}ht) according to\nrj=Trσjρ(t) =/an}bracketle{tΨ(t)σj|Ψ(t)/an}bracketri}ht;j= 0,1,2,3 ;⇒Trσ0ρ=Trρ=/an}bracketle{tΨ(t)|Ψ(t)/an}bracketri}ht=r0(15)\n43 Atomic state purity operator and mixed state measure\nAs presented in section 2 above, the JC and aJC models are pure sta te bipartite systems in which the atom\nand field mode subsystems are left in mixed states described by the r espective reduced density operators\nobtained in general form in equation (11). In this section, we provid e a simple consolidated derivation of\nthe state purity operator to determine the mixed state measure a nd the associated entanglement quantifiers\nof the atom in the JC and aJC models. The state purity operator is de fined as the square of the reduced\ndensity operator of the atom.\nSquaringρin equation (11), applying standard algebraic identity ( a·/vector σ)(b·/vector σ) = (a·b)σ0+i/vector σ·(a×b)\ngiving (r·/vector σ)(r·/vector σ) =r2σ0and eliminating r·/vector σfrom the result using equation (11), we obtain the atomic\nstate purity operator ρ2in the form\nρ2=r0ρ−Mσ0;M=1\n4(r2\n0−r2) ;r=|r|=/radicalBig\nr2\n1+r2\n2+r2\n3 (16)\nThis is an important relation, which reveals that the atomic state pur ity operator ρ2is composed of a\ncompletely pure state part ρand a completely mixed state part Mσ0, where we interpret the coefficient M\nas the mixed state measure.\nTreating the completely mixed state part in equation (4) as the comp letely mixed state density operator\nρmixprovides the definition of the mixed state measure Mas in equation (16) through the trace in the form\n(notingTrσ0=TrI= 2)\nρmix=Mσ0;M=1\n2Trρmix (17)\nExpressing the reduced density operator ρin equation (11) in the matrix form provides the general definition\nof the mixed state measure as the determinant according to\nρ=1\n2/parenleftbigg\nr0+r3r1−ir2\nr1+ir2r0−r3/parenrightbigg\n;M= detρ (18)\nWe obtain the state purity measure Pdefined as the trace of the purity operator ρ2in equation (16) in the\nform\nP=Trρ2:P=r2\n0−2M (19)\nwhere we have used Trρ=r0from equation (15). The mixed state measure Mthus defines the state purity\nmeasure and characterizes the dynamical evolution of the atomic s tate purity. The value M= 0 at|r|=r0,\ngivingP=r2\n0, characterizes a pure state, while M=1\n4r2\n0at|r|= 0, giving P=1\n2r2\n0, characterizes a\nmaximally entangled state, where we note that the normalized state caser0= 1 gives standard values of the\npurity measure ( r0= 1,P= 1,1\n2) for pure or maximally entangled states.\nIn [1], Gea-Banacloche used the time evolving state purity measure P=Trρ2to study the dynamical\nevolution of the atomic state between pure and maximally entangled s tates in the normalized ( r0= 1)\nbipartitestateofthe JCmodel. Evolutionofthe puritymeasurerev ealedapuredisentangledstatedeveloping\nin the middle of the collapse region of the atomic spin state population in version. This interesting dynamical\nfeature is demonstrated through the evolution of the degree of p urity, concurrence and spin state excitation\nnumber in both JC and aJC models in Fig.1 , Fig.2 below.\nHaving established that Messentially defines the state purity measure, we now determine its u nderlying\nphysical meaning by demonstrating how it is related to all the other c haracteristic elements of state purity,\nincludingconcurrence,tangleandthedegreeofpurity,whicharein terpretedasquantifiersoftheentanglement\nof formation of the bipartite system.\n3.1 Characteristic elements of state purity measure\nFactoring the bracket in the definition in equation (16), we express the mixed state measure in the form\nM=ε−ε+;ε−=1\n2(r0−|r|) ;ε+=1\n2(r0+|r|) (20)\nwhereε−,ε+are the eigenvalues of the reduced density operator ρas determined in the Appendix.\n5Next, adding and subtracting r2\n0inside the bracket in equation (16) and reorganizing as appropriate gives\nalternative equivalent relations\nM=1\n4r2\n0−λ−λ+;λ−=1\n2/parenleftbigg\nr0−/radicalBig\nr2\n0−r2/parenrightbigg\n;λ+=1\n2/parenleftbigg\nr0+/radicalBig\nr2\n0−r2/parenrightbigg\n(21)\nM=1\n2(r2\n0−|ε|2) ;ε=1√\n2(r0+i|r|)⇒ε=√\nPeiϕ; tanϕ=|r|\nr0(22)\nwhereP=1\n2(r2\n0+r2) in equation (22) is the state purity measure obtained earlier in equa tion (19).\nIt follows from equations (20) , (21) , (22) that the mixed state me asureMis a composite of the\ncharacteristic elements ε∓,λ∓,ε(ϕ= tan−1|r|\nr0), which have been variously identified as basic quantifiers\nof entanglement or nonclassicality of the atomic state in the normaliz ed state (r0= 1) JC model [4-6 , 20 ,\n21].\nWe observe that, in [4-6], P L Knight and collaborators used the eigen valuesε∓obtained here in the\ngeneral form in equation (20) to define the von Newmann entropy\nS=−ε−log2ε−−ε+log2ε+ (23)\nto study the evolution of the atomic state purity in the JC model. The evolution of the von Newmann\nentropy revealed precisely the same dynamical features determin ed through the purity measure in the earlier\nstudies of Gea-Banacloche [1-3]. Noting that the evolution of the vo n Newmann entropy essentially follows\nthe form of evolution of ε−over the same time ranges, Phoenix and Knight [1] interpreted the e igenvalueε−\nas the basic measure of atomic state purity in the JC model.\nOn the other hand, we identify the characteristic element λ−obtained here in general form in equation\n(21) as the atomic state nonclassicality quantifier N(ρ) recently derived through the Wigner-Yanase skew\ninformation in [20] and applied in studying atomic nonclassicality in the JC model in [21]. In the normalized\nstater0= 1, the nonclassicality quantifier N(ρ) was determined [20 , 21] in the form\nr0= 1 :λ−=1\n2/parenleftBig\n1−/radicalbig\n1−r2/parenrightBig\n;N(ρ) =λ− (24)\nBelow, we establish that the characteristic elements λ∓, now identified with the nonclassicality quantifier\nN(ρ) =λ∓, may be interpreted as redefinitions of the concurrence of the bip artite system (equation (27)).\nIn contrast, the complex characteristic element εintroduced here in equation (22), may be interpreted\nas the state purity complex amplitude and its argument ϕis the phase of the state purity measure. The\nstate purity complex amplitude has never been determined in earlier s tudies. As defined in equation (22),\nthe argument defined by r0tanϕ=|r|, interpreted here as the phase of the state purity measure, dire ctly\ndescribes the evolution of the Bloch sphere, characterizing the dis tribution of the atomic states within or on\nthe surface of the Bloch sphere. The evolution of the state purity phase satisfies the inequality 0 ≤tanϕ≤1,\ngiving 0 ≤ϕ≤1\n4π, which follows from the property that the length |r|of the 3-component Bloch radius\nvectorr= (r1, r2, r3) takes a range of values 0 ≤ |r| ≤r0. Note that, in general, the phase ϕcan take a\nspectrum of values ϕ=qπ(q= 0,1,2,3,...) satisfying tan ϕ= 0 which characterize a maximally entangled\nstate orϕ= (2q+1)1\n4πsatisfying tan ϕ= 1 which characterize a pure state.\n3.2 Concurrence and tangle\nTaking the trace of the state purity operator ρ2in equation (16) once again and reorganizing as appropriate,\nwe obtain an important relation\n2M=r2\n0−Trρ2⇒4M=C2;C=/radicalBig\n2(r2\n0−Trρ2) (25)\nafter introducing the concurrence Cof the bipartite system [11 , 14]. Concurrence was introduced as a\nquantifier of the entanglement of formation of a pure state bipart ite qubit system by Hill and Wootters in\n[8 , 9] and determined in the explicit form in equation (25) by Rungta, e t al in [11], noting r0=|Ψ(t)|Ψ(t)/an}bracketri}ht\ntakes value r0= 1 for normalized bipartite state.\nEquations (18) , (25) provide the definition of concurrence in term s of the mixed state measure Mand\nthe determinant of the reduced density operatot ρin the form\nC= 2√\nM ⇒ C = 2/radicalbig\ndetρ (26)\n6According to equations (16) , (26), the vanishing of Mat|r|=r0meansCvanishes, leaving the system in a\npure disentangled state, while the evolution of Mto1\n4r2\n0at|r|= 0, yielding C=r2\n0, leaves the system in a\nmaximally entangled state.\nUsing (r2\n0−r2) = 4M=C2in equation (21) reveals that the nonclassicality quantifiers λ∓can be\ninterpreted as redefinitions of concurrence according\nλ∓=1\n2(r0∓C) (27)\nwhich means that the nonclassicality quantifier N(ρ) =λ−(r0= 1) determined in [20 , 21] is essentially the\nconcurrence of the pure state bipartite system.\nIn [13], Coffman, Kundu and Wootters (CKW) introduced tangle τ, defined as the square of concurrence,\nas the appropriate quantifier of the entanglement of formation of a bipartite system. Equations (25) , (26)\nprovide the tangle of the pure state bipartite system in the precise form obtained by Coffmann, Kundu and\nWootters as\nτ=C2:τ= 4M ⇒τ= 4detρ (28)\nIt follows from equations (19) , (26) , (28) that, the state purity measure Pand the established quantifiers\nof the entanglement of formation of a bipartite system such as con currence C, tangleτ=C2, are defined in\nterms of the mixed state measure M, which according to equations (20) , (21) , (22), is a composite of t he\ncharacteristic elements of the state purity measure. In addition, equation (22) reveals another property that\nthe state purity measure P, defined by Min equation (19), has a hidden phase ϕ= tan−1|r|\nr0, which directly\ndescribes the evolution of the Bloch sphere through the states of the atom. The state purity phase may thus\nbe a more fundamental quantifier of state purity or entanglement , as we demonstrate through the degree of\npurity derivable from M.\n3.3 Degree of state purity and complementarity relation wit h concurrence\nReorganizing the definition of the mixed state measure Min equation (16) and introducing the definition of\nϕfrom equation (22), we obtain the relation\nM=1\n4r2\n0(1−tan2ϕ) ; 0 ≤tanϕ≤1⇒0≤ϕ≤1\n4π (29)\nwhich reveals that the mixed state measure Mis determined by phase ϕof the state purity measure. The\nrelations in equations (26) , (28) mean that concurrence Cand tangle τ=C2are also determined by the\nphaseϕ.\nIntroducing r0=Trρ,M= detρfrom equation (18) in equation (29), we obtain the definition of the\nphase of the state purity measure in the form\ntanϕ=/radicalBigg\n1−4detρ\n(Trρ)2(30)\nwhichtakespreciselythesameformasthedegreeofpolarizationof light, where ρcorrespondstothecoherence\nmatrix [22 , 23]. In a correspondinginterpretation, we now identify t anϕas the degree of purity of the atomic\nstate.\nSubstituting the concurrence Cfrom equation (26) into equation (29) or (30) provides an importan t\nproperty that the degree of purity tan ϕand concurrence are complementary quantifiers of the purity or\nentanglementoftheatomicstatesatisfyingthecomplementarityr elationinthenormalizedstate Trρ=r0= 1\ntan2ϕ+C2= 1 (31)\nIt follows easily from the range of values 0 ≤tanϕ≤1 in equation (29) and the complementarity relation in\nequation (31) that both degree of purity and concurrence take c omplementary values in the range {0,1},\ne.g., (tanϕ= 1,C= 0) , (tanϕ= 0,C= 1). The complementarity propertymeans that the degreeofpur ity\nand concurrence evolve in reverse order, one increasing and the o ther decreasing with time as demonstrated\nin Fig.1 , Fig.2 in the JC and aJC dynamics below.\n7Using 1−C2= tan2ϕfrom the complementarity relation in equation (31), the binary entr opy function\nH(1\n2(1±√\n1−C2)) introduced in [7 , 8 , 9] as the entanglement of formation of the bip artite system is\nobtained in terms of the degree of purity in the form\nH=−1\n2(1+tanϕ) log21\n2(1+tanϕ)−1\n2(1−tanϕ) log21\n2(1−tanϕ) (32)\nThe derivation of the degree of purity, the complementarity relatio n with concurrence and the associated\nbinary entropy function completes the specification of essentially a ll the characteristic quantifiers of the\natomic state purity or entanglement provided within the definition of the mixed state measure M. We may\ntherefore interpret the mixed state measure as the natural univ ersal quantifier of the atomic (subsystem)\nstate purity or entanglement of formation of the bipartite atom-fi eld system.\nIn the present article, we interpret the degree of purity tan ϕand concurrence C, both varying within the\nrangeofvalues {0,1}governedbythe complementarityrelationinequation(31), asthe b asiccomplementary\nquantifiersofthepurityorentanglementoftheatomicstate. Wep resenttheinterestingcharacteristicfeatures\nof the degree of purity and concurrence of the atomic state in the JC and aJC models in Fig.1–Fig.4 below.\n3.3.1 Evolution of the degree of purity and concurrence in th e JC and aJC models\nAccording to the effective definition of the degree of purity tan ϕin equation (22), exactly equal to the form\nin equation (30) using equation (18), we only need to determine the e xplicit forms of the components of\nthe Bloch radius four-vector obtained in equations (13) , (14) to d emonstrate the evolution of the degree of\npurity and concurrence in the JC , aJC models.\nThe JC Bloch radius four-vector components are obtained in explicit form by substituting the definitions\ngiven in equations (7) , (9) into equation (13), giving\nr0= 1 ;r3=∞/summationdisplay\nn=0Pn+1s2\nn+1sin2(Rn+1t)−∞/summationdisplay\nn=0Pn(cos2(Rnt)+c2\nnsin2(Rnt))\nr1=−2∞/summationdisplay\nn=0/radicalbig\nPn+1Pnsn+1sin(Rn+1t)( sin(ωt)cos(Rnt)+cncos(ωt)sin(Rnt) )\nr2= 2∞/summationdisplay\nn=0/radicalbig\nPn+1Pnsn+1sin(Rn+1t)( cos(ωt)cos(Rnt)−cnsin(ωt)sin(Rnt) ) (33)\nThe aJC Bloch radius four-vector components are obtained in explic it form by substituting the definitions\ngiven in equations (8) , (9) into equation (14), giving\nr0= 1 ;r3=∞/summationdisplay\nn=0Pn−1s2\nnsin2(Rnt)−∞/summationdisplay\nn=0Pn(cos2(Rn+1t)+c2\nn+1sin2(Rn+1t))\nr1= 2∞/summationdisplay\nn=0/radicalbig\nPnPn−1snsin(Rnt)( sin(ωt)cos(Rn+1t)−cn+1cos(ωt)sin(Rn+1t) )\nr2= 2∞/summationdisplay\nn=0/radicalbig\nPnPn−1snsin(Rnt)( cos(ωt)cos(Rn+1t)+cn+1sin(ωt)sin(Rn+1t) ) (34)\nwhere the evaluation r0= 1 in each case establishes the property that the respective JC , a JC state vectors\n|Ψgα(t)/an}bracketri}ht,|Ψgα(t)/an}bracketri}htare normalized according to equation (15). The important feature which emerges here is\nthat the coherence components r1,r2are modulated by the periodically time varying field mode frequency-\ndependent factors sin( ωt) , cos(ωt) from the respective free evolution global phase factors e−iωnt,e−iω(n+1)t\nof the general solutions in equations (7) , (8).\nThe normalization property r0= 1 now means that the degree of purity takes the simple form tan ϕ=\n|r|=/radicalbig\nr2\n1+r2\n2+r2\n3, which directly describes the evolution of the Bloch sphere between the pure states at\n|r|= 1 and the maximally entangled (totally mixed) states at |r|= 0. Using r1,r2,r3from equations\n(33) , (34) provides the explicit form of the degree of purity, which we substitute into the complementarity\nrelation in equation (31) to determine the corresponding concurre nce in the JC , aJC models. Choosing\n8the coherent field mode eigenvalue α= 7 (mean photon number n=α2= 49) to compare with Gea-\nBanacloche’s original studies [1], we have plotted the degree of purit y tanϕ(blue color), concurrence C(red\ncolor) and spin excitation number1\n2(1 +r3) (green color) at resonance β= 0 and off-resonance choosing\nβ= 60 in Fig.1-Fig.4 in the JC , aJC models. In general, the plots over scale d timeτ=gtreveal that the\nmodulation of the coherence components r1,r2by the global phase factors cos ωt, sinωtdoes not affect the\ndynamical evolution, in agreement with the standard assumption th at only the interaction component of the\nHamiltonian generates measurable dynamics.\nUnder resonance β= 0, Fig.1 , Fig.2 in the JC , aJC models show that the degree of purity (b lue color)\nfalls rapidly from the initial pure state at tan ϕ= 1 to a momentary totally mixed (maximally entangled)\nstate at tan ϕ≈0, then rises gently to a very closely pure disentangled state at tan ϕ≈1 in the middle of the\ncollapse region of the spin excitation number (green color). On the o ther hand, the concurrence (red color)\nrisesrapidly from the initial pure state at C= 0 to a momentarytotally mixed (maximally entangled) state at\nC ≈1, then falls gently to a very closely pure disentangled state at C ≈0 in the middle of the collapse region,\nprecisely at about half the collapse time of the spin excitation number where the corresponding maximum\nvalue of the degree of purity occurs. The evolution in opposite sens e of the two complementary quantifiers\nof state purity or entanglement simultaneously reveal that, within a very short time after the interaction\nbegins, the atomic state collapses rapidly to a momentary maximally en tangled state then evolves gently to\nan approximately pure disentangled state after about half the colla pse time of the spin excitation number.\nThis interesting feature of the dynamical evolution of the atomic st ate, exhibited here by the degree of purity\nand concurrence, agreesprecisely with the originalresults of Gea -Banaclocheexhibited through the evolution\nof the state purity measure Trρ2[1-3] and the results of P L Knight and collaborators exhibited throu gh\nthe evolution of the eigenvalue ε−and von Newmann entropy S[4-6] in the JC model. Approximate forms\nof the disentangled pure state emerging at half the collapse time of t he spin state population inversion were\ndetermined in these original studies.\n0 10 20 30 40 50t0.20.40.60.81.0JC/MinusSTATEQUANTIFIERS\nFigure 1: Resonance evolution of atomic state degree of purity Tan ϕ(BLUE), concurrence C) (RED) and\nexcitation number (GREEN) in the JC model at α= 7 ;β= 0 ;f= 10−7over scaled time τ=gt\n0 10 20 30 40 50t0.20.40.60.81.0aJC/MinusSTATEQUANTIFIERS\nFigure 2: Resonance evolution of atomic state degree of purity Tan ϕ(BLUE), concurrence C) (RED) and\nexcitation number (GREEN) in the aJC model at α= 7 ;β= 0 ;f= 10−7over scaled time τ=gt\nWe have established that as the red-sideband detuning parameter βis increased to large values, off-\nresonance dynamics in both JC and aJC models is characterized by pe riodic evolution of the degree of purity\n(blue color) with peaks (maximum values) at the upper value tan ϕ= 1, while the concurrence (red color)\nand the spin excitation number (green blobs on the zero-axis) each evolves periodically with deeps (minimum\nvalues) at the lower value C= 0 ,1\n2(1+r3) = 0 as shown in Fig.3 , Fig.4 for parameter values α= 7 ,β= 60\n,f= 10−7. The periodicity is perfect such that the red deeps at the minimum va lues of the concurrence\n9touch the corresponding green blobs of the spin excitation number on the zero-axis. The periodic evolution\nof the degree of purity, concurrence and the spin excitation numb er around their correspondingupper / lower\nvalues signifies the property that increasing the detuning paramet erβtowards larger values progressively\ndrives the dynamical evolution of the atom to a disentangled pure st ate. We find at very large values β≥175\n(withα2= 49), the atom is in a perfectly pure state specified by tan ϕ= 1 ,C ≈0 ,1\n2(1+r3) = 0. These\nfeatures of the dynamical evolution to a pure disentangled state a t large values of the detuning parameter\nβagree precisely with the results of earlier studies of the JC model in [5 ]. The physical interpretation of\nthis important dynamical feature follows from the definition of the d imensionless parameters β=δ\ng,f=ω\ng,\nwhich specify the atom-field coupling regimes in the JC , aJC interactio ns. Increasing the parameters β,fto\nvery large values essentially drives the system to very weak-couplin g regimes where the atom-field interaction\nis too weak to generate state transitions, leaving the bipartite sys tem in a disentangled pure state.\n0 100 200 300 400 500 600 700t0.20.40.60.81.0JC/MinusSTATEQUANTIFIERS\nFigure 3: Off-resonance evolution of atomic state degree of purity Tanϕ(BLUE), concurrence C(RED) and\nexcitation number (GREEN) in the JC model at α= 7 ;β= 60 ;f= 10−7over scaled time τ=gt\n0100 200 300 400 500 600 700t0.20.40.60.81.0aJC/MinusSTATEQUANTIFIERS\nFigure 4: Off-resonance evolution of atomic state degree of purity Tanϕ(BLUE), concurrence C(RED) and\nexcitation number (GREEN) in the aJC model at α= 7 ;β= 60 ;f= 10−7over scaled time τ=gt\nEven as we have described important features of the evolution of t he atomic state in both JC and aJC\nmodels by varying only the dimensionless red-sideband detuning para meterβ, which specifically arises in the\nJC interaction in equation (1), we must recall an underlying dynamica l property that the aJC interaction\nis specified by the blue-sideband frequency detuning δ=ω0+ωas defined in equation (2). Hence, in\nspecifying the general resonance and off-resonance conditions a pplicable to both JC and aJC dynamics,\nwe have redefined the aJC blue-sideband frequency detuning δin terms of the JC red-sideband frequency\ndetuningδ=ω0−ωin equation (1) according to\nδ=δ+2ω;δ=βg;ω=fg⇒δ= (β+2f)g (35)\nwhere the dimensionless parameters β,fare defined in equations (5) , (6). It follows that the JC dynamics\nis characterized only by the parameter β, while the aJC dynamics is characterized by both βandf. Hence,\nat resonance, the field-atom frequency ω=ω0is effectively eliminated from the Rabi frequency Rn+jin the\nJC dynamics in equation (7), but remains as a residual detuning δr= 2fg,f >0 in the Rabi frequency\nRn+jin the aJC dynamics in equation (8). In the plots in Fig.1-Fig.4, we have s etf= 10−7to minimize its\ncontribution in the aJC relative to the JC dynamics. However, if we va ryf >0 to large values, then even\nat resonance β= 0, the atomic state evolution in the aJC can be driven to a disentang led pure state faster\nthan in the JC, which will strictly follow the form of resonance evolutio n in Fig.1. In general, increasing\nboth parameters β,fdrives the atom to the pure disentangled state faster in the aJC co mpared to the JC\nevolution.\n10Apart from the property that the aJC process is faster than the corresponding JC process, it has emerged\nin the corresponding plots in Fig.1 , Fig.2, similarly, in Fig.3 , Fig.4, that the dynamical evolution of the\natomic state takes the same form in the JC and aJC models. The degr ee of purity and the concurrence, which\nquantify state purity or entanglement, evolve in the same form in th e two models. We observe that in the\nJC interaction the two-level atom couples to the rotating component , while in the aJC interaction the atom\ncouples to the counter-rotating component of the same quantized electromagnetic field. Both interaction\nmechanisms have conserved excitation numbers and are characte rized by transitions between well defined\nbasic qubit states according to equations (1) , (2) , (5) , (6). The main difference between the two processes\nis that the JC interaction generates red-sideband transitions cha racterized by difference frequency δ=ω0−ω\nequivalent to alternate emission or absorption by the atom or field mo de, while the aJC interaction generates\nblue-sideband transitions characterized by sum frequency δ=ω0+ωequivalent to simultaneous emission or\nabsorption by both atom and field mode. The dynamical evolution gen erated thus takes the same form. The\nsimilarity in the forms of dynamical evolution may also be associated wit h the property that the JC and aJC\ninteraction Hamiltonians are duality conjugates, related by a unitar y duality symmetry transformation.\n3.4 Concurrence and the Bloch radius four-vector\nIntroducing the concurrence C= 2√\nMfrom equation (26) in equation (16) gives the important relation\nC2= (r2\n0−r2)⇒ C2= (r2\n0−r·r) ;r=|r| (36)\nwhich we recognize as the four-vector mathematical relation for t he invariant square of the length of the\nBloch radius four-vector R= (r0,r) introduced in equation (11). This leads to the interpretation that\nthe square of concurrence is equal to the square of the length of the Bloch radius four-vector defined in a\nfour-dimensionalspacetime frame. Introducing standard four- vectorindex µ= 0,1,2,3, we express the Bloch\nradius four-vector in the contravariant Rµand covariant Rµforms\nRµ= (r0,r) ;Rµ= (r0,−r) ;r= (r1, r2, r3) ;µ= 0,1,2,3 (37)\nThe squared concurrence C2in equation (36) is then obtained as the invariant squared length of t he Bloch\nradius four-vector in the spacetime covariant mathematical form\nC2=RµRµ=RµRµ⇒ C=|R|=/radicalBig\nr2\n0−r2 (38)\nFor normalized state r0= 1, we obtain\nr0= 1 :Rµ= (1,r) ;R2=RµRµ= 1−r2;C=/radicalbig\n1−r2 (39)\nIt follows that concurrence is fully defined by the Bloch radius four- vector. Introducing the degree of purity\nin the form r2\n0tan2ϕ=r2from equations (22) , (30) into equation (38) providesthe complem entarityrelation\nin equation (31) in the general form\nC2+r2\n0tan2ϕ=r2\n0 (40)\nwhich reduces to the standard form in equation (31) for the norma lized stater0= 1.\n4 Conclusion\nWe haveestablished that in a bipartite system of a two-levelatom int eracting with a single mode ofquantized\nelectromagnetic field in a JC , aJC model or with a spin-1\n2particle in a qubit-qubit model, the purity or\nentanglement of the state of the atom is described by a state purit y operator obtained as the square ( ρ2)\nof the reduced density operator ( ρ) of the atom. The state purity operator ρ2is composed of a completely\npure state part and a completely mixed state part. The mixed state part is characterized by a mixed state\nmeasure, which can be expressed in various equivalent forms to pro vide all the elementary quantifiers of the\npurity or entanglement of the atomic state. The form of the mixed s tate measure automatically yields the\ncomplementarity relation which unifies the degree of purity and conc urrence as complementary quantifiers\nof the state purity or entanglement. In addition, a simple interpret ation that concurrence is fully defined by\nthe Bloch radius four-vector is derivable from the general form of the mixed state measure, such that the\nsquare of concurrence is obtained in covariant form in a four-dimen sional spacetime frame.\n11We have provided a qualitative picture by plotting the degree of purit y, concurrence and the atomic spin\nexcitation number to demonstrate the dynamical evolution of the a tomic state over scaled time in the JC\nand aJC models. In resonance dynamics, the evolution of the degre e of purity and concurrence between\nthe corresponding extremal values in the range {0,1}reveals that, the atomic state collapses rapidly to\na momentary totally mixed or maximally entangled state within a very sh ort time after the interaction\nbegins, then evolves gently to a disentangled pure state in the middle of the collapse region of the spin\nexcitation number, after which it falls gently to the revival point. In off-resonance dynamics, increasing the\ndetuning parameters to very large values drives the degree of pur ity, concurrence and spin excitation number\nto periodic evolution along the respective extremal maximum or minimu m values, signaling the evolution of\nthe atom to a perfectly disentangled pure state where the degree of purity essentially takes the maximum\nvalue 1, while the concurrence and spin excitation number each take s the corresponding minimum value 0.\nBy definition, increasingthe detuning parametersprogressivelydr ives the atom-field system to weak-coupling\nregimes where beyond some critical large value, the interaction is to o weak to generate transitions and the\nsystem remains in a perfectly pure state where the degree of purit y is exactly 1, while the concurrence and\nspin excitation number are exactly 0.\n5 Acknowledgement\nI thank Maseno University for providing facilities and a conducive wor k environment during the preparation\nof the manuscript.\n6 Appendix : Eigenvectors and eigenvalues of the reduced den sity\noperator\nRewritingthereduceddensityoperatorinequation(11)intheform (σ0=I,σx=s++s−,σy=−i(s+−s−))\nρ=1\n2(r0I+ˆR) ;ˆR=r·/vector σ=r3σ3+/radicalBig\nr2\n1+r2\n2(e−iφs++eiφs−) ;φ= tan−1r2\nr1(A1)\nitiseasilyestablishedthatthestatetransitionoperator ˆRactingontheatomicspingroundstate |g/an}bracketri}htgenerates\ncoupled qubit states {|g/an}bracketri}ht,|φg/an}bracketri}ht}satisfying state transition operations\nˆR|g/an}bracketri}ht=|r||φg/an}bracketri}ht;ˆR|φg/an}bracketri}ht=|r||g/an}bracketri}ht;|φg/an}bracketri}ht=−cosθ|g/an}bracketri}ht+sinθ e−iφ|e/an}bracketri}ht; tanθ=/radicalbig\nr2\n1+r2\n2\nr3(A2)\nSimple linear combination of the qubit states give the eigenvectors |ψ±\ng/an}bracketri}htofˆRand the density operator ρ,\nsatisfying eigenvalue equations\n|ψ±\ng/an}bracketri}ht=|g/an}bracketri}ht±|φg/an}bracketri}ht:ˆR|ψ±\ng/an}bracketri}ht=±|r||ψ±\ng/an}bracketri}ht;ρ|ψ±\ng/an}bracketri}ht=ε±|ψ±\ng/an}bracketri}ht (A3)\nwhere the density operator eigenvalues ε±are exactly the characteristic elements of state purity or entan-\nglement obtained in equation (20), which have been used to define th e atomic state von Newmann entropy\nSin equation (23). It is also important to note that the Bloch radius ±|r|are eigenvalues of the qubit state\ntransition operator ˆR. Starting from the atomic spin excited state |e/an}bracketri}htalso provides the same eigenvalues.\nThe eigenvectors |ψ±\ng/an}bracketri}htin equation (A3) are easily normalized as\n|ψ+\ng/an}bracketri}ht= cos1\n2θ|g/an}bracketri}ht+sin1\n2θe−iφ|e/an}bracketri}ht;|ψ−\ng/an}bracketri}ht= sin1\n2θ|g/an}bracketri}ht−cos1\n2θe−iφ|e/an}bracketri}ht (A4)\ntakingessentiallytheform, with phase φdefinedinequation(A1), agreeingwiththeapproximateeigenvecto rs\nand phase determined within the initial collapse period in the JC model in [4], particularly at cos(1\n2θ) =\nsin(1\n2θ) =1√\n2. Notice that the phase φis significantly different from the phase ϕof the state purity measure\nwhich we introduced in equation (22).\n12References\n[1] J Gea-Banacloche 1990 Collapse and revival of the state vector in the Jaynes-Cummings model : an\nexample of state preparation by a quantum apparatus, Phys.Rev.L ett.65, 3385\n[2] J Gea-Banacloche 1991, Phys.Rev. A44, 5913\n[3] J Gea-Banacloche 1992 A new look at the Jaynes-Cummings model for large fields : Bloch sphere\nevolution and detuning effects, Opt. Commun. 88, 531\n[4] SJDPhoenix, PLKnight1991,Establishmentofanentangledatom -fieldstateintheJaynes-Cummings\nmodel, Phys.Rev. A44, 6023\n[5] V Buzek , H Moya-Cessa , P L Knight , S J D Phoenix 1992, Schroedin ger-cat states in the reso-\nnant Jaynes-Cummings model : collapse and revival of oscillations of the photon-number distribution,\nPhys.Rev. 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Comp. 1, 27\n[15] K V Antipin 2020, Lower bounds on concurrence and negativity f rom a trace relation, Mod. Phys.\nLett.A35, 2050254\n[16] H Choi , J S Kim 2015, Negativity and strong monogamy of multi-pa rty quantum entanglement beyond\nqubits, Phys. Rev. A92, 042307\n[17] M F Cornelio 2013, Multipartite monogamy of the concurrence, P hys. Rev. A87, 032330\n[18] Y K Bai , F Y Xu , Z D Wang 2014, General monogamy relation for th e entanglement of formation in\nmultiqubit systems, Phys. Rev. Lett. 113, 100503\n[19] H Li , T Gao , F Yan 2022, Tighter monogamy relations in multipartit e quantum systems, arXiv :\n2205.11972 [quant-ph]\n[20] H Dai , S Luo 2019, Information-theoretic approach to atomic spin nonclassicality, Phys. Rev. A100,\n062114\n[21] H Dai , S Fu , S Luo 2020, Atomic nonclassicality in the Jaynes-Cum mings model, Phys. Lett. A384,\n126371\n[22] E Wolf 2008, Can a light beam be considered to be the sum of a comp letely polarized and a completely\nunpolarized beam ?, Opt. Lett. 33, 642\n[23] E Wolf 2007, Introduction to the theory of coherence of polarization of l ight, Cambridge University Press\n13"    },    {        "title": "2207.04365v1.Semimetallic_spin_density_wave_state_in_iron_pnictides.pdf",        "content": "arXiv:2207.04365v1  [cond-mat.str-el]  10 Jul 2022Semimetallic spin-density wave state in iron pnictides\nGarima Goyal and Dheeraj Kumar Singh∗\nSchool of Physics and Materials Science, Thapar Institute o f Engineering and Technology, Patiala-147004, Punjab, Ind ia\n(Dated: July 12, 2022)\nWe examine the existence of semimetallic spin-density wave states in iron pnictides. In the experimentally\nobserved metallic spin-density wave state, the symmetry-p rotected Dirac cones are located away from the Fermi\nsurface giving rise to tiny pockets and there are also additi onal Fermi pockets such as one around Γ. We find\nthat the location of a pair of Dirac points with respect to the Fermi surface exhibits significant sensitivity to the\norbital splitting between the dxzanddyzorbitals. Besides, in the presence of orbital splitting, th e Fermi pockets\nnot associated with the Dirac cones, can be suppressed so tha t a semimetallic spin-density wave state can be\nrealized. We explain these finding in terms of difference in t he slopes and orbital contents of the bands which\nform the Dirac cone, and obtain the necessary conditions dep endent on these two and other parameters for the\ncoexisting Dirac semimetallic and spin-density wave state s. Additionally, the topologically protected edge states\nare studied in the ribbon geometry when the same are oriented either along xoryaxes.\nI. INTRODUCTION\nIron-based multiband superconductors have attracted con-\nsiderable attention in recent times because of their comple x\nband structure [1, 2] and variety of phases they can exhibit i n-\ncluding unconventional superconductivity, nematic order and\nother novel phases [3, 4]. However, a renewed interest has\nbeen generated largely because of the signatures of topolog -\nical states obtained in this class of superconducting mater i-\nals [5, 6].\nZero-energy Majorana bound states were found to exist\non the surface of superconducting iron chalcogenides as in-\ndicated by the scanning-tunneling microscopy (STM) [7, 8].\nThese zero-energy bound states were attributed to the band\ninversion between the bands dominated by the pnictogen pz\nand iron dxz/dyzorbitals [5, 9, 10]. For similar reasons, it was\nsuggested that the iron-based superconductors (IBS) could ex-\nhibit a band topology similar to that of a topologically insu la-\ntor (TI) or Dirac semimetal (DS), which was substantiated by\nthe laser-based spin- and angle-resolved photoemission sp ec-\ntroscopies (ARPES) [5].\nEvidence of Dirac cone in the four-fold rotational symme-\ntry broken metallic spin-density wave (SDW) state have been\nobtained through ARPES [11] and quantum oscillation mea-\nsurements [12], which was predicated to be gapless [13]. The\nSDW state, with a collinear or striped magnetic order, con-\nsists of chains of magnetic moments pointing along the same\ndirection while the moments of the neighboring chains are\naligned along the opposite directions [14, 15] [Fig. 1(a)]. This\nstate, with an ordering wavevector ( π,0), is considered widely\nto be a consequence of the inherent Fermi-surface instabili ty\nas there exists a very good nesting in between the hole- and\nelectron-pockets around Γat(0,0)and X at ( π,0) points, re-\nspectively [16–19].\nThe gaplessness of the SDW state arises from the symme-\ntry and band topology [13]. In the unordered state, the band\ntouching around (0, 0) (and around an additional point ( π,π)\nin the two-orbital model) leads to vorticities equal to ±2 for\n∗Electronic address: dheeraj.kumar@thapar.eduthe hole pockets. The vorticity corresponds to a complete ro -\ntation of the spinor defined in terms of two orbitals dxzand\ndyz, which is also reflected in the contribution of different or-\nbitals to the hole pocket [Fig. 1(b)]. It vanishes, on the oth er\nhand, for the electron pockets around ( π,0) or (0,π) as the\npockets are dominated by a single orbital. As a result, there is\na vorticity mismatch for the hole and electron pockets, whic h\nsubsequently leads to a gapless SDW state accompanied with\nan even number of Dirac cones, despite the presence of a good\nnesting.\nThe robustness of the Dirac cones and nodes originates\nfrom three symmetries associated with the metallic SDW\nstate: collinearity of the magnetic order, inversion symme -\ntry about an iron atom and another symmetry which com-\nbines together the time reversal and inversion of magnetic m o-\nments [13]. These cones in the SDW state are not far away\nfrom the Fermi level [11]. Thus, there is a strong possibilit y\nof obtaining a coexisting DS and SDW states by tuning pa-\nrameters accessible through experiments. A DS state is char -\nacterized by a linear band crossing of conduction and valenc e\nbands at the Fermi level [20, 21]. The band-crossing point,\nalso known as Dirac point (DP), is four-fold degenerate. The\nmassless fermion in the vicinity of the DP, with several nove l\nelectronic behavior [22–24], is described by the Dirac equa -\ntion [20, 21].\nThe search for materials or phases hosting DS besides\ngraphene, which turns into a topological insulator (TI) be-\ncause of a small spin-orbit couplings, has gathered momentu m\nlately [25]. Potential existence of DS has been indicated in 3D\ncompounds such as β-cristobalite BiO 2[26], BiZnSiO 4[27],\nCd3As2[28–30] etc. and efforts are being made to explore\nthis state in new 2D systems as well [31, 32].\nIn this paper, we investigate coexisting DS and SDW states\nof iron pnictides. Among several parameters whose varia-\ntion do not affect the symmetries required for the stable Dir ac\ncones, we show that the location of DPs with respect to the\nFermi surface can be tuned by the orbital splitting between\nthedxzanddyzorbitals in addition to the band filling. Ex-\nperimentally, while the latter can be controlled by charge c ar-\nrier doping, the former can be achieved by applying in-plane\nstress on the sample. Another consequence of orbital splitt ing\nis that the additional Fermi pockets, apart from the ones ass o-2\n-1-0.5 0 0.5 1\n-1-0.5 0 0.5 1(b)\nQky/π\nkx/πdxzdyz\nFIG. 1: (a) Schematic representation of spin arrangements i n the\n(π,0)SDW state. (b) Fermi surfaces in the unordered state within\nthe two-orbital model. The nesting vector Q= (π,0)connects the\nhole pocket at Γ= (0,0)to the electron pocket at X= (π,0). The\ncolor palette shows the variation of orbital-charge densit ies for the\ndxzanddyzorbitals.\nciated with the Dirac cones, may disappear so that the result -\ning state has only Dirac cones crossing the Fermi level lead-\ning to tiny Fermi pockets or DPs. In the latter case, the state\nis semimetallic SDW state, for which we calculate necessary\nconditions dependent on orbital splitting and other parame -\nters. In addition, linearlized dispersions are obtained ne ar the\nDirac points and the nature of associated edge states is also\nstudied.\nII. MODEL AND METHOD\nIron pnictides have a quasi-two dimensional crystal struc-\nture where the square lattices formed by Fe and As atoms are\ninterlaced together so that the As atoms are positioned a lit tle\nabove and below the center of each square plane of Fe atoms.\nTherefore, a crystallographic unit cell consisting of two F e\nand two As atoms is formed. The Fe 3 dorbitals hybridize\nwith the As 4 porbitals, which increases the complexity of\nband structure. Bandstructure calculations indicate that the\nmajor contributions to the density of state (DOS) at the Ferm i\nlevel comes from dxz,dyzanddxyorbitals [18]. In the follow-\ning, we consider the tight-binding Hamiltonian based on two\norbitals dxzanddyz, which is given by [17]\nHk=∑\nij∑\nµ,ν∑\nσtµν\nijd†\niµσdjνσ−δ∑\ni,σ(d†\nixzσdixzσ−d†\niyzσdiyzσ),\n(1)\nwhere tµν\nijare the intra- and inter-orbital hopping parameters\nfor the first and second nearest neighbor. d†\niµσ(diµσ)denotes\nthe creation (annihilation) operators for the orbital µat site\niand spin σ. The second term with parameter δtakes into\naccount the orbital splitting (OS) between the two orbitals ,\nwhich has been observed to exist up to a very high temperature\neven beyond the SDW transition temperature [33, 34].The standard on site Coulomb interaction terms are\nHi=U∑\niµniµ↑niµ↓+/parenleftbigg\nU′−J\n2/parenrightbigg\n∑\niniµniν\n−2J∑\niSiµ·Siν+J∑\ni,σd†\niµσd†\niµ¯σdiν¯σdiνσ (2)\nThe first and second terms describe the intra- and inter-orbi tal\nCoulomb interaction, respectively, where niµσ=d†\niµσdiµσand\nniµ=∑σσ′d†\niµσdiµσ. The third and fourth terms take into\naccount the Hund’s coupling and the pair hopping, where\nSj\niµ=∑σd†\niµσσj\nσσ′diµσ′and ¯σdenotes the spin anti-parallel\ntoσ. The relation U=U′−2Jis ensured to keep the rota-\ntional symmetry intact.\nVarious interaction terms in the Hamiltonian can be mean-\nfield decoupled in the SDW state, which yield terms bilinear\nin the electron creation (or annihilation) operator for the SDW\nstate. These terms, when the magnetic moments are oriented\nalong zdirection, are\nHi\nmf=U\n2∑\nkµσ/parenleftbigg\n−sσmµ+nµ/parenrightbigg\nd†\nkµσdkµσ\n+(U′−J\n2)∑\nk,µ/negationslash=ν,σnνd†\nkµσdkµσ\n−J\n2∑\nk,µ/negationslash=ν,σσsmνd†\nkµσdkµσ. (3)\nnµandmµare the orbital-resolved charge density and sub-\nlattice magnetization for the orbital µ, respectively. sandσ\nare equal to 1 (-1) for A (B) sublattice and ↑spin (↓spin),\nrespectively.\nAfter combining the kinetic, OS, and meanfield decoupled\ninteraction parts, the Hamiltonian for the ( π, 0) SDW state in\ntwo sublattice basis is\nHm f=∑\nkσΨ†\nkσ(ˆTkσ+ˆMkσ)Ψkσ, (4)\nwhere the matrix elements Tµν\nkσand Mµν\nkσ=−s∆µν+\n5J−U\n2nµδµνare the kinetic and interaction part contributions.\nsis+/−onA/Bsublattice. The electron-field operator is\nΨ†\nkσ= (d†\nAk1σ,d†\nAk2σ,···d†\nBk1σ,d†\nBk2σ,···)and the exchange\nfields is 2 ∆µν=U m µδµν+Jδµν∑µ/negationslash=νmν. The charge den-\nsitynµand magnetization mµare calculated self consistently\nby diagonalizing the Hamiltonian matrix present in Eq. 4.\nIII. TWO-ORBITAL MODEL\nIn the two-orbital model, the meanfield SDW Hamiltonian\nin the two sublattice basis Awith spin up and Bwith spin down\nis given by\nHMF=∑\nk,σΨ†\nkσ/parenleftigg\nHσ\nαα(k)Hσ\nαβ(k)\nHσ†\nαβ(k)Hσ\nβ β(k)/parenrightigg\nΨkσ (5)3\nwith Ψ†\nkσ=(d†\nAασ,d†\nBασ,d†\nAβ σ,d†\nBβ σ), where the sub- or super-\nscript αandβare used to denote the orbitals dxzanddyz,\nrespectively, throughout. The matrices in Eq. 5 are\nHσ\nαα(k)=/parenleftbiggεαα\ny−σ∆α−δ+Nα εαα\nx+εαα\nxy\nεαα\nx+εαα\nxy εαα\ny+σ∆α−δ+Nα/parenrightbigg\n(6)\nand\nHαβ(k)=/parenleftigg\n0εαβ\nxy\nεαβ\nxy 0/parenrightigg\n. (7)\nThe intra- and inter-orbital hopping parameters along xandy\ndirections for different orbitals are given by\nεαα\nx=−2t1coskx,εβ β\nx=−2t2coskx\nεβ β\ny=−2t1cosky,εαα\ny=−2t2cosky\nεαα\nxy=εβ β\nxy=−4t3coskxcosky\nεαβ\nxy=−4t4sinkxsinky (8)\nThe hopping parameters t1andt2link similar orbitals corre-\nsponding to the nearest neighbor σandπbonds respectively,\nwhile the the next-nearest neighbour hopping parameters t3\nandt4denote the overlap amplitude between two similar and\ndissimilar orbitals, respectively. Various hopping param eters\nconsidered for our calculations are t1= -1.0, t2= 1.3, t3=t4\n= -0.85 [17]. Herefrom, we set |t1|as the unit of energy. The\nexchange field ∆α/βand charge-density dependent Nα/βare\ngiven by\n∆α/β=(Um α/β+Jmβ/α)/2,Nα/β=(5J−U)nα/β/2 (9)\nwhere mα/βandnα/βare magnetization and charge density\nfordxz/yzorbital, respectively.\nWe will first examine the parameter space in the theory of\nSDW state to search for the DS state coexisting with SDW\nstate. All the on-site Coulomb interaction parameters are\nfixed unless stated otherwise. We have chosen U/|t1|∼4 or\nU∼W/3 with Wbeing the bandwidth [41] and J∼0.18U\nnearly in the middle of the range 0 .15U/lessorsimilarJ/lessorsimilar0.25Uin accor-\ndance with various estimates [36, 37]. The chemical potenti al\nis fixed throughout so that the band filling is n=2.\nA. Bulk dispersion\nFig. 2 shows the quasiparticle dispersion and Fermi surface\nin the SDW state for different values of OS parameter δ. For\nδ=0 [Fig. 2(a)], there exist two pairs of Dirac cones D1and\nD2,D1along the Γ-X and D2along X′-M directions. Former\nis below the Fermi level while the latter one is above it. The\norbital distribution of the Fermi pockets are largely simil ar\nfor both the pair of Dirac cones. As noticed, the face of the\npockets towards ky=0 is dominated by dxzorbital whereas by\ndyzorbital towards ky=±π/2 [Fig. 2(b)].\nOne of the parameters which can control the location of\nDirac cones with respect to the Fermi level is the band fill-\ning. However, it will shift both the Dirac cones either up ordown together so that the Fermi pocket associated with one of\nthe cones with increase in size while the other will decrease .\nThus, the DS-SDW state cannot be obtained by merely doping\ncharge carriers. On the other hand, we find that by increasing\nOS parameter δ,D1andD2simultaneously can be pushed up\nand down, respectively, so that both the associated DPs may\napproach the Fermi level together, as required to achieve th e\ncoexisting DS and SDW states [Fig. 2(c)]. Fig. 2(d) shows\nthe Fermi pockets obtained for δ∼0.2, where both DPs as-\nsociated with D1andD2can be seen at the Fermi level as\nindicated by the Fermi pockets turning into the Fermi points\nin the reduced-Brillouin zone.\nThe shifting up or down of the DPs can be understood with\nthe help of slope of the bands dominated either by dxzorbital\nor by dyzorbital, which cross each other to generate the DPs.\nForD1, the ratio rof the absolute value of the slope of the\nbands dominated by dxzorbital and dyzorbital is r>1. Thus,\na positive δ, which lowers the energy of dxzorbital and elevate\nthe energy of dyzorbital, will shift D1upward. Similarly, it\nmay be noted that r<1 for D2. Therefore, when the energy\nofdxzorbital is lowered, the band dominated by dxzorbital\nwill shift down, which will, in turn, bring down D2. Both D1\nandD2approach Fermi level when the splitting δincreases.\nIn this way, the location of Dirac points can be controlled wi th\nthe help of OS parameter δ.\nFig. 3 shows self-consistently obtained orbital-resolved\nmagnetization and charge density as functions of OS param-\neter δ. As expected, the dxzanddyzorbital charge density\nincreases and decreases with δ, respectively. However, the\nbehavior of magnetization is in contrast with what is expect ed\nconventionally. When the orbital filling continues to incre ase\nbeyond unity, the magnetization is expected to decrease. Si m-\nilarly, when the orbital filling continues to drop below unit y\nthen the magnetization is expected to rise. On the contrary,\none notices that the dxzorbital magnetization mxzfirst in-\ncreases slightly and then becomes nearly constant whereas myz\nshows a relatively sharper decline. This may arise because o f a\nsubtle interplay between bandstructure and correlation ef fect,\nas the largest interaction U∼4 is in the intermediate coupling\nregime.\nThe self-consistently obtained curves denoting the exis-\ntence of DS-SDW state for different Us are shown in the J-δ\nspace [Fig. 3(c)]. It may be noted that for all Us considered,\nJincreases with δwithin the range 0 .15U/lessorsimilarJ/lessorsimilar0.25U. The\nrange of Jis dependent on U, for this reason, Jis in the unit\nof|t1|in the plot. We note that the range of δshrinks as U\nincreases, which implies more sensitivity to any change in δ\nfor higher U. However, for U=4 a value close to various\nestimates, we find a relatively broad range of δwith value\nextending from δ=0.18 to 0.27.\nB. DS-SDW conditions\nIn order to obtain the DS-SDW state, the self-consistency\nshould be achieved subjected to conditions, which will be\ndiscussed below. First of all, we focus along the direction\nky=0 orπin the Brillouin zone, where Hαβbecomes a null4\n-1-0.5 0 0.5 1\n(a)\nU = 4.0\nn = 2.0\nδ = 0.0Ek\n-1-0.5 0 0.5 1\n(b)ky/πdxzdyz\n-1-0.5 0 0.5 1\n(0,0)(π/2,0)(π/2,π)(0,0) (0,π)(π/2,π)(c)\nδ = 0.215Ek\nkdxz dyz-1-0.5 0 0.5 1\n-0.5  0 0.5(d)ky/π\nkx/π\nFIG. 2: Energy bands in the SDW state with U=4 and J=0.18U\nwhen the OS parameters are (a) δ=0.0 and (c) δ=0.215. The\nvarying color scheme used for the dispersion curve represen ts orbital-\ncharge density. Corresponding Fermi surfaces are shown in fi gures\n(b) and (d), respectively. The same color scheme also shows t he\ndominating orbital along the Fermi surfaces.\n 0.8 0.9 1 1.1\n 0 0.1 0.2(a) U = 4.0mµ\nδxz\nyz\n 0.2 0.22 0.24 0.26\n 0 0.1 0.2(b) U = 4.0nµ\nδ 0.6 0.9 1.2 1.5\n 0.2 0.24 0.28(c)J\nδU = 4.0\nU = 4.5\nU = 5.0\nU = 5.0\nFIG. 3: (a) Orbital magnetization and (b) charge density as f unctions\nof OS parameter δbetween dxzanddyzorbitals in the ordered state\nforU=4 and J=0.18U. (c) Self-consistently obtained DS-SDW\nstates for various Us inJ-δspace for band filling n=2.\nmatrix. Therefore, HMFtakes a block diagonal form with\ntwo block Hamiltonians Hαα(k)andHβ β(k)corresponding\nto each orbital with size 2 ×2 matrices, which can be readily\ndiagonalized. The energy eigenvalues are given by\nEx\ns±=(εss\ny±δ+Ns−µ)±/radicalig\n(εssx+εssxy)2+∆2s (10)\nwith setting ky=0 orπ.srefers to αorβorbitals. Superscript\nxdenotes the fact that the energy is essentially a function of\nkxonly for ky=0 or π, while subscript smerely indicates\nthat the dispersion is dominated by either of the orbitals. T wo\ndispersions Ex\nα+andEx\nβ−cross each other at the Fermi level\nwhen Eα+=Eβ−=0, which requires\ncosk0\nx∓=/radicalbig\n(∓2t2−δ+Nα−µ)2−(∆α)2\n2(t1±2t3)\n=∓/radicalig\n(∓2t1+δ+Nβ−µ)2−(∆β)2\n2(t2±2t3).(11)\nHere, k0\nx−andk0\nx+give the locations of DPs associated with\nthe Dirac cones D1andD2along Γ-X and X′-M directions,\nrespectively. It may also noted that k0\ny−=0 and k0\ny+=π.C. Linearized dispersion\nUsing Taylor expansions of Ex\nα+andEx\nβ−around DPs along\nkxwith the help of Eq. (9), one obtains linearized dispersion\nEx\ns∓=cx\ns∓qx, where the constant cx\ns∓is given by\ncx\nα∓=∓2(t1±2t3)2\n|∓2t2−δ+Nα−µ|sin2k0\nx\ncx\nβ∓=±2(t2±2t3)2\n|∓2t1+δ+Nβ−µ|sin2k0\nx. (12)\nNote that the subscript −and+incx\nα∓refers to the DPs D1\nandD2, respectively.\nHMFis not in the block-diagonal form along ky-direction.\nMoreover, the Hamiltonian for ↑-spin electron is two-fold de-\ngenerate at the DPs. Therefore, the degenerate perturbatio n\ntheory yields the following corrections to the energies nea r\nthe DPs along ky\nEy\n±=±/parenleftbigg∆α−eα\nbα+∆β+eβ\nbβ/parenrightbigg\nεαβ\nxy (13)\nwhen εαβ\nxyis very small. Note that the subscript shas been\ndropped here because the band along kyis not far away from\nan even mixture of both the orbitals. bα=εαα\nx+εαα\nxyandeα=/radicalbig\nb2α+∆2α. The linear dependence of the energy dispersion\nnear the DP is readily obtained from εαβ\nxy, where sin qycan be\napproximated by qyfor small qyso that Ey\ns±=ds±qy.ds±is a\nconstant given by\nds±=±4t4sink0\nx(fα∓+fβ∓), (14)\nwhere\nfα∓=∆α−|∓ 2t2−δ+Nα−µ|/radicalbig\n(∓2t2−δ+Nα−µ)2−∆2α\nfβ∓=∆β+|∓2t1+δ+Nβ−µ|/radicalig\n(∓2t1+δ+Nβ−µ)2−∆2\nβ. (15)\nUpper and lower sign correspond to the Dirac cones D1and\nD2, respectively. Eqs. 11-15 provide the conditions for the\ncoexistence of Dirac semimetallic and SDW state as well as\nthe linear energy dispersion in the vicinity of DPs.\nD. Edge states\nAn edge state may exhibit behavior which is different from\nthe bulk band as in the case of topological insulator where\nthe bulk bands are gapped while charge transport occurs by\nthe topologically protected surface states [38]. In iron pn ic-\ntides, there exists localized edge states in the high temper ature\nphase even without long-range order [39]. It may be noted\nthat these edge states, nearly degenerate, are not associat ed\nwith any Dirac cones, which are absent in the paramagnetic\nstate. For a given ky, these edge states are bonding and anti-\nbonding mixture of states localized at the edges of ribbon/s trip5\n-8-4 0 4(a)Ek\nky\n-8-4 0 4(b)Ek\nky\n-8-4 0 4\n-π -π/2 0 π/2 π(c)Ek\nkx\nFIG. 4: Energy bands for the DS-SDW state for a ribbon of width\n(a)W= 50, (b) W= 51 considered along x-direction and (c) W= 50\nconsidered along y-direction for U=4 and J=0.18U, for which,\nthe orbital charge densities are nα=1.14 and nβ=0.86 whereas the\nmagnetic-exchange fields are ∆α=0.58 and ∆β=0.53.\nextending along ydirection. They result from the fact that a\none-dimensional Hamiltonian for a given kycan be deformed\ncontinuously to one which has topologically protected wind -\ning number, while the edges states are preserved in the defor -\nmation process.\nNext, we examine the edge states in the DSM-SDW state\nwith ribbons oriented either along xorydirections. First, a\nribbon of width W(Watomic sites) lying along ydirection is\nconsidered so that kyis a good quantum number. The ribbonHamiltonian with dimension 2 W×2Wis given by\nHRbx(k)=\nH+H′O···\nH′†H−H′···\nO H′†H+···\n............\n(16)\nwhere\nH±=/parenleftbiggεαα\ny∓∆α−δ+Nα−µ 0\n0 εβ β\ny∓∆β+δ+Nβ−µ/parenrightbigg\nand\nH′=/parenleftbigg\n−t1−2t3cosky−2it4sinky\n−2it4sinky−t2−2t3cosky/parenrightbigg\n.\nSimilarly, when the ribbon’s length is oriented along xdirec-\ntion so that kxis a good quantum number, the Hamiltonian\nHRby(k)with size 4 W×4Whas a form similar to that of HRbx.\nHowever, H±andH′have size 4 ×4 instead. These matrices\nare given by\nH+=H−=H=/parenleftbigg\nhAAhAB\nhBAhBB/parenrightbigg\nwith\nhAA(k)=/parenleftbigg\n−∆α−δ+Nα−µ 0\n0 −∆β+δ+Nβ−µ/parenrightbigg\nand\nhAB(k)=/parenleftbiggεαα\nx 0\n0εβ β\nx/parenrightbigg\nwhile\nH′(k)=\n−t2 0−2t3coskx−2it4sinkx\n0 −t1−2it4sinkx−2t3coskx\n−2t3coskx−2it4sinkx−t2 0\n−2it4sinkx−2t3coskx 0 −t1\n.\nThere exists nearly degenerate two edge states for the rib-\nbon in the paramagnetic state which may or may get clearly\nseparated in the presence of magnetic order depending on\nwhether Wis odd or even. Fig. 4 show the edge state dis-\npersions in the DS-SDW state. There are two edge states\nlocalized at the edges of ribbon when Wis even. Another\npartially visible edge-state like dispersion stands out cl early\nfrom the bulk bands. These edge states in the DS-SDW state\nare dispersing unlike those in graphene, which are flat [40].\nOne of the edge-state dispersion crosses the Fermi level whi le\nthe other one not. The clear separation of edge dispersions\nis missing when Wis an odd, which is simply a consequence\nof the same magnetic-exchange fields at both the edges. It\nmay be noted that they are always degenerate only at kx=0\nandky=±π. Fig. 4(c) shows the dispersion in a ribbon ori-\nented along the xdirection. As expected, they are symmetric\nabout kx=−π/2 an π/2 because of the two-sublattice struc-\nture along the ribbon extending infinitely along x. None of the\nedge-state dispersion is flat and both can be noticed to cross\nthe Fermi surface.6\n-0.3 0 0.3\n     (0,0) ( π/2,0) ( π/2,π ) (0,0) (0, π )              \n(b)\nδ = 50meV-0.3 0 0.3\n                   \n(a)\nU = 1.5eV\nJ = U/4\nδ = 0meV\nFIG. 5: Electron dispersion in the (π,0)SDW state of five-orbital\nmodel of Graser et. al. when (a) δ=0 (b) δ=50meV , where\nU=1.5eVandJ=0.25U. The hole pocket at Γdisappears in the\npresence of OS δ=50meV .\nIV . DS-SDW STATE IN FIVE-ORBITAL MODEL\nFinally, we discuss DS-SDW state in a five-orbital model\nwhich describes the band structure more realistically. We c on-\nsider the five-orbital model due to Graser et. al. . In the un-\nordered state, it has similarities as well as differences fr om\nthe two-orbital model. There is a hole pocket around Γand\nan electron pocket around X while the hole pocket around M\nis absent. The electron pocket is mostly dominated by dxyor-\nbital [17, 18].\nFig. 5 shows the electronic dispersion in a self-consistent ly\nobtained SDW state for U=1.5eV and J=0.25Uwith and\nwithout orbital splitting. The ratio U/W∼0.25 with W∼6eV\nbeing bandwidth is in accordance with various estimates [41 ].\nThe OS parameter δis taken to be 50meV , which is nearly\nof similar order in observed in the experiments. The evidenc e\nof the splitting already present in the high-temperature ph ase\ncomes from transport [42], ARPES [43, 44] and magneto-\ntorque measurements [34].\nIt may be noted that the unlike the two-orbital model, there\nis a large hole pocket around Γand a tiny hole pocket located\nnot far away from Γalong Γ-X. With increasing OS parameter\nδ, which lowers the energy of dxzorbital, the dxz-dominated\nbands such as the hole pocket around Γwill be pushed down\nbelow the Fermi level as shown in Fig. 5 (a) and (b). There\nare two important differences from two-orbital models with\nregard to the Dirac cone. First, the difference in the magnit ude\nof slopes of the crossing bands at DPs are not as large as in\nthe two-orbital model. Second, along kx, one of the crossing\nbands is largely dominated by dxzwhile the other one by dxy\norbital unlike the dyzorbital. Therefore, when δis changed,\nthe corresponding shift in the positions of DPs is relativel y\nsmall in comparison to the two-orbital model. However, the\nDPs are already very close to the FS even in the absence of\nOS. Thus, the important role of OS δis to suppress the holepocket around Γpoint, which is necessary to obtain the DS-\nSDW state.\nV . SUMMARY AND CONCLUSIONS\nIn the two-orbital model, the pair of Dirac cones in the\nSDW state are located away from the Fermi surface. While\none pair is above, the other one is below. The separation of\nthese pairs with respect to the Fermi surface can be minimize d\nwith the help of OS. Inclusion of OS pushes up the Dirac cone\nfound below the Fermi level and pushes down the one above it,\nwhich is possible because of a sharp difference in the slopes of\nthe crossing bands dominated largely by a single orbital. Fo r\na given OS parameter, an overall small shift away from the\nFermi surface may also occur despite both pairs of DPs being\nat the same energy level. However, such a shift can be over-\ncome by either doping holes or electrons. The signature of\nshifting of DPs can be observed through various experiments\nsuch as transport measurements, quantum oscillation, ARPE S\netc. [42].\nIn the current work, a detailed study of the DS-SDW state\nwas carried out in the two-orbital model because of its sim-\nplicity, as it allows for obtaining conditions for the DS-SD W\nstate in an analytical form. However, DS-SDW state can also\nexist in a more realistic five-orbital model in a region of int er-\naction and OS parameters space, which is illustrated for suc h\na particular set within the range in accordance with various\nestimates. However, there are several important differenc es\nfrom the two-orbital model in terms of orbital content of the\nDirac cone and presence of hole pocket around Γpoints. An-\nother major difference exists in terms of the number of pairs\nof Dirac cones. There is only one pair of Dirac cones close\nto the Fermi level in the five-orbital model [11], for which, i t\nmay relatively be easier to bring the DPs at the Fermi level.\nOverall, we find that, the OS pushes the hole pocket around Γ\nbelow and secondly it may also shift the DPs. As a result, it\nis not only possible to control electronic properties inclu ding\nthe charge transport by tuning the OS, thus, in turn, regulat e\nthe contribution of Dirac cone but it is also feasible to real ize\nDS-SDW state.\nThe OS parameter plays a crucial role in obtaining the DS-\nSDW state, whose inclusion here has been motivated by its\npresence in the high-temperature state with tetragonal sym -\nmetry and nematic order [45–48]. The nematic phase is\nmarked by the presence of a short-range order with broken\nfour-fold rotation symmetry [49] and is observed at a temper -\nature T>TSDW in several IBSs [50]. It was noted that the\nlattice anisotropy in the orthorhombic symmetry cannot giv e\nrise to a splitting as large as ≈60meV between the dxzanddyz\norbitals. This perhaps indicates that the OS, which persist s\ninto the high temperature tetragonal phase, in addition to t he\northorhombic symmetry or magnetic order, could be the key\nfactor behind the anisotropic electronic behavior. Theref ore,\nthere is no reason why such a splitting should be ignored in\nthe low-temperature phases such as the SDW state. Although\nan additional splitting will also be generated by the SDW sta te\nitself as it breaks the four-fold rotation symmetry. Experi men-7\ntally, OS can be controlled by applying appropriate pressur e\nsuch as an in-plane stress [51].\nIn conclusions, we have examined the possible existence of\nspin-density wave state with Dirac points at the Fermi level\nin iron pnictides such that the state is essentially a Dirac\nsemimetal without any other band crossings. We find that such\na state indeed can be obtained in the presence of finite OS.\nWhile the interaction parameters are not tunable experimen -\ntally, charge carrier doping may shift entire bands up or dow n,\nthe OS of otherwise degenerate dxzanddyzorbitals, on the\nother hand, can suppress other bands crossing the Fermi leve l\nwhile leaving the Dirac points in the vicinity of Fermi surfa ce.\nThus, the OS, tunable experimentally with the help of in-pla ne\nmechanical stress, can be used to modify the electronic stat es\nnear the Fermi surface in order to control the electronic pro p-erties as well as in obtaining DS-SDW state. 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Fisher, Science 329, 824 (2010)."    },    {        "title": "2207.05465v1.Local_and_global_ordering_dynamics_in_multi_state_voter_models.pdf",        "content": "Local and global ordering dynamics in multi-state voter models\nLuc\u0013 \u0010a Ramirez\u0003\nInstituto de F\u0013 \u0010sica Interdisciplinar y Sistemas Complejos, IFISC (CSIC-UIB),\nCampus Universitat Illes Balears, E-07122 Palma de Mallorca, Spain and\nDepartamento de F\u0013 \u0010sica, Instituto de F\u0013 \u0010sica Aplicada, Universidad Nacional de San Luis-CONICET,\nEj\u0013 ercito de Los Andes 950, D5700HHW, San Luis, Argentina\nMaxi San Miguelyand Tobias Gallaz\nInstituto de F\u0013 \u0010sica Interdisciplinar y Sistemas Complejos, IFISC (CSIC-UIB),\nCampus Universitat Illes Balears, E-07122 Palma de Mallorca, Spain\nWe investigate the time evolution of the density of active links and of the entropy of the dis-\ntribution of agents among opinions in multi-state voter models with all-to-all interaction and on\nuncorrelated networks. Individual realisations undergo a sequence of eliminations of opinions until\nconsensus is reached. After each elimination the population remains in a meta-stable state. The\ndensity of active links and the entropy in these states varies from realisation to realisation. Making\nsome simple assumptions we are able to analytically calculate the average density of active links\nand the average entropy in each of these states. We also show that, averaged over realisations, the\ndensity of active links decays exponentially, with a time scale set by the size and geometry of the\ngraph, but independent of the initial number of opinion states. The decay of the average entropy is\nexponential only at long times when there are at most two opinions left in the population. Finally,\nwe show how meta-stable states comprised of only a subset of opinions can be arti\fcially engineered\nby introducing precisely one zealot in each of the prevailing opinions.\nI. INTRODUCTION\nOne of the most popular classes of models of opinion\ndynamics is that of so-called voter model (VM) [1{5].\nVMs describe populations of individuals, who are each\ncharacterised by their discrete opinion state, and where\nthe principal mechanism of change is imitation (i.e., one\nindividual copies the opinion state of another individual).\nVMs are not only a paradigmatic model of opinion forma-\ntion, they are also of interest in statistical physics. They\noperate out of equilibrium, and have absorbing states and\ncertain symmetries, de\fning an interesting universality\nclass [1, 6].\nIn the most simple version of the VM each individual\ncan take one of two opinion states. Individuals are as-\nsumed to reside on the nodes of an interaction network\n(this includes the special case of all-to-all interaction).\nAt each step an individual is chosen at random and then\ncopies the state of one randomly chosen neighbour on\nthe network. This simple system has Z2-symmetry, and\ntwo absorbing `consensus' states (if all individuals hold\nthe same opinion, no further change is possible). The\nfeatures of most interest to physicists include the time\nrequired to reach absorption, and the coarsening dynam-\nics during the process leading to consensus [4{11].\nMany variants of the voter model have been introduced\nin order to capture di\u000berent features of social interaction.\nExamples are the so-called noisy voter model in which\n\u0003Electronic address: luciaramirez@i\fsc.uib-csic.es\nyElectronic address: maxi@i\fsc.uib-csic.es\nzElectronic address: tobias.galla@i\fsc.uib-csic.esindividuals can change opinion spontaneously [12{15], or\npopulations including so-called `zealots'. These are in-\ndividuals who are less prone to changing opinion than\nregular agents, or who never change opinion [16{19]. A\nfurther variation, which we focus on here, is the so-called\nmulti-state voter model (MSVM) [20]. These are voter\nmodels in which there are more than two possible opin-\nion states, and consequently multiple absorbing states.\nThe path to one of these states involves a sequence of\nsuccessive extinctions of opinions. One main distinction\nis between models in which the di\u000berent opinion states\nare ordered in some way (representing e.g. the political\nspectrum) versus models in which all states are equiva-\nlent. We here focus on the latter case.\nExisting literature on MSVM with equivalent states\nincludes in particular work on consensus and extinction\ntimes [20{22]. For the case of all-to-all interaction the\nauthors of Ref. [20] derived analytical expressions for ob-\njects such as the mean consensus time, and the mean\nnumber of di\u000berent states in the population as a func-\ntion of time. Ref. [20] also contains numerical studies of\nthe model in low-dimensional lattices.\nThe authors of Ref. [21], among other results, further\nprovided closed-form expressions for all moments of the\nconsensus time for uniform initial distributions on the\nall-to-all interaction. Baxter et al. [22] obtain a solution\nfor a model describing neutral genetic drift at a single\nlocus with multiple alleles. This model, while set up in a\nbiological context, is mathematically very similar to the\nMSVM in an all-to-all geometry, and the bulk of the ideas\nand results therefore carry over. It should be noted that\nthe authors of [22] work in the di\u000busion approximation.\nMany of the results in the existing literature concern\nquantities such as the consensus time, the time until thearXiv:2207.05465v1  [physics.soc-ph]  12 Jul 20222\nextinction of the di\u000berent opinions, or the properties of\nthe stationary state (the latter becomes non-trivial if\nspontaneous opinion changes are added to the dynam-\nics, as this removes the absorbing states).\nThe question how consensus is reached in MSVMs, and\nwhat the coarsening process before absorption looks like,\non the other hand, appear to have received relatively lit-\ntle attention. In this work, we therefore study the VM\nwith a general number of initial opinion states, and with\na focus on the dynamics before consensus is reached.\nWe focus on the cases of a complete graph (all-to-all\ninteraction), uncorrelated networks such as Erd os{Renyi\ngraphs, and scale-free networks. Throughout our paper\nscale-free networks are generated using the Barab\u0013 asi{\nAlbert growth process of preferential attachment [23],\nand have a degree distribution which decays as p(k)\u0018\nk\u00003. We will simply refer to these as Barab\u0013 asi{Albert\nnetworks. The assumption of an uncorrelated network is\nthen valid [10, 11].\nWe focus on two key quantities in our analysis. One\nis the familiar `density of active links' [6, 10{12, 24{26],\nthat is the proportion of connected pairs of agents that do\nnot share the same opinion. This quantity characterises\nthe organisation of individuals on the graph. A low den-\nsity of active links indicates the presence of domains of\nindividuals of the same state (neighbours tend to be in\nthe same opinion states). The pattern is more scattered\nif the density of active links is high [27]. Additionally, we\nlook at the entropy of the distribution of agents across\nthe di\u000berent opinion states. This is a global measure\nof order (it does not make use of pairs of neighbours),\nindicating how dispersed the individuals are across the\ndi\u000berent opinions.\nThe time evolution of the density of active links in the\ntwo-state model has been studied on complete graphs\n[7, 25], and on uncorrelated graphs [10{13, 24]. Aver-\naged over an ensemble of realisations, an exponential de-\ncay of the mean density of active links with time is here\ntypically found. The decay time is proportional to the\npopulation size, N, for complete graphs and Erd os-Renyi\nnetworks [9, 25]. On Barab\u0013 asi{Albert networks the time\nscale is proportional to N=lnN[8].\nThe behaviour in individual realisations is quite dif-\nferent. Both in the all-to-all scenario and on graphs one\n\fnds that single runs of the two-state VM typically set-\ntle to quasi-stable `plateau' of the density of active links,\nbefore a sudden \ructuation then takes the system to con-\nsensus [10, 28]. The density of active links at the plateau\ndi\u000bers in the cases of all-to-connectivity or networks. It\nalso depends on the initial proportion of agents in the\ntwo opinion states [11].\nIn the two-state model consensus is reached after a\nsingle extinction of an opinion. The multi-state model\non the other hand undergoes a sequence of extinctions.\nOur main objective is to study how this a\u000bects the time\nevolution of the density of active links, and of the entropy.\nWe address this both at the ensemble level (i.e., as an\naverage over realisations), and on the level of individualruns.\nThe remainder of the paper is set out as follows. In\nSec. II we de\fne the model. Sec. III then focuses on\nthe time evolution of the density of active links and of\nthe entropy at the level of an ensemble average. The\nphenomenology of individual realisations is studied in\nmore detail in Sec. IV. We \fnd a sequence of meta-stable\nstates, and characterise some of the statistical features\nof these states. We then use this to establish how the\nensemble-level behaviour can be understood from that of\nsingle realisations. In Sec. V we then proceed to show\nthat zealots can be used to `engineer' steady states of\nmixed opinions similar to the meta-stable states found in\nindividual realisations in Sec. IV. Finally, Sec. IV con-\ntains a summary and our conclusions.\nII. MODEL DEFINITIONS\nA. Setup and interaction network\nThe model describes Nindividuals, who can each be\nin one ofMdiscrete states or opinions. We label individ-\nualsi= 1;:::;N and states \u000b= 1;:::;M . During the\ncourse of the dynamics each individual can interact with\nits nearest neighbours on a static network. We use the\nnotationcijfor the adjacency matrix of the undirected\ninteraction network. We have cij=cji= 1 if individu-\nalsiandjare neighbours, and cij=cji= 0 otherwise.\nWe also set cii= 0. We will use the notation j2ito\nindicate that jis among the neighbours of i. We write\nkifor the degree of the node representing individual i,\ni.e.,ki=P\njcij. The total number of links in the graph\nisE=P\ni<jcij. In the complete graph ( cij= 1 for all\ni6=j) one hasE=N(N\u00001)=2.\nB. Dynamics and transition rates\nThe variable si(t)2f1;:::;Mgrepresents the state\nof individual iat timet. At the start of the dynam-\nics (t= 0) the states of all individuals si(t= 0) are\ninitialised. Di\u000berent initial conditions can here be cho-\nsen. We typically consider homogeneous initial condi-\ntions [20], that is a con\fguration in which the same num-\nber of agents in each opinion state are randomly dis-\ntributed in the nodes of the network. For the model on\nnetworks, these individuals are placed on the graph at\nrandom.\nThe dynamics then proceeds through pairwise interac-\ntions between individuals. An interaction of individual\niwith individual j2iconsists of an imitation process,\ni.e.,icopies the opinion state of j. It is important to no-\ntice that, although the interaction network is undirected,\neach interaction event is directed. An interaction of i\nwithjis not the same as an interaction of jwithi.\nThe rates with which interactions between agent iand3\njoccur are given by\nTij=cij\nki=8\n<\n:1=kij2i\n0j =2i: (1)\nIn this setup time is continuous, and measured in units\nof Monte Carlo steps (`generations' in the language of\npopulation dynamics), i.e. O(N) imitation events take\nplace in the population per unit time.\nWe denote the number of individuals holding opinion\n\u000bbyn\u000b, and we write n(t) = [n1(t);:::;nM(t)]. We also\nintroducex\u000b=n\u000b=Nas the fraction of individuals in\nopinion state \u000b.\nC. Density of active links\nAt each point in time the links on the interaction net-\nwork can be grouped into links that are `active' or `inac-\ntive' respectively. A link ( i;j) is said to be active when\nthe two nodes at its ends are in di\u000berent states ( si6=sj),\notherwise the link is inactive.\nIt is further useful to introduce the fraction of links of\ntype\u000b\f, where\u000b;\f2f1;:::;Mg. This is the fraction\nof links which have an agent in state \u000bat one end, and\nan agent in state \fat the other end. Suppressing the\ntime-dependence of the fsi(t)gwe then have\n\u001a\u000b\f=1\nEX\ni<jcij\u0000\n\u000esi;\u000b\u000esj;\f+\u000esj;\u000b\u000esi;\f\u0001\n(2)\nfor\u000b6=\f, and with \u000ethe Kronecker delta. We recall\nthatEis the total number of links in the graph. The\ntotal density of active links in the system is then\n\u001a=X\n\u000b<\f\u001a\u000b\f=1\nEX\ni<jcij(1\u0000\u000esi;sj): (3)\nThe overall rate of events in the population leading\nto state changes is proportional to the fraction of links\nthat are active (an imitation process involving individu-\nals connected by an inactive link will not result in any\nopinion change). The density of active links therefore\ncharacterises the amount of (potential) `activity', and in-\ndicates how far the system is from reaching an absorbing\nstate.\nThe density of active links can also be seen as a mea-\nsure of disorder in the con\fguration of opinions on the\ninteraction network. Consider a particular con\fguration\nnof individuals in the di\u000berent opinion states. If the indi-\nviduals were located at random nodes on the graph with\nno particular order, then the probability that a randomly\nchosen link is active is \u001arandom = 2P\n\u000b<\fn\u000bn\f=[N(N\u0000\n1)]. This is also the fraction of active links on a com-\nplete graph, \u001aCG(n), given thefn\u000bg. If there is order\nin the network (i.e., the neighbours of a node in state \u000b\nalso tend to be in state \u000b), then the density of links willbe lower than \u001aCG(n). At an absorbing state one has\ncomplete order, \u001a= 0. Thus, \u001a(n) indicates the amount\nof disorder in a con\fguration of the networked system.\nThe behaviour of this quantity in time can be used to\ncharacterise the coarsening dynamics. We stress that the\ndensity of active links arises from a local de\fnition of dis-\norder (the state of an individual is compared with that\nof its neighbours).\nD. Entropy\nWe now introduce a second measure of disorder,\nnamely the entropy of a particular con\fguration. This\nis de\fned as\nS=\u0000X\n\u000bx\u000blnx\u000b; (4)\nwhere we recall that x\u000b=n\u000b=N. This de\fnition makes\nno use of neighbourhood relationships between nodes.\nHence entropy as a measure of disorder has a global char-\nacter.\nStates of maximum entropy are those for which x\u000b=\n1=Mfor all\u000b, i.e., states with equally many individuals\nin each opinion state. This leads to S= lnM. If the\nsystem has reached consensus ( x\u000b= 1 for one value of \u000b,\nandx\f= 0 for all \f6=\u000b), we have S= 0. This is the\nstate of maximal order.\nE. Master equation for the model on a complete\ngraph\nIn an all-to-all geometry all individuals are equivalent\nin terms of their position on the graph, and the system is\ntherefore fully speci\fed by n= (n1;:::;nM). The rates\nin Eq. (1) become Tij= 1=(N\u00001) for alli6=j. This in\nturn means that the total rate with which individuals in\nstate\u000bin the population are converted to individuals in\nstate\fis\nT\u000b!\f(n) =n\u000bn\f\nN\u00001: (5)\nThe dynamics of the system is then described by the\nmaster equation\nd\ndtP(n) =X\n\u000b6=\f(E\u000bE\u00001\n\f\u00001)[T\u000b!\f(n)P(n)];(6)\nwhereE\u000bis the `raising operator' acting on functions\nofnby increasing the argument n\u000bby one,E\u000bf(n) =\nf(n1;:::;n\u000b+ 1;:::;nM).\nIII. CHARACTERISATION OF THE\nCOARSENING PROCESS AT THE LEVEL OF\nTHE ENSEMBLE AVERAGE\nTo set the scene we will \frst focus on the time evolution\nof averaged quantities. By this we mean an average over4\nrealisations of the stochastic voter model dynamics, each\nrealisation with a di\u000berent initial condition. We will refer\nto this as an `ensemble average', and use angle brackets\nh\u0001\u0001\u0001i to describe it.\nA. Evolution of the average density of links\n1. Complete Graph\nIn the basic case of all-to-all interactions one has, for\nany function f(n),\nd\ndthfi=X\nnf(n)d\ndtP(n); (7)\nwithd\ndtP(n) as in Eq. (6).\nFrom Eqs. (1) it is also clear that the rate for events\nconverting individuals from state \u000bto state\fis the same\nas that for the reverse event. Therefore, the ensemble\naverage ofx\u000bis constant in time,\nd\ndthx\u000bi= 0: (8)\nIf an individual of type \u000badopts opinion \fin an event,\nthen we have n\u000b!n\u000b\u00001 andn\f!n\f+ 1. Given that\n\u001a\u000b\f= 2n\u000bn\f=[N(N\u00001)] this means that \u001a\u000b\fchanges\nby 2(n\u000b\u0000n\f\u00001)=[N(N\u00001)]. We therefore have\nd\ndth\u001a\u000b\fi=2\nN(N\u00001)\u001c\nT\u000b!\f(n)\u0002(n\u000b\u0000n\f\u00001)\n+T\f!\u000b(n)\u0002(n\f\u0000n\u000b\u00001)\u001d\n=\u00002\nN\u00001h\u001a\u000b\fi: (9)\nWe conclude\nh\u001a\u000b\f(t)i=h\u001a\u000b\f(0)ie\u0000t=\u001c; (10)\nwith a time scale \u001cgiven by [25]\n\u001c=N\u00001\n2: (11)\nFrom this and using Eq. (3) we have\nh\u001ai=h\u001a(0)ie\u0000t=\u001c: (12)\nWe note that \u001cis independent of the number of opinion\nstatesM.\nFor homogeneous initial conditions ( n\u000b=N=M for all\n\u000b), we have\nh\u001a(0)i=(M\u00001)N\n(N\u00001)M\u00191\u00001\nM; (13)\nwhere the approximation applies for N\u001d1.\nIn Fig. 1 we con\frm the validity of Eq. (12) in simula-\ntions. The expression in Eq. (11) is veri\fed in Fig. 2.\n0 2000 4000 6000 8000 100000.11\n<ρ>\ntN=4000 M=2 M=4\nN=6000 M=2 M=4\nN=10000 M=2 M=4FIG. 1: Average density of links in the MSVM with all-to-all\ninteraction and homogeneous initial conditions (see text) for\ndi\u000berent values of initial number of opinions Mand di\u000berent\nsystem sizes N. Symbols show results from numerical simula-\ntions, averaged over 5000 realisations. Lines are the analytical\nprediction in Eq. (12).\n20004 0006 0008 0001 00001000300050002\n0004 0006 000800010000100020003000400050006000 M = 2 \nM = 3 \nM = 4τ\nN\nT\nhis is for the CG. The red line is (N-1)/2. H\nere it is not  necessary to use a log-log scale  \nFIG. 2: Decay time scale \u001cin the MSVM on a complete\ngraph for di\u000berent values of initial number of opinions M, as\na function of the system size N. Markers are obtained from\n\ftting an exponential curve to simulation data for h\u001a(t)i. The\nsolid line shows Eq. (11).\n2. Pair approximation for the two-state model on\nuncorrelated networks\nIn a networked geometry it is not straightforward to\nobtain closed laws for the time evolution of macroscopic\naverage quantities such as the density of active links.\nThis is because the state of the system is no longer\nfully described by the numbers n1;:::;nM. The correla-\ntions that build up between nodes in the network are one\nkey element distinguishing the voter process on networks\nfrom that with all-to-all interaction.\nAnalytical progress for the model on networks is possi-\nble as an approximation. One prominent approach is the\nso-called `pair approximation' [11, 13, 24, 29], capturing\ncorrelations between nearest neighbour nodes, but not\nbetween nodes which are further apart on the graph. The\npair approximation is known to capture the behaviour\non uncorrelated networks to a good accuracy [11{13, 24].\nThese networks can have an arbitrary degree distribu-\ntion, but the degrees of nodes are uncorrelated, including\nthe degrees of the nearest neighbour [30] (the probabil-5\nity that any two nodes are connected therefore only de-\npends on their degrees). This includes Erd os{Renyi and\nBarab\u0013 asi{Albert networks.\nWithin the pair approximation the following expres-\nsion can be found for the average density of active links\nin the VM with two opinion states [11, 13],\nh\u001a(t)i= 2hx1(0)[1\u0000x1(0)]ik\u00002\nk\u00001e\u0000t=\u001cfort>t?;(14)\nwherekis the mean degree of nodes in the network. The\nquantityx1(0) is the initial fraction of individuals in opin-\nion state 1. If this fraction is \fxed then the average\nh\u0001\u0001\u0001i on the right-hand side of Eq. (14) can be removed.\nWithin the pair approximation the time scale \u001cis given\nby [11]\n\u001c=(k\u00001)k2N\n2(k\u00002)k2: (15)\nIn this expression, k2is the second moment of the de-\ngree distribution of the interaction network.\nIt is important to note that Eq. (14) is only valid after\na short transient of duration t\u0003. During this transient,\nthe average density of active links reduces from its initial\nvalue 2hx1(0)[1\u0000x1(0)]iby a factor of ( k\u00002)=(k\u00001).\nUnlike the decay time \u001c, the time scale t?does not in-\ncrease with N, i.e., we have t?=O(N0) [11].\nThe time evolution of the average density of links in\nthe two-state VM in Eq. (14) consists of three main\nfactors: (i) the initial average density of active links,\n2hx1(0)[1\u0000x2(0)]i, one would obtain in an all-to-all\ngeometry for the same distribution of initial propor-\ntions of agents in the two opinion states, (ii) a factor\n(k\u00002)=(k\u00001) accounting for the network geometry, and\n(iii) exponential decay.\n3. Pair approximation for the MSVM\nWe will now use a simple argument to show that this\ngeneral structure carries over to the multi-state model.\nThe only change required is to adapt the expression for\nthe initial density of active links to the case of multiple\nopinion states, given initial proportions of agents in these\ndi\u000berent states.\nThe argument is based on the fact that the dynamics of\none single opinion in the multi-state voter model can be\nunderstood from a reduction to a two-state model. This\nwas \frst proposed in the context of the VM in Ref. [5]\nand later used also in Ref. [14]. The same ideas can also\nbe found in earlier work by Kimura and Littler in the\n\feld of genetics [31, 32].\nIf the focus is on the dynamics of one single opinion\n(say\u000b= 1) then it is not necessary to resolve the dif-\nferent other opinion states. Instead what is relevant for\nan individual to change out of state \u000b= 1 is that they\ninteract with an individual in anyof the other states.Similarly, an individual changes into state \u000b= 1 if they\nwere previously in any other state and interact with an\nindividual in state 1. For the purposes of studying indi-\nviduals in state \u000b= 1 it is not required to know what\nthese other states were. This means that all other opin-\nions (\f= 2;:::;M ) can be amalgamated into one single\nopinion state. This then reduces the model to a two-\nstate voter process, one state represents opinion \u000b= 1\nand the second state stands for `all other opinions'. We\nlabel these states as + and \u0000respectively. The dynam-\nical rules in Eq. (5) are such that this reduced models\nfollows the dynamics of a two-state voter model.\nSuppose now, we start the multi-state voter process\nfrom homogeneous initial conditions, x\u000b(0) = 1=Mfor\n\u000b= 1;:::;M , and focus on a particular opinion. We\nthen havex+(0) = 1=M, andx\u0000(0) = 1\u00001=M. The\nreduced model is therefore a two-state voter model with\ninhomogeneous initial conditions.\nUsing Eq. (14) we have the following average density\nof active links in this reduced model,\nh\u001ared(t)i= 21\nM\u0012\n1\u00001\nM\u0013k\u00002\nk\u00001e\u0000t=\u001c: (16)\nWe note that this is the average fraction of links be-\ntween states + and \u0000in the reduced model. In the\noriginal model (before the reduction) this corresponds\nto the fraction of links connecting an individual in state\n\u000b= 1 with an individual in anyother state \f6= 1,\ni.e.,\u001ared=P\n\f>1\u001a1;\f. Carrying out the ensemble av-\nerage, and exploiting the symmetry between states in\nthe MSVM with homogeneous initial conditions we haveDP\n\f>1\u001a1;\fE\n=DP\n\f6=\u000b\u001a\u000b;\fE\nfor any \fxed \u000b. Hence, the\nresulting average density of active links in the MSVM is\nh\u001a(t)i=1\n2X\n\u000b6=\fh\u001a\u000b\f(t)i=M\n2h\u001ared(t)i: (17)\n(In this expression both \u000band\fare summed over). Us-\ning Eq. (16) we then obtain\nh\u001a(t)i=\u0018(M;k)e\u0000t=\u001cfort>t?; (18)\nwith\n\u0018(M;k)\u0011\u0012\n1\u00001\nM\u0013k\u00002\nk\u00001; (19)\nfor the multi-state voter model with homogeneous initial\nconditions. The decay time \u001cis as in Eq. (15). As before,\nthe exponential law in Eq. (18) is only valid after a short\ninitial transient of duration t?=O(N0).\nSimilar to the two-state model, we notice that the pre-\nfactor\u0018(M;k) in Eq. (18) is made up of the network-\nspeci\fc factor ( k\u00002)=(k\u00001), and the density of active\nlinks 1\u00001=Min Eq. (13), resulting from a con\fgura-\ntion in which equally many individuals hold each opin-\nion (x\u000b= 1=M8\u000b), and in which these individuals are\nplaced on the network at random. The network-speci\fc\nfactor only depends on the mean degree.6\n4. Test against simulations and metastable state in the\ncoarsening dynamics\nWe now test the predictions of the previous section in\nnumerical simulations. These were carried out on un-\ncorrelated networks with di\u000berent degree distributions,\nspeci\fcally the Erd os{Renyi and Barab\u0013 asi{Albert ensem-\nbles.\nWe \frst verify the validity of Eq. (18). We show\nthe time-evolution of the average density of active links,\nh\u001a(t)i, from simulations in Fig. 3, along with the analyt-\nical predictions from the pair approximation. As seen in\nthe \fgure, satisfactory agreement is found. In Fig. 3(b)\nwe re-plot the same data as in panel (a), but on a double\nlogarithmic scale. This allows us to focus on the early\nstages of the time evolution. As also appreciated in the\ninset, the average density of active links h\u001aiquickly de-\ncays to a value of \u0018= 0:6, in-line with the prediction of\nEq. (19).\nThis initial decay of the density of active links is a\nsignature of a rapid growth of local clusters of individ-\nuals with the same opinion state on the graph. As in\nthe binary VM [11], this initial decay is not described by\nthe exponential law for h\u001a(t)ifrom the pair approxima-\ntion. After this initial phase the system is in a partially\nordered metastable state, characterised by a density of\nactive links \u0018(M;k). The system remains in this state\ninde\fnitely in the limit of in\fnite system size, we note\nthat\u001c!1 forN!1 in Eq. (15). If there are \fnitely\nmany individuals in the network, then the system will\neventually exit this state, triggered by \ructuations. Fur-\nther ordering then occurs on a time scale of \u001c, and the\naverage density of active links decays exponentially.\nWe now make some further observations about the par-\ntially ordered state. As seen in Fig. 4(a), the density of\nactive links in this initial metastable state, \u0018(M;k), in-\ncreases with the mean degree of the network, k, and with\nthe number of opinion states, M. From Eq. (19) we \fnd\n\u0018(M;k)\n\u0018(2;k)= 2\u0012\n1\u00001\nM\u0013\n; (20)\nas con\frmed in Fig. 4(b). We note the limiting value\n\u0018(M;k)=\u0018(2;k)!2 forM!1 .\nFinally, we brie\ry discuss the time scale \u001cin Eq. (18),\nwhich is given by the expression in Eq. (15), and does not\ndepend on the number of opinion states M. As such the\ntime scale in the multi-state model is the same as that in\nthe conventional voter model with two opinion states. We\nnote that the network structure enters not only through\nthe mean degree k, but also through the second moment\nk2of the degree distribution. As a consequence the ex-\nponential decay of h\u001a(t)iat \fxed network size Nand\nmean degree is slower on an Erd os{Renyi graph than on\na Barab\u0013 asi{Albert network, see Figs. 3 and 5. As in the\ntwo-state model we have \u001c/Nfor complete graphs and\nER networks, but \u001c/N=lnNfor largeNin the BA\nnetwork [8, 10, 11].\n11 01 001 0001 00000.10.20.30.40.50.60.705 0001 00000.20.40.60.81\n1 01 000.50.60.71\n1 01 000.50.550.60.6550001 00000.10.20.30.40.50.60.7<\nρ>ξ (M = 4) = 0.6<ρ>(\nb)(a)t\n N = 4000    BA  ER \nN = 6000    BA  ER \nN = 10000  BA  ERFIG. 3: (a) Time-evolution of the average density of active\nlinks in the multi-state voter model with M= 4 for di\u000ber-\nent systems sizes N. Markers show results from simulations,\nstarted from random homogeneous initial conditions, and av-\neraged over 5000 realisations. We show data for Erd os{Renyi\ngraphs (ER, orange open symbols) and for Barab\u0013 asi{Albert\nnetworks (BA, black full symbols). All networks have mean\ndegreek= 6. Solid lines are the analytical prediction in\nEq. (18). Panel (a) is on linear-log scale. In panel (b) we\nshow the same data on double logarithmic scale, the density\nof active links shows a plateau \u0018\u00190:6, in-line with Eq. (19).\nThe lifetime of the plateau is \fnite for \fnite populations and\nincreases with N. The inset provides a further zoom-in, high-\nlighting the initial decay of the density of active links from its\ninitial value to \u0018= 0:6.\nB. Time-evolution of average entropy\nIn numerical simulations we have also investigated\nthe behaviour of the average entropy, hS(t)i=\n\u0000P\n\u000bhx\u000blnx\u000biover time. Simulation results are shown\nin Fig. 6. At long times the data is consistent with an\nexponential decay of the form\nhS(t)i=\u0018S(M;N )e\u0000t=\u001c;fort&t2; (21)\nsee Fig. 6(a) and Fig. 7. The decay time scale \u001cis the\nsame as that for the average density of active links [e.g.\nEq. (18)]. The time t2from which (approximately) the\ndecay of the mean entropy becomes exponential can be\nestimated as the mean time at which only at most two\nopinions are left in the population. This time can be\napproximated analytically (see Sec. IV C below). When\nM= 10, we \fnd t2\u00190:38Nfor complete graphs, and\nt2\u00190:07Non Barab\u0013 asi{Albert networks. These time7\n05 1 01 51.01.21.41.61.82.00.40.60.8 \nk = 4   k = 6 \nk = 8   k = 10 \nCGξ(M)/ξ(M=2)M\n(b)(a) \nk = 4   k = 6 \nk = 8   k = 10 \nCGξ(M,k)\nFIG. 4: (a) Density of active links, \u0018(M;k), in the meta-\nstable state as function of Mfor di\u000berent values of k. Markers\nshow simulations with homogeneous initial conditions for the\ncomplete graph (CG, orange stars), and for Erd os{Renyi net-\nwork with di\u000berent mean degrees (the model on the Barab\u0013 asi{\nAlbert network has the same plateaux values as on Erd os{\nRenyi graphs). This data is obtained by measuring h\u001a(t)i, and\nthen \ftting to an exponential decay. Lines are from Eq. (19),\nshown also for non-integer values of Mfor optical convenience.\nPanel (b) shows \u0018(M;k)=\u0018(2;k) as function of M. Symbols\nare the simulations from panel (a), the solid line is Eq. (20).\n2000600010000140001800020004000600080001000012000B\nA k = 6:  \nM = 2 \nM = 3 \nM = 4 \nEq. (15)τ     ER k = 6: \nM = 2 \nM = 3 \nM = 4 \nEq. (15)N\nFIG. 5: Time scale \u001cof the exponential decay of the average\ndensity of active links for Erd os{Renyi (ER) graphs (empty\nsymbols) and Barab\u0013 asi{Albert (BA) networks (full symbols).\nThe black and gray lines corresponds to Eq. (15) for ER ( k=\n6) and BA ( k= 6), respectively. In-line with Eq. (15) \u001c\nis independent of M, i.e., the time scale \u001cin the MSVM is\nidentical to that in the two-state model. Simulation data is\nobtained from measuring h\u001a(t)i, and a subsequent \ft to an\nexponential decay.points are indicated in Fig. 6(a). We stress that the devi-\nations from an exponential for t.t2are not short-lived\nas those for the average density of active links on net-\nworks. Instead we \fnd that t2scales linearly with the\npopulation size N.\nFocusing on short times (of order N0) in Fig. 6(b) we\nnote the absence of an initial drop of the mean entropy on\nnetworks. Instead the mean entropy remains at its initial\nvalue for homogeneous initial conditions, S(0) = lnM.\nThis is in contrast with the behaviour of the average den-\nsity of active links, compare with Fig. 3(b).\nThis suggests the following picture of the coarsening\nprocess on networks:\n(1) Once released from the homogeneous initial condi-\ntion the system \frst undergoes a quick local relaxation\nprocess during the time interval up to t=t?=O(N0).\nIn this phase the (average) density of active links reduces\nfrom its initial value 1 \u0000M\u00001to the value \u0018(M;k) given\nin Eq. (19). The average entropy however, remains un-\nchanged,hS(t)i=S(0) = lnM. This indicates that\nduring this phase some local ordering takes place on the\nnetwork (hence the reduction in h\u001ai), but that the pro-\nportions of individuals in the di\u000berent opinion states do\nnot materially change across the graph as a whole. One\npossible explanation is the formation of local domains.\nIn each domain a particular opinion starts to outnumber\nthe other opinion states. However, di\u000berent domains do\nnot `communicate', hence the e\u000bects of the local ordering\naverage out across the system.\n(2) After this initial phase the system tends towards\nconsensus, andh\u001a(t)idecays exponentially, with a time\nscale\u001c=O(N). The decay of the average entropy is not\nexponential until time t\u0019t2=O(N).\n(3) At long times ( t&t2), when only two opinions are\nleft in the system, the average entropy hS(t)ialso decays\nexponentially, on the same time scale \u001cas the average\ndensity of active links.\nWe will discuss this further in Sec. IV C.\nIV. PATH TO CONSENSUS IN INDIVIDUAL\nTRAJECTORIES\nWe now ask how representative the average behaviour\ndiscussed in Sec. III is of the dynamics of individual re-\nalisations. We \frst focus on the density of active links,\nand subsequently study the entropy of the distribution\nof individuals across the di\u000berent opinion states.\nA. Evolution of the density of active links for\nindividual realisations\n1. Sequence of plateaux in individual realisations\nFig. 8 illustrates the time evolution of the density of\nactive links for individual realisations of the model with8\n0.00.20.40.60.81.01.21.40.111\n1 01 001 0001 00000.1111 01 001 0001 0000120\n100020003000400050006000700080009000100000.1102 5005 0001231\n1 01 001 00023t\n N = 6000    BA (k = 8)  CG \nN = 10000  BA (k = 8)  CG(a)<\nS>t\n/N(\nb)<\nS><S> = log 10M = 10 N = 15000  SF   CG  ER  <\nS> = log 10\nFIG. 6: Time evolution of the average entropy hSifor a sys-\ntem started from homogeneous initial conditions and M= 10.\nPanel (a) shows results on complete graphs (CG) and on\nBarab\u0013 asi-Albert networks (BA, mean degree k= 8) on linear-\nlog scale. Markers are from simulations (averaged over 5000\nrealisations). Lines are Eq. (21) with \u001cgiven by Eq. (11) for\nthe complete graph, and Eq. (15) for the BA networks. Ver-\ntical dashed lines indicate the time t2beyond which there are\ntypically at most two opinions present in the population (see\nSec. IV C). Panel (b) shows hSion a doubly logarithmic scale\nfor the CG (black diamonds), BA networks ( k= 8, purple\ntriangles), and Erd os-Renyi graphs ( k= 8, orange squares),\nall forM= 10 and system size N= 15000. The horizon-\ntal dashed line is S=S(t= 0) = lnM. The inset shows a\nfurther zoom-in, and highlights that on networks there is no\nshort-time drop of the entropy from the initial value.\nM= 3 opinion states on a complete graph [panel (a)],\nand on an Erd os{Renyi graph [panel (b)].\nThe density of links is \frst found to \ructuate around\nan initial plateau (on networks this is preceded by a short\ntransient.) For the complete graph this plateau is located\nat a point consistent with the initial value of the density\nof active links for homogeneous initial conditions, \u001a=\n1\u0000M\u00001[see Eq. (13)]. For the Erd os{Renyi network the\ndensity at this plateau is consistent with that predicted\nby Eq. (19).\nSubsequently, the density of active links falls to a sec-\nond plateau, where it then spends some time before a\n\fnite-size \ructuation takes the system to one of the ab-\nsorbing states, where \u001a= 0.\nUnlike the initial plateau, the value of the density of\nactive links at this second plateau di\u000bers from realisation\nto realisation (see Fig. 8). The process leading to this\n05 0001 00001 50002 0000200040006000800010000N\nBA k = 8:  \nM = 2  \nM = 4  \nM = 10 \nEq.(15)τER k = 8:M\n = 2M\n = 4M\n = 10 \nEq.(15)CG:M\n = 2M\n = 4M\n = 10FIG. 7: Decay time of the average entropy for the complete\ngraph (CG), and for Erd os{Renyi graphs (ER) and Barab\u0013 asi{\nAlbert networks (BA) with mean degree k= 8 when, in av-\nerage, there are two opinion states left. Markers are from \fts\nof simulation data for hS(t)ito an exponential at large times.\nThe black solid line correspond to Eq. (15), evaluated for the\nER graph where hence \u001c\u0019N=2 fork= 8, identical to the\nresult for the complete graph. The gray line is from Eq. (15)\nevaluated for BA graphs with k= 8.\n0.00.20.40.60\n1 5003 0004 5006 0000.00.20.40.6(a)t\n(b)ρ\nFIG. 8: Time evolution of the density of active links for three\nindividual realizations indicated by di\u000berent colors. Panel (a)\nis on a complete graph ( M= 3;N= 6000), panel (b) for\nan Erd os{Renyi graph ( k= 6,M= 3,N= 6000). Sim-\nulations were started from homogeneous initial conditions.\nThe dashed lines indicate the values obtained from Eq. (13)\n[h\u001a(0)i= 0:66] and Eq. (19) [ \u0018(M= 3;k= 6) = 0:66], for the\ncomplete graph and Erd os{Renyi graphs, respectively.\nintermediate plateau can be better understood from the\ninspection of the evolution of the number of agents in\neach opinion state in Figs. 9(b) and 10(b). As shown,\nopinions go extinct one after the other. Each extinction\ntakes the density of active links \u001ato a new plateau (at\na value lower than that of the previous plateau), until\nconsensus is reached and \u001a= 0.\nWhenKextinctions have taken place in the model9\n0.00.30.60.91.21.50\n5 0010001500200025000.00.20.40.60.81.0(a)(\nb)CG N = 3000 \nρ   St\n x1  x2  \nx3  x4\nFIG. 9: Panel (a): Evolution of the density of active links\n(purple curve), of the entropy (green curve) for the model\non a complete graph. Panel (b) shows the fraction of agents\nin each opinion state. Data is from a single simulation with\nM= 4 andN= 3000. The vertical dotted lines indicate the\npoints in time at which the \frst and second opinions go ex-\ntinct respectively. Dashed lines represent the plateau value \u0018\nconsidering a voter model with `4', `3', and `2' opinion states\nand the corresponding initial conditions; for the second and\nthird plateau, the value was computed considering the num-\nber of agents in the surviving opinions once the extinction of\nopinions `4' and `2' took place.\nwith initially Mopinions, we are left with a MSVM with\nL=M\u0000Kstates. Crucially however, the initial con-\ndition for this model with Lstates is the result of the\nprevious dynamics up to the time when the K-th extinc-\ntion takes place. This initial condition in turn determines\nthe value of the density of active links at the subsequent\nplateau. Given that di\u000berent realisations result in di\u000ber-\nent con\fgurations at the time of the extinctions the en-\nsuing plateau densities also vary across realisations. This\nis the reason for the spread of plateaux in Fig. 8.\n2. Density of active links at the di\u000berent plateaux\nWe now proceed to calculate the mean density of ac-\ntive links in the model with initially Mopinions at the\n\frst point in time where only Lopinions are left. Sup-\npressing any possible dependence on M, we will denote\nthis quantity byh\u001aiL.\nWithout loss of generality we assume that the Lopin-\nions left in the population are \u000b= 1;:::;L . For an all-\nto-all interaction We then have\n\u001a=1\n2LX\n\u000b=1[2x\u000b(1\u0000x\u000b)]; (22)\n0.000.300.600.901.200\n1 00020003000400050000.000.250.500.751.00ER k = 6 N = 6000 \nρ   S(a)(\nb)t\n x1  x2  x3FIG. 10: Panel (a): Evolution of the density of active links\n(purple curve) and of the entropy (green curve) for the model\non Erd os-Renyi graphs. Panel (b) shows the fraction of agents\nin the di\u000berent opinion states. Data is from a single simula-\ntion withM= 3 andN= 6000. The graph has mean degree\nk= 6. The vertical dotted line indicates the time at which\nthe \frst opinion state becomes extinct. Dashed lines repre-\nsent the plateau value \u0018considering a voter model with `3'\nand `2' opinion states and the corresponding initial condi-\ntions; the second plateau value was obtained considering the\nnumber of agents in the surviving opinions once opinion state\n`2' disappears.\nwherePL\n\u000b=1x\u000b= 1. Each term 2 x\u000b(1\u0000x\u000b) accounts for\nlinks involving individuals of type \u000bconnected with in-\ndividuals of any other opinion, and the overall pre-factor\n1=2 corrects for double counting. After taking an average\nover realisations we thus have\nh\u001aiL=*LX\n\u000b=1x\u000b(1\u0000x\u000b)+\n=LZ1\n0dxPL(x)x(1\u0000x); (23)\nwhere the quantity PL(x) is the distribution of the frac-\ntion of agents found in a particular opinion state when\nonlyLopinions are left. We have used the fact that, by\nsymmetry, no opinion state is preferred over any other.\nThe distribution PL(x) can be obtained making the\nfollowing hypothesis. We assume that, at the extinction\npoint leaving only Lopinions in the system, all con\fgu-\nrations with x\u000b\u00150 andPL\n\u000b=1x\u000b= 1 are equally likely,\ni.e., the distribution of ( x1;:::;xL) at that time is as-\nsumed to be\nPL(x1;:::;xL) = [(L\u00001)!]\u0002\u000e LX\n\u000b=1x\u000b\u00001!\n;(24)\nwhere\u000e(\u0001) is the Dirac delta function.10\nThe distribution PL(x) in Eq. (23) is the single-variable\nmarginal of PL(x1;:::;xL),\nPL(x1) = (L\u00001)!Z\ndx2\u0001\u0001\u0001dxL\u000e LX\n\u000b=1x\u000b\u00001!\n= (L\u00001)(1\u0000x1)L\u00002: (25)\nFor further details see Appendix A.\nWe can then directly calculate h\u001aiLin Eq. (23),\nh\u001aiL=L\u00001\nL+ 1: (26)\nThe argument so far applies to complete graphs, where\nthe initial condition is known to directly set the typical\ndensity of active links at the plateau that follows (see\nSec. III). As also discussed in Sec. III, the system under-\ngoes a brief transient when it is started on an uncorre-\nlated graph, and the subsequent plateau value of h\u001aiis\nobtained applying a multiplication factor ( k\u00002)=(k\u00001).\nFor uncorrelated graphs we therefore predict\nh\u001aiL=k\u00002\nk\u00001L\u00001\nL+ 1: (27)\n3. Test against simulations\nWe now test these predictions against simulations.\nFirst, we verify the validity of our hypothesis of a \rat dis-\ntribution for ( x1;:::;xL) at the \frst point in time when\nthere are only Lopinions in the population. Fig. 11 shows\nsimulation results for the marginal distribution for x\u000bfor\ndi\u000berent choices of MandLon complete graphs and on\nErd os{Renyi networks. As can be seen from the \fgure,\nthese simulations are consistent with the predictions of\nEq. (25).\nWe next introduce the concept of restricted ensemble\nat a given time. This is the ensemble of trajectories\nthat, at this time, have precisely Lsurviving opinions.\nIn Fig. 12 we show the average of the density of active\nlinks over this restricted ensemble. The data in panel\n(b) was obtained by \frst averaging over the restricted\nensemble at any given time, and subsequently an aver-\nage over time is performed. Fig. 12(b) thus demonstrates\nthat the average density of interfaces at the \frst point in\ntime at which there are only Lopinions in the system is\ngiven by Eqs. (26) and (27) for all-to-all interaction and\non networks respectively, and that this average density of\nactive interfaces is then maintained by the system until\nthe next extinction occurs.\nB. Connection to exponential decay of the\nensemble averaged density of links\n1. Sequence of `jumps' in the model with multiple opinions\nWe have described the dynamics at the ensemble level\nand at the level of individual realisations. We now pro-\n0120\n.00 .20 .40 .60 .81 .00123x\nαP(xα)(\nc)(b)(a) \nCG, M = 5, L = 4 \nP(xα) = 3 (1 - xα)2012 ER, M = 5, L = 2 \nP(xα) = 1 \nCG, M = 4, L = 3 \nP(xα) = 2 (1 - xα)FIG. 11: Marginal distribution PL(x\u000b) for the fraction of\nagents in any one opinion at the \frst point in time when\nLopinions are left in a model with initially M > L states.\nGray bars are from numerical simulations ( N= 5000, av-\neraged over 5000 realisations), the solid lines are Eq. (25).\nPanel (a) is for Erd os-Renyi networks, panels (b) and (c) for\ncomplete graphs.\nceed to a characterization in terms of the restricted en-\nsembles that we have just introduced.\nThe average density of active links, h\u001aidecays expo-\nnentially, as indicated in Eqs. (10) for complete graphs,\nand in Eq. (14) for uncorrelated networks. The decay\ntime scales are given in Eqs. (11) and (15) respectively.\nIndividual realisations can be characterised as under-\ngoing a sequence of `jumps' in the density of active links,\nfrom one plateau to another, as illustrated in Fig. 13.\nEach of these jumps is associated with the extinction of\nan opinion. As we have seen the density of active links\nalong the sequence of plateaux di\u000bers across realisations.\nWe have established that the average density of active\nlinks at the plateau at which Lopinions are left in the\npopulation is given by Eq. (26) for complete graphs, and\nby Eq. (27) on uncorrelated networks.11\n01 0002 0003 0000.00.20.40\n123456789100.000.250.500.75(a)t\n<ρ>2 = 0.28<ρ>2<ρ>2 = 0.33 \n                CG:  M = 6, L = 2, N = 2000  \nBA  ER:  M = 5, L = 2, N = 6000 (\nb)<\nρ>L \nCG \nER k = 6 \nBA k = 8L\nFIG. 12: (a) Evolution of the density of links in realisations for\nwhich only L= 2 opinion states are left in the system for the\nCG, and the ER and BA networks with k= 8. The plateau is\nlocated ath\u001aiL=2\u00190:33 for the CG and at h\u001aiL=2\u00190:28 for\nthe networks, in good agreement with Eq. (26) and Eq. (27),\nrespectively. Panel (b) shows the location of the intermediate\nplateaux,h\u001aiLas a function of Lin a model with initially M=\n15 states. Markers are from simulations on CG (triangles), on\nBA networks (squares), and on ER networks (circles). Lines\nare from Eqs. (26) and (27) respectively.\n2. Exponential decay in the ensemble on complete graphs\nWe would now like to connect the observations at en-\nsemble level with those at the level of realisations. In\norder to do this we need information about the typical\nresidence time in the di\u000berent plateaux. We \frst discuss\nthis for the model on the complete graph.\nFollowing [20] the mean time the system spends in the\nmetastable state with Lopinions can be estimated as\nthe di\u000berence between the mean consensus times in voter\nmodels with initially LorL\u00001 states respectively. The\nconsensus time from a state with Kopinions in the model\nwith all-to-all interaction is given by [20]\nhTN(K)i=NK\u00001\nK; (28)\nfor a system of size N, and assuming that all initial condi-\ntions are equally likely. We then have the following mean\nresidence time in the state with exactly Lopinions,\nh\u0001tiL=hTN(L)i\u0000hTN(L\u00001)i\n=N\nL(L\u00001): (29)\nFIG. 13: Illustration of the jump process di\u000berent realisa-\ntions of the MSVM undergo. Each jump is associated with\nthe extinction of one opinion. The density of active links in\nthe resulting sequence of plateaux di\u000bers from realisation to\nrealisation (as indicated by the scatter of markers), the mean\ndensity in the plateaux is given by Eqs. (26) or (27) respec-\ntively. In the \fnal state ( L= 1) consensus has been reached,\nand hence\u001a= 0. The residence time in each level varies from\nrealisation to realisation as well, the mean time spent in level\nLis given by Eq. (29). The illustration is for a model with\ninitiallyM= 5 states. For homogeneous initial conditions\none then has \u001a= 4=5.\nWe also know that the mean density of active links at\nthe plateau with Lopinions ish\u001aLi= (L\u00001)=(L+ 1)\n[Eq. (26)]. Therefore the mean change of the density of\nlinks when transitioning from the plateau with Lopinions\nto that with L\u00001 opinions is\nh\u0001\u001aiL!L\u00001=h\u001aiL\u00001\u0000h\u001aiL\n=\u00002\nL(L+ 1): (30)\nUsing these results for h\u0001\u001aiL!L\u00001,h\u0001tiLandh\u001aiLwe\nconclude\nh\u0001\u001aiL!L\u00001\nh\u0001tiL=\u00002\nNh\u001aLi: (31)\nThe left hand side is a proxy for the time derivative of\nh\u001aiwhen there are exactly Lopinions in the system. The\nmean density of active links in this situation is h\u001aiL.\nTherefore we haved\ndth\u001aiL=\u0000(2=N)h\u001aiL, and we re-\ncover the exponential decay in Eq. (10). The time scale\nof the decay matches that in Eq. (11), up to the replace-\nmentN!N\u00001 (which becomes irrelevant for large\nN).\nThe exponential decay law [Eq. (12)] for the average\ndensity of active links in the ensemble can therefore be\nrecovered from the picture of jumps in Fig. 13, and is\na consequence of the speci\fc relation between the level\nspacings and the mean residence time in each level.12\n24 6 8 1 020003000<\nTN(K)>(\nK-1)/KER N = 2000k\nFIG. 14: Test of Eq. (32) for the model on Erd os-Renyi\nnetworks with N= 2000. Markers represent the ratio\nhTN(K)i=[(K\u00001)=K] from simulations. The line is [ k\u0000\n1)k2]=[(k\u00002)k2], wherek2=Np(1\u0000p), withp=k=(N\u00001).\n3. Uncorrelated networks\nA similar argument applies on uncorrelated networks.\nTheh\u001aiL, and hence theh\u0001\u001aiL!L\u00001, are then multiplied\nby the common factor ( k\u00002)=(k\u00001), see Eq. (27). This\ntherefore drops out in Eq. (31). In order to recover the\nexponential decay in Eq. (18) with a decay time as in\nEq. (15) we then need to show that the result of [20]\ngeneralises to uncorrelated graphs as follows,\nhTN(K)i=NK\u00001\nK\u0002(k\u00001)k2\n(k\u00002)k2: (32)\nThis can be demonstrated using a reduction to an ef-\nfective two-state model, and the properties of the two-\nstate VM on uncorrelated graphs, where it is known that\nthe relevant time scales undergo re-scaling by a factor\n[(k\u00001)k2]=[(k\u00002)k2] relative to the case of all-to-all\nconnectivity [11]. This is described in more detail in Ap-\npendix B. The validity of Eq. (32) is demonstrated in\nFig. 14.\nC. Evolution of the entropy for individual\nrealisations and in the ensemble average\n1. Absence of simple exponential decay at ensemble level\nIn Figs. 9 and 10 we also observe intermediate plateaux\nin the time evolution of entropy for individual realisa-\ntions. In Fig. 15(a) we therefore proceed as we did for\nthe density of active links and perform averages over re-\nstricted ensembles of realizations with \fxed Lsurviving\nopinions,L < M , at di\u000berent times. In the example in\nthe \fgure, we \fnd that hSiL=2\u00190:5.\nAnalytically, we \fnd\nhSiL=\u0000LZ1\n0dxPL(x)xlnx\n=HL\u00001; (33)where theHL=PL\n`=1(1=`) are the harmonic numbers.\nThe distribution PL(x) is given in Eq. (25). The predic-\ntion in Eq. (33) is tested against numerical simulations\nin Fig. 15(b).\nWe can now use this to understand in more detail why\nthe decay of the entropy at ensemble level does not follow\nan exponential law initially, but becomes exponential at\nlong times. From Eq. (33) we \fnd h\u0001SiL!L\u00001= 1=L.\nUsing Eq. (29) we then have\nh\u0001SiL!L\u00001\nh\u0001tiL=\u0000L\u00001\nN(HL\u00001)hSLi: (34)\nInterpreting the left-hand side again as a time derivative,\nwe therefore have\nd\ndthSi=\u00001\n\u001cS(L)hSi (35)\nin the time regime when there are Lopinions left in the\nsystem, with\n\u001cS(L) =NHL\u00001\nL\u00001: (36)\nfor complete graphs, and \u001cS(L) =NHL\u00001\nL\u00001\u0002(k\u00001)k2\n(k\u00002)k2on\nuncorrelated graphs.\nThe pre-factor\u00001=\u001cS(L) on the right in Eq. (35) ex-\nplicitly depends on L. In the corresponding equation (31)\nfor the average density of active links the pre-factor is in-\nstead constant. This is why the average density of links\ndecays exponentially, and the average entropy does not.\nForL= 10,\u001cs(L) evaluates to (approximately) 0 :21N,\nforL= 9 to 0:23N, forL= 8 to 0:25Nand so on.\nThis demonstrates that the relative decay of the average\nentropyhS(t)islows down as more and more opinions\nbecome extinct.\nWe note that at any one time tdi\u000berent realisations of\nthe MSVM process will be in states with di\u000berent num-\nbers of opinions Lleft in the population. It is therefore\nnot straightforward to aggregate the mechanics of the\ndecay of entropy for a \fxed value of L[Eq. (35)] into a\nglobal picture for the behaviour of entropy at the ensem-\nble level. This is at variance with the decay of the average\ndensity of links. Given that all \u001cs(L) scale linearly in N,\nwe can however conclude that all timescales governing\nthe decay of entropy are O(N), as also demonstrated in\nFig. 6 (a).\n2. Regime at large times\nOne can identify a regime in which most realisations\nwill either have reached consensus or in which the pop-\nulation contains only two opinions. In a model with ini-\ntiallyMopinions this will be the case at times which are\ngreater than\nt2\u0011MX\nL=3h\u0001tiL; (37)13\nThis is the sum of average times spent in states with\nM;M\u00001;:::; 3 opinion states respectively, i.e., it is the\nmean time until only two opinion states are left in the\nsystem.\nWhenL= 2 opinions are left, we \fnd \u001cS(L) =N=2\nfor the complete graph in Eq. (36). In this long-time\nregime, we then haved\ndthSi2=\u0000(2=N)hSi2, leading to\nan exponential decay with the same time scale as that\nfor the average density of active links. On uncorrelated\nnetworks this time scale is multiplied by the factor ( k\u0000\n1)k2=[(k\u00002)k2].\nWe can interpret this as the dynamics of a two-level\nsystem, see Fig. 16 for an illustration. Each realisation\nwill either have two opinions left in the population, or\nhave reached consensus. There is hence an active state\n(two opinions) and a passive consensus state. Realisa-\ntions in the active state have positive densities of active\nlinks and positive entropies. There is some scatter across\nrealisations, averages among active realisations are given\nbyh\u001ai2>0 andhSi2>0, respectively. In the consen-\nsus state both \u001aandSare zero. Each realisation is \frst\nin the upper level ( L= 2), and then reaches consensus\nthrough a sudden \ructuation. This can be thought of as\na Markovian jump process. Realisations transition from\nthe upper to the lower level independently, with rate 2 =N.\nThe population of the upper level hence decays exponen-\ntially in time. This then leads to the exponential decay\nof bothh\u001aiandhSi.\nV. STABILISING INTERMEDIATE STATES\nTHROUGH THE INTRODUCTION OF ZEALOTS\nThe sequence of plateaux discussed in previous sec-\ntions is a consequence of long-lived metastable states in\nindividual realisations. Eventually a \fnite system leaves\neach of these states, and proceeds to the next plateau\nand \fnally to absorption. In this section, we now seek to\nengineer a MSVM in which these intermediate states are\nstable inde\fnitely. We do this by adding zealots to the\npopulation, that is, agents who do not change their opin-\nion state during the dynamics [17{19]. If such agents are\nadded for more than one opinion state, then the system\nno longer has any absorbing states, and the dynamics\ncontinues inde\fnitely. The question we address here is if\nand how a con\fguration of zealots can be chosen so as to\nstabilise the meta-stable states in Sec. IV.\nA. Competing zealots\nWe consider a population of Nconventional agents and\nZzealots. As before n\u000bis the number of agents in each\nopinion state \u000b= 1;2;:::;M (not including zealots). We\nwritez\u000bfor the number of zealots for opinion state \u000b.\nRegular agents can change their opinion by interacting\nwith another regular agent or a zealot. Zealots never\nchange their opinion.\n2002 2004 2006 2000.250.500.752\n345678910012(a)<\nS>2t\nM = 3, L = 2, N = 10000  \nER  BA  CG<S>2 = 0.5(\nb)<\nS>LL\n CG \nER k = 8 \nBA k = 6FIG. 15: (a) Time evolution of the entropy, at any time aver-\nages only over realisations for which exactly L= 2 opinions\nare present in the population. Simulations are for the CG,\nER, and BA graphs of size N= 10000 and initially M= 3\nopinion states. The plateau is located at hSiL= 0:5 in agree-\nment with Eq. (33). (b) Symbols show simulation results for\nhSiLas a function of Lin a model with initially M= 15\nopinions. Markers are from simulations for the CG (squares),\nER graphs (triangles) and BA networks (circles). These are\nobtained from performing a time average on data such as the\none in panel (a). The solid line is the analytical prediction in\nEq. (33).\nFIG. 16: Illustration of the dynamics when only at most two\nopinions are left in the population. Some realisations are in\nthe active state (two opinions). Each such realisation is in\na meta-stable state with di\u000berent plateau values for \u001aandS\nacross realisations. Transitions to consensus occur with rate\n\u001cS(L= 2)\u00001= 2=N, leading to exponential decay of h\u001aiand\nhSiwith a decay time scale N=2.14\n11 01 001 0001 00000.30.40.50.60.70.82\n4 6 8 1 00.20.40.60.80\n2 4 6 8 1 00.40.6(a)t\nM = 2<ρ><\nρ> = 0.33M = 5M\n = 4M\n = 3<\nρ><\nρ>L<ρ>   Eq. (26)<\nρ>L<ρ>   Eq. (27)(b)L\n<\nρ>(c) \n<ρ>, L = 3  <ρ>, L = 2z\nFIG. 17: (a) Time evolution of the average density of active\nlinks (complete graph) when there are two zealots of di\u000ber-\nent opinions. At \fxed number of opinions M, the size of the\npopulation was varied ( N= 2000;4000;and 6000). The hor-\nizontal dashed line indicates the value h\u001ai= 0:33 obtained\nfrom setting L= 2 in Eq. (26). (b) Filled symbols show the\nstationary density of active links, h\u001ai, in a model with one\nzealot in each of Ldi\u000berent opinion states. Squares are for\na complete graph, circles for ER networks. Open symbols\nshow the density h\u001aiLin a model with initially M= 15 states\n(and no zealots). Stars are for a complete graph, triangles\nfor ER networks. The lines are from Eqs. (26) and (27) re-\nspectively. (c) Simulation results for the stationary density of\nactive links in a model with Lopinions and zzealots in each\nof these opinions. Simulations are on complete graphs with\nsystem size N= 4000.\nWe \frst focus on the model on a complete graph. The\ntransition rates for this model are then\nT\u000b!\f=n\u000b(n\f+z\f)\nN+Z(38)\nSuppose now, we start the VM with initially Mstates,\nand with two zealots of two di\u000berent opinions. The re-\nmainingM\u00002 opinions will go extinct eventually. In\nFig. 17(a) we show the time evolution for h\u001aifor thissystem. The stationary density of active links is found\nash\u001ai\u0019 0:33 in simulations, irrespective of the initial\nnumber of opinions M. This density of active links is\nconsistent with that at the plateau for L= 2 in the model\nwithout zealots in the previous section [Eq. (26)].\nIf, more generally, we populate Lopinions with one\nzealot each, then, as seen in Fig. 17 (b), the stationary\ndensity of active links also agrees with the plateau value\nh\u001aiLin the model without zealots. This is found both on\nthe complete graph and on uncorrelated networks.\nThe data in panel (c) of Fig. 17 shows that the agree-\nment in (a) and (b) between the density of active links of\nthe model with one zealot in each of Lopinions and the\nplateauh\u001aiLin a zealot-free model with initially M >L\nopinions only holds when there is precisely one zealot in\neach of the Lopinions. We further corroborate this in\nthe next two subsections.\nB. Flat stationary distribution of opinions when\nthere is one zealot in each opinion state\n1. Analytical treatment of the model on a complete graph\nWe can demonstrate analytically that the stationary\ndistribution [in the space ( n1;:::;nL)] of the model with\nzzealots in each of Lopinion states on the complete\ngraph is \rat if and only if z= 1. To do this we consider\nthe master equation\ndP(n)\ndt=X\n\u000b6=\fP(n\u000b+ 1;n\f\u00001)(n\u000b+ 1)[(n\f\u00001) +z]\nN+Z\n\u0000X\n\u000b6=\fP(n)n\u000b(n\f+z)\nN+Z; (39)\nwhere the notation P(n\u000b+ 1;n\f\u00001) is a shorthand\nforP(E\u000bE\u00001\n\fn), with the raising operator E\u000bde\fned\nin Sec. II E. Direct algebra shows that P(n) = const. is\na stationary solution of Eq. (39) if and only if z= 1.\n2. Numerical evidence for the model on networks\nWhile an analytical solution for the model on networks\nis not easily available, simulation results are consistent\nwith the assertion that placing one zealot in each opin-\nion state leads to a \rat stationary distribution in the\nspace of (n1:::;nL). We show the marginal stationary\ndistribution P(x\u000b) for the model with zzealots in each\nofLopinions in Fig. 18. For z= 1 these marginals are\nwell described by the expression in Eq. (25), which is in\nturn derived from the assumption of a \rat distribution\nP(x1;:::;xL). Panels (b), (d), and (f) on the other hand\ndemonstrate that the marginals are markedly di\u000berent\nwhenz6= 1.15\n0120\n120\n.00.20.40.60.81.00123L\n = 4 z = 4x\nα(f)(d)(b)0\n123L\n = 3 z = 3L = 2 z = 20\n.00.20.40.60.81.00123x\nαP(xα)(\ne)(c)(a)L\n = 4 z = 1012L = 2 z = 1L\n = 3 z = 1\nFIG. 18: Marginal distribution PL(x\u000b) for the fraction of\nagents in any one opinion for a MSVM with zzealots in each\nofLopinions. Gray bars are from numerical simulations for\nER graphs of size N= 1000 averaged over 5000 realisations.\nThe solid lines are the prediction of Eq. (25), derived from\nthe assumption of a \rat stationary distribution.\nVI. SUMMARY AND CONCLUSIONS\nIn summary, we have investigated the approach of\nmulti-state voter models to consensus. We have distin-\nguished between the behaviour of individual realisations\nand that of the ensemble average, and we have compared\ndescriptions in terms of local and global properties. We\nhave described local order by the density of active links,\nand global order by a time-dependent entropy. On net-\nworks, we \fnd that local ordering can occur without\nglobal ordering. There is an initial coarsening process\nup to a time t?, in which the average density of active\nlinks decays to a value \u0018(M;k) on networks, while the\naverage entropy remains constant. The time scale t?is\nindependent of the size of the system. This initial decay\nis not observed in a complete graph where there is no\ndistinction between local and global order.\nAt the level of an ensemble average, we further \fnd\nthat the density of active links decays exponentially after\ntimet\u0003, with a time constant \u001cwhich diverges with sys-\ntem size and which is independent of the number of ini-tial opinions. This is observed both on complete graphs\nand on uncorrelated networks. As in the model with two\nopinion states, the amplitude and time scale of the ex-\nponential decay depend on the structure of the graph.\nThe amplitude and the time scale can be characterised\nanalytically using a pair approximation combined with a\nreduction to an e\u000bective two-state dynamics.\nThe behaviour of the average entropy is more intricate.\nWe \fnd non-exponential behaviour up to approximately\nthe time at which only two opinions survive. For larger\ntimes the entropy decays exponentially with the same\ntime constant \u001cas the density of active links. All time\nscales associated with the decay of the average entropy\ndiverge with the system size. The time scale t?(which\nis of orderN0) has no particular signi\fcance for the av-\nerage entropy. For large systems, the average entropy\ntherefore remains at its initial value for very long times.\nOn networks this is in contrast with the density of active\nlinks, which remains at a value associated with partial\nlocal ordering and which is lower than the initial density\nof active links.\nIndividual realisations undergo a sequence of extinc-\ntions of opinion states, akin to a `jump process'. We\n\fnd that each realisation remains in plateau values of the\nactive links and entropy between successive extinctions.\nThe location of these plateaux varies from realisation to\nrealisation, and is determined by the con\fguration of the\nsystem at the time of the preceding extinction. The non-\nequilibrium ensemble average does not provide a proper\ndescription of this process, the individual realisations are\nnot self-averaging.\nTo describe the ordering process we therefore introduce\nrestricted ensembles of realisations which, at a \fxed time,\nhave a given number of surviving opinions. Averaging\nover these restricted ensembles in simulations allows us\nto study the mean plateau values exhibited by individual\nrealisations. The average location of the plateaux can\nalso be estimated analytically. The picture we develop is\nconsistent with a maximum spread of con\fgurations at\nthe time of the intermediate extinctions. Using results\nfrom Ref. [20] for the mean time between extinctions we\nare also able to recover the exponential decay of the av-\nerage density of active links from the jump process for\nindividual realisations.\nThe restricted ensembles are a useful concept, and al-\nlow us to identify partially ordered states along the way\nto consensus. These states are very long lived in large\npopulations, but eventually the system exits from these\nstates. Arguably, the construction of these ensembles of\nrealisations with Lopinions in the system is also some-\nwhat arti\fcial. We have however shown that these par-\ntially ordered states can be engineered as genuine sta-\ntionary states. This is achieved through the introduction\nof precisely one zealot in each of the Lopinions.\nIn closing, we think that the combination of global\nand local measures of order provides an interesting way\nof looking at the coarsening dynamics in models of opin-\nion formation. Our work also highlights that the time-16\nevolution of individual realisations can be very di\u000berent\nfrom that of the ensemble average. Randomness can sig-\nni\fcantly in\ruence individual trajectories, and the e\u000bects\ncan last for substantial amounts of time. In other words,\n`history is contingent' [33]. The average path to consen-\nsus by opinion extinction is therefore of limited use for\nthe description of a given historical empirical occurrence.\nAcknowledgements\nWe acknowledge funding from the Spanish Ministry\nof Science, Innovation and Universities, the Agency AEI\nand FEDER (EU) under the grant PACSS (RTI2018-\n093732-B-C22), and the Maria de Maeztu program for\nUnits of Excellence in R&D (MDM-2017-0711). We ac-\nknowledge enlightening discussions with Ra\u0013 ul Toral.\nAppendix A: Normalisation and marginals of the\ndistribution PL(x1;:::;xL)\n1. Simplex in Kdimensions\nFor anyK2Nandy\u00150, we de\fne\nSK(y) =f(x1;:::;xK) :x\u000b\u00150;KX\n\u000b=1x\u000b\u0014yg;(A1)\nas the simplex of size yinKdimensions. We de\fne its\nvolume inK-dimensions\nVK(y) =Z\nSK(y)dx1:::dxK; (A2)\nand we have VK(y) =yKVK(1) by simple scaling.\nTo evaluate VK(1) we note\nVK(1) =Z1\n0dx1Z\nSK\u00001(1\u0000x1)dx2:::dxL\n=VK\u00001(1)Z1\n0dx1(1\u0000x1)K\u00001\n=VK\u00001(1)\nK: (A3)\nWe also have V1(1) = 1. By induction therefore, VK(1) =\n1=K!, and\nVK(y) =yK\nK!: (A4)\n2. Normalisation of PL(x1;:::;xL)\nWe start from the de\fnition\nPL(x1;:::;xL) =A\u000e(x1+:::+xL\u00001); (A5)where thex\u000bare required to be non-negative, and where\nAis the appropriate normalisation constant. UsingR\ndx1:::dxLPL(x1;:::;xL) = 1, we have after integrat-\ning overxL,\nA\u00001=Z\nSL\u00001(1)dx1:::dxL\u00001\n=VL\u00001(1) =1\n(L\u00001)!: (A6)\nTherefore,A= (L\u00001)!.\n3. Single-variable marginal of PL(x1;:::;xL)\nWe now calculate the single-variable marginal of\nPL(x1;:::;xL),\nPL(x1) =Z\ndx2:::dxLPL(x1;:::;xL): (A7)\nUsing the normalisation constant in Eq. (A6) we have\nPL(x1) = (L\u00001)!\u0002Z\ndx2:::dxL\u000e(x1+x2+\u0001\u0001\u0001+xL\u00001)\n= (L\u00001)!Z\nSL\u00002(1\u0000x1)dx2:::dxL\u00001\n= (L\u00001)!VL\u00002(1\u0000x1): (A8)\nHence, using Eq. (A4), we \fnd\nPL(x1) = (L\u00001)!\u0002VL\u00002(1\u0000x1)\n= (L\u00001)(1\u0000x1)L\u00002: (A9)\nAppendix B: Extinction and consensus times in\nall-to-all geometries and on graphs\nIn Sec. IV B of the main text we use a result for the av-\nerage consensus time of the multi-state voter model with\nKopinion states on an all-to-all geometry, and when ini-\ntial conditions are chosen at random with \rat distribu-\ntion from the simplex de\fned byPL\n\u000b=1x\u000b= 1. Speci\f-\ncally,\nhTN(L)i=NL\u00001\nL: (B1)\nThis was previously reported in Starnini et al. [20].\nStarnini et al. derive this from the backward Fokker-\nPlanck equation of the multi-state model (valid in the\nlimit of large, but \fnite N), and using a separation ansatz\nexploiting the exchange symmetry between the di\u000berent\nopinion states.\nWe use this consensus time to obtain the mean time,\nhtiL!L\u00001that elapses in a model with initially Lopinions\nuntil the \frst extinction, assuming again a uniform distri-\nbution of initial conditions in the simplexPL\n\u000b=1x\u000b= 1.17\nThe main purpose of this appendix is to justify the\ngeneralisation of the expression for the consensus time\nhTN(L)ito uncorrelated networks, Eq. (32). For conve-\nnience we repeat this expression here,\nhTN(L)i=NL\u00001\nL\u0002(k\u00001)k2\n(k\u00002)k2: (B2)\nOur argument is based on two principles: (a) hTN(L)i\ncan be obtained from the so-called `mean conditional con-\nsensus time' of a suitable two-state model. This applies\nboth in the case of all-to-all interactions and on networks.\n(b) The mean conditional consensus time for the two-\nstate model in turn can be calculated (in the limit of\nlarge, but \fnite N) from the backward Fokker{Planck\nequation (BFPE) of the respective model. It is known\nthat the Fokker{Planck equation for the two-state model\n(and hence also the BFPE) on an uncorrelated network\ncan be obtained from that for the model with all-to-all\ninteractions by a re-scaling of time by a factor of(k\u00001)k2\n(k\u00002)k2\n[11]. This is also discussed in [8, 9] although the factor\nis slightly di\u000berent as explicitly acknowledged in [11].\nWe now address (a) and (b) in turn.\n1. Reduction to two-state model\nSuppose the dynamics of the L-state model is started\nfrom an initial condition x= (x1;:::;xL) (withP\n\u000bx\u000b=\n1) at timet= 0. Writing pC(x;t) for the probability that\nconsensus (on any opinion) has been reached by time t,\nwe have\npC(x;t) =LX\n\u000b=1f\u000b(x;t); (B3)\nwheref\u000b(x;t) is the probability that consensus on opin-\nion\u000boccurs by time t. We note that the events on the\nright in Eq. (B3) are all mutually exclusive.\nThe arrival time distribution at consensus (on any\nopinion) is then dpC(x;t)=dt, and the density of arrivals\n(per time) at consensus on opinion \u000bisdf\u000b(x;t)=dt.\nWe note that\nq\u000b(x)\u0011Z1\n0dt0f\u000b(x;t0) (B4)\nis the probability that consensus occurs in opinion \u000b(as\nopposed to another opinion). We haveP\n\u000bq\u000b= 1 (con-\nsensus occurs with certainty eventually), and in general\nq\u000b<1 for any one opinion \u000b(i.e., thef\u000bare not nor-\nmalised probability densities).\nWe are interested in mean arrival times, so we calculate\nhTN(L;x)i \u0011Z1\n0dt0t0d\ndt0pC(x;t0)\n=X\n\u000bZ1\n0dt0t0d\ndt0f\u000b(x;t0):(B5)This is the mean time to consensus (on any opinion).\nWe have explicitly indicated the starting point x. The\naverage in this expression is only over realisations of the\ndynamics, but not over the starting point.\nWe next note that the mean consensus time, condi-\ntional on arrival at consensus on \u000b, is given by\nhT\u000b(x)i=R1\n0dt0t0d\ndt0f\u000b(x;t0)\nq\u000b(x): (B6)\n(we suppress the dependence on LandN). Hence,\nhTN(L;x)i=X\n\u000bq\u000b(x)hT\u000b(x)i: (B7)\nConsensus occurs at \u000bwith probability q\u000b(x), and given\n\u000bthe mean time for this consensus is hT\u000b(x)i.\nThe key observation is now that, similar to the argu-\nment in Sec. III A 3, f\u000b(x;t) (and hence also q\u000b(x)) can\nbe obtained from looking at a two-state version of the\nmodel, in which one opinion is \u000band where all other\nopinions are amalgamated into one second opinion state.\nThis idea was used in the context of the voter model in\nRef. [5] and later in Ref. [14]. Similar principles had pre-\nviously been proposed for multi-allele models in the \feld\nof genetics [31, 32].\nWe note that the above argument applies for all-to-\nall interaction and for networks. We did not make any\nassumptions on the interaction network in deriving any\nof the relations up to and including Eq. (B7). What\ndoes change in going from all-to-all interaction to net-\nworks is the functional form of object such as pC(x;t)\nandf\u000b(x;t), but not the relations between these quanti-\nties.\nAll-to-all interaction. As an illustration we now show\nhow the reduction to a two-state model can be used to\ncalculate the mean conditional consensus times hT\u000b(x)i,\nand from these, to derive Eq. (B1) (valid for all-to-all\ninteractions).\nIf the two-state model is started with a proportion x\nof agents in opinion 1 (and 1 \u0000xin opinion 2), then the\nmean consensus time conditioned on consensus in opinion\n1 is\nhT1(x)i=\u0000N1\u0000x\nxln(1\u0000x): (B8)\nThis is a well-known result, see for example [9].\nUsing this, and the reduction of the multi-state model\nto the two-state model we then have\nhT\u000b(x)i=\u0000N1\u0000x\u000b\nx\u000bln(1\u0000x\u000b): (B9)\nWe also note that q\u000b(x) =x\u000b(this follows from the\nfact that the dynamics of the model preserves the time-\naveragehx\u000b(t)i).\nHence, inserting into Eq. (B7),\nhTN(L;x)i=\u0000NLX\n\u000b=1(1\u0000x\u000b) ln(1\u0000x\u000b) (B10)18\nWe next use this to derive Eq. (B1). To do this we\nneed to average over x, with a \rat measure in the sim-\nplex de\fned by x\u000b\u00150 andP\n\u000bx\u000b= 1. Exploiting the\nsymmetry of the model we \fnd from Eq. (B10),\nhTN(L)i=\u0000NLZ1\n0dx PL(x)(1\u0000x) ln(1\u0000x);(B11)\nwith the marginal PL(x) = (L\u00001)(1\u0000x)L\u00002in Eq. (25).\nCarrying out the integral we have\nhTN(L)i=NL(L\u00001)\nL2=NL\u00001\nL; (B12)\ni.e., we recover Eq. (B1).\n2. Extension to uncorrelated networks\nWe now discuss the extension of Eq. (B1) to uncorre-\nlated networks, i.e., we derive Eq. (B2). All results apply\nin the pair approximation.\nBased on the reduction argument in the previous sec-\ntion of this Appendix it is su\u000ecient to show that themean conditional consensus time of the two-state model\nin Eq. (B9) generalises to\nhT\u000b(x)i=\u0000N(k\u00001)k2\n(k\u00002)k21\u0000x\u000b\nx\u000bln(1\u0000x\u000b):(B13)\nThis time,hT\u000b(x)i, in turn is derived from the back-\nward Fokker{Planck equation describing the model in the\nlimit of large, but \fnite N. This is standard in the case\nof all-to-all interaction (see e.g. [9]).\nThe crucial observation is now that the (forward)\nFokker-Planck equation of the two-state model on un-\ncorrelated graphs is obtained from that for the two-state\nmodel with all-to-all interactions by a simple re-scaling\nof time by a factor [( k\u00001)k2]=[(k\u00002)k2]. This is dis-\ncussed in [11], see for example Eq. (12) in this reference\n(see also [8, 9]).\nThe same re-scaling then also applies to the backward\nequation, and it then follows immediately that all time\nscales derived from the backward equation also undergo\nthe same re-scaling. This then leads to Eq. (B13).\n[1] T. M. Liggett, Interacting Particle Systems, Springer,\nNew York, (1985).\n[2] P. Cli\u000bord, and A. Sudbury, Biometrika 60, 581 (1973).\n[3] R. Holley, and T. Liggett, Ann. Probab. 3, 643 (1975).\n[4] C. Castellano, S. Fortunato, and V. Loreto, Rev. Mod.\nPhys. 81, 591 (2009).\n[5] S. Redner, C. R. Phys. 20, 275 (2019).\n[6] I. Dornic, H. Chat\u0013 e, J. Chave, and H. Hinrichsen, Phys.\nRev.Lett. 87, 045701 (2001).\n[7] E. Ben-Naim, L. Frachebourg, and P. L. Krapivsky, Phys.\nRev. E 53, 3078 (1996).\n[8] V. Sood and S. Redner, Phys. Rev. Lett. 94, 178701\n(2005).\n[9] V. Sood, T. Antal, S. Redner, Phys. Rev. E 77, 041121\n(2008).\n[10] K. Suchecki, V. M. Egu\u0013 \u0010luz, and M. San Miguel, Phys.\nRev. E 72, 036132 (2005).\n[11] F. Vazquez, V. M. Egu\u0013 \u0010luz, New J. Phys. 10, 63011\n(2008).\n[12] A. Carro, R. Toral, and M. San Miguel, Sci. Rep. 6, 24775\n(2016).\n[13] A. F. Peralta, A. Carro, M. San Miguel, and R. Toral,\nChaos 28, 075516 (2018).\n[14] F. Herrer\u0013 \u0010as-Azcu\u0013 e and T. Galla, Phys. Rev. E 100,\n022304 (2019).\n[15] F. Vazquez, E. S. Loscar, and G. Baglietto, Phys. Rev.\nE100, 042301 (2019).\n[16] N. Khalil, M. San Miguel, R. Toral, Phys. Rev. E 97,\n012310 (2018).[17] N. Khalil and T. Galla, Phys. Rev. E 103, 012311 (2021).\n[18] M. Mobilia, Phys. Rev. Lett. 91, 028701 (2003).\n[19] M.Mobilia, A. Petersen, and S. Redner, 2007, J. Stat.\nMech. P08029 (2007).\n[20] M. Starnini, A. Baronchelli, R. Pastor-Satorras, J. Stat.\nMech. P10027 (2012).\n[21] W. Pickering and C. Lim, Phys. Rev. E 93, 032318\n(2016).\n[22] G. J. Baxter, R. A. Blythe, and A. J. MacKane, Math.\nBiosci. 209, 124 (2007).\n[23] A.-L. Barab\u0013 asi and R. Albert, Science 286, 509 (1999).\n[24] E. Pugliese and C. Castellano, EPL (Europhysics Let-\nters), 88(5):58004, (2009).\n[25] C. Castellano, V. Loreto, A. Barrat, F. Cecconi, and D.\nParisi, Phys. Rev. E 71, 066107 (2005).\n[26] C. Castellano, D. Vilone, and A. Vespignani, Europhys.\nLett. 63, 153 (2003).\n[27] H. Kauhanen, D. Gopal, T. Galla, and R. Berm\u0013 udez-\nOtero, Sci. Adv. 7, eabe6540 (2021).\n[28] M. San Miguel, V. M. Egu\u0013 \u0010luz, R. Toral, and K. Klemm,\nComput. Sci. Eng. 7, 67 (2005).\n[29] R. Dickman, Physical Review A 34,4246{4250, 12 (1986).\n[30] S. Dorogovtsev, Lectures on complex networks. Phys. J.\n9, 51 (2010).\n[31] M. Kimura, Evolution 9, 419 (1955).\n[32] R. A. Littler, Math. Biosc. 25, 151 (1975).\n[33] Z. D. Blount, R. E. Lenski, J. B. Losos, Science 362,\neaam5979 (2018)."    },    {        "title": "2208.03086v1.QCD_in_the_cores_of_neutron_stars.pdf",        "content": "QCD in the cores of neutron stars\u0003\nOleg Komoltsev,\nFaculty of Science and Technology, University of Stavanger, 4036 Stavanger,\nNorway\nReceived August 8, 2022\nI discuss why state-of-the art perturbative QCD calculations of the\nequation of state at large chemical potential that are reliable at asymptot-\nically high densities constrain the same equation of state at neutron-star\ndensities. I describe how these theoretical calculations a\u000bect the EOS at\nlower density. I argue that the ab-initio calculations in QCD o\u000ber signif-\nicant information about the equation of state of the neutron-star matter,\nwhich is complementary to the current astrophysical observations.\n1. Introduction\nThe equation of state (EOS) of the dense matter at zero temperature is a\nnecessary input for the neutron-stars (NS) physics. Theoretical calculations\nof the EOS can be done only at the two opposite (low- and high-density)\nlimits. At the low-density limit the matter can be described within the\nchiral e\u000bective \feld theory (CET) [1, 2]. Those calculations are reliable up\nto around nuclear saturation density ns= 0:16=fm3. On the other hand we\ncan access the EOS using perturbative Quantum Chromodynamics (pQCD)\nat the asymptotically high densities, above \u001840ns[3,4]. Central densities of\nmaximally massive neutron stars are around 4 \u00008ns, which is not reachable\nwithin CET or pQCD. Therefore, there are no tools in our possession to\ncompute EOS of the cores of NS from the \frst principles.\nHowever, we can obtain an empirical access to the cores of NSs using\nrecent astrophysical observations. The most important probes of NS physics\nare the discovery of massive NSs [5{7], mass - radius measurements [8,9], and\nthe gravitational-wave and multi-messenger astronomy [10,11]. Utilizing all\nconstraints coming from astrophysical observation as well as \frst principle\ncalculations narrows down dramatically the range of possible EOSs, which\nallows us to use the densest objects in the Universe to test independently\nvarious beyond standard model scenarios and/or general relativity.\n\u0003Presented at Quark Matter 2022\n(1)arXiv:2208.03086v1  [nucl-th]  5 Aug 20222 main printed on August 8, 2022\nMajority of the EOS studies extrapolate CET EOS up to NS densities\n5-10nsand conditioning it with the observational inputs. The results di\u000ber\nfrom the works that include high-density limit and interpolate between two\norders of magnitude. The qualitative di\u000berence is in the softening of the\nEOS happening around \u000f\u0018750 MeV/fm\u00003, which can be interpreted as\nquark matter cores inside the most massive NS [12].\nIn this work I answer why and how pQCD input o\u000bers signi\fcant infor-\nmation about the EOS at NS densities. I \fnd that pQCD input propagates\nnon-trivial constraints all the way down to 2.2 nsjust by using solely ther-\nmodynamic stability, consistency and causality [13]. In addition the com-\nplementariness of the pQCD input to the astrophysical observations was\nstudied in [14]. I show that pQCD is responsible for the softening of the\nEOS at the NS densities. Therefore, it is essential to include pQCD input\nin any inference study of the EOS.\n2. Setup\nAll technical details as well as analytical formulas are presented in [13].\nIn this section I describe the conditions I use, in particular stability, consis-\ntency and causality, and the resulting propagation of the pQCD input down\nto lower densities. Let us start with the baryon density nas a function of\nthe chemical potential \u0016as shown in \fg.1a. The goal is to \fnd all possible\nlines that connect endpoint of CET results (dark blue line in the bottom\nleft corner) with the \frst point of pQCD calculations (purple line in the\nupper right corner) using 3 conditions.\nThe \frst condition is thermodynamic stability, which implies concavity\nof the grand canonical potential @2\n\u0016\n(\u0016)\u00140. At zero temperature \n( \u0016) =\n\u0000p(\u0016), which implies that the number density is monotonically increasing\nfunction of the chemical potential @\u0016n(\u0016)\u00150.\nThe second condition is causality { the sound speed cannot exceed the\nspeed of light c2\ns\u00141. This provides constraints on the \frst derivative of the\nnumber density with respect to the chemical potential\nc\u00002\ns=\u0016\nn@n\n@\u0016\u00141: (1)\nFor each point on the \u0016\u0000nplane we can calculate the least allowed slope\ncoming from causality, which is represented by the arrows in the \fg.1a.\nThis cuts upper (lower) region of the plane, because any points from the\narea above (below) orange line c2\ns= 1 cannot be connected to pQCD (CET)\nin a casual way.\nThe third condition is thermodynamic consistency. In addition to nand\n\u0016we need to match pressure pat the low- and high- density limits. Themain printed on August 8, 2022 3\n{μ0,n0}pQCDCETcs2=1Integral\nconstraints\nΔpmax(μ0,n0)\nΔpmin(μ0,n0)\n1.0 1.5 2.0 2.501234567\nBaryon chemical potential μ[GeV ]Baryon density n[fm-3]\n(a)\n [GeV/fm3\n]\n01234\np [GeV/fm3]0.00.51.01.52.0n [fm3]\n0.00.51.01.52.02.5Consistent\nIn tension\nNot consistent (b)\nFig. 1: ( a) - Baryon density as a function of chemical potential. Simultane-\nous ful\flment of thermodynamic consistency, stability and causality narrows\ndown the allowed region (white area) of the EOS. ( b) - Zero temperature\nEOSs from CompOSE database plotted with the allowed area (gray shape)\narising from the new constraints in \u000f\u0000p\u0000nspace. Consistent/in ten-\nsion/not consistent means that EOS is consistent with integral constraints\nfor all/some/none X values in a range [1,4].\npressure is giving by the integral of the number density\nZ\u0016QCD\n\u0016CETn(\u0016)d\u0016=pQCD\u0000pCET= \u0001p: (2)\nThis implies that the area under the curve for any EOS is \fxed by our\ninput parameters. For each arbitrary point \u00160;n0we can construct the\nEOS that maximize/minimize the area under the curve \u0001 pmax=min (\u00160;n0)\nshown as a green/blue dashed line in the \fg.1a. If \u0001 pmax(\u00160;n0)<\u0001p\nthen any EOS that goes through the point \u00160;n0does not have enough area\nunder the curve. This discards the region in the lower right corner in the\n\fg.1a under the red line called \"integral constraints\". If \u0001 pmin(\u00160;n0)>\n\u0001pthen any EOS that goes through the point \u00160;n0has too much area\nunder the curve. This cuts area in the upper left corner above the red\nline. The integral constraints can be obtained without any assumptions of\ninterpolation function in a completely general and analytical way.\nWe can map the allowed region from \u0016\u0000nto\u000f\u0000pplane. The results\nof such mapping is shown in the \fg.2. The green envelope corresponds to\nthe the white area in the \fg.1a restricted by the causality and the integral\nconstraints. The shapes of allowed region with and without pQCD are4 main printed on August 8, 2022\n10nsexcluded by\npQCD\n5ns3ns\n2nspQCD\nCETIntegral\nconstraintsCausality\nconstraints\nCausality\nconstraintsIntegral\nconstraints\n500 1000 5000 104101001000104\nEnergy densityϵ[MeV/fm3]Pressure p[MeV/fm3]\nFig. 2: Constraints on the \u000f\u0000pplane coming from low- and high-density\nlimits. Shapes outlined by solid black line are the allowed areas for \fxed\nnumber density without pQCD. Blue shapes are allowed regions after im-\nposing pQCD input.\nshown for the \fxed number density n= 2,3,5 and 10 ns. This explicitly\nshows how pQCD input can propagate information down to lower density\nstarting from 2.2 ns. And, strikingly, at 5 nsit excludes 75% of otherwise\nallowed area.\nUsing the new constraints we can check the consistency of publicly avail-\nable EOSs. Results for all zero temperature EoSs in \f-equilibrium from the\npublic CompOSE database [15, 16] are shown in the \fg.1b. Almost all of\nthe EOSs start to be inconsistent with pQCD input at some density within\nthe provided range.\n3. Bayesian inference of EOS\nWith the construction described above we can propagate information\nfrom ab-initio QCD calculations down to NS densities, where we already\nhave constraints from astrophysical observations. To understand if the new\nconstraints from pQCD go beyond the constraints coming from the NS\nmeasurements we construct a Bayesian-inference framework. This was done\nin [4], where we generate a large ensemble of di\u000berent EOSs using Gaussian-\nprocess regression. We anchor the ensemble to CET calculations and ex-\ntrapolate it up to 10 ns, where we impose pQCD input as a blue shape from\n\fg.2. We condition ensemble sequentially with the astrophysical observa-\ntions. With this setup we can turn on and turn o\u000b pQCD input in order to\nstudy its e\u000bect on our posterior after imposing astrophysical observation.\nThe results are present in \fg.3. The reduction of the pressure (greenmain printed on August 8, 2022 5\narrow on the right plot), which is caused by the QCD input, happens be-\nfore the density reaches its maximal central value. In another words, the\nprediction of QCD input is the softening of the EOS that happens inside\nthe most massive neutron stars.\nFig. 3: Left plot shows the sample of 10k EOSs. The coloring represents the\nlikelihood after imposing all observations as well as pQCD input. Right plot\nshows 67 %-credible intervals conditioned with the di\u000berent astrophysical\nobservations and high-density limit. The gray band shows 67 %-credible\ninterval for the maximal central energies density reached in NSs.\n4. Conclusion\nIn this work, I show how QCD calculations at asymptotically high den-\nsities can propagate information down to lower densities using solely ther-\nmodynamic consistency, stability and causality. This information o\u000bers a\nsigni\fcant constraints to the EOS at NS density, which is complementary\nto the current astrophysical observations. In addition, I show that the pre-\ndiction of QCD input is the softening of the EOS that happens in the most\nmassive NSs. An easy-to-use python script is provided to check consistency\nof the EOS with pQCD input, available on Github [17].\nIn order to achieve accurate determination of the EOS it is crucial to\nutilize all available controlled measurements and theoretical calculations.\nThis strategy either helps us to understand the matter of the densest objects\nin the Universe or \fnd a discrepancy between di\u000berent inputs, which allows\nus to use NS as a tool for fundamental discoveries.\nREFERENCES6 main printed on August 8, 2022\n[1] I. Tews, T. Kr uger, K. Hebeler, and A. Schwenk, \\Neutron matter at next-to-\nnext-to-next-to-leading order in chiral e\u000bective \feld theory,\" Phys. Rev. Lett. ,\nvol. 110, no. 3, p. 032504, 2013.\n[2] C. Drischler, K. Hebeler, and A. Schwenk, \\Chiral interactions up to next-to-\nnext-to-next-to-leading order and nuclear saturation,\" Phys. Rev. Lett. , vol.\n122, no. 4, p. 042501, 2019.\n[3] T. Gorda, A. Kurkela, P. Romatschke, M. S appi, and A. Vuorinen, \\Next-\nto-next-to-next-to-leading order pressure of cold quark matter: Leading loga-\nrithm,\" Phys. Rev. Lett. , vol. 121, no. 20, p. 202701, 2018.\n[4] T. Gorda, A. Kurkela, R. Paatelainen, S. S appi, and A. Vuorinen, \\Soft Inter-\nactions in Cold Quark Matter,\" Phys. Rev. Lett. , vol. 127, no. 16, p. 162003,\n2021.\n[5] P. B. Demorest, T. Pennucci, S. M. Ransom, M. S. E. Roberts, and J. H. T.\nHessels, \\A two-solar-mass neutron star measured using Shapiro delay,\" Na-\nture, vol. 467, pp. 1081{1083, 2010.\n[6] J. Antoniadis et al. , \\A massive pulsar in a compact relativistic binary,\" Sci-\nence, vol. 340, no. 6131, p. 1233232, 2013.\n[7] E. Fonseca et al. , \\Re\fned Mass and Geometric Measurements of the High-\nmass PSR J0740+6620,\" Astrophys. J. Lett. , vol. 915, no. 1, p. L12, 2021.\n[8] M. C. Miller et al. , \\The radius of PSR J0740+6620 from NICER and XMM-\nNewton data,\" Astrophys. J. Lett. , vol. 918, no. 2, p. L28, 2021.\n[9] T. E. Riley et al. , \\A NICER view of the massive pulsar PSR J0740+6620 in-\nformed by radio timing and XMM-Newton spectroscopy,\" Astrophys. J. Lett. ,\nvol. 918, no. 2, p. L27, 2021.\n[10] B. P. Abbott et al. , \\GW170817: Observation of gravitational waves from a\nbinary neutron star inspiral,\" Phys. Rev. Lett. , vol. 119, no. 16, p. 161101,\n2017.\n[11] ||, \\Multi-messenger observations of a binary neutron star merger,\" Astro-\nphys. J. Lett. , vol. 848, no. 2, p. L12, 2017.\n[12] E. Annala, T. Gorda, A. Kurkela, J. N attil a, and A. Vuorinen, \\Evidence for\nquark-matter cores in massive neutron stars,\" Nat. Phys. , vol. 16, no. 9, pp.\n907{910, 2020.\n[13] O. Komoltsev and A. Kurkela, \\How Perturbative QCD Constrains the Equa-\ntion of State at Neutron-Star Densities,\" Phys. Rev. Lett. , vol. 128, no. 20, p.\n202701, 2022.\n[14] T. Gorda, O. Komoltsev, and A. Kurkela, \\Ab-initio QCD calculations impact\nthe inference of the neutron-star-matter equation of state,\" 4 2022.\n[15] S. Typel, M. Oertel, and T. Kl ahn, \\CompOSE CompStar online supernova\nequations of state harmonising the concert of nuclear physics and astrophysics\ncompose.obspm.fr,\" Phys. Part. Nucl. , vol. 46, no. 4, pp. 633{664, 2015.\n[16] M. Oertel, M. Hempel, T. Kl ahn, and S. Typel, \\Equations of state for super-\nnovae and compact stars,\" Rev. Mod. Phys. , vol. 89, no. 1, p. 015007, 2017.\n[17] \\https://github.com/OKomoltsev/QCD-likelihood-function.\""    },    {        "title": "2210.03807v2.A_convolution_formalism_for_defining_spatial_densities_of_hadrons.pdf",        "content": "NT@UW-22-17\nA convolution formalism for defining densities of hadrons\nAdam Freese1,∗and Gerald A. Miller1,†\n1Department of Physics, University of Washington, Seattle, WA 98195, USA\nWe clarify the meaning of spatial densities of hadrons. A physical density is given by the expec-\ntation value of a local operator for a physical state, and depends on both internal structure and the\nhadron’s wave packet. In some particular cases, the physical density can be written as a convolution\nbetween a density function that depends on internal structure but not wave packet, and a smearing\nfunction that depends on wave packet but not internal structure. We show that the light front den-\nsities often encountered in the literature have this property but that instant form densities do not.\nFor hadrons prepared in broad wave packets, physical instant form densities approximately obey\nsuch a convolution relation, with Breit frame densities as the apparent internal densities. However,\nthere is an infinite series of relativistic corrections to this convolution formula.\nI. INTRODUCTION\nThere has lately been renewed interest and debate about the proper manner for describing the internal structure\nof hadrons in terms of spatial coordinates. This debate is especially pertinent now with the increasing focus on the\nenergy momentum tensor and its associated densities in the literature (see e.g. Refs. [1–3]). A major focus of the\nupcoming Electron Ion Collider [4–6] is to provide a spatial picture of the distributions of partons in hadrons, making\nit vital that an accurate method for obtaining this spatial picture is used.\nFor a long time, authors have primarily calculated relativistic densities through three-dimensional Fourier transforms\nof form factors, giving a result referred to as the Breit frame density [7]. For almost as long, this approach has been\ncriticized [8–10] for failing to follow from the elementary definition of a density in relativistic field theory (i.e.,\nthe expectation value of a local operator). While the majority of authors continue to use the Breit frame density\nuncritically, multiple solutions to addressing the issue of relativistic spatial densities have been advanced.\nThere are currently three established camps on the question of relativistic spatial densities. The first camp uses\nthe formalism of Wigner phase space distributions to justify the Breit frame densities as a slice of a six-dimensional\nphase space distribution when the hadron momentum is zero [2].\nThe second camp obtains densities using expectation values of local currents in light front coordinates, integrating\nout the coordinate x−to produce a two-dimensional density in the transverse plane [3, 11, 12]. Within this formalism,\none often localizes the hadron’s wave packet [3, 9] or works with Dirac delta functions [10, 11, 13] to remove dependence\non the wave packet.\nThe last camp attempts to define three-dimensional densities in instant form coordinates in a similar manner to the\nsecond camp, namely by taking the expectation value of a local current for a physical state and localizing the wave\npacket. This idea was first proposed by Fleming in 1974 [8], but the idea has recently been revived [14–16].\nFleming’s proposal was motivated by the fact that, in non-relativistic quantum mechanics, localizing the center-of-\nmass wave packet of a composite system does reproduce the internal charge density of the system as conventionally\ndefined. Without a clear notion of what constitutes the internal charge density of a relativistic bound system, using\nthis procedure as a generalized definition for the internal densities seems reasonable. However, the approach exhibits\nseveral shortcomings, including the need to assume spherical symmetry of the wave packet during localization [8, 14–\n18], suggesting lingering wave packet dependence. Additionally, densities associated with the energy-momentum\ntensor diverge [17, 18], a manifestation of the uncertainty principle.\nThe purpose of this work is to propose an alternative definition of internal densities for relativistic composite\nsystems, which like Fleming’s proposal reproduces known results when applied to the non-relativistic domain. Within\nthis approach, we analyze when wave packet localization reproduces the internal densities and when it does not. In\nparticular, we follow a similar vein to recent work by Li et al. [19], with a focus on separating physical densities into\nwave packet dependent and independent pieces. Ultimately, we find that localization does reproduce our results for\na handful of light front densities, but not for a class of cases we refer to as “compound densities,” which includes all\ndensities in instant form coordinates.\nThis work is organized as follows. In Sec. II, we provide our definitions and the physical motivation behind them.\nIn Sec. III, we obtain light front densities within the constraints laid out in the previous section. These include the\n∗afreese@uw.edu\n†miller@uw.eduarXiv:2210.03807v2  [hep-ph]  24 Jul 20232\nelectromagnetic current and spatial distributions associated with the energy-momentum tensor, reproducing previous\nfindings where they exist but without requiring wave packet localization. In Sec. IV, we examine three-dimensional\ndensities in instant form coordinates, showing that exact three-dimensional internal densities do not exist within our\nproposed formalism. In the special case of diffuse wave packets we find an infinite tower of internal densities that\nmake increasingly small contributions, with the Breit frame density appearing as the leading term. We summarize\nand conclude in Sec. V.\nII. INTERNAL AND PHYSICAL DENSITIES\nA physical density is defined as the expectation value of a local operator for a physical state |Ψ⟩in Hilbert space:\nρphys(x) =⟨Ψ|ˆO(x)|Ψ⟩. (1)\nWe assume |Ψ⟩is a single-hadron state. The physical density will contain influences both from the internal structure\nof the hadron, and from the wave packet that it’s prepared in. The goal of hadron structure research is to isolate\ninternal properties of hadrons that are independent of artifacts of state preparation. It is thus vital to determine\nunder what circumstances a purely internal hadron density can be defined.\nIt is moreover prudent to understand how the internal densities contribute to the physical densities defined in\nEq. (1). The simplest scenario that can occur is that the physical density corresponds to an internal density being\nsmeared over space at any particular fixed time τ(which could be time in any of the forms of relativistic dynamics [20]):\nρphys(x, τ) =Z\ndnxSΨ(R, τ)ρint.(x−R). (2)\nHereSΨ(R, τ) is a smearing function that depends on the wave packet but not on internal structure, whereas ρint.(b)\ncontains no wave packet dependence. We have notated nspatial dimensions for generality; we will have n= 3 for\nnon-relativistic densities and n= 2 for relativistic light front densities. We shall refer to the density in a scenario\nsuch as Eq. (2) as a simple density .\nEq. (2) has a clear physical interpretation: ρint.(b) describes the distribution of some quantity (such as charge or\nenergy) a displacement bfrom the barycentric position Rof the hadron, and SΨ(R, τ) gives a distribution for the\nhadron’s barycenter itself. The convolution of both then gives the physical density at a position x=R+b. We shall\nshow in Sec. II A how such a formula naturally arises in the context of non-relativistic quantum mechanics.\nIn an old work that has found recent attention, Fleming [8] noticed that the non-relativistic charge density took the\nform of Eq. (2), with the smearing function being the probability density SΨ(R) = Ψ∗(R)Ψ(R). If the wave packet\nis localized in this case, the physical density approaches the internal density. In fact, this will always happen when\nthe physical density is a simple density and the smearing function is the barycentric probability density. Fleming\nextrapolated beyond this simple case and postulated that localizing the wave function would always produce an\ninternal density.\nHowever, localization will not reproduce the desired internal densities whenever the physical density does not obey\nEq. (2). There will be cases that the physical density is actually a sum of different internal structures, each weighted\nby a different smearing function:\nρphys(x, τ) =Z\ndnxS(1)\nΨ(R, τ)ρ(1)\nint.(x−R, τ) +Z\ndnxS(2)\nΨ(R, τ)ρ(2)\nint.(x−R, τ) +. . . (3)\nHere, ρ(1)\nint.(x−R, τ) and ρ(2)\nint.(x−R, τ) correspond to different internal spatial distributions, which are each independent\nof the wave packet Ψ. We shall refer to such a density as a compound density . In the light front formalism (Sec. III),\nwe shall see that the energy density and stress tensor are examples of compound densities. We shall also see—\nassuming convergence of the series—that instant form densities are always compound densities with infinitely many\ncontributions, as previously found by Li et al. in Ref. [19].\nA. Example from non-relativistic quantum mechanics\nLet us consider a simple non-relativistic example of where Eq. (2) naturally arises. We consider two particles with\nmasses m1andm2bound to a system with mass M=m1+m2−ε. The wave function for the barycentric position3\nis Ψ(R, t) and the wave function for the relative position is ψ(r)eiεt. If the positions of the particles are r1andr2,\nthe barycentric and relative coordinates are defined:\nR=m1r1+m2r2\nm1+m2(4a)\nr=r2−r1. (4b)\nWe define a density for some quantity Q, which may be electric charge or mass for instance, and for which particles\n1 and 2 carry an amount q1andq2respectively. The density is:\nρQ(x, t) =Z\nd3RZ\nd3r|Ψ(R, t)|2|ψ(r)|2n\nq1δ(3)(x−r1) +q2δ(3)(x−r2)o\n. (5)\nTo integrate out the delta functions, we must use the inversion of Eq. (4), which gives:\nρQ(x, t) =Z\nd3R|Ψ(R, t)|2(\nq1\f\f\f\fψ\u0012m1+m2\nm2(R−x)\u0013\f\f\f\f2\n+q2\f\f\f\fψ\u0012m1+m2\nm1(x−R)\u0013\f\f\f\f2)\n. (6)\nThis has the form of Eq. (2), with:\nSΨ(R, t) =|Ψ(R, t)|2(7a)\nρint.(b) =q1\f\f\f\fψ\u0012\n−m1+m2\nm2b\u0013\f\f\f\f2\n+q2\f\f\f\fψ\u0012m1+m2\nm1b\u0013\f\f\f\f2\n. (7b)\nIf, at some moment of time t0we have |Ψ(R, t0)|2=δ3(R), then we obtain ρQ(x, t0) =ρint.(x). This is a generic\nfeature of simple densities where the smearing function is equal to the probability distribution.\nIII. INTERNAL DENSITIES IN LIGHT FRONT COORDINATES\nIn this section, we demonstrate that the light front densities encountered in the literature [2, 3, 9, 11, 21–26] are\nfaithfully reproduced in our formalism. We also determine how they relate to their associated physical densities.\nAlthough much of the previous literature tended to use a localization procedure with a specific wave packet form (or\nsimply used delta function wave packets) to obtain these results, we will show that the results are in fact independent\nof the wave packet.\nFor simplicity, we will focus exclusively on spin-zero targets. This work is meant to provide conceptual clarity and\nrigor to the research program of hadron densities rather than to be an exhaustive catalogue. Higher-spin targets with\ngeneral polarization will contain complications from helicity-flip contributions [22, 23, 25, 27], so densities that are\nsimple for spin-zero targets will often be compound for higher-spin targets.\nA. General expression for physical densities\nWe will derive general expression for the physical light front densities of spin-zero targets, without making any\nspecific assumptions about the wave packet.\nThe Lorentz-invariant completeness relation for states in the Hilbert subspace of single-particle spin-zero states is:\nZdp+d2p⊥\n2p+(2π)3|p+,p⊥⟩⟨p+,p⊥|= 1. (8)\nWe define the momentum space wave function as:\nΨ(p+,p⊥) =1p\n2p+⟨p+,p⊥|Ψ⟩, (9a)\nso that it obeys the normalization condition:\nZdp+d2p⊥\n(2π)3\f\f\fΨ(p+,p⊥)\f\f\f2\n= 1. (9b)4\nWe define the position-space wave function through a Fourier transform:\nΨ(z−,z⊥) =Zdp+d2p⊥\n(2π)3Ψ(p+,p⊥)e−ip+z−e+ip⊥·z⊥, (10a)\nso that it obeys the normalization condition:\nZ\ndz−Z\nd2z⊥\f\f\fΨ(z−,z⊥)\f\f\f2\n= 1. (10b)\nUsing the completeness relation twice with momenta pandp′, integrating out x−, and defining a change of variables:\nP⊥=1\n2\u0010\np⊥+p′\n⊥\u0011\n(11a)\n∆⊥=p′\n⊥−p⊥ (11b)\nP+=p+=p′+(11c)\nwe obtain:\nρLF(x⊥) =ZdP+d2P⊥\n2P+(2π)3Zd2∆⊥\n(2π)2⟨Ψ|P+,p′\n⊥⟩⟨P+,p′\n⊥|ˆO(0)|P+,p⊥⟩\n2P+⟨P+,p⊥|Ψ⟩e−i∆⊥·x⊥. (12)\nRewriting in terms of the position-space wave functions gives:\nρLF(x⊥) =Z\ndR′−Z\nd2R′\n⊥Z\ndR−Z\nd2R⊥ZdP+d2P⊥\n(2π)3Zd2∆⊥\n(2π)2e−iP+(R′−−R−)eiP⊥·(R′\n⊥−R⊥)\ne−i∆⊥·\u0012\nx⊥−R⊥+R′\n⊥\n2\u0013\nΨ∗(R′−,R′\n⊥)⟨P+,p′\n⊥|ˆO(0)|P+,p⊥⟩\n2P+Ψ(R−,R⊥).(13)\nTo proceed, the dependence on P+andP⊥must be removed from the matrix element. This can be accomplished by\nnoting (for instance):\ni\n2\u0010\n∇(R)\ni− ∇(R′)\ni\u0011\neiP⊥·(R′\n⊥−R⊥)=Pi\n⊥eiP⊥·(R′\n⊥−R⊥)\ni\n2\u0010\n∇(R)\ni− ∇(R′)\ni\u0011\ne−i∆⊥·\u0012\nx⊥−R⊥+R′\n⊥\n2\u0013\n= 0.\nThus any factors of Pi\n⊥that appear in the matrix element can be transformed into differences of gradients, and\nintegration by parts can be used to move the gradients to act on the barycentric wave functions. A similar trick can\nbe applied to P+, with the end result being that the following substitutions can be made:\nPi\n⊥→ −i\n2← →∇i (14a)\nP+→+i\n2← →∂−, (14b)\nwith the two-sided derivatives to be placed between the barycenter wave function and its conjugate. With these\nsubstitutions made, it is possible to do the P+andP⊥integrals, which produce delta functions, and these delta\nfunctions can be eliminated by doing the R′−andR′\n⊥integrals. Performing these steps gives:\nρLF(x⊥) =Z\ndR−Z\nd2R⊥Zd2∆⊥\n(2π)2e−i∆⊥·(x⊥−R⊥)Ψ∗(R−,R⊥)⟨P+,p′\n⊥|ˆO(0)|P+,p⊥⟩\n2P+Ψ(R−,R⊥),(15)\nwhere any P+orPi\n⊥appearing in this formula should be understood in terms on the substitutions of Eq. (14). To\nproceed any further, specific local operators must be considered.5\nB. Electromagnetic current\nLet us first consider electromagnetic current of a spin-zero hadron. The matrix element of the electromagnetic\nfour-current jµ(0) between momentum kets is:\n⟨P+,p′\n⊥|jµ(0)|P+,p⊥⟩= 2PµF(t), (16)\nwhere F(t) is the electromagnetic form factor, and:\nt= (p′−p)2=−∆2\n⊥. (17)\nFor the plus component of this current—the light front charge density—the physical density takes a simple form:\nj+\nLF(x⊥) =Z\ndR−Z\nd2R⊥\f\fΨ(R−,R⊥)\f\f2Zd2∆⊥\n(2π)2F(−∆2\n⊥)e−i∆⊥·(x⊥−R⊥). (18)\nThis constitutes a simple density in the form of Eq. (2), with an internal density:\nj+\nint.(b⊥) =Zd2∆⊥\n(2π)2F(−∆2\n⊥)e−i∆⊥·b⊥, (19)\nwhich is the standard result [12], and a smearing function:\nP(R⊥) =Z\ndR−\f\fΨ(R−,R⊥)\f\f2, (20)\nwhich is the transverse probability density for the barycentric position. We have thus obtained the standard result\nfor the light front charge density of a spin-zero target, but without making any assumptions about the hadron wave\npacket nor localizing it. Note, however, that if P(R⊥) =δ(2)(R⊥), then j+\nLF(x⊥) =j+\nint.(x⊥), which is why previous\ntreatments with localized states obtained the correct internal charge density.\nIt is possible to also obtain the transverse current. One need only use µ= 1,2 instead of µ= +. It will be necessary\nto use the rules of Eq. (14):\n⟨P+,p′\n⊥|ji\n⊥(0)|P+,p⊥⟩\n2P+=Pi\n⊥\nP+F(t)→ −← →∇i← →∂−F(t). (21)\nSince R−is integrated out in defining smearing functions, it is possible to turn the← →∂−into an expectation value:\n−← →∇i← →∂−F(t)→−i← →∇i\n2\u001c1\nP+\u001d\nF(t). (22)\nWe find thus that:\nj⊥(x⊥) =\u001c1\nP+\u001dZ\nd2R⊥V⊥(R⊥)j+\nint.(x⊥−R⊥) (23)\nV⊥(R⊥) =Z\ndR−Ψ∗(R−,R⊥)−i← →∇⊥\n2Ψ(R−,R⊥). (24)\nThe transverse electromagnetic current is thus also a simple density, and in fact involves the same internal density as\nthe electric charge density. The only difference is how the smearing function distributes the internal density.\nC. Momentum densities and energy density\nWe next consider momentum and energy densities, which are given by matrix elements of the energy-momentum\ntensor (EMT). Matrix elements between momentum kets are [1]:\n⟨P+,p′\n⊥|Tµν(0)|P+,p⊥⟩= 2PµPνA(t) +∆µ∆ν−gµν∆2\n2D(t), (25)6\nwhere A(t) and D(t) are called gravitational form factors [1, 28, 29], and tis given in Eq. (17). The momentum and\nenergy densities are given by considering µ= + specifically, for which:\n⟨P+,p′\n⊥|T+ν(0)|P+,p⊥⟩\n2P+=PνA(t) +g+ν∆2\n⊥\n4P+D(t). (26)\nThe momenta densities are those with ν= +,1,2, while the energy density is given by ν=−and is the only density\nto which D(t) contributes.\nTheP+density is the most straightforward, since the P+multiplying A(t) just becomes an expectation value. The\nphysical density is given by:\nT++\nLF(x⊥) =⟨P+⟩Z\nd2R⊥P(R⊥)T++\nint.(x⊥−R⊥), (27)\nwhere the smearing function is defined in Eq. (20) and the internal density is given by:\nT++\nint.(b⊥) =Zd2∆⊥\n(2π)2A(−∆2\n⊥)e−i∆⊥·b⊥, (28)\nwhich is the standard result [2, 3]. Although it was unnecessary to consider a localized wave packet to obtain this\nresult, if we consider a localized state P(R⊥) =δ(2)(R⊥), then T++\nLF(x⊥) =⟨P+⟩T++\nint.(x⊥), which is why Ref. [3]\nobtained the correct result.\nTheP⊥density is similarly straightforward. The substitution rule of Eq. (14a) must be used, which ends up giving:\nT+i\nLF(x⊥) =Z\nd2R⊥Vi\n⊥(R⊥)T++\nint.(x⊥−R⊥), (29)\nwhere the smearing function was defined in Eq. (24). So far, the situation is similar to what we saw for the electro-\nmagnetic current: the intrinsic P+density can be convolved with a different smearing function to obtain either the\nphysical P+density or the physical P⊥density. The former is in fact proportional to the barycentric probability\ndensity.\nLet us look at the energy density (i.e., the P−density) next. The matrix element contains a factor P−, which is\ngiven by:\nP−≡p−+p′−\n2=M2+P2\n⊥+1\n4∆2\n⊥\n2P+. (30)\nThe appropriate matrix element thus takes the form:\n⟨P+,p′\n⊥|T+−(0)|P+,p⊥⟩\n2P+=M2+P2\n⊥+1\n4∆2\n⊥\n2P+A(t) +∆2\n⊥\n4P+D(t)\n→\u001c1\n2P+\u001d\u0012\u0012\nM2−1\n4← →∇2\n⊥+1\n4∆2\n⊥\u0013\nA(t) +∆2\n⊥\n2D(t)\u0013\n,(31)\nwhere we used the substitution rule in Eq. (14a). To proceed, we need to define both a new smearing function:\nW−(R⊥) =Z\ndR−Ψ∗(R−,R⊥)−← →∇2\n⊥\n4Ψ(R−,R⊥), (32)\nand a new internal density:\nT+−\nint.(b⊥) =Zd2∆⊥\n(2π)2\u0012\u0012\nM2+∆2\n⊥\n4\u0013\nA(−∆2\n⊥) +∆2\n⊥\n2D(−∆2\n⊥)\u0013\ne−i∆⊥·b⊥. (33)\nWith this, we find the physical energy density to be a compound density:\nT+−\nLF(x⊥) =\u001c1\n2P+\u001d\u0012Z\nd2R⊥W−(R⊥)T++\nint.(x⊥−R⊥) +Z\nd2R⊥P(R⊥)T+−\nint.(x⊥−R⊥)\u0013\n. (34)\nThis compound density exhibits a curious structure. The first term can be interpreted as a kinetic term and the\nsecond as a dynamical term. The smearing function W−(R⊥) in the first term arises from P2\n⊥in the momentum-\nspace matrix element, and describes the kinetic motion of the barycenter. The convolution with the internal P+density7\nthen distributes the barycentric kinetic motion over the constituents. Because of this first term, if P(R⊥) =δ(2)(R⊥),\nthen T+−\nLF(x⊥) diverges. This is in effect a manifestation of the uncertainty principle [10].\nThe second term in Eq. (34) encodes every contribution to the hadron energy aside from barycentric kinetic\nmotion. In models of internal hadron dynamics that frame the hadron as a many-body system, this includes masses\nof constituents, kinetic energy of the constituents relative to the center-of-momentum, and potential energy. This can\nbe seen by considering the general form of the light-front Hamiltonian for an N-body system:\nH=NX\ni=1m2\ni+p2\ni⊥\n2p+\ni+V . (35)\nHere, Vis the potential energy, and the absolute momenta are related to relative momenta and light front momentum\nfractions by:\np+\ni=xiP+(36a)\npi⊥=xiP⊥+ki⊥, (36b)\nand these obey sum rules:\nNX\ni=1xi= 1 (36c)\nNX\ni=1ki⊥= 0. (36d)\nUsing these definitions and the sum rules, it follows:\nH=P2\n⊥\n2P++1\n2P+NX\ni=1m2\ni+k2\ni⊥\nxi+V . (37)\nTheP2\n⊥/(2P+) term of course corresponds to the first term in Eq. (34), and thus the remaining terms in Eq. (37)—\nwhich are exactly the aspects of the internal energy we have listed—must correspond to the remaining second term of\nEq. (34). It is for this reason that we identify T+−\nint.(b⊥) as the internal energy density. In fact, the presence of D(t)\nin this density, which has a well-established connection to internal stresses [1–3, 30], helps indicate the dynamical\nnature of this term. Of course, one must bear in mind that quantum chromodynamics, as a quantum field theory, is\nan infinite-many body theory, and that hadrons have an indefinite number of constituents. To be sure, the derivation\nof Eq. (34) did not require us to assume that the hadron has a finite number of pointlike constituents, but considering\nsuch an approximate scenario helps give intuition into the breakdown of the energy density we have presented.\nIt is worth mentioning that the internal internal energy density is equivalent to the light front energy density in\nthe Drell-Yan frame in the formalism of Wigner phase space distributions, which is discussed in Ref. [2].\nD. Stress tensor\nThe light front stress tensor is given by Tij\nLF(x⊥) with i, j= 1,2. As remarked previously in Ref. [3], the stress\ntensor contains contributions from hadron flow and a “comoving” piece. Accordingly, the stress tensor constitutes a\ncompound density in the manner of Eq. (3). We will now derive the exact form of this compound density.\nThe relevant matrix elements for the stress tensor are:\n⟨P+,p′\n⊥|Tij(0)|P+,p⊥⟩\n2P+=Pi\n⊥Pj\n⊥\nP+A(t) +∆i\n⊥∆j\n⊥−δij∆2\n⊥\n4P+D(t), (38)\nfor which the substitution rules of Eq. (14) must be used. The presence of two factors of P⊥in front of A(t) requires\ndefining a new smearing function:\nWij(R⊥) =Z\ndR−Ψ∗(R−,R⊥) \n−i← →∇i\n⊥\n2! \n−i← →∇j\n⊥\n2!\nΨ(R−,R⊥), (39)8\nwhich is related to the smearing function in Eq. (32) by:\nW−(R⊥) =δijWij(R⊥). (40)\nIf we define the internal stress tensor as:\nTij\nint.(b⊥) =1\n4Zd2∆⊥\n(2π)2\u0010\n∆i\n⊥∆j\n⊥−δij∆2\n⊥\u0011\nD(−∆2\n⊥)e−i∆⊥·b⊥, (41)\nthen the physical stress tensor can be written as a compound density:\nTij\nLF(x⊥) =\u001c1\nP+\u001d\u0012Z\nd2R⊥Wij(R⊥)T++\nint.(x⊥−R⊥) +Z\nd2R⊥P(R⊥)Tij\nint.(x⊥−R⊥)\u0013\n. (42)\nThe quantity Tij\nint.(b⊥) is (aside from a factor P+) identical to the pure stress tensor of Ref. [3]. The first and\nsecond terms in Eq. (42) can be identified as a flow tensor and comoving stress tensor respectively [3, 31], so that the\nphysical stress tensor essentially takes the classical continuum form [31]. It is interesting to note that while diagonal\nelements of Tij\nint.(b⊥) correspond to internal or static pressures as seen by observers comoving with the motion of the\nhadron, diagonal elements of Tij\nLF(x⊥) instead correspond to dynamic pressures, which includes impulses that would\nbe imparted by the hadron’s motion and which is akin to radiation pressure (see discussion in Refs. [3, 26, 31, 32]).\nE. Regarding wave packet localization\nDeriving Eq. (34) for the energy density required foregoing localization of the wave packet. For a localized wave\npacket, the energy density actually diverges, which can be understood in terms of the uncertainty principle [10].\nLet us consider what might happen if we attempt to localize the wave packet by a procedure similar to that in\nRefs. [3, 8, 14, 15], but as in Ref. [33] in particular, we absorb the divergence into a normalization constant. The wave\nfunction is localized by a scaling transformation:\nΨ(R−,R⊥)→1\nσΨ\u0012\nR−,R⊥\nσ\u0013\n, (43)\nwhich transforms the smearing functions as:\nP(R⊥)→1\nσ2P\u0012R⊥\nσ\u0013\n(44a)\nW−(R⊥)→1\nσ4W−\u0012R⊥\nσ\u0013\n. (44b)\nUsing the variable change Y⊥=σR⊥, the energy density can be written:\nT+−\nLF(x⊥) =\u001c1\n2P+\u001d\u0012Z\nd2Y⊥P(Y⊥)T+−\nint.(x⊥−σY⊥) +1\nσ2Z\nd2Y⊥W−(Y⊥)T++\nint.(x⊥−σY⊥)\u0013\n.(45)\nDefining the normalization:\nN∞= lim\nσ→01\nσ2\u001c1\n2P+\u001dZ\nd2Y⊥W−(Y⊥), (46)\nthe energy density in the limit σ→0 becomes:\nT+−\nLF(x⊥)− − − →\nσ→0N∞T++\nint.(x⊥). (47)\nThe dynamical term was dropped because it remains finite in the σ→0 limit, and therefore is dominated by the\nkinetic term. The problem should be immediately clear: by localizing the compound density and keeping only the\ndominating term, we lose important information about the internal structure of the hadron. In fact, the internal\ndensity corresponding to dynamics is precisely what was lost, and the remaining kinetic term is actually trivial and\nuninteresting in this context. On the other hand, organizing the physical density into a power series in σwill lead to\nadditional complications, since σappears within the internal densities. In particular, derivatives of the internal P+9\ndensity will mix with the internal P−density at order σ0, so collecting order σ0terms together as done in Refs. [17, 18]\ndoes not reproduce the internal P−density as defined in our formalism.\nThis example should prove a cautionary tale about the localization procedure. In several special cases—namely\ncharge and P+density—the internal light front densities were obtained from localization because these are simple\ndensities. As we shall see in Sec. IV, however, none of the physical instant form densities associated with the\nelectromagnetic current or energy-momentum tensor are simple. We are thus skeptical of the instant form densities\nthat arise from wave packet localization.\nF. Numerical illustrations\n1. Soft wall holographic pion\nThe formulas we obtained above are fully general, and do not require the application of light front dynamics.\nNonetheless, it may be especially enlightening to examine the densities in a light front Hamiltonian model, especially\nsince having a light front Hamiltonian allows an explicit breakdown of the energy density into kinetic and potential\nenergy pieces.\nAs a simple example, we consider Brodsky’s and de Teramond’s model of the pion in soft wall holographic QCD [34].\nAt large Q2, or small transverse separations, the system is approximately described by the light front Hamiltonian:\nHinternal =k2\n⊥\nx+k2\n⊥\n1−x+κ4x(1−x)y2\n⊥−2κ2, (48)\nand the pion ground state is described by the internal wave function:\nψ(x,y⊥) =κ√πp\nx(1−x)e−1\n2κ2x(1−x)y2\n⊥, (49)\nwhere κ= 0.375 GeV. We have followed Soper [35] in denoting the transverse separation as y⊥, and reserve b⊥for\nthe actual transverse distance from the barycenter to the transverse coordinate—i.e., for the impact parameter. The\ntransverse separation is related to the actual transverse positions via:\ny⊥=\u0012r1⊥−R⊥\n1−x\u0013\n=b1⊥\n1−x=−\u0012r2⊥−R⊥\nx\u0013\n=−b2⊥\nx. (50)\nThe charge density is the most straightforward case to consider. We assume a π+to avoid a trivially zero result.\nThere is an up quark with charge Quand a down antiquark with charge −Qd, and the physical charge density is just\ngiven by the sum of the single-particle distributions weighted by their charges:\nj+\nLF(x⊥) =Z\ndR−Z\nd2R⊥\f\f\fΨ(R−,R⊥)\f\f\f2Z1\n0dxZ\nd2y⊥\f\f\fψ(x,y⊥)\f\f\f2n\nQuδ(2)(x⊥−r1⊥)−Qdδ(2)(x⊥−r2⊥)o\n,(51)\nwhere Qu= 2/3 and Qd=−1/3. Comparing to Eq. (18) and Eq. (19), the internal internal charge density is:\nj+\ntrue(b⊥) =Z1\n0dx(\nQu\n(1−x)2\f\f\f\fψ\u0012\nx,b⊥\n1−x\u0013\f\f\f\f2\n−Qd\nx2\f\f\f\fψ\u0012\nx,b⊥\nx\u0013\f\f\f\f2)\n. (52)\nThe integral can be done analytically for the wave function in Eq. (49), and the result is:\nj+\nint.(b⊥) =κ2\nπeκ2b2\n⊥\u0010\nE1(κ2b2\n⊥)−E2(κ2b2\n⊥)\u0011\n, (53)\nwhere\nEn(z) =Z∞\n1dte−zt\ntn(54)\nis the generalized exponential integral function [36]. The result is plotted in the left panel of Fig. 1.\nBy a similar token, the internal P+density is given by:\nT++\nint.(b⊥) =Z1\n0dx(\nx\n(1−x)2\f\f\f\fψ\u0012\nx,b⊥\n1−x\u0013\f\f\f\f2\n+1−x\nx2\f\f\f\fψ\u0012\nx,b⊥\nx\u0013\f\f\f\f2)\n, (55)10\nwhich can be evaluated using the wave function in Eq. (49) as:\nT++\nint.(b⊥) =κ2\nπeκ2b2\n⊥\u0010\n2E1(κ2b2\n⊥)−4E2(κ2b2\n⊥) + 2E3(κ2b2\n⊥)\u0011\n. (56)\nThis result is also plotted in the left panel of Fig. 1.\nThe last internal density we consider is the internal energy density. Now, it is straightforward to find the part of\nthe internal energy density associated with quark kinetic energy, using just the substitution rule k⊥→ −i\n2← →∇(y), since\nthis energy is spatially attached to the quark in question:\nT+−\nkin(b⊥) =−Z1\n0dx(\n1\nx(1−x)2ψ∗\u0012\nx,b⊥\n1−x\u0013← →∇2\n4ψ\u0012\nx,b⊥\n1−x\u0013\n+1\n(1−x)x2ψ∗\u0012\nx,b⊥\nx\u0013← →∇2\n4ψ\u0012\nx,b⊥\nx\u0013)\n.(57)\nFor the wave function in Eq. (49), we note:\n−1\n4ψ∗(x,y⊥)← →∇i← →∇jψ(x,y⊥) =1\n2κ2x(1−x)δijψ∗(x,y⊥)ψ(x,y⊥), (58)\nand the kinetic energy density can be explicitly found to be:\nT+−\nkin(b⊥) =2κ4\nπeκ2b2\n⊥\u0010\nE2(κ2b2\n⊥)−E3(κ2b2\n⊥)\u0011\n. (59)\nThe potential energy cannot be assigned to either quark, however, so some additional hypothesis about its spatial\ndistribution is needed to proceed. The y2\n⊥term has the form of an elastic potential, so we hypothesize that the\ncorresponding energy is distributed along a string or flux tube between the quarks. The energy density operator so\ndefined is thus:\nˆT+−\nstring(s⊥(τ)) =λZ1\n0dτ(˙s⊥(τ))2δ(2)(x⊥−s⊥(τ)), (60)\nwhere s⊥(τ) parametrizes the location of the string in the transverse plane and λis the string energy density. For a\nstraight line,\ns⊥(τ) =r2⊥+τy⊥. (61)\nTo reproduce the y2\n⊥term, an energy density λ=κ4x(1−x) is sufficient. For lack of a better hypothesis for the −2κ2\nterm, we distribute this energy evenly over the same string. This makes the necessary string energy density:\nλ(x,b⊥) =κ4x(1−x)−κ2\ny2\n⊥. (62)\nThe internal energy density associated with this string is then:\nT+−\npot(b⊥) =Z1\n0dxZ1\n0dτb2\n⊥\n(τ−x)4\f\f\f\fψ\u0012\nx,b⊥\nτ−x\u0013\f\f\f\f2\nλ\u0012\nx,b⊥\nτ−x\u0013\n. (63)\nIt is more difficult to obtain an exact analytic result for this integral than for the other integrals we’ve considered.\nAn easy form to work with numerically for the result is:\nT+−\npot(b⊥) =κ4\nπ(\neκ2b2\n⊥\u0010\nE2(κ2b2\n⊥)−E3(κ2b2\n⊥)\u0011\n−3√π\nκb⊥Zπ/2\n0dθsin2θcos2θerfc(κb⊥tanθ))\n, (64)\nwhere erfc is the complementary error function [36]:\nerfc(z) =2√πZ∞\nzdz e−z2. (65)\nThe results for the internal densities in this model are plotted in Fig. 1. The left panel includes both the charge\nand momentum densities, both of which are normalized to 1. The charge density is broader than the momentum\ndensity. The right panel includes the internal energy density, as well as its breakdown into potential and kinetic energy\ncontributions. An interesting feature of the energy density is that it can be negative, in contrast to the momentum\ndensity which is positive-definite.11\n0.0 0.5 1.0 1.5 2.0\nb⊥(fm)0.00.51.01.52.02πb⊥ρint.(b⊥) (fm−1)Charge density\nMomentum density\n0.0 0.5 1.0 1.5 2.0\nb⊥(fm)−0.5−0.4−0.3−0.2−0.10.00.12πb⊥T+−\nint.(b⊥) (fm−1)\nEnergy\nKinetic energy\nPotential energy\nFIG. 1. Internal pion densities in Brodsky’s soft wall holographic model. (Left panel) charge and momentum densities. (Right\npanel) energy density, including its breakdown into kinetic and potential energy.\n2. Phenomenological multipole model\nAlthough illuminating, a light front Hamiltonian model is not necessary to obtain the light front densities. In\nfact, quantum chromodynamics is a quantum field theory and its bound states thus have an indefinite number of\nconstituents, so any few-body model of hadron structure will necessarily be incomplete. The definitions for light\nfront densities do not require assuming the pion to be a two- or even N-body system, however—they require only the\nPoincar´ e-invariant form factors to be provided.\nThe form factors A(t) and D(t) can be obtained from phenomenology. Multipole forms are reasonable for the form\nfactors, based on analyticity rules for the scattering amplitude. For the pion in particular:\nA(t) =1\n1−t/m2\nf2(66a)\nD(t) =−1\n(1−t/m2\nf2)(1−t/m2σ). (66b)\nThe monopole form for A(t) is supported by large- Ncphenomenology [37], and the presence of an f2(1270) pole is\nsuggested by spin-two meson dominance as a gravitational analogy to vector meson dominance in electromagnetic\nform factors. Moreover, mf2= 1270 MeV reproduces the pion radius reported by Kumano et al. ’s analysis of Belle\ndata [38], as well as the proton radius suggested by Kharzeev’s analysis of GlueX data [39] (using a dipole form for the\nproton’s A(t) form factor). The presence of an additional σpole in D(t) comes from dressing of the graviton-quark\nvertex [40], and the value D(0) = −1 comes from a low-energy pion theorem [1, 41, 42]. Using mσ= 630 MeV\nreproduces the D(t) slope reported in Ref. [38].\nFrom these multipole forms of the form factors, we obtain an internal momentum density:\nT++\nint.(b⊥) =m2\nf2\n2πK0(mf2b⊥), (67)\nwhere K0(x) is a modified Bessel function of the second kind [36]. The internal energy density is given by:\nT+−\nint.(b⊥) = \nm2\nπ−m2\nf2\n4!\nm2\nf2\n2πK0(mf2b⊥) +m2\nf2\n4δ(2)(b⊥) +1\n4πm2\nf2m2\nσ\nm2\nf2−m2σ\u0010\nm2\nσK0(mσb⊥)−m2\nf2K0(mf2b⊥)\u0011\n.(68)\nIn the left panel of Fig. 2, we compare the pion energy density obtained with this phenomenological multipole\nparametrization to the result found using the soft wall holographic model. The energy densities look radically different,\nsuggesting a need for additional measurements of hard exclusive reactions to further experimentally constrain the\npion’s gravitational form factors. A curious feature that models have in common is that the internal energy density\nbecomes negative at short distances.12\n0.0 0.5 1.0 1.5 2.0\nb⊥(fm)−1.5−1.0−0.50.00.52πb⊥T+−\nint.(b⊥) (fm−1)Multipole model\nHolographic model\n0 2 4 6 8 10\nb⊥(fm)0.000.050.100.150.200.250.300.352πb⊥T+−\nphys(x⊥) (GeV/fm)σ= 2 fm\nσ= 1.5 fm\nσ= 1 fm\nσ= 0.5 fm\nFIG. 2. Pion energy densities associated with the phenomenological multipole model. (Left panel) compares the internal\nenergy density in the multipole model to the soft wall holographic model. (Right panel) presents the physical pion energy\ndensity (see Eq. (34)) for a Gaussian wave packet for several wave packet widths and with 2 P+=mπ.\nIn the right panel of Fig. 2, we present the physical energy density using a Gaussian wave packet of the form:\nΨ(R⊥) =1√\n2πσ2e−1\n4σ2R2\n⊥, (69)\nalong with the phenomenological multipole model at several values of the wave packet width σand with 2 P+=mπ\nfor concreteness. It can be seen clearly that the physical density does not approach the internal density as the wave\npacket is localized, and is qualitatively quite different. This is a stark manifestation of how wave packet localization\nwill not always distill the desired aspects of a hadron’s internal structure.\nIV. INTERNAL DENSITIES IN INSTANT FORM COORDINATES\nTo investigate physical densities in instant form coordinates in a wave packet independent way, we need an analogous\nformula to that in Eq. (15). We shall proceed to obtain such a relation. The procedure we follow here closely mirrors\nthe derivation first reported in Ref. [19], but with a different convention for defining the position-space wave function\n(which in both our approach and in Li’s is not a true wave function).\nThe Lorentz-invariant completeness relation for states in the Hilbert subspace of single-particle spin-zero states is:\nZd3p\n2Ep(2π)3|p⟩⟨p|= 1. (70)\nWe define the momentum space wave function as:\nΨ(p) =1p\n2Ep⟨p|Ψ⟩, (71a)\nso that it obeys the normalization condition:\nZd3p\n(2π)3\f\fΨ(p)\f\f2= 1. (71b)\nWe similarly define the position-space wave function:\nΨ(r) =Zd3p\n(2π)3Ψ(p)eip·r, (72a)13\nso that it obeys the normalization condition:\nZ\nd3r\f\fΨ(r)\f\f2= 1. (72b)\nIt should be noted that this position space wave function is normalized differently than that of Li et al. in Ref. [19].\nSeveral of our results thus differ cosmetically from Li’s, but are in agreement when the difference in conventions for\nthe position space wave function is accounted for. Following an analogous process to that outlined in Sec. III, it can\nbe shown that:\nρphys(x) =Z\nd3R′\n⊥Z\nd3R⊥Zd3P\n(2π)3Zd3∆\n(2π)3eiP·(R′−R)e−i∆·\u0010\nx−R+R′\n2\u0011\nΨ∗(R′)⟨p′|ˆO(0)|p⟩\n2p\nEpEp′Ψ(R). (73)\nAs with the light cone case, we need a substitution rule for Pwhen it shows up in the matrix element. The appropriate\nrule can be shown to be:\nP→ −i\n2← →∇. (74)\nWherever P0shows up, it is necessary to write it out in terms of P:\nP0=Ep+Ep′\n2=1\n2\ns\nM2+\u0012\nP−∆\n2\u00132\n+s\nM2+\u0012\nP+∆\n2\u00132\n, (75)\nand then use Eq. (74) on each instance of Phere. With such substitutions implicitly applied, the instant form\nequivalent of Eq. (15) is:\nρphys(x) =Z\nd3R⊥Zd3∆\n(2π)3e−i∆·(x−R)Ψ∗(R)⟨p′|ˆO(0)|p⟩\n2p\nEpEp′Ψ(R), (76)\nwhere any Pare to be implicitly understood in terms of Eq. (74). This is the furthest we can go without looking at\nspecific local operators. Obtaining a simple or finite compound density will depend on the matrix element factorizing\nin its ∆andPdependence, or else breaking down into a finite number of terms that factorize as such. We will\npresently see that this does not happen.\nA. Electric charge and energy densities\nThe appropriate matrix element for the electric charge density is:\n⟨p′|j0(0)|p⟩= 2P0F(t) =\ns\nM2+\u0012\nP−∆\n2\u00132\n+s\nM2+\u0012\nP+∆\n2\u00132\nF(t). (77)\nIn instant form coordinates, tis given by:\nt= (p′−p)2=−∆2+ 2\nM2+P2+∆2\n4−s\u0012\nM2+P2+1\n4∆2\u00132\n−(P·∆)2\n. (78)\nThe physical charge density thus takes the form:\nj0\nphys(x) =1\n2Z\nd3R⊥Zd3∆\n(2π)3e−i∆·(x−R)Ψ∗(R) r\nM2+1\n4\u0010\n−i← →∇−∆\u00112\n+r\nM2+1\n4\u0010\n−i← →∇+∆\u00112!\n\u0014\u0010\nM2−1\n4← →∇2+1\n4∆2\u00112\n+1\n2(← →∇·∆)2\u00151/4\nF\n−∆2+ 2\nM2−1\n4← →∇2+1\n4∆2−s\u0012\nM2−1\n4← →∇2+1\n4∆2\u00132\n+1\n2(← →∇·∆)2\n\nΨ(R).(79)14\nThe problem is immediately clear: it is impossible to write this as a finite number of terms whose Rand∆dependence\nfactorizes. At best, we could expand the expression as an infinite formal series in← →∇and obtain an infinitely compound\ndensity. Convergence of the series will depend on the wave packet taking a specific form, which is precisely what we\nneed to avoid to obtain a truly internal density.\nThe instant form energy density is similarly unfactorizable. The relevant matrix element is [1]:\n⟨p′|T00(0)|p⟩=1\n2\ns\nM2+\u0012\nP−∆\n2\u00132\n+s\nM2+\u0012\nP+∆\n2\u00132\n2\nA(t) +∆2\n2D(t), (80)\nwhere tis again defined in Eq. (78). This makes the physical instant form energy density:\nT00\nphys(x) =Z\nd3R⊥Zd3∆\n(2π)3e−i∆·(x−R)Ψ∗(R) \n(Ep+Ep′)2\n4p\nEpEp′A(t) +∆2\n4p\nEpEp′D(t)!\nΨ(R), (81)\nwhere we have neglected to explicitly write the substitution rules in order to keep the formula compact. However, a\nquick glance at Eq. (79) should convince the reader that this expression is not factorizable into either a simple density\nin the manner of Eq. (2), nor a finite compound density in the manner of Eq. (3).\nIn fact, we shall see below, that the instant form densities are at best infinitely compound densities , but convergence\nof this series depends on the form of the wave packet.\nB. Diffuse wave packets and Breit frame densities\nWe have shown that physical densities are infinitely compound in instant form coordinates. Nonetheless, it may\nbe worth clarifying the approximate form of the physical densities when they are expanded as a formal series in the\nderivative, assuming that this expansion is controlled by a small parameter. This can be the case for diffuse wave\npackets, for which the derivative of the wave function is smaller than the wave function.\nLet us consider the first few terms in the physical charge density when expanded out in powers of the derivative.\nThe result is an infinite series of internal densities weighted by different smearing functions. Li et al. [19] refer to\nthe resulting tower of internal densities as multipole moment densities. Odd powers of the derivative cancel in the\nexpansion, so we will consider the up to the second power. The leading terms take the form:\nj0\nphys(x) =Z\nd3RP(R)j0\nBreit(x−R) +Z\nd3RQij(R)jij\nnew(x−R) +. . . , (82a)\nwhere the smearing functions:\nP(R) = Ψ∗(R)Ψ(R) (82b)\nQij(R) = Ψ∗(R) \n−i← →∇i\n2M! \n−i← →∇j\n2M!\nΨ(R) (82c)\nand internal density functions:\nj0\nBreit(b) =Zd3∆\n(2π)3F(−∆2)e−i∆·b(82d)\njij\nnew(b) =Zd3∆\n(2π)3∆i∆j\n1 +∆2\n4M2 \n1\n8M2F(−∆2)\n1 +∆2\n4M2+F′(−∆2)!\ne−i∆·b(82e)\nappear. These of course are only the first two terms in an infinite series, and we reiterate that the convergence of\nthe series depends on the wave packet being diffuse. This series expansion for the physical charge density, along with\nan explicit result for the “new” term, was first reported by Li et al. in Ref. [19]. The apparently different result we\nobtain for the “new” term is due to a difference in convention for defining the position-space wave function.\nA similar expansion can be made for the instant form energy density:\nT00\nphys(x) =Z\nd3RP(R)T00\nBreit(x−R) +Z\nd3RQij(R)Qij\nnew(x−R) +. . . , (83)15\nwhere the standard Breit frame density is given by [1]:\nT00\nBreit(b) = 2 M2Zd3∆\n(2π)31√\n4M2+∆2\u0012\nA(−∆2) +∆2\n4M2h\nA(−∆2) +D(−∆2)i\u0013\ne−i∆·b, (84)\nand the new higher-order internal structure by:\nQij\nnew(b) =M2Zd3∆\n(2π)31√\n4M2+∆2(\nδij\u0012\nA(−∆2)−∆2\n4M2+∆2D(−∆2)\u0013\n+ 2∆i∆j\u0012\nA′(−∆2) +∆2\n4M2+∆2\u0014D(−∆2)\n4M2+∆2+D′(−∆2)\u0015\u0013)\ne−i∆·b.(85)\nFor both the charge density and energy density, the leading term for diffuse wave packets is a convolution between\nthe probability density of the barycenter and the standard Breit frame density. This suggests that, after all, the Breit\nframe densities do actually encode an internal hadron density. However, their interpretation as an internal density\nrelies on an assumption about the wave packet—namely, that it is spatially diffuse.\nThrough the consideration of diffuse wave packets, we have reproduced the finding of Ref. [19] that Breit frame\ndensities provide a leading-order description for the internal densities of hadrons in diffuse wave packets. Moreover,\nby providing the next-to-leading order corrections, we have also provided a means to numerically test the breakdown\nof their applicability. We shall numerically study this breakdown in Sec. IV C.\nC. Numerical illustration of Breit frame density breakdown\nWe will illustrate the domain of applicability of Breit frame densities using a simple wave packet—a Gaussian with\naverage momentum P:\nΨ(R) =1\n(2πσ2)3/4e−R2\n4σ2eiP·R. (86)\nFor such a wave packet:\n−1\n4Ψ∗(R)← →∇i← →∇jΨ(R) =\u0012\nPiPj+δij\n4σ2\u0013\nΨ∗(R)Ψ(R), (87)\nmeaning that the expansion considered in Sec. IV B is valid when both |P|is small (a slow wave packet) and σis\nlarge (a diffuse wave packet). The second-order smearing function for this wave packet is given by:\nQij(R) =\u0012PiPj\nM2+δij\n4σ2M2\u0013\nP(R). (88)\nWe shall consider a case with proton-like kinematics, using M= 940 MeV and simple multipole models for the form\nfactors:\nF(−∆2) =1\n(1 +∆2/m2ρ)2(89a)\nA(−∆2) =1\n(1 +∆2/m2\nf2)2(89b)\nD(−∆2) =−1\n(1 +∆2/m2\nf2)3, (89c)\nwith mρ= 776 MeV and mf2= 1270 MeV. Although the proton is a spin-half particle, this calculation is meant only\nfor illustrative purposes.\nFor a numerical example, we consider wave packets with zero average momentum. The leading approximations for\nthephysical instant form charge density are presented in Fig. 3 for two wave packet widths. The more diffuse packet\nhas a width equal to the Compton wave length:\nλc=2π\nM≈1.3 fm, (90)16\n0 1 2 3 4 5\nr(fm)0.00.10.20.30.44πr2j0\nphys(r) (fm−1)σ=λc\nBreit frame only\nBreit frame only + NLO\n0 1 2 3 4 5\nr(fm)0.00.20.40.60.81.01.24πr2j0\nphys(r) (fm−1)σ=λc/(2π)\nBreit frame only\nBreit frame only + NLO\nFIG. 3. Leading approximations of the physical instant form charge density for a spin-zero target with proton-like properties,\ncalculated according to Eq. (82). Both wave packets use Eq. (86) with P= 0.\n0 1 2 3 4 5\nr(fm)0.00.51.01.52.04πr2T00\nphys(r)Wave packet with σ=λc\nBreit frame only\nBreit frame + NLO\n0 1 2 3 4 5\nr(fm)02468104πr2T00\nphys(r)Wave packet with σ= ¯λc\nBreit frame only\nBreit frame + NLO\nFIG. 4. Leading approximations of the physical instant form energy density for a spin-zero target with proton-like properties,\ncalculated according to Eq. (83). Both wave packets use Eq. (86) with P= 0.\nand the more localized packet has a width of the reduced Compton wavelength, smaller by a factor 2 π. For the more\ndiffuse packet, corrections to the charge density from structures beyond the Breit frame density are negligible. For\nthe more localized packet, however, there are significant corrections to the physical density from the first higher-order\nterm. This indicates that the Breit frame charge density encodes one aspect of the internal distribution of charge in\nthe hadron, but does not tell the entire story, since there are corrections when the hadron wave packet is localized to\nsmaller distance scales than its Compton wavelength.\nWe next repeat this exercise for the physical instant form energy density in Fig. 4. Just as with the charge density,\nthe Breit frame density dominates other aspects of internal hadron structure for sufficiently diffuse wave packets, and\na width as large as the Compton wavelength is sufficient. For smaller wave packets, however, the Breit frame density\nis no longer sufficient to reproduce the physical density, indicating that the Breit frame density gives only a partial\npicture of a hadron’s internal energy distribution.\nThere is one significant difference between Fig. 3 and Fig. 4: the higher-order term in Fig. 4 changes the normal-\nization. In fact, for the smaller wave packet, it should: a more localized wave packet means a greater uncertainty\nin momentum, and accordingly, a greater expectation value for the total energy. The integral of the physical energy\ndensity should give the expected value of the energy, but integrating the leading term in Eq. (83) only gives the mass.\nIt may be tempting to interpret the leading term of Eq. (83) as a mass density and the corrections as kinetic energy\ndensities, but this would be erroneous. There is no “kinetic charge” in the charge density, yet there were higher-order17\ncorrections to the instant form charge density. The higher-order corrections to the energy density certainly contain\nbarycentric kinetic energy, but it is unclear how to separate these from higher-order corrections to the mass density.\nIsolating a barycentric kinetic energy for the light front energy density (see Sec. III) was possible only because of the\nGalilean subgroup, which does not manifest in instant form coordinates.\nV. SUMMARY AND CONCLUSIONS\nIn this work, we proposed and developed a formalism for identifying internal densities of hadrons in a fully wave\npacket independent manner. Physical densities are identified as matrix elements of local operators, which necessarily\nmix internal structure with wave packet dependence. When the dependence on internal structure and wave packet\ncan be cleanly separated within a convolution relation—or the physical density can be written as a sum of such\nconvolutions—an internal density can be defined.\nWhen using light front coordinates with x−integrated out, internal densities can always be identified, owing to\ninvariance of the remaining coordinates under the Galilean subgroup. The scenario is thus analogous to the situation in\nnon-relativistic quantum mechanics. For several simple cases such as the charge density and P+density, the physical\ndensity approaches the internal density when the wave packet is localized, explaining why previous derivations of\nthese quantities through localized wave packets obtained the correct results.\nBy contrast, when using instant form coordinates, it is not possible in general to separate wave packet and internal\nstructure dependence. A special case occurs when the hadron is prepared in a diffuse state (i.e., broad in coordinate\nspace), in which case the physical density can be expanded as an infinite series of convolutions between internal\ndensities and wave packet dependent smearing functions—a result previously found by Li et al. [19], who refer to\nthe tower of internal densities as “multipole moment densities.” For sufficiently broad wave packets, on the order\nof the Compton wavelength or wider, the physical density is dominated by a convolution between the barycentric\nprobability density and the conventional Breit frame density. Thus, the Breit frame density can be identified as\none of infinitely many internal densities that describe hadron structure in instant form coordinates, but one which\ndominates for diffuse wave packets. Breit frame densities therefore have a legitimate claim to describe anaspect of\nhadron structure, but they do not provide a complete description.\nACKNOWLEDGMENTS\nWe would like to thank Yang Li for illuminating discussions on the topics covered in this paper. This work was\nsupported by the U.S. Department of Energy Office of Science, Office of Nuclear Physics under Award Number\nDE-FG02-97ER-41014.\n[1] M. V. Polyakov and P. Schweitzer, Int. J. Mod. Phys. A 33, 1830025 (2018), arXiv:1805.06596 [hep-ph].\n[2] C. Lorc´ e, H. Moutarde, and A. P. Trawi´ nski, Eur. Phys. J. C 79, 89 (2019), arXiv:1810.09837 [hep-ph].\n[3] A. Freese and G. A. Miller, Phys. Rev. D 103, 094023 (2021), arXiv:2102.01683 [hep-ph].\n[4] D. Boer et al. , (2011), arXiv:1108.1713 [nucl-th].\n[5] A. Accardi et al. , Eur. Phys. J. A 52, 268 (2016), arXiv:1212.1701 [nucl-ex].\n[6] R. Abdul Khalek et al. , (2021), arXiv:2103.05419 [physics.ins-det].\n[7] R. Sachs, Phys. Rev. 126, 2256 (1962).\n[8] G. N. 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Lett. 45, 688 (1980)."    },    {        "title": "2210.09762v2.Two_dimensional_hydrodynamic_simulation_for_synchronization_in_coupled_density_oscillators.pdf",        "content": "arXiv:2210.09762v2  [nlin.PS]  24 Mar 2023Two-dimensional hydrodynamic simulation for synchroniza tion in coupled density\noscillators\nNana Takeda, Hiroaki Ito, and Hiroyuki Kitahata∗\nDepartment of Physics, Chiba University, Chiba 263-8522, J apan\n(Dated: March 27, 2023)\nAdensityoscillator isafluidsysteminwhichoscillatory flo woccursbetweendifferentdensityfluids\nthrough the pore connecting them. We investigate the synchr onization in coupled density oscillators\nusing two-dimensional hydrodynamic simulation and analyz e the stability of the synchronous state\nbased on the phase reduction theory. Our results show that th e antiphase, three-phase, and 2-2\npartial-in-phase synchronization modes spontaneously ap pear as stable states in two, three, and four\ncoupled oscillators, respectively. The phase dynamics of c oupled density oscillators is interpreted\nwith their sufficiently large first Fourier components of the p hase coupling function.\nI. INTRODUCTION\nSynchronizationofcoupled limit-cycle oscillatorsis ob-\nserved in various systems such as electrical, chemical,\nand biological systems [1–3]. Synchronization among\nmany oscillators makes the oscillation more robust in\nsome systems, for example, an ensemble of pacemaker\ncells in a heart beating, whereas it can be harmful in\nother systems, for example, an ensemble of neurons in an\nepileptic seizure [3]. Therefore, it is important to under-\nstand the mechanism of synchronization and to control\nthe synchronous behavior as needed. Many efforts have\nbeenmadetodescribesynchronizationtheoretically. One\nof the well-known methods is a phase reduction theory,\nwhich approximately describes the behavior of weakly\ncoupled oscillators using one variable (i.e., phase) for\neach oscillator [2]. The phase coupling function char-\nacterizing the interaction in coupled oscillators has been\ninvestigatedforvarioussystems[4]. Forfluid systems, for\nexample, the oscillatory convection in a Hele-Shaw cell,\nflow around the beating flagella, and the cylinder wake\nwere studied, and their synchronization under hydrody-\nnamic interaction was discussed with phase reduction [5–\n7]. The relation between synchronization and energy ef-\nficiency in fluid systems was also studied to understand\nhow the emerging synchronization mode is selected as an\noptimal state [8–10]. The energy efficiency can be eval-\nuated from some criteria with energy dissipation, such\nas viscous dissipation and the amount of fluid transport.\nThe investigationsof the phase dynamics and energy effi-\nciency will be effective for understanding the underlying\nmechanism of synchronization phenomena in fluid sys-\ntems.\nA density oscillator is a typical fluid system that ex-\nhibits limit-cycle oscillations. It consists of the lower\ndensity fluid in the outer container and the higher den-\nsity fluid in the inner container with a pore at the bot-\ntom. In appropriate conditions for the fluid densities\nand the dimensions of the containers, the downward flow\nof the higher density fluid and the upward flow of the\n∗kitahata@chiba-u.jplower density fluid through the pore alternate periodi-\ncally. This oscillatory phenomenon was first reported\nby Martin in 1970 [11]. Subsequently, experimental and\ntheoretical studies on a density oscillator were reported\nfrom various aspects, such as the mechanism of the oscil-\nlatory flows [12–21], the bifurcation phenomena [18, 22–\n24], and the response to external force [25, 26]. In cou-\npled density oscillators, which consist of multiple inner\ncontainersin a common outer container, varioussynchro-\nnization modes are observed depending on the coupling\nstrengthanddetuningoftheintrinsicfrequencies[14, 19–\n21,27–30]. Foralmostequalintrinsicfrequencies,thetwo\ncoupled oscillators exhibit the antiphase synchronization\nmode [14, 20, 27, 30], the three coupled ones exhibit the\nthree-phase synchronization mode (also called the rota-\ntion mode) [14, 20, 29, 30], and the four coupled ones\nexhibit several synchronization modes depending on the\ncoupling strength [29, 30] in experiments. Several the-\noretical models for the oscillatory flows reproduced the\nsynchronizationmodes[14,19–21,30]. Horie et al.repre-\nsented the time variation of the water level using a com-\nbination of exponential functions and investigated syn-\nchronization modes in two, three, and four coupled oscil-\nlatorswith the phasemodel [30]. Theymainlyfocusedon\nclustering states depending on the parameters, and did\nnot perform detailed analysis of the stability for all the\npossible synchronization modes. In order to understand\nthe criteria for selection of the emerging synchronization\nmode, we need to compare hydrodynamic behaviors be-\ntweenstable and unstable synchronizationmodes, aswell\nas to analyze the stability of them.\nIn the present study, we investigate the hydrodynamic\nbehavior and phase dynamics of coupled density oscilla-\ntors. We take the following three approaches. First, we\nperform the two-dimensional hydrodynamic simulation\nfor coupled density oscillators consisting of two, three,\nand four inner containers and observe the synchroniza-\ntion phenomena. The time evolutions of the water levels\nand the phase differences are calculated for stable and\nunstable synchronization modes. Second, the phase re-\nsponse to the perturbation is measured by the hydrody-\nnamic simulation for a single density oscillator. Based on\nthephasereductiontheory,thedynamicsofthephasedif-2\nferences between coupled oscillators is analyzed. Third,\na linear stability analysis is applied for the fixed points of\nthe phase differences corresponding to stable and unsta-\nble synchronization modes. We evaluate the stabilities of\nthe fixed points generically with the Fourier components\nof the phase coupling function. We finally discuss the\ncriterion for selection of the synchronization mode.\nII. SIMULATION MODEL\nWe perform the two-dimensional hydrodynamic simu-\nlation for coupled density oscillators consisting of niden-\ntical inner containers ( n= 1,2,3,4). The simulation\nmodel is extended from the model for a single density os-\ncillatorintroducedinourpreviousstudy[24]. Figure1(a)\nshows the schematic drawing of coupled density oscilla-\ntors. The widths of the inner and outer containers, din\nanddout, are fixed. The calculation areas are fixed in-\nside the fluid and separated by the walls for each inner\ncontainer. The calculations are performed in respective\nareas. The noscillators interact only through the com-\nmon pressure at the lower boundaries of the calculation\nareas; this pressure corresponds to the water level in the\ncommon outer container. Figure 1(b) shows the ith cal-\nculation area ( i= 1,2,...,n). The width and length of\nthe pore through which the fluid passes are 2 aand 2b, re-\nspectively. Thedistancebetweentheupperboundariesof\nthe walls and the calculation area is Hupper, and the dis-\ntance between the lower boundaries of the walls and the\ncalculation area is Hlower. The origin of the coordinates\nis set at the center of the pore, and the calculationarea is\nset as−din/2≤x≤din/2,−b−Hlower≤y≤b+Hupper.\nFor theith calculation area, the Navier-Stokes equation\nρ(i)/bracketleftbigg∂v(i)\n∂t+(v(i)·∇)v(i)/bracketrightbigg\n=−∇p(i)+µ∇2v(i)+ρ(i)g,\n(1)\nand the incompressible condition\n∇·v(i)= 0, (2)\nare adopted, where ρ(i)=ρ(i)(x,y,t) is the density,\nv(i)=/parenleftBig\nv(i)\nx(x,y,t),v(i)\ny(x,y,t)/parenrightBig\nis the velocity, p(i)=\np(i)(x,y,t) is the pressure, µis the viscosity, and g=\n(0,−g) is the acceleration of gravity. The normalized\nconcentration c(i)=c(i)(x,y,t) is calculated using the\nadvection-diffusion equation\n∂c(i)\n∂t+∇·(c(i)v(i)) =D∇2c(i), (3)\nwhereDisthediffusion coefficient. We definethe density\nρ(i), which depends on the concentration c(i), as\nρ(i)=ρlow+c(i)(ρhigh−ρlow), (4)\nwhereρhighandρloware the densities of the higher and\nlower density fluids, respectively. The water levels in theHupper \nHlower 2a\n2b\n(0,0) \nxy(a) (b) din dout \n2\ni = 1 i = 2 i = ndout \n2\nFIG.1. Two-dimensional modelofcoupleddensityoscillato rs.\n(a) Schematic drawing of coupled density oscillators consi st-\ning ofninner containers. The red broken rectangles represent\nthe calculation areas. (b) Details of the calculation area. The\nhatched area shows the bottom wall of the inner container.\nith inner container y(i)\nin(t) and the common outer con-\ntaineryout(t) are associated with the pressures p(i)\nupper(t)\nandplower(t) at the upper and lower boundaries of the\nith calculation area, respectively, as\np(i)\nupper(t) =ρhighg/parenleftBig\ny(i)\nin(t)−b−Hupper/parenrightBig\n,(5a)\nplower(t) =ρlowg(yout(t)+b+Hlower),(5b)\nwhereplower(t) is common for all oscillators. The fluid\ndensities in the inner and outer containers should change\nwith oscillations, but the change is so small that they are\nassumed to be constants ρhighandρlowin Eqs. (5a) and\n(5b), respectively. The changes in the water levels are\nobtained from the amount of fluid passing through the\npore per unit time Q(i)(t) as\ndy(i)\nin\ndt=Q(i)\ndin, (6a)\ndyout\ndt=−/summationtext\niQ(i)\ndout, (6b)\nQ(i)(t) =/integraldisplaya\n−av(i)\ny(x,b,t)dx. (6c)\nThe boundary conditions are set as follows. At the\nboundaries on the walls, the velocity follows the nonslip\nboundary condition as v(i)=0, and the concentration\nfollows the Neumann boundary condition as ∇⊥c(i)= 0,\nwhere∇⊥denotes the spatial derivative in the direction\nperpendicular to the boundary. At the upper and lower\nboundaries inside the fluid, v(i)andc(i)follow the Neu-\nmann boundary conditions as ∇⊥v(i)=0and∇⊥c(i)=\n0, respectively, and the pressure follows Eqs. (5a) and\n(5b).\nIn the initial state, the higher and lower density fluids\nare stationary ( v(i)=0) and are not mixed with each\nother (c(i)= 0 aty < bandc(i)= 1 aty≥b). The ini-\ntial water level in the outer container yout(0) =yout,0is\nfixed in all simulations. For the single oscillator ( n= 1),\nthe initial water level in the inner container is set from\nthe balance with the hydrostatic pressure by the outer\ncontainer fluid as y(1)\nin(0) =b+ (ρlow/ρhigh)(yout,0−b).\nFor the coupled oscillators ( n≥2), the initial water lev-3\nels in the inner containers y(i)\nin(0) are varied as control\nparameters.\nTo numerically solve the Navier-Stokes equation with\nthe incompressible condition in Eqs. (1) and (2), the\nmarker-and-cell method was adopted [31, 32], where the\nPoisson equation obtained from the divergence of Eq. (1)\nwas calculated for the pressure. We ignored the gradient\nof the density, ∇ρ(i), in the Poisson equation since it is\nsufficiently small. The explicit method was adopted for\nthe advection-diffusion equation in Eq. (3). Each calcu-\nlation area was divided into 200 ×240 meshes, and the\nspatial mesh size was set to dx=dy= 0.005. The time\nwas evolved with the time step dt= 0.0002. The param-\neters were set as follows: a= 0.03,b= 0.05,din= 1,\ndout= 6,g= 10,µ= 1/300,D= 0.0001,ρhigh= 1.2,\nρlow= 1,Hupper=Hlower= 0.55, andyout,0= 10.05.\nIII. SIMULATION RESULTS\nWe performed the hydrodynamic simulation for the\ncoupled density oscillators with identical intrinsic fre-\nquencies and coupling strengths between oscillators for\nn= 2,3,4.\nIn the simulation for the two coupled oscillators ( n=\n2), an upward flow and a downward flow alternately\noccurred in each oscillator, and several periods later,\nthe oscillations of flows in both oscillators synchronized;\nwhen an upward flow started in one oscillator, a down-\nward flow started in the other oscillator. The time se-\nries ofy(1)\ninandy(2)\ninare shown in Fig. 2(a-1). To de-\nscribe the synchronization quantitatively, we calculated\nthe phase difference between two oscillators. Here, a\nphaseφ(0≤φ <2π) is defined in proportion to time;\nφ= 0 corresponds to the time when an upward flow\nstarts and φ= 2πcorresponds to when the next upward\nflow starts [see Fig. 3(a)]. The phase difference between\ntwo oscillators is represented as ∆ φij=φj−φi, where\nφiis the phase of the ith oscillator. The phase difference\n∆φ12converged to the constant value ∆ φ12=π, i.e., the\nantiphase mode as shown in Fig. 2(a-2).\nWe also realized the in-phase mode (∆ φ12= 0), which\nhas not been observed in experiments, by setting the\nidentical initial conditions for two oscillators. Due to\nthe symmetric procedure in the numerical calculation,\nthe initial phase difference ∆ φ12= 0 was kept, even if\nit was unstable. The symmetry of the initial conditions\nwas controlled by the water levels: the antiphase mode\nappeared from y(1)\nin(0)/negationslash=y(2)\nin(0), and the in-phase mode\nappeared from y(1)\nin(0) =y(2)\nin(0).\nThree coupled oscillators ( n= 3) converged to the\nthree-phase mode with the equivalent phase differences\n2π/3 as shown in Fig. 2(c). By controlling the sym-\nmetry of the initial conditions for a part or all of the\noscillators, we realized the partial-in-phase mode from\ny(1)\nin(0) =y(2)\nin(0)/negationslash=y(3)\nin(0) and the all-in-phase mode\nfromy(1)\nin(0) =y(2)\nin(0) =y(3)\nin(0) as shown in Figs. 2(d)and 2(e), respectively.\nFour coupled oscillators ( n= 4) converged to the 2-2\npartial-in-phase mode, where two pairs of in-phase oscil-\nlators synchronized with the phase difference πas shown\nin Fig. 2(f). It took a longer time to converge to the\nconstant phase differences compared with the antiphase\nmode (n= 2) and the three-phase mode ( n= 3). By\ncontrolling the symmetry of the initial conditions for a\npartoralloftheoscillators,werealizedthe2-2partial-in-\nphasemode [Figs. 2(g) and 2(h)], the 3-1partial-in-phase\nmode [Fig. 2(i)], and the all-in-phase mode [Fig. 2(j)].\nHere, the 3-1 partial-in-phase mode consists of a set of\nthree in-phase oscillators and an oscillator with a differ-\nent phase from the other three.\nIV. ANALYSIS OF PHASE DYNAMICS\nWe analyzed the stabilities of the synchronization\nmodes from the viewpoint of phase dynamics. We first\ninvestigated the single density oscillator subjected to a\nperturbation to measure the phase response for the limit-\ncycle oscillation ( n= 1). Figure 3(a) shows the time\nseries of the outer water level youtwithout perturbation\nand the corresponding phase φ. The perturbation of the\nouter water level ∆ youtis introduced at t=t0, and the\ntime evolution of youtwith the perturbation is given in-\nstead of Eq. (6b) as\ndyout\ndt=−Q(1)\ndout+∆youtδ(t−t0). (7)\nThe phase shift ∆ φis determined by subtracting the\nphaseφ0in the unperturbed system from the phase φ′in\nthe perturbed system after sufficiently long time as\n∆φ=φ′−φ0. (8)\nIn the simulation, the perturbation was introduced 15\nperiods after the initial state, and the phase shift was\nmeasured 15 periods after the perturbation was intro-\nduced to ensure sufficient relaxation. The phase response\ncurve ∆φ= ∆φ(φ,∆yout) represents the phase shift as\na function of the phase φat which the perturbation is\nintroduced and the perturbation amplitude ∆ yout. For a\nsufficiently small perturbation, the phase response curve\nis expected to be linear to the perturbation amplitude\n∆youtas\n∆φ(φ,∆yout) = ∆youtZ(φ), (9)\nwhereZ(φ) is the phase sensitivity function. Figure 3(b)\nshows the phase shift ∆ φplotted against the perturba-\ntionamplitude ∆ yout, andthe linearregionwhereEq.(9)\nholds was confirmed below ∆ yout∼0.0002. Then, we\nmeasured the phase sensitivity function Z(φ) with the\nperturbation amplitude ∆ yout= 0.0001 as shown in\nFig. 3(c). The phase sensitivity function Z(φ) is posi-\ntive, and the phase is preceded at 0 /lessorsimilarφ/lessorsimilarπ/3,4π/3/lessorsimilar4\n200 250 240 210 230 220 \n200 250 240 210 230 220 \n200 250 240 210 230 220 \n200 250 240 210 230 220 \n200 250 240 210 230 220 200 250 240 210 230 220 \n200 250 240 210 230 220 \n200 250 240 210 230 220 900 950 940 910 930 920 \n700 750 740 710 730 720 0 250 200 50 150 100 300 \n0 250 200 50 150 100 300 \n0 250 200 50 150 100 300 \n0 250 200 50 150 100 300 \n0 250 200 50 150 100 300 0 250 200 50 150 100 300 0 250 200 50 150 100 300 0 250 200 50 150 100 300 0 1000 800 200 600 400 \n0 800 200 600 400 t\ntt\nt\ntt\ntt\nt\ntt\ntt\nt\ntt\ntt\nt\nt0.00 0.01 \n–0.01 \n0.00 0.01 \n–0.01 0.00 0.01 \n–0.01 \n0.00 0.01 \n–0.01 \n0.00 0.01 \n–0.01 0.00 0.01 \n–0.01 \n0.00 0.01 \n–0.01 0.00 0.01 \n–0.01 \n0.00 0.01 \n–0.01 \n0.00 0.01 \n–0.01 2π \nπ\n0\n2π \nπ\n02π \nπ\n02π \nπ\n0\n2π \n2π/3 \n02π \nπ\n0\n2π \nπ\n02π \nπ\n02π \nπ\n0\n2π \nπ\n0yin (i) – yin (i)yin (i) – yin (i)yin (i) – yin (i)yin (i) – yin (i)\nyin (i) – yin (i)yin (i) – yin (i)yin (i) – yin (i)yin (i) – yin (i)yin (i) – yin (i)\nyin (i) – yin (i)Δϕij Δϕij Δϕij Δϕij Δϕij\nΔϕij Δϕij Δϕij Δϕij Δϕij(a-1)\n(b-1)\n(c-1)\n(d-1)\n(e-1)(a-2)\n(b-2)\n(c-2)\n(d-2)\n(e-2)(f-1)\n(g-1)\n(h-1)\n(i-1)\n(j-1)(f-2)\n(g-2)\n(h-2)\n(i-2)\n(j-2)yin (1)(0) > yin (2)(0)\nyin (1)(0) = yin (2)(0)\nyin (1)(0) > yin (2)(0) > yin (3)(0)\nyin (1)(0) = yin (2)(0) > yin (3)(0)\nyin (1)(0) = yin (2)(0) = yin (3)(0)yin (1)(0) > yin (2)(0) > yin (3)(0) > yin (4)(0)\nyin (1)(0) = yin (2)(0) > yin (3)(0) > yin (4)(0)\nyin (1)(0) = yin (2)(0) > yin (3)(0) = yin (4)(0)\nyin (1)(0) = yin (2)(0) = yin (3)(0) > yin (4)(0)\nyin (1)(0) = yin (2)(0) = yin (3)(0) = yin (4)(0)\nΔϕ12 Δϕ13 Δϕ14 i = 1 i = 2 i = 3 i = 4 4π/3 \nFIG. 2. Synchronization in coupled density oscillators. [( a),(b)]n= 2, [(c)–(e)] n= 3, and [(f)–(j)] n= 4. [(a-1)–(j-1)] Time\nseries of y(i)\nin−¯y(i)\nin, where ¯y(i)\ninis the time average of y(i)\nin. The grey bold solid, red narrow solid, green bold dashed, an d blue\nnarrow dashed lines represent i= 1,2,3,and 4, respectively. The symmetry of the initial water level y(i)\nin(0) is shown above\neach panel. [(a-2)–(j-2)] Corresponding phase differences ∆φij=φj−φi. The red circles, green crosses, and blue triangles\nrepresent ∆ φ12, ∆φ13, and ∆φ14, respectively.5\nφ/lessorsimilar2π. In contrast, Z(φ) is negative, and the phase is\ndelayed at π/3/lessorsimilarφ/lessorsimilar4π/3. These behaviors could be\ninterpreted as follows. Due to an increase in the outer\nwaterlevel bythe perturbation, the upwardordownward\nflow ran out earlier for Z(φ)>0, whereas it kept longer\nforZ(φ)<0, compared to those without the perturba-\ntion. The change in the sign of Z(φ) might be associated\nwith the acceleration of the outer water level d2yout/dt2\n(=ωdf(φ)/dφ) [see Fig. 3(d)]. An increase in the outer\nwater level by the perturbation can precede the phase if\nthe upward flow is getting strong or the downward flow\nis getting weak. In contrast, it can delay the phase if\nthe upward flow is getting weak or the downward flow is\ngetting strong.\nThe phase dynamics of nweakly coupled identical os-\ncillators can be analyzed from the phase response of a\nsingle oscillator under the small perturbation based on\nthe phase reduction theory [2]. The dynamics of the ith\noscillator under the small perturbation from other oscil-\nlators is described as\ndφi\ndt=ω+Z(φi)/summationdisplay\nj/negationslash=if(φj), (10)\nwhereωis the intrinsic frequency of the limit-cycle os-\ncillation, and f(φi) =−Q(1)(φi)/doutis the periodic\nperturbation by the ith oscillator to youtas shown in\nFig. 3(d). The periodic time series of Q(1)was obtained\nfrom the time series 15 periods after the initial state for\nn= 1. Equation (10) is approximatedby taking the time\naverage of the second term on the right-hand side over a\nperiod as\ndφi\ndt=ω+/summationdisplay\nj/negationslash=iΓ(∆φij), (11)\nwhere Γ( φ) is the phase coupling function:\nΓ(φ) =1\n2π/integraldisplay2π\n0Z(θ)f(φ+θ)dθ. (12)\nFigure 3(e) shows the phase coupling function Γ( φ) cal-\nculated from Z(φ) andf(φ) in Fig. 3(c) and 3(d), re-\nspectively. Γ( φ) looks like a sinusoidal function although\nZ(φ) andf(φ) seem to include higher harmonics. This\nreason is discussed in detail in Appendix A. Here, we\nconsider the Fourier series\nΓ(φ) =a0\n2+∞/summationdisplay\nk=1(akcos(kφ)+bksin(kφ)),(13)\nwhereakandbkare the Fourier cosine and sine coeffi-\ncients of Γ( φ), respectively. The coefficients akandbk\nfork≤6 calculated from the discrete points in Fig. 3(e)\nare shown in Fig. 4. For the following calculation includ-\ning Γ(φ), we use the Fourier series expanded up to sixth\norder. The phase coupling function represented by the\nFourier series is also shown with a line in Fig. 3(e), which\nwell fits the discrete points.200 250 210 220 230 240 \nt2\n0\n–2 0π2π yout  – yout  ×10 –3 ϕ\n2 4 6 0 8 10 ×10 –4 0\n–8π –4π –2π \n–6π \nΔyout Δϕ×10 –2 \n0 π π/2 3π/2 2π \n0 π π/2 3π/2 2π 0 π π/2 3π/2 2π ϕ\nϕ ϕ–2 2\n–1 1\n0×10 2Z(ϕ)\n01\n–1 f(ϕ)×10 –3 \n01\n–1 Γ( ϕ)×10 –1 (a) \n(e) (d) (c) (b) \nFIG. 3. Phase description of a density oscillator. (a) Defi-\nnition of the phase φ. The black line shows the time series\nofyout−¯youtscaled on the left axis, where ¯ youtis the time\naverage of yout. The red line shows the corresponding phase\nφscaled on the right axis. (b) Dependence of the phase shift\n∆φon the perturbation amplitude ∆ yout. The perturbation\nwas introduced at φ=πin a cycle, where the phase shift\nwas sufficiently large. The slope of the line was determined\nby the fitting from five points for ∆ yout≤0.0001. (c) Phase\nsensitivity function Z(φ) obtained from the perturbation with\n∆yout= 0.0001. (d) Periodic perturbation f(φ) to the wa-\nter level in the outer container. (e) Phase coupling functio n\nΓ(φ). The points represent the discrete data calculated from\n(c) and (d). The line shows the fitting curve by the Fourier\nseries up to sixth order.\nThe dynamics of the phase difference ∆ φ12in two cou-\npled oscillators is derived as\nd∆φ12\ndt= Γ(−∆φ12)−Γ(∆φ12). (14)\nThe time derivative of ∆ φ12is expressed as a function\nof ∆φ12, and the phase portrait is obtained as shown in\nFig. 5. The fixed points, where d∆φ12/dt= 0 holds, are\n∆φ12= 0 and π. The derivative d∆φ12/dtis positive\nat 0<∆φ12< πand negative at π <∆φ12<2π.\nTherefore, the antiphase mode corresponding to ∆ φ12=\nπis stable, whereas the in-phase mode corresponding to\n∆φ12= 0 is unstable.\nThe dynamics of the phase differences ∆ φ12and ∆φ136\n1 2 4 3 6 5 1 2 4 3 6 510 –1 \n10 –2 \n10 –3 \n10 –4 \n10 –5 \n10 –6 10 –1 \n10 –2 \n10 –3 \n10 –4 \n10 –5 \n10 –6 \nk k|ak|\n|bk|(a) (b) \n0ak > 0\nak < 0bk > 0\nbk < 0\nFIG. 4. Fourier components of Γ( φ) shown in Fig. 3(e). (a)\nFourier cosine coefficient ak. (b) Fourier sine coefficient bk.\nThe signs of akandbkare represented by solid circles (posi-\ntive) and open circles (negative).\n0 π π/2 3π/2 2π \nΔϕ12 dΔϕ12 /dt \n0.00 0.05 0.10 \n–0.05 \n–0.10 0.15 \n–0.15 \nFIG. 5. Phase portrait in the (∆ φ12,d∆φ12/dt) plane for two\ncoupled oscillators. It indicates that the antiphase mode f or\n∆φ12=πis stable, whereas the in-phase mode for ∆ φ12= 0\nis unstable.\nin three coupled oscillators are derived as\nd∆φ12\ndt= Γ(−∆φ12)+Γ(∆φ13−∆φ12)\n−Γ(∆φ12)−Γ(∆φ13), (15a)\nd∆φ13\ndt= Γ(−∆φ13)+Γ(∆φ12−∆φ13)\n−Γ(∆φ12)−Γ(∆φ13). (15b)\nThe stabilities of the phase differences are visualized\non the vector field in the (∆ φ12,∆φ13) plane as shown\nin Fig. 6. There are three synchronization modes sat-\nisfyingd∆φ12/dt=d∆φ13/dt= 0: (i) the all-in-\nphase mode for (∆ φ12,∆φ13) = (0,0), which is an\nunstable star node; (ii) the partial-in-phase mode for\n(∆φ12,∆φ13) = (α,0),(0,α),(2π−α,2π−α), which are\nunstable saddle points; and (iii) the three-phase mode\nfor (∆φ12,∆φ13) = (2π/3,4π/3),(4π/3,2π/3), which are\nstablespirals. Here, αsatisfiesΓ( α) = 2Γ(−α)−Γ(0)and\n0< α <2π.\nThedynamicsofthephasedifferences∆ φ12, ∆φ13, and0 π π/3 2π/3 4π/3 5π/3 2π \nΔϕ12 0π\nπ/3 2π/3 4π/3 5π/3 2π Δϕ13 \n0.00 0.05 0.10 0.15 0.20 0.25 0.30 Phase velocity \nFIG. 6. Vector field of ( d∆φ12/dt,d∆φ13/dt) in the\n(∆φ12,∆φ13) plane for three coupled oscillators. The un-\nstable star nodes (square), the unstable saddle points (di-\namond), and the stable spirals (circle) represent the all-\nin-phase, partial-in-phase, and three-phase modes, respe c-\ntively. The phase velocity/radicalBig\n(d∆φ12/dt)2+(d∆φ13/dt)2in\nthe (∆φ12,∆φ13) plane is shown as a color gradient.\n∆φ14in four coupled oscillators are derived as\nd∆φ12\ndt= Γ(−∆φ12)+Γ(∆φ13−∆φ12)\n+Γ(∆φ14−∆φ12)\n−Γ(∆φ12)−Γ(∆φ13)−Γ(∆φ14),(16a)\nd∆φ13\ndt= Γ(−∆φ13)+Γ(∆φ12−∆φ13)\n+Γ(∆φ14−∆φ13)\n−Γ(∆φ12)−Γ(∆φ13)−Γ(∆φ14),(16b)\nd∆φ14\ndt= Γ(−∆φ14)+Γ(∆φ12−∆φ14)\n+Γ(∆φ13−∆φ14)\n−Γ(∆φ12)−Γ(∆φ13)−Γ(∆φ14).(16c)\nThere are four synchronization modes satisfying\nd∆φ12/dt=d∆φ13/dt=d∆φ14/dt= 0: (i)\nthe all-in-phase mode for (∆ φ12,∆φ13,∆φ14) =\n(0,0,0), (ii) the 3-1 partial-in-phase mode for\n(∆φ12,∆φ13,∆φ14) = (β,0,0),(0,β,0),(0,0,β),(2π−\nβ,2π−β,2π−β), (iii) the 2-2 partial-in-phase mode\nfor (∆φ12,∆φ13,∆φ14) = (0 ,π,π),(π,0,π),(π,π,0),\nand (iv) the four-phase mode for (∆ φ12,∆φ13,∆φ14) =\n(π/2,π,3π/2),(π/2,3π/2,π),(π,π/2,3π/2),(π,3π/2,\nπ/2),(3π/2,π/2,π),(3π/2,π,π/2). Here, βsatisfies\nΓ(β) = 3Γ( −β)−2Γ(0) and 0 < β < 2π. Instead\nof visualizing the three-dimensional dynamics of the\nphase differences, we investigate the time variation\nof the phase differences from arbitrary initial states.7\nEquations (16a)–(16c) were discretized with the Euler\nmethod, and the time evolutions of ∆ φ12,∆φ13, and\n∆φ14were calculated with the time step dt= 0.01.\nThe numerical calculation indicated that the system\nconverged to the 2-2 partial-in-phase mode as shown in\nFig. 7(a), where the magnified image around the initial\nstate is shown in Fig. 7(b). Figure 7(c) shows the phase\ndifferences ∆ φ13and ∆φ24during the same time range\nas in (b). Two antiphase pairswere rapidly formed in the\ntime scale of t∼101, whereas the system converged to\nthe 2-2partial-in-phasemode in thetime scaleof t∼104.\nThus we consider the phase dynamics of the two an-\ntiphase pairs (∆ φ12,∆φ13,∆φ14) = (∆φ12,π,∆φ12+π).\nEquations (16a)–(16c) are reduced to\nd∆φ12\ndt= Γ(−∆φ12)−Γ(∆φ12)\n+Γ(π−∆φ12)−Γ(∆φ12+π),(17a)\nd∆φ13\ndt=d∆φ24\ndt= 0. (17b)\nThe phase portrait in the (∆ φ12,d∆φ12/dt) plane for\nthe four coupled oscillators with two antiphase pairs is\nshown in Fig. 7(d). The fixed points, where d∆φ12/dt=\n0 holds, are ∆ φ12= 0,π/2,π, and 3π/2. The derivative\nd∆φ12/dtis positive at π/2<∆φ12< π,3π/2<∆φ12<\n2πand negative at 0 <∆φ12< π/2,π <∆φ12<3π/2.\nTherefore, the 2-2 partial-in-phase mode corresponding\nto∆φ12= 0andπisstable, whereasthefour-phasemode\ncorresponding to ∆ φ12=π/2 and 3π/2 is unstable.\nWe compare the analysis of the phase dynamics with\nthe simulation results in Sec. III. The antiphase ( n= 2),\nthree-phase ( n= 3), and 2-2 partial-in-phase ( n= 4)\nmodes obtained from the simulations with asymmetric\ninitial conditions were found to be stable, whereas the\nother modes obtained from the simulations with sym-\nmetric initial conditions for a part or all of the oscillators\nwere found to be unstable.\nThe difference in the period between different synchro-\nnization modes in the simulation results can be explained\nbased on the phase reduction theory. Assuming the syn-\nchronization with the constant period Tand integrating\nthe phase equation in Eq. (11) over T, we obtain\nT=1\n1+/summationtext\nj/negationslash=iΓ(∆φij)\n2πT0T0, (18)\nwhereT0= 2π/ωis an intrinsic period. It is found that\nthesynchronizationperiod Tislongerorshorterthanthe\nintrinsic period T0for negative or positive/summationtext\nj/negationslash=iΓ(∆φij),\nrespectively. The periods obtained from the simula-\ntion and those estimated by Eq. (18) are shown in Ta-\nble I. Simulation results indicate that synchronization\nperiods are longer or shorter than the intrinsic period\nfor/summationtext\nj/negationslash=iΓ(∆φij)<0 or/summationtext\nj/negationslash=iΓ(∆φij)>0, respec-\ntively, which qualitatively agrees with the evaluation by\nEq. (18). The quantitative difference in periods between\nthe simulation and the estimation could be attributed to\nthe following two factors. The first is a slight difference(a) (b) \n–2 1\n–1 \n0 π π/2 3π/2 2π \nΔϕ12 02dΔϕ12 /dt 0π2π \n0 10000 30000 \nt\n×10 –4 20000 \n(d) 0π2π Δϕij \n0 10 30 \nt20 40 50 Δϕij \nt0 10 30 20 40 50 0π2π Δϕij (c)Δϕ12 \nΔϕ13 \nΔϕ14 \nΔϕ13 \nΔϕ24 \nFIG. 7. Behavior of the phase differences ∆ φ12, ∆φ13, and\n∆φ14in four coupled oscillators. (a) Time series of the\nphase differences from the initial state (∆ φ12,∆φ13,∆φ14) =\n(π/3,7π/6,8π/5). (b) Magnified image for 0 ≤t≤50\nin (a). (c) ∆ φ13and ∆φ24during the same time range\nas in (b). The red, green, blue, and magenta lines repre-\nsent ∆φ12, ∆φ13, ∆φ14, and ∆φ24, respectively. (d) Phase\nportrait in the (∆ φ12,d∆φ12/dt) plane for the four coupled\noscillators with two antiphase pairs (∆ φ12,∆φ13,∆φ14) =\n(∆φ12,π,∆φ12+π). It indicates that the 2-2 partial-in-phase\nmode for ∆ φ12= 0 and πis stable, whereas the four-phase\nmode for ∆ φ12=π/2 and 3π/2 is unstable.\nin amplitudes of coupled oscillators from the single os-\ncillator in the simulation as shown in Table I. They are\nassumed to be equal in the phase reduction theory. The\nsecond is the time averaging from Eq. (10) to Eq. (11) in\nthe phase reduction.\nV. LINEAR STABILITY ANALYSIS\nA linear stability analysis is applied to the fixed points\nof the phase differences between coupled oscillators in\nEqs. (14), (15), and (16) for n= 2,3,and 4, respectively.\nWe describe the stabilities of the fixed points generically\nwith the Fourier components akandbkof the phase cou-\npling function Γ( φ) in Eq. (13).\nFor the two coupled oscillators, we consider a fixed\npoint ∆φ∗\n12and the perturbation η12(t) from a fixed\npoint, which is defined as η12(t) = ∆φ12(t)−∆φ∗\n12. The\nperturbation approximately follows\ndη12\ndt=λη12, (19)\nwhereλisthederivativeoftheright-handsideinEq.(14)\nwith respect to ∆ φ12at the fixed point ∆ φ12= ∆φ∗\n12.8\nTABLE I. Period and amplitude of the oscillation in synchron ization. The first, second, third, fourth, fifth, and sixth co lumns\nshowthenumberofcoupledoscillators, thesynchronizatio nmode, theperiodobtained from thesimulation, theperiode stimated\nby Eq. (18) with T0and Γ(φ) obtained from the simulation, the sign of/summationtext\nj/negationslash=iΓ(∆φij), and the amplitude obtained from the\nsimulation, respectively.\nNumber of Synchronization Period by Estimated Sign of Ampli tude\noscillators mode simulation period/summationtext\nj/negationslash=iΓ(∆φij) by simulation\n1 13.2 0.0104\n2 antiphase 14.4 13.9 negative 0.0112\n2 in-phase 12.5 12.5 positive 0.0096\n3 three-phase 15.3 14.3 negative 0.0111\n3 partial-in-phase 14.1 13.7 negative 0.0102( i= 1,2), 0.0121( i= 3)\n3 all-in-phase 12.1 12.0 positive 0.0089\n4 2-2 partial-in-phasea14.4 13.9 negative 0.0112\n4 3-1 partial-in-phase 13.8 13.2 negative 0.0095( i= 1,2,3), 0.0129( i= 4)\n4 all-in-phase 11.8 11.4 positive 0.0082\naAmong the three simulation results in Figs. 2(f)–2(h), the s imulation result of Fig. 2(h) is adopted.\nFor the three coupled oscillators, we consider a\nfixed point (∆ φ∗\n12,∆φ∗\n13) and the perturbation ( η12,η13)\nfrom a fixed point, which is defined as ( η12,η13) =\n(∆φ12,∆φ13)−(∆φ∗\n12,∆φ∗\n13). The perturbation approx-\nimately follows\nd\ndt/parenleftbigg\nη12\nη13/parenrightbigg\n=A/parenleftbigg\nη12\nη13/parenrightbigg\n, (20)\nwhereAis the Jacobian matrix of the right-hand\nsides in Eqs. (15a) and (15b) at the fixed point\n(∆φ12,∆φ13)=(∆φ∗\n12,∆φ∗\n13).\nFor the four coupled oscillators, we consider a\nfixed point (∆ φ∗\n12,∆φ∗\n13,∆φ∗\n14) and the perturba-\ntion (η12,η13,η14) from a fixed point, which is\ndefined as ( η12,η13,η14) = (∆ φ12,∆φ13,∆φ14)−\n(∆φ∗\n12,∆φ∗\n13,∆φ∗\n14). The perturbation approximately\nfollows\nd\ndt\nη12\nη13\nη14\n=A\nη12\nη13\nη14\n, (21)\nwhereAis the Jacobian matrix of the right-\nhand sides in Eqs. (16a)–(16c) at the fixed point\n(∆φ12,∆φ13,∆φ14)=(∆φ∗\n12,∆φ∗\n13,∆φ∗\n14).\nThe stability of the fixed point can be evaluated from\nthe eigenvalues λof the Jacobian matrix A, which indi-\ncate the growth rates of the perturbation. In our simu-\nlation, the first Fourier components a1andb1are much\nlarger than the higher-order components ( k≥2) [see\nFig. 4]. Thus a1andb1most contribute to the stabil-\nity if they are included in the real parts of the eigen-\nvalues. Therefore, we discuss the stability only with\na1andb1in the eigenvalues. The results of the linear\nstability analysis for various synchronization modes are\nshown in Table II, where the eigenvalues, the types of\nthe fixed points, and the stabilities evaluated from thenumerical result of a1andb1are indicated. We obtained\nα= arctan/parenleftbig\n−6a1b1/(a2\n1−9b2\n1)/parenrightbig\nfor the partial-in-phase\nmode (n= 3) and β= arctan/parenleftbig\n−4a1b1/(a2\n1−4b2\n1)/parenrightbig\nfor\nthe 3-1 partial-in-phase mode ( n= 4) only with a1and\nb1. The results of the linear stability analysis suggest\nthat if the antiphase mode is stable in the two coupled\noscillators (i.e., b1is negative), the three-phase mode is\nalso stable in the three coupled oscillators as long as the\nfirst Fourier sine coefficient b1is sufficiently larger than\nthe higher-order ones.\nThe eigenvalues are zero for the four-phase and 2-2\npartial-in-phase modes for n= 4 due to the neglect of\nthe higher-order components. In this case, the higher-\norder components should be considered to evaluate the\nstability. WiththenumericalresultoftheFouriercompo-\nnents for k≤6, it wasfound that the 2-2partial-in-phase\nmode is stable, whereas the four-phase mode is unstable.\nHere, the eigenvalues including the higher-order compo-\nnents are shown in Appendix B. The results of the linear\nstability analysis are consistent with the stabilities ob-\ntained in Sec. IV.\nOkuda applied a linear stability analysis to the sym-\nmetric cluster states with equivalent phase differences\nin globally coupled oscillators, where the eigenvalue was\ndescribed with the Fourier components [33]. The sym-\nmetric cluster states, where each cluster consists of an\nequal number of oscillators, include the synchronization\nmodes obtained in our study, except for the partial-in-\nphase (n= 3) and 3-1 partial-in-phase ( n= 4) modes.\nTheeigenvaluesforthe symmetricclusterstatesobtained\nin our study correspond to those described by Okuda.9\nTABLE II. Linear stability analysis for the fixed point of the phase difference in coupled oscillators only with the first Fo urier\ncomponents a1andb1. The first, second, third, fourth, and fifth columns show the n umber of coupled oscillators, the synchro-\nnization mode, the eigenvalue λ, the type of the fixed point, and the stability of the fixed poin t evaluated from the numerical\nresult of a1andb1, respectively.\nNumber of Synchronization mode Eigenvalue Fixed point type Linear\noscillators stability\n2 antiphase 2 b1 node stable\n2 in-phase −2b1 node unstable\n3 three-phase 3( b1±ia1)/2 spiral stable\n3 partial-in-phase 3 b1,−9b1(a2\n1+b2\n1)/(a2\n1+9b2\n1) saddle unstable\n3 all-in-phase −3b1(duplicate) node unstable\n4 four-phase 0a,2(b1±ia1)\n4 2-2 partial-in-phase 0a(duplicate) ,4b1\n4 3-1 partial-in-phase 4 b1,−8b1(a2\n1+b2\n1)/(a2\n1+4b2\n1)(duplicate) saddle unstable\n4 all-in-phase −4b1(triplicate) node unstable\naa1orb1is not included in the eigenvalue.\nVI. DISCUSSION\nWe compare our simulation results with experimen-\ntal results reported in previous studies. In the exper-\niments of two coupled oscillators, the antiphase mode\nwas observed [14, 20, 27, 30]. In the experiments of three\ncoupled oscillators, the three-phase mode was observed\n[14, 20, 29, 30]. In our simulation, the antiphase and\nthree-phase modes were obtained for n= 2 and n= 3,\nrespectively, which agreed with the experimental obser-\nvations. In theexperiment offourcoupledoscillators,the\nfour-phase mode was observed for the stronger coupling,\nand the 2-2 partial-in-phase mode was observed for the\nweaker coupling [29]. Since the weak coupling was as-\nsumed in our study, our simulation for n= 4, where\nthe 2-2 partial-in-phase mode appeared, agreed with this\nexperimental observation.\nIn addition to the 2-2 partial-in-phase mode, the 2-1-1\npartial-in-phase mode was also observed for the weaker\ncoupling [30]. Here, the 2-1-1 partial-in-phase mode con-\nsists of a pair of in-phase oscillatorsand the other two os-\ncillators with different phases. This mode did not appear\nwithin oursimulationand analysisbased onthe phasere-\nduction theory. In our study, we consider the interaction\nonly through the common pressure among coupled iden-\nticaloscillators. Intheexperiment, thehydrodynamicin-\nteraction could also work between neighboringoscillators\ndepending onthe arrangementofthe innercontainers. In\naddition, the oscillators in the experimental system are\nnot exactly identical due to the experimental error or\nthe external noise. These factors could be related to the\nemergence of the 2-1-1 partial-in-phase mode only in the\nexperiment.\nRegardingtherelationbetweentheintrinsicperiodand\nthe synchronization period in two coupled oscillators, it\nwas reported that the period in the antiphase mode was\nalmost twice the intrinsic period in the experiment [28].It was also reported that the period in the in-phase mode\nwas shorter than the intrinsic period and decreased with\nan increase in the number of coupled oscillators in the\nnumerical calculation using the model with the ordinary\ndifferential equations [21]. Our simulation results and\nthe evaluation by Eq. (18) qualitatively agreed with the\nresults in these studies. To quantitatively compare our\nsimulation result with the experimental one, the depen-\ndence of the synchronization period on the parameters\nsuch as the intrinsic period or coupling strength needs to\nbe investigated, which is left as future work.\nWe finally discuss the change in the water level as the\ncriterionforselectionofthesynchronizationmode. Inthe\nantiphase ( n= 2), three-phase ( n= 3), and 2-2 partial-\nin-phase ( n= 4) modes, which are stable, the changes\ninyoutare expected to be canceled out among the oscil-\nlators with different phases. We confirm the correlation\nbetween the change in youtand the stability of the syn-\nchronization modes obtained in the simulation. Figure 8\nshows the time series of youtin Figs. 8(a-1)–8(c-1) and\nthe total distances of the changes in the water levels l(t)\nin Figs. 8(a-2)–8(c–2) for the ncoupled oscillators, where\nl(t) is defined as\nl(t) =/integraldisplayt\n0/vextendsingle/vextendsingle/summationtext\niQ(i)(t′)/vextendsingle/vextendsingle\ndoutdt′. (22)\nThe slope of lwas evaluated as the rate of the absolute\nchange in yout. For the two coupled oscillators ( n= 2),\nthe slope of lin the antiphase mode, which was stable,\nwas much smaller than that in the in-phase mode, which\nwas unstable, as shown in Fig. 8(a-2). For the three cou-\npled oscillators ( n= 3), the slope of lin the three-phase\nmode, which was stable, was also smaller than those in\nthe partial- and all-in-phase modes, which were unstable,\nas shown in Fig. 8(b-2). For the four coupled oscillators\n(n= 4), the slope of lin the 2-2 partial-in-phase mode,\nwhich was stable, was also smaller than those in the 3-110\n0.004 \n0.002 \n0.000 \n–0.004 –0.002 \n200 210 220 230 240 250 \nt\n200 210 220 230 240 250 \nt0.004 \n0.002 \n0.000 \n–0.004 –0.002 200 100 300 00.4 \n0.2 \n0.0 0.3 \n0.1 \ntl\n200 100 300 0\nt0.4 \n0.2 \n0.0 0.3 \n0.1 0.5 l\n200 210 220 230 240 250 \nt0.006 \n0.003 \n0.000 \n–0.006 –0.003 \n200 100 300 0\nt0.4 \n0.2 \n0.0 0.3 \n0.1 0.5 0.6 l(a-1) (a-2) \n(b-1) (b-2) \n(c-1) (c-2)yout  – yout  yout  – yout  yout  – yout  2-2\n3-1\nallthree\npartial\nallanti\nin \nFIG. 8. Comparison of the water level in the outer container\nyoutamong the synchronization modes for (a) n= 2, (b)\nn= 3, and (c) n= 4. [(a-1)–(c-1)] Time series of yout−¯yout,\nwhere ¯youtisthetimeaverageof yout. [(a-2)–(c-2)] Timeseries\nof the total distance of the change in the water level l. Among\nthethree simulation results indicating the2-2partial-in -phase\nmode in Figs. 2(f)–2(h), the simulation result of Fig. 2(h) i s\nadopted.\npartial- and all-in-phase modes, which were unstable, as\nshown in Fig. 8(c-2). As a result, the simulation results\nindicated that the slope of lin stable synchronization\nmode was the smallest of the obtained synchronization\nmodes for all n. It is noted that, for n= 3, the slope\noflin the partial-in-phase mode was almost equal to\nthat in the three-phase mode [Fig. 8(b-2)], whereas the\noscillatory amplitudes in these modes were significantly\ndifferent [Fig. 8(b-1)]. Thus, another criterion that is\nrelated to the oscillatory amplitude could be considered\nfor the stable synchronization modes, besides the small\nabsolute changes in yout.\nVII. CONCLUSION\nWe performed two-dimensional hydrodynamic simu-\nlation for ncoupled identical density oscillators ( n=\n1,2,3,4). The antiphase ( n= 2), three-phase ( n= 3),\nand 2-2 partial-in-phase ( n= 4) synchronization modesappeared, which agreed with the experimental observa-\ntion reported in previous studies. The all-in-phase and\npartial-in-phase modes were also realized by setting the\nidentical initial conditions. The stabilities of the syn-\nchronization modes were analyzed based on the phase\nreduction theory, where the phase response to the per-\nturbation was obtained from the simulation for a single\ndensity oscillator. The antiphase ( n= 2), three-phase\n(n= 3), and 2-2 partial-in-phase ( n= 4) modes were\nfound to be stable, whereas the other modes were found\nto be unstable. The linear stability analysis with the\nFourier components of the phase coupling function well\nreproduced these stabilities. We numerically confirmed\nthat the stable synchronization modes indicated smaller\nabsolute changes in the water levels in the outer con-\ntainers than unstable modes, which could be one of the\ncriteria for selection of the synchronization mode in the\ncoupled density oscillators.\nOur simulation for coupled density oscillators is suit-\nable to investigate various synchronization phenomena\nby changing the parameter or boundary condition. We\nexpect that studies on the hydrodynamic behavior and\nphase dynamics of coupled density oscillators, including\nthe present study, will contribute to further understand-\ning of the synchronization phenomena in fluid systems.\nACKNOWLEDGMENTS\nThis work was supported by JST SPRING, Grant No.\nJPMJSP2109 (N.T.), by JSPS KAKENHI Grants No.\nJP19H00749, No. JP21K13891 (H.I.), No. JP20H02712,\nNo. JP21H00996, and No. JP21H01004 (H.K.), and by\nthe Cooperative Research Program of “NJRC Mater. &\nDev.” No. 20224003 (H.K.). This work was also sup-\nported by JSPS and MESS Japan-Slovenia Research Co-\noperative Program Grant No. JPJSBP120215001 (H.I.),\nand JSPS and PAN under the Japan-Poland Research\nCooperative Program No. JPJSBP120204602 (H.K.).\nAPPENDIX A: FOURIER COMPONENTS OF\nZ(φ)ANDf(φ)\nWe examine the dominant Fourier components of Z(φ)\nandf(φ) and how they relate to Γ( φ). We consider the\nFourier series\nZ(φ) =p0\n2+∞/summationdisplay\nk=1(pkcos(kφ)+qksin(kφ)),(A1)\nf(φ) =r0\n2+∞/summationdisplay\nk=1(rkcos(kφ)+sksin(kφ)),(A2)\nand compare the magnitudes of Fourier components for\nZ(φ) andf(φ) in Fig. 9, which are calculated from the\ndiscrete points in Fig. 3(c) and 3(d), respectively. It was\nfound that the third ( k= 3) Fourier components are11\n1 2 4 3 6 5 1 2 4 3 6 510 3\n10 2\n10 1\n10 0\n10 –1 10 3\n10 2\n10 1\n10 0\n10 –1 \nk k|pk|\n|qk|(a) (b) \n0pk > 0\npk < 0qk > 0\nqk < 0\n10 –2 \n10 –3 \n10 –4 \n10 –5 \n10 –6 \n10 –7 10 –2 \n10 –3 \n10 –4 \n10 –5 \n10 –6 \n10 –7 |rk|\n|sk|(c) (d) \n1 2 4 3 6 5 1 2 4 3 6 5\nk k0rk > 0\nrk < 0sk > 0\nsk < 0\nFIG. 9. Fourier components of Z(φ) andf(φ). (a) Fourier\ncosine coefficient pkofZ(φ). (b) Fourier sine coefficient qk\nofZ(φ). (c) Fourier cosine coefficient rkoff(φ). (d) Fourier\nsine coefficient skoff(φ).\nthe second largest. We derive the Fourier cosine and sine\ncoefficientsofΓ( φ),akandbk, respectively, usingthoseof\nZ(φ) andf(φ) asa0=p0r0/2,ak= (pkrk+qksk)/2 (k≥\n1), andbk= (pksk−qkrk)/2 (k≥1) based on Eq. (12).\nThe amplitudes of the k-th harmonic oscillations for k≥\n1,Zk,fk, and Γ k, forZ(φ),f(φ), and Γ( φ), respectively,\nare given as\nZk=/radicalBig\np2\nk+q2\nk, (A3)\nfk=/radicalBig\nr2\nk+s2\nk, (A4)\nΓk=/radicalBig\na2\nk+b2\nk\n=1\n2/radicalbig\n(pkrk+qksk)2+(pksk−qkrk)2\n=Zkfk\n2. (A5)\nThe product of Zkandfkgives the scale of Γ kby\nEq. (A5). The ratio of Γ 1and Γ 3estimated by the\nFouriercomponentsof Z(φ)andf(φ)inFig.9isΓ 3/Γ1=\n(Z3/Z1)(f3/f1)∼10−1×10−1= 10−2, which is so small\nthat Γ(φ) looks like a sinusoidal function. Therefore, weobtained nearly sinusoidal Γ( φ) from nonsinusoidal Z(φ)\nandf(φ).\nAPPENDIX B: EIGENVALUES FOR n= 4IN A\nGENERAL CASE\nWe calculate the eigenvalues λof the Jacobian matrix\nAin Eq. (21) for the four-phase and 2-2 partial-in-phase\nmodes (n= 4) with all Fourier components akandbk\nof Γ(φ). The eigenvalues for the four-phase mode are\nderived as\nλ= 2/parenleftBigg∞/summationdisplay\nk=1Akbk±i∞/summationdisplay\nk=1Bkak/parenrightBigg\n,8∞/summationdisplay\nk=1Ckbk,(B1)\nwhere\nAk=\n\nk(k= 2m−1)\n0 (k= 4m−2)\n−2k(k= 4m),m= 1,2,..., (B2)\nBk=\n\nk(k= 4m−3)\n−k(k= 4m−1)\n0 (k= 2m),m= 1,2,..., (B3)\nCk=\n\n0 (k= 2m−1)\nk/2 (k= 4m−2)\n−k/2 (k= 4m),m= 1,2,....(B4)\nThe eigenvalues for the 2-2 partial-in-phase mode are de-\nrived as\nλ= 4∞/summationdisplay\nk=1(−1)k+1kbk,−8∞/summationdisplay\nk=1Dkbk(duplicate) ,(B5)\nwhere\nDk=/braceleftBigg\n0 (k/negationslash= 2m)\nk/2 (k= 2m),m= 1,2,.... (B6)\nWith the numerical result of bkfork≤6 shown in\nFig. 4, the real parts of the eigenvalues are calculated as\nRe(λ) =−1.3×10−1,1.2×10−4for the four-phase mode\nand Re(λ) =−2.7×10−1,−2.2×10−4for the 2-2 partial-\nin-phase mode. Therefore, the 2-2 partial-in-phase mode\nis stable, whereas the four-phase mode is unstable.\n[1] A. T. 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Okuda, Physica D 63, 424 (1993)."    },    {        "title": "2211.07543v1.Scaling_of_the_Non_Phononic_Spectrum_of_Two_Dimensional_Glasses.pdf",        "content": "Scaling of the Non-Phononic Spectrum of Two-Dimensional Glasses\nLijin Wang1, Grzegorz Szamel2, and Elijah Flenner2\n1School of Physics and Optoelectronic Engineering, Anhui University, Hefei 230601, China and\n2Department of Chemistry, Colorado State University, Fort Collins, Colorado 80523, USA\n(Dated: November 15, 2022)\nLow-frequency vibrational harmonic modes of glasses are frequently used to understand their\nuniversal low-temperature properties. One well studied feature is the excess low-frequency density\nof states over the Debye model prediction. Here we examine the system size dependence of the\ndensity of states for two-dimensional glasses. For systems of fewer than 100 particles, the density\nof states scales with the system size as if all the modes were plane-wave-like. However, for systems\ngreater than 100 particles we \fnd a di\u000berent system-size scaling of the cumulative density of states\nbelow the \frst transverse sound mode frequency, which can be derived from the assumption that\nthese modes are quasi-localized. Moreover, for systems greater than 100 particles, we \fnd that the\ncumulative density of states scales with frequency as a power law with the exponent that leads to\nthe exponent \f= 3:5 for the density of states independent of system size.\nA striking feature of glasses is the excess of low-\nfrequency vibrational harmonic modes over that pre-\ndicted by Debye theory. Many computational stud-\nies analyzed low-frequency vibrational modes in glasses\nwith the goal to characterize the excess harmonic modes,\nwhich are frequently referred to as non-phononic modes.\nSince the same low-temperature properties are found in\ndi\u000berent glasses, one is naturally tempted to look for\nuniversal features of the excess modes, independent of\nthe model and the glass formation protocol. Impor-\ntantly, these features should hold in the thermodynamic\nlimit. In many studies the implicit hypothesis is that\nthe excess modes density Dex(!) follows a power law,\nDex(!) =Aex!\f.\nTwo methods are generally employed to determine \f.\nOne approach is to separate modes by their participa-\ntion ratio with the assumption that the low-frequency ex-\ncess modes have a signi\fcantly smaller participation ra-\ntio than the extended, plane-wave-like modes [1{3]. This\nmethod was used to argue that \f= 4 in three dimensions\n[2, 3]. However, this method fails in two dimensions since\nit is impossible to choose a suitable participation ratio\ncuto\u000b to di\u000berentiate between the excess modes and the\nplane-wave-like modes in two dimensions.\nThe second method is to study small systems [4{6].\nThe frequency range of the \frst band of plane-wave-like\nmodes scales as L\u00001whereLis the simulation box size.\nThe assumption is that the modes below this band of\nplane-wave-modes represent the excess modes and one\ncan determine the scaling of Dex(!) from studying the\nharmonic mode spectrum for frequencies signi\fcantly be-\nlow those of the \frst band. While conceptually straight\nforward, there are some subtle issues that must be kept\nin mind. First, every simulated glass has a slightly di\u000ber-\nent shear modulus that results in a slightly di\u000berent fre-\nquency expected for the \frst plane-wave-like mode, !T1,\nfor an ideal elastic body. In addition, the modes are not\npure plane waves and there is a distribution of frequen-\ncies around !T1for each glass. For these reasons one can\nhave a rather large range of frequencies where the \frst\nplane-wave-like modes exist. If one does not separate theplane-wave-like modes (like in the \frst method), one has\nto be careful to infer Dex(!) using only modes that do\nnot correspond to plane-wave-like modes.\nAnother di\u000eculty is that there is a reported \fnite size\ne\u000bect in\ruencing the exponent \fin three-dimensional\nglasses [6]. It has been reported that \f < 4 for small\nsystems and \fequals 4 for large enough systems [6]. Re-\ncall that the plane-wave-like modes move to lower fre-\nquencies with increasing system size, and thus there is\na smaller frequency range where the excess modes spec-\ntrum is found. In addition, in two dimensions there are\nvery few excess modes [3, 7] and a very large number\nof glass realizations are needed to obtain good enough\nstatistics to determine an accurate value for \f. Analysis\nof\fin two dimensions is complicated by a reported \fnite\nsize e\u000bect in the pre-factor Aex,Aex\u0018(logN)(\f+1)=2\nwithNthe number of particles in the system [5].\nIn a previous publication we stated that there was no\nclear evidence that there is a \fnite size e\u000bect for \fin\nsimulations of two-dimensional glasses [7]. In contrast,\nLerner and Bouchbinder argued very recently [8] that \f\nincreases with increasing system size and is four for a\nlarge enough (N=1600) aged glass.\nHere we present analysis of our results of the cumula-\ntive density of states I(!) =R!\n0+D(!0)d!0, whereD(!)\nis the full density of states, that lead to our conclusions\nthat there is no system size dependence of \fand examine\nin detail some \fnite size e\u000bects of the density of states\nobserved in two-dimensional glasses. We also argue that\nthere is no clear evidence of systematic system size de-\npendence observed in the data presented Ref. [8].\nFor an ideal elastic body, the frequency of the \frst\ntransverse wave is given by !T1=p\n(G=\u001a)(2\u0019=L), where\nGis the shear modulus, \u001ais the mass density, and Lis\nthe linear size of the sample. The frequency of the \frst\nlongitudinal wave is given by !L1=p\n(G+K)=\u001a(2\u0019=L),\nwhereKis the bulk modulus. We start our analysis of\nthe IPL-10 system discussed in Ref. [7] for systems of\nN\u0014100. Shown in Fig. 1 is the density of states D(!) =\n(2N\u00002)\u00001P\nn\u000e(!\u0000!n) forN= 36, 64, and 100. The\nfrequencies !nare obtained by diagonalizing the HessianarXiv:2211.07543v1  [cond-mat.dis-nn]  8 Nov 20222\nN = 36\nN = 64\nN = 100(a)D(ω)\n00.010.020.030.040.05\nω0 10 20 30 40\nωT1ωL1N = 36\nN = 64\nN = 100(b)D(ω)\n00.010.020.030.040.05\nωN0 20 40 60 80 100\nFIG. 1: (a) The density of states of small two-dimensional\nglasses with Npaticles. There is a distinct system size\ndependence with a shoulder at low frequencies and peaks\nshowing up at Ndependent frequencies. (b) The density of\nstates versus !p\nN. The shoulder is shown to correspond\nto!T1=p\nG=\u001a(2\u0019=L) and the \frst peak corresponds to\n!L1=p\n(G+K)=\u001a(2\u0019=L).\nmatrix. For these system sizes there are two distinct\nfeatures at low frequencies, a shoulder and a peak. In\nFig. 1(b) we scale the frequency byp\nN\u0018Land \fnd\nthat the two low frequency features align. The vertical\nlines are the frequencies of ideal plane waves, !T1(red)\nand!L1(blue). Fig. 1(b) suggests that the shoulder is\nassociated with the \frst transverse wave and the peak is\nassociated with the \frst longitudinal wave.\nTo determine the scaling exponent of the density of\nstates,\f, we examine the cumulative density of states\nI(!). Unlike the density of states, the cumulative density\nof states does not depend on the binning algorithm. The\nfrequency dependence of the cumulative density of states\nfollows from the scaling of D(!0) for values of !0less than\n!. Since the hypothesis is that the lowest frequencies\nbelow!T1representDex(!), we should be able to infer\nDex(!) from the low frequency behavior of I(!).\nWe examined the scaling of I(!) for very small systems\nofN\u0014100 particles. Shown in Fig. 2(a) is I(!) versus\n!plogN, which is a scaling predicted from continuum\nelasticity due to a r\u00001decay of the non-phononic modes\nat large distances r[5, 10]. There is a systematic devia-\ntion from this scaling. In contrast, we \fnd nearly perfect\nN = 36\nN = 64\nN = 100(a)I(ω)10−610−510−410−310−210−1\nωlog N1 2 5 10\nN = 100\nN = 64\nN = 36β = 3.38β = 3.614 modes(b)\nωT1I(ω)[2N-2]10−410−310−210−11101\nωN5 10 20FIG. 2: (a) Scaling of the cumulative density of states pro-\nposed in Ref. [10]. There are small deviations from this scaling\nfor these small system sizes. (b) Expected scaling of the cu-\nmulative density of states if all the modes are plane-wave-like.\nThis scaling results in better data collapse than the scaling\nproposed in Ref. [10] for these small system sizes. The red ver-\ntical line is the frequency !T1expected for an ideal transverse\nwave. For an ideal amorphous elastic body, I(!)(2N\u00002) = 4\nat!T1. The dotted green line is a \ft to the N= 36 parti-\ncle results and the dashed green line is a \ft to the N= 64\nparticle results; the corresponding \fin the \ftting function\nI(!)[2N\u00002]\u0018(!p\nN)\f+1is indicated in the legend. Both\n\fts are from 3\u0014!p\nN\u00146.\ndata collapse of I(!)[2N\u00002] versus!p\nN, Fig. 2(b),\nwhich is a scaling motivated by the system size depen-\ndence of Debye theory. We are counting the average num-\nber of modes up to !p\nNand seeing how many fall be-\nlow!T1. For an ideal elastic body, I(!)[2N\u00002] = 4 at\n!T1p\nN(vertical red line). For these very small systems\nwe do not \fnd any evidence of excess modes and do not\nbelieve that the excess mode scaling can be inferred from\nsystems less than 100 particles. This analysis also high-\nlights the di\u000eculty in determining the frequency range\nto use to study excess (non-phononic) modes.\nWe \ftI(!)[2N\u00002] versus!p\nNfor 3\u0014!p\nN\u00146\nforN= 36, 64 and 100 to a(!p\nN)\f+1. We \fnd that\n\fvaries from 3 :38 forN= 36 to 3:61 forN= 64, and\nthat\f= 3:48 forN= 100. We show the \fts for N= 36\nandN= 64 in Fig. 2(b). It is impossible to determine\nfrom the \fts if one value of \fshould be preferred over3\nN = 30K\nN = 3200\nN = 1600\nN = 786\nN = 196(a)I(ω)\n10−910−810−710−610−510−410−3\nωlogN0.1 0.2 0.5 1 2\nN = 30K\nN = 1600β= 3.5β= 4(b)I(ω)\n10−910−810−710−610−510−410−3\nωlogN0.1 0.2 0.5 1 2\nFIG. 3: (a) Scaling plot of the cumulative density of states\nfor system sizes 196 \u0014N\u001430 000 for the IPL-10 system [7].\nUnlike for smaller systems, the I(!) =f(!plogN) scaling\nproduces reasonable data collapse at small frequencies. There\nis some deviation at low frequencies, but this cannot be sta-\ntistically ruled out. (b) Direct comparison for systems of N=\n1600 andN= 30 000. We \ft the N= 1600 particle system\nfrom 0:2\u0014!plogN\u00140:8 toI(!plogN)\u0018(!plogN)\f+1.\nThe dotted orange line corresponds to a \ft with \f= 4 \fxed,\nwhile the dashed orange line corresponds to a \ft with \f= 3:5\n\fxed.\nanother. Ideally, we would want to greatly expand the\n\ftting range to claim a speci\fc power law, but it is very\ndi\u000ecult to \fnd a large range of frequencies in which a\npower law is obeyed.\nWe note that the I(!) =f(!plogN) scaling is obeyed\nfor 100< N\u001430 000, Fig. 3(a). Since the f(!plogN)\nscaling is derived from continuum elasticity [5, 10], it\nis unsurprising that this scaling breaks down for small\nsystems. There are some deviations from scaling at the\nsmallest frequencies, but we believe that better statis-\ntics would improve the overlap. In Fig. 3(b) we show\nonly theN= 1600 and N= 30 000 results. We \ft\ntheN= 1600 results for 0 :2\u0014!plogN\u00140:8 to\nI(!plogN)\u0018(!plogN )\f+1where we \fxed \f= 4 (or-\nange dotted line) and \f= 3:5 (orange dashed line).\nThe\f= 3:5 \ft provides a better description of the\nN= 30 000 results. Note that we cannot statistically\nrule out\f= 4 on this analysis alone, but we will show\nthat\f\u00193:5 is consistent with all our results for systems\nN = 3200\nN = 1600\nN = 786\nN = 196IPL-10I(ω)/ω4.5\n10−310−2\nω0.1 1FIG. 4: The cumulative density of states for the IPL-10 sys-\ntem for di\u000berent system sizes studied in Ref. [7]. Note that\nthis \fgure is a reproduction of Fig. 2(b) of the supplemental\nmaterial of Ref. [7].\nover 100 particles.\nShown in Fig. 4 is I(!)=!4:5(corresponding to \f= 3:5)\nfor the IPL-10 system for di\u000berent system sizes discussed\nin Ref. [7]. The low frequency data suggests that \f\u00193:5\nfor every system size except for N= 786. As pointed out\nin Ref. [8], the N= 786 particle result shows an upturn at\nsmall frequencies suggesting a \fdi\u000berent than the other\nsystem sizes. We \fnd that this system size for the IPL-\n10 system is an outlier and it does not provide su\u000ecient\nevidence of a \fnite size e\u000bect. As we show below in Fig. 6,\nwith improved statistics we \fnd that \fforN= 786 is\nconsistent with our results for other systems and other\nsystem sizes for the same system.\nTo determine the exponent \ffor systems of more than\n100 particles we \ft log[ I(!)] = (\f+ 1) log(!) +a. For\neach system size we \ft three to four frequency ranges\nto obtain\fand average \fobtained from these \fts. The\nuncertainty is chosen to include each value of \ffor a given\nsystem and system size. We exclude the results for the\nLennard-Jones systems presented in Ref. [7] since there\nis a systematic downturn at small !for each system size\nthat is not seen for the other systems. Shown in Fig. 5\nare the results for the remaining systems analyzed in Ref.\n[7] (closed symbols).\nFigure 5 does not show any clear, systematic system\nsize dependence of the exponent \f. For the harmonic\nsphere system (HARM), \fdoes appear to grow with sys-\ntem size, but the growth is within the uncertainty of the\ncalculation. The two inverse-power-law systems (IPL-10\nand IPL-12) show no systematic change in \fwith system\nsize.\nFor the data presented in Fig. 5 there is one outlier,\nand that is the IPL-10 system with 786 particles, see also\nFig. 4. We increased the number of glass samples for this\nsystem and system size from the 0.79 million samples\nused in Ref. [7] to 2.4 million samples. Shown in Fig. 6\nisI(!)=!4:5forN= 786 with the improved statistics.\nWe can see that the upturn starts at lower !, where the\nstatistics are poor, and there is a larger range of !where4\nIPL-12\nHARM\nIPL-10β\n3.03.23.43.63.84.0\nN200 500 1000 2000\nFIG. 5: The exponent \fobtained from \fts of the cumulative\ndensity of states I(!) for the systems studied by Wang et al.\nin Ref. [7]. The horizontal line is the average value of \f.\n0.79 Million\n2.4 MillionN = 786IPL-10I(ω)/ω4.5\n0.0020.0040.006\nω0.1 1\nFIG. 6: Comparison of the scaled cumulative density of states\nfor the IPL-10 system with N=786 for ensemble sizes of 0.79\nmillion con\fgurations (open symbols) and 2.4 million con\fg-\nurations (closed symbols).\n\f\u00193:5 is a reasonable description of the data. We redid\nthe \ftting for \ffor this system and the result is shown\nin Fig. 5 as the open circle. With the improved statistics\n\fis consistent with the other results.\nThe change of \fwith increasing the number of glass\nsamples highlights the di\u000eculty of obtaining an accurate\nvalue of\ffor two-dimensional systems. There are so\nfew excess modes that many glass samples are needed to\nobtain accurate results to do a direct calculation of the\ndensity of states. This was pointed out in Ref. [11] where\nthey utilized non-linear mode analysis [9, 12] to analyze\ntwo-dimensional systems.\nWe \fnd that all systems where N > 100 result in\n\f\u00193:5 for all our systems and we cannot rule out \f\u00193:5\nforN\u0014100 with our data alone. We note that we do\nnot believe that one can obtain the scaling of the non-\nphononic density of states for N < 100. We also \fnd that\nthese conclusions are consistent with the data presented\nin Ref. [8]. To argue for a \fnite size e\u000bect the authors\nof Ref. [8] \ft the density of states, as opposed to the cu-\nmulative density of states, and obtained di\u000berent valuesof\ffor the power law Dex(!) =Aex!\f. However, when\nthe density of states is rescaled there is very convincing\ndata collapse, which suggests that all the results corre-\nspond to the same function and \fis the same for each\nsystem size. To see convincing evidence of a system size\ndependence of \fone would need to see deviations from\nthe data collapse at low frequencies where we expect that\nthe plane-wave-like modes do not contribute to the den-\nsity of states, and these deviations are not present in the\ndensity of states presented in Ref. [8].\nWe also don't \fnd clear evidence for any system size\ndependence of \fin the cumulative density of states shown\nin Fig. 2 of Ref. [8]. We argued above that the scaling\nof the non-phononic density of states cannot be inferred\nfrom systems of 100 particles or less. If we focus on the\nsystems of greater than 100 particles we \fnd that Fig.\n2(d) of Ref. [8] shows that \fis the same for N= 196\nandN= 1600 poorly annealed system. According to\nFig. 1(c) of Ref. [8] \f\u00193:6 for these systems. This\nvalue of\fis consistent with our results as well as the\nlow frequency scaling of the cumulative density of states\nfor theN= 196 and N= 1600 well-annealed system of\nRef. [8].\nAll the systems discussed in this note and in Ref. [8]\nare fairly small, and it is possible to study much larger\nsystems. The di\u000eculty with larger systems is that the\n\frst plane-wave-like modes get pushed to lower frequen-\ncies and the frequency range where only excess modes are\nexpected to be found becomes smaller. This di\u000eculty is\ncombined with the very limited number of non-plane-\nwave-like modes in two-dimensions. We examined a\n30 000 particle system for the same IPL-10 model as used\nin Ref. [7] and found inconclusive results for the value of \f\nat the lowest frequencies examined, see Fig. 3(b). These\nresults were similar to those for three-dimensional sys-\ntems in Ref. [13] in that \fwas smaller than 3.5 at lower\nfrequencies. This puzzling result persisted despite hav-\ning about 0.23 million con\fgurations of 30 000 particles.\nExamination of these large systems deserves more work.\nNon-linear excitation analysis has been used as a proxy\nto measure exponent \fin two-dimensional glasses [9{\n12]. Many properties of glasses are determined by the\nharmonic modes [14{17], and thus it is important to un-\nderstand how the speci\fc features of the harmonic modes\nin\ruence these properties. We believe it would be ideal\nto directly measure \fusing harmonic modes, but \fnd\nthat this is a very di\u000ecult task in two dimensions. The\nresults presented here strongly suggest that a harmonic\nmodes-based calculation of \ffor modes below the \frst\nsound mode results in \f\u00193:5 rather than \f= 4:0 for\ntwo-dimensional glasses.\nTwo-dimensional solids and liquids have characteristics\ndi\u000berent than their three-dimensional counterparts [7,\n18]. One characteristic for two-dimensional glasses is the\nvery small density of excess modes, which makes study-\ning the excess modes di\u000ecult. There is evidence that\nthe density of the excess modes scales as !2for excess\nmodes above !T1for two-dimensional systems [7]. Since5\n!T1!0 in the thermodynamic limit, it is unclear that\nthe scaling of the modes below !T1is important in the\nunderstanding of the behavior of two-dimensional glasses.\nFuture work needs to focus on the in\ruence of excess har-\nmonic modes on the low-temperature, thermodynamic\nproperties of glasses.\nWe thank E. Lerner and E. Bouchbinder for useful dis-\ncussions. We also thank E. Lerner and E. Bouchbinder\nfor sharing some of their data published in Ref. [8], which\nwe found to support the conclusions presented here. L.Wang acknowledges the support from National Natural\nScience Foundation of China (Grant No. 12004001), An-\nhui Projects (Grant Nos. S020218016 and Z010118169),\nHefei City (Grant No. Z020132009), and Anhui Univer-\nsity (start-up fund). E.F. and G.S. acknowledge support\nfrom NSF Grant No. CHE-2154241. We also acknowl-\nedge Hefei Advanced Computing Center, Beijing Super\nCloud Computing Center, and the High-Performance\nComputing Platform of Anhui University for providing\ncomputing resources.\n[1] H. R. Schober and B. B. Laird, Phys. Rev. B 44, 6476\n(1991).\n[2] L. Wang, A. Ninarello, P. Guan, L. Berthier, G. Szamel,\nand E. Flenner, Nat. Commun. 10, 26 (2019).\n[3] H. Mizuno, H. Shiba, and A. Ikeda, Proc. Natl. Acad.\nSci. U.S.A. 114, E9767 (2017).\n[4] E. Lerner, G. D uring, and E. Bouchbinder, Phys. Rev.\nLett. 117, 035501 (2016).\n[5] G. 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